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
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