Contents
Chapter 1： Background and Signi.cance ..................... 1
1.1 Framework of This Book............................................... 1
1.2 Polycrystalline and Single-Crystal Plasticity ......................... 3
1.3 Size Effect on Crystal Plasticity at Micron and Submicron Scales..............................................5
1.3.1 Size Effect Observed in Material Experiments ................................................ 5
1.3.2 Size Effect of Yield Stress ........................................... 7
1.3.3 Strain Burst and Dislocation Avalanches........................................................12
1.3.4 Size Effect of Submicron Crystal Under Cyclic Loading.............................. 14
1.3.5 Size Effect of Deformation Morphology of Compressed Micropillars .............................................. 16
1.4 Method to Bridge Size Effect............................................ 17
1.4.1 Supersurface From Macro to Micron..................................17
1.4.2 Nonlocal Crystal Plasticity .................................................... 19
1.4.3 Discrete Dislocation Dynamics Simulation Method............. 21
1.5 Content of This Book .................. 22
Part 1 Continuum Dislocation Mechanism-Based Crystal Plasticity ......27
Chapter 2： Fundamental Conventional Concept of Plasticity Constitution .............. 29
2.1 Introduction ............................................. 29
2.2 One-Dimensional Plasticity ............................................ 30
2.2.1 Isotropic Hardening.................................................... 30
2.2.2 Kinematic Hardening ............................ 33
2.2.3 Rate-Dependent Plasticity............................................... 35
2.3 Multiaxial Plasticity ....................................................... 37
2.3.1 Hypoelastic-Plastic Materials .................................... 37
2.3.2 Small Strain Plasticity .................................................. 41
2.4 J2 Flow Theory Plasticity ............................................... 42
2.4.1 Kirchhoff Stress Formulation of J2 Flow Theory Plasticity .......................... 42
2.4.2 Extension to Kinematic Hardening ............................ 44
2.4.3 Large Strain Viscoplasticity......................................... 46
Contents
2.5 Rock-Soil Constitutive Model .......................................... 47
2.5.1 Mohr-Coulomb Constitutive Model ...................... 47
2.5.2 Drucker-Prager Constitutive Model..................... 49
2.6 Gurson Model for Porous Elastic-Plastic Solids.............. 51
2.7 Corotational Stress Formulation ..................... 54
2.8 Summary ............................................ 56
Chapter 3： Strain Gradient Plasticity Theory at the Microscale............................ 57
3.1 Size Dependence of Material Behavior at the Microscale.................................57
3.2 Couple Stress Theory...............................................60
3.2.1 Couple Stresses ............................................ 60
3.2.2 Rotation and Rotation Gradient..................................... 62
3.2.3 Virtual Work Principle..................................................64
3.2.4 Constitutive Relation of Couple Stress Strain Gradient Plasticity Theory ................................... 65
3.2.5 Principles of Minimum Potential Energy and Minimum Complementary Energy .......................................... 66
3.2.6 Equivalent Stress and Equivalent Strain ................... 67
3.3 Stretch and Rotation Gradient Theory ................. 70
3.3.1 Strain Gradient Tensor............................71
3.3.2 Decomposition of Strain Gradient Partial Tensor h0 and Total Equivalent Strain ESG............................... 72
3.3.3 Constitutive Relation of Stretch and Rotation Gradient Strain Gradient Plastic Theory ........................... 76
3.4 Microscale Mechanism-Based Strain Gradient Plasticity Theory ......................................... 78
3.4.1 Experimental Law for Strain Gradient Plasticity Theory...................... 79
3.4.2 Motivation for Microscale Mechanism-Based Strain Gradient Plasticity Theory..................................81
3.4.3 Microscale Computation Framework ................... 82
3.4.4 Dislocation Model.......................................................84
3.4.5 Constitutive Equation of Mechanism-Based Strain Gradient Plasticity Theory...........................................84
3.4.6 Size of Cell Element at the Microscale .............. 86
3.4.7 Mechanism-Based Strain Gradient Plasticity Predicts Stress Singularity at Crack Tip ........................... 88
3.5 Summary ........................................ 89
Chapter 4： Dislocation-Based Single-Crystal Plasticity Model .................... 91
4.1 Introduction ...................................... 92
4.2 General Constitutive Model for Single Crystals.................92
4.2.1 Basic Kinematics of Crystal Plasticity............... 92
4.2.2 Slip Rate and Dislocation Density Evolution .................. 95
4.2.3 Plastic Stress Required for Dislocation Motion .............. 102
4.2.4 Update of Cauchy Stress in Single-Crystal Plasticity ........ 103
Contents
4.3 Higher-Order Dislocation Dynamics-Based Crystal Plasticity Model .................... 104
4.3.1 Governing Equations of Macroforces .................................... 104
4.3.2 Governing Equations of Microforces ................... 105
4.3.3 Coupling of Macroscopic and Microscopic Equations.......... 108
4.4 Size and Bauschinger Effect in Passivated Thin Films.........109
4.4.1 Two Hardening Mechanisms Caused by Geometrically Necessary Dislocations .......................................... 109
4.4.2 Model Description........................ 110
4.4.3 Size Effect of Passivated Thin Films Under Tension......... 112
4.4.4 Bauschinger Effect of Passivated Thin Films During Unloading ....................... 113
4.5 Summary ........................................ 118
Chapter 5： Revealing the Size Effect in Micropillars by Dislocation-Based Crystal Plasticity Theory ..................................... 121
5.1 Introduction ........................... 121
5.2 Strain Burst and Size Effect in Compression Micropillars .............................. 122
5.2.1 Stochastic Crystal Plasticity Model................... 122
5.2.2 Determination of Size-Dependent Slip Resistance....................124
5.2.3 Strain Bursts at Small Scales ......................................... 130
5.2.4 Application to the Compression of Single-Crystal Ni Micron Pillars ........................................................ 133
5.3 Size-Dependent Deformation Morphology of Micropillars.............................. 137
5.3.1 Simulation Setups ....................................................... 138
5.3.2 Size-Dependent Deformation Morphology ................................................... 140
5.3.3 Role of Short-Range Back Stress..................................... 143
5.3.4 Critical Transition Size ....................................................... 143
5.3.5 Discussions of Material Softening .................................. 147
5.4 Summary .................................... 149
Chapter 6： Microscale Crystal Plasticity Model Based on Phase Field Theory............................................... 151
6.1 Introduction ........................................... 151
6.2 Theoretical Model............................................153
6.2.1 Basic Equations of Crystal Plasticity Theory............................153
6.2.2 Phase Field Description of Plastic Slip....................................... 154
6.2.3 Stored Energy and Dissipated Energy ....................................... 155
6.2.4 Principle of Virtual Power..........................157
6.2.5 Coupled Balance Equations...........................158
6.2.6 Finite Element Discretization ............................. 159
6.3 Computational Demonstrations............................ 160
6.3.1 Dislocation Near a Free Surface ....................... 161
6.3.2 Dislocation in an Anisotropic Material...................162
6.3.3 Dislocation Near a Bimaterial Interface ............ 163
Contents
6.4 Applications to Heteroepitaxial Structures ................ 165
6.4.1 Critical Shell Thickness of Core-Shell Nanopillars ...... 166
6.4.2 Dislocations in Heteroepitaxial Thin Films ........... 169
6.5 Summary .............................................. 172
Part 2 Discrete Dislocation Mechanism-Based Crystal Plasticity ..........................................175 Chapter 7： Discrete-Continuous Model of Crystal Plasticity at the Submicron Scale ................................. 177
7.1 Discrete Dislocation Dynamics ............................................. 177
7.1.1 Dislocation Kinetic.............................. 179
7.1.2 Dislocation Interactions and Topology Update......................... 181
7.1.3 Dislocation Cross-Slip .................................... 182
7.1.4 Current Three-Dimensional Discrete Dislocation Dynamics Simulations .................................. 183
7.2 Coupling Discrete Dislocation Dynamics With Finite Element Method.............................................185
7.2.1 Superposition Method...................................185
7.2.2 Discrete-Continuous Model ......................... 187
7.3 Improved Discrete-Continuous Model .................... 189
7.3.1 Ef.cient Regularization Method .................. 189
7.3.2 Image Force Calculation.................... 198
7.3.3 Finite Deformation.................................203
7.4 Application to Heteroepitaxial Films ................................... 209
7.4.1 Thermoelastic Calculation to Determine Internal Stress Field........................................................209
7.4.2 In.uence of Substrate Thickness on Dislocation Behavior...................................... 210
7.5 Application to Irradiated Materials .................................. 213
7.6 Summary ......................................... 217
Chapter 8： Single-Arm Dislocation Source SAS-Controlled Submicron Plasticity ............................... 219
8.1 Introduction .................................. 219
8.2 Single-Arm Dislocation Source Mechanisms at Submicron Scales......................................221
8.3 Single-Arm Dislocation Source-Controlled Strain Burst and Dislocation Avalanche ................................. 222
8.4 Description of Single-Arm Dislocation Source-Controlled Plasticity ............. 226
8.4.1 Single-Arm Dislocation Source-Controlled Dislocation Density Evolution ...................................... 226
8.4.2 Effective Single-Arm Dislocation Source Length ............................... 230
8.4.3 Single-Arm Dislocation Source-Controlled Flow Stress ..................... 232
8.5 Summary ......................................... 237
Contents
Chapter 9： Con.ned Plasticity in Micropillars........................................ 239
9.1 Insights into Coated Micropillar Plasticity ...................................... 240
9.1.1 Stress-Strain Curves in Coated and Uncoated Pillars ......................... 240
9.1.2 Dislocation Source Mechanism in Coated Micropillars........................241
9.1.3 Back Stress in Coated Micropillars...................................... 244
9.1.4 Evolution of Mobile and Trapped Dislocation ............................... 245
9.2 Implications for Crystal Plasticity Model .................................... 247
9.3 Theoretical Models for Coated Micropillars........................... 252
9.3.1 Dislocation Density Evolution Model..............................253
9.3.2 Prediction of Stress-Strain Curve...................................255
9.4 Brief Discussion on Coating Failure Mechanism .......... 257
9.4.1 High Hoop Stress of Coated Layer......................................258
9.4.2 Transmission Effect of Dislocations Across Coating ......................... 258
9.5 Summary ................................... 261
Chapter 10： Mechanical Annealing Under Low-Amplitude Cyclic Loading ........................................ 263
10.1 Introduction ........................ 263
10.2 Cyclic Behavior of Collective Dislocations.......................264
10.3 Cyclic Instability of Dislocation Junction............................... 268
10.3.1 Glissile Dislocation Junction ............................... 268
10.3.2 Sessile Dislocation Junction ........................... 272
10.4 Cyclic Enhanced Dislocation Annihilation Mechanism ............................. 274
10.5 Dislocation Density In.uenced by Cyclic Slip Irreversibility ................................ 275
10.6 Critical Size for Mechanical Annealing................... 278
10.7 Summary .................................. 280
Chapter 11： Strain Rate Effect on Deformation of Single Crystals at Submicron Scale....................... 281
11.1 Introduction .......................................... 281
11.2 Strain Rate Effect on Flow Stress in Single-Crystal Copper Under Compression Loading...................................282
11.2.1 Strain Rate Effect of Submicron Copper Pillars Under Uniaxial Compression .................................... 283
11.2.2 Strain Rate Effect of Dislocation Evolution in Copper Cubes Under Hydrostatic Pressure............................289
11.3 Strain Rate Effect on Dynamic Deformation of Single-Crystal Copper Under Tensile Loading .................................... 298
11.3.1 Resolution of Discrete Dislocation Dynamics.......................299
11.3.2 Coupling Dislocation Dynamics Plasticity With Finite Element..........................................300
11.3.3 Model Description and Simulation Results ........................ 303
Contents
11.4 Shock-Induced Deformation and Dislocation Mechanisms in Single-Crystal Copper ........................... 313
11.4.1 Dynamic Mechanical Behavior Corresponding to Dislocation Microstructure....................................313
11.4.2 Dynamic Multiscale Discrete Dislocation Plasticity Model .................... 315
11.4.3 Coarse-Grained Homogeneous Nucleation Model ....... 316
11.4.4 Shock-Induced Plasticity at the Submicron Scale .......................... 320
11.4.5 Discussion and Conclusion.............................327
11.5 Summary ............................................. 328
內容試閱：
近几十年，金属材料力学行为的多时空尺度理论、计算和实验研究取得了突飞猛进的发展，在原子尺度和连续介质尺度均成果丰硕。但是，关联原子尺度与连续介质尺度的力学问题，却成为力学理论和计算继续取得突破的关键。例如，在20世纪90年代，著名的细铜丝扭转和薄梁弯曲实验展示了几十微米至几百微米的强度尺寸效应，诞生了应变梯度塑性理论。21世纪初，几百纳米至几十微米直径金属单晶柱的压缩实验展示了强度和变形的尺寸效应，然而，横截面上不存在应变梯度，这些实验现象形成了跨纳米至细观尺度的新的力学挑战性问题，这些问题正是本课题组的研究初衷。本书所展示的基于位错机制的晶体塑性理论和计算模型汇集了面对这一问题国内外的相关研究进展，重点介绍了本课题组近十几年来的研究成果。
本书是国际上首部诠释在100纳米至10微米尺度的晶体塑性力学专著。书中的第一部分展示了基于位错机制的连续化晶体塑性理论，包括应变梯度理论、可考虑尺寸效应和变形间歇性的晶体塑性理论、微尺度相场理论等。第二部分展示了基于位错机制的离散化晶体塑性模型，包括位错动力学与有限元耦合计算方法，揭示了在微纳米尺度下的强度和变形的尺寸效应，界面、温度和应变率效应等机制。书中展示的理论和计算方法一方面可以应用于微尺度下金属材料的反常规力学行为，如强度的尺寸效应、变形的间歇突跳、高温下的退火硬化等，进而指导微器件的设计、缺陷调控及可靠性评估等，对微尺度材料与结构的应用具有重要意义。另一方面，金属材料中的变形和失效过程如断裂、剪切失稳等发生在微尺度，相关理论和计算方法为建立基于内在微观机制的塑性力学和断裂力学理论模型提供了新的思路。
自主创新的不变初心、挑战难题的使命担当、均无向隅的精诚合作、顶天立地的奋斗精神，是支撑我们进行科学研究和完成这部专著的动力。同时，特别致谢黄克智院士始终不渝的鼓励和支持，多次与我们在塑性理论和固体本构关系方面进行有益的讨论。感谢清华大学出版社的石磊主任和戚亚编辑在出版上的帮助。
庄茁，柳占立，崔一南
2019年11月，于清华园
i
Preface
This book aims to provide a comprehensive introduction to the theoretical models and computational methods of dislocation mechanism-based crystal plasticity at the micron and submicron scales 100 nme100 mm. The popular notion that smaller is stronger has accentuated the need to introduce a size effect. Contrary to the prevalent notion that prestrain leads to hardening and annealing leads to softening, a small-scaled material exhibits softening when it is prestrained and anneal hardening when the temperature is increased. It is also interesting that a variety of a typical mechanical behaviors are observed in the crystal when the material size is reduced from the macroscopic to the micron to submicron scales： for example, size-dependent yield stress and intermittent plastic .ow during compression tests of micron pillars. The widely observed intermittent plastic .ow at small scales prompts a paradigm shift away from traditional well-de.ned continuous and determined plastic .ow behavior. These unconventional features have attracted a great deal of academic attention worldwide in the engineering mechanics and materials community. Much research work has been conducted. This book attempts to provide insights into understanding micron and submicron plasticity and to present a theoretical framework and simulation efforts.
This book is structured in two parts. Part I focuses on continuum dislocation mechanism-based crystal plasticity. In contrast, Part II focuses on discrete dislocation mechanism-based crystal plasticity. Part I, including Chapters 2e6, presents the continua dislocation mechanism-based crystal plasticity theory and computations, which include the fundamental concept of conventional crystal plasticity theory without a size effect at macroscale; the strain gradient crystal plasticity theory based on the Taylor law dislocation mechanism considering a size effect at micron scale; the developed dislocation-based crystal plasticity model within the framework of continuum mechanics theories by introducing the dislocation mechanism revealed by experiments and dislocation dynamic simulations at the submicron scale; and the phase-.eld theory of crystal plasticity. Part II, including Chapters 7e12, describes the discrete dislocation mechanism-based theory and computations at the submicron scale, which includes the single-crystal plasticity theory and the discrete-continuous model of crystal plasticity by coupling three-dimensional discrete dislocation dynamics and the .nite element method. Three kinds of plastic deformation mechanisms for submicron pillars are systematically presented： single-arm dislocation source-controlled plastic .ow in micropillars, con.ned plasticity in coated micropillars, and dislocation
iii
Preface
starvation under low-amplitude cyclic loading in micropillars. Flow stress is deduced according to the single-arm dislocation source mechanism, which is a signi.cant achievement in crystal plasticity theoretical work at a submicron scale. Two other inter-esting issues regarding crystal plasticity related to discrete dislocation evolution are described. One is dislocation nucleation and multiplication at a high strain rate, with examples of compression micropillars. The other is the temperature effect for the disloca-tion annihilation mechanism considering a void diffusion-based dislocation climb model and helical dislocations based on a coupled glide-climb model.
The plasticity theory formula, computational modeling, and experiment data of crystalline materials at the micron and submicron scales are discussed in this book. The focus is mainly on dislocation motion-controlled plastic behavior in crystals. Understanding the properties of crystalline materials to be captured by an appropriate constitutive model is the key to modeling large plastic deformation. Many books describe crystal plasticity theory models, computation methods, and experiment data at the macroscale. However, few books are dedicated to describing crystal plasticity covering the micron and submicron scales; thus, it may be that this is the .rst book on both continua and discrete dislocation mechanism-based crystal plasticity theories and computations at these scales.
There are more than 600 related articles listed in references, more than 25 of which were published by our group. This book originated from doctoral thesis research work during 2004e2018 at the Department of Engineering Mechanics, Tsinghua University in Beijing. The contributors were Yu Guo, Xiaoming Liu, Zhanli Liu, Junfeng Nie, Zhaohui Zhang, Xuechuan Zhao, Yuan Gao, Yinan Cui, Peng Lin, Liyuan Wang, Jianqiao Hu, and Fengxian Liu. Specially, Prof. Zhanli Liu and Dr. Yinan Cui are coauthors of this book.
We sincerely appreciate Prof. Keh-Chih Hwang, who gave us a lot of encouragement and fruitful discussion to understand theory models and constitutive relations in solid mechanics.
We would like to thank Mr. Lei Shi and Ms. Ya Qi at Tsinghua University Press for helping publish this book.
Zhuo Zhuang, Zhanli Liu, Yinan Cui
Department of Engineering Mechanics, School of Aerospace Engineering, Tsinghua University, Beijing, China
iv
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