Field Theory of Multiscale Plasticity 9781108836609, 9781108874069

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Field Theory of Multiscale Plasticity This unique book provides a concise and systematic treatment of foundational material on dislocations and metallurgy and an up-to-date discussion of multiscale modeling of materials, which ultimately leads to the field theory of multiscale plasticity (FTMP). Unlike conventional continuum models, this approach addresses the evolving inhomogeneities induced by deformation, typically as dislocation substructures like dislocation cells, as well as their interplay at more than one scale. This is an impressively visual text with many and varied examples and viewgraphs. In particular, the book presents a feasible constitutive model applicable to crystal plasticity-based finite element method (FEM) simulations. It will be an invaluable resource, accessible to undergraduate and graduate students as well as researchers in mechanical engineering, solid mechanics, applied physics, mathematics, materials science, and technology. Tadashi Hasebe is Associate Professor of Mechanical Engineering at Kobe University, Japan, and is an expert in a very wide range of engineering fields, more specifically, in metallic materials, including high-temperature strength, impact engineering, plastic forming technology, high-energy rate forming, theory of elasto-plasticity, and micromechanics. Professor Hasebe likes to incorporate experimental, mathematical, and numerical perspectives in his works.

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Published online by Cambridge University Press

Field Theory of Multiscale Plasticity T A D ASH I H ASEBE Kobe University, Japan

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Shaftesbury Road, Cambridge CB2 8EA, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 103 Penang Road, #05–06/07, Visioncrest Commercial, Singapore 238467 Cambridge University Press is part of Cambridge University Press & Assessment, a department of the University of Cambridge. We share the University’s mission to contribute to society through the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108836609 DOI: 10.1017/9781108874069 © Tadashi Hasebe 2024 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press & Assessment. First published 2024 Printed in the United Kingdom by TJ Books Limited, Padstow Cornwall A catalogue record for this publication is available from the British Library Library of Congress Cataloging-in-Publication Data Names: Hasebe, Tadashi, 1965– author. Title: Field theory of multiscale plasticity / Tadashi Hasebe, Kobe University, Japan. Description: First Edition. | New York : Cambridge University Press, 2024. | Includes bibliographical references and index. Identifiers: LCCN 2022041278 | ISBN 9781108836609 (hardback) | ISBN 9781108874069 (e-book) Subjects: LCSH: Plasticity. | Field theory (Physics) Classification: LCC TA418.14 .H364 2024 | DDC 620.1/1233–dc23/eng/20230202 LC record available at https://lccn.loc.gov/2022041278 ISBN 978-1-108-83660-9 Hardback Cambridge University Press & Assessment has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

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Contents

Preface page xv Acknowledgments xviii

Part I  Fundamentals 1

2

1

Dislocation Theory and Metallurgy 1.1 Elasticity versus Plasticity 1.2 Fundamentals of Dislocations 1.2.1 Overview 1.2.2 Types of Dislocation 1.2.3 Stress Field around Dislocations 1.2.4 Elastic Strain Energy of Dislocations 1.2.5 Dislocation Processes (Important Features) 1.3 Crystallography 1.3.1 Crystal Systems (Structures) 1.3.2 FCC versus BCC 1.3.3 Slip Systems in FCC and BCC 1.3.4 RSS and the SF 1.3.5 Dislocation–Dislocation Interactions Revisited: Interaction Matrix 1.4 Miscellaneous 1.4.1 Twin 1.4.2 Texture and Pole Figure 1.4.3 Stereographic Projection and Standard Triangle 1.4.4 Crystal Dislocations versus CD Dislocations Appendix A1  Energy Landscape for Dislocation Pairs

3 3 4 4 11 15 20 23 45 45 48 51 56

Dislocation Dynamics and Constitutive Framework 2.1 Overview: Unified-type Crystal Plasticity Constitutive Equation 2.2 Strain-Rate Dependency of Flow Stress 2.2.1 FCC versus BCC Revisited 2.2.2 Thermal and Athermal Obstacles 2.2.3 Examples of Obstacles 2.3 Dislocation Velocity versus Stress

86 86 87 87 89 91 93

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58 65 65 69 74 75 79

vi

3

4

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2.4

Physically Based Constitutive Framework 2.4.1 Arrhenius-type Rate-controlling Equation 2.4.2 Expression of Apparent Activation Energy 2.4.3 Force–Displacement Diagram 2.5 Representation of Obstacles 2.6 Kocks–Mecking Model 2.7 Unified Constitutive Equation 2.8 Experimental Evaluation of Activation Energy 2.8.1 Activation Volume 2.8.2 Activation Enthalpy 2.8.3 Experimentally Evaluated Activation Volume 2.9 Deficiencies of Inappropriate Strain-Rate Dependency Appendix A2  Systematic Prestrained-Impact Experiments

97 97 98 100 103 107 108 113 114 115 117 123 130

Dislocation Substructures: Universality of Cell Structures 3.1 Universality of Cell Structures 3.1.1 Overview 3.1.2 Dislocation Cells among Others 3.2 Similitude Law for Dislocation-Cell Size 3.3 Strain-History Effects: Proportional versus NP Strain Paths 3.4 Strain-Rate and Temperature-History Effects 3.4.1 FCC versus BCC 3.4.2 Strain-Rate History Effect versus Proportional Strain-History Effect 3.4.3 Strain-Rate History Effect versus NP Strain-History Effect for BCC Metals 3.4.4 Strain-Rate History Effect versus NP Strain-History Effect for FCC Metals 3.5 Dislocation Substructures under Hypervelocity Impact 3.5.1 Phenomenology 3.5.2 Strain Rate/Temperature Effect Revisited 3.6 Effect of Surface: Surface versus Bulk 3.7 Dislocation Substructures versus Fatigue 3.7.1 High-cycle Fatigue 3.7.2 Low-cycle Fatigue 3.8 Structural Stability and Mechanical Roles of Dislocation Cells 3.9 Long-range Internal Stress Field 3.10 Dislocation Substructures under Finite Deformation 3.10.1 Phenomenology 3.11 Dislocation Cell versus Subgrain Appendix A3  Some Comments on the Plasticity of UFG Materials

139 139 139 143 154 161 172 172

Single Crystals versus Polycrystals 4.1 Complexities in Single Crystals

232 232

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174 178 180 182 182 184 190 193 193 203 207 211 215 215 225 227

Contents

4.2 4 .3 4.4 4.5

5

6

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Some Comments on the Lower Mobility of Screw Dislocation in BCC Metals 237 4.2.1 Complexity in the Screw-Dislocation Core 237 4.2.2 Core Structure of Screw Dislocation in α -Fe242 Effect of Active Slip Systems on a Yield Surface 244 Conventional Studies and Coarse-Grain Plasticity 248 Collective Effects of Grain Aggregates as Additional Features 255 4.5.1 Implications: Granular Materials 255 4.5.2 Implicative Experimental Results 259

Part II  Theoretical Backgrounds: Description and Evolution

265

Overview of Field Theory of Multiscale Plasticity 5.1 Field Theory of Plasticity 5.2 Images of “Fields” 5.3 Ingredients of Field Theory 5.3.1 Three Key Features 5.3.2 Three Basic Field Theories 5.3.3 Three Important Hierarchical Scales 5.4 Field Triangle: Three Well-defined Field Theories 5.4.1 Differential Geometrical Field Theory 5.4.2 Gauge Field Theory 5.4.3 Quantum Field Theory 5.5 New Features: Toward a Complete Field Theory of Plasticity 5.5.1 Inhomogeneity 5.5.2 Duality 5.5.3 Interacting Incompatibility 5.6 A Tentative Vehicle for FTMP-based Simulations: A Crystal Plasticity-based Constitutive Model 5.6.1 Constitutive Framework 5.6.2 Hardening Law and Field-Theoretical Strain-Gradient Terms 5.6.3 Derivation of Field-Theoretical Strain-Gradient Terms 5.6.4 Field-Theoretical Strain-Gradient Terms in Multiple Scales Appendix A5.1  Why Is a “New” Field Theory Necessary? Appendix A5.2  Brief Overview of “Continuum Mechanics” and “Crystal Plasticity”

267 267 269 270 270 271 272 274 274 275 275 276 276 276 277

Differential Geometrical Field Theory of Dislocations and Defects 6.1 Brief Review of Differential Geometry 6.1.1 General Relativity and Riemannian Geometry 6.1.2 Curvilinear Coordinates 6.1.3 Metric Tensor 6.1.4 Parallelism and Connection 6.1.5 Torsion Tensor

304 304 304 306 308 309 311

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Contents

6.2

6.3

6.4

6.5

6.6

6.7

7

6.1.6 Curvature Tensor 311 6.1.7 Einstein Tensor 313 Non-Riemannian Plasticity 315 6.2.1 Background and Overview 315 6.2.2 Basic Quantities in Non-Riemannian Plasticity 316 6.2.3 Dislocation-Density Tensor 323 6.2.4 Incompatibility Tensor 325 6.2.5 Resolution into Slip Systems 327 6.2.6 Explicit Representations of Dislocation-Density and Incompatibility Tensors 330 6.2.7 Dual Structure between Strain Space and Stress Space 334 6.2.8 Torsion in Stress-Function Space 338 6.2.9 Relationship between mij and α ij 339 6.2.10 Curvature in Stress-Function Space 339 Disclination and Curvature Tensor 340 6.3.1 New Physical Interpretation of Incompatibility 341 6.3.2 Components of the Disclination-Density Tensor 343 Fundamental Equations for the Dislocation Field 346 6.4.1 General Description 346 6.4.2 Some Miscellaneous Comments about the Dislocation Line 350 Physical Images of DG Quantities 352 6.5.1 Dislocation-Density Tensor: GN Dislocations 352 6.5.2 Incompatibility Tensor-driven Inhomogeneity Evolution 353 Physical Images of Incompatibility and Stress-Function Tensors Analogous to Granular Media Representation 355 6.6.1 Graph Representation of Granular Assembly 355 6.6.2 A Comment on Quantum Stress and the Stress-Function Tensor 359 Theory of Interaction Fields 360 6.7.1 Relative Deformation and Interaction Fields 360 6.7.2 Expressions for DG Quantities 362 6.7.3 Extension to Multiple Scales 365 6.7.4 Some Remarks on Multiple-scale Formalism 367 6.7.5 Summary 370

Gauge Field Theory of Dislocations and Defects 7.1 Fundamentals of Gauge Theory 7.1.1 Maxwell’s Equation as a Classical Gauge Theory 7.1.2 Electromagnetic Gauge Theory (Commutative) 7.1.3 Yang–Mills Gauge Theory (Noncommutative) 7.1.4 Expansion of Gauge Theory 7.2 Yang–Mills-Type Kadic ́–Edelen Gauge Theory 7.2.1 Gauge Field Theory of Dislocations and Defects

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371 371 371 374 378 380 382 382

Contents

7.2.2 7.3 8

9

Interrelationship between Gauge Theory and DG Field Theory 7.2.3 Chern–Simons Gauge Term Derivatives of Gauge Theory 7.3.1 Euler–Lagrange Equation 7.3.2 Examples of E–L Equations 7.3.3 Euler–Lagrange Equations for Dislocation and Defect Fields 7.3.4 Energy-Momentum Tensor 7.3.5 Physical Meaning of the Energy-Momentum Tensor 7.3.6 Energy-Momentum Tensor for the Gauge Field of Dislocated and Defected Elastic Media 7.3.7 Application Examples of the Energy-Momentum Tensor

ix

387 389 391 392 393 396 402 405 407 411

Method of Quantum Field Theory 8.1 Brief Reviews 8.1.1 Quantum Mechanics versus Quantum Field Theory 8.1.2 Brief Review of Quantum Mechanics 8.1.3 Matrix Formulation of Quantum Mechanics 8.1.4 Schrödinger Picture versus Heisenberg Picture 8.1.5 Quantum Mechanics versus QFT 8.1.6 Second Quantization 8.2 Equivalence of Quantum Mechanics and Statistical Mechanics 8.2.1 General Framework 8.2.2 Feynman Path Integral 8.3 Application of QFT Method to a Many-Dislocation System 8.3.1 Brief Overview of the Underlying Idea 8.3.2 Application of QFT Method to a Many-Dislocation-System Cell Formation

416 416 416 417 419 420 422 423 427 427 429 431 431

Part III  Applications I: Evolution of Inhomogeneity in Three Scales

437

Identification of Important Scales 9.1 Introduction: New Hierarchical Recognition 9.2 Three Important Scales 9.3 Independent Mechanisms of Field Evolutions 9.3.1 Scale A: Dislocation-Substructure Level 9.3.2 Scale B: Intragranular Level 9.3.3 Scale C: Grain-Aggregate Level 9.3.4 Description of Field Evolutions 9.4 Application Examples of Three Scale-Based Approaches 9.4.1 Three Projects 9.4.2 Outlines of the Three Projects 9.4.3 Details and Some Tentative Results

439 439 441 446 447 449 450 450 453 453 454 459

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Contents

10

Scale A: Modeling and Simulations for Dislocation Substructures 10.1 Overview 10.2 Evaluation of the Hamiltonian for the Dislocation Field 10.2.1 Preparation 10.2.2 Second Quantization of Dislocation Field 10.2.3 Statistical-Mechanics Evaluation of Dislocation System Based on Method of QFT 10.3 Effective Action for the Coupling Term and the Elastic Term 10.4 Evaluation of the Effective Hamiltonian for the Total System 10.5 Physical Interpretation of the Coupling Term 10.6 Elimination of the Elastic Field 10.7 Time-Dependent GL Equation 10.8 Numerical Scheme for Cell-Formation Simulation 10.8.1 Analytical Conditions 10.9 Simulation Results and Discussion 10.9.1 Reason for Cellular Morphology and the Origin of the Similitude Law 10.9.2 Effect of Internal Stress Distribution 10.9.3 Elastic Strain Energy and Incompatibility-Tensor Field 10.10 Important Implications from the Simulation Results 10.10.1 Overview 10.10.2 Supplements 10.10.3 Cell Formation as Thermal Process 10.10.4 Stress-Supporting Structure and Reservoir Appendix A10  Critical Review on Cell Models

483 483 484 484 487

Scale B: Intragranular Inhomogeneity 11.1 Overview 11.2 Evolution and Description 11.3 Constitutive Equations Equipped with Field-Theoretical Strain-Gradient Terms 11.4 Preliminary Simulation Results 11.4.1 Surrogate Model for Exaggerated Effects 11.4.2 Contribution of the Incompatibility-Tensor Field to Stress Response 11.5 Simulation Results: Evolving Intragranular Inhomogeneity 11.5.1 Modulation Structuring and Misorientation 11.5.2 Modulation Developments in Stress and Strain Fields 11.5.3 Evolution of Modulating Stress and Stress Fields 11.5.4 Mechanical Roles of the Incompatibility Field 11.5.5 The Effect of Evaluation Size and Discretization 11.5.6 Morphology and Overwriting 11.5.7 Orientation Dependency of Modulated Structures 11.5.8 The Effect of Projection Direction on Morphology

545 545 546

11

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496 507 509 510 512 517 518 519 520 525 529 530 532 532 533 534 535 536

547 549 549 550 553 553 558 561 565 567 574 576 577

Contents

11.6 Simulation Results with Disclination Fields 11.6.1 Model Description 11.6.2 Simulation Results 11.6.3 Summary 11.7 Effect of Initially Introduced Strain Distribution: Toward Rational Modeling of Arbitrary Microstructures 11.7.1 Preliminary Simulations 11.7.2 Application Example: Modeling Lath Martensite Structures 11.7.3 Future Scope: Modeling Deformation Twinning 11.8 Concluding Remarks 12

13

xi

580 580 580 584 584 584 586 591 597

Scale C: Modeling and Simulations for Polycrystalline Aggregate 604 12.1 Introduction: Polycrystalline Plasticity 604 12.2 Anticipated Features of Polycrystalline Plasticity: The SSS Hypothesis 605 12.3 Single-Phase Model 606 12.3.1 Simulation Model 606 12.3.2 Sneak Preview of the Simulation Results 608 12.3.3 Simulation Results for Single-Phase Models 611 12.4 Application to DP Alloy Models 624 12.4.1 Dual-Phase Models Vf ~25% 625 12.4.2 Dual-Phase Models Vf ~15% 637 12.4.3 Vf ~50% DP Model 643 12.5 Effect of “Grain Morphology” 647 12.5.1 Model Description: Distributed Grain-Size Models 647 12.5.2 Roles of Grain-Size Distribution: The Macro-Response 650 12.5.3 The Roles of Duality 654 12.6 Duality Revisited 657 12.6.1 Overview 657 12.6.2 Examples of Applications 660 12.7 Summary 664 Appendix A12  Effect of Transgranular Inhomogeneity on Bauschinger Behavior of DP Polycrystalline Aggregates 664

Part IV  Applications II: Stability and Cooperation

679

Cooperation of Multiple Inhomogeneous Fields 13.1 Field Equation and Stability 13.1.1 Field Equation for Instability 13.1.2 Euler–Schouten Relative-Curvature Tensor 13.1.3 Derivation of Field Equation for Stability 13.1.4 Application Examples 13.2 Preliminary Simulation for Interaction Fields 13.2.1 Overview: Materials as Complex Systems 13.2.2 Analytical Models and Procedures

681 681 682 682 685 690 692 692 694

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14

15

Contents

13.2.3 General Expressions for the Three-Scale Problem 13.2.4 One-Dimensional Expression for the Three-Scale Problem 13.2.5 Application Example 13.2.6 Analytical Results and Discussion 13.2.7 Toward System Stability/Instability Evaluation 13.2.8 Conclusion 13.2.9 Other Application Examples 13.3 Stability of Dislocation-Cell Structure 13.3.1 Problem Description 13.3.2 Simulation Setting 13.3.3 Results and Discussion 13.3.4 Summary 13.4 Global–Local Structure of Stress and Strain Fields 13.4.1 Preliminary Simulations 13.4.2 Detailed Simulations 13.4.3 Energy Spectrum Similarities for Turbulent Flow

696 699 701 702 706 709 709 714 714 715 717 724 725 725 739 743

Outlooks: Some Perspectives on New Multiscale Solid Mechanics 14.1 Application of “Small-World” Concepts 14.1.1 Overview of SWNs 14.1.2 “Specificity” versus “Universality” Viewed from SWN 14.1.3 The “Complex System” versus “Simple System” in Terms of “Stability” 14.1.4 Another Form of SWN 14.2 Global Analysis 14.2.1 Homeomorphism 14.2.2 Gauss–Bonnet Theorem 14.2.3 Atiyah–Singer Index Theorem 14.2.4 Seiberg–Witten Theory 14.3 Bioinspired Mechanics 14.3.1 Irreducible Structure of Life 14.3.2 Tight Coupling versus Loose Coupling Appendix A14

748 748 748 749 752 754 756 756 756 759 761 763 763 765 772

Flow-Evolutionary Law as a Working Hypothesis 774 15.1 Introduction 774 15.2 Working Hypothesis for Flow-Evolutionary Law 774 15.3 Field-Theoretical Fluctuation-Dissipation Hypothesis 778 15.4 Application Examples: 1 780 15.4.1 Bauschinger Strain ε B in DP Model ((V f )HP = 50% ) 780 15.4.2 Surface Roughness Ra in DP Model ((V f )HP = 15% ) 782 15.4.3 Strain Distribution Compared with Experiment via EBSD–Wilkinson Technique 785

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Contents

15.4.4 Comparison of Evolved Dislocation Substructures and Bauschinger Behavior 15.4.5 Virtual Experiment on Single Crystals with Instability-Triggered Ductile Fracture 15.4.6 Detecting Onset of Deformation Twinning 15.5 Application Examples: 2: Discrete Dislocation Systems 15.5.1 Instability of Cell Wall in Scale A 15.5.2 Stored Energy in 2D Walls 15.5.3 Collapsing 3D Walls 15.5.4 Unified Evaluation of GNBs 15.5.5 Dumping Effect of Dislocation Segments on Reducing Apparent Elastic Modulus 1 15.5.6 Dumping Effect of Dislocation Segments on Reducing Apparent Elastic Modulus 2 15.6 Summary: Toward Practical Applications Appendix A15  Comments on Strain-Gradient Plasticity

xiii

786 788 793 794 794 797 799 801 801 803 807 807

References809 Author Index833 Subject Index835

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Preface

For a long time, one of the dreams of researchers working in the field of mechanics of materials has been to construct a viable theory that seamlessly connects various length scales. Hopefully this will stretch from electrons to engineering structures, covering deformation and strength properties, leading ultimately to the development of a new, practically feasible framework applicable to material/structural designs based on it. Among others, spontaneous types of substructuring/patterning that are known to occur in deformation-induced manners, in particular, are always the steepest hurdles to overcome in multiscale approaches. This is in that sense that they include processes from out-of-equilibrium states accompanied by energy dissipation or require external treatments such as those via phase-field simulations, in addition to the deformation analyses; such processes are not only generally too large for atomistic treatments or even a discrete dislocation scheme to explicitly deal with, but also too small and/or complex for conventional continuum-mechanics-based approaches. No theory promising this scope, or even pointing in this direction, has been proposed hitherto, except some optimistic views based on multimillion atomistic and massive dislocation-based simulations, or slogans that start with a roar and end with a whimper based on “information-passage” types of merely enumerated schemes. These views, against their wills maybe, have taught us that there still remains quite a long way for us to go, and many things for us to do, at least until peta-flop-CPU (central processing unit)-driven super-parallelized computations become available. Further serious problems would also be the lack of “methodology” or “principles” for examining and processing the vast amount of information generated by simulations down to the smallest detail, even within such a dreamlike environment. What we need right now is a bird’s-eye (overhead) view of the whole materials system in terms of, for example, plasticity, beyond a cause-and-effect (causal)-based premise, including the descriptive/predictive capabilities of spontaneous patterning of deformation-induced events. Furthermore, real relationships are not time-ordered in general, taking place concurrently, sometimes synchronously, and, at the same time, self-consistently to a large extent, but not always in a one-way manner from micro to macro, particularly if such self-substructuring prevails. If this was not the case, the so-called information-passage type multiscale schemes could work all the time, although this seems to be insufficient in many practical situations.

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xvi

Preface

Turning to the theoretical aspects of these problems, one will notice that, although several schools of thought have proposed candidate theories, the theories have two things in common. First, many of them seem to have not demonstrated their descriptive capabilities in an explicit manner beyond the complexities of their mathematical formulations. The second commonality is manifested in what E. Kröner mentioned in the summary of his lecture at Stuttgart in 1997, regarding the differential geometric formalism of plasticity, namely that “this geometry permits an elegant formalism of dislocation theory, but not yet in such a way that elasto-plasticity becomes easier” (Kröner, 1998). This may be paraphrased such that a new theory should have an ability to convert complicated contents in the conventional counterparts into much ­simpler ones, or else it does not deserve the name. I believe that these two things need to be kept in mind. This book aims to provide not only an overview of my own field theory that has the potential to solve many of these problems, but also rather idiosyncratic perspectives, and their specific ideas, that are seriously directed to successful multiscale modeling and simulations of the plasticity of solids. In the book I try not just to present an encyclopedic list of previous theories, models, and numerical schemes proposed over various scale levels. Rather than attempt to cover every mathematical detail, the book tries to furnish a concise minimum set-up necessary to understand the basic concepts, which I believe will greatly assist in opening the reader’s eyes to the “landscape” of the field so that they may be able to tackle multiscale-related problems as quickly as possible and, further, to study them in greater depth. The topics chosen for the “Fundamentals” are also directed to aspects that conventionally are not particularly highlighted, regardless of their significance, and, furthermore, the close interrelationships among notions that seem to have no interconnections at first glance, with special emphasis on application aspects. All are intended to be directly/indirectly used in practical situations, not just for the researcher’s curiosity. For some entities, such as differential geometry, gauge theory, and quantum field theory (QFT), which may not be familiar for researchers working in the field of mechanics of materials, I have tried to include a brief introduction in a manner that is as concise as possible. In a lecture, one usually elaborates the picture layouts for systematic, intuitive, and instantaneous understanding, often different from the way one does when writing. One of my attempts was to combine those two merits, that is, theatrical presentation of the text contents. According to my experience, this seems to be effective not only in presenting complex concepts but also in explaining intricate algebraic derivation processes full of equations by extracting the essence. Two- or three-dimensional interrelationships can be expressed, simultaneously highlighting the important portions, for example, quantities, terms, and so on. To this end, I sometimes intentionally made more than one description of the same content. The book is organized into four parts – Fundamentals (Part I), Theoretical Backgrounds (Part II), and two parts on Applications (Parts III and IV). Part I describes basic notions about dislocations and crystallography in a somewhat unique fashion emphasizing, for example, the distinction between face-centered cubic

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Preface

xvii

(FCC) and body-centered cubic (BCC) metals (Chapter 1), dislocation dynamics as a basis of constitutive framework construction (Chapter 2), universal features and roles of dislocation cell structures (Chapter 3), and distinction between single crystals and polycrystals in contrasting fashions (Chapter 4). Part II provides three major foundations of the current field theory – differential geometry (Chapter 6), gauge theory (Chapter 7), and method of QFT (Chapter 8) after giving an overview of field theory (Chapter 5). In Part III, after identifying three important scales in polycrystalline plasticity and their overviews (Chapter 9), the up-to-date application results and the achievements in the individual scales – A, B, and C (Chapters 10, 11, and 12) – are presented in some detail. Part IV addresses further extension of the field theoretical scopes (Chapter 13) including outlooks (Chapter 14). An up-todate working hypothesis called “Flow-Evolutionary Law” as the core of the field theory of multiscale plasticity (FTMP) is presented in Chapter 15, together with its 12 application examples. The text materials, especially many of the pictures and figures, have been compiled in a systematic manner while teaching a graduate course entitled “Multiscale Solid Mechanics” at the Department of Civil & Environmental Engineering, at the University of California, Los Angeles (UCLA) in the 2006 spring quarter. Since most of the students taking the course were unfamiliar with “dislocations” and “physical metallurgy,” I constructed the pictures and figures about the fundamentals for that course, these now form the basis of the “Fundamentals” part. I have presented graduate-course lectures on “Multiscale Solid Mechanics” at Kobe University since 2004, in which I have attempted to give an “experimental” series of lectures providing new and not well-documented scopes and perspectives together with concepts that are unfamiliar to the mechanical-engineering students. The purpose was to provide them with intensive environments and experiences and to share a process of constructing novel concepts and theoretical frameworks out of nothing by not just relying upon their own discipline, because, especially in Japan, students are too accustomed to “learn” ready-made, well-organized concepts and theories unquestioningly (without feeling any doubt) and without having constructive arguments among themselves. This, I feel, has historically stopped us producing novel concepts and theories to cover more situations. So, the reactions of the students and the in-class discussions based on those were very stimulating and useful. Those experiences are revisited in the rest of the chapters. This book is actually a written version of such a series of trials.

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Acknowledgments

I am deeply indebted to Masateru Ohnami, who opened the door for my research career and further gave me an initial hint for the field theory construction, who himself once had promoted the application of the non-Riemannian plasticity to engineering problems, and has published a book titled Introduction to Micromechanics (in Japanese), based on achievements of a research activity on mechanics of generalized continua in Kyoto (GCM). This book has been a bible to me especially for constructing the differential geometrical part of the field theory, detailed in Chapter 6. Ohnami has also published two books in English, entitled Plasticity and High Temperature Strength of Materials: Combined Micro- and Macro-Mechanical Approaches (Elsevier, 1988) and Fracture and Society (Ohmsha, 1992). I am grateful to Ali S. Argon for his constructive comments on the early version of the draft manuscript of the book. He hosted my sabbatical leave from 2000 through 2001, giving me a great opportunity to coorganize a series of four special symposia at the Department of Mechanical Engineering, at the Massachusetts Institute of Technology (MIT) in spring 2001. He is a man of real knowledge, and I always enjoyed discussions with him during my stay there and every time I visited him afterwards. The content provided in Appendix A10 is one of the topics I produced as a contribution to the symposia. Thanks also go to the never-failing enthusiasm toward research of F. A. McClintock, who encouraged me to cultivate a new field by telling me that he hoped someone would develop a “flow-evolutionary law” in the place of conventional “stress–strain” relationship during the symposia. The “flow-evolutionary” law proposed in Chapter 15 is named after this. I would like also to thank Yoshihiro Tomita, who has been encouraging my activity from its early stage, which has given irreplaceable incentives. I started to collaborate with him in 2003, which was the start of the series of simulations on polycrystalline aggregates in Scale C, extended in Chapter 12, two years before I moved to Kobe University to be his colleague there. I would like to acknowledge Jiun-Shyan Chen for providing me a valuable opportunity to teach a graduate course on “Multiscale Solid Mechanics” at UCLA in the spring quarter of 2006. The materials from this course form a major foundation of the “Fundamentals” and “Theory” parts. I benefitted from many fruitful technical discussions with Nasr Ghoniem at UCLA as well as Nicolas Kioussis at California State University, Northridge (CSUN) particularly on the Project #1 in Chapter 9, collaborations with M. Sugiyama and the colleagues in Nippon Steel, Co. Ltd. on Project #2, and those with T. Wakai and S. Onizuka at the Japan Atomic Energy Agency (JAEA) on Project #3.

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Acknowledgments

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While writing the book, I benefited from many discussions with colleagues in the field, but particularly with Yoji Shibutani, Kazuyuki Shizawa, Tetsuya Ohashi, and Yuji Nakasone. I would like to acknowledge the important contributions made by numerous colleagues in Doshisha University and Kobe University, Japan. Last but not least, I should like to mention my wife, Fumi, and my treasures, Io, my daughter, and Maasa, my son, for their continuous support, patience, and understanding. I have so often been inspired by their young and prodigious talents.

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Part I

Fundamentals 1 2 3 4

Dislocation Theory and Metallurgy Dislocation Dynamics and Constitutive Framework Dislocation Substructures: Universality of Cell Structures Single Crystals versus Polycrystals

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https://doi.org/10.1017/9781108874069.002 Published online by Cambridge University Press

1

Dislocation Theory and Metallurgy

After a brief discussion of “plasticity” and “elasticity,” the chapter presents a minimal set of basic concepts about “dislocations.” Following an overview of dislocation theory, specific notions such as the “Lomer–Cottrell (LC) sessile junction” and “stacking fault energy” (SFE) are detailed. These are exceptionally important to gain a comprehensive understanding of many of the characteristics of dislocation–dislocation interactions and their strengths in particular. The next part of the chapter provides a simple introduction to metallurgy, especially to crystallographic structures, placing special emphasis on the substantial distinction between face-centered cubic (FCC) and body-centered cubic (BCC) structures, which is expected to further understanding of the associated contrasting features between the two.

1.1

Elasticity versus Plasticity Let us take steel as an example to clarify the distinction between “plasticity” and “elasticity,” although their names are similar. Figure 1.1.1 presents a relatively diverse range of plasticity-related mechanical properties, such as yield stress, maximum tensile stress, hardening characteristics, and ductility measured by uniform elongation. Two to three orders of difference can be found, for example, in the yield stress, that is, from tens of MPa up to a few GPa. The elastic properties such as Young’s modulus, however, are not basically altered; they usually have values of around 200 GPa, even for alloys containing a number of alloying elements, those associated with metallurgical microstructures produced via heat treatments, and those with different crystal structures, that is, FCC austenite (SUS304 [or type 304] stainless steel, for instance). Such insensitivity is attributed to the origin of the elastic deformation in metals. The elastic modulus is a manifestation of the resistance against the interatomic bonding force that displaces the composing atoms, which is substantially determined by electronic interactions (hence, based on quantum mechanics). Thus, it cannot be controlled (altered) in principle. Plasticity, however, can be artificially controlled relatively easily, because it is carried by the motion of dislocations. Further contrasting features are also indicated in Figure 1.1.1, and are summarized in Figure 1.1.2, including (a) “controllable versus uncontrollable,” (b) compressive versus incompressive, and (c) conservative versus dissipative. The former is manifested as Poisson’s ratio in elasticity, which is around   0.3 for metals, allowing

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Dislocation Theory and Metallurgy

Figure 1.1.1  Schematic comparison of stress–strain curves for various steels of mild to high-

strength types in terms of elastic and plastic properties. Typical contrasting features between elasticity and plasticity, (a) and (b), are also indicated.

Figure 1.1.2  Comparison between “elasticity” and “plasticity,” consolidated into three typical

categories, where category (a) leads to the “dislocation” concept that follows.

volumetric change (e.g., tension by 1 in one direction results in −0.3 in the other two directions, respectively, resulting in a volumetric strain of  v  0.4, that is, an increase in volume). Plasticity, on the other hand, exhibits volume constancy in general, simply because it is brought about by shear deformation, or, more precisely, slip deformation of the crystal lattice. For the latter, (b), more than 90% of the work done by plastic deformation, measured by the area swept by a stress–strain curve, is known to be dissipated into heat, meaning that plasticity is a kind of nonequilibrium (farfrom-​equilibrium) process, whereas elastic strain energy is fully recoverable. Note that these twofold aspects of the elastoplasticity in metals play crucial roles when we think about evolutionary aspects of inhomogeneities in the present field theory of multiscale plasticity (FTMP) (cf. Chapter 15).

1.2

Fundamentals of Dislocations

1.2.1 Overview Figure 1.2.1 presents the reason why the notion of dislocation arose. The concept itself had been introduced many years before it was micrographically confirmed (observed),

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1.2  Fundamentals of Dislocations

5

Figure 1.2.1  The extremely large gap between ideal strength for shear deformation and

experimental observation for yield stress triggered the birth of dislocation theory (Taylor, 1934a). This story resembles that for fracture mechanics (Griffith, 1921).

in order to explain why the empirically observed yield stress is much smaller than the estimate based on the ideal strength under shear (Orowan, 1934a, 1934b; Polanyi, 1934; Taylor, 1934a, 1934b; Yamaguchi, 1928). The ideal strength needed for shear deformation to occur predicts one-fifth of the shear modulus, whereas experiments show 0.5–10 MPa for pure metals, resulting in a three- to four-order difference in magnitude. In the 1930s, Taylor, Orowan, and Polanyi independently advocated the concept of dislocation to explain this gap. Also, we must remember that V. Volterra, an Italian mathematician, introduced essentially the same notion purely within the framework of the mechanics of continua (Love, 1944; Volterra, 1907). This can be acknowledged as the origin of the notion of continuously distributed (CD) dislocations (cf. Figure 6.2.22). This story is quite similar to the case of the birth of “fracture mechanics”; A. A. Griffith, an English aeronautical engineer, proposed the basis for this in 1921 (Griffith, 1921). Figure 1.2.2 indicates the motion of a dislocation, which is often compared to a crawling inchworm. Another intuitive metaphor is a row of wrinkles in a carpet. In order to shift the carpet position, what we need to do is to transfer the “wrinkle row” to the edge of the carpet, resulting in a shift of the carpet position by the amount of the wrinkle-row width. Moving the entire carpet, even a millimeter, requires extraordinary strength, as you can imagine. The most important thing here is that plastic deformation is carried by the motion of dislocations. Accordingly, the stress needed for plastic deformation to occur can be completely replaced by that needed for moving the dislocations, which are generally preexisting within crystalline samples. Another significant factor to bear in mind is that the plasticity (i.e., the resistance against plastic deformation) can easily be altered by introducing various “obstacles”

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Figure 1.2.2  Schematics of a dislocation as a carrier of plastic deformation, often compared to

a crawling inchworm or wrinkles in a carpet.

Figure 1.2.3  Schematic drawing of strengthening mechanisms showing various kinds of

resistance against dislocation motion using motor vehicles as examples (inspired by an illustration by Tanino, 1966).

to obstruct dislocation movements. This means that the plasticity can be artificially controlled (or we should say, freely controllable), which is substantially different from “elasticity” that is uncontrollable in principle, as stated in Section 1.1 (see Figure 1.1.2(a)). Figure 1.2.3 illustrates representative obstacles against the motion of dislocations in a cartoonish manner (adapted from Tanino, 1996). They are (b) impurities or solute atoms for solution hardening, (c) small precipitates for precipitation hardening, (d) dislocations for dislocation hardening (resembling traffic congestion), and (e) grain boundary (GB) for GB hardening.

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1.2  Fundamentals of Dislocations

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Figure 1.2.4  Schematic illustration of strengthening mechanisms via (a) solid solutions

(interstitial and substitutional types) and (b) precipitations (coherent and incoherent types), together with (c) schematics about interstitial-free (IF) steel, and (d) the attendant improved stress–strain relationship with respect to “nonsmooth” yielding.

Practically, especially in high-strength steels and metals, these strengthening mechanisms are combined to achieve the desired strength properties. More detail on (b) and (c) can be found in Figure 1.2.4(a) and (b), respectively; in Figure 1.2.4(a) two types of solution, interstitial and substitutional types, are schematically depicted, while two types of precipitate are indicated in Figure 1.2.4(b), coherent and incoherent. Figure 1.2.5 shows a schematic indicating the Cottrell atmosphere typical in commercially pure Fe or low-carbon steels, where carbon and nitrogen atoms tend to gather around (below) an edge dislocation – below because of the larger space due to the extra half plane of an atomic layer intruding from above. Under stress, trapped dislocations must break away to start moving, causing nonsmooth yielding in mild steels (Figure 1.2.4(c)), which can also trigger inhomogeneous postyield plasticity, for example, Lüders elongation, which is normally undesired in practical situations. Note that, for steels, even a ppm order of C and/or N can cause such “yielding” phenomena in Fe and steels. Since the complete removal of them is quite difficult practically, they are inactivated by adding Ti and/or Nb to form TiC/NbC, which anchors these interstitials. Such steel is known as IF steel (Figure 1.2.4(c)) and is further detailed in Section 1.4.2. Looking again at Figure 1.2.4(a)–(d), we can review the previous discussion, together with schematic stress–strain curves (d) indicating nonsmooth and smooth yielding, corresponding to that associated with the Cottrell atmosphere (Figure 1.2.5)

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Figure 1.2.5  Schematic of Cottrell atmosphere for iron around an edge dislocation, where the

insertion of an extra half plane from the top is illustrated for the purpose of emphasizing the induced compressive/tensile stress field above/below.

and IF steels, respectively, from which we also recognize the importance of impurities in metals and alloys in terms of their roles and thus their pertinent controls. By appropriately combining the strengthening methods (b) through (e) in Figure 1.2.3, along with well-controlled heat treatment processes, we can design “microstructures” to achieve desired strength-related mechanical properties. A wide variety of such metallurgical microstructures for Fe and steels, together with typical stress–strain curves, are presented in Figure 1.2.6(a) and (b), while representative structural factors such as characteristic sizes are also indicated in Figure 1.2.6(a) (Tomota, 2001). A noteworthy example in the present context is the lath martensite structures in high Cr heat-resistant ferritic steels, for example, Mod. 9Cr-1Mo steel, where the alloys are strengthened not only by lath martensite structures with high dense dislocations (Figure 1.2.6(a), (e), and (d)) and the associated block/packet structures (Figure 1.2.6(a) and (e)), but also by precipitations via V/Nb additions, that is, MX/ M23C6 (Figure 1.2.6(a) and (c)), together with W/Mo solid solutions (Figure 1.2.6(a) and (b)). As noted, all the strengthening methods are combined to achieve an excellent high-temperature creep strength. One more thing worth mentioning concerns the hierarchically emerging nature of the strengths as summarized in Figure 1.2.7, revealed via the multisized indentation technique (NIMS, 2003). As can be seen, the macro-strength indicated by the broken line cannot be achieved until the indentor size becomes large enough, whereas, for example, the scale corresponding to the minute lath with high dense dislocations does not support the strength alone. This implies there exists an intimate interplay among the composing hierarchical scales for achieving the macroscopically observed strength, meaning “the partial sum is not necessarily the whole.” A schematic stress–strain relationship is also shown on the right, whose stress levels are built up from the base strength. Therefore, modeling such complex material systems requires a “genuine” multiscale perspective, which is tackled in Chapter 9 and based on the FTMP discussed in the present book.

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1.2  Fundamentals of Dislocations

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Figure 1.2.6  Typical metallurgical microstructures for Fe and steels (a) as a variety of

combinations of strengthening mechanisms in Figure 1.2.3, together with typical stress–strain curves (b) for representative microstructures.

Let us return to “dislocations.” Figure 1.2.8 is a micrograph showing an atomistic image of a dislocation (dislocated region). A Burgers circuit encircling the disturbed region is drawn, showing that the Burgers vector b surely exists. This proves or corroborates the single-edge dislocation existing within the circuit. It should be noted that the notion of a dislocation “line” is conceptual in the sense that it is not a substantial object, but a “state” of being disturbed in a background (crystalline) field existing a priori. This implies that the notion of “field” is suitable for describing dislocations, as will be frequently mentioned throughout the book. A multilevel image of a dislocation field is presented in Chapter 5 (Figure 5.1.1) in terms of three representative field theories.

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Figure 1.2.7  Example of appropriate combinations of all the strengthening methods in

Figures 1.2.3 and 1.2.4, that is, high Cr ferritic steel composed of martensite lath/block/packet structures embedded within prior austenitic grains, whose hierarchically emerging strengths are revealed via multiscale indentation tests (NIMS, 2003).

Figure 1.2.8  Atomistic image demonstrating the existence of dislocation, where the Burgers

circuit encircling a dislocated region is shown to produce closure failure measured by the Burgers vector (courtesy of Prof. Julia R. Greer).

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1.2  Fundamentals of Dislocations

1.2.2

11

Types of Dislocation There are three types of dislocation – edge, screw, and mixed. The former two are more important than the third, since the last is expressed by combinations of the other two. Various ways to draw or schematize an edge dislocation exist, as shown in Figure 1.2.9. Here we provide several typical examples found in conventional textbooks. Some emphasize an extra half plane (b) and some highlight the Burgers circuit or closure failure (c). Some are suitable for stress-field calculation (a). It is important to observe that the direction of the dislocation line l is always perpendicular to the Burgers vector, b, that is, l ⊥ b, which is the mathematical definition of the edge dislocation component. One can note that the intuitive image of the edge dislocation is quite tangible compared with the screw counterpart. However, as will be discussed later, the stress field produced around an edge dislocation is much more complex than that for the screw dislocation, and, furthermore, is difficult to calculate. The former has both the normal and shear components of stress, whereas the latter has shear components only. Screw dislocation is often exemplified by “stair-case steps.” Figure 1.2.10 shows various schematics depicting a screw dislocation. The important thing again is the relationship between the directions of dislocation line and the Burgers vector, that is, they must be mutually in parallel, l // b. This allows screw dislocations to cross slip onto other intersecting planes sharing the same Burgers vector. Also the screw dislocation is a good example for understanding the topological nature of dislocations in terms of multivaluedness. (This is also true for the edge dislocation, but the present

Figure 1.2.9  Various representations of an edge dislocation.

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Figure 1.2.10  Various representations of a screw dislocation.

case is more intuitive.) The starting point, after encircling the Burgers circuit, does not coincide with the ending point, as can easily be confirmed with any one of the examples shown in the figure. This emphasizes that dislocation is a topological object or imperfection. It is also important for understanding the differential geometrical aspects of dislocation theory (to be detailed in Chapter 6). This closure failure means torsion of the space in the context of differential geometry. Another type of dislocation is termed a “mixed” component. Figure 1.2.11 displays various schematics. It is usually very difficult to gain a clear image of the mixed portion of a dislocation. Fortunately, the mixed components can always be resolved into edge and screw components and expressed by combinations of the two. As is understood from the so-called Volterra operation, explained in Figure 1.2.12, dislocation can be viewed as a region in a medium (crystalline body) where there is a boundary between “slipped” and “nonslipped” regions. Since the “slipped” region has experienced shear deformation, the distribution of shear strain for a dislocation loop becomes similar to that represented in Figure 1.2.11(b). You can notice the “strain gradients” where dislocation lines exist; this will be revisited in Section 6.5.1 (Figure 6.5.1) in Chapter 6. One should notice the difference between real objects and dislocation as an excitation of a medium. As schematically shown in Figure 1.2.13, the screw dislocation line, for example, moves perpendicularly to the direction of the force (shear stress), which seems to be against the rule of common-sense mechanics. The edge component, on the other hand, moves in the same direction as the applied (shear) force in accordance with our intuition. The former resembles the act of peeling a poster off a wall from one corner, where the boundary between the already-peeled off and unpeeled regions corresponds to the screw dislocation. Figure 1.2.14 provides more tangible images for both cases.

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Figure 1.2.11  Schematics of a mixed dislocation (a), together with a dislocation loop and corresponding strain distribution (b), where regions with finite strain gradient detect a ­dislocation line.

Figure 1.2.12  Schematic illustration explaining the “Volterra operation,” providing another representation of the dislocation loop that can be well described mathematically based on micromechanics.

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Figure 1.2.13  Comparison of moving directions between edge and screw components against external shear stress.

Figure 1.2.14  Atomic lattice-based representation of moving dislocations against external shear stress comparing the directions between them for edge and screw components (Smallman, 1970).

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1.2  Fundamentals of Dislocations

1.2.3

15

Stress Field around Dislocations One important feature of dislocations is their long-range nature, that is, inversely proportional to the distance from them r, ∝ 1 / r . Not only does this make numerical treatments difficult, for example, the “cutting-off” treatments in discrete dynamics simulations, but it also provides the origin of the long-range stress field evolved within dislocation s, differentiating them from other substructures, to be discussed in Chapter 3. Here, we deal with the most fundamental case of single straight dislocation lines. For straight dislocations, it is relatively easy to find the stress fields around them in the sense just discussed, although the edge dislocation requires some elaborate techniques. This is largely due to the absence of the self-stress in the straight segments, whereas, for curved dislocation lines, the treatments become extremely complicated, as briefly mentioned in Section 6.4.

1.2.3.1

Screw Dislocation To obtain the stress field around a straight screw dislocation, a typical setup, together with a cylindrical polar coordinate system (r , θ , x3 ) , is depicted in Figure 1.2.15, where the displaced cylinder along the axial direction coincides with x3 is prepared, containing a coaxially extending screw-dislocation line. The displacement field in the x3 direction is easily expressed as



u3screw 

x  b b tan 1  2  (1.2.1)  2 2  x1 

screw screw Otherwise, u= u= 0. From Eq. (1.2.1), the strain field and, thus, stress field, 1 2 can be obtained. They are given, respectively, as

Figure 1.2.15  Cylindrical polar coordinates introduced around a straight screw dislocation, used for finding the stress field therein.

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 ijscrew

 0  0   0  SYM 

b 1 4 r   (1.2.2) 0  0 

 ijscrew

 0  0  0  SYM 

b 1  2 r   , (1.2.3) 0  0 

and



where µ is the shear modulus and b the magnitude of the Burgers vector, b ≡ b .

1.2.3.2

Edge Dislocation The inset in Figure 1.2.16 shows the setup for obtaining the stress field for a straightedge dislocation, which is basically the same as that for the screw dislocation. Since no displacement in the x3 direction exists, we can assume the plane strain condition. In this case, even if it is straight, one needs a special sort of technique, because the treatment of the displacement field is not straightforward as in the screw counterpart: There is a jump at    which prevents us from expressing u1 by a simple function (unlike in the screw case), because doing so violates the stress-equilibrium condition. To cope with this, it is necessary to use an elaborated stress-function method, in which we need to seek a suitable form of the stress function that satisfies the biharmonic equation, that is,  4   0 with the Laplacian

Figure 1.2.16  Cylindrical polar coordinates introduced around a straight-edge dislocation, to be used for finding the stress field therein.

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1.2  Fundamentals of Dislocations

2 



17

2 1  1 2  , (1.2.4)  r 2 r r r 2  2 2

 2 1  1 2  4  2   2    (1.2.5) r r r 2  2   r





for the cylindrical coordinates. For the edge dislocation problem, the following form of the stress function can be used: b  (r , )  R(r ) ( )  r sin  ln r 2(1  ) (1.2.6) 1/ 2 b  y ln x 2  y 2 , 2(1  ) from which we readily obtain





 b sin    rr      2(1  ) r  . (1.2.7)       b cos  r  r 2(1  ) r



In Eq. (1.2.6), R(r ) ( ) is intended to emphasize the separable nature of the stress function, in this case via respective functions of r and θ . Since the present case satisfies the plane-strain condition, another component is given by  zz   ( rr    ). Note that a more sophisticated method based on micromechanics is also available (Mura, 1963). For the anisotropic case, refer, for example, to Asaro et al. (1973) and Willis (1970). The stress field around a straight-edge dislocation is rewritten in matrix form as

 b sin     2 (1  ) r  edge   ij     SYM  



 b cos 2 (1  ) r  b sin   2 (1  ) r

    0  , (1.2.8)   b sin      (1 ) r  0

where not only the shear component (off-diagonal) but also the normal components (diagonal) exist. Detailed derivation processes can be found in Hirth and Lothe (1982), Kato (1999) and Suzuki (1967). The corresponding expressions to Eqs. (1.2.3) and (1.2.8) with respect to the Cartesian coordinates are given by

 ijscrew



and

  b x2 0   0 2 x12  x22    b x1  0  2 x12  x22  0 SYM

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    (1.2.9)    

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Figure 1.2.17  Stress field around straight screw and edge dislocations with respect to the Cartesian coordinate system.



 b x2 (3 x12  x22 )  2 2 2  2 (1  ) ( x1  x2 )   ijedge      SYM  

b x1 ( x12  x22 ) 2 (1  ) ( x12  x22 )2 b x2 ( x12  x22 ) 2 (1  ) ( x12  x22 )2

     , (1.2.10) 0    b x2    (1  ) x12  x22  0

respectively. These results are summarized in Figures 1.2.17 and 1.2.18. One of the most important aspects for us to recognize about the stress fields produced by a dislocation is its long-range nature, for example, decaying in proportion to 1 / r (see Eqs. (1.2.3) and (1.2.8)). This actually introduces many computational complexities in dealing with dislocation-dislocation interactions (see also Section 1.4.4). The corresponding strain fields are

 ijscrew



 b x2 0   0 2 4 x1  x22   b x1  0  4 x12  x22  0 SYM

and

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    (1.2.11)    

1.2  Fundamentals of Dislocations

19

Figure 1.2.18  Stress field around straight screw and edge dislocations with respect to the ­cylindrical polar coordinate system.





 b x2 (2  3 ) x12   x22   2 (  2  )( x12  x22 )2   edge  ij     SYM 



b x1 (xx12  x22 ) 2 (1  ) ( x12  x22 )2



2 2 b x2 (2   ) x1   x2 2 (  2  )( x12  x22 )2



 0     0  (1.2.12)  0 

The displacement field is obtained when the strain field in Eq. (1.2.12) is integrated as







siin   (1  2 )   2(1  ) sin  ln r  4(1  )   cos     b  (1  2 ) cos  cos  ln r     sin   (1.2.13) 2  2(1  ) 4(1  )    0  

uredge   edge  u  uedge   z 

For Cartesian coordinates, the corresponding expression is x1 x2   1 x2   tan x  2(1  )( x 2  x 2 ) 1 1 2 u1edge    2 2  edge  b  1  2 x x   1 2 (1.2.14) ln( x12  x22 )  u2   2 2 4(1  )( x1  x2 )   edge  2  4(1  ) u  3    0

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Figure 1.2.19  Hydrostatic stress distribution around screw and edge dislocations. Nonzero hydrostatic stress for edge components brings about interactions with point obstacles.

1.2.3.3

Stress Contours Let us consider the hydrostatic stress  m  ( 11 22  33 ) / 3 for both the components as a representative. Figure 1.2.19 shows the hydrostatic stress fields for both the dislocations. Screw dislocation always yields zero hydrostatic stress, that is,



 mscrew  0, (1.2.15) simply because of the absence of the normal component of stress, as in Eqs. (1.2.3) or (1.2.9). For edge dislocation, on the other hand, the hydrostatic stress is calculated as



 medge  

 b  1    x2 (1.2.16) 3  1   x12  x22

One can immediately notice that this is in proportional to the σ 33 component. Therefore, the hydrostatic stress field around a straight-edge dislocation, displayed in the inset of Figure 1.2.19, is identical to that for σ 33, as far as the profile is concerned. As can be imagined from the atomic configuration, we have a compressive stress field in the upper region, due to the insertion of an extra atomic layer, and a tensile stress field in the lower region. The latter tends to attract interstitials, as discussed in Figure 1.2.5 in the context of the Cottrell atmosphere.

1.2.4



Elastic Strain Energy of Dislocations Based on the stress field we have just obtained, we can evaluate the strain energy for the dislocations (Figure 1.2.20). For a straight screw dislocation extending infinitely along the x3 direction, we have 1 E screw    ijscrew ijscrew dx 2 V  R  1 2   d  dr  dx3  screw screw , (1.2.17) x3 x3 r 0  0 2   b2   R 1 2   d  dr  dx3  2 2   r0 2 0  8 r 

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1.2  Fundamentals of Dislocations

21

Figure 1.2.20  Strain energy evaluation for a straight screw dislocation.

Figure 1.2.21  Strain energy for a disclination dipole, demonstrating its mathematical equivalence to that for an edge dislocation line, together with that for a single disclination line as well as its stress field. 

Since  dx3 is proportional to the length, we can find an explicit expression per unit  length as

E screw  0

 b2  R  ln   , (1.2.18) 4  r0 

where R and r0 are the upper and lower bounds in the radial integral with r0 ∼ 5b corresponding to the core radius. Note that since, with R  , E screw logarithmically 0 diverges, we need to set a cut-off radius, normally taken as being a mean free path of the order of 10 µm. Here the subscript “0” shows that the quantity is represented per unit length, that is, E screw ≡ E screw / L . 0 Note that the same logarithmic-type strain energy representation can be obtained for a disclination dipole, as concisely summarized in Figure 1.2.21, although a single disclination line produces the strain energy in proportion to the square of the sample

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Figure 1.2.22  Rough estimation of strain energy of dislocations, together with that for core

regions.

size, that is, R 2, which strongly inhibits its existence within metallic crystals in general. Here, Ω represents the Frank vector that specifies disclinations, corresponding to the Burgers vector against dislocations. More details are mentioned in Chapter 6, in the context of differential geometry. Similarly, for a straight-edge dislocation we have

E edge  0

R  b2 ln   ; (1.2.19) 4 (1  )  r0 

this also relates to per unit length. Since both cases yield the same form, we may roughly express them together (as in Figure 1.2.22):

E disloc   b2   0.5  1 (1.2.20) 0 The energy of the core region can normally be regarded as about 10% of E disloc . 0 Hence, it can be taken into account altogether in E disloc by replacing r0 ~ 5b with 1b. It is important to remember that the energy of dislocations, whatever the types, are given in proportion to b2. Figure 1.2.23 provides an example of the energy of a screw dislocation for Cu. We have E screw  2.5  10 9 J per unit length, and 6.4  10 19 J per atom. This roughly 0 corresponds to 4 eV, which is much larger than that for a vacancy (∼1 eV), meaning the dislocations are thermodynamically unstable within a crystal (Kato, 1999). Note that 1 eV  1.602 176  10 19 J. For a mixed component, a simple superposition can be utilized, since there are no overlapping components of stress fields between the two, as can be confirmed by

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1.2  Fundamentals of Dislocations

23

Figure 1.2.23  Example of strain energy for Cu.

Figure 1.2.24  Strain energy for straight mixed dislocation per unit length, given as a simple superposition of those for edge and screw components, rationalized due to the nonoverlapping of stress-field components between the two.

Eqs. (1.2.2) and (1.2.3). Figure 1.2.24 shows the process to obtain the corresponding energy per unit length, that is, E mixed  E edge cos2   E screw sin 2  0 0 0

 R  (1.2.21)  b2 1  cos2  ln   , 4 1   r0  where θ is the angle between the two components.

1.2.5

Dislocation Processes (Important Features)







This subsection presents and details several important dislocation processes. One immediate feature that must be pointed out is that the dislocations are created and annihilated. We will discuss the multiplication (i.e., creation) due to the Frank–Read mechanism, and the annihilation due to cross slip (Figure 1.2.25), which is followed by the LC junction formation. In particular, for understanding the latter two, the SFE associated with dislocation dissociation is defined and its significances are emphasized. The

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Figure 1.2.25  Schematic drawing of pair annihilation of two dislocations of opposite sign, together with a possible process yielding the reaction via two screw dislocations cross slipped onto a common slip plane sharing the same Burgers vector.

cross slip is considered to be responsible for efficiently reducing dislocation density during the course of deformation, called “dynamic recovery.” This is further responsible for the dislocation cell formation, extensively discussed in Chapter 2.

1.2.5.1

Frank–Read Source and Multiplication Schematics of the Frank–Read source (Frank and Read, 1950) and associated multiplication mechanisms are presented in Figure 1.2.26. The multiplication process from the Frank–Read source is one of the most important mechanisms, among other possibilities. Consider a pinned segment (see Figure 1.2.16(a)) with a length L that, under the force f   b (step 1), starts bowing out (step 2) until the critical configuration (a half-circular arc [step 3]) is reached, after which the segment becomes unstable and continues to expand spontaneously. The critical shear stress τ cr corresponding to the critical configuration is given by



 cr 

2T 2 b  , (1.2.22) bL L

where T is the line tension of the dislocation segment evaluated as T   b2 with α a proportional factor. The critical stress is also called Orowan stress. If expansion continues, the curved segments tend to go around the pinning points from both sides (step 4) until they meet and react on the reverse side to leave an expanding loop (step 5). By repeating this process, dislocations can multiply. This can occur wherever similar pinned segments exist. This series of processes is schematized in Figure 1.2.26(b). Figure 1.2.26(c) provides an example of a double cross-slip event, which is considered one of the possible mechanisms for enhancing the Frank–Read multiplication process (sites for the Frank–Read source to occur). Examples of the experimentally observed Frank–Read source, in Si (via chemical etching) (Dash, 1957) and in age-hardened Ni-Fe alloy (via transmission electron microscopy [TEM]) (Murr, 2015, 2016) are presented in Figures 1.2.27 and 1.2.28, respectively. Here, the F–R source in Si yields an anisotropic shape, reflecting its anisotropy in the slip systems, whereas, for the Ni-Fe FCC alloy, almost isotropically

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Figure 1.2.26  Multiplication mechanism via the activation of a Frank–Read source based on the bowing-out behavior of a dislocation segment (a) and (b), together with a possible process by which this can occur (c), that is, a double cross slip, which can generate a pinned segment of a screw dislocation.

Figure 1.2.27  Example of experimentally observed Frank–Read source in silicon, showing successive generations of anisotropically expanding dislocation loops, which are delineated by chemical etching (Dash, 1957).

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Figure 1.2.28  Example of experimentally observed Frank–Read source in age-hardened Ni-Fe alloy, yielding circular multiplication of dislocation loops (Murr, 2015, 2016).

Figure 1.2.29  Example of a simulated series of snapshots for proliferating dislocation loops under shear stress on the [111] slip plane in FCC metal, based on the discrete dislocation dynamics method.

expanding dislocation loops are observed. Figure 1.2.29 displays a series of simulated snapshots of proliferating dislocation loops based on the discrete dislocation dynamics method. A closely related important interaction of a dislocation against second-phase particles (e.g., relatively large precipitates) to the bowing-out mechanism is the Orowan process (Orowan, 1984). A schematic is given in Figure 1.2.30, where bow-out dislocation segments around spherical particles ultimately leave loops of dislocation behind. These dislocation loops can further act as obstacles against subsequent

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Figure 1.2.30  Schematic illustration of the Orowan mechanism, depicting the dislocation interaction, with second-phase particles leaving behind Orowan loops. When a dislocation line tries to pass through the array of particles, it bows out around them to form loops, which will act as further obstacles against subsequent dislocations.

dislocation motions that pass through, increasingly enhancing the effective diameter of the particles, thus, with decreasing interparticle spacing, this efficiently contributes to strengthening; this is regarded as one of the important strengthening mechanisms in alloyed metals and is known as the Orowan mechanism. Figure 1.2.31 shows a series of snapshots for the Orowan process, simulated by utilizing the discrete dislocation dynamics method together with a precipitate model introduced in Yamada et al. (2008), demonstrating the formation of double Orowan loops. Notice that, in the double Orowan loops, the inner loops have slightly shrunk in diameter due to the stress field of the loop-forming dislocations, and, at the same time, the second loops are elongated in the stressing direction. The former can ultimately lead to the collapse of the particles as the Orowan process continues. More specifically, this process can take place rather exclusively against incoherent or partially coherent precipitates with relatively large interspacing (or the order of 100 nanometers). On the other hand, dislocations can cut through smaller and coherent precipitates by shearing. A comprehensive summary of this distinction is shown in Figure 1.2.32 (Sugimoto et al., 1991). Note that distinctions between the coherent and incoherent precipitates are shown in Figure 1.2.4(b) and revisited in Section 1.4.4 (Figure 1.4.16). For further details about strengthening by alloying in general, including more sophisticated and advanced treatments as well as their experimental verifications, refer to a comprehensive monograph by Argon (2012).

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Figure 1.2.31  Simulated series of snapshots of the Orowan mechanism up to the second Orowan loop formations, based on discrete dislocation dynamics, utilizing a precipitate model proposed in Yamada et al. (2008).

Figure 1.2.32  Comparison of mechanism for dislocation-precipitate interactions (Sugimoto et al., 1991). Adapted with permission of the publisher (Asakura Publishing Co.).

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1.2.5.2

29

Partial Dislocations and Stacking Fault SFE is one of the most important notions in understanding the diversity of mechanical behaviors of metals, especially those with an FCC structure, because it substantially controls the dislocation motions in terms of their interactions, such as junction formations and associated strain hardening, cross slip and resultant dynamic recovery, and further, attendant substructure evolutions. Also a sharp distinction can be found between FCC and BCC metals in the light of SFE. A dislocation can be split into two partial dislocations. Figure 1.2.33 shows an example for FCC metals, where the reaction is expressed as



a a a 101   1 12    211 , (1.2.23) 2 6 6 where a is the lattice constant. This cases shows Shockley’s partial dislocations. This 2 2 2 dissociation is energetically favorable when the SFE is absent, that is, b  b1  b2 , as confirmed by simple arithmetic. We must not, however, ignore the SFE to be added to the right-hand side of the inequality, as shown in Figure 1.2.34. This ultimately decides whether the dissociation takes place or not. The SFE is the interfacial energy for the imperfect stacking sequence of atomic layers situated between the extended dislocations (termed leading and trailing partials, respectively). The table in Figure 1.2.33 lists the values of SFE for typical metals. Roughly speaking, FCC

Figure 1.2.33  Dissociated dislocations (called partial dislocations) with a stacking fault located between them, which demands additional energy for creating a planer defect due to the imperfection of a stacking sequence of atomic layers called SFE. The table lists examples of SFE for typical metals.

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Figure 1.2.34  Schematic drawing of dissociation of perfect dislocation into a pair of Shockley partial dislocations for FCC metals, which is energetically favorable in the absence of ­stacking fault.

metals have relatively small SFE, while BCC metals yield extremely large values. Aluminum is known to have the largest values, of about 100–200 mJ m−2, among FCC metals, while Cu is recognized as having relatively small SFE, that is, 40 mJ m−2: They are frequently referred to as typical FCC metals with large and small SFE in the literature. The smallest SFE for FCC metals goes to Cu-Si alloys (e.g., Cu-8.8at% Si) with 3–5 mJ m−2 or less (e.g., Murr, 1975), followed by aluminum-bronze (Cu-about 10% Al alloys) and α-brass (Cu-less than 35% Zn alloys) with less than 10 mJ m−2, and austenitic stainless steels (e.g., 18–8 or type 304) with around 10–13 mJ m−2. It is commonly recognized that BCC metals basically do not have a stacking fault because it is not energetically favorable. The values of the SFE are extremely large in comparison even with that for Al. This means that dislocations in BCC metals substantially do not (or never) extend. Figures 1.2.35 and 1.2.36 show a stacking fault viewed from the top, each indicating shifts in the stacking sequence of atoms above and below, and raised electron density distribution in the SF (Suzuki, 1967), respectively. The inset in Figure 1.2.35 is a table listing the extended widths w for Cu and Al, comparing the values for edge and screw components (Karashima, 1972). Screw component tend to have larger SFE than the edge for both the metals, since the SFE is inversely proportional to the extended width, that is, w



 a 2  2  3    , (1.2.24) 16 SFE  3(1  ) 

where γ SFE stands for the SFE of the material concerned.

1.2.5.3

SFE and Cross Slip The SFE is closely related to the cross-slip process by the screw dislocations (component). Figure 1.2.37 shows a schematic of the cross-slip process for an extended screw

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Figure 1.2.35  Schematic drawing of stacking fault illustrated as an imperfect stacking sequence of atomic layers (Suzuki, 1984). Adapted with permission of the publisher.

Figure 1.2.36  Representation of a stacking fault as a region with high electron density in Cu (Suzuki, 1967). Adapted with permission of the publisher (Agne Publishing Co.).

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Figure 1.2.37  Schematic drawing of the cross-slip process. Dissociated dislocation must be constricted once in order to realize the cross slip onto another plane, after which double kinking can promote the further motion of the cross-slipped segment.

dislocation. When a cross slip occurs from one slip plane to another (referred to as the cross-slip plane), the extended partial dislocations, each having the Burgers vectors slightly deviating from the line direction, must constrict once in order to change the glide plane, because the Burgers vectors must be parallel to the dislocation line to share the slip planes. Energetically, this cannot take place all at once but can occur partially (Figure 1.2.37(a)), and the constricted part will proceed to the cross-slip plane (Figure 1.2.37(b)). Once this happens, the kinking mechanism can help advance the cross-slipped segment further along the cross-slip plane where the pair of kinks of the cross-slipped segment spread laterally, as depicted by open arrows (Figure 1.2.37(c)). Figure 1.2.38 displays an example of simulated results by discrete dislocation dynamics, providing a series of snapshots for a cross-slip process. This process is assisted by the resolved shear stresses (RSSes) both on the primary and crossslip planes and particularly the help of thermal vibrations, which will be detailed in Chapter 2 in the context of the thermal activation mechanism. Since this mechanism requires shrinkage of the extended dislocations into a perfect one, the frequency is substantially controlled by the SFE. As is shown in the table in Figure 1.2.39, the ease or difficulty of the cross slip is measured by the SFE. Metals with smaller SFE yield lower cross-slip frequency, whereas those with higher SFE exhibit higher frequency. Among FCC metals, Al yields the largest frequency of the cross-slip events, while Ag and Cu show greatly restricted cross slip due to their relatively small values of SFE. BCC metals, on the other hand, coupled with relatively larger number of slip systems than FCC metals, have a propensity to yield extremely frequent cross slips during plastic deformation.

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Figure 1.2.38  Simulated cross-slip process by discrete dislocation dynamics.

Figure 1.2.39  Relationships between the SFE and dislocation processes, that is, cross-slip and dislocation–dislocation interaction strength. Small SFE yields less frequency of cross slip and large interaction strength, whereas large SEF results in higher frequency of crossslip phenomena and weaker interaction strength. The table lists typical values of SEF for representative metals including not only FCC but also BCC.

Figure 1.2.40 shows an atomistic simulation result for cross slip from the primary plane (first line) to the cross-slip plane (second line), demonstrating that it takes place quite spontaneously given the initial and final states (Rasmussen et al., 1997). The bottom image in the figure displays an experimentally observed image of the crossslip process (Robertson and Fivel, 1999).

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Figure 1.2.40  Live images of cross-slip processes obtained in atomistic simulation (top) (Rasmussen et al., 1997). and that observed experimentally via TEM (bottom) (Phillips, 2001). Adapted with permission of the publishers (APS Publishing Co. for the simulation results, while Cambridge Univ. Press for the TEM micrograph).

One of the manifestations of such trends is the slip-line geometries that emerge on the sample surfaces, as schematically shown in Figure 1.2.41(a)–(c) (Takamura, 1999), where comparison is made among α-brass, Al, and α -Fe. They show sharp contrast: α-brass exhibits straight slip lines due to highly restricted cross slip, while Al shows occasional direction changes in the slip lines as the sign of cross slips. Further, α -Fe exhibits wavy slip lines due to restlessly occurring cross slips together with indefinite slip planes. Figure 1.2.41(d) illustrates such equivocally wandering slip behavior, called “pencil glide” because the slip trace looks like a hexagonal cylinder wall of a pencil.

1.2.5.4

Dislocation–Dislocation Interactions Dislocation interactions are important ingredients in understanding the hardening phenomena in terms of the metals’ responses. Even in 100% pure metals, many complications exist because of the complexities associated with the variation of the interactions and the resultant reaction products. This chapter does not intend to address the state-of-the-art of dislocation interactions, which have seen large advances in their reporting recently, coupled with massive and direct atomistic or dislocation dynamics’ simulations (e.g., Bulatov et al., 2006), but rather to concentrate on fundamental but often overlooked issues of importance.

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Figure 1.2.41  Schematic illustration of slip lines typically observed on aluminum, alpha-brass, and iron sample surfaces, together with pencil glide for iron (Takamura, 1999). Adapted with permission of the publisher (Kyoto University Press).

One may gain the impression that even a single interaction process of dislocations involves many details. Although the elucidation of such details is certainly relatively important, what is paramount is why such microscopic details do not have much effect on the macroscopic response, showing, in a sense, a sort of “universality,” rather than “specificity.” A candidate mechanism for this “specificity–universality” transition problem will be given in Chapter 5.

1.2.5.5



LC Sessile Junction For understanding hardening mechanisms, junction formations are the most important reaction between dislocations. The LC reaction and the resultant LC junction (or lock) formation are of particular importance for FCC metals, among others, simply because it is the strongest, that is, it is a sessile lock yielding maximum strength. Since the reaction takes place between two leading partial dislocations belonging to different planes, as depicted in Figure 1.2.42, the SFE also plays a decisive role in terms of both the frequency and the strength. As indicated in the figure, the smaller the SFE is, the higher the strength, but with lower frequency. This reaction is expressed as a  121  a 211   a  110. (1.2.25) 6 6 6

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Figure 1.2.42  Formation mechanism of an LC junction as a result of the reaction between two leading partial dislocations. Since the reaction product has a resultant Burgers vector with the “third” direction, it becomes sessile.

Figure 1.2.43  Geometrically tractable representation of the formation of an LC junction.

As a result of the reaction, a junction segment having a different Burgers vector a/ 6 [110] from the parent dislocations is produced, which makes the lock sessile (Figure 1.2.43). Figures 1.2.44 and 1.2.45 provide two more schematics representing the LC junction. An intersection of dislocations on the primary and the conjugate systems is illustrated in the figures. Figure 1.2.46 presents simulation results for the LC junction formations produced by Shenoy et al. (2000) based both on molecular dynamics (atomistic; not shown here) and dislocation dynamics (linear elasticity), comparing the configurations between Al and Ag. In Al, the slightly extended dislocations due to high SFE react to form an LC

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Figure 1.2.44  Schematics of the LC junction formed between two intersecting dislocations on the primary and conjugate slip systems, where the close-up view indicates the reaction product having the “third” Burgers vector a/6[.11̄0]

Figure 1.2.45  Simulated LC junction formation process by discrete dislocation dynamics.

sessile junction along the intersection line of the two slip planes, while Ag exhibits a similar LC junction but with largely extended dislocations because of low SFE. The excellent agreement between the atomistics and continuum elasticity implies that the configuration is determined so as to lower the strain energy, excluding the core region where the linear elasticity is considered to become inaccurate. This implies that the configuration of the LC junction is basically dominated by the linear elasticity. Figure 1.2.47 shows a TEM micrograph of an LC junction observed in stage II for Cu-15at%Al (Karnthaler and Winter, 1975). The configuration of the simulated result for Al in Figure 1.2.46 agrees well with the experimentally observed one for Cu alloy, even for the stair-rod shape at the edge of the junction. To summarize this discussion, we now understand many of the mechanical properties of FCC metals, which can be relatively easily captured if we focus on the “SFE,” as overviewed in Figure 1.2.48. Here, Cu and Al are taken as representatives of low and high SFE, respectively (Figure 1.2.48(d)), as they normally exhibit mutually contrasting mechanical properties, manifested as the hardening characteristics appearing

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Figure 1.2.46  Simulated LC junctions based on linear elasticity-based dislocation dynamics for two typical FCC metals with high and low SFE, that is, Al and Ag. The configurations are demonstrated to agree nicely with those via atomistic simulations (Shenoy et al., 2000, p. 1491). Adapted with permission of the publisher (APS Publishing Co.).

Figure 1.2.47  Experimentally observed LC junction for Cu-15at%Al during stage II hardening (Karnthaler and Winter, 1975). Adapted with permission of the publisher (Elsevier Science & Technology Journals).

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Figure 1.2.48  Overview of the dislocation-based unified perspective for macroscopic mechanical properties in FCC metals as a summary from the viewpoint of SFE.

in the stress–strain responses, yet both, nevertheless, belong to the same FCC family. A schematic comparison of their stress–strain relations under monotonic tension is presented in Figure 1.2.48(e), emphasizing the difference in the hardening moduli. With large enough SFE, as in Al, one may safely ignore the extension of dislocations into partials (Figure 1.2.48(a)), resulting in frequent cross slip and subsequent pair annihilations (dynamic recovery) (Figure 1.2.48(b)), while easily dissociable LC junctions scarcely contribute to strain hardening (Figure 1.2.48(c)). The reverse is true for Cu with small enough SFE, that is, less frequent cross slip tends to hinder dynamic recovery on one hand, while fully extended dislocations are apt to form strong LC sessile locks that ultimately enhance strain hardening on the other. Some practical examples of such SFE-based views are presented in both Appendix A2 and Chapter 3 (Figure 3.3.13), which may provide strong leverage for justifying the above views. In the former, a systematic series of experiments on the coupling effects between the nonproportional (NP) strain history and the strain rate, including the impact loading regime, are extensively discussed, while, in the latter, a systematic set of variations observed in the evolved dislocation cell structures strongly depending on the SFE are discussed based on experimental results under NP cyclic straining. It should be noted that understanding BCC metals is not that simple, unlike the FCC case described previously, but a good start for tackling the issue is provided in the “FCC versus BCC” perspective.

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1.2.5.6

Jog Formation Let us mention “jog,” which may become important, particularly in understanding hardening in BCC metals, as summarized in Figure 1.2.49. Jog is a product of the orthogonal intersecting of two dislocation lines (one of the forest intersection), characterized as a step formed on a dislocation line in the out-of-slip plane (those formed on the same slip plane are called “kinks” [see Figure 2.2.7]). There are four kinds of such orthogonal intersection depending on the combinations of edge and screw components, that is, “edge against edge or screw” and “screw against edge or screw.” The intersection expected to be exceptionally important in BCC metals (in terms of hardening) is that formed between two screws (i.e., on a screw dislocation against a screw segment), as schematically shown in Figure 1.2.50. The intersecting screw dislocation line leaves “jog” on it. Since the “jog” portion of the dislocation segment

Figure 1.2.49  Formations of jog and kink as a result of intersecting edge dislocation (a) and screw dislocation (b) against edge and screw forests, respectively.

Figure 1.2.50  The jog-drag mechanism as a result of the intersection of two mutually perpendicular screw dislocations. One of the dislocations yields a “jog” with the nature of an edge component, which cannot continue to glide except a climb motion, where vacancies are provided from the surroundings, which induces resistance against the dislocation motion.

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Figure 1.2.51  Schematics of “jog drag” showing the sequence of motion of a jogged screw dislocation line (a), followed by (b) bowing out, (c) formation of edge dipoles, and (d) resultant jog dragging leaving vacancy rows behind.

is perpendicular to the Burgers vector b1 (which is conserved), it acts as an “edge” component that cannot be slipped along the same slip plane any further. For it to glide further, vacancies must be brought from somewhere to replace the “excessive” atoms (this process is thus “nonconservative,” in contrast to other dislocation motions, and is called a “climb” motion). Therefore, the glide motion of a screw jog must leave an array of vacancies along the trace, as depicted on the right-hand side of Figure 1.2.50 (this is called “jog drag”). The “jog-dragging” process will also produce an edge dipole, as illustrated in Figure 1.2.51, since dislocations tend to bow out due to the line tension, while the jog is highly resistant compared with the other portions of the screw dislocation line. Note that the reverse motion of the screw jog produces an array of “interstitial” atoms instead of vacancies. Examples of discrete dislocation-based simulation results for the “jog-related” processes, including “jog drag,” are displayed in Figure 1.2.52, that is, (a) formation of a jog dipole, (b) the jog-dragging process, and (c) bypassing after jog dipole formation. Continuous stressing against (a) results in (c), further acting as a Frank–Read source that leads to multiplication. These overall observations remind us of a versatile aspect of the “jog-related” processes critical to many aspects of plasticity. Note that the process just described is of further importance when we look into a fatigue-crack initiation mechanism from persistent shear band (PSB) ladder structures (under high-cycle fatigue, see Section 3.7.1), because the “jog-dragging” accompanying-edge dipoles produced within the interladder wall regions can produce a number of vacancies, as will be briefly discussed in Appendix A9.

1.2.5.7

About Dislocation Density Let us consider afresh an intuitive image of “dislocation density” for metals, by focusing on two typical extreme cases (Kato, 1999), as displayed in Figure 1.2.53: For well-annealed pure metals, we normally have   109  1010 m 2 , whereas for deformed states the values reach 1014 − 1015 m −2 .

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Figure 1.2.52  Simulated “jog-related” processes by discrete dislocation dynamics: (a) jog dipole formation, (b) jog-dragging process, and (c) bypassing a formed jog dipole.

Figure 1.2.53  Examples of typical dislocation density for fully annealed and work-hardened samples, together with the commensurate total length of the dislocation line assumed to be contained in a unit cube. See also Figure 1.4.2.

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Figure 1.2.54  Intuitive images of 2D and 3D dislocation density, that is, mean spacing for 2D and total length per unit volume for 3D definitions, based on which we evaluate typical values of the smallest and largest dislocation density, as listed in Figure 1.2.53.

For an intuitive image of “dislocation density,” let us use two quantities – “mean spacing” and “total length” per unit volume (m 3). The former is directly related to the two-dimensional (2D) definition of ρ , while the latter to the 3D definition. Figure 1.2.54 schematically illustrates how we define “dislocation density” in the context of 2D and 3D images, respectively. As can be readily understood from the schematics illustrated in Figure 1.2.54, the mean spacing of dislocations can be roughly estimated from the density via  1/ 2, assuming uniform distribution. The dislocation density of 1010 m −2 corresponds to lspacing  10 m (Figure 1.2.53). This is of the order of the grain size in conventional polycrystalline metals, meaning few dislocations included within a crystal grain. In sharp contrast, the density of 1015 m −2 is commensurate with lspacing ≈ 30 nm, roughly corresponding to 100 atoms. From this fact one can discover the maximum density that crystals can contain to be around 1016 m −2. For the total length, the former yields Ltotal = 10 km, while the latter Ltotal =106 km (Figure 1.2.53). This is equivalent to several times the distance to the moon from the Earth (which is about 3.8 ×10 4 km). A typical example of the highest dislocation density is that in martensite or bainite structures. Figure 1.2.55 shows a TEM picture of lath martensite observed in Fe-0.6%C (Maki et al., 1979), where the black contrasted regions depict high-density

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Figure 1.2.55  TEM micrograph of a martensite lath structure as an example of one of the ­highest dislocation densities.

Figure 1.2.56  TEM micrograph presenting an example of intense dislocation debris with the highest density observed in the proximity of the bainite/austenite transformation front (Bhadeshia, 2001). Reprinted with permission of the publisher (Cambridge University Press).

dislocations. They are basically introduced for accommodation reasons when martensitic phases with body-centered tetragonal (BCT) structures abruptly emerge during quenching and intrude into the FCC-structured matrix phase (austenite). A close up of such an “intruded” front can be found in Figure 1.2.56 (not for a “martensitic” transformation, but for a “bainitic” one) (Bhadeshia, 2001), where extremely high-density debris of dislocations have been introduced at the austenite (γ )-bainitic ferrite (α ub) interface for the purpose of accommodating the attendant incompatibility. The corresponding situations for the lath martensite formation process are schematically illustrated in Figure 1.2.57, where lath-shaped martensitic phases are nucleated and subsequently grown from the prior austenite (γ ) boundaries, ultimately evolving into lath block/packet structures. Note that thus-introduced high-density dislocations are pinned by minute precipitates and/or solute atoms introduced separately, without which most of them do not remain anchored. Figure 1.2.58 shows a set of measured plots of dislocation density ρ as a function of the transformation temperature, including those not only for the martensite and bainite, but also for some ferritic phases (Bhadeshia, 2001).

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Figure 1.2.57  Schematic illustration of the lath martensite formation process under austenite/ martensite transformation.

Figure 1.2.58  Variation of dislocation densities in martensite, bainite, acicular ferrite, and ferrite, with transformation temperature (Bhadeshia, 1997, 2001). Adapted with permission of the publishers (Routledge and Cambridge University Press).

1.3 Crystallography 1.3.1

Crystal Systems (Structures) Crystal structure is typically classified into three systems as schematically depicted in Figure 1.3.1, that is, FCC, BCC, and hexagonal close-packed (HCP) structures. As

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Figure 1.3.1  Typical crystal lattice structures for FCC, BCC, and HCP metals, together with

representative slip systems in the Miller index notation.

Figure 1.3.2  Various atomistic representations of typical crystal lattice structures of FCC, BCC,

and HCP, together with respective coordination numbers and packing factors. This schematic is adapted by permission (© Iowa State University Center for Nondestructive Evaluation).

atomically represented in Figure 1.3.2, the FCC and HCP are the closest-packed systems with the maximum density of atoms. The isotropic bonding state of the outer-shell electrons of the metals results in these structures, while a slight distortion of the bonding structure due to additional anisotropy tends to lead HCP. The two structures differ in the order of stacking of the atomic layers. ABAB … stacking produces the HCP, while the

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Figure 1.3.3  Distinction between FCC and HCP structures in terms of packing sequences

of atomic layers (both are the closest-packing structures) (Shimura, 2000). Adapted with permission of the publisher (Asakura Publishing Co.).

Figure 1.3.4  Comparison of stacking sequence of atoms between FCC and BCC structures.

ABCABC … stacking corresponds to the FCC, as depicted in Figure 1.3.3. As one can readily understand, these are the ways to stack bolls (atomic layers) in the densest manner. The BCC, on the other hand, is a loosely packed system, as a result of the directionality in the electronic boding states, normally reflecting, for example, the d-bands for transition metals. Typical metals yielding this system are Mo, W, Ta, Nb, and Co. They have partially “covalent-type” bonds, making them relatively high-temperature resistant. The difference in the atomic stacking sequence between FCC and BCC is presented in Figure 1.3.4.

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1.3.2

FCC versus BCC In this section, distinctions between FCC and BCC metals in terms of plasticity are discussed. In my experience, many researchers will tend to answer such a question as “what is the substantial difference between FCC and BCC?” in an obvious context. One of the most frequent answers is likely to be about “the number of slip systems.” Some might mention “interactions among dislocations,” especially in recent years. While these answers are true, they are rather insubstantial and rather secondary at most, in the sense that they are derivable. A more fundamental difference would be FCC and BCC’s mutually “dual” construction of atomic structures, on which almost all the specifics are derived, from dislocation core structures to contrasting rate and temperature dependencies. Needless to say, this originates from the electronic structures, governed ultimately by quantum mechanics, however it is not always necessary to proceed down this path, unless chemistry is explicitly involved, as in Fe, which is revisited in Section 4.2.2 (citing a work reported in Chen et al., 2008). The dual-atomic constitutions manifest themselves as the dual constructions of the slip systems, that is, {110} for FCC and {111} for BCC, as depicted in Figure 1.3.5. Note that, as is widely known, for BCC other planes containing orientations may also be slip planes, for example, {112}, {123}, and so on, but they are excluded here for simplicity. As can be seen, BCC and FCC combinations are totally opposite; the slip planes in the FCC are the slip directions in the BCC and vice versa. The same is true for “twin” deformations (see Section 1.4.1). The dual construction in the slip system is primarily due to the dual-atomistic configurations between the FCC and BCC, whose dual interrelationship is defined in the context of the reciprocal lattice (Kittel, 1953). Figure 1.3.6 provides such comparisons, summarizing the interrelationship between the FCC and BCC lattices. The reciprocal lattice to the FCC lattice agrees with the BCC lattice, and vice versa. (Note that the reciprocal of a simple cubic lattice is also a simple lattice.) In other words, the

Figure 1.3.5  Dual constitution of FCC and BCC crystal structures; the nature of crystalline

plasticity substantially differs between the two.

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Figure 1.3.6  Dual constitution of FCC and BCC metals in terms of the atomic structure, known

as Wigner–Seitz primitive cells in real space, and their reciprocal relationship, termed as the first Brillouin zone, which are mutually transferable via Fourier transform.

Fourier transform of the FCC lattice generates the BCC lattice, and the reverse is true. In the field of solid state physics, we often use the “Wigner–Seitz cell” representation in the Bravais lattice for determining the first Brillouin zone based on the reciprocal space representation, since they coincide with the Bragg-reflected wave vectors. A similar procedure with attendant “duality” is found in Section 6.6.1, where we discuss graph theory-based representation of granular assemblies It should be noted that the “Wigner–Seitz cell” is equivalent to the “Voronoi cell,” constructed (drawn) via the Voronoi tessellation procedure. Since the FCC lattice is close packed, accordingly the dual-BCC lattice is not; indeed, it may safely be stated to be “loose.” From an atomistic point of view, such looseness in the packing structure of BCC metals can cause a situation whereby the closest-packed plane is not clearly identified, unlike in the FCC, and hence several “nearly” closest-packed planes can coexist. This “looseness” in packing structure is the very reason for the resulting multiple slip systems (unidentifiable slip planes) in BCC metal. Furthermore, it is the source of the often observed complexity and variety in the mechanical responses peculiar to BCC metals as well as their much higher Peierls–Nabarro (PN) stress than FCC. Immediate examples are BCC metal’s strong strain rate and temperature dependencies on the stress response and the complexities in the core structure of the screw dislocation for BCC metals in general. (These will be discussed in later chapters, especially Chapter 4.) In the case of α -Fe, this is, as a matter of fact, not a natural-born BCC metal, as is pointed out and discussed in some detail in Section 4.2.2. The specific origin of the BCC structure, coupled with the complexity in the screw core structure (see Section 4.2.1), is considered to be a crucial source of the extreme varieties in the mechanical properties of this metal. Furthermore, combined with the fact that the high temperature austenitic phase (γ -Fe) with FCC structure is within easy reach on the phase diagram

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Figure 1.3.7  Intriguing correspondence in {111} and {100} pole figures representing rolling

textures between FCC copper in RD and BCC Fe-3Si in ND, as a manifestation of the “dual constitution” of slip systems (Rollett and Wright, 1998). Adapted with permission of the publisher (Cambridge University Press).

via appropriate heat treatments, the complexity regarding α -Fe also provides us with such a fertile spectrum of metallurgical microstructures (such as pearlite, martensite, and bainite structures). Appreciating α -Fe this way is quite important for recognizing the phenomenology and fully understanding the necessity of multiscale approaches in the present context. One prominent consequence of the already mentioned duality in slip-system constructions is the difference in the textures to be evolved, for example, under cold rolling (details about “texture” and “pole figure” are given in Section 1.4.2). Figure 1.3.7 compares and pole figures between FCC metal (Cu) and BCC alloy (Fe-3%Si) cold-rolled up to 80% reduction in thickness, where those with RD (rolling direction) for the former and ND (normal direction) for the latter are indicated (Kocks et al., 2000; Rollett and Wright, 1998). Note that the transposition of the RD and ND ensures the two sets of pole figures for FCC and BCC correspond well. This dual-texture evolution is regarded as an eloquent manifestation of the previously discussed dual construction in the atomic structures between FCC and BCC. Note that slight differences may be largely due to the geometrical effects of the elongated crystal grains along the RD, which differentiate the transposed case from the reference. As conjectured from this duality in the rolling textures, we suppose some type

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Figure 1.3.8  Typical {100} pole figure representing γ-fibered texture, specific to rolled BCC

iron and IF steels, comparing experiments and a corresponding schematic.

of indigenous preferred orientation should exist in relation to the crystal structure, for example, FCC has its own inherently preferred orientations that will evolve, although the texture in FCC metals depends quite strongly on the SFE (see Figure 1.4.10). This is also true for BCC metals. Iron (Fe) or high purity steels such as IF steel, among ­others, are well documented to yield {111}– γ -fibered textures, where the {111} plane tends to be directed in parallel to the sheet sample surface, together with the preferred orientations in between and in RD (also expressed as that with strong || ND fiber). The typical γ-fibered texture for IF steels, for example, is manifested as the pole figures displayed in Figure 1.3.8, where (b) representative and (c) schematic (or ideal) {100} pole figures are compared. Since the representative γ-orientations of {111} and {111} are located periodically along a concentric circle on the pole figure, as shown in (c), we can understand why the experimentally observed pole figures become like the one demonstrated in (b), that is, with an isotropically converging concentration of the preferred orientation. The γ -fibered texture is revisited in Section 1.4.2, with another representation (via orientation distribution function [ODF]). As easily imagined based on the above argument, FCC metals, having the dual-crystal structure relative to BCC metals, should give rise to totally different textures from the above, for example, β -fibered texture; it will be extremely difficult for them to yield the γ -fiber. (Typical rolling textures for FCC metals are shown in Section 1.4.2.)

1.3.3

Slip Systems in FCC and BCC FCC metals have 12 independent slip systems, as represented in Figure 1.3.9, that is, four independent slip planes with each containing three slip directions, making the total number of the slip systems 12, where the combination of slip plane and

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Figure 1.3.9  Slip systems for FCC metals, consisting of four crystallographically equivalent

slip planes, each having three independent slip directions, based on Schmid–Boas notations.

direction is called “slip system.” The four slip planes are often referred to as primary (B), conjugate (D), cross-slip (C), and critical (A), respectively, according to their configurations and roles viewed from the primary system (B), while the slip directions are labeled by numbers corresponding to the six independent directions, that is, from 1 through 6. The combination of them (A–D and 1–6) is referred to as the Schmid–Boas notation (Schmid and Boas, 1950 [1968]), which is widely used in the literature for identifying each slip system. For example, “B2” denotes the slip system on the primary plane (B) with direction 2. Figure 1.3.10 displays the corresponding vector representations, where m(α ) and s(α ) indicate the unit vectors for the slip plane normal and the slip direction (i.e., that of the Burgers vector), respectively, with the superscript (α) denoting the slip system, that is, α = 1, 2, …, 12 for FCC metals. These notations are further used in the mathematical treatment of crystal plasticity, partially mentioned in conjunction with the Schmid factor (SF) (e.g., Figure 1.3.17). As is inferred from the “FCC versus BCC” arguments in Section 1.3.2, BCC has a dual construction of slip systems in contrast to that of FCC crystals. The closest-packed plane for BCC crystals is {110} and we normally regard the associated slip systems of the {110} type as the representative system. Figure 1.3.11 displays the 12 independent slip systems of the {110} type for BCC metals, that is, six equivalent slip planes with two slip directions each, summarized in Figure 1.3.11 in terms of the Schmid–Boas notation. The corresponding vectorial representations are given in

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Figure 1.3.10  Explicit expressions of slip systems for FCC in Figure 1.3.9 via unit vectors for slip planes and slip directions (i.e., the Burgers vector) of FCC metals.

Figure 1.3.11  Slip systems for BCC metals ({110} type only), consisting of six crystallographically equivalent slip planes, each having two independent slip directions, based on Schmid–Boas notations, yielding dual construction, in contrast to those of FCC counterparts shown in Figure 1.3.9.

Figure 1.3.12. From the figure, one can confirm the “duality” relative to FCC, especially by looking at the Schmid–Boas notation, that is, “A–D” denotes the slip directions, while “1–6” expresses the slip plane, based on which we readily obtain a set of ( ) ( ) expressions for BCC from that of FCC via simple transpositions of sfcc  mbcc and ( ) ( ) mfcc  sbcc . Here, to avoid confusion, the subscript “fcc/bcc” is attached to the plane normal and direction vectors. As pointed out in Section 1.2, BCC-constituting atomic structures are relatively loosely packed compared to FCC’s (compare the packing factors listed in Figure 1.3.2), which gives rise to difficulty in distinguishing the most densely stacked planes from others, that is, there exist plural dense planes very close to the densest

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Figure 1.3.12  Explicit expressions of slip systems for BCC in Figure 1.3.11 via unit vectors for slip planes and directions (i.e., the Burgers vector), yielding dual construction in contrast to those for FCC metals in Figure 1.3.10.

Figure 1.3.13  Three typically recognized slip planes for BCC metals containing the common slip direction in the orientation.

one. This serves as the major reason for the indefinite slip planes often observed in BCC metals, combined with the extremely large SFEs, manifested typically for α -Fe as “wavy” slip-traces (c) and “pencil glide” (d) in Figure 1.2.41. Figure 1.3.13 summarizes a set of generally postulated variations of the slip planes for BCC structures, that is, {110}, {112}, and {123}, where the most densely packed

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Figure 1.3.14  SF map applicable both to FCC and BCC crystals, together with schematically drawn typical stress–strain curves in the case of FCC metals.

direction is uniquely identified to be , providing the definite slip direction of BCC metals. This can be visually confirmed also in Figure 1.3.13, where these three planes intersect on the common line, and hence they are called to form the zone. The independent slip planes for {112} and {123} types are partially shown in the figure, having 12 and 24 slip systems, respectively. Therefore, BCC metals have 48 slip systems altogether if we assume all the contributions from the above three kinds including the {123} types. Since the SF is measured by ( s( )  m( ) )sym, as detailed in Section 1.3.4, we notice from the earlier discussion about duality in slip systems between FCC and ( ) ( ) ( ) ( ) BCC, namely ( sfcc  mfcc )sym  ( sbcc  mbcc )sym, that two crystal systems’ SFs coincide, as far as the {110} types for BCC are concerned. Figure 1.3.14 shows the contour map of the SF, commonly applicable to both the crystal structures. The map is drawn on the standard triangle of the stereographic projection (see Figure 1.4.13 for details), together with the number of active slip systems under tension in the prescribed orientations. The orientations [001], [111], and [011], located at the apices of the triangle, are highly symmetric, which are followed by orientations along the edges (with two equivalent slip systems), as explicitly specified in (a). The orientations inside the triangle, on the other hand, yield low symmetry, manifesting a limited number of slip-system activities, for example, single slip. Some immediate examples of the corresponding stress–strain curves are schematically shown in (b), showing a highly symmetric multiple slip orientation [111] (six equivalent slip systems) and a

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Figure 1.3.15  A do-it-yourself kit of cubic structure model containing all the fundamental planes of {100}, {110}, and {111} (Shimura, 2000). Adapted with permission of the publisher (Asakura Publishing Co.).

single-slip orientation [123], together with a moderately symmetric orientation along the triangle edge [-112] (two slip systems), for FCC metals, as examples. Single-slip orientations tend to yield two- to three-stage hardening, as also detailed in Figure 3.1.6 in conjunction with evolving dislocation structures, and revisited in Figure 4.1.3 in the context of “single crystal versus polycrystal” plasticity. Regarding the SF, such low-symmetric orientations are apt to take relatively larger values, stemming from the biased activities of their slip systems, with a maximum of 0.5 realized at [149], followed by, for example, 0.497 at [136], and so on, located inside the triangle, as a natural consequence. For easy understanding of the slip-system configurations for FCC and BCC crystals, Figure 1.3.15 provides a do-it-yourself kit for representing all the typical planes including and , in addition to the cube planes for cubic structures (Shimura, 2000). The completed drawing is shown in the upper left.

1.3.4

RSS and the SF For discussing slip-system activities and/or slip-system-wise shear deformations, we use the RSS. Figure. 1.3.16 illustrates how we define the RSS together with the associated SF. The force F acting on the cylindrical sample, as in the figure, is projected onto the slip plane in the slip direction, inclined by φ from the loading axis, that is,

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Figure 1.3.16  Schematics showing how to obtain the RSS, where the SF is defined as the ­measure for evaluating slip activities.

F cos f . By dividing it by the area of the slip plane Aslip  A / cos  , we obtain the RSS, that is,



F cos  F  cos  cos  (1.3.1) Aslip A

Since F / A   , we finally have



   cos  cos  . (1.3.2) Here, the prefactor cos  cos    is called the SF, which measures the RSS. By using this factor, we can distinguish the activity of the slip systems, for example, “primary” or “secondary,” and “active” or “inactive.” The equation for finding the RSS can be generalized into the one for the tensorial definition, which serves as the foundation of the kinematics of crystal plasticity (e.g., Asaro et al., 2003; Khan and Huang, 1995; Nemat-Nasser, 2004). Figure 1.3.17 displays the process and the definition. The SF is generalized as the Schmid tensor, indicating 3D slip-system constitutions, defined as 1 Pij( )  si( ) m(j )  s (j ) mi( ) , (1.3.3) 2 1 P ( )   s( )  m( ) sym   s( )  m( )  m( )  s( )  , (1.3.4) 2 where si(α ) and m(jα ) are unit vectors representing the slip direction and slip plane normal belonging to the slip system specified by the superscript (α), respectively. The subscript “sym” in Eq. (1.3.4) denotes “symmetrization.” By using the Schmid tensor, we can calculate the RSS via







  Pij( ) ij , (1.3.5)



  P ( ) :  tr ( P ( ) .  ), (1.3.6)

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Figure 1.3.17  Three-dimensional generalization of the SF, referred to as the Schmid tensor or direction tensor, which is constructed by the tensor product of two unit vectors representing slip direction and slip plane normal, together with the corresponding relationship for calculating the RSS. Two expressions, via direct and index notations, indicated for both SF and RSS, are presented.



1.3.5

where “:” denotes the scalar product for tensors. The rewriting of the second expression in Eq. (1.3.6) is simply due to a computational reason, because algorisms for calculating the multiplication P ( ). are simpler than directly computing P ( ):  . It is also known that BCC metals do not always obey the Schmid law, often referred to as the “non-Schmid effect,” requiring additional calculations (Ito and Vitek, 2001). Some related topics are discussed in Section 4.2 in relation to the complexity of screw core structures peculiar to BCC metals. When the SF is summed up over multi-orientations for the purpose of representing polycrystal versions of the relationship between σ and τ , this is called Taylor factor, that is, M for   M . Roughly, it has been reported that M = 3.06 for FCC metals and M = 2.83 for BCC metals (Figure 1.3.17). The skew-symmetric part of s( )  m( ), defines the spin tensor, to be used in the kinematics of finite crystal plasticity formulation for expressing deformation-induced lattice rotations, that is, 1 Wij( )  si( ) m(j )  s (j ) mi( ) , (1.3.7) 2 1 W ( )   s( )  m( ) skew   s( )  m( )  m( )  s( )  , (1.3.8) 2 where “skew” indicates “skew (anti)-symmetrization.”





Dislocation–Dislocation Interactions Revisited: Interaction Matrix Figure 1.3.18 displays an example of the interaction matrix for FCC metals, classifying the kinds of pairwise interaction for the arbitrary combinations of the slip systems. In the case of FCC metals, such interactions are expressed by a 12 × 12 matrix. Here,

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Figure 1.3.18  Example of an interaction matrix for FCC metals expressing kinds and strengths of dislocation–dislocation interactions, also explaining how to view the interaction matrix, for example, from the primary slip system denoted by thick double-pointed arrow in Thompson’s tetrahedron on the right.

the Schmid–Boas notation (see Figure 1.3.9) is used to symbolize the slip systems. The inset shows a 2D representation of Thompson’s tetrahedron, expressing the constitution of the FCC slip systems, together with the interactions specified with respect to the primary slip system, indicated by a broad doubled-pointed arrow. The interactions among dislocations in FCC metals have empirically been classified into five kinds, that is, (O) self-hardening, (H) Hirth lock or reaction-producing jogs, (C) coplanar junction among those on the same slip plane, (G) glissile junction, and (S) LC sessile junction. The last one is described in detail in Section 1.2.5.5. The interaction associated with the LC sessile junction formation exhibits the maximum strength, which is followed by the glissile lock. There have been arguments about the others (Bassani and Wu, 1991; Francoisi et al., 1980). Also explained in Figure 1.3.18 is how to read the interaction matrix. If the primary system denoted by O be the reference, which corresponds to B5 according to the Schmid–Boas notation (corresponding to the slip system along the lower edge of the primary plane in Figure 1.3.9), we need first to find it in the row. As can be confirmed in the figure, “B5” is located on the eighth row from the top. Then we can identify all the interactions with arbitrary others by looking at this row. For example, the interaction with D1 is found to be “S,” meaning LC sessile junction formation, and so on. The values of the components in the above interaction matrix, that is, the strengths of the interactions, can be identified (or at least evaluated) by a series of experiments which make up the latent hardening test.

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Figure 1.3.19  Schematics of the latent hardening test consisting of two-step loading (tensile) tests, by which the components in the interaction matrix can be evaluated (Wu et al., 1991). Adapted with permission of the publisher (Royal Society Publishing).

Schematics of the latent hardening test are shown in Figure 1.3.19, where a twostep test on a single crystal specimen is performed. The details of the test are as follows. The first test is conducted on a parent specimen normally directed in single slip, and the second test follows it on a child sample machined from the parent with a specified direction from its stress axis, such that the interaction between the primary slip system activated in the first test and the latent systems start operation in the second test can be measured. The graph in the figure shows an example of the output results, where shear stress–strain curves for the primary and the secondary tests are indicated, from which the latent hardening ratio (LHR) is obtained as

LHR 

s . (1.3.9) p

Here, τ p and τ s are the values of flow stress in the primary and secondary tests, respectively, where A2 or D6 orientation is assumed in the secondary tension against a B2 primary orientation, corresponding to LC junction and no junction formations, respectively, as extrema. There are some arguments about how to determine τ s since the secondary curve contains a number of subtleties. Usually the backward extrapolation is employed, as in the figure. Examples of experimentally observed responses for Cu are displayed in Figures 1.3.20 (Wu et al., 1991) and 1.3.21 (Jackson and Basinski, 1967), respectively. In the former (Figure 1.3.20), the backward extrapolation procedure is indicated for evaluating the secondary flow stress. In Figure 1.3.21, on the other hand, three secondary orientations of B2, A2, and D1 are chosen against the B4 primary counterpart, corresponding to G (glissile junction) in common in the interaction matrix f listed in Figure 1.3.18. As observed, D1 and A2 exhibit marked stress increase around the reyielding, whereas B2 shows no additional hardening but is demonstrated to smoothly continue the primary stress curve. Since B4 → B2 is classified as C

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Figure 1.3.20  Example of the latent hardening test for a Cu single crystal (Nemat-Nasser, 2004; Wu et al., 1991). Adapted with permission of the publishers (Cambridge University Press and Royal Society Publishing).

Figure 1.3.21  Examples of latent hardening test results with various orientations in the secondary experiments for a single crystal Cu (Jackson and Basinski, 1967). Adapted with permission of the publisher (Canadian Science Publishing).

(coplanar junction) in f , the present result implies no distinction among the three coplanar slip systems in their contributions to the strain hardening at least for Cu, meaning that the C component in f is supposed to take the value of 1.0, that is, no extra/additional contribution to the self-hardening.

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Figure 1.3.22  Variations of LHR with (a) shear stress in the primary test (Francoisi et al., 1980) and (b) normalized SFE (Francoisi, 1985), respectively, where the maximum values of LHR are used for each material for the latter (b). The primary system for all the tests is B4, yielding formations of sessile (A6), glissile (A2), and coplanar junctions (B2), respectively. Adapted with permission from the publisher (Elsevier Science & Technology Journals).

It is noteworthy that the components of the interaction matrix f are not constant in general, but roughly yield a decreasing function of the primary shear strain, as shown in Figure 1.3.22(a). From the comparison of LHR among the secondary orientations, A6, A2, and B2, not only can we confirm the strength order of the dislocation interactions as S (sessile) > G (glissile) > C (coplanar), but we also learn that all these interactions follow the same decreasing trend for both Al and Cu. It is further worth noting that a rough negative correlation of the LHRs for S, G, and C with increasing SFE exists, as observed in Figure 1.3.22(b). This corroborates the summarized overview concerning SFE-hardening behavior relationship for FCC metals given in Figure 1.2.48. We next consider the case of BCC metals, although they have been quite limited, in contrast to FCC metals, primarily due to complexity relating to BCC indefinite slip-system activities (cf. Figures 1.2.41 and 1.2.13). Among others, Nakada and Keh (1966) systematically investigated – for single crystal Fe, choosing [111 ] as the primary system direction – the effects of a Burgers vector combination, the amounts of prestrain, and temperature, concluding the LHR tends to vary between 1.2 and 1.4, roughly independent of those factors. Figures 1.3.23 and 1.3.24 show examples of their results. In Figure 1.3.23, what we can readily confirm is the markedly pronounced additional hardening at yielding compared to the FCC cases shown earlier, followed by similar flow responses regardless of the secondary slip systems (oriented in #1[ 111 ], #2 [ 1 11 ], #3 [1 11 ], and #4 [111]). In Figure 1.3.24, on the other hand, an exceptionally high rate of additional hardening is found for the reloaded curve #4′, in which two secondary slip systems, where double slip with [1 11 ] and [1 1 1 ] directions are activated simultaneously, distinctly differ from FCC. Figure 1.3.24 provides a comparison of results for the case of “coplaner” latent hardening, where the RSS was designated to be zero on the latent system during the first test, and, similarly, the

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700

Fe-48

SHEAR STRESS Kg/mm2

600

2 3

1

500

4

400

2

3

P

300

P

1 2 3 4

4

200

P

100

0

1 0.10

0.20

Fe Single Crystal

0.30

0.40

SHEAR STRAIN

[ 111] [ 111] [111] [111]

0.50

Nakada & Keh (1966)

Figure 1.3.23  Examples of latent hardening tests for a BCC Fe single crystal with various secondary orientations, denoted as 1 to 4 on the stereographic projection (Nakada and Keh, 1966). Adapted with permission of the publisher (Elsevier Science & Technology Journals).

600

with Double Slip

4’

1' 2' 3' 4'

SHEAR STRESS (Kg/mm 2)

500

400

Fe-42

1’ 3’ 2’

300

200

A

A

A

4’ 3’ 1’

100

0

0.24

0.08

[ 111] [ 111] [111] [111] & [111]

0.12 0.16 0.20 SHEAR STRAIN

Fe Single Crystal

0.24

2’

0.28

Nakada & Keh (1966)

Figure 1.3.24  Examples of latent hardening tests for a BCC Fe single crystal with various secondary orientations, denoted as 1′ to 4′ on the stereographic projection, with 4′ being double-slip oriented (Nakada and Keh, 1966). Adapted with permission of the publisher (Elsevier Science & Technology Journals).

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Figure 1.3.25  Interaction matrix for BCC Fe integrated from literatures, that is, with reactions classified by Madec and Kubin (2004), and values evaluated by Francoisi (1983).

RSS on the primary system was zero during the second test, meaning latent hardening in the absence of secondary activation of the slip systems. Relatively large coplaner interaction is observed, which is another feature of BCC metals. Figure 1.3.25 provides an example of the interaction matrix for BCC Fe which will be used in the simulations that follow in the present book. The values of the components are determined in an integrated manner and refer to several data sources, that is, the classification of the interactions is based on a series of dislocation dynamics simulations (Madec and Kubin, 2004; Tang et al., 1999), and the values themselves specified in the list in the right are from those evaluated by Francoisi (1983) in the latent hardening tests, where the slip systems are limited to the {110} and {112} families, that is, B5 through A2 and B5′ through A2′, respectively. Since there is no distinction in the reaction of the two dislocations, the interaction matrix in this case is symmetric. Note that in the following series of simulations we used “1.0” for components not available in the literature. Figure 1.3.26 displays another source of the interaction matrix for BCC metal (Cuitinõ et al., 2001) where a specific interaction, that is, between a moving edge dislocation and a stationary screw dislocation, forming jogs as a result of the reaction, is considered (see also Figure 1.2.49(a)). Cuitinõ et al. evaluated all the 24 × 24 interactions in terms of the formation of energy based on energy and mobility considerations for Ta. Since the reactions of an edge dislocation against a screw counterpart

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Figure 1.3.26  Interaction matrix for BCC Ta representing normalized jog formation energy numerically evaluated (Cuitinõ et al., 2001).

are assumed, the matrix becomes asymmetric, in contrast to that in Figure 1.3.25. Sorting the original matrix by the number, from 1 through 24, used in Figure 1.3.25, we notice some noteworthy features that are roughly classified into four submatrices corresponding to the reactions for ( 1) {110} edge against {110} screw (upper left), (2) {110} edge against {112} screw (upper right), (3) {112} edge against {110} screw (lower left), and (4) {112} edge against {112} screw (lower right), exhibiting mutually common trends in the component structure, with some minor exceptions, for example, periodically crossing cater-corner bands of “1.0” or “-” (shaded in the figure), and intercorrespondence of the values between “1.5 and 3.2” and “2.4 and 1.8” (these can be mutually converted by swapping). These common features imply qualitatively similar contributions of the {110} and {112} family dislocations to the jog formations.

1.4 Miscellaneous 1.4.1 Twin Another mode of plastic deformation is “twinning.” There are two types of twin in terms of their forming mechanisms, that is, annealing twin and deformation (or mechanical) twin. The former is introduced to reduce the energy of the system as a

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Figure 1.4.1  Schematics of twinning in FCC metals.

part of “static recovery” in the absence of stress or deformation, while the latter takes place in order to accommodate the imposed deformation under stress. This subsection basically deals with the latter – deformation twin. The deformation twin is considered to be a major deformation mode for HCP metals, and also for some FCC metals with low SFE less than 25 mJ m−2 and BCC metals under high strain rates and low temperatures. Even in FCC metals with intermediate SFE around 50–70 mJ m−2 (such as Cu and Ni), where most of the plastic deformation can be carried out by the dislocation motion, mechanical twinning can take place, for example, under impact or hypervelocity impact loading, as partially shown in Figure 3.5.2. It is known that the critical stress for the onset of mechanical twinning for FCC metals is in proportion to the SFE. This means that small SFE metals and alloys (such as α-brass and SUS304 [austenitic stainless steels]) tend to exhibit mechanical twinning quite easily. Figures 1.4.1 and 1.4.2 detail the crystallography of the twinning for FCC and BCC metals, respectively. As can be confirmed by comparing the two, putting them side by side, as in Figure 1.4.3, there also exists a “dual” constitution, that is, {112} for FCC and {111} for BCC. It is worth comparing this with the case of “slip” in Section 1.3.2 or Figure 1.3.5. For BCC metals, another distinction should be kept in mind, which is that between twin and antitwin directions, depending on the directionality of deformation due to its geometrical constitution, causing, for example, tension–compression asymmetry. Note that the emission of a partial dislocation from a GB in nanocrystalline aggregates, leading to the formation of twins across the grain, has been considered to be responsible for the outset of plastic deformation in such nanocrystal samples (e.g., Asaro et al., 2003; Van Swygenhoven et al., 2002).

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Figure 1.4.2  Schematics of twinning in BCC metals.

Figure 1.4.3  Duality in twin deformation modes between FCC and BCC metals as a

consequence of their dual constitution of crystallographic structures.

Figure 1.4.4 presents a schematic comparison of dislocation-based representations of “twin” and “microband (MB)” between FCC and BCC (Murr et al., 1996), where we can find some similarity between the twin and MB, except for the case of twin for FCC. Comparisons between deformation twins and MBs formed under oblique shock/ hypervelocity impact loading for Cu can be found in Figure 1.4.5, which demonstrates the micrographically resembling morphologies between them. It is worth noting that

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Figure 1.4.4  Schematics of deformation twins and MBs, comparing FCC and BCC metals and

demonstrating the similarity of the two deformation modes (Murr et al., 1996, p. 131, figure 8). Adapted with permission of the publisher (ASM International).

Figure 1.4.5  Comparison of deformation twin and MB formed on a polycrystalline Cu surface:

(a)(d) twins in oblique shocked copper targets with dG = 375 µm and 141 μm, respectively, and (b)(c) MBs below hypervelocity impact crater in copper targets with dG = 763 µm and 35 µm, respectively (Murr et al., 1996, p. 124, figure 2). Adapted with permission of the publisher (ASM International).

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the arrays of dislocations illustrated there will not always be “lattice” dislocations, but more likely CD ones (namely, “virtual” – see Section 1.4.4 for details). This tends to mean researchers experienced in TEM observations may not always accept (or be rather skeptical about) such dislocation-based representations. The twinning in FCC can be interpreted quite distinctly from that in BCC, as a serial stacking of plural “stacking faults” sandwiched between pairs of partial dislocations. Therefore, the formation mechanism is greatly attributed to the motion of leading and trailing partial dislocations. Since, basically, both the twins and MBs are well described as CD dislocations to be introduced via accommodation of a sort of “excessive” local deformation, extended use of the “incompatibility tensor”-based model will be effective for descriptions of them, as discussed in Chapter 11 in the context of application to single crystal pure Mg.

1.4.2

Texture and Pole Figure Another important item to be added to Part I, “Fundamentals,” is “texture,” especially rolling texture, that is, recrystallization texture and deformation texture. Figure 1.4.6 provides a schematic drawing of the rolling process of sheet metal, together with the attendant deformation of crystal grains rotated toward and elongated into the RD. This is accompanied by the developments of “preferred orientations” of grains in addition to their significant shape changes. The term “texture” refers formally to the former, the preferred orientation, not the latter. However, the actual texture, presented via the pole figure or ODF, inevitably includes the effect of such morphological aspects of the composing grains, together with likely occurring intragranular inhomogeneous deformations manifested as various forms of deformation structure (cf. Chapter 3). Representative rolling textures are schematically summarized in Figure 1.4.7, that is, cube, copper, brass, γ, and gross orientations, expressed via hkl uvw as rolling direction rolling plane , together with their relationships with the Bunge-type Euler angles (1 , , 2 ). Euler angles are described in some detail in Appendix A1.7.

Figure 1.4.6  Schematic illustration of rolling process and attendant “texture” development with

preferred orientation.

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Figure 1.4.7  Schematics showing representative rolling textures, that is, cube, gross,

and γ ­orientations, together with copper and brass orientations, represented as {rolling plane}.

For scrutinizing the texture, we generally need the pole figure, or more precisely, the ODF for their 3D totality. Figure 1.4.8 concisely explains how we obtain a pole figure, while Figure 1.4.9 provides an overview of the ODF, both assuming texture presentations. The pole figure is a 2D stereographic representation of the orientation of a “selected” plane normal (pole) with respect to the sample reference frame, often utilized to describe texture, for example, for a rolled sheet metal, on which a set or group of all the equivalent specific crystallographic orientations of the crystal phases involved in the targeted sample are stereographically projected. In Figure 1.4.8, a pole is taken as an example with respect to a sample reference frame, specified as RD, ND, and transverse direction (TD). Here, a single crystal cube located at the center of a projection sphere is depicted, from which three cube directions of [100], [010], and [001] are ultimately projected onto the projection plane, resulting in the pole figure (a). By repeating the process, we obtain the corresponding plot (b) and the contour plot (c), indicating the intensity of such distributions of projected points. For the ODF, on the other hand, Figure 1.4.9 displays presentations in (a) full 3D, (b) 2D, and (c) a selected 2D Euler angle space (1 , , 2 ), together with typical examples of the rolling textures (c) and (d), respectively, specifying typical  ,  , and γ fibered textures. Special emphasis is placed on the γ fibered texture, to be discussed as follows. As shown already in the context of “FCC versus BCC” (in Section 1.4.1), the dual constitution of the atomic structures is also manifested as distinct textures of high contrast between them (see Figure 1.3.7). Typical textures for FCC metals observed

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Figure 1.4.8  Schematic drawing showing how to obtain and view the pole figures for a

textured sheet via cold rolling, taking an example of {100} poles of a cubic single crystal. A stereographic projection of directions onto a projection plane is illustrated, together with rolling, transverse, and normal directions of the rolled sheet sample, indicated as RD, TD, and ND, respectively (Hatherly and Hutchinson, 1979).

Figure 1.4.9  An overview of the ODF presented in Euler angle space, (1, , 2 ), with an emphasis on the γ fibered texture, typical to Fe and low-carbon steels such as IF steel. (a) 2D presentation on the (1,  ) plane at 2  45 (Urabe and Jonas, 1994, p. 437, figure 4), (b) 3D view of (b) (Hirsch and Lücke, 1988a, p. 142, figure 5(b)), and (c) that for FCC metals, schematizing typical rolling textures (Hirsch and Lücke, 1988b, p. 2869, figure 3). Adapted with permission of each publisher (Iron and Steel Institute of Japan, Hindawi, and Elsevier).

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Figure 1.4.10  Typical pole figures in FCC metals, referred to as “pure copper” type and “α-brass” type (Hu et al., 1952). Used with permission of The Minerals, Metals & Materials Society.

in cold rolling are depicted in Figure 1.4.10, that is, (a) pure Cu type and (b) αbrass type compared on {111} pole figures (Hu et al., 1952), where the former is for 95%-rolled Al, while the latter is 95%-rolled for 70–30 brass. The BCC metals, on the other hand, have their own counterpart, widely known as γ -fibered texture (see Figure 1.3.8). It is worth noting that the difference observed between (a) and (b) in Figure 1.4.10 is largely attributed to the difference in the SFE, where type (b) is considered to be greatly affected by the alternative deformation mode, that is, twinning (Section 1.4.1). The former type of pole figure (described as (123)[ 412] and (146)[ 211]) has been widely observed for FCC metals with relatively high SFE, whereas the latter type (described as (110)[ 112]) for those with low SFE such as brass and silver (Ag). Transition reported from the former toward the latter seems to be very informative, realized by decreasing the rolling temperature down to −196°C (e.g., Hu and Goodman, 1963), demonstrating that the high-intensity areas near the center of the former tend to split, eventually becoming closer to the former as a function of temperature. These authors discussed the close relationship with SFE change. For the case of the γ -fibered texture for BCC metals mentioned in Section 1.3.2 in the context of high r-value, the pole is often used where the intensity tends to be concentrated in the circular region, as schematically illustrated in the top right of Figure 1.4.11 for the case of IF steels. With this type of texture, the r-value can reach 2.5 or more.

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Figure 1.4.11  Schematics showing a typical texture called γ-fiber found in mild steels such as IF steels, where the C-impurity level is significantly reduced down to a few ppm or less, with the residuals being anchored by adding Ti or Nb as a form of TiC/NbN, and so on.

Figure 1.4.12  Definition of r-value (also referred to as the Lankford value) as a measure of deep drawability.

Incidentally, the γ -fibered texture has been known to be suitable for press forming, especially in deep drawing, because of its superior thinning resistance manifested as high r-value (Lankford value or plastic anisotropy parameter [Wagoner and Chenot, 2001]). The r-value, as schematically explained in Figure 1.4.12, is defined by the ratio of width strain ε w to the thickness strain ε t , that is,

r

 w  ln(w / w0 ) w with  (1.4.1) t  t  ln(t / t0 )

As understood from Eq. (1.4.1), the large r-value means to yield, for example, a large transverse deformation with a relatively small thickness reduction. In the metal-forming technology, this provides an index of deep drawability, since the large

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r-value yields easier shrink-type flanging deformation, allowing larger amount of materials to be drawn into a die cavity (the ability to allow circumferential shrinkage at the flange part against the reducing thickness). The high r-value in the γ -fibered texture (∼2.5 or larger) basically stems from the intrinsic slip-system constitutions to BCC metals as described earlier. As can be imagined, with the {111} plane in parallel to the sheet surface, the deformation through thickness direction is restricted because the slip direction is normal to the surface of the sheet, as schematically illustrated in Figure 1.4.11. It is interesting to note that some researchers, assuming the same mechanism will work, have tried to realize the γ -fibered texture in certain kinds of aluminum alloy (FCC) for the purpose of enhancing the press formability. As readers may readily notice, this will probably not be possible due to the following three reasons: (1) the γ -fibered texture is indigenous to BCC structures, (2) the relationship between the γ -fibered texture and high r-value is peculiar to BCC metals, and (3) the slip plane to be aligned in parallel to the blank sheets ought to be {110} for FCC metals according to the above logic.

1.4.3

Stereographic Projection and Standard Triangle One of the standard ways of describing crystallographic orientations explicitly is via stereographic projection. This is also called the “inverse” pole figure, since in this case the crystallographic axes are taken as the reference, instead of the targeted sample, as in the pole figure (see Figure 1.4.8 in Section 1.4.2). A way to obtain the stereographic projection is schematically given in Figure 1.4.13, where a two-step projection is indicated, that is, (a) from a cube (of atomic structure) to an enveloping sphere, and (b) from the sphere to a tangent circle. The resultant projection is shown in (c), where a standard triangle out of the 24 geometrical equivalents is highlighted. The standard triangle is composed of the three representative orientations of , , and . Note that the term “stereographic projection” itself stands for the projection of a sphere onto a plane, corresponding to the above process (b), which preserves local angles (referred to as the conformal transformation [or map]) but not length or area. Arbitrary crystallographic orientation is represented as a dot in the standard triangle, as exemplified in Figure 1.3.14(a). It should be noted that the dot in the standard triangle, however, does not contain all the crystallographic information because it represents just a single direction with respect to the cube axes. Based on Euler angles, it is represented by two angles, for example (, 2 ). Therefore, we need one more piece of information about the direction to fully identify the 3D configuration of the crystallographic orientation, for example, (1 , , 2 ), as in the ODF described earlier (Figure 1.4.9). Figure 1.4.14 summarizes the stereographic projections described in Sections 1.4.2 and 1.4.3, by compactly combining Figures 1.4.8 and 1.4.13, emphasizing the obtaining processes up to the resultant pole figure and the standard triangle, respectively. Here, the cubic structure model provided as the do-it-yourself kit in Figure 1.3.15 is

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Figure 1.4.13  Schematic illustration of the process for obtaining stereographic projection of directions in a cubic crystal, where a standard triangle is highlighted in (c) ((a) and (b) are reconstructed from Marder, 2000, p. 33, figure 2.17). Adapted with permission of the ­publisher (John Wiley & Sons).

used as the cube to be projected in the place of the one situated in Figure 1.4.13(a), for the sake of easier comparisons.

1.4.4

Crystal Dislocations versus CD Dislocations There is often confusion among researchers about the distinction between “CD dislocations” and “lattice or crystal dislocations.” They are, as a matter of fact, essentially distinct concepts (Shiotani, 1989; Yokobori, 1968; Yokobori and Ichikawa, 1967). As pointed out in Section 1.2.1, the notion of “dislocations” (of the CD kind) were initially introduced within the continuum mechanics framework (Love, 1944; Volterra, 1907) with no explicit correspondence to the “crystal” dislocations. Furthermore, the concept had been used in solving mechanics problems of continuum solids (not based on crystalline plasticity). While the crystal dislocations have a finite Burgers vector roughly commensurate with the lattice constant of the crystal considered, the CD dislocations have an infinitesimal Burgers vector. Unless at least the slip systems are specified and projected onto it, we cannot interpret the CD dislocations as the crystal counterparts. The CD dislocations can be imaginary, and do not always need to correspond to (or be attributed to) the “crystal” ones.

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Figure 1.4.14  Summary overview of two stereographic projections in Figures 1.4.8 and 1.4.13, that is, pole and inverse pole figures, comparing how to obtain and how to view the pole figures, taking an example of {100} poles of a cubic single crystal.

Figure 1.4.15  Comparison between interfacial dislocations and misfit dislocations, where the former can be mimicked by CD dislocations with an infinitesimal Burgers vector, thus normally yielding a long-range stress field, whereas the latter is represented by an array of isolated crystal dislocations with a finite Burgers vector introduced so as to relax the misfit situation.

Figure 1.4.15 provides an example eloquently comparing the two cases, which considers the interfaces of two atomic structures with slightly different lattice constants, like those with coherent precipitates (comparison of precipitates between coherent

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Figure 1.4.16  Comparison of precipitate-matrix interfaces between coherent and incoherent types.

Figure 1.4.17  Application of the notion of CD dislocations to the crack problem (mode I type) (Bilby et al., 1963).

and incoherent types is given in Figure 1.4.16). The left denotes CD dislocations accommodating the misfit caused by the difference in the lattice spacing between the two phases, whereas the right is for the same interface but has been relaxed via an array of “crystal” dislocations. The former is called “interfacial” dislocations, while the latter “misfit” dislocations. As can be readily understood, the former case does not actually include “crystal” dislocation at all, meaning the illustrated dislocation array is imaginary. Instead, such a situation is well described via a stress field based on the dislocation theory. Another example where such “imaginary” arrays of dislocations can work nicely as a model is “crack” (known as the BCS [Bilby-Cotterell-Swinden] model [Bilby et al., 1963]). Figure 1.4.17 displays a crack model via CD dislocations under mode I loading. The long-range nature of the stress field (as mentioned in Section 1.2.3.2) as well as the singularity of dislocations (in the present example, edge dislocations) are

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Figure 1.4.18  Analytical solution of the crack problem given in Figure 1.4.17 in the form of the distribution function of CD dislocations.

Figure 1.4.19  A crack expressed by CD dislocations interacting with emitted discrete dislocations for evaluating COD (Vitek, 1976b).

the reasons this model works well. As you can see, the assumed dislocation array is nothing more than the imaginary one, situated so as to satisfy both the crack-tip stress field and the boundary conditions. The corresponding distribution of dislocations to the crack is obtained by solving a kind of integral equation (1st Fredholm type) with respect to the distribution function f ( x ) (inset of Figure 1.4.17), and is explicitly shown in Figure 1.4.18. Application examples of this technique to the fatigue-crack propagation problem can be found in Homma (1989) and Homma et al. (1984). Figure 1.4.19 displays an example of such applications to crack-tip problems, for the purpose of evaluating the crack-opening displacement (COD), where interactions with a discrete array of dislocations are considered (Vitek, 1976b). Elaborate achievements in dislocation theory-based fracture mechanics have been published as a monograph by Johannes Weertman (1996).

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A1.1  Derivation of Peach–Koehler Equation (Formula)

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Appendix A1  Energy Landscape for Dislocation Pairs A1.1

Derivation of Peach–Koehler Equation (Formula) Forces acting on dislocations are called the “Peach–Koehler (P–K) force” and the formula for calculating them is generally referred to as the “Peach–Koehler (P–K) equation.” We show in the following a standard derivation process for the P–K equation. Note that the expression for the P–K force is a specific version of the energy-momentum tensor that applied to a dislocation line, just like the J-integral against a crack tip. We derive the P–K equation along this line in Chapter 7, in the context of the gauge fieldbased formalism of dislocations and defects; reference is also make to the J-integral. Consider a straight dislocation line (the unit vector along with it is represented as ξ ) with the Burgers vector b (inclined to it at an angle θ ). Displacing the dislocation line by δ r, we measure the area swept by the operation as



ds    dr. (A1.1.1) Under the application of stress σ , the corresponding force acting on the dislocation line is calculated as



f    ds      dr  . (A1.1.2) Since the resultant shear displacement should be b, the work done by the above operation is given by



 W  f  b

     dr    b,

(A1.1.3)

 b      dr  



  b       dr    b       dr

(A1.1.4)

   b     dr.

To obtain the last line, we used the invariance of the operation under cyclic permutations, that is, a   b  c    a  b   c   abc   bca   cab . Therefore, the force per unit length of dislocation line is obtained as

f    b    . (A1.1.5) This is called P–K equation, defining the P–K force (cf. Eq. (7.3.65) in Section 7.3.7). The corresponding index notation is expressed as



fk kji  jl bl i   ijk  jl bl i . (A1.1.6) From this result, we immediately learn that the force on dislocation lines f always acts perpendicularly to them ξ , that is, f   . From projections of the P–K force, f , on the appropriate directions, we readily obtain the corresponding components (Hirth and Lothe, 1982). For the climb component, for example, it should be the direction both normal to b and ξ , which can be calculated by taking the inner product with the unitary vector of ( b  ), that is,

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fclimb  f 

( b   )   b      ( b   ) (A1.1.7)  b b

Similarly, for the glide component we obtain

fglide  f 

  ( b   )   b        ( b   ) , (A1.1.8)  b b

because this should be normal to both ξ and ( b  ). If the Burgers vector b is inclined to the dislocation line (ξ ) at an angle θ , it is made up of two components, that is, the edge component, bedge  b sin  , normal to ξ , and the screw component, bscrew  b cos  , parallel to ξ . The glide plane is defined as that which contains both the dislocation line and the Burgers vector. Another important thing we should know is that applied stress σ can be replaced by more generalized stress, or the linear superposition of various contributions or origins. Such stress typically includes those produced by or associated with: (1) self-interaction of a curved dislocation, (2) image force in the presence of the free surface, and (3) interactions with other dislocations and defects.

   ext   self   image   int (A1.1.9)



Correspondingly, we can deal with the P–K forces independently as f  fext  fself  fimage  fint (A1.1.10)



A1.1.1

Examples of P–K Force Consider a dislocation line   [0, 0,1]T . Assuming edge and screw dislocation lines b = [b, 0, 0]T and b = [0, 0, b]T , respectively, we have, explicitly,



e1 e2 e3 T T f0   11b,  12 b,  13 b    0, 0,1   11b  12 b  13 b , 0 0 1 T   12 b,   11b, 0  for the edge dislocation. f0   31b,  32 b,  33 b    0, 0,1 T

T

  32 b,   31b, 0  for the screw dislocation. T

A1.2

Force Acting on a Parallel Dislocation Pair Consider two straight-edge dislocation lines   [0, 0,1]T aligned in parallel. Assume they have bA = [bA , 0, 0]T and bB = [bB , 0, 0]T , respectively, and are situated d apart. The interaction energy between the two edge dislocations A and B is given by



edge edge edge Eint 0    A :  B dx (A1.2.1) V

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Rewritten in terms of the eigenstrain, this becomes edge edge *edge Eint dx 0    A : B V



V









*edge edge *edge     Aedge dx 21  B 21   A 12  B 12

(A1.2.2)

Here, the eigenstrain components for the edge dislocation B is given by (see Eq. (A1.5)) bB *edge  B*edge   x2  d  H   x1  x  , (A1.2.3) 21   B12  2 while the stress component is





2 2  bA x1 x1  x2 (A1.2.4)  x1 , x2   x1 , x2   2 1   x 2  x 2 2 2 1 Substituting them into Eq. (A1.2.2), we have

 Aedge 21

  Aedge 12





 bA bB   x1 ( x12  x22 )  ( x2  d ) H ( x1  x ) dx1dx2 2 (1  )   ( x12  x22 )2 2 2  bA bB x x1 ( x1  d ) dx1  2 (1  )  ( x12  d 2 )2

edge Eint 0 





 d2 x  bA bB  x x13  dx   2 2 2 1  2 2 2 dx1  . 2 (1  )  ( x1  d ) ( x1  d ) 

The above integration can be analytically performed, provided the lower bound  is replaced by a reasonable value −R, as,

A1.3



edge Eint 0 

2 2  bA bB  1  x1  d ln   2 (1  )  2  R 2  d 2

Elastic Strain Energy for Dislocations Elastic strain energy for dislocations is given in two ways as 1 1 E disloc    :  e dx     :  * dx, (A1.3.1) 2 V 2 V where ε * represents the eigenstrain introduced by the dislocations. Correspondingly, σ in the second expression indicates the induced internal stress field. By rewriting the elastic strain ε e using ε * in the first expression, that is,  e     *, we have 1  2 V 1    2 V

E disloc 

 d2 d 2   2 2  2 2   x1  d R  d  





:    * dx 1  u  :   dx    :  * dx 2 V  x 

(A1.3.2)

Here,  :  in the first line can be replaced by  : (u / x) (u / x   is the ­distortion tensor that is generally asymmetric) because of the symmetry of σ . The first term is further rewritten as follows by performing integration by parts:

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1  2 V

1 1     u  :   dx    uds     udx 2 s 2 V  x   x  1 1   n    uds   div  udx (A1.3.3) s 2 V 2  0.

Since we are looking at the internal stress that yields zero at the surface, the first term in the second line vanishes. The second term, on the other hand, also becomes zero because div  0 in the absence of body force. Therefore, we ultimately obtain the expression Eq. (A1.3.1) from Eq. (A1.3.2).

A1.4

Elastic Interaction Energy for a Dislocation Pair For elasticity, the interaction between two dislocation fields can be evaluated by the linear superposition (the superposition principle if linear elasticity) as 1 E disloc    A   B  :  Ae   Be dx 2 V 1 1 1 1    A :  Ae dx    B :  Be dx    A :  Be dx    B :  Ae dx (A1.4.1) 2 V 2 V 2 V 2 V  E Adisloc  E Bdisloc  E int







Here, the interaction energy is defined as 1 1  A :  Be dx    B :  Ae dx 2 V 2 V (A1.4.2) e    A :  B dx    B :  Ae dx ,

E int 



V

where  A :  Be

A1.5



  B :  Ae

V

is used. Using the eigenstrain formalism, we alternatively have





E int     A :  B* dx     B :  *A dx . (A1.4.3) V

V

Examples of Dislocations in Eigenstrain Representation Let us consider a straight-edge dislocation on the x1 − x2 plane. We can introduce it by the following operation. A cut is made in the x1 < 0 region (Figure 1.2.16) first, and then the opposite surfaces of the cut are welded to restore the continuity after displacing them relative to each other by b. This results in an edge dislocation situated at the origin with the Burgers vector b = (b, 0, 0). This procedure is the so-called Volterra operation. Since the displacement caused by this operation is u = (b, 0, 0), the strain components are given as *edge *edge b  21  12   ( x2 ) H   x1 , (A1.5.1) 2

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A1.6  Energy Landscape of Edge Dislocation Pairs

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otherwise zero. Here, δ ( x ) is the Dirac delta function and H ( x ) represents the Heaviside step function. They are defined respectively as  (at x  0)  ( x)   0 (at x  0)



a

a  ( x)dx  1 (a 0), (A1.5.2)

and 0 (at x  0)  H ( x )  1 / 2 (at x  0) . (A1.5.3) 1 (at x  0) 



There is a following relationship between the two functions, that is,

A1.6

 ( x) 





x dH ( x )  H ( x )    ( x )dt (A1.5.4)  dx

Energy Landscape of Edge Dislocation Pairs Consider a pair of straight-edge dislocations on the x1 − x2 plane, as depicted in the inset in Figure A1.6.1, where relative angles θ1 and θ 2 are ranged from 0 to 180°, as well as the mutual distance between them. Here,   180 corresponds to

Figure A1.6.1  Energy landscape for pairwise configurations of edge dislocations.

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vacancy-type dipole configuration, while   0 represents the monopole alignment. The former, with the 45° alignment, yields the minimum energy, as shown in the bottom-left region in Figure A1.6.1, whereas the latter takes the maximum, as displayed in the middle-upper region. With the mutual distance tending to 0 for the dipole configuration, the energy decreases down to 0, as shown in the bottom-right region, demonstrating roughly a double-well-type energy landscape responsible for the pairwise interaction. This is assumed in Chapter 10 for the derived effective theory with respect to the annihilated field.

A1.7



Euler Angles Euler angles can uniquely specify 3D rotations of the targeted coordinates with respect to a reference coordinate system by using three angles. In metallurgy, there are two conventions for defining the Euler angles, that is, Bunge and Roe (Nagashima, 1984). Throughout the book, we use the Bunge-type definition, represented conventionally by (1 , , 2 ). Note, the Roe convention uses ( , ,  ) instead. An easy way to understand Euler angles is to break down the associated coordinate transformation process into three steps, as illustrated in Figure A1.7.1, that is, from ( X A , YA , Z A ) to ( X B , YB , Z B ). Since each step is simply a rotation about an axis, all we have to consider is the order of the axes. For the Bunge convention, we choose Z → X → Z . This is the very reason for this style to use 1    2 as the notation. Note that, in the Roe style, the choice is Z → “Y ” → Z, thus a simple set of conversions holds between the two conventions, as 1     / 2,    , 2     / 2 . The three steps are as broken down sequentially as follows, as displayed in Figure A1.7.1(a) and (b). The first step is the rotation about Z A-axis, specified by the angle ϕ1, given by XA   x   cos 1 sin 1 0   X A           y     sin 1 cos 1 0   YA   1  YA  , (A1.7.1) Z   z   0 0 1   Z A   A    which transforms from ( X A , YA , Z A ) to ( x ′, y′, z ′). Likewise, since the second step refers to the rotation about the “ X A-axis,” specified by Φ, we have



0 0   x   x  1  x           y   0 cos  sin    y    y  , (A1.7.2)  z   0  sin  cos    z    z         expressing the transformation from ( x ′, y′, z ′) to ( x ′′, y′′, z ′′). Lastly, the third step is again the rotation about the z ′-axis with angle ϕ2, which is given by the same form as that for the first step, that is,



 X B   cos 2     YB     sin 2 Z   0  B 

sin 2 cos 2 0

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0   x   x       0   y  2  y  . (A1.7.3)  z   1   z    

A1.7  Euler Angles

85

Figure A1.7.1  A series of processes obtaining Euler angles (Bunge style).

By combining the above three processes, Eqs. (A1.7.1) to (A1.7.3), we define the Euler angles of Bunge type as

 X B   2  1  X A  , (A1.7.4)

 X B   cos 2     YB     sin 2 Z   0  B 



sin 2 cos 2 0

0  1 0 0   cos 1 sin 1 0   X A      0  0 cos  sin     sin 1 cos 1 0   YA  1  0  sin  cos    0 0 1   Z A   cos 2 cos 1 sin 2 cos  sin 1 cos 2 sin 1 sin 2 cos  cos 1 sin 2 sin      sin 2 cos 1 cos 2 cos  sin 1  sin 2 sin 1 cos 2 cos  cos 1 cos 2 sin    sin  sin 1  sin  cos 1 cos    XA     YA  Z   A

(A1.7.5)

Notice that the first two steps determine the [001] direction away from the referential Z A − X A (ND-TD) plane, as conformed in Figure A1.7.1(c). Therefore, if we assume we skip the first step, the [001] direction stays on the Z A − X A (ND-TD) plane, while the [100] direction coincides with the X A (RD) axis. Based on this consideration, we obtain a simple schematic, as inserted in Figure 1.3.14(d) and Figure 1.4.14(d), indicating a crystal orientation with respect to the cube axes by using the remaining two angles (, 2 ). This is convenient for quick and intuitive recognitions of crystal orientations, for example, on standard triangles (Figure 1.3.14(c) and Figure 1.4.14(c)). In this case, of course, the other two crystal orientations are indefinite, and should be specified by ϕ1.

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2

Dislocation Dynamics and Constitutive Framework

2.1

Overview: Unified-type Crystal Plasticity Constitutive Equation This chapter provides a physically sound foundation for constructing a constitutive framework by considering statistical mechanics-based dislocation dynamics. Figure 2.1.1 illustrates a large gap that has been unfilled for a long time, causing a serious rift among researchers in different fields that has not been addressed to date. Continuum mechanics researchers have often used a power-law-type constitutive equation (Hutchinson, 1976; Pan and Rice, 1983) because Eq. (2.1.1) is easy and convenient to handle computationally, although it is just phenomenological having no physically clear meaning in it:



 ( )



  ( )  a ( )  ( ) g

   ( )   ( )   g

1

1

m , (2.1.1)  

where m represents the strain-rate sensitivity coefficient, with m → 0 the strain-rate independent type specified by  ( )  g ( ) at  ( )  a ( ) . The relationship with the constitutive behavior we want to describe in the present context is schematically shown in Figure 2.1.2, allowing the power-law-type model to mimic a part of it at most. Even if we utilize a statistical mechanics-based constitutive framework, as a matter of course, there could be still substantial differences between individual and group motions of dislocations. This is, however, not very serious compared with the use of the phenomenological counterpart, in which all the physics-based mechanisms are excluded from the start. In the statistical mechanics’ sense, dislocations move in a crystal with the assistance of thermal fluctuations at finite temperatures under the help of externally applied stress. Such a process can thus be described by statistical mechanics-based dislocation dynamics. The probability of finding the mean velocity of dislocations against obstacles under external stress is given by the Arrhenius-type equation, which provides the basis of constitutive modeling. The details of the formalism and its applicability are given in this chapter, after some important phenomenology and basic notions have been presented.

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2.2  Strain-Rate Dependency of Flow Stress

87

Figure 2.1.1  Large gap between atomistic pictures of plasticity and a phenomenological

constitutive equation of the power-law type conventionally used in crystal plasticity-based simulations.

Figure 2.1.2  Relationship between the constitutive model ultimately used in our study and the

conventionally used power-law model.

2.2

Strain-Rate Dependency of Flow Stress

2.2.1

FCC versus BCC Revisited FCC and BCC metals have mutually similar stress responses with respect to the strain-rate dependencies, but with distinct underlying microscopic controlling mechanisms. Figure 2.2.1 displays examples of experimental results for Cu and Fe/carbon steels, as representatives of FCC and BCC metals/alloys, respectively, that is, strain-rate-dependent stress–strain curves (top row) and strain-rate versus flow-stress

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Figure 2.2.1  Examples of experimentally observed stress–strain curves (top row), together with

strain-rate versus flow-stress diagrams (bottom row), comparing Cu and Fe/carbon steels with apparently mutually similar trends. The latter for Cu is from Follansbee and Kocks (1988). Adapted with permission of the publisher (Elsevier).

diagrams (bottom row), where the latter for Cu (0.9999 purity) is from Follansbee (1986), and covers a wider range of impacts strain rate, exceeding 10 4 s−1 . Figure 2.2.2, on the other hand, shows the same set of diagrams but emphasizes the contrast between them in terms of simulated stress–strain curves and the attendant schematic strain-rate–flow-stress diagrams. As indicated in the figure, FCC metals tend to yield an enhanced hardening rate with a smaller difference around yield stress, whereas BCC metals show noticeably enhanced stress rise at high strain rates around the yielding, with smaller (even reducing) hardening rate with increasing strain. Regarding the    or    diagram, both metals yield a similar trend, that is, linear and moderate increase in flow stress with ε or γ up to around 103 s−1, followed by a rapid and drastic increase in σ or τ after 10 4 s−1 , but with different constitutions, that is, from dislocation processes and drag mechanisms for FCC, and dislocation processes and the Peierls overcoming mechanism for BCC, respectively. Thus, the constitutive equations for metals are required for expressing these wide ranges of strain rate and temperature effects as well as their material dependencies, particularly the clear distinction between FCC and BCC metals. Note that these requirements are not confined

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2.2  Strain-Rate Dependency of Flow Stress

89

Figure 2.2.2  Schematics of strain-rate-dependent stress–strain behaviors of typical FCC and

BCC metals, showing similar but distinct underlying rate-controlling mechanisms.

to the high strain rates but should also apply to static or quasi-static cases; see Section 1.4.3. It should be noted that there remain many uncertainties about the mechanism for the rapid increase in the flow stress exceeding 103 s−1 , a representative interpretation of which will be mentioned in the last part of Section 2.7. The current book, however, assumes this rather tentative view for simplicity.

2.2.2

Thermal and Athermal Obstacles Figure 2.2.3 is a famous diagram denoting strain-rate- and temperature-dependent mechanisms of plastic deformation, that is, those against dislocation motion, originally presented by Rosenfield and Hahn (1974) and later modified by Perzyna (1974). There are roughly three regimes for the rate-controlling mechanisms, that is, athermal (regime I), thermally activated (regime II), and damping (regime IV) mechanisms. Figure 2.2.4 is also a famous diagram, by Campbell and Ferguson (1970), which provides an example of the strain-rate and temperature dependencies of the lower yield stress for low-carbon steels. BCC metals, including Fe and carbon steels, generally exhibit remarkably large strain-rate and temperature dependency of the flow behavior. At high enough strain-rate range (over 103 s−1), and equivalently low temperatures, in the context of statistical mechanics, one can observe noticeable sensitivity to strain rate in Figure 2.2.4, which is one of the salient features of BCC metals.

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Figure 2.2.3  Rate-controlling mechanisms map of temperature versus strain-rate diagram

(Perzyna, 1974).

Figure 2.2.4  Lower yield stress–strain-rate diagram for mild steel (Campbell and Ferguson, 1970).

Adapted with permission of the publisher (Taylor & Francis).

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2.2  Strain-Rate Dependency of Flow Stress

91

The governing mechanism of special importance is the thermal activation processes for overcoming various kinds of thermal obstacle (regime II). The second most important mechanism is viscous damping, such as phonon drag due to lattice vibration or electron scattering (region IV). The thermal activation mechanism is controlled by the corresponding activation energy ∆G for overcoming thermal obstacles (also referred to as “short-range” obstacles), and is generally given as a reducing function of the effective stress τ *, that is, G( * ).

2.2.3

Examples of Obstacles Thermal obstacles are roughly classified in two groups, as listed in Figure 2.2.5, that is, intrinsic and extrinsic kinds. The Peierls potential belongs to the former, while point obstacles, small precipitates, and forest dislocations are categorized in the latter. The Peierls potential is the major rate-controlling mechanism for BCC metals, whereas forest cutting is responsible for the strain hardening in FCC metals, as is alluded to in Figures 2.2.1 and 2.2.2. The athermal obstacles, on the other hand, are those that cannot be overcome by thermal activation, that is, provided ∆G  kT . They are usually large obstacles, such as second-phase particles and precipitates, and long-range obstacles, for example, grain boundaries and coherent precipitates. Figure 2.2.6 schematically illustrates various kinds of obstacle existing within crystalline metals (Vohringer, 1989).

Figure 2.2.5  Twofold classification of obstacles against dislocation motion; that is, intrinsic and

extrinsic obstacles/thermal and athermal obstacles.

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Figure 2.2.6  Schematic illustration of various kinds of defect within crystalline materials

(Meyers, 1994; Vohringer, 1989; Wang et al., 2019). Adapted with permission of the publishers (Wiley-Interscience, Elsevier, and Academic Press).

Stress in the present context is often organized into thermal (or effective) and athermal (or internal) parts, that is,

   * (T , , structure)   i (structure), (2.2.1) where the former is given as a function of strain rate, temperature, and substructures, while the latter depends solely on the microstructures. Note that this expression, commonly employed by materials scientists and metallurgists, is not always convenient for materials mechanists to use, as they generally need to ensure a reciprocal relationship with respect to strain rate γ in their constitutive modeling. The expression given by the form of Eq. (2.2.1) cannot be solved for γ when plural mechanisms are simultaneously considered. This problem will be mentioned later in relation to the “Kocks–Mecking model.”

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2.3  Dislocation Velocity versus Stress

93

Figure 2.2.7  Schematics showing the Peierls potential in terms of a force–distance diagram,

together with a corresponding dislocation motion from one equilibrium position to the next.

Figure 2.2.7 shows a schematic of the PN force responsible for the Peierls overcoming process, together with the force (stress)–distance diagram. The Peierls overcoming process is shown in detail in Figure 2.2.7, where the formation, and the following (subsequent) transverse extension, of a double kink is considered to be the major mechanism, referred to as the double-kink mechanism. The double-kink profile at its equilibrium determines the activation volume and thus the activation energy for the process.

2.3

Dislocation Velocity versus Stress Figure 2.3.1 provides a typical example of the relationship between applied shear stress and the velocity of dislocations for a LiF single crystal measured by the etchpit technique (Johnston and Gilman, 1959). As can be confirmed in the figure, edge dislocations are always faster than screw components. Up to the intermediate stress level, the velocity can be approximated by a power-law function of the effective stress, that is,





 

v  v0  *

m

, (2.3.1)

while for a larger stress regime, the relationship is well described by an exponential function such as v  v0 exp  D  * , (2.3.2)

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Figure 2.3.1  Dislocation velocity versus resolved shear stress for an as-grown LiF single

crystal, together with a schematic of measurement method via etch-pit technique (Johnston and Gilman, 1959, p. 152; Meyers, 1994). Adapted with permission of the publishers (American Institute of Physics and Wiley-Interscience).

where D is a constant. We can also confirm from the diagram the definite upper limit of the dislocation velocity corresponding to the shear wave velocity cshear  G /  , where ρ is the density of the material. The inset in Figure 2.3.1 is a schematic of the measurement method using the etchpit technique, where the mean velocity is evaluated by dividing the distance between the pits by the stress pulse duration ∆t. The inset shows large and small pits, each indicating the initial and ending positions of a dislocation, respectively, marked before and after imposing a stress pulse. Figure 2.3.2 provides an example of a micrograph with a number of etch pits, indicating dislocation movements before and after the test (Marukawa, 1967). The etch-pit technique allows us to measure the rate dependency of dislocations for various materials. Figure 2.3.3 (left) shows an applied shear stress versus dislocation velocity representation for various materials (Nadgornyi, 1988; Takeuchi, 2013), together with a schematic diagram (Figure 2.3.3, right), classifying three regimes of the distinct rate-controlling mechanisms. These are those associated with the thermal activation mechanism (regime I), phonon drag (regime II), and the relativistic effects (regime III). Also, five material categories are schematically indicated (Figure 2.3.3, right), from which we can roughly distinguish the following trends:

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2.3  Dislocation Velocity versus Stress

95

Figure 2.3.2  Example of dislocation velocity measurement by using etch-pit technique in a Cu

single crystal (Marukawa, 1967, p. 506) (courtesy of K. Marukawa’s family).

Figure 2.3.3  Dislocation velocity-resolved shear-stress relationships for various materials

(left) and their schematic representations (right), indicating three regions with distinct rate-controlling mechanisms (Maeda and Takeuchi, 2011; Meyers, 1994; Takeuchi, 1991). Data on the right is adapted from Takeuchi (2013, p. 80) and Nadgornyi (1988) with permission of the publishers (Uchida Rokakuho and Permagon).

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( 1) FCC metals: Located far upper left, readily accelerated to the phonon-drag region. (2) BCC metals: Located far lower right (contrastive to FCC metals), mainly activated thermally, at most. (3) HCP metals: Basically similar to FCC metals. (4) Ionic crystals: Occupy transcendently the diagram from the bottom left to the top right, covering all the three rate-controlling mechanisms. (5) Semiconductors: Similar velocity range to BCC metals but need larger range of external stress, activated thermally. Regarding Figure 2.3.3, S. Takeuchi compiled data presented in Nadgornyi (1988) with painstaking elaboration to construct the unified diagram. Similar versions can be found in Haasen (1996, p. 283), Takeuchi (2013, p. 80), and Meyers (1994). Figure 2.3.4 is a schematic viewgraph about the drag effect on the rapid flow-stress rise at around the strain rate of 104 s−1. There has been a controversy over interpretations of the mechanism responsible for this abrupt increase in flow stress, that is, one group roughly ascribes the increase to the strain-rate-dependent substructure evolution and the other group believes it to be based on the phonon drag. The former is closely connected to the notion of mechanical threshold stress (MTS), as will be encountered in Section 2.7, which represents the flow stress at absolute zero temperature, thus reflecting the evolving or evolved substructures. Note that measured MTS actually follows the same trend as the    diagram, whereas a strain-rate change test at a strain-rate range exceeding 5,000 s−1 demonstrated instantaneous response of the flow stress, during which substructure evolution cannot be taking place. Also worth noting is that this sort of argument should consider the associated temperature rises if it is to be done rigorously (e.g., Meyers, 1994). Figure 2.3.5 shows an example of the relativistic effect on the stress field around a screw dislocation. The relativistic effect is considered to become non-negligible as the dislocation velocity approaches cshear (Meyers, 1994). The stress field becomes

Figure 2.3.4  Schematics showing contribution of phonon-drag mechanism to rapid flow-stress

increase over a 103–104 s−1 strain-rate regime.

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97

Figure 2.3.5  Example of the relativistic effect on a stress field around a screw dislocation,

where initially circular stress fields are noticeably compressed along the direction of motion (Meyers, 1994).

extremely flattened when the dislocation is accelerated up to 99% of the shear wave velocity, while no drastic change takes place yet at a velocity ratio of 0.5.

2.4

Physically Based Constitutive Framework

2.4.1

Arrhenius-type Rate-controlling Equation A physically based constitutive framework is defined using statistical mechanics. The derivation process is summarized in Figure 2.4.1. The mean velocity of dislocations v is expressed by the Arrhenius-type equation, describing the probability of finding v against the activation energy under the given temperature T (absolute temperature), that is,



 G  v  v0 exp    , (2.4.1)  kT  where ∆G is the apparent activation energy for the rate-controlling process of interest. Meanwhile, v0 is the velocity at G  0; v0  Lb * where L is the mean free path of dislocations b and υ * is the modified Debye frequency representing the ground vibrational frequency of dislocations estimated as  *   D  (b / 4l* ), with υ D being the Debye frequency (Kocks et al., 1975) and b the magnitude of the Burgers vector. Here, the activation energy is further given as a function of the effective stress τ *,

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Figure 2.4.1  Overview of the statistical mechanics-based constitutive equation, where mean

velocity given by the Arrhenius equation as a function of apparent activation energy is substituted into Orowan’s equation.

that is, G  G( * ). Equation (2.4.2) will be derived by envisaging the overcoming process of a dislocation segment under external stress based on statistical mechanics detailed in Section 2.4.3. By substituting the above mean velocity into Orowan’s equation,

   m bv, (2.4.2) we have the framework for constructing the constitutive equation, that is, the    relationship



 G( * )    0 exp    , (2.4.3) kT   with 0   m bv0 . This is the basis for the descriptions that follow.

2.4.2

Expression of Apparent Activation Energy Figure 2.4.2 describes the apparent activation energy, reduced with the help of work done by the effective stress τ * , that is,



G  G0  b  a* d * , (2.4.4) where a* ≡ l* d * is called the activation area, having a physical image of the area swept by a dislocation, with l* and d * representing the length of a dislocation segment responsible for the process of interest and the width of the obstacle to be considered. Similarly, v*  b  l* d *  b  a* is referred to as the activation volume simply because it has a dimension of volume but no direct physical image. Here, the effective stress τ * is also termed thermal stress, in contrast to the internal (athermal) stress denoted as τ i .

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2.4  Physically Based Constitutive Framework

99

Figure 2.4.2  Apparent activation energy and its modeling as a reducing function of effective (thermal) stress, where its general expression is based on the MTS concept (Kocks et al., 1975).

Figure 2.4.3  Derivation process of Orowan’s equation.

These are interrelated via  *     i . A convenient form that can approximate a wide variety of obstacles, proposed by Kocks et al. (1975), is q



  * p   G  G0 1   *   , (2.4.5)          with p and q being parameters to specify the obstacle, provided 0 ≤ p ≤ 1 and 1 ≤ q ≤ 2. Here τˆ* represents the effective stress at T = 0K, called the MTS. Note that MTS is not always constant, but can be subject to change as the microstructures evolve during plastic deformation. Given all the information about the substructural changes and attendant history, MTS is, in a sense, one of the most important notions in expressing all the history effects. A schematic explaining the derivation of Orowan’s equation is given in Figure 2.4.3. When  m  0, we have an equation one can readily understand. By substituting Eq. (2.4.5) into Orowan’s equation (2.4.2), we have an explicit framework for the constitutive equation (corresponding to Eq. (2.4.3)),

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Dislocation Dynamics and Constitutive Framework



G   0 exp  0

kT

  * p  1         *      

q

. (2.4.6)



Further, by using the expression for the effective stress given in Eq. (2.2.1), we have





G   0 exp  0

kT

  1      i    *  

   

p q 

  

. (2.4.7)



Thus, we finally have the constitutive relation in the form of a    relationship.

2.4.3

Force–Displacement Diagram Figure 2.4.4 is a schematic drawing of the forest-cutting process as one of the important thermal activation mechanisms controlling strain-hardening properties. For single-phase pure metals, especially FCC, this process is considered to be responsible for determining the hardening characteristics. The details can be delineated through the interaction matrix for the dislocation–dislocation interactions mentioned in the “latent hardening test” (Section 1.3.5). Figure 2.4.5 shows force–displacement diagrams, which are convenient for describing thermal obstacles expressed as a function of the effective stress. In the figure, force–displacement diagrams for both short- and long-range obstacles, together with their combined version, are schematically shown. The long-range obstacle can be characterized as



∆G  kT (2.4.8) in terms of the activation energy, where k is Boltzmann’s constant. Figure 2.4.6 describes the force–displacement diagram in detail, with the area ∆G0 being the thermally activated motion of a dislocation. Let us imagine a situation where

Figure 2.4.4  Schematics of the forest-cutting mechanism (Meyers, 1994).

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101

Figure 2.4.5  Schematics of force–displacement diagrams for long- and short-range obstacles,

together with their combined version (JIM, 1985; Meyers, 1994).

Figure 2.4.6  Details of a force–displacement diagram for the thermally activated motion of

dislocation, together with a corresponding free energy diagram, where a dislocation under external force (stress) is halfway up the mountain waiting for additional energy to be provided by the thermal vibration of atoms in order to overcome an obstacle.

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Figure 2.4.7  Details of a force–displacement diagram for the thermally activated motion of

dislocation, together with its description based on statistical mechanics, leading ultimately to an Arrhenius-type expression of the rate-controlling process for finding mean dislocation velocity.

a dislocation is located at the foot of the mountain waiting for the thermal energy to be provided by the thermal vibration of atoms in order to overcome an obstacle. Under an application of external stress, the work done by it can push the dislocation halfway up the mountain, as shown in Figure 2.4.6. Now the rest of the energy to be thermally provided is

G  G0  Wp , (2.4.9) where Wp represents the work done by the external stress. The quantity ∆G is the net activation energy actually required to overcome the obstacle, and thus is called “apparent” activation energy, given as a function of effective stress τ *. From Eq. (2.4.9), one can see that the apparent activation energy is a kind of Gibbs free energy, as is also shown in the figure. Using this “apparent” activation energy, we can evaluate the mean velocity of dislocations passing through the prescribed obstacle, as visually presented in Figure 2.4.7. Since the probability for a dislocation segment to overcome the obstacle with the help of thermal energy is proportional to the Boltzmann factor,



 G  exp    , (2.4.10)  kT  we then have the explicit expression for the mean velocity of dislocations as

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2.5  Representation of Obstacles



103

 G  v  v0 exp    . (2.4.11)  kT  Thus, we can derive the most fundamental equation identical to Eq. (2.4.1), called the Arrhenius-type equation. From the preceding discussion, now we know that the dislocation motion ( v or γ ) is describable with explicit expressions for G  G( * ), depending on the kinds of obstacle responsible for the rate-controlling processes.

2.5

Representation of Obstacles Examples of various kinds of short-range obstacle are depicted in Figure 2.5.1, that is, (a) jog drag, (b) bowing-out through second particles, (c) Peierls overcoming, and (d) cross-slip processes. Table 2.1 lists examples of typical values of the quantities characterizing the activation energy for representative mechanisms, that is, nonconservative motion of jog, the PN barrier, clime, cross-slip, and dislocation intersections. Since the nonconservative motion of jog and the clime mechanisms are dominated by the diffusion process, the activation energy for these is associated with self-diffusion. Only the Peierls overcoming process yields constant activation volume, that is, v*  v*( ), because it is characterized as the intrinsic obstacle among others. This will be revisited later with

Figure 2.5.1  Various thermal activation processes: (a) jog dragging, (b) bowing out, (c) Peierls overcoming, and (d) cross slip via partial constriction of extended dislocations against stacking fault energy, where the effective length of the dislocation segment is highlighted in each case (JIM, 1985).

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Table 2.1  Various thermal activation mechanisms and corresponding activation volumes. Adapted from Tomota (2001). Mechanisms

d*  b 

l*  b 

v*  b 

Remarks

Forest intersection Peierls–Nabarro overcoming Cross-slip Nonconservative motion of jog Climb

≤1 ~1 ~1 ~1 ~1

>103 1–20 10–102 10–103 ~1

103–104 1–20 10–102 10–103 ~1

v*  v* ( ) v*  v* ( ) v*  v* ( ), G  GSD v*  v* ( ), G  GSD

G*: Apparent activation energy for deformation. GSD: Apparent activation energy for self-diffusion.

regard to the experimental evaluation of activation volume v* in connection with the “strain-rate-change test.” Figure 2.5.2 displays three specific forms of ∆G and their corresponding force–displacement diagrams, that is, hyperbolic, sinusoidal, and rectangular shapes. They are characterized by two parameters, p and q, each yielding 0 ≤ p ≤ 1 and 1 ≤ q ≤ 2, and a pair of values, = p 1= / 2, q 3 / 2, which have been widely employed in the literature. Some representative cases are shown in Eq. (2.5.1): 2    *  12 

  

G  G0 1   *    

   

2

*  G  G  1    . 0

(2.5.1)   *   

*  

G  G  1    0 *

    



Note that in Figure 2.5.2, the rectangular shape that the third element of Eq. (2.5.1) produces coincides with the Lindholm-type activation energy, G  G0  v* (   * ) (Lindholm, 1965, 1968), one of the simplest equations expressing the τ * dependency. Figure 2.5.3 provides an example of the explicit form of ∆G for the cross-slip process, which is empirically found in Escaig (1968), given as

2    ECS ln   , (2.5.2)  SFE   CS  Here γ SFE expresses the stacking fault energy, which substantially dominates the process, as discussed in detail in Section 1.2.5.3. With this expression, the frequency of the cross slip to take place is given by

G  

E2



CS

    SFECSkT  G  . (2.5.3)  0 exp    0     kT    CS 

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2.5  Representation of Obstacles

105

Figure 2.5.2  General expression of apparent activation energy based on the MTS model,

together with three representative shapes of thermal barriers, that is, hyperbolic, sinusoidal, and rectangular, where d* indicates barrier width.

Figure 2.5.3  Explicit expression for the apparent activation energy for the cross slip given as a

function of effective stress and SFE of material of interest γSFE.

Note that many things still remain unidentified about the details of the cross-slip processes, for example, in terms of expressing ∆G , which deserve further investigation, including those based on molecular dynamics and dislocation dynamics. For summarizing this subsection, we display afresh the major rate-controlling mechanisms for BCC metals in Figure 2.5.4 and for FCC metals in Figure 2.5.5,

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Figure 2.5.4  Summary of the major rate-controlling Peierls overcoming mechanism for BCC

metals, which can be expressed by the same mathematical formula, based on statistical mechanics, as that for FCC metals’ major rate-controlling mechanism.

Figure 2.5.5  Summary of the major rate-controlling mechanism for FCC metals, that is,

the dislocation process (forest cutting), which can be expressed by the same mathematical formula, based on statistical mechanics, as that for BCC metals’ major rate-controlling mechanism.

demonstrating distinct underlying processes against dislocation motions, that is, dislocation processes for FCC metals and Peierls overcoming for BCC counterparts, but with essentially the same thermal activation mechanism to be expressed by the identical mathematical expression described earlier, which can cover a wide range of strain rate and temperature, allowing us ultimately to construct an unified constitutive modeling framework, to be mentioned in Section 2.7 and further detailed in Chapter 5.

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2.6  Kocks–Mecking Model

2.6

107

Kocks–Mecking Model Figure 2.6.1 shows the Kocks–Mecking model (Kocks, 1976; Mecking and Kocks, 1981), which is applicable to cases where several distinct processes must be considered simultaneously. The figure shows an example considering two distinct thermally activated processes due to the Peierls overcoming and solution hardening. The generalized form of the Kocks–Mecking model is expressed as



 n   ( * )n   ( i )n , (2.6.1) with n = 1 for weak obstacles and n = 2 for strong obstacles. The normalized form with n = 1 is given as



  * i    0 

(2.6.2) ˆI* ˆD* ˆi ( )  sI (, T )  sD (, T )  , 0 0 

where two distinct thermal activation mechanisms, that is, those associated with Peierls potential and solution hardening, and a single internal stress, are assumed in the second line. Here sI (γ, T ) and sD (γ, T ) are given, respectively, via



/p  1/ qI 1 I 

s (, T )  1   kT ln 0 I    

I      G0  b3

  . (2.6.3)  1/ pD / q 1 D 

  kT  

sD (, T )  1    ln 0 D  3

    G0  b   

Figure 2.6.1  Schematics showing the Kocks–Mecking model for combining Peierls

overcoming and solution-hardening mechanisms (Tomota, 2001).

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Figure 2.6.2  Schematics of extended Kocks–Mecking model for use in a crystalline

plasticity-based constitutive equation (Hasebe et al., 2014).

The expressions in Eq. (2.6.3) can be derived by combining Eqs. (2.4.3) and (2.4.6), and solving for the effective stress. Figure 2.6.2 presents an idea to utilize the Kocks–Mecking model in the crystal plasticity-based constitutive equation, which is not straightforward because the Kocks–Mecking model cannot be solved for the strain rate γ in general. To solve this dilemma, we need to choose one effective stress component and transpose it to the left-hand side, regarding the relationship as a function only of it, that is,

  ( 1*   2*     n* )   i



  1*    ( 2*     n* )   i .

(2.6.4)

For the sake of constitutive modeling in crystal plasticity, the variable components given as a function of strain should be distinguished from others. This will be called as the “chief” effective stress hereafter.

2.7

Unified Constitutive Equation Based on these arguments (Eq. (2.6.4)), we can construct a physically sound constitutive equation of the form

J (D )

­ § (D ) *(D )  :(D ) ° ¨ W  W1 ®1  K (D ) ° ¨© ¯

ª « 'G J0 exp «  0 kT « ¬

­ § (D ) *(D ) *(D ) (D ) ° ¨ W  (W 2  ˜˜˜  W n )  : ®1  K (D ) ° ¨© ¯

· ¸ ¸ ¹

p ½º

q

ª « 'G  J 0 exp «  0 kT « ¬

°» ¾» ° ¿»¼ · ¸ ¸ ¹

p ½º

q

(2.7.1)

°» ¾» , °» ¿¼

where K (α ) and ( ) are the drag stress and back stress, respectively (Figure 2.7.1). When choosing the Peierls overcoming process as the secondary rate-controlling mechanisms for BCC metals, as displayed in Figure 2.7.2, the dislocation process

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Figure 2.7.1  Overview of a unified-type constitutive equation based on dislocation dynamics

combined with an extended Kocks–Mecking model, together with a brief description of a hardening evolution model characterized by hardening ratio Qαβ (Hasebe, 2006).

Figure 2.7.2  Explicit form of a unified-type constitutive equation considering a contribution

from the Peierls mechanism, where effective stress for the Peierls overcoming process is rewritten by using normalized activation energy for parameter determination.

should normally be regarded as the chief effective stress because it varies with strain, while Peierls mechanism does not. We have, then,





 G disloc (  )   0 exp  0 kT

*( a )  Peierls

 *Peierls

  ( ) *( ) ( )      Peierls   1    K ( )   

  1   kT ln 0 P    b3 g0 P ( a )  

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p 

q



 , (2.7.2)  

1/ qP 1/ pP

   

  

, (2.7.3)

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Dislocation Dynamics and Constitutive Framework

Table 2.2  Example of parameters list for Eq. (2.6.2) for various materials including HCP, FCC, and BCC metals and alloys for the Kocks–Mecking representation of constitutive models available from the literature (Tomota, 2001). HCP

BCC

Parameters α -Ti

Ti–Fe–O

SUS310S

IF

σa

50

300

65+1,000 ε

60+120 ε 0.4 60+120 ε 0.4

65+130 ε

g0 P

0.292

0.25

0.14

0.12

0.12

0.19

pP

1.0

1.0

0.5

0.5

0.5

0.5

qP

2.0

1.67

1.5

1.0

1.2

1.5

σˆ P

905

1,375

675

932

978

1,073

g0 S

1.5

1.5

0.26

0.45

0.5

1.0

pS

0.5

0.59

0.5

0.5

0.5

0.5

qS

1.5

2.0

1.5

1.5

1.5

1.5

σS

1,200

1,400

1,000

300

320

400

Θ0

3,200

2,400

1,400

3,200

3,200

3,800

8

8

8

108

8

108

ε 0P ε 0S



FCC

10

10 10

7

10

10 10

7

10

8

10

LC

10 10

8

10 10

P-alloyed 0.4

*( ) by considering  2( )   Peierls in the second line of Eq. (2.7.1). Here, the normalized activation energy can further be expressed as G0Peierls   (T )b3 g0 P , as its explicit form. Example of the parameters applicable to Eqs. (2.7.2) and (2.7.3) are listed in Table 2.2 (Tomota, 2001). The list includes not only BCC steels, but also FCC and HCP metal and alloys. The values for various materials can also be found in Nemat-Nasser (2004). To additionally consider the phonon-drag mechanism (intrinsic damping effect), we should consider the mean velocity of moving dislocations under the two situations, that is, overcoming the thermal obstacles and running in between them against the viscous resistance (Follansbee et al., 1984), as schematized in Figure 2.7.3. The mean velocity is expressed as L v , (2.7.4) tta  t pd

where L represents the mean spacing between thermal obstacles, and tta and t pd are, respectively, the mean waiting time needed for overcoming an obstacle and to reach another obstacle after that. Particularly for FCC metals under sufficiently high strain-rate impact loading, for example, over 5´103 s−1 or higher, the phonon drag becomes non-negligible, as mentioned in Section 2.3 (also see Figure 2.3.4). In the following, we derive a final form of the unified constitutive equation considering this mechanism as well. As indicated in Figure 2.3.4, the balance equation for the force on a moving dislocation segment f   * b against viscosity due to the phonon drag is given by

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Figure 2.7.3  Derivation process of the unified-type constitutive equation considering both

thermal activation and phonon-drag mechanisms through mean velocity for the two processes.

Figure 2.7.4  Derivation process of a complete framework of the constitutive equation, covering

a wide range of strain rate involving the hypervelocity impact regime, to be used as the vehicle for the present FTMP approach.



 b  Bv pd , (2.7.5) where B represents the viscosity coefficient for the process. Here, an analogy to the linear (Newtonian) viscous fluids is used. Thus, the mean running time in between the thermal obstacles against the phonon-induced resistance t pd is given by



t pd 

L BL  * . (2.7.6) v pd  b

For the thermal activation process, on the other hand, the mean waiting time tta is given as

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 G( * )  tta   *1 exp   . (2.7.7)  kT 



Combining Eqs. (2.7.6) and (2.7.7) into Eq. (2.7.5), and using the thus obtained mean velocity v for Orowan’s equation    m bv, we have

1

  G( * )  BL     bL  * 1 exp    *  . (2.7.8)   kT   b 



Based on this, we ultimately have



with

1 ­ q ª º ­ § W c(D ) · p ½ ° (D ) ° ° A SRW c(D ) «« W c(D ) BSR exp ®1  ¨ (D ) ¸ ¾  CSR »» °°J ® °¯ © K ¹ °¿ «¬ »¼ , (2.7.9) ° °W c(D ) { W (D )  W *(D )  :(D ) Peierls °¯

G0disloc BL * , CSR  A SR   m bL *, BSR  kT b for the complete form of our constitutive equation, where the specific form for the activation energy G( * ) expressed by Eq. (2.4.5) is utilized. The derivation process up to the point of substituting the explicit form of G( * ) is summarized in Figure 2.7.4. The utility of the present unified constitutive model is briefly discussed in Section 2.9. Further details about the framework are provided in Section 5.6.1, together with an implementation of the FTMP-based models. Let me briefly mention the MTS model proposed by Follansbee and Kocks (1988), as an opposing interpretation of the rapid stress rise exceeding  103 s1. They advocated the concept of MTS as a more suitable internal variable than the ordinarily used “strain,” for “strain” does not effectively consider the underlying substructure evolutions. In other words, “strain” itself does not essentially contain information about the “strain-rate history” and the “temperature history” effects, because they are associated with, for example, the deformation-induced dislocation substructures (see Section 3.4 for details). The MTS model is expressed as p



    ˆ  ˆ      ( T )     (T ) 

p

1/ q   kT 0   1   ln   (2.7.10)   G0  b3     

where σˆ is the MTS, defined as the flow stress at temperature 0K. We can notice that, as it basically has the same form as Eq. (2.6.3), the idea is fully based on the thermal activation mechanism. With the use of the MTS concept, the experimental results presented in Figure 2.3.4 (compared at an off-set strain of 0.15) are recorrelated as those in Figure 2.7.5, where strain-rate effects almost disappear, even above   103 s 1, showing that the rapid stress increase observed in Figure 2.3.4 is just an “apparent” trend, and accordingly, the rate-controlling mechanism should be based on thermal activation, at least below   10 4 s 1. This interpretation seems to be reasonable, for

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2.8  Experimental Evaluation of Activation Energy

113

Figure 2.7.5  Rearranged strain-rate-stress diagram by utilizing MTS model (Follansbee,

1985; Follansbee et al., 1984; Wang et al., 2019). Reused with permission of the publisher (Academic Press).

the phonon-drag-based model overestimates the onset of the stress rise. The relationship with the substructure evolutions, including the MTS-based interpretation, is revisited in Section 3.5.1. In any case, since the MTS represents the stress states at 0K to be dominated both by the strain and strain rate (and temperature), it literally describes the dislocation substructures themselves that evolved during the loading processes responsible for the attendant flow-stress levels.

2.8

Experimental Evaluation of Activation Energy Some of the thermal activation characteristics are experimentally measurable. In  this section, we will see how they can be measured and what they are like. The theoretical backgrounds (e.g., Kato, 1999) are provided first, and are followed by some experimental results including some of my own for both FCC and BCC metals. The activation energy is a Gibbs-type free energy, as pointed out previously, given as a function of the effective stress (see Figure 2.8.1) as



 S * (T )dT  v*( * )d * . (2.8.1) Taking the total differentiation, as shown in Figure 2.7.5, leads to



 G*   G*  * dG*    dT   *  d  T   (2.8.2)   *  T

 

 S * (T )dT  v*  * d * ,

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Figure 2.8.1  Activation energy given as a function of effective stress and absolute temperature,

where its total derivative is shown, from which we derive definitions of activation entropy and activation volume.

Figure 2.8.2  Explicit expression of activation volume as a function of strain-rate ratio and

flow-stress difference. With this equation, activation volume can be experimentally measured by carrying out a strain-rate change test.

where S * (T ) is the activation entropy and v*(τ * ) is the activation volume, respectively, defined as  G*   G*  * * S * (T )     and v ( )   *  , (2.8.3)  T  *   T



where ( ) * and (∂ )T mean the states with  *  const and T = const , respectively.

2.8.1

Activation Volume We will express v*(τ * ) with respect to the stress and strain rate, as illustrated in Figure 2.8.2. Based on Eq. (2.4.3), the activation energy is written as



 G*       G*  kT ln       0 exp     . (2.8.4)  0    kT  

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2.8  Experimental Evaluation of Activation Energy

115

By substituting this into Eq. (2.8.3), we have   ln   0   v* ( * )  kT   . (2.8.5) *   T



In practice, the partial differentiation can be replaced by the differential form, that is, v*  kT



ln  2 1 

 2  1 

. (2.8.6)

Therefore, the activation volume is obtained by measuring the stress change  2   1 by performing a strain-rate jump test with γ1 « γ2.

2.8.2

Activation Enthalpy Next, let us consider expressing the activation “enthalpy” (not activation “energy”), as shown in Figure 2.8.3. Similarly to Eq. (2.4.3), we first have  G*      G*  k ln        0 exp     . (2.8.7)  0  T   kT  



Taking a partial differentiation of this with respect to 1 / T holding τ * constant, we have *     ln   0    * 1  G kT   G    T   1 T   * (2.8.8)   1 T   * 



 G*  TS * ,





because ¶G* / ¶ 1 T 

*

in the second term of the right-hand side, reads

Figure 2.8.3  Procedure for deriving the relationship for obtaining activation enthalpy as a

function of strain-rate ratio and difference in the inverse temperature.

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Figure 2.8.4  Procedure for deriving the relationship for obtaining activation enthalpy as a

function of stress and temperature differences.



 G*   G*  T   S*  T 2 .       1 1  T T T        *    * 



Since H * º G* + TS * renders enthalpy altogether, we have the explicit expression for the activation enthalpy via

  ln   0     ln  0    H *  kT    kT   . (2.8.9)   1 T   *   1 T   * The corresponding difference version is obtained as



 ln  1 2   H *  kT   . (2.8.10)  1 T2   1 T1   Based on Eq. (2.8.10), we can evaluate H * by performing a temperature change test T1 « T2 to measure the corresponding strain-rate change. Note that in order to obtain the activation energy, we need to know the entropy at the given temperature. Another way to obtain the activation enthalpy H * is as follows (Figure 2.8.4). We rewrite Eq. (2.8.9) as



  ln   0     ln   0   2 H *  k    kT     T   1 T   *

    . (2.8.11)  T 

By using Eq. (2.8.6), we can further rewrite the above equation as

   H *  Tv*    . (2.8.12)  T 

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2.8  Experimental Evaluation of Activation Energy

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Thus, H * can be also evaluated from the change in stress in the temperature jump test. The corresponding form via difference is given by

2.8.3

H *  Tv* .

 2  1  . (2.8.13) T2  T1 

Experimentally Evaluated Activation Volume Figure 2.8.5 is an example of the activation volume v* for single-crystal and polycrystalline Al, presented as a function of strain (Lindholm, 1968). As mentioned in relation to Table 2.1, the activation volume v* yields variation with strain in the case of FCC metals. The figure demonstrates that v* decreases as deformation proceeds, meaning that the activation mechanism is attributed to the interaction with evolving dislocation substructures (roughly corresponding to increasing dislocation density). In a comparison between the single-crystal and polycrystal data, we confirm that, with a few exceptions, the activation mechanism is the same for both cases. For experimentally measuring the activation volume, as mentioned in Section 2.8.2, we normally perform the strain-rate jump tests with γ1 « γ2 , assuming the corresponding stress change  2   1, whose process is schematized in the inset of Figure 2.8.6 (top row). Figure 2.8.6 also presents a representative set of examples

Figure 2.8.5  Activation volume for high-purity Al (99.995%) in both single-crystal and

polycrystal cases as a function of strain (Lindholm, 1968).

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Figure 2.8.6  Experimental measurement of the activation volume: (a) schematics, and

examples of the experimentally observed variations with strain for (b) FCC copper and (c) BCC iron and tungsten.

of experimentally obtained activation volume, presented as variations of a function of strain, comparing (b) FCC and (c) BCC metals. As readily confirmed by comparing (b) and (c), FCC metals exhibit decreasing v* with deformation, as mentioned previously, whereas that for the BCC metals tends to stay unchanged if deformation proceeds, regardless of temperature. These contrasting features between FCC and BCC metals are schematically summarized afresh in Figure 2.8.7. The basically constant v* in BCC metals is attributed to its main rate-controlling mechanism, that is, the Peierls (or PN) overcoming process (see Figure 2.2.7 and Figure 2.5.1(c)), which is an intrinsic mechanism that does not change even when deformation proceeds. In FCC metals, on the other hand, while the PN potential is negligibly small, the major rate-controlling process is that based on interactions with forest dislocation, referred to as dislocation processes in general (see Figure 2.5.1(b)), which is essentially deformation dependent, as briefly stated earlier. Accordingly, the value ranges can thus be roughly estimated to be 102~103/b3 for FCC and 10~102/b3 for BCC (see also Table 2.1), corresponding to the areas that are swept by a dislocation to overcome (e.g., Figure 2.2.7) times the magnitude of the Burgers vector b (thus, having the dimension of “volume”). Based on this physical meaning for v* , we also intuitively understand its temperature dependence, aside from that described in Eq. (2.8.6), by assuming the difference in the critical configurations of overcoming dislocation segments. Namely, the lower the temperature is, the smaller

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119

Figure 2.8.7  Schematics overviewing the variation of activation volume with deformation

(strain) for FCC, BCC, and ultrafine-grained (UFG) materials, emphasizing relationships with representative rate-controlling mechanisms.

the area swept by an overcoming dislocation at its critical configuration becomes, resulting in smaller activation volume for both FCC and BCC metals. Let us see the individual details of the above experimental results schematically overviewed in Figure 2.8.7. They were obtained under impact loading as well as a low-temperature condition (T = 77K at liquid nitrogen temperature), obtained by my group. The impact tests have been conducted using the split Hopkinson pressure bar (SHPB) method (also referred to as the Kolsky bar method) for the strain-rate range of 600–2,000 s−1. The materials targeted are commercially pure Fe and carbon steels with various carbon content, that is, S15CK, S45C, and S55C, together with commercially pure Ta, all having a BCC structure. Figures 2.8.8 through 2.8.10 summarize the results, each representing the base data including the stress–strain curves, together with the corresponding flow-stress–strain-rate diagrams on a log-log plot (Figure 2.8.8), activation volume changes for Fe/steels (Figure 2.8.9), and Ta (Figure 2.8.10), respectively. Strain-rate dependency illustrated in Figure 2.2.2 can be confirmed from the results in Figure 2.8.7, where Ta exhibits a similar trend to the steels and Fe on the    plot. Strain-rate change tests are conducted for the static conditions at both RT and T  =  77K. The results for steels at RT, comparing the carbon contents, are shown in Figure 2.8.9. First, all the steels including Fe exhibit basically constant v* with increasing strain. Second, we observe different values of v* depending on the carbon content, that is, smaller v* with larger C content. Fe, with smallest C, shows the maximum value of v*, while S45C (0.45wt%C) yields the minimum (since the data for S55C are not shown here), inversely corresponding to the flow-stress levels observed in Figure 2.8.8. This stems basically from the reduction in the activation distance with

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Figure 2.8.8  Base experimental data for Fe, representative carbon steels, and Ta, from which

the variations of activation volumes presented in Figures 2.8.9 and 2.8.10 are measured.

increasing C. Increasing C content, in the present case, results in a larger volume fraction of the perlite phase to be introduced, which noticeably enhances the resistance against dislocation motion due to the reduced activation distance. Figure 2.8.10 shows the results of the strain-rate jump tests for BCC Ta, comparing the effect of test temperature, where the set of variations of the activation volume is redisplayed from Figure 2.8.6. We also observe the constant v* for both the temperatures, that is, RT and 77K, while the lower temperature yields smaller v*. This temperature dependency of the activation volume v* is easily understood from Eq. (2.8.6), and will be quantitatively confirmed afresh later. The current framework for the dislocation dynamics based on statistical mechanics is employed as a basis for our constitutive modeling. The reason is twofold. One is its well-documented physical ground applicable to a wide range of strain rate and temperature (including low-temperature creep), which is essential for our purposes, as pointed out on many occasions. The other reason concerns data acquisition. There exists a relatively systematic series of data on this formalism, as already presented in Table 2.2 in conjunction with the Kocks–Mecking model. One recent prominent example is the data obtained by Nemat-Nasser and his colleagues (Nemat-Nasser, 2004). We will see how effective they are in the following discussion.

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121

Figure 2.8.9  Experimental results of a strain-rate change test for Fe, together with the obtained

variation of activation volumes for Fe and carbon steels as a function of strain (Hasebe and Imaida, 2003).

Figure 2.8.11 presents a thermal activation mechanism-based constitutive equation (1D version), that is, 1/ p



  kT  1/ q  * *    m 1  ln 0        G0 

 a0 n , (2.8.14)

together with a table listing the material parameters identified by Nemat-Nasser et al. (Nemat-Nasser, 2004; Nemat-Nasser and Issacs, 1997), for commercially pure Ta. Based on the parameters identified by them, after making a slight modification to fit our data, we set up a simplified constitutive equation for Ta. Figure 2.8.12 makes comparisons between the calculations via the constitutive equation identified in this way and the experimental data under various strain rates as well as temperature conditions. It is clearly demonstrated that there is an excellent agreement between the two for all the conditions. This emphasizes the effectiveness of the present framework: A wide range of  ,T  -dependent stress–strain responses can be expressed by a single set of parameters in a unified manner. The excellent agreement between the experiments and Eq. (2.8.14) shown in Figure 2.8.12 also clearly demonstrates the equivalence of temperature T and strain rate ε. As pointed out earlier, the temperature range to be covered by the same set of data would be a low-temperature–high-stress creep regime.

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Figure 2.8.10  Experimental results of strain-rate change test for Ta at room temperature (RT) and 77K, together with the obtained variation of activation volumes as a function of strain (Hasebe and Imaida, 2003).

Figure 2.8.11  A simplified constitutive equation used for demonstrating the effectiveness of the statistical mechanics-based formalism, together with a list of material parameters via Nemat-Nasser et al. for pure Ta (Hasebe and Imaida, 2003).

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123

Figure 2.8.12  Comparison of stress–strain curves with various strain rates and temperature for pure Ta, demonstrating the powerful applicability of the present constitutive framework (Hasebe and Imaida, 2003).

It is also noteworthy that, for ordinary creep, that is, at high temperatures, a similar framework can be utilized but with distinct activation mechanisms such as diffusion. The diffusion in this context is classified roughly into three types  – lattice, grain boundary, and pipe – corresponding to Nabarro–Herring creep, Coble creep, and dislocation (or power-law) creep, respectively (Frost and Ashby, 1975; Weetman and Weetman, 1965). In the dislocation creep case, the diffusion takes place along the dislocation core, which is responsible for the low-temperature creep mentioned earlier.

2.9

Deficiencies of Inappropriate Strain-Rate Dependency Let me present some intriguing examples demonstrating the inadequacy of the widely used constitutive framework. As mentioned in Section 2.2.1 in conjunction with Figure 2.1.2, FCC metals exhibit a sharp increase in flow stress exceeding a strain-rate range of 103 s−1. Figure 2.9.1 displays examples of numerical results of    diagrams for Cu- and Al-type models, obtained based on crystal plasticity-based analyses by using Eq. (2.7.9), where the sharp stress rise is due to the phonon-drag term, that is, CSR, in the present case.

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Figure 2.9.1  Simulation results indicating strain-rate dependency of flow stress for FCC metals

with Cu and Al types, expressing rapid increase in flow stress over 103 s−1.

Figure 2.9.2 shows a dynamic-explicit FEM (finite element method)-based simulation result for high-velocity tension of a cylinder for a polycrystalline model. High velocity is applied to the top edge of the cylinder while the bottom is fixed. Use is made of the conventionally employed power-law-type equation (Hutchinson, 1976; Pan and Rice, 1983),

 ( )



  ( )  a ( )  ( ) g

   ( )   ( )   g

1

1

m , (2.9.1)  

with m expressing the strain-rate sensitivity exponent. At the limit m ® 0 , the equation arrives at the rate-independent type with  ( )  g ( ) at  ( )  a ( ). Note that although this is rate dependent, it is not always suitable for expressing a wide range of strain rate, particularly for those over 103 s−1, as pointed out at the beginning of this chapter. Figure 2.9.2(a) shows the result of the deformed cylinder simulation based on Eq. (2.9.1), yielding an unrealistically concentrated deformation at the impact edge (top edge) of the cylindrical specimen, resulting in unrealistic elongation there like thick malt syrup. Corresponding time histories of the equivalent stress at the top edge and middle of the cylindrical sample are presented in Figure 2.9.3(a), where that for the top edge increases rapidly leaving the latter behind. Under high-velocity tension, a tensile stress wave generated at the pulled edge is propagated through the sample toward the other end at a finite speed (less than or equal to the longitudinal wave velocity) and then reverberates over it. So, a high enough velocity tension (thus corresponding to high strain rates over 103 s−1 and more) results in a delay in the deformation within the sample depending on the site, causing strain localization as in the present case. In the real materials response, however, this

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Figure 2.9.2  Example of simulation results comparing the deformation behaviors of a

cylindrical sample under impact tension with that of the power-law-type conventional constitutive equation and the present model.

phenomenon is restricted (at least in FCC metals such as Al) via rapid additional hardening at the strain-rate range, as shown in Figure 2.9.1. Consequently, we have larger elongation than that in the static loading condition. Such enhanced ductility is called “hyperplasticity” (Daehn et al., 1995, 2000; Hasebe et al., 1998) as opposed to the famous “superplasticity.” Figure 2.9.2(b) displays a result reproducing such a situation, where the complete form of the unified constitutive equation Eq. (2.9.2) is used within the same numerical setting as the above, that is, 1



( ) ( )  A SR

   ( )   ( )   ( ) exp BSR  1  ( )   CSR , (2.9.2) ( )  ( ) K  K

  K 

which has been derived based on the procedure given in Figures 2.4.1 and 2.8.10. This model accommodates two distinct rate-controlling mechanisms of dislocation motion, that is, thermal activation and phonon drag. The latter is assumed here to be responsible for the sharp flow-stress increase in this model, whereas the associated temperature rise and the attendant thermal conduction, and so on are disregarded. With the appropriate consideration of the rapid hardening at very high strain-rate region, somehow the localization of deformation at the impacted edge is promptly restricted by the rapid hardening, causing a shift of the localized region to the next region down from the top. The repetition of this process toward the bottom of the

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Figure 2.9.3  Example of simulation results comparing time histories of equivalent stress at the

top edge and middle portions of a cylindrical sample under impact tension with that of the power-law-type conventional constitutive equation and the present model.

cylindrical sample results ultimately in enhanced elongation. Comparison between the time histories of equivalent stress at the top edge and the middle are also presented in Figure 2.9.3 (on the right), where a sharp stress rise takes place upon impact tension at the top edge, but it asymptotically and rapidly approaches the stress level in the middle part of the sample. Figure 2.9.4 provides the same result as Figure 2.9.2, but under impact compression, where basically the same trend can be observed. One further example concerns high strain-rate phenomena, but in relation to sheet metal forming. I conducted a series of forming experiments on thin sheet metals or foils (0.1–0.3 mm thickness), making them into flat-headed cup shapes (see the inset of Figure 2.9.5 at the top right) by means of hydrospark-forming (also referred to as the electrohydraulic-forming) method (Hasebe and Imaida, 1996, 2007; Yokoi et al., 1995). The method utilizes an underwater wire explosion and attendant impulsive pressure wave including the underwater shockwave within the pressure vessel to shape the blank (sheet metals) into desired shapes without using a rigid paunch unlike in the conventional technology. The high-energy release rate in the technique yields high strain-rate forming, exhibiting “hyperformability” (Daehn et al., 1995, 2000; Hasebe, 2018) as a consequence of the earlier-mentioned “hyperplasticity.” Figure 2.9.5(a) shows the resultant thickness strain distribution measured from the center of the products obtained by hat-shape bulging. The figure shows three conditions with different waveforms of the impulsive pressure wave, which can

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Figure 2.9.4  Example of simulation results comparing deformation behaviors of a cylindrical

sample under impact compression with that of the power-law-type conventional constitutive equation and the present model.

Figure 2.9.5  Simulation results on thickness distribution for flat-headed deep drawing by the

hydrospark-forming method, with and without considering the effect of a high strain rate and comparing with experimental results.

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affect the deformation mode of the blank and thus controls the formability. But the formability issue is another story, so I will not go into further detail here. Figure 2.9.5(b) shows simulated results obtained based on finite difference code (modified SALE – Simplified Arbitrary Lagrangian Eulerian), where a phenomenological constitutive equation of a rate-dependent type is used, with an additional term proportional to ε mimicking the rapid increase in the flow stress exceeding 103 s −1 due to the phonon-drag effect, that is,

   p  A pn   pm exp    B p , (2.9.3)   where B is a controlling parameter of the contribution of the second term. The simulation results with B = 2,000 well reproduce the experimental thickness distributions for the three conditions. Without the additional term, that is, B = 0, on the other hand, corresponding to the case with Eq. (2.9.1), this results in unrealistic elongation and thus extreme thinning, as shown in Figure 2.9.5(c). This result also indicates the necessity of using the appropriate constitutive equation in terms of the strain-rate dependency in simulations of this type. Making use of “hyperformability,” the forming technique can be applied not only to difficult-to-form product shapes but also to microforming thin foils. An example for the former that eloquently demonstrates “hyperformability” is shown in Figure 2.9.6 (Balanethiram and Daehn, 1993) where comparison is made between hydrostatic forming and the hydrospark (or electrohydraulic) technique for a conical-shaped bulging. The hydrospark-forming method yields outstanding formability, where bulging almost reaching the top is achieved. A similar example but for demonstrating also the latter, that is, microforming, which is my own work, is displayed in Figure 2.9.7(a), where pyramidal microbulging using a die prepared by the Vickers indentation is performed, demonstrating that a large enough charging

Figure 2.9.6  Comparison of conical budging between hydrostatic (left) and hydrodynamic

(electrohydraulic or hydrospark: right) forming techniques, with the latter demonstrating “hyperformability” (Balanethiram and Daehn, 1993). Adapted with permission of the publisher (Elsevier Science & Technology Journals; courtesy of Prof. Glenn S. Daehn).

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Figure 2.9.7  Example of pyramidal microbulging, using the hydrospark-forming method for

Al foil with 5 µm thickness, succeeding in sharp-tip forming without breakage at sufficient charging energy. The pyramidal die is prepared via the Vickers indentation, with an indentation depth of 38 µm.

energy allows successful forming close to the tip of the pyramid without causing breakage. An extension of that line is the microbulging of complicated shapes and dimensions. Such an example is presented in Figure 2.9.8, comparing FIB (focused ion beam)-fabricated dies and the formed products, using Al foil of 5 µm thickness. Figure 2.9.9 summarizes a set of examples of such effective usages, where a series of microbulging through microembossments are displayed (Hasebe and Imaida, 2007). Here, the hydrospark-forming method is used to achieve such seamless microforming from 1,000 through a few μm representative scale orders. It is noteworthy that the smallest scale includes the embossment of a hologram image, indicated on the top left, where a piece of a pencil board made of PVC was used as the die. As can be confirmed, an iridescent pattern has been clearly reproduced on aluminum foil, extending over a several centimeter-squared area. This indicates that the fine grooves on the order of 1µm for reproducing the hologram image have been successfully embossed on the Al foil (confirmed by SEM, not shown here).

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Figure 2.9.8  Example of microbulging for complicated product shapes, where microgear

forming is attempted using the hydrospark-forming technique.

Figure 2.9.9  Overview of “seamless” microforming and working by the hydrospark-forming

method (Hasebe and Imaida, 2007).

Appendix A2  Systematic Prestrained-Impact Experiments We often refer to the mechanical properties of a material of concern based on a single stress–strain curve, normally under a uniaxial loading condition, which can practically be any one of the properties. It is, however, just one limited aspect of them, with most of the “main parts” left unrevealed: There exists, at least, strain history, strain rate, and temperature effects that can greatly alter the stress responses. Among others,

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Figure A2.1  Schematics of NPIT test.

Figure A2.2  Summary of NPIT test results for four typical FCC metals.

the NP strain-history effect that is discussed in Chapter 3, as well as the strain-rate effect mentioned in the present chapter is apt to be overlooked, although they can greatly alter the stress–strain response.

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In this appendix, we concisely summarize a systematically conducted series of experimental results of my own, aiming at revealing the coupling effects between nonproportional/reversed proportional impact tests (NPIT and RPIT, respectively) strain histories, and strain rates, covering the impact loading range for a representative set of metals (all are annealed polycrystals). Schematics of the tests and summary of the results are presented in Figures A2.1 through A2.11, together with those for representative Al alloys (#5000 and 6000) in Figures A2.12(a) and (b).

Figure A2.3  Summary of NPIT test results for BCC and HCP metals.

Figure A2.4  Summary of NPIT test results for BCC metals, examining the negative strain-rate history effect via impact compression followed by static torsion.

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Appendix A2  Systematic Prestrained-Impact Experiments

Figure A2.5a  Results of the variable strain-rate Bauschinger test (i.e., RPIT) for pure Al.

Figure A2.5b  Summary of the variable strain-rate Bauschinger tests (RPITs) for pure Al.

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Figure A2.6a  Results of the variable strain-rate Bauschinger test (RPIT) for pure Cu.

Figure A2.6b  Summary of the variable strain-rate Bauschinger tests (RPITs) for pure Cu.

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Appendix A2  Systematic Prestrained-Impact Experiments

Figure A2.7  Results of constant strain-rate Bauschinger test (const. RPIT) for FCC metals.

Figure A2.8  Results of constant strain-rate Bauschinger test (const. RPIT) for BCC metals

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Figure A2.9  Summary of constant strain-rate Bauschinger tests (const. RPITs) for BCC metals.

Figure A2.10  Summary of NP prestrained impact tension (NPIT) test for FCC metals

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Figure A2.11  Comparison of NPIT results among FCC, BCC, and HCP metals

Figure A2.12(a)  Summary of NPIT results for Al alloys (#5000).

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Figure A2.12(b)  Summary of NPIT results for Al alloys (#6000).

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Dislocation Substructures Universality of Cell Structures

3.1

Universality of Cell Structures

3.1.1 Overview Metallic materials are known to be strongly rate and history dependent, where the history includes strain, strain rate, strain path, and temperature. The history effects manifest themselves as, for example, additional hardening and softening in terms of the stress response. Since the materials inevitably experience various histories during both forming/working processes into desired shapes or structures and during operations, as schematized in Figure 3.1.1, modeling of these history effects has been one of the most challenging and difficult problems in the field of mechanics of materials. Most of these histories are brought about by dislocation substructures with cellular morphology evolving during plastic deformation. In other words, the information about the loading “history” is stored in the evolved/evolving dislocation cells. Dislocation substructures, as well as their evolutionary aspects, may be the most unique features of crystalline metals. These features are basically absent in other engineering materials such as ceramics, polymers, and composite plastics. They evolve quite spontaneously, like living organisms. It may safely be said that it is the only phenomenon that exhibits self-organization over the hierarchical scales. At the same time, their importance in discussing or determining macroscopic responses such as hardening has tended to be overlooked by researchers, who seem to be interested only in the “patterning”-related aspects or who stick to conventional crystal plasticity (ignoring substructure evolutions). Roughly speaking, the dislocation substructures can be classified into two kinds, as illustrated in Figure 3.1.2, that is, the mechanically necessary type and the geometrically necessary (GN) type. The latter follows Hansen and his colleagues (Hansen, 2001, 2004), who have systematically and extensively examined in detail the deformation structures observed in a relatively large strain regime. Such kinds of substructures are normally accompanied by relatively large misorientation across the boundaries, thus yielding a sort of lowest energy configuration, in the absence of longrange stress. The former, on the other hand, is named by the author himself based on his experience, and this category contains mainly the cell structures to be extensively detailed in this chapter. The cell structures, as will be argued later, are often accompanied by long-range stress fields in the cell interior regions, thus remaining in a high-energy state. These situations are summarized in Figure 3.1.3, with respect to the

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Figure 3.1.1  Schematic illustration for a process –from raw materials to sheet sampling into

structural components – emphasizing the histories that the material has experienced and their effects on its mechanical responses, most of which are attributed to the substructure change of dislocations.

Figure 3.1.2  Classification of dislocation substructures, that is, mechanical and geometrical

types. The former includes cells and veins evolved even under small strains, as during lowcycle fatigue, while the latter has many others available under relatively large deformation, normally accompanied by large misorientation.

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Figure 3.1.3  Comparison of dislocation walls in terms of “flow-evolutionary” perspectives (cf.

Chapter 15), among geometrically necessary bands (GNBs), and cell and lath walls.

composing dislocation walls, from the FTMP-based perspective, that is, the incompatibility tensor η KK versus elastic strain energy fluctuation δ U e (see Chapter 15 for details), together with the martensite lath wall, for comparison. From this rather simplified view, the two types of dislocation structures are largely distinguished by the strain energy stored within (between) the walls. A series of experimental observations shows that the formation of cell structures is a universal feature common to most of the plastically deforming crystalline metals, as summarized in Figures 3.1.4 and 3.1.5. Either under high or low strain rates, including conditions under impact and even under hypervelocity impacts, high and low temperatures, monotonic and cyclic straining, proportional and NP loading conditions, we commonly observe dislocation cells (cellular dislocation substructures) having the size of the order of submicron to micrometer. A detailed manifestation of such universal features can be found in the well-documented relationship referred to as the “similitude” law, that holds between the inverse cell size and the flow stress. A number of data obtained by various researchers on various metals and alloys have been shown to fall on a single master curve if the cell size and the stress are normalized by the Burgers vector and the elastic modulus, respectively, to be argued in detail in Section 3.2. This implies that the hardening properties (that determine the flow-stress response) are substantially dominated by the “elastic properties” despite being a plasticity-dominant response, while they are insensitive to detailed dislocation constitutions. The secret of this universality and its significance will be disclosed in the present book, transcending the other theories and models proposed hitherto (see Chapter 10 for related discussions). There, the cellular morphology, including the size of the cells (thus automatically connected with the origin of the “similitude”) as well as their

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Figure 3.1.4  Features of dislocation-cell structure yielding a universality, including the famous

“similitude law.” Since the structure is formed under stress and at the onset of dynamic recovery, the origin is expected to be “mechanically necessary” rather than incidental.

Figure 3.1.5  A list demonstrating the ways in which the cell structure is “universal.” The

structure can be observed under diverse mechanical and thermal conditions.

roles, are all demonstrated to be determined “mechanically” rather than “incidentally” at same time. Also, the universality implies, in the presence of cell structures, the universality of the macroscopic properties (the flow-stress level, at least), despite a number of microscopic specificities. From this, we may assume that cell structures tend to act as “reservoirs” of the renormalized or absorbed microscopic degrees of freedom and information during plastic deformation for the material system concerned, stored as a form of “dislocation substructure.” The stored information is not always lost but can be released upon request, resulting in, for example, instability outset and/or fracture. This is a genuine secret: The “toughness” of metallic materials is due to their excellent “energy-absorbing” abilities over those of other engineering materials, serving further as a clue for us to link “deformation” to “fracture” modeling in a seamless manner. Therefore, the scale associated with the dislocation substructure provides us

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with the most important essential key to unfold multiscale mysteries and obstacles. This scale level will be called “Scale A” throughout this book; the features, as well as the interrelations with the other scales (Scales B and C) are overviewed in Chapter 9 and discussed in detail in Chapter 10. An intuitive image showing Scale A in the above context is schematically presented in Figure 9.3.1, where the exclusive significance of the scale, particularly associated with the dislocation-cell structures to be extensively dealt with hereafter, is visually emphasized.

3.1.2

Dislocation Cells among Others As mentioned earlier, the “cellular” morphology in dislocation substructures is a universal feature of plastically deforming metallic materials. Figure 3.1.6 shows a set of quick examples of cell structures universally formed in pure metals ((a) FCC and (b) BCC), alloys (c) low-carbon austenitic steel (FCC), and (d) ultra-low-carbon steel (BCC)), and (e) at high temperature. As readers can confirm, all exhibit dislocation substructures with cellular morphology. In what follows, after overviewing

Figure 3.1.6  Examples of dislocation-cell structures observed for FCC and BCC metals, alloys, and those under high temperature: (a) Cold-rolled Cu (Humphreys and Hatherly, 2004), (b) pure iron (Mino et al., 2000), (c) austenitic stainless steel in tension (Oliferuk et al., 1995) (courtesy of Professor Wieslaw A. Swiatnicki), (d) ultra-low-carbon steel in tension (Kojima, 2001), and (e) hot-rolled aluminum (McQueen, 1988). Adapted by permissions of the publishers/societies, respectively.

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dislocation substructures typically evolving during the course of deformation (early, intermediate, and late stages), we look at a variation of the experimentally observed cell structures in order to appreciate the common aspects as well as differences.

3.1.2.1

Three-stage Hardening and Dislocation Substructures in FCC Metals Figure 3.1.7 illustrates the typical three-stage hardening response of FCC single crystals. Stage I yields “easy glide,” taking place in a single (primary) slip system generating relatively uniformly distributed edge dipoles. This is followed by stage II, where the secondary slip systems start activating and the dislocations are mutually entangled and eventually organize into cellular modulation. This stage exhibits linear hardening with a slope of the order of   /300 regardless of materials, where the dislocation density increases due to multiplication, making the stress linearly increase with strain. At stage III, the tangled dislocations are organized into cell structures at the onset of dynamic recovery. The dynamic recovery is considered to be one of the key processes responsible for the cellular patterning (cell formation), where massive annihilation of the dislocations, probably due to cross-slip (and/or other possible mechanisms), takes place. For further deformation, stage IV and sometimes even stage V are observed. They are observed under shear or rolling mode, and are normally absent in tension. Stage IV is characterized by the further evolution of the cell structure, where the cell walls become more distinct via the elimination (annihilation) of redundant dislocations and the misorientation across the walls increases to more than a few degrees.

Figure 3.1.7  Typical three-stage hardening of a single slip-oriented FCC copper together with

corresponding dislocation substructures.

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Figure 3.1.8  Dislocation substructures for stages I and II. For stage I, in the easy-glide regime

where a single-slip system is activated, aligned edge dipoles are depicted via etch pits, while for stage II, entangling dislocations under a multiple-slip condition are presented via TEM.

Figure 3.1.9  Dislocation substructures for stage I via TEM showing arrays of edge dislocations

in a primary slip system (Swann, 1963, p. 158). Adapted with permission of the publisher.

The distinction between stages IV and V is not always obvious (it is rather ambiguous) in the sense that the definition and distinction depend on researchers. The substructure in stage IV is analogous to the subgrain structure observed in the later stages of low-cycle fatigue. Figures 3.1.7 to 3.1.10 display the details of the dislocation substructures for stages I, II, III, and IV (Argon, 1996). Figure 3.1.8(a) is a micrograph showing an etch-pit image of the edge dipoles corresponding to the dislocations intersecting on the sample surface during easy glide in stage I (Cu, RT). Figure 3.1.9 also focuses on stage I but via a TEM observing edge-dislocation arrays in the primary slip system for Cu. Figure 3.1.8(b) is a close up of entangled dislocation lines tending to aggregate

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Figure 3.1.10  Dislocation substructures for stages III and IV via TEM. Stage III yields cell structure, while in stage IV is observed a relatively large strain regime, yielding subgrain structure.

to form a tangled structure with nearly cellular morphology via TEM in stage II. Figure 3.1.10(a) presents an example of a well-developed cell structure in stage III for a [001]-oriented Cu single crystal under tension. We observe elongated cells in the loading direction. A typical TEM image of substructure in stage IV is shown in Figure 3.1.10(b), where we see sharper walls with clearer contrast than those for the cell structure in Figure 3.1.10(a). The sharper black-and-white contrast indicates large misorientation existing across the walls. A transient behavior captured via in situ TEM (ultra-high voltage on relatively thick foil samples) observation from stages II to III is presented in Figure 3.1.11 (Fujita, 1967). We observe not only the increase in dislocation density but also an aggregation process into near-cellular morphology. Figures 3.1.12 through 3.1.15 collect various dislocation substructures independently observed by different researchers. Figure 3.1.12 includes also typical variations other than cell, that is, PSB ladder embedded within vein, labyrinth under cyclic

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Figure 3.1.11  In situ observations of the entangling process of dislocations into cellular morphology via high-voltage TEM (Fujita, 1967; Karashima and Kainuma, 1975; Koda, 1993). Reprinted with permission of the publishers (Physical Society of Japan, Maruzen, and Corona Publishing Co.).

Figure 3.1.12  Various types of dislocation substructure – PSB ladder embedded within veins, cell, labyrinth, and cellular structures developed during fatigue (Ackerman et al., 1984; Suresch, 1998). Adapted with permission of the publisher.

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Figure 3.1.13  Various appearances of cell structure in Al and Cu. Al tends to yield a less fuzzy cell wall, reflecting higher SFE and lower melting temperature, and resulting in cell morphology closer to that of a subgrain. Cu, on the other hand, shows tangled and thus fuzzy cell walls with smaller misorientation (Nishiyama and Koda, 1975). (TEMs by S. Karashima, N. Kainuma, and T. Ogura.) Reprinted with permission of the publisher.

loading, and uniform distribution (see also Section 3.7) in low temperature (180°C) for α-Fe (BCC). The latter also resembles those evolved under impact loading (see Chapter 2). What is worth mentioning is that they commonly have a characteristic size of the order of micrometers (more specifically, ranged from submicron to a few micrometers), as readers can confirm. Let me emphasize again that such a “length” scale does not exist, at least explicitly, in the individual dislocations (having one definite scale, that is, the magnitude of the Burgers vector of the order of subnanometers).

3.1.2.2

Variations in Dislocation-Cell Structures As already previewed in Figure 3.1.6, the “cells” we have been discussing exhibit various appearances. Even in the same material, the detailed morphology can vary depending on the deformation mode, rate, temperature, and other factors. Also, there exists material dependency. Let us examine the material dependency of the cellular morphology by referring to some typical TEM pictures in the literature. Figure 3.1.13 compares typical micrographs between Al and Cu. Because of high SFE (cf. Section 1.2.5.1), Al exhibits sharp and clear cell walls with slight

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misorientation across them, where excessive dislocations have been eliminated. Cu, on the other hand, yields still-tangled dislocations for the walls. In Al, sometimes it is difficult to differentiate the cells from subgrains. The comparison between the two structures is mentioned in Section 3.11. Since cell formation takes place at the onset of dynamic recovery, metals with higher SFE, such as Al, are regarded as “cell-forming” materials. Even metals with very low SFE, however, can exhibit cell formation, at least under cyclic loading, as will be examined for austenitic stainless steels (e.g., SUS304 steel) in Section 3.3. For the relationship between SFE and dynamic recovery (due to cross slip), see Section 1.2.5. In BCC metals, in sharp contrast to FCC metals, the dislocations are rarely extended into partials (equivalently, stacking faults are rarely observed) because of their significantly higher SFE (see Figure 1.2.33 in Section 1.2.5.2). On the other hand, the mobility of the screw dislocations responsible for the “dynamic recovery” is much smaller than the edge component. The former has a positive effect on the cell formation, while the latter exhibits a negative effect. These contradictory effects tend to make cell formation in BCC metals rather sensitive to the strain rate, temperature, alloying elements, and introduced metallurgical microstructures. Higher purity Fe is expected to behave rather like Al (in terms of cell formation), whereas high-carbon steels will become closer to, for example, Cu. Figures 3.1.14 and 3.1.15 show examples of the cells evolved within Fe and a carbon steel, respectively. Fe, in this case, exhibits a shellwork-like pattern, which has been obtained at 25°C (Embury, 1971). The sharp boundaries resemble those observed in Al (Figure 3.1.13) as expected. This is probably a consequence of the positive effect led by the easy cross-slip nature of BCC Fe (thus yielding noticeable “dynamic recovery”) coupled with multiple slip operations (characterized as “pencil glide”). This morphology, however, is not always found: Restrictions of the cross-slip events will alter the wall morphology. The cross-slip frequency seems to be sensitively influenced by the kinds of impurity atoms added. Steels, probably because the alloying elements introduced have these negative effects (possibly for restricting cross slip), usually exhibit cells with fuzzier walls, as shown in Figure 3.1.15 (under cold rolling). The micrographs also indicate further developments of the structure into subdivided morphology at large enough rolling ratios. Note that dislocation substructures in finite deformation stages are separately discussed in Section 3.10. A notable example of the previously mentioned “impurity-sensitive” cell formability is Cu-added steels under fatigue. Some details are given in Appendix 9, in relation to the outline of an ongoing project of mine. In this case, not only the content but also its form within the steel drastically change the dislocation substructures to be evolved. Cu addition as a solid solution yields a 2D vein structure of dislocations instead of 3D cells, whereas an addition as small precipitates of the order of a few nanometers results in uniform distribution without structuring. These substructural changes also totally alter the cyclic stress response – from cyclic hardening to cyclic softening via stable response, and they ultimately affect the fatigue-crack initiation processes following slip banding and extrusion/intrusion morphology.

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Figure 3.1.14  Example of cell structure in Fe deformed at RT, presenting shellwork-like morphology (Embury, 1971).

The most typical variations in the deformation or loading mode are “monotonic” and “cyclic.” Figure 3.1.16 compares cell structures for polycrystal Al between that under cyclic (fatigue) and monotonic (static) loading (Karashima and Kainuma, 1975), and those for polycrystal Cu under cyclic loading between that with and without prestrain (30%) (Weiss and Maurer, 1968). Intriguingly, almost identical cell structures are observed for Al, with sharp walls, and similar size and morphology. A large difference, on the other hand, is seen for Cu, where the prestrained result yields tangled and fine cells in sharp contrast to that under cyclic loading without prestrain, showing well-developed and coarser cells, which resemble the Al results. Another example demonstrating the effect of preloading history on cyclically evolved cells

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Figure 3.1.15  The evolution process of dislocation-cell structure in steel under cold rolling.

can be found in Figure 3.1.17 (Feltner and Laird, 1967), where a comparison is made between those from annealed (that is, with no preloading) and cold-rolled (after 5% reduction in diameter) for polycrystalline Cu at room temperature. As readily confirmed, the evolved cell structures almost coincide, both in morphology and size, meaning they are basically independent of the initial condition, even for Cu with mid/low SFE. Note that well-developed cells with unambiguous walls after cyclic straining basically agree with those observed in Figure 3.1.17(b). The cell size, by the way, depends on the straining amplitude and the temperature, as exemplified in Figure 3.1.18 (Feltner and Laird, 1967), and increases with decreasing strain amplitude and increasing temperature (not shown here). Generally speaking, more cells are formed in the smaller strain ranges under cyclic loading than under monotonic loading. This is simply because the dynamic recovery responsible for the cell formation tends to be enhanced under cyclic conditions.

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Figure 3.1.16  Comparison of cell structures for polycrystal Al between that under cyclic (fatigue) and monotonic (static) loading (top) (Karashima et al., 1967) and those for polycrystal Cu under cyclic loading between that with and without prestrain (30%) (bottom) (Weiss and Maurer, 1968). Adapted with permission of the publishers.

The previous discussion considered the dislocation-cell structures for metals and alloys with low/medium SFE. For those with extremely low SFE, however, we rarely observe “cells,” but they tend to yield planar or sparse distributions of dislocations rather than tangled ones, with quite different morphologies, as displayed in Figure 3.1.19 (Nishiyama and Koda, 1975), at least under monotonic loading conditions. This is simply because of the highly restricted cross-slip processes caused by largely extended dislocations, that is, stacking faults. Note that the stripes observed in Figure 3.1.19 (left) indicate the stacking faults. Cyclic stressing/straining, however, can lead to the formation of cells, but in a confined manner, as in highly stressed/severely strained zones such as near fatigue cracks. Figure 3.1.20 displays representative dislocation substructures evolved during high-strain amplitude fatigue (Hatanaka, 1974), emphasizing the effect of the SFE. Cellular morphologies are found even for low-SFE metals such as 18–8 stainless steel under cyclic conditions, whereas the lowest SFE alloy, that is, Cu-7.5%Al (aluminum-bronze), tends to yield slip-band structures and never yields cells, quite similar to those under monotonic loading shown in Figure 3.1.19 for Cu-8%Al. Such a situation, however, is subjected to change when it comes to the crack vicinity. This aspect is separately considered in Section 3.7.

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Figure 3.1.17  Another example of comparison for cyclically strained cell structures in Cu at RT between annealed and weakly cold rolled (5% reduction in diameter), exhibiting, at least apparently, independency of the initial condition (Feltner and Laird, 1967). Adapted with permission of the publisher.

Figure 3.1.18  Variation of cyclically evolved cells with straining amplitude (Feltner and Laird, 1967). Adapted with permission of the publisher.

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Figure 3.1.19  Typical deformation structures observed in extremely high SFE metals and alloys, that is, aluminum-bronze (Cu-8%Al) and α-brass (Cu-30%Zn), normally not yielding dislocation cells, but planar or sparse arrays of dislocations (Nishiyama and Koda, 1975). (TEMs by S. Karashima, N. Kainuma, and T. Ogura.) Reprinted with permission of the publisher.

3.2

Similitude Law for Dislocation-Cell Size A well-known and well-documented manifestation of the universality of the dislocation-cell structures is the so-called similitude law, sometimes referred to as the “similitude principle” (Kuhlmann-Wilsdorf, 1999a). Figure 3.2.1 shows a famous representation of the relationship between cell size and the stress normalized by the Burgers vector b and shear modulus µ , respectively, for (a) α -Fe and (b) Al by Raj and Pharr (1986). They collected a set of data from published works and made the correlations to examine statistically the effectiveness of the law. Figure 3.2.2 also displays the similitude correlation with the normalized shear stress and cell size ( / b versus dcell in this arrangement) (Staker and Holt, 1972). Intriguingly, not only the linear relationship between the two but also the correlation holds, regardless of the materials, that is, Al, Cu, α -Fe, and their alloys, correlated by a single master curve.

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Figure 3.1.20  Dislocation substructures developed under cyclic loading (fatigue) at high-strain amplitudes, comparing five metals and alloys with various SFE (Grosskreutz, 1963; Hatanaka, 1974). Adapted with permission of the publishers.

These correlations will remind the reader of another well-known empirical Hall– Petch (H–P) relationship where the stress (either normal stress or shear stress) is inversely proportional to the square root of the mean grain size. If the stress is plotted with the inverse root of the cell size 1/ dcell instead of 1/dcell , we also have another good correlation, as in Figure 3.2.3(a), whose agreement is commensurate with that plotted against 1/ dcell, as in Figure 3.2.3(b). Then the question is “why can the similitude relationship (stress versus inverse cell size 1/ dcell ) be considered in general as more appropriate than the H–P-type correlation?” One possible answer to this question would be the fact that the former tends to yield negative values of stress at the intercept on the vertical axis in Figure 3.2.3(a), which seems to be unrealistic, although the data may not always have to be extrapolated to that range. One more intriguing empirical fact is the proportionality between the cell size dcell and the mean spacing of the wall-constructing dislocations l , that is, dcell  g  l . Figure 3.2.4 shows the schematics. By combining this relationship l  1/  w with the well-known Bailey–Hirsh relationship  w   b  w , we can regain the similitude relationship. This implies that the similitude law is just an indirect, and merely an apparent, relationship derivable from other more substantial relationships, having no direct physical significance on the cell size itself. Also, one should notice that there

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Figure 3.2.1  Similitude law representing a universal feature of a dislocation cell (or subgrain),

whose size is well correlated with normalized stress regardless of materials (Raj and Pharr, 1986). Adapted with permission of the publisher (Elsevier Science & Technology Journals).

is no physically clear mechanism for that results in the dcell  g  l relationship. Note that this relationship has not been well documented to my knowledge compared with the similitude law. The dislocation-cell size has reported to be affected by the initial grain size, too. Figure 3.2.5 shows a variation of dcell with strain observed in a cold-rolled Cu (Gracio, 1995, pp. 97–104). Larger initial grain size yields larger cell size, while smaller initial grain size results in smaller cell size. This stems roughly from interactions with the intragranular inhomogeneity where other types of deformation structures such as MBs are formed in order to accommodate the imposed geometrical constraints (see Section 3.10). Since smaller grains are expected to be subject to higher constraint, the interactions will promote cell evolution. Contrary to this, Oliferuk et al. (1995) observed almost mutually commensurate cell sizes between fine- (8 µm) and coarse(80 µm) grained samples for an austenitic stainless steel (18.58Cr, 17.3Ni, and 0.05C wt%) under tension at RT, as shown in Figure 3.2.6, after relatively large straining (0.22–0.25). This was reported in an attempt to discuss the grain-size effect on the energy storage rate (Oliferuk et al., 1995). Note that the material they chose is the one with the lowest SFE. In the 1970s, Fujita et al. conducted a systematic series of studies examining the interrelationship between the grain size and the cell size of various materials, including

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Figure 3.2.2  Another representation of the similitude law, demonstrating the universal

arrangement of various data for various metals and alloys with a single master curve (Staker and Holt, 1972, p. 577, figure 8). Adapted with permission of the publisher (Elsevier/ University of Toronto Press).

polycrystalline Al (Fujita and Tabata, 1973), Cu, and Cu-13%Al alloy (Miyazaki and Fujita, 1978), where an inability of the H–P relation as the single parameter to correlate the yield/flow stress is also pointed out; Fujita and Tabata (1973) with various temperatures (77K through 473K), while Miyazaki and Fujita (1978) in the context of t / dG (specimen thickness/grain size) perspectives (see Section 4.5.2). They confirmed basically the same trend as discussed earlier, that is, the smaller the grain size, the smaller the evolved cell size becomes, and vice versa. Representative TEMs are displayed in Figure 3.2.7, while the correlation between the initial grain size dG and the corresponding cell size dcell is presented in Figure 3.2.8, comparing three temperature conditions (Fujita and Tabata, 1973). From Figure 3.2.8, what is noteworthy, among other things, is that the correlations do not yield a linear relationship, but we commonly observe a saturating tendency as the initial grain size increases, probably toward those for the single-crystal samples. Based on the findings in their series of studies, Fujita et al. ultimately suggested replacing the H–P relation by the use of the mean free path of dislocations, which is commensurate with the inverse cell size for Al (Fujita and Tabata, 1973) and the slip-band length for Cu (Miyazaki and Fujita, 1978). Note that the substructures evolved for Cu-13%Al are mostly confined to slip bands instead of cells because of the extremely small SFE (less than 10 mJ m−2; see Section 1.2.5.2, as mentioned earlier).

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Figure 3.2.3  Comparison of two graphs of 0.2% proof stress-pure Al between an H–P plot and

one with inverse cell size, resulting in comparable correlations (Kimura, 1998).

Figure 3.2.4  Schematic drawing for another empirically observed relationship between cell

size and mean dislocation spacing. Combining it with the Bailey–Hirsch relationship, one can derive the similitude law.

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Figure 3.2.5  Initial grain-size dependency of cell size for Cu under cold rolling, implying effect

of intragranular inhomogeneity on cell evolution (Gracio, 1995). Adapted with permission of the publisher (Elsevier).

Figure 3.2.6  Comparison of cell structures for a low-C austenitic stainless steel strained by 0.22

and 0.25 between samples with initial grain sizes of 8 mm and 80 mm (Oliferuk et al., 1995) (high-resolution micrographs courtesy of Professor Wieslaw A. Swiatnicki). Adapted with permission of the publisher.

The amount of data as well as the variation of tested materials, however, has still been relatively limited, although the results themselves are already quite informative in modeling cell evolution and identifying the mechanism of the formation. Further systematic observations, hopefully including the effects of strain rate, temperature, and their histories, as well as the NP strain paths, along similar lines to those provided in Section 3.3, where systematic series of studies have been conducted, are keenly anticipated.

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Figure 3.2.7  Effect of initial grain size on the evolved dislocation-cell structure for

polycrystalline Cu (at 20% tensile strain, 293K) (Fujita and Tabata, 1973). Adapted with permission of the publisher (Elsevier).

Figure 3.2.8  Relationship between initial grain size and the evolved dislocation-cell size,

comparing three testing temperatures, 77, 293 (RT) (see Figure 3.2.6), and 473K for polycrystalline Cu (at 20% tensile strain, 293K) (Fujita and Tabata, 1973). Adapted with permission of the publisher (Elsevier).

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3.3

161

Strain-History Effects: Proportional versus NP Strain Paths The roles of dislocation cells on the history effects are clearly manifested as the additional hardening observed in NP cyclic-straining experiments, particularly for metals and alloys with small SFE. This section presents the well-documented interrelationships between NP strain paths and the concomitant dislocation-cell structures, which ultimately leads us to conclude that the NP strain history is written and stored in the evolving dislocation-cell structures, possibly as a form of strain energy. Figure 3.3.1 displays a striking example of the NP strain-history effect on the cyclic stress response, where seven strain paths are compared, that is, two proportional paths (push–pull and reversed torsion) and five NP paths (star-shaped, rectangular, and square, and two cruciform; see Tanaka et al., 1985a). The material tested is austenitic stainless steel (SUS316), that is, one of the typical materials with lowest SFE. Cyclicstraining tests are conducted under a plastic strain amplitude-controlled condition at room temperature. The graph represents variation of the cyclic stress amplitude with cumulative plastic strain. The details of these strain paths are presented in Figures 3.3.2 and 3.3.3, which are described later. The rectangular and circular paths are also referred to as a 90° out-of-phase strain path.

Figure 3.3.1  Variation of cyclic stress for austenitic stainless steel with strain path, showing

markedly large additional hardening depending on nonproportionality (Tanaka et al., 1985a), together with dislocation substructures expected to be evolved under typical paths, that is, push–pull (P), reversed torsion (T), and cruciform (alternating P and T) (Nishino et al., 1984). Adapted with permission of the publisher.

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Figure 3.3.2  Schematics of proportional and NP strain paths assuming a typical biaxial

testing, that is, combined push–pull and reversed torsion on a thin-walled tubular specimen, represented on the Mises equivalent strain circle.

Figure 3.3.3  Schematics of NP strain paths for alternating push–pull and reversed torsion path

(cruciform path) and 90° out-of-phase-type paths (square and circular) represented on the Mises equivalent strain circle, together with corresponding waveforms.

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Clearly demonstrated in Figure 3.3.1 is the dramatically enhanced additional hardening in the cyclic stress responses, strongly depending on the NP strain paths, whereas the results under the two proportional paths almost coincide. The NP strain paths exhibit significantly rapid cyclic hardening and much larger saturated stress amplitude than those under proportional loading. It has been reported that the circular path yields the maximum additional hardening among the bi-axial NP paths, while the star-shaped path results in significantly delayed saturation among others. This drastically enhanced additional hardening strongly depends on the NP strain path, that is, nonproportionality, being substantially dominated by the dislocation substructures, especially cells, evolved during the cyclic deformation. The insets in Figure 3.3.1 are TEM pictures of the typical dislocation substructures for the three paths, that is, push–pull, reversed torsion, and cruciform paths. They are for SUS304 stainless steel at high a temperature (Nishino et al., 1986). (The additional hardening in the cyclic stress response and these micrographs do not directly correspond; however, we may consider for them as corresponding, at least qualitatively.) The proportional paths, that is, push–pull and reversed torsion, yield ladder-like and labyrinth patterns, respectively, while the NP cruciform path exhibits a well-developed cell structure. Note that the morphologies of the substructures for the proportional strain paths, that is, ladder and labyrinth patterns, are brought about by the slip-system constitution for the observed region within the crystal grain. We may observe also cellular morphology if we choose regions or grains with multiple slip conditions. What really matters in the present context is the mean “size” or the “spacing” (implying that the “morphological” factors are not important as far as the additional hardening for a “single” path is concerned). One can readily understand that different cell sizes exhibit different degrees of additional hardening in the cyclic stress response. This is also true for monotonic types of loading conditions. Figures 3.3.4 and 3.3.5 show the evolution process of the dislocation substructures for the three strain paths, that is, push–pull, reversed torsion (proportional), and cruciform (NP). We confirm at an early stage the coarser and fuzzy patterns that gradually evolve into refined and sharper substructures with cellular or wall morphologies as the number of cycles proceeds. Correspondingly, the crystalline lattices are distorted, manifesting themselves in growing misorientations. Note that they can be measured as the expansion of the Laue spots via the X-ray diffraction technique. Details of the NP strain paths are shown in Figures 3.3.2 and 3.3.3. Here we assume a typical biaxial testing method which can be relatively easily carried out and thus has been widely employed, that is, combined push–pull and reversed torsion on a thinwalled cylindrical (tubular) specimen (strain controlled), where a material element in the gauge section of the specimen is subject to combined tension/compression and shear strain under a nearly plane stress state. The inset represents a strain circle on the Mises basis in the    / 3 space, on which the Mises equivalent strain ε eq is constant (ε eq is used in this chapter instead of ε for expressing the equivalent strain, which will be used in the other chapters).

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Cyclic NP Loading

Mode

N1

N3

N2

P 1μm

1μm

1μm

1μm

1μm

1μm

1μm

1μm

1μm

T

APT Figure 3.3.4  Evolution process of dislocation substructure under low-cycle fatigue comparing

three strain paths, that is, push–pull, reversed torsion (proportional), and cruciform (NP), where gradually sharpening and refined cells or walls are observed

The abscissa of the Mises strain circle represents axial strain while the ordinate corresponds to the shear strain. Therefore, proportional-type combined strain paths are expressed by inclined lines on the circle. The corresponding strain waveform, along with time, to be applied to the specimen is given in the inset. Figure 3.3.3 shows the NP strain paths for the three typical cases, that is, cruciform, rectangular, and circular paths. With the cruciform path, push–pull and reversed torsion strain are alternatively applied, whereas, with the rectangular and circular paths, the two modes of strain are continuously applied. The latter two are also called 90° out-of-phase loading, because, as can be confirmed in the strain waveform, there will always be a delay in the phase by 90° between the push–pull and reversed torsion. For the former, the direction of the principal strain axis is intermittently or periodically changed between the axial direction and 45° from it. For the latter two, on the other hand, the principal strain axis direction is altered continuously, which makes the nonproportionality maximum. Another interesting feature of the dislocation substructures associated with their stability is a strain history-dependent overwriting of the substructures. Such interactions

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Figure 3.3.5  Evolution process of dislocation substructure under low-cycle fatigue comparing

NP and proportional loading conditions, together with schematics of correspondingly distorting lattice (Nishino et al., 1984). Reprinted with permission of the publisher.

among dislocation substructures with distinct morphologies and sizes can be examined by introducing a mode change in the cyclic stress amplitude. Figure 3.3.6 shows examples of the results, where the strain path changes from NP to proportional paths (i.e., cruciform to push–pull [APT to P]) and that between two proportional paths (i.e., push–pull to reversed torsion [P to T]) are conducted for SUS304 stainless steel (Nishino et al., 1984). When the push–pull loading (P) is applied following the cruciform strain path (APT), the previously constructed fine cell structure tends to be overwritten by relatively coarser ladder-like structures during the subsequent push–pull loading, which accordingly results in a transition of the cyclic stress response from the level corresponding to the former path, to approaching that closer to the latter, as shown in the figure. Note that the NP strain history tends to persist even after the strain path change, and is not overwritten thoroughly. By way of contrast, for the path change “P to T,” a temporary rise in the cyclic stress is observed immediately after (or at the onset of) the path change, but eventually the stress amplitude reaches the original level. This is simply because of the substantially similar cyclic stress levels common to the two paths, whereas the brief stress rise is caused rather exclusively by the interaction between the two substructures.

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Figure 3.3.6  Effect of mode change on cyclic stress amplitude, that is, from NP to proportional

paths (cruciform to push–pull) and among proportional paths (push–pull to reversed torsion) for SUS304 stainless steel (courtesy of Professor S. Nishino). (Nishino et al., 1984; Ohnami, 1992).

These results imply that the level of the additional hardening in the cyclic stress response is mainly dominated by the characteristic “size” of the evolving dislocation substructures rather than the morphology (“cellular” in this case). Therefore, as far as the modeling for the additional hardening response (macroscopic) is concerned, we do not have to consider morphological details. This rationalizes a simplified treatment, as in the following simulation. Figures 3.3.7 to 3.3.9 show a series of simulation results on additional hardening due to NP cyclic loading over various strain paths (Hasebe et al., 1997). In Figure 3.3.7(a), the variation of cell size with number of cycles is compared among four strain paths including push–pull (proportional), cruciform, rectangular, and circular paths (NP), while Figure 3.3.7(b) compares the saturated flow stress normalized by that for push–pull loading among the strain paths, comparing also between simulated and experimental values. We notice that there is a consistent relationship between the cell size and the flow stress at saturation; smaller cell size yields larger flow stress, and vice versa. The same analyses are performed in Figure 3.3.8 but for more complicated strain paths, that is, stair-step strain paths. Comparison is made among four different numbers of steps, that is, 0 (coinciding with 45° proportional

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Figure 3.3.7  Simulation results showing strain path-dependent cell size evolution and flow

stress compared with experiments for cruciform and 90° out-of-phase-type paths based on the crystal plasticity-based model, where dislocation–dislocation interactions are taken into account.

Figure 3.3.8  Simulation results showing strain path-dependent cell size evolution and flow

stress compared with experiments for stair-step-type paths based on a crystal plasticity-based model, where dislocation–dislocation interactions are taken into account.

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Figure 3.3.9  Simulation results showing cyclic stress and shear stress-axial stress diagrams

comparing various strain paths.

path), 4, 2, and 1, where the nonproportionality increases in this order, as can be intuitively understood from the figure. The diagram showing flow-stress ratio versus strain paths demonstrates that higher nonproportionality causes larger additional hardening, as is expected. Note that a slight difference can be found between the simulation and experiment in the transition region. The experimental result indicates more pronounced transition from P to NP, exhibiting a slight increase in the flow stress for the four-step path compared with that for the proportional loading (zero-step path), following rapid (or steep) increase in the additional hardening with increasing number of stair-steps. Figure 3.3.9 shows: (a) Cyclic stress amplitude with number of cycles for the four paths, and (b) hysteresis loops for the stair-step NP paths obtained in the simulation. The figures confirm satisfactory agreements between the simulation results and experiments, not only in the amount of additional hardening but also in the cyclic hardening response to stress. Note that in the hardening evolution model, certain details about the interactions among dislocations belonging to different slip systems, together with their past activities, are taken into account, based on the results obtained in a latent hardening test (see Section 1.3.5) The details about the constitutive equation, including the hardening evolution model, will be given in Chapter 5. Figures 3.3.10 to 3.3.12 present an example of an extended investigation on the effect of a wider range of NP strain paths on substructure evolution and additional hardening (Kida, Itoh et al. 1992). Strain-controlled low-cycle fatigue tests at high temperature for SUS304 stainless steel were conducted employing the 13 NP strain paths (Cases 1 to 13) shown in Figure 3.3.10 in addition to push–pull loading

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Figure 3.3.10  Systematically designed 13 NP strain paths for combined push–pull and reversed torsion tests on SUS304 stainless steel (courtesy of Professor T. Itoh) (Itoh, 1992).

Figure 3.3.11  Transmission electron micrographs showing evolved cell structures corresponding to the NP strain paths in Figure 3.3.10 (10 among 13) for combined push– pull and reversed torsion tests on SUS304 stainless steel (Kida et al., 1997). Adapted with permission from the publisher (Blackwell Publishing Ltd.).

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Figure 3.3.12  Cyclic stress range correlated with dislocation-cell size with a slope close to −0.5, implying a similar relationship to similitude law SUS304 stainless steel under various NP strain paths in Figure 3.3.9 (courtesy of Professor T. Itoh) (Kida et al., 1997). Adapted with permission of the publisher (Blackwell Publishing Ltd.).

(proportional) as the reference (Case 0). As partially shown in Figure 3.3.11, the increasing nonproportionality roughly results in finer cell size, where stacking faults are also observed because of the low SFE. In such cases, the failure life does not well correlate with the cyclic stress range, as shown in Figure 3.3.12, falling with an extremely large scatter of the data that deviate significantly from the reference (the results for Case 0) on the conservative side. On the other hand, Figure 3.3.12 presents a relationship between the cyclic stress range and the averaged cell size, demonstrating a good correlation. Since the slope is found to be nearly 0.5, a similitude-like relationship is also shown to hold between the cell size and cyclic stress level for NP loading conditions. It is also noteworthy that the effect of nonproportionality on substructure evolution and its contributions to the stress response is strongly material dependent. Figure 3.3.13 provides an example of systematically conducted experimental observations, comparing cell structures evolved under proportional (push–pull) and NP (cruciform or APT) cyclic loading conditions for various FCC polycrystalline materials with different SFE (Itoh, 1992). Here, all the tests are conducted at room temperature. Finer cell sizes are generally observed under NP strain paths,

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Figure 3.3.13  Comparisons of cell structures between those evolved under P and NP loading for various FCC metals and alloys with different SFE (Itoh et al., 1992). Adapted with permission of the publisher.

Figure 3.3.14  Correlation between cell size ratio and cyclic stress ratio, comparing four FCC metals/alloy in Figure 3.3.13, where the ratios of those in NP to P loading are considered, with ∆σNP/P measured from the corresponding yield stresses (Itoh et al., 1992). Adapted with permission of the publisher.

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while the distinction between those under NP and P loading becomes rather unclear for the higher SFE such as in Al, resulting in comparable flow-stress level even under NP loading. The associated intriguing correlation is displayed in Figure 3.3.14, where the cell size ratio (the inverse square root 1/ dcell ) is plotted against the corresponding stress ratio  R. Here, the “ratio” indicates that of NP loading to the P counterpart, while an effective stress amplitude measured from yield stress in P loading in each material is used for evaluating  R . The correlation eloquently demonstrates that an organic interrelationship should exist among evolved cell size, the attendant stress level, and the SFE, strongly implying that the “similitude”-like relationship discussed in Section 3.2 should hold also for cyclic P/NP loading conditions.

3.4

Strain-Rate and Temperature-History Effects

3.4.1

FCC versus BCC In my opinion, a decisively important aspects for us to focus on when considering dislocation cells is the fact that they are also quite sensitive to their strain-rate/temperature histories. Combining the strain-history effects, together with the strong material dependencies involving FCC/BCC and SFE distinctions, discussed earlier, dislocation-cell structures may safely be regarded as a “curriculum vitae (CV)” of materials, to say the least. Here, we separately discuss the effect of strain rate, temperature, and their histories. Basically, the strain-rate effect is equivalent to the temperature effect according to statistical thermodynamics, as long as the averaged motions of dislocations are controlled by thermal-activation processes. Details can be found in Chapter 2. Figure 3.4.1 schematically shows strain-rate and “strain-rate history” effects for FCC and BCC metals, together with the corresponding dislocation substructures that are likely to evolve, depicting not only the strain-rate dependence but also a marked contrast between FCC and BCC metals. The difference in the evolved dislocation substructure is directly attributed to the “strain-rate history” effect together with its material dependency. FCC metals normally yield finer and denser dislocation cells under higher strain rates such as impact loading, leading to a “positive” history effect (i.e., additional hardening), whereas BCC metals tend to exhibit a planer type or more scarce dislocation structure, as presented in the inset at top right, resulting in “negative” history, that is, a softer response of the flow stress. Experimental results demonstrating both “positive” and “negative” strain-history effects are compared in Figure 3.4.2, for OFHC Cu (FCC) and mild steel (BCC), respectively. These are obtained under strain-rate “jump” torsion tests at room temperature, where stored torsional strain energy is instantaneously released to realize the “quasi-static-to-impact” transition (Eleiche and Campbell, 1976), together with the strain-rate history of each test as well as that of the corresponding adiabatic temperature rise evaluated by assuming all the plastic work is converted into heat. They also systematically examined the effect of test temperature, with six levels including RT,

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3.4  Strain-Rate and Temperature-History Effects

Figure 3.4.1  Schematics comparing strain-rate and strain-rate history effects and attendant

dislocation substructures between FCC and BCC metals. FCC metals reveal a “positive” strain-rate effect while BCC metals yield a “negative” effect, each contributing to the dislocation substructure that will evolve.

Figure 3.4.2  Example of contrasting “positive” and “negative” strain-rate history effects

for FCC and BCC metals/alloys, demonstrated in experimental results on oxygen-free high-thermal conductivity (OFHC) copper and mild steel, respectively, together with the corresponding strain-rate and temperature histories (Eleiche and Campbell, 1976).

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Dislocation Substructures

Figure 3.4.3  Typical dislocation substructures for BCC metals (ultra-low-carbon steel) under

high strain rates (right), compared with that under static loading (left) (Kojima, 2001), where cell formations are significantly restricted, to yield sparsely distributed low-dense dislocations responsible for the “negative” strain-rate (and temperature) history effects. Adapted with permission from the Iron and Steel Institute, Japan.

ranging from −150°C to 400°C, revealing basically the same trends, except the case for mild steel at 200°C, which shows higher yield and flow stresses than the dynamic one, accompanied by serration, due to dynamic aging. Putting aside this exception, the authors concluded that softening, which must have been brought about by the adiabatic heating during high rate deformation, can only have a small effect. Figure 3.4.3 compares dislocation substructures for an ultra-low carbon steel (0.0022wt%C), as a typical BCC metal/alloy, between that developed under static and impact tension (Kojima, 2001), demonstrating that cell formations are significantly restricted during impact loading, which eloquently explains the reason why BCC metals/alloys tend to exhibit “negative” strain-rate effects, as schematized in Figure 3.4.1. Since low temperature conditions exhibit quite similar dislocation substructures to those under high strain rates, they were encapsulated in “the strain-rate history effects” in the present subsection.

3.4.2

Strain-Rate History Effect versus Proportional Strain-History Effect Let us focus on the Bauschinger effect for the purpose of systematically examining the material-dependent strain-history effects via stress responses, as an effective manifestation of the evolved dislocation cells (substructures) introduced during prestraining. The materials chosen here are commercially pure Al and Cu as FCC typical, and commercially pure Fe and some low-/medium-carbon steels as the BCC category. Roughly, two tests are performed, that is, those under constant strain-rate and variable strain-rate conditions in precompression and the tension that follows, involving the impact of the strain-rate range to evaluate Bauschinger stress as a typical measure of the permanent softening. The strain rate approximately ranged from 3 × 10−3 s−1 through to 2 × 103 s−1 at room temperature.

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Figure 3.4.4  Dynamic Bauschinger effect under compression/tension with constant strain

rates for Al and Cu as representatives of large and small SFE metals, respectively, showing variation of Bauschinger stress (as a measure of permanent softening) with strain rate, which reveals contrasting trends between Al and Cu.

Figures 3.4.4 and 3.4.5 summarize the results obtained, together with a variation of the Bauschinger stress ratio,  B / ref , with prestrain rate, comparing Al and Cu (Figure 3.4.4), and prestrain and restrain rate combinations for Al (Figure 3.4.5) and Cu (Figure 3.4.6), respectively. For the constant strain-rate tests in Figure 3.4.4, Al and Cu are shown to exhibit distinct trends depending on the difference in the SFE. In Al, slight but increasing Bauschinger stress is observed, simply because of more densely developed dislocation cells at high strain rates (see the inset schematics), attributing to the enhanced “back stress,” which corresponds to “kinematic hardening” when rephrased in the potential theory of plasticity, whereas the normally evolved cells at the static condition have much less ability to yield sufficient back-stress field, resulting in relatively low Bauschinger stress. Cu, on the other hand, can produce a larger back-stress field due to small SFE, resulting in relatively large Bauschinger stress even under static-loading states. Compared with Al, however, the back stress tends to be overwhelmed by the attendant growing drag-stress field (“isotropic hardening”) associated with the denser cells as strain rate increases, leading ultimately to decreasing Bauschinger stress with strain rate as an overall trend. The distinctions between Al and Cu based on those in the SFE, seen earlier, are apt to be more pronounced when it comes to the variable strain-rate Bauschinger tests. Figure 3.4.5 for Al basically shows similar results as in Figure 3.4.4 for constant prestrain- and restrain-rate cases, whereas Figure 3.4.6 for Cu is subject to significant changes from the counterpart in Figure 3.4.4 in that a high restrain-rate condition tends to exhibit a “negative” Bauschinger effect, that is, larger flow-stress

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Figure 3.4.5  Dynamic Bauschinger effect under compression/tension with variable strain rates

for Al as a representative of a large SFE metal, showing variation of Bauschinger stress with strain rate, from which relatively similar variations to the constant strain-rate counterpart (broken line) can be confirmed.

Figure 3.4.6  Dynamic Bauschinger effect under compression/tension with variable strain rates

for Cu as a representative of a mid/small SFE metal, showing variation of Bauschinger stress with strain rate, which demonstrates the Bauschinger effect tending to become negative as the prestrain rate increases, probably due to the attendant denser cell structures.

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Figure 3.4.7  Correlation of the Bauschinger stress ratio with plastic–work ratio to that in the

p p pre preloading (compression) Wten Wcom , comparing all the results, including those presented p p pre in Figures 3.4.5–3.4.7 between Al and Cu, where Wten Wcom = 1 corresponds to the constant strain-rate tests in Figure 3.4.4. Good correlation is confirmed regardless of the materials, implying a sort of universal role for the evolved dislocation cells (substructures) to be played on the strain/strain-rate history effects.

results even under reversed loading. This apparently anomalous phenomenon is simply brought about by the raised flow stress in the reloading (i.e., drag stress in isotropic hardening) against a poorly evolved back-stress field associated with the cells developed during prestraining in static/lower strain-rate conditions. The inset schematics for the dislocation cells (bottom-center) may hopefully help with understanding the latter situation, which illustrate cells likely to be developed in prestraining. Regardless of whether contributing to the isotropic hardening (drag stress) or kinematic hardening (back stress), the intensity of the developed dislocation cells seems to control the Bauschinger stress, as judged from the just discussed series of results. So, considering the plastic work as a tentative macroscopic measure for representing the intensity of developed cells, we plot the Bauschinger stress ratio p p pre with it in Figure 3.4.7, where the ratio to that in the preloading, that is, Wten Wcom , is evaluated. There is found to be good correlation between the two,  B / ref and p p pre Wten Wcom , regardless of the materials (solid/open circles for Cu/Al) and the strainrate combinations, although the relationship is not always linear and has some scatter. The plot also clearly indicates the transition of the Bauschinger stress from “positive” to “negative,” that is,  B / ref  0 for Cu, observed in Figure 3.4.6, prop p pre vided the plastic–work ratio Wten Wcom becomes less than unity. Also confirmed p p pre is that Wten Wcom = 1 corresponds to the results in the constant strain-rate tests (Figure 3.4.4), located in the middle of the transition, showing us how such transition proceeds.

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Figure 3.4.8  Dynamic Bauschinger effect under compression/tension with constant strain

rates for BCC Fe and some carbon steels, showing a variation of the Bauschinger stress (as a measure of permanent softening) with strain rate, which reveals a common decreasing trend with strain rate as well as an increasing Bauschinger effect with increasing carbon content, that is, perlite volume fraction against ferrite phase, with the former possibly due to the restricted cell formations at high strain rates.

For BCC metals and alloys, on the other hand, we anticipate what may be an opposite trend to the FCC cases presented hitherto, for the substructure itself tends to become less capable of producing large enough back stress responsible for the Bauschinger effect. Figure 3.4.8 summarizes the same plots as in Figure 3.4.4, that is,  B / ref as a function of strain rate, but for BCC pure Fe and three low-/high-carbon steels, that is, S15CK, S45C, and S55C, together with the corresponding stress–strain diagrams for Fe and S55C as representatives. As we expected, a common decreasing trend with increasing strain rate is observed, exhibiting almost zero Bauschinger effect being approached for Fe at the high enough strain rate. Also to be noted is that the  B / ref  ln  diagrams are simply parallel-shifted upward as the carbon content increases, yielding a finite Bauschinger effect even at a large strain-rate regime. This eloquently indicates the back-stress field to be brought about by the metallurgical microstructures, possibly perlite (see the inset schematic) in the present context, whose contribution is considered to be commensurate with the shifted amounts on the diagram.

3.4.3

Strain-Rate History Effect versus NP Strain-History Effect for BCC Metals Let us make doubly sure of the “negative” strain-rate effect in BCC metals/alloys via a different series of experiments. Here, we again discuss an NP type of strain history

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Figure 3.4.9  Nonproportional tests for BCC Fe and carbon steel with static pretorsion followed

by static/impact tension, designated for examining a “positive” strain-rate history effect in the NP context.

to see to what extent the additional hardening occurs in the BCC samples of concern. For an effective comparison, two modes of NP prestrain are introduced, that is, statically and dynamically, with which one can quite easily guess the outcome. When the NP prestrain is given statically, we expect “normal” cell structures to be developed, probably leading to commensurate additional hardening no matter what the following strain rate is. For the case with dynamically introduced NP strain history, on the other hand, what we anticipate is that the corresponding substructure will be quite incompetent to produce sufficient additional hardening. So, let us look at the results. Figure 3.4.9 is a summary of a series of test results for the former, that is, with a statically introduced NP strain history, where static torsion is applied, followed by static/impact tension, on commercially pure Fe and two carbon steels of 15CK and S45C. Corresponding shifts are made on the abscissa to the NP prestrains in each diagram. As anticipated, we observe relatively large additional hardening, roughly regardless of the strain rate and the carbon content. Figure 3.4.10, on the other hand, presents the results for the latter, where “impact” compression followed by “static” torsion tests are conducted on Fe and S45C (same materials as in the former). We confirm similar additional hardening in the static–static condition for both the materials, however, this tends to diminish as the restrain rate increases. More precisely, the additional hardening in the S45C sample almost vanishes at a strain rate of 2000 s−1.

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Figure 3.4.10  Nonproportional tests for BCC Fe and carbon steel with impact precompression followed by static torsion, designated for examining a “negative” strain-rate history effect in the NP context.

The pure Fe sample, on the other hand, exhibits a rather pronounced additional hardening in the present static–static condition, resulting, accordingly, in greatly reduced but still-remaining additional hardening even at the same high restrain-rate regime.

3.4.4

Strain-Rate History Effect versus NP Strain-History Effect for FCC Metals For FCC metals, we also need to look at the effect of SFE in discussing the strain-history effect, as viewed in the last part of Section 3.3. Figure 3.4.11 compares the results corresponding to Figure 3.4.9 (with static NP strain history via torsion straining) for typical FCC metals with different SFE, that is, commercially pure Al, Ni, Cu, and α-brass. Major trends observed in the present context include diminishing NP strain-history effect with increasing reloading strain rate for large SFE metals as Al and Ni, with, inversely, rather enhanced NP strain-history effect with increasing strain rate for small SFE metals as Cu and α-brass. The former can roughly be understood by the inertial unzipping of the high-velocity dislocation segments against the forests, where the probability of the occurrence of inertial unzipping is reduced with decreasing SFE that dominates the strength. The latter strongly implies a certain positive “coupling effect” between the NP strain history and the strain rate when the SFE is small enough. This seems to be due to the “positive” strain-rate effect in FCC metals pointed out first in Section 3.4.1, in the sense that dislocation populations to be interacted with the statically introduced forests are simply multiplied compared with those in static conditions. What is intriguing, in this respect, is that those for the

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Figure 3.4.11  Nonproportional tests for FCC metals with various SFE, with static pretorsion followed by static/impact tension, designated for examining the “positive” strain-rate history effect in the NP context.

BCC cases in Figure 3.4.9 basically do not show such a coupling effect, where the “negative” strain-rate effect is accompanied by reducing interacting populations but with unaltered interaction strength. For a quantitative comparison of these results, the net stress rise, measured by the difference between the observed and the estimated raised stress ratio, is correlated with the reloading strain rate in Figure 3.4.12(a). Here, the estimated values mean those calculated by the simple sum of the raised stresses solely due to the NP strain history and the strain rate, respectively. What is clearly demonstrated is the transition of the coupling effect from the “positive” to “negative” regions as SFE increases. If the slopes of the correlations are plotted against SFE, we have a linear relationship, as shown in Figure 3.4.12(b), meaning that such a coupling effect may be treated simply as a function of SFE, at least apparently. Combined with a series of TEM observations presented in Figure 3.3.13 and 3.3.14, although they do not relate specifically to the present experiments, we can safely conclude that the above complex-looking

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Figure 3.4.12  Flow-stress increase ratio as a function of SFE, comparing: (a) The four FCC metals with distinct SFE in Figure 3.4.11, together (b) with the stress rise ratio as a function of SFE.

strain-rate history–NP strain-history materials dependency in FCC metals are all represented ultimately via evolved dislocation-cell structures, which would make unified modeling much simpler and practically feasible, as will be argued in Chapter 5. Note that, including all these tests, a systematic series of NP dynamic and dynamic Bauschinger tests are listed in Appendix 3, where the results on FCC/HCP metals/ alloys with different SFE, together with some Al alloys, are given. Also note that the HCP metals chosen, that is, Ti and Zn, exhibit anomalous trends, respectively, that differ from both the FCC and BCC materials, possibly due to an additional mode of plasticity – “deformation twinning” – which, while interesting, is outside the scope of this book.

3.5

Dislocation Substructures under Hypervelocity Impact

3.5.1 Phenomenology Under hypervelocity impact, materials are significantly hardened (or strengthened) while a relatively small amount of plastic deformation is caused. This hardening is also dominated by substructures evolved inside the material. Figure 3.5.1 shows a map of pressure versus pulse duration, displaying the evolved dislocation substructures in the impacted Ni samples (Kuhlmann-Wilsdorf, 1999a). Note that under such high-velocity impacts, not only dislocation substructures but also twin structures are likely to be formed, even in FCC metals with relatively high SFE. Figure 3.5.2 shows a similar result to Figure 3.5.1 for Ni (Meyers, 1994), summarizing the evolved structures with many twins. Under higher stress with relatively longer pulse duration, the metal tends to yield distinct structures to that in Figure 3.5.1, where deformation twins tend to prevail over the observed region.

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Figure 3.5.1  Dislocation substructure map under hypervelocity impact for Ni as functions of

pressure and pulse duration (Kuhlmann-Wilsdorf, 1999a; Murr and Kuhlmann-Wilsdorf, 1978), where cellular structuring is emphasized. Adapted with permission of the publishers.

Figure 3.5.2  Dislocation substructure map for Ni under hypervelocity impact as a function of

pressure and pulse duration (Meyers, 1994), where deformation twins are frequently observed, in contrast to Figure 3.5.1. Adapted with permission of the publisher (Wiley-Interscience).

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Figure 3.5.3  Remarkable additional hardening for OFHC copper brought about by shock

loading in comparison with annealed counterpart (Meyers, 1994). Adapted with permission of the publisher (Wiley-Interscience).

Figure 3.5.3 compares stress–strain curves for a shocked OFHC Cu with annealed OFHC Cu under quasi-static tension (Meyers, 1994), where the amount of predeformation due to the shock loading for the former is taken into account in the abscissa. We observe a drastic strain-rate history effect with around 100 MPa flow-stress increase. This is also attributed to the rate-dependent evolution of the dislocation substructures. Figure 3.5.4 shows TEMs for dislocation substructures in deformed Cu under a 10 GPa shock pulse, with various prestrains (Gray, 1992). The sample with the smallest residual strain (less than 2%) shows a conventionally observed cell structure. The residual strains, on the other hand, can drastically change the substructural morphology into folded lamellar patterns. This means that the dislocation substructures themselves are also strongly history dependent. Figure 3.5.5 shows a highly deformed shear band and the corresponding substructure for polycrystalline Ta under a high strain-rate shear exceeding  105 s1 up to   910% (Nemat-Nasser, 2004). What is emphasized in the paper is that the highly heterogeneous dislocation substructure is in sharp contrast to the macroscopically uniform strain field within the shear band. That is, they are definitely “unobservable” from meso- and macroscopic scales, but nevertheless play crucially significant roles, as corroborated earlier. This is also an indispensable aspect to be kept in mind when tackling multiscale plasticity problems.

3.5.2

Strain Rate/Temperature Effect Revisited Let us review afresh the strain-rate (temperature) history effects, in terms not only of those described in Section 3.4, but also of the hypervelocity context. It is evident from the previous discussion that the evolved dislocation substructures serve manifestations

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Figure 3.5.4  Conspicuous effects of residual strain on shocked substructures of dislocations

for Cu, demonstrating an extended strain-history effect even in a hypervelocity deformation regime (Gray, 1992). Adapted with permission of the publisher (Springer).

Figure 3.5.5  Highly deformed shear band and corresponding underlying dislocation

substructure for Ta under extremely high strain rate over 105 s−1 (Nemat-Nasser et al., 1998). Adapted with permission from the publisher (Elsevier Science & Technology Journals).

of the history of how and to what extent a material has loaded or strained. This statement will remind readers of the concept of the MTS mentioned in the last portion of Section 2.7, where the MTS represents the net stress for the evolved dislocation

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Figure 3.5.6  Summary of the MTS model, together with an experimental data correlation based

on which the MTSs are evaluated (Follansbee, 1985).

substructures (at 0K, precisely). Figure 3.5.6 summarizes the MTS concept, displaying an experimental data correlation for FCC Cu (bottom), based on which the MTSs are evaluated by extrapolation to 0K (Follansbee, 1986), together with a set of schematic dislocation substructures supposed to be developed depending on the imposed strain rate (left), as well as the model (Eq. (2.7.10)) derived from the Arrhenius-type formalism (top; also see Section 2.7). What is noteworthy in the present context is the fact that the MTS is no other than an eloquent manifestation of the strain-rate-dependent dislocation substructures, putting aside the true mechanism for the rapid stress rise in the impact region. In other words, “the raised stress is accompanied by the commensurate dislocation substructure that supports it.” One tangible rearrangement of the experimental results that corroborates this, by Follansbee et al. who proposed the concept, is presented in Figure 3.5.7, where an MTS diagram is overplotted on Figure 2.7.5 (left). Since the MTS diagram involves results categorized in the range of the hypervelocity impact (Section 3.5.2), that is, under 10 GPa shock (Follansbee, 1986), the raised stress over   10 4 s1 regime (assumed to lie within the range 105 s−1 to 107 s−1) is concluded to be accompanied by commensurate residual-dislocation substructures. This may answer the problem about “whether the chicken or the egg comes

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Figure 3.5.7  Mechanical threshold-stress strain-rate diagram for Cu including data under shock

deformed at a shock pressure of 10 GPa (Follansbee, 1986), overplotted on flow stress–strain rate diagram (Figure 2.2.1: Bottom left). Adapted with permission of the publisher (Taylor & Francis Group).

first,” at least partially, for under such a condition there is almost no chance for the substructures to be fully evolved instantaneously, strongly implying that “the chicken came first” in the hypervelocity impact regime. From the previous discussion, the following scenario seems to be realistic. Dislocations that steeply accelerated under shock loading are heavily subjected to the viscous phonon drag resistance or other possible mechanisms, resulting in a sharp stress rise in the first place (“the chicken”). Under the greatly raised stress, dislocations are activated or newly generated if needed (particularly at shock fronts in the case of hypervelocity impacts), and then eventually or ultimately relaxed into lower (not always the “lowest”) energy configurations to yield the observed substructural morphologies of dislocations that take a little more time (“the egg”) as in Figures 3.5.2, 3.5.3, or 3.5.4. This means that what we evaluate are the MTSs that are commensurate with these later-developed substructures (indeed, they cannot be evaluated in practice until reloading tests are performed). For verifying the above surmise, strain jump tests in the strain-rate region exceeding  ´10 4 s1 would be effective. Sakino conducted such a series of experiments (sudden strain-rate reduction tests) covering 1  10 4 s1 to 4  10 4 s1 of strain-rate range in a systematic manner, combined with strain jump tests for 6061-T6 Al, which can be found in Sakino (2021). He reported that instantaneously reduced flow stresses are demonstrated to almost coincide with those under corresponding constant strainrate tests (Figure 3.5.8). If the substructure evolutions have taken place first, that is, before the deceleration, the flow stress ought to exhibit higher flow-stress levels. This

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Figure 3.5.8  Schematics of strain-rate reduction tests at a strain-rate range exceeding 104 s−1 (a), together with typical detected stress response (b), flow-stress response (c), and a summary of results (d), demonstrating that steeply decelerated flow stresses coincide with those under constant strain-rate data but not exhibiting a strain-rate history effect (Sakino, 2021). Adapted with permission from the publisher (JSMS).

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Figure 3.5.9  Measured flow-stress responses at very high strain rates of 104–105 s−1 as a

function of strain rate for (a) pure Al (5n) (Sakino, 2018) and (b) Fe-0.01%C (Sakino, 2020), exhibiting also the contrastive strain-rate dependencies over 104 s−1 between FCC and BCC metals/alloys. Adapted with permission from the publisher (JSMS).

means the substructure evolutions are not “the chicken” but should be “the egg,” that is, they are “born” later. There still exist, however, two deficiencies in using the phonon drag model for the microscopic constitutive equation; one is the negative temperature dependency, yielding larger flow stress at higher temperature, compared with experimental results (Sakino, 2003), and the other is the “much steeper” increasing rate of flow stress. For the former, one needs to also consider the temperature-dependent viscous coefficient, while the latter should additionally assume spatiotemporal accelerations of dislocation segments that bring about a sort of “differential effect,” which can resultantly lower the increasing rate in the macroscopic stress responses. The latter, of course, is not normally contained in the microscopic constitutive model itself, but requires appropriate spatial discretization for reproducing it. It is noteworthy that, as demonstrated in Figure 3.5.9(a) by Sakino (2018) on high purity Al(5n), the data in the strain-rate range from 104 to 105 s−1 exhibit a sharp enough flow-stress rise, largely deviated from the extended line via the thermal-activation mechanism, to strongly imply the activation of the phonon drag mechanism in that regime. What also should be kept in mind is the fact that these arguments are only true for FCC metals/alloys, and do not apply to BCC cases, in the sense they tend to exhibit a “negative” strain-rate history effect, as described in Section 3.4. To the best of my knowledge, however, to date no attempt has been made to discuss the MTS for BCC metals/alloys. Two things are at least expected in this case: (1) The inverse trend will result in the MTS (the ordinate in Figure 3.5.6), that is, smaller MTS at higher strain rate; and (2) the “scenario” ought to be revised, considering, for example, an assumption that only a fraction of the raised stress via mechanism other than Peierls overcoming can contribute to the substructure evolutions. The latter is taken into account when constructing the constitutive equation presented in Chapter 5. It should also be noted, as stated in Section 2.2.1 (see also Figure 2.2.2), that the thermal-activation mechanism

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applies even to the strain-rate range exceeding 104 s−1 for BCC metals, only with a transition of the detailed mechanisms from that of dislocation processes to the Peierls overcoming. This is corroborated by the “dislocation velocity versus stress” diagram Figure 2.3.3 in Section 2.3, and, further, direct experimental results for Fe-0.01%C reported by Sakino, (2020) shown in Figure 3.5.9(b), where the data reaching 105 s−1 strain rate are demonstrated to be located roughly on the extended line evaluated based on the thermal-activation mechanism by Aono et al. (1981) below 104 s−1. Data approaching the 107 s−1 shear-strain rate are reported in Huang and Clifton (1985) for OFHC Cu via inclined plate pressure-shear test (on plate or deposited layer specimen), exhibiting steeply increasing shear stress toward one order of magnitude, in a similar manner to those in Figure 3.5.9(a).

3.6

Effect of Surface: Surface versus Bulk Figure 3.6.1 demonstrates a distinction in the evolved dislocation substructures between the surface region and the bulk of a sample for a BCC metal (Fe under cyclic deformation at T = 77 K) (Ivanova and Terentyev, 1975). The micrographs clearly show that the surface regions have a propensity to exhibit conspicuous development of dislocation substructures in comparison with the bulk region. FCC metals, however, seem to have totally different trend, as shown in Figure 3.6.2 (Ivanova and Terentyev, 1975), where Cu under cyclic loading exhibits almost the same morphology of the evolved cell structure regardless of the observed position in the cross-section of the specimen, although this may not always be the case, as presented in the following. Fourie (1968) conducted a series of systematic experiments and attendant micrographic observations regarding the surface effect on the flow stress and dislocation substructures in conjunction with the size effect for a single-crystal Cu. Figure 3.6.3

Figure 3.6.1  Effect of surface on substructure evolution for BCC iron under cyclic loading at

a low temperature (−196°C), where notable differences between surface and bulk regions are shown (Ivanova and Terentyev, 1975).

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Figure 3.6.2  Effect of surface on substructure evolution for FCC copper under cyclic loading,

where basically no differences between surface and bulk regions are observed (Ivanova and Terentyev, 1975).

Figure 3.6.3  Effect of surface on single-crystal Cu under monotonic tension till the end of

stage II, together with stress distribution through thickness, showing the growth of a large flow-stress gradient toward the surface, which demonstrates a more rapid hardening rate in the interior than the surface (Fourie, 1968). Adapted with permission of the publisher.

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Figure 3.6.4  Effect of surface on substructure evolution for polycrystalline Cu deformed to 1%

strain (Miyazaki et al., 1979), associated with a systematic series of experiments with various t / dG (grain size/thickness) ratios, discussed in Section 4.5.2. Adapted with permission of the publisher.

summarizes the representative results. He observed a considerably large stress gradient extending below the surface, accompanied by an increase in the cell size near the surface and a decrease in dislocation density. Note that it has been pointed out that the mean moving distance of the edge dislocations plays a crucially important role in the size effect rather than in the screw counterparts. Figure 3.6.4 is a similar example, albeit for a polycrystalline Cu sample, but at an early stage of deformation of 1% strain, where dislocation substructures are not yet well developed. Miyazaki et al. (1979) conducted a systematic series of experiments by preparing various t / dG (thickness to initial grain-size ratio) to examine the effect of the specimen surface on the stress responses as polycrystalline aggregates (see Section 4.5.2 for further details). The surface region (a) tends to yield relatively inhomogeneous distribution of dislocations with some biased around

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the intergranular triple points, whereas ubiquitously distributed dislocations are observed in the bulk region (b). This basically implies the same trend should take place in polycrystal samples as in the single-crystal cases in terms of “surface versus bulk.” In conclusion, the effect of the surface on the dislocation substructure evolution, in close relation to the size effect, is undoubtedly important in the sense that it substantially controls the surface undulations or roughening developed during metal forming or working processes as well as crack-initiation events in fatigue, even if we confine ourselves to bulk-related problems, excluding those related to thin films. For a rational treatment and computational modeling, further extensive studies, including more systematic examinations into the set of accumulated data, would be necessary.

3.7

Dislocation Substructures versus Fatigue

3.7.1

High-cycle Fatigue Fatigue is a phenomenon in which cracks are initiated and gradually grow (propagate) within the materials under cyclic loading (stressing or straining), ultimately leading to failure (or fracture). In high-cycle fatigue, the cracks are usually initiated on the surface of the samples, from specific portions subjected to intensive slips, called “persistent slip bands” (PSBs). Distinctions between the surface and the bulk (interior), extensively discussed earlier, become particularly significant in this respect. The underlying dislocation substructure is known as a PSB ladder, shown in Figure 3.7.1 (Mughrabi et al., 1979). The slip bands are normally accompanied by two important phenomena, as summarized in Figure 3.7.1, that is, (a) intrusion/extrusion formations on the material’s surface (sometimes grown into protrusions), and (b) a specific dislocation pattern of a well-organized kind referred to as “PSB ladder” as the attendant internal substructure of the PSBs embedded within the matrix filled with another patterned dislocation distribution called “vein.” Further details of the PSB ladder structure are depicted by a series (set) of tangible schematic illustrations in Figure 3.7.2 (Okuda, 1985). An example of a DD-based simulation result corresponding to Figure 3.7.2 for the PSB ladder is presented in Figure 3.7.3, exhibiting bowed-out screw segments from the edge walls, moving back and forth, passing each other within the channel region. Also shown in Figure 3.7.2 is the corresponding surface, which tends to be subjected to roughening called “intrusion and extrusion,” ultimately acting as initiation sites of fatigue cracks in general. The intrusions/extrusions are often grown into bundles, referred to as “protrusions,” as evidenced in a series of snapshots displayed in Figure 3.7.4 (Ma and Laird, 1989a), with examples of a fully grown protrusion (after 4.5 × 104 cycles) extended over the sample surface. This is further superimposed by intrusions/extrusions on it (a), and a magnified view of such a PSB protrusion (after 1.2 × 105 cycles) (b) of about 30 μm wide, containing more than 20 lamellae of PSBs,

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Figure 3.7.1  Persistent shear-band ladder structure as the underlying dislocation substructure of

PSB embedded within a matrix called vein structure, showing a TEM image for a Cu crystal fatigued at γpl =10−3 at room temperature. Reprinted, with permission, from Mughbrabi et al. (1979).

Figure 3.7.2  Three-dimensional schematic drawing of PSB ladder and vein structures formed

during high-cycle fatigue at a limited range of strain amplitude. Both are composed of arrays or bundles of edge dislocation dipoles on a single-slip system (Okuda, 1985, p. 280, figure 15.7 and p. 281, figure 15.9). Adapted with permission of the publisher.

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Figure 3.7.3  Example of a simulated PSB ladder wall structure by the discrete DD method,

showing to-and-fro motions of bowed-out screw-dislocation segments in the channel region.

Figure 3.7.4  Propagating/growing protrusion observed in single slip-oriented single-crystal Cu

(Ma and Laird, 1989a). Adapted with permission of the publisher.

as shown in Figure 3.7.5 (Ma and Laird, 1989a). The most important thing here is the fact that all of these are ultimately attributed to the underlying PSB ladder structures (Figures 3.7.1 to 3.7.3), from which fatigue cracks are favorably initiated, as demonstrated in Figures 3.7.4(c) and (d) (Ma and Laird, 1989a). Vivid TEMs for developing and collapsing processes are shown in Figures 3.7.6 and 3.7.7, respectively (Tabata et al., 1983, p. 841). As shown, both the ladder and vein structures are composed of collections (or bundles) of straight-edge dislocation dipoles with a single Burgers vector extending in parallel to the primary slip plane ({111} in the case of FCC metals), and they are basically 2D substructures different

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Figure 3.7.5  Fully developed protrusions (a) and (b) on the surface of a single slip-oriented

single-crystal Cu, together with fatigue cracks initiated from the edge of protrusions (c) and (d) (Ma and Laird, 1989a). Adapted with permission of the publisher.

from the cell (structures) discussed earlier (they are extended in three dimensions in nature). Between the ladder walls are the veins, whose spacing ranges from 0.75 to 1 µm, and which are referred to as “channels,” through which the flow-carrying dislocations pass back and forth all the time during cyclic deformation. Since the bowing-out segments of dislocations between the walls have screw components, interactions among the components frequently take place within the channel regions. The passing stress for that process is considered to determine the critical stress τ cr at threshold. Mughrabi (1981) has discussed the role of the “passing stress” between two screw segments in terms of determining τ cr. Experiments focusing on the PSB ladder formation – plastic strain amplitude­controlled cyclic tests – have generally been conducted on single slip-oriented specimens, as illustrated in Figure 3.7.8, which mimic the corresponding grain facing on the surface of a polycrystalline counterpart under stress-controlled fatigue that is responsible for slip banding and the eventual intrusion/extrusion formations. Please notice the difference between the two kinds of hysteresis loops as in the insets. Figure 3.7.9 is a typical example of the hysteresis loops for the former (RSS τ R versus resolved shear strain γ R is shown here) for a FCC single-crystal sample oriented for single

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Figure 3.7.6  Transmission electron micrograph showing the development of a PSB ladder

structure, where a quite gradual emergence of ladder-like morphology from a matrix vein structure is observed (Tabata et al., 1983).

Figure 3.7.7  Example of the dissociation process of a dislocation wall structure, showing in situ

observations during recovery processes for [111]-oriented fatigued specimen to 7,000 cycles at 350°C: (a) Not annealed, and (b) annealed for 5 mins, (c) 10 mins, and (d) 13 mins (Tabata et al., 1983).

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Figure 3.7.8  Schematic illustration explaining the relationship between stress-controlled

high-cycle fatigue and a typical experiment targeting PSB ladder formation on single-crystal samples, which is generally conducted under a plastic strain-controlled condition against single-slip orientations, mimicking the deformation states in a grain on the surface of a corresponding polycrystalline sample.

Figure 3.7.9  Typical example of cyclic hysteresis loops in a plastic shear–strain-controlled test

for a single slip-oriented single-crystal specimen, eventually reaching a saturation stage.

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Figure 3.7.10  Schematic illustration of a cyclic stress–strain relationship obtained via a locus of the tips of saturation stress–strain curves, together with representative values of plastic shear– strain amplitudes and shear stress for the plateau regime for various FCC metals and alloys.

slip, showing rapidly growing cyclic hardening under plastic strain-controlled ( pl controlled) loading (Suresch, 1998). The hysteresis loop eventually reaches a saturation at N s cycles specified by a peak tensile stress τ s and an associated shear strain, γ ts. Figure 3.7.10 shows saturation stress–strain curves where those obtained under various levels of  pl are overplotted. By connecting the tips (a locus of the tips) of the overlapped hysteresis loops in Figure 3.7.10, we have a “saturation stress–strain curve.” Figure 3.7.11 presents a schematic drawing of the saturation cyclic stress– strain relationship,  sat   pl , together with the corresponding dislocation substructures to be evolved in each regime. We observe a plateau regime B, ranged between pl,A  B and  pl,BC , which yields PSB formation. Here,  pl, A  B is the minimum shear strain below which PSBs do not form (accordingly no crack initiation takes place, thus corresponding to the “fatigue limit” and  pl, A  B is labeled the “threshold”), while  pl,BC is the maximum above which multiple slip occurs, yielding cell structures. Within the plateau regime B, the plastic deformation of the tested sample is carried by to-and-fro motions of the ladder-composing dislocations. A labyrinth structure is observed in the transition regime from B to C, wherein the secondary slip system starts operating.

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Figure 3.7.11  Schematic illustration of a cyclic stress–strain relationship showing three regimes, A, B, and C, together with corresponding dislocation substructures, that is, PSB ladders embedded within veins (single slip), labyrinth (duplex slip), and cell (multiple slip) structures (Suresch, 1998).

It is noteworthy that the previous discussion is substantially affected by the crystallographic orientation. A standard triangle, together with representative orientations and the corresponding number of equivalent slip systems, is schematically shown in Figure 1.3.14. One can notice that orientations along the edges of the triangle yield multiple slip, and that single-slip orientations can be found within it. The PSB formation basically occurs in single slip-oriented grains, otherwise labyrinths or cells would be formed from the start.

3.7.1.1

High-cycle Fatigue for BCC Metals Distinct from FCC metals, described in Section 3.7.1, BCC metals/alloys seem to allow us to have no unified views around the fatigue processes (Suresch, 1998), especially in terms of dislocation substructures. Such complexity, observed in the BCC fatigue processes, is ultimately rooted in the “loose” atomic packing structure (see Sections 1.3.1 and 1.3.2), which then causes extremely high SFE (Section 1.2.5.3), lower screw mobility and associated tension–compression asymmetry (Section 4.2.1), and much stronger sensitivities to strain rate and temperature than their FCC counterparts (refer to Section 4.1). Although, to my knowledge, no systematic intercorrespondence between fatigue processes and the dislocation substructures has been revealed for BCC metals/alloys, some organic relationships do seem to exist, at least according to some systematically designated experimental observations, in the sense that the evolved/evolving dislocation substructures tend also to be highly rate/temperature sensitive, commensurate with the relatively complex cyclic stress–strain responses. Examples of such attempts follow. Eifler and Macherauch (1990) conducted a systematic series of stress– strain-controlled fatigue tests (under push–pull conditions) for a set of selected plane carbon/low-alloyed steels over a wide range of temperatures (Mughrabi and Christ,

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Figure 3.7.12  Variation of cyclic plastic strain amplitude with number of cycles for normalized SAE 4140 plane carbon steel under a stress-controlled push–pull test with  /2  345 MPa, f = 5 Hz, together with the correspondingly evolved dislocation substructures (Eifler and Macherauch, 1990). Adapted with permission of the publisher.

1997), including SAE 1010–1080 and SAE 4140 in normalized, quenched and/or tempered states (all are BCC ferritic steels with 0.1–0.8 wt% carbon content). Figure 3.7.12 shows an example of the results for normalized SAE 4140, conducted under stress amplitude-controlled conditions ( /2  345 MPa with f = 5 Hz), where the attendant variation of the cyclic plastic strain amplitude  p / 2 with the number of cycles is displayed, together with the correspondingly changing dislocation substructures. As demonstrated in the figure, after rapidly progressing cyclic softening, a peak is reached at N = N1 where dislocation walls are already formed, which eventually change into cell structures (N = N 2 ) associated with slowly growing cyclic softening, followed ultimately by cyclic hardening accompanied by well-developed cells (N = N 3). Here, in stress-controlled tests, cyclic softening/hardening manifests itself as the increasing/decreasing plastic strain amplitude  p / 2 (Eifler and Macherauch, 1990). Note that similar trends are also observed for normalized SAE 1045, implying no marked influence of the alloying elements of Cr and Mn on the cyclic deformation behavior. What is noteworthy is that these relatively complex cyclic deformation features are remarkably sensitive to the test temperature, as summarized in Figure 3.7.13, where cyclic deformation curves are displayed as a function of temperature, ranging from 20°C to 600°C, together with representative evolved dislocation substructures (Eifler and Macherauch, 1990). The plastic strain amplitude curves exhibit

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Figure 3.7.13  Summary of cyclic deformation curves for normalized SAE 1045 as a function of temperature, together with representative dislocation substructures (Eifler and Macherauch, 1990). Adapted with permission of the publisher.

quasi-elastic incubation periods before the onset of cyclic softening (20°C ≤ T ≤ 200°C) with rapid softening reducing toward zero plastic strain amplitude (250°C ≤ T ≤ 400°C), and anomalous cyclic hardening associated with the dynamic strain aging above 450°C. The attendant dislocation substructures are correspondingly diversified, from walls (50°C), cells (150°C), veins, and bundle-like structures (250°C), to large block cells at 600°C. It should be noted that, in the case of strain-controlled tests, the above cyclic softening/hardening appears as the inverse trend, that is, it manifests as decreasing/increasing cyclic stress amplitude. Two things may be tentatively pointed out based on this. One is that the evolved/ evolving dislocation substructures in high-cycle fatigue are apt to yield wider and much more diverse variations strongly dependent on the stressing/straining and temperature conditions than their FCC counterparts. The second is that such complexity seems to be rather mild compared with the kaleidoscopically haphazard stress–strain responses. The second perspective serves as an informative point of view for the FTMP-based multiscale modeling of materials (MMMs), which places special emphasis on the evolving inhomogeneity.

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Figure 3.7.14  Cell structures commonly observed near fatigue-cracked regions even for extremely

low SFE alloys, that is, (c) Cu-8%Al (aluminum-bronze) (Karashima et al., 1967) and (d) Cu-30%Zn (α-brass) (Karashima et al., 1968). Compare with Figure 3.1.19 (left and right) and Figure 3.1.20(d). Adapted with permission from the publisher (Japan Institute of Metals).

3.7.2

Low-cycle Fatigue Low-cycle fatigue has not been well documented compared with its high-cycle counterpart in terms of the relationship between the evolving dislocation substructures and fatigue-crack initiation/propagation, partially because of the complexity additionally introduced due to higher stress and larger strain states. So the description here will be somewhat biased, based on the limited range and amount of data in the literature. For low-cycle fatigue, unlike high-cycle fatigue, PSBs are not formed because of the nearly multiple slip conditions taking place almost everywhere in the sample from the start, from which the formation of cell structure or the like becomes rather dominant (see Section 3.3). Accordingly, fatigue cracks are rarely initiated from the surface, but tend to emerge from inside the bulk, probably associated, to a greater or lesser extent, with the evolved dislocation substructures. Furthermore, crack propagations may be affected more or less by the fluctuating stress fields possibly accompanying them. Let us look at a couple of examples that may give us some information about the relationship between cell development and fatigue-fracturing processes, both of which are obtained under “high”-cycle fatigue. Near fatigue-crack regions, even for high/extremely low SFE metals and alloys, can yield cell formations. Figure 3.7.14 compares dislocation substructures evolved near

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Figure 3.7.15  Refined cell structures observed in the vicinity of a fractured surface for a single-crystal Cu, where minimum size is about 0.1 μm. Also observed was that the width of striation coincided with the size of a few cells (Lukas et al., 1969). Adapted with permission of the publisher (Taylor & Francis Group).

fatigue cracks among for metals/alloys (Karashima et al., 1967), including Cu-30%Zn alloy (α-brass) and Cu-8%Al alloy (aluminum-bronze) (Karashima et al., 1968) as representatives of alloys with low and the lowest SFE. As one can see, they also show cell structures, although that for Cu-8%Al alloy yields rather fuzzy morphology. If they are compared with Figure 3.1.20, especially for (d), we find a totally different morphology: Unidirectional marked slip bands in Figure 3.1.20(d) versus relatively fuzzy but equiaxial cells in Figure 3.7.14(d). Figure 3.7.15 shows a cell structure near a fractured surface (Lukas et al., 1969). Significantly refined cells down to 0.1–0.2 µm are observed in the proximity of the fractured surface. This extreme refinement of the cells is probably due to the higher intensity of stress in the plastic zone around the crack, whose sizes are found to correspond to the striation widths. The role of the evolved cells over the fracture is worthy of further investigation along such lines. Another intriguing observation concerns the associated misorientation developments. Figure 3.7.16 plots the misorientation distribution measured away from the fracture surface, exhibiting relatively large intercellular orientation differences reaching 14°, rapidly decreasing toward a saturation at the plastic zone boundary. Since such rotation modes can be an effective energy reducer in the region, the result implies non-negligible contributions to the toughness, too.

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Figure 3.7.16  Misorientation change as a function of the distance from the fracture surface for a single-crystal Cu fatigued in Figure 3.7.15 (Lukas et al., 1969). Adapted with permission of the publisher (Taylor & Francis Group).

Figure 3.7.17  Dislocation structures observed in fatigued Al 1100-O: (a) Before fatigued, (b) 6 µm away from the fractured surface, and (c) 2.5 mm away from the crack tip (Grosskreutz and Shaw, 1967).

Another case is overviewed in Figure 3.7.17, which is on fatigued Al 1100-O (Grosskreutz and Shaw, 1967), also under high-cycle fatigue. Similar to the previous discussion, cell structures are well developed near the fracture surface compared to the distant regions away from the crack tip. In this case, however, the authors reported that no correlation between cell size and striation width was found. A similar but more

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Figure 3.7.18  Systematic series of TEM observations for a fatigue-crack region, as a function of distance from the crack tip, demonstrating the transition of dislocation substructures toward refined cells (Wilkins and Smith, 1970); (c) and (d) adapted with permission of the publisher.

systematic series of observations is summarized in Figure 3.7.18 (Wilkins and Smith, 1970) conducted on Al alloy (0.5%Mg, 0.004%Cu), from which we doubly confirm the strong crack-tip field-driven cell formations, where the crack tip is approached from (a) through (e). To be noted here are the subgrain-like cell formations, observed immediately behind the crack tip (c), accompanied by lower density than that in (d) as well as relatively large misorientation across the walls, providing evidence that recovery had effectively taken place near the crack-tip region, which can be a possible mechanism for promoting the cell formations in the vicinity. It may be an important question whether the cell determines the local stress leading to the crack initiation there or, conversely, the internal stresses intensified in the presence of the crack promote the cell refinements. It is indeed a question of whether the egg or the chicken comes first. (A similar thing happens in the context of MTS. See Section 3.5.2.) Perhaps they take place concurrently, exchanging information about the fluctuating stress and strain fields in between. A related investigation considers a linear relationship found between cell (subgrain) size and the stress-intensity factor (Ogura et al., 1975). As shown, the fractography has revealed that the spacing of the striations observed on fracture surfaces sometimes coincides with the cell size,

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Figure 3.7.19  Typical dislocation cells observed in fully fatigued Cu, as a terminal substructure representing fatigued states (Karashima and Kainuma, 1975). Reprinted with permission of the publisher (Maruzen).

implying a close interrelationship between the two, but sometimes do not. Although two rather controversial cases have just been described regarding the influence of the developed cell structures on the “crack propagation” via the associated striation morphology, there can be no doubt about their non-negligible effects on fatigue-failure processes. It is worth noting that fatigued metals tend to eventually reach well-developed cell structures regardless of the histories, ultimately resulting in failure. Figure 3.7.19 shows an example of the shellwork-like cell structure often observed in fully fatigued metals under relatively high-stress amplitudes (Karashima and Kainuma, 1975), which may be suitably referred to as “subgrain.” This type of dislocation substructure may safely be considered as a manifestation of the “end destination” for the fatigued metals. Given that viewpoints of this sort are both quite intriguing and informative, a systematic series of investigations combining experimental observations and multiscale analyses are eagerly expected.

3.8

Structural Stability and Mechanical Roles of Dislocation Cells Figure 3.8.1 is an intriguing as well as implicative experimental observation of a transition of dislocation substructures observed during a load reversal for polycrystalline Al, together with the corresponding macroscopic stress–strain response (Hasegawa et al., 1975). Once formed, the cell structure at point A is completely dissociated when the load is reversed to point C, followed by the reconstruction of a cell structure at D. The dissociation or collapse of the cell structure is considered to correspond to a plateau-like region on the stress–strain curve, accentuating the Bauschinger effect in this material.

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Figure 3.8.1  The construction, dissociation, and following reconstruction of dislocation-cell

structures during loading–reversed loading processes for polycrystalline Al (Hasegawa et al., 1975). Adapted with permission of the publisher (Elsevier Science & Technology Journals).

Although this result is quite informative, we should interpret it with caution in terms of material dependency. The material tested is Al with a high SFE, thus yielding the weakest dislocation interactions among FCC metals. Therefore, there may be a possibility that the result is peculiar to Al, whose cell structure might be easily dissociated. This provides another important problem about the stability/instability of dislocation-cell structures against, for example, mode changes in stressing or straining, including load reversal, because the experimental facts imply that the cells are “not” always stable, at least mechanically. Some literature has demonstrated the stability by showing no visible change of the morphology even under heating. So, here is an important question: “What determines the stability or, inversely, instability of the dislocation cells including the material dependency?” Figure 3.8.2 shows a similar example, but for IF steel (having BCC) under a change in the straining direction from simple shear to tension along the specimen axis (Nesterova et al., 2001), whereby a temporal stress rise upon the mode change is clearly confirmed on the macroscopic stress–strain curve, immediately followed by a drop and continuously following gradual relaxation to the referential level. The left side of Figure 3.8.2 shows the typical dislocation substructure obtained during

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Figure 3.8.2  Intersection of dislocation wall structures during mode change from simple shear

to tension for IF steel, bringing about an abrupt stress rise immediately followed by a drop and asymptotic response (Nesterova et al., 2001). Adapted with permission from the publisher (ASM International).

tension, while the right depicts the corresponding micrographs after the stress drop, exhibiting penetrations of newly introduced dislocation sheets through the intersecting previously developed wall structures. This observation implies the following sequential events brought about by the mode change: The intersections between the dislocation walls with mutually different directions cause the noticeable stress rise, but this is halted by the initial penetrations leading to the stress drop due to a structural instability. After the penetrations, relatively moderate interactions continue to take place until the new substructure prevails over the old one. Note that the overwriting phenomena under cyclic straining is provided in Figure 3.3.6, together with the attendant changes in the stress response. Descriptions of those substructural changes, that is, dynamic interactions such as overwriting and penetrations, and the attendant stress responses normally accompanied by “strain softening,” are quite difficult to achieve, and usually require elaborate modeling, which tends to be ad hoc. A successful description is one of the important goals of current multiscale plasticity. An attempt toward natural descriptions of such aspects will be presented in Chapter 13.

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Figure 3.8.3  Comparison of cell structures of Al before and after annealing (Humphreys and

Hatherly, 2004). Adapted with permission of the publisher (Elsevier).

Figure 3.8.4  Sudden removal of an LC junction during static recovery, leading to a collapse

of surrounding dislocation structures (Prinz et al., 1982). Adapted with permission from the publisher (Elsevier Science & Technology Journals).

Let us next look at the thermal stability of the cell structures. Figure 3.8.3 compares TEMs of a cell structure before and after annealing (2 m at 250°C) (Humphreys and Hatherly, 2004). The two micrographs clearly demonstrate that the cellular morphology is basically maintained even after annealing. Tangled and fuzzy dislocations of “redundant” kinds are cleaned up by annihilation or rearrangement into sharper boundaries with finite misorientation across the walls.

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Figure 3.8.5  Hypothetical process of glide collapse due to the removal of LC junctions, which

have pinned wall structures via frequent dynamic attacks of fluctuating dislocation flux under stress (dynamic recovery).

There are two pieces of important information about the dislocation cells to be noted from these observations. The first is that the dislocation cells are ­thermodynamically ­stable to a large extent (at least to the extent of keeping the cellular morphology unchanged). The second is that this is still not an energy-minimized structure, because of the ­misorientation growth that has taken place during anneal, which should be a consequence of an ­energy-release process. The latter implies that a certain amount of (non-negligible) excessive energy has been stored in the cellular morphology, in the cell interior regions, probably as a form of the long-range internal stress field (LRSF), since the excessive energy to be released is undoubtedly the elastic strain energy ­accompanied by distortion of the crystal lattice. The existence of the LRSF will be discussed in Section 3.9. Figure 3.8.4 uses a sequence of TEM pictures to demonstrate the sudden removal of an LC junction (b) and the following collapse of the surrounding tangled dislocations (c) during static recovery (Argon, 1996; Prinz et al., 1982). This has motivated A. S. Argon to propose a notion referred to as “glide collapse” as an alternative mechanism to the cross slip, which is considered to be responsible for the dynamic recovery (Argon, 2009). Figure 3.8.5 schematizes the process, where LC junctions pinning and thus stabilizing the dislocation wall, are removed by frequent attacks of dislocation fluxes, causing the eventual collapse of the wall. The collapse is expected to be immediately followed by a dissociation of the entangled dislocations around the junctions, possibly leading to a massive annihilation and/or rearrangements.

3.9

Long-range Internal Stress Field This section provides some evidence of the LRSF within cells and discusses its mechanical roles. As we have seen, much of the information about the history effects,

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Figure 3.9.1  Schematic drawing of a mechanism for producing an LRSF in a cell interior

region due to piled-up redundant dislocations (Mughrabi et al., 1986).

including strain, strain rate, and temperature, is reflected in the evolved cell structures. Consequently, most of the microscopic pieces of information are stored in the cell structure. Then a new question arises: How are they stored? I have postulated that the microscopic degrees of freedom, including the information about the histories, are stored as a form of LRSF in the cell interior region. I assert that the LRSF developed within the dislocation cells (in the cell interior regions) plays a decisively important role in the dislocation cell-formation process itself (in ultimately determining the morphology and size) as well as cell stability. The grounds for this will be detailed in Chapter 8, in conjunction with some simulation results for cell formation in Chapter 10. The existence of long-range internal stress has been pointed out and experimentally measured by Mughrabi et al. (1986). They also proposed a tangible model for it ­considering a [001]-oriented single-crystal Cu under tension, as depicted in Figure 3.9.1. Piled-up dislocations belonging to two inclined slip planes against the cell wall produce the resultant dislocations, with another resultant Burgers vector, with the third direction coinciding with the loading direction (or in parallel to the walls), thus yielding compressive stress field in the cell interior region. They proposed this “opposite-signed” stress field in the cell interiors was the source of the LRSF and also led to the production of back stress. Note that different models explaining the origin of LRSF are also found, for example, in Sedláček (1995). Recent examinations along this line can be found in Jackobsen et al. (2006) and Levine et al. (2011), providing more-or-less consistent results. Figure 3.9.2 indicates an experimental result by Mughrabi et al. (1986) measuring the relative change in the lattice parameter, ∆a / a, both in the walls and interiors of a cell via the X-ray line-broadening technique. It is verified that, as they theoretically predicted, the cell interior region exhibits a negative ∆a / a , meaning a “compressive” field even under tension, whereas the cell walls yield a positive value of ∆a / a.

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Figure 3.9.2  Variations of the lattice constant with stress measured by the X-ray line-

broadening technique, demonstrating the existence of a long-range stress field (oppositesigned stress field) in the cell interior regions (Mughrabi et al., 1986).

The stress–strain curves for cell-wall and cell-interior regions are schematically depicted in Figure 3.9.3 (left). The figure also shows a corresponding elasticity-based model based on micromechanics (right). In the latter, the long-range stress field can be expressed in terms of change in the shear modulus between the two regions, which serves as the basis for modeling it in the context of the field-theory-based modeling of cell formation in Chapter 10. As confirmed earlier, the LRSF within the cell interiors manifests itself in the opposite-signed stress field to the applied stress. This “opposite-signed stress field” has also been confirmed in a computer simulation based on DD (Devincre and Kubin, 1997). The result is shown in Figure 3.9.4, where a simulated dislocation configuration sliced in parallel to (111) plane (a) is indicated together with the corresponding RSS field on the plane (b) by black-and-white contrasts. The black contrast indicates the same sign of the RSS as the applied shear stress, while the white contrast is the opposite-signed RSS. The coarsely dislocated regions in the figure, for example the central region, exhibit white contrast (negative signed), whereas the dense dislocation regions are black (positive signed), agreeing with the previously mentioned experimental results. Associated simulation results of great interest to Figure 3.9.4 are presented in Figure 3.9.5, where the effect of “cross slip” on (a) the growing dislocation density and (b) the resultant stress–strain curves is shown (Kubin and Devincre, 1999), with the cross-slip probability off/on in their DD (dislocation dynamics) simulations for Al crystal labeled as “A/B.” It may seem to be strange at first glance since the cross slip should reduce the dislocation density ρ , responsible for “dynamic recovery,” but a larger increasing rate of ρ (a), and the corresponding higher stress level (b), are

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Figure 3.9.3  Schematic stress responses in a cell wall and cell interior regions (Mughrabi et

al., 1986) and a corresponding elasticity-based micromechanical model, where the long-range stress field is expressed in terms of a change in shear modulus between the two regions.

Figure 3.9.4  Example of simulation results based on DD, showing an opposite-signed stress

field in cell interior regions, which corroborates Mughrabi’s model for an LRSF, obtained under [100] stressing with constant strain rate at 300K for an Al single crystal of size 15 µm3 after strain γ = 0.003 (Devincre and Kubin, 1997). Reprinted with permission of the publisher (Elsevier).

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Figure 3.9.5  Simulated variations of (a) dislocation density and (b) stress response with strain,

comparing the effect of cross slip (Kubin and Devincre, 1999).

achieved, essentially due to substructuring toward “cells,” as in Figure 3.9.5(b). Here, “double cross slip” triggered efficient multiplications of dislocations that tended to act as the underlying mechanism. A saturation, on the other hand, is eventually reached without cross slip, since the system is incapable of exhibiting “multiplication,” thus provides much less dislocation “storage.” The LRSF existing in the cell interiors over at least a few micrometers is considered to be a direct reason for the substructures of that scale range to evolve so as to oppose or support the imposed stress in a concurrent manner, just like the hard coherent precipitates embedded in matrix metals. This is neither “geometrically necessary” nor has it been formed “incidentally,” but is rather “mechanically (or physically) necessary” and is formed “inevitably,” serving as a key element of the appropriate construction of the multiscale plasticity framework. This also can be an answer to the question about the lower limit of the cell size (0.1 m).

3.10

Dislocation Substructures under Finite Deformation

3.10.1 Phenomenology Under large deformation, like deformation under cold rolling, also known as finite strain/deformation, we have a much larger variety of substructures (tentatively named deformation structures) that evolve than under relatively small or moderate strain ranges as discussed earlier, because the effects of geometrical constraints accompanying the dimensional changes of the sample, coupled with the crystallography, considerably overlap the inhomogeneity evolutions. A typical TEM is presented in Figure  3.10.1, observed under 80% cold rolling for low-carbon steel (Doyama and Yamamoto, 1988), where we can confirm not only submicrometer-order cells elongated along the rolling direction but also some higher-order structures that divide crystal grains in a somewhat hierarchical manner.

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Figure 3.10.1  An example of a “deformation structure” via a TEM, observed in 80% coldrolled low-carbon steel (Doyama and Yamamoto, 1988). Adapted with permission of the publisher (University of Tokyo Press).

Figure 3.10.2  Schematic illustration of step-by-step development processes of deformation substructures observed under cold rolling, comparing two typical types, that is, those for (a) high and/or medium SFE such as Al and Cu, and (b) low and/or extremely low SFE such as α-brass and austenitic stainless steels (Higashida and Morikawa, 2008). Adapted with permission of the publisher (Iron and Steel Institute of Japan).

In short, active and sometimes dynamic interactions with the inhomogeneities in the upper scale (grain-aggregate order: Scale C) and lower scale (mechanically necessary types of dislocation substructures like cells: Scale A) tend to significantly enhance the complexity in the morphological aspects of the deformation structures, as schematically illustrated in Figure 3.10.2 (Higashida and Morikawa,

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Figure 3.10.3  Schematics showing hierarchical deformation structures in heavily deformed polycrystalline metals (Humphreys and Hatherly, 2004). Adapted with permission of the publisher (Elsevier).

2008). Therefore, it is important for us to regard most of the complexity observed therein as being ostensible and quite capricious, otherwise we will get stuck in (or be infatuated with) deformation structures’ endlessly emerging diversity. Note that extensive studies about the deformation structures have been conducted historically in the light of the subsequent recrystallization processes, for they are known to act as the nucleation sites of new crystals (Humphreys and Hatherly, 2004), and, in more recent years, in relation to the fabrications of UFG and/or nanocrystalline (NC) materials via severe plastic deformation, they tend to become the embryos of such new grains (see, for example, Altan, 2006; Lowe and Valiev, 2000). As stated, there has been some confusion among researchers regarding how to classify and name such hierarchically emerging “deformation structures,” especially for metals and alloys with relatively high SFE (termed collectively Cu type). Accordingly, we see attendant variations, together with similar schematics to that displayed in Figure 3.10.2(a), in the context of “grain subdivisions.” Figure 3.10.3 is one such example (Humphreys and Hatherly, 2004), and it emphasizes somewhat different aspects of deformation structures, termed “deformation bands.” The expression “deformation band” seems to encapsulate similar terms such as “matrix band,” “transition band,” and “MB.” A schematic that summarizes deformation structures, including them in rather general context schematized with respect to cold-rolled sheets, is shown in Figure 3.10.4 (Kawasaki and Matsuo, 1984), whereas one that emphasizes the “matrix band” and “transition band” is illustrated in Figure 3.10.5 (Doyama and Yamamoto, 1988), together with brief descriptions given by the respective authors. An example of a transition band observed in 50% cold-rolled polycrystalline Fe (BCC) (Dillamore et al., 1972) is shown in Figure 3.10.6, where elongated cells aligned in parallel are confirmed within about a 3 μm total width (2

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Figure 3.10.4  Schematics of “deformation structures” in the context of cold-rolled sheets (Kawasaki and Matsuo, 1984).

Figure 3.10.5  Schematic of “deformation structures” illustrating further detailed underlying structures of a “matrix band” and a “transition band” (Doyama and Yamamoto, 1988). Adapted with permission of the publisher (University of Tokyo Press).

to 5) that separates a matrix band (1 and 6), bringing a resultant misorientation about 30° across the transition band. In the moderate deformation stage we often clearly observe MBs and dense dislocation walls (DDWs), as indicated in Figure 3.10.2(a), which have been reported to develop under multiple slip conditions when more than two slip systems on the same slip plane are activated. Figure 3.10.7 displays afresh the schematics illustrating MBs and DDWs that delineate the cell-structured regions, together with a TEM for Ni at

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Figure 3.10.6  Example of a transition band (2–5) separating a matrix band (1 and 6) observed in 50% cold-rolled polycrystalline Fe (Dillamore et al., 1972).

Figure 3.10.7  Schematics of deformation substructures to be observed under finite strain, together with a TEM for Ni under torsion (εeq = 3.5).

 eq  3.5, which demonstrates some MBs crossing from upper left to lower right of the micrograph (links between the Mises equivalent strain ε eq and nominal strain for various conventional deformation processes are summarized in Table 3.1 [Tsuji, 2006]). Figure 3.10.8 displays another TEM for MBs, observed for cold-rolled Al (Hansen

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Table 3.1  Correspondence between equivalent strain and nominal strain for various conventional deformation processes. Nominal strain × 100% Equivalent strain (ε eq or ε )

Tension (e)

Compression (re )

Rolling (rr )

Extrusion or wire drawing (rA )

0.5 1 2 3 4 … 8 10

64.9% 172% 639% 1900% 5360% … 298,000% 2,200,000%

39.3% 63.2% 86.5% 95.0% 98.2% … 99.97% 99.995%

35.1% 57.9% 82.3% 92.6% 96.9% … 99.90% 99.98%

39.3% 63.2% 86.5% 95.0% 98.2% … 99.97% 99.995%

Shaded area indicates maximum strain available by them, above which special techniques for achieving severe plastic deformation become necessary (Tsuji, 2006).

Figure 3.10.8  Example of MBs observed in 15% cold-rolled Al (Hansen and KuhlmannWilsdorf, 1986). Reprinted with permission of the publisher (Elsevier Science & Technology Journals).

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and Kuhlmann-Wilsdorf, 1986), in which MBs’ underlying structures are revealed. According to Kimura (1998), MBs observed therein seem to be markedly refined and elongated cells, essentially the same as those discussed in the previous sections. If this is the case, MBs can simply be regarded as “modulated” dislocation cells as a consequence of their interactions with the intragranular inhomogeneous deformation fields (Scale B). DDWs are also understood, in the same way as MBs, to be banded regions organized into more intensely concentrated elongated cells. It is also worth noting, as was pointed out at the beginning of this subsection, that all these variations in the deformation structures should be regarded as more or less superficial, in spite of their apparent diversity transcending a relatively large scale range, in the sense that they are all formed geometrically necessarily. Based on this idea, they have recently been labeled GNBs by the research group at Risø National Laboratory led by Niels Hansen (2004), alongside MBs, DDWs, cell-band walls, lamellar boundaries, and so on, whereas dislocation cells are termed incidental dislocation boundaries (IDBs), as annexed to the schematics in Figures 3.10.1(a) and 3.10.7. I think that this is a wise as well as bold decision, like the so-called Occam’s razor cutting out annoying classification processes. This, at the same time, is of great significance in the multiscale modeling perspective, in the sense that it requires us to describe the essence of the various deformation bands that allows “naturally selected” substructuring, without relying on ad hoc models, such as those with complicated rule-based case divisions. The FTMP-based modeling schemes that address these items and some preliminary simulation results, combined with the conventional crystal-plasticity framework, are discussed in Chapters 5 and 11, respectively. An overview of the directly corresponding simulation results can be found in Figure 3.10.9, where hierarchically emerging deformation structures are confirmed to evolve spontaneously along with the progressing deformation. Here, an FCC single-crystal sample is compressed to 40% in (11̅0)[111] orientation with   2.0  10 3 /s. A series of dislocation-density norm snapshots and corresponding cross-sectional misorientation distributions are displayed in Figure 3.10.9, together with a tentative set of interpretations of the obtained deformation structures. We confirm that secondary-banded structures eventually started to grow, inside which tertiary substructures further emerged, each with own scale and misorientation ranges, roughly corresponding to matrix bands, transition bands, and MBs. Another feature worth mentioning is secondary banded structures’ relatively clear orientation dependencies in terms of their subtle morphologies. Figure 3.10.10 is a schematic illustration of the orientation-dependent grain subdivisions for FCC metals, accompanied either by banded or by nonbanded structures (Hansen, 2004), while Figure 3.10.11 arranges the experimental observations using a slightly updated classification of the three categories (Hansen, 2001; Huang and Winther, 2007). We confirm three representative orientation regimes in the standard triangle corresponding to three distinct structural morphologies, evidently attributed to the number of active slip systems. Similar but slightly different orientation dependence in the deformation

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Figure 3.10.9  An example of an FTMP-based CP (crystal plasticity)-FEM simulation exhibiting hierarchically emerging deformation structures, that is, dislocation-density counter and corresponding cross-sectional misorientation distribution.

Figure 3.10.10  Schematic drawing of orientation-dependent deformation structures observed in FCC metals, together with TEM images obtained in experiments. Comparison is also made with FTMP-based simulation results of dislocation-density tensor (norm) for horizontally pulled single-crystal models (courtesy of Dr. X. Huang).

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Figure 3.10.11  Experimentally revealed orientation-dependent deformation substructures (Hansen, 2001; Huang and Winther, 2007). Adapted with permission of the publishers.

structure in terms of the morphological aspects has also been confirmed recently for BCC metals such as pure Fe and low-carbon steels (N. Tsuji, 2005, personal communication). One of the prominent distinctions between GNBs and IDBs is the misorientation across boundaries and spacing, that is, the former always yields much larger values than the latter. Figures 3.10.12 and 3.10.13 compare evolutionary aspects of the misorientation and spacing between two structures along with the increasing Mises strain. IDBs yield misorientation of around 1° even at their maximum and spacing of about 1.0 µm. GNBs, on the other hand, exhibit greatly extended misorientation to the order of 10° while the spacing is reduced to a commensurate size with IDBs. As pointed out at the beginning of this subsection, these deformation-induced structures that have evolved in relatively large strains are generally attributed to the prescribed geometrical constraints, categorized, in my opinion, as Scale B, that is, intragranular inhomogeneity, which will be describable based on continuum mechanics-based kinematics. This will be discussed in Chapter 11. The similar grain size-dependent stress–strain responses for UFG samples between IF steel (BCC) and Al (FCC), as shown in Figure 3.10.14 (Tsuji, 2006), and the scaling laws found in (or holding for)   D (lamellar spacing) and    (misorientation) over a wide range of strain (Hughes, 2001) support this hypothesis (another universality: Geometry based).

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Figure 3.10.12  Transmission electron micrograph of GNB and IDB structures observed in 10% cold-rolled Al, together with a corresponding sketch specifying misorientations across GNBs (Liu et al., 1998). Adapted with permission from the publisher (Elsevier Science & Technology Journals).

Figure 3.10.13  Comparison of GNBs and IDBs with respect to strain-induced misorientation between the walls and the interwall spacing (Liu et al., 1998). Adapted with permission from the publisher (Elsevier Science and Technology Journals).

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Figure 3.10.14  Comparison of stress–strain curves on nominal basis between ARB (accumulated roll bonding)-processed IF steel and Al (Tsuji, 2006). The increasing number of cycles indicated in the graphs represent the number of rolling processes, roughly corresponding to decreasing grain size. Both materials exhibit quite similar stress–strain responses regardless of crystal structure.

3.11

Dislocation Cell versus Subgrain In Figure 3.11.1 a comparison is made between dislocation cells and subgrains, which have been often confused by researchers and in the literature. This is especially the case with regard to Al, which, because of its high SFE together with its comparatively lower melting temperature, yields a cell structure with sharper walls and small entangled dislocations of redundant kinds (see Figure 3.1.13 [left]). Hence, it is sometimes

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Figure 3.11.1  Comparison features between dislocation-cell structure and subgrains.

Figure 3.11.2  Schematics of a subgrain-formation process (Argon, 1996).

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Figure 3.11.3  Close-up view of well-equilibrated subgrain boundaries observed in Al-5at% Zn after creep, as well as after additional annealing at 523K, showing a process toward further well-organized networks of dislocations (Mekala et al., 2011) (high-resolution images courtesy of Doctor W. Blum and Professor P. Eisenlohr). Reprinted with permission of the publisher.

very difficult to differentiate cell structures from subgrains (the distinction is not always very evident). Figure 3.11.2 is a schematic illustration of a formation process of subgrains (Argon, 1996; Takeuchi and Argon, 1976), where the walls are constructed by well-aligned arrays of edge dislocations, accompanied by no long-range stress. Figure 3.11.3 is a micrograph showing well-equilibrated subgrain boundaries for an Al-Mg alloy consisting of dislocation networks; essentially, this is the lowest energy configuration.

Appendix A3  Some Comments on the Plasticity of UFG Materials A3.1

Effect of Specimen Geometry When evaluating the mechanical properties of materials, it is also important to consider the effect of specimen geometry on the measured stress–strain curves. Under monotonic ten̄ sion, we sometimes experience different stress–strain responses for the same materials, for example, between those responses obtained using a solid cylindrical specimen or a thin-sheet specimen. Such a difference becomes significant, especially in materials with higher yield strength and relatively small work hardening,

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Figure A3.1.1  Effect of specimen shape on the stress–strain curve for UFG copper, compared with that for annealed copper. UFG copper is fabricated by ECAP.

for these are very sensitive to instability characteristics. UFG materials are a case in point, as can be confirmed in Figure 3.10.14. Typical examples of the distinction will often appear at the onset of “necking,” causing apparent softening on the nominal stress basis. Figure A3.1.1 compares stress–strain curves of a UFG Cu (commercially pure Cu: C1020) between two specimen geometries – solid cylinder and thin sheet. Here, those for annealed Cu (conventional polycrystalline) are also compared. Both the UFG materials are fabricated based on equal channel angular pressing (ECAP), where the latter is machined from a cylindrical rod into the sheet shape after the process. One can observe the similar maximum stress level but with a totally different flow response. The former (cylinder) yields a plateau-like flow exhibiting 30–40% of strain (uniform elongation or ductility), while the latter (sheet) rapidly shows stress drop similar to the results in Figure 3.10.14 (produced by ARB). In the cylindrical specimen, a delay of the necking takes place and results in the relatively large ductility. Note that relatively small difference is observed for the annealed Cu, implying insensitivity to the specimen shape. This observation serves as a kind of cautionary note, especially with regard to interpretations of experimental data toward, for example, constitutive modeling and simulations.

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A3.2

229

Evolved Microstructures and Deformation Mechanisms Another thing I would like to mention is the mechanism of the plastic deformation for UFG materials in terms of the effect of the evolved dislocation substructures. To examine this, my group conducted a series of experiments under various strain rates and temperatures using ECAP-processed UFG Cu and Al. Figure A3.2.1 demonstrates strain-rate-dependent stress-strain curves for (a) annealed and (b) UFG Cu, also comparing the temperature effect. Similar strain rate/temperature dependence is confirmed for the two materials, but the salient differences between them are scrutinized below. A TEM study on UFG Al under static/impact tension (monotonic and proportional) has revealed two intriguing features. They are: (1) No substructural evolution is observed under either static or impact tension, with few dislocations within the grains; and (2) grain refinement or subdivision seem to take place under impact tension, which is absent under static loading. Figure A3.2.2 displays the TEMs of typical microstructures of UFG Al before and after the tests (static and impact), comparing with those for annealed Al. We can confirm evidently refined crystal grains within which no dislocations are included for the UFG sample after the impact test. These observations imply an essentially different plastic deformation mechanism for the UFG Al from that for conventional polycrystalline counterparts. To identify the mechanism, we also examined the activation volume with strain based on the strain-rate jump test (see Section 2.8.1 for details and examples). Figure A3.2.3 compares the activation volume as a function of strain and temperature (RT and 77K) with those for annealed Cu. Two important findings are derived based on the results; UFG Cu exhibits: (1) Around an order smaller activation volume than the annealed counterpart, and (2) an almost constant activation volume with increasing strain at RT, while there is a slightly decreasing trend under low temperatures (77K). The former indicates a distinct mechanism for the plastic deformation, probably via grain boundary sliding, while the latter implies a temperature-dependent mechanism.

Figure A3.2.1  Effect of strain rate and temperature on stress–strain curve for UFG copper, comparing with that for annealed copper. UFG copper is fabricated by ECAP.

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Figure A3.2.2  Typical microstructures of UFG (a), (c), (e) and polycrystalline aluminum (b, (d), (f); before tests (a, b), after static tensile tests (c, d), and after impact tensile tests (e, f) (courtesy of Doctor Miyamoto, Doshisha University, 2005).

Figure A3.2.3  Nonproportional strain-history effect for ECAP-processed UFG copper. Pretorsion straining followed by static tension is applied to a tubular specimen.

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Figure A3.3.1  Variation of activation volume with strain for ECAP-processed UFG copper, comparing temperature effect.

Particularly, the low temperature condition is assumed to inhibit grain boundary sliding, which tends to activate alternative mechanisms, ultimately causing, for example, grain subdivision or refinement, as observed earlier under impact loading. Note that if the rate-controlling mechanism for plastic deformation is thermal activation, a high strain rate is considered to be equivalent to low temperature (refer to Section 2.4 for the theoretical basis and Figure 2.8.12 for an example).

A3.3

Role of Dislocation Substructures If substructure evolutions take place during pretorsion straining, we expect to have additional hardening in the following stress response, either in static and/or impact conditions. Actually, for annealed Cu (conventional polycrystalline), we observe additional hardening, as shown in FigureA3.3.1(a). In sharp contrast, the UFG Cu exhibits no additional hardening, where the following flow-stress curve coincides with that without pretorsion, as in FigureA3.3.1(b). Therefore, we can conclude, at least indirectly, there is no substructural evolution and associated dislocation–dislocation interaction during plastic deformation in UFG Cu.

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4

Single Crystals versus Polycrystals

4.1

Complexities in Single Crystals Like the question about “FCC versus BCC” posed in Chapter 1 (see Section 1.3.2), there is another interesting question of a very fundamental kind that few might be able to answer clearly and appropriately: What is (are) the substantial distinction(s) between single crystals and polycrystals in terms of plasticity? Of course, secondary and tertiary factors, such as the effect of “textures,” should be excluded. This question is extensively addressed in this section. Historically, researchers have tried to seek a way (methodology) to estimate polycrystalline responses in terms of the stress–strain curve from corresponding information for the composing single crystals, not only in plasticity but also in elasticity (e.g., to estimate overall elastic moduli as a polycrystalline aggregate). Since this is, in a sense, simply a problem of “how to sum up” the elements, as history has seen only a sum has discovered as a result of such attempts. Along the same lines are studies on the effect of crystal-grain geometries (shape and size distributions), where so far, to my knowledge, no noticeable effect nor clear conclusions have been identified. Let us look at some phenomenology before going into details. Figure 4.1.1 shows schematics comparing single crystals with the corresponding polycrystals for Cu, LC steel, and Mg as representatives of FCC, BCC, and HCP metals, respectively (Kato, 1999). As a matter of course, single crystals are strongly orientation dependent. Single-crystal Cu exhibits a famous three-stage hardening, but we should recognize that this is peculiar to the single-slip oriented samples: The duplex or multiple-slip orientations yield rather similar stress–strain responses to that of polycrystalline Cu, as depicted in the figure. This is basically true also for HCP Mg. This example is eloquent of the substantial distinction between single crystals and polycrystals in terms of the effect of dislocation interactions, which can sensitively alter the single crystal’s behavior. Figure 4.1.2 demonstrates the orientation-dependent stress–strain responses for single crystals (after Honeycomb, 1968). Several representative crystallographic orientations are chosen (curves 5, 6, and 7) and a comparison is made among them. Also the grain-size dependency is compared for polycrystalline Al in curves 1–4. As is confirmed in the inset, 5 and 6 correspond to multislip orientations, that is, 111 and 001 , respectively. They yield no stage I, immediately followed by stage II hardening, unlike curve 7, which shows a typical three-stage hardening. On the other hand,

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Figure 4.1.1  Typical stress–strain curves for single crystals and polycrystals, comparing FCC,

BCC, and HCP metals (Kato, 1999).

Figure 4.1.2  Orientation-dependent stress–strain curves for single-crystal Al, where the

single-slip orientation yields the well-known three-stage hardening (after Honeycomb, 1968).

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Figure 4.1.3  Single crystal versus polycrystal showing typical stress–strain curves for FCC

and BCC single crystals at various temperatures temperatures (Berner and Kronmuller, 1965; Koda, 1993; Kumagai et al., 1990); Adapted with permission of the publishers (Corona Publishing Co., and the Japan Institute of Metals and Materials), respectively.

curves 5 and 7 yield no Stage I, showing Stage II from the beginning, and are quite similar to those for polycrystalline curves. What is of further importance is the complicated behavior observable in pure Fe single crystals (with BCC structure, i.e., α -Fe), which is extremely sensitive to strain rate and temperature, in addition to crystallographic orientation. This sensitivity stems directly from the complexity of the distinct dislocation behaviors between its edge and screw components, simply speaking, in terms of mobility. Figure 4.1.3(b) is a schematic diagram for temperature-dependent stress–strain curves of a single slip-oriented ultra-high-purity (UHP) Fe, comparing with those for single slip-oriented Al (both are single crystals) (Kimura, 1998; Kumagai et al., 1990). Readers can readily find the haphazard response with temperature in UHP-Fe, whereas FCC-Al yields mutually similar and systematic variation over a prescribed range of temperatures. For T = 300K, we confirm a stage I-like hardening response resembling that for UHP-Fe single crystals, followed by a faint but stage II-like linear region of hardening until stage III-like saturation is reached. For T = 333K, by way of sharp contrast, albeit at a slightly higher temperature than the previous case, the stress–strain curve exhibits “softening,” which is absent at T = 300K. Figure 4.1.4 provides more recent experimental results by another research group, for UHP-Fe single crystal (Suzuki et al., 2005), comparing the two temperatures, T = 300K and 333K. The new results basically agree with the previous observation, convincing us that the anomalous stress response is a universal feature of high-purity Fe or ferrite.

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Figure 4.1.4  Peculiar temperature dependence of the stress–strain response in high-purity

Fe. T = 300K (RT) yields slight work hardening, whereas T = 333K exhibits slight work softening (Suzuki et al., 2005). Adapted with permission of the publisher.

The “anomalous” phenomena can be explained by considering the large mobility difference between the edge and screw components of dislocations in BCC Fe, together with their strong temperature dependency, coupled with a distinction between primary and secondary slip activities (Kimura, 1998). Figure 4.1.5 schematizes the temperature dependency of the effective stress, comparing the edge and screw components, and the primary and secondary systems of slip. Note that, at a high enough temperature, the mobility of the edge and screw dislocations coincide (see the extreme right). This is expressed here by a linearly decreasing effective stress τ * with increasing temperature for every type of dislocation. Hence, there should exist intersections between the secondary edge and primary screw lines in between 300K and 333K, as shown in Figure 4.1.2. At a low enough temperature, say T = 250K, the application of stress will result, first, in movement of the primary edge dislocation, followed, in order, by the secondary edge and the primary screw, and will end with the secondary screw. Whether the multiplications of primary screw and primary edge occur first makes a difference. At T = 300K, multiplication of the primary screw takes place first, as a consequence of the interaction with previously activated secondary-edge dislocations, causing hardening due to the lower mobility of the screw components, whereas

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Figure 4.1.5  Schematic diagram of temperature-dependent effective stress for high-purity iron,

where distinctions among primary and secondary as well as edge and screw components are made, specifying dislocation activity.

T  =  333K yields activity of the primary screw before the secondary edge, causing multiplication of the latter, resulting in “softening” because of its much higher mobility. (The F–R source is activated, producing a mobile population.) Figure 4.1.6, on the other hand, draws schematics of dislocation loops for FCC and BCC, where the latter yields a significantly elongated loop shape because of much larger mobility for the edge components than that for the screw segment. Because of such a large difference in mobility for BCC, the dislocation loops will extend in the manner illustrated in the figure, leaving long screw dipoles behind. These screw dipoles will act as obstacles against the subsequent motions of the secondary edge and screw segment, which can promote the previously mentioned mechanism. This example eloquently demonstrates the complexities of single-crystal behaviors for pure iron: It seems that they are too complex to express mathematically in a compact form of equations. What is worse, the above response is extraordinarily sensitive to impurity levels, even of the ppm order. Because the addition of impurity atoms can completely change the dislocation behaviors, the stress–strain response of the system is also subject to drastic change. Regarding the previous discussion, we notice that single-crystal behaviors are much more complicated than those of their polycrystalline counterparts. Actually, most of these complexities are peculiar to single crystals. The corresponding polycrystalline samples tend to behave in a somewhat simpler manner, because most of the complexities cease to occur due to so many constraints when the composing single crystal grains are embedded within a medium (surrounded by many crystal grains);

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Figure 4.1.6  Comparison of dislocation-loop shapes between FCC and BCC metals.

what generally remains is v , the mean velocity of dislocations passing through the environment as the statistical response. This can more or less rationalize the statistical mechanics-based construction of the constitutive framework in polycrystalline plasticity, as is argued in Chapter 2. Let me remind readers of Zagoskin’s cartoon, shown in Figure 8.2.4, and provide a quote from him: “[an] infinite number of particles is almost as easy to handle as one, and much, much easier than, say, 3, 4 or 7” (Zagoskin, 1998, p. 2). This statement seems to be true also for polycrystalline plasticity, in a somewhat different context.

4.2

Some Comments on the Lower Mobility of Screw Dislocation in BCC Metals

4.2.1

Complexity in the Screw-Dislocation Core Historically, two mechanisms have been advocated to explain the lower mobility of screw dislocations in BCC metals. One is concerned with complexity in the core structure, while the other is based on the atomic geometry peculiar to the BCC structure. Here, the former is briefly mentioned, although this mechanism has been in dispute, especially in recent years.

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4.2.1.1

Complexity in the Core Structure in BCC Because of the looseness of the atomic packing and associated slip-system constitutions (see Section 1.3.2), the core structure of the screw dislocation in BCC metals can take many configurations. Two extreme cases have been recognized, that is, nonpolarized (with sixfold symmetry) and polarized (with threefold symmetry, dissociated either on the 110 or 112 planes), as schematically shown in Figure 4.2.1. This is ultimately conjectured to depend on the underlying electronic structures, that is, more precisely, those related to the unfilled d-orbitals, which have made the atomic structure for the transition metals BCC in the first place. Such complex core structures are considered to be one of the reasons for the lower mobility and the so-called non-Schmid effect, in conjunction with the twinning–antitwinning slip asymmetry, because they require nonglide components of the stress tensor for the dislocations to move. The extended core structure was first suggested by Hirsch (1960) and confirmed by being found in atomistic simulations by Vitek et al. (1970), who also invented how to express the core structure two-dimensionally – called a differential displacement map – as exemplified in the following. Figure 4.2.1 displays the two typical cases of the extended a/2 111 screw-dislocation cores via the differential displacement map, where the arrows indicate the relative displacements of two atoms along the dislocation line (vertical to the picture) scaled by b/3 (b is the Burgers vector). The left has a sixfold

Figure 4.2.1  Two extreme core structures in BCC metals, that is, nonpolarized and polarized.

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Figure 4.2.2  Examples of nonpolarized (V, Nb, Ta) and polarized (Cr, Mo, W) cores of

screw dislocation in BCC metals (left-hand row). Core structures of edge dislocation are also depicted for comparison (right-hand row) (Duesbery and Vitek, 1998). Adapted with permission of the publisher.

structure and is called “nonpolarized” (or equivalently “degenerated” and “unextended”), while the right yields a threefold structure (spreading into three 110 planes of the 111 zone) and is referred to as polarized (or equivalently “nondegenerated” and “extended”). Numerical examples of the polarized and nonpolarized cores of an a/2 111 screw dislocation are compared in Figure 4.2.2, between transition metals roughly classified into VB and VIB groups, together with the associated edge components (magnified 15 times for the visualization), simulated based on (static) atomistic calculations utilizing Finnis–Sinclair-type interatomic potentials (Duesbery and Vitek, 1998). Those in Group VB, such as V, Nb, and Ta, yield nonpolarized cores, whereas those belonging to Group VIB, such as Cr, Mo, and W, show a fully polarized counterpart. It is noteworthy that the latter is accompanied by non-negligible edge components, as in the right-hand column. To examine these, Duesbery and Vitek argued there was an interrelationship with the asymmetrized γ (generalized stacking fault energy) surface for those in Group VIB. As pointed out earlier, the transition metals are characterized by unfilled d-orbitals, where the vacant d(eg ) -orbital promotes BCC structuring, such as in Group VB, with p6 d 3(eg )s 2 or p6 d 4 (eg )s1 . The same is true for Group VIB, with p6 d 4 (eg )s 2 or p6 d 4 (eg )d1(t2 g )s1, but those in this group also include a d(t2 g )-orbital that tends to stabilize FCC structuring by spreading

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Figure 4.2.3  Schematics of an extended/polarized screw-dislocation core with a threefold

structure, responsible for a non-Schmid effect, further bringing about lower mobility and tension–compression asymmetry in the stress response.

toward 12 110 directions, which is anticipated to be more or less responsible for the above distinctions. A possible mechanism for lowering the mobility of a screw dislocation, with a core extended into three directions, is illustrated in Figure 4.2.3. For the dislocation to move, say, to the right, two of the extended portions must be constricted first, which requires additional shear stresses that are on the nonglide planes. Therefore, such a polarized screw core is likely to exhibit significantly lower mobility than the edge counterpart, and it will violate the Schmid law (Ito and Vitek, 2001; Vitek et al., 2004) as mentioned earlier (see also the inset equation in Figure 4.2.3). At the same time, the motion of the screw dislocation acts like a wedge, resulting in the “tension– compression asymmetry” observed in experiments. From the “polarity”-based viewpoint, this occurs as in Figure 4.2.4, where polarized and nonpolarized screw cores are schematized by a triangle and a clockwise windmill, respectively (Takeuchi, 2013). The polarized cores tend to exhibit “zig-zag” routes as they move, seeking adjacent stable configurations depending on the twinning/antitwinning directions. Since the “twinning” slip coincides with a clockwise rotation direction of the windmill (polarized core), it requires lower resolved shear stress to move than the antitwinning slip, leading to plastic asymmetry. It should be noted that Edagawa et al. (1997) revealed that a simple line-tension model exhibiting Peierls overcoming by a kink-pairing mechanism can reproduce

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Figure 4.2.4  Schematized simulation results indicating asymmetric movements of

screw-dislocation cores depending on their polarized/nonpolarized nature, combined with twinning/antitwinning directionalities (Takeuchi, 2013).

this plastic anisotropy (tension–compression asymmetry) well, which clearly demonstrates that the core structure (polarized/nonpolarized) does not always matter. Combining with another more intrinsic mechanism for low mobility, shown in Section 4.2.1.2, the mechanism under discussion seems to be less plausible at present, although such symmetry/asymmetry core structures can, of course, greatly affect detailed motion, particularly those leading to cross-slip frequency of screw dislocations in BCC crystals.

4.2.1.2

Intrinsic Atomic Structure in BCC Another theory for explaining the low mobility attributes it to the atomic configuration in the BCC structure itself and, thus, is more intrinsic (Suzuki, 1967). In contrast to Section 4.2.1.1, which discusses its material dependency, this theory should hold in common to all materials with BCC atomic lattices, regardless of the polarity of the screw-core structures. Figure 4.2.5 schematizes the situation that BCC crystals encounter when a screw dislocation exists. As the screw core moves to the right, the surrounding three atomic rows (specified by an arrow C → A → B ) must align at the same time to minimize the interatomic spacing, simply for geometrical reasons, as illustrated in Figure 4.2.5 (bottom right). This situation maximizes the potential energy. The

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Figure 4.2.5  Atomic structure of BCC crystal and displacements of atoms at a

screw-dislocation core.

other case, indicated by B → A → C , on the other hand, maintains the interatomic spacing, causing no energy rise (bottom left). The advantage of this theory over the others (based on the extended core, described earlier) is its applicability even to metals yielding a nonpolarized screw core. Recent fully ab initio calculation has revealed that the screw core of α -Fe is not extended, having a sixfold (isotropic) structure (Chen et al., 2008; Frederiksen and Jacobsen, 2003). Even so, α -Fe exhibits very low mobility of the screw dislocation. The extended core theory cannot explain this case.

4.2.2

Core Structure of Screw Dislocation in α-Fe The Fe atom has an  Ar  3d 6 4 s 2 electronic structure, where the unpaired as well as spin-polarized electrons in the unfilled d-band play substantial roles in generating the magnetism and, consequently, the BCC structure (α -Fe: Ferrite) at ambient temperatures. The magnetic force based on the 3d 6 electrons produce the covalent-like directional bonding that stabilizes the BCC structure. Therefore, at a high enough temperature (T > 911°C) under which the 3d 6 electrons can form pairs, the FCC structure (γ -Fe: Austenite) is reestablished, because metallic bonding of the spherically symmetric kind reflecting the s-outer-shell electrons becomes dominant.

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Figure 4.2.6  Electronic structures and associated crystalline structures in Fe.

(Above A4 the dissociation of the spin takes place due to thermal fluctuation, yielding once more the BCC structure, δ-ferrite.) Figure 4.2.6 summarizes the relationship between the electronic structures of Fe and the associated crystal structures. On the other hand, Cu has a fulfilled d-band ( Ar  3d 8 4 s1), meaning that the valence electrons have spherical symmetry reflecting the s-band, leading to an FCC crystal structure. So we may say that Fe is not an inborn BCC metal but a sort of “acquired” or “enforced” one, different from others like Mo, Nb, and Cr, while Cu is a perfect FCC metal. Let us take an intriguing example which is considered to be closely related to the previously discussed nature of Fe. Consider a case where Cu atoms are added to the screw-core regions of Fe. Based on the above recognition, it can be postulated that if the s-band electrons in Cu atoms have a certain non-negligible effect on the d-band electrons in Fe responsible for its magnetism (thus for the BCC structuring) at the screw-core region, the core symmetry will be subject to change. An example of the ab initio-based simulation results are displayed in Figure 4.2.7, where the addition of three atomic rows of Cu is demonstrated to totally alter the screw-core structure of α -Fe, that is, from nonpolarized to fully polarized. Moreover, such core structural change is shown to be accompanied by a “meta-stable” configuration under shear stress. For details, refer to Chen et al. (2008). Note that these results are a part of an ongoing project, discussed in Chapter 9.

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Figure 4.2.7  Example of ab initio-based screw-core simulation for α -Fe, with and without a Cu cluster added to the core (Chen et al., 2007).

4.3

Effect of Active Slip Systems on a Yield Surface Predictions of the stress–strain response of polycrystalline aggregates based on those of the composing single crystals have partially focused on yield surfaces. A famous approach for the yield locus prediction, both for textured and randomly oriented BCC polycrystal metals based on crystalline plasticity, can be found in work by Logan and Hosford (1980). This section does not intend to give a review of such attempts, but tries to examine some implicative results in the present “SC versus PC” context. The accuracy of the strain-rate effect assumed in the constitutive equation greatly influences the shape of the yield surface. Abe et al. (1996) conducted a systematic series of crystal plasticity-based point-wise simulations for polycrystalline FCC metal to evaluate the yield surface by utilizing the power-law-type rate-dependent constitutive model (Hutchinson, 1976; Pan and Rice, 1983), that is, 1



 ( )



1

  ( )    ( )  m  a ( )  ( )   ( )  , (4.3.1)  g   g 

where m is the strain-rate sensitivity exponent. The rate-independent version can be regained at m → 0. Abe revealed that m tends to act as a parameter controlling the activity of slip systems, that is, the number of active slip systems increases with increasing m, as shown in the inset in Figure 4.3.1. The difference in m can noticeably

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Figure 4.3.1  Simulated yield surfaces based on crystalline plasticity using the power-law-type

rate-dependent constitutive equation, with the strain-rate sensitivity exponent being regarded as a parameter to control the slip activity (Abe et al., 1996). Adapted with permission of the publisher.

change the shape of yield surface, that is, from that close to the Tresca type to the Mises type as m increases, as depicted in Figures 4.3.1 for polycrystal cases and 4.3.2 for single crystals with representative orientations. The rate-dependent activity of slip systems is helpful for us when revisiting which constitutive equation should be used in multiscale simulations, for example, the strain-rate effect must be precisely taken into account even for static loading conditions (see also Section 2.9). Figure 4.3.3 summarizes some averaging schemes; from the simplest Taylor model to FEM via a self-consistent model based on micromechanics which satisfies both the stress equilibrium and strain compatibility. A comprehensive overview of this line can be found in Takahashi (1999). One of the well-known versions along such lines is the Kröner–Budiansky–Wu (KBW) model. Takahashi concluded that the FEM results yield a close stress response to the KBW model (Takahashi, 1999). It is noteworthy that a fairly recent FEM-based study (Okumura and Ohno, personal communication, 2005) has revealed that a softer response tends to result in finer mesh divisions in each grain, even for the same number of composing grains and the same orientation distribution. This is probably attributed to the enhanced intragranular accommodation, albeit still stiffer than reality. This may probably

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Figure 4.3.2  Simulated yield surfaces in a 2D principal stress space, based on crystalline

plasticity using the power-law-type rate-dependent constitutive equation for three representative orientations (right-hand row) and comparing to those for single crystal (left). Here the former is presented as a function of rate sensitivity parameter m, regarded as that for controlling the slip activity (cf. Figure 4.3.1) (Abe et al., 1996). Adapted with permission of the publisher.

be due to the lack of appropriate underlying degrees of freedom responsible for dislocation-substructure evolutions (in Scale A). In a more realistic view, each grain must be heterogeneously deformed so the deformation may be compatible among others, as schematically shown in Figure 4.3.4. The overlaps are compensated via GN dislocations (Ashby, 1970), which can generally enhance the hardening in the overall response. Evidently, the approaches discussed earlier lack this viewpoint. Such a “correction,” however, is not sufficient for capturing the appropriate substructural degrees of freedom, thereby we have further accommodations against such intruded local deformation. As is discussed in Chapter 11, sufficient accommodations can even result in a “softening” of the overall response.

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Figure 4.3.3  Overview of conventionally attempted polycrystalline plasticity in order to

achieve rational evaluation of the overall stress–strain response (Takahashi, 1999). Adapted with permission from the publisher.

Figure 4.3.4  Comparison between virtually uniform deformation modes of individual crystal

grains and accommodated counterparts when embedded in a polycrystalline sample where GN dislocations are introduced (Ashby, 1970).

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4.4

Conventional Studies and Coarse-Grain Plasticity This subsection highlights some crucial aspects in studies on coarse-grained samples that discuss polycrystalline plasticity, as shown by the samples displayed in Figure 4.4.1 (Teodosiu et al., 1992), and, furthermore, a crucial drawback of the conventional crystal plasticity-based finite element method (CP-FEM, hereafter). What should be kept in mind when using coarse-grained samples, referred to as oligo-crystals, is tangibly manifested in a statement by Stuwe (1997, p. 10): “Surface grains feel no constraint on one side. They are softer than grains in the interior as has been shown in various experiments. This should be kept in mind when considering ‘oligo-crystals’ like those treated in the chapter by Teodosiu.” The sample he mentions in this statement is shown in Figure 4.4.1, in a form of FEM mesh, which was incorporated with a CP constitutive model for pure Cu (Teodosiu et al., 1992). As will be shown later, if the sample is Al, such a coarse-grained sample may behave rather similarly to the polycrystalline counterparts, because metals/alloys with relatively large SFE tend to yield a small affected zone (see discussions in Section 4.5.2). This inversely means that, in the previous case, which is concerned with pure Cu that has relatively small SFE, the composing grains tend to behave differently from those in the “polycrystalline” counterparts, implying they do not always provide the same responses. If this is the case even in the grain interiors, the responses will differ even more greatly at the GB regions, which are frequently major targets of this type of study. Regarding CP-FEM itself, it is undoubtedly a powerful tool for simulating the precise elastoplastic behavior of metallic materials, but, as shown later, it has no intrinsic

Figure 4.4.1  Typical coarse-grained sample in a FE mesh version, termed oligo-crystals,

together with comments by Stuwe (1997) (Teodosiu et al., 1992). Adapted with permission of the publisher.

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Figure 4.4.2  Example of simulation results on a coarse-grained sample, compared to

experimental observations in terms of slip lines (Teodosiu et al., 1992). Adapted with permission of the publisher.

ability to describe evolving substructures. Figure 4.4.2 compares a simulation result for the model displayed in Figure 4.4.1 with corresponding experimental observations of the deformed sample, exhibiting marked slip-line traces (Teodosiu et al., 1992). Satisfactory agreement between the two results seems to be achieved; however, what ought to be recognized is that the simulation does not explicitly reproduce the slip-line patterns itself. What Teodosiu et al. (1992) elaborated for such a visualization is the finite element (FE)-wise plots of the three most active slip systems, each with a length proportional to the accumulated glide in the system. Namely, what we see therein are not “slip lines” of discrete kinds, as in the experiment, but rather a “discretized” contour of slip-system activities. Inversely speaking, such “averaged” fields of local plasticity can be well represented by CP-FEM, in general. A similar but more recent example can be found in Zhao et al. (2008), where 3D surface-deformation oligo-crystal Al is targeted, utilizing both high-resolution electron backscatter diffraction (EBSD) measurements and up-to-date CP-FEM simulation in 3D. It should be noted that they “intentionally” used the oligo-crystal sample for examining the surface effect in their study. Figure 4.4.3 displays the FE model (a) and an example of the deformed state in simulation (b). Good agreement was

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Figure 4.4.3  Investigation of an oligo-crystal sample, combining 3D CP-FEM simulation and high-resolution EBSD measurements: (a) An FE model, and (b) a deformed state in simulation, comparing axial surface strain between simulation and experiment (Zhao et al., 2008). Adapted with permission of the publisher.

found in the axial surface strain between the simulation and experiment, as demonstrated in (c) of Figure 4.4.3, whose result is essentially commensurate with that in Figure 4.4.2. What is noteworthy here is the intragranularly developed substructures, shown in Figure 4.4.4, accompanied by banded/lamellar-shaped misorientation distributions in roughly two scales, that is, of the orders of 1,000 and a few to several 10 micrometers, respectively. These are measured experimentally via EBSD with 5° resolution, with the latter reflected also in the {111} pole figure (top right) as much larger scatter than in the simulation. Not all of them are reproduced in the simulation, leading the authors to state in conclusion that this is “beyond the capabilities of CP-FEM models” (Zhao et al. (2008), p. 2296). A noteworthy fact is that the above “drawback” of the CP-FEM in reproducing deformation-induced intragranular band-like structures is not the problem found with “mesh resolution.” Eloquent evidence of the latter is presented in Figure 4.4.5, where ultrafine mesh divisions are introduced in a seven-grained sample under tension. No substructure is formed even under severely compressive plastic straining, a result that stems from the lack of underlying degrees of freedom of necessity in the conventionally employed constitutive models. This problem is revisited in conjunction with Figure 3.10.7, as well as in Chapter 11. Also to be noted in the present context is the effect of the shapes of the composing grains on the macroscopic stress–strain curves. A number of attempts to scrutinize the morphological effects of composing grains have been conducted by researchers, including this author (cf. Chapter 12), from the square and/or regular hexagonal shapes to those fabricated based on Voronoi tessellation, and even those accurately copied from the targeted micrographs for modeling their realistic

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Figure 4.4.4  Misorientation developments in a grain (grain #6 in Figure 4.4.3) revealed via high-resolution EBSD scanning, together with comparison of {111} pole figures between those in experiment and simulation (Zhao et al., 2008). Adapted with permission of the publisher.

morphologies. What we expected was something similar to the tangible differences that are caused by morphological aspects of the composing grains, where they deviated from the simplest square or hexagon, on the stress responses, for example, due to the attendant inhomogeneous stress–strain distributions in local plasticity. The results, however, almost always tended to end up with a discouraging output in the sense that the obtained stress–strain curves basically coincide. Although some differences are, of course, detected given the detailed observations of the stress/strain contours to a certain extent, they tend to be “averaged out” from the macroscopic responses. One of the systematically designated studies that nicely demonstrates this is found in Terada et al. (2006), although their main target was to find the source of the H–P relation. To achieve this, they first examined the effect of grain shapes on the stress– strain response, as summarized in Figures 4.4.6, 4.4.7, and 4.4.8. Note that, in their study, the homogenization method was utilized, assuming the periodic boundary conditions in all directions of the samples, so that the surface effects are eliminated. Quite eloquently, essentially no difference appears among regular hexagonal, greatly flatted hexagonal, and realistic grain shapes, based on which the authors tentatively concluded that the heterogeneities in the grain-aggregate level do not contribute significantly to the H–P effect. A candidate feature of “polycrystalline plasticity” beyond these perspectives is extensively discussed in the following, and clarified in Chapter 12.

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Figure 4.4.5  Ultrafine meshed CP-FEM under compression (Watanabe and Terada, personal

communication, 2006) (courtesy of Doctor I. Watanabe and Professor K. Terada).

Terada et al. attempted to discover the origin of the H–P relation, which is a sort of macroscopically observed scale effect, in two ways using microscopic views, that is, by introducing 1 / dG -dependent and GN-equivalent strain-gradient terms in the hardening law, respectively. They revealed that the former raises the yield stress level as a matter of course, while the latter mainly contributes to the increase in hardening rate, which had been widely recognized previously. This is informative, especially in the present context, in the sense that such macroscopic stress responses, including the H–P effect, seem to be dominated more by intragranular phenomena rather than those related to the “polycrystalline aggregates.” A more direct approach can be found in a series of studies led by Berbenni (Figure 4.4.7) (Perrin et al., 2010; Richeton et al., 2009), where an extended version of Eshelby’s inclusion model is utilized to take the effect of discrete slip lines into account, consequently introducing nonuniform strain distributions within individual grains, which are mimicked by a sphere, as schematized in the inset of Figure 4.4.8 (above right). What motivated these researchers were the experimentally observed intragranularly evolved slip lines that tend to emerge on the surface of some crystal grains of polycrystalline samples (“spatial heterogeneity” of plastic flow, in their

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Figure 4.4.6  The effect of grain shapes on the macro-stress–strain response (Terada et al.,

2006) (courtesy of Professor K. Terada and Doctor I. Watanabe).

Figure 4.4.7  An attempt to take intragranular heterogeneity into account. An extended

Eshelby’s model is proposed, to assume stacking dislocation loops within a sphere-shaped crystal grain that allows nonuniform plastic strain within (a), leading the model to express a grain-size-dependent stress–strain response, as in (b) and (c) (Richeton et al., 2009). Adapted with permission of the publisher.

words). They constructed a sort of direct model that simply mimicked them, as schematized in Figure 4.4.7(a), where dislocation loops are assumed to be piled up discretely against a spherical-shaped GB region, in a layered manner with spacing h. As displayed in Figure 4.4.7(b), the polycrystalline analysis using the model can express the grain size-dependent stress response, that allows an “H–P-like” relation to be reproduced (Figure 4.4.7(c)). Furthermore, when the spacing “h” is adjusted for the experimental observation for the Ni they tested, the “spatial heterogeneity” in a grain can be roughly reproduced, as shown in Figure 4.4.8. Here, comparison of the cross-sectional misorientation distributions in a targeted grain are made with a

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Figure 4.4.8  Related attempt to Figure 4.4.7 expressing intragranular misorientation

distribution via the extended Eshelby model, where comparisons among cross-sectional distributions are made (Perrin et al., 2010). Adapted with permission of the publisher.

constant and the adjusted spacing h, demonstrating that the “adjusted h” case yields a better estimate in the sense that it reproduces the misorientation gradient in the GB region well. It should be noted that, even with such elaborate modeling, the finer orientation fluctuation, corresponding to the slip-line patterns argued in relation to Figure 4.4.2, “cannot” be described. There is thus a definite limitation of the CP-FEM, considering all the above. Another area in which a number of researchers have been and still are studying concerns the “hierarchy” from a single crystal all the way up to a polycrystalline assemblage, through which they explore the “key” to understanding “polycrystalline plasticity” (Phillips, 2001; Randle, 1994); that is, (a) a single crystal, (b) a bicrystal containing one GB, (c) a tricrystal with three GBs and a triple point, (d) a bamboo crystal, (e) a coarse-grained sheet metal, and (f) a bulk polycrystal. Most of the researchers seem to get stuck at (b) or (e), never reach (f), and only end up with purely macroscopic mean-field approaches, such as those displayed in Figure 4.3.3, or else extended versions of purely phenomenological potential-based classical plasticity. A way to break through this stagnation is desperately needed. This will be revisited in Chapter 12, where some advanced views are discussed.

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4.5

Collective Effects of Grain Aggregates as Additional Features

4.5.1

Implications: Granular Materials

255

In Section 4.4, a dangerous aspect of using “oligo-crystals” and the like was pointed out when discussing genuine polycrystal plasticity. This can be paraphrased that something is missing in the finding associated with “ologo-crystal” plasticity. Then, what on earth would be missing there? One may readily raise a question about “collective effects,” possibly brought about by a “large number” of grains (another “collective behavior” in MMM to that for dislocations, to be referred to as Scale C in FTMP). Assuming, for example, a 1 × 1 × 1 cm 3 polycrystalline sample, of the size of a sugar cube, which is composed of cubic crystal grains of 10  10  10 m 3 , we can instantly calculate the number of crystal grains in the cube to be 1012 (( (10 2 )3 / (10  10 6 )3). With such large number of grains, it would not be surprising that there are some special features that cannot be captured by the averaged properties. Since polycrystalline forms are quite common in most metals and alloys, we ought to identify what are those in this context are like. Let us consider this from a somewhat different point of view, that is, granular media. Figure 4.5.1 provides an example of visualized stress chains via a polarizing plate based on photoelasticity (Behringer, 2008; Geng et al., 2001; Hayakawa, 2003).

Figure 4.5.1  Visualized stress chains grown from the bottom of a container via the

photoelasticity technique, together with model discs mimicking granular assembly ((a) and (b)) and (c) their self-restructuring to cope with the application of a pointed load. Adapted from (Geng et al., 2001) Adapted with permission of the publisher.

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Photoelasticity discs are statically situated two-dimensionally in a container, mimicking a granular medium, as depicted in Figure 4.5.1(a). A corresponding stress state is seen in Figure 4.5.1(b), where stress chains are visualized as developed (grown) from the bottom, like dendrites. This observation clearly indicates that the pressure caused by the particles’ weight against, for example, the bottom surface of the container, is not quite uniform but is greatly fluctuating. The averages of the two, of course, should coincide. Note that, in recent years, such aspects have been termed “load chain,” in conjunction with atomistic simulations. Figure 4.5.1(c) shows the effect of point loading on the stress-chain structure. It is clearly demonstrated that the stress-chain structures are reconstructed so as to oppose the imposed load concentration. What is of importance here is that the effect of such a localized load tends to be redistributed over the assembly, that is, “dissipation” or “dispersion” of local stress spontaneously takes place in a context-dependent manner. It is worth reminding readers that this example is a “static” case, where the fluctuation effect would be still smaller or the minimum in comparison with those under dynamic conditions. In flowing particles, for example, the fluctuation will probably be significantly enhanced, due to dynamic effects, including complex interparticle interactions (i.e., the interactions would be nonlinear). Since these “fluctuations,” coupled with “dynamic effects,” are strongly nonlinear phenomenon, unpredictable responses of the medium can take place as a result. Figure 4.5.2

Figure 4.5.2  Photograph capturing a critical moment of a collapsing silo due to unexpectedly

large stress fluctuation.

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Figure 4.5.3  Example of a simulated stress-chain structure (Bardet, 1994), where contact forces

are visualized. Reprinted with permission of the publisher.

is one such example, showing a shocking photograph of a collapsing grain silo, probably due to the strong stress fluctuations caused by the inhomogeneous stress transmission (along the stress chains) coupled with the dynamic effects of the interacting grains. The structure would have been designed conservatively based on structural mechanics for the walls to endure the pressure from the granular assembly afforded a sufficient margin, but probably not taking account of the “fluctuating” nature at all, which might have been an “undreamt-of” event for the engineers who designed the structure. Numerical studies on discrete element (particle) methods have also revealed a similar chain structure (Bardet, 1994; Iwashita and Oda, 1999) as shown in Figure 4.5.3. In the figure, contact forces under isotropic stressing are depicted by the thick lines, while the contact lengths are overplotted by thin terminated lines. We find a similarity between the stress chain visualized experimentally in Figure 4.5.1(b) and the numerical results in Figure 4.5.3, although the imposed stress states differ from each other. A more familiar example of such stress chains will be found in the situation illustrated in Figure 4.5.4(a). Here, a small sand mountain is about to be held down from above by a hand – something the readers probably have experienced in childhood

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Figure 4.5.4  Stress-chain formation in a sand mountain in compression acting as a

“load-transmitting structure,” and the dual “deformation-transmitting structure” due to further deformation (a), together with graph theory-based representation as a good contrast case (b).

when playing in the sand. Your hand will soon feel a sort of solid resistance against the developed stress chain, indicated by the networks in the figure, termed here “load-supporting structure.” By pushing further, you may deform the sand mountain a little more without collapsing it, but deformation tends to take place along dual “deformation-transmitting structures” as if avoiding the stress chain structure. Such mathematical or numerical treatment is available based on the “graph-theoretical” representations, as shown to contrast in Figure 4.5.4(b), whose details are given in Section 6.6 in relation to the DG field theory (Chapter 6), and will be revisited in Chapter 12 in the context of polycrystalline plasticity. The questions here are: “Shouldn’t we consider such things also for polycrystalline aggregates?” and “What would be the difference between the gross deviations of the fluctuating stress field (assuming this exists) and of conventional polycrystalline plasticity?” (They are basically already “normalized” by averaging out the microscopic pieces of information, including those that should not do.) If non-negligible fluctuations exist, the polycrystalline media contain, even at the macroscopic or mesoscopic level, quite inhomogeneous states of stress and strain as forms of “fluctuation,” especially in space, as predicted in Figure 12.1.1. That is, there exist “highly stressed” grains forming stress-supporting structures (SSS) and “highly strained” grains mainly carrying the plastic flow (FCS or flow-carrying structure) of the sample. A significant feature of this viewpoint is not only SSS but also its dual FCS as

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the breakdown of the fluctuation, and, furthermore, they are mediated by enhanced “rotation” structures. The answers to our questions will be given in detail in Chapter 12, including the role of “rotation.” It is also worth noting that basically the same thing is taking place in dislocation systems, leading to the role of “dislocation cells,” detailed in Chapter 3.

4.5.2

Implicative Experimental Results Regarding the effect of the number of composing grains, there exists a quite implicative experimental finding. Miyazaki et al. (1979) conducted a systematic series of experiments and obtained a number of valuable pieces of information. They prepared samples with various thickness-to-grain size ratios (t / dG) for metals, including Al, Cu, Cu-13%Al alloy, and Fe (commercially pure Fe), and conducted tensile tests on them. Variation of stress with t / dG is shown in Figure 4.5.5 for (a) Al, (b) Cu, (c) Cu-13%Al alloy, and (d) Fe, where t is the initial thickness of the specimen and dG the initial mean grain size. Small t / dG (≥1) means a small number of crystal grains contained in the specimen, especially in with regard to the through-thickness direction.

Figure 4.5.5  Results of a systematically designed series of experiments for identifying the

effects of the number of grains and absolute size on the macroscopic flow-stress response for various materials (Miyazaki et al., 1979). Adapted with permission of the publisher (Elsevier Science & Technology Journals).

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Normalized Stress

Single Crystals versus Polycrystals

Normalized Stress

260

1.0

Al

0.5

G * = 180m m

20% 10% 5%

0

Yield Stress

0

5

1.0

Cu

0

10

Normalized Stress

Normalized Stress

0.5

Cu-13at%Al 20% 10%

0

G * = 40mm

Yield Stress

0

10

20

30

40

50

Specimen Thickness / Grain Size

Yield Stress

10 15 Specimen Thickness / Grain Size

Specimen Thickness / Grain Size

1.0

G * = 65mm 20% 10% 5%

0.5

0

5

1.0

0.5

0

Fe suy sly

G * = 25mm

0

10 20 30 40 50 Specimen Thickness / Grain Size

Figure 4.5.6  Results as for Figure 4.5.5 but on a normalized basis, clarifying the size of the

affected zone, which depends heavily on materials (Miyazaki et al., 1979). Adapted with permission of the publisher (Elsevier Science & Technology Journals).

The stress yielding a small value at smaller t / dG range gradually increases with increasing t / dG until saturation is reached for all the materials. The ratio of the saturation (maximum) stress to the minimum stress at around t / dG = 0 doubles (100%). We can then confirm a variation of the transition range in t / dG for saturation with materials. Al exhibits saturation already at t / dG = 5, meaning multigrained samples will behave like polycrystalline samples with sufficiently many grains. Also the grain-size dependency is confirmed. Smaller grain yields larger dependency of the stress response on t / dG . Figure 4.5.6 is the normalized version of Figure 4.5.5, which emphasizes the effect of t / dG and its material dependency as just described. Figure 4.5.7 provides a schematic image of the constraint effect on the grain-wise deformation behavior between small and large t / dG (i.e., coarse-grained sample and fine-grained sample, respectively) (Miyazaki et al., 1979). For the coarse-grained sample, a grain at an arbitrary place (e.g., A0 ) can relatively freely deform, affecting or being affected by only its immediate neighbors (in other words, not affecting, e.g., grain B0), whereas, in the fine-grained sample, the effect of a deforming grain (e.g., A0) can influence remote grains (e.g., B0) due to mutual constraints. The latter situation means that there exists a sort of “affected zone” extending over grains.

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4.5  Collective Effects of Grain Aggregates as Additional Features

261

Figure 4.5.7  Schematic drawing of mutual constraints in polycrystalline aggregate, comparing

coarse-grained and fine-grained samples (Fujita and Miyazaki, 1978). Adapted with permission of the publisher (Elsevier).

Figure 4.5.8  Evaluated affected zone size as a function of grain size comparing various

materials (Miyazaki et al., 1979), Adapted with permission of the publisher (Elsevier Science & Technology Journals).

Figure 4.5.8 compares the affected zone, which measures the remote effect, among the four materials as a function of grain size. The affected zone here is evaluated from the above experimental results coupled with a series of numerical calculations (Miyazaki et al., 1979). Cu and Cu alloy show strikingly large grain-size

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262

Single Crystals versus Polycrystals

Figure 4.5.9  Summary of the results in Figures 4.5.5 and 4.5.6, demonstrating the effect of the

number of grains on the macroscopic stress response and attendant remote effect, manifested as the finite size of the affected zone.

dependence (rapidly decreasing the affected zone with dG) as well as a relatively large affected zone. Al, on the other hand, shows the opposite trend to this, that is, a small affected zone and small grain-size dependence on it. Fe exhibits a relatively large affected zone with small grain-size dependency. These results imply, with caution, that at least for Cu and Fe, the coarse-grained samples possibly behave quite differently from their polycrystalline counterparts, thus are not always representatives of them. This will make the determination of the representative volume elements (RVEs) difficult for polycrystalline simulations, which can be much larger than one expects. Figure 4.5.9 summarizes the above results. Again, emphasis should be placed on the experimental fact that the number of composing grains and their sizes have strong effects on the mechanical response of the sample. Not only single crystals and bicrystals, but also multigrained samples with coarser grain sizes will not be appropriate models for corresponding polycrystalline aggregates, at least in terms of the stress response. These features of polycrystalline plasticity are revisited in Chapter 12 from a field-theoretical viewpoint within the context of the “collective behavior” of grains. Based on the observations thus far, we can hypothesize some unique features of the polycrystalline aggregates, as depicted in Figure 4.5.10, which will be substantiated in Chapter 12. There, the mechanism responsible for the “remote effect,” expressed as the “affected zone” in the

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4.5  Collective Effects of Grain Aggregates as Additional Features

263

Figure 4.5.10  Hypothetical schematics for polycrystalline plasticity in the light of collective effects of composing grains – to be discussed in detail in Chapter 11.

preceding text, will be also elucidated based on it, in addition to the new feature, that is, fluctuating fields in potential inferred from Section 4.5.1. I included this chapter, titled “Single Crystals versus Polycrystals,” in the “Fundamentals” part of the book because I regard this topic, which has tended to be overlooked so far, to be fundamental to an appropriate understanding of the multiscale aspects of plasticity.

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Part II

Theoretical Backgrounds Description and Evolution 5 6 7 8

Overview of Field Theory of Multiscale Plasticity Differential Geometrical Field Theory of Dislocations and Defects Gauge Field Theory of Dislocations and Defects Method of Quantum Field Theory

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5

Overview of Field Theory of Multiscale Plasticity

The hierarchical nature of materials requires us to tackle related scientific and engineering problems from a diverse range of viewpoints, extending over broad scale levels, that is, from microscopic levels (quantum mechanics, atomistics, and dislocations) to macroscopic levels (grain aggregates, continuum media, and structures) via mesoscopic levels (dislocation ensembles and substructures, metallurgical microstructures, and crystal grains). Although many difficulties seem to exist in taking into account all of these and solving the problems related to them, we will notice that most of the complexities attributed to the evolutionary aspects of various inhomogeneities appear in multiple scales. Therefore, a theory that can directly deal with the “evolving inhomogeneity” can be a suitable framework for our purposes. What the FTMP provides is such a theoretical framework. This chapter intends to overview the FTMP in terms of the key concepts (keywords), the basic theories, and the fundamental hierarchical recognition (i.e., the identification of important scales). These will be followed by the introduction of several new features that I have found and introduced afresh. Practically, the theory is applied using the crystal-plasticity formalism-based framework as a tentative and convenient vehicle. So, the constitutive framework, together with some detailed models for the evolution equations therein, are also presented in the present chapter, that is, strain-gradient terms for the dislocation-density and incompatibility tensors.

5.1

Field Theory of Plasticity Since the most important concept in the FTMP is threefold, the theory is formally (or virtually) represented by a triad of the three subconcepts, that is, three key features, three basic field theories, and three important hierarchical scales. The triad thus produces a sixth-rank tensor in 6D space (Figure 5.1.1) if each is represented by a tensor-like expression. That is,



FTMP      , (5.1.1)

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Overview of Field Theory of Multiscale Plasticity

Figure 5.1.1  Multifold images of “field theory” and “FTMP.”

with

   Evolution, Description, Cooperation      Differential Geometriical FT, Gauge FT, Quantum FT  ,    Scale A, Scale B, Scale C   each of which is “not” vector but tensor in nature, as will be presented again in the following. Here, Λ represents the three key features associated with the construction of the field theory, that is, “evolution,” “description,” and “cooperation,” while Ω stands for three basic field theories forming a “field triangle,” that is, “DG field theory,” “gauge field theory,” and “quantum field theory (QFT),” and Σ refers to

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5.2  Images of “Fields”

269

three important hierarchical scales, that is, Scales A, B, and C (see Chapter 9 for details). As the index notation, we express the above as FTMPijklmn   j  kl  mn , (5.1.2)

where

 11  j    21   31

12  22  32

13   11   23   kl  21  31  33 

12 22 32

13   11   23   mn  21  31 33 

12 22 32

13  23  . (5.1.3) 33 

Note that they are “not” symmetrical in general. An example of the explicit form of Σ or Σ mn represents the matrix for the incompatibility terms in the interaction-field formalism, as given in Eq. (13.2.37) in Section 13.2.8, which is generally asymmetric.

5.2

Images of “Fields” The image of the “field” in the present context is multifold, as schematically shown in Figure 5.2.1. The most primitive image is the dislocated field as an elementary excitation of a crystalline (atomic) field (background field given a priori), which is “quantized” like a vortex line. This essentially is the same image as the particle picture in QFT. Major differences are in the dimension (1D particle in normal QFT versus 2D string in the current context), and the nature of the background field, which cause strong interaction with the elementary excitation of a tensorial kind. Including such interaction with the background field (which is an elastic field), the field representing the elementary excitation can be expressed as a gauge field introduced via the inhomogeneous broken translational symmetry of the Lagrangian functional of an elastic medium. The continuum mechanics-based counterpart of such a field is the plastic distortion tensor, the curl of which gives the definition of the dislocation-density tensor, further corresponding to the torsion of a differentiable manifold. Similarly, the other types of defects, including disclinations, are expressed as a gauge field with locally broken rotational symmetry, corresponding to the curvature tensor in DG language. The continuum mechanics-based description of it is the incompatibility tensor, given as the double curl of strain tensor. Note that, in principle, the latter is always reduced to (torn into) a “dislocated field,” which is called “tearing of the torsion tensor” in differential geometry (Kondo, 1960), while the curvature tensor field has broader meaning than the torsion field. A more generalized and even advanced image of the “field,” on the other hand, is the “inhomogeneous” fields, not limited to the dislocation/defect fields described earlier; this concept can be extended to evolving inhomogeneities of general kinds. The notion is applicable to all the scales of significance, that is, dislocation substructures (Scale A), intragranular heterogeneities (Scale B), and grain aggregate-order inhomogeneities (Scale C).

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5.3

Ingredients of Field Theory

5.3.1

Three Key Features



   Evolution, Description, Cooperation  The present field theory is substantially intended to deal with “inhomogeneously evolving” fields in terms of stress and strain (or deformation). Figure 5.3.1 identifies the three main concepts of the field theory denoted by Λ in Eq. (5.1.1), that is, (a) evolution, (b) description, and (c) cooperation of the fields. They are concerned with the following, respectively. a. How and why do the inhomogeneities (field fluctuations) in respective scales emerge and evolve? b. How can thus emerged or evolving inhomogeneous fields be mathematically expressed or described? c. How and why do the evolving inhomogeneities mutually interplay or cooperate? Conversely speaking, “field theory” in the present context must answer the above questions appropriately. Obviously the conventional framework of “theory of plasticity,” including its crystalline-plasticity versions is almost useless, whereas numerical techniques by themselves are quite insufficient and still powerless to deal with and simulate the above aspects completely. To this end, at least, one would readily notice that we are required to access or invent totally new perspectives, some of which might not exist in previous research. This book, especially in what follows, makes an attempt to provide the answers in a unified manner, based on field theory.

Figure 5.3.1  Three key notions of FTMP.

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5.3  Ingredients of Field Theory

5.3.2

271

Three Basic Field Theories    Differential Geometrical FT, Gauge FT, Quantum FT  Figure 5.3.2 represents the major ingredients supporting the framework of well-​ documented field theories, that is, three fundamental field theories, represented as Ω in Eq. (5.1.1), forming a “field triangle.” They are 1 . Differential Geometrical Field Theory. 2. Gauge Field Theory. 3. Method of QFT. Their details are provided in the following chapters, that is, Chapters 6, 7, and 8, respectively. Some descriptions are additionally given in Section 5.4. Here, we overview each field theory, its interrelationships, and its connections with the three key features Λ. The interrelationship among the three field theories is also indicated in Figure 5.3.2. The DG field theory (1), founded on the celebrated theoretical framework known as “non-Riemannian plasticity,” and constructed by K. Kondo (1955), is responsible for (b) the description of the inhomogeneously evolving or evolved fields. Based on the formalism, two tensors, that is, torsion and curvature, give a complete description for the geometrical aspects of inhomogeneity in any scale. Furthermore, by considering the relative deformation between any two arbitrary scales, one may look into the interactions among different scales from a geometrical viewpoint. One of the driving forces for field evolutions would be the “collective” or “synergetic” effects normally brought about by the extremely large number of the interacting elementary units, for example, dislocations. Such aspects can be rationally dealt with by (3) the method of QFT, thus partially yielding (a) evolution. Both theories, (1) and (3), which are essentially independent, can be bridged by another field theory, that is, gauge field theory (2). The gauge theory for dislocations and defects was constructed by Kadić and Edelen (1983) based on the Yang–Mills formalism (Yang and Mills, 1954),

Figure 5.3.2  Field triangle, consisting of three “field theories” as the bases of FTMP.

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Overview of Field Theory of Multiscale Plasticity

while the QFT was successfully applied to the many-electron system to rigorously derive the Ginzburg–Landau (GL) equation (which is basically a phenomenological theory) responsible for the superconductivity close to the critical temperature from a microscopic BCS theory (Gor’kov, 1959). In the latter, the starting point is to write down the Hamiltonian for many-electron systems, including their pair interaction. The gauge theory gives a mathematically sound basis for the dynamical aspects of the DG quantities in (1), provided the correspondences between the two theories are assumed, for example, torsion and curvature tensors in (1) and gauge fields for dislocated and defected states in (2). More importantly, the theory describes how the applied/stored elastic strain energy in the targeted system should be converted to the dislocation/defect degrees of freedom in the mechanically equilibrium states. This process is generally irreversible, thus representing “field evolution.” On the other hand, the gauge theory also provides a point of departure for the use of QFT (3), for it tells us how to write down the gauge-invariant Lagrangian in an unambiguous manner, from which we rewrite the corresponding Hamiltonian. What remains is (c) cooperation, whose treatment is, in a sense, most the important but, at the same time, most difficult portion of the theory construction. However, one of the bases can be provided based on (1), termed “interaction fields” as mentioned earlier (details are given in Section 6.7). Also, stability-related topics are discussed in Chapter 13, but will require further elaboration. Some potential keys or clues for solving this final puzzle are introduced in Chapters 13 and 14 as “outlooks.” Chapter 15 introduces a “flow-evolutionary law” that is expected to embody the notion of “duality” responsible for the evolutionary aspects of inhomogeneous fields in general. Several application examples are also presented in that chapter.

5.3.3

Three Important Hierarchical Scales    Scale A, Scale B, Scale C  Figure 5.3.3 illustrates the three key scales that are critically important for creating a successful multiscale plasticity model, indicated as Σ in Eq. (5.1.1), that is, Scales A, B, and C, further details of which are given in Chapter 9. In order to capture the essence of the multiscale plasticity-related phenomena, at a minimum these three scales of inhomogeneities have to be considered. They are the orders of (A) dislocation substructure (substructure-order inhomogeneity), (B) crystal grain (intragranular inhomogeneity), and (C) aggregate of crystal grains (transgranular inhomogeneity), all of which can be mathematically interrelated via the interaction formalism to be constructed in detail in Section 6.7. Roughly, their individual roles can be expressed as “reservoir,” “absorber,” and “regulator,” respectively. To understand the secret of the hierarchical structure of polycrystalline plasticity in particular, the significance of two key hierarchies, (A) and (C), must be emphasized. Both relate to the “collective effects” of the respective elementary units, that is, dislocations and crystal grains, respectively. The former leads to substructure evolutions such as dislocation cells, whose phenomenology was presented in Chapter 3

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5.3  Ingredients of Field Theory

273

Figure 5.3.3  Three key scales for successful multiscale plasticity modeling based on FTMP.

(the theoretical treatment will be dealt with in Chapter 10 based on (3) method of QFT, to be detailed in Chapter 8). The latter has not been well documented thus far; I attempted to get to the heart of the problem in Chapter 4 and will tackle this again, from the field-theoretical point of view, in Chapter 12. Scale B, on the other hand, affects both Scales A and C, and, at the same time, is affected by them, thus acting as the “absorber” or “generator” of additional inhomogeneities. Such roles and features will be clarified in Chapter 11. Note that, practically, the best use of Scale B has been in controlling mechanical properties, for example, by alloying and introducing metallurgical microstructures. Figure 5.3.4 is an example of the type of “time versus space” scale diagram frequently found in papers and articles mentioning “multiscale modeling of materials (MMM)” in more-or-less identical versions. As one can see, this graph simply lists current concepts or theories, neither providing or even implying any information about “scale-bridging” ideals. By looking at this picture, one would readily think of an “information-passage”-type of scheme or MMM approach, probably overwhelmed by the extremely large “gaps” both in temporal and spatial scales, lying between the ab initio (quantum mechanics or even atomistics) all the way up to the structural scales. For a breakthrough, totally different viewpoints, departing from such a stereotyped recognition, are undoubtedly indispensable. In this sense, I strongly recommend that readers not start any discussions about, for example, multiscale plasticity from this diagram or the like, unless significant modifications or alterations based on additional points of view are made.

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Overview of Field Theory of Multiscale Plasticity

Figure 5.3.4  A conventional view of multiscale problems, including plasticity, leading to

“information-passage”-type approaches.

5.4

Field Triangle: Three Well-defined Field Theories

5.4.1

Differential Geometrical Field Theory From a mathematical point of view, the use of “torsion” and “curvature” tensors can achieve a complete description of any type of inhomogeneity regardless of scale. Therefore, differential geometry provides the most important part of the “description” of the inhomogeneous fields, including the microscopic pictures of dislocations and disclinations. The evolutionary aspect of them, however, is not contained there in a self-contained manner. Based on this notion, we have explicit expressions of them in a continuum-mechanics framework, that is, dislocation-density and incompatibility tensors given by the curl of the distortion tensor and the double curl of the strain tensor, respectively. The use of these “strain gradients” has many advantages from the mechanics point of view, as follows: 1. Strain gradients can be a natural extension of the classical or conventional framework (continuum mechanics).

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5.4  Field Triangle

275

2. “Length scales” are introduced as a natural consequence of “strain gradients” inclusion in the theory (in the sense of generalized continuum mechanics [Kröner, 1958]). This has attracted much attention in recent years in conjunction with “miniaturization” trends. 3. Nonlocality would be key to understanding complex phenomena in general. Interactions among different scales are naturally contained via “curvature” or an “incompatibility” tensor field. 4. Well-developed foundations about “theory of relativity” can be utilized in many aspects, for example, interpretations of the quantities and identities, and derivations of associated equations. 5. The quantities, that is, dislocation-density and incompatibility tensors, are measurable based on their physical images in the present context. They can be further utilized in practice for, for example, nondestructive damage evaluations.

5.4.2

Gauge Field Theory Another field theory based on “gauge” formalism has a more abstract framework than the DG field theory described earlier. Hence, this theory is more powerful in the sense that it can provide all the dynamic information, including equations of motion (EOMs) and field evolutions, with mathematical rigor. The features are summarized as follows: 1. Analytical mechanics is the framework within which everything is deduced from the Lagrangian density of the system with mathematical rigor. All the information about the symmetry of the system is incorporated in a visible fashion. 2. Bridging of DG field theory with the QFT method is achieved, where the interaction with the background elastic field can additively be taken into account. (The constructed Lagrangian density maintaining the DG interpretations of the gauge fields can be transformed into Hamiltonian, to be used in QFT.) 3. The framework has more sources, such as an inexhaustible source of information, once the theory is constructed within this formalism.

5.4.3

Quantum Field Theory The significance of the use of QFT is twofold. One is its natural expression of the discrete (or second-quantized) picture of dislocations as the elementary excitation of the background crystalline-ordered field (see Figure 5.2.1), in conjunction with the gauge formalism that gives the foundation. The other is its mathematical (or formal) equivalence to statistical mechanics, which is missing in the former two field theories. This viewpoint is indispensable for tackling many dislocation problems, for example, leading to cell formation, after the corresponding effective theory is derived. As stated in Section 5.4.2, the foundation, such as the Hamiltonian representation, is given by the gauge field theory, where the invariance of the Lagrangian under prescribed symmetry transformation determines everything.

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5.5

New Features: Toward a Complete Field Theory of Plasticity In addition to the original threefold concept, the current FTMP features three more new notions based on the findings obtained in the systematic investigations. The three are displayed in Figure 5.5.1 and briefly described in the following.

5.5.1 Inhomogeneity Present field theory provides a unified framework for the description and evolution of inhomogeneous fields, as pointed out in Section 5.2. Let us look anew at “inhomogeneity” as a new notion. In particular, the evolutionary aspects of the inhomogeneities are encapsulated by the two novel notions of “duality” and “interacting incompatibility” (see Sections 5.5.2 and 5.5.3), respectively. By introducing a model incorporating the interaction-field notion (see Section 6.7) into the hardening law (Eq. (5.6.4) in Section 5.6), both new features can be accommodated in the framework. Examples of duality and the interaction field are discussed throughout Chapter 11 and in Section 13.2, respectively. Further details of the two are presented in the following sections.

5.5.2 Duality “Duality” is a generic concept governing the field evolutions and dictating the evolutionary aspects of the respective (given) inhomogeneities in terms of nonlocal energy conversions between the elastically stored state and dissipative plastic flow. Note that conversions will not always be one way: Cooperative effects of localized plastic flows at several regions can promote energy storage in neighboring regions. Simulation results implying this are presented in Chapter 12, in the context of inhomogeneous fields along the polycrystalline-aggregate scale (Scale C). This notion can be extended to other material systems, such as nanocrystals, polymers, and metallic glasses, as far as energy conversion or redistribution takes place, and it causes mesoscopic flows during the course of deformation.

5.5.3

Interacting Incompatibility Interaction fields based on the relative deformation between two arbitrary scales, for example, micro- and macro-scales, will be discussed in Chapter 6, which allows us to deal here with interscale coupling problems with mathematical rigor. It will be shown that only the curvature tensor, or equivalently, the incompatibility-tensor field, can explicitly account for the interplay. An appropriate extension of this formalism is highly expected to lead to the invention of a methodology for overall system stability/ instability evaluations. A preliminary attempt is made in Chapter 13 (see 13.2). The incompatibility-tensor field model that will be introduced in Section 5.6.3, coupled with interaction-field formalism, is expected to have broad applicability. Three examples of the usages are indicated in Figure 5.5.2.

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5.6  A Tentative Vehicle for FTMP-based Simulations

277

Figure 5.5.1  New notions in FTMP.

Figure 5.5.2  Threefold usages of the incompatibility tensor in FTMP.

5.6

A Tentative Vehicle for FTMP-based Simulations: A Crystal Plasticity-based Constitutive Model As a tentative vehicle for applying the field-theoretical concepts overviewed in the previous sections to practical applications, the kinematical foundations of crystalline plasticity (e.g., see Asaro, 1983b; Khan and Huang, 1995; Nemat-Nasser, 2004) are utilized. The constitutive equation to be used, however, has its basis in the statistical mechanics-based DD provided in Chapter 2, with which wide ranges of strain rate and temperature can be taken into account in a unified manner.

5.6.1 5.6.1.1

Constitutive Framework Shear Stress–Shear Strain Relationship The current field-theoretical notions and quantities, including the interaction-field framework, can be easily implemented into the crystalline-plasticity model through strain-gradient terms (Aoyagi and Hasebe, 2007). I have proposed a constitutive

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Figure 5.6.1  Constitutive framework based on DD, derived in Chapter 2, applicable to a wide

range of strain rates and temperatures, as well as to both FCC and BCC metals.

model applicable both to FCC and BCC metals based on statistical mechanics-based DD, as displayed in Figure 5.6.1, which is discussed in more detail in Section 2.7. The explicit form is given by



  ( )  *( )    Peierls  ( )  (  )  ( )  *  *( ) BSR exp  1    ASR   K ( )

    ( ) *( )  *  ( )   Peierls  ( ) 

p q

   

1



 CSR

, (5.6.1)



with

G0disloc BL * , CSR  , (5.6.2) A SR   m bL *, BSR  kT b where K (α) and () are drag stress and back stress, respectively, responsible for isotropic and kinematic types of hardening. In Eq. (5.6.2),  m , L,  * , b, and B refer to mobile dislocation density, mean flying distance of dislocations, the modified Debye frequency, the magnitude of the Burgers vector, and the damping coefficient due to, for example, phonon drag, respectively, while G0disloc   (T )b3 g0disloc stands for the activation energy for dislocation processes at T = 0K, with g0disloc being the normalized one and µ (T ) the temperature-dependent shear modulus. For BCC metals, we can set *( ) CSR = 0, whereas for FCC we may normally assume  Peierls  0. The exponents p and q

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5.6  A Tentative Vehicle for FTMP-based Simulations

279

in Eq. (5.6.1) are the parameters specifying the thermal obstacle of interest, provided 0 ≤ p ≤ 1 and 1 ≤ q ≤ 2 . In the current case, a pair of values = p 1= / 2, q 3 / 2 is used for representing dislocation processes. Furthermore, in Eq. (5.6.1),   (  ) / 2 *( ) represents the Macaulay parentheses, with  Peierls expressing the effective stress for the Peierls overcoming process, given by

*( )  Peierls

  kT   1   Peierls 3 ln 0 Peierls   g0 b  ()  

1/ qPeierls

   

1/ pPeierls

   

, (5.6.3)

where g0Peierls , γ0 Peierls , pPeierls , and qPeierls are parameters for the thermal activation process via the Peierls overcoming event of dislocations. Equation (5.6.1) can express stress–strain responses both for FCC and BCC metals over a wide range of strain rate and temperature, including impact-loading conditions. Different trends between FCC and BCC metals in the stress response in terms of strain rate, that is, similarly increasing flow-stress levels but due to distinct mechanisms, are well described, as confirmed. Not only these, but also the associated dislocation-​substructure evolutions, represented by “effective cell” size defined in Eq. (5.6.10) are simulated. Figure 5.6.2 displays examples of the stress–strain curves (top), together with variations of the effective cell size with strain (bottom), for FCC and BCC metals using the present model, comparing responses between those under static and impact loading. Compare these with Figures 3.4.1 and 3.4.2 in Section 3.4. Here, larger effective cell size dcell, as in the FCC under static and the BCC under impact, mean coarser dislocation cells. For the BCC case, the situation corresponds to the micrograph shown in in Figures 3.4.1 and 3.4.2 (above right).

5.6.1.2



Drag-Stress Model

The hardening evolution models are introduced through drag stress K (α ) and back stress ( ) (Hasebe et al., 1999), with an overview displayed in Figure 5.6.3. A physical image for modeling the drag stress is schematically shown in Figure 5.6.4, where the interactions of dislocation against forests (or equivalently, interactions among dislocations mostly belonging to different slip systems, represented by the interaction matrix f ) are regarded as being responsible for the additional hardening. The quality of the additional hardening, that is, “history,” is characterized by the effective cell size dcell, to be defined later via the hardening ratio Q . The time evolution of the drag stress defines the instantaneous hardening modulus and is assumed to be expressed as follows: K ( )  Q H ( )  ( ) , (5.6.4) where H(γ ) is the referential hardening modulus with no history and no interaction among slip systems, while Q denotes the hardening ratio that evolves with the histories and the interactions. In order to express the nonlocal actions associated with the evolutions of inhomogeneities based on dislocation-density and incompatibility-tensor fields, strain-gradient terms F(α (α ) ) and F( ( ) ) are additively introduced into Q as detailed in Section 5.6.2. Figure 5.6.5 is an overview of the drag-stress model,

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Figure 5.6.2  Examples of simulated stress–strain curves of FCC Cu and BCC Fe, simulated by

using the constitutive equation in Figure 5.6.1, comparing two typical strain rates, static and impact, together with effective cell-size evolutions, which are also strain-rate dependent.

Figure 5.6.3  Overview of the hardening evolution model composed of drag stress and back

stress, with the latter characterized by a hardening ratio in which dislocation interactions as well as field-theoretical strain-gradient terms are incorporated.

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Figure 5.6.4  Schematic drawing of the drag-stress evolution model. Here, dislocation–

dislocation interactions, especially those that result in junction formation, are considered to be responsible for additional hardening, which is simultaneously associated with the evolution of dislocation cells.

emphasizing the feature that can take into account interaction-field formalism (given in Section 6.7) through the incompatibility term F( ( ) ). Here, the construction of K (α ) is depicted in Figure 5.6.5, that is,

 ( )  ref K ()     ( ) , (5.6.5)  ref  ( ) where τ ref represents the referential flow stress without history effects, so that  ref /  ( )  H ( ) defines the referential hardening modulus. The hardening ratio Q   ( ) /  ref expresses the ratio of the flow-stress increase to the referential level, measuring the additional hardening, whose basic form is assumed to be given by



Q   f S , (5.6.6) where f represents the dislocation-interaction matrix (see Section 1.3.5), and S expresses the history matrix. This is further given as an increasing function of plastic work done by the effective stresses that are responsible for dislocation processes, for example,



p*   W   () S  tanh   Wsat   

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 ( ) with Wp *  C Wp* , (5.6.7)

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Figure 5.6.5  Construction of a drag-stress model, where all the information about the evolving

dislocation substructures as a result of dislocation–dislocation interactions is contained in the hardening ratio Q .

(β ) where Wp* defines the plastic work done by the effective stress for dislocation processes only (excluding other contributions), for example,



Wp(* )   W p(* ) dt

*( ) with W p(*)   ( )   Peierls   () . (5.6.8)

Here, C  represents a coefficient measuring the correlation between the two slip systems α and β , in terms of dislocation–dislocation interactions, that is,

1 if  ( )  0 C    , (5.6.9) 0 if  ( )  0 which means that only the stationary (inactive) slip systems are assumed to have the latent hardening effects on the dislocations that belong to the currently active slip systems.  As is understood from this, if there exists no strain history, that is, Wp*  0, the hardening ratio coincides with the unity, that is, Q   , resulting in the evolving drag stress to be that for the reference, that is, K ( )  H ( )  ( ) . The hardening ratio Q (Figure 5.6.6) is further used to evaluate the effective cell size dcell, because, in principle, it contains all the information about the dislocation cells evolved (responsible for the additional hardening). Therefore, we assume that the quantity characterizes an effective size of dislocation cells, which is defined here as





1 2



1

 N  2 ( ) or dcell  k   Q Q  , (5.6.10)   1    where k represents the initial cell size, normally coinciding with a fraction of the grain size. The former definition is normally used, while the latter is adapted when slip system-wise information is required. 1  dcell  k  Q Q  N 

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Figure 5.6.6  Schematics of drag-stress model giving hardening coefficient h  Q H ( ). The evolution equation is composed of the hardening ratio multiplied by the referential hardening modulus. Interactions among three significant scales, A, B, and C, are incorporated in the hardening ratio.

5.6.1.3

Back-Stress Model The back-stress model, on the other hand, is schematized in Figure 5.6.7, where the evolutionary process of an internal stress field produced by “redundant” dislocations piled up against the cell walls is considered. The evolution equation for ( ) is given by )  (sgn  





A  x N



* dcell  xN  a

2

1

K ( ) K sat

, (5.6.11)

( )

where 1 K / Ksat is the attenuating term against the growth of the drag stress * K (α ), with Ksat being the saturation value of K (α ), while dcell ≡ dcell / 2 and x N(α) are the mean moving distances of dislocations evaluated by multiplying l*  1 / (  m b) with  (); a is the cut-off distance. The integration of Eq. (5.6.11) with respect to time gives

N 1 A ) )  (sgn (sgn   dt   , (5.6.12) N a

where the attached subscript “sgn” indicates the sign of piled-up dislocations where “+” is assigned for forward loading and “−” for reversed loading. The second term counts the contribution from the already piled-up dislocations against the cell walls. The sum of the two contributions ultimately gives the back stress introduced in Eq. (11.3.1), that is,

( )  ( )  ( ) . (5.6.13) Note that the “pile-up” of dislocations is not only a literal term, but also include the kinds of dislocations schematized in Figure 3.9.1 (Mughrabi et al., 1986).

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Figure 5.6.7  Schematic drawing of the back-stress model, assuming a pile-up of dislocations

forming a long-range stress field of the order of a cell size.

5.6.1.4

Effective Cell Size dcell The effective cell-size model defined in Eq. (5.6.10) is a model that virtually represents the characteristic length scale in Scale A, since its direct reproduction (or treatment) is considered to be beyond the present continuum mechanics-based modeling framework, even with the use of “incompatibility-tensor” field-related modeling, which requires separate treatment, as detailed in Chapter 10. The dcell model can also effectively be made use of for modeling or expressing the underlying phenomena and the attendant interactions with them, for example, in modeling underlying highly dense dislocation structures, as Scale A, within lath martensite blocks modeled and simulated in Scale B (see Project #1 in Chapter 9). The idea of evaluating this via the reciprocal of the norm of the hardening ratio Q stems from the “similitude” relationship   1 / dcell discussed in Section 3.2. This means that where the flow-stress level is accurately reproduced, the corresponding cell size can be roughly estimated. Since our constitutive framework Eq. (5.6.1) can appropriately evaluate the flow-stress level over a wide strain rate and temperature range, this assumption seems to be effective, at least for a rough estimate. Examples of the dcell evaluation are presented in Figures 5.6.1, 5.6.2, and 5.6.8, respectively. The latter is a rearranged set of simulation results from Figures 3.3.7– 3.3.9 for the current purpose, demonstrating cyclic-stress responses under various nonproportional strain paths versus evaluated variations of the effective cell size with the number of cycles for FCC austenitic stainless steel (SUS304).

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Figure 5.6.8  Simulated effective cell-size variations during cyclic straining under

nonproportional loading. Rearranged from Figures 3.3.7 through 3.3.9.

In the former, the strain rate as well as the material-dependent dislocation-​ substructure evolutions are qualitatively reproduced, that is, smaller cell formations  under  high strain rate for FCC metals, and “coarser” dislocation-structure developments under impact loading for BCC metals. Here, the “coarser” structure is expressed as a “much larger” effective cell size. Although one can readily understand that the raised flow stress for FCC metals directly contributes to a smaller effective cell size, as readily understood from the figure, that for BCC metals does not. This is because the raised flow stress in BCC metals is due mostly to the Peierls overcoming *( ) process, that is,  Peierls (Eq. (5.6.3)), meaning what contributes to the cell evolution is *(  ) ( )    Peierls in Eq. (5.6.1) (more precisely, via the effective plastic work defined in Eq. (5.6.8)), resulting in relatively restricted cell formation. In Figure 5.6.8, on the other hand, strain path-dependent dcell variations with accumulated plastic-strain cyclic stress are well described, corresponding to well-reproduced cyclic-stress responses. Note that the “similitude”-like relationship, holding also for the nonproportional cyclic straining, is displayed in Figure 3.3.12. Since the path-dependent cyclic-stress responses, in this simulation, simply stem from the path-dependent dislocation–dislocation interactions as a result of the use of the interaction matrix f in Eq. (5.6.6), we may safely say that the evolving as well as the morphological aspects of the dislocation-cell structures can be ignored, at least as far as the additional hardening characteristics are concerned.

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5.6.2

Hardening Law and Field-Theoretical Strain-Gradient Terms As mentioned previously, the present constitutive model incorporates the contributions of α (α ) and  ( ) through the hardening ratio Q as (Figure 5.6.3), Q    f S   F ( ( ) ; ( ) ), (5.6.14)





where f again denotes the dislocation-interaction matrix. No summation is taken in the last term in the right-hand side of Eq. (5.6.14). Here, Fk ( k(  );k(  ) )  Fk ( k(  ) )  Fk (k(  ) ) expresses the field-theoretical “strain-​ gradient terms,” given respectively by Aoyagi and Hasebe (2007), as 1/ 2   ( )   1 k  ( )   , (5.6.15) F ( ( ) )  k  dcell  p  b    1

( ) F ( ( ) )  k  dcell  sgn( ( ) ) 



1/ 2

k  ldefect ( )    p  b 

, (5.6.16)

where p , p are coefficients related, with the contributions of  k(  ) and k(  ), to the change in the effective cell size dcell , while ldefect represents the characteristic length of the defect field considered, for example, ldefect = b, for dislocation dipoles, and ldefect  10 6 m for dislocation substructures like cells. Here  k(  ) and k(  ) are the resolved components of ij and ij , respectively, defined as ( )

and





 k( )    k  ti( ) s (j )  si( ) s (j ) ij (5.6.17)



k

k( )   sgn(k ) k ( )



k

( )



(5.6.18)

 ti( ) s (j )  si( ) s (j )  si( ) m(j ) ij ti( )  ijk s (j ) mk( ),

s (jβ ) ,

mk(β )

where with being the unit vectors in the slip direction and slip plane normal, respectively. Note that the first and second terms of  k(  ) respectively correspond to the edge and screw components of dislocation density. The same is true for k(  ) where the third is called the disclination type. The effective cell size in this case is also evaluated by Eq. (5.6.10) via the hardening ratio Q . Therefore, the effects of the dislocation-density and incompatibility terms are manifested, for example, as “modulated” cell-size distributions in simulation results, as schematically shown in Figure 5.6.9.

5.6.3

Derivation of Field-Theoretical Strain-Gradient Terms The explicit forms of Eqs. (5.6.15) and (5.6.18) were obtained as follows. Figures 5.6.10 and 5.6.11 summarize the derivation processes, respectively. Since the hardening ratio Q physically dictates the inverse of the effective cell size, in the present context of characterizing the mean dislocation-free path, as understood from Eq. (5.6.10), the strain-gradient terms to be introduced should have the same dimensionality, with the effect of either enhancing or reducing it. For the dislocation density, we can evaluate the mean spacing of dislocations via

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Figure 5.6.9  Image of a “modulated” effective-cell size distribution brought about by the

introduction of FT (field theoretical) strain-gradient terms. The cellular structure displayed is a simulated result based on the derived model for Scale A in Chapter 10.

Figure 5.6.10  The derivation process of the dislocation-density term is basically given as a function of the first strain gradient (a curl of distortion tensor).

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Figure 5.6.11  The derivation process of the incompatibility term is basically given as a function of the second derivative of strain (double curl of strain tensor), where not only the magnitude but also the sign is taken into account, which corresponds to the sign of curvature (positive or negative).

( ) ldisloc 



 1 ( )   b    

1/ 2

, (5.6.19)

since (1 / b) α (α ) counts “geometrically necessary” types of dislocation density. (α ) Assuming the proportionality between the cell size and ldisloc to be based on the empirical facts (refer to Figure 3.2.4), that is, ( ) ( )  dcell  p  ldisloc , (5.6.20)



with pα being the proportionality factor, we arrive at Eq. (5.6.15). The same is true also for the incompatibility tensor, except one additional parameter ldefect and the sign of  ( ). The former is introduced for the dimensionality reason, while the latter takes account of the accommodation ability of the quantity, dictating how the dislocations are redistributed to relax the excessive deformation, as described in Section 2.2, that is,

( )  dcell 

( ) p  ldefect 

l  p  sgn  ( )  defect  ( )   b 





1/ 2

. (5.6.21)

By combining the contributions given by Eqs. (5.6.20) and (5.6.21) to the effective cell-size evolution, we have,



1

1

( ) ( ) Q    f S   k  dcell   dcell

 , (5.6.22)

leading to the final expression in Eq. (5.6.10). Note that the effect of the evaluation method of the derivative for obtaining F( ( ) ) distribution is extensively discussed in Aoyagi et al. (2008).

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The application examples of the incompatibility tensor and the associated strain-gradient terms after being implemented in the crystalline-plasticity framework given earlier are presented in Chapters 11 and 12.

5.6.4

Field-Theoretical Strain-Gradient Terms in Multiple Scales When we consider multiple scales, that is, more than two scales, we simply introduce the corresponding incompatibility terms based on the interaction-field formalism. Since the term f S in Eqs. (5.6.6) and (5.6.14) is responsible for the effective cell size dcell in Scale A, the additionally introduced F ( ( ) ; ( ) ) in Eq. (5.6.14) are basically for Scale B or C. Figure 5.6.6 includes also the case with such multiple scales.

Appendix A5.1  Why Is a “New” Field Theory Necessary? A5.1.1

What Are the Deficiencies in Conventional Views?

A5.1.1.1 Treatment of “Inhomogeneity” Some might think that the framework of the conventional crystal plasticity-based field theory would be sufficient even for tackling multiscale problems. But, looking at the subjects introduced in this book afresh, like those associated with the three representative scales and their interplay (in Chapters 10, 11, and 12), especially in terms of evolutionary aspects, will ensure many readers recognize the numerous deficiencies and limitations of conventional approaches. Let us confirm first some apparent drawbacks of the conventional field theory of continua, and then look at the reasons why the new viewpoints that are presented in this book are necessary. First of all, there is a clear limitation in the classical field theory of continuum mechanics (Truesdell and Noll, 2003) in terms of describing substantially “inhomogeneous” fields, including “discontinuities.” The primary variables therein are stress and strain, which are defined locally. So, they are basically only suited for describing rather uniformly deforming states. The material body describable by such formalism is called a simple body, in which the motion of the body is simply expressed as a function of the deformation-gradient tensor, that is, F  X / x, with X and x being position vectors at reference and current configurations, respectively. Based on such formalism, the fields of dislocations and defects, for example, are expressed indirectly only via their distributions. Also, they contain no length scale characterizing further microscopic degrees of freedom, such as those that stem from microstructures. Historically, these aspects had been coped with by extending the notion to higher orders, as briefly described in the following. For the extension directions of the continuum mechanics, Kröner (1968) pointed out three indices (mechanics of generalized continua) as schematically shown in Figure  A5.1.1. They are: (1) Polarity, (2) nonlocality, and (3) configurational

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Figure A5.1.1  Generalized mechanics of continua in terms of Kröner’s three indices of

extension.

complexity (e.g., Ohnami, 1980, 1988). There have always been, however, questions about the “physical meanings” of extra concepts and parameters introduced in the frameworks, for example, additionally introduced characteristic length in (1) and the cut-off order for the higher-order terms in (2). We have recently witnessed a revival of the index (2) in terms of “strain-gradient plasticity” or, more specifically, “GN dislocations,” in conjunction with a need to express empirically measured “scale effects” especially emphasized either in plastic instability or in small scales. Although more than a decade has passed since their rise, they have not yet evolved as a decisive theory for tackling complex problems in multiscale plasticity. The remaining (3), on the other hand, include the other two, at least conceptually. A well-known example of (3) is the non-Riemannian plasticity theory (Kondo, 1955), based on differential geometry, which forms one of the bases of the current field theory. Although the concepts introduced there have been well documented, almost no practical extension or applications have been made to date. Such a state of affairs was concisely expressed by Kröner (1998), who mentioned differential geometry-based formalism when he concluded his special lecture, held in Stuttgart, with: “This geometry permits an elegant formalism of dislocation theory, but not yet in such a way that elasto-plasticity becomes easier.”

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A new theory should make many things that had been difficult to treat much easier. This is one of the most important aspects for us to consider when judging whether a “new” theory or concept deserves special attention or not. The second drawback of conventional theories concerns the inability to describe evolving inhomogeneities. Essentially, to my knowledge there is no generalized way to describe such evolutionary aspects of inhomogeneity, which seems to me another reason why the “non-Riemannian plasticity” theory has not attracted much attention. This stems from the fact that the geometry itself contains no information about the evolutionary aspects of the deformation fields, and does not address why and how they should evolve. Noticing this, Kondo struggled to construct the right “field equation” based on which field evolutions associated with the curvature tensor ought to take place. The representative outcome of this struggle is presented in Chapter 13. The field equation for the DG quantities presented in Section 13.1 is such an example, however it is still too abstract. Typically, this shortcoming would be directed to the treatment of “statistical mechanics”-related aspects, especially of nonequilibrium kinds involving “dissipation,” leading to self-organizing dislocation substructures such as cells, as partially pointed out by Kröner (2001). The third deficiency is a lack of scope in the treatment of the “complexity” of the targeted systems. Materials systems can be regarded as “complex” for they generally yield nonpredictable mechanical responses (especially in plasticity) due to multiscale nonlinear interactions. One often sees a number of disputes about “how to identify or deal with ‘meso’ scales” without touching this important aspect at all. This problem is discussed in Section A5.1.1.2.

A5.1.1.2 Treatment as a “Complex” System Let me pose another question, which may be crucially important for thinking about MMMs: Why do we need a “multiscale” approach, in the first place, for example, for viewing, analyzing, modeling, or simulating plasticity of materials? One of the reasons would be because MMMs ought to be regarded as problems in “complex” systems (some related arguments are provided in Section 9.1). A “complex” system is a system with nonlinear interactions among the constituting elements and/or hierarchical scales, and with feedback loops that can or may ultimately generate nonpredictabilities. If this is the case for the material systems of interest, the minimum requirements for the MMMs are almost self-evident. They are, inclusion of multiple scales more than “three” and the nonlinear interactions among them. The nonlinearity should go relate to the interactions themselves and the evolutionary processes of “inhomogeneity” in individual scales, both in terms of kinematics and physics (or mechanics). The “evolutionary” aspect of the inhomogeneity is particularly important in this context, because it differentiates the current problem from other complex ones. Moreover, it is always “dissipative” in terms of energy.

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Another question that I would like to ask is the following: If you accept the above statements, do you think the conventional frameworks (theory and modeling schemes) are still sufficient to tackle our problems? If the answer is “no,” what are the major drawbacks? Asking these questions may clarify what we really need to further MMMs.

A5.1.2

What Is the Key for Breakthroughs? Revisiting Phenomenology Dislocation cells (Scale A) are structures with characteristic lengths of the order of sub- to a few micrometers, accompanied by heterogeneous deformation (strain field) and concomitant fluctuating internal stress fields (the latter will be particularly important because they are “invisible”). What complicates matters is that this scale is more or less continuously connected with the intragranular scale of inhomogeneities (Scale B). On the other hand, polycrystalline aggregates (Scale C) can be the source of another inhomogeneity via the cooperative effects of composing crystal grains, which differentiate polycrystalline plasticity from that of its single-crystal counterpart. We can note that all of these aspects are associated with “inhomogeneities” in deformation-induced kinds of stress and strain fields. What is more, there generally exists “interplay” among those plural scales, as mentioned earlier. These facts, in addition to their crucial importance in many of the practical problems, lead us to construct a new theoretical framework beyond the classical and recent approaches. You will notice that there has never been a sophisticated way to deal with them together in a satisfactory manner, despite tremendous efforts for many decades by excellent researchers all over the world.

A5.1.3

Challenges in This Book The current FTMP aims to overcome the deficiencies pointed out in the preceding text, especially those in Section A5.1.1.1, simultaneously taking account of the individual phenomenology that apparently seems to be specific to the scale levels. The specifics to achieve that are concisely presented in Section 5.3. Such a perspective clearly differentiates the present theory from many others. The present book contains several hypothetical arguments, based on new findings of my own obtained through ongoing research and research projects, rather than well-documented ones, which are ultimately aggregated into a working hypothesis called “flow-evolutionary law,” together with its ample application examples, as provided in Chapter 15. The “well-documented” arguments are less mentioned because they can be found elsewhere. We can now identify a general principle for inhomogeneity in terms of the description, evolution, and cooperation among scales. The reasonable interpretations and explanations of them are now more or

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less available in terms of the evolutionary aspects of the multiple-scale inhomogeneities, and also to solve a dilemma concerning, for example, the “specificity versus universality” controversy. Understanding of this will need some sort of epistemological leverage, such as those supplied in Chapter 14. The loose coupling mentioned in 14.3.2, for instance, is embodied as “role sharing” (a manifestation of duality) in the upper scales.

A5.1.4

Bird’s-eye Maps of This Book Figures A5.1.2 through A5.1.5 present a series of bird’s-eye maps for the current book, from “fundamentals” all the way up to the “flow-evolutionary law” to be detailed in Chapter 15, via “theory” chapters. Chapters relating mainly to simulation results (Chapters 10 through 12) and those with extended content (Chapters 13 and 14) are excluded from the maps, as is Chapter 8, which is more or less solely related to Chapter 10. The content of the current chapter is presented in Figure A5.1.2, which converges toward the “microscopic constitutive equation” that will be used in most of the FE simulations presented and argued within this book.

Figure A5.1.2  From dislocation interactions to hardening law (drag-stress model), coupled with effective cell size via a latent hardening test and “similitude law.”

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Figure A5.1.3  From DD to a unified constitutive framework via a Kocks–Mecking model and concept of MTS.

Figure A5.1.4  From “non-Riemannian plasticity” to FTMP via an incompatibility-tensor model explicitly implemented into the hardening law of the crystal plasticity-based constitutive framework through strain-gradient terms.

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Appendix A5.2  Continuum Mechanics and Crystal Plasticity

295

Figure A5.1.5  From “role sharing” or “duality” to “flow-evolutionary law” via an extension of the incompatibility tensor into 4D space–time and an extended use of the energy-momentum tensor.

Appendix A5.2  Brief Overview of “Continuum Mechanics” and “Crystal Plasticity” As described in Section 5.6, we utilize the well-developed framework of the continuum mechanics-based crystal-plasticity kinematics/kinetics as a vehicle for explicitly calculating FTMP-related quantities/models. In this appendix, continuum mechanics and crystal-plasticity formalism are presented as concisely as possible. Roughly, continuum mechanics is organized into three parts: (1) Kinematics, (2) Kinetics, and (3) constitutive equation. For crystal plasticity, after introducing a slightly complex concept regarding Objectivity as well as elastic-plastic decomposition applicable to finite strains, what should be minimally given are: (1) The crystal-plasticity version of the latter, and (2) the constitutive framework that rigorously takes into account the objective stress rate.

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A5.2.1

Continuum Mechanics This subsection provides brief descriptions of the kinematics, kinetics, and constitutive equation, which are followed by the elastic-plastic decomposition proposed by E. H. Lee (1969), as a bridge to “crystal-plasticity” kinetics and formalism. For kinematics, the main subject is about how to describe “deformation,” leading ultimately to defining the strain tensor (Figure A5.2.1). Simply by comparing the material line vectors, dX and dx, which express the vicinity of a material point before and after the motion of the body (at reference and current configurations, respectively), we can explicitly evaluate to what extent the material body has been deformed during the motion (e.g., time interval t). More specifically, the difference between their norms, that is, the scalar products, as dx  dx  dX  dX , is evaluated to calculate the “net” length change (which eliminates the extra information about “rotation”), providing us with the measure of strain. All that is left is to normalize this by dX ⋅ dX or dx ⋅ dx, resulting in the strain tensors, referring, respectively, to the Green–Lagrange E and the Almansi–Euler A strain tensors. Here, it is convenient to define the deformation-gradient tensor F  x / X to express them, once we understand the derivation process explained earlier (see bottom of Figure A5.2.1). In the current configuration, we further define the velocity gradient tensor L by comparing the velocity difference dv with respect to the spatial interval dx, that is, L  v / x , with its symmetric part D defining the strain-rate tensor, and with the skew-symmetric portion W giving the spin tensor. The relationship with F is given by L  F  F 1. For kinetics, on the other hand, we have to skip over “what is force” at the outset, since it is beyond the “classical” mechanics category, requiring QFT (more precisely, gauge field theory). Somehow, the point of departure here is the laws of physics, that is, the conservation of linear momentum p = f among others, where the “force vector f ” is involved on the right-hand side against the change rate of the linear momentum on the left (Figure A5.2.2). Here, the stress vector t , as the traction defined per unit area, is postulated as a function of the surface, represented by the unit plane normal n, in addition to the body force b, that is, f   tda    bdv, with ρ being the mass a v density of the material, which is called Cauchy’s stress principle. In order to specify the stress vector t, the conservation of linear momentum p = f is applied afresh to an infinitesimal tetrahedron, as a minimum set-up that contains an arbitrary surface represented by n against the well-defined other three. At the zero-volume limit, we ultimately obtain the relationship between t and n as t   T  n, thus the Cauchy stress tensor σ is defined. Furthermore, by substituting the expression for t with respect to σ back into the right-hand side of the conservation of linear momentum, we have the x  div T   b. Note that, in Figure A5.2.2, EOM of a deformable body, that is,   the material time derivative D / Dt   / t  v   / x is used instead of the usual time differentiation. What must be added is that as-derived σ is not symmetric, until it is asked to satisfy the conservation law of angular momentum as well, resulting ultimately in  T   . Therefore, in the case where the couple stress m exists, the skew-symmetric

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Figure A5.2.1  “Kinematics” aspects of continuum mechanics.

Figure A5.2.2  “Kinetics” aspects of continuum mechanics.

part  skew  (1 / 2)(   T ) does not vanish and :  should balance with divm, which forms the foundation of the Cosserat theory (see Section 6.2.8). The third part simply interrelates the kinematics and kinetics (Figure A5.2.3), more specifically, the stress and the strain tensors, and is called the constitutive equation, through which we can specify the kind or class of materials to be treated. The most fundamental as well the simplest equation is for a (linear) elastic body   C e:  . The explicit form called “generalized Hooke’s law” is easily obtained when we assume the spatial isotropy in elasticity, simply by considering the linear combination of the three e fourth-rank isotropic tensors, Cijkl   ij  kl  2  ik  jl , where the symmetry with respect to (i,j) and (k,l) has been already taken into account. From this, we readily know that the number of independent elastic constants for isotropic elasticity is 2. The final form

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Figure A5.2.3  “Constitutive equation” aspects of continuum mechanics.

is given as  ij   ij  kk 2  ij (in index notation) or    (tr ) I  2  (in direct nota 1 tion), together with its corresponding inverse relationship  ij   ij   kk  ij and E E  1    (tr )I, respectively. Likewise, the elastic anisotropy is specified through E E e Cijkl explicitly in general, reflecting the symmetry of the crystal lattice of interest. Notice that all of these derivation processes are based on purely mathematical ones, for the framework is referred to as rational continuum mechanics. One of the most advantageous aspects for that can be found in the objectivity arguments, indispensable for extending the framework to the finite deformation regime. The objectivity requires any tensorial quantities, for example, A, to be transformed, as per A*  Q  A QT , with Q representing the orthogonal tensor, that is, QT  Q  I . Finite plasticity must be considered in general, in addition to infinitesimal elasticity. For the stress-rate tensor to be objective (Figure A5.2.4), we must extend the definition of the time derivative, for example, Jaumann rate,     W      W , where the continuum-spin tensor W represents the same quantity as that defined in the kinetics. Practically and historically, there have been controversies concerning the choice of such an objective stress rate since the 1960s and 1970s, however, the Jaumann rate has been widely used to date. When dealing with elastoplasticity, we also ought to consider elastic–plastic decomposition, since, in a finite deformation regime, the additivity of the elastic and plastic strain rates is not always self-evident. A widely used version of this is that proposed by Lee (1969), given simply as F  F e F p (Figure A5.2.5), where the “stress-release state” is further introduced as the third configuration, from the “elastoplastically deformed” current state, as illustrated in Figure A5.2.5(a). However, as is well known, this does not rigorously validate the additive decomposition of the velocity gradient when derived from D  ( F F 1 )sym; however, one can safely assume D  De + D p, provided the elasticity is infinitesimal, that is, F e  I  ue /   I .

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Figure A5.2.4  “Objective” stress rate, confirming that the Jaumann rate satisfies objectivity.

Figure A5.2.5  Elastic-plastic decomposition by Lee (1969).

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Figure A5.2.6  From “classical” plasticity to “crystal” plasticity.

A5.2.2

Crystal Plasticity “Crystal plasticity” simply regards “crystallographic slip,” which microscopically attributes to dislocation motion, as the elementary process of plastic deformation (Figure A5.2.6), whereby slip system-based anisotropy is automatically expressed. Moreover, if combined with the rate-dependent microscopic constitutive equation (slip system-wise shear/stress–shear strain relationship), there is no awkward procedure to follow when choosing the slip systems. Therefore, we can dismiss the annoying “yield condition/criterion,” together with its subsequent behaviors. What is more, if utilizing the “objectivity” framework described earlier, lattice rotations are also naturally expressed.

A5.2.2.1 A Crystal-Plasticity Version of “Elastic-Plastic Decomposition” The continuum-mechanics framework discussed earlier can be utilized in the crystal-​ plasticity case, as long as care is taken that the crystal lattice is not altered by the dislocation glide (crystallographic slip), as rather pronounced in Figure A5.2.7. With this, the elastic contribution F e is replaced by F*, which also contains the contribution of the rigid-body rotation, which reads, F  F* F p. (In the corresponding elastic-plastic decomposition explained in Figure A5.2.5, the rigid-body rotation is assumed to be included in F p instead of F e.) Expressing the arbitrary slip system α via (s , m ), that is, the slip (Burgers’ vector) direction and the slip plane normal, we can specify the slip independence of the crystal lattice by m  s   m*( ) s*( )  0. Care must be also taken, however, to ensure that since the unitariness of the two quantities do not hold rigorously under transformations m*( )  m( )F *1 and s*( )  F*  s( ), renormalizing them or replacing F* by the orthogonal tensor R* (based on the polar decomposition F*  R*  U * with U *  I) may affect simulation results to some extent in practice.

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Figure A5.2.7  Elastic-plastic decomposition in crystal plasticity as the bases of the kinematics.

A5.2.2.2 Constitutive Framework Using an Objective Stress Rate Last but not least we consider the constitutive framework, whose point of departure is the finite deformation version of the elastic law of its rate form, given as ( (*J ) )  C e: d * (Hill and Rice, 1972; Khan and Huang, 1995), where  (*J )  J represents the Kirchhoff stress tensor, with J  det F  0 /  Jacobian, and ( (*J ) )       trd as its Jaumann rate (corotational rate seen by an observer who rotates with the lattice), using the relationships J  trd  trd * ( tr d p = 0: The volume constancy in plasticity), while the slip system-wise quantities are denoted by the corresponding lower-case letters, for example, D → d and W → w. The reason the Kirchhoff stress is used, instead of the Cauchy stress, is mainly numerical, in that the corresponding FE (tangent) stiffness matrix becomes symmetric by utilizing it, making the treatments much simpler and easier (e.g., Nemat-Nasser, 2004). For our purpose, one more step is needed, as summarized in Figure A5.2.8, where the Jaumann rate of the Kirchhoff stress tensor for plasticity (given as the difference between those for the total and elastic parts) is ­ calculated as  p p *  * * ( ( J ) )  ( ( J ) )   w w    w w   w    w , where (τ ( J ) )  means the rate seen by an observer rotating with the material. At the end of the day, we N  R   C e : P       o have  ( J )  C e :d   R ( )  ( ) , with  , where we use        1    W      W

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Figure A5.2.8  Derivation of a crystal plasticity-based constitutive framework.

dp

N

N

 1

 1

 P    and w p   W    with the Schmid tensor P    ( s*( )  m*( ) )sym

(see Section 1.3.4) as well as W    ( s*( )  m*( ) )skew, while the slip system-wise shear rates    are separately evaluated via the microscopic constitutive equation, as described in Section 5.6.1. Note that, with regard to using such rate-dependent formulations for    , we do not have to distinguish between active/inactive slip systems, because each    is taken to solely depend on the current resolved shear stress    and the hardening modulus. For J =1 (incompressibility) in the above, the equation simply reads

o

N

  Ce :d   R ( )  ( )  1

 R   C e : P       with  ,          W      W

which makes the stiffness matrix asymmetric, when used from the start, as indirectly pointed out earlier. While some uncertainties remain unclarified, it is well known that the crystal-plasticity framework overviewed here describes reality to a large extent, especially average orientation-dependent stress redistributions and crystal rotations. Having said that, it is also well documented that the framework is incapable of reproducing the substructures that ought to evolve intragranularly. This deficiency seems to be a serious weak point, among others, in the light of multiscale simulations of materials (see Chapter 3). As argued throughout this book, FTMP can substantially solve this problem, specifically with the help of additional underlying degrees of freedom provided by the incompatibility tensor.

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You may wonder why no one has identified such a prominent descriptive capability innate in the incompatibility tensor for describing the deformation-induced inhomogeneous fields to date. Such a state of affairs may be well described by Kröner’s statement earlier in this chapter. The reason, however, may be fairly simple: That the “dislocation-torsion” correspondence has been too tangible to necessitate any other extensions that make it complicated. Of course, there surely exists an underlying reason why mechanics people tend to overlook the importance of dislocation substructures, such as those discussed in Chapter 3, as they often place extra emphasis on texture formation without taking care of GNBs.

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6

Differential Geometrical Field Theory of Dislocations and Defects

6.1

Brief Review of Differential Geometry Descriptions of inhomogeneity, including dislocations and defects based on differential geometry, form the basic core of the FTMP. This chapter first provides the basic notions of differential geometry necessary for understanding “non-Riemannian plasticity.” Fundamental concepts and quantities are presented next, followed by some new features peculiar to the current FTMP.

6.1.1

General Relativity and Riemannian Geometry Figure 6.1.1 has portraits of Albert Einstein (1879–1955), who invented general relativity, and Georg Friedrich Bernhard Riemann (1826–1866), who created Riemannian geometry. The notion of general relativity is schematized in Figure 6.1.2, where the globe (the Earth) distorts the enveloping space. General relativity asserts that this distortion, measured by the curvature of space, is the very source of gravity (gravitational force). The inset equation in Figure 6.1.2 explains the famous “equation of gravity,” equating the curvature tensor with the energy-momentum tensor, both of which are second-rank tensors. This equation is acknowledged as one of the most beautiful in physics because it presents the interrelationship between “geometry” and “physics (mechanics)” in a relatively simple form. (I have noticed recently that basically the same equation seems to hold for plasticity-related phenomena, which are discussed later.) In what follows, a minimum set of information for readers to understand the differential geometrical (DG) field theory of plasticity will be given. For this purpose, let us first think about a curved space equipped with a “metric” (Section 6.1.2). What we need after introducing the metric is to define the differential operation (Section 6.1.3). To this end, a concept of “parallelism” must be defined (Section 6.1.4), because, in order to find derivatives, one must compare the quantity of interest at two adjacent locations in space. Further details refer to, for example, the introductory parts of the texts on general relativity (see Schutz, 1985).

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Figure 6.1.1  Albert Einstein and Georg Friedrich Bernhard Riemann – founders of the theory

of relativity and Riemannian geometry, respectively.

Figure 6.1.2  Schematics of general relativity (theory of gravity), together with Einstein’s

equation for gravity, where curvature of space–time is equated with matter field.

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6.1.2

Curvilinear Coordinates In curvilinear-coordinate systems, in contrast to Cartesian coordinates, two sets of systems can be independently defined, that is, contravariant and covariant,  i xi j  g  x g j  . (6.1.1)   g  xi g  i x j j 



One can intuitively understand why the two sets coexist to represent a quantity by referring to the inset of Figure 6.1.3 (upper left), where a vector A is decomposed into bases in two independent ways. In the Cartesian case, on the other hand, the two ways coincide, as depicted in the bottom left. The names “contravariant” and “covariant” components are quite confusing. The reasons for these can be understood as follows (Figure 6.1.3). The “contravariant” vector follows the transformation law for the total differentiation of, say, a function, that is,

df 

f f f dx1  dx2  dx3 , (6.1.2) x1 x2 x3

or linear coordinate transformations, as well as those for infinitesimal changes:



x1 x x d x1  1 d x2  1 d x3 . (6.1.3) x1 x2 x3  The “covariant” vectors, on the other hand, follow the same rule as that of the variable transformation in terms of partial differentiation, that is, dx1 

 x  x  x  .  1  2  3 x1 x1 x1 x1 x2 x1 x3 (6.1.4)  This is also the same as the transformation rule for a basis (vector). Namely, the vectors that are transformed in the same manner as for the basis are “covariant” (as long as we consider the covariant basis as a reference). Note that, as shown in these examples, “contravariant” vectors are rather more familiar than “covariant” vectors. The prefix “contra” in the contravariant vectors means the following. One may notice from Eq. (6.1.3) that the components of the “transformed” vector, that is,  x1 / x1 , x1 / x2 , x1 / x3  , are scaled by the new coordinates  x1 , x2 , x3 , and are, roughly speaking, “inversely” proportional to the new coordinates. For the “co” variant case, the corresponding relationship is given as being “proportional.” Thus, in total we have three ways for expressing tensors in the curvilinear coordinates, that is, contravariant, covariant, and mixed, as shown in Figure 6.1.4, that is,

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6.1  Brief Review of Differential Geometry

Figure 6.1.3  Contravariant and covariant components of a vector defined in the

curvilinear-coordinate system.

Figure 6.1.4  Three ways of expressing a tensorial quantity in the curvilinear-coordinate

system–contravariant, covariant, and mixed components – indicated via combinations of upper and lower indices.

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 ij x i x j  : Contravariant T    T x x   x  x  . (6.1.5) T : Covariiant Tij  i x x j   i x i x   T : Mixed T j   x x j 



Note that components of vectors (covariant vectors) transform contravariantly, while components of covectors (contravariant vectors) transform covariantly. In  other words, vectors are covariant, and covectors are contravariant, whereas the components of vectors are contravariant and the components of covectors are covariant.

6.1.3

Metric Tensor The metric tensor gij is the most foundational quantity in Riemannian and non-Riemannian geometry, from which all the quantities are calculated. The sketch summarizing this is given in Figure 6.1.5. The tensor is a function that tells us how to compute the distance between any two points in a given space. The square of the distance is computed as



i j = ds 2 g= gij dxi dx j , (6.1.6) ij dx dx

where gij and gij are the covariant and contravariant components of the metric tensor. For a flat space (i.e., Euclidian space), the metric tensor coincides with Kronecker’s delta δ ij (which is 1 for i = j and 0 for i ≠ j ), that is,

ds 2   ij dx i dx j , (6.1.7) which defines the Cartesian coordinates.

Figure 6.1.5  Definition of the metric tensor, which measures the distance in curved space, and

which can also be used for raising and lowering the indices.

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The metric tensor is also used to raise and lower the indices. For an arbitrary vector,

6.1.4

A  gij A j

and

A j  gij A , (6.1.8)

Parallelism and Connection Figure 6.1.6 explains how to define “parallelism,” leading to the definition of the covariant derivative, taking vector quantity Ai ( x ) as an example. Let us consider a parallel displacement by an infinitesimal distance dx k along the coordinate. The “parallelism” cannot be defined globally in the curved space, because the direction of Ai ( x ) is subject to change along a curved coordinate axis in general. Therefore, a correction to the excessive directional change due to the curvature of the coordinate during dx k must be made, which is given by a linear summation of Ai ( x ) multiplied by coefficients, that is, Γijk A j ( x )dx k . Therefore, the vector apart by dx k and parallel to Ai ( x ) is expressed as



Ai ( x  dx ) / /  Ai ( x )  ijk A j ( x )dx k , (6.1.9) where Γijk are the coefficients specifying how Ai ( x ) at the two adjacent points are interconnected, so that they are called “coefficients of connection.” (Γijk are also

Figure 6.1.6  Defining “parallel displacement” by introducing a “coefficient of connection” in

terms of infinitesimal displacement (top), which enables a differentiation procedure in curved space, leading us to define the “covariant derivative.”

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Figure 6.1.7  Transformation of coefficients of connection together with the definition of torsion

tensor. The former corroborates that the quantity is not a tensor. Antisummarization produces the latter.

called Christoffel symbols of the second kind in Riemannian space and are known as affine connections, as well.) Now we can define the derivative of Ai ( x ) in the curved space as i A j ( x )  lim



A j ( x  dx )  A j ( x  dx )

dx 0



dx j

(6.1.10)

 i A j ( x )  ikj Ak ( x ),

j

where ∇i A ( x ) is called the covariant derivative. The covariant derivative is ­tensorial because it is transformed as a tensor. Note that the operation itself is called covariant differentiation. Since, in differential geometry, the most fundamental quantity is the metric ­tensor, gij or gij, as mentioned in Section 6.1.3, the coefficients of connection Γijk are expressed by using them. In the case of Riemannian space, that is, without torsion, they become

ijk 

k 1 kl g   j gli   i glj   l g ji     , (6.1.11) 2 i j 

where the extreme right is called the Christoffel symbol. It is important to know that the coefficients of connection are not tensors. Figure 6.1.7 summarizes this situation in terms of the transformation of coordinates. Under the coordinate transformation between x µ and x i , Γijk are transformed into Γijk via

ijk 

x i x x  x i  2 x     , (6.1.12) x  x j x k x x j x k

where there arises a nonzero second term at the right-hand side, indicating that Γijk are not tensors. It may be interesting to imagine what would happen if Γijk were tensorial, as the theory of general relativity asserts that Γijk represent gravitational forces. Since tensors do not change their form regardless of any coordinate transformations, ijk  0 at one

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place means ijk  0 there forever. That would be very inconvenient, or impossible. In other words, thanks to the fact that Γijk are nontensorial, gravitational forces, that is, Γijk , can vanish only when and where we choose a coordinate system appropriately, consistent with our experiences (as in a falling object or vehicle in an amusement park or in an airplane flying through turbulence).

6.1.5

Torsion Tensor Taking the skew-symmetric part of Γijk with respect to indices j and k, we can derive a tensor out of this (called antisymmetric connection), that is,       i jk   Bi B j Bk     Bi B j Bk      Bi B j Bk   



(6.1.13)

 Skl.. j , where [ ] means to take the skew-symmetric part with respect to the indices enclosed    in it. Since the second term on the right side vanishes, Bi B j Bk   0 via the commutable nature of the differentiation with respect to the indices j and k. The tensor thus obtained is called a torsion tensor.

6.1.6

Curvature Tensor The curvature tensor, as the name implies, describes how the space is curved. As illustrated in Figure 6.1.8, curved spaces can change the direction of a vector when a parallel displacement along a line is performed. Therefore, the noncommutativity of the covariant derivatives for the two different routes represents how the region encircled by the routes is curved. This operation gives the definition of the curvature tensor, that is, for an arbitrary vector quantity Ak

Figure 6.1.8  Definition and derivation of a curvature tensor, which measures the

noncommutativity of the covariant derivative of a vector quantity between two distinct routes.

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Figure 6.1.9  Alternative derivation processes of torsion and curvature tensors based on

commutation operations, against a scalar or vector quantity with respect to infinitesimal translation and rotation, respectively.

[i ,  j ] Ak   i  j   j i  Ak





  i ljk   j lik  lim  kmi  lim  kmj Ak (6.1.14)





Rijlk Ak .

Therefore,

Rijlk   i ljk   j lik  ljm  kmi  lim  km j  2  [ k il ] j  i j lm k ] .  

(6.1.15)

The above process is revisited in Figure 6.1.9 in a more rigorous fashion, together with that for the torsion tensor described in Section 6.1.5, where the discrepancies of a scalar quantity A and a vector quantity Ak , caused by commutations of the infinitesimal translational and rotational operations, define the torsion and curvature of the space of concern, respectively. Figure 6.1.10 provides a typical example of the operation for measuring curvature on a spherical surface, which is also intuitively tangible. As can be easily confirmed, after performing a parallel displacement along the triangle constructed on the great circles on the surface, the vector is rotated by 90°. This measures the curvature for the

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Figure 6.1.10  Typical example of a measuring process of curvature performed on a spherical surface, together with the corresponding definition of the curvature tensor.

triangle region, and the curvature tensor is defined per unit area. In other words, the curvature tensor mediates between a vector quantity of interest and its rotation for a prescribed region, that is,

 Ak   lik Al dx i  Rlijk Al dx i dx j . (6.1.16)



For the Riemannian space without torsion, the curvature tensor is rewritten, in terms of the Christoffel symbol given in Eq. (6.1.11), as   n   n   p  ... n Rklm  2  [ k     . (6.1.17)   l ]m   [ k p   l ]m 



6.1.7





Einstein Tensor It is clear that the curvature tensor Rijkl is a fourth-rank tensor, having four indices. The curvature tensor used in the equation of gravity, however, is actually a second-rank one called the Einstein tensor (Figure 6.1.11), defined as 1 Gij  Rij  gij R, (6.1.18) 2 where Rij and R are Ricci curvature and scalar curvature, respectively, derived by contractions of Rijkl as n ij R= ij R= ji Rinj and R = g Rij . (6.1.19)

The reason for using this instead of Rijkl is because of its divergenceless nature, that is,

 j Gij  0. (6.1.20) Satisfying this condition means that Gij is a conserved quantity. The derivation process of the Einstein tensor Eq. (6.1.18), as well as the proof for Eq. (6.1.20), are outlined in Figure 6.1.12, where the point of departure is the Bianchi identity, given as

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Figure 6.1.11  The definition of the Einstein tensor, derived from the curvature tensor, which is used in the equation of gravity in general relativity.

Figure 6.1.12  Outline of the derivation process for the Einstein tensor from the Bianchi identity, which ultimately produces the unique combination of Ricci and scalar curvatures satisfying the divergenceless condition.

l l Rijk  Rljki  Rkij  0, (6.1.21)



which reflects the symmetry of the curvature tensor, that is, Rijkl   R jikl   R jilk  Rklij . Taking a covariant differentiation of Eq. (6.1.21), we have l l l i Rmjk   j Rmki   k Rmij 0



 [ i Rlm jk ]  0.

(6.1.22)

This leads to

1    j  Rij  gij R   0, (6.1.23) 2   coinciding with Eqs. (6.1.20) and (6.1.18). The important thing in here is that Eq. (6.1.18) defines the unique combination of Rij and R that satisfies Eq. (6.1.20), as can be understood from the derivation process. It should be noted that the incompatibility tensor ηij , introduced in the following, is identical to this Einstein tensor.

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6.2  Non-Riemannian Plasticity

6.2

315

Non-Riemannian Plasticity On the basis presented in the preceding sections, we now can proceed to the DG field theory for plasticity. The fundamental framework was constructed by Kazuo Kondo and his colleagues. Here, we not only revisit the basic content, but also some extensions, rearrangements, and reinterpretations that will be made especially for our purposes.

6.2.1

Background and Overview Figure 6.2.1 presents a portrait as well as an article explaining the achievements of Kazuo Kondo (1911–2001), the originator of “non-Riemannian plasticity” and also a founder of the Research Association of Applied Geometry (RRAG) in Japan (Eringen, 1981; Ohnami, 1992). It reads: He is particularly famous for originating a framework called non-Riemannian plasticity. It established a unified theoretical framework connecting the microscopic lattice ­imperfections or defects within metallic materials and the macroscopic behaviors such as yielding, by regarding the distributed dislocations existing in crystalline media as a source of torsion of the material space.1

Figure 6.2.1  A description of the achievements of Kondo, a founder of non-Riemannian

plasticity, which forms the basis of “DG field theory.” 1

Author’s translation from the original article, www.u-tokyo.ac.jp/content/400004698.pdf, p. 30.

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Note that this description is slightly inaccurate to my mind, because it does not mention anything about the curvature of the space, for it is the concept that he emphasized on every other occasion, as will be discussed later.

6.2.2

Basic Quantities in Non-Riemannian Plasticity Since the essentials of the plasticity of metallic materials are motions and interactions of crystallographic imperfections, that is, dislocations and defects existing within crystalline space, it is natural and rational for us to use DG language in describing them. As schematically illustrated in Figure 6.2.2, the “torsion” of the crystalline space corresponds to dislocations, while the “curvature” describes all the other kinds of defects. Such defects include both simple ones like dislocation dipole, vacancy, foreign atoms, and precipitates, and also complicated fields, including dislocation substructures such as ladders, veins, labyrinths, and cells; furthermore, generalized inhomogeneities will be extensively discussed in Chapter 11. Let us draw on physical images and corresponding mathematical descriptions of torsion and curvature tensors in the present context. Figure 6.2.3(a) illustrates the naturalization process proposed by Kondo (1955), where the defected state of a

Figure 6.2.2  Physical images of crystalline fields specified by torsion tensor and curvature

tensor. The former corresponds directly to the existence of “dislocation(s)” as one recalls “the Burgers circuit,” while the latter embraces wider ranges of imperfections of global kinds, for example, not only aligned dislocations as dislocation dipoles and multipoles in general, but also other types of defects such as vacancy, impurity atoms, and precipitates.

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Figure 6.2.3  Physical image and processes that identify the torsion and curvature of a

medium, where a defected real state is released into a natural state: (a) A circuit enclosing imperfections is subject to closure failure (torsion) and/or rotation of material vector (curvature), and (b) by cutting the medium into pieces of material elements of the Euclidean kind, coordinates between the two states are interrelated via transformation between holonomic and nonholonomic coordinates.

plastically deformed crystal, referred to as real state, is released into small pieces without imperfection, such that each piece occupies Euclidian space, called the natural state. Since the topological correspondence between the coordinates dxκ in the a real state and  dx  in the natural state is not uniquely determined, the relationship between the two states, that is,

 dx a 

x a  dx  Ba dx , x

is not a total differentiation, but is regarded as the transformation between holonomic and nonholonomic coordinate systems. This natural state (nonholonomic) can be characterized by the translation and the rotation of a material vector v, that is, ∆x and ∆v, respectively, with respect to the holonomic coordinates, displaced in parallel along a circuit enclosing the defected region in real state, as depicted in Figure 6.2.3(b). The former corresponds to the existence of “torsion,” while the latter means having “curvature.” It is noteworthy that all kinds of imperfections, including dislocations and disclinations, can be rigorously described by arbitrarily combining the torsion and curvature tensors. This allows us a

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Figure 6.2.4  Physical images of real and natural states, where the former corresponds to duplex

grains in a plastically deformed state and the latter yields the following two states: (a) The two crystals get separated but with the same (mutually parallel) crystallographic orientations, and (b) the two crystals rotated independently to have different crystal axes. The former is described by “torsion” while the latter expressed as being “curved.”



complete mathematical setting to express dislocations and defects of any kind, providing a sound basis for the “description” part of the FTMP (some details are presented in Chapter 5 [see esp. Figure 5.3.1]). Figure 6.2.4 is a schematic drawing showing a distinction between “torsion” and “curvature” in terms of a crystalline aggregate (duplex grains). Since curvature measures the rotation of a material vector depending on the route to get there, there arises indeterminacy of grain orientations at the place, such a situation had been criticized as being “no-more crystal” by European researchers, for example, B. A. Bilby, an English physical metallurgist, at the time. Torsion, on the other hand, which had been accepted in contrast to curvature, connotes a “closure failure,” but this situation means “no-more continuum,” demonstrating that researchers’ criticism against the use of curvature tensor in plasticity was imperfect. By definition, defects represented by the torsion tensor, for example dislocations, are categorized as “topological defects,” whereas those expressed by curvature tensor are classified as “metrical defects.” This distinction, and the interrelationship between the two, will be of substantial importance in discussing “local versus global” aspects of multiscale plasticity, as mentioned in Chapter 14. Figure 6.2.5 explains the terminologies, Riemannian, non-Riemannian, and distant parallelism, depending on the nature of the space considered. When only a torsion ten... n sor exists, without curvature, that is, Rklm = 0 , the space is called distant parallelism. Since a torsion tensor in a crystalline space represents dislocated state, the theory in the distant parallelism space coincides with the dislocation theory. Non-Riemannian plasticity, constructed in the non-Riemannian space with both torsion and curvature ... n coexisting, that is, Skl.. j ≠ 0 and Rklm ≠ 0, can thus comprehend a much broader range of defect theories. Metric tensor in the strain space gives the measure of strain, as denoted in Figure 6.2.6. The difference in the metric tensor before and after deformation gives the strain tensor. Thus, the strain tensor is defined as 1  ij   gij   ij  (6.2.1) 2

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Figure 6.2.5  Classification of space in terms of torsion and curvature tensors.

Figure 6.2.6  The definition of a strain tensor in DG field theory is particularly clear. Since

the metric tensor in strain space provides a gauge to compare amounts of change in the local coordinates before and after deformation, the calculation provides nothing but the measure of strain. In the above, a flat space is chosen for the initial state (reference).





when measured from the initial state of being an Euclidian flat space. In the case when we take already distorted spaces as a referential configuration, Eq. (6.2.1) can be extended as 1  ij  gij  gij0 , (6.2.2) 2 with a metric tensor gij0 for the referential state. Figure 6.2.7 revisits the definition of the strain tensor in terms of the co/contravariant arguments presented in Figure 6.1.3. Figures 6.2.8 and 6.2.9 present the rest of the fundamental quantities in non-​ Riemannian plasticity. As already presented in Section 6.2.1, the covariant derivative of the space is defined, by using the coefficients of connection Γ klj , as





x i  dx i   klj x l dx k . (6.2.3) Torsion and curvature tensors are defined, as shown earlier, respectively, as



Skl.. j  [jkl ] , (6.2.4)



... n Rklm  2 [ k ln]m  [nk p lp]m  . (6.2.5)  

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Figure 6.2.7  Metric-based definition of the strain tensor with respect to the covariant/

contravariant bases described in Figure 6.1.3.

Figure 6.2.8  Overview of quantities and their interrelationship in non-Riemannian p­ lasticity (i.e., DG field theory). The coefficient of the connection introduced in the covariant derivative, which is given via metric tensor, provides definitions of torsion and curvature tensors, each measuring dislocation density and defect density. They are ­further reduced to second-ranked dislocation-density tensor and incompatibility tensor, respectively.

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Figure 6.2.9  Curvature tensor, defined via first differentiation of coefficients with respect to space–time coordinates. The coefficients of connection comprise a Christoffel symbol of the second kind and a combination of torsion tensors. For torsionless space (i.e., Riemannian space), the quantities are given by the former alone, called Levi-Civita parallelism.





The coefficients of connection Γ klj for the non-Riemannian space, that is, with t­ orsion, are given as k ijk     Sij..k  S .jk.i  S .kij , (6.2.6) i j  with the Christoffel symbol  k  1 kl    g   j gli   i glj   l g ji  . (6.2.7) i j  2 The curvature tensor for the torsionless space, that is, Skl.. j = 0, is expressed by Kijkl . The interrelationship among the three quantities (Figure 6.2.9) is given as



Kijkl  Rijkl  2[ l Si ] jk  2[ l Sk ]ij  2[ l S j ]ki (6.2.8)  Rijkl  2[ l S jk i ]  2[ j S li k ] . Here, the second line is reached following the procedure shown in Figure 6.2.9, based on the symmetric nature of Skl.. j. Contractions of these higher-order tensors, considering the symmetry, result in well-known second-rank tensors. These are called “dislocation-density tensor” and “incompatibility tensor,” given, respectively, by a curl of distortion tensor and double curl of strain tensor, that is,



.. j ij  ikl  k ljp  21 ikl Skl , (6.2.9)

ij  ikl  jmn  k  m  lnp 

1 ... n ikl  jmn Rklm ( g  det( gij )). (6.2.10) 4g

The important thing here is that these quantities are expressed as gradients of distortion or strain tensor in the context of continuum mechanics, meaning the theory intrinsically requires “strain gradients” at least up to the second order.

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Figure 6.2.10  Relationship among curvature tensors in non-Riemannian space and Riemannian space (i.e., Riemann–Christoffel kind in Levi-Civita parallelism), and torsion tensor, together with their contracted quantities, that is, incompatibility and dislocation-density tensors.

As demonstrated later, the dislocation-density tensor “includes” but is not limited to the recently popular concept of “GN” types of dislocations. Note that the energy duals of these quantities mathematically introduce their physical counterparts, that is, couple-stress tensor and stress-function tensor (Hasebe and Imaida, 1999), as will be discussed later. From Eq. (6.2.8), we have for Skl.. j = 0,

Kijkl  2[ l S jk i ]  2[ j S li k ] , (6.2.11) as shown in Figure 6.2.10. This corresponds to



 ij  ikl  k jl  jkl  k il  2(i kl  k  j )l .



This expression is revisited in Section 6.2.3 (Eq. (6.2.26)).

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323

Dislocation-Density Tensor Torsion of a crystalline space corresponds to dislocations, as can be understood via definition of the Burgers vector expressing the amount of the closure failure of a Burgers circuit encircling a dislocated region. As shown in Figure 6.2.11, the Burgers vector is expressed as bk   dxk     klp dxl C

C



   lmn  m  nkp dSl ,

(6.2.12)

S

where Stokes’ theorem  lkp dxl   lmn  m  nkp dSl is used to derive the second line. C S On the other hand, by definition, bk    lk dSl . (6.2.13)



S

By equating Eqs. (6.2.12) and (6.2.13), we have

 lk  lmn  m  nkp , (6.2.14)



which gives the definition of the dislocation-density tensor. The physical image of the dislocation-density tensor is illustrated in Figure 6.2.12. Since the first and the second indices represent the directions of the dislocation line (that penetrates the plane) and the Burgers vector, respectively, the diagonal (e.g., α11) and the off-diagonal (α12 , α 23 , ...) components express the edge- and the screw-​ dislocation components, respectively. We can also rewrite the dislocation-density tensor in terms of elasticity, that is, by utilizing the elastic distortion tensor βije in place of βijp , as illustrated in Figure 6.2.13.

Figure 6.2.11  Direct derivation process of the dislocation-density tensor based on its physical

picture.

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Figure 6.2.12  Physical image of the dislocation-density tensor.

Figure 6.2.13  Relationship between two expressions of the dislocation-density tensor in terms of the plastic distortion tensor and elastic counterparts.

For that, by making the replacement  nkp 

 u e   lk  lmn  m  k   nk   xn  (6.2.15) 2  uk e   lmn  lmn  m  nk . xm xn



Since lmn

un e   nk , we have xk

 2 uk  0, we finally obtain xm xn e  lk  lmn  m  nk . (6.2.16)

Let us consider a single dislocation expressed by a dislocation-density tensor for visualizing a more detailed image (see Figure 6.2.13)

bk   lk dSl , (6.2.17) where the first index l represents the normal to the plane through which the dislocation concerned penetrates, thus the direction of the dislocation line, and the second index

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Figure 6.2.14  Derivation and its physical meaning for the continuity condition of a dislocationdensity tensor.

k is that of the Burgers vector. Figure 6.2.25 (discussed later) illustrates the physical image of each component in α lk . Figure 6.2.14 shows the derivation process of the constraint condition for the dislocation-density tensor, which is concerned with the conservation of the Burgers vector. Consider a dislocation loop penetrating a plane. Since the net Burgers vector should be zero, we have

S lk dSl  0  V llk dV  0, (6.2.18)



where the Green–Gauss divergence theorem is used. For the above to hold for arbitrary volume elements, we obtain

 l lk  0 (6.2.19) This is referred to as the conservation of the Burgers vector, indicating that dislocations do not have the end point within the media considered except the surfaces. This can be easily confirmed as



V llk dV  V lmn l  m  nk dV  0 (6.2.20) p

because the partial derivatives ∂ l ∂ m are commutative, that is,  l  m   m  l .

6.2.4

Incompatibility Tensor To understand the incompatibility tensor, it will be helpful to learn first about the compatibility condition in classical continuum mechanics (3D), following that in the mechanics of materials (2D). They are given, respectively, by



pki qlj  k  l  ij  0 or curl curl  0, (6.2.21)



 2211  12 22  21112  0, (6.2.22)

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Figure 6.2.15  The definition of the incompatibility tensor looks complicated and difficult

to understand. This schematic intuitively explains the incompatibility tensor deduced from the “compatibility condition” of total strain by splitting it into elastic and plastic parts.

which ensures the uniqueness of the corresponding displacement field is determined from the given strain field. The incompatibility, hence, measures the deviation from the “compatible” state in terms of displacement field. In other words, it represents the degree of “indeterminacy” of the displacement vector for the prescribed region, whose image coincides with that for the “curvature” in space provided in Section 6.1.6. Figure 6.2.15 provides another explanation of the meaning of the incompatibility tensor. With the decomposition of strain into elastic and plastic parts, comes the following equation





ikl  jmn  k  m  lne   lnp  0





ikl  jmn  k  m  lnp

  ikl  jmn  k  m  len .

(6.2.23)

Both sides can have finite (nonzero) values, and be called incompatibility tensors, defined both in terms of plastic and elastic strain tensors, that is,

ij ikl  jmn  k  m  lnp , (6.2.24)



ij   ikl jmn  k  m  lne . (6.2.25) Figure 6.2.16 lists various expressions for the dislocation-density and incompatibility tensors found in the literature. The incompatibility tensor is also rewritten via the dislocation-density tensor, ij   ikl  k jl , or more explicitly,





sym

1 ij   (ikl  k jl   jkl  k il ) (6.2.26) 2   (i kl  k  j )l , where “sym” and ( ) represent symmetrization with respect to the indices enclosed with those within being kept out.

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Figure 6.2.16  Various expressions and notions for dislocation-density and incompatibility

tensors.

6.2.5

Resolution into Slip Systems The dislocation-density tensor and the incompatibility tensor have simply been the concepts within the continuum-mechanical framework (phenomenological), and so have no physical correspondence to crystal dislocations and defects within themselves. Application examples along this line are given in 1.4.4. However, we are fortunately able to attain corresponding physical interpretations by specifying ­crystallographic information such as that associated with slip systems and the Burgers vector. Figures 6.2.17 and 6.2.18 show resolutions of the dislocation-density tensor and incompatibility tensor in several physically sound directions associated with a prescribed slip system. Thus, for example, for the dislocation-density tensor, we can distinguish between edge and screw components as follows,

 

 

( )  edge  t ( )  s( ) :    ( )  screw  s( )  s( ) :  



( )  edge  ti( ) s (j ) ij  , (6.2.27)  ( )  screw  si( ) s (j ) ij

where t ( )  s( )  m( ) or ti( )  ijk s (j ) mk( ) denotes the line direction in the case of dislocation. The total density can be evaluated, for example, as ( )

( )

( )

2  total   edge2   screw , (6.2.28)

or else,





 k( )    k  ti( ) s (j )  si( ) s (j ) ij . (5.6.17) ( )

k

Note, we use Eq. (5.6.17) for our modeling and simulations throughout the book, which is revisited in Section 5.6.2. For the incompatibility tensor, essentially the same can be done, but, due to its generic features, another projection toward the normal direction to the given plane will be possible:

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Figure 6.2.17  Projections of the dislocation-density tensor into slip systems, through which its edge and screw components are explicitly evaluated.

Figure 6.2.18  Projections of the dislocation-density tensor into slip systems, through which its edge, screw, and disclination-related components are explicitly evaluated.

  

 

( ) edge  t ( )  s( ) :    ( ) ( ) ( ) : screw  s  s  ( ) discl  m( )  s( ) :  





( ) edge  ti( ) s (j )ij   ( ) ( ) ( ) screw  si s j ij . (6.2.29)  ( ) ( ) ( ) discl  mi s j ij

More details will be discussed in Section 6.3.2 in conjunction with the description of the “disclination” field. In Eq. (6.2.29), the last component is expressed tentatively ( ) as discl assuming disclination. By analogy to Eq. (6.2.28), we may express the total incompatibility as ( )

total  sgn( H ) H , (6.2.30)

with

H   sgn(k ) k ( )

k

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( )

2

, (6.2.31)

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Figure 6.2.19  Method for the explicit evaluation of edge and screw components of GN dislocation density, calculated directly from the spatial gradient of slip strain in each slip system, divided by the magnitude of the Burgers vector (Ohashi and Sawada, 2002). This allows a physically sound interpretation of GN dislocation density as that of “crystal” dislocations.

or else, more simply as,

k( )   sgn(k ) k ( )





( )

k



 ti( ) s (j )  si( ) s (j )  si( ) m(j ) ij ,

(5.6.18)

where k = edge, screw, discl represent the components. It should be noted that, unlike the dislocation-density tensor, the distinction in the sign needs to be considered for the incompatibility tensor. Direct evaluation from the slip strains in each slip system was first proposed by Ohashi (1997), as presented in Figure 6.2.19, ( )  ( ) 1 ( )     s  GN,edge i xi b  , (6.2.32)  ( ) 1 ( )   ( )  GN,screw  b ti xi 



which has served as the basis of the ideas presented earlier about the projection of strain-gradient quantities. He also proposed to use the norm of the two components as a measure of the GN dislocation density, that is, ( )

GN 





( )

GN,edge

   2

( )

GN,screw

 , (6.2.33) 2

together with the line directions identified as

( )

lGN 

1 ( )

GN



( )

GN,edge t

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( )

( )

 GN,screw s

( )

. (6.2.34)

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By using these interpretations, he successfully reproduced a dislocated state around a cuboidal precipitate in terms of dislocation-loop formations (Ohashi and Sawada, 2002).

6.2.6

Explicit Representations of Dislocation-Density and Incompatibility Tensors Let us make some effort to familiarize ourselves with the notions of dislocation-density and incompatibility tensors by viewing several examples. More advanced and more sophisticated images will also be given in Section 6.5. Figure 6.2.20 presents what the strain gradients corresponding to dislocation density and incompatibility (given by Eqs. (6.2.9) and (6.2.10)) are like. Here, four kinds of 1D plastic strain ε 22 (or distortion β 22 ) distributions are initially given, that is, lump and trapezoidal shapes (T. Ohashi, 1998, personal communication) as well as their inverted (upside-down) images. In this case, the dislocation density and incompatibility are simply obtained as



 32  1  22 (6.2.35)  2 2 33  1  22  1  22 Characteristic distributions of dislocation density and incompatibility are observed each corresponding to the given strain variation in Figure 6.2.20. In particular, the distributions of incompatibility yield intriguing profiles, implying important contributions to additional hardening and softening depending on the incompatibility sign when used in the hardening law of a constitutive equation. We have a rather broad negative region for the lump-shaped strain distribution, whereas localized regions with sharp fluctuation result from trapezoidal one. Some intuitive illustrations assuming bi- and tricrystals are given in Figure 6.2.21. Since there should arise mutually different deformation modes for the comprising crystal grains, we normally have “incompatibility” around the interfaces or boundaries. To accommodate these, arrays of dislocations must be introduced therein. For the tricrystal case, with the deformation modes of individual grains assumed there, we expect to have a “disclinated” state around the triple point (Ohashi, 1990), which will be expressed by the incompatibility tensor, as discussed in Section 6.3. Figure 6.2.22 show a classification of defects by Volterra (1907) including not only edge- and screw-dislocation types, but also disclinations (details of disclination types of defects and their mathematical descriptions are presented in Section 6.3). As one can readily notice, the defect type displayed in the bottom right (referred to as “wedge” disclination) of the figure corresponds to the above-mentioned state in the context of the tricrystal. Let us next look at an explicit example of incompatibility distribution corresponding to the case of lump-shaped strain or distortion distribution in Figure 6.2.20, which can be regarded as that mimicking a shear-banded region. The component-wise expressions for both the dislocation-density and incompatibility tensors are given in Figures 6.2.23 and 6.2.24. For comparison, the Laplacian of strain tensor is presented in

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Figure 6.2.20  One-dimensional examples of first and second derivatives of strain with typical profiles, each corresponding to dislocation density and incompatibility, supposing their future uses in constitutive modeling.

Figure 6.2.21  Intuitive images of dislocated or disclinated states expressed in terms of a continuous dislocation distribution using examples of bi- and tricrystals.

Figure 6.2.25; this has been frequently used in the context of plastic instability (Aifantis, 1984; Zbib and Aifantis, 1988) for the purpose of regularizing the targeted problems. Figure 6.2.26 compares the results of incompatibility tensor (η33 component) with the Laplacian  2 12 evaluated from a given γ 12 distribution for a shear-banded region

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Figure 6.2.22  Volterra’s defects together with corresponding components of the curvature

tensor.

Figure 6.2.23  Component-wise expressions of the dislocation-density tensor and the incompatibility tensor.

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Figure 6.2.24  Variation of decompositions for the incompatibility tensor with respect to the physical meanings, that is, edge versus screw, spherical versus deviatoric, and rotation versus deformation (see Figure 6.3.2 for details), in index notations (left column) and direct notations (right column).

Figure 6.2.25  Laplacian of strain tensor defined also as the second derivative, often used in the context of regularization of plastic instability, which resembles but is basically different from the incompatibility shown in Figures 6.2.23 and 6.2.24.

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Figure 6.2.26  Two-dimensional comparison of expressions for shear-banded fields between the incompatibility tensor and Laplacian of strain. They are calculated based on the 2D shear-strain distribution given in the far left (courtesy of Doctor M. Uchida).

(M. Uchida, 2005, personal communication). Band-like regions with the negative sign of η33 sandwiched by positive-signed bands are observed, corresponding to the 1D analysis in Figure 6.2.20. Also, we find similarity in the morphologies between η33 and  2 12, although the latter shows relatively isotropic distribution stemming from its definition (Figure 6.2.25).

6.2.7

Dual Structure between Strain Space and Stress Space Figure 6.2.27 shows the dual construction of strain and stress spaces. In non-​ Riemannian plasticity, the strain space is considered transcendentally and the stress or stress-function space is defined based on the energy duality. The spaces are constructed as metric, torsion, and curvature, following the order of differentiation with respect to the space–time coordinates. For the strain space, these are strain, dislocation density, and incompatibility. We define stress as the energy dual of strain, as is well known. Similarly, for dislocation density and incompatibility, we introduce “dual dislocation α ij ” and “stress function χij ” as their energy duals, according to Minagawa (1962) and Amari (1962). The stress-function tensor is regarded as a physical action against the growth of incompatibility in this context, similar to the case of stress against strain. Dual dislocation reads the same, that is, a physical action resisting the increase in dislocation density. There are twofold dualities, one being “geometrical correspondence” and another being “physical” or “mechanical correspondence.” It is noteworthy that, for the stress

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Figure 6.2.27  Dual construction of strain and stress space based on non-Riemannian plasticity, possessing twofold duality, that is, geometrical and physical.

Figure 6.2.28  Derivation process of the stress-function tensor as an energy dual (work conjugate) of the incompatibility tensor.

(function) space, the metric is the stress-function tensor, and its second differentiation generates the stress tensor corresponding to the curvature of the space. This is in sharp contrast to the strain space case, where the strain tensor acts as the metric, most fundamental quantity, the first derivative of which generates dislocation-density and the second differentiation yields the incompatibility tensor. According to the construction, the stress tensor is already a nonlocalized quantity. Figures 6.2.28 and 6.2.29 summarize the derivation processes of the “stress-​ function” tensor and the “dual-dislocation” tensor based on the energy duality mentioned earlier, respectively.

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Figure 6.2.29  Definition of a dual-dislocation tensor as an energy dual (work conjugate) of the dislocation-density tensor, together with an expression of the stress-function tensor.

As shown in Figure 6.2.28, a point of departure is

Wp

1 1  ij  ijp dV   ijij dV , (6.2.36) 2 2

where W p represents the plastic work, so that χij denotes the work conjugate of the incompatibility tensor ηij . Substituting the definition for ηij in terms of the plastic strain tensor, that is, Eq. (6.2.24), into the above equation, we have, 1 ij (ikl  jmn  k  m  lnp ) dV 2 (6.2.37) 1 p   (ikl  jmn  k  m  nl ) ij dV . 2

Wp



For the derivation of the last line, integration by parts, in addition to a boundary condition for the surface integral to vanish, is used. By comparing the above with Eq. (6.2.36), we finally reach  ij  ikl  jmn  k  m  nl , (6.2.38) giving a definition of the stress-function tensor. For a dual-dislocation tensor, similarly, from



Wp





1  ji ij dX 2

or  ij  

W p (6.2.39)  ji

we have the explicit expression

 ij  2 ikl  k lj . (6.2.40)



Figure 6.2.30 is a comparison of the analogous constructions of the incompatibility tensor and stress-function tensor. The double contraction of the fourth-rank curvature tensor defines the second-rank incompatibility tensor as

ij 

1 ... n ikl  jmn Rklm ( g  det( gij )), (6.2.41) 4g

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Figure 6.2.30  Generalization of the stress-function space based on an analogy with strain-space construction. The Einstein tensor of curvature in the stress-function space gives the definition of the stress tensor, which automatically satisfies the equilibrium condition as a consequence of its nondivergent nature.



which is further rewritten as the double curl of strain tensor at infinitesimal strain, that is, ij  ikl jmn  k  m  lnp . (6.2.42) By analogy, we can constitute a corresponding expression for the stress-function tensor to the incompatibility–curvature relationship, that is,







 ij 





1 . . .n ikl  jmn  klm   det   ij  , (6.2.43) 4

based on the mathematically equivalent definition for stress function to that for incompatibility tensor, that is,  ij  ikl  jmn  k  m  nl (6.2.44) The expression given in Eq. (6.2.43) accentuates the fact that stress is regarded as the . . .n in the stress-function space, which is a rather abstract notion without a curvature Σ klm clear physical image. A somewhat intuitive image of it will be given in Section 6.6.1. The incompatibility tensor is also the Einstein tensor in the strain space, derivable from the curvature tensor via 1 ij  Rij  gij R. (6.2.45) 2

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Thus defined, the Einstein tensor, as presented previously, automatically satisfies the divergenceless condition  iij  0, (6.2.46)



corresponding to the Bianchi identity for the curvature tensor. Similarly, we also have the corresponding expression of stress tensor as an Einstein tensor of the curvature in the stress-function space as

1  ij  ij   ij , (6.2.47) 2 where Σij , Σ are the Ricci curvature and scalar curvature of the stress-function space, respectively, defined by ... k ij   ji  ikj ,    ij ij . (6.2.48)



Thus, given the stress tensor, by nature, must satisfy Eq. (6.2.49),  i ij  0. (6.2.49)



which coincides with (is nothing more than) the stress equilibrium condition.

6.2.8

Torsion in Stress-Function Space Let us consider a physical meaning of the torsion tensor defined in the stress-function space from a purely mathematical point of view (Minagawa, 1968). Let the torsion tensor be formally defined and expressed as Sij.. k . From the first Bianchi identity, as shown in Figure 6.2.31, we have, for the curvature and torsion tensors, ...i .. i .. m .. i [ lkj ]  2[ l Skj ]  4 S[ lk S j ]m . (6.2.50)



By taking contraction with respect to l and i, for the purpose of deriving the Ricci ...i curvature out of Σ[ lkj ] on the left-hand side, we finally obtain





[ ji ]   m S ji.. m  2[mj Si ] . (6.2.51)

Since

1   ji   ji   ji  , (6.2.52) 2    ji : Symmetric 

the right-hand side reads [ ji ]   [ ji ] . (6.2.53)



This expresses the skew-symmetric part of stress tensor, which must be balanced with the gradient of couple stress in the polar media, otherwise it will vanish. For this to be nontrivial, we can interpret S ji.. m  2[mj Si ] in the right-hand side as the couple stress, that is,











. .i mj  2 S ji.. m  2[mj Si ] . (6.2.54)

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Figure 6.2.31  Physical meaning of a torsion tensor in stress-function space. From the Bianchi identity, the balance equation for couple stress with the antisymmetric part of the stress tensor is derived, meaning non-Riemannian plasticity intrinsically embraces couple-stress theory.

Therefore, Eq. (6.2.51) is rewritten as 2 [ ji ]   m  ji. . m , (6.2.55)



which reproduces the balance equation for the moment (couple) stress in the Cosserat continuum theory. From Eq. (6.2.54), we can define the “moment-stress” or “couple-stress” tensor by mij 



6.2.9

1 imn mnj . (6.2.56) 2

Relationship between mij and α ij Figures 6.2.32 and 6.2.33 present a process to examine the relationship between a couple-stress tensor mij and a dual-dislocation tensor α ij , defined by Eqs. (6.2.39) or (6.2.40). Based on the content provided therein, we can conclude that they express basically the same concept, that is,

 ij  mij . (6.2.57)



6.2.10

Curvature in Stress-Function Space From the second Bianchi identity . .n . . .i [ m [ lk. . ].ji  2 S[ ml  k ]nj , (6.2.58)

we can derive

. . n . . . i kj i m. i   mi S[ mi  k ]nj  , (6.2.59)

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Figure 6.2.32  Relationship between the couple stress as a natural manifestation of the torsion tensor in a stress-function tensor and a dual-dislocation tensor obtained as the energy dual of a dislocation-density tensor.

Figure 6.2.33  Relationship between the couple-stress and the dual-dislocation tensors, demonstrating their formal equivalence.

where the contraction with respect to the indices k and j are taken after multiplying the metric tensor γ ji by both sides of Eq. (6.2.58), while  ij  ij  (1 / 2) ij , as in Eq. (6.2.47). Since the right-hand side of Eq. (6.2.59) is a higher-order term with respect to γ ji and S ji. . k , it becomes negligible when they are small enough. In such a case, we have

i m. i  0. (6.2.60) This is a generalized stress equilibrium condition corresponding to Eq. (6.2.49).

6.3

Disclination and Curvature Tensor As is pointed out in conjunction with Figure 6.2.22 (Volterra defects), the curvature tensor covers not only dislocation-based translational defects but also disclination-like rotational defects. Figure 6.3.1 classifies the disclinations. The incompatibility tensor

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Figure 6.3.1  Schematic of disclinations roughly classified into two kinds just as in dislocations,

that is, wedge and twist types, depending on the relationship between the Frank vector and the disclination-line directions. The latter can be further distinguished as a splay or twist type. See also Figure 6.2.22.

can be decomposed into pure deformation and pure rotation parts, and the latter will correspond to the disclination field (Aoyagi and Hasebe, 2007). In what follows, we describe explicitly how the incompatibility tensor can distinguish the rotational nature from its definition of its translational part.

6.3.1

New Physical Interpretation of Incompatibility The incompatibility tensor can be decomposed into its pure rotational and translational parts (Aoyagi and Hasebe, 2007; Kleinert, 1989), as summarized in Figures 6.3.2 and 6.3.3, where both the index and direct notation versions are presented, respectively. Using the plastic distortion tensor βljp, the dislocation-density tensor α ij is given by

 ij   ikl  k ljp , (6.3.1)



where ∈ikl is the permutation symbol. When we decompose the plastic distortion tensor into components of strain ε ljp and rotation ωljp , Eq. (6.3.1) becomes

 ij   ikl  k ( ljp  ljp )

  ikl  k  ljp  ikl ljm  k wmp 

 ikl  k  lpj

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  ij  k wkp

  j wip ,

(6.3.2)

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Figure 6.3.2  Decomposition of the incompatibility tensor into pure rotation and pure deformation parts, in index notation, where the former coincides with the disclination-density tensor.

Figure 6.3.3  Decomposition of the incompatibility tensor into pure rotation and pure

deformation parts, in direct notation.

where ωljp is rewritten by the corresponding axial vector wip, that is, ljp  ljm wmp . From Eq. (6.3.2), the trace of the dislocation-density tensor is written in the form

 ii   ikl lij  k w jp

 2 kj  k w jp 

(6.3.3)

2 j w jp .

In Eq. (6.3.2), transposing the first term in the right-hand side to the other side and substituting Eq. (6.3.3), we obtain

ikl  k  lnp   n wip   in  k wkp   in 1   (6.3.4)   n wip    in   in kk  . 2   Note that the characters are replaced in Eq. (6.3.4). The terms in the parenthesis of Eq. (6.3.4) correspond to contortion tensor K ni , defined by Nye 1953) as



1 K ni   in   in kk . (6.3.5) 2

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p Meanwhile, the incompatibility tensor ηij is written as the following form with ε nq ,



p ij  ipq  jmn  p  m nq . (6.3.6)

The result of multiplying the left-hand side of Eq. (6.3.4) by  jmn  m coincides with the right-hand side of Eq. (6.3.6). Therefore, the incompatibility tensor takes the form

ij  jmn  m  n wip  jmn  m K ni . (6.3.7) Introducing the disclination-density tensor (Kleinert, 1989; Minagawa and Ogata, 1990)



ij    jmn  m  n wip (6.3.8) into Eq. (6.3.7), the incompatibility tensor is expressed by the use of Θij and K ni as



6.3.2

ij  ij  jmn  m K ni . (6.3.9)

Components of the Disclination-Density Tensor The disclination-density tensor is defined by



ij  li  j , (6.3.10) where li denotes the direction of disclination line and Ω j the Frank vector (Kleinert, 1989). Considering the relationship between li and Ω j , the disclination-density tensor can be further decomposed into wedge, splay, and twist components, as shown in Figure 6.3.4. For the slip system with slip direction si(α ) and normal direction of slip plane mi(α ), if we let the disclination plane and the disclination line correspond to the slip plane and ti( )  ijk s (j ) mk( ) as in Figure 6.3.4, respectively, each component of disclination density is obtained as a mapping of Θij to each slip system, such that



) (wedge  ti( )t (j )ij   ( ) ( ) ( ) splay  ti s j ij . (6.3.11)  ( ) ( ) ( )  twist  ti m j ij

Figure 6.3.4  Schematics showing how one can explicitly evaluate the three components of the

disclination tensor classified in Figure 6.3.4 from the pure rotation part of the incompatibility tensor.

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Figure 6.3.5  The continuity equation in the presence of disclination, and its component-wise

expression.

Examples of the three types of disclination density evaluated in a simulation are given in Section 11.5. Explicit components of Θij in terms of the axial vector wip are summarized as a matrix representation as (Figure 6.3.5)



 ( 2  3   3  2 )w1p  ij   ( 2  3   3  2 )w2p  p  ( 2  3   3  2 )w3

( 3 1 1 3 )w1p ( 3 1  1 3 )w2p ( 3 1  1 3 )w3p

(1 2  2 1 )w1p   (1 2   2 1 )w2p  . (6.3.12)  (1 2   2 1 )w3p  

Alternatively, by using  ijp   i w jp , we can rewrite Eqs. (6.3.10) and (6.3.12) as ij  ikl  k  ljp

p p  3 21   2 31    1 3p1   311p  p p   211  1 21

p p  3 22   2 32 p 1 32   312p p  212p  1 22

p p   3 23   2 33  (6.3.13) p 1 33   313p  . p   213p  1 23 

With the existence of such rotational defects, the equation of continuity for dislocation-​ density tensor α ij, that is, Eq. (6.2.19), is altered as (Figure 6.3.6)

 i ij   jkl  kl , (6.3.14) with its component-wise representation



 111   2 21   3 31  (23  32 )  112   2 22   3 32  (31  13 ) . (6.3.15)            (   ) 12 21  1 13 2 23 3 33 Let us consider a 2D case with ( x1 , x2 ), where the surviving component is 33  (1 2   2 1 )w3p . As one can readily learn from Eq. (6.3.15), there is no contribution from Θ33 to α ij field. This is a natural consequence of the 2D character of

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Figure 6.3.6  Overview of the derivation process of the disclination-density tensor.

the dislocation-density field; it automatically satisfies  i ij  0 (since what survive in this case are α 31 and α 32, they make no contribution to the right-hand side of Eq. (6.3.15)). On the other hand, for the translational part  jmn  m K ni , 1 K 23   2 K13  1 32  2 31 holds. Regarding the relationship between Θij and Kij , one has from  jij  0,

 j ij   jmn  j  m K ni  0 (6.3.16)   j ij   jmn  j  m K ni . Similarly, all the components vanish in the 2D case in Eq. (6.3.16). Note that an example of the redistribution of ηij field into Θij and Kij degrees of freedom is mentioned in Section 11.5, based on a simulation result. Another definition of the disclination-density tensor can be found as





1 1 ijq R jil...k  ijq  j  l  ikp . (6.3.17) 2 2 The derivation process of this is summarized in Figure 6.3.6, where the rotation of a frame due to the existence of curvature is explicitly considered (Lardner, 1974; Ohnami, 1980). As demonstrated in Figure 6.3.7, this definition is equivalent to that of the incompatibility tensor ηij , which can be easily confirmed by contracting with respect to indices k and l after multiplying ∈lkm. We finally have dlkq 

dij ikl  jmn  l  m  knp , (6.3.18) where dqm  2 lkm dlkq . This coincides with the definition of the incompatibility tensor, Eq. (6.2.24).

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Figure 6.3.7  Relationship between the disclination-density tensor and the incompatibility

tensor, showing their equivalence.

6.4

Fundamental Equations for the Dislocation Field

6.4.1

General Description Historically, the incompatibility tensor has been utilized in the context of calculating the internal stress field produced by continuously distributed dislocations based on isotropic elasticity (Kröner, 1958). Figure 6.4.1 summarizes the basic equations for the process, that is,



div T  0    curl curl  e    curl  sym . (6.4.1)   e 1    (tr ) I   E E  As discussed earlier, the stress field is given with respect to the stress-function tensor as

  curl curl  . (6.4.2)



Kröner employed the following stress function

 

1    (tr  ) I  , (6.4.3)   2G  1  2 

together with a fixing condition for the stress-function tensor, such that the stress components are uniquely determined, that is,

div   0. (6.4.4)



By thus substituting the given stress tensor Eq. (6.4.3) into the incompatibility equation (Eq. (6.4.1) second line) through the constitutive equation (Eq. (6.4.1) first line), we finally reach the following equation (Beltrami equation), after rather lengthy algebraic calculations,  2  ( x)   ( x) (6.4.5)

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Figure 6.4.1  A set of fundamental equations for obtaining the internal stress field based on the

dislocation-density tensor field via Kröner’s stress-function tensor.

Figure 6.4.2  Schematic showing the derivation process of a biharmonic-like equation for

ultimately obtaining the stress field from given dislocation-density distribution.

(some details of the derivation process are presented in Figure 6.4.2). As one may notice, the above equation resembles the biharmonic equation, 2    ( x)  0, to be solved in the conventional theory of elasticity, in which the “incompatibility” condition is replaced by the compatibility condition for strain, that is, curl curl  e  0, in the second line of Eq. (6.4.1).

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Figure 6.4.3  Explicit evaluation of the stress-function tensor as a solution of biharmonic-like

differential equations derived in Figures 6.4.1 and 6.4.2. A special solution is available based on the Green function method (Kröner, 1958).

Figure 6.4.4  Derivation of Poisson’s equation for the hydrostatic component of the

stress-function tensor, brought about by the hydrostatic component of the stress tensor.



For evaluating the stress-function tensor explicitly, we may use the Green function method, as overviewed in Figure 6.4.3. The special solution for Eq. (6.4.5) is given explicitly as 1  kl (x)   x  x kl (x)dV , (6.4.6) 8 V  by utilizing the Green function (Figures 6.4.4 and 6.4.5), and is rewritten via the dislocation-density tensor α as



 kl (x)  

1  lmn   nk (x) m RdV  , (6.4.7)  sym V  8 

The special solutions are thus given by a volume integral form, based on which any stress field can be calculated from given dislocation-density distributions. The explicit expression for the stress field is written as

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Figure 6.4.5  An example of the stress-function tensor calculation based on the scheme in

Figure 6.4.4. The result is revisited in Figure 15.4.6.

V kl (x)



ª P §1· D rn (xc) « lmn G kr w m ¨ ¸ ³ c ' V 4S ©R¹ ¬



1 º Rmn G kl ’ 2  w k w l w m R » dV c, 1 Q ¼







(6.4.8)

with R  x  x . In deriving the above,  2 R  2 / R is used. In the case where we obtain the stress field for a dislocation line, as is required in formulating DD, it is useful to introduce a replacement, dl   nk (x)dS  bn dlk , (6.4.9)





in Eq. (6.4.8). By doing so, the volume integral can be rewritten by a line integral, that is, 1 lmn bn   m Rdlk  kl (x)   , (6.4.10)  sym 8 from which we can derive



V kl (x)



P bn 4S

ª



§1·

³ * «¬ 2w m ¨© R ¸¹(l mn dlkc )

­ ½ º 1 §1·  rmn ®G kl w m ¨ ¸  w m w k w l R ¾ dlrc » . 1 Q ©R¹ ¯ ¿ ¼

(6.4.11)

This equation provides the foundation of DD. For details of a more conventional derivation process, together with the anisotropic version in terms of elasticity (which is

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much more complex and needs mathematical techniques and a numerical scheme for calculating it), readers can refer to the recent detailed book by Ghoniem and Walgraef (2008). An alternative method for obtaining the stress-function tensor from Eq. (6.4.5) is presented in Figure 6.4.4, where the equation is reduced to Poisson’s equation when we limit the discussion to the trace (hydrostatic component). This limitation seems to be unusual; however, it will be useful in discussing field fluctuation in terms of “duality,” the details of which will be presented in Chapter 12. Figure 6.4.5 is an example of the  m  1 / 3   kk contour, obtained by solving  m   kk to give a  m distribution (shown on the left-hand side), which is for a dual-phase (DP) alloy model, further discussed in Section 15.4.1.

6.4.2

Some Miscellaneous Comments about the Dislocation Line The displacement field in a crystal containing a dislocation loop (Mura, 1963); Volterra, 1907) (see Figure 6.4.6) is given by ui ( x )    D ejlmn bm



S

 Gij ( x, x )nn dS ( x ), (6.4.12) xl

which is referred to as Volterra’s displacement formula. Here Gij ( x, x ′) represents Green’s function, giving a point-force solution for the EOM of an elastic medium, that is, e  j ji  fi  0  Dijkl  l ulk  fi  0. (6.4.13)



The corresponding distortion tensor to Eq. (6.4.12) is obtained as ui, p ( x)   D ejlmn bm  Gij ,l p ( x  x)nn dS ( x)



S e   D jlmn bm pql

 C

Gij ,l p ( x  x)dx ,

(6.4.14)

where the explicit form of Gij ( x, x ′) (Mura, 1963) is given by

Gij ( x  x) 

1 1 16 (1 ) x  x

(xi  xi)(x j  x j )   (3 4 ) ij   . (6.4.15) (x  x)2  

The line integral form over a closed dislocation loop obtained by de Wit (1960) reads (Figure 6.4.7),

ui  

bi 4

1



1



 C Ak dlk  8  C ikl bl R, pp  1  kmn bn R,mi  dlk , (6.4.16)

from which we can calculate the stress field for a dislocation loop as

 ij  

 4



1



 C  R,mpp ( j mn dli )  1  kmn  R,ijm dev dlk . (6.4.17)

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351

Figure 6.4.6  Mura’s formula using Green’s function for a general curved dislocation line

(Mura, 1963).

Figure 6.4.7  Comparison of expression for the displacement field of a curved dislocation line

between Mura’s formula and that derived by de Wit (1960), together with the corresponding stress fields.

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6.5

Physical Images of DG Quantities

6.5.1

Dislocation-Density Tensor: GN Dislocations It is relatively easy to display physical images about the dislocation-density tensor, especially in the “GN dislocation” context. Figure 6.5.1 depicts one of the most understandable illustrations (T. Ohashi, 1996, personal communication). Consider a shearstrain distribution first, which indicates an already slipped region (left two boxes) and a nonslipped region (the extreme right box), and the transition region located in between. Obviously, dislocations must be in the transition region only where the gradient of shear strain has finite value. The density of such dislocations is counted as ( / x ) / b. Since the transition region must have been introduced so as to accommodate the unbalanced plastic strain between the regions on either side, the above should be called GN dislocations. Hence,



1  . (6.5.1) b x Figure 6.5.2 provides another example concerning GN dislocations. The top of the figure shows a state with a uniform plastic shear strain. In such a case, “statistically stored” (SS) dislocations, with exactly equally signed populations, are assumed to have carried the plastic deformation. Further uniform plastic deformation yield increase in SS dislocation density, as in the middle. Imposing a distributed plastic strain means that GN dislocations intrude from the left-hand side, as depicted in the bottom picture. Figure 6.5.3 is a 1D representation of dislocation-density (equivalent to GN dislocation) tensor α and incompatibility tensor η , expressed by the first and second derivatives, respectively. The two-step shear-strain distribution shown in the figure results in two corresponding humps for α and two sets of continuously aligned negative–positive (down-and-up) humps for η. The hump for α can be interpreted as the

GN 

Figure 6.5.1  Intuitively tangible physical image of “GN” dislocations defined as the spatial

gradient of strain (T. Ohashi, 1996, personal communication).

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Figure 6.5.2  Relationship between “SS” and “GN” dislocations. The former comprises

opposite-signed dislocations of equal number, producing no strain gradient on average, while the latter is composed of one-signed populations causing spatial gradient of strain.

Figure 6.5.3  Physical image of an incompatibility field along the same lines as for a

dislocation-density field equivalent to “GN” dislocations, shown in Figures 6.5.1 and 6.5.2.

density of one-signed dislocations, whereas the down-and-up hump corresponds to a pair of dislocation rows of negative and positive signs. As displayed in Figure 6.2.22, mathematically all the types of defects, including ... n dislocations and disclinations, are describable via the curvature tensor Rklm , that is, the incompatibility tensor. In what follows, I attempt to provide a tangible image of the incompatibility tensor in a multiscale perspective.

6.5.2

Incompatibility Tensor-driven Inhomogeneity Evolution By combining all the above primitive images for α and η out of the derivatives of strain, we can gain a slightly more advanced image of the two quantities. This subsection provides such an example.

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The torsion and curvature tensors defined earlier can play pivotal roles in describing not only dislocation and defect fields but also generalized inhomogeneously deforming fields in any scale levels, for example, dislocation substructures in grainsize orders and SSS in grain-aggregate orders. Since the inhomogeneous fields are evolved ultimately as a consequence of motions and interactions of crystallographic imperfections, including both dislocations and defects, their generalized physical images are also attributed ultimately to the redistributions of dislocations and defects. Introducing the incompatibility tensor to, for example, the hardening law in an “appropriate” manner, as will be detailed in Chapter 11, has been demonstrated to soften the overall responses, and is associated with additionally introduced “modulations” in the deformation field (Aoyagi and Hasebe, 2007). This stems from supplementary accommodation mechanisms that are embodied by the incompatibility field. An intuitive physical image of this accommodation process is given in Figure 6.5.4, schematizing a dislocation rearrangement into a kink-band configuration as one of the possibilities. When a region of a material is subjected to localized shear (right-hand side of Figure  6.5.4(a)), a nonuniform plastic deformation should be introduced, resulting in a localized strain gradient around the interfacial region between, before, and after the passage of one-signed dislocations, producing locally an “incompatible” deformation. Figure 6.5.4(b) illustrates the corresponding “virtual” configuration, which is conventionally interpreted as compensated by “GN” dislocations (e.g., Fleck and Hutchinson, 1997). To lower the strain energy of the whole system, however, further rearrangement of the dislocations will be necessary, to accommodate the locally intruded “incompatible” deformation. Such a process can be achieved by introducing an incompatibility-tensor field. An example of the accommodation is illustrated in Figure 6.5.4, exhibiting a kink-band formation. This roughly corresponds to a bandlike pattern produced in the incompatibility distribution extending plus and minus signs, which is obtained in the second gradient of the locally introduced strain. The “appropriate manner” in the previous paragraphs means the following. In taking account of the incompatibility tensor, for example, in the hardening law, one must consider the sign in addition to its magnitude. Clearly, the incompatibility distributions extending both positive and negative signs are indispensable, to take the earlier mentioned accommodation processes into account, without which additional hardening may result by doubly counting the contribution. The modulated patterns that emerge in the incompatibility field are thus introduced; their morphological features, such as directionality, depend on the crystallography, that is, crystal orientation and the associated number of active slip systems (Aoyagi and Hasebe, 2007). The above image can also capture another important feature of the incompatibility tensor-based description of inhomogeneous fields in the field theory of plasticity. The use of the incompatibility tensor at a certain scale level allows us to account for the  underlying “smaller scale” degrees of freedom that are not usually considered in the model being used. In the example in Figure 6.5.4, a redistribution of dislocations is expressed via an incompatibility field without explicitly introducing or modeling dislocation degrees of freedom.

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355

Figure 6.5.4  Practically feasible image of an incompatibility-tensor field, evaluated initially

based on given strain distribution, whose profile foretells how a locally intruded deformation should be accommodated as a result of the redistribution of dislocations. A kink-band formation is taken as an example.

6.6

Physical Images of Incompatibility and Stress-Function Tensors Analogous to Granular Media Representation

6.6.1

Graph Representation of Granular Assembly For a continuum description of the fluctuating fields, as a result of the collective effects to be discussed in Chapter 12, use is made of graph theory (Hasebe, 2004a), based on which we can attain another physical image of the incompatibility tensor ηij and the stress-function tensor χij . Figure 6.6.1 illustrates schematics of a particle and a void graphs based on the graph theory, responsible for force and deformation transmissions respectively (Oda and Iwashita, 1999; Satake, 1978). This viewpoint can be a leverage for the new notions to be examined in Chapter 12, that is, SSS and its dual FCS (see Section 12.2) as sources of field fluctuations, ultimately to be related to ηij and/or χij . The particle graph is constructed by connecting the centroids of all the particles, where the particles, contact points, and voids in the assembly correspond to points (centroids of particles), branches, and loops, respectively. If we consider the oriented branches, called branch vectors, the graph is called an oriented graph. The void graph,

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Figure 6.6.1  Schematics of particle and void graphs based on the graph theory.

on the other hand, is a twin to the particle graph, where the centroids of the voids are interconnected. A branch in the void graph is called a dual branch, whose orientation is given by a counter-clockwise rotation of the corresponding branch for the particle graph (Oda et al., 1999; Satake, 1978). There exists a constraint condition called “Euler’s formula” expressing a sort of invariant associated with the graph representation. For a number of particles, contact points N c, and voids N v, the following relationship should hold:

N p  N c  N v  1. (6.6.1) A particle in the particle graph should be regarded, for example, in the context of polycrystalline aggregates, as treated in Chapter 12, as an aggregate of hardened grains rather than an individual grain, whereas a void is that of softened grains. For more details about the graph theory in the context of mechanics of granular materials, refer to Satake (1978) and Oda and Iwashita (1999). Void force FV and void displacement UV are, respectively, related by particle force FP and particle displacement U P through the fundamental matrices D PC and L PC (referred to as incidental and loop matrices, respectively) as



 FP   ( D PC LCV )FV  0 , (6.6.2)  UV   ( LVC D CP )U P  0

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6.6  Images of Incompatibility and Stress-Function Tensors

357

Figure 6.6.2  Schematic of incident and loop matrices in a graph-theoretical representation of

particles and voids.

which express balance of force and compatibility of displacement, respectively. Here = D PC LCV 0= , LVC D CP 0 (6.6.3)



should hold for the two fundamental matrices as the constraint conditions (identities) (Oda and Iwashita, 1999). Details of the incident and loop matrices are presented in Figure 6.6.2. Their definitions are given as follows, that is, for the incident matrix,



 1 : If particle p is positively connected at contact c;  i.e., branch vector I is oriented away from p. c  D PC  1 : If particle p is negatively connected at contact c; . (6.6.4)  i.e., branch vector I c is oriented toward p.   0 : Otherwise, and for the loop matrix,



 1 : If branch vector I c is included in loop v  and orientations of I and loop v coincide. c   . (6.6.5) LVC  1 : If branch vector I c is included in loop v  and orientations of I c and loop v do not coincide.   0 : Otherwise, Figure 6.6.3 provides tangible illustrations of particle and void graphs, attempting to visualize the situations where the above equilibrium and compatible conditions hold, respectively.

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Figure 6.6.3  Schematics showing the equilibrium condition for particle force and the

compatibility condition for void displacement, expressed by means of incident and loop matrices based on graph theory.

There is a mathematical correspondence between the fundamental matrices in graph theory and the differential operators, such that Eq. (6.6.3) corresponds to

 0, ()  0, (6.6.6)



= curl grad 0= , div curl 0. (6.6.7) Therefore, we also have corresponding expressions to Eq. (6.6.2) as



         curl curl or  , (6.6.8)           curl curl which coincide with the definitions of stress function (Eq. (6.2.44) with respect to elastic strain tensor) and incompatibility tensor (Eqs. (6.2.21) or (6.2.42)), respectively. These intercorrespondences can provide motivation to introduce the incompatibility tensor and stress-function tensor into the continuum description of the inhomogeneous fields. As shown in Figure 6.6.4, the stress-function tensor, corresponding to FV , characterizes the deviation of the stress field from its equilibrium. Fluctuating stress, which is averaged out macroscopically to be zero, can be regarded as this sort. Therefore, this can be a suitable mechanical parameter describing why and how stress fluctuation evolves. The incompatibility tensor, on the other hand, is given as a second derivative (more precisely, double curl) of the strain tensor, defining, as it stands, the degree by which the strain-compatibility condition breaks.

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359

Figure 6.6.4  Relationship between the incompatibility tensor and stress-function tensor,

emphasizing their new interpretations as measures of field fluctuations in strain and stress, respectively, in the sense of local deviations from strain compatibility and stress equilibrium.

The quality as well as the quantity of the stress fluctuation can be discussed in terms of the correlation function of  



6.6.2

1  ( x) ( x  x ) dx. (6.6.9) V V  The correlation length ξ will be used in explicitly evaluating χ , to be discussed in the following section. Here ξ is expected to coincide with or to be in proportion to the “affected zone” and is determined as a result of micro–macro-interactions through collective behavior. Explicit evaluation of the correlation function of stress field is provided in Chapter 15 in a somewhat different context. Further decomposition of   into deviatoric and hydrostatic components can clarify the roles for “local plasticity” and the SSS evolution, respectively: G( ( x)) 

1 I           m I       m . (6.6.10) d d   The stress and strain field fluctuations in terms of polycrystalline plasticity will be discussed extensively in Chapter 12.

A Comment on Quantum Stress and the Stress-Function Tensor Since the second differentiation with respect to space generates the stress tensor (in the above context, inhomogeneous stress), the stress-function tensor is regarded as a sort of potential function of the inhomogeneity in terms of stress. There is an analogous notion in quantum mechanics (Pauli, 1933). It is called “quantum stress” or “electronic stress” (Figure 6.6.5), defined as



 ij 

2  *i  j  i  j *  i j *  i * j , (6.6.11) 4m





where ϕ and ϕ * are the wave function and its Hermitian conjugate. This equation asserts that the second spatial derivative of the wave function provides the stress tensor. In this case, the wave function representing electronic states is regarded as a source of force.

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Figure 6.6.5  Schematics for obtaining quantum or the electronic-stress tensor from the wave

equation (Pauli, 1933).

6.7

Theory of Interaction Fields

6.7.1

Relative Deformation and Interaction Fields Let us consider two scales as a simplest case, that is, coordinates in macro- and microscales, to be denoted by xi , Xi and xi , Xi , and so on, respectively, as shown in Figure 6.7.1. Here, i, j =1, 2, 3. All the other quantities referring to the fine scale will be expressed by attaching a single “bar” to the macroscopic counterparts. The quantities in the interaction field, on the other hand, will be similarly expressed by attaching a “tilde” instead. We introduce a relative deformation between the two scales for the purpose of considering the field interactions between them in the DG sense. The relative deformation is defined as a transformation between the line elements dxi and d xi for a current configuration (Ikeda, 1975), that is,





xi dx j  Aij dx j , (6.7.1) x j where Aij defines the relative deformation between the two scales. The inverse relationship is given by d xi 

dxi 

xi d x j  Aij1d x j . (6.7.2) xj

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361

Figure 6.7.1  Relative deformation between micro- and macro-metric fields as a fundamental

setting for deriving interaction fields.

This exists only when the determinant of the relative deformation tensor satisfies

J int  det(Aij ) 

 xi  0. (6.7.3) x j

This insures the coexistence of the two scales during field evolution. The transformation of the line vectors before and after deformation is expressed in terms of a deformation-gradient tensor for both scales, namely,



xi  dxi  X dX j  Fij dX j  j , (6.7.4)   x d xi  i dX j  Fij dX j  X j



 Fij  ij  ij . (6.7.5)   Fij  ij  ij Here, βij , βij denote distortion tensors for the respective scales. The covariant derivative for the relative deformation is given by



DX k  dX k  ijk X i dx j  ijk X i d x j (6.7.6) k k i  dX k  ijk X i dx j  im Amj ) X i dx j . X Amj dx j  dX k  (ijk  im So if we write the above by expressing



DX k  dX k   ijk X i dx j, (6.7.7)

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then the coefficient of connection in the interaction field is defined as k  ijk  ijk  im Amj . (6.7.8)



If we assume affinely interacting deformation fields, the relative deformation tensor is simply expressed by a scalar quantity, that is, a meaning of “scale ratio” characterizing a spatial degree of separation between the two scales, Aij  e 1 ij with e 1  l l , multiplied by Kronecker’s delta (identity tensor). Considering this to be the simplest case, we rewrite Eq. (6.7.8), using the scale ratio, as k  ijk  ijk  im  e 1 mj  ijk  e 1ijk . (6.7.9)



Substituting Aij  e 1 ij into Eq. (6.7.2), we have a relationship between the two-scale deformation-gradient tensors, that is, Fij   im nj Fmn Fij . (6.7.10)



6.7.2

Expressions for DG Quantities For the two-scale interaction field, the differentiation (Figure 6.7.2) is defined as      i  . (6.7.11)  Aim   e 1 xi  xm xi  xi



Therefore, by definition, the coefficient of connection (Figure 6.7.3) is given as



    ijk  Flk  i Fjl  Flk   e 1  xi  xi

Fjl Fjl  (6.7.12)  e1 Flk  Fjl  Flk  xi xi 

Thus, we have the total coefficient of connection given as a summation of those in the macro- and micro-fields, that is,  ijk  ijk  e 1ijk , (6.7.13) which agrees with Eq. (6.7.9). The skew-symmetric part of the coefficient of connection, as in Eq. (6.1.13), gives the torsion tensor in the interaction field, as shown in Figure 6.7.3. Sijk   [kij ]  [kij ]  e 1[kij ]  Sijk  e 1Sijk . (6.7.14)



The curvature tensor, on the other hand, is also given by using Γ ijk based on Eq. (6.7.13). Substituting Eq. (6.7.13) for Eq. (6.1.15) (see Figure 6.7.4), we have R lij...k  2  [ l  ik] j   [kl m im] j   





 2  [ l  e 1[ l 

  ik] j  e1ik] j    [kl m  e1[kl m   im] j  e1im] j 



 2 [ l ik] j  [kl m im] j  e 2 [ l ik] j  [kl m im] j 







e 1 [ l ik] j  [ l ik] j  [kl m im] j  [kl m im] j  . 

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(6.7.15)

6.7  Theory of Interaction Fields

363

Figure 6.7.2  Definition of differentiation in relative deformation setting by using scale ratio.

Figure 6.7.3  Coefficients of connection in the interaction field and an attendant definition of the

torsion tensor, where no interaction component is introduced.

Equation (6.7.15) demonstrates that there explicitly exists an interaction between the two scales for the curvature tensor field. The interaction term, given in the last line of Eq. (6.7.15), is expressed by the anticommutation of sequential differentiations with respect to the two scale coordinates xi and xi . The torsion tensor, as in Eq. (6.7.14), on the other hand, yields no interscale coupling, expressed as a summation of the individual-scale quantities. Note that this is similar to the situation for the Finslerian space-based descriptions of crystalline imperfections (Amari, 1962). The significance of these results (Figure 6.7.5) is the following. Explicit interscale interactions can come into play only through the curvature tensor field as far as the previously discussed framework asserts. Since the curvature tensor can describe inhomogeneities on any scale level, we now have a definite way to be able to deal with interscale couplings and the consequent evolutions of multiple inhomogeneously deforming fields within the framework of continuum mechanics in an explicit manner. Next, we derive the corresponding continuum mechanics-based expressions to the torsion and curvature tensors obtained earlier (Figure 6.7.6). As shown in Eqs. (6.2.9) and (6.2.10), single and double contractions of the torsion and curvature tensors respectively produce two second-rank tensors called dislocation-density and incompatibility tensors, that is,

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Figure 6.7.4  Derivation process of the explicit expression of curvature tensor in the interaction

field, where not only curvatures in macro- and micro-fields, but also the interaction term, are introduced.

Figure 6.7.5  Summary of interaction-field formalism, starting from relative deformation and

ending with explicit expressions for torsion and curvature tensors.

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365

Figure 6.7.6  A contracted version of interaction-field formalism, leading to explicit

expressions of the dislocation-density and incompatibility-tensor fields, corresponding to the torsion and curvature tensors, respectively.

ij  ikl  k ljp  e 1ikl  k ljpij  ikl  k ljp  e 1ikl  k ljp







 ikl  k ljp  e 1 k ljp ,

(6.7.16)

and

 ij  ikl  jmn  k  m  p  e 2 ikl  jmn  k  m  p ln ln



 e 1ikl  jmn  k  m  lnp  k  m  lnp







ikl  jmn  k  m  lnp  e 2  k  m  lnp  e 1

(6.7.17)



 k  m  lnp

 k  m  lnp

.

It should be noted again that Eq. (6.7.16) implies that we cannot take into account any interscale coupling explicitly based only on the conventionally and widely used “GN” dislocation densities whose definition is identical to the dislocation-density tensor.

6.7.3

Extension to Multiple Scales The previously obtained framework for the interaction field can be readily extended to multiple fields with more than three scales in a straightforward manner. In what follows, Eqs. (6.7.16) and (6.7.17) are extended to a three-scale interaction field. In expressing such multiple scales, the direct notation, together with sub- or superscripts standing for the scale levels, will be used in place of index notation for simplicity in the following. Considering two Scales A and B, we rewrite Eqs. (6.7.16) and (6.7.17) in the direct notation as

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1    curl B  Bp  eBA curl A  Ap . (6.7.18)

and





2 1   curl B curl B Cp  eBA curl A curl A Ap  eBA curl B curl A Ap  curl A curl B  Bp . (6.7.19)

Let us consider three scales as a first extension. Let the three scales be called A, B, and C, from small to larger scales, with Scale B regarded as a reference scale. Since the same relationships should hold between arbitrary scale pairs, namely, A and B as 1 well as B and C, substitution of the curl operation curl B  curl B  eBA curl A, is made in Eqs. (6.7.18) and (6.7.19). After some algebraic calculations (Figure 6.7.7), the three-scale expressions of the interaction field are derived as 1 1    eBC curlC  Cp  curl B  Bp  eBA curl A  Ap (6.7.20)

and

2 2   eBC curlC curlC  Cp  curl B curl B Bp  eBA curl A curl A Ap









1 1 curl B curl A Ap  curl A curl B  Bp  eBC curl B curlC  Cp  curlC curl B  Bp  eBA







2 1 curl A curlC  Cp  curlC curl A  Ap ,  eBA  eAC

(6.7.21)

1 1 1  eCA  eBA is used in the second line of Eq. (6.7.21). Note that the product where eBC l l 2 1 of scale ratios in the third line reads eBA  eAC  C2A . Eqs. (6.7.20) and (6.7.21) can lB be rearranged in a more compact form,

and

 C   curlC  pC  1  C 1  A   B   eBC   B  eBA     curl B  pB (6.7.22)  A pA   curll A  2  C  B 2  A   eBC    eBA  1 1 C pC   curlC curlC   eCB curlC curl B pB  eCA curlC curl A pA  B 1 1 pC pB pA (6.7.23)   eBC curl B curlC   curl B curl B  eBA curl B curl A .   A  e 1 curl curl  pC  e 1 curl curl  pB  curl curl  pA A A AC A C AB A B 



Expressions for multiple fields with more than four scale levels can be similarly derived simply by repeating the same process as for three scales. As shown in Figure 6.7.8, the corresponding matrix expressions to Eqs. (6.7.22) and (6.7.23) are given as  C  curlC  B       0  A   0   

and

 C   curlC curlC  B   1     eBC curl B curlC  A  e 1 curl curl A C    AC

0 curl B 0

pC 0       0    pB  (6.7.24) curl A    pA   

1 eCB curlC curl B curl B curl B 1 eAB curl A curl B

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1 eCA curlC curl A   pC    1 eBA curl B curl A   pB  . (6.7.25) curl A curl A   pA 

6.7  Theory of Interaction Fields

367

Figure 6.7.7  Extension to three-scale interaction fields, referred to as Scales A, B, and C.

Figure 6.7.8  Matrix representation of the three-scale incompatibility-tensor field.

6.7.4

Some Remarks on Multiple-scale Formalism The off-diagonal components in the curl matrices, for example those of Eq. (6.7.25), represent interscale field couplings, without which independency of each scale cannot be rationalized. The curl matrix for incompatibility, Eq. (6.7.25), is asymmetric in gen1 1 1 1 1 1  eBC , eCA  eAC , eBA  eAB eral because of the asymmetry of the scale ratios, that is, eCB , and also due to a loss of information caused by coarse-graining, for example, during the alternating curl operations with respect to the distinct scales. A pair of off-diagonal components in the curl matrix can only become symmetric when two scales coincide.

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The loss of information can be measured by the anticommutation relation of the curl operations, that is, by a change in the order of curl operations.

curl I , curl J   curl I curl J  curl J curl I , (6.7.26) where, in the present case, I , J express either A, B, or C . Therefore, the noncommutativity of the curl operations with respect to the distinct two scales gives a measure of the field interaction, that is, the interscale correlation:



[curl I , curl J ]  curl I curl J  curl I curl J . (6.7.27) For the same distribution of ε p , we generally have [curl I , curl J ] = 0 when spatially different scales are considered, making the curl matrix symmetric if there is no information loss. In real situations, for example those associated with polycrystalline plasticity, as will be discussed in Section 13.2, we expect to have [curl I , curl J ] ≠ 0 for the assumed Scales A, B, and C, corresponding to the dislocation substructures, intragranular and transgranular inhomogeneities, respectively, because of their distinct physical and geometrical origins and the change rates of field evolutions. The origins are as follows – that is, collective behavior of dislocations and concomitant patterning for Scale A (e.g., with cellular morphology) and accommodation of deformation for Scales B and C. Moreover, Scale B yields modulated distributions, having more or less directionalities in stress and strain fields due to accommodation of the deformation based on intragranular crystallography, that is, crystal orientation and associated redistribution of dislocations (e.g., with lamellar morphology). Meanwhile, Scale C exhibits distributions associated with transgranular accommodation of inhomogeneity not yielding modulated patterns in general (e.g., with nearly random morphology). Figure 6.7.9 illustrates a set of intuitive images for systems enriched with the interaction fields both in micro- and macro-perspectives, where the former takes into account the effect of the surrounding regions that can mimic a sort of embedded conditions, while the latter accounts for microscopic perturbations, both via the relevant interaction terms of the incompatibility tensor, respectively. Here, the effect of energy-momentum fluctuations on a scale level are converted to the other scales through the interaction-incompatibility fields, as schematized in Figure 6.7.10 with respect to Scales A, B, and C. Extending this simple thought further allows us to draw a picture, given also in Figure 6.7.10, referred to as “duality network.” The dual relationship between the incompatibility tensor and the fluctuation of the energy-momentum tensor is derived in Chapter 15, and is called the flow-evolutionary hypothesis. It should be noted that this formalism can be readily extended to the space–time framework based on which the coupling or interaction among distinct temporal scales are explicitly dealt with simultaneously. This is also intriguing as well as providing challenging open questions that we must face when accomplishing the multiscale modeling of materials. The temporal aspects of interscale coupling are closely related to the evolutionary features of multiple fields of interests, which are partially involved in the constitutive descriptions to be modeled in the analysis.

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Figure 6.7.9  Schematics illustrating intuitive pictures about the use of the interaction fields in

two representative situations, that is, microscopic and macroscopic systems.

Figure 6.7.10  An extended view of interaction-field formalism with respect to the “flow-evolutionary hypothesis” derived in Chapter 15, where a sort of energy-momentum flow constructs a “duality network” that exchanges multiple-scale pieces of information via the incompatibility-tensor field.

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Also worthy of note is that the present formalism is not confined to continuum mechanics but can be applied to discrete pictures of dislocations and defects, for example, based on discrete dislocations as well as atomistics as far as the incompatibility tensor can be evaluated. For discrete dislocation representations, plastic distortion or strain-tensor distribution is difficult to evaluate in a straightforward manner, however, the dislocation-density tensor can be evaluated from the given configuration, from which components of the incompatibility tensor are explicitly calculated. Some specific examples utilizing this idea are found in 15.5 of Chapter 15.

6.7.5 Summary This section has described in detail a concept and the associated derivation process of the interaction fields, applicable to multiple-scale problems in general based on the field theory of plasticity. Relative deformation between two distinct scales, for example, macro- and micro-fields, are considered based on which all other DG quantities, that is, the coefficients of connection, torsion tensor, and curvature tensor, are derived. Thus obtained two-scale interaction formalism is extended to multiple scales with more than three levels. Demonstrated are explicit coupling in the curvature tensor field, equivalent to the incompatibility-tensor field, and no coupling in the torsion, that is, the dislocation-density tensor field. The interaction fields are implemented in the crystal plasticity-based constitutive equation through which we can tackle multiscale problems in the light of interactions in an explicit manner.

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7

Gauge Field Theory of Dislocations and Defects

7.1

Fundamentals of Gauge Theory One of the main advantages of using gauge formalism as a field theory is its sophisticated mathematical structure, based on analytical mechanics. Everything about the system dynamics that we need can be derived by rote from a Lagrangian density of the system, which itself can be uniquely determined based on the prescribed symmetry underlying the physical phenomenon we want to describe. In our case, we can find how the dislocation and defect fields should be incorporated into the continuum theory of elasticity, with direct correspondences to the DG counterparts introduced in Chapter 6. Also, the formalism can provide us with a bridge between the DG pictures and the method of QFT discussed in Chapter 8 via the Lagrangian density. This section summarizes the gauge theory for both the commutative and noncommutative types by using examples of electromagnetism and the Yang–Mills field, respectively. We start out with electromagnetism based on Maxwell’s equations as a classical gauge theory to which many of the readers will be familiar. This is followed by a more advanced version of gauge formalism which gives a basis of quantum electromagnetic dynamics (QED) as a prototype of the commutative gauge theory, where a purely mathematic process can derive all the equations identical to the classical Maxwell equation. The noncommutative Yang–Mills gauge theory, which has basically the same mathematical constitutions as the commutative type, is discussed in Section 7.1.3.

7.1.1

Maxwell’s Equation as a Classical Gauge Theory Figures 7.1.1 and 7.1.2 summarize Maxwell’s equations for the classical electromagnetism as a typical example of the classical gauge theory. Maxwell’s equations are a set of four partial differential equations describing the properties of the electric and magnetic fields, composed of the following four equations:



   E  em     E   B  t , (7.1.1)   B  0  E    B  jem  t 

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Figure 7.1.1  Classical representation of Maxwell’s equations, together with the notion of gauge

potential coupled with gauge transformation.

Figure 7.1.2  Pictorial presentation of Figure 7.1.1 for Maxwell’s equations.

where B is the magnetic field and E is the electric field. The first and the fourth lines of Eq. (7.1.1) relate them to their sources, that is, the charge density ρem (scalar) and the current density jem (vector). Some supplementary descriptions are provided in Figure 7.1.3 for obtaining explicit expressions for B and E. The third line of Eq. (7.1.1), expressing “absence of magnetic current,” indicates that B should be further given as a curl of a vector field; while substituting this into the fourth line of Eq. (7.1.1), the modified Ampère’s law, we attain an expression for E as the gradient of a scalar field, added by the change rate of the above vector field. Therefore, by setting the vector and the

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373

Figure 7.1.3  The interrelationship between Maxwell’s equations and gauge potential-based

formalism.

scalar potentials as A( x ) and φ ( x ), respectively, the magnetic and electric fields are further expressed as  B   A   A . (7.1.2)  E     t



This shows that the vector and scalar potentials, A and φ , are more fundamental quantities than B and E. As can be readily confirmed in Eq. (7.1.1) together with Eq. (7.1.2), Maxwell’s equation is invariant under the transformations

 A  A  A      , (7.1.3)         t where χ is an arbitrary function. These operations are referred to as “gauge transformation.” This is the very reason why the theory of electromagnetism is regarded as a classic gauge theory. Let us proceed to a more abstract but compact expression of Maxwell’s equations. Figure 7.1.4 displays the preparation for it and the output. By rewriting the potentials introduced in Eq. (7.1.2) into a four-vector form A  ( , A), and simi larly by expressing the charge and current densities (sources) with jem  ( em , jem ), Maxwell’s equations can be concisely rewritten as   F   jem , (7.1.4)

with

F     A   A  2 [  A ] . (7.1.5)

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Figure 7.1.4  Unified representations of vector/scalar potentials and currents, each referred

to as four-vector and four-current, respectively. Consequently, Maxwell’s equations can be rewritten in compact form with respect to the field strength (curvature).

Here, F  is called the field strength (which coincides with Eq. (7.1.11) in Section 7.1.2). The gauge transformation given in Eq. (7.1.3) can be rewritten completely as

A  A   A     . (7.1.6) From Eq. (7.1.4), two of Maxwell’s equations, that is, the third and fourth lines of Eq. (7.1.1), are reproduced, as presented in Figure 7.1.5. The first two equations in Eq. (7.1.1) are derived from the Bianchi identity, that is,



  F   F     F   0, (7.1.7) as is also shown in Figure 7.1.5.

7.1.2

Electromagnetic Gauge Theory (Commutative) In gauge theory, the point of departure is always a Lagrangian (density). Just by imposing the invariance of the Lagrangian under a gauge transformation associated with the intrinsic symmetry of the field variable, we can derive all equations via purely mathematical operations. This is quite powerful in the sense that the formalism can be applied to a broad range of situations as far as such intrinsic symmetry can be identified. In the case of electromagnetism, as summarized in Figures 7.1.6 and 7.1.7, the Lagrangian for a matter field is given as a function of wave function and its derivative

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375

Figure 7.1.5  Evaluation of the explicit form of Maxwell’s equations from the field equation

based on gauge formalism for electromagnetism, together with the Bianchi identity.

Figure 7.1.6  Overview of the gauge theory of electromagnetism. A point of departure is to

write down the Lagrangian, against which the invariance under local gauge transformation via U(1) gauge is imposed. The gauge field is introduced in order to satisfy this requirement in the form of minimal replacement via covariant derivative. The field strength is defined from the gauge potential, called curvature.

(from which the Schrödinger equation is derived, corresponding to the Euler–Lagrange (E–L) equation via action S   [ x ] dr [see Section 7.3.1]), that is,

[ x ]  i † ( x ) t ( x ) 

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2  i † ( x ) i ( x ), (7.1.8) 2m

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Figure 7.1.7  Substitution of the covariant derivative into the Lagrangian of the matter field,

together with construction of the gauge-invariant Lagrangian for electromagnetism defined by a quadratic of the curvature, completing the electromagnetism minimally coupled with the matter field within the gauge formalism.

where ψ (x) expresses the wave function and ψ † (x) is its Hermitian conjugate. Under local gauge transformation

 (x)   (x) G(x)  (x) or  ( x)    ( x)  ei ( x ) ( x), (7.1.9) with a commutative gauge group U (1)  ei ( x ), the Lagrangian must be invariant. For this condition to be satisfied, the derivative must be replaced as





e A ( x), (7.1.10) c where i, e, and c are the imaginary unit, the electric charge, and the speed of light, respectively, and   h / 2 the reduced Planck constant. Here, Aµ ( x) represents the gauge field (the phonon field, in the context of QED), with the role of absorbing the additional terms caused by the local gauge transformation, Eq. (7.1.9). The righthand side of Eq. (7.1.10) is called the covariant derivative, as in the differential geometry in Chapter 6. The operation expressed in Eq. (7.1.10) is called “minimal replacement.” The field strength is evaluated from the gauge field Aµ ( x) and is given by     i

F  2[  A ] , (7.1.11) which agrees with Eq. (7.1.5) in Section 7.1.1. Again, this is also derived from the noncommutability of the covariant derivative. With the above replacement, we have a gauge-invariant Lagrangian as follows,

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[ x ]  i † ( x ) t ( x ) (7.1.12) 2  e e      i  i Ai ( x)  † ( x )   i  i Ai ( x))  ( x ) .  2m  c c    Note that this specifies how the electromagnetic field as a gauge field interacts with the matter field (i.e., the background field) described by ψ (x). For the gauge field, the gauge-invariant Lagrangian is also constructed as



1 F F . (7.1.13) 16 By combining Eqs. (7.1.8) and (7.1.13), we have a total Lagrangian for the matter EM [ x ] 

[ x ]  [ x ]  EM [ x ]. (7.1.14)

From the action

SEM   EM [ x ] dr, (7.1.15)



we derive the field equation (E–L equation), based on a variational principle (Figures 7.1.8 and 7.1.9), that is,  SEM  0, as A   (  A )  jEM , (7.1.16)



1 2 2 2 is the d’Alembertian operator, with   being the c 2 t 2 x i x i Laplacian. Maxwell’s equations are all derived from Eq. (7.1.16) together with the Bianchi identity with respect to F (see Figures 7.1.8 and 7.1.9). where    2 

Figure 7.1.8  Relativistic version of Maxwell’s equation derived as a field equation based on the

variation principle.

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Figure 7.1.9  Electromagnetism as a consequence of local gauge transformation via U(1) for

the matter field represented by the wave function. The introduced gauge field A generates the electromagnetic field via curvature.

7.1.3

Yang–Mills Gauge Theory (Noncommutative) If the gauge group (Lie group) to be used is commutative, that is, ab = ba , then the theory is called commutative or Abelian gauge theory, whereas for the “noncommutative” gauge group, that is, ab ≠ ba, the framework is referred to as noncommutative or non-Abelian gauge theory. In the electromagnetism theory shown earlier, the gauge group U(1) is commutative, so the theory is commutative. Yang and Mills (1954) constructed the basis of what is known today as the Yang– Mills gauge theory by extending the concept into a noncommutative gauge group, SU(2), for the purpose of dealing with the proton and neutron fields in a unified manner (Figures 7.1.10 and 7.1.11), that is,



   Pr oton    p  , (7.1.17)  n   Neutron  as the doublet matter field with proton ψ p and neutron ψ n. Similar to the case of electromagnetism, in order to make the Lagrangian invariant



M ( n ( x ),   n ( x )) (7.1.18) under the local gauge transformation of the matter field ψ n ( x )



  n ( x )  exp(ien ( x )) n ( x ) , (7.1.19)     A  ( x )  A  ( x )    ( x ) the theory requires the partial derivative   l ( x ) to be replaced by the covariant derivative. This replacement is given as

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Figure 7.1.10  Yang–Mills gauge theory as a noncommutative-type gauge formalism. The gauge

group is replaced by a non-Abelian Lie group; otherwise all the procedures for constructing the framework are the same as the commutative counterpart.

Figure 7.1.11  Original version of the Yang–Mills gauge theory proposed in 1954 as a theory for the doublet field of proton and neutron via the SU(2) gauge group of isotropic spin-gauge rotations analogous to that of electromagnetism.



 D ( x) l    l ( x)  iA  ( x)(t )l m m ( x), (7.1.20) where A  ( x ) is the gauge field, which transforms as in the second line of Eq. (7.1.19), and Λ( x ) is an arbitrary function. The field strength (curvature) is similarly defined by



F     A   A  C  A A , (7.1.21)

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and the associated gauge-invariant Lagrangian is constructed as 1    g F  F   . (7.1.22) 4



Here t β in Eq. (7.1.20) is a set of constant matrices obeying  [t , t  ]  iC t , (7.1.23)



 with C being a set of real constants, referred to as the structure constants of the  assumed gauge group (in the present case, C  ). The total Lagrangian of the matter interacting with the gauge field is given by



7.1.4 7.1.4.1

1    g F  F    M ( , D ). (7.1.24) 4

Expansion of Gauge Theory Gauge Theory in Theoretical Physics As mentioned earlier, gauge formalism is a very powerful method for constructing field theories in a mathematically closed manner. In the field of theoretical physics, it actually is believed to provide a unified framework for all forces existing in the universe, that is, the strong force, the weak force, the electromagnetic force, and even the gravitational force, as schematically illustrated in Figure 7.1.12. The gauge groups necessary for theory constructions are SU(3), SU(2), U(1), and the general coordinate transformation group, respectively. The distinction between commutative and noncommutative gauge theory depends on the nature of the gauge group to be used. The corresponding fields indicating particles responsible for transmitting the forces

Figure 7.1.12  Gauge theory as a unified framework of all the forces of this world, even including gravitational force. The gauge fields in the respective cases correspond to the particles responsible for transmitting the forces.

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Table 7.1  Comparison between “gauge theory” in physics and “fiber-bundle theory” in mathematics. Gauge theory

Fiber-bundle theory

Gauge type Gauge Gauge group Gauge transformation Gauge field   Gauge potential Field strength Yang–Mills field Wave function Phase factor Integrability condition Action (functional)

Principal fiber bundle Local trivialization Structure group Bundle automorphism Fiber bundle  Connection  Curvature Yang–Mills connection Section of fiber bundle Parallel displacement The Bianchi identity Yang–Mills functional

are gluons, weak bosons, photons, and gravitons, respectively. Interested readers can refer to the literature (Weinberg, 1995, 1999; Uchiyama, 1956). It is also worth noting that the gauge theory is mathematically equivalent to the fiber bundle theory, where the fiber at each point of a base space corresponds to the gauge field (in this case, the “connection” is termed an Ehresmann connection). The interrelationships among the terminologies are listed in Table 7.1.

7.1.4.2

Intuitive Image of a Gauge Field Figure 7.1.13 shows an intuitive image of a local gauge transformation in contrast to a global one (Icke, 1995), where the “rotation” operation against a square piece of paper is taken as a tangible example. The invariance of the Lagrangian in this context corresponds to the state where the frame does not rotate, but the paper does. When the paper (background field) has many wrinkles, like tissue paper, these wrinkles can absorb a localized rotation without causing overall rotation of the paper. This such ability corresponds to the gauge field. Note that, based on the second quantization process, we can derive a corresponding “particle” picture from this (see Chapters 8 and 10). In the gauge formalism for dislocations and defects described next, the background field is a crystalline-ordered space that consists of a regular array of atoms microscopically. Since the interatomic spacing is flexibly changeable, a locally introduced translation of the lattice constant order can be absorbed by the background atomic structure without causing structural change in most parts of the system. This local transformation is nothing more than a “dislocation.” This operation, however, distorts the lattice around it, resulting in a relatively spatially long affected zone called the stress field. In this context, this is expressed as the interaction with the background field (elasticity).

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Figure 7.1.13  Intuitive image of a gauge field via a square of tissue paper, which can absorb local rotation without this affecting the whole configuration. Thus, the gauge is also understood as an additional degree of freedom (in the form of an auxiliary field [AF]) through which interactions with the background field are specified (Icke, 1995).

7.2

Yang–Mills-Type Kadic´–Edelen Gauge Theory This section presents the gauge formalism for dislocations and defects constructed by Kadić and Edelen (1983). Crystalline space has a structure where the translational symmetry is already spontaneously broken. Dislocation can be regarded as a topological defect embedded within such a crystalline-ordered space, where the translational symmetry is locally broken. Thus it is also called “translational” defect. Other types of defects, such as dislocation dipoles, impurity atoms, and precipitates, are classified as “rotational” defects, where the rotational symmetry is locally broken. They are introduced via the invariance of the Lagrangian for elastic media under local symmetry transformations (gauge transformation). As described in Chapter 6, they are expressed respectively as the torsion and curvature of the space from the DG point of view, providing corresponding images to the gauge representations. Note that a different gauge field theory of plasticity can be found in Kleinert (1989).

7.2.1

Gauge Field Theory of Dislocations and Defects The torsion of the space is accompanied by the closure failure of the circuit enclosing it, referred to as the Burgers vector, so it also can be viewed as a state where the translational symmetry is locally broken. The broken translational symmetry is defined by the inhomogeneous action of a translational group T (3) in the gauge formalism. Similarly, rotational defects are expressed by SO(3), corresponding to all the other types of defects, as schematically shown in Figures 7.2.1 and 6.2.2. In gauge field

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Figure 7.2.1  Dislocations and defects (dislocation dipoles) regarded as translational and

rotational defects by which translational and rotational degrees of freedom are locally broken, each represented via gauge groups of T(3) and SO(3), respectively.

theory for dislocations and defects, the invariance of the Lagrangian of the system is primarily required under the local symmetry transformation specified by gauge groups G = T (3)  SO(3). Here the gauge groups are expressed by the Lie group. Figure 7.2.2 presents the definition of a Lie group, that is, a group which is also a differentiable manifold on which smooth maps are defined. In order to maintain the invariance of the Lagrangian, as shown in Figure 7.2.3, we must introduce the covariant derivative Da x1 in place of the normal derivative ∂ a x1, that is,

 a x i  Da x i   a x i   ij x j Wa  ai  Bai , (7.2.1) where  a x i  x i / X a and sub- or superscripts in the equations represent a, b = 1, 2, 3, 4 (4 means time), A, B = 1, 2, 3 (spatial coordinates). This is called “minimal coupling” and it uniquely specifies the type of interaction between the dislocation field and the background crystalline-ordered field. Since the covariant derivative is acting on the coordinate cover x i , Eq. (7.2.1) defines the distortion tensor Bai . Thus, introduced new fields can absorb the additional terms caused by the local transformation, and these are called gauge fields. These gauge fields directly correspond to the coefficient of connection in the context of DG field theory (Hasebe et al., 1998), and give the definitions of dislocation and defect fields, respectively. Field strengths for the gauge fields, referred to as curvature, are defined by







 j i  Dab  2[ abi ]   i j W[a bi ]  Fab x  , (7.2.2)        Fab  2[ a Wb ]  2C Wa Wb

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Figure 7.2.2  Definition of a “group” which becomes a “Lie group” if it is a differentiable

manifold on which smooth maps are defined.

Figure 7.2.3  A gauge-theoretical construction of dislocation and defect fields (Yang–Mills-type

noncommutative theory or non-Abelian gauge theory).

where C  is the structure constant of Lie algebra, given by the alternating symbol  for SO(3). Taking the quadratic-invariant forms of the field strengths (Figure 7.2.4), the gauge-invariant Lagrangian densities for dislocation and defect fields are obtained as



1  i ac bd j  disloc   2  ij Dab k k Dcd , (7.2.3)  1  ac bd     C F g g F  ab cd  defect 2

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Figure 7.2.4  The derivation of the gauge-invariant Lagrangian for elastic body, dislocation, and

defect fields. A summation of the three gives the total Lagrangian of the dislocated system.

where kab and gab are given by



0  0   1 0 0  1 0 0  0 1 0   0  0 1 0 0  kab   , gab   . (7.2.4)  0 0 1 0   0 0 1 0       0 0 0 1 / y  0 0 0 1 /  The parameters ς and y represent the speeds of the dislocation and defect fronts, respectively. The strain tensor is defined by using the distortion tensor given in Eq. (7.2.1),



E AB 





1 i BA ij BBj   AB . (7.2.5) 2

The explicit form is given in Figure 7.2.5. By using E AB , we can define the gauge-invariant Lagrangian density for an elastic body containing dislocations and defects (dislocated and defected elastic body) as

elastic  T ( B4i )  V ( E AB ) (7.2.6) 1 1 e  0 B4i  ij B4j  DABCD E AB ECD , 2 2 e where DABC is the elastic constant tensor and ρ0 is the mass density of the crystal e considered. For the isotropic case, DABCD   AB CD  2  AC  BD with λ and µ being the Lamé constants, as shown in Figure 7.2.6. Therefore,

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Figure 7.2.5  Explicit expression for the strain-tensor field based on gauge theory, with which

the gauge-invariant Lagrangian for an elastic body is constructed.

Figure 7.2.6  Rewriting of the gauge-invariant Lagrangian for an elastic body in the case of

isotropic elasticity by using the Lamé constants or bulk modulus, where the latter distinction between compressive (volumetric) and incompressive parts is pronounced.

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Figure 7.2.7  Explicit expression of the gauge-invariant Lagrangian for dislocation with respect

to the dislocation field, together with the corresponding Hamiltonian obtained via Legendre transformation.

 





 

2 1 1 0 B4i  ij B4j   E AB  2  E AB E AB 2 2 (7.2.7) 2 1 1 j i    0 B4 ij B4  K E AA  2  E AB E AB , 2 2

elastic 



  E AB  (1 / 3)EKK  AB represents deviwhere K    (2 / 3) is bulk modulus and E AB atoric strain. Combining Eqs. (7.2.3) and (7.2.7), we can construct the total Lagrangian density (Figure 7.2.7) as

  elastic  s1disloc  s2 defect , (7.2.8) where si are coupling constants, having a dimension of energy, to be determined based on the creation energy of unit dislocation and defect fields.

7.2.2

Interrelationship between Gauge Theory and DG Field Theory As one may notice, there is an analogy in the quantities between gauge theory and DG field theory (non-Riemannian plasticity) presented in Chapter 6. This situation is similar to the relationship between gauge theory and general relativity (Weinberg, 1999). Figures 7.2.8 and 7.2.9 compare the two theories in terms of the major quantities. In

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Figure 7.2.8  Correspondence between notions in gauge field theory and non-Riemannian

plasticity (DG field theory).

Figure 7.2.9  Comparison between notions in gauge field theory and non-Riemannian plasticity

(DG field theory), specifying both equivalence and difference. The most fundamental quantity in the former is “connection” as a gauge field, whereas for the latter it is “metric,” from which the “connection” is derived.

Figure 7.2.8, the left-hand column lists the definitions of curvatures (field strengths) of the dislocation and defect fields in gauge theory, that is,

i  Dab  2[ abi ] , (7.2.9)        Fab  2 [ a Wb ]  C Wa Wb 

whereas the right-hand column contains those of the dislocation-density tensor and the curvature tensor (identical to the incompatibility tensor), that is,

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  ij ikl  k lje  . (7.2.10)  ...n p  n n   Rklm  2 [ k l ]m  [ k p l ]m 



In the first line of Eq. (7.2.9), the interaction term with the defect gauge field Wbα is ignored. The equivalence between the quantities is evident, where φbi corresponds to the elastic distortion tensor βije, while Wbα corresponds to the coefficients of connection Γijk . Figure 7.2.9 is another version of the comparison, not only for the torsion and curvature tensors but also for the metric tensor and the coefficients of connection, where we can recognize two differences between the two concepts. In DG field theory, the most fundamental quantity is the metric tensor, the first derivative of which gives the definition of the coefficients of connection as ijk  (1 / 2)g kl   j gli   i glj   l g ji . In gauge theory, on the other hand, the connection is not expressed in terms of a more fundamental quantity (field). Another difference concerns the dislocation field. In DG field theory, the quantity is defined basically as the torsion of the (crystalline) space, given by the skew-symmetric part of the coefficients of connection, that is, Sij..k  [kij ], whereas in gauge theory, the curvature (i.e., field strength) of the connection for the dislocation field defines the corresponding quantity. Rigorously speaking, the “connection” in DG field theory means the affine connection, while in gauge theory the “connection” expresses that of internal degrees of freedom brought about by breaking local symmetry.

7.2.3

Chern–Simons Gauge Term As a feature of gauge theory, the construction of the Lagrangian allows flexibility to add arbitrary terms (additional degrees of freedom) as long as they are gauge invariant. One which I have considered using to express the entangling behavior of dislocation lines is the Chern–Simons (CS) gauge term. It is given, via differential form, as 2   CS(a )   3 Tr  a  da  a  a  a  dx. (7.2.11) R 3  



for the noncommutative (non-Abelian) gauge field a. Figure 7.2.10 provides schematics of the term. The Feynman path integral of the form

Lk (Ci , C j )  

aA( R3 )

Ci (a) C j (a ) exp  CS(a ) a. (7.2.12)

gives the correlation function of the two particles χCi χC j . The summation thus obtained Lk (Ci , C j ) over the region coincides with a topological invariant called the Gauss linking number, that is,

Q   Lk (Ci , C j )  i, j

dxi  dx j  ( xi  x j ) 1 bi b j  . (7.2.13)  3 8 i, j x x CC i

j

i

j

One of the immediate examples of the application concerns the Biot–Savart law for electromagnetism, as shown in Figure 7.2.11. Examples of the linking number are presented in Figures 7.2.12 and 7.2.13.

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Figure 7.2.10  Overview of the CS gauge field, which has many topologically unique and intriguing features. Feynman path integration of a product of two fields multiplied by the CS term produces a topological invariant that coincides with the Gauss linking number defined on 3D manifold.

Figure 7.2.11  Intuitive image of the Gauss linking number in the context of electromagnetism, known as the Biot–Savart law.

The CS gauge concerns correlations between two or three particles or strings, ultimately giving rise to a topological invariance regarding the entanglements. So, it is useful to consider applications that will describe or model tangling behaviors of dislocations that frequently take place during plastic deformation. More general discussions about topological invariance in terms of “local versus global” relationships of geometrical fields is given in Section 14.2. What Eq. (7.2.12) implies is that the topological nature of the tangled dislocations is not affected at all by their configurational details but solely by how many times they

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Figure 7.2.12  Examples of two different linking numbers.

Figure 7.2.13  Link and knot, together with a comparison of two braids that look different but have the same linking number (homotopically equivalent).

are tangled or linked. This argument seems to be similar to the conclusion derived in Chapter 11, where the LRSF developed in cell-interior regions determines the cell size and morphology regardless of their details.

7.3

Derivatives of Gauge Theory One of the advantages of using gauge-theoretical formalism, as pointed out in Section 7.1, is that it provides many derivatives in a mathematically formal and rigorous manner, even beyond our intuition, since it essentially follows Lagrangianbased analytical mechanics. The examples given here include field equations and the energy-momentum tensor.

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The former tells us, in our context, how dislocation and defect degrees of freedom are reflected in the EOM of the system as well as their individual field equations. The latter can provide information about, for example, generalized forces acting on the inhomogeneous fields, represented by x i ,  i and Wa . In what follows, we begin with the general and fundamental aspects regarding the EOM (field equations), followed by three detailed examples (taken from outside plasticity).

7.3.1

Euler–Lagrange Equation The E–L equation, derived based on the variational principle, gives the EOM of a system. The conventional process of the derivation in analytical mechanics is presented in Figures 7.3.1 and 7.3.2. In field theory, the E–L equation can be obtained in basically the same way (Figure 7.3.2), and is referred to as the field equation. The variational principle with respect to the field variables is applied as shown in Figure 7.3.3. The variation of the action is evaluated as

 S   d 4 x     ,           d 4 x   ,    V





V

     d4 x      V     



    (7.3.1)  .    V d 4 x          

The second term in the second line vanishes due to the boundary conditions for  . Thus we have      S   d4 x       0. (7.3.2)   V      For this to hold for arbitrary  , we finally obtain the E–L equation as



     0. (7.3.3)    

Figure 7.3.1  Overview of the derivation process of the field equations in the framework of

gauge theory. The E–L equation derived from action based on the variation principle with respect to a field variable gives the corresponding field equation (EOM).

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Figure 7.3.2  Brief outline of how to obtain the E–L equation based on the variational principle

in the normal setting of analytical mechanics.

Figure 7.3.3  Derivation outline of the E–L equation in gauge field theory, following the same

process as in Figure 7.3.1 for the field-theoretical Lagrangian and attendant action functional.

7.3.2

Examples of E–L Equations Let us look at some typical examples of the derivation processes of E–L equations for representative cases: The Klein–Gordon equation, the Schrödinger equation, and the equation for elastic wave (Takahashi, 1997). Figures 7.3.4–7.3.6 show outlines of the derivation processes for each case. For deriving the Klein–Gordon equation (Figure 7.3.4), that is, the relativistic wave equation for bosons, we start out from the Lagrangian (density)









1        2  . (7.3.4) 2 The action integral is constructed, via the space–time integral of Eq. (7.3.4), as 1 S   d 4 x [ x ] . (7.3.5) c V4 [ x ]  

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Figure 7.3.4  Example of the derivation process when using the E–L equation to obtain the

Klein–Gordon equation.

Figure 7.3.5  Example of the derivation process when using the E–L equation to obtain the

Schrödinger equation.

Figure 7.3.6  Example of the derivation process when using the E–L equation to obtain the

equation for the elastic strain wave within an elastic medium.

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The variation of the action reads  [ x ] 1 [ x ]   S   d 4 x     , (7.3.6)   c V4      1   d 4 x  x 2  . (7.3.7) c V4





For satisfying  S  0 with arbitrary  , we finally have

  x2  ( x) 0 . (7.3.8)



For the Schrödinger equation (Figure 7.3.5), the Lagrangian density has the same form as that in Eq. (7.1.8),



2  i † ( x ) i ( x ). (7.3.9) 2m 1 Similarly, the variation of the action integral S   d 4 x[ x ] is evaluated as c V4 [ x ]  i † ( x ) t ( x ) 

S 

 †   1 2 2  2 2  †  4  d x  ( x ) i      ( x )  ( x ) i    ( x ) . (7.3.10)         t t c V4 2m 2m      

From  S  0 with  ( x ) and  †( x ) being arbitrary and independent of each other, we obtain a set of Schrödinger equations, that is,  2 2   ( x)  i t ( x )   2m . (7.3.11)  2   i  † ( x )     2 † ( x ) t  2m



The Lagrangian for an isotropic elastic body (Figure 7.3.6) is identical to that given in Eq. (7.2.6):

[ x ] 

1 1  t ui  t ui  (   ) i ui  j u j   i u j  i u j , (7.3.12) 2 2

where ρ is the density of the elastic medium,  and  are the Lamé constants, and ui = ui (x) is the displacement of the mass element measuring deviations from the equilibrium positions. The E–L equation in this case reads



[ x ] [ x ] [ x ] [ x ] [ x ]     t j ui   ui ui  t ui  j ui 

  2t ui





(7.3.13)

 (   ) i  j u j   ui . 2

Based on the variational principle, the last line is equated with 0, giving

 2t ui  (   ) i  j u j  2 ui . (7.3.14) This is the equation for elastic waves within elastic media. For the derivation process of the equation for electromagnetic field, that is, Maxwell’s equations, visit Eqs. (7.1.5)–(7.1.9).

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7.3.3

Euler–Lagrange Equations for Dislocation and Defect Fields In what follows, we derive the field equations for the gauge fields treated here, that is, for dislocations and defects with respect to the field variables x i ,  i and Wa , respectively. Figure 7.3.7 outlines the process. The action is constructed as





S    x i ,  i , Wa dx . (7.3.15)



By imposing  S  0, we finally obtain  j   A  ab A   4 pi   A i   ai W4 pi  WA  j  Fab R j  1 b  ab . (7.3.16)   a Ri  Z j 2  1 b  ab   ab   a G  C Wa G  2 J







The details of each case are presented in the following subsections.

7.3.3.1

Field Equation with Respect to x i The derivation process is summarized in Figure 7.3.8. Each term of the E–L equation      0 is obtained as i x  ( a x i ) j   Baj  Dab   j x i Baj x i Dab x i (7.3.17)



 j  Z aj Wa  j i  R ab j Fab   i

and     Z ia , (7.3.18) i ( a x )  a Bai

respectively, where R ab j  tion in this case becomes

 j Dab

ba with R ab j   R j is used. Therefore, the E–L equa-

 ab  a Z ia  Z aj Wa  j i   j i Fab R j . (7.3.19)

Using notions Z iA   iA and Z i4 = pi , we can rewrite the above more explicitly as





 ab  4 pi   A iA   aij W4 pi  WA  jA  Fab R j . (7.3.20)

This is the field equation (EOM) for the dislocated and defected elastic body. As is found from the obtained equation, the effects of the dislocation and defect fields are reflected, on the right-hand side, as a body-force-like contribution.  In the absence of the defect field, that is, Wa  0 and Fab  0, we regain the EOM for an elastic body, that is,

 4 pi   A iA  0.   4 pi   A iA . (7.3.21)

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Figure 7.3.7  Overview of the derived field equation for the gauge theory of dislocations and defects. We finally have three EOMs corresponding to three field variables, that is, xi, f i, and Waα.

By rewriting this equation using substitutions from the isotropic elasticity relationship, that is,  iA  ( iA jB  2  i j  AB )  j uB and the momentum pi    4 u, we can arrive at the same expression as that presented in Eq. (7.3.14) for elastic media (elastic waves). Even without the dislocation field, we have





 4 pi   A iA   aij W4 pi  WA  jA . (7.3.22) The defect field is shown to act as a body-force-like contribution in the field (Eq. (7.3.22)). This implies that the existence of the curvature or incompatibility-tensor field in Chapter 6 will have a nonnegligible effect on the kinetics of the system considered, while the torsion or dislocation-density tensor field does not.

7.3.3.2

Field Equation with Respect to φai Figure 7.3.9 briefly displays the derivation process. We present the variation with respect to the dislocation-gauge field φai as



 

  i  ia  i  Dab  0, (7.3.23) i  a Dab

i i where  ia and δ Dab represent the variations in φ ia and Dab , respectively. Since i i i i j  Ba   a x   j x Wa  a , the first term in Eq. (7.3.23), that is, /  ia , is written as



  B bk  b      i  Z ia , (7.3.24) i i k i a  a B b  a B b B a i and, for ∂ / ∂Dab , we have



  s1 ij k ac k bd Dcdj  s1 ij k ac k bd [ cdj]  2 Riab . (7.3.25) i Dab

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Figure 7.3.8  Equation of motion for gauge fields with respect to xi.

Figure 7.3.9  Equation of motion for gauge fields with respect to φ i .

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Figure 7.3.10  Gauge-fixing condition and corresponding explicit field equation with respect to φ i .

Substitutions of Eqs. (7.3.24) and (7.3.25) for Eq. (7.3.23) lead to i    Z ia ia  2 Riab Dab  0. (7.3.26)



i i Here, for the induced variation δ Dab , we read  Dab   a (ai )   i j Wa ai . Since



 a ( Riabai )   a Riabai  Riab  a (ai ), we can rewrite Eq. (7.3.26) as

  ab i  a ab   Z i  2 a Ri  2 Ri   j Wa  ai

   Z ia  2 a Riab  2 Riab i j Wa ai   a ( Riabai )

(7.3.27)

 0, where  a ( Riabai ) vanishes on the boundary. Since Eq. (7.3.27) must hold for arbitrary ai we finally have





1 B  aB aB i    a R j  Ri   j Wa   2  j . (7.3.28)     R A4  R A4 i W   1 p i j A j  A j 2 For the systems with the dislocation field only, that is, Wa  0, we have 1 b ab i   a R ab j  Ri   j Wa  Z j 2

1 B  aB   a Ri   2  j  a Riab   . (7.3.29)   R A4  1 p j  A i 2 Let us rewrite Eq. (7.3.29) as a more explicit expression by imposing a gauge-fixing condition, as shown in Figure 7.3.10. Let us use the “pseudo-Lorentz gauge,” that is, 1  a (k abbi )   A Ai   44i  0. (7.3.30) y 1  Z bj 2

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With it, we obtain



  i 1 B 1 BD  A  s1 ji    A   4  4  D   j y 2    . (7.3.31)   s1    A   1     i  1 p j A 4 4 4  y ji  y 2    Equation (7.3.31) provides a set of wave equations for a dislocated system; these are used in the second quantization process in Chapter 8 when the method of QFT is utilized based on the foundations derived in the present framework. Note that the system is uniformly hyperbolic provided y > 0 (Kadić and Edelen, 1983). Without the stress field σ Bj in the first line of Eq. (7.3.31), the equation is reduced to that for the free dislocation field, which has plane-wave solutions, as will be utilized in Chapter 10 for the second quantization associated with the QFT method.

7.3.3.3

Field Equation with Respect to Waα Figure 7.3.11 summarizes the process of the derivation in this context. The E–L equation with respect to Waα is obtained by following a procedure similar to that in Section 7.3.3.2. The variation is written as       Wa    Fab  Wa Fab



  Ja Wa  Gab Fab  0,

(7.3.32)

Figure 7.3.11  Equation of motion for gauge fields with respect to Waα.

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where we put

Gab 



Ja 

  s2 g ca g dbC Fcd  Riab i j x j , (7.3.33)  Fab  Wa

 Fab





  i j Z ia x j  2 Riabbj . (7.3.34)

Here, the second terms on the extreme right-hand side of Eq. (7.3.33) and (7.3.34) come from the partial derivative of the total Lagrangian  with respect  j i i to Dab  2[ abi ]   i j W[a bi ]  Fab x and that of Dab with respect to Waα , respectively.





α  The variation in Fab , that is,  Fab , reads    Fab   a ( Wa )  C Wa  Wa .



   )   a Gab  Fab  Gab  a ( Fab ) for the second term in the By using  a (Gab Fab right-hand side of the second line of Eq. (7.3.32), together with the vanishing condi )  0, we have, from Eq. (7.3.32), tion on the boundary, that is,  a (Gab Fab   a Gab  C Wa Gab 



1 b J . (7.3.35) 2

To further rewrite this, let us introduce Gab  s2



defect , Gab  Gab  Riab i j x j . (7.3.36)  Fab

Substituting this for Eq. (7.3.35), eliminating ∂ a Riab by using Eq. (7.3.28) and performing some algebraic calculations making use of the commutation relationship for the generating matrices  i j , that is,  i C   j  [ i j ,  i j ]   i k  k j   i k  k j



we finally reach the EOM for the defect field,   a Gab  C Wa Gab 



1 b J , (7.3.37) 2

where

Ja  2 i j Riab Bbj . (7.3.38) Here, Jαa means a physical action resisting the growth of the Waα field. For more details about the derivation process of Eq. (7.3.37), refer to Kadić and Edelen (1983). An overall summary of thus derived gauge field theory-based EOMs for dislocations and defects, that is, the final forms of those presented in Figures 7.3.7 and 7.3.8 through 7.3.11, are listed in Figure 7.3.12, showing their general expressions, together with the corresponding specific forms, for example, using the notions Z iA   iA and Z i4 = pi.

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Figure 7.3.12  Summary of gauge field theory-based EOMs for dislocations and defects (disclinations).

7.3.4

Energy-Momentum Tensor Lastly, we consider the energy-momentum tensor and discuss its potential applications by providing some examples from the past. The derivation will be made from a general point of view. Let us start with Noether’s theorem, based on which we find a tangible definition of the energy-momentum tensor as well as its clear (unobstructed) derivation process. Noether’s theorem asserts a profound interconnection between the symmetry of a physical system and the corresponding conservation law (Noether, 1971). Figure 7.3.13 is a portrait of Emmy Noether (1882–1935), who proved this beautiful theorem in theoretical physics. Figure 7.3.14 summarizes examples of the correspondences between the symmetries and the conservation laws: The symmetry with respect to the spatial translation of the action of a physical system, for example, corresponds to the conservation of linear momentum. Likewise, the symmetries in rotation and temporal translation are interrelated with the conservations of angular momentum and energy, respectively. Note that the system must be described via the Lagrangian in order for the theorem to be applied, otherwise no corresponding conservation law has to exist. From the symmetry of the action integral with respect to the space–time translation, we derive the notion of “energy-momentum tensor” in the context of the corresponding conservation law. Figure 7.3.15 concisely presents the derivation process (Lifshitz and Landau, 1979). The symmetry of the system with respect to the space–time translation is expressed as

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Figure 7.3.13  Portrait of Emmy Noether, a German mathematician, famous for Noether’s theorem, acknowledged as one of the most beautiful theorems in mathematics.

Figure 7.3.14  Schematics depicting the substantial relationship between “symmetry” and the “conservation law.”

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Figure 7.3.15  Outline of the derivation process of the energy-momentum tensor in analytical

mechanics.

  0, (7.3.39) x b





with b = 1, 2, 3, 4 (4 means time t). The left-hand side of the above condition is rewritten as   ei  ( aei )   . (7.3.40) x b ei x b ( aei ) x b From the E–L equation with respect to the field variable ϕei , we have    a i e x



      0 i  a  i  e x   (  a e ) 

   . (7.3.41)  i    (  a e ) 

Substituting the above for the right-hand side of Eq. (7.3.40) results in    ei  ( bei )    i b ( aei ) x a  ( ae )  x (7.3.42)    ei   a . x  ( aei ) x b 

   a b x x



( aei )  2ei ( bei )   is used for the second term, in order to derive x b x a x b x a the last line. Here,

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By the way, the left-hand side of Eq. (7.3.40) can be rewritten as



     ba a  a  ba . (7.3.43) b x x x Therefore, by comparing the right-hand sides of Eqs. (7.3.42) and (7.3.43), we finally have  x a

  ei    ba    0, (7.3.44)  i b  ( ae ) x 

which indicates the conservation of a current, referred to as Noether’s current,

 ei   ba . (7.3.45) i b ( ae ) x

Tba 

This is called energy-momentum tensor based on its physical meaning. As indicated earlier, Tba is divergenceless (or yields the vanishing of the divergence), that is, Tba   a Tba  0, (7.3.46) x a



which is equivalent to the statement that the integral ∫ Tba dSa, producing a vector, is conserved. This also has an important meaning in applying the concept to the mechanics of materials, as will be argued in Section 7.3.4. Figure 7.3.16 summarizes this conclusion. Where there are coexisting plural fields, the extension is straightforward, as depicted in Figure 7.3.17, that is,

Tba   (k)

(k)  bei(k)   ba , (7.3.47) ( aei(k) )

which will be used to calculate the dislocation- and defect-gauge fields later in this chapter.

7.3.5

Physical Meaning of the Energy-Momentum Tensor Figure 7.3.18 is a summary depicting the physical interpretations of the spatiotemporal components of Tba . For b = 4, corresponding to time t, Eq. (7.3.46) becomes



T44 T4A   0. (7.3.48) x 4 x A The integration over a volume leads to



 T4A T44 dv    dv    n AT4A da, (7.3.49)  v x A a t v where the Green–Gauss divergent theorem is used to derive the extreme right-hand side of the equation. Since T44 expresses the energy density, as is explicitly discussed in Section 7.3.6, the left-hand side refers to the change rate of energy, called energy density in dv,

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Figure 7.3.16  Energy-momentum tensor and attendant conservation law.

Figure 7.3.17  Energy-momentum tensor obtained within the gauge field theory of dislocations and defects.

Figure 7.3.18  Physical meaning of the conservation equation for the energy-momentum tensor with respect to the temporal components, yielding balance equations of energy and momentum in the form of “change rate versus flux.”

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whereas the extreme right exhibits the “energy flux” across da (x A-surface). Therefore, Eq. (7.3.49) represents “the conservation of energy.” For a = 4, on the other hand, Eq. (7.3.46) becomes TB4 TBA   0. (7.3.50) x 4 x A



Similarly, the integration of Eq. (7.3.50) leads to  TBA dv    n ATBA da (7.3.51) TB4 dv     v x A a t v



Since TB4 is the momentum density, the left-hand side indicates the change rate of momentum, while the extreme left represents the “momentum flux” in the x B-direction across da (x A-surface), meaning “the conservation of momentum.”

7.3.6

Energy-Momentum Tensor for the Gauge Field of Dislocated and Defected Elastic Media Based on Eq. (7.3.47), the energy-momentum tensor for our case, that is, the gauge field theory of dislocations and defects (Figure 7.3.19) (Kadić and Edelen, 1983), is given as Tba Tba elastic  Tba disloc  Tba defect , (7.3.52)

where



elastic  a  b x i   ba elastic  Tb elastic  i  (  x ) a   a disloc disloc  bei  s1  bWei  s1 ba disloc . (7.3.53)  Tb disloc  i ( ae ) ( a Wei )   defect  bWei  s2 ba defect  Tba defect  ( a Wei )  In what follows, taking the elastic term of the energy-momentum tensor as an example, the component-wise expressions are explicitly obtained. The processes are depicted in Figures 7.3.20 through 7.3.23. For the temporal component (Figure 7.3.20), we have Lelastic  4 x i   44 Lelastic T44 elastic   ( 4 x i ) 



Lelastic i B4  Lelastic B4i

1 1 e 0 B4i  ij B4j  DABCD E AB ECD 2 2  Helastic . 

Here, the relation

elastic   B4i B4i

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(7.3.54)

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Figure 7.3.19  Energy-momentum tensor for the gauge field theory of dislocations and defects comprising three terms for elasticity, dislocation (translational defect), and defect (rotational). The last is expected to be used as a parameter for describing evolutionary aspects of inhomogeneous fields.

Figure 7.3.20  Temporal component of the energy-momentum tensor in the gauge field theory of dislocations and defects for the elasticity term, together with the physical meaning, that is, energy density (coinciding with the Hamiltonian).

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Figure 7.3.21  Spatiotemporal component of the energy-momentum tensor in the gauge field theory of dislocations and defects for the elasticity term, together with the physical meaning, that is, energy flux.

is used. The above derivation indicates that the temporal component T44 elastic coincides with the Hamiltonian of the system, Helastic  K  W , which is identical to that obtained via the Legendre transformation, as in Figure 7.2.7. Therefore, the pure temporal component refers to the energy density of the system under consideration. Space–time components T4Aelastic and TA4 elastic (Figures 7.3.21 and 7.3.22) can be evaluated them as T4Aelastic  



elastic  4 x i   4A elastic  ( A x i ) elastic i B4 EiA

(7.3.55)

   iA B4i , TA4 elastic  



elastic  A x i   A4 elastic  ( 4 x i ) elastic i A B4i

 pi

, (7.3.56)

ui x A

respectively. In the above, elastic elastic e   DAiCD  0 B4i  Z i4  pi E DC    iA and A Ei B4i

are used.

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Figure 7.3.22  Spatiotemporal component of the energy-momentum tensor in the gauge field theory of dislocations and defects for the elasticity term, together with the physical meaning, that is, momentum density.

Figure 7.3.23  Spatial component of the energy-momentum tensor in the gauge field theory of dislocations and defects for the elasticity term, together with the physical meaning, that is, momentum flux.

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For spatial components (Figure 7.3.23), Lelastic  B x i   BA Lelastic TBAelastic   ( A x i ) 



Lelastic B i EiA

   iA 

(7.3.57)

u B   BA W, x i

which coincides with the Eshelby stress, as will be revisited in Section 7.3.7. Figure 7.3.24 summarizes the results via matrix representation. From the divergenceless condition, that is,  a Tba  0, (7.3.58)



we have a balance equation for the generalized force, that is,

 a Tba elastic   a Tba disloc   a Tba defect (7.3.59)  Fb elastic  Fb disloc  Fb defect . The explicit expressions (Figure 7.3.25) are given as follows:

Fa elastic   iB  aBi  pi  a4i

  i j x j  iB  a WB  pi  a W4   i j Fcb Ricb  a x j



 B i i i j b

Fa disloc   i  aB  pi  a4    j x Z i  a Wb

  k i Rkec 2We  aci   i j x j  a Wc (7.3.60) 

 i  i  i   a 2W[ e c ]   e 2 x  a Wc   a x Fec



Fa defect   Jb  a Wb

 C 2C C  Gbc Wb  a Wc  Gbc  a Wb Wc  .























 





7.3.7



Application Examples of the Energy-Momentum Tensor Figure 7.3.26 shows the energy-momentum tensor by Eshelby (1951, 1975).





Pjm  W  jm  T ji 

ui , (7.3.61) x j

which is called the Eshelby tensor. The line integral of Pjm along the path encircling an elastic singularity, Fm   n j Pjm ds, (7.3.62) 

gives a vector specifying the forces acting on the singularity. Here n j is the unit plane normal satisfying ds j = n j ds . As one can understand readily, based on the earlier mentioned generalized descriptions of the energy-momentum tensor (see Section 7.3.6),

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Figure 7.3.24  Summary of the energy-momentum tensor in the gauge field theory of dislocations and defects for elasticity term, together with the physical meanings of the components.

Figure 7.3.25  Resultant balance (or conservation) equation derived from the energy-momentum tensor in the gauge field theory of dislocations and defects.

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Figure 7.3.26  The energy-momentum tensor in the context of that proposed by Eshelby, the integral of which defines a force acting on the singularities encircled, which is interpreted as the generalization of the J-integral in fracture mechanics (Eshelby, 1951; Ohnami, 1980, 1988).



this force is a conserved quantity and is attributed automatically to its origin. The inset shows some situations to which the concept can be applied (Ohnami, 1980, 1988). Figure 7.3.27 shows a more simplified but much more famous application to the crack-tip region, called the J-integral, proposed by Rice (1968), which has formed the sound basis of fracture mechanics. It is a 2D version of the Eshelby tensor (Eq. (7.3.62)), with Eq. (7.3.61) which has a form of line integral along the path that encircles a crack tip counterclockwise. u   J   Wdy T  ds . (7.3.63)  x  The path independency of the quantity is obvious (self-evident, in a sense) considering the nature of the energy-momentum tensor. A number of variations of the above, to be applied to specific cases, have been proposed. A noteworthy one is the dynamic J-integral (Figure 7.3.28), applicable, for example, to cracks extending at high velocity (Nishioka and Atluri, 1983),  ui   W  K  nk  ti ds xk    ij W    ui u   ui    u i i   ij   dV . V V xk xk  xk   xk

Tk*  

  C



(7.3.64)

Note that since the energy-momentum tensor is defined by space–time coordinates, it naturally involves the dynamics of the targeted system, and it seems straightforward to derive the above expression from the energy-momentum tensor. Consider a straight dislocation line in an unbounded medium. The force acting on the dislocation is given as

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Figure 7.3.27  The J-integral as a 2D version of the Eshelby energy-momentum tensor applied to a crack tip. A derivation process is briefly shown (Rice, 1968).

Figure 7.3.28  The dynamic J-integral as an extension of Rice’s J-integral as applied to dynamically propagating cracks (Nishioka and Atluri, 1983).

f   T 



u ds x

  T  du

(7.3.65)



  T  b. With T     , the above equation represents the P–K equation.

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The vector Fm defined by Eq. (7.3.62) is also referred to as a “configurational force” (Epstein and Elzanowski, 2007; Eshelby, 1975; Phillips, 2001), a generalized driving force responsible for the evolution of inhomogeneity, including defects and interfaces. Another possible way to make use of the energy-momentum tensor in describing the evolutionary aspects of the inhomogeneities is to combine it with the incompatibility tensor, another quantity for describing inhomogeneously evolving deformation fields. Some details of this will be discussed in Chapter 15. I have recently postulated a candidate form of the “flow-evolutionary law” as

ij   Tij , where κ measures the energy conversion rate. The left-hand side expresses the current state of inhomogeneity, which measures the potential evolution of the field (in the future), while the right measures the energetic change, including the release of inhomogeneously stored strain energy or dissipation. Note that this relationship is hypothetical, so it requires further validation via practical demonstration. An extended discussion can be found in Chapter 15.

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Method of Quantum Field Theory

8.1

Brief Reviews

8.1.1

Quantum Mechanics versus Quantum Field Theory As seen in Section 1.2.5, dislocations are created from and annihilated into the background crystalline field. Also they tend to behave collectively rather than individually in most practical situations, as demonstrated in Chapter 3, and they interact with each other in a complex manner. To my knowledge, the one and almost the only suitable language (or mathematical tool) for describing such objects in a sophisticated way is quantum “field” theory (QFT). In other words, we are required to assume a “field” that has an ability to create and annihilate the dislocations from it a priori. There seems to be confusion or misunderstanding regarding the first principle of matter among researchers with nonphysics backgrounds, who tend to believe “quantum mechanics” is the first principle. But there is a more fundamental theory about matter, which is based on the notion of “fields” rather than discrete “particles,” that is, QFT (Figure 8.1.1). More detail about the distinction between the two will be given in a later section. This genuinely fundamental theory – QFT – can be used to effectively describe dislocation fields in particular, as their collective dislocation fields in a rather simplified manner. This chapter provides an introduction and overview of QFT to impart a basic understanding. Quantum field theory is a theory that can describe not only the motion and interaction of particles but also their “annihilation” to and “creation” from the background field. (QFT regards a field a priori, from which “particles” come into existence.) This view seems suited to describing dislocations, which can also be regarded as a particle or a string embedded within a crystalline-ordered field. Another feature of QFT is that the framework is mathematically identical to that of statistical mechanics (this is also true for quantum mechanics). This means that the theory can naturally and essentially deal with “many-particle” or “many-body” systems. Therefore, one can intuitively notice that the theoretical framework is suitable for dealing with collective behaviors of dislocations in a sophisticated manner as well. Details about the linkage between an application of the present formalism and the treatment of a collective behavior of interacting dislocations are provided in Chapter 10, as the application for Scale A.

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Figure 8.1.1  Relationship between quantum mechanics and QFT.

8.1.2

Brief Review of Quantum Mechanics Figure 8.1.2 shows the Schrödinger equation for a one-particle system, representing the time evolution of the wave function y( r , t ) , that is,



ih

  pˆ  y(r , t ) =Hˆ y(r , t ) =  + V (r )  y(r , t ), (8.1.1) t 2m  

where Hˆ represents the Hamiltonian operator. The formal solution of this equation is written as

 i ˆ  y( r, t )  exp  Ht  y(r, 0), (8.1.2)  h  where h is Planck’s constant. The multiplier exp(−iH t /h) in Eq. (8.1.2) is the time-evolution operator, expressing (dominating) the temporal evolution of the wave function. The square of the wave function refers to the probability of finding or detecting a particle at the position and time ( r, t ) 2

P( r, t ) = y( r, t ) . (8.1.3)



Therefore, P( r, t ) must satisfy the completeness condition, that is,

d

3

2

r P( r, t )   d 3 r y( r, t )  1. (8.1.4)



The Schrödinger equation for an N-particle system (Figure 8.1.3) is considered to be  ih y  x1 , x2 ,, xN ; t   H y  x1 , x2 ,, xN ; t  . (8.1.5) t What we have to do is to simply extend this by adding to the number of particles



r1 ,, rN y(t )  y( r1 ,, rN , t ). (8.1.6) Also, for the Hamiltonian, we can extend as



N

H   Hi . (8.1.7) i 1

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Figure 8.1.2  Brief review of quantum mechanics: the Schrödinger equation and wave function.

Figure 8.1.3  Brief review of quantum mechanics: distinction between boson and fermion in an

N-particle quantum system, serving as the basis of quantum statistical mechanics.

In the N-particle system, we can include the statistical nature of the particles c­ onsidered, which introduces the distinction between boson and fermion, defined by an even and odd permutation of coordinates for two arbitrary particles:

y( r1 ,..., ri ,..., rj ,.. rN , t ) y( r1 ,..., rj ,..., ri ,.. rN , t )   y( r1 ,..., ri ,..., rj ,.. rN , t ) This provides the basis for quantum statistical mechanics.

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(Bosoon) . (8.1.8) (Fermion)

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8.1.3

419

Matrix Formulation of Quantum Mechanics For the matrix formulation, bra and ket vectors in the Hilbert space have been invented and introduced by Dirac for representing functions, for example, f ( r ) and g( r ) (Figure 8.1.4). Based on this setting, the inner product is expressed as *

g f   d 3 r g* ( r ) f ( r )  f g ,



(8.1.9)

where * represents the Hermitian conjugate, while arbitrary operators Aˆ are expressed as ˆ (r) . g Aˆ f   d 3 r g*( r ) Af



(8.1.10)

By introducing the orthonormal basis satisfying i j   i, j , we can express, for example, a function f ( r ) in the Hilbert space as

 f  i i f  i , (8.1.11)   f f i i   i  where the completeness condition i i  1ˆ must be satisfied.

 i

Space coordinates are defined as rˆ r = r r , which satisfies

r  r   ( r  r ) and  d 3 r r r  1. (8.1.12) Similarly, for momentum, we define this as pˆ p = p p , with pˆ 



 satisfying i r

p p   ( p  p) and  d 3 p p p  1. (8.1.13) Here   h / 2 is called Dirac’s constant or the reduced Planck constant.

Figure 8.1.4  Brief review of quantum mechanics: matrix representation introducing bra- and

ket-vectors in Hilbert space.

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Figure 8.1.5  Brief review of quantum mechanics: commutative relation between space

­coordinates and momentum operators, which deduces Heisenberg’s uncertainty principle.

The commutation relation is introduced between the two as well as among them (Figure 8.1.5), that is,

[r , r  ]  i ,  and [r , pˆ  ]  i (8.1.14)



[r , pˆ  ]  r pˆ  pˆ  r (8.1.15) from which we can derive Heisenberg’s uncertainty principle, that is, rˆ pˆ   . (8.1.16)



8.1.4

Schrödinger Picture versus Heisenberg Picture There are roughly two ways in quantum mechanics to describe the time evolution of the system, that is, the Schrödinger picture and the Heisenberg picture, as summarized in Figure 8.1.6. In the Schrödinger picture, the wave function (state vector) is regarded as being time dependent. Equation (8.1.2) corresponds to that in the picture, in which we have i



i

  Ht  y(t )  H y(t ) with y(t )  e  y(0) (8.1.17) t

where all the operators for an arbitrary physical quantity, including the Hamiltonian ˆ it ˆ are treated as being time independent. If we express an arbitrary operator by A, H, is stationary, that is,

i

dAˆ = 0. (8.1.18) dt

In the Heisenberg picture, on the other hand, the operators (fields) representing physical quantities, for example, position and momentum, are considered to be time

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Figure 8.1.6  Brief review of quantum mechanics: relationship and equivalence between the

Schrödinger picture and the Heisenberg picture.

dependent, while the wave function is regarded as being time independent. The corresponding expression in the Heisenberg picture is i



 y ( r )  0, (8.1.19) t

while the arbitrary physical quantities must obey the Heisenberg EOM given by

i

dAˆ H (t ) = [ Hˆ , Aˆ H (t )] (8.1.20) dt

The relationship between the two pictures is represented, in terms of the expectaˆ as tion of an arbitrary physical quantity A, ˆ ˆ  (i / h ) Hˆ t y(t) Aˆ y(t)  y(0) e(i / h ) H t Ae y (0) .



 y(0) Aˆ H (t ) y(0)

(8.1.21)

Here, the operator is interrelated as

ˆ ˆ  (i / h ) Ht ˆ Aˆ H (t )  e(i / h ) Ht Ae . (8.1.22)

Note that by using the time differentiation of Eq. (8.1.22) one can regain Eq. (8.1.20). There is another picture, known as “interaction picture,” which more or less combines the above two pictures, and in which the time evolution of the system is shared by the wave function and the operator. If the Hamiltonian is decomposed into free and interaction parts, that is,

Hˆ  Hˆ 0  Hˆ int (t ), (8.1.23) the time evolutions of the wave function and an arbitrary operator Aˆ I (t ) are expressed, respectively, as

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i

 ψΙ (t )  Hˆ int (t )ψI (t ), (8.1.24) t

i

dAˆ I (t ) = [ Hˆ 0 , Aˆ I (t )], (8.1.25) dt

where the subscript I stands for “interaction.” As seen earlier, the interaction part Hˆ int (t ) is responsible for the time evolution of the wave function yΙ (t ), while the free part Hˆ 0 undertakes that of the operator Aˆ I (t ). The above three pictures, that is, the Schrödinger, Heisenberg, and interaction pictures, are equivalent in the sense that they yield the same expectation value for a given physical quantity, as in Eq. (8.1.21).

8.1.5



Quantum Mechanics versus QFT Figure 8.1.7 summarizes comparisons among one-particle quantum mechanics, N-particle quantum mechanics, and QFT (Nagaosa, 1995). We notice that in the QFT we regain the identical form of the EOM for the field for the one-particle Schrödinger equation, that is,  ih yˆ ( x, t )  H yˆ ( x, t ), (8.1.26) t

Figure 8.1.7  Overview of the interrelationship among quantum mechanics, N-particle quantum

mechanics, and QFT in terms of the fundamental EOM (Nagaosa, 1995).

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where yˆ ( x, t ) is the operator representing a quantized field, which must satisfy the corresponding commutation relation [y ( x, t ), y ( x, t )]  i ( x  x) (8.1.27)



Note that the substantial difference between the ordinary one-particle Schrödinger equation and that in QFT is in the nature of y( x, t ) and yˆ ( x, t ), that is, the former is a c-number function while the latter is a q-number operator.

8.1.6

Second Quantization To extract a “particle” picture from the “field” considered, we introduce two special operators, that is, creation and annihilation operators, Aˆ n and Aˆ n† , respectively, as shown in Figure 8.1.8. They are given the ability to create or annihilate one particle from the given field via



 yˆ ( r )   Aˆ n ( r )  n . (8.1.28)  ˆ† ( ) y r    Aˆ n†n* ( r ) n  The commutation relations for these operators are defined as



[ Aˆ n , iAˆ m† ]  i n,m and [ Aˆ n , Aˆ m ] = [iAˆ n† , iAˆ m† ]  0 (8.1.29) Therefore, the commutations relation for the fields become



[ψˆ ( r ), ψˆ † ( r )]   φn( r )φn* ( r )   ( r  r )  n (8.1.30)   [ψˆ ( r ), ψˆ ( r )]  [ψˆ † ( r ), ψˆ † ( r )]  0 

Figure 8.1.8  The second quantization procedure, introducing creation and annihilation operators, based on which we can distinguish a discrete “particle” picture from a continuous “field.”

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Figure 8.1.9  An example of the process for adding one particle to the many-particle system by

the application of the creation operator.

The number of particles, that is, the particle density, can be counted by the number operator, defined as

nˆ ( r ) = yˆ †( r )yˆ ( r ), (8.1.31) which has eigenvalues of nˆ ( r )  0, 1, 2, ...,  . In the second quantization formalism, a particle is described as the excitation of the field that is created and annihilated (Nagaosa, 1995). Figure 8.1.9 shows an example of the process for creating one particle from the field via the creation operator Aˆ n . Figure 8.1.10 is an illustration of an intuitive image of the field description of a moving particle (Nagaosa, 1995), based on an analogy to an advertising illumination board with a number of electric lamps embedded on it. By turning the lamps on and off successively, we can express a particle in motion. As can be imagined, in the case of a dislocation, the illumination board is replaced by a crystalline-ordered field composed of a number of atoms, and a turned-on lamp corresponds to the dislocated region, both yielding higher energy than the background. One of the advantages of using the QFT formalism is the ease of expressing the interactions among particles. As displayed in Figure 8.1.11, the pair interaction of particles can be expressed simply by combining the creation and annihilation operators of the field, that is,

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Figure 8.1.10  Intuitive image of the motion of a particle based on field representation, by repeatedly applying creation and annihilation operators.

Figure 8.1.11  The second quantization-based representation of pair interaction by using ­creation and annihilation operators.



yˆ †(r)yˆ †(r ) v( r  r )yˆ(r )yˆ(r), (8.1.32) where details of the interaction are specified by a potential function v( r  r ). It is important to remember that an “infinite number of particles is almost as easy to handle as one, and much, much easier than, say, 3, 4 or 7” (Zagoskin, 1998, p. 1). Figure 8.1.12 shows cartoons expressing such a situation, with (b) taken from

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Figure 8.1.12a  Cartoon representation of many-particle problems, cautioning that dealing

with a finite-numbered system makes the problem much more difficult to solve, implying the overwhelming strength of field-based treatments, emphasizing the mathematical presentations compared in Figure 8.1.7.

Figure 8.1.12b  Cartoon representation of many-particle problems, cautioning that dealing with

a finite-numbered system makes the problem much more difficult to solve, citing a phrase from Zagoskin (1998). Reprinted with permission of the publisher (Springer).

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Zagoskin (1998). This is especially useful for researchers, including myself, who have been struggling with direct simulations based on, say, discrete models of dislocations and crystal grains corresponding to the latter, that is, finite-numbered systems, requiring us to introduce more and more computational power. It might be useful for us to carefully reexamine and reconsider the significance of the approaches we are relying upon from this perspective.

8.2

Equivalence of Quantum Mechanics and Statistical Mechanics

8.2.1

General Framework The reason we should use the “quantum” field, which is basically “classical” approach, to describe dislocation fields, is the equivalence of mathematical to statistical mechanics, which we are able to make use of in treating many-dislocation systems. Figure 8.2.1 indicates the interrelationship between the two frameworks. They can be mutually converted via the formal replacement t  i , where τ represents the inverse temperature   1 / kT and is called imaginary time. Here k is the Boltzmann constant. The time-evolution operator in quantum mechanics corresponds to the density matrix (statistical operator) in statistical mechanics. Because of the additional degree of freedom associated with the imaginary time τ , there is important correspondence between the two systems, for example, a “d-dimensional quantum system of interacting particles” is equivalent to a “(d+1)-dimensional

Figure 8.2.1  Schematic drawing representing formal (mathematical) equivalence between a

d-dimensional quantum system and a (d+1)-dimensional classical system via imaginary time formalism (Nagaosa, 1995).

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Figure 8.2.2  Brief description of statistical mechanics, emphasizing the relationship between

the partition function and Helmholtz free energy.

classical system of interacting strings,” as schematically drawn in Figure 8.2.1. This correspondence allows dramatically simplified treatments of, for example, interacting dislocations in 3D, which requires many complications inherited due to their line-like nature. Figure 8.2.2 summarizes the fundamentals of statistical mechanics. Let us consider a system expressed as H n = En n , (8.2.1)





where H is the Hamiltonian of the system and En are the eigenvalues that represent the corresponding eigenstates of energy. Then, the probability for the state in the canonical distribution is given by 1 Pn  e   En . (8.2.2) Z Here, Pn must satisfy

 Pn 1 (the completeness condition). The system is charactern

ized by the partition function (or sum over states) defined by Z   e   En

n

  n e   H n  Tre   H ,

(8.2.3)

n

where the sum for all possible states is calculated. All of the statistical information of the system can be derived from the partition function, for example, Helmholtz free energy is obtained as

F   kT ln Z . (8.2.4) Note that, to obtain an effective theory about dislocation patterning the evaluation of the partition function for the coarse-grained system is indispensable, as will be discussed in detail in Chapter 10.

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8.2.2

429

Feynman Path Integral An alternative as well as intuitively tangible way to formulate or construct quantum mechanics is that invented by Richard P. Feynman (1972), whose portrait is displayed in Figure 8.2.3. It is known as the “Feynman path integral.” A schematic of the derivation process is presented in Figure 8.2.4.

Figure 8.2.3  Portrait of Richard P. Feynman, who reformulated quantum mechanics via the

Feynman path integral.

Figure 8.2.4  Overview of the Feynman path integral as another formalism of quantum

mechanics, focusing on the time-evolution operator instead of solving the wave equation.

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Method of Quantum Field Theory

His idea was to focus on the time-evolution operator rather than directly solve the Schrödinger equation to find the wave function, that is,

y( xN , t N )   dxG( xN , t N : x1 , t1 )y( x1 , t1 ). Sliced into minute pieces of time intervals ∆t, the motion of a particle during the time interval can be dealt with as the superposition of all the classical paths, each weighted with the Boltzmann factor e iH t / h. Ultimately, the time-evolution operator is written by the Feynman path ­integral, that is,



G( xN , t N : x1 , t1 )  x N e

i  H ( t N  t1 ) h

x1

i   x ( t ) x x(t )exp  S  x(t )  , h  x ( t )  x 

(8.2.5)

where the action is given as

t

S  x(t )   dtL  x(t ). (8.2.6) t

In an imaginary timeframe, that is, with t  i conversion, we can formulate, in the same manner, the partition function in statistical mechanics, as indicated in Figure 8.2.5:

Figure 8.2.5  The Feynman path integral in the imaginary timeframe, leading to the formulation

of the partition function in statistical mechanics.

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Figure 8.2.6  The Matsubara Green function (or temperature Green function)

for a statistical-mechanics formulation.

Z  Tre   H   dx x  e   H x



 1 h  (8.2.7)     (  )  x ( )exp  d H x     x (  h )  x ( 0 )  h 0 .   Note that, in this case, the integral of the Hamiltonian of the system rather than the Lagrangian inspires the action. It also should be noted that, in the Feynman path integral, the Lagrangian or Hamiltonian that appears in the action no longer has a q-number function but a c-number function, so no care needs to be taken with regard to the ordering of their operations. The price that must be paid, instead, is to perform an infinite-dimensional integral. The use of an imaginary timeframe to discuss statistical mechanical issues is also known as Matsubara formalism or the Matsubara Green function method. This is briefly summarized in Figure 8.2.6 (Abrisokov et al., 1963).

8.3

Application of QFT Method to a Many-Dislocation System

8.3.1

Brief Overview of the Underlying Idea



We are now ready to make use of the QFT method to address our problems with large numbers of interacting dislocations. Before entering into detail, let us look at some of my preliminary ideas about the application of this method. A detailed derivation process for the cellular patterning of dislocations (see Chapter 3), based on a combination of the gauge formalism introduced in Chapter 7 and the method of QFT in the present chapter, together with the associated simulation results and discussion, is given in Chapter 10. I first noticed an analogy between the superconductivity (e.g., Tinkham, 1996) and dislocation patterning in, for example, the PSB ladder, as schematically illustrated in Figures 8.3.1 and 8.3.2 (for further detail about the PSB ladder see Chapter 3). In superconductivity, the energy gap responsible for the emergence of the superconductivity

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Figure 8.3.1  The QFT-based derivation of the GL equation for superconductivity.

Figure 8.3.2  Analogy between superconductivity due to Cooper-pair formation and PSB-ladder

patterning due to dislocation-dipole formation.

appeared as a result of the condensation of Cooper pairs of electrons, separating a new grand state from the original state of the system. Since the electrons are fermion, they normally do not condensate into a single state. By forming the Cooper pairs, however, they can yield condensation as in boson (called Bose–Einstein condensation). In the case of dislocations, first, each of them is long range in terms of the stress and strain fields. Second, they also have signs (positive or negative) that are mutually repulsive, so that a pair of opposite-signed dislocations tends to form a dipole to lower the interaction energy, resulting in a relaxation of the long-range stress field. For the former (electrons), this condensed state produces a state of superconductivity, whereas, for the latter (dislocations), the condensed state results in patterned configurations (or distributions with lower energy) instead of uniform distribution, for example, PSB

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ladder and vein structures. Since the gap equation essentially captures the microscopic details of the mechanism and, at the same time, is the origin of the GL equation for superconductivity (Gor’kov, 1959), I expected to also be able to derive a GL-type evolution equation for the dislocation patterning by following the same procedure (Hasebe and Imaida, 1998). For the ladder-like and vein pattern formations, Walgraef and Aifantis (1985) have carried out pioneering work by utilizing the reaction-diffusion equation for two dislocation populations, based on the analogy to chemical reactions such as the Belousov– Zhabotinsky (BZ) reaction. Figure 8.3.3 presents schematics of the reaction-diffusion simultaneous equations for activator  and inhibitor Ψ h, that is,



   2  t  F   ,  h  Da   , (8.3.1)     h  G   ,   D  2   h h h  t where Da and Dh denote the diffusion coefficients for the respective concentrations, while F   ,  h  and G   ,  h  represent the reaction functions. When a critical condition of Da / Dh  1 is met, this set of equations is known to yield spatiotemporal patterning due to Turing instability, as exemplified in the inset (bottom) element in Figure 8.3.3. Inspired by the correspondence between the activator/ inhibitor and the mobile/immobile dislocation density, that is, ρ m and ρi , Walgraef and Aifantis proposed the following model for a typical sort of dislocation patterning, such as PSB vein/ladder structure observed in fatigue condition (cf. Figure 3.7.1), that is,



  m  2 m  b i i2  m  Dm   t x 2  2  i  g(  )  b   2   D  i (8.3.2) i i i m i  t x 2 g( i )   a  i  i  . Here, b and γ are coefficients, while  represents spatial average. The physical interpretations of Eq. (8.3.2), for example, the diffusion term, cubic term, and the coefficients introduced there, as well as its microscopic origin, however, have not been made clear. My plan for this deficiency is to identify it based on the QFT method, as outlined in Section 8.3.2. If the patterns are to be governed by the same processes as those assumed in reaction-diffusion equations in general, the major mechanism for the patterning to occur is system bifurcations, as depicted in Figure 8.3.4. According to the linear and weakly nonlinear perturbation analyses against Eq. (8.3.2), the ladder and vein patterns are the results of pitchfork and transcritical bifurcations of the dislocated system, respectively. We can derive a single GL-type differential equation based on a potential energy functional by obtaining the effective action functional based on the method of QFT

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Figure 8.3.3  Reaction-diffusion-type simultaneous differential equation yielding ladder-like

patterning as a dissipative structure (Walgraef and Aifantis, 1985).

Figure 8.3.4  Bifurcation diagram for dislocation patterning based on linear perturbation and

weak nonlinear analyses of reaction-diffusion equations.

presented earlier. However, we need one more piece of evidence to identify the resultant morphology of the targeted substructures. To this end, we need to treat the internal stress field as a manifestation of the interaction with the background field.

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Figure 8.3.5  Example of ladder-like pattering based on reaction-diffusion equations

(Haken, 1977). Adapted with permission of the publisher (Springer).

Note that, before recognizing the contribution by Walgraef and Aifantis, I had encountered a reaction-diffusion equation yielding ladder-like patterns toward a simulation in a biosystem, that is, a creation of tentacles of hydras (Figure 8.3.5) (Haken, 1977; Meinhardt and Gierer, 1974).

8.3.2

Application of QFT Method to a Many-Dislocation-System Cell Formation One of the most prominent outputs of the field theoretical approach in connection with the current QFT-based method is a successful reproduction of dislocation-cell structures (Hasebe, 2006; Hasebe and Imaida, 1998) based on an effective theory rigorously derived from a dislocation theoretical-microscopic theory. The complete details of the process are provided in Chapter 10. Here, use was made of the QFT method, as an equivalent mathematical formalism to statistical mechanics, to achieve rigorous derivation of an effective theory through systematic “coarse-graining” and renormalization of scales. The point of departure of the derivation is to write down the Hamiltonian of the dislocation system embedded in a crystalline space, which is determined based on the gauge field theory given in Chapter 7. Figure 8.3.6 displays an outline of the derivation procedure, starting from the FP dislocation-theoretical first-principle Hamiltonian of the system H total , which is given as a sum of pure elastic, pure dislocation, and their coupling terms. Second,

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Figure 8.3.6  Overview of the derivation of the effective theory, yielding cellular patterning of

dislocations based on the QFT method, details of which are provided in Chapter 10.

the quantized form of the Hamiltonian for the pure dislocation system incorporating dislocation-pair interaction by a quartic term ˆd†ˆd ˆd†ˆd  is considered. Regarding the pair interaction as pair-annihilation processes responsible for dynamic recovery, the energy-expectation value for the interaction   g d d  is introduced and is used as an order parameter (OP) for the coarse-grained system. That is, the ultimate effective action, or equivalently, a GL-type potential energy functional, is given as a function of Ψ. Note that we must also eliminate the elasticity field u such that the system satisfies the stress-equilibrium condition to reach the final equation. Examples of 2D simulation results based on the derived equation are shown in Figure 10.9.1. They provide many implications, as will be discussed in Chapter 10: (1) a dislocation interaction relating to pair annihilations and their collective effect leads to modulation in dislocation distribution instead of uniform distribution, and (2) a long-range internal stress field plays a key role in the evolution of cellular morphology, and directly determines the size, without which “far-from-cell” structures will possibly result.

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Part III

Applications I Evolution of Inhomogeneity in Three Scales 9 1 0 11 12

Identification of Important Scales Scale A: Modeling and Simulations for Dislocation Substructures Scale B: Intragranular Inhomogeneity Scale C: Modeling and Simulations for Polycrystalline Aggregate

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9

Identification of Important Scales

This chapter intends to identify the critical scales for achieving successful multiscale modeling and simulations of metallic materials in plasticity. To that end, we will look at the hierarchical structure from the viewpoints provided in Part I. The breaking down of the hierarchical scales in plasticity into finite numbers of representative scales of critical importance, in combination with the extensive use of FTMP-based models, is expected to make the practically important but difficult-to-solve problems much easier for us to tackle. As examples, three of my own in-progress research projects are outlined next. The contents provided will effectively lead us to the next three chapters in Part III, where FTMP-based approaches, models, and perspectives are applied to identified individual scales.

9.1

Introduction: New Hierarchical Recognition Let us reconsider the hierarchical system for polycrystalline metallic materials in plasticity in view of the content provided thus far, especially that in Chapters 3 and 4. As an alternative to a conventional recognition of the hierarchy, as displayed in Figure 5.3.4, we have Figure 9.1.1, where multiple scales starting from that associated with a single dislocation line, which is essentially in the atomistic scale, all the way up to that commensurate with test specimens, are illustrated. One can see that there are two highlighted scales: A collection (or an aggregate) of dislocations (a self-organized substructure) and of composing crystal grains, respectively. Both are associated with the “collective” behaviors of the composing elements, and lead to qualitatively new inhomogeneities. Hereafter, they will be called Scale A and Scale C, respectively. They are thus related to the inhomogeneities associated with dislocation substructures and grain aggregates, respectively. The scale located in between, on the other hand, is that for crystal grains (more specifically, intragranular inhomogeneity), and is called Scale B. These “inhomogeneities” inevitably lead to subsequent evolutions of thus-emerging “new scales” in respective scale levels. Figure 9.1.2 emphasizes the “scales” in terms of their emergence/evolution and effects on mechanical properties, and illuminates their interrelationships with the three “field theories” to be detailed in the following. To my knowledge, the significance of Scales A and C seems to have been overlooked or, at least, has not been discussed in most conventional textbooks to the level

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Figure 9.1.1  Hierarchical recognition of polycrystalline plasticity, where two scales emerge,

due to the collective effects of dislocations and crystal grains, are highlighted.

Figure 9.1.2  Interrelationship among three field theories, where the differential geometrical

field theory for describing dislocation and defects and the QFT method are linked by the gauge field theory.

of Scale B. This fact partially motivated me to elaborate a new framework of field theory, which differentiates the contents of the present book from many others. In order to emphasize the importance of these two scales, A and C, I introduced in Chapters 3 and 4 the “universality of cells” and “single crystals versus polycrystals” as the major

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ingredients of the Fundamentals part. This is, I believe, one of the unique features of the current monograph. Identifying these important scales enables us to clear away a lot of extraneous stuff such that we can focus on what really matters in accomplishing successful MMMs. Figure 9.1.2 presents a new schematic emphasizing these important scales, A, B, and C, where typical simulation results are attached to each scale. Figure 9.2.4 is similar to Figure 9.1.1, but puts more emphasis on the inhomogeneity that evolves in each scale. Another important perspective that we should recognize concerns the treatment of the targeted materials as one of the “complex” systems. If the material of interest is complex enough, we should look at at least “three” rather independent hierarchical scales interacting with each other, unless the system can be reduced to a “one-body problem” that does not need a multiscale approach from the start, or alternatively, unless a simple “information-passage”-type scheme will work. For visualizing to what extent such a scheme is unsatisfactory, a matrix-based representation will be useful. This is revisited in Section 9.2 (see Figure 9.2.5). Some detailed and concrete treatments are extensively argued in Section 13.2 in the context of the interaction-field formalism constructed in Section 6.7. In what follows, the three key scales are detailed individually. Also, some practical examples using the current perspective are presented. They are based on my own ongoing research projects. Here, the emphases will be placed on why and how such multiple-scale identifications and treatments are necessary and how they can be implemented.

9.2

Three Important Scales In order to capture the essence of the multiscale plasticity-related phenomena, at least three scales of inhomogeneities have to be considered, as pointed out in the previous section. Figure 9.2.1 reposts Figure 5.3.3, which symbolically represents a triangle consisting of the three key scales for multiscale modeling of polycrystalline plasticity. They are (A) dislocation substructures, (B) crystal grain, and (C) an aggregate of crystal grains, for example, an SSS described earlier. To aid understanding, it is important to recognize that the two “collective behaviors” lying between the microand macro-scales, as depicted in Figure 9.1.1, that is, Scales A and C, in a sense determine the hierarchical nature of the whole system. The interrelationship among the three scales is presented in Figure 5.3.3. Roughly speaking, each of them has its own role in the context of multiscale plasticity, as depicted in Figure 9.2.2: Scale A is the “reservoir” of the microscopic pieces of information as a form of strain energy, Scale B acts as an “absorber,” and Scale C works as the “regulator,” where macroscopically imposed stress or energy is classified as either a storage or carrier of plastic flow. As mentioned in Section 9.1, Scales A and C contribute to the collective behavior of the composing elements, that is, dislocations for Scale A and crystal grains for Scale C. Therefore, such qualitatively new characteristic scales spontaneously emerge

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Figure 9.2.1  Three key scales for successful MMMs in plasticity.

Figure 9.2.2  Multiple-scale correlations based on FTMP, together with distinct physical images

of an incompatibility-tensor field, emphasizing the specific roles of the three representative scales, that is, (A) reservoir, (B) absorber, and (C) regulator, mutually interacting during the course of elastoplastic deformations.

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Figure 9.2.3  Hierarchical scales for polycrystalline plasticity emphasizing three important

scales, A, B, and C, each with representative simulation results detailed in Chapters 10, 11, and 12, respectively.

therein, and they essentially come to control the upper-/lower-scale phenomena. Scale B, on the other hand, affects both Scales A and C, and, at the same time, is affected by them. Practically, Scale B has been used to control mechanical properties by alloying and introducing metallurgical structures, often accompanied by banded morphologies, termed GNBs. Figures 9.2.3, 9.2.4, and 9.2.5 provide further variations for representing the three scales. Figure 9.2.3 is similar to Figure 9.1.1, but is accompanied by a typical simulation result for each scale. Here, Scale B is further broken down into two types: Geometric (or noncrystallographic) and crystallographic types. Without considering microscopic degrees of freedom for the substructuring in this scale, only the former will result, as in any of the conventional crystal plasticity-based finite element (CPFE) simulations. In Figure 9.2.4, on the other hand, the emphasis is placed on the “inhomogeneities” to be evolved in each scale (those based on the simulation results). In the figure the indicated typical scale lengths roughly correspond to the individual sizes of the assumed representative volumes in simulating the evolving inhomogeneities:  5 m for Scale A, 10  20 m for Scale B, and 500  700 m for Scale C. Furthermore, they are illustrated as embedded within the “scale-free-like” hierarchical stress-power spectrum that will be extensively discussed in Section 13.4. This scale-free-like nature

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Figure 9.2.4  Three representative scales in conjunction with the “scale-free” relationship

numerically observed in the stress distribution (cf. Section 13.4), representing respective simulated contours, together with cross-sectional stress distributions.

of the stress spectrum implies that polycrystalline aggregates should have strong and nonnegligible interconnections in “local–global” structures from the start. Also mentioned in conjunction with Scale A is a connection with the core structure of dislocations (indicated tentatively as Scale “Λ”). When I wish to discuss the interscale interactions in more detail, the variation in Figure 9.2.5 will be used. Here, all the scales, from electrons to the macro-structure scale, are indicated by a matrix-like representation where the off-diagonal components indicate interactions among scales, and arrows specifying the directions of the flow are shown instead of individual components. The thickness of the arrows qualitatively indicates the strength: The thicker the arrow is, the stronger the interaction is considered to be. Note that if all the interactions are symmetric, the matrix may be diagonalized by an appropriate transformation. If this is the case, the treatments of the individual scales become completely independent. Some detailed and concrete treatments are extensively argued in Section 13.2 in the context of the interaction-field formalism constructed in Chapter 6. From Figure 9.2.5, readers should note that there exists a relatively isolated subsystem stretching from the electronic scale to atomistic (dislocation) scale. This means, for example, the electronic characters are carried to the upper scales through the scales of

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Figure 9.2.5  Interaction field-based perspective of simulations taking account of three scales

via an evolutionary matrix scheme for polycrystal plasticity to be treated as a complex system.

atomistics or dislocations only. Under such circumstances, information-passage-type descriptions become effective within the subsystem. Practically, the “empirical potentials” employed in molecular dynamics methods are a typical example, where the targeted energy landscape obtained in ab initio-based calculations in terms of variations in the interatomic configurations is approximated by some simple equations (analytical functions). Another example is the PN model for describing dislocation cores based on γ (gamma)-surface calculations. There, the ab initio-based energy landscape for the γ-surface of a targeted metal is obtained first and then, based on it, the core of a dislocation is expressed via continuously distributed dislocations. In this sense, such problems are not regarded as complex, even though many complicated computational and technical difficulties do exist. In sharp contrast to these, among Scales A, B, and C there exist strong (i.e., nondegenerated) interactions. Allowing for this, the system-wise treatment can be safely modeled as indicated in Figure 9.2.6, that is, the system can be composed mainly of Scales A, B, and C in terms of their evolving inhomogeneities. It should be noted that there are also some direct connections from the scale of dislocations in a group to the macro-scale based on statistical mechanics, that is, via “mean velocity” and “dislocation–dislocation interactions.” These can be taken into account through the crystal plasticity-based constitutive framework

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Figure 9.2.6  Conventional recognition of multiscale modeling and simulation of materials in

the light of analytical methodologies, anticipating information-passage-type procedures.

detailed in Section 5.6.1. In this case, crystal plasticity is simply regarded as a “vehicle,” in which FTMP-based concepts can be incorporated in practically feasible manner. Some detailed and concrete treatments of this are extensively argued in Section 13.2 in the context of the interaction-field formalism constructed in Section 6.7.

9.3

Independent Mechanisms of Field Evolutions As far as the multiscale aspects of plasticity, especially in polycrystalline metallic materials, is concerned, we can identify several important scale levels in terms of the evolutions of inhomogeneous fields, for example, orders of (A) dislocation substructures, (B) intragranular, and (C) transgranular (grain-aggregate) deformations. Interestingly, they seem to evolve quite independently or self-consistently, motivating researchers in the related fields to use rather arbitrary bases of their own when discussing multiscale plasticity. Careful observation of the practical phenomena, however, enable us to distinguish the definite existence of complex interactions and correlations among individual scale evolutions of inhomogeneities. Therefore, of major concern here are the evolutionary aspects, not only at individual- (multiple-)scale levels but also with regard to their interplay during plastic deformations.

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Note that the behavior of single crystals (plasticity) is sometimes much more complicated than that for polycrystals, as extensively discussed in Chapter 4, although this statement might seem paradoxical at first glance. This is because of sensitivity to the purity/ impurity levels and the complexities associated with individual motions of dislocations, crystallographic orientation (single slip/multislip), and boundary conditions for the specimen geometry being used. Most of the complications are absent, or, at least, nearly negligible in polycrystalline counterparts. A generalized version of this problem concerns “bare elementary processes versus elementary processes embedded within a hierarchy.”

9.3.1

Scale A: Dislocation-Substructure Level For Scale A, the collective behavior of interacting dislocations at high density, together with the long-range internal stress field, are responsible for cellular patterning (see Chapter 10 for details). The former, where massive pair annihilations are crucial interactions in a dual process of multiplication, is responsible for the modulation evolution (nonuniformity in the distribution), whereas the latter, having larger spatiotemporal scales than the former, dominates the 3D cellular morphology as well as the size. A statistical mechanics-based approach is required for the former, but the latter is additionally indispensable for the cellular morphology. We already have two spatiotemporal scales in here. The similitude law to be mentioned next is a consequence of the latter. The similitude law and various observations over a wide range of metals and alloys under various loading conditions have been well documented to date (see Section 3.2), which implies the cellular structure has universal features regardless of the microscopic details, for example, they can be tangled, aligned, or even be a bunch of small loop patches. A tremendous number of other microscopic details, for example, interaction strength or crystallography, in submicrometer down to nanometer scales, are renormalized into such a finite number of mesoscopic features. Also, my theoreticaland simulation-based approach essentially suggests the elasticity origin of the “similitude” relationship and the cellular morphology, which is dominated by the LRSF. The significance of this “universality” is the fact that there exists at least one cushion on the way from the microscopic degrees of freedom up to the macro- properties. This seems to act as a “reservoir” for many of the microscopic degrees of freedom, making the macro-properties rather insensitive to the microscopic details, as observed in reality. Once the cell structure is formed, the inhomogeneous field can be expressed by way of a continuum mechanics-based quantity, that is, the incompatibility tensor or, more simply, a single parameter such as the effective cell size dcell (Eq. (5.6.10)). Figure 9.3.1 provides a new hierarchical recognition for plasticity based on the previous discussion, where Scale A (the dislocation-substructure scale) plays key roles in many respects (to be discussed extensively in later chapters). Most of the microscopic “specificities,” such as electronics, atomistics, individual dislocation natures, interaction strengths, and other details about dislocations, are renormalized in Scale A as a form of LRSF in the cell interiors, as will be extensively discussed in Section 10.10. In between the micro-“specificities” and macro-responses, “weak ties” are introduced

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Figure 9.3.1  New hierarchical recognition for plasticity, where Scale A (the

dislocation-substructure scale) plays key roles in many respects, for example, “specificity versus universality,” “loose coupling,” and “weak ties” for system stability.

via the “reservoir” in Scale A. This also means the interrelations are “loosely coupled.” The two keywords, “weak ties” and “loose coupling,” are discussed in Sections 14.1.3 and 14.3.2, respectively. Based on these arguments, it may safely be said that 70 to 80% of the microscopic information is stored before going up to larger scales ultimately to determine the macroscopic properties. This situation may be rephrased as follows, that is, more than half of them, at least, generate a huge amount of “redundancy” in the sense that they do not directly or explicitly contribute to the macroscopic response. They are, however, as has repeatedly been emphasized, vitally important to understand what the targeted materials systems are in terms of the multiscale context. What is of exceptional importance and worth pointing out is that information about microscopic details “does not disappear” there but is simply “stored,” and can be released whenever needed. (Thus we should refer to the scale as a “reservoir.”) If the microscopic pieces of information remain stored during the course of deformation, the corresponding macro-properties are called “stable,” whereas if any of them are released to affect the macro-response, the properties may be referred to as “unstable.” An example of “stable” properties is the (macro-)stress–strain curve. Examples of “unstable” properties, on the other hand, are those related to “softening,” including transient softening in the Bauschinger effect, slip banding, shear banding (known to be the result of plastic instability), damage evolutions, and any other processes leading ultimately to fracture.

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Reconsidering from this viewpoint, we notice that many past studies seem to try to evaluate or predict the macro-stress–strain response (curve) based on microscopic set-ups such as atomic systems or discrete dislocation systems, which have “stable” rather than “unstable” properties. I would argue that this approach is quite inefficient, in the sense that we do not always need such detailed information if we are merely reproducing the stress–strain curves of the targeted materials or systems. This fact eloquently demonstrates the significance of both defining multiscale problems before starting modeling and simulations, and placing such problems in an appropriate order of importance.

9.3.2

Scale B: Intragranular Level For Scale B, geometrical constraints from surrounding grains (since all of them try to deform in different ways) are dominant, and field evolution is largely due to the accommodation process of excessively introduced inhomogeneous deformation. Thus, the associated substructures are expressed as “GN types” that can be dealt with by using the continuum mechanics-based framework (neither individual dislocation motion nor interaction details are always necessary). Therefore, the role of this scale in the multiscale plasticity of the system is as an “absorber.” Section 3.10 discusses the basic features of the dislocation substructures observed in this scale, where distinctions from that in Scale A, that is, dislocation-cell structure of mechanically necessary kinds, are also clarified. The inhomogeneous fields evolved in Scale B are generally manifested as “misoriented banded” structures (Figure 9.3.2), where the

Figure 9.3.2  Comparison with TEM images obtained in experiments by Winther, where

the norm of the dislocation-density tensor for horizontally pulled single-crystal models are displayed as simulation results, together with simulated misorientation distributions across A– A′.

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“misorientation” across the developed bands has the critical role of “absorbing” the excessive deformation, as described earlier. There also may be interactions with Scale A, which will be examined in Section 13.2 based on the interaction-field notion (see also Section 6.7). The incompatibility-tensor-based description of the accommodation process and the resultant structures are shown in Figure 9.2.1(b), where band-like structures (modulations) extending over the positive and negative signs of η evolve. An intuitive schematic of this process is given in Section 6.5.2 (Figure 6.5.4). Further details, based on a series of simulations as well as modeling examples based on the current field theory, are provided in Chapter 11. A key to capture the current scale correctly would be the “missing” degrees of freedom represented by the incompatibility-tensor field, coupled with the “flow-evolutionary law” introduced in Chapter 15. This perspective has, in the end, been the most important aspect for FTMP.

9.3.3

Scale C: Grain-Aggregate Level For Scale C, the collective behaviors of the composing crystal grains are expected to bring about transgranular inhomogeneity, as implied in Chapter 4. Since the current scale can be examined via a series of direct FE simulations on polycrystalline-​ aggregate models, many new implications could be derived from the results via direct observations of the obtained results, for example, “duality” and the associated remote effect, which can further be applied to other scales. The effects of grain morphology (including the size distribution) and/or heterogeneities (second phases) introduced are expected to appear activated via these new features: In many practical situations, they will enhance those effects. In this respect, the viewpoint is quite important in practice. “Duality” was found initially for this scale between the field fluctuations in hydrostatic stress and deviatoric strain components (see Chapter 12). This new notion is proposed as a key driving force for field evolutions in general. The fluctuating fields are manifested as an SSS and an FCS, respectively, which ultimately activate the “remote effect.” This also means a sort of “role sharing” between hardworking grains and others, further affecting the inhomogeneity evolutions in Scale B and Scale A. Since the macroscopically imposed energy of the system is roughly divided into SSS and FCS in the first place in terms of the roles of the composing crystal grains, the scale can be regarded as a “regulator” for other scales. Thus, the above “duality” is rephrased as “role sharing” between SSS and FCS, as presented in Figure 9.3.3. Note that an attempt to generalize this concept is made in Chapter 15, embodied as a “flow-evolutionary” hypothesis, where an extended use of the incompatibility tensor is made.

9.3.4

Description of Field Evolutions The various types of field evolutions can be mathematically expressed by the “incompatibility tensor” in a unified manner, as overviewed in Chapter 5. Some detailed

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Figure 9.3.3  Fluctuating stress and stress fields as a consequence of the “collective behavior”

of a large number of composing crystal grains, resulting in “role sharing” between an SSS and an FCS, termed as “duality,” which ultimately gives rise to a system-wise function as a “regulator.”

presentations of the incompatibility tensor are given in Chapter 6, especially in Section 6.3.2. Once this is done, all possible interactions among the scales of interest are also clarified and described in the mathematical context. Explicitly, this can be done in the manner summarized in Figure 9.3.4, where the evolving inhomogeneities in Scales A, B, and C are all incorporated in the hardening ratio of the hardening law to be used. Details of the formalism are described in Chapter 5. Also shown in lower part of Figure 9.3.5 is an image of the interscale cooperation in terms of a “scalefree”-like perspective, discussed in Chapter 13. The next step would be to propose a concept or scheme for measuring the evolution of the whole system in the light of interaction-controlled stability/instability. A preliminary attempt along this line will be given in Section 13.2.8, based on the interaction-field notion. Hypothetical but more advanced treatments are discussed in Chapter 15, where a general law for the field evolutions is proposed and partly verified. At this point, we should notice the necessity of more dynamic intersections among the cutting-edge theories and concepts in contemporary physics and mathematics. It may be valuable to revisit those individually obtained so far in detailed investigations from this point of view. Some potential ways to do so are introduced in Chapter 14. For example, the “topological invariants” presented in Section 14.2 may characterize the incompatibility field for individual scales. This notion is tentatively employed in evaluating a measure for representing the inhomogeneity in each scale in Section 13.2.8, as mentioned earlier.

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Figure 9.3.4  Crystal plasticity-based constitutive framework and attendant hardening model

illustrating how multiple-scale descriptions are taken into account.

Figure 9.3.5  Crystal plasticity-based constitutive framework and attendant hardening model

illustrating how multiple-scale descriptions are taken into account, together with a combined image of hierarchical recognition for three-scale correlations with the “global–local” nature of polycrystalline material systems.

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9.4  Examples of Three Scale-Based Approaches

9.4

Application Examples of Three Scale-Based Approaches

9.4.1

Three Projects

453

Let us look at how the three-scale perspective described previously would work by considering some of the application examples. I have been working on some practical aspects of multiscale plasticity based on the current FTMP in collaboration with industry and a research institute. Figures 9.4.1(a) and 9.4.1(b) show three project-based research subjects that I have been leading, based on the three scale-based notions identified and detailed previously. They are: Project #1: Prediction of long-term degradation of creep strength of high Cr steels for a fast breeder reactor (FBR). Project #2: High-precision prediction of springback properties for HSS. Project #3: Modeling and simulation of control of fatigue-crack initiation for Cu-added steels. Each project is broken down further into the subproblems of the three fundamental scales, Scales A, B, and C. The divisions are, of course, one of the possibilities of the problem setting; however, doing so seems to provide new insights to seriously solve the targeted issue in the multiscale perspectives.

Figure 9.4.1(a)  Pictorial presentation of the three projects based on the three key scales, A, B,

and C.

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Figure 9.4.1(b)  Keyword-based presentation of the three projects based on the three key scales,

A, B, and C.

In what follows, an outline of each project is provided, followed by the modeling guideline and some results of simulations conducted so far.

9.4.2

Outlines of the Three Projects

9.4.2.1

Outline for Project #1: Long-Term Degradation of Creep Strength of High Cr Steels (Inhomogeneous Recovery-Triggered Accelerated Creep Rupture) This project aims at the development of an evaluation method for the long-term creep damage and strength of high Cr ferritic heat-resistant steels (high Cr steel, hereafter), planned to be used for the heat-exchange piping of an upcoming FBR in Japan. One of the critical problems concerns the evaluation of the long-term degradation of the creep strength due to the inhomogeneous recovery of the microstructure. Since the design life of the plant is supposed to be over 500,000 hours (about 60 years) at a temperature of 550°C, the material to be used in the time range needs to be maintenance free. Since it is difficult to experimentally evaluate such a long-term behavior of the material’s strengths, even with stress- and temperature-accelerated conditions, designing simulation-based evaluation techniques is practically important. Figure 9.4.2 displays an overview of the project, comprised of the three key scalebased modeling and simulations, A, B, and C. The high Cr steel itself has complex hierarchical microstructures of a multiscale kind, composed of martensite laths, lath blocks, and lath packet structures. The three key scales are: Scale A: Highly dense

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Figure 9.4.2  Pictorial presentation of the project on long-term creep damage evaluation of high

Cr steel for FBR piping, composed of three-scale modeling and simulations.

dislocations pinned by minute precipitates, as a major ingredient of the lath martensite; Scale B: Lath block structures made up of collections of laths with the same variants; and Scale C: Lath packets as aggregates of the lath blocks with different variants.

9.4.2.2

Outline for Project #2: High-Precision Prediction of Springback Properties (Modeling Transient-Softening Behavior) This project aims to provide high-precision prediction of HSS for automotive bodies. For a precise prediction of the springback behavior, we must appropriately evaluate the Bauschinger behavior as well, in terms not only of permanent but also transient softening, since the blank sheet metal is subjected to bending–anti-bending-type load reversal at the flange part when it is deep-drawn into a desired shape. The prediction of the Bauschinger behavior beyond the use of phenomenological models is still one of the most difficult problems in industry (e.g., Wagoner and Chenot, 2001). An overview of the project is schematized in Figure 9.4.3. The smallest scale (i.e., Scale A) in this project would be a single segment of a dislocation pinned at both ends. Such a pinned dislocation segment will bow out under applied shear stress. The load reversal results in, as one can imagine, the segment returning to the original position without causing permanent deformation. This means that the response of the system should be “reversible,” regardless of the nonelasticity set-up. It will, however, result in apparent reduction in the elastic (shear) modulus.

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Figure 9.4.3  Pictorial presentation of the project on high-precision springback prediction of

HSS for automotive use, further composed of three-scale modeling and simulations.

Scale B in this project is for the intragranular scale, where GN-type dislocation substructures are formed and act as sources of the back stress in this scale. Examples of the inhomogeneity evolved in the scale will be given in Chapter 11. No contribution to the irreversibility in the deformation, however, has yet been identified, except in one case using a coarse-meshed FE model, referred to in Section 11.6.2. Scale C focuses on the grain-aggregate scale, where collective behavior-induced role sharing between the SSS and FCS (for details, see Section 12.3.3.4) is expected to produce back-stress fields in this scale. Regarding the output for Scale C, simulation results on the Bauschinger behavior for DP polycrystalline models are extensively examined in Appendix 12 and, additionally, in Chapter 15, in the context of the “flow-evolutionary law.”

9.4.2.3

Outline for Project #3: Cu-Added Steels in Fatigue (Fatigue-Crack Initiation Process) Cu additions have been reported to drastically change the dislocation substructures to be evolved in steels from cellular to planer ones, ultimately yielding finer surface intrusion/extrusion that delay fatigue-crack initiation. Figure 9.4.4 shows a rough sketch of this project, with twofold objectives, that is, the effect of Cu addition on the dislocation-substructure evolution and modeling fatigue-crack initiation from PSB.

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Figure 9.4.4  Pictorial presentation of project on the control of fatigue for Cu-added steels,

further composed of three-scale modeling and simulations.

The problem is roughly divided into three subjects: (A) Ab initio calculation of change in the electron density distribution of Fe considering the effect of magnetism (spin degrees of freedom must be taken into account) in the presence of a Cu atom. The crossslip energy for Fe screw dislocations in this context will also be examined; (B) evolution of dislocation substructures and their effects on the intrusion/extrusion-forming process; and (C) crack-initiation modeling and simulation from slip bands with intrusion/extrusion. Note that (A) requires a special scheme to deal with such highly nonsymmetric configurations of atoms with an impurity atom. Emphasis is placed on one of the ultimate objectives of the project, which is to develop a practically feasible computational model for simulating fatigue-crack initiation events at threshold, in connection with the evolving dislocation substructures during cyclic straining, thus on (B) and (C) in Figure 9.4.4. See Section 3.7.1 for relationships among slip bands under cyclic loading, accompanying dislocation substructures, and fatigue. Since the background of this project, that is, information about Cu-added steels, is not well documented, more detail are given in the following. It has been observed experimentally that the addition of Cu drastically changes the dislocation substructures in steels. Steels that are not strengthened by Cu generally exhibit a well-developed 3D cell structure of dislocations of the micrometer order, accompanied by coarse intrusions/extrusions at the steel product surfaces with the corresponding width, leading ultimately to crack initiation. The addition of Cu

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Figure 9.4.5  Overview of the fatigue problem with Cu-added steels. Cu addition greatly affects

not only the evolved dislocation substructures but also the cyclic hardening properties, and ultimately fatigue-crack initiation (Yokoi et al., 2001, 2004). Figures used with permission.

drastically changes the dislocation substructures to be evolved, that is, from 3D cell to rather uniform distributions in 2D, accompanied by finer intrusions/extrusions on the surface, leading to delayed crack initiation and, consequently, much enhanced high-cycle fatigue life. The solid solution form of Cu results in a 2D vein structure, whereas the precipitate form (of the order of several 10 nanometers) yields uniformly distributed planer dislocations. Cu also drastically changes the cyclic properties from cyclic hardening to softening. Figure 9.4.5 summarizes a series of experimental results demonstrating the above phenomena. Such a drastic change in dislocation substructures can be utilized as a key factor to control the fatigue properties of steels in order to yield the optimal performance for, for example, automotive and power-plant applications. No alternatives to Cu, however, have been found to date, despite intensive trial-and-error efforts, so we look at the quantum- (or electronic-)level effects of Cu on, for example, the magnetism-based BCC atomic structure of Fe, or the core structure of screw dislocations in Fe. The latter will significantly affect “cross-slip” events responsible for dynamic recovery (and thus also responsible for 3D cell formation). This is strongly implied from the fact that with the addition of Cu, the formation of 3D dislocation-cell structures is evidently inhibited, yielding 2D substructures.

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Figure 9.4.6  Transition from “slip bands” to a “crack” in terms of “critical slip distance”

(Kishimoto, 2004) (courtesy of Professor Y. Naka).

Another experimental examination that is relevant to our project is displayed in Figure 9.4.6, where the transition from “slip bands” to “a crack” has been investigated in detail by utilizing atomic force micrograph (AFM) observations coupled with interrupted fatigue tests on several materials. They found that a marked rate change takes place when a critical slip distance measured on the surface intrusions is reached, indicating the “transition” to a crack (Kishimoto, 2004). Furthermore, the critical slip distance is material dependent, that is, 100  nm for SUS304, 168  nm for HSS, and 380 nm for α-brass.

9.4.3 9.4.3.1

Details and Some Tentative Results Tentative Results for Project #1 Scale A, in this case, is set for highly dense dislocation structures initially pinned by small precipitates (carbo-nitrides and carbides, for example, MX and M23 C6) or impurity atoms (substitutional type), which are simulated via DD (Yamada et al., 2008). Some of the results are displayed in Figures 9.4.7 and 9.4.8. Here, the effect of the initially introduced number of precipitates (Figure 9.4.7), stress condition

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Figure 9.4.7  Examples of simulation results for Scale A of Project #1 regarding stability/

instability evaluation of highly dense dislocations within a martensite lath via DD using coarsening precipitate models. The result compares the effect of a number of precipitates, accompanied by coarsening, on the recovered configuration of dislocations.

(Figure  9.4.8), and the assumed precipitate-dislocation interaction types (inset diagram of Figure 9.4.8) are examined in detail. One of the results worth mentioning is that the combination of “repulsive” interaction and the “with stress” condition tends to yield the smallest variance for the final precipitate distribution even after coarsening, as demonstrated in Figure 9.4.8, in terms of the comparison of the variance of the precipitate size among four cases (see inset diagram on the right-hand side). Scale B is used to model the lath martensite blocks with the same variant, by making use of the incompatibility-tensor-based description of microstructure degrees of freedom that are provided in Chapter 11. A summary of lath block modeling along this line is presented in Figures 9.4.9(a) and 9.4.9(b), where simply by introducing an initial eigenstrain distribution based on the Bain lattice correspondence with a periodic rectangular wave shape in FEM analysis, for example, under biaxial compression, we can reproduce the lath block wall structures that accommodate misorientation that roughly satisfy the K–S (Kurdjumov–Sachs) variant (Figure 9.4.10). Here, the reproduced lath wall is made up of the screw components of the dislocation-​density tensor, which is also commensurate with an experimental observation (S. Morito, 2012, personal communication).

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Figure 9.4.8  Comparison of precipitate distributions among four precipitate-dislocation

interaction conditions: Combinations of “attractive” or “repulsive,” “stressed” or “nonstressed.” The combination of the “repulsive” and “stressed” conditions exhibits the smallest variance of precipitates.

Figure 9.4.9  Lath martensite block modeling as a preliminary example of the effect of initial

strain/distortion distribution. Bain lattice correspondence-based eigenstrain distribution followed by equibiaxial compression, results in screw components of dislocation-density developments, commensurate with a high-resolution TEM observation that reveals a lathe boundary wall consisting of a screw network (Morito, 2012).

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Figure 9.4.10  Lath martensite block modeling as a preliminary example of the effect of initial strain/distortion distribution. The obtained orientation distribution in conjunction with Figure 9.4.9, exhibits twist boundaries reaching more than 10°, and compares the conditions of three block widths.

Scale C relates to a recovery modeling of the lath packet structures, combined with the earlier mentioned lath block model in Scale B. An interaction field-based recovery model, presented in Figure 9.4.11, is additionally introduced, where incompatibility-​ tensor-related field interactions with Scale A are taken into account by utilizing the A-lath effective cell-size model dcell in Section 5.6.1.4 to provide information about highly dense dislocations and their recovery, treated earlier. This interaction field-based recovery model is ultimately controlled by the interaction term  BA  -  ∇ B × A sym

st 1 B A ¶t  sym , corresponding to the spatial and spatioand the recovery term R º eBA temporal interaction fields, respectively. Here, a series of creep analyses on the embedded packet models are conducted by using the FTMP–CP-FEM, by arbitrarily combining the lath block models fabricated in Scale B, whose hierarchical constructions are displayed in Figure 9.4.12 (ruptured results are shown). Simulated creep curves under two stress conditions are shown in Figure 9.4.13, comparing conditions with and without the interaction term η BA. Inhomogeneous recovery-triggered accelerated creep rupture under a low stress condition (100 MPa) is successfully reproduced, with the interaction field η BA taken into account, whereas no difference resulted under a high stress condition (200 MP). Figure 9.4.14 displays a stress versus time-to-rupture diagram plotted on the experimental counterpart. A systematic

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Figure 9.4.11  Interaction field-based “recovery” model, considering interactions with Scale A-lath A via the effective cell-size model dcell , ultimately comprised of two key factors of the interaction term η BA and the recovery term expressed concisely as R .

Figure 9.4.12  Hierarchical constructions of (A, B) single lath block, (B′) single packet, and (C) embedded packet model, where ruptured contours are shown.

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Figure 9.4.13  Simulated creep curves under two stress conditions (high and low) on an e­ mbedded packet model in Scale C, comparing the effect of the interaction term η BA, which demonstrates that inhomogeneous recovery-triggered acceleration in creep rupture occurs only in the low stress condition.

Figure 9.4.14  Comparison of stress versus time-to-rupture curves among three conditions.

series of examinations reveal that the simultaneous contributions of η BA and R are required for the accelerated degradations (Figure 9.4.15), in which a sort of “negative feedback” is activated to significantly enhance the fluctuating dislocation-density field in Scale A during the loop cycle (Figure 9.4.16).

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Figure 9.4.15  Effect of the (η BA , R ) combination on accelerated creep-rupturing behavior,

demonstrating the necessity of simultaneous contributions.

Figure 9.4.16  Negative feedback mechanism against a fluctuating dislocation-density field activated by the combination of (η BA , R ), ultimately bringing about accelerated creep rupture under a low-enough stress condition.

Let me add some more comments on the modeling of the lath martensite structures. Modeling lath martensite structures aimed at simulating not only the creep deformation behavior of the alloys that make up the structure but also the attendant rupture processes, including those promoted by, for example, inhomogeneous recovery, is quite challenging. Based on continuum mechanics, there exist at least three hurdles to clear before achieving this: (1) Modeling a “lath block” structure that satisfies: (a) crystallographic requirements such as the K–S variant; and (b) accommodates recovery modeling during creep.

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(2) Only by expressing the “raised strength” as an accumulation from the “base strength,” we can simulate the “deteriorated states” after recovery. The most basic stress riser is the introduced highly dense dislocation structures associated with the lath martensite in Scale A, which are pinned by minute precipitates and solute atoms. (3) Modeling “recovery” processes, ultimately stemming from those in Scale A, due to inhomogeneous coarsening of the precipitates. The third hurdle is the most difficult and, at the same time, most elusive, because of lack of knowledge about the detailed controlling mechanisms beyond the phenomenology, for example, inhomogeneous recovery near the block boundaries due to coarsening of the M23 C6 carbides. For the raised-strength hurdle, a systematic series of studies by the National Institute for Materials Science (NIMS), Japan, revealed a “scale-dependent” emergence of the strength of high Cr steels, Japan, based on variable-size Vickers tests, as summarized in Figure 9.4.17. This means, for instance, the total strength of the material is not determined by the martensite lath structure alone, but is achieved by combinations with the hierarchically introduced block, packet, and prior-austenitic grain structures, which must be built up on the base strength.

Figure 9.4.17  Scale-dependent strengthening mechanism revealed variable-size Vickers tests, an important modeling guideline for multiscale modeling of martensite lath structures (NIMS, 2003).

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Figure 9.4.18  Stress–strain relationship assumed in simulations that yield raised strength, that is, emergence of scale-dependent strengthening corresponding to Figure 9.4.17, achieved using the FTMP-based modeling scheme.

The model used in this project satisfies these basic requirements, as demonstrated in Figure 9.4.18. In particular, the raised strength corresponding to the scale-dependent emergence of the strengthening mechanism has been achieved by FTMP-based strain-gradient terms, particularly the incompatibility term with interaction-field formalism for Scales B and A.

9.4.3.2

Tentative Results for Project #2 Scale A modeling in this project was concerned with discrete dislocation segments that yield bowing-out and back motions under loading and unloading via shear stress. Such nearly reversible motions apparently reduce the elastic modulus when measured macroscopically. A systematic series of DD simulations for two typical configurations (horizontal [Case A] and vertical [Case B] to the shear-stress direction) with various numbers of the segments were conducted and evaluated based on FTMP schemes. Representative results are shown in Figure 9.4.19, comparing those for Cases A, B, and a theoretical prediction (Ihara et al., 2020). Since these sorts of phenomena can take place everywhere dislocation structures exist, the attendant mechanism is responsible for the empirically observed apparent reduction in the elastic modulus during elastoplastic deformation, as well as the so-called early reyielding to a large extent. An example of the FTMP-based evaluation, that is, via a duality diagram, will be presented in Chapter 15. Simulation results for further complex situations accompanied by pinning/unpinning by precipitates are summarized in Figure 9.4.20, where the residual strain differs from the above cases. The Scale B approach includes irreversible stress responses caused by substructural evolutions in single-crystal samples. One of the targets is a work-hardening stagnation

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Figure 9.4.19  Nearly reversible stress–strain responses brought about by bowing-out dislocation segments, resulting in an apparently reduced elastic modulus.

Figure 9.4.20  Nearly reversible stress–strain responses for cases with pinning/unpinning events as opposed to bowing-out dislocation segments.

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Figure 9.4.21  Stress plateau caused in conjunction with an evolved dislocation substructure, together with successfully reproduced orientation-dependent dislocation substructures and attendant stress–strain responses.

manifested as a “stress plateau” that appeared in a load-reversal response under simple shear for one of the γ -oriented ({111}) single-crystal Fes, as shown in Figure 9.4.21(a). Since another sample ({111}-oriented sample) does not show the “plateau” in the reversed stress response, it can be concluded that the difference comes from the distinct dislocation substructures evolved during the forward loading, as depicted in the figure via the corresponding TEMs. FTMP-implemented CP-FEM results are compared in Figure 9.4.21(b), together with simulated stress–strain curves. What is clearly demonstrated is that successfully reproduced dislocation substructures can reproduce the distinction in the reversed-stress responses, while those in the forward loading almost coincide, agreeing with the experimental trend (Hasebe et al., 2014). Some related simulation results are presented in Figure 9.4.22, giving a comparison of the developed misorientations as a function of imparted strain between the two γ orientations. Therein lies another, and far-reaching, strength of the FTMPbased scheme absent in any others; that is, not only the morphological aspects of the evolving dislocation substructures, but also the attendant misorientation developments across the walls. Scale C of this project targets irreversible stress–strain responses, with particular emphasis on transient-softening behaviors. A series of preliminary simulations imply that reproducing the “smooth” transient behaviors widely observed in experiments is

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Cross-sectional Misorientation Developments (Simulated)

Figure 9.4.22  Variation of simulated misorientation with strain for {111} and {111} during simple shear loading, corresponding to Figure 9.4.21(b).

quite difficult, at least if using simple modeling settings. Figure 9.4.23 summarizes a set of simulation results on 613-grained models with various grain-size distributions, without introducing any strain-gradient terms in the hardening law used. Regardless of the grain-size distribution, both almost coinciding stress–strain curves and only a slight transient region result. Other examples use the three models in Figure 9.4.24, that is, two seven-grained models with different resolutions, and a 613-grained model,

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Figure 9.4.23  Simulated stress–strain curves during tension compression for 613 polycrystal models with various grain-size distributions (without an incompatibility term).

Figure 9.4.24  Comparison simulated stress–strain curves during tension compression among three models. An incompatibility term is used in the hardening law.

referred to as Models 1, 2, and 3, respectively. Although these cases use the incompatibility term F   in the hardening law, they basically yield the same tendency as in Figure 9.4.23. Among them, however, Model 1 exhibits a relatively pronounced stress plateau-like response in the reversed loading. As confirmed in the attendant

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Figure 9.4.25  Comparison of the incompatibility contours among three models, together with cross-sectional distributions at points A, B, and C.

incompatibility contours and the cross-sectional distributions from point A to C displayed in Figure 9.4.25, Model 1 yields the largest change in the incompatibility between points B and C, that is, during reversed loading, which is responsible for the emergence of the pronounced stress plateau. To reproduce more “realistic” transient Bauschinger behavior, at least at present, we seem to need to introduce greatly exaggerated developments of inhomogeneity during forward loading. One eloquent example of this can be found in the DP models used in Appendix 12, with 50% hard-grain volume fraction, together with rather unrealistically enhanced strength ratios for the soft grains of 10, 50, and 100. Figure 9.4.26 redisplays the case for a strength ratio of 50, together with an attendant duality diagram and the unifiedly correlated Bauschinger strain, with the reciprocal of the duality coefficient measured on the diagrams at the maximum tension (see Chapters 12 and 15 for more details). This indicates that, during forward loading, the conversion rate of the inhomogeneously stored strain energy into the growth of the incompatibility, that is, the field of inhomogeneity, dominates transient-softening characteristics. There seem to be two possibilities at the present time for solving the question of how to reproduce realistically smooth “transient” softening behavior: (1) a much larger number of composing grains (over 613 grains) so that their collective behavior can embody marked SSS, as discussed in Chapters 4 and 12, and (2) coupling with the orientation-dependent intragranular dislocation-substructure evolutions that were discussed in the Scale B category, including organic interactions between the Scales B and C. This deserves further systematic investigation.

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Figure 9.4.26  Simulated pronounced “transient-softening” behavior in a DP polycrystal model with a relatively large volume fraction of the hard grains, and exaggerated strength ratios of 10, 50, and 100 against that of the soft grain. Redisplayed in Chapter 15.

9.4.3.3

Outline for Project #3: Cu-Added Steels in Fatigue The Scale A category includes not only (A-1) ab initio-based simulations for the screw-dislocation core of Fe with Cu precipitates, but also (A-2) molecular dynamics and DD simulation for dislocation interactions with Cu precipitates including the cutting events. A summary of the former is redisplayed in Figure 9.4.27. In order to examine the effect of copper addition on the dynamic recovery, its effect on the screw dislocation-core structure is investigated, since the cross slip by screw dislocations is considered to be responsible for dynamics recovery (moreover, cell structures are not formed in the absence of “dynamics recovery”). A systematic series of ab initio simulations reveal that the addition of three atomic rows of Cu totally alters the screw core structure of α -Fe, that is, from nonpolarized to fully polarized. Furthermore, such core structural change is shown to be accompanied by a “meta-stable” configuration under shear stress. For details, refer to Chen et al. (2008). Note that these results are mentioned in Chapter 4. The Scale B category is further split into subsubjects as (B-1) a discrete DD simulation of the detailed ladder walls under cyclic straining/stressing aiming mainly at examining the dislocation interactions in the channel region ultimately leading to vacancy formations; and (B-2) diffusion analyses of the vacancies in conjunction with a PSB ladder structure.

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Figure 9.4.27  Example of ab initio-based screw core simulation for α -Fe, with and without a

Cu cluster added to the core (Chen et al., 2008).

Simulation results for (B-1) are presented in Figures 9.4.28 to 9.4.31. Dynamic interactions among bowed-out screw segments passing to and fro in the channel region yield loop debris formations as the cross-slip event is activated, as revealed in the detailed DD simulations shown in Figure 9.4.28, driving the system response on the duality diagram from linear to nonlinear and complex ones, as exemplified in Figure 9.4.29. These loop debris eventually collapse into vacancies (Figure 9.4.30), whose assumed distributions are compared in Figure 9.4.31. It is, thus, clear that the PSB ladder wall structure tends to act as a vacancy generator, whose efficiency is controlled by the wall density. Regarding (B-2), the representative results obtained in the simulations are listed from Figures 9.4.32 to 9.4.34, basically demonstrating a role that assists the groove extension on the surface along the PSB edges. The vacancy formation and their

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Figure 9.4.28  Detailed simulation by DD for examining dislocation interactions in the channel region of the PSB ladder structure, comparing two cases of edge dipole densities, together with a variation of cross-slipped nodes with time steps.

Figure 9.4.29  Examples of duality diagrams: Case 1 shows nearly a linear response, while Case 2 partially yields a cyclic response, corresponding to dislocation interactions, including minute loop formations.

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Figure 9.4.30  Variation of equivalent stress contour with time steps for Case 2, demonstrating vacancy formations from collapsed minute loop debris.

Figure 9.4.31  Variation of number of vacancies with simulation time steps, together with anticipated vacancy distributions, comparing Case 1 and Case 2.

diffusion toward the surface along the band should be separately modeled and simulated. For the former, that is, the information about the vacancy formation is obtained partly from (B-1), as described earlier. For the latter, on the other hand, a diffusion equation proposed by Repetto and Ortiz (1997) is tentatively used, that is,

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Figure 9.4.32  Vacancy-diffusion analysis for surface-groove formation, assuming elastic strainenergy distribution corresponding to the PSB ladder pattern.

Figure 9.4.33  Effect of assumed elastic strain-energy distribution (PSB ladder pattern) against a free sample surface on vacancy flux, comparing four cases.

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Figure 9.4.34  Comparison of predicted receded surface profiles due to vacancy outflow among four cases in Figure 9.4.33, demonstrating insensitivity to the wall position against the sample surface.



cvacancy   cvacancy  0  D   0 cvacancy   0W   sv , (9.4.1) RT   with a tensorial form of the diffusion coefficient



D º D lattice I + Dpipe , where Dpipe is the coefficient for pipe diffusions. Note that Dpipe b2  D s Ä s is assumed in Repetto and Ortiz (1997). Here, several strain energy distributions are assumed (Figure 9.4.33) to seek desirable conditions for the vacancies to be diffused in order to efficiently produce grooves at the surface–band intersections. The vacancy flow associated with the walls turns out to have a dominant effect on the surface-groove formation, whereas the wall position against the surface is insensitive to it. FE simulation results show that the strain energy distribution after 10 cycles is similar to the assumed one, both in terms of the ups and downs at the wall-channel regions, and additionally have an increasing trend toward the surface. We confirm two things: (1) increasing strain energy approaching the surface, and (2) alternating high and low strain energy corresponding to the ladder wall and channel regions, respectively. These trends turn out to be desirable for both vacancy concentration and for diffusion toward the surface.

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The vacancy flux is given by

cvacancy   jvacancy  D   0 cvacancy   0W . (9.4.2) RT   From the corresponding velocity vvacancy = jvacancy /cvacancy , the attendant velocity for the surface to recede is evaluated as



vnsurface  jvacancy  nsurface . (9.4.3) Four cases are assumed in terms of the relative position of the wall against the surface (Figure 9.4.34). Since the vacancy flow will be sensitively affected by the condition, it is expected to control the groove-formation behavior. The simulation results are displayed in Figures 9.4.33 and 9.4.34; the former represents the variation of vnsurface with time, while the latter shows virtual processes of the surface grooving. As demonstrated, the wall position has a dominant effect on the vacancy flow-driven groove formation. With the given strain energy distribution, the grooves are basically formed at the band edges, except Case (a), in which the wall is facing the surface. The closer the wall is located from the surface, the faster the surface recedes. Cases (a) and (d) seem to be equivalent but they will differ when an additional gradient in the strain energy is superimposed. The Scale C category targets “crack-initiation” processes, evolved from slip bands, typically accompanied by the PSB ladder structure, as dealt with in Scale B. These processes are further categorized into three subdivisions: (C-1) those from a PSB ladder in single slip-oriented single-crystal samples, as the most fundamental problem, (C-2) those from other morphologies of dislocation substructures such as cells, and (C-3) those for polycrystal cases. Regarding (C-1), since we have made satisfactory progress, we can present a preliminary report. Figures 9.4.35 to 9.4.38 provide a preliminary collection of results, showing the formation of an inclined banded region extending from the bottom right-hand edge of the sample, which is a single-crystal sample subjected to cyclic straining in the single-slip orientation [125]. We employ a plastic-strain amplitude-controlled condition with  p / 2  6.0 ´10 3. The test material is assumed to be BCC Fe with {110} 12 slip systems, with no special treatment prescribed to the banded and matrix regions, unlike those in Repetto and Ortiz (1997). This evolved band resembles the so-called persistent slip band (PSB) consisting of edge-dislocation dipoles with ladder-like morphology, although the ladder-like pattern in the current simulation slightly deviates from the band direction. The incompatibility contour, Figure 9.4.35(a), is obtained after projecting all the F(    ) terms on that of the primary slip system. As shown, this projection procedure allows cancellations of the modulated pattern in the matrix region across the sample except the banded region, which distinguishes the laddered slip band from the matrix in the η-contour. For further demonstrating that the banded region carries the major fraction of the imposed deformation, the sample is pulled after interrupting the cyclic straining (Figure 9.4.37). The result is displayed in Figure 9.4.35(c), exhibiting “landslide-like” localized shear deformation taking place along the band, which proves that the banded region has been exclusively softened during the cyclic straining.

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Figure 9.4.35  Cyclically straining single slip-oriented single-crystal model under a plasticstrain amplitude-controlled condition, demonstrating a spontaneously emerging PSB ladderlike substructure, showing contours of (a) an incompatibility term projected on the primary slip system, (b) a corresponding dislocation-density norm, and (c) the result of ­tentatively pulled sample after PSB formation.

Also worth noting is that not only the banded area but also the matrix yield negligible misorientation, at least less than 1.0°, which is commensurate with the real situation where the dislocation structures are basically organized into edge dipoles (Suresh, 1998). A quick verification of this is given in Figure 9.4.36, where contours of the edge and screw components, that is, α 31 and α 22 of the dislocation-density tensor, are compared, as is a snapshot of the cross-sectional distribution (the partner components, α 32 and α11 [not shown here] also show similar trends, respectively). The two contours are basically aligned and alternatively vary, corresponding to the wall-constructing edge dipoles and bowing-out screw segments in the channel region, respectively. The ultimate objective of the above simulation in our project is, of course, not merely the reproduction of the PSB ladder structure but to reproduce crack-initiation processes, that is, to simulate transition processes from a groove to be formed at the edge of the PSB in the sample surface into a “crack.” Here, the completion of the transition is judged by a fully developed stress singularity at the grove root along the PSB. Figure 9.4.37 shows variation of the surface profile with straining cycles, where the eventually emerged “groove” is under extension. The analyses are continued on

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Figure 9.4.36  Example of some detailed analysis of the emerging PSB ladder-like structure in Figure 9.4.35, in terms of component-wise mutual distribution of the dislocation-density tensor. Alternatingly aligned distribution and variation of edge and screw components of dislocation density are confirmed.

Figure 9.4.37  Simulated surface-groove formation at the PSB-surface edge region, and evolution toward a “crack.”

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Figure 9.4.38  Further surface-groove extension by a restart analysis from Figure 9.4.37, together with developing stress distribution from the groove root, demonstrating its approach to a “crack,” that is, HRR singularity.

a localized model by utilizing a “cut-and-paste” operation that maintains the final surface profile in Figure 9.4.37. The results are summarized in Figure 9.4.38, demonstrating further extension of the groove (not shown here), while the stress distribution along the PSB measured from the groove root progressively approaches that of a crack, where comparisons with elastic and HRR (Hutchinson-Rice-Rosengren) solutions are made.

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10 Scale A: Modeling and Simulations for Dislocation Substructures

10.1 Overview As we have seen in Chapter 3 (for “universality” of dislocation cells), many of the microscopic “specificities” are renormalized into a limited number of degrees of freedom at the dislocation-substructure scale (Scale A), especially into those with “cellular” morphology, essentially extending over 3D crystalline space. Therefore, as a critical step toward successful multiscale plasticity, we are required to answer the following questions about 3D cell structure: “Why do we need the 3D ‘cellular’ morphology?”; “What is the substantial role of 3D cell structure, especially with regard to mechanical properties?”; “Why does well-documented ‘universality,’ manifested as a similitude law, hold?”; and “How are microscopic degrees of freedom (information) stored and when will they be released?” As reviewed in Appendix A10, no satisfactory theoretical approach and model has been proposed, partially because of the difficulty of 3D treatments of dislocation lines, including self- and mutual interactions. The recent advent of rapidly growing computer technology (including both central processing unit (CPU) speed and sophisticated parallelization schemes) and associated large-scale massive simulations, coupled with well-developed numerical techniques with both intelligence and brute force (e.g., Kubin, 1996), have been tackling the related problems. Even so, it seems to take a lot of time (of the order of 10 years?) to solve and answer the above questions appropriately and satisfactorily. From a practical point of view, we cannot wait until then. We must seek alternative approaches. The first goal of this chapter is to rigorously derive an effective theory governing dislocation-substructure evolutions, particularly cellular patterning, from a dislocation theory-based microscopic description of the Hamiltonian through a rational “coarse-graining” procedure provided by the QFT method (see Chapter 8). Second, after presenting some representative simulation results, an extensive series of discussions on cell-formation mechanisms and mechanical roles are presented. The Yang–Mills-type gauge formalism proposed by Kadić and Edelen (1983), with the T(3) gauge group, is used first to construct the microscopic expression of the Lagrangian density, taking into account both the dislocation field and its interaction with the background elastic field (see Chapter 7). The partition function expressed by the Feynman path integral in the imaginary time formalism is evaluated to ultimately obtain an effective theory of the form of GL potential-energy functional, from which time-dependent Ginzburg–Landau (TDGL) equations can be derived.

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Readers who are not interested in the derivation details and want to proceed to the results can skip to Section 10.6, where the final explicit form of the GL-type potential-energy functional is given.

10.2

Evaluation of the Hamiltonian for the Dislocation Field

10.2.1 Preparation As preparation for the second quantization of the dislocation field, we will derive the corresponding Hamiltonian from the gauge field-based Lagrangian. To this end, the canonical conjugate moment of the dislocation field and associated gauge-fixing conditions are required, as shown in Figure 10.2.1. In the following, we consider the dislocation field only, that is, with the T(3) gauge group. In this case, the covariant derivative defined in Eq. (7.2.1) is reduced to

Da x i   a x i  ai  Bai . (10.2.1)



Accordingly, the strain tensor Eq. (7.2.5) is expressed explicitly as 1 E AB  ( A x i   Ai ) ij ( B x j  Bj )   AB 2 (10.2.2) 1   A x i  B x i   AiBi  2 i( A  B ) x i   AB . 2 From the field strength of the dislocation-gauge field, that is,



i Dab  2 [ abi ] , (10.2.3)











the gauge-invariant Lagrangian density for the dislocation field (Eq. (7.2.3)) is given explicitly as 1 i disloc    ij Dab k ac k bd Dcdj 2 1    ij  abi   bai k ac k bd  cdj   dcj (10.2.4) 2  2 k ac k bd [ abi ][ cdi ] .









The Hamiltonian density for the dislocation field can be obtained via a Legendre transformation of the corresponding Lagrangian density, given in Eq. (10.2.4),



Hdisloc   iK Ki  Ldisloc   iK  Ki   K  K 4i  Ldisloc (10.2.5) 1 i K 1    K  i   iK  K 4i  [ K Li ][ K Li ] , 2 4 with K running over 1, 2, 3. Here, π Ki represents the canonical conjugate momentum to the dislocation field φKi , defined by



 Ki 

Ldisloc   D4i K . (10.2.6)  4Ki

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Figure 10.2.1  Preparation step for second quantization for dislocation field, where the canonical conjugate of dislocation field, together with fixing conditions, is provided.

Also, we have





disloc i   D44  0, (10.2.7) i  44 because the Lagrangian density is independent of the time derivative of φ4i due to the i antisymmetric nature of Dab in Eq. (10.2.3). Equation (10.2.7) gives the primary constraint condition. The secondary constraint is given by

 4i 

disloc   4  0. (10.2.8) i D4 K The corresponding gauge-fixing conditions to Eqs. (10.2.7) and (10.2.8) are  K  iK   K

4i ( x)  0 and  K Ki ( x)  0. (10.2.9) The second equation is called the “Coulomb gauge” condition, and confines φ Ai into the transverse (solenoidal) component. This confinement reflects the fact that the disi i located fields are the place where local shear occurs. Let φaT and π kT be the dislocation and its conjugate fields, respectively, satisfying Eq. (10.2.9) hereafter. From the Lagrangian density obtained earlier, the corresponding Hamiltonian density is given by





 total

elastic  s1disloc ,

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(10.2.10)

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where elastic, denotes the Hamiltonian density for elastic body, derived as 1 1 e 0 B4i  ij B4j  DABCD E AB ECD 2 2 (10.2.11) 2 1 1  E AB  .  0 B4i  ij B4j  K E AA  2  E AB 2 2

elastic 











Let us further derive the explicit form of elastic. For that purpose, we employ the explicit forms of the strain tensor Eq. (10.2.2), as depicted in Figure 10.2.2. Rewriting Eq. (10.2.2) with respect to the displacement vector ui  xi  Xi , we have

 SYM  12 AiBi   i( A B)ui (10.2.12) 1   AB   AB    AiBi   i( A  B )ui , SYM 2

E AB   ( A uB )   AB



where “SYM” expresses symmetrization. The deviatoric and volumetric components are thus given respectively as 1     AB    AB   AiBi   i( A  B )ui  E AB SYM 2 (10.2.13)   E AA   AA  1  Ai Ai   i( A  A)ui .  2

 



For the kinetic energy term in Eq. (10.2.11), only the pure elastic component remains because the dislocation-related components all vanish due to the gauge-fixing condition 4i  0 introduced previously, that is, B4i   4 x i  4i   4 x i. Note that an important difference from the electromagnetic gauge theory (Sugita et al., 1998; Weinberg, 1995) is the nonlinearity of the dislocation field in Da x i with respect to the gauge field φai, while the electromagnetic field is linear. This produces coupling terms to be included in the Hamiltonian describing the interactions between the dislocation field and the background elastic field via terms such as  iA iA   A u A. For our purposes, we use the volumetric component,  iA iA   A u A, only as the coupling term considering its physical interpretation, which will be expressed hereafter as  iA iA   u . Substitution of Eq. (10.2.13) for Eq. (10.2.11) allows us to divide elastic into three components, that is, pure dislocation, pure elastic, and their coupling:









2 1 1  E AB  0 B4i  ij B4j  K E AA  2  E AB 2 2 1 1 1 2   AB   K iA iA   u    others   0  4 x i ij  4 x j  K  u   2  AB 2 2 2 pure        , (10.2.14) couple disloc elastic

elastic 





′ , consisting of higher-order terms with respect where (others) are expressed as disloc to φ iA.

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Figure 10.2.2  Explicit form of the strain tensor obtained based on gauge field theory in Chapter 7.

Both total and deviatoric components are given.



10.2.2

pure Combining all the dislocation-related Hamiltonian components into disloc , that is,    , the total Hamiltonian density for our system is written as disloc disloc

pure   s1 disloc



 total

pure   disloc

pure   elastic

couple . (10.2.15)

Second Quantization of Dislocation Field

10.2.2.1 Canonical Quantization Based on the Lagrangian density obtained earlier, we consider a second quantization of the dislocation field, that is, for disloc . The overview of this process is summarized in Figure 10.2.3. This provides an important basis for the application of the method of QFT provided in Chapter 8 to the dislocation-ensemble system. The canonical quantization procedure (Sugita et al., 1998; Weinberg, 1995) is applied here to the pure Hamiltonian density of the pure dislocation field disloc in Eq. (10.2.15). The Poisson brackets to be held between the dislocation and its conjugate fields are

ai ( x),  bi ( x)P  ib ab 3 ( x  x) and (10.2.16) ai ( x), bi ( x)P   ai ( x),  bi ( x)P  0, which are necessary for the canonical quantization procedure given later. Here, b ­represents the magnitude of the Burgers vector. Since the choice of this gauge violates

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Figure 10.2.3  Derivation of the Hamiltonian of the dislocation field as a first step toward canonical second quantization.

the commutation relation between  aai and π bi given above, the equation should be rewritten as

ai ( x),  bi ( x)P  ib   ab  a 2b   3 ( x  x), (10.2.17) where the longitudinal components of φai and π bi are eliminated from the original relation. Here,  2   a  a denotes the Laplacian. Let us express the transverse comi i ponents of φai and π bi by φaT and π bT in the following. In the Coulomb gauge, the Hamiltonian density for the dislocation field Eq. (10.2.5) becomes



i Hdisloc   iTK KT  Ldisloc j i i .   KT  iTK   K  AT  K  AT

(10.2.18)

Figure 10.2.4 displays the derivation process of the second quantized-dislocation field from the gauge field-theoretical Hamiltonian Eq. (10.2.18) to be carried out here. The field equation (E–L equation) for the dislocation gauge is obtained based on the variation principle with respect to φai (Kadić and Edelen, 1983) (see Section 7.3.3.2) as

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Figure 10.2.4  Plane-wave solution of the field equation (stress free) for the dislocation field, which gives the second-quantized field of dislocation.

  i 1 1 B BD  A  s1 ji    A   4  4  DT   j y 2    . (10.2.19)   s1    A   1     i  1 p j A 4 4  4T  y ji  y 2   



The left-hand side of the second equation should vanish because of the gauge-​fixing condition given in the first part of Eq. (10.2.9). For the free dislocation field, which is obtained from the E–L equation for Ldisloc with respect to φai , we have  i  A 1 i    A   4  4  BT  BT  0, (10.2.20) y  



where    A  A  (1 / y) 4  4 gives the d’Alembertian. Assuming a plane-wave 2 i i ­solution KT ( x) = ˆKT ( k )eik  x to Eq. (10.2.20), we have  k0   k 2  0, where k0 ≡ k . The second quantized-dislocation field is thus given by

i ( x)    d 3 k ˆKT i 1

b

 2 

3

k0





 i(r ) K(r ) ˆd(r ) ( k )e ik  x  ˆd(r )† ( k )eik  x , (10.2.21)

where ε i(r ) and ε K(r ) are polarity vectors specifying normal directions to the Burgers vector and slip direction, respectively. For ε K(r ) , on the other hand, r takes 1 through 3 that describe edge or screw components of dislocation. The former can be set to unity

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Figure 10.2.5  The second quantized-dislocation field and its canonical conjugate.

in the present context, that is,  i(r )  1, without a loss of generality if we confine the dislocation field to a certain slip plane. i ( x) by introducing commutation Next, let us obtain the canonical conjugate of φˆKT relations, as indicated in Figure 10.2.5. The coefficients φˆd(r ) and φˆd(r )† in Eq. (10.2.21) represent the creation and annihilation operators for the dislocation field. With these operators, the commutation relations Eq. (10.2.17) become     



ˆd(r ) ( k ), ˆd( s )† ( k )   ib rs 3 ( k  k )   , (10.2.22) r s ( ) ( ) ˆd ( k), ˆd ( k )   ˆd(r )† ( k ), ˆd( s )† ( k )   0    

kK kL i ( x) is is used. The canonical conjugate of φˆKT where  K(r ) ( k ) L(r ) ( k )   KL   k2 given by i i ( x)  ˆKT ( x) ˆ KT

   d 3 k i 1

b

 2 

3

k0



 (10.2.23)

 i(r ) K(r )ik ˆd(r ) ( k )e ik x  ˆd(r )† ( k)eik x .

Substituting Eqs. (10.2.21) and (10.2.23) for Eq. (10.2.18), we have the second-quantized Hamiltonian for the free dislocation field (Figure 10.2.6), that is,

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Figure 10.2.6  Derivation of the Hamiltonian for the second-quantized field of dislocation.

0   d 3 xdisloc Hˆ disloc



j i i  iTK   K  AT   d 3 x  KT  K  AT

 i, j







1 d kd k   ik0 K(ik) ˆd(i ) ( k )e ik x  ˆd(r )† ( k)eik x d3 x (2 )3  2 k0 2 k0   ik   ( j ) ˆ ( j ) ( k )e ik x  ˆ ( j )† ( k )eik x 3

 ikK  L(ik)

3



0 Kk 

 ˆ





(i )  ik x ˆ (i )†  d ( k )eik x d ( k )e



d

d



 

ikK  L( kj ) ˆd( j ) ( k )e ik x  ˆd( j )† ( k )eik x ,  (10.2.24)  i, j



1 d 3 kd 3 k  3 d x  2k0 2k0 (2 )3 





†  ( j) (i ) ( j ) (i ) (r )† -i ( k  k) x ( j )   d ( k )d ( k)ei ( k  k)x   k0 k0  Kk Kk  d ( k ) d ( k)e  kK kK  L(ik) L( kj ) ˆd(i ) ( k )ˆd(i )† ( k)ei ( k  k)x  ˆd( j ) † ( k )ˆd( j ) ( k)ei ( k  k)x  





(10.2.25)



Performing the spatial integral 1 (2 )3

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  i, j



d 3 kd 3 k 3  ( k  k) 2 k0 2 k0





† ( j)  (i ) ( j ) (i )  i ( k  k) x ( j )  (r ) †   d ( k )d ( k)ei ( k  k)x  k0 k0  Kk Kk  d ( k ) d ( k)e kkK kK  L(ik) L( kj ) ˆd(i ) ( k )ˆd(i )† ( k)ei ( k  k)x  ˆd( j ) † ( k )ˆd( j ) ( k)ei ( k  k)x  





(10.2.26)





 

d3 k k02 K(ik)  K( jk) ˆd(i ) ( k )ˆd( j ) ( k )  ˆd(r )† ( k )ˆd( j )† ( k ) 2 k0  k2 L(ik) L( kj ) ˆd(i ) ( k )ˆd(i )† ( k )  ˆd( j )† ( k )ˆd( j ) ( k )

 

d3k 2 k0

i, j

i



 



 k  k  ˆ 2 0

2

(i ) ˆ (i )† ˆ ( j )† ( k )ˆ ( j ) ( k ) d ( k )d ( k )  d d

  d3 x    d k k0ˆd( j )† ( k )ˆd( j ) ( k )   k , 0 i  2 2  

 (10.2.27)

3

where  K(ik)  K( jk)   ij is used in obtaining the third line from the second. Since the second term of the final equation can be neglected (Figure 10.2.6), we finally have

3

0 Hˆ disloc   d 3 k  bk0 ˆd(r )† ( k )ˆd(r ) ( k ). (10.2.28) r 1

Thus, the dislocation field in the second-quantized picture is described as the excitation of the crystalline background field that is created and annihilated (cf. Figure 5.1.1 for a corresponding image).

10.2.2.2 Expression for Dislocation–Dislocation Interaction In the second-quantized formalism defined earlier, pair interactions between two dislocation fields with opposite sign can be expressed by a quartic term, such as

pair H disloc   dr ˆd† ( r )ˆd† ( r )ˆd  ( r )ˆd  ( r ). (10.2.29)

The sign attached to the subscript d represents that of the Burgers vector, so the interaction described by the term is regarded as a pair interaction of two dislocations of opposite sign, for example, dipolar formation, junction formation, or pair annihilation. Figure 10.2.7 illustrates an intuitive image of this pair interaction, together with possible interpretations, in the present context. Here, emphasis is placed on the contrast between “pair annihilation” and the others. For the cellular patterning of dislocations, we consider pair annihilation rather than the other kinds of interactions. As one can see, this can drastically simplify the theoretical treatment, because the latter inevitably produce various products as a result of reactions. Combining Eqs. (10.2.26) and (10.2.27), gives us the second-quantized Hamiltonian for the pure dislocation system as a point of departure for the quantum field calculation:

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Figure 10.2.7  Pair-wise interactions among dislocations, in terms of a contrast between “annihilation” and “creations,” with the latter including all other kinds of reactions such as elastic interaction, dipole formation, and junction and jog formations. The energy-expectation value of such pair interaction (paring potential) can be regarded as an AF representing the interaction.



pair 0 Hˆ disloc  Hˆ disloc , (10.2.30)  Hˆ disloc



3  ˆ0 ˆ (r )† ˆ (r ) H  d r  disloc   0 d ( r )d ( r ) , (10.2.31) r 1   Hˆ pair  g dr ˆ † ( r )ˆ† ( r )ˆ ( r )ˆ ( r )  disloc  d d d d





where 0  s1k0 b, which will be further given explicitly as 0   * b2 / 4 ln  r1 / r0  per unit length, with  *   for screw and  *   / (1  ) for edge components.

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10.2.2.3 Second-quantized Expression of the Total Hamiltonian As summarized in Figure 10.2.8, by using the just-obtained second-quantized dislocation-field expression, the second-quantized version of the total Hamiltonian for the whole system is given as pure pure Hˆ total  Hˆ disloc  Hˆ elastic  Hˆ couple , (10.2.32)

where



3 ­ ˆ pure ˆ (r )† ˆ (r ) ˆ† ˆ† ˆ ˆ ° H disloc ³ dr ¦ [0 Id Id  g ³ dr Id Id Id Id  r 1 ° °° 1 1 2 º ª pure i j c H AB c » . (10.2.33) ® Hˆ elastic ³ dr «¬ U0 w 4 x G ij w 4 x  2 K  ’˜u  2 PH AB 2 ¼ ° ° 1 r r r r ( )† ( )† ( ) ( ) drK Iˆd Iˆd  Iˆd Iˆd ˜  ’˜u ° Hˆ couple 2³ °¯

^

^

`

`

10.2.2.4 Process Overview To derive an effective Hamiltonian of the dislocation ensemble from the microscopic version, including dislocation–dislocation interactions, next we should calculate the partition function of the system, from which any of the thermodynamical properties

Figure 10.2.8  Total Hamiltonian for a dislocated elastic medium containing contributions from an interacting pair of dislocations, as the microscopic expression of a point of departure from the current use of the QFT method.

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Figure 10.2.9  Partition function via the Feynman path integral in an imaginary timeframe, expressed as the exponent of the action functional as a function of the Hamiltonian given in Figure 10.2.8, further rewritten by the product of three contributions, that is, pure elastic, pure dislocation, and their coupling.

can be obtained. The partition function is expressed as a Feynman path integral of an exponent of the action functional (Kleinert, 1989; Nagaosa, 1995), which is further given by a temporal integral of the Hamiltonian of the system in the imaginary time formalism. The partition function for us to evaluate is given via the Feynman path integral of the exponent of the action functional formed with the Hamiltonian Eq. (10.2.32) in the imaginary time (see Eq. (8.2.7)), as presented in Figure 10.2.9, that is,





Z   d d u e  S (d , u)    d d u exp    d H total (d , d , u)  ,   0

(10.2.34)

 pure   d d u exp    d H elastic (u)   0  (10.2.35)   pure  exp    d H disloc (d )   exp    d H couple (d , u)  ,  0   0 

where   1 /  is the inverse temperature, with τ indicating imaginary time  t  i . Note that in the imaginary time Feynman path integral, these q-number field operators φˆd† , φˆd are expressed by c-number fields, that is, φd , φd (see Section 8.2 and Figure 8.2.5). The coupling term, between the dislocation field and the background elastic field, will play an essentially important role in describing the “long-range stress field” caused by redundant dislocations. For the theory to be hermitic, the annihilation of a dislocation pair must be coupled with the creation, so we wrote d d   d d       u 

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in the third line of Eq. (10.2.33). This situation can be taken into account later by assuming the conservative system in constructing the GL equation to be discussed in Section 10.2.3.2, where the annihilation of the OP field is continuously associated with its creation. A detailed physical interpretation of the coupling term will also be provided in Section 10.5. The LRSF, which is considered to be essential for the cell-formation process, can be taken into account by the coupling term. For intuitively visualizing this, let us consider a simple model, presented originally by Mughrabi (1983), for a [100]-oriented single crystal in tension. By the array of the redundant dislocations accumulated along the cell wall, a misfit-like LRSF is produced in the cell-interior region, which is equivalent to the case where there is a difference in the elastic constants between the two phases based on elasticity theory. This misfit-like effect can be dealt with by introducing the OP dependency of the elastic constants, that is,   ref   , as detailed in Section 10.5. For the pure dislocation part, we apply the method of QFT to rigorously obtain the coarse-grained version of the corresponding Hamiltonian, as shown earlier. For the interaction part, on the other hand, here we tentatively employ a mean field approximation on the dislocation field to gain the corresponding effective term.

10.2.3

Statistical-Mechanics Evaluation of Dislocation System Based on Method of QFT

10.2.3.1 Derivation of Effective Hamiltonian In order to perform a statistical thermal average for dislocation ensembles, use is made of the method of QFT (Abrisokov et al., 1963; Chaikin and Lubensky, 2000; Nagaosa, 1995). The overview of the process is depicted in Figure 10.2.10, where renormalization of the microscopic degrees of freedom associated with the dislocation field is shown to lead to the effective theory. The method has a certain generality and has been successfully applied to reproduce the phenomenological GL theory for superconductivity (Abrisokov et al., 1963; Zagoskin, 1998), and so on. Note that the procedure given in this section is essentially the same as that for superconductivity, for example, provided by Nagaosa (1995). The partition function for a many-body system is expressed as the trace of an imaginary time-evolution operator, and can be rewritten by the Feynman path integral. For the case of the grand canonical ensemble, that is, pure



Z disloc  Tre   ( Hdisloc  chem N )   d d e  S (d ,d ) ,

(10.2.36)

where N  dd and µchem make up the chemical potential for the boson-representing energy needed for the addition (or removal) of one particle to (from) the system. S (φd , φd ) is the action functional given as the integral of the Hamiltonian instead of the Lagrangian for the present imaginary time formalism, that is,





0

0

pure Sdisloc (d , d )   d  drd  d   d H disloc . (10.2.37)

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Figure 10.2.10  The partition function for the pure dislocation part, together with a strategy for rigorous evaluation utilizing the S–H transformation in interpreting the pair-interaction term. By introducing an AF as a paring potential and then integrating out the microscopic degrees of freedom (dislocation field in the present case), ultimately an effective theory, given as a function of the AF, is derived.

Here, the first term represents the phase term, and the Hamiltonian in the second term for the grand canonical ensemble is given by

pure pair 0 H disloc  H disloc  H disloc , (10.2.38)



3  0 (r ) (r )  H disloc   dr  0  chem d ( r, )d ( r, ) (10.2.39) r 1   H pair  g dr  ( r, ) ( r, ) ( r, ) ( r, ). d d d  disloc  d

This partition function cannot be evaluated analytically due to the nonlinear term d d d d  included in the pair Hamiltonian. In order to perform the integration, an auxiliary field (AF) which couples to the dislocation fields φdφd should be introduced, in conjunction with the Stratonovich–Hubbard (S–H) transformation (Ivanchenko and Lisyansky, 1995; Nagaosa, 1995).

10.2.3.2 Stratonovich–Hubbard Transformation Figures 10.2.11 through 10.2.14 provide schematics of the S–H transformation. By introducing an AF and by making use of the nature of the Gaussian integral, we can rewrite any quartic term into a combination of the quadratic expressions of the AF in addition to the original one. The S–H transformation enables us to evaluate integrals of the type



 e

x4

dx,

which cannot be calculated analytically. The integrals of the form, on the other hand,

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Figure 10.2.11  Schematics of the S–H transformation as an extension of Gaussian integration.



 e



 ax 2

dx 

 a



are obtainable, and are known as Gaussian integrals. Introducing an AF A ≡ x 2, as 4 shown in Figure 10.2.10, we can rewrite the integrand e x as

2

eA 

1





 e 

 x 2  2 Ax

dx,

where  x 2  2 Ax  ( x  A)2  A2 and  e ( x  A) dx   are used.  A more practically useful example is presented in Figure 10.2.12, where the S ( )  (a / 2) 2  (b / 4) 4 type of action functional is assumed as the exponent (known as that for φ 4-theory). Here, an AF    2 is introduced for the S–H transformation. Accordingly, we can break down the expression into the following three contributions,

2

 1   1  i  exp   2  , exp   2  , and exp  2  .  2  2   4b  The first two are quadratic with respect to φ and ψ , respectively. The interaction initially expressed via the quartic term φ 4 is now converted to the third coupling term exp (i / 2) 2 , that is, through the interaction with the AF Ψ. It should be noted that, in utilizing the S–H transformation, we must redefine the AF as a physically measurable quantity with a sense of the OP in terms of this new variable, which could ultimately produce the GL field.

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Figure 10.2.12  Application of the S–H transformation to the partition function by introducing an AF; the original quartic term expresses interaction through the AF.

The AF to be introduced in the present study is regarded as a paring potential of the dislocation fields of opposite sign, defined as

  g d  ( x)d  ( x) , (10.2.40) which physically means the energy-expectation value for the pair-annihilation process that occurs between two dislocations of opposite sign. This process is interpreted in the present context as the dynamic recovery responsible for the dislocation-cell formation. The conjugate process to this is the creation of a dislocation pair of opposite sign, that is,



  g d† ( x)d† ( x) , (10.2.41) the physical image of which is the generation of a dislocation loop from, for example, a Frank–Read source (or multiplication). As explicitly denoted in the derivation overview in Figure 10.2.13(a), integrating out the dislocation fields to leave the AFs, Ψ and Ψ, leads us to obtain the effective theory with respect to the auxiliary fields. We finally have a partition function in terms of the effective action, that is,



Z disloc     e  Sdisloc (  , ) . (10.2.42) eff

This action provides a GL-type free-energy functional, from which the TDGL equation is obtained, that is,

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Figure 10.2.13(a)  Overview of the derivation process of the effective theory via the method of

QFT, where the interaction (coupling) term is evaluated separately based on the mean field approximation.

Figure 10.2.13(b)  A schematic showing how the additional terms can be separately evaluated

within the present formalism, where additional contributions are expressed as H int .



eff  Fdisloc (,  )  Sdisloc (,  ). (10.2.43)

Figure 10.2.13(b) illustrates how one can treat the interaction term, which is represented collectively as H int , in two separate ways. Figure 10.2.14 gives another overview of the derivation process in terms of the flow from the gauge-invariant eff Lagrangian total all the way to the final effective theory FGL via the second-quantized ˆ Hamiltonian total that requires the method of QFT described in Chapter 8.

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Figure 10.2.14  Another overview of the derivation process in terms of Lagrangian/Hamiltonian

formalism.

10.2.3.3 Derivation Process via the S–H Transformation The detailed procedure for deriving the effective action for the dislocation-ensemble system is discussed in this section. Using the S–H transformation, we can rewrite the interaction term in Eq. (10.2.32) with Eq. (10.2.39) as



 exp   g  d  drd†  r,  d   r,  d†  r,  d   r,     0 (10.2.44)    1     exp    d  dr    d†d   d†d   . 0



g 

Accordingly, the partition function Eq. (10.2.32) is rewritten in the form Z disloc   d d  e



 Sdisloc d , d ,  , 



 (10.2.45)    d d   exp    d  drd    0  chem  d   0

    1  
exp   d  dr    exp  d  dr d d   d d  , 0  0

g  







where the pair interaction d d d d  is regarded, at this stage, as substituted with the new interaction terms d†d  and d†d  through the AFs Ψ and Ψ. We can

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then integrate the dislocation fields from this equation, leading us to obtain the effective theory as a function of Ψ and Ψ. Introducing the Fourier transform of the dislocation fields, that is,    r,     V 1/ 2 eikr n   k,   d d n , (10.2.46)  1/ 2  ik r n e d  k, n   d  r,     V 



we substitute these fields into Eq. (10.2.39). Here, ωn denotes the Matsubara frequencies, that is, n  2 n / , with n being an integer. The partition function is written as an imaginary-time functional integral Z disloc   d d  e

where







Sdisloc d , d , ,    d 0

 dr

d



 Sdisloc d , d ,   k ,   k 

 q, n     0  chem  d  q, n 

1   q, l    q, l  g  q, n  d   q  k, l  n  d   k, n   q, n  d   k, n  d   q  k, l  n .



 , (10.2.47)

(10.2.48)



In what follows, we perform coarse-graining by renormalizing the energy, successively eliminating the microscopic degrees of freedom that are unnecessary in the macroscopic effective description. A series of the processes are schematically summarized in Figures 10.2.15 to 10.2.17. For this purpose, we integrate out the high-energy components  k  D, where ωD denotes the Debye frequency, which gives the highest energy corresponding to the smallest length scale in the present context. By expanding Eq. (10.2.48) in q and ωl , and using the 0th-order term, we have



S d , d , , 

   1g  0   0     ,k  , n







n

(in   k ) d ( k, n )d ( k, n )

k

 k D

 (0 )

V  (0)

V



d  ( k,  n )d  ( k, n )



d  ( k, n )d  ( k,  n )

n ,  k  k D

n ,  k k  

1

 g  0   0 

n , k



 d   k, n    

n , k  d    k, n  

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  in   k   1  0     V

  0  V

 d   k, n  .

   k, n   in   k  d 

 (10.2.49) 1

Figure 10.2.15  Overview of derivation process #1 (Eqs. (10.2.45) to (10.2.49)): Substitution of the Fourier transform of dislocation fields into the partition function and then expansion of it is made in the wave-number space to take the 0th-order term.

Figure 10.2.16  Overview of derivation process #2 (Eqs. (10.2.50) to (10.2.52)): By integrating out the dislocation field, we have a self-consistent equation for renormalizing energy (or equivalently, renormalizing scale) for obtaining the effective theory (action).

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Scale A: Modeling and Simulations for Dislocation Substructures

Figure 10.2.17  Overview of derivation process #3 (Eqs. (10.2.55) to (10.2.65)): By using the renormalization equation, Eq. (10.2.52), in the vicinity of the critical point, we can explicitly evaluate the effective action as an expansion of the auxiliary (OP) field, taking into account both homogeneous and inhomogeneous contributions.

By integrating out the dislocation fields and performing the integral over obtain





Z disloc  d d e   e



 Sdisloc d , d ,   k ,   k 

eff  Sdis loc    k ,   k  

we



(10.2.50) ,

where the integrand is written as     in   k eff  1  Sdisloc    0 ,   0   e  exp   0   0   ln det   1 n  k  g  0      V

 1 1 1  exp    2 2  g V

n k n k   1 

exp   0   0   .  geff 

 k   D,

  0    V   in   k    

1

    0   0      (10.2.51)

The effective energy geff is thus defined self-consistently as

1 1 1   geff g  V

1

   2   2  a T  . (10.2.52) n  k

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n

k

10.2  Evaluation of Hamiltonian for Dislocation Field

The second term is calculated as 1 1   0   V n k n2   k2 

  0 



1

  d  2   2   n n

D





n D

(10.2.53)

n  2 n   n  ,   

1 2 n 00 n

 0 

505

D , T where n  2 n / is used to give the last line (see Figure 8.2.5). In the vicinity of the critical temperature Tc , by taking the first term of the Taylor expansion of the logarithmic term, we have  0 ln

a(T )  q(Tc )   ln



Tc T  Tc   . (10.2.54) Tc Tc

Let us now derive the effective action, which is further assumed to be given as the sum of the inhomogeneous and homogeneous parts, as shown in Figure 10.2.17, where the OP fields Ψ( q) and Ψ( q) are nonuniformly distributed in the former, that is,





eff- in hom eff Sdisloc  q  , q   Sdisloc   q  ,  q    Sdiseff-hom loc  ,   . (10.2.55)

Through the inhomogeneous part, the spatial fluctuation effect can be taken into account. For obtaining the explicit expression for the inhomogeneous part, assuming that Ψ( q) and Ψ( q) change slowly in space, only terms up to the second order in ( q) and ( q) are taken into account, that is, eff- in hom Sdisloc    a( q)  q(0) q   q  , (10.2.56)



q

where a  q  is given by a  q  a 0  

1 V 1 

  i

n D k

1 1  n   k  q in   k

d 3 q  1 1   3  i    i   n D  2   n k  n kq

(10.2.57)  .



Expanding q and reaching the second order, we have a  q  a 0

1 E

d 3 q w 2 ­° 1 1 ¦ ³ 3 wq2 ® iZ  [ ˜ iZ  [ Zn ZD  2S n k ¯° n k  q

1 E

­ d3q ° 1 ¦ ³ 3® 2 Zn ZD  2S ° iZ  [ iZn  [ k ¯ n kq

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2

½° ¾ ¿° 2½ §q· ° ¨m¸ ¾ © ¹ ° ¿

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Scale A: Modeling and Simulations for Dislocation Substructures



D 2   chem 0  d  2  D  m n D n2   2





 1   chem 0  , 3 2 m n D n

(10.2.58)

3

 chem   0   m 2 n 0 2 n chem 1   3  ,  0 m  T 2 16







1 is the Riemann zeta function. Therefore, substituting the final  n 1 n

where  ( )  

result of Eq. (10.2.58) into Eq. (10.2.56), we have the explicit form of the inhomogeneous part, that is,

eff- in hom Sdisloc   q  ,  q   

 (3) 0 chem   q   q  . (10.2.59) 16 ( T )2 m

For the homogeneous part, by placing  ( q)  ,  ( q)   for simplicity in Eq. (10.2.49), we have





  in   k eff-hom Sdisloc  ,    gV     ln det  1 n  k     V

  , (10.2.60)  in   k   1

V

 V       ln n2   k2    ln  1  2 2  g n  k  k  n   k  efff- in hom( 0 ) eff- in hom( 2 ) eff- in hom( 4 )  Sdisloc  0, 0   Sdisloc  ,    Sdisloc  ,    . 





Here, the third term of the second line is expanded as

    1  1   ln 1   2   2      2   2   2 2  k n k   k  n k  n   k



     . (10.2.61) 

2



2

Therefore, the second-order term of the action is given by

eff-hom( 2 ) Sdisloc  ,     



2 k n

1  . (10.2.62)   k2

Using Eq. (10.2.54), that is, a(T )  q(Tc )    T  Tc  / Tc, we can write Eq. (10.2.62) explicitly in the vicinity of the critical temperature as

eff-hom( 2 ) Sdisloc  ,     0

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T  Tc  . (10.2.63) Tc

10.3  Effective Action for the Coupling and Elastic Terms

507

Similarly, the fourth-order term is given by

eff- in hom( 4 ) Sdisloc  ,   



 

n D k

V 2



n D

 V     4 





1

n2  k2 

 

D

 D



n D

2

d 1

n

3

  

2



1



n2

  k2

   

2



2

   

2

(10.2.64)

2  (3)  V      . 32  T 2

Combining all the terms, that is, Eq. (10.2.55) with Eqs. (10.2.59), (10.2.63), and (10.2.64), we finally have effective action with respect to the OP fields. By setting  ( r )   ( r ) we have  (3) 1 chem 2 eff Sdisloc  r   r  , r    dr 0  16 2 m ( T ) 







2  (3) 4 (10.2.65) 1 T  Tc .  r     r   32  T 2 Tc  Therefore, we obtain the GL free-energy functional as 

eff Fdisloc   r     1Sdisloc   r  







2 1 4  (10.2.66) 2 1    dr  S1   r    S2  r   S3  r  , 2 4   where Si denotes coefficients, that is,

S1 

 (3) 1 chem  (3) 1 T  Tc , S2  , S3  . (10.2.67)  2 16 ( T ) 32  T 2 m Tc

Here, µchem  µ b2 , m  ρ0 b2, ς (3)  1.20, and 1/ ( T )2 10 6 , while (T − Tc ) / Tc  1 is near the critical state. When looking at the domain morphologies near the critical state, we may assume S1  S2  S3  1 to be the normalized values without loss of generality.

10.3

Effective Action for the Coupling Term and the Elastic Term We evaluate the effective action for the coupling term H couple (φd , φd , u) in Eqs. (10.2.32) and (10.2.33). Figure 10.3.1 briefly summarizes the process. The partition function is given by



 Z couple   d d u exp    d H couple (d , d , u)  , (10.3.1)  0 

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Scale A: Modeling and Simulations for Dislocation Substructures

Figure 10.3.1  Schematics showing treatments of the coupling (interaction) term and the pure elastic term. For the former, the mean field approximation for the paring terms is obtained by considering the separable time constant of the interaction of the annihilated field with the background elastic field from individual dislocation processes. The latter, on the other hand, is kept unchanged.

where the action for the coupling term is defined as 



Scouple   d H couple 0





1    d  dr  K d d  d d   u   ,   2 0

(10.3.2)

based on the third line of Eq. (10.2.33). Here we simply introduce a mean field approximation by using the pairing potential introduced earlier, in the present context, that is,   g d  ( x)d  ( x) and   g d† ( x)d† ( x) . The physical interpretation of the coupling term chosen above will be given in Section 10.6. The effective action for the coupling term is formally given by

Z couple   d d u exp  Scouple (d , d , u)  eff  exp  Scouple (,, u)  .

(10.3.3)

Therefore, the corresponding free-energy functional is obtained as

eff Fcouple   r  , u    1Scouple



(10.3.4) 1 dr  K  r    r    u  .  2

pure To the pure elastic component H elastic (u), on the other hand, the saddle-point approximation will be applied later on. Therefore, let us now simply define



pure pure Selastic (u)   d H elastic (u) 0







1 2 1

  AB   (10.3.5)    dr  0  4 x i ij  4 x j  K   u   2  AB 2 2 pure

 Felastic (u).

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10.4  Evaluation of the Effective Hamiltonian for the Total System

509

The free-energy functional is obtained as





1 1

eff   AB   Felastic  u    dr  0  4 xi ij  4 x j  K  u 2  2  AB 2 2  (10.3.6) eff

 dr felastic  u , eff where felastic  u  denotes the free-energy density for the elastic part.

10.4

Evaluation of the Effective Hamiltonian for the Total System Combining the free-energy functional expressions for dislocation, elastic, and coupling components, given respectively by Eqs. (10.2.66), (10.3.4), and (10.3.6), we have





1 1  eff F   r  , u    dr  S1 2  S2  2  S3  4  felastic  u  S4    u , (10.4.1) 2 4   with 1 1 eff   AB  . felastic  u   0  4 xi ij  4 x j  K  u 2  2  AB 2 2





Figure 10.4.1 displays the final result derived so far. In the corresponding effective action Seff for the total system, the saddle-point approximation is further applied

Figure 10.4.1  Summary of the effective theory in the form of GL-type potential energy as a

functional of the “annihilated” field and the elastic field represented by the displacement vector.

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Scale A: Modeling and Simulations for Dislocation Substructures

to the elastic field in order to eliminate it. Taking the functional differentiation of Eq. (10.4.1) with respect to the displacement field u and putting equal to zero, we have a corresponding classical solution of the action, which physically means mechanical equilibrium. The detailed process is given in Section 10.6.

10.5

Physical Interpretation of the Coupling Term As described earlier, the coupling term S4    u  introduced in the present study in Eq. (10.4.1) expresses the LRSF expected to be produced by redundant dislocations at the cell–cell wall boundaries (see Section 3.9). We will use a simple model for this that was introduced by Mughrabi et al. (1986) as a visually tangible example. Figure 10.5.1 illustrates the schematics that Mughrabi provided for a [100]-oriented Cu single crystal in tension, where redundant dislocations with the resultant Burgers vector bres (they are the products of two edge dislocations piled up at the intersecting points) produce a “compressive” stress field in the cell-interior region. This situation is mechanically similar to the case with a coherent precipitate embedded in an elastically softer matrix having a positive misfit strain. We will call this the “misfit-like effect” hereafter. The misfit-like LRSF is thus produced in the cell-interior region, which is equivalent to the case where there is a difference in the elastic constants between the two phases, based on micromechanics (referred to as the “equivalent inclusion problem”) (Mura, 1982). Note that the Burgers vector in the former takes a discrete value, that is, b , whereas there is an infinitesimally small Burgers vector in

Figure 10.5.1  A model for the long-range stress field by Mughrabi et al. (1986).

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10.5  Physical Interpretation of Coupling Term

511

the latter, as in the continuously distributed dislocations. This misfit-like effect can be dealt with by introducing the OP dependency of the shear modulus, as shown in the following. Figure 10.5.2 shows a schematic of the elastoplastic stress–strain responses for the cell wall and the interior region, also used by Mughrabi et al. (1986), where the two regions have the same elastic properties but with different plasticity; the wall region has a larger yield stress than the interior region. Reinterpreting this in terms e e of elasticity, where the misfit-like strain is given by    wall   cell , instead of p p    cell   wall, we can express the situation by using the difference in elastic modulus (shear modulus) between the two regions. Let τˆ be a mean shear stress based on Mughrabi’s composite model, that is, ˆ  fwall wall  fcell cell , where fwall , fcell are the volume fractions of the cell wall and interior regions, respectively. The mean shear stress is further decomposed into ˆ  ˆ  ˆ . We then have e e    wall   cell  1 1  ˆ     cell  wall



(10.5.1)   ,   ˆ   wall cell 

where   cell  wall denotes the shear modulus difference. Furthermore,  is evolved with an accumulation of the redundant dislocations of the Burgers vector bres, so it is also expressed as   N bres . (10.5.2)



Equating Eq. (10.5.1) with Eq. (10.5.2), we can express the apparent difference in the shear moduli  as N bres  ˆ



       wall cell wall cell ˆ 

  N bres . (10.5.3) 

Here, since   cell , wall, we can approximate 2 wall cell  cell  ˆ 2 . (10.5.4)

So we have

wall cell  cˆ, (10.5.5) ˆ where c′ is a proportionality constant. Also we can regard N bres    b. (10.5.6)



Also we can regard the number of redundant dislocations N with Ψ, because the redundant dislocations should be compensated immediately by the annihilated field in the conservative system. Using Eq. (10.5.5) and Eq. (10.5.6) to Eq. (10.5.3), we explicitly express the apparent reduction in the shear modulus as   cbˆ   (10.5.7)

Therefore, we have

cell  wall  cbˆ  (10.5.8)

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Scale A: Modeling and Simulations for Dislocation Substructures

Figure 10.5.2  Representation of the coupling term for expressing the long-range stress field developed via redundant dislocations.

In the present context, µcell corresponds to µ , while µwall represents the referential shear modulus and is expressed as µref hereafter. Expressing   cbˆ in Eq. (10.5.8), the Ψ-dependent shear modulus is defined by

  ref     (10.5.9)



where we can generally assume    ref.

10.6

Elimination of the Elastic Field As was mentioned in Section 10.4, for the effective action Seff of the total system, the saddle-point approximation is applied with respect to the elastic displacement field u in order to eliminate it. Taking functional differentiation with respect to the displacement field and putting it as equal to zero, we have a corresponding classic solution of the action, which physically means mechanical equilibrium, that is,



 Seff  , u  u

 0   j ji  0  u  usad . (10.6.1)

Elimination of the elastic field is considered to be equivalent to the saddle-point approximation setting u = usad. The former half of the process is outlined in Figure 10.6.1, while the latter is displayed in Figures 10.6.2 and 10.6.3.

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10.6  Elimination of Elastic Field

513

Figure 10.6.1  Elimination procedure of elastic field from effective theory – 1: derivation of 0th-order solution.

Figure 10.6.2  Elimination procedure of elastic field from effective theory – 2a: derivation of first-order correction.

The explicit form of Seff  , u  is obtained by combining Eq. (10.4.1) with Eq. (10.5.9), that is,

1 1  Seff  , u     dr  S1 2  S2  2  S3  4  S4    u  2 4  1 1 2   0  4 x i ij  4 x j  K  u   2( ref    ) ij  ij  . (10.6.2) 2 2 





Let us evaluate the left-hand side of Eq. (10.6.1) by using Eq. (10.6.2). Since it is a functional derivative, we can rewrite it as

 Seff  , u  u

 Seff  , u    i  . (10.6.3)  ( i u j )   

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Figure 10.6.3  Elimination procedure of elastic field from effective theory – 2b: details of first-order correction.

The first, second, third, and fifth terms vanish since they are independent of u. The other terms, on the other hand, become i





i

 S4    u   ( i u j )

 ( ref    ) ij  ij   ( i u j )

 S4  i  ,  i



 K u   K  u , 1 2

2

i

 ( i u j )

 2 i ( ref    ) ij



 2 ref  i  ij  2   i (  )ij where div   i   i  i   2 is used. So, we finally have

 Seff  , u 



u

0 

2   S4  i    K  ref   i   u   2 ref  2 u  2  j   ij   0. (10.6.4) 3   Let us evaluate the perturbation of u, that is, δ u, in terms of       by solving Eq. (10.6.4) (Onuki, 2002). Here, Ψ represents the spatial average of Ψ. We can rewrite Eq. (10.6.4) as



S4  i    ref  i    u   2 ref  2 u  2  j   ij   0, (10.6.5) where ref  K  (2 / 3) ref refers to the referential first Lame’s constant. Taking the divergence of the above Eq. (10.6.5), we have



 2 ref    u   S4  2  i  j   ij   0, (10.6.6) where  i ( 2 u)  0 is used.

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10.6  Elimination of Elastic Field

515

The 0th-order solution of Eq. (10.6.5), to be expressed as δ u(0 ), is obtained when we include   0 (    ref ) into Eq. (10.6.6), that is,

 









 2 ref   u(0 )  S4   0   ref   u(0 )  S4   0.



Solving this for δ u(0 ) , we have 1  u(0 )  S4 ref  . (10.6.7)



Here, χ is a potential function satisfying the following Poisson’s equation, with the OP fluctuation   being the source, that is,  2     (10.6.8)



The corresponding volumetric strain to δ u(0 ) , that is,   u(0 ), is given by 1 2   u(0 )  S4 ref   (10.6.9) 1  S4 ref  .



The deviatoric strain of the 0th order is evaluated as



 ij (0 )   ij   i u(0 )



dev

1   ij  2 S4 ref  i  j  

(10.6.10) dev

,

where   expresses the average deviatoric strain corresponding to the applied shear stress τˆ , while “dev,” attached to the parentheses in the second term, stands for the deviatoric part, that is,

 i  j  dev   i  j  d1 2 ij   ,

  where d denotes the dimension of the problem concerned. Using  ij (0 ) for  ij in Eq. (10.6.5), we can evaluate the first-order correction of the volumetric strain. From Eq. (10.6.6) (Figures 10.6.2 and 10.6.3), we have











ref  2   u (1)    i  j  ij (0 )  0. (10.6.11)



By solving this for   u (1) , we formally have  1 i j   u (1)    ref   ij (0 ) . (10.6.12) 2 Substituting Eq. (10.6.10) into Eq. (10.6.12),



1   u (1)    ref



 

ij

1  2 S4 ref  i  j  

dev

 ,



By using

i  j 

 2 





   1  i j 2 i j    ref  2  ij   2 S4 ref 2    i  j  dev .     i  j 

2

 ij  0,

i  j 

2



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i  j 2

     i  j  

dev

,

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Scale A: Modeling and Simulations for Dislocation Substructures

i  j 

2

 i  j  dev   i  j  dev . 2

and for evaluating the first and second terms, respectively, we ultimately have

1   u (1)  ref  i  j  

dev

2   ij  2 S4 ref   i  j  

2 dev

. (10.6.13)

Therefore, the coupling term is finally expressed as

2 Fcouple   2 S4 ref  dr     i  j  

2 dev



2

2 S4 ref

 i  j   dr   2   dev







 i  j   gLRSF  dr   2   , (10.6.14)   dev

2

2



2 where gLRSF   2 S4 ref represents the coefficient measuring the strength of the LRSF based on the “misfit-like” effect. The contribution of the external stress to the energy is given by the first term in Eq. (10.6.13), that is,  i  j  1 Fext   ref  dr  2    ij 1   ref  dr  i  j     ij

  gref  dr   i  j     ij . (10.6.15)

Note that, for the case of anisotropic plasticity, the contribution can be taken into account in  ij by replacing it with

N

 si() m(j) (), where si(α), m(jα) are unit vectors in

 1

the slip direction and slip plane normal for the (α )-slip system, respectively, and  () is the slip strain on the slip system directly corresponding to τˆ in τˆ . The final explicit form of the GL-type potential-energy functional for the governing equation for dislocation-cell formation is given as the sum of the above-obtained individual contributions. That is,

FGL  Fdisloc  Fcouple  Fext , (10.6.16)



  2 1 2 1 4 Fdisloc   dr  S1     2 S2   4 S3     2 , (10.6.17)

F g dr    i  j   dev couple LRSF   Fext   gext  dr   i  j    j 2 1 and gext   ref are the measures for their contributions. where gLRSF   2 S4 ref

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10.7  Time-Dependent GL Equation

10.7

517

Time-Dependent GL Equation From the previously obtained GL potential-energy functional, we can obtain any type of TDGL equation. The general procedure is illustrated in Figure 10.7.1. Considering the dynamic recovery during plastic deformation, we assume here the conservative system with respect to the OP field Ψ, that is, the local annihilation of dislocation pairs is immediately compensated by creation of the pairs. The rate of time evolution of the GL functional should be negative, that is, dF   r  





dt

 0. (10.7.1)

For conservative cases (systems), the OP must additionally satisfy the following continuity condition  ( r )  ( r )   j  0     j, (10.7.2) t t where j is the flux of the OP field. The time derivative of the GL free-energy functional is then rewritten as dF   r  



dt

  dr

 F   r    r   t  r 

  F   r      dr     j  0,  r   

(10.7.3)

where Eq. (10.7.2) is used to derive the last line. In order to satisfy the last inequality, the flux j must be nonpositive. Assuming j is linear with respect to F   r  , we can set

Figure 10.7.1  Derivation of the TDGL equation for a conservative system, which defines an evolution equation for the OP field.

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Scale A: Modeling and Simulations for Dislocation Substructures

  F   r    j   L r    , (10.7.4)  r   



with L r   0 being the coefficient. Substituting Eq. (10.7.4) into Eq. (10.7.2), we have   F  ( r )    ( r )    L   t  ( r )   (10.7.5)   F  ( r )   2  L  .   ( r ) 



The equation of motion for the OP field is generally given by Eq. (10.7.5), where the differential operator L2 is Onsager’s coefficient. L ( r )  L0  L2  2 with L0 = 0 for the conservative OP and L0 ≠ 0 for the nonconservative case. When the Gaussian noise term is added, the above equation coincides with a Langevin-type equation.

10.8

Numerical Scheme for Cell-Formation Simulation To solve the governing equation, Eq. (10.7.5), with Eq. (10.6.14), we use the finite-difference method. The Laplacian operator is written as 2 



2 2 2   2 x j xi xi2 x 2j

 i

2 2 .  xi i  j x j xi



The governing equation to be solved in a dimensionless form is given as       2    3 2  gLRSF   bij2   bii2    i j  t  i   2    2 (bij ) 4   (bii ) g   2 gLRSF    2 , (10.8.1)  ext  2 2 2  i  j x j xi  x  x  x  i  j j i i i  

where

 2   ij bij    2   .  xi x j d   



This equation is further rewritten, in a more explicit form, in terms of the finite-difference discretization, as

   2  1  2    3  t









 2 gLRSF   x  y      y  z      z  x   2

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2

2

10.8  Numerical Scheme for Cell-Formation Simulation



 

 

519



2 2 2 2 gLRSF  2x    2y    2z    2x   2y    2y   2z    2z   2x  3







 4gLRSF  x  y   x  y    y  z   y  z    z  x   z  x  



  

(10.8.2)



          2z  2x     2z  2x     4 gext   2x  2y    2y  2z    2z  2x   . 



2 gLRSF  2x  2y   2x  2y    2y  2z   2y  2z  3

The appropriate combinations of the central and backward-difference schemes, as well as the choice of their orders of calculations, are important to achieve a stable solution in the present simulation. The first part of Eq. (10.8.2), that is,





 





2 2 3   1      2 gLRSF   x  y      y  z      z  x    2 2 2 2   gLRSF  2x    2y    2z    2x   2y    2y   2z    2z   2x   , (10.8.3) 3 

 

 

2

2

2





is approximated by the central difference method. For the fourth part, that is,





4 gext  2x  2y    2y  2z    2z  2x  ,



the central difference scheme also is used. The second part,

4gLRSF  x  y   x  y    y  z   y  z    z  x   z  x   , (10.8.4) is broken down into three steps as follows,







i  j      i  j    i  j    i  j   . (10.8.5) The backward-difference method is first used for the i  j   i  j  in the above. For the third term, 2 gLRSF 3



                    , (10.8.6)              2 x

2 y

2 x

2 y

2 z

2 x

2 z

2 x

2 y

2 z

2 y

2 z

the process is divided into three steps as follows

i2 2j      i2 2j   i2 2j    i2 2j   . (10.8.7) By combining these finite-difference schemes, we can obtain the solutions for Eq. (10.8.2) in each time step. Note that, to solve Poisson’s equation, Eq. (10.6.8), the Fourier transform method is used both for the 2D and 3D cases.

10.8.1

Analytical Conditions Both 2D and 3D simulations are performed based on the finite-difference method described earlier. For 2D simulations, 2562 divisions per unit simulation cell

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corresponding to 6.4  6.4 m 2 and 643 divisions corresponding to 1.6  1.6  1.6 m 3 for 3D are used, respectively. In these cases, the minimum element unit for a dislocated region is regarded as containing a number of dislocations. In the simulation, what we trace is the evolution of the annihilated regions from an initial state, where a random distribution of dislocated regions with a fixed density is given in general (in the simulations, they are given as the volume fraction of the annihilated/dislocated regions). The initial dislocation density is assumed to reflect the given strain history measured by the plastic work W p (or the stored strain energy U e ) associated with the attendant flow-stress level τˆ . Therefore, the density of dislocations corresponds to the volume fraction of the annihilated phase Ψ ann in the present series of simulations. Here, we consider two representative cases of   4.3  1014 m–2, (4.8 or 9.6 × 1014 ) –2 m , or 7.0 × 1014 m–2, (8.0 or 1.6 × 1015 )m–2, each corresponding to the volume fractions of  ann  30% and 50% annihilated (or annihilated dipolar) regions, respectively. Note that the former corresponds to a mildly deformed state, while the latter is close to the maximum dislocation density (ρmax 1016 m–2). The given stress state, on the other hand, is reflected in the simulations through the coefficient of the cou2 pling term, that is, gLRSF  2 S4 ref  ˆ , where a proportional relationship between the external stress and the averaged internal stress τˆ is assumed.

10.9

Simulation Results and Discussion Figures 10.9.1 to 10.9.7 show simulated 3D cellular structures reproduced by solving the present model (partial 2D results are included). Figure 10.9.1 is an overview of the findings obtained based on the current simulation results, where not only 2D but also 3D simulations with and without external stress conditions are conducted. The extreme upper left displays a comparison with a TEM (cf. Figure 3.1.12; Mughrabi et al., 1979), demonstrating excellent reproduction of the cellular morphology. Not only that, the effect of the LRSF on the domain formation strongly suggests the crucial origin of both the size and morphology. Since the LRSF is based on the elasticity, as argued in Section 10.5, the LRSF origin of the domain size and morphology ultimately lead us to see the LRSF as the very origin of the “similitude law,” together with its “material independency,” discussed in Section 3.2 (Figures 3.2.1 and 3.2.2). It should be noted that the LRSF is measured by the value of gLRSF (see Eq. (10.6.14)) Two representative cases with typical dislocation densities are considered in Figures 10.9.2 to 10.9.5, that is,   4.3  1014 m–2 (i.e., black domain fraction of   30%) and 7.0 × 1014 m–2 (  50%). Figures 10.9.2 to 10.9.5 provide the results for gLRSF = 0, 0.15, 0.3, and 0.5, respectively. Also shown in the middle rows are the corresponding cut diagrams and in the right-hand rows the annihilated field distributions at each time step. A smaller dislocation density with sufficiently small gLRSF , that is,   30% and gLRSF = 0, tends to yield “isolated” domains with near spherical shape. This case corresponds to a “vein”-like pattern in 2D, which is made up of

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Figure 10.9.1  Breakdown of 2D and 3D simulation results for dislocation-cell formation.

Figure 10.9.2  Three-dimensional simulation results for dislocation-cell formation (gLRSF = 0).

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Figure 10.9.3  Three-dimensional simulation results for dislocation-cell formation (gLRSF = 0.15).

Figure 10.9.4  Three-dimensional simulation results for dislocation-cell formation (gLRSF = 0.3).

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Figure 10.9.5  Three-dimensional simulation results for dislocation-cell formation (gLRSF = 0. 5).

bundles of edge-dislocation dipoles that are free of the LRSF, that is, nearly satisfying gLRSF = 0. For 3D simulation results, on the other hand, there seems to be no corresponding structure in reality, except aggregates of minute dislocation loops (which are also LRSF free). With increasing values of gLRSF, the dislocated domains (black) tend to be interconnected three-dimensionally, and are ultimately formed into a “cellular” structure at large enough g LRSF, that is, gLRSF ≥ 0.3 in the 3D case. The evolving processes in 2D are compared in Figure 10.9.6, where those in the lower two rows are obtained under external stresses, yielding directional growth of the domains. Figure 10.9.7 summarizes the 3D results, including the effect of external stress. Figure 10.9.8 displays the evolution of the mean domain size (to be referred to as cell size dcell hereafter) with simulation time steps, comparing the effect of gLRSF for the two dislocation densities, that is,   50% and 30%, obtained in (a) 2D and (b) 3D simulations, respectively. From these, we confirm that larger gLRSF results in the smaller dcell. Due to the pinning effect via LRSF, dcell depending on the given gLRSF. Note that the temporal evolution processes themselves are regarded as virtual and do not correspond directly to real cell-formation processes: Only the final results at saturation matter, which give an SSS against given external stress levels. Figure 10.9.9 correlates the reciprocal of the average cell size 1 / dcell with gLRSF for both the 2D and 3D results at two representative time steps of t = 100 and 1,000.

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Figure 10.9.6  Two-dimensional simulation results for dislocation-cell formation.

Comparison of the gLRSF −1/ dcell diagrams between the 2D and 3D versions demonstrates that the 3D results always yield smaller cell size dcell with given gLRSF than that for 2D. Namely, the cell size becomes smaller (more efficiently evolved) in 3D for given stress states. This implies that 3D cells are more efficient at providing an SSS than 2D versions. Note this argument about dimensionality just scratches the surface and requires further examination.

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Figure 10.9.7  Comparison of 3D simulation results for dislocation-cell structures in analytical

conditions.

10.9.1

Reason for Cellular Morphology and the Origin of the Similitude Law As demonstrated previously, the LRSF (elastic misfit-like effect in the context of the present simulations) plays a decisive role in determining both the size and the morphology of the resultant structures to be evolved. When a balance between the elastic inhomogeneity and the misfit-like effect is reached, the growth of the annihilated domain is terminated. In other words, “domain pinning” takes place there. The larger

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Figure 10.9.8  Comparison of 2D and 3D simulation results in terms of time evolution among different gLRSF.

Figure 10.9.9  Relationship between gLRSF and the inverse of the mean cell size dcell, comparing 2D and 3D results at t = 100 and 1,000 simulation time steps.

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Figure 10.9.10  Comparison of the simulated domain morphologies eloquently demonstrates the crucial role of LRSF. Redisplayed from Figure 10.9.7.

gLRSF yields a smaller cell size, which is a possible origin of the “similitude law,” holding between the applied stress and reciprocal of the cell size (cf. Section 3.2). Since gLRSF is proportional to the applied shear stress in the present model, the linear relationship (proportionality holding) between the reciprocal of the average cell size and gLRSF (Figure 10.9.9, as mentioned previously) can be a direct reason for this. Figure 10.9.10 redisplays a comparison of simulated domain morphologies between analytical conditions with and without considering the LRSF via gLRSF. As clearly confirmed, without the LRSF the domain tends to evolve into either liquid-like phase separations for low density, or soup-like, within which spherical beans are distributed. The comparison of the three results eloquently demonstrates the importance of the LRSF for both the cellular morphology and the size to be achieved, without which “far-from-cellular” morphologies will possibly result (cf. Figures 10.9.7(a), (b), and (c)). Similar to the case of binary alloys with elastic misfit (Onuki and Furukawa, 2001, p. 452), the size of the domain is determined so as to balance the interfacial energy between the dislocated and annihilated phases with the elastic inhomogeneity of free energy. In the present case, letting the average cell size be dcell , each of them is roughly given by 2  d   Esurface  TL  4  cell   2     3 4  Einhom  gLRSF      3  3



2  K   2 S4 ref   2

3

 dcell   2   

3

cb ˆ 

4  dcell  3  2 

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3

2   4  dcell   K  3 ref   3  2  (10.9.1)    

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Figure 10.9.11  Summary of the “origin of similitude law” argument.

where TL denotes the surface tension on dislocation lines, being proportional to µ b2 . As the relationship between them is comparable, we have, for dcell,

dcell 

TL . (10.9.2) bˆ

Assuming TL  k  b2, and substituting this into Eq. (10.9.2), we have

dcell  k  b

1 1  ˆ  k  b . (10.9.3) ˆ dcell

The above equation coincides with the “similitude” relationship (see Section 3.2). Figure 10.9.11 summarizes the current argument, showing again the plot of gLRSF versus the reciprocal of the mean cell size 1 / dcell, where a linear relationship between the two quantities is confirmed, as is predicted from Eq. (10.9.3). Note that the elastic misfit-like effect in the present context is taken into account through the coupling term between the annihilated field and the background elastic field (Section 10.5), that is, S4   u , ultimately giving rise to 2



 i  j  Fcouple  gLRSF  dr   2   , (10.9.4)   dev where  u   v is the volumetric component of strain (see Section 10.6 for the derivation process). This implies the hydrostatic stress σ m , the energy dual of ε v , makes a significant contribution to stabilizing the cellular morphology. This has been partially confirmed by recent discrete dislocation-based simulations, where neither the shear or the hydrostatic LRSF alone stabilize the dislocation walls (see Section 13.3).

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The modulation in the dislocation (or equivalently “annihilated”) field is essenpure tially attributed to the pure dislocation term disloc , which has been “coarse-grained,” based on the method of QFT. This clearly indicates that the “collective or corporative behavior” of a large number of dislocations with pair interactions leads to such structural modulation rather than uniformity. However, it is very important to recognize that this alone does not (or never) yield “cellular” patterning, as argued earlier.

10.9.2

Effect of Internal Stress Distribution The effect of internal stress distributions on the cell structures to be evolved can be taken into account via distributed gLRSF in the present simulations. Figure 10.9.12 shows two simple examples of the simulation results where band-like gLRSF distributions are set. As demonstrated, corresponding distributions of the cell structures are obtained, that is, fine–coarse–fine and coarse–fine–coarse distributions, respectively. Such distributions are considered to correspond to those of dcell in our field theory-based model. It is worth comparing them with, for example, a result shown in Figure 11.5.22, where an incompatibility-induced modulation in dcell is presented. If we allow a correspondence between the dislocation density and the local shear stress, via, for example, the Bailey–Hirsch relation (Bailey and Hirsch, 1960), we can roughly estimate the corresponding shear-stress distribution. Here, we discuss the normalized basis (from –1 to +1), without mentioning the absolute values of the

Figure 10.9.12  Two-dimensional simulation results, with internal stress distribution represented by gLRSF values.

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Figure 10.9.13  The evaluated stress distribution across a simulation cell.

stress. This normalized shear-stress distribution, along the broken line for a 2D simulation result, is shown in Figure 10.9.13. Comparison with an experimental measurement (also normalized) by Mughrabi (1983) is shown in Figure 10.9.14, demonstrating a good agreement. A 3D representation of the simulated stress distribution is given in Figure 10.9.15.

10.9.3

Elastic Strain Energy and Incompatibility-Tensor Field Distribution of the elastic strain energy can be obtained by



1 2 2 1 4 Fdisloc  r    r    r    r  , 2 4 where the elasticity field u has been renormalized in the coefficients. For an evaluation of the incompatibility-tensor distribution, we first need the dislocation density-tensor

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Figure 10.9.14  The stress distribution across a cell interior compared with the experimental measurement.

Figure 10.9.15  Three-dimensional distribution of the stress field for Area B.

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distribution corresponding to the targeted cell structure, which can be evaluated from the obtained distribution of the dislocated region (instead of the  r  regions). Since, in the current series of simulations, no distinction between the edge and screw components is explicitly made, here we assume the isotropy of the dislocation-density tensor, plus fluctuation from it, that is, 0 0     ij   0    0  . (10.9.5)  0 0    



From Eq. (10.9.5), we have, for the incompatibility tensor,





 0  3  22  11  1 ij  0 2 SYM 



 

 

 2 11   33   3  0  1  1  33   22   0  2   SYM 0 

 2  21  . 0  (10.9.6)

Note that since the intensity of ηij is determined by  , which is set as arbitrary here, the absolute values are not essential in the present context. So our discussions will be limited to the qualitative trends, based on the normalized distributions.

10.10

Important Implications from the Simulation Results

10.10.1 Overview One can immediately see that the choice of the annihilation of two dislocations, instead of entanglement or junctions, given here for specifying the interaction has greatly simplified the theoretical treatment. If we use another approach, we must take into account unnecessary details and variations of arbitrary dislocation-pair configurational information. Also this implies that pair annihilation via any possible mechanisms, for example, cross slip, is the key dominant factor for cell formation, which is consistent with experimental observations. Note that the “massive” annihilation of dislocations in the present context directly corresponds to the “dynamic recovery.” The results obtained in this study further emphasize two important facts: (1) The cell structure is not always an energy-minimum configuration but is stabilized (pinned) quite mechanically with the help of the LRSF. It also means the structure can become unstable easily if some configurational changes of the wall-constructing dislocations to those free from LRSF occur. (2) Therefore, we can say that the cell structure concerned here is not formed “incidentally” but is rather “mechanically necessarily.” As is discussed in Section 10.10.3, the cell structure is also regarded as a sort of SSS in Scale A. The existence of the LRSF in the cell-interior regions means that a significant lattice distortion occurs there, expressed here as “misfit-like effect.” Since the primary

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role of the LRSF is to pin the cellular domain, preventing it from dissociation (further coarsening, in the simulation), the cell wall will collapse upon the release or the loss of LRSF, possibly resulting in either a collapse of the structure itself or a sharpening of the cell wall (i.e., recovery) accompanied by growing misorientations across the wall (for relaxing the distortion). This can be a model for the evolution of growing misorientation transformed from the cell, interpreted as an evolving process driven via “duality.” Related discussions are found in Sections 10.10.3 and 13.3 based on a series of discrete dislocation simulations on the stability of dislocation-wall structures. As clarified earlier, both the statistical mechanics and the conventional mechanics are necessary to understand the cell’s structural evolution and its stability, neither of these can be missing when constructing a theory or a model. A review of the models proposed hitherto (see a critical review in Appendix A10 for further details) seems to show that these important factors are missing or excluded in them, as they provide only a one-sided aspect of the phenomena, for example, clustering or modulation. It is highly likely that these findings will be confirmed in the near future via discrete dislocation dynamics-based superparallelized, massive, direct calculations (simulations) or multimillion atomistic simulations, which are currently being globally promoted.

10.10.2 Supplements In what follows, some supplementary information is given to enhance readers’ understanding of the current field-theoretical approach. First of all, the gauge formalism given at the beginning of this chapter (Section 10.2) is essentially equivalent to the theory of “continuously distributed dislocations” and as such contains at least the elasticity part of the dislocation theory. (This is also applicable when describing a “single” or “discrete” dislocation line.) The stress field for infinitely long screw and edge dislocations can be reproduced within this framework (Osipov, 1991). Regarding the QFT-based derivation process, the essence here is that, once a rational framework is introduced, many of the derivations can become “formal” solely based on pure mathematical operations (Ivanchenko and Lisyansky, 1995). Although most of the algebraic processes themselves are not always physically tangible, we cannot follow the interrelationships among the pieces without them. For the second quantization, since the process itself is also rather formal, an appropriate physical interpretation with respect to the use of the “S–H transformation” is required to keep the physical meaning. For the dislocation field, the field equation (Eq. (10.2.19) or (10.2.20)), derived based on the variational principle within gauge formalism, plays a crucial role in this context. The current approach and the obtained simulation results include several important and indispensable procedures, which are definitely original and completely new. (1) The use of the gauge formalism for obtaining the second-quantized form of a dislocation system enables the use of the method of QFT, without which it is difficult to rationalize the explicit form of the Hamiltonian as the point of departure.

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(2) Once the second-quantized dislocation field is written in terms of the creation and annihilation operators, any types of interactions (not limited to pair-wise) can be easily and explicitly written down, as in the quartic term in Eq. (10.2.27) expressing pair interactions of dislocations of opposite sign, although this is one of the simplest forms of this sort. (Note that the additions of the interaction terms can break the underlying gauge symmetry.) (3) In the derivation of the GL-type potential-energy functional, the appropriate physical interpretation for the AF is also essential, without which there is no appropriate interpretation of the final equation. The choice of the “annihilated field,” that is, the nondislocated elasticity field, is one of the most important steps toward successful cell modeling in the present context. (4) The line-like characteristics of dislocations are not explicitly maintained in the second-quantized form of the dislocation as well as that in the gauge formalism just like in the continuously distributed dislocation-based pictures. But this improves when we use the QFT method because of the additional degree of freedom along the “imaginary time” line, such that dislocation fields will interact as string-like objects rather than particles (see Figure 8.2.1). Although the time evolution is excluded in constructing the second-quantized form (Eq. (10.2.21)), it is reinstated in Eq. (10.2.46), with which any complex interactions among dislocations such as entanglements can, in principle, be expressed. (5) The coarse-graining process is essential for the pure dislocation part of the Hamiltonian. Only through this process can we find out why the system of interacting dislocations prefers “modulated” distributions rather than uniformity. Also to be noted is that this alone does not simply lead to “cellular” patterning, as we have seen: LRSF modeling through the coupling term is critical. (6) The use of statistical mechanics here (via the method of QFT) is motivated by the fact that the cell formation is a thermally activated process.

10.10.3 Cell Formation as Thermal Process (7) Cell formation is evidently a thermal process, differentiating it from other kinds of dislocation patterning like 2D clustering and vein-ladder types. This is understood from the fact that the “dynamic recovery” is responsible for the formation, without which the evolution is significantly (or evidently) restricted. During the dynamic recovery, the strain-hardening rate reaches a dynamic equilibrium, where the dislocation density remains constant (meaning a dynamic balance between “multiplication” and “annihilation”). It has been well documented that cell formation occurs at the outset of dynamic recovery, whose major elementary process is believed to be the cross slip (i.e., one of the thermal activation processes). The temperature dependency of the threshold stress τ III for stage III hardening in single-crystal FCC metals (Argon, 1996, 2009) eloquently demonstrates this fact.

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(8) The absence or restriction of cross slip has been experimentally demonstrated to restrict cell formation. Indirect evidence can be found in an experimental study of fatigue on some steels with/without Cu addition (strengthening), where the addition of Cu is demonstrated to strongly restrict the cell formation, probably due to restrictions of cross slip. See Section 9.4 for more details. This provides more evidence for the close connection among cross slip, dynamic recovery, and cell formation.

10.10.4 Stress-Supporting Structure and Reservoir In the real processes, the cellular patterning of dislocations takes place concurrently with the increasing flow stress that is being externally applied. One of the experimental observations eloquently showing this is related to MTS (see Chapters 2 and 3). For an extreme example, a significantly enhanced flow-stress level, under hypervelocity impact or shock loading in pure Cu, is accompanied by commensurate MTS. This implies that the dislocation substructures are instantaneously evolved (immediately following the multiplications) for compensating (or supporting) the raised flow-stress level, probably due to intrinsic resistance such as the phonon drag (see Section 2.7). Allowing the above hypothesis, we have two distinct ways of dealing with the dislocation cell-related phenomena. One is to evaluate the cell size from a given flow-stress level without looking either into the evolving processes or the morphological details of the substructures, as we did in Section 5.6.1.2. The other is the approach presented in this chapter, focusing on how and why the “cellular” morphology is configured as it is, not on its “actual” evolving process. Both approaches are regarded as being equivalent, and will be tentatively called the equivalent hypothesis for dislocation cell. This makes the treatment of Scale A dramatically easier, in the sense, for example, that we do not have to pay much attention to the morphological aspects in modeling the constitutive equation as far as the overall flow response is concerned. I pointed out in Chapter 9 that the dislocation-cell structures responsible for Scale A act as a “reservoir” of the microscopic pieces of information. Such pieces of information are stored as the LRSF, more or less regardless of the configurational details of wall-constructing dislocations. In other words, as far as the wall-constructing dislocations can produce the appropriate LRSF, the cell structure is stabilized. If something causes the LRSF to deteriorate, the cellular morphology will not be able to be maintained, resulting in its collapse and, accordingly, the release of the stored microscopic information. This is regarded as a problem concerning the stability/instability of dislocation cells, and is discussed in Section 13.3. The LRSF in the present context can be rephrased as “elastic strain energy.” This accordingly reads “when the stored-elastic strain energy inside cells is released, collapse of the cell structure would result.” So, we may say that pieces of information about microscopic details and/or past loading histories are stored as a form of “elastic strain

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energy.” In this context, what we need to do is to evaluate the “quality” of the strain energy stored in the targeted inhomogeneity. Some of the preliminary examples about how strain energy is stored in dislocation substructures (i.e., Scale A) are given later. Figure 15.4.25 shows three typical dislocation-wall models with the same dislocation density   5.2  1015 m 2 but consisting of different arrangements, that is, dipolar, one-signed, and a configuration following the Mughrabi model (Mughrabi et al., 1986) (see Figure 3.9.1). The third arrangement is expected to effectively generate the LRSF in the channel region, and, thus, can be a model for the cells. The average strain energies for the interwall regions are roughly 0.05  0.4  10 3, 0.035, and 0.4 Jm–2, respectively. This means that the stored strain energy can range across at least a three-order magnitude even for a fixed dislocation density. When released, it is consumed, for example, for the elimination of redundant populations accompanied by growing misorientation across the walls (see Figure 3.8.3) or collapse of the walls (see Figure 3.8.1) in the case of the cells. The above situation is generalized in the context of the “duality” concept (see Section 12.6) and, further, is embodied as the working hypothesis “flow-evolutionary law,” to be discussed in Chapter 15. The point here is that the inhomogeneously stored strain energy is dynamically converted to dissipative local plastic flows, and vice versa, and the processes themselves act as driving forces for the further evolution of inhomogeneities.

Appendix A10 Critical Review on Cell Models A10.1 Introduction A wide variety of dislocation patterns are formed during plastic deformation within crystal grains of metallic materials (Kubin, 1996). Cellular patterning is one of the most commonly observed phenomena among them in many pure metals and alloys over wide ranges of strain rate and temperature, and the metals’ histories. Since the size and morphology of the cells directly dominate the macroscopic hardening behavior of metals, a full understanding of the cell-formation process and mechanism has been one of the most important and challenging problems in plasticity. The cell-formation process contains several complex factors such as the following: (i) multiple slip activities, with more than two Burgers vectors, (ii) both short- and long-range dislocation–dislocation interactions including, for example, sessile junction formation, (iii) dynamic recovery due to cross slip or other possible mechanisms, (iv) the presence of an externally applied stress field, and (v) LRSF (Mughrabi, 1983) relating to the cell structure itself. Another important factor often overlooked is the “collective effect” brought about by an extremely large number of interacting dislocations, which inevitably leads to a many-body problem. The “collective effect” sometimes creates qualitatively new phenomena which are absent in the microscopic level, implying that identification of the elementary key process does not simply lead to an understanding of the evolutionary mechanism.

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A10.2

537

Common Shortcomings of Previously Proposed Theories and Models There are several common shortcomings to most of the theories and models proposed hitherto, for example: (1) appropriate “coarse-graining” through which the “collective effect” is appropriately taken into account is not considered or is ignored in evaluating macro- or mesoscopic equations governing pattern formation, and (2) discussions regarding the possibility of obtaining a modulated dislocation structure due to growing instability in the initial distribution, mainly based on the linear stability analysis, without explicitly mentioning the final morphology of the cell structure, are limited. The former is due to confusion between a microscopic description of a theory, including elementary processes like dislocation-pair interactions, and the coarse-grained version with respect to macroscopic variables. In the latter case there is a possibility that researchers may look at a modulated pattern unlike that from a cell, such as those indicated in the two left-hand distributions in Figure 10.9.1.

A10.2.1 Holt’s Model Holt’s attempt (Holt, 1970) (Figures A10.2.1 and A10.2.2) is undoubtedly a pioneering work for modeling cell formation. This model is based on the analogy of a clustering of atoms during the spinodal decomposition of a supersaturated solid solution yielding certain modulated structures. As has been pointed out (Kubin, 1996), dynamic recovery accompanying annihilation, applied stress, and other important factors are all missing in this model. Holt considered an ensemble of straight and parallel screw dislocations of both signs where the number is conserved. According to his conclusion, elastic interaction alone can lead to the cellular patterning in the absence of other factors. The most serious issue with this attempt, however, is the lack of an explicit coarse-graining procedure, though the coarse-graining process solely provides the right way to evaluate the macroscopic expression of the governing equation, yielding patterning. Instead Holt used a microscopic expression of energy for dislocation-pair (elastic) interaction, simply multiplying the distribution function fu (r , ρ ), that is,



E  Es  EIu (  )  Gb2  R0  ln    Es  (A10.2.1) 4   r  ,  2 R0 R0 Gb   EIu (  )    1/2 2 rfu (r ,  ) 4 ln r dr where Es and EIu ( ρ ) denote self and interaction energy, respectively. The distribution function is expanded with respect to a fluctuating part of the dislocation density  and the resultant expression of EIu ( ρ ) is substituted into the equation of continuity. Holt then carried out a stability analysis on the equation’s solution and showed that the uniform distribution of dislocation is unstable. He interpreted that this instability leads to the clustering of dislocations into a cellular structure similar to phase decomposition. To evaluate the characteristic wavelength corresponding to the cell size,

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Figure A10.2.1  Summary of Holt’s model for cell patterning.

Figure A10.2.2  Features and shortcomings of Holt’s model for cell patterning.

an explicit form of fu (r , ρ ) is needed and he then assumed R0  rc  K 0  01/ 2 . Once fu (r , ρ ) is known, dislocations are aggregated so as to reduce the interaction energy EIu ( ρ ) of the system. This assumption accordingly results in a cell size proportional to 0 1/ 2 because, as Kubin (1996) found out, this artificially introduces a cut-off, causing artificial self-screening. We should like to know how fu (r , ρ ) can be determined or evaluated. Another weak point of this model is related to (2) in the introductory section (A10.2). As Hasebe (2006) pointed out, an additional factor is needed for cellular patterning, as is discussed in Section 10.9.1.

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A10.2  Common Shortcomings to Previous Theories/Models

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A10.2.2 Walgraef–Aifantis Model Reaction–diffusion-type equations were originally proposed as a model for the competing processes between activator and inhibitor yielding spatiotemporal patterning, for example, chemical reactions, and usually had the form of simultaneous differential equations. Walgraef and Aifantis (1985) (see also Aifantis, 1987) applied the model to dipolar/multipolar patternings like the PSB ladder, considering competition between mobile and immobile species of dislocation density. Although no direct modeling of the cell-formation process has been made, the model is worth mentioning here because the approach was the first attempt to introduce the concept of dissipative structures (advocated originally by Prigogine) to the dislocation-patterning problem. So this is also a pioneering work. This approach appears to be quite a powerful phenomenological tool, applicable even to far-from-equilibrium states where no potential energy function exists, and it may be able to reproduce almost all the existing patterns. According to this model, dipolar/multipolar patterning emerges as a result of bifurcations of the system (Turing instability). However, a rigorous derivation or justification from a microscopic theory is strongly required for it to be a convincing model, because by setting an appropriate set of competing species we may have any type of existing patterns, for example, multiplication and annihilation for cellular structures, without accommodating an external stress field. But such a process is full of artifacts. Also, the inclusion of the long-range stress field is not straightforward. The weak point is thus the lack of physical background in determining all the terms (especially the cubic term) and their coefficients in the equations. The derivation from microscopic theory has been attempted based on the field theory (Hasebe, 2006; Hasebe and Imaida, 1998), and is discussed throughout the main text.

A10.2.3 Kratochvil’s Model Kratochvil (1990a, 1990b) made use of a synergetic model for a more detailed discussion of the patterning than provided by the Walgraef–Aifantis model. For the cellular patterning, he suggested an internal bending model. But this may be more suited as a model for the subgrain formation rather than the cell, since the cell does not always accommodate misorientations across the cell walls. These two cases should be treated separately. The model itself is intriguing, containing nonlocal effect, and so on; however, there is no space to go into detail here.

A10.2.4 Hähner’s Model This model (Hähner, 1996) (Figures A10.2.3 and A10.2.4) incorporates the stochastic nature of cellular patterning. A Langevin-type phenomenological equation with respect to dislocation densities from which one can derive the corresponding Fokker– Planck equation is used as a point of departure, that is,

 i   1     (   ) W , (A10.2.2) where δ W indicates the Gaussian noise term satisfying   W (0) W (t )    (t ). Hähner pointed out that the deterministic part of the equation (up to the second term at the right-hand side of the equation) never yields any change in dislocation distribution,

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but the multiplicative noise term brings about instability, and hence a kind of phase transition from a uniform distribution to a modulated one occurs. This is called noise-induced phase transition. He derived the Fokker–Planck equation in terms of the transition probability of the dislocation distribution from the above equation, and came up with a stability analysis based on the steady-state solution. When the noise intensity reaches a certain value, noise-induced phase transition occurs. The broadened distribution function implies that a cell structure with no definite size has a fractal nature. The discussion here is limited to the probability function space, that is, it merely discusses indirectly the possibility of dislocation aggregates producing quasi-periodic structures in the probability space. Therefore, late-stage morphology is not mentioned and there is the possibility of merely looking at structures with far-from cellular morphology similar to Holt’s model. Also, this approach starts from the already coarsegrained set of equations with respect to the dislocation density. According to this model, the driving force for cellular patterning is intrinsic strain-rate fluctuations maintained by the external mechanical work which cause a noise-induced phase transition. An interesting feature of this model is the introduction of a “fluctuation–dissipation theorem” in plastic flow through the strain-rate sensitivity S, which holds between flow stress and strain rate, that is,

( eff )2  S  int , S 

  eff  ln 

. (A10.2.3)

According to this theorem, a relatively small strain-rate sensitivity for FCC metals may result in large fluctuation of stress state, which may easily trigger the phase transition, whereas BCC metals exhibit no transition because of the large S, resulting in small stress fluctuation below the transition temperature. However, this assumption does not seem to hold in general. For a high strain-rate range over 1000 s–1, where the

Figure A10.2.3  Summary of Hähner’s model for cell patterning (Hähner, 1996).

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541

Figure A10.2.4  Features and shortcomings of Hähner’s model for cell patterning.

strain-rate sensitivity dramatically increases even for FCC metals, we can nevertheless experimentally observe well-organized cellular structures (see discussions in 3.4 and 3.5). Regarding this, however, there is a possibility that evolution mechanisms between the two conditions might be substantially different, but it needs to be confirmed if this assumption is justified.

A10.2.5 El-Azab’s Formalism El-Azab (2000) has proposed an unique statistical-mechanics framework using a Liouville-type kinetic equation for the distribution of dislocations in the phase space (Figures A10.2.5 and A10.2.6), that is,

 (i )  (i )  t  v   v  v   ( x, v, , t )   S ( x, v, , t ), (A10.2.4)   where  (i ) ( x, v, , t ) represents the distribution function in the phase space and S (i ) ( x, v,θ , t ) is the scalar source term. The acceleration of dislocation lines v is obtained by solving the equation of motion,



fgt(i )  Bv  sgn( fgt(i ) ) f p    (v, v,  ). (A10.2.5) This formalism includes detailed descriptions of dislocation interactions (multiplication, junction formation, and cross slip) in the scalar source term S (i ) ( x, v,θ , t ). Stress equilibrium and strain compatibility are both taken into account. The derived set of kinetic equations is basically microscopic and no explicit coarse-graining procedure is included. In this sense, as El-Azab himself pointed out, this formalism is a continuum version of discrete DD, which needs to be directly solved. He suggests solving the set of equations numerically by, for example, a lattice Boltzmann or kinetic Monte Carlo technique.

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Figure A10.2.5  Summary of El-Azab’s model for cell patterning (El-Azab, 2000).

Figure A10.2.6  Features and shortcomings of El-Azab’s model for cell patterning.

Advantages over the discrete methods are unclear as yet. The coarse-graining process is not appropriately taken into account, although he did state that some phenomenological models can be reproduced after appropriate interpretations of the distribution function. However, this corresponds to a mean field approximation and is not always an appropriate coarse-graining process. In this approach, source terms can be explicitly evaluated in a rigorous manner to a certain extent. This is one of the merits of this model over Groma’s model, presented next.

A10.2.6 Groma’s Formalism Groma (1997) and Groma et al., (2000) attempted to construct a rigorous statistical-mechanics framework that considers the hierarchical nature of the phenomena, starting from equations of motion of individual dislocations elastically interacting each other (Figures A10.2.7 and A10.2.8). He wrote down a set of hierarchical differential equations introducing different order dislocation-density functions, which is known as a kind of BBGYK (Bogolyubov–Born–Green–Kirkwood–Yvons) equation. This equation is rigorous but in general cannot be solved unless higher-order

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Figure A10.2.7  Summary of Groma’s model for cell patterning (Groma, 1997).

Figure A10.2.8  Features and shortcomings of Groma’s model for cell patterning.

terms are explicitly determined because endlessly higher-order terms are introduced in the equation. The lowest-order equation, for example, has the form

2 (r1 , r2 )  1 (r1 ) 1 (r2 )  D(r1 , r2 )   (r1 )      (r1 , r2 )    (r1 , r2 )F (r1  r2 )dr2 . (A10.2.6) t r1 Here ρ2 (r1 , r2 ) represents the two-body density function. Groma neglected the correlation function D(r1 , r2 ) without justifying it in order to solve the above equation self-consistently. This assumption corresponds to the so-called mean field approximation. Based on this assumed equation, he concluded that the elastic interaction between dislocations never produces any instability in the distribution, and a source term g( ρ ,....) satisfying dg / d   0 is necessary for the instable growth of the solution, yielding patterning. Explicit descriptions of the dislocation interactions to be introduced in the source term, however, are unclear. The first version is limited to a simple 2D case of a parallel edge-dislocation system in a single-slip configuration, however, generalization to a 3D case is largely straightforward.

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Figure A10.3.1  Summary of theoretical approaches for cellular patterning of dislocations,

comparing abilities to describe important aspects.

A10.3 Summary In this appendix, the representative models of cellular patterning of dislocations proposed so far have been critically reviewed in the light of their features and shortcomings (Figure A10.3.1). It has been found that what are commonly missing concern an appropriate “coarse-graining procedure” and “a key factor that is responsible for the cellular morphology itself.” The latter in particular, which is “long-range internal stress,” seems to play crucial roles, as clarified in the main text.

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11 Scale B Intragranular Inhomogeneity

11.1 Overview Typical inhomogeneities to be evolved in this scale level are the deformation- induced structures, as presented in Section 3.10, normally yielding lamellar or band-like morphologies accompanied by relatively large “misorientation” across the bands or the walls, for example, MBs, DDWs, and matrix and transition bands (Hansen and Kuhlmann-Wilsdorf, 1986; Hughes and Hansen, 2000). Since the misorientation is introduced for the grain of interest to accommodate the imposed geometrical constraint from its surroundings, these substructures are roughly categorized as “geometrically necessary” types of bands (GNBs), according to N. Hansen and his coworkers (Bay et al., 1989; Liu, et al., 1998), in contrast to the dislocation cells in Scale A (that are “mechanically necessary”). The evolving misorientation is considered to be an effective energy minimizer. Namely, the geometrical accommodation-based energy minimization will be the major “driving force” for the inhomogeneity evolution in this scale, which is substantially different from those in Scales A and C; their inhomogeneities are basically driven by the “collective behaviors” of dislocations and crystal grains, respectively. This means that we need another method to describe the evolutionary aspects of the current scale, to be discussed in Section 11.2. As understood from this, Scale B is a scale that mediates the other two scales, that is, Scales A and C. It receives much information from Scale A and consumes it while coupling with the inhomogeneity evolution in Scale C through accommodation processes of its own. The role of this scale, therefore, may be expressed as “absorber.” Whereas the evolution of inhomogeneity in Scale A tends to harden the material response, thus tending to raise the flow-stress level, that in Scale B has the role of reducing it, so that it also relaxes the inhomogeneity in Scale C accordingly. Scale B, as a result, is rather elusive compared with the other two; both of which have more or less clear evolutionary laws or the like, respectively, as discussed in Chapters 10 and 12. Conversely, an appropriate control of Scale B in terms of inhomogeneity can be the key to control the overall mechanical properties of the targeted material systems by balancing the inhomogeneities of both Scales A and C. Another type of inhomogeneity observed in Scale B is “slip bands,” which are formed as a result of plastic instability. Slip bands will not be discussed in this chapter, although they play a key role in looking into, for example, the damage evolutions of this scale and the attendant fatigue-crack initiation processes. Some descriptions of

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them, in conjunction with the underlying dislocation substructures, that is, PSB ladder and vein, can be found in Section 3.7.1. “Shear bands” are similar kinds of inhomogeneity that are produced due to plastic instability as well, but of a macroscopic kind, so that they are not always confined by the crystallography. We frequently observe these penetrating even through GBs, extending over plural grains possessing different crystallographic orientations. This type should be dealt with separately from the others, in terms, for example, of its interaction with grain aggregate-order inhomogeneity in Scale C (Chapter 12). It should be noted that the band-like structures in Scale B also exhibit universality both in terms of their morphological aspects and their role as a carrier (or absorber) of severe plastic deformations, in a somewhat different sense from the dislocation cells in Scale A (i.e., the “similitude law”). The universality manifests itself, for example, as a form of a scaling law against the decreasing wall spacing and increasing misorientation with strain (e.g., Hughes, 2001; Sethna et al., 2003). It has been pointed out that the models to express the phenomena do not always require any dislocation-related information at all, implying the phenomena’s macroscopic origins rather than microscopic details (with any dislocation processes disregarded). These facts rationalize somewhat distinct treatments for describing the evolutionary features of the current scale, solely based on continuum models. Besides the previously mentioned various complexities in the deformation structures observed in the scale, many of the deformation structures are expected to be modeled or reproduced by the field theory-based approaches introduced in this book. For a demonstration, this chapter discusses the inhomogeneity evolutions in Scale B based on FE-based simulations, and incorporates the incompatibility-tensor field model in its constitutive framework, provided in Section 5.6. Having started by showing preliminary simulation results, some advanced outcomes are presented, including modeling of metallurgical microstructures (e.g., martensite lath block and packet) as a further extensions.

11.2

Evolution and Description Unlike the cellular substructures in Scale A, which require elaborate statistical mechanics-based treatments of the “collective” effects, plus a physical modeling of the long-range stress field, to reproduce the “cellular” morphology (see Chapter 10), the inhomogeneity in Scale B can be described by continuum mechanics-based models in principle, but great care needs to be taken when dealing with the “underlying microscopic degrees of freedom.” It is critically important to recognize that models without appropriate substructural degrees of freedom are incapable of adequately expressing the accommodation processes occurring there, resulting in “sometimes-similar-but-substantially different” patterns and/or “much harder” stress responses, for example. To avoid such shortcomings, my group introduced an “incompatibility tensor”-based model into the constitutive framework (Aoyagi and Hasebe, 2007; Hasebe, 2006), as briefly revisited in Section 11.3. The primitive but intuitively

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11.3  Constitutive Equations

547

tangible image of this accommodation mechanism via the incompatibility tensor is demonstrated in Figure 6.5.4. Let us confirm afresh the advantages for utilizing the incompatibility tensor for describing the inhomogeneities. The incompatibility tensor in the present context has the following several prominent features: (1) It is suitable for describing inhomogeneity in general, attributing to its physical meaning, that is, the curvature tensor that characterizes imperfections manifested as distortion of a given space with mathematical rigor. (2) By appropriately setting the information about the crystallographic degrees of freedom, that is, regarding which planes and directions are to be projected, any kinds of crystallographic accommodations can be expressed easily as well as appropriately. (3) The evolution or development of misorientation can be also expressed naturally without ad hoc models trying to reproduce this. Decomposition into pure deformation and pure rotation components also explicitly facilitate this mechanism, although studies about this have been inconclusive. (4) By implementing it into a constitutive (hardening) model, the “duality” can be automatically taken into account as a driving force for the evolution of the inhomogeneity, although the accuracy depends strongly on the hardening model itself. We always have to remember that the incompatibility tensor is not just denoting lattice curvatures due to the existence of dislocations, but is a manifestation of the curvature tensor of the background space (“curved” crystalline space) that acts as a potential energy source against the evolving inhomogeneity. See also Chapter 15 for “flow-evolutionary law,” which further provides a principle for the evolution of inhomogeneity in general via an interrelationship between the incompatibility tensor and the energy-momentum tensor.

11.3

Constitutive Equations Equipped with Field-Theoretical Strain-Gradient Terms Let us review briefly the fundamental set-up for describing and simulating the inhomogeneities to be evolved in the present scale. As examined earlier in the chapter, what we need is to introduce the appropriate substructural degrees of freedom, via the incompatibility tensor, in the constitutive model to be used in the simulations. The constitutive equation used throughout this chapter is that derived in Chapter 2 (and detailed in Chapter 5 [see Section 5.6]), and given by



1 ­ q ª º ­ § W c(D ) · p ½ ° (D ) ° ° A SRW c(D ) «« W c(D ) BSR exp ®1  ¨ (D ) ¸ ¾  CSR »» °°J K ¹ °¿ ® °¯ © «¬ »¼ , (11.3.1) ° °W c(D ) { W (D )  W *(D )  :(D ) Peierls °¯

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with * *( )  Peierls   Peierls



  1   kT ln 0 P    b3 g0P  ( )  

1/ qP 1/ pP

   

  

, (11.3.2)

where A SR   m bL *, BSR  G0disloc / kT , and CSR  BL * / b . Here, K (α ) is the resistance force, ( ) the back stress, τ m* the critical resolved shear stress at absolute zero temperature, k the Boltzmann constant, T the temperature, ∆G the activation energy, and a the referential strain rate. The evolution models for drag stress K (α ) and back stress ( ) (Hasebe et al., 1999) are given respectively by  ()   Acell K ()  Q H ( )  ( ) and 



d

( ) * cell  x N





2

, (11.3.3)

where H(γ ) represents the hardening modulus for a referential stress–strain curve. The hardening ratio Q , through which the contributions of α (α ) and  ( ) are accounted for, is given by





Q  f S   1  F ( ( ) ; ( ) ) . (11.3.4)



From this Q , we further evaluate the effective cell size, defined as

dcell

1   k  Q Q  N  



1 2

. (11.3.5)

The strain-gradient terms (see Section 5.6.2 for the derivation process) are given as F ( ( ) ) 



k p

 ( ) , (11.3.6) b

and F ( ( ) )  sgn( ( ) )



k p

ldefect ( )  , (11.3.7) b

where b denotes the Burgers vector and k , pα , and pη are the material constants. So α (α ) and  ( ) are respectively written in the forms as mappings of α ij and ηij into an α slip system.

 ( )  si( ) m(j ) ij , (11.3.8)

and

 ( )  si( ) m(j )ij . (11.3.9)

In order to investigate further the effects of the pure rotational and pure deformation-based degrees of freedom on the evolving incompatibility field η (F( ( ) )), that is, Θij and  jmn  m K ni (see Eq. (6.7.6) in Chapter 6), the corresponding terms of Kη and Θ (F ( K( ) ) and F(( ) )) are obtained by substituting K( ) and ( ) into Eq. (11.3.6), where K( ) and ( ) are given by mapping to each slip system in the same manner as in Eq. (11.3.8), that is,

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F ( K( ) )  sgn( K( ) )

k p

ldefect ( ) K , (11.3.10) b

F (( ) )  sgn(( ) )

k p

ldefect ( )  , (11.3.11) b

and with

K( )  si( ) m(j )  jmn  m K ni , (11.3.12)

and

( )  si( ) m(j ) ij . (11.3.13)



As understood from this construction of the hardening model, all the information about the inhomogeneities, including those induced by the strain-gradient terms, ultimately come down to the evolution of the effective cell size dcell in Eq. (11.3.5). Hence, the quantity can be used as a convenient indicator of the evolving inhomogeneous field.

11.4

Preliminary Simulation Results

11.4.1

Surrogate Model for Exaggerated Effects In order to check the basic properties (descriptive capabilities) of the incompatibility term in Eq. (11.3.4), first its specific contributions to the stress response of single-crystal and multicrystal models are examined, comparing those from the dislocation-density term. To this end, in a simplified and an exaggerated manner, Eqs. (11.3.6) and (11.3.7) are replaced, respectively, by a hyperbolic tangent function, that is,  ( ) Fplm ( ( ) )  tanh     sat



  , (11.4.1) 

and

  ( ) Fplm ( ( ) )  sgn( ( ) ) tanh   sat 

  , (11.4.2)  

where α sat and ηsat are the parameters to control the contributions of the respective terms to the hardening ratio Q . With the nature of the hyperbolic tangent function “tanh,” the above strain-gradient terms are ranged as

0  Fplm ( ( ) )  1 and  1  Fplm ( ( ) )  1, (11.4.3) respectively. Their contributions to the hardening ratio Q are freely controlled by changing the saturation values α sat and ηsat , according to the levels of α (α ) and  ( ) to be developed.

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11.4.2

Contribution of the Incompatibility-Tensor Field to Stress Response Figure 11.4.1 shows the obtained stress–strain curves for a single-crystal model with Cu-type material parameters oriented in a multislip direction, where either Fplm (α (α ) ) or Fplm ( ( ) ) are introduced into the hardening law with varying α sat or ηsat. As is expected, Fplm (α (α ) ) enhances the hardening (increase in the flow-stress level), which is similar to past cases with GN dislocation-density-inspired models found in the literature. What is of great interest is the result with Fplm ( ( ) ), demonstrating significantly enhanced “softening” even with a negative hardening rate or stagnation. This is due to the effectively activated “accommodation” of the excessive deformation (straining) introduced by the addition of the incompatibility term Fplm ( ( ) ) to the hardening law, as we expected. As illustrated in Figure 6.5.4 in a schematic manner, Fplm ( ( ) ) has a role in “redistributing” the imposed excessive strain (deformation) into a lower energy configuration, whose microscopic interpretation is the corresponding redistribution or introduction of both-signed dislocations. Figure 11.4.2 shows the superposed effects of both Fplm (α (α ) ) and Fplm ( ( ) ) on the stress response. For the left-hand side where the effect of Fplm (α (α ) ) with sat  15 is presented, for example, we can confirm such a combined effect, that is, slight work-hardening stagnation even with additional hardening. This clearly demonstrates the role sharing between Fplm (α (α ) ) and Fplm ( ( ) ), that is, enhanced hardening and enhanced softening. This aspect is discussed further in Section 11.4.3. Figure 11.4.3 gives the same results as Figure 11.4.2 but for a multicrystal model composed of 23 grains. We observe even more pronounced additional hardening or

Figure 11.4.1  Variation of the stress–strain curve of a single-crystal Cu-type model with fieldtheoretical strain-gradient terms F (α (a ) ) or F (h(a ) ), comparing the effect of the degrees of their contributions measured by asat and hsat , respectively, representing enhanced hardening in the former and additional softening in the latter.

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Figure 11.4.2  Effect of the field-theoretical strain-gradient terms F (α ( a ) ) and F (h( a ) ) on the

stress–strain curve of a single-crystal Cu-type model, comparing different contributions of the two terms measured by asat and ηsat .

Figure 11.4.3  Effect of the field-theoretical-strain-gradient terms F(α ( a ) ) and F (h( a ) ) on

the stress–strain curve of a multicrystalline Cu-type model (with 23 grains), comparing different contributions of the two terms measured by α sat and hsat , showing similar but more pronounced effects to those in the single-crystal case.

softening with the inclusion of Fplm ( ( ) ) or Fplm ( ( ) ), compared with the above single-crystal case. These enhanced effects are simply due to higher mutual constraints among the composing grains than in the single-crystal case, which generally

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Figure 11.4.4  Comparison of stress response with arbitrary combination of F(α ( a ) ) and F (h( a ) )

between single-crystal and multicrystal models of Cu type, demonstrating markedly enhanced effects in the latter.

enhances the inhomogeneity in the distortion or strain distribution. From the present output we learn more about the ability of the present strain-gradient models, especially the incompatibility term Fplm ( ( ) ), to act as “absorbers”: The larger the constraints imposed, the more a model’s accommodation ability functions, which can be confirmed once again by comparing the combined effect of Fplm ( ( ) ) on the overall response between single-crystal and multicrystal models in Figure 11.4.4. As mentioned at the end of Section 11.3, the effective cell size dcell defined by Eq. (11.3.5) is an indicator of the evolving inhomogeneous field in Scale A. Let us reexamine the above evolutionary trends of the inhomogeneity induced by Fplm (α (α ) ) or Fplm ( ( ) ) from this viewpoint. Figure 11.4.5 shows the effect of Fplm (α (α ) ) and/or Fplm ( ( ) ) on the variation of dcell with strain for the 23-grain model, with slip system-wise and grain-wise bases. It can be observed that there are relatively wider variations for the slip system-wise responses than for the grain-wise counterparts. Also, we confirm clear correspondences to the enhanced hardening and softening (depicted in, for example, Figure 11.4.3). So, Fplm (α (α ) ) tends to speed up cell evolution, exhibiting a rapid drop of dcell , whose trend stays almost the same in the grain-wise presentation Fplm ( ( ) ). On the other hand, it is demonstrated to cause conspicuous delay or slowing down of the effective cell-size evolution in an irregular manner in some of the slip systems, corresponding to their softening behavior in the stress response. In contrast, viewed from the grain-wise perspective, it tends to obey quite mutually similar behavior. The same is true for the combined case, that is, Fplm ( ( ) )  Fplm ( ( ) ). This case yields a relatively large variation in dcell on the slip system-wise basis, while it conspires to yield a rather converged variation on the grain-wise basis. The latter almost coincides with the case with Fplm (α (α ) ) only.

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11.5  Simulation Results: Evolving Intragranular Inhomogeneity

553

Figure 11.4.5  Variation of the effective cell size with strain obtained in simulations on 23-grained model, comparing the effects of strain-gradient terms with (top) slip system-wise and (bottom) grain-wise plots.

The contrasts between the two presentations mean that such local fluctuations in the effective cell size (corresponding to those in the local stress–strain fields) within a grain, especially those observed with Fplm ( ( ) ), are more or less averaged out when viewed from the grain-wise basis, conspiring to exhibit a relatively milder response as a whole grain. These results again emphasize the importance of the current incompatibility-based multiscale perspectives, allowing us to look at multiscale inhomogeneities with distinct levels of fluctuations, as pointed out, for example, in Section 3.5, and, simultaneously, demonstrate the outstanding ability of the present model Fplm ( ( ) ) to describe them.

11.5

Simulation Results: Evolving Intragranular Inhomogeneity

11.5.1

Modulation Structuring and Misorientation Let us now examine afresh the contributions of the strain-gradient terms Fplm (α (α ) ) and Fplm ( ( ) ) to the intragranular inhomogeneity evolution. To that end, let us introduce the original forms of the strain-gradient models (Eqs. (11.3.6) and (11.3.7)) instead of their exaggerated versions (Eqs. (11.4.1) and (11.4.2)). They are introduced into the constitutive framework through the hardening law, Eqs. (11.3.3) and (11.3.4).

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Figure 11.5.1  Comparison of contour diagrams for dislocation-density and incompatibility terms for FCC multigrain FE model under tension. Modulation patterns are observed for the latter, by way of contrast to the former, where intensity goes mainly to the GB regions.

Here we use first the same 23-grained model as in the previous section, together with Cu-type material parameters. Twelve slip systems for FCC structures are considered, and 30% tension is applied under a plane stress condition. Figure 11.5.1 compares contours between dislocation-density and incompatibility terms F(α (α ) ) and F( ( ) ) (referred to as α - and η -terms hereafter, for simplicity). The η -term yields modulated band structures extending both positive and negative signs in some of the grains, whereas the α -term tends to take higher values exclusively near the GBs. The latter coincides with the trend that has been reported so far associated with GN dislocations. For the former, on the other hand, such band structures have rarely been recognized and discussed. One more intriguing feature is that the modulations are always accompanied by misorientation across the bands (Figure 11.5.2). Figure 11.5.3 displays an orientation contour diagram, together with the corresponding η -term contour for a 23-grained model under 30% tension. The grains exhibiting modulation bands are highlighted. The correspondences are clearly confirmed. Further careful examination reveals that the grains exhibiting modulations yield multiple slips on single-slip planes. Figure 11.5.4 lists the slip plane normal against the tensile axis via ( ,  ) for each grain after 30% tension, where the left-hand table indicates those for the primary slip system, while the right-hand those for the secondary slip system. Here, the angles ( ,  ) for the banded grains are shaded in the tables. We can easily notice that such

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11.5  Simulation Results: Evolving Intragranular Inhomogeneity

555

Figure 11.5.2  Comparison between incompatibility-induced modulated patterns for multislipped grains and experimentally observed MBs, pointing out some similarities.

Figure 11.5.3  Orientation contour diagram, together with the grains exhibiting modulation bands in Figure 11.5.1 (emphasized by thick lines), implying misorientations are exclusively developed across the modulated bands.

grains always yield the same ( ,  ), meaning the primary and secondary slip systems are on the same slip plane. The inset illustrates the primary and secondary slip directions for each grain, whereas black lines express the initial primary slip direction before deformation. This observation eloquently implies that the structures are not incidental kinds but have a definite crystallographic origin.

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Figure 11.5.4  Primary and secondary slip-plane directions and slip planes for all the grains in the multigrained model in Figure 11.5.2, where those exhibiting “modulations” are highlighted.

Since the variation in the η -term contributes to the local additional hardening and softening depending on its sign, and ultimately the effective cell-size distribution in the present model (see, for example, Figure 11.5.8, discussed later), the band structure can be interpreted as one of the GNBs categorized in Section 3.10. If the width is sharp enough, the corresponding structure will be MBs or even DDWs. In such cases, MBs and DDWs are regarded as refined cells aligned in band-like morphologies. Note that detailed dislocation configurations have been identified in Hong et al. (2013) and we reproduced and examined them based on the FTMP scheme in Ihara and Hasebe (2019). Figure 11.5.5(a) compares slip system-wise variations (left) between α - and η -terms with strain for the 23-grained model used earlier, together with their grainwise averaged versions (right). We confirm a sharp contrast between the two cases, similar to the trends for the exaggerated models in Figure 11.4.5: The α -term basically yields monotonic increase with increasing strain for all the slip systems, whereas the η -term varies irregularly depending on the slip system; some even change signs during the course of deformation. This temporally haphazard evolution of the η -terms is regarded as an indication of the ongoing evolution of the intragranular inhomogeneity, which “dynamically” accommodates the growing excessive local deformations. Since the irregular variation into positive and negative regions extends quite symmetrically, the slip system-wise irregularity tends to be averaged out as a whole grain. This means

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Figure 11.5.5  Slip system-wise and grain-wise variations of incompatibility (a) and dislocationdensity (b) terms with strain for 12 slip systems in an FCC model, together with extracted comparisons for (b) between modulated and nonmodulated grains (in Figure 11.5.2) on the grain-wise basis (c).

that what the incompatibility field tends to enhance is mostly the “fluctuation” part of the inhomogeneity (besides the overall softening). Therefore, the corresponding “grain-wise” variation (right-hand side) yield is one order smaller compared to that for the slip system-wise counterpart. Here, the grains that show modulated patterns are indicated by broken lines. A close look at the grain-wise version (Figure 11.5.5(b)) reveals that the modulated grains yield relatively smooth variations, demonstrating they are apt to absorb the excessive deformation of the surrounding nonmodulated grains, allowing them to deform rather “freely.” This is another manifestation of the “duality” that was pointed out earlier. In sharp contrast, the α -term in the grain-wise basis behaves similarly to the slip system-wise counterpart, although the variation is milder. As will be argued later, the dislocation-density field also plays a role in enhancing the incompatibility-induced inhomogeneities. The role sharing between the two, that is, the α - and η -terms, is also an important aspect of current field theory-based strain-gradient models.

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11.5.2

Modulation Developments in Stress and Strain Fields Let us examine next the effect of incompatibility-induced modulation developments on the stress and strain distributions. For this purpose we will use the same multigrained sample but containing some harder grains, a model of a DP alloy ((V f )HP  25%,( f )HP /( f )SP  4 ), which is expected to enhance the inhomogeneity in the soft grains. Here, the same Cu-type material parameters are used, together with 12 slip systems for the FCC structure and crystallographic orientations allocated to the composing grains. The model is schematically shown in the inset of Figure 11.5.6. Figure 11.5.6 shows cross-sectional distributions of deviatoric stress (equivalent stress) and strain (equivalent strain) along the broken line across the sample. Here, a comparison is made among four cases, that is, those with the α -term, with the η -term, with both the α - and η -terms, and without them (reference). We can clearly confirm the role of the η -term, especially in the left-hand soft grain, which did not yield modulation in Figure 11.5.2 for the case without hard grains. The stress and strain fields are modulated into band-like distributions so as to cope with the constraints due to the hard grains located on both sides, while the modulation pattern reflects that in the η -term itself. Since the band-like pattern does

Figure 11.5.6  Typical example of deviatoric stress and strain distributions along a broken line of a DP model shown in the inset. Incompatibility is demonstrated to yield highly “modulated” or fluctuating stress and strain fields, while the dislocation density-tensor term plays a role to enhance it. An FCC Cu-type model with 12 slip systems is used.

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not exist in the case without hard grains, as in Figure 11.5.2, the present modulation is purely induced by the inhomogeneity due to the hard grains, thus, has no crystallographic origin. Also confirmed is that the η -term has a softening effect on the average stress level, exhibiting the smallest stress response. The α -term, in contrast, is apt on average to enhance the stress level, without essentially causing change in the distribution. The role sharing between the α - and η -terms, which has been already confirmed by the exaggerated models in Section 11.4.2, is demonstrated to work also in the present case. Considering both terms simultaneously, they can have the combined effect of accommodation-induced softening and enhanced hardening of the GN type. In the deviatoric strain distribution, we observe rather enhanced flow in the targeted soft grain, whereas reduced flows result in hard grains. This trend becomes more obvious as the strength ratio (σ f )HP / (σ f )SP increases. This, together with the above trend for σ , implies that the η -term promotes conversions of elastic strain energy into inhomogeneous (or local) plastic flow, meaning the quantity acts as a driving force of the inhomogeneity. A similar phenomenon is found in Chapter 12 for Scale C, to be termed “duality,” and is discussed extensively there. Figure 11.5.7 displays the corresponding contours of the deviatoric stress σ and strain ε to the above cross-sectional distributions (Figure 11.5.6). Essentially the same trends can be observed as those in Figure 11.5.6. The band-like structures, absent in the case with the α -term only, emerge with the introduction of the η -term, which are pronounced when the effect of the α -term is overlapped (extreme righthand side). Note that the incompatibility-induced modulations with crystallographic origin observed in Figure 11.5.2 (in the model with no hard grain) are also confirmed in both the stress and strain contours, although the contour levels are slightly adjusted for the targeted grain. Figure 11.5.8(a) and 11.5.8(b) display the cross-sectional distribution of the effective cell size dcell corresponding to Figure 11.5.7, where the evolution process is also shown. Corresponding contours of dcell will be presented in Figure 11.5.10, clearly showing the associated distributions with those of σ . Introducing the η -term also results in a modulation in the effective cell size, while the α -term tends to enhance the cell evolution as a whole (accelerates the refinement), directly corresponding to the enhanced overall flow-stress level. As pointed out in Section 11.5.1, the regions with smaller dcell correspond to those with refined cells. Furthermore, since such regions, frequently yielding banded-like morphology, are accompanied by misorientation from the surrounding areas, they are regarded as “blocks” of refined cells. Therefore, depending on the morphology, including the sharpness of the transition from the adjacent regions, the incompatibility-based emerging structures can be categorized in either transition bands, MB, DDW, or geometrically necessary dislocations (GND). Note that the evolutionary aspects of dcell are discussed in Section 11.5.4. As demonstrated earlier, introducing the η -term into the hardening law can allow us to simulate intragranular accommodation processes and resultant band-like dislocation substructures, accompanied by fluctuating internal stress fields as well as

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Figure 11.5.7  Corresponding contour diagrams to Figure 11.5.6 for deviatoric stress and strain, where the band-like structure caused by the incompatibility term is pronounced by the addition of the dislocation-density term. A DP model with FCC Cu-type parameters and 12 slip systems is used.

misorientation. As understood from the softened overall stress responses, to be discussed in Section 11.5.4, the structuring yields a lower energy state. Further field quantitative examinations on the mechanical role of the η -term are given there.

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Figure 11.5.8  Corresponding cell-size contour to Figure 11.5.7 (top) showing emergence of band-like refined cell region regarded as MBs or transition bands, which tend to be accentuated by the addition of the dislocation-density term, together with (bottom) variation of the cross-sectional distribution with increasing strain.

11.5.3

Evolution of Modulating Stress and Stress Fields Let us examine next the evolutionary process of the field fluctuations of stress and strain due to the η -term by using a higher resolution model. A seven-grained model, with 96 × 96 crossed-triangular elements is used, together with the Cu-type material parameters, as previously. Figure 11.5.9(b) and 11.5.9(c) show examples of evolving deviatoric stress and strain fields under tension, respectively, where the variations of a cross-sectional distribution with strain are displayed. The figures also show corresponding contours of σ

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Figure 11.5.9  Development processes of (a) an incompatibility term, (b) incompatibilityinduced deviatoric stress, (c) deviatoric strain, and (d) hydrostatic stress-fluctuation modulations, where modulation grows from early-stage fluctuation without changing its wavelength. An FCC Cu-type model with 12 slip systems is used.

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Figure 11.5.9  (cont.)

and ε at 30% nominal strain. One can notice that the wavelengths of the modulations seem to be determined already in the early stage of deformation, and they basically persist even at the later stage. It should be noted that, in the case when the dimension of the sample drastically changes, such as under compression and rolling, the wavelengths will be subject to change in proportion to the proceeding deformation. Distribution of the η -term is also displayed in Figure 11.5.9(a). We clearly confirm that the modulated structures in the incompatibility field are basically fixed already at u / L = 2.5%, especially in the central grain. The η -term distribution itself yields roughly a vertically upside-down modulation pattern to that for σ , while it approximately agrees with the second derivative of ε , as we understand from its definition. Therefore, we conclude that the modulation evolutions in the stress and strain fields are the direct consequence of the growth of the incompatibility field. The grains in both sides, on the other hand, show rather flexible evolutions of modulation structure, especially in the later stage of deformation (20–30%), exhibiting even abrupt sign changes of the η -term. This is probably because they face traction-free surfaces. Such sudden sign changes in the η -term do not affect the overall σ and ε distributions. Figure 11.5.9(d) displays the corresponding evolving distribution of the hydrostatic stress σ m to Figure 11.5.9(b) for σ . Basically, a similar evolutionary trend is observed, except in the two-sided grains, where negative growth of σ m takes place locally. What we can confirm is a correspondence for such rather “anomalous” evolution in σ m with that of the η -term. Figure 11.5.10 compares magnified variations of the σ m distribution with those of the η -term for the side grains. Abrupt sign changes take place locally (one in the right grain and two in the left grain) in the incompatibility field between 20% and 30% nominal strain near the regions where the negative growths of σ m occur. I have confirmed that local changes of the sign of the η -term are frequently accompanied by those of the hydrostatic stress, similar to this case. This correspondence is interpreted as follows. Inhomogeneously accumulated (or stored) elastic strain energy

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Figure 11.5.10  Comparison of locally anomalous growth of hydrostatic stress with attendant abrupt sign change in incompatibility term in a later stage of tensile deformation.

(in the form of [or manifested as] a fluctuating σ m field) is irregularly released into the vicinity, causing inhomogeneous local plastic flow in the surrounding regions (manifested as haphazard variations in the incompatibility field). This is considered to be an indication of the “duality” which is extensively discussed in Chapter 12 in the context of inhomogeneity evolution in Scale C. One of the most important features of the present incompatibility field model can be derived based on these observations. It is the automatic incorporation of the “duality” mechanism, achieved simply by introducing the η -term into the hardening law (moduli). Doing so, we can naturally take into account arbitrary spontaneous accommodation mechanisms and reproduce attendant evolutions of inhomogeneity, in context-dependent manners, as far as the projection directions of the incompatibility tensor employed are appropriate. Moreover, post-process evaluations of the duality are also possible. Separate treatments of the duality-related properties are extended in Chapter 15, where an explicit relationship between the incompatibility tensor and the energy-momentum tensor, called “flow-evolutionary law,” is derived and applied to several simulation results.

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565

Mechanical Roles of the Incompatibility Field Figure 11.5.11 compares the overall stress–strain response among the three cases, with either of the α - and η -terms and without them. There is an obvious overall softening in the flow-stress response due to the inclusion of the η -term; on the other hand, we have additional hardening with the α -term, which has been confirmed in the context of “size effect” via GN dislocation density. Therefore, as mentioned previously, the substructuring via the η -term into band-like patterns leads the system into a lower energy state than others without the term. Figure 11.5.12 compares variations of the effective cell size dcell with strain, comparing four cases with and without the α - and η -terms, where dcell for all the slip systems is plotted (averaged slip system-wise). Distinction between the two cases, with the α -term and with the η -term, can be found in comparison with that without the strain-gradient terms. With the η -term, the evolution of dcell is delayed as a whole, while enhanced evolution (i.e., faster decrease of dcell ) occurs with the α -term. These trends are the same as those observed in Figure 11.4.5. Note that, more precisely, the refining cells and less-evolving cells correspond to the negative and positive signs of the evolving modulation structures of the η -term, respectively.

Figure 11.5.11  Effect of the strain-gradient terms on the overall (macroscopic) stress–strain response, together with variations of dislocation-density and incompatibility terms with strain (slip system-wise). The incompatibility term tends to enhance softening, while the dislocation-density term promotes additional hardening.

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Figure 11.5.12  The effect of the strain-gradient terms on effective cell-size evolution. The incompatibility term tends to delay the outset of cell evolution while the dislocation-density term promotes refinements.

These results clearly demonstrate the mechanical role of the incompatibility tensor (the η -term) as the absorber of excessive deformations, as is expected based on the schematic drawing of Figure 6.5.4 (where the corresponding modulation [bandlike pattern] is also drawn). The absorption of such high amounts of deformation is promoted based on the “duality” mechanism, as can be indirectly confirmed in the inset (b) of Figure 11.5.11, which exhibits irregular and mutually haphazard variations of the η -terms, with strain for each slip system. This is in marked contrast to the α -term, which shows monotonic and mutually similar behaviors (inset (a) of Figure 11.5.11). The irregular changes in the η -terms stem from the indeterminate evolution of the duality fluctuations in the stress and strain fields during the course of deformation. This is an eloquent manifestation of the evolving “inhomogeneity” in this scale. The “absorption” is evidently associated with “energy dissipation” in this scale, because the “excessive deformation” in this context is generally transferred to local plastic flows. As is pointed out in Section 13.4, where a global–local nature of the stress and strain spectral distributions for a polycrystalline sample is examined, treatments of the energy dissipation at the smallest scale play a critical role in appropriately

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simulating multiscale plasticity. This implies the necessity of considering interactions with Scale A and smaller levels in order to appropriately take into account finer-level energy dissipation to avoid unrealistic stagnation.

11.5.5

The Effect of Evaluation Size and Discretization Readers may be interested in the effects of the resolution of the FE discretization assumed and the evaluation size of the higher derivatives on the emerged “modulating” structures. Since obtaining the incompatibility-tensor field requires second spatial differentiations, those simulation conditions may have a critical influence on the accuracy as well as the physical interpretations of the produced substructure itself. The effect of the number of FE mesh divisions on the simulated incompatibility-​ tensor field has been examined in several respects by the author. To be displayed here is an example associated with a substructuring of GNDs under wedge indentation in single-crystal Cu, which has been observed (measured) experimentally (Kysar et al., 2007). The simulation model is shown in Figure 11.5.13, where the half region under wedge indentation is drawn. Here, the motion of the indenter is mimicked by adding

Figure 11.5.13  Analytical model for wedge indentation against single-crystal Cu, mimicking experiments by Kysar et al. (2007) for EBSD-based GND measurement.

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Figure 11.5.14  Examples of simulated contours of the incompatibility term, comparing two mesh divisions. Essentially no difference is observed in modulated structures regardless of the mesh size.

Figure 11.5.15  The dislocation-density contour compared with an experimental result by Kysar et al. (2007). Adapted with permission of the publisher (Pergamon [Elsevier]).

nodal displacements step by step, as schematically shown in the inset. Figure 11.5.14 compares the simulation results with two resolutions, that is, with 96 × 96 and 48 × 48 crossed-triangle elements, for otherwise the same simulation. We observe essentially

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no difference between the two. As far as the mesh size is comparable with, for instance, the widths of the emerging modulations, we can safely conclude the independency of the evolved patterns of the FE mesh division. Note that the banded structure appearing in the incompatibility brings about a corresponding pattern in the dislocation-density distribution, which is demonstrated to be similar to that observed in the experiment by Kysar, as compared in Figure 11.5.15. A more intuitively tangible set of examples is displayed in Figure 11.5.16, where two simple sheared single-crystal Fe samples in typical γ -orientations are compared for the purpose of demonstrating not only mesh independency (top two rows) but also real-size dependency (bottom three rows). The sample-size dependent substructuring is critically important, among other factors, in MMMs in general. Refer also to Figure 9.4.21 and Section 9.4.3.2 for a related discussion.

Figure 11.5.16  Examples of simulated contours of the dislocation-density norm for two γ -orientations of Fe under simple shear (Hasebe et al., 2014), demonstrating (a) mesh independency and (b) size dependency.

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Figure 11.5.17  Analytical model and conditions for incompatibility simulation with different evaluation sizes for strain gradients (derivative).

The evaluation size of the spatial derivative, on the other hand, can possibly affect the simulated substructures, particularly in terms of their refinements. Finer mesh divisions with smaller evaluation sizes are expected to be used to capture the finer details of substructures. Figure 11.5.17 displays the model employed for examining this aspect, again with two resolutions, that is, 96 × 96 and 48 × 48 crossed-triangle elements, together with two evaluation sizes for the derivatives of rGR=2.0 and 4.0µm, that is, those commensurate with four quadrangular elements in diameter, respectively. Here, a represents the evaluation size instead of rRG. Figures 11.5.18 and 11.5.19 provide cross-sectional distributions for the α - and η -terms and the corresponding contour diagrams, comparing the two conditions. Evidently, finer modulations result from the smaller evaluation size for the η -term, but the morphology, including the directionality of the bands, is not basically affected, as depicted in the contours. The result with the larger evaluation size looks similar, but the contour becomes a bit blurred. It should be noted that, setting the same evaluation size, for example, eight elements for 96 × 96 and four elements for 48 × 48 divisions, results in the same modulated patterns, as can be seen in the results in Figure 11.5.16. Figure 11.5.20 compares the resultant effective cell-size distribution, correspond1/ 2 ing to the above results, evaluated via dcell  k *  Q Q  . We can confirm the

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Figure 11.5.18  Effect of evaluation size of gradients on inhomogeneity evolution due to dislocation-density and incompatibility terms.

Figure 11.5.19  Contour diagrams for the incompatibility field showing the effect of evaluation size for derivatives on inhomogeneity evolution due to strain-gradient terms. A smaller evaluation size yields finer modulation but with a similar morphology.

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Figure 11.5.20  Corresponding contour diagrams to Figure 11.5.19 but for effective cell size.

corresponding modulation also in the dcell, meaning alternating structures of fine and coarse dislocation cells, as a result of the evolved inhomogeneity in Scale B. Figure 11.5.21 makes a direct comparison between the η -term and dcell distributions, showing a correspondence between substructures in Scales A and B. The effect of the evaluation scheme for the higher-order derivatives, as well as its accuracy, has been also extensively discussed by my colleagues and I (Aoyagi et al., 2008) based on the reproducing kernel particle method (RKPM), together with a FEM-based method. We concluded that the morphological features are not essentially affected by the evaluation method, nor are the parameters introduced in the η -term. A summary of the results is presented in Figure 13.2.3, where comparisons among different evaluation sizes for both RKPM- and FEM-based analyses are made. Based on those results, it is also revealed that the number of modulations is continuously changed, with an increasing evaluation size for the gradient evaluation, but also show systematic variation, as summarized in Figure 11.5.22 (see also Figure 13.2.2). This implies the possibility of automatic capturing different physical meanings for the substructures to be simulated simply by changing the evaluation size.

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Figure 11.5.21  Comparison of contours between incompatibility term and effective cell size, where the modulation that emerges in the former is confirmed to bring about corresponding modulated cell-size distribution.

Figure 11.5.22  Variation of number of incompatibility-induced modulated bands with evaluation size for derivatives of strain, together with corresponding contours, comparing results for FEM- and RKPM-based analyses. Insets show definitions of evaluation size for two methods. Results obtained by a corresponding 1D analysis (as in Figure 13.2.3) are also overplotted.

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11.5.6

Morphology and Overwriting Figure 11.5.23 displays the η -term contours obtained under tension and simple shear, respectively, where mutually different morphologies depending on the loading mode can be confirmed. For making the comparison easier, another comparison is made on the extreme right, based on the same sample shape, where the deformed samples have been virtually deformed back to roughly that in the initial configuration (or dimensions). From the figure, distinction in the morphology between the two is clearly confirmed, especially in the central grain. Now, let us see what will happen in the η -term distribution if the loading mode is changed during the course of deformation, for example, from tension to simple shear. Figure 11.5.24 schematizes an analytical condition corresponding to this, where 10% tension followed by 20% simple shear is applied. The result is shown in Figure 11.5.24, demonstrating that an “overwriting” of the simulated substructures actually takes place as a consequence of the interactions between the two incompatibility-induced structures. In this case, the “overwriting” results in dissolution of the once-evolved modulations within the central grain. Figure 11.5.25 depicts the corresponding misorientation contour, also showing its mutual cancellation in the central grain.

Figure 11.5.23  Schematics of simulation for examining the “overwriting” effects of an incompatibility-based inhomogeneous field, where the substructure evolved during tension is overwritten by that under simple shear. Pictures to the right are distorted-back versions of the above for making the comparison easier.

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Figure 11.5.24  Result of “overwriting simulation” attained under 10% tension followed by 20% shear. The incompatibility-induced modulated pattern in the in central grain evolved during tension is offset by that produced following the shear deformation.

Figure 11.5.25  Overwritten incompatibility distribution and corresponding contour of lattice

rotation.

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11.5.7

Orientation Dependency of Modulated Structures As has been shown so far, the modulation structures that emerge via the η -term depend naturally on the deformation mode. Also, the orientation dependency is expected to exist as in real materials. Figure 11.5.26 displays examples of the simulation results for the typical crystallographic orientations of , , and for an FCC Cu model, together with the experimental observations by Risø National Laboratory, Denmark, led by N. Hansen (see, for example, Liu et al., 1998). Orientation dependency as well as similar morphologies to the experiments are confirmed, although the directionality of the simulated structures differ from the micrographs, probably due to the inappropriate projection direction of the incompatibility tensor in evaluating the η -term. Here, a projec( ) ( ) ( ) tion via ms  mi( ) s (j )ij was used instead of ts  ti( ) s (j )ij and ss  si( ) s (j )ij . Further examination of the effect of the projection direction can be found in Section 11.5.8. Updated simulation results exhibiting orientation-dependent evolutions of dislocation substructures evolved during tension, together with cross-sectional misorientation distribution along the lines A–A ′, are compared in Figure 11.5.27, where the three projection directions are simultaneously used, although no direct comparisons with experiments are made. It is noteworthy that the descriptive capability allowing

Figure 11.5.26  Orientation dependence of deformation structures for FCC metals, comparing schematics and simulation results of incompatibility distribution, TEMs, and their traces. Schematics and experimental observations belong to Risø National Laboratory.

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Figure 11.5.27  Simulated orientation-dependent substructure evolutions for single-crystal samples, together with corresponding cross-sectional distributions of misorientation.

associated misorientation developments is also one of the strengths of the current FTMP-based CP-FEM, yet is essentially absent in conventional counterparts without using the incompatibility tensor.

11.5.8

The Effect of Projection Direction on Morphology Figure 11.5.28 compares the resultant η -term distributions among three distinct pro( ) ( ) ( ) jection directions, that is, edge  ti( ) s (j )ij , screw  si( ) s (j )ij , and discl  mi( ) s (j )ij . Here, we put ti( )  ijk s (j ) mk( ) for the line direction of the defect of interest. In this case, a single-crystal Fe under simple shear in the {111} direction is taken as an example. We can clearly observe different morphologies, especially in terms of the directionality, demonstrating a strong dependency of the patterns to be evolved on the projection direction, even for the same sample. The projection direction for the η -term will also affect other incompatibility-​ based components as well as the α -term. Figure 11.5.29 displays results with ( ) ( ) edge  ti( ) s (j )ij and screw  si( ) s (j )ij projections, comparing four resultant distributions, that is, the α - and η -terms, contortion, and disclination components of the η -term (see Section 11.6). Evidently, the evolved morphology of the η -term resembles the others. ( ) Although we have used discl  mi( ) s (j )ij thus far, there exists room for the choice of the projection. The choice of the projection direction will be particularly important for the incompatibility-tensor model, because it determines the accommodation modes of the excessive deformation, ultimately controlling the overall stress response of the model used. As is described in Section 11.2, the η -term is obtained by a projection of the

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Figure 11.5.28  Examples of the variation of incompatibility-based modulation morphology, depending on the projection direction of the incompatibility tensor under simple shear, for {111}-oriented single-crystal Fe.

incompatibility-tensor distribution, evaluated in every time step of the simulation, from the plastic strain distribution developed in the previous step via the hardening evolution equation. The dislocation-density tensor, on the other hand, is expected to have a relatively smaller effect because it only contributes to the hardening aspect of the response. Since there is no reasonable ground about which direction to choose for the incompatibility tensor at this juncture, we need to use experimental observations to seek the best conditions. Conversely, we can also say that, including additional degrees of freedom in terms of rotation, as is discussed in the following, the arbitrariness, at the moment at least, can render it possible for us to describe or model various types of microstructures. They can include not only deformation structures, as those discussed earlier in this chapter, but also lath martensite and twins, as will be discussed in Section 11.6. Comparisons with some recent experimental observations for single-crystal Fe ( ) (BCC) under pure shear (Hasebe et al., 2014) suggest that we use ts  ti( ) s (j )ij for reproducing the emerging morphology of the dislocation substructures of this scale.

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Figure 11.5.29  Examples of the variation of incompatibility-based modulation morphology in respective strain-gradient terms, depending on the projection direction of incompatibility tensor under simple shear for {111}-oriented single-crystal Fe.

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11.6

Simulation Results with Disclination Fields

11.6.1

Model Description As discussed in Section 6.3, the incompatibility tensor can be decomposed into pure deformation and pure rotation parts. The latter expresses the disclination field, which is further broken down into three components, as detailed in Section 6.3.2, that is, wedge, twist, and splay. This section examines the individual evolutions as well as their interplay during the course of deformation. Figure 11.6.1 shows the computational model used here, where the plane stress condition is assumed with 12 slip systems on {110} planes of BCC polycrystal. The initial crystal orientations are depicted also in the figure. Tension is applied up to 20% nominal strain based on FEM. A traction-free condition is assumed for both sides of the model.

11.6.2

Simulation Results Figure 11.6.2 shows distributions of the incompatibility-based terms, that is, η , K , and Θ, for the primary slip system of the central grain at a nominal strain of 19%. The distributions of η - and K -terms look alike, however, slight differences can be observed: The η-term exhibits basically band-like substructures, whereas a lattice-like morphology is rather pronounced for the K term. This implies that the pure deformation component of incompatibility without a rotation, that is,  jmn ∂ m K ni , will possibly be responsible for cell-like modulations. The pure rotation component of incompatibility Θ, on the other hand, yields one order smaller magnitude compared with the

Figure 11.6.1  Analytical model used for examining the effects of decomposed components of the incompatibility tensor, that is, pure deformation and pure rotation parts on the deformation structure (modulation) to be evolved. BCC Fe with 12 {111} slip systems under a plane stress condition is assumed.

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11.6  Simulation Results with Disclination Fields

Figure 11.6.2  Contours of incompatibility terms comparing total, pure deformation (contortion), and pure rotation (disclination) components. Tension at 20% is applied.

Figure 11.6.3  Comparison of three disclination densities evaluated from the pure rotation components of the incompatibility tensor.

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others simply due to the small amount of lattice rotation in this deformation range under a simple tension condition. Distributions of each component of the disclination-density tensor Θij are shown in Figure 11.6.3. Although they roughly show similar distributions to that of Θij , the twist component exhibits relatively clearer modulation among the three, which corresponds to in-planar evolution of the lattice rotation, probably leading to tilt-angle boundary formation. Figure 11.6.4 shows a variation of cross-sectional distributions of the terms of α , η, K, and Θ for the central grain, along the broken line (top left-hand side). The variation of the η-term in Figure 11.6.4 demonstrates a growing modulation with

Figure 11.6.4  Evolution of field-theoretical strain-gradient terms along with deformation, where cross-sectional distributions of the central grain are displayed.

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Figure 11.6.5  Comparison of variations of the averaged incompatibility, contortion, and disclination terms, with nominal strain, demonstrating “redistribution” of the defect degree of freedom into pure deformation and pure rotation.

increasing nominal strain. This growing modulation arises basically from a positive feedback effect induced by the incompatibility field via the hardening law, producing the modulations directly from given strain distributions. We also observe a slight slowing down of the growth rate and coarsening of the modulated pattern at around u/L = 0.10. Correspondingly, the growths of  jmn ∂ m K ni and Θij are accelerated and the modulations are refined, implying that redistribution of the incompatibility into pure deformation and rotation components has taken place during deformation. This is also considered to be a manifestation of the “duality” mechanism, promoting in this case the local rotation via incompatibility evolution. To quantitatively confirm this trend, spatial averages over the grain are taken for the three terms and the variations with nominal strain u / L are compared in Figure 11.6.5. We can observe monotonic increase for F( ( ) ) after u / L = 9%, whereas F ( K (α ) ) takes a declining turn at u / L = 12% . In response to this, F(( ) ) starts increasing again. Figure 11.6.6 compares distributions of equivalent strain at nominal strain of 0.19 between two conditions, that is, with and without η -term. We find that the incompatibility-induced modulation also affects the distribution of strain. Similar modulation can be found in the stress distributions, although this is not shown here. Thus, further induced modulation in the stress field can be a possible model for a complicated fluctuating internal stress field. It should be noted that such modulation is never observed in simulation results without the η -term, shown in the right-hand side of Figure 11.6.6.

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Figure 11.6.6  Comparison of equivalent strain contours with and without the incompatibilitytensor term.

11.6.3 Summary This section discussed a possible decomposition of the incompatibility tensor into pure deformation and rotation components. The rotation component is further resolved into three kinds of disclinations, that is, wedge, twist, and splay. Preliminary simulation results, based on the CP-FEM, demonstrates growing modulation of intragranular substructures as a result of the positive feedback of incompatibility-induced fluctuation in the strain distribution, bringing eventually about redistribution of them into rotational fields.

11.7

Effect of Initially Introduced Strain Distribution: Toward Rational Modeling of Arbitrary Microstructures

11.7.1

Preliminary Simulations As was pointed out in Section 11.5.3, the incompatibility tensor-based description of inhomogeneity is expected to be used to express a wide variety of microstructures in an efficient and, at the same time, rational manner. This section presents a set of

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Figure 11.7.1  Analytical model used to investigate the effect of initial strain/distortion distribution on incompatibility field evolution, where two typical profiles, trapezoidal and lump shape, are considered.

preliminary simulations along this line, with two typical kinds of initial strain (distortion) distributions. Figure 11.7.1 summarizes the computational conditions, including multigrained model, orientation map, and the assumed initial strain distributions. Here two typical types are chosen, that is, trapezoidal (Case 1) and lump (Case 2) shapes, in terms of the transverse cross-sectional distribution for the central grain. Figure 11.7.2 provides an example of the simulated contours of η , K, and Θ for Case 1 at 3% and 12% elongations. We observe two sets of persisting sharp bands corresponding to the initially given strain distribution from the early stage of deformation. The cross-sectional variations of the four terms, including that for dislocation density with proceeding deformation, are displayed in Figure 11.7.3. From the profiles at u /L = 0.0, we can check that the strain-gradient evaluations in the simulation are appropriate for both the α - and η -terms (Cf. Figure 6.2.20). For example, it can be seen for the η -term that naturally evolving modulations gradually superimpose on the initial strain-induced (based) bands and eventually prevail over the region, but the two bands are not overwritten completely, as confirmed in the corresponding contour. Figure 11.7.4 shows corresponding results to Figure 11.7.3 for the distributions of the effective cell size dcell, equivalent stress σ , and strain ε , respectively. The influence of the bands due to the initial strain distribution is clearly confirmed, especially

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Figure 11.7.2  Contours of the incompatibility terms at 3% and 12% nominal strain, demonstrating sharp-banded regions corresponding to the initially introduced trapezoidal strain distribution, which are persistent even at 12% strain.

in dcell and σ . This result implies the promising applicability of the incompatibility tensor-based expression of inhomogeneity, coupled with initial strain distribution, to a rational and effective modeling of complex metallurgical microstructures, like lath martensite, considered to be accompanied by a significantly fluctuating stress field. This will be confirmed via a preliminary simulation. Figures 11.7.4 and 11.7.5 show the same results for the simulations, except for Case 2, that is, the “lump-shape” initial strain distribution. In this case, we can reproduce a slip-band-like morphology via the incompatibility tensor-based model, together with the attendant inhomogeneous stress and strain fields (Figures 11.7.6 and 11.7.7). The emphasis should again be on the ability of the present model to mimic not only the morphological features but also the attendant complex fluctuating stress fields in an easy and effective manner, which I believe is completely lacking in the models proposed along this line hitherto.

11.7.2

Application Example: Modeling Lath Martensite Structures As stated earlier, the incompatibility-based description of inhomogeneities can be extended to the modeling of metallurgical microstructures. Examples are those introduced in steels via well-controlled heat treatments, for example, pearlite, martensite,

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Figure 11.7.3  Evolution of the field-theoretical strain-gradient terms (cross-sectional distribution), clearly reflecting the initial strain/distortion distribution (trapezoidal shape).

Figure 11.7.4  Corresponding variations of effective cell size, with equivalent stress and strain to the strain-gradient terms in Figure 11.7.3, together with their contours at 12% nominal strain. Highly localized refined effective cells and attendant stress peaks are confirmed.

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Figure 11.7.5  Contours of the incompatibility terms at 3% and 12% nominal strain, demonstrating sharp-banded regions corresponding to the initially introduced trapezoidal strain distribution (lump shape).

Figure 11.7.6  Evolutions of the field-theoretical strain-gradient terms (cross-sectional distribution), clearly reflecting the initial strain/distortion distribution (lump shape).

and bainite structures. Let us look at an example of the model for lath martensitic structures typically found in high Cr steels, which is used for creep analysis. (This is a part of the ongoing research project discussed in Chapter 9.) Note that simulating

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Figure 11.7.7  Corresponding variations of the effective cell size, with equivalent stress and strain to the strain-gradient terms in Figure 11.7.6, together with their contours at 12% nominal strain.

the formation processes themselves, such as those based on phase-field models, is another story, which is not discussed here. Interested readers can refer, for example, to Khachaturian (1983). The martensite laths are produced during the quenching process from the austenitic state (referred to as  phase: FCC). For the rapidly transformed phases (into BCT) with locally intruded shear deformation, geometrical accommodations are required. Hence, a huge amount of dislocations are introduced at the interfaces (see Figure 1.2.55 for an eloquent TEM). Normally, since impurity atoms and/or small precipitates pin those highly dense dislocations, preventing them from escape or annihilation, the developed microstructure becomes extremely hard in terms of plasticity. At the same time, for energetic accommodations, misorientations satisfying, for example, K–S variant relations, are introduced. One of the typical micrographs is shown in Figure 11.7.8 (EBSD-based image). Normally, the structure has a three-scale hierarchy, composed of lath, block (aggregates of laths with same variants), and packet structures (aggregates of lath blocks with different variants), while aggregates of the packets are embedded in prior austenite grains. For mimicking the above lath martensite block structures, an initial strain distribution, assuming the intruded shear deformation, is introduced, as depicted in the inset of Figure 11.7.9. The model is pulled up to 10% nominal strain so that the incompatibility-tensor field can evolve to accommodate the artificially introduced

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Figure 11.7.8  Typical EBSD-based micrograph for a lath martensite structure in high Cr steel, together with misorientation distribution (courtesy of JAEA).

strain. The resultant structure is displayed also in Figure 11.7.9, demonstrating the attendant stress fluctuation and misorientation. The close-up in the figure demonstrates the misoriented lattice with a kink band-like morphology. These artificially developed distributions are employed afresh as the initial condition, by using their “fluctuation” parts (i.e., the averaged values are subtracted) for the creep analyses. A more advanced version of the modeling lath block structure is found in Figure 11.7.10, demonstrating that appropriate initial strain (distortion) distribution gives rise to more realistic lath block walls as twisted boundaries. Figure 11.7.11 shows an example of the simulated diagram for creep strain rate versus simulation time steps for a lath block model (shown in the inset), comparing conditions with and without the initial stress fluctuations. For comparison, a condition with artificially enhanced initial stress fluctuation (amplitude three times larger than in the original one) is also considered. The increasing amplitude of the initial stress fluctuation is demonstrated to lower the creep rate, meaning larger creep strength. A quantitative comparison of the trend is made in the histogram on the right, looking at creep strains at 1,000 simulation time steps on the creep rate–time steps diagram. Note that, including the effect of the initial misorientation, the contributions to the

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Figure 11.7.9  Modeling of a lath martensite block structure based on a field theory-based incompatibility-tensor field model combined with initial strain (distortion) distribution. The lath block structure with fluctuating internal stress and misorientation mimicking the K–Svariant relation is easily constructed.

creep behavior are strongly orientation dependent, which needs further investigation together with experimental verification. This lath block modeling can be directly extended to those of lath packets. Figure 11.7.12 provides an example of the same results as Figure 11.7.11, but for a lath packet model. Taking account of the initial fluctuating fields is demonstrated to delay the creep deformation, meaning enhancement of strength. The model presented here is still at a primitive stage. More advanced (updated) models, yielding properties as satisfactory as the lath martensite block model, will be presented in future publications from our group.

11.7.3

Future Scope: Modeling Deformation Twinning Simply by introducing twinning degrees of freedom into the projection directions of the incompatibility-tensor field in evaluating the η -term, we expect to be able to model the deformation twinning in a similar manner to the slip deformation discussed so far, that is,



( ) ( ) twin  sitwin ( ) m twin ij , j

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Figure 11.7.10  Attempts toward appropriate misorientation developments that satisfy the K–S variant as well as the screw-dislocation network as the underlying structure of the lath boundaries.

Figure 11.7.11  Example of the simulated creep response (creep rate versus time steps) of a lath block model, comparing with and without initial stress fluctuation.

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Figure 11.7.12  Example of simulated creep response (creep rate versus time steps) of a lath packet model, comparing conditions with and without initial stress fluctuation and misorientation. The inclusion of both factors is demonstrated to enhance creep strength (yielding smallest creep rate) of the model structure. (α ) where m twin , sitwin (α ) denote the twinning plane normal and twinning direction, j respectively. A candidate form of the explicit equation of the twin evolution is ( ) twin  ( ) prev twin  Q  twin ,

with

twin Q

  sgn



( ) twin

 aF

( ) twin



1



( ) F twin twin Fsat



.

Here, the additional parameter α is introduced to control the reference if necessary. The contribution of twin-induced strain is assumed to be operative when a critical condition such as  twin   crtwin is met. Here, τ crtwin represents the critical shear stress for the onset of twinning. Figure 11.7.13 summarizes the twinning model for the case of a = 0, together with the decomposition basis for the deformation-gradient tensor with that for an intermediate configuration F twin , through which the contribution of twinning is taken into account. An advantage of this model would be that the accommodations by twinning can be automatically made whenever and wherever it is necessary. We will need, however, to set the criteria for the outset of the activities in order to reproduce the orientation or texture-dependent experimental stress–strain responses.

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Figure 11.7.13  Field theory of multiscale plasticity-based twinning model, where the incompatibility tensor projected onto the twinning plane and direction is used to express alternative degrees of freedom to that of slip.

Figure 11.7.14  Example of simulation results, using the FTMP-based twinning model, where the incompatibility tensor projected onto the twinning planes and directions is used to express alternative degrees of freedom to that of slip (Kelley and Hosford, 1968a, 1968b).

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Figure 11.7.15  A tentative scheme for modifying lattice rotation via the pure rotation part of the incompatibility tensor.

Figure 11.7.16  Contour diagrams of the incompatibility term for twinning F(η twin ), rotation

angle, and strain on a {101̄2} twinning plane, comparing twinning models with and without modified lattice rotation.

Figure 11.7.14 shows an example of simulation results that reproduce the experimental results of Kelley and Hosford (1968) quite nicely. However, there is a deficiency in the current model – large as well as abrupt lattice rotations brought about by twinning are not reproduced appropriately. Although the evolving twinned regions

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Scale B: Intragranular Inhomogeneity

can induce orientation change, this is not enough for a twinning model. To solve this, we tentatively introduce a rotation modification scheme by making use of a physical meaning of the curvature tensor, which causes rotation of vectors. The scheme is overviewed in Figure 11.7.15, where the pure rotation part of the incompatibility tensor (see Figure 6.3.2), is used to modify the lattice rotation. Figure 11.7.16 compares the result with and without the modification scheme, showing the contours of the incompatibility term for twinning F tw , twinning strain, and rotation angle. In particular, the development of the rotation angle is drastically improved, that is, from 3.12 to 36.4 degrees at maximum, where the modified result well reproduces the realistic lattice rotation. In addition to this, the modified scheme alters all the field evolutions from “emerging” type to “propagating” type. Comparisons of the evolved rotation contours with experiment (EBSD maps) (Chapuis and Driver, 2011) are displayed in Figure 11.7.17, for Ori.A and Ori.E respectively, showing good agreement in the respective morphologies. As a tentative conclusion, it is demonstrated that the present modification scheme for the rotation significantly improves not only the lattice rotation itself but also its evolving mode and, furthermore, the evolved morphologies.

Figure 11.7.17  Comparison of the morphology of the evolved rotation field between an experiment and a simulation with the modified rotation scheme as in Figure 11.7.15 (Chapuis and Driver, 2011). Adapted with permission of the publisher.

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11.8  Concluding Remarks

11.8

597

Concluding Remarks In concluding the chapter, let us clarify afresh the capability of the present FTMPbased η -model, as distinct from any other models, including those accommodated with GN dislocation. For this purpose, we take one of the most fundamental and simplest targets, that is, a tension in a single-slip orientation, for example [123]. As schematically depicted in Figure 11.8.1(a), a specimen thus pulled is supposed to yield slip lines that prevail, with associated “inevitable” lattice rotations. When this situation is simulated by the conventional CP-FEM, the result will probably be the one shown in (c), where “conventional” here means those cases without the present η -model. Even when the elaborate set of constitutive equations and models implemented into FE codes with ultrahigh-resolution 3D mesh is used, this will probably achieve the same or a similar result. Our simulation output, on the other hand, is shown in (b), and reproduces the slip lines associated with lattice rotations quite nicely. Figure 11.8.2 uses the same exercise to compare three cases, including a case with the dislocation-density term F(α ), that is, commensurate with GN-related degrees of freedom. Here, contours of: (a) incompatibility, (b) dislocation-density norm, (c) strain-energy fluctuation, and (d) the corresponding histogram showing the associated slip activities (bottom), measured by slip system-wise summation of slip rate, are compared. As suspected, basically the previous situation is not improved at all, even with F(α ). The major reason for this incompetence in the two cases is attributed to the lack of the sufficient accommodation ability against the imparted excessive deformation. As

Figure 11.8.1  Tension in a single-slip orientation [123], comparing: (a) schematic illustrations including a virtual state without central axis constraint; (b) simulated contours in progress with F(η ); and (c) those without gradient terms, that is, based on conventional crystal plasticity.

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Figure 11.8.2  Comparison among simulated results for single-slip tension with and without incompatibility term F(η ), with respect to: (a) incompatibility; (b) dislocation density (norm); and (c) strain-energy fluctuation, together with histograms for slip activity.

virtually illustrated in the middle of Figure 11.8.1(a), the single-slip specimen under tension would deform in the off-axial direction when without the axial constraint. To cope with such restriction, lattice rotations have to take place appropriately, or else the secondary slip system should operate to a large extent. This is the case for the two cases without the η -model, exhibiting the degraded single-slip condition confirmed in Figure 11.8.2(d), manifested as the intersecting slip-line-like patterns extending from both side edges of the samples that emerge in the contours (a) and (b) in Figure 11.8.2. The case with the η -model, on the other hand, has the ability for such axial constraint to be accommodated by producing “modulations,” like the cases we have discussed so far in this chapter, which can effectively “absorb” the associated excessive strain energy, as clearly confirmed in Figure 11.5.3. This is also a tangible manifestation of the “dual” relationship between η KK and δ U e (Hasebe et al., 2014). In the present case, the “modulations” are interpreted as “slip lines,” while in the previous cases discussed in this chapter, such modulations were interpreted as dislocation structures. Conversely, this means that the accommodation ability of the current η -model is not limited to the reproduction of dislocation-related deformation structures, but bears a sort of universality in describing inhomogeneities in general, that is,

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Figure 11.8.3  Comparison between simple shear experiments and simulation results for two

γ -oriented BCC Fe samples (Hasebe and Sugiyama, 2014).

as far as appropriate degrees of freedom are prescribed, the model can work accordingly to them, resulting in reproductions of the corresponding desired inhomogeneous structures. The next example is shown in Figures 11.8.3–11.8.6. Figures 11.8.3 and 11.8.6 are reproduced from Hasebe and Sugiyama (2014) and rearranged for the present context. Figure 11.8.3(a) and (a’) summarize the targeted experimental results, that is, TEMs of two contrasted dislocation substructures observed for two representative γ -orientations of BCC Fe, that is, {111} and {111}, and the attendant stress–strain curves under simple shear-reversed shear loading, respectively.

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Figure 11.8.4  Variation of simulated misorientation with strain for {111} and {111} during simple shear loading in Figure 11.8.3(b’).

In particular, the vertically evolved dislocation substructure in {111} exhibits work-hardening stagnation during the load reversal, manifested as a “stress plateau,” which is absent in the case of {111}, yielding horizontally evolved substructural morphology. The corresponding simulation results, α , utilizing the FTMPbased η -model, are displayed in Figure 11.8.3(b) and (b’), respectively, showing satisfactory agreements both for the evolved morphologies of the dislocation substructures (b) and the stress–strain responses (b’), including the stress plateau, that is, B’, B, and C, on them.

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Figure 11.8.5  Variation of duality coefficient κ with shear strain evaluated from simulations (a)

corresponding to Figure 11.8.4(b), together with correlation of simulated misorientation with κ , comparing {111} and {111} (b).

Figure 11.8.6  Simulated dislocation-density contours and stress–strain curves corresponding to Figure 11.8.3 (b) and (b’), based on conventional CP-FEM.

Furthermore, the reason for the mutually distinct behaviors, together with the plateau-like stress response in {111}, is visually explained, based on the simulation results, via the corresponding duality diagrams in Figure 11.8.3(c’), together with a supplementary series of strain energy-fluctuation contours in (c). The vertically evolved morphology of the dislocation substructure for {111} tends to maintain excessively stored strain energy within the modulated structure even during the load reversal, A to B’, and the thus-stored strain energy is eventually released rather abruptly, B to C, as delineated by broken lines in Figure 11.8.3(c) and (c’). In sharp

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contrast, the energy storage-release process tends to take place quite smoothly in the case of {111}, yielding relatively large recovery in η KK on the duality diagram, that is, A through B followed by smooth transition to C in Figure 11.8.3(c’). This simply stems from the horizontal morphology of the evolved structure, which allows inflow/outflow of the stored strain energy through the traction-free surfaces on both sides during the loading reversal. Figure 11.8.4(a) shows cross-sectional misorientation distributions as a function of progressing shear strain, comparing {111} and {111} in Figure 11.8.3(b) during forward loading. Here, those perpendicular to the banded structures are presented as representatives, that is, along the horizontal and vertical cross-sections, respectively. The result demonstrates similar marked misorientation developments for the two cases. One can confirm, however, that {111} exhibits relatively smaller misorientation growth, up to the order of few degrees at maximum, than {111}. This trend, including the absolute values of the developed misorientations, is commensurate with experimental quantitative observations (M. Sugiyama, 2014, personal communication). The averaged variations for both the cross-sectional distributions during the whole processes of deformation, that is, both during loading and the following reversed loading, are compared in Figure 11.8.4(b). There, {111} yields larger misorientation than {111}, accompanied by a rather pronounced temporal drop immediately after the shear reversal. The misorientation development along with that of the modulated substructures plays a key role in the energy storage-release processes just discussed, ultimately controlling the Bauschinger behavior. To show this explicitly, we evaluate the duality coefficient κ from simulated duality diagrams (Figure 11.8.3 (c’)). Figure 11.8.5(a) compares the variation of κ with progressing shear strain via load reversal, demonstrating a similar trend to that of the averaged misorientation in Figure 11.8.4(b), whose correlation with κ is also compared in Figure 11.8.5(b). One can see a direct correspondence between the increasing κ and the growing misorientation, including distinction between {111} and {111}. Since the duality coefficient κ measures the ratio of the excessively stored elastic strain energy and the attendant local plastic flow, manifested as the incompatibility tensor, as the definition stands one can clearly confirm an organic interrelationship among the developing misorientation, the energy storage-release properties, and the resultant Bauschinger behaviors. In this case, the misorientations increase in proportion to κ until   1.6 (indicated by a down arrow), after which that for {111} branches out and begins to stagnate at around 0.4 degrees, while that for {111} continues to grow. The stagnation in this context means the conversion of the strain energy to the incompatibility field evolution, resulting in the Bauschinger behavior manifested as a “stress plateau” for {111}. To make doubly sure of the exceptional capability of the present FTMP-based η -model, Figure 11.8.6 displays the corresponding α to the above two cases obtained “without” using the η -model and the α -model, that is, via a conventional CP-FEM. As we suspected, basically no modulation emerges in the dislocation-density contours,

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resulting in no specific Bauschinger behavior, that is, the stress-strain responses for the two γ -orientations almost coincide. This means that all of the previous discussions are useless for the conventional CP-FEM. Conversely speaking, this result eloquently and strongly supports the interpretation associated with Figure 11.8.3(c) and (c’), together with Figures 11.8.4 and 11.8.5, that is, the evolved misoriented modulations, including their directionality, are the key to control the experimentally observed rather complex Bauschinger behavior. Emphasis should again be placed on the fact that the reproduction of such modulated substructures, accompanied by misorientation across them, is crucial, otherwise the complex macroscopic stress responses will never be captured.

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12 Scale C Modeling and Simulations for Polycrystalline Aggregate

12.1

Introduction: Polycrystalline Plasticity Until quite recently, discussions on “polycrystals” have been quite concentrated on or confined to how to realistically evaluate the averaged (macroscopic) stress–strain response, focusing on, for example, relaxed constraints, even with FEM simulations. While the deficiencies of coarse-grained samples, sometimes referred to as “oligo-crystals,” containing much smaller numbers of grains than reality, has, however, been repeatedly pointed out in simulating polycrystal behavior based on FEM, few have established what the “deficiencies” really are. Using these accounts, some potentially intriguing features that have not been pursued before were presented in Chapter 4. I have proposed a new perspective envisaging that the fluctuations of the grainwise stress and strain fields can be an essential feature of polycrystalline plasticity (Hasebe, 2004b, 2006). This chapter discusses this and other new perspectives (often the latest achievements), as related to Scale C and the attendant theory and modeling, for polycrystalline materials, including nanocrystals, based on field theory. Emphasis here is placed on the collective effects brought about by a large number of composing grains on the meso- and macroscopic deformation behavior of polycrystals, as presented in Figure 12.1.1 in the context of the hierarchy of polycrystalline plasticity. For this purpose, a series of systematically designed FE simulations have been conducted. The following sections will be devoted to identifying the collective behavior of crystal grains. This has been examined and identified mainly by my group. A preliminary image of the collective effect is given in Figure 12.2.1, where “role sharing” between the supporting stress and the carrying flow (plastic deformation) is schematized. They are expected to generate the fluctuations in the stress and strain fields (i.e., “inhomogeneity” in Scale C), whose experimentally observable manifestation would be the surface “roughening” developed (grown) during plastic deformation. Local softening as a dual of local hardening, on the other hand, can be a source of damage accumulation, such as slip banding, whose embryos are macroscopically “invisible” in general. These aspects are thus of great importance in the MMMs in general.

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12.2  Anticipated Features of Polycrystalline Plasticity

605

Figure 12.1.1  Viewpoints of field-theoretical polycrystalline plasticity, considering a key feature of the “collective effect” for two kinds of aggregates, dislocations and crystal grains.

12.2

Anticipated Features of Polycrystalline Plasticity: The SSS Hypothesis Figure 12.1.1 illustrates the hierarchical structure of polycrystal plasticity, emphasizing two key scales (Scales A and C), both relating to the “collective effects” of the composing elements, that is, dislocations and crystal grains. The former (Scale A) leads to the substructure evolution and has been dealt with in Chapter 10. The latter (Scale C) has not been well documented so far and will be discussed in this chapter, probably in detail for the first time. Figure 12.2.1 shows a hypothetical schematic of the collective effects of grains, that is, the transgranular mesoscopic SSS, where the attendant stress fluctuations with long wavelengths over several or several 10 grains will probably be developed. If this takes place, we expect to have not only “enhanced hardening” but also “enhanced softening,” with the latter leading to, for example, damage accumulation as an embryo of the fracture (such as void nucleation and crack initiation). One of the manifestations of such fluctuating fields would be, as mentioned in Section 12.1, the surface roughness of the order of the wavelength of the SSS developed during plastic deformation. As will be clarified in this chapter, a counterpart of the SSS is the FCS responsible for local and global plastic flow; the dynamic relationship between the SSS and FCS will be called the “duality,” where a sort of energy conversion takes place along with its storage and dissipation. Since the characteristic of the duality basically determines how the macroscopically imposed stress or energy is to be redistributed into microscopic degrees of freedom (i.e., into Scales B and A), the role, among others, of the current scale would be as a regulator.

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Figure 12.2.1  Anticipated features of polycrystalline plasticity associated with the collective behavior of crystal grains, leading to mesoscopic structure evolutions, that is, the SSS and the FCS.

12.3

Single-Phase Model

12.3.1

Simulation Model To identify the key factors for modeling a polycrystalline aggregate, especially the “collective” effect of a large number of composing grains on the mesoscopic deformation behavior, including the SSS formations, a systematically designed series of FE simulations are conducted. Figure 12.3.1 shows the FE polycrystalline models used in the present study. The models have a common representative area, containing 23 grains with a same-orientation distribution surrounded by different numbers of grains. The total numbers of grains are 23, 77, 163, 613, 1,073, and 1,661. All the grains here are assumed to have the same hexagonal shape to eliminate additional factors, for example, size distribution and morphology, other than the “number” effect. Each grain is divided into 576 to 64 triangular elements, as listed in the inserted table in Figure 12.3.2. A two-slip-system plane-strain crystal-plasticity model is used to introduce the highest constraint to the grain deformations for the purpose of maximizing the collective effect. The constitutive equation and related models employed in the analysis are listed in Figure 12.3.3. The boundary conditions for the present analysis are illustrated in Figure 12.3.2. Both sides of the models are set to be traction free (so as to discuss the effect of surfaces, as in the “oligo-crystals”). Tension up to 30% nominal strain

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Figure 12.3.1  Systematically designed FEM polycrystal models for investigating the effect of the “number of composing grains” on the deformation behavior of polycrystalline aggregates. The central region contains 23 grains with a fixed-orientation distribution in common, referred to as the “sensing area.”

Figure 12.3.2  Mesh divisions for the polycrystal models in Figure 12.3.1, together with boundary conditions.

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Figure 12.3.3  Constitutive model used in the FE simulation on polycrystalline plasticity.

is applied with a laterally traction-free condition. Note that, to capture the essence, we do not always need realistic and detailed models. So, here we use one of the simplest models with a minimal setting, that is, a hexagonal shape as a basic granular shape, and a widely used power-law-type constitutive framework, essentially with no strain-gradient effect (or, equivalently, size effect). The GB is defined simply as the interface between two adjacent regions with different crystal orientations, with neither mutual sliding nor any chemical effects. All of these are assumed in order to exclude side effects in the models being employed as much as possible. Figure 12.3.4 shows macroscopic stress–strain curves for all the models. The number effects, similar to those discussed in Section 4.5.2, are naturally reflected in the simulated responses, that is, the polycrystal model with N = 23 yields the smallest flow stress and the stress level increases with the increasing number of composing grains until saturation is reached at around N = 431.

12.3.2

Sneak Preview of the Simulation Results In what follows, the main discussion will focus on grain-wise stress–strain responses, where averages of the stress and strain fields for each grain are taken, in order to focus on the effect of the collective behavior of the composing grains. Figure 12.3.5 schematically illustrates the obtained results, showing stress–strain curves for the individual grains (thin curves) together with the averaged response (thick curves). A scatter in the stress responses for the individual grains, which is greatly pronounced as the number of grains increases while the macroscopic stress– strain curve stays basically unchanged, can be seen. Also demonstrated (see the inset at the top) is that even with the same combination of neighboring grains in the common area, the stress response of grains in the area tend to show different trends depending

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Figure 12.3.4  Averaged stress–strain curves for the polycrystalline models, where the effect of the surface is naturally expressed.

on the number of surrounding grains and their randomness. This means that the stress response of a grain is not determined only by its immediate neighbors, but is greatly affected by grains in the distance. This implies that there exists a “remote effect” activated through the SSS. Therefore, the representative region or volume on which every mesoscale simulation is based cannot be determined as easily as one might expected and is substantially controlled by the macroscopic information. Figure 12.3.6 schematizes the relationship between “macro-” stress and “micro-” stress from this perspective. The macro-stress is thus expressed formally as

 macro (x)   micro (x)   micro (x), (12.3.1) where Ο represents the spatial average and   micro ( x) expresses the stress fluctuation. Since the fluctuation tends to vanish under spatial average   micro ( x)  0, we have



 macro (x)   micro (x) . (12.3.2)

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Figure 12.3.5  Preview of the collective effects of crystal grains on the polycrystalline behavior, characterized by “field fluctuations.”

Figure 12.3.6  Relationship between macro-stress and micro-stress in terms of field fluctuations.

This means that, as far as the macroscopic response-like overall hardening behavior (macroscopic response) is concerned, fluctuations do not always have to be considered because they are macroscopically “invisibly” being averaged out. Such a situation, however, should no longer be known as “multiscale.”

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Figure 12.3.7  Grain-wise stress–strain curves for polycrystalline models, demonstrating growing fluctuations in stress and strain fields with an increasing number of grains.

12.3.3

Simulation Results for Single-Phase Models

12.3.3.1 Growing Fluctuations in Stress and Strain Fields Figure 12.3.7 presents the obtained results where grain-wise stress–strain curves are overplotted for N = 23, 77, 163, and 613 as the representatives. Growing fluctuations both in the stress and strain fields   and  , with increasing number of grains, are clearly observed, while the macroscopic stress–strain curve stays almost unchanged. Since the morphological effects are basically eliminated in the present models, the results imply that field fluctuations are a natural consequence of the collective behavior of the composing grains. One may suspect that this is just a result of the orientation dependence of the stress–strain response to come up with many grains. But this is not the case, as is clarified in the following. Let us consider further decompositions of the stress and strain fields into deviatoric and hydrostatic components for the purpose of clarifying the essential roles of the above field fluctuations (Figures 12.3.8 and 12.3.9), that is,

       m I      I  m . (12.3.3)          v I      I  v Figure 12.3.8 has corresponding results to Figure 12.3.7 but for deviatoric stress  y versus deviatoric strain  y . We observe relatively small variation in the deviatoric

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Figure 12.3.8  Grain-wise deviatoric stress–strain curves for polycrystalline models, showing small scatter in the stress components and large variation in the strain counterpart.

component  y in contrast to the total stress σ y in Figure 12.3.7. This should suffice to demonstrate that the growing stress fluctuation with increasing number of grains observed in Figure 12.3.8 is not simply due to the variation in crystal orientations of the composing grains. The deviatoric component of strain  y , on the other hand, yields pronounced variation, especially at N = 613. Regarding the hydrostatic stress σ m and volumetric strain ε v, we need to plot them separately, for example with respect to their nominal counterparts, that is,  n  u / L and σ n, respectively, as schematically presented in Figure 12.3.9, because they are proportional to each other, resulting in overlapping straight lines. Figure 12.3.10 displays the grain-wise hydrostatic stress responses as a function of nominal strain ε n, while Figure 12.3.11 shows the corresponding variation for volumetric strain with nominal stress. In Figure 12.3.10, a growth of the fluctuation in σ m is observed with an increasing number of grains, in sharp contrast to the deviatoric counterpart. The fluctuation goes not only to positive but also to negative values, and becomes more pronounced after N = 431. Therefore, we can tentatively conclude that the growth of fluctuation in the stress field with an increasing number of grains is mainly attributed to the hydrostatic component. This may seem to be rather paradoxical because the hydrostatic component of stress is considered normally to be indifferent to the metal plasticity. Such a rather unanticipated result, as a matter of fact, plays a crucially important role in “understanding”

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Figure 12.3.9  Schematic drawing of arrangements for hydrostatic stress σ m and volumetric

strain ε v with their nominal counterparts, respectively, that is, u / L and σ n.

Figure 12.3.10  Grain-wise variations of hydrostatic stress with an increasing number of grains, clearly demonstrating significant increase of field fluctuation.

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Figure 12.3.11  Grain-wise variations of volumetric strain with an increasing number of grains, showing corresponding fluctuation to the hydrostatic stress response in Figure 12.3.10.

polycrystalline plasticity, especially in the context of multiscale perspectives, which will be discussed later in this chapter. In Figure 12.3.11, on the other hand, the volumetric strain ε v also shows growing fluctuation with increasing number of grains corresponding to the case of σ m. Since the magnitude of ε v, however, is one to two orders smaller than the total strain, the effect should be negligible. Therefore, the strain-field fluctuation due to the collection of grains is concluded to stem mainly from the deviatoric component. The corresponding contour diagrams to Figures 12.3.8 ( ) and 12.3.11 ( m) are shown in Figures 12.3.12 and 12.3.13, respectively. The black and gray grains respectively indicate those having higher values of   and σ m.

12.3.3.2 Orientation Dependency of Fluctuating Fields For corroborating these arguments, as well as the conclusion that was derived from them, another arrangement is performed on each of the components, including the grainwise lattice rotation θ , that is, the plots of σ , σ m, ε , and θ against the initial crystal orientations of every grain. Here, equivalent stress and strain are employed in the place of the deviatoric components, considering the multiaxial stress and strain states. Figure 12.3.14(a–d) shows the results for (a) σ , (b) σ m, (c) ε , and (d) θ , respectively, in terms of variations with the initial crystal orientation θ 0 with respect to the specimen

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Figure 12.3.12  Contour of deviatoric strain comparing number of composing grains, where those with higher values are accentuated.

Figure 12.3.13  Contour of hydrostatic stress comparing number of composing grains, where those with higher values are highlighted.

axis. For all the diagrams, we roughly acknowledge the periodic changes in the quantities of interest along with the crystal orientation, denoting their orientation dependencies (except σ m). The hydrostatic stress σ m, as shown in Figure 12.3.14(b), exhibits an exceptional trend to the others; it is conspicuously scattering almost without a trace of periodicity, in strong contrast to the deviatoric stress σ presented in Figure 12.3.14(a), which is clearly follows the orientation dependency. The deviatoric strain ε , as in Figure 12.3.14(c) also noticeably fluctuates, although not so much as that for σ m. The lattice rotation θ yields relatively small fluctuations, and it is basically orientation dependent.

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Figure 12.3.14(a)  Variations of deviatoric stress for all the grains with initial crystal orientation,

with respect to the tensile axis.

Figure 12.3.14(b)  Variations of hydrostatic stress for all the grains with initial crystal

­orientation, with respect to the tensile axis.

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Figure 12.3.14(c)  Variations of deviatoric strain for all the grains with initial crystal orientation,

with respect to the tensile axis.

Figure 12.3.14(d)  Variations of lattice rotation for all the grains with initial crystal orientation,

with respect to the tensile axis.

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Figure 12.3.15  Summary of collective behavior of crystal grains for single-phase polycrystalline models, where growing fluctuations in stress and strain fields, each attributing to hydrostatic and deviatoric components, are observed

12.3.3.3 Brief Summary of the Results Figure 12.3.15 concisely summarizes the results we have just obtained on growing field fluctuations in the stress and strain fields. Comparatively small and negligible fluctuation in the deviatoric stress   can be seen, whereas large fluctuation becomes noticeable in the deviatoric strain component  . On the other hand, quite large fluctuation in the hydrostatic stress σ m is demonstrated to exist. Therefore, the origin of the stress fluctuations is attributed to the hydrostatic component. The fluctuation of the volumetric strain ε v is elastically proportional to that of the hydrostatic stress; however, the magnitude is negligibly small compared with the total strain. So the major contributor to the strain fluctuation is the deviatoric component  . These situations are expressed in Figure 12.3.16, where the contributions of  m and   are highlighted as major features of polycrystalline plasticity.

12.3.3.4 Remote Effect Let us next examine the affected zone (AZ) of a composing grain in terms of the grainwise stress–strain response. To this end, the stress–strain curve of the grain located in the center of the N = 1,073 model is monitored, while gradually extending the size of the surrounding region with a fixed combination of orientation distribution otherwise randomly changed. If the AZ is smaller than the fixed region size, N fixed, the central grain’s response is not altered by the random orientation changes on the outside.

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Figure 12.3.16  Summary of field fluctuations for stress and strain, where dominant contributions from hydrostatic and deviatoric components, respectively, are emphasized.

Figure 12.3.17  The remote effect in polycrystalline aggregate. The stress–strain response of the central grain with a systematically increasing number of surrounding grains is examined to specify the AZ.

Figure 12.3.17 shows the variation in the stress–strain response of the central grain in a fixed-orientation sensing area. What is shown there is that the stress–strain curve is influenced by the randomness in the orientation distributions of grains outside the fixed region. The scatter in the stress–strain response due to the randomness tends to converge as the fixed region size increases, but it still remains even at N fixed = 613. https://doi.org/10.1017/9781108874069.016 Published online by Cambridge University Press

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This result clearly demonstrates that the averaged stress–strain response of the grain is evidently influenced by the grains located outside the orientation-fixed region, verifying the existence of a finite AZ, at least in the model used. (The quantitative comparison is made in Figure 12.4.2, together with the results for DP models.) These observations clearly imply that the stress response of a grain is not only determined by its immediate neighbors but is greatly affected by grains in the distance, meaning that there exists a remote effect. This is another feature of polycrystalline plasticity absent in the single-crystal counterparts. Since all the morphological factors are eliminated in the simulation model, the remote effect is concluded to be a natural consequence of the large enough numbers of composing grains, that is, a collective effect. As will be shown in the following sections, the remote effect tends to be enhanced when additional factors like grain-size distribution and all the other morphological effects, including second phases, are introduced, just as in the realistic situations. Then, the question is “how can we specify the RVE?” Some have suggested the minimum size as the criterion with which the isotropy of the overall stress-strain response is available for checking the validity of the RVE in the case of polycrystalline samples. However, as will be demonstrated later, the same average response is available even with essentially different mesocopic inhomogeneities leading to sometimes distinct results in the context of multiscale plasticity. Therefore, while the criterion does depend on the purpose of the simulation, in most cases more elements need to be taken into account. Before working out an answer to the question, let us examine the mechanism producing the remote effect more closely. Figure 12.3.18 shows a rearrangement of the results shown in Figure 12.3.17 with respect to the hydrostatic component. These rearranged results reveal that the remote effect is mainly due to the hydrostatic stress component, implying a close connection between the fluctuating stress fields discussed previously and the present remote effect. Figure 12.3.19 compares deviatoric stress as well as hydrostatic stress responses of three arbitrary grains with similar crystal orientations. As one can see, the deviatoric component shows mutually similar responses, depending on the given crystal orientation regardless of the embedded positions within the aggregate, whereas, in sharp contrast, the hydrostatic stress response yields essentially irregular behaviors regardless of the prescribed orientation. Also, the deviatoric strain, on the abscissa, exhibits relatively large variations. These trends are consistent with those observed in the previous section in the context of growing field fluctuations. From all these observations, the following can be conjectured at the present stage: (1) The increasing number of grains composing an aggregate inevitably introduces excessive constraints to the grains three-dimensionally, which require them to be flexible when accommodating their elastoplastic deformation. (2) When (1) is viewed from the grain-wise perspective, the excessive constraints must be consumed either by the deviatoric or volumetric components of the stress and strain as a whole.

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Figure 12.3.18  Variation of the hydrostatic stress response with an increasing number of surrounding grains, which is considered to be responsible for the remote effect demonstrated in Figure 12.3.16.

Figure 12.3.19  Comparison of hydrostatic stress and deviatoric responses among grains with similar crystal orientations but located in arbitrary positions within a polycrystalline model.

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(3) The hydrostatic (volumetric) component can consume the excessive constraints of the stress since it does not participate in either local or overall plasticity, thus has an additional degree of freedom. For the strain, on the other hand, the deviatoric components should be put into play because the volumetric counterparts can accommodate only a negligibly small fraction of its excessive constraints. Therefore, these two components fluctuate as a natural consequence of the increasing mutual constraints introduced by a large number of composing grains. (4) So (3) can take place rather arbitrarily, since both the hydrostatic stress and deviatoric strain components act as “additional degrees of freedom” in terms of the prescribed macroscopic elastoplastic deformation. Therefore, their distributions are determined so as to “effectively” and “efficiently” carry out the above accommodation processes. This will result in relatively well-organized and self-organizing “transgranular”-type mesoscopic structure evolutions based on energetics (to minimize or at least lower the energy of the whole system), leading to formations of, for example, an SSS and a complimentary FCS. (5) The significance of the mesoscopic structure in (4) is twofold. First, it should be (or can be) flexibly dissociated and/or reconstructed depending on the macroscopically imposed stress and strain conditions, including the boundary conditions. Second, once the meso-structures are formed, they can carry (or transfer) local pieces of information to remote places through those structures to share (or redistribute) them. A manifestation of the latter would be the remote effect discussed earlier in this section. Therefore, we can tentatively conclude that the remote effect is essentially brought about by the fluctuating hydrostatic stress field as a consequence of the collective behavior of a large number of (interacting) grains, probably activated through the SSS. The present conjecture is extensively corroborated in the following in the context of the effects of second-phase and grain-size distributions. Note that any coarsegrained models will fail to capture these important aspects of polycrystalline behavior and, furthermore, the existence of the remote effect makes it difficult to identify the “representative volume” with periodic boundary conditions.

12.3.3.5 Role Sharing or “Duality” Figure 12.3.20 displays the contour distributions of hydrostatic stress, deviatoric strain, and grain rotation for a model with N = 613, while the contour distributions are overlapped in Figure 12.3.21 for making the comparison easier. In Figure 12.3.21 only the grains having values above certain thresholds are shown for simplicity. The structure evolution, like the SSS, is not always obvious at this stage, probably because of an insufficient number of grains in the present model. Comparison of the former two distributions indicates that they do not generally coincide, but rather developed complimentarily, mediated by rotation. This may be regarded as a sort of “role sharing” between stress support and deformation transmission, which is another essential feature of polycrystalline plasticity. (Cf. the graph representation of granular assemblies in Chapter 6.) Details of this “role sharing,” including its actual “role” in determining the overall deformation behavior, will be discussed later.

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Figure 12.3.20  Contour diagrams for hydrostatic stress, deviatoric strain, and lattice rotation, where the grains with larger values are highlighted.

Figure 12.3.21  Comparison of the distributions of hydrostatic stress, deviatoric strain, and lattice rotation, where the grains with larger values are highlighted.

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From the above discussions, it is tentatively concluded that the fluctuating hydrostatic stress field  m emerges as a result of constraint caused by the large number of grains and it tends to be organized into SSS, which is promptly followed by the complimentary FCS that attributes to the fluctuating deviatoric strain field  . Since the morphological effects are basically eliminated in the models used in this section, the growing field fluctuations and the attendant mesoscopic structure developments are some of the natural consequences of granular assemblages, peculiar to polycrystalline aggregates. Introducing grain-size distributions is demonstrated to yield the same trend, as will be discussed later.

12.4

Application to DP Alloy Models The viewpoint of the “field fluctuations” presented in Section 12.3 can be a generic notion common to all polycrystals. As was emphasized in Hasebe (2004b), one of the most prominent and strongest features of the field theory, especially regarding the “field-fluctuation” viewpoint, is its broad applicability to much more complicated and realistic situations beyond the single-phase case. As illustrated in Figure 12.4.1, the immediate examples would be DP alloy models containing hard grains (phase) with various volume fractions and morphologies. The concept can be easily extended to multiphase alloys, essentially in the same manner.

Figure 12.4.1  Field fluctuation applicable to complex cases with grain-size distributions and second phases with various volume fractions. For the latter, DP alloy models are used in the following discussions.

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Figure 12.4.2  Dual-phase models used in simulations with roughly two types of hard-grain distributions, together with various volume fractions.

Introducing such “heterogeneity” is expected to significantly enhance the field fluctuations, which will be used to identify the specific features of such complex-phased polycrystalline media over those on single-phase polycrystals, already presented in previous sections. Figure 12.4.2 displays the DP models, employed in the present series of simulations, with various volume fractions of hard grains, that is, V f =15%, 25%, 35%, 50%, and 70%, with roughly two morphologies, that is, random and regular (block). The flow-stress ratio of the hard grains to the soft grains (matrix) is ranged as  HG /  SG  3, 5, 7, and 10. In what follows, however, we will confine ourselves to a discussion on  HG /  SG  10, because otherwise the tendency is to show no clear distinction regarding the effect of hard-phase (HP) morphologies, which are hidden behind unnecessary complexities.

12.4.1

Dual-Phase Models Vf ~25%

12.4.1.1 Basic Examination and  m versus  

A DP model where hard grains are introduced is also examined to see the effect of artificially introduced heterogeneities on the field fluctuations. Here a flow-stress level 10 times larger than the matrix grains (i.e.,  HG /  SG  10) is assumed and the discussion is limited to 25% volume fraction with 613 grains, as illustrated in Figure 12.4.3 (cases with other volume fractions will be discussed). As shown in the figure, two types of HP distribution are considered, that is, uniform or regular (Model 1) and nonuniform or biased (Model 2) distributions.

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Figure 12.4.3  Dual-phase models with a 25% volume fraction of hard grains. Two types of ­distributions are considered, that is, uniform distribution (Model 1) and nonuniform ­distribution (Model 2).

Figures 12.4.4 through 12.4.7 present the results obtained for Model 1, showing grain-wise stress–strain curves, deviatoric stress–deviatoric strain curves, hydrostatic stress  n , and volumetric strain  n relationships, respectively. Comparing with those for the single-phase model discussed earlier, the DP model exhibits even larger field fluctuations both in stress and strain (see Figure 12.4.4). As can be observed in the figure, the enhanced fluctuations are especially present in the soft grains in the present case. Figure 12.4.5 verifies that the fluctuation in the stress field is not due to the deviatoric component, since there is small variation even at N = 613. Fluctuation in the hydrostatic stress field  m, as presented in Figure 12.4.6, on the other hand, more than doubles in comparison with that in the single-phase model (Figure 12.3.9). Therefore, as in the case of the DP model, stress fluctuation is attributed to the hydrostatic component, that is,  m. Note that the larger  m in the present case can be intuitively understood by imagining a situation where soft grains sandwiched by hard grains are compressed hydrostatically, even under tension. For the strain field, on the other hand, we also confirm relatively large variation in deviatoric component, that is, even larger   than in the single-phase model (Figure 12.3.8). Additionally, the volumetric component of strain  v becomes nonnegligibly large in this case, as shown in Figure 12.4.7, because of the previously mentioned enhanced  m, more than doubling. This larger  v can be one of the key features of DP alloys and,

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Figure 12.4.4  Grain-wise stress–strain curves for a DP polycrystalline model (Model 1),

demonstrating pronounced fluctuations in stress and strain fields with an increasing number of grains.

Figure 12.4.5  Grain-wise deviatoric stress–deviatoric strain curves for a DP polycrystalline model (Model 1), showing small scatter in the stress components and relatively large variation in the strain counterpart.

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Figure 12.4.6  Grain-wise variations of hydrostatic stress with increasing number of grains for a DP polycrystalline model (Model 1), demonstrating noticeably enhanced field fluctuation.

Figure 12.4.7  Grain-wise variations of volumetric strain with increasing number of grains, for a DP polycrystalline model (Model 1), showing nonnegligibly large fluctuation corresponding to hydrostatic stress response in Figure 12.4.6.

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Figure 12.4.8  Grain-wise stress–strain curves for a DP polycrystalline model (Model 2), demonstrating further pronounced fluctuations in the stress and strain fields even for hard grains (encircled), with an increasing number of grains.

if so, is important in modeling DP polycrystals because it will affect, for example, void nucleation and growth behavior, ultimately controlling the ductile fracture. It is also shown with Model 2 that introducing nonuniformity in the HP distribution can greatly change the field fluctuations. Figures 12.4.8 to 12.4.11 show the same results as Figures 12.4.4 to 12.4.7, but for Model 2. Model 2 exhibits even larger fluctuations of the hydrostatic stress as well as deviatoric strain than Model 1. A number of issues should be pointed out regarding the contrast between Model 2 and Model 1. First, in the grain-wise stress–strain curves (Figure 12.4.8), the enhanced field fluctuation in Model 2 also impacts the hard grains (marked by a circle). This means not only the soft phase (SP) but also the HP tend to carry the imposed deformation when the distribution of the HP deviates from uniformity. Even in this case, the deviatoric stress does not fluctuate very much, as confirmed in Figure 12.4.9. Regarding the hydrostatic stress field (see Figure 12.4.10), the fluctuation in Model 2 is conspicuously larger than in Model 1, extending over both positive and negative values. Accordingly, the contribution of the volumetric strain also becomes noticeable, as in Figure 12.4.11.

12.4.1.2 Brief Summary of the Results Based on these examples, the features of DP polycrystalline aggregates can be encapsulated, as in Figure 12.4.12. In contrast to the case of single-phase polycrystals in

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Figure 12.4.9  Grain-wise deviatoric stress–strain curves for a DP polycrystalline model (Model 2), showing small scatter in the stress components and relatively large variation in the strain counterpart.

Figure 12.4.10  Grain-wise variations of hydrostatic stress with increasing number of grains for a DP polycrystalline model (Model 2), demonstrating conspicuously enhanced field fluctuation.

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Figure 12.4.11  Grain-wise variations of volumetric strain with increasing number of grains for a DP polycrystalline model (Model 2), showing nonnegligibly large fluctuation corresponding to the hydrostatic stress response in Figure 12.4.11.

Figure 12.4.12  Summary of field fluctuations for stress and strain in a DP polycrystalline model with a 25% HP, where an additional contribution from volumetric strain to from the dominant contributions of hydrostatic and deviatoric components, respectively, are emphasized

Figure 12.3.16, there is also a large fluctuation in the volumetric strain  v in addition to  m and  . Figures 12.4.13(a) and 12.4.13(b) summarize all the results of the field fluctuations in the single-phase and the DP models so far, in terms of the grain-wise responses in a schematic form. As is clarified, the enhanced fluctuations in  m and attendant nonnegligible  v are the key features of the DP models, at least

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Figure 12.4.13(a)  Summary of field fluctuations for stress and strain in single-phase and DP

polycrystalline models, showing schematics of grain-wise stress–strain curves.

for V f = 25%. What also needs to be emphasized is that the nonuniformity in the HP distribution can markedly enhance the field fluctuations in the DP model. As a consequence, even the hard grains must carry a certain nonnegligible fraction of the plastic flow, causing enhancement of the  m also for the hard grains.

12.4.1.3 Stress-Supporting Structures and FCS Developments and Duality Figure 12.4.14 denotes development processes of the SSS and FCS, showing a series of snapshots of contours of  m and  . The SSS emerges first at an early stage of the deformation ( u / L = 0.06) and is grown by connecting the HP regions, eventually transcending longitudinally over the sample as deformation proceeds. The FCS, on the other hand, starts developing from the surface on the left side and grows complimentarily in the SP regions following the SSS. Since the development of the FCS seems to be taking place in the maximum shear direction, unlike the SSS, which grows in parallel to the loading direction, we notice that the “duality” between the two structures cannot be explained simply by graph representation, as argued in Chapter 6. This process implies how the “duality” distribution is formed.

12.4.1.4 Remote Effect via Vf ~25% DP Model

Figure 12.4.15 shows a set-up for measuring the AZ for a DP model with V f = 25% hard grains. Regions with fixed distributions of crystallographic orientation and HP, surrounded by randomly oriented grains, are systematically expanded, as schematically illustrated in the figure (shaded areas), while the total number of grains is fixed

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Figure 12.4.13(b)  Summary of field fluctuations for stress and strain in single-phase and DP

polycrystalline models, showing schematics of grain-wise stress–strain curves, together with corresponding deviatoric stress–strain and hydrostatic stress–volumetric strain plots. Enhanced fluctuations both in the deviatoric and volumetric components of strain components, as well as the hydrostatic stress responses in DP models (especially in Model 2), are emphasized by double-headed arrows.

to be 1,073. A further two cases are considered with this setting, that is, one with variation in randomness of orientation distribution only with fixed HP (Case 1) and the other with variations in both orientation and hard-grain distributions (Case 2). Responses of the grain in the center are plotted to examine the effect of the surrounding information. As shown in Figure 12.4.16 for Case 1, variations in the central-grain response to the randomness of the surrounding region are clearly observed and the variation tends to converge as the size of the central fixed region increases. Similarly to the single-phase model shown in Figure 12.3.17, the stress–strain response of a grain is again evidently affected by distant grains, not simply determined by neighboring grains, as for a DP model. The variations are evidently larger than those in the single-phase model; they do not converge even at N fixed = 613. Figure 12.4.17 shows the results for Case 2. No large difference can be seen between the two cases. This means that HP distribution is mainly responsible for the remote effect in the DP models. Note that the two effects, that is, HP and crystal-orientation distributions, become comparable, as a matter of course, when the strength ratio σ HG / σ SG is relatively small, say,  HG /  SG  5.

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Figure 12.4.14  Contour diagrams for hydrostatic and deviatoric strain for a 25% DP model, showing the development of an SSS and a complimentary FCS, which form a “duality” distribution.

Figure 12.4.15  Dual-phase model (25% HP) for identifying an AZ with increasing fixed HP and an orientation region, otherwise hard-grain distribution (Case 1) or both hard-grain and orientation distributions (Case 2) are randomly changed.

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Figure 12.4.16  Variation of stress–strain response for central grain (Case 1) with randomness in hard-grain distribution outside the fixed region, where the fixed region size is systematically increased from 23 to 613 grains.

Figure 12.4.17  Variation of the stress–strain response for central grains (Case 2), with randomness both in hard-grain and orientation distributions outside the fixed region, where the fixed region size is systematically increased from 23 to 613 grains.

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Figure 12.4.18  Variation of AZs with fixed region sizes comparing Cases 1 and 2, where the former is measured by averaged SD of the varying flow stress over the whole strain region, while the latter is represented by the number of grains contained in the fixed region. The result for the single-phase polycrystalline model is also plotted.

For more quantitative discussion, the standard deviation (SD) for the stress variations in Figures 12.4.16 and 12.4.17 are plotted against the fixed region size in Figure 12.4.18. Here, the result for the single-phase model is also plotted for comparison. What can be confirmed first is not only the significantly larger fluctuation (SD) for DP models but also much later saturation, meaning a larger AZ. The two cases, that is, Cases 1 and 2, are commensurate with each other. The single-phase model yields the AZ saturated at around N fixed = 300, whereas for the DP models the saturation is reached after N fixed = 500, with larger SD of the flow stress. The value of N fixed at saturation can be roughly regarded as the “AZ,” within which the remote effect is operative. This clearly demonstrates the enhancement of the remote effect via the introduction of second phases. Furthermore, as one can readily imagine considering the discussion on the HP distribution in Section 12.4.1.1, the AZ can be further enhanced when a nonuniformity or “biased” distribution of the HP is introduced, as is likely to take place in practical situations. As will be discussed extensively in the following, this remote effect due to “field fluctuations” strongly depends on the volume fraction and morphology of the HP. For example, the Block-type regular distributions, and the negative of the block pattern (Block-negative, hereafter) model, to be presented in the context of the Bauschinger effect in Appendix A12, yields one of the maximum fluctuations and the effect even on the macroscopic (thus averaged) SS responses. This resembles the dislocation-cell structure discussed in Chapter 3, where the effective and efficient ways of a SSS are naturally realized and, at the same time, provide a source of back stress. Figure 12.4.19 shows the same set of results as in Figure 12.3.19 but for the DP model with V f = 25%, comparing hydrostatic and deviatoric stress responses for three arbitrary grains with similar crystal orientations. While the responses are similar, the

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Figure 12.4.19  Comparison of hydrostatic stress and deviatoric responses among grains with similar crystal orientations but located in arbitrary positions within the DP polycrystalline model (with 25% HP).

hydrostatic stress varies irregularly in a more pronounced manner, not depending on the given crystal orientation, while the deviatoric stress yields a mutually similar and monotonic response depending on the crystal orientation. Therefore, as has been discussed already for the single-phase model, together with the visualized “duality” and its developing process in Section 12.4.1.3, hydrostatic stress fluctuation also plays the key role in controlling the overall collective effects in polycrystalline plasticity for DP models.

12.4.2

Dual-Phase Models Vf ~15%

12.4.2.1 Role of Duality via Vf ~15% DP Model Figure 12.4.20 shows DP models with 15% hard-grain volume fraction. Four types of hard-grain distributions with regular arrangements, having different degrees of clustering, are prepared in order to examine their effects on field fluctuations and its attendant duality, as well as the remote effect. They are roughly categorized into two types, that is, one is isolated (referential) and the other is clustered distributions. The latter is further organized into those with two-, three-, and four-grain clusters, respectively. By using these models, together with the same crystal plasticity-based constitutive equation as in the previous section, tension up to 30% nominal strain is applied, under a transversely traction-free boundary condition (to examine the role of the “duality” on the evolution of surface roughness).

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Figure 12.4.20  Comparison of duality distribution (a) for DP polycrystalline models with 15% HP, together with macroscopic stress–strain responses (b) and corresponding surface profiles (c).

The inset in the bottom left shows the overall (macroscopic) stress–strain curves comparing the four models. As is confirmed in Figure 12.4.20(b), the difference in the hard-grain distribution (degree of clustering) has almost no impact on the macroscopic response, meaning the effect of the hard-grain distributions are macroscopically “invisible.” The corresponding grain-wise responses are given in Figure 12.4.21, where the curves for the hard grains are indicted by black, while those for the soft grains are in gray. In the figure, subtle differences can be observed, especially in the hard-grain response. The reference (isolated) model, for instance, shows smaller   for the hard grains than the others. Major differences are found in the “duality” distribution, as depicted in Figure 12.4.20(a), where the grains with relatively larger  m and   are highlighted, respectively. One can observe distinct trends in the duality distributions depending on the degree of clustering. The reference (isolated) model yields longitudinally interconnected   relatively uniformly spreading over the sample with comparatively smaller  m. As the degree of clustering increases,   distribution tends to be localized (disconnected), while  m starts extending so as to wrap around the clustered hard grains. The four-grain-clustered model evidently exhibits localized duality distribution in this sense, leading further to quite localized deformations at several places in the sample. In other words, the “inhomogeneity” is enhanced.

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Figure 12.4.21  Grain-wise stress–strain curves for DP polycrystalline models with a 15% HP, comparing different degrees of clustering in the hard-grain distribution.

This “spread-to-localized” transition of the duality distribution with an increasing degree of clustering manifests itself as a change in the surface-roughening behavior, as depicted in the inset of Figure 12.4.20(c), where the surface profiles are compared. The isolated model results in the smoothest side surface even after 30% tension, while the clustered models, for example, that with four grains, show undulating surface profiles approximately corresponding to the wavelength of the duality distribution that is also clustered. These results eloquently teach us about the role of the “duality,” that it is a “driving force” of the mesoscopic (meaning the grain-aggregate scale order) inhomogeneity or flow evolutions. Quantitative discussions will be given in Appendix 12 based on the FTMP, while the most recent quantitative evaluation of such inhomogeneity and its effect on the surface roughness evolution will be mentioned in Chapter 15.

12.4.2.2 Duality versus Remote Effect Let us also discuss the relationship between “duality” and the remote effect by taking the same DP model as the previous subsection, that is, that with V f =15% as a tangible example. Figure 12.4.22 shows two DP models out of the four in Figure 12.4.20, with slightly different HP distributions from each other, that is, one with isolated and the other with duplex-clustered hard grains. By utilizing the same scheme as used in Section 12.4.1.4, the AZs for the two models are compared. Figure 12.4.22 displays the results. Here, the stress–strain curve of the central grain is monitored while the orientation-fixed region is extended step by step, that is, (a) N fixed = 77, (b) 163, and (c) 432 grains as examples.

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Figure 12.4.22  Comparison of the AZ for DP models between two distributions of 15% hard grains, that is, isolated and duplex clustered, where the stress–strain curve of the central grain is monitored.

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Figure 12.4.22  (cont.)

One can confirm a convergence of the stress–strain response at N fixed = 432 for the duplex-clustered model, whereas there a scatter for the isolated hard-grain model is still observed. This indicates that the isolated model has a larger AZ, yielding a stronger remote effect than the duplex-clustered model. Figure 12.4.23 shows a summary of the results of the relationship between “duality” distribution and the “AZ.” Dispersed duality distribution (like that in the isolated hard-grain model) yields a larger AZ (i.e., a larger remote effect), whereas the clustered distribution (like that in the four-grain-clustered model) tends to have a smaller (i.e., isolated) AZ. Therefore, we can tentatively conclude that “duality” can have a critical role in controlling the evolution of the associated “inhomogeneity.”

12.4.2.3 Brief Summary Figure 12.4.24 schematically illustrates a summary of the findings for the DP models with V f =15% HP, in terms of “dispersed” versus “localized,” with regard to the relationship between the duality distribution (the dual distributions of hydrostatic stress and deviatoric [equivalent] strain) and some material properties, that is, SFE, grain size, and elastoplastic anisotropy. Materials with relatively small SFE such as Cu and austenitic stainless steels, small grain sizes, or large elastic plastic anisotropy are expected to yield dispersed

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Figure 12.4.23  Summary of simulation results on DP models with a 15% HP, showing the relationship between “duality” distribution and remote effect. The dispersed duality distribution tends to have a larger AZ, and vice versa.

Figure 12.4.24  Schematics of the collective effects of the crystal grains for a DP model with relatively smaller volume fraction of HP. Smaller SFE as well as finer grain size, for example, are expected to yield dispersed duality, leading to globally uniform deformation with smaller surface roughness.

duality distribution (see the left-hand side of Figure 12.4.24), similar to the uniformly ­distributed HP model. On the other hand, for materials with large SFE such as Al, large grain sizes, or smaller anisotropy in elastoplasticity we probably have rather localized duality distributions (see right-hand side of Figure 12.4.24), similar to the DP model with clustered HP distributions. For the localized duality case, the difference in the strength (or the resistance against plasticity) among the regions in a medium tends to cause additional local flows, as schematically indicated in the right-hand side of Figure 12.4.24. This is roughly the mechanism for the evolution of inhomogeneity in this scale.

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Comparison of the two typical cases clearly shows that, even with the same hard-grain volume fraction, we can have essentially different duality distributions depending on the hard-grain morphology. The dispersed duality clustering of the hard grains results in clustered duality distributions of both hydrostatic stress and deviatoric strain, and it resultantly produces, for example, rougher surface morphology as a consequence of the attendant localized (mesoscopic) flow being induced. Interestingly, the corresponding overall (averaged) stress–strain curves almost coincide regardless of these mesoscopic differences, which tends to mislead us to conclude that the response of DP models is not basically influenced by the differences in the hard-grain distribution or morphology. This perspective, as has been pointed out previously, is no longer worth treating as a “multiscale” problem anymore. Note that, as will be further evidenced later, the above conclusion can be altered when the volume fraction of the HP increases, that is, the morphological factors of the HP can significantly change both the mesoscopic inhomogeneity evolutions and the macroscopic stress response. The latter might also be misleading in the sense that the “duality”-based standpoint can be overlooked.

12.4.3

Vf ~50% DP Model Let us next examine the case with relatively larger volume fraction of the HP, that is, (V f )HP = 50%, where the macroscopic stress–strain response can be affected by the HP morphology, in contrast to the previous case (15% HP volume fraction, exhibiting negligible effect on the stress–strain curves). The purpose, again, is to identify the role of “duality” in such cases.

12.4.3.1 Duality versus Macro-Response Figure 12.4.25 shows two DP models, both with a 50% hard-grain volume fraction and typical HP patterns, that is, random and regular (lattice-like) distributions of the hard grains, to be referred to as Random and Block models, respectively. For such a case with large volume fraction, the macroscopic stress–strain response is greatly affected by the HP distribution, in contrast to that for the V f =15% HP model (Figure 12.4.20(b)). The regular distribution (Block model) exhibits larger flow-stress level than the Random model. Grain-wise responses of the deviatoric (Mises equivalent) stress σ and hydrostatic stresses σ m are respectively plotted against the averaged deviatoric (Mises equivalent) strain ε in Figure 12.4.26, comparing the two models. There is a relatively small difference (variation) in the deviatoric component of stress σ , whereas marked difference is observed in the hydrostatic component, as is always the case. Larger  m belongs to the SP in the Block model, while the growth of ε is restricted, resulting in higher flow-stress response. In sharp contrast, the larger ε is concentrated in the SP in the Random model, which evidently contributes to the smaller flow-stress response. The HP tends to support larger  m in this case.

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Figure 12.4.25  Comparison of grain-wise stress–strain curves between the Block and Random models with V f = 50%, where those for hard grains and soft grains are separately indicated.

Figure 12.4.26  Comparison of grain-wise stress response between the Block and Random models

with V f = 50%, for deviatoric and hydrostatic components, respectively. The former shows mutually similar responses with small difference, whereas the latter yields marked distinction.

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Figure 12.4.27  Relationship between macro-response and duality in field fluctuations. The Block model tends to hold (store) large hydrostatic stress in the soft-grain matrix, whereas the Random model is apt to release the stress without storing, leading to the distinction between the macro-stress responses.

Figure 12.4.27 again contrasts the two cases for σ m, emphasizing a distinguished feature of the SP behaviors (in terms of hydrostatic stress response), together with the averaged (macro-)stress–strain curves. It may be interesting to imagine a process where the Block-type morphology of the HP is continuously transformed to that for the Random model. Thereby the stored  m (as a form of strain energy, indicated by a two-directional arrow) in the SP will be successively released into local plastic flow measured by ε (expressed by a right arrow). This is what exactly the “duality” concept, which is exclusively discussed in Section 12.6, asserts.

12.4.3.2 Summary and Comments As seen in this section, the manifestation of the “duality” is sensitively dependent on the volume fraction of the HP for DP polycrystalline aggregates. Figure 12.4.28 displays a series of simulation results on 0.2% proof stress σ 0.2 as a function of (V f )HP, where two typical cases of regular (Block-type) and random distributions with  HG /  SG  10 are shown. As is well known, extreme cases of strain constant (Voigt model) and stress constant (Reuss model) assumptions yield the upper and lower bounds, respectively. There exists a transition from the lower bound to the

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Figure 12.4.28  Variation of overall stress level with volume fraction of HP, comparing the Block and Random models. The Block model exhibits a transition of the macro-stress from stress-constant type to strain-constant type at around (V f )HP = 45%.

upper bound at around (V f )HP = 40  50% for the Block model. Around that volume fraction range, the percolation limit is gradually reached in terms of the HP distribution, therefore the transition can take place. Note that the model with the negative HP distribution to the Block model yields greatly enhanced flow stress with completely interrupted FCS by the fully connected SSS. (This situation resembles the dislocation-cell structure detailed in Chapter 3 and was revisited in Chapter 10.) The effect of such distinct “duality” on the flow inhomogeneity (and the attendant irreversible stress response) is exemplified in Appendix A12. We should also note that this trend about the transition is only true for cases when the strength ratio between HP to SP (σ HG / σ SG) is large enough. Figure 12.4.29 summarizes an experimentally observed presentation of (V f )HP versus σ 0.2 (Ohnami, 1980; Tamura et al., 1973), together with corresponding examples of the micrographs (Ohnami, 1980; M Tokizane and T. Yamanouchi, 1978, personal communication). As is shown, the simplest mixture law holds for small enough  HG /  SG   2 and 3 , however, the experimental data tends to deviate from it at around  HG /  SG  4. The transition discussed previously does not appear until σ HG / σ SG reaches 7. The last trend resembles the simulation result in Figure 12.4.27, especially for the Block model, implying the real morphology of the HP can be best mimicked by the regular distribution rather than the random one (compare the simulation models for (V f )HP = 50 and 70% [Block] and the micrographs for (V f )HP = 55 and 80%).

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Figure 12.4.29  Summary of the experimental observations systematically carried out for DP steels with arbitrary combinations of ferrite, austenite, and martensite phases yielding various volume fractions and strength ratios (after Ohnami, 1980).

12.5

Effect of “Grain Morphology”

12.5.1

Model Description: Distributed Grain-Size Models Let us next discuss the roles of grain-size/shape distributions in polycrystalline plasticity (Figure 12.5.1). A huge number of studies have been carried out and reported on the subject by extensively utilizing FEM-based models, some with rectangle or cubic grains, some with hexagonal, others with Voronoi tessellation models (e.g., Barbe et al., 2001a, 2001b). Few such studies, however, have succeeded in clarifying or identifying the roles of grain-size/shape distributions in polycrystalline plasticity. We have already clarified basic and essential features of the polycrystalline media in plasticity, as shown earlier in this chapter, to be: (1) growing fluctuations in hydrostatic stress and deviatoric strain, (2) duality between the two fluctuating fields in conjunction with the mesoscopic structuring, that is, SSS and FCS, and (3) the remote effect activated through those mesoscopic structures. One of the advantages of this new viewpoint is its applicability to more complex or realistic situations, like those with grain-size and shape distributions or inhomogeneities due to second phases, from the unified perspective of “field fluctuation” and its “duality,” as schematically represented in Figure 12.5.2.

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Figure 12.5.1  Field fluctuation applicable to complex cases with grain-size distributions and second phases with various volume fractions. In what follows, we make a preliminary attempt against the effect of grain-size distribution, in the light of how the field fluctuations are affected by the deviation of the grain shapes from their hexagonal counterpart.

Figure 12.5.2  Schematics representing the application of the notions of “field fluctuations” and “duality” when investigating the effects of grain-size distribution, including morphological factors.

This section intends to clarify the distributions’ actual roles on macro- and mesoscale plasticity, based on a series of systematic FE simulations. The constitutive equation used and the analytical conditions are basically the same as those indicated in Section 12.3.1. The difference is only in the grain-size distributions, as detailed in what follows. Figures 12.5.3 and 12.5.4 provide a set of schematics of the models with ­distributed grain size used in this section, which is expected to additionally affect the field fluctuations, especially  m and  . Roughly, two types of grain-size distributions are assumed to simplify the discussion, that is, global and local. The former (Figure 12.5.3) is further classified into two types, that is, those with increasing or decreasing grain diameter

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Figure 12.5.3  Polycrystal models IV, IF, DV, and DF with global grain-size distributions, together with corresponding distribution maps.

Simulation Models (Local Distributions) Model A

Model B

Model C

Figure 12.5.4  Polycrystal models A, B, and C with local grain-size distributions, together with corresponding distribution maps.

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Figure 12.5.5  Finite-element mesh divisions for distributed grain-size models, where the same resolution is introduced regardless of grain size.

(Models IV and IF and Models DV and DF). The latter (Figure 12.5.4), on the other hand, are those with localized size changes (Models A to C). They are: (1) increasing grain size from central part to the periphery, IF and IV, and (2) decreasing grain size, DF and DV, where IF and DF have regular hexagonal grains in the fixed region, while IV and DV do not. Here, “I” and “D” stand for “increasing” and “decreasing,” respectively, while “V” and “F” indicate “varied” and “fixed” grain sizes for the common 23-grained region, respectively. Radial distributions of the grain size are also indicated below each model, representing the distinction among the models, where averages of the grain sizes within each of the three layers from the center are employed. Finite-element polycrystal models comprising 613 grains are employed in this series of simulations. Figure 12.5.5 illustrates the mesh division of the models, using an example of the IV type, where the number of elements in each grain is set to be the same regardless of the relative size. The boundary condition assumed in the simulation is depicted in Figure 12.5.6. For all the models, the same relative crystal-orientation distribution is allocated, together with the 23-grain sensing area embedded in the central part. The inset shows the three representative cases accentuating the sensing area, that is, reference, IV, and DV models. So far, we have studied a polycrystalline model with 613 grains with a hexagonal grain shape, not only as a rational model for examining the “collective effects” but also as this is efficient in terms of computational time. So, here we use the same model as a reference. Note that some results on a Voronoi model are given in 13.4.1.

12.5.2

Roles of Grain-Size Distribution: The Macro-Response Figure 12.5.7 redisplays all the models considered here, together with their macroscopic or overall stress–strain curves in the top-left inset, exhibiting almost coinciding responses regardless of the mutually different grain-size distributions, as we have anticipated.

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Figure 12.5.6  Boundary condition for FEM analysis on distributed grain-size models, together with sensing area, with the same crystal-orientation setting for three representative models.

Figure 12.5.7  Distributed grain-size polycrystalline models used in the analysis, that is, global (IV, IF, DV, and DF) and local (A, B, and C) distributions, together with corresponding distribution maps. Their macroscopic stress–strain responses almost coincide.

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Figure 12.5.8  Overall averaged stress–strain response for distributed grain-size models, comparing three representative models.

For now, let us focus on the three models among them, that is, the reference model, and the IV and DV models, for a detailed discussion. Their stress–strain curves are plotted afresh in Figure 12.5.8. Clearly seen there is the almost perfect coincidence again of the stress–strain responses regardless of the introduced grain-size distributions. This may tentatively lead us to conclude the overall or averaged stress–strain response on grain-size and shape distribution is independent. Actually, similar conclusions have been reported by many researchers thus far, as already pointed out in Chapter 4. Let us, in what follows, scrutinize the field fluctuation-based properties of these. The grain-wise stress–strain curves are shown in Figure 12.5.9, comparing the three models, where no noticeable difference can be found among them, except some grains yielding marked softening (e.g., in the DV model). Figure 12.5.10 compares those for the hydrostatic stress components (top) and the deviatoric strain components (bottom), in which we can find pronounced effects of the collective behavior of composing grains in the previous subsections, each plotted against the nominal counterpart. A close look at the two sets of comparisons reveals slight differences, especially in grains that yield stress–strain responses compared with the referential model. To delve into this deeper, let us next focus on the central grain, which is shown in Figure 12.5.11. We can see, in this case, that the differences are clearly visible,

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Figure 12.5.9  Grain-wise stress–strain curves for distributed grain-size models, comparing those of three representative models.

Figure 12.5.10  Corresponding grain-wise responses of the hydrostatic components of stress (top) and the deviatoric components of strain (bottom) to Figure 12.5.9, plotted, respectively, against the nominal counterparts.

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Figure 12.5.11  Grain-wise hydrostatic stress responses as a function of nominal strain for the central grain in the sensing area of distributed grain-size models, comparing three representative models.

especially for the DV model, in contrast to the reference and IV models, whereas those for the total stress–strain curves are still negligibly small. From this, we confirm in this case that the grain-size distribution can also affect the stress–strain-field fluctuations, especially in the hydrostatic stress and deviatoric strain components. For making doubly sure, we reexamine this case from a different perspective in the next subsection, that is, the roles of duality between the fluctuating fields.

12.5.3

The Roles of Duality Let us examine further the effect of the grain-size distribution on the duality by looking at hydrostatic stress and deviatoric strain redistributions in representative grains. Look at all the models with distributed grain size displayed in Figure 12.5.7, together with the overall (averaged) stress–strain curves, comparing those obtained for all the models. We confirm that even the localized distribution models (Models A to C) coincide with the others, regardless of the variation in the grain-size distribution introduced here, meaning all of them are macroscopically “equivalent.” The global

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Figure 12.5.12  Comparison of hydrostatic and deviatoric stress responses for arbitrary grains among nine models.

size-distributed models exhibit relatively large deviations of the stress and strain distributions from those for the constant size model, whereas the localized distribution models yield small deviation. Details of the interrelationship between the fluctuations in hydrostatic stress and deviatoric strain are extensively discussed in what follows. One of the typical results is displayed in Figure 12.5.12, where the stress responses of grains located in the same relative positions are compared among the models. Hydrostatic stress commonly tends to yield irregular change during the course of plastic deformation in general, whereas the deviatoric stress curves yield monotonic increase regardless of the models. Figure 12.5.13 picks up Model C’s hydrostatic stress contours together with the surrounding grains at three marked points on the inset  m   curve (for the central grain [black]). From the change in the counter, redistribution of high σ m region (i.e., larger  m) within the central grain from upper left to the middle (1 to 2), and then from the central grain to the surrounding grains (2 to 3), are observed. In particular, the latter process corresponds to the decrease in σ m as confirmed in the  m   curve (hydrostatic stress response). This observation is interpreted as follows. The stored σ m within the central grain tends to support the higher stress level up to 2. Upon the release of it to the surroundings (2 to 3), softening takes place (3).

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Figure 12.5.13  Detailed hydrostatic stress response of a grain for Model C showing irregular behavior, together with the corresponding hydrostatic stress contours of surrounding grains.

Figure 12.5.14  Detailed hydrostatic stress response of a grain for Model IV showing irregular behavior, together with the corresponding hydrostatic stress contours and deviatoric strain responses of surrounding grains.

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Figure 12.5.14 shows another example of the hydrostatic stress behavior of the specified grain in Model IV, together with the deviatoric strain responses of the surrounding six grains. The figure implies that the saturation of the hydrostatic increase corresponds to rate changes in deviatoric strain of the surrounding grains. The irregular response in hydrostatic stress stems from heterogeneous deformation of the underlying grain, mainly attributed to the deviated grain shape from hexagon, while redistributions of the hydrostatic stress occur upon relaxation of the heterogeneously distributed intragrain stress, which consequently promotes additional plastic deformations in the surrounding grains. These two observations are direct evidence of the duality, together with its role in driving the evolutions of inhomogeneity, supporting the current hypothesis.

12.6

Duality Revisited

12.6.1 Overview Summarizing all the previous discussions and associated findings, we conclude that we have three keys to distinctively characterize “what the polycrystalline plasticity is,” that is: (1) growing fluctuations of  m and   , (2) the remote effect, and (3) duality, as listed in Figure 12.6.1. As (2) is to be activated through (1), while (1) is driven by (3), the “duality” can then be recognized as the most critical feature. The “duality,” argued so far as a conversion of elastically stored energy to local plastic flow and the inverse process as well, is the driving force of the mesoscopic flows, or equivalently, evolving inhomogeneity, as summarized in Figure 12.6.2. This can be a universal mechanism common to many other situations beyond polycrystalline aggregates (Scale C). Figure 12.6.3 illustrates an oversimplified schematic of the “duality” in the present context, where an elastically expanded or shrunken crystal grain located in the center inevitably induces shear deformations of the surrounding grains to accommodate this situation, that is, local plastic flow. As can be easily understood, the former is represented by  m, while the latter is attributed to  . Such a process is reinterpreted as a sort of “energy flow” or “conversion” from elastically stored states to local plastic flow, that is, dissipation, termed as “duality of fluctuations,” as in Figure 12.6.1. It has a multifold “duality,” as listed in Figure 12.6.4. First of all, based on the analogy to grain assemblies, SSS versus FCS is well described by the graph representation in terms of incompatibility and stress-function tensors (see Section 6.6.1). Figure 12.6.5 displays an SSS map in terms of the AZ size (abscissa) and the collective effect (ordinate), where two diagonally located extreme conditions of “smaller grain size with deformation-induced small inhomogeneity” and “larger grain size with large inhomogeneity” are depicted. First, emphasis is placed, for the current case, on the fact that a particle corresponds to a cluster of grains with enhanced hardening, whereas a void corresponds to a cluster of grains with enhanced softening. Second, SSS and FCS work conjugately with each other, albeit in a sense indirectly as one is conservative and the other is dissipative, whereas they are geometrically complimentary. Third,

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Figure 12.6.1  Summary of the essential features of polycrystalline plasticity caused by the collective behavior of a large number of composing grains.

Figure 12.6.2  Hypothesis of the “duality” in field fluctuations, where elastically stored energy is converted (dissipated) into local plastic flow, and its inverse process as well, and is regarded as the “driving force” of inhomogeneity evolution.

Figure 12.6.3  Schematic drawing for the mechanism of “duality,” where an excessively expanded or shrunken grain (region) is assumed to be compensated by additional shear deformation in another region (this can be in a remote place).

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Figure 12.6.4  Threefold physical meaning of “duality,” that is, in terms of graph-theoretical, thermodynamical, and continuum-mechanical perspectives, together with an FTMP-based view associated with the “flow-evolutionary” hypothesis detailed in Chapter 15.

Figure 12.6.5  A map presenting SSS in terms of the size of the AZ (abscissa) and the

collective effect (ordinate), where “SSS” is expressed by aggregates of hexagons with various sizes. Two extreme conditions of “smaller grain size with deformation-induced small inhomogeneity” and “larger grain size with large inhomogeneity” are located diagonally, as indicated.

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“elasticity (reversible and compressible) versus plasticity (irreversible and incompressible)” means that, in the classical theory of plasticity, hydrostatic stress σ m does not affect plasticity due to constancy of the volume. Because of this, with  m  0 the component can fluctuate more or less freely, being a “reservoir” of additional degrees of freedom. All of these “duality” perspectives are ultimately consolidated into a working hypothesis called “flow-evolutionary law” and proposed in Chapter 15, followed by a number of related simulation results.

12.6.2

Examples of Applications Let us look at two examples in which the “duality” perspective can be a major mechanism responsible for the evolution of inhomogeneity in Scale C, that is, the grain-aggregate scale. Duality-driven plastic flow is schematized in Figure 12.4.24 in this connection, where two representative cases are illustrated, that is, dispersed duality yielding relatively uniform deformation and localized duality bringing about localized flow. Numerical examples of these two are found in Figure 12.4.20 (to be revisited in Figure 15.4.7), that is, Model 1 for the former and Model 4 for the latter, resulting in differences in the surface profiles, further quantitatively evaluated based on the flow-evolutionary law in Figure 15.4.8. In what follows we provide a couple of practical examples relating to such aspects. Lüders banding is well known to be greatly pronounced for finer-grained, well-annealed mild steels. We now provide an interpretation of this phenomenon in the light of “duality.” Finer grains mean the sample is comprised of a larger number of crystal grains, so this case would yield a larger  m even before yielding a fine SSS that effectively activates the remote effect when a number of surrounding grains become involved, as schematized in Figure 12.6.6. Enhancement of  , which is the dual of  m , is restricted owing to the large critical resolved shear stress (CRSS) common to low-carbon steels based on pinning by carbon solutions (Cottrell atmosphere [see Figure 1.2.5]). Once the unpinning occurs at a certain place in a specimen, it triggers an avalanche of   across the specimen, where the width of the band will coincide with the wavelength of the SSS or the AZ (Fujita and Miyazaki, 1978; Miyazaki and Fujita, 1979). A schematic illustration of such series of processes is presented in Figure 12.6.7. This is the Lüders band and it is propagated throughout the specimen, during which the flow stress stagnates (from point A to B, called lower yield stress), with a speed controlled by the energy-momentum tensor, according the flow-evolutionary law (Chapter 15). The stress starts to rise again after the Lüders band prevails over the gauge part of the specimen. Note that the large CRSS resulted from low mobility of the screw dislocation, originating from its atomic arrangement or the complex core structure (see Section 4.2 for more detail). Another example of an extremely large number of crystal grains is the UFG or nanocrystalline materials, which are basically the same as the above case with “localized duality,” in Figure 12.6.5. For further details about the related experimental

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Figure 12.6.6  Schematics depicting the “collective behavior” of crystal grains causing Lüders banding. A sample with finer grains, consequently comprising a larger number of grains, is apt to yield such collective effects, resulting in noticeable Lüders banding behavior.

Figure 12.6.7  Schematics of Lüders banding attributed to SSS formation as a result of the collective behavior of composing crystal grains aggregates, corresponding to the Lüders elongation on the stress–strain curve.

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Figure 12.6.8  Experimentally observed dimples of the order of 10 to 20 μm regardless of the grain size, comparing UFG Cu fabricated by ECAP and annealed Cu under tension.

results, refer to Appendix A3. This case is expected to yield much clearer role sharing between  m and  . Large  m tends to gravitate to grain interiors because of the large long-range stress field. Consequently,   will concentrate on GBs and is expected to help drive GB sliding, because deformation within each grain tends to be restricted due to the long-range stress field. The long-range stress field within the grains is considered to originate from the so-called nonequilibrium GB (see Tucker and McDowell, 2011; Valiev, 2007), but this requires a separate discussion. One manifestation of this greatly enhanced field fluctuation in stress and strain would be the formation of dimples of the order of several 10 micrometers observed in the fractured surfaces. Figure 12.6.8 shows such an example of the fractured surface, observed in UFG Cu fabricated by the ECAP method under tension. Normal-sized dimples are found in the micrograph, whose morphology as well as size resembles that for annealed Cu. It has been reported that several molecular dynamics-based simulations on nanocrystalline aggregates (e.g., Shimokawa et al., 2005) demonstrate mesoscopic flow of an order much larger than the assumed grain size, as shown in Figure 12.6.9, corroborating this conjecture, although the grain size in this case is roughly two orders smaller. Here, trajectories of composing nanocrystalline grains

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Figure 12.6.9  Example of the cooperative behavior of composing grains in a nanocrystalline segregate bringing about fluid-like local flows due to GB sliding, migration, and rotation, observed in a molecular dynamics simulation (Shimokawa et al., 2005).

of 5 nm diameter are traced and indicated by solid arrows, where GB sliding (more precisely, local sliding, migration, and rotation) principally carries the plastic deformation. This can be also interpreted as being caused by the “collective behavior” of grains driven by “duality,” described earlier. Reexaminations of those results from the current viewpoints are highly likely since they will not only be able to verify the present hypothesis but will also prove quite valuable toward reaching a unified understanding of the associated phenomena. Recently, a concept called plaston has been proposed, which tries to capture a new mode of plastic deformation, including ones like those just described (Tsuji et al., 2020). It was motivated by a systematic series of studies by Tsuji about the excellent strength-ductility properties of UFG metals and alloys; based on this, the cooperative activation of the aggregate of atoms is considered to be responsible for a new carrier of plasticity. Such a mode of plasticity due to deformation-induced excitation of the collection of composing atoms in their context is considered to be commensurate with the previously described duality-driven plastic flow. Therefore, the flow-evolutionary perspective given in Chapter 15 is expected to cover this case too, superadding a potential reason and the underlying mechanism as to why and how such extra modes can be induced.

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12.7 Summary This chapter provides the first substantial approach toward modeling “polycrystalline” plasticity beyond the “averaged” perspective, and clarifies the deficiencies of the conventionally used “coarse-grain” or “oligo-grain” models in simulating general features of polycrystalline aggregates in the following respects: (1) Stress and strain fields greatly fluctuate as the number of grains increases. Stress fluctuation is essentially attributed to the hydrostatic components whereas strain fluctuation is mainly due to deviatoric components. (2) The stress–strain response of a grain is not determined by the information of its immediate neighbors but is greatly affected by remote grains, implying a remote effect. (3) The remote effect caused by the collective effect of a large number of grains makes it difficult to identify the RVE using periodic boundary conditions. Therefore, in the identification of the RVE, the AZ responsible for the remote effect should be considered so as not to disregard the associated phenomena. (4) Inhomogeneities additionally introduced by the grain-size distributions and the second-phase distributions can generally enhance the field fluctuations and, accordingly, affect the duality characteristics. These changes sometimes affect the macroscopic averaged stress–strain response and sometimes do not. (5) Applicability of the present scope based on the field fluctuations has been briefly examined by taking well-known examples, that is, Lüders banding and GB sliding in UFG and nanocrystalline materials. Ultimately, understanding how to control the evolution of the duality distribution would be substantially important in the strength design of polycrystalline materials.

Appendix A12  Effect of Transgranular Inhomogeneity on Bauschinger Behavior of DP Polycrystalline Aggregates A12.1 Synopsis The effect of transgranular inhomogeneity on Bauschinger behavior, that is, transient and permanent softening, including early reyielding, is here systematically investigated for DP polycrystal models based on crystal plasticity-based simulation. Particular emphasis is on the latter, represented by Bauschinger strain ε B. Three HP distributions with a volume fraction of 50% are taken as examples; they exhibit distinct Bauschinger behaviors corresponding to different duality distributions of fluctuating stress and strain fields. Random and lamellar-like HP models exhibit larger transient and small permanent softening, whereas a block-like HP model shows the opposite trends, that is, Bauschinger behavior with smaller transient and larger permanent softening. Detailed correspondences between the duality and Bauschinger strain

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A12.2  Background and Problems

665

are extensively examined. Note that the material provided here is linked with Project #2 in Chapter 9 for Scale B.

A12.2

Background and Problems High-strength steels (HSSs) used for automobile bodies yield relatively large springback during stamping processes. It is generally very difficult to predict springback behavior, especially for HSSs, however, because these steels have complex microstructures (see Akiyama et al., 2002; Chun et al., 2002; Cleveland and Ghosh, 2002). No practically satisfactory physics-based model has been developed to date (Cleveland and Ghosh, 2002) in spite of the industry’s strong need. Furthermore, HSSs are subject to complex loading histories, including load reversal, for example, bending and antibending at the bead portions (Chun et al., 2002) where the blank material passes through. Therefore, the Bauschinger behavior plays a crucial role in springback evaluation. The Bauschinger behavior is roughly organized into two types of softening, that is, transient softening and permanent softening, where the former includes early reyielding (Yoshida et al., 2002). These Bauschinger behaviors are known to be sensitive to the inhomogeneities in the stress and strain fields at multiple scale levels. The Bauschinger behavior is considered to be caused by the back stress developed internally during forward loading; however, its physical origin is not always clear, thus its mathematical modeling has been quite unsatisfactory. In particular, we must admit that the evaluation or prediction of reversal behavior is much more difficult than for behavior under forward loading. It is important to recognize, however, that the back-stress field can evolve in plural scale levels stemming from individual physical origins half-independently. We must certainly take into account the following three levels of inhomogeneity as their origins. They are: (A) dislocation substructures, (B) intragranular substructures, and (C) grain aggregate-level inhomogeneity. In HSSs, for instance, (C) is expected to be responsible for a major part of the Bauschinger behavior, since the second (hard) phase of the order of the grain size has been introduced therein. The intention in this appendix is to clarify the effect of inhomogeneity in the grain-aggregate scale (Scale C) on the Bauschinger behavior of polycrystals from a novel viewpoint based on the FTMP. The Bauschinger behavior, that is, transient and permanent softening including early reyielding phenomena, crucially affects the springback characteristics during the stamping processes. For this purpose, a crystal plasticity-based FE simulation is carried out on DP polycrystalline models with three HP distributions. Duality fluctuations between hydrostatic stress and deviatoric strain fields are extensively investigated based on the FTMP in order to examine the relationship between the evolving transgranular inhomogeneities and the Bauschinger behavior, that is, transient and permanent softening including early reyielding during load reversal.

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Figure A12.3.1  Dual-phase alloy models with Vf = 50% examining transient softening in the Bauschinger effect under tension compression for three representative hard-grain distributions, that is, Random, Block, and Block-negative models.

A12.3

Analytical Models and Procedure We consider DP polycrystalline models composed of 613 grains with the hard grains having various volume fractions. The model is discretized into 96 × 96 crossed-​ triangle elements, where each grain is divided into 64 elements, as depicted in Figure 12.5.5. Figure A12.3.1 shows the three types of DP polycrystalline models used in the present simulation with distinct distributions of the hard grains, that is, Random, Block-type regular distributions, and Block-negative. The effect of the HP distribution on the Bauschinger behavior is investigated in detail, especially concerning transient softening including the early reyielding phenomena (Yoshida et al., 2002), in view of the field theoretical perspectives (Hasebe, 2006). The hard grains are allocated by 10, 50, and 100 times-larger yield and flow stresses than those for the soft grains, that is, ( f )HP / ( f )SP  10, 50, and 100. Here, a volume fraction of 50% is taken as an example. Two slip systems and plane strain are assumed to maximize the inhomogeneous deformation (Hasebe, 2004a, 2004b). Tensile deformation is applied up to 5% nominal strain, followed by a load reversal back to the 0% at constant strain rate of 10−3 s−1. Each grain size is set to be 23.53 µm and the distribution of the crystal orientation is given at random. The material for this study was assumed to be pure copper. One may feel such large (σ f )HP / (σ f )SP is unrealistic. This is to clarify the distinction among the three models in terms of inhomogeneity effects on the Bauschinger behavior. We thus intentionally exaggerate the strength difference

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A12.3  Analytical Models and Procedure

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Figure A12.4.1(a)–(c)  Comparison of stress–strain curves during tension and following

unloading and compression among three hard-grain distributions, for strength ratios of 10, 50, and 100, demonstrating pronounced Bauschinger strain as the strength ratio increases.

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Figure A12.4.1(a)–(c) (cont.)

between the hard and soft grains, that is, ( f )HP / ( f )SP  50 and 100 are also considered so that the underlying mechanisms become tractable, because, even with ( f )HP / ( f )SP  10, which is already emphasized more than reality, the distinctions are merely conceivable.

A12.4

Results and Discussion

A12.4.1 General Trends Figure A12.4.1 displays macroscopic stress–strain curves during forward loading followed by unloading and further reversed-loading processes, for the three (σ f )HP / (σ f )SP conditions, comparing three DP models. We can observe clearly the transient softening following early reyielding as well as permanent softening for all the cases, although the constitutive models employed here include neither the backstress model nor the associated kinematic hardening law resulting in such an irreversible response. The transient softening can be characterized by the Bauschinger strain ε B, counting the amount of deviation from the elastic line, while the permanent softening can be represented by Bauschinger stress σ B, corresponding to the normalized stress reduction in the saturated region. Here ε B at zero macro-stress is compared among the three models.

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A12.4  Results and Discussion

Figure A12.4.2(a)–(c)  Grain-wise hydrostatic stress responses during forward and reversed

loading, together with residual hydrostatic stress at unloaded state, comparing three DP models.

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Figure A12.4.2(a)–(c) (cont.)

A12.4.2 Grain-Wise σ m Responses To scrutinize how the above transient softening is brought about, we examine the grain-wise hydrostatic stress responses, as we did earlier in this chapter. Figure A12.4.2(a)–(c) compares grain-wise  m   y curves among the three models during loading (left column) and unloading (middle column) processes, together with those at which the macro-stress are zero (right column), that is,   0. The gray lines represent those for the hard grains, while the black lines are for the soft grains. First of all, we observe mutually distinct trends among the three models, becoming pronounced with increasing (σ f )HP / (σ f )SP, that is, the Random model with the largest spread in the positive  y for the soft grains, the Block model yielding negative spread in  y for the soft grains, accompanied by their largest spread in σ m , and the Block-negative model exhibiting a markedly large spread in σ m for the hard grains during loading, causing earlier reyielding for the soft grains due to a thus-developed back-stress field in the unloading process (indicated by thin arrows). Since the widely spreading σ m and/or  y for the soft grains mean that the accommodations are effectively functioning, the accommodation processes are complementarily reflected in the σ m behaviors. Therefore, we may safely focus solely on the latter, ultimately  m, for measuring the “inhomogeneities” that promote early outset of local plastic deformation in the grain-aggregate scale order (i.e., Scale C).

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A12.4.3 Mechanism Producing Distinct Bauschinger Strains As discussed in this chapter, in polycrystalline aggregates in loading, the relationship between field fluctuations in σ m and  y is called “duality,” and this is responsible for the growth of the inhomogeneous fields. As shown earlier, there exist marked differences in the “duality” among the Random, Block, and Block-negative models. This implies that there is a distinction in the growth of inhomogeneous fields during unloading among the three models, and the difference is responsible for the difference in the Bauschinger behavior (particularly in ε B). Let us look at the stress–strain responses to understand the origin of such distinct trends in the following. As we readily find from Figure A12.4.1, the Block-negative model exhibits a distinct trend from the other two, showing noticeable “two-step” yielding behavior in the forward-loading curve, that is, the first has a similar stress level as the other models, but the second one has larger stress. The stress level at 5% tension is comparable to that for the Block model, and the following unloading–reversed-loading curve almost overlaps it with more pronounced transient softening. For ( f )HP / ( f )SP  50, three important points can be made based on the simulated responses. One is the conceivably large deviation of the result for the Block-negative model from the other two, in which the “two-step” yielding behavior is more pronounced. Second, is that we have very different unloading–reversed-loading behaviors between the Random and Block models, but similar forward-loading curves. Third, the slope of the unloading–reversed-loading curve of the Block-negative model, which is much smaller than the elastic line of the slope corresponding to Young’s modulus, is almost identical to that between the first and second yield points. This will result in the minimum slope of the unloading–reversed-loading curve yielding the largest ε B. For ( f )HP / ( f )SP  100, even more pronounced trends can be observed for these three aspects. Figure A12.4.3 schematically illustrates a stress–strain response for DP polycrystals, demonstrating the mechanism bringing about transient softening, as clarified earlier. The features are summarized as follows: (1) The stress–strain curve initially follows the elastic line with the slope equal to Young’s modulus (both the phases are assumed to have the same Young’s modulus in the present case) as a matter of course, until the first yield point is reached, when the soft grains yield while the hard grains still deform elastically. (2) The slope after that point should then be equal to the average between that of the plastic curve for the soft grains and Young’s modulus for the hard grains. (3) The slope lasts until the second yield point is reached, where the hard grains also start yielding, after which the slope becomes identical to the plastic curve common to the two phases. (4) Upon unloading, the curve should follow the elastic line, at least immediately after the load reversal, until the soft grains start yielding in the reverse direction (compression in the present context) with the help of the “back-stress” field, which has developed during the forward loading.

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Figure A12.4.3  Schematics showing how the Bauschinger strain is determined for DP models

with 50% HP volume fraction.

(5) The slope will continuously decrease as the yielding of the soft grains proceeds, followed by that of the hard grains, until the minimum value is reached. The minimum slope coincides with that in the second curve in the forward loading (the slope in (2)). Note that the amount of transient softening that has taken place as a result of this series of processes can be measured by the Bauschinger strain ε B, meaning the deviation from the perfectly elastic response during reversed loading.

A12.4.4 Measure for Bauschinger Strain Based on the observations in (2) and (3) in the preceding list, we now understand the mechanism of the transient softening as far as the present analytical set-up is concerned, so that we can identify the factors that dominate or control the unloading–reversed-loading behavior producing finite Bauschinger strain ε B. Again from Figure A12.4.3, we can readily notice that ε B should be proportional to the maximum stress before unloading, which is expressed as σ max hereafter. Also of note is that the smaller slope in the process (5) yields larger ε B. Since the origin of the back stress, which is responsible for the process (5), is the early reyielding of the soft grains due to the “inhomogeneity” developed in conjunction with the fluctuating hydrostatic stress field, mostly in the hard grains in the present case, we can expect to have a correlation between ε B and the amount of the spread of σ m , to be expressed as   m  0. Figure A12.4.4 displays correlations of the Bauschinger strain ε B with (a) maximum stress σ max, and (b)  m and  m

 0

 0

  m

  0

  m

 0

, where  m

  0

are the average hydrostatic stress with positive and negative signs,

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Figure A12.4.4  Correlations of the Bauschinger strain with (a) the spread of hydrostatic stress at

unloaded state and (b) the maximum overall stress at maximum tension.

respectively, for all the hard grains in a macroscopically unloaded state, that is,   0. Here  m  0 is tentatively employed as a measure for representing the spread in σ m. Good correlations can be found for both the cases as far as the individual models are concerned. No systematic trend, however, is observed in terms of the distinction among the three models. Let us consider finally the contributions of the above two measures simultaneously. As simplest examples, a weighted average and the geometrical average are taken. Figure A12.4.5 shows the results for both the new measures presented on a log–log plot. We can see excellent correlations for both cases, where single master curves can approximate all the data regardless either of the model used or the strength ratios (σ f )HP / (σ f )SP, although relatively large scatters exist in the smaller region of both measures, which corresponds to smaller (σ f )HP / (σ f )SP . This implies that the larger (σ f )HP / (σ f )SP can dictate the distinction of the “inhomogeneity” leading to that in ε B more clearly, as can be imagined.

A12.4.5 Duality From the field theoretical perspective, the inhomogeneity evolved in arbitrary scales is describable by the incompatibility tensor (Hasebe, 2008, 2009a). We also reexamine the previous sections by evaluating the incompatibility field in the context of “duality,” although the present series of simulations does not take duality into account in the constitutive equation employed.

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Figure A12.4.5  Correlations of the Bauschinger strain with new measures, that is, (a) arithmetic

mean, and (b) geometrical mean of residual hydrostatic stress range and maximum overall stress at maximum tension.

The notion of “duality” hypothesizes an organic relationship between elastically stored excessive strain energy and the inhomogeneity caused by local plastic deformation, measured by the incompatibility tensor, as will be detailed in Chapter 15. The corresponding incompatibility F(η ) and strain energy U e (not fluctuation) contours among the three DP models are compared in Figure A12.4.6. One thing to be pointed out therein is again a rather peculiar trend in the Block -negative model compared to the others, that is, weak F(η ) and contrastingly distinct U e distributions. This means an exclusively large amount of elastic strain energy is stored in the hard grains, with relatively homogeneous deformation, as understood from the trend described with respect to Figure A12.4.2. The plot interrelating the two quantities, that is, η KK versus δ U e, is called the “duality diagram.” Figure A12.4.7 (top left) displays a representative set of duality diagrams for the three DP models, where the solid and open circles denote the maximum stress points and final states after reversed loading, respectively, each corresponding to that marked on the stress–strain curve at the top right. The ratio of the incompatibility (trace) in the ordinate to the strain energy fluctuation in the abscissa defines the duality coefficient    KK /  U e, expressing the conversion rate of the excessive strain energy to the dissipative local plastic flow. We may anticipate such a conversion rate is closely related to the emergence of the Bauschinger strain ε B, as follows. A small conversion rate means relatively large residual (hydrostatic) stress remains unrecovered, resulting in large ε B , and vice versa. The graph on the lower

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Figure A12.4.6  Comparison of contour diagrams of strain energy and the incompatibility term

among three DP models, depicting different levels of inhomogeneity and associated energy storage.

Figure A12.4.7  Duality diagrams comparing three hard-grain distributions for the strength ratio

of 50 as representative, together with the correlation of the Bauschinger strain with the duality coefficient obtained at maximum tension on each duality diagram.

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Figure A12.4.8  Typical correlation functions, comparing the fluctuations of hydrostatic stress

and equivalent strain, together with a corresponding evaluation of the duality coefficient.

right-hand side of Figure A12.4.7 shows the correlation of ε B with the reciprocal of κ , that is,  1, giving excellent fit with a single master curve. The generalized discussions about the “duality” to be extended to “flow-evolutionary law” are given in Chapter 15, where we revisit the current topic with more advanced perspectives.

A12.4.6 Duality in Terms of Hydrostatic Stress Fluctuation



The effectiveness of the use of the hydrostatic component of stress  m and the deviatoric component of strain  for representing and examining the fluctuating stress and strain fields, respectively, have been clarified and then discussed in detail throughout this chapter. We lastly examine again the evaluation of the duality coefficient κ from this viewpoint. One way to visualize or effectively quantify such fluctuating fields is to evaluate the correlation functions. Figure A12.4.8 compares the autocorrelation functions for the fluctuation of σ m, that is,  m m , and for the deviatoric strain    . As readily confirmed, the former  m m tends to be longitudinally connected, giving rise to an SSS, whereas the latter    , on the other hand, exhibits an “X-shaped” distribution due to the attendant transverse deformation, yielding an FCS. The “duality” extensively argued in the present chapter asserts a complimentary relationship between the two, and the energy conversion between the two would drive the field evolutions. This idea leads us to propose a new evaluation scheme for the energy conversion rate, that is, “duality coefficient,” via its correlation function, as 1 1    d 3 x   m ( x ) m ( x ) , m   which is analogous to the case of evaluating transport coefficients based on the famed “fluctuation-dissipation theorem.” An example of such an evaluation in practice is

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Figure A12.4.9  Correlation function-based evaluation of the duality coefficient, comparing

three DP distributions, together with the Bauschinger strain, and well correlated with the thusevaluated inverse duality coefficient.

demonstrated in Figure A12.4.9 (bottom right), resulting in a successful correlation of all the data, commensurate with the result we have in Figure A12.4.6, directly obtained from the duality diagram. Here, we assume “the correlation norm multiplied by the correlation length” is used for obtaining the explicit value of the correlation function. More systematic discussion about this can be found in Section 15.4.1, where an equivalent way to the above via the correlation function of the incompatibility-tensor field, that is,  , is also attempted, and its commensurate effectiveness is evaluated.  KK KK

A12.5 Summary This appendix has dealt with the effect of inhomogeneity on the Bauschinger behavior for DP polycrystalline models with a 50% volume fraction of the HP. A CP-FE analysis of 613-grained DP polycrystalline models with three hard-grain distributions under loading and following reversed loading, was undertaken. Different transientand permanent-softening responses were observed rather sensitively depending on the HP distribution, and were shown to be related with the duality in the fluctuating fields between the hydrostatic stress σ m and deviatoric strain  y in soft and hard phases. The large spread of σ m for the HPs in the Random and Block-negative models resulted in a relatively large transient softening characterized by the Bauschinger strain, whereas

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the large spread for the SP in the Block model yielded small transient and large permanent softening. The measure for predicting the Bauschinger strain based on the duality was proposed. Note that the situation assumed in the Block-negative model is similar to the case of Figure 10.5.2, that is, cellular dislocation substructure in Scale A. So the cell structure is expected to have the role of maximizing the strength to efficiently and effectively support the imposed stress and, at the same time, act as a strong source of back stress (equivalent to Mughrabi’s composite model). This appendix next discussed the effect of inhomogeneity of the grain-aggregate order on the Bauschinger behavior for DP polycrystal models with a 50% volume fraction of the HP. FE analyses were made on 613-grained DP polycrystalline models with three hard-grain distributions, that is, Random, Block, and Block-negative models. Distinct transient-softening responses that depend on the HP distribution were observed, and were further shown to be related to the duality in the fluctuating fields between hydrostatic stress σ m and deviatoric strain  y in soft and hard phases. The large spread of σ m for the HPs in the Random and Block-negative models resulted in a relatively large transient softening characterized by larger Bauschinger strain, whereas the large spread for the SP in the Block model yields small transient softening and thus smaller Bauschinger strain. Lastly, a measure for predicting the Bauschinger strain based on FTMP was tentatively proposed. Based on a working hypothesis about the duality that asserts a process of energy conversion from the elastically stored state to local plastic flow (which is dissipative), and the inversion process as well, the relationship between elastic strain-energy fluctuation and the incompatibility-tensor field at maximum tensile stress was extensively examined. Linear relationships are found to hold for all the models (regardless of the model), implying the effectiveness of the hypothesis. Furthermore, the duality coefficient is demonstrated to be precisely predicted based on the autocorrelation function of the fluctuating hydrostatic stress field, whose process has been inspired by the famed “fluctuation-dissipation theorem.”

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Part IV

Applications II Stability and Cooperation 1 3 Cooperation of Multiple Inhomogeneous Fields 14 Outlooks: Some Perspectives on New Multiscale Solid Mechanics 15 Flow-Evolutionary Law as a Working Hypothesis

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13 Cooperation of Multiple Inhomogeneous Fields

In Chapter 5, we mentioned FTMP in relation to three keywords (evolution, description, and cooperation); this chapter will deal with the last of these three, that is, cooperation, in relation to stability. The first three topics discussed in this chapter are: (1) field equation and stability, (2) preliminary simulation for the interaction field, and (3) stability of the dislocation-cell structure. The fourth topic concerns the “local versus global” nature of interscale cooperation, that is, (4) the global–local structure of stress and strain fields, which is also related to the topics that will be discussed in Chapter 14. The first two topics are based on the differential geometrical field theory (non-Riemannian plasticity) provided in Chapter 6. In (1), a field equation responsible for the system transition from elasticity to plasticity is derived, which has a similar form to that of the buckling of a plate, specifying the limit condition for stress or incompatibility. Meanwhile, (2) provides an example of the interaction formalism constructed in Section 6.7 to address a three-scale problem in 1D. A primitive version of the system-stability analysis is constructed, based on the simulation result. Topic (3) focuses on the structural stability of dislocation-cell walls based on direct simulations, using the (discrete) DD. Artificially constructed dislocation walls are examined in terms of “stability/instability” to ultimately identify the controlling parameter for these. This is expected to give leverage to the conclusion in Chapter 10, where it was argued that the long-range stress field (LRSF) developed in the cell-interior regions determines the morphology, and consequently the stability, of the cell structure itself.

13.1

Field Equation and Stability We derive a field equation, following Kondo (1955), in the context of non-Riemannian plasticity. It is about the stability of a dislocated or defected elastic medium given boundary conditions. Kondo considered an analogy 2D Plate : 3D Flat-Space Material = Bucking Limit : Yielding

as depicted in the inset of Figure 11.13.1. We can notice that, at fin du jour, this attempt is equivalent to that aiming at finding the critical condition for energy conversion to occur from an elastically stored state to that yielding local dissipation, namely the notion of “duality,” introduced in the context of polycrystalline plasticity

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in Chapter 11. This, therefore, has already been embodied by the “flow-evolutionary” law that will be proposed in Chapter 15, given as a relationship between the incompatibility tensor and the fluctuation of the energy-momentum tensor.

13.1.1

Field Equation for Instability A derivation overview for the field equation is depicted in Figure 13.1.1, where the analogy to the plate buckling is illustrated. The total energy of the system is generally given as a summation of elastic and plastic contributions, that is, U  U e  U p , (13.1.1)



where each contribution is given by



Ue 

1 ij 1   ij dV    m  v dV , (13.1.2)  2 2



... k U p  U p Sij..k , Rijl



  l ijkl Rijkl dV 

(13.1.3) 1   dV , ij ij 2

and where dV = g dX and k ijkl are tensorial quantities representing a physical action against the growth of the curvature field. The quantity k ijkl should correspond to the stress-function tensor χij (see Section 6.2.4 for details). What we consider in the present context is a critical condition for the transition of the following two states. The one is about elasticity, during which the strain energy is stored. This is called the “interior” transition. The other yields energy dissipation associated with plasticity, and is called the “exterior” transition. The interior and exterior transitions are schematized in Figure 13.1.2, where two typical examples are presented, that is, creation of a dislocation (bottom left) and evolution of dislocation cells (bottom right). The stability condition for the system in terms of the transition from interior to exterior states is specified by

 U   U e  U p   0. (13.1.4) By substituting the explicit expressions for Eqs. (13.1.2) and (13.1.3) into this condition, after some algebraic calculations we finally reach a field equation of the form



 k  i ( Bijkl  l  j w)   i ( ij  j w)  0, (13.1.5) together with the associated boundary conditions. A detailed derivation process to obtain the above equation is provided in the following sections.

13.1.2

Euler–Schouten Relative-Curvature Tensor As preparation for the explicit derivation of the field equation, we will introduce another kind of curvature tensor, referred to as the “Euler–Schouten relative-curvature”

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Figure 13.1.1  A set of field equations derived from DG field theory, providing an outline of the derivation process of a field equation as a stability/instability condition associated with “duality” between elasticity and plasticity.

Figure 13.1.2  Examples of interior and exterior transitions as precursors of the “duality”-based stability/instability evolution.

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Figure 13.1.3  Definition and its expression in small disturbances of the Euler–Schouten relative-curvature tensor.

. . tensor  H  . This is a curvature measuring the “relative” amount of how much the space is curved out into the enveloping space (a Riemannian space is immersed as a continuous curved figure in the enveloping [Euclidian] space of higher dimensionality). Figure 13.1.3 gives a summary for defining this, together with the relationship . . . with the ordinary curvature tensor K . Let the enveloping space be the 6D Euclidian, whose Cartesian coordinates are given by w  w ( x ), where xκ are arbitrary coordinates belonging to the 3D subspaces (Riemannian space). Here, w  w ( x ) represents the “deviation” or the “displacement” out of a subspace into the enveloping space. Therefore, the relative curvature to the enveloping space is defined as a (second) covariant derivative of wα with respect to xκ , that is,



. . H       w . (13.1.6)

This quantity acts as a vector with respect to the enveloping space wα (the upper index), and, at the same time, as a second-rank tensor with respect to the subspace xκ (the lower indices  and  ). . . The curvature tensor in the Levi–Civita connection is expressed in terms of H  as

. . . K  2 H[. .  H . ] , (13.1.7) . . . . . . where K represents the Riemannian part of the R–C curvature tensor R , that is, . . . .. with S  0 (the contribution of the torsion tensor being excluded). In this case, K is expressed by the Christoffel symbol alone as



             . . .  K  2 [   , (13.1.8)    with i      ]       ]     . . . . . . .. With S , K and R interrelated via

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Figure 13.1.4  Relationship between the Euler–Schouten relative-curvature and R–C curvature tensors in a 3D manifold.







. . . . . . R  K  2[ l Si ] jk  2[ l Sk ]ij  2[ l S j ]ki . (13.1.9)

Figure 13.1.4 summarizes the relationship among three cases associated with the . . ... k and Rijl . An important fact to be noted here is that the Euler– combination of H  . . Schouten relative curvature H  has more general features than the R–C curvature ... k tensor Rijl in terms of measuring global imperfections.

13.1.3

Derivation of Field Equation for Stability

13.1.3.1 Plastic Contribution







Figures 13.1.5–13.1.9 provide an outline and summary of a derivation process of the field equation for stability. Let us start with writing down the plastic part of the energy contribution δ U p for a defected system. Here we solely consider the curvature . . . tensor-related imperfections K , disregarding the torsion counterparts. From Eq. (13.1.3), the variation of plastic energy (Figure 13.1.5) can be written as 1  U p    l K gdX , (13.1.10) 2 . . . where l represents the physical action against the growth of K : . . . . . .  . .  α K  2 H[  H  ] with H       w . With w being a scalar in regard to the immersed Riemannian space, we have   w    w  . (13.1.11)        [   ] w  [    ] w  [  ]    w    . . Further, under small disturbances, we can assume H  to be . . H       w      w . (13.1.12)

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Figure 13.1.5  Derivation process of field equation #1a: Variation of plastic energy expressed via the Euler–Schouten curvature tensor and corresponding physical resistance against its growth (Eqs. (13.1.10)–Eqs. (13.1.14)).

Figure 13.1.6  Derivation process of field equation #1b: Variation of the plastic strain energy (Eq. (13.1.15)).

By substituting these expressions into Eq. (13.1.10), we have, approximately,

 U p    l [   w   ]  w dX





  l [  ]     w    w     w     w dX



 l 

 [  ]

l

 [ ]

    w     w dX 

1 B     w    w dX , 2

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(13.1.13)

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Figure 13.1.7  Derivation process of field equation #2a: Elastic strain energy and its variation (Eqs. (13.1.17)–Eqs. (13.1.22)).

Figure 13.1.8  Derivation process of field equation #2b: Elastic strain energy and its variation (continued).

where





B  2 l [  ]  l  [ ] . (13.1.14)



is introduced in the last line. By performing the integration by parts, we have

U p 





1    w B    w dS 2 S (13.1.15) 1      w  B    w dX . 2 V



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Cooperation of Multiple Inhomogeneous Fields

Here, the second term (Figure 13.1.6) is further rewritten as









1 1  w  B    w dS    w    B    w dX . (13.1.16)  2 S 2 V

13.1.3.2 Elastic Contribution

For the elastic contribution, strain energy density (per unit volume) is defined as 1 1 ue       D     . (13.1.17) 2 2 The variation of the above ue is given as



 ue  D     (13.1.18) 1          g . 2 Therefore, the elastic energy variation is given by

 U e    ue gdX





(13.1.19) 1    g gdX ,  2

where 1  0 4   g  g  2   w   w  O(w ) . (13.1.20)    1  w  w  O(w 4 )     4 



Substituting these into Eq. (13.1.19), neglecting the higher-order terms, we rewrite



 







1      w   w dX 4 V 1        w   w    w    w dX (13.1.21) 4 V 1        w   w dX . 2 V

 Ue 





By performing the integration by parts, we finally have 1 1  U e    w     w dS    w      w dX . (13.1.22) 2 S 2 V





13.1.3.3 Field Equation Finally, substituting all the contributions, that is, Eqs. (13.1.15) and (13.1.22), into the variation condition  U   U e  U p   0, we can derive the field equation. The ­derivation process is briefly shown in Figures 13.1.9 and 13.1.10. Here, variations with respect to  w and    w are considered. From the former, the field equation is obtained as

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Figure 13.1.9  Derivation process of field equation #3: Final stage where explicit expressions for variations of the elastic and plastic energy are substituted into total energy variation, from which the field equation, together with boundary conditions, is derived.

Figure 13.1.10  Field equation together with boundary conditions.









   B    w       w  0. (13.1.23)



Similarly, from the latter, the associated boundary conditions are obtained as follows:

 B    w  0 . (13.1.24)      w      w  0   B is a material constant tensor, defined by Eq. (13.1.4).





Here, B

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Cooperation of Multiple Inhomogeneous Fields

13.1.3.4 Isotropic Case For the isotropic case, the field equation and the attendant boundary conditions, given in Eqs. (13.1.23) and (13.1.24), are reduced to a simpler form. Figure 13.1.11 shows schematics of the derivation process. With replacements, . . w  w  H       w  H  . (13.1.25)

and



  

G11 B  H  H  H 22 33  11   G   B  H  G 22  B  H 22  H33  H11   33  G  B H33  H11  H 22   we have the corresponding field equation to Eq. (13.1.23) as

  , (13.1.26) 

Bw       w  0, (13.1.27)



and the boundary conditions to Eq. (13.1.24) as



 2 w (1  ) B 2   Bw  0   . (13.1.28)  3 (2   ) B  w  (1   ) B  w     w  0    3  Here, B is the scalar version of B , while κ is a material constant supposed to represent a Poisson ratio-like quantity defined in the plastic region instead of the elastic region.

13.1.4

Application Examples Figure 13.1.12 shows the application examples of the field equation obtained above for both strain and stress spaces. For the strain space (see left-hand side of Figure 13.1.12), Kondo considered a transition where a material yields into plasticity from elasticity (Kondo, 1968; Minagawa, 1968). This is the case, in the strain space, where the disturbance w corresponds to incompatibility and the field equation provides a critical condition for the stress field. Namely, when the stress reaches a critical value σ cr , the incompatibility w starts increasing. This is interpreted as a proliferation of dislocation loops at yielding, more specifically, Lüders banding during Lüders elongation in mild steels, where the stress remains constant (lower yield stress). In this context, the field equation is regarded as the one governing “hardening.” When the same thing is viewed from a standpoint in the stress (function) space (right-hand side of Figure 13.1.12), we have a similar equation to Eq. (13.1.23) for the field, that is,



 k  i (C ijkl  l  j v)   i ( ij  j v)  0, (13.1.29)

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691

Figure 13.1.11  Isotropic versions of the field equation together with the boundary conditions.

Figure 13.1.12  Application examples of field equations in strain- and stress-function spaces, that is, hardening evolution for the former and damage evolution for the latter.

where v corresponds to the stress. When the incompatibility reaches a critical value ηcr , the stress v starts increasing. Minagawa (1968) interpreted this as “fatigue,” because the incompatibility evolves while the stress keeps a constant value (during cyclic loading). Hence, the incompatibility means fatigue damage, more specifically,

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Cooperation of Multiple Inhomogeneous Fields

Figure 13.1.13  Application example of Figure 13.1.12 in an FTMP-based simulation, showing the variation of evolved patterns with increasing/decreasing gradient-evaluation range.

crack initiation and the ensuing propagation. In a step-wise point of view, this can be regarded as “damage evolution” of a general kind, not confined to fatigue. In the present FTMP-based FE simulations, detailed in Chapter 11, such incremental applications of the field equations can be roughly mimicked by increasing/decreasing the gradient-evaluation size rGR (e.g., Figure 11.5.22), as exemplified for a seven-grain model in Figure 13.1.13 on a duality diagram (refer to Figure 15.2.4). Decreasing rGR (upper-right) tends to convert the stored strain energy into incompatibility-based degrees of freedom which are dissipative, exhibiting rapidly refining substructures, possibly resulting in hardening evolution. Increasing rGR (lower-left), on the other hand, leads to a successive energy release, yielding the attendant incompatibility drops, resulting in coarsening of the substructure, that is, damage evolution.

13.2

Preliminary Simulation for Interaction Fields

13.2.1

Overview: Materials as Complex Systems Materials can be viewed as an example of “complex systems,” where nonlinear interactions among multiple scales do exist together with feedback loops among them,

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693

making it more than the sum of its parts and extending far beyond the “reductionist” perspective (Hasebe, 2009a; Phillips, 2001). To model such complexes, we need, at least, to deal with the “interactions” among plural scales (more than three scales [e.g., Xie, 1994]), together with the evolutionary aspects of the individual inhomogeneities. Fortunately, our experiences have shown that each scale seems to have its own evolutionary rule or similar for the inhomogeneous fields, based on rather distinct physical and geometrical origins. Let us take examples by choosing three important scale levels in polycrystalline plasticity of metallic materials in terms of inhomogeneous field evolutions, that is, the scales of (A) dislocation substructures, (B) grain size, and (C) grain aggregates (Hasebe, 2004a, 2004b, 2006). Scale A, for dislocation substructures, the collective effects of interacting dislocations at high density in a sense of statistical mechanics, is responsible for dislocation clustering, while the LRSF evolving concurrently tends to determine the resultant morphology and size of the pattern, especially in the case of cellular structures (Hasebe, 2006). For Scale B, on the other hand, the field evolution in it is mainly attributed to the geometrical constraints imposed by the external load, together with those from the surrounding grains, which is describable using continuum mechanics-based formalism (Aoyagi and Hasebe, 2007). It should be noted that the morphology of the substructures to be evolved is essentially dominated by the crystallographic information, except shear bands, which are, rather, influenced by macroscopic stress states and thus can penetrate plural crystal grains and not always be terminated at the GBs. Field evolutions in Scale C are attributed, to a large extent, to the collective behaviors of the crystal grains composing the polycrystalline aggregate, in terms of role sharing and duality (Hasebe, 2004a, 2004b, 2006). In this case, unlike the dislocations in Scale A, the statistical-mechanics framework cannot be directly applied because of the nondistinguishability of the crystal grains, each of which has its own shape, size, and crystallographic character. These rather distinct mechanisms for individual-scale field evolutions have encouraged the simple “information-passage”-type modeling perspectives, which are more or less effective in many situations. But questions arise: What are the interactions like, and when and how they are activated? Here is the motivation for us to go beyond “reductionistic” perspectives, prompting the conventional multiscale modeling to grow into practically feasible stages. In Section 6.8, a theoretical framework for describing the interfield correlations among multiple scales was developed and implemented into a crystalline plasticity-based constitutive model (Hasebe, 2009a, 2009b). The output there can provide us with a new standpoint to examine multiscale crystalline-plasticity problems in the light of “interaction” among plural scales. In this section, an example simulation on a three-scale problem is given to show what the interactions are like, together with some potentially tractable future uses based on it. A crystalline plasticity-based simulation result is used for the present simulation, assuming three scales of great significance in polycrystalline plasticity, that is, the Scales A, B and C, exemplified earlier.

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Cooperation of Multiple Inhomogeneous Fields

13.2.2

Analytical Models and Procedures Figure 13.2.1 shows the 1D strain distribution used as an initial condition in the present study. The cross-sectional strain distribution along a line shown in the inset has been obtained via an FE analysis on a multigrained model (Aoyagi et al., 2008), where the plots indicate the FEM results, while the interpolated line is based on the RKPM (Chen et al., 1996; Liu et al., 1995). The change in the gradient-evaluation size can change the contour of the incompatibility-tensor field, represented here by F( ( ) ) (η term), as systematically compared in Figure 13.2.2, together with the RKPM-based results with changing half width at half maximum (HWHM) of the kernel function used. The three scales, A, B, and C, chosen here are indicated in Figure 13.2.3, where the number of modulations in the central grain along the transverse cross-section is plotted against the evaluation range (size) of the second derivative. The corresponding continuous presentation of the cross-sectional η distributions is displayed in Figure 13.2.4, with emphasis on those for Scales A and B. In what follows, for dealing with Scale A, the effective cell-size model is used to evaluate the strain distribution and its derivatives, assuming the periodicity commensurate with dcell for the strain, whose details are given later. This treatment will be

Figure 13.2.1  Plastic strain/distortion distribution obtained in field theory-based FEM analysis and interpolated by RKPM employed in 1D interaction-field simulation.

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Figure 13.2.2  Examples of the incompatibility terms evaluated based on FEM, compared with those obtained via RKPM approximation (Aoyagi et al., 2008).

Figure 13.2.3  Variation of number of modulations in the incompatibility distribution with evaluation size of derivative, together with three scales, A, B, and C, chosen for field-interaction analysis.

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Figure 13.2.4  Continuous variation of incompatibility field (1D) with evaluation size, where those in Scale A and Scale B are compared.

practically reasonable because we normally find it difficult to directly simulate cell structures in Scale A within FEM-based simulations even via the FTMP scheme. In the following subsection, we will first derive general expressions for the threescale interaction field with the effective cell-size-based Scale A field applicable to full 3D situations. Then, its reduced version, applied to a 1D problem, will be given.

13.2.3

General Expressions for the Three-Scale Problem To evaluate the distortion and strain distributions in Scale A, we virtually assume the following sinusoidal form with a periodicity corresponding to the effective cell size dcell in a point-wise manner, as schematically depicted in Figure 13.2.5. Tensorial forms of Scale A distortion and strain are assumed to be given as



 2   ijeA   ijecell sin  ij x A  and  ijeA   ijeA , (13.2.1) d  sym  cell  where x A represents Scale A coordinates whose values are given randomly. So  ijecell and  ijecell are the amplitudes of the cell size-order elastic distortion and the correij sponding elastic strain fluctuations, while dcell is evaluated using the effective cell (α ) size for each slip system, that is, dcell, by multiplying a direction tensor and taking summation over all the slip systems considered, namely

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Figure 13.2.5  Schematic drawing for the treatment of Scale A in the present simulation.

N





ij ( ) dcell   si() m(j) dcell , (13.2.2)





where, si(α) and m(jα) indicate unit vectors in the slip direction and slip plane normal for the (α ) slip system, respectively. Equation (13.2.2) was obtained by equating

 ()  K 

and

b ( ) dcell

. (13.2.3)





e  ()  Pij() ijcell  Pij() Dijkl  klecell . (13.2.4)





Here, Pij(α) is the Schmid tensor defined by Pij()  si() m(j) (see Section 1.3.4). sym The corresponding elastic distortion and strain amplitudes ije cell ,  ije cell, which appear in Eq. (13.2.1), are estimated based on linear elasticity, together with the similitude relationship for the resolved shear stress with the reciprocal of dcell , as



( ) e ije cell   ije cell  Dmnij Pmn



1

K 

b ( ) dcell

. (13.2.5)

Note that the above-assumed sinusoidal form of the distortion or strain itself does not correspond directly to Scale A fluctuations but its derivatives at each point do characterize them, that is, as the dislocation-density and incompatibility tensors.

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By using the distortion or strain tensor expressed by Eq. (13.2.1) for Scale A, we calculate explicitly the dislocation-density and incompatibility tensors, respectively, as

 ijA  ikl  kA ljeA  2  ijecell  2   (13.2.6)  2  cos ij x A  .  ikl  kA  ljecell sin  lj x A   ikl  ij   d   dcell    cell   dcell

and



ijA   ikl  kA jlA

sym

 2  e  cell eA  ikl jmn  kA  mA  ln  ikl jmn  kA  mA  ln sin  ln x A   dcell  2     2  2 

  ikl  jl   ejlcell sin  jl x A  . d  d 

 cell   cell 



(13.2.7)

sym

For Scales B and C, we will conduct direct evaluation of the derivatives, that is,

 ijB  ikl  kB ljpB ,





ijB   ikl  kB Bjl

sym ikl jmn  kB  mB lnpBn ,

(13.2.8)

and

 ijC  ikl Ck ljpC ,





ijC   ikl Ck  Cjl

sym ikl jmn Ck Cm lnpCn .

(13.2.9)

For the interaction terms, for example, between Scales A and B, see  2  ijBA   ikl jmn  kB  mA  lneA   ikl jmn  kB  mA  lne cell sin  ln x A   dcell  B A   ikl  k  jl (13.2.10)





sym

  2   2  ejlcell   ikl  kB cos jl x A  . jl d  dcell

 cell  sym and



ijAB ikl  jmn  kA  mB  lnpB   ikl  kA Bjl

sym . (13.2.11)

int It should be noted that, in the second interaction component η AB , the further differentiB ation of α jl with respect to Scale A coordinates, which have been virtually introduced, cannot be explicitly evaluated. To this, we tentatively assume a locally (point-wise) sinusoidal variation of α Bjl with a periodicity corresponding to the evaluation range of differentiation in Scale B ∆x B, plus its additional finer fluctuation with respect to Scale A in the present chapter, that is,



 2   2   Bjl (x A )   Bjl sin  B x A    Bjl (x A )   Bjl sin  B x A    jlA . (13.2.12)   x   x Assuming Scale A fluctuation  ljB (x A ) in Eq. (13.2.12) to be commensurate with α ljA, that is,  ljB (x A )   ljA, we finally have, for Eq. (13.2.11),

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13.2  Preliminary Simulation for Interaction Fields

   2  ijAB   ikl  kA Bjl sin  B x A   ikl  kA jlA  x   sym





2 (ikl  jlB )sym

x B

 2  cos B x A   ijA.

x  

699

(13.2.13)

Similarly, for the interaction between Scales C and A, we have  2  ijCA   ikl  jmn Ck  mA  lneA   ikl  jmn Ck  mA  lne cell sin  ln x A    ikl Ck  jlA sym  dcell    2   2  ejlcell (13.2.14)   ikl Ck cos jl x A 

jl   dcell  dcell  sym



and



ijAC  ikl  jmn  kA Cm  lnpC   ikl  kA Cjl



sym

 2    ikl  kA Cjl sinn  C xC  ikl  kA jlA  (13.2.15)  x 

sym





2 (ikl Cjl )sym x

C

 2  cos C x A  ijA .  x 

For another interaction between Scales B and C, we have

ijBC ikl  jmn  kB Cm  lnpC . (13.2.16)

and

ijCB ikl  jmn Ck  mB  lnpB . (13.2.17)



Combining Eqs. (13.2.6)–(13.2.17), we finally have the total incompatibility tensor for the interaction field. The total dislocation-density and incompatibility tensors in their direct as well as index notations are given by 1 C 1 A ij  eBC  ij   ijB  eBA  ij



1 C 1 A   eBC    B  eBA 

and

2 C 2 A ij  eBC ij ijB  eBA ij











(13.2.18)



2 1 1 1  eBA ijBA ijAB  eBC ijBC ijCB  eBA  eAC ijAC ijCA

 

2 C 2 A eBC   B  eBA  1 1 BA AB  eBC  eBA   







(13.2.19)

1  BC  CB   eBA2  eAC  AC  CA  ,

respectively.

13.2.4

One-Dimensional Expression for the Three-Scale Problem The 1D constitutive equation employed in this study is







  D1 1 c1sgn()  ln(1 c2 ), (13.2.20)

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where D1 , c1, c2 are constants. This is a reduced version of the constitutive equation for a crystalline-plasticity model, that is,   ()   ()  f  ()  , (13.2.21) K 

with



1

1  2 K  Q H ( )  and dcell  k  Q Q  , (13.2.22) N   where H(γ ) represents the hardening modulus for a referential stress–strain curve, and Q indicates the hardening ratio, further given as  ( )

 ( )





Q   1  F ( ( ) ) , (13.2.23)

with

1/ 2

k  ldefect ( )    . (13.2.24) p  b  Equation (13.2.20) is evaluated, in the present study, from a given initial strain distribution that has been obtained in the FE analysis on a multigrained model (Aoyagi et al., 2008) and interpolated based on RKPM. In the present case, Scale B is set to be the reference, so the total incompatibility in the interaction-field representation is written as F ( ( ) )  sgn( ( ) ) 

2  2    eBC C  B  eBA  A . (13.2.25)



Substituting Eq. (13.2.20) into Eq. (13.2.25), we finally have 2 1 int 2 1 int 2 1 int   eBC C  eBC  BC  B  eBC eCA BA  eBA  A  eBA CA . (13.2.26)



For Scale A in the 1D case, the effective cell-size model is used to evaluate the strain distribution and its derivatives, assuming the periodicity commensurate with dcell for the strain, instead of its tensorial counterparts given in Eq. (13.2.1), namely, Here,



 2 A   Ap   Ap0 sin  x  . (13.2.27)  dcell 

b Kb   cell   cell  , (13.2.28) dcell dcell where K is the proportionality constant. For Scales B and C, direct differentiations based on the central finite-difference scheme are utilized with regard to the respective evaluation ranges, ∆x B and ∆xC ,

 K 

 B ( xi )  2B Bp and C ( xi )  C2  Cp . (13.2.29) Similarly, for the interaction terms between Scales B and C, we have



p int   BC ( xi )   B C  C   BC , (13.2.30)  int p  CB ( xi )  C  B B  C  B

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where the derivatives are evaluated based on the finite-difference method, that is, central difference for the spatial derivative and forwarded difference for the time derivative,  B Bp 



C  Bp 



( Bp )i  xB / 2  ( Bp )i  xB / 2  ( Cp )i  xC / 2  ( Cp )i  xC / 2 x B  xC ( Bp )i  xC / 2  ( Bp )i  xC / 2 xC

, (13.2.31)

,

and  BC 



(C )i  xB / 2  (C )i  xB / 2 x B

, C  B 

( B )i  xC / 2  ( B )i  xC / 2 xC

. (13.2.32)

For Scale A, the incompatibility is obtained by the second derivative of Eq. (13.2.27), that is, 2

 A ( xi ) 2A Ap





 Ap0

 2   2  xi  . (13.2.33)   sin   dcell   dcell 

Therefore, the interaction terms become  int  Ap0   2 B   2 B   cos  xi -1   xi 1   cos   BA   B A ( xi )  B dcell x   dcell   dcell  

 . (13.2.34)   2  2   2   int  AB   A   B  sin  x B xi     B x B cos  x B xi   

 







and



13.2.5

 int  Ap0   2 C   2 C   cos  xi -1   xi 1   cos  CA  C  A ( xi )  C d dcell x   cell   dcell    , (13.2.35)   2  2   2   int  AB   A   C sin  xC xi     C xC cos  xC xi   

 

respectively.









Application Example Here we consider steels in fatigue, focusing on the effect of dislocation substructures on the fatigue properties that evolve during cyclic deformation. It has been shown experimentally (Yokoi et al., 2004) that steels yielding a well-developed 3D cell structure of dislocations exhibit extrusion/intrusion at the specimen surface with the corresponding width to the average cell size of the order of submicrometer to micrometer, ultimately leading to crack initiation. By way of sharp contrast, steels yielding 2D substructures, that is, that with vein or planer morphology, tend to be accompanied by finer extrusions at the sample surface that can significantly delay the crack initiation in comparison with the former case. Furthermore, the cyclic properties are also changed drastically from cyclic hardening to softening.

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This is a striking example of the multiscale properties observed in a certain class of steels that definitely need to be interpreted and solved from a “multiscale” perspective, such as that used in the present study. Note that such drastic changes in the dislocation substructures and the concomitant fatigue properties are reported to be caused by the addition of copper, depending on its form within the ferrite matrix, for example, solid solution and nano-sized precipitates (Yokoi et al., 2004). The connection with electronics, that is, ab initio-based viewpoints and approaches, has been extensively discussed elsewhere (Chen et al., 2008), where the addition of copper atoms is shown to drastically change the core structure of a screw dislocation of BCC iron from nonpolarized to fully polarized.

13.2.6

Analytical Results and Discussion

13.2.6.1 Results Three scales, A, B, and C, are considered as an example case of the application; each being assumed to correspond to the scale levels of dislocation substructure (A), crystal grains (B), and their aggregate (C), respectively. The characteristic scales commensurate with the respective evaluation size of the derivative operations set here are (A) 0.5, (B) 1.76, and (C) 8.8 µm, respectively, as depicted in Figure 13.2.3. Therefore, the scale ratios for the present case are



lC l l l 2 1  5, eBA  A  0.28, eBA  eAC  C2A  2.5. (13.2.36) lB lB lB In what follows, use is made of the incompatibility terms for the individual scales, A, B, and C, together with the pair interactions between arbitrary two-scale levels, that is, BC, BA, and AC. Here, they are evaluated separately via 1 eBC 

F ( scale )  sgn( scale ) escale   scale , (13.2.37) where “scale” indicates the distinction of scales, for example, scale = A for Scale A and scale = BC for interscale of B and C, while escale expresses the corresponding −2 −1 2 1  eAC . scale ratio, for example, eBA , eBA , eBA Figure 13.2.6 displays the obtained distributions for the respective incompatibility terms, that is, individual-scale incompatibility and interaction terms. Scale A exhibits finer fluctuations than the others, extending both positive and negative values reflecting the wavelength of the underlying effective cell size dcell. Scales B and C show similar distribution to each other. Their interaction terms, BC and CB, yield slight difference between the two, which is caused by artificially introduced information loss in the differentiation operations (Eq. (13.2.30)) for the purpose of meeting the present aim. Large asymmetry, on the other hand, is observed in the interaction terms for the int int int int  η AB and ηCA  η AC . In the latter two cases, A–B and A–C scale pairs, that is, η BA Scale A fluctuation in the strain tends to be averaged out during the differentiation with respect to the larger scale, that is, B or C. Figure 13.2.7 shows the total incompatibility distribution F(η ) for the interaction field, comparing with that without interaction (referred to as “referential”).

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Figure 13.2.6  Obtained incompatibility terms for Scales A, B, and C, together with interaction

terms.

It is shown that the field interactions evidently enhance the total field fluctuation where finer modulations are superimposed on the referential field. In the present case, as can be understood from the comparison among the results displayed in Figure 13.2.6, the contribution from Scale A plays a prominent role, especially through the interaction int int ) and F (ηCA ). This implies the importance of the field fluctuations in terms F (η BA the dislocation-substructure order in evaluating mesoscopic damage evolution, for example, ultimately leading to crack initiation in fatigue. It must be emphasized that such microscopic information normally tends to be averaged out in the upper-scale simulation based on conventional crystal plasticity. Let us examine next the effect of Scale A fluctuation on the interaction field. We consider three cases with different average sizes of the effective cell dcell mimicking three distinct dislocation substructures to be evolved during high-cycle fatigue, for example: (1) three-dimensionally well-organized cell, (2) two-dimensionally developed vein, and (3) uniformly distributed planer array of dislocations. These are assumed here to be modeled by introducing small, intermediate, and large effective cell sizes, respectively, in the present context; that is, dcell = 0.5, 1.5, and 5 μm. The latter two cases, for simplicity, employ the same dcell (x) distribution as the first one but multiplied by magnification factors, with the amplitude leaving dcell constant. Figure  13.2.8(a)

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Figure 13.2.7  Comparison of the total incompatibility-term distribution with and without

interaction.

compares the assumed cell-size distributions with the three dcell , where the result for dcell  0.5 m was obtained in the FE analysis in Aoyagi et al. (2008). Figure 13.2.8(b) shows a comparison of the Scale A incompatibility term F ( A )  sgn( A )  A obtained for the three dcell cases, while Figure 13.2.8(c) displays the resultant total incompatibility terms F(η ). It is clearly demonstrated that the difference in dcell controls the fluctuation levels in the Scale A field, ultimately influencing the levels in the total incompatibility field. The largest cell size, that is, dcell  5 m, brings about minute fluctuation in F (η A ) with small enough amplitude, resulting in a negligible effect on the total incompatibility F (η A ), while the smallest size, that is, dcell  0.5 m, causes larger and finer field fluctuations which significantly modulate the F(η ) distribution, especially in its frequency. The average size of dcell  1.5 m, on the other hand, exhibits intermediate trends in both F (η A ) and F(η ) between the above two cases. These results demonstrate the relatively high sensitivity of Scale A field fluctuation to the overall response in terms of the overall field fluctuation. Corresponding stress distributions to this total incompatibility are compared in Figure 13.2.9, demonstrating the noticeable effect of dcell on the resultant fluctuation in the stress field. Significantly enhanced fluctuations can be observed, where the small enough dcell evidently further modulates the stress field with respect to the scale of the order of dcell , keeping the overall distribution profile unaltered. The large enough dcell , in sharp contrast, results in much smaller and, accordingly,

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Figure 13.2.8  Effect of Scale A fluctuation on total incompatibility distribution: (a) cell-size distributions with three average sizes, (b) the corresponding incompatibility-term distributions for Scale A, and (c) the resultant total incompatibility distributions comparing the three cases.

negligible effect on it. Also, we can notice that the amplitude of the averaged response is slightly reduced with the increasing contribution of Scale A field fluctuation. This is due to the additional degrees of freedom in deformation to accommodate the imposed inhomogeneity, which tends to soften the stress response, as mentioned earlier. Practically, these results for the stress response with enhanced fluctuation are expected to be closely related to rougher surface undulations, for example, during fatigue (see Yokoi et al., 2004). Therefore, we can conclude that the introduction of the interaction field among multiple scales captures the essential effects of the microscopic fluctuations, for example, those in the dislocation-substructure order, normally absent in conventional crystal plasticity-based models and simulations, on the inhomogeneous field evolutions in the upper scales, which will be macroscopically ­averaged out. Note that, below Scale A, dynamical effects associated with moving and interacting individual dislocations are expected to become dominant, generating “temporal” field fluctuations rather than, or in addition to, the “spatial” fluctuations discussed previously. The contribution of such dynamical effects can in principle be taken into account by combining separate and direct simulations based on, for example, discrete DD (Ghoniem et al., 1999; Kubin et al., 1992; Zbib et al., 1998). An example of a related discrete dislocation-based discussion on the effect of stress-field fluctuation on the stability of dense dislocation structures can be found in Yamada et al. (2008).

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Figure 13.2.9  Comparison of the stress distribution among different interaction conditions in terms of average cell size in Scale A.

13.2.7

Toward System Stability/Instability Evaluation One of the ultimate aims of the present study is to develop a novel methodology able to evaluate the stability/instability of the complex systems of polycrystalline materials in plasticity, explicitly considering interactions among multiple scales accompanied by inhomogeneous fields. In this section, a candidate framework will be presented, based on the interaction-field formalism given earlier. In the interaction-field theory, everything about the field evolution can be characterized by



2 C 1 CB 2 1 CA   F (eBC  ) F (eBC  ) F (eBA  eAC  )   B   1 BC F ( ) F (eBA1  BA )   F ( )   F (eBC ) 1 2 A 2 1 AC AB eBA F (eBA  )    eAC F ( ) eBA F ( ) (13.2.38)     FC FCB FCA       FBC FB FBA ,       FAC FAB FA 

which is further introduced in the incompatibility term of the hardening model. A preliminary analysis will be made by utilizing the matrix in Eq. (13.2.38), which will be called the evolutionary matrix hereafter. Each component in the evolutionary matrix is tentatively evaluated by a signed spatial average, that is,

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Figure 13.2.10  Schematics of the overall evaluation for system stability/instability based on the “Ph–S trajectory” in terms of the Scale B incompatibility field, as an example of a three-scale problem.



 Fscale  sgn F ( B() )

2 1 N F ( B() ) . (13.2.39)  N

The evolution of the system will be virtually simulated by a continued product of the evolutionary matrix, as summarized in Figure 13.2.10, and can be characterized by a    Fscale phase–space (Ph–S) trajectory, that is, Fscale , where

•

n

          Fscale  n  Fscale   Fscale 

n 1

. (13.2.40)

From the results shown in Figure 13.2.7, we can evaluate the evolutionary matrix as

 8.7 7.5 13.0      Fscale 6.5 3.0  . (13.2.41)     7.0  2.2 0.12 8.8  As may be readily found, the above matrix is asymmetric and cannot be diagonalized. The corresponding Ph–S trajectory with reference to Scale B, that is, FB  FB, is displayed in Figure 13.2.11(a). Variations of FBη and FBη with power n are shown in Figure 13.2.11(b). Oscillation takes place initially and is followed by limit cycle-like behavior as n increases. This implies that the system will oscillate in the Scale B order as cyclic deformation proceeds. Let us next examine some virtual cases with different off-diagonal terms. If there is no interaction, all the off-diagonal components should vanish, that is,



 8.7     Fscale     0  0

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0 0  6.5 0  . (13.2.42) 0 8.8 

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Cooperation of Multiple Inhomogeneous Fields

Figure 13.2.11  Results of a virtual system-stability analysis based on the evolutionary matrix, comparing cases with and without field interactions: (a) Ph–S trajectories, that is, FBη versus FBη , and (b) variations of FBη, and with FBη , the number of simulation steps.

This case corresponds to a system composed of linearly interacted and thus completely separate multiple scales, with no unpredictable response. The discriminant of the characteristic equation in this case is Ddiscr  22.0  0, meaning the equation has three distinct real roots. The phase diagram and variations of FBη and FBη for this case are also shown in Figure 13.2.11(a) and 13.2.11(b), respectively. No limit cycle-like oscillation takes place in this case, and a convergence is reached soon, implying that the system is stable in the absence of field interaction. Several other examples have been examined based on slightly modified evolutionary matrices. Variations such as spiral converging, spiral diverging, and zigzag are observed, depending on differences in the off-diagonal components. These results imply that even a slight difference in the off-diagonal components can result in a totally different system response. The possibility of chaotic responses will be pursued in the next step. Also, it should be noted that the above is just a “toy” problem of a preliminary kind. The full 3D polycrystalline systems may exhibit totally different and more complex responses, since Scales B and C have qualitatively different incompatibility fields based on their origins, as exemplified previously. Further details of these results will be presented in future publications.

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13.2.8 Conclusion The current subsection has applied mathematical formalism to describe the multiple field interactions constructed in Section 6.8 as applied to a simplified three-scale problem assuming a multigrained plasticity of steels under fatigue, where three orders of dislocation substructures (A), grain size (B), and grain aggregates (C) were considered. A 1D constitutive model, together with plastic strain distribution obtained in FEM-RKPM analysis, was used for the simulation. Field fluctuations both for individual scales and the interaction terms were explicitly obtained and their roles on the evolution of the overall system were discussed. In the present context, Scale A fluctuations were demonstrated to be highly influential in the upperscale field evolutions, especially in Scale B, which is consistent with experimentally reported results. Furthermore, a candidate scope for the system-stability analysis was proposed, based on an evolutionary matrix evaluated from the incompatibility distributions.

13.2.9

Other Application Examples A scheme similar to the one in the previous subsections, that is, another 1D simplified simulation, has been applied to a problem about the stability/instability of the hierarchical lath martensite structure (see Project #1 in Section 9.4.2.1 for details), that is, the scales of (A) high-dense dislocation wall for martensite laths, (B) lath block/ packet structures, and (C) aggregates of lath packet or prior γ grains. The corresponding three-scale representation is displayed in Figure 13.2.12, which will be utilized in further Ph–S trajectory-based stability/instability system evaluations.

Figure 13.2.12  A concept of multiscale problems of complexity, further reduced to a three-scale problem as a minimum setting.

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Figure 13.2.13  System response under full three-scale interaction, yielding a “stable” response of the targeted system, represented in terms of the “Ph–S trajectory for Scale B” as converged to zero.

Figure 13.2.14  Examples of a system response that exhibits “instabilities” on the “Ph–S trajectory” in terms of the effect of Scale A and interactions with it.

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Figure 13.2.15  Examples of a system response that exhibits “instabilities” on the “Ph–S trajectory” in terms of the effect of Scale B and interactions with it, together with Ph-S trajectories that show asymptotic stabilities.

Figure 13.2.13 displays an example of the basic Ph–S trajectory for this problem, assuming the presence of full three-scale interactions, yielding a “convergence” of the Scale B behavior, meaning that the system is “stable.” With this regarded as the referential case, we can play around by altering the composing scale and interaction combinations to see the effects on the system response. Figures 13.2.14–13.2.16 list the obtained results, classified in terms of the presence/absence of Scales A and/or B as well as the associated interactions. The absence of Scale A, together with the attendant interactions, tends to bring the system toward an unstable state, meaning the presence of Scale A significantly contributes to system stability (Figure 13.2.14). On the other hand, the role of Scale B is the major one. It is self-evident that it will result in complete “system failure,” as found the Ph–S trajectories shown on the top row in Figure 13.2.15, whereas, with interactions with Scale A, the system regains stability, as demonstrated in the bottom row, albeit in an asymptotic manner. Inversely, a lack of major interactions is also apt to keep the system stable,

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Figure 13.2.16  Other conditions apparently show “stable” system responses, under no-interaction conditions, that is, substantially single-scale problems.

simply due to their “stationary” nature of such isolated-state systems, as exemplified in Figure 13.2.16. These are a set of examples of simplified virtual problems, from which we can virtually see how such a scheme would be effective in visually and systematically evaluating the stability/instability aspects of the targeted problems. When applying the Ph–S trajectory scheme to FE simulation results like those argued in Chapter 11, see Figure 13.2.17, where single-crystal models in tension with four typical orientations are exemplified. Here, region-wise averaged trajectories are compared in respective models, demonstrating similar trends in the sense that the trajectories are apt to follow “clockwise-rotating” loci. For the [001] sample, also shown is the point-wise version of the Ph–S trajectory presentation (bottom left). Limit cycle-like behaviors, on the other hand, are exemplified in Figure 13.2.18, and are obtained in discrete DD simulations mimicking the PSB ladder structure, representing back-and-forth motions of screw dislocations in the channel region.

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Figure 13.2.17  Simulated Ph–S trajectories for FCC single-crystal models under tension, using FTMP-based CP-FEM.

Figure 13.2.18  Simulated Ph–S trajectories for discrete dislocation systems mimicking PSB ladder walls under cyclic loading, yielding “limit cycle”-like behaviors.

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Figure 13.2.18  (cont.)

13.3

Stability of Dislocation-Cell Structure

13.3.1

Problem Description As discussed and clarified in Chapter 10, the dislocation-cell structure is formed as a result of the collective effects of interacting dislocations, that is, pair annihilations, with the substantial help of the long-range internal stress field (LRSF), developed

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concurrently. The conclusion there implies that the cellular structuring is insensitive to the detailed configuration of the wall-constructing dislocations (WDs). However, speaking of the stability of the cell structure, details are expected to emerge to play crucial roles. To demonstrate this hypothesis is effective, we need to clarify the relationship between the LRSF and the wall stability via detailed simulations equipped with explicit configurations of the WDs. This section makes an attempt to clarify the mechanism determining the structural stability/instability of dislocation-cell walls based on DD simulations, in light of the LRSF. The detailed behavior of the WDs and their relationships with the LRSF distribution produced between the walls, both under relaxation and tensile stresses, are extensively investigated.

13.3.2

Simulation Setting A systematic series of DD simulations was designed to examine the stability of the cell wall structures. Figure 13.3.1 shows the simulation model used throughout the study. Here, a DD code (MDDP-suite: Multiscale Dislocation Dynamics Plasticity) developed by Zbib et al. (1998) is utilized. As shown in the figure, a single dislocation wall is artificially constructed and situated in the center of a unit simulation cube (here, “cell” is not used to avoid a confusion with the dislocation “cell”) with 0.1 µm width. The side lengths of the simulation cube are 1.10 µm (4000b) × 0.551 µm

Figure 13.3.1  Schematics showing the dislocation cell-wall model used in the simulation, with DD examining the relationship between the long-range stress field (LRSF) that develops in the cell-interior region and structural stability/instability.

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(2000b) × 1.10 µm (width × height × depth), such that the wall spacing (cell size) becomes 1.0 µm. The simulation time step is set to be t  2.534  10 11 s. A periodic boundary condition is applied to all three directions. In this study, pure Cu with FCC structure is assumed, with the Burgers vector b  2.756  10 10 m. The wall model is assumed to further consist of four dislocation-networked walls, as detailed in Figure 13.3.2, that is, two orthogonally crossing straight dislocations,

Figure 13.3.2  Detailed construction of cell wall model assumed in simulation, composed of two networked walls, referred to as WDs, sandwiched by redundant dislocation (RDi) walls responsible for producing the LRSF.

Figure 13.3.3  Comparison of the LRSF developed via artificially constructed dislocation walls among Models 1 to 4, with different dislocation densities.

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Figure 13.3.4  Schematics of dislocation models composed of identical dislocation configurations but with different combinations of the Burgers vector for the WDs, that is, Cases 1 to 6.

WD, and two redundant dislocations (RDi). The RDi walls, which are immobilized, are introduced such that the LRSF can be developed in between the walls, in the sense of Mughrabi’s model presented in Chapter 10. The WDs are sandwiched by a pair of RDis. The crystallographic constitutions of the wall model are also depicted in Figure 13.3.3, while the LRSFs to be produced by the RDis are demonstrated in Figure 13.3.3 as functions of presumed dislocation density and the cell size (wall spacing) dcell. The dislocation density in the analyses that follow is set as   5.583  1013 m 2, commensurate with moderately deformed states. For the WDs, six combinations of the Burgers vectors are further assumed, as illustrated in Figure 13.3.4, in order to investigate the effect of the LRSF on the wall stability of a fixed dislocation configuration (thus having the same density). These combinations will be referred to as Cases 1 through 6.

13.3.3

Results and Discussion The stress fields in the channel regions (left- and right-hand sides), for example, hydrostatic stress σ m and equivalent stress σ , are displayed in Figures 13.3.5 and 13.3.6 respectively. Different stress distributions are observed even for the same dislocation configuration, and they are basically asymmetric. The right-hand side of the wall for Case 1, for instance, exhibits well-developed long-ranged stress distribution both for both σ m and σ 23 in all the cases, whereas the left-hand side, in sharp contrast, yields a poorly constructed stress field. Figure 13.3.7 displays the side view of the simulation cube, showing snapshots of the collapsing walls at 300 time steps, comparing the six cases. Corresponding to the different stress distributions in the channel regions shown in Figure 13.3.6, each case exhibits its own collapsing behavior. The right-hand side of Case 1, for example, is quite stable in response to the well-developed LRSF, while the left-hand side shows

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Figure 13.3.5  Comparison of the hydrostatic stress distribution in the interwall region among the six cases.

a marked propensity to collapse. Similar trends are confirmed for the other cases, that is, a wall with well-developed LRSF for “both” σ m and σ tends to be stable (left-side channel for Case 6), otherwise unstable behaviors result. For a more quantitative discussion, the distributions of the dislocation nodes are plotted in Figure 13.3.8 for both sides of the walls. The node distribution is regarded in a similar way to the dislocation distribution. The SD and the maximum moving (reaching) distance of dislocations away from the wall (xmax ) are evaluated for all the cases. These two quantities are used here as indices measuring the instability. In the present context, the SD is interpreted as an index representing the scatter of the collapsing wall, whereas xmax expresses the maximum translation distance of the

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Figure 13.3.6  Comparison of the shear stress distribution σ23 in the interwall region among the

six cases.

wall-composing dislocation segments measured from the initial position. On the other hand, the LRSFs for σ m and σ are measured here via the spatial average σ m and σ 23 , respectively, represented by the area of the respective stress distribution (profile) in the channel region. Figure 13.3.9(a) and (b) correlate the SD with σ 23 and σ m , respectively. The σ 23 –SD diagram shows moderately good correlation. For the σ m –SD, on the other hand, we can confirm a weak inverse correlation between the two, as demonstrated in Figure 13.3.9(b), although it is not so obvious. The same diagrams, but for xmax,

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Figure 13.3.7  Comparison of snapshots of WDs at 300 simulation time steps among all the cases, demonstrating various collapsing behaviors depending on the Burgers vector combinations in the initial states. The right-hand side of the wall in Case 1 and the left-hand side in Case 6 show stable configurations of dislocations yielding small amounts of scatter and translation.

are given in Figure 13.3.10. We observe a similar relationship for σ 23 –xmax (Figure 13.3.10(a)); however, almost no correlation seems to hold for  m  xmax (Figure 13.3.10(b)). Let us take account of the contribution of σ m to the wall stability, because we can acknowledge it to be nonnegligible based on these results. Since the negative sign of σ m is expected to yield a positive contribution, that is, pushing the dislocations against the wall, the signed value of σ m , measured from the initial position (x = 0 on the abscissa in Figures 13.3.9(b) and 13.3.10(b)), is taken for each case as a modification factor. We tentatively define an index for representing the LRSF as

 LRSF   23    m , (13.3.1) where α is a parameter specifying the contribution of σ m . With   5, Figures 13.3.9(c) and 13.3.10(c) depict the amount of the shifts via Eq. (13.3.1) for all the data points from those in Figures 13.3.9(a) and 13.3.10(a) respectively. This modified

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Figure 13.3.8  Comparison of dislocation-node distribution among the six cases at t = 300 steps, representing the quantitative behavior of collapsing dislocations. The peaks denoted by arrows roughly correspond to translated WDs.

Figure 13.3.9  Correlation of the SD of the dislocation-node distribution with parameters representing the LRSF developed in the channel region at t = 300 steps.

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Figure 13.3.9  (cont.)

result is shown in Figure 13.3.9(c), resulting in a better correlation than the previous one (Figure 13.3.9(a)). Therefore, the stability of the dislocation wall, as far as the current simulation condition is concerned, is concluded to be controlled by the LRSF developed in the channel region, with additional contribution from the hydrostatic component. Figure 13.3.10(c) presents the modified results with σ LRSF . In this case, unlike the results for SD, the correlation is rather deteriorated by the modification. Since xmax expresses the amount of translation of WDs rather than the scatter, the difference between the two correlations implies the role of σ m is resistance against random

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Figure 13.3.10  Correlation of the maximum moving distance of dislocation nodes away from the initial wall position, with parameters representing the LRSF developed in the channel region at t = 300 steps.

collapse. On the other hand, σ 23 is responsible for the restriction of the translation rather than otherwise. The modified correlations for SD and xmax are compared afresh in Figure 13.3.11, where stable walls with sufficiently large σ LRSF seem somehow to be available for both.

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Figure 13.3.10  (cont.)

Figure 13.3.11  Comparison of the two correlations in Figures 13.3.9(c) and 13.3.10(a), with the proposed parameter representing the LRSF in the cell-interior region.

13.3.4 Summary Details of the WDs, using the largest density model with a constant simulation cube size, are systematically studied by changing the combinations of the Burgers vectors.

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Distinct internal stress distributions are observed depending on the Burgers vector combinations even for the identical wall configuration, leading to totally different collapsing behaviors of the walls. The relationships between internal stress distribution and wall stability/instability are investigated based on the statistical analysis of the dislocation-node distributions. Component-wise examinations reveal that there exists a good correlation between the mean value of the shear stress component and the maximum moving distance of WDs out of the wall region under tensile stress. Also, the SD is well correlated with the mean shear stress modified with that of hydrostatic component, implying the significantly important roles that the internal stress fields are playing in terms of wall stability. The above two correlations imply the following features. Among the LRSFs, the deviatoric component tends to resist the translational deviation of dislocations from the wall, while the hydrostatic component inhibits their scattering. In the real situation, a larger σ m will promote sharpening of the wall, whereas a smaller σ m will result in fuzzy boundaries. Above all, it is explicitly demonstrated that “quality,” like the LRSF, rather than “quantity,” such as dislocation density, controls the stability of the dislocation walls. Note that this issue is revisited in 15.5.1.

13.4

Global–Local Structure of Stress and Strain Fields Ohashi has found that a long wavelength in the stress field showing a “scale-free”like power-spectrum distribution exists in elastic polycrystalline analysis (Ohashi and Sawada, 2002). This is due to a longitudinally developed SSS along the grains with a larger Young’s modulus under tension. In sharp contrast, the spectrum distribution for transverse distribution yields a weak peak at the wavelength smaller than the specimen width. In this section, we will discuss global–local scale-free-like cooperation in polycrystalline elastoplasticity.

13.4.1

Preliminary Simulations

13.4.1.1 Variation in Grain-Wise Stress–Strain Responses Figure 13.4.1 displays the models employed in the current examination, based on crystal plasticity assuming a two-slip system model (same as that employed in Chapter 12). Roughly, two types are prepared. One examines the effect of granular morphology containing 78 grains, whereas the other a globally distributed inhomogeneity composed of 613 grains (they are used in Chapter 11). The former includes the reference model with a regular hexagonal grain shape, and that with Voronoi tessellated grains. The latter is further divided into single-phase and DP models, respectively, with global grain-size distribution and with second-phase (hard grains) distribution. The fluctuation characteristics argued in Chapter 12 for the present Voronoi models are examined first. Figure 13.4.2 compares grain-wise  m  curves for the hexagonal

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Figure 13.4.1  Finite-element polycrystal models for global–local analysis.

Figure 13.4.2  Comparison of grain-wise hydrostatic stress–deviatoric strain responses between hexagonal (referential) and Voronoi models, with different orientation distributions.

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Figure 13.4.3  Effect of a grain behavior on the surrounding grains’ stress–strain response (six grains numbered from 1 through 6), where the crystal axis orientation of the central grain varies from 0° to 150° against the stress axis.

and Voronoi models, among three orientation distributions. Note the conspicuous and irregular fluctuations in the Voronoi models, whereas there are relatively moderate variations in the hexagonal models. A major reason for the difference comes from the deviations of the grain shapes in the Voronoi models from the regular hexagon models, as the latter tend to enhance the inhomogeneity. To verify this, the following two cases are examined, that is, one grain versus surrounding grains, in terms of the resultant variation of the stress–strain responses due to orientation distribution. Figure 13.4.3 demonstrates the effect of a grain’s behavior on the surrounding grains (six grains numbered from 1 through 6), where the crystal axis orientation of the marked grains change from 0° to 150° against the stress axis. With such an orientation change for the central grain, the stress–strain response for each of the surrounding grains is also subject to change, as shown in the figure. Evidently, the Voronoi model yields larger variations in the stress responses than the hexagonal grain model. Figure 13.4.4 shows similar results to Figure 13.4.3, but focusing on the effect of the orientation change in the surrounding grains on the response of the marked grain. Also confirmed is a larger variation for the Voronoi than the hexagon. Figure 13.4.5 examines the duality response of a grain and its surroundings in terms of  m  responses. As was discussed in Section 12.5.3, the irregular hydrostatic-stress-response behavior of a grain is synchronized with the behavior of its neighboring grains.

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Figure 13.4.4  Effect of the orientation change in the adjacent grains on the response of the marked grain.

Figure 13.4.5  The relationship between the hydrostatic stress response of a grain with irregular change and those of the surrounding grains for the Voronoi polycrystal model.

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13.4.1.2 Spectral Analysis and Global Nature of the Stress Field Figure 13.4.6(a) provides an example of the spectral analysis to be argued here; the figure displays a set of component-wise stress distributions along a longitudinal cross-section of a polycrystalline model in tension (obtained for a 613-grain model: Details can be found in Chapter 12). The Fourier transform of the stress distributions gives the corresponding power-spectrum distributions, as indicated on the left-hand side, plotted against, in this case, the wave number per grain k / d . The wave number ratio k / d = 1 corresponds to the case where the wavelength of the stress fluctuation coincides with the grain size. Therefore, the regions with k / d > 1 and k / d < 1 refer to intra- and transgranular sizes, respectively, as schematically illustrated in Figure 13.4.6(a) (bottom). It is important to note two things in the present context. First of all, the longitudinal stress distribution is demonstrated to exhibit a scale-free relationship with k / d ; while, second, both the deviatoric and hydrostatic components yield basically the same “scale-free” natures, allowing us to discuss without caring about the distinction between the components. With this in mind, a schematic representing the three-scale perspective appears, similar to that in Figure 13.4.6(b), which implies the three scales are not always fully independent but should be intercorrelated. Note that, for strain, Figure 13.4.6(b) intriguingly yields no scale-free distribution, contrastive to the stress counterparts, exhibiting a “characteristic scale” in the region k / d > 1, which will be revisited later in Section 13.4.2. What the “scale-free” relationship in the stress distribution means in the first place is that macroscopically imposed stress is successively redistributed into smaller degrees of freedom, at least down to the subgrain-size order. Simultaneously, this implies some “indistinguishable” interscale relationships for the polycrystalline plasticity, as

Figure 13.4.6(a)  An example of the spectral distribution of stress in polycrystalline aggregates based on FE simulation.

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Figure 13.4.6(b)  An example of the spectral distribution of strain, corresponding to Figure

13.4.6(a).

Figure 13.4.7  Schematics of the scale-free-based perspective of the three-scale interaction fields.

pointed out earlier, that is, the nonexistence of a “characteristic scale” in this case. Many implications result, which are discussed separately in the next subsection. Such multiple inhomogeneous fields schematized in Figure 13.4.7 can practically be treated in a unified manner within the crystalline plasticity-based framework, in combination with the field-theoretical interaction fields based on the incompatibility tensor, allowing cross-scale interplay among Scales A, B, and C, as extensively examined in Section 13.2.

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Before discussing this further, let us overview some experimental phenomenology regarding the “scale-free” nature of this relationship, together with the plasticity-​ related research examples. Figure 13.4.8(a) is an example of the famed “1/ f -type noise” phenomena, known as “flicker noise” and observed in electric circuits (Musha, 1977). Figure 13.4.8(b) presents a general schematics of the scale-free relationships in the power spectrum versus frequency plot, together with an example line pattern yielding 1/ f noise in comparison with the fully regular pattern. In the case of white noise (meaning fully random), we have a constant power-spectrum distribution, as shown in the figure. Note that in practical situations, as in numerical simulations of stochastic sorts, we normally use “Gaussian noise” for mimicking this, which has a probability-density function with normal distribution. One of the significant aspects of such “scale-free” characteristics in the noise is the fact that the scale-free noise implies the underlying physical process accompanied by correspondingly long temporal correlations, that is, time constant should exist. Namely, a larger time constant than the measurement interval means we may assume we are measuring fluctuations of an unsteady not a steady process. Another noteworthy aspect to be stated here concerns “the self-organized criticality (SOC)” (Bak et al., 1988) into which dissipative (nonequilibrium) dynamical systems are spontaneously evolved in general, accompanied by spatial and temporal power-law scaling behavior as their fingerprints. Such spatial/temporal scale-invariant characteristics, or universality, like those in the critical point of equilibrium second-order phase transitions, strongly imply that we cannot rely only on fine-tuning models in order to understand/model genuine aspects of the

Figure 13.4.8(a)  An example of power-spectrum distribution for the “1/ f -type noise” phenomenon, which is known as “flicker noise,” observed in an electric circuit (Musha, 1977). Reprinted with permission of the publisher.

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Figure 13.4.8(b)  Schematic diagram of the power-spectrum distribution for 1/ f -type noise and the corresponding line pattern. Power spectrum of white (Gaussian) noise is also shown for comparison.

targeted phenomena. Parenthetically, the scale-free distributions have been widely observed in numerous natural as well as man-made phenomena, such as earthquake frequencies known as the Gutenberg–Richter law for the magnitude versus the occurrence, economic balance referred to as the Pareto principal that states 80% of outcomes roughly come from 20% of the population (also called 80/20 rule), and even in linguistics, holding between wording frequencies and their ranks, from which Zipf’s law was derived, which applies to various rank–frequency distributions. One of the most tangible as well as the simplest examples of the above SOC is found in experiments with sandpiles at critical angle of repose (Bak et al., 1988; Grumbacher et al., 1992), based on which not only can a power-law relationship be deduced in a practically feasible manner, but also one can intimately “feel” what the “critical situation” is like (onset of “avalanche,” in this case). Similar “scale-free” aspects to those just discussed have been argued in recent years with regard to metal plasticity-related phenomena, both in terms of temporal (Argon, 2013; Miguel et al., 2001a) and spatial processes (Hähner et al., 1998; Zaiser et al., 2004). The former has targeted, for example, strain bursts/avalanches (Figure 13.4.9(a)), while the latter focused on dislocation-cell size (Figure 13.4.9(b)) and surface roughness. Both yield a common fractal character, even with similar fractal exponents of 1.5–2.0. Although relationships with the SOC have been pointed out, neither the underlying mechanisms nor their significance in plasticity have been clarified, to my knowledge.

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Figure 13.4.9  Examples of plasticity-related “scale-free” distributions for: (a) temporal (Miguel et al., 2001a), and (b) spatial (Székely et al., 2002) processes. Adapted with permission of the publishers.

Getting back to our polycrystal-plasticity simulations, we first overview the variation in the stress-spectral distribution with the model specifications, that is, grain shapes, grain-size distributions, and second-phase distributions, based on a preliminary series of simulations, which is followed by some future scopes, before proceeding to detailed simulations in Section 13.4.2. It should be noted that neither the incompatibility tensor nor the interaction fields are taken into account in the following simulations. For examining the effect of grain shapes, Figure 13.4.10 compares the stress power-spectrum distribution based on the simulation results for the Voronoi tessellated-grain models (A, B, and C, constructed from different sets of embryonic initial points) with that for the hexagonal-grained counterpart, where the average size of the composing grains are used to evaluate k / d for the Voronoi models. For now, we will use normal coordinates for the ordinate, instead of logarithmic ones, for the differences to be identifiable. Basically there is no significant difference between the two cases; however, one can observe relatively larger values of Iσ y over the whole region of k / d with rather improved linearity for the Voronoi models than the hexagonal case. This is considered to simply stem from the enhanced fluctuations in the stress field in the Voronoi models, as argued previously (see, for example, Figures 13.4.4 and

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Figure 13.4.10  Stress power-spectrum distribution for Voronoi models compared with that for the hexagonal model.

Figure 13.4.11  Stress power-spectrum distribution for 613-grain models with grain-size distribution (IV and DV) compared with that for the hexagonal model.

13.4.5). The result implies sufficient field fluctuations as a requisite of the power-law distribution to be developed. The effect of grain-size distribution is examined via the 613-grain models used in Section 12.5.2 (IV and DV models, see also Figure 12.5.3), by comparing them with

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Figure 13.4.12  Stress power-spectrum distribution for 613-grain DP models with block and random HP distributions compared with those for the hexagonal model.

the hexagonal model. Figure 13.4.11 gives the result obtained, showing the effect of grain-size distribution on the power-spectrum diagram. The inset displays the macroscopic (overall) stress–strain curves for the three cases overplotted together, emphasizing the negligible effect on them. The power-spectrum distribution clearly yields difference in the region k / d  1, that is, in the low-frequency (wave-number) region, reflecting the global distribution of grain sizes, as can be anticipated. In particular, the DV model is shown to exhibit larger slope, that is, a larger exponent, whereas the IV model yields a smaller exponent than the hexagonal model. This means that inhomogeneities in the grain-aggregate scales (Scale C) possibly dominate the power-law exponent. This is simply because the low-frequency region can sensitively contribute to the overall slope on the power-spectrum diagrams. Last but not least, we focus on the effect of additionally introduced heterogeneities. Figure 13.4.12 indicates the effect of the second phase on the distribution of the stress power spectrum. The comparison is made for the three hard-grain distributions, that is, transverse lamellar, block, and random with V f = 50% (see Section 12.4), at   0.3, where the macroscopic stress levels are commensurate with each other. The periodicities of the hard grain distributions are evidently reflected in the power spectra; the lamellar model has a sharp peak at k / d = 0.5, corresponding to the periodicity of the lamellar, while the block yields a peak at around k / d 0.3 commensurate with the hard-grain block size. These k / d values refer to the characteristic length scales for these two cases, respectively, based on which the corresponding RVE can be readily identified as the basis of unit-cell representations. The random model, in sharp contrast to the other two cases, exhibits an 1 / f -type relationship like the single-phase models with no noticeable stress peak. There are two points here worth noting. One is that there still exist scale-free-like trends if we ignore the peaks at the characteristic

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Figure 13.4.13  Summary of the preliminary spectral analyses for polycrystal models.

scales. The second is that the existence of the characteristic scales, however, reduces the overall slopes, resulting in a smaller power-law exponent as a whole. The latter means that at the same time there arises stagnation of the imposed stress at around the characteristic scales, bringing about, for example, stress concentrations and attendant damage accumulations. Figure 13.4.13 schematically summarizes this series of preliminary results. Here, a macro-model without inhomogeneous stress distribution is also indicated as case 0 for comparison, supposed to yield a single-scale spectrum. What is of great significance that needs to be emphasized is the anticipated effect on the slope of the spectral diagram, leading ultimately to the fractal exponent. Also to be noted is that the macroscopic overall response of the stress cannot always capture those effects of meso-scale inhomogeneities, as tangibly exemplified in Figure 13.4.11. In other words, marked changes in the fluctuating stress fields can occur even while the same or similar macroscopic responses remain unaltered. From a modeling/simulating point of view, this can be rephrased as follows: (1) Scale-free-like spatial stress distributions are reproducible rather easily, as far as polycrystalline plasticity is concerned, at least when we explicitly express the composing grains. (2) Detailed conditions, such as the grain shape, size, and their distributions, can affect the slope of the scale-free relationship (i.e., exponent), while the overall trends remain relatively unaffected. One more thing to be pointed out here is: (3) The scale order for k / d to be extended depends on the model size and the resolution. When we simply superpose these results together, the diagram of the stress power spectrum will be obtained, as in Figure 13.4.14(a), and will ultimately be combined based on the interaction fields.

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Figure 13.4.14  Schematics of hierarchical model-based simulation composed of N = 7 (either with or without the strain-gradient [incompatibility] term), 77- (with either hexagonal or Voronoi shapes), and 613-grain models.

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Figure 13.4.15  Polycrystalline models for detailed spectral analysis with different grain representations.

Figure 13.4.16  Schematics showing the data process for obtaining the power-spectrum distribution for stress.

Based on this finding, we may now envisage a more likely multiscale perspective than that shown in Figure 3.4.6(b). Figure 13.4.14(a) and 13.4.14(b) provides an example of such an updated schematic of hierarchical model-based simulations, where we tentatively suggest using polycrystalline samples composed of N = 7 (either with or without the strain-gradient [incompatibility] term), 77 (with either hexagonal or Voronoi shapes), and 613-grain models.

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13.4.2

739

Detailed Simulations For a further detailed examination of the scale-free-like stress-fluctuation field of polycrystalline models, four polycrystalline models are prepared. Figure 13.4.15 shows those models, referred to as Models A, B, C, and D. Model A corresponds to the previous N = 613 model with a hexagonal grain shape, while a square grain shape is assumed in Models B and C, with different stacking modes. Model D also assumes a polycrystalline aggregate but with each element (one crossed-triangle element) being regarded as one grain. Similar to the condition set out in Section 13.4.1, a two-slip model based on CP-FEM analysis is carried out under tension up to 10% nominal strain. Figure 13.4.16 explains the data processing. Fourier transform is performed on stress and strain distributions along the 96 longitudinal cross-sections to obtain the averaged power-spectrum distribution. In the present case, the elastic anisotropy is also taken into account (which is absent in the previous simulations), so that the evolving process of the scale-free-like power spectrum from the purely elastic stage can be examined. Figure 13.4.17 shows the simulated stress and strain power-spectrum distributions overplotting the results at the representative steps, that is,   0.01 (perfectly elastic),

Figure 13.4.17  Stress and strain power-spectrum distributions for the hexagonal (referential) polycrystal model, where results at   0.01% (perfectly elastic), 0.1%, 1%, and 10% are compared.

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0.1%, 1%, and 10%. For the stress at   0.01, an almost linear relationship is confirmed on the log–log plot. This trend basically persists even at  10 when plastic deformation prevails over the sample, although the slope tends to be reduced as the deformation proceeds. We also confirm a slight disturbance showing “stagnation” in the k / d > 1 region, especially for   1. The strain power spectrum, on the other hand, exhibits distinct trend from the stress case, where saturation of Iε y is clearly observed in the region k / d < 0.5, yielding nearly white noise (Figure 13.4.18). The reason for this, together with that for the stress to yield scale-free-like distribution, may be roughly and intuitively understood by looking at their contours as presented in Figure 13.4.19. Namely, the stress (longitudinal, that is, in tensile (y)-direction) tends to be connected longitudinally. The strain (in tensile direction), on the other hand, is apt to exhibit “X”-shaped distribution, simply due to the simultaneously induced transverse deformation (Poisson’s contraction, in the case of elasticity), absent in the stress counterpart, resulting in less-correlated distributions manifested ultimately as a white-noise-like spectrum. When we use the equivalent strain, as in the case of Figure 13.4.6(b) shown previously, it may become a “peak” indicating a characteristic length because the strain components are mixed up. The “stagnation” observed in k / d > 1 for the stress power-spectrum distribution with relatively large plasticity probably stems from the inappropriate treatments of the substructure degrees of freedom in the intragranular region of the model, in addition to the insufficient number of mesh divisions therein. The imposed stress is apportioned in succession to the smaller degrees of freedom, but it tends to get stuck at around k / d =1 because of those reasons. Namely, the plastic deformation mimicked by the model cannot afford to absorb the excessive elasticity anymore. For coping with this, we can utilize the incompatibility-based model that allows intragranular substructure formations as well, as demonstrated in Chapter 11, and as schematized in Figure 13.4.14(a) as Scales A and B. This situation is similar to the case of “turbulence,” where the vortices (or eddies) are successively transferred into smaller ones down to the minimum scales until dissipated into heat. The numerical treatments of turbulence thus require a commensurate fine-enough simulation mesh. If the resolution is not appropriate, energy stagnation takes place, which tends to produce unrealistic responses from the flow, as will be mentioned later. In our context, such energy stagnation is thought to be processed in Scales B and A, as pointed out in the previous subsection, that is, as driving forces of the substructure evolutions. The contrasting features between the stress and strain contours can also be viewed by using the correlation functions. Figure 13.4.20 displays autocorrelation functions for σ y and ε y corresponding to Figures 13.4.17 and 13.4.18, obtained via a process based on Wiener–Khinchin’s theorem, which interrelates the special density and the correlation function (see Figure 13.14.21 for the derivation process). Relatively larger but rapidly decreasing correlations as deformation proceeds toward plasticity are found for the stress, whereas the strain tends to yield smaller as well as almost constant correlation. It is noteworthy that, as will be demonstrated in Sections 15.4.1 and 15.4.2, the correlation functions of the hydrostatic stress and/or the incompatibility

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13.4  Global–Local Structure of Stress and Strain Fields

Figure 13.4.18  Comparisons of stress and strain power-spectrum distributions among four polycrystal models.

Figure 13.4.19  Comparison between stress and strain contours for four polycrystal models.

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Figure 13.4.20  Comparisons of the autocorrelation functions for stress and strain among four polycrystal models.

Figure 13.4.21  Relationship between spectral density and the autocorrelation function, known as Wiener–Khinchin’s theorem.

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tensor are successfully utilized to quantitatively evaluate the Bauschinger strain (Figure 15.4.2) and the surface roughness (Figure 15.4.8) for DP polycrystalline models, in conjunction with the flow-evolutionary law discussed therein.

13.4.3

Energy Spectrum Similarities for Turbulent Flow We can notice some similarities between the “local–global nature” of the spectral structure of stress and strain for polycrystalline aggregates discussed earlier and the energy-cascade structure of turbulence. This subsection briefly discusses the turbulence in terms of the energy spectrum. Figure 13.4.22 is an example of typical turbulent flow generated downstream of a lattice (Van Dyke, 1982), where large eddies (swirls) having sizes of the order of the lattice spacing situated upstream are successively evolved into small ones as they travel downstream, finally yielding uniform distribution. In other words, the energy injected at the lattice is redistributed in succession into smaller-sized eddies as a result of nonlinear interactions (cascaded), and ultimately dissipated as heat at the smallest scale where the viscous effects become dominant. Figure 13.4.23 is an example of the schematics illustrating the energy-cascade mechanism via the successive division of eddies. This is the earliest and simplest of such mechanism, known as K41 (Frisch, 1995; Kolmogorov, 1941a, 1941b, 1941c, 1941d).

Figure 13.4.22  Example of uniform turbulent flow generated downstream of a lattice (Photograph by T. Coke and H. Nagib) (Frisch, 1995; Van Dyke, 1982). Reprinted with permission of the publisher (Cambridge University Press).

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Figure 13.4.23  Simplest example of the energy-cascade mechanism in terms of successive divisions of eddies, from largest size at injection down to smallest at dissipation, where fulfilling space is assumed by Kolmogorov (Frisch, 1995).Reprinted with permission of the publisher (Cambridge University Press).

If we plot the energy spectra against the size (wave number), schematically it looks like Figure 13.4.24 (in normal plot) or Figure 13.4.25 (in log–log plot). The energy-spectral distribution exhibits a universal feature of the uniform turbulent flow, which is composed of three regions in terms of energy, that is, the injection, flux, and dissipation. The wave number-dependent energy is defined as

E ( k, t ) 

k2 k



3

 u j ( k, t )

2

, (13.4.1)

k k j 1 k   k k  2 2

where u j ( k, t ) is the Fourier component of the velocity of a fluid given as a function of wave vector k and time. Theoretical studies about well-developed turbulent flow have been started from an approach via dimensional analysis based on the self-similarity hypothesis of the velocity field by Andrey Nikolaevich Kolmogorov (1903–1987) (Frisch, 1995). Figure 13.4.26 presents the schematics of Kolmogorov’s approach and includes his portrait. Kolmogorov considered that the energy spectrum of a turbulent flow must be

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13.4  Global–Local Structure of Stress and Strain Fields

Figure 13.4.24  Schematics of the energy-spectral distribution for uniform turbulence as a function of the wave number in a normal plot.

Figure 13.4.25  Schematics of the energy-spectral distribution for uniform turbulence as a function of the wave number in a log–log plot.

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Cooperation of Multiple Inhomogeneous Fields

Figure 13.4.26  Summary of the energy-spectrum function for turbulence proposed by Kolmogorov and Kolmogorov’s similitude law, together with his portrait.



specified by the viscosity ν and the energy-dissipation rate ε , and proposed an explicit expression for this, purely based on the dimensional analysis, as  k  E ( k, t )   5 / 4  1 / 4 e   . (13.4.2)  kK  Here, e(k / kK ) is a dimensionless function for the wave-number dependency, with kK being the Kolmogorov wave number defined as kK  ( /  3 )1/ 4 , below which the viscosity-based dissipation prevails. This is called Kolmogorov’s similitude. Surprisingly, this simple expression can explain the overall trend of the energy-spectral distribution that stems from the energy cascade as eddies, as indicated in Figure 13.4.27. In the figure, since the ordinate represents the normalized energy by  1/ 4 5 / 4, while the abscissa denotes the normalized wave number by kK  ( /  3 )1/ 4 , the slope of the linear region results in –5/3. This is why this theory for E ( k, t ) is called “Kolmogorov’s –5/3 (minus five-thirds) law.” For a large enough Reynolds number, the relationship can be approximated as



E (k )  CK  2 / 3 k 5 / 3 , with CK = 1.4 1.8 . Further models, such as the β model, consider a mixture of the active and inactive regions of large eddies, and more recent multifractal models explicitly take into account

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13.4  Global–Local Structure of Stress and Strain Fields

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Figure 13.4.27  Correlation of the experimental data with various Reynolds numbers on Kolmogorov’s similitude representation (Kida and Goto, 1997; Kida and Yanase, 1999). Adapted with permission from the publisher.

the intermittency of the twirling flow (Frisch et al., 1978). Landau and Lifshitz (1987) pointed out the growing inhomogeneity, that is, fluctuation in the energy-dissipation rate, with decreasing spatial scale, leading to spatiotemporal intermittency. In simulations of turbulence, treatments of the energy-dissipation region become crucially important because the failure of this will cause unrealistic eddy evolutions due to the stagnation of energy within this. The energy to be dissipated at the smaller eddies should normally be replaced by an appropriate model. The above situation is quite the same as our polycrystalline plasticity issue in the sense that the energy stagnation in the k/d