Multiscale Analysis of Deformation and Failure of Materials

Contents......Page 9
About the Author......Page 23
Series Preface......Page 25
Preface......Page 27
Abbreviations......Page 29
1.1.1 Property-structure Relationship at Fundamental Scale......Page 31
1.1.3 Upgrading Products Based on Material Structure-property Relationships......Page 32
1.1.4 Exploration of In-depth Mechanisms for Deformation and Failure by Multiscale Modeling and Simulation......Page 33
1.2.2 Classification Based on Multiscale Modeling Schemes......Page 34
1.2.3 Classification Based on the Linkage Feature at the Interface Between Different Scales......Page 35
1.3.2 Links Between the Two Classes of Multiscale Analysis......Page 36
1.4 Examples in Formulating Multiscale Models from Practice......Page 37
1.4.1 Cyclic Creep (Ratcheting) Analysis of Pearlitic Steel Across Micro/meso/macroscopic Scales......Page 38
1.4.2 Multiscale Analysis for Brittle-ductile Transition of Material Failure......Page 40
1.5 Concluding Remarks......Page 42
References......Page 43
2.1.1 Characteristics, History and Trends......Page 45
2.1.2 Application Areas of Atomistic Simulation......Page 46
2.1.3 An Outline of Atomistic Simulation Process......Page 47
2.2.1 The Relation Between Interatomic Force and Potential Function......Page 49
2.2.2 Physical Background and Classifications of Potential Functions......Page 50
2.3 Pair Potential......Page 51
2.3.1 Lennard-Jones (LJ) Potential......Page 52
2.3.2 The 6-12 Pair Potential......Page 53
2.3.3 Morse Potential......Page 54
2.3.4 Units for Atomistic Analysis and Atomic Units (au)......Page 55
2.4.1 Motion Equation of Particles......Page 57
2.4.2 Verlet Numerical Algorithm......Page 59
2.4.3 Velocity Verlet (VV) Algorithm......Page 60
2.5 Geometric Model Development of Atomistic System......Page 61
2.6.1 Periodic Boundary Conditions (PBC)......Page 65
2.6.2 Non-PBC and Mixed Boundary Conditions......Page 66
2.7.2 Nvt Ensemble......Page 67
2.7.3 Npt Ensemble......Page 68
2.8.2 Data Analysis Based on Statistical Mechanics......Page 69
2.9 Statistical Simulation Using Monte Carlo Methods......Page 70
2.9.1 Introduction of Statistical Method......Page 71
2.9.2 Metropolis-Hastings Algorithm for Statics Problem......Page 72
2.9.4 Adsorption-desorption Equilibrium......Page 73
2.10 Concluding Remarks......Page 80
References......Page 81
3.1.1 Applications of High-performance Ceramics......Page 83
3.1.2 Ceramic Atomic Bonds in terms of Electronegativity......Page 84
3.2.2 Born-Mayer and Buckingham Potentials......Page 85
3.3 Shell Model......Page 86
3.4.1 Basic Assumptions......Page 88
3.4.2 General Methods in Determining Potential Parameters......Page 89
3.4.3 Three Basic Methods for Potential Parameter Determination by Experiments......Page 90
3.5 Applications in Ceramics: Defect Structure in Scandium Doped Ceria Using Static Lattice Calculation......Page 91
3.6.1 Background......Page 94
3.6.2 Structure and Defect Mechanisms of YAG Garnets......Page 95
3.6.3 Simulation Method and Results......Page 96
3.7.1 MD Simulation of the Motion of Oxygen Ions in SOFC......Page 98
3.8.1 Introduction of the Abell-Tersoff Bonder-order Approach......Page 101
3.8.2 Tersoff and Brenner Potential......Page 102
3.9 The Atomistic Stress and Atomistic-based Stress Measure......Page 105
3.9.2 The Computation Form for the Virial Stress......Page 106
3.9.3 The Atomistic-based Stress Measure for Continuum......Page 108
3.11.1 Basic EAM Formulation......Page 109
3.11.2 EAM Physical Background......Page 111
3.11.3 EAM Application for Hydrogen Embrittlement......Page 112
3.11.4 Modified Embedded Atom Method (MEAM)......Page 113
3.11.5 Summary and Discussions......Page 115
3.12 Constructing Binary and High Order Potentials from Monoatomic Potentials......Page 117
3.12.2 Determination of Parameters in Morse and Exponential Potentials for Unlike Atoms......Page 118
3.12.3 Determination of Parameters in EAM Potentials for Alloys......Page 119
3.13 Application Examples of Metals: MD Simulation Reveals Yield Mechanism of Metallic Nanowires......Page 120
3.14 Collecting Data of Atomistic Potentials from the Internet Based on a Specific Technical Requirement......Page 122
3.14.2 Physical and Chemical Vapor Deposition to Produce Ceramics Thin Coating Layers on Steel Substrate......Page 123
3.14.3 Technical Requirement for Potentials and Searching Results......Page 124
3.14.4 Using Obtained Data for Potential Development and Atomistic Simulation......Page 125
Appendix 3.A Potential Tables for Oxides and Thin-Film Coating Layers......Page 126
References......Page 131
4.1 Determination of Uranium Dioxide Atomistic Potential and the Significance of QM......Page 135
4.2 Some Basic Concepts of QM......Page 136
4.3 Postulates of QM......Page 137
4.4 The Steady State Schrödinger Equation of a Single Particle......Page 143
4.5 Example Solution: Square Potential Well with Infinite Depth......Page 144
4.5.1 Observations and Discussions......Page 145
4.6.1 General Expression of the Schrödinger Equation and Expectation Value of Multi-body Systems......Page 146
4.6.2 Example: Schrödinger Equation for Hydrogen Atom Systems......Page 147
4.6.3 Variation Principle to Determine Approximate Ground State Energy......Page 148
4.7 Three Basic Solution Methods for Multi-body Problems in QM......Page 149
4.7.2 An Approximate Method......Page 150
4.8 Tight Binding Method......Page 151
4.9.1 Hartree Method for a Multi-body Problem......Page 153
4.9.2 Hartree-Fock (HF) Method for the Multi-body Problem......Page 154
4.10 Electronic Density Functional Theory (DFT)......Page 155
4.11.1 Energy Linkage Between QM and Atomistic Simulation......Page 157
4.11.3 Using Spline Functions to Express Potential Energy Functions......Page 158
4.11.4 A Systematic Method to Determine Potential Functions by First-principle Calculations and Experimental Data......Page 159
4.12 Concluding Remarks......Page 160
Appendix 4.A Solution to Isolated Hydrogen Atom......Page 161
References......Page 162
5.1 Introduction......Page 163
5.1.4 Plan for Study of Concurrent Multiscale Methods......Page 164
5.2 The Geometric Model of the GP Method......Page 165
5.3.1 Two Imaginary Domains Next to the Scale Boundary......Page 168
5.3.3 Mechanisms for Seamless Transition......Page 169
5.3.4 Linkage of Position Vectors at Different Scales by Spatial and Temporal Averaging......Page 170
5.4 Verification of Seamless Transition via 1D Model......Page 171
5.5 An Inverse Mapping Method for Dynamics Analysis of Generalized Particles......Page 176
5.6 Applications of GP Method......Page 180
5.7 Validation by Comparison of Dislocation Initiation and Evolution Predicted by MD and GP......Page 181
5.8 Validation by Comparison of Slip Patterns Predicted by MD and GP......Page 185
5.9 Summary and Discussions......Page 186
5.10.1 MAAD Concurrent Multiscale Method......Page 189
5.10.2 Incompatibility Problems at Scale Boundary Illustrated with the MAAD Method......Page 190
5.10.4 Coupling Atomistic Analysis with Discrete Dislocation (CADD) Method......Page 191
5.10.6 Embedded Statistical Coupling Method (ESCM) with Comments on Direct Coupling (DC) Methods......Page 192
5.10.7 Conclusion......Page 193
References......Page 194
6.1 Introduction......Page 197
6.2 Principle of Minimum Potential Energy of Solids and Structures......Page 198
6.2.2 Work Potential......Page 199
6.3.1 Discretization of Continuum Domain BC into Finite Elements......Page 200
6.3.2 Using Gaussian Quadrature to Calculate Element Energy......Page 201
6.3.3 Work Potential Expressed by Node Displacement Matrix......Page 202
6.3.4 Total Potential Energy P Expressed by Node Displacement Matrix......Page 203
6.3.5 Developing Simultaneous Algebraic Equations for Nodal Displacement Matrix......Page 205
6.4.1 Formulation of Representative Atoms and Total Potential Energy in the QC Method......Page 208
6.4.2 Using Interpolation Functions to Reduce Degrees of Freedom......Page 209
6.4.3 Model Division......Page 210
6.4.4 Using the Cauchy-Born Rule to Calculate Energy Density Function W from Interatomic Potential Energy......Page 211
6.4.5 The Solution Scheme of the QC Method......Page 213
6.4.8 Ghost Force......Page 214
6.5.1 Energy-based Non-local QC Model (CQC(m)-E)......Page 217
6.6 Applications of the QC Method......Page 218
6.6.1 Nanoindentation......Page 219
6.6.2 Crack-tip Deformation......Page 220
6.6.5 Polarizations Switching in Ferroelectrics......Page 222
6.7 Short Discussion about the QC Method......Page 223
Part 6.3 Analytical and Semi-analytical Multiscale Methods Across Atomic/Continuum Scales......Page 224
6.8.1 Mathematical Definition of Deformation Gradient F(X)......Page 225
6.8.2 Determination of Lattice Vectors and Atom Positions by the Cauchy-Born Rule through Deformation Gradient F......Page 226
6.8.3 Physical Explanations of Components of Deformation Gradient......Page 227
6.8.4 Expressions of F and ε Components in Terms of Displacement Vector......Page 228
6.8.5 The Relationship Between Deformation Gradient, Strain and Stress Tensors......Page 230
6.9.1 Application of the Cauchy-Born Rule in a Centro-symmetric Structure......Page 231
6.9.2 Determination of Interatomic Length rij and Angle θijk of the Crystal after Deformation by the Cauchy-Born Rule......Page 232
6.9.3 A Short Discussion on the Precision of the Cauchy-Born Rule......Page 234
6.10.2 Interatomic Potentials Used for Atom/Continuum Transition......Page 235
6.10.3 The Atomistic-based Continuum Theory of Hydrogen Storage......Page 236
6.10.4 Atomistic-based Continuum Modeling to Determine the Hydrogen Density and Pressure p......Page 241
6.10.5 Continuum Model of Interactions Between the CNT and Hydrogen Molecules and Concentration of Hydrogen......Page 242
6.10.6 Analytical Solution for the Concentration of Hydrogen Molecules......Page 246
6.10.7 The Double Wall Effects on Hydrogen Storage......Page 247
6.11 Atomistic-based Model for Mechanical, Electrical and Thermal Properties of Nanotubes......Page 248
6.11.2 Mechanical Properties......Page 249
6.11.3 Electrical Property Change in Deformable Conductors......Page 250
6.11.4 Thermal Properties......Page 251
6.12 A Proof of 3D Inverse Mapping Rule of the GP Method......Page 252
References......Page 253
7.1.1 Interface Design of the DC Multiscale Models......Page 257
7.1.2 Connection and Compatibility Between Atom/Continuum at the Interface......Page 258
7.2.1 Energy-based and Force-based Formulation......Page 259
7.2.2 Constitutive Laws in the Formulation......Page 260
7.3.1 Partitioning and Coupling of Model Region......Page 261
7.3.2 System Energy and Hamiltonian in Different Regions......Page 263
7.3.3 Handshake Region Design......Page 264
7.4 Force-based Formulation of Concurrent Multiscale Modeling......Page 265
7.5.1 Realization of Force-based Formulation for CADD/FEAt......Page 266
7.5.2 Basic Model for CADD......Page 267
7.5.3 Solution Scheme: A Superposition of Three Types of Boundary Value Problems......Page 268
7.6.1 The Internal Force and Equivalent Mass of a Dynamic System......Page 270
7.6.2 Derivation of the FE/MD Coupled Motion Equation......Page 272
7.6.3 Numerical Example of the Coupling Between MD and FE......Page 274
7.6.4 Results and Discussion......Page 275
7.7 Bridging Domains Method......Page 276
7.8 1D Benchmark Tests of Interface Compatibility for DC Methods......Page 278
7.9.1 The Benchmark Computation Test......Page 281
7.10 The Embedded Statistical Coupling Method (ESCM)......Page 284
7.10.3 MD/FE Interface......Page 285
7.10.4 Surface MD Region......Page 287
References......Page 288
8.1.1 Schematic View of Hierarchical Multiscale Analysis......Page 291
8.2.1 Basic Assumption......Page 293
8.2.2 Introduction to Self-consistent Schemes (SCS)......Page 294
8.2.3 Weakening Constraint Effect of Aggregate on Inclusion with Increase of Plastic Deformation......Page 295
8.2.4 Quantitative Linkage of Variables Between Mesoscopic and Macroscopic Scales......Page 296
8.3.1 Several Basic Elements of Continuum Plasticity Theory......Page 297
8.3.2 Description of Continuum Plasticity Theory Within Deviatoric Stress Space......Page 298
8.4.1 Internal Variable Theory Expressed by a Mechanical Model......Page 300
8.4.2 Calculation of Back Stress Rij in Terms of Plastic Strain......Page 302
8.4.3 Expressing Elastoplastic Constitutive Equations for Each Constituent Phase......Page 303
8.5.1 Developing Meso-cell (Inclusion) Constitutive Equations......Page 304
8.5.2 Bridging Micro- and Macroscopic Variables via the Meso-cell Constitutive Equation......Page 305
8.6 Determining Size Effect on Yield Stress and Kinematic Hardening Through Dislocation Analysis......Page 306
8.6.2 Expressing Size Effects on Yielding and Hardening Behavior by Dislocation Pile-up Theory......Page 307
8.6.4 Equating Dislocation-obtained Shear Stress Increment with that Obtained by Continuum Plasticity Theory......Page 309
8.6.5 Explicit Expressions of Size Effects on Tangential Modulus and Kinematic Hardening Behavior......Page 310
8.7.2 Numerical Procedure for the Iterative Process......Page 311
8.7.3 How to Carrying on the Volume Averaging of Meso-cell Plastic Strain to Find Macroscopic Strain......Page 312
8.8 Experimental Study on Layer-thickness Effects on Cyclic Creep (Ratcheting)......Page 313
8.9.1 General Features of the Numerical Simulation......Page 314
8.9.2 Determination of Basic Material Parameters......Page 315
8.9.4 Comparison Between the Results of Three-scale Multiscale Simulation with Data of Cyclic Experiments......Page 316
8.10 Findings in Microscopic Scale by Multiscale Analysis......Page 318
8.11.1 Methods for Bridging Three Scales......Page 321
8.11.6 The Formulation and Important Role of Residual Stress......Page 322
Appendix 8.A Constitutive Equations and Expressions of Parameters......Page 323
Appendix 8.B: Derivation of Equation (8.12e) and Matrix Elements......Page 325
References......Page 327
9.1.1 The Role of Multiscale Analysis in Materials Design......Page 329
9.1.2 Issues of Bottom-up Multiscale Modeling in Deductive Material Design Process......Page 330
9.2 Introduction to Temporal Multiscale Problems......Page 331
9.2.2 Brief Introduction to Methods for Temporal Multiscale Problems......Page 332
9.3 Concepts of Infrequent Events......Page 334
9.4.1 Minimum Energy Path (MEP) and Saddle Point......Page 335
9.4.2 Nudged Elastic Band (NEB) Method for Finding MEP and Saddle Point......Page 336
9.4.3 Mathematical Description of the NEB Method......Page 340
9.4.4 Finding MEP and Saddle Point for a 2D Test Problem of LEPS Potential via Implementation of the NEB Method......Page 341
9.5.2 Examples and Impact (1): Strain Rate and Temperature Effects on Dislocation Nucleation from Free Surface of Nanowires......Page 348
9.5.3 Examples and Impact (2): Departure Between Plasticity and Creep Based on Activation Energy and Activation Volume......Page 350
9.5.4 Examples and Impact (3): Findings for Mechanisms of High Strength and High Ductility of Twin Nanostructured Metals......Page 351
9.5.5 Other Methods in Extending Time Scale in Atomistic Analysis......Page 352
9.6.1 Hierarchical Structure of Protein Materials......Page 354
9.6.2 Large Deformation and Dynamic Characteristics of Protein Material......Page 355
9.6.3 At Molecular (Nano) Scale: Molecular Dynamics Simulation of Dimer and the Modified Bell Theorem......Page 356
9.6.4 Unique Features of Deformation, Failure and Multiscale Analysis of Biomaterials with Hierarchical Structure......Page 358
9.7.2 At Atom-nano and Submicron scale: Selection of Implant Chemical Composition Based on Maximum Bonding Energy......Page 359
9.7.3 At Mesoscopic Scale (μm): Cell Adhesion Strength is Calculated and Characterized......Page 360
9.7.4 Discussion......Page 366
Appendix 9.A Derivation of Governing Equation (9.11) for Implicit Relationship of Stress, Strain Rate, Temperature in Terms of Activation Energy and Activation Volume......Page 367
References......Page 368
10.1.1 UNIX Operating System......Page 373
10.1.2 UNIX Shell Commands......Page 374
10.2 A Simple MD Program......Page 378
10.2.1 Five Useful Commands of Fortran 90......Page 379
10.2.2 Module and Subroutine......Page 383
10.2.3 Using crystal_M_simple.f90 to Create Initial Configuration......Page 384
10.2.4 Use multi1.f90 to Run a Molecular Dynamics Calculation......Page 385
10.3 Static Lattice Calculations Using GULP......Page 386
10.3.2 Input File Structure and Running GULP......Page 387
10.3.3 Structure Optimization and Output File Structure......Page 389
10.3.4 Determining Potential Parameters by Fitting Calculations......Page 392
10.3.5 Shell Model......Page 394
10.3.6 Defect Calculation......Page 395
10.4.1 Gnuplot......Page 397
10.4.2 Visual Molecular Dynamics (VMD)......Page 401
10.4.3 AtomEye......Page 404
10.5.1 Introduction......Page 407
10.5.3 General Features of DL_POLY_2 Files......Page 408
10.5.4 Compile and Run......Page 409
10.5.5 Units of Measure......Page 411
10.5.6 Input Files of DL_POLY......Page 412
10.5.7 Output Files......Page 414
10.5.8 Data-Processing for Variable Evolution Versus Time by the ela_STATIS.f90 Code......Page 416
10.5.9 Useful Tools for Operating and Monitoring MD Simulations......Page 417
10.6 Nve and npt Ensemble in MD Simulation......Page 419
10.6.1 Nve Simulation with DL_POLY......Page 420
10.6.2 Npt Simulation with DL_POLY......Page 423
10.6.3 Data Post-processing via STATIS and HISTORY Output Files......Page 424
10.7 Non-equilibrium MD Simulation of One-phase Model Under External Shearing (1)......Page 427
10.7.1 Features and Procedures of MD Simulation Under Shearing Strain Rate......Page 428
10.7.2 Preparation for Input Files and Running 3D npt Equilibration......Page 429
10.7.3 Post-processing Analysis for Equilibration Data......Page 433
10.8.1 Bi-periodic nvt Equilibration in 2D_EQUI_nvt......Page 434
10.8.2 Reference Position Calculation via Producing MEAN.xyz......Page 436
10.8.3 MD simulation Under Shearing Rate on the Top Layer......Page 438
10.8.4 Data Analysis Using ela_history_2009.f90 for Shearing......Page 439
10.8.5 Tips to Reduce Error When Using ela_history_2009.f90......Page 441
10.9.1 Dimensional Equilibration of the Individual Phase......Page 442
10.9.2 Developing the Initial Configuration for the Two-phase Model......Page 444
10.9.3 Run the 3D_npt Equilibration in the INI_CONF_coating Directory......Page 446
10.9.4 Non-equilibrium Simulation of the Coating Layer Under Top Shearing Strain Rate......Page 447
10.9.5 Post-data Processing to Determine the Displacement of the Coating Layer Under a Given Shearing Rate......Page 448
10.10.1 Introduction......Page 451
10.10.2 A Simple Simulation Using VMD and NAMD......Page 452
10.10.3 Post-processing Data Analysis......Page 455
10.11 Stretching of a Protein Module (1): System Building and Equilibration with VMD/NAMD......Page 456
10.11.1 Preparation of the Initial Configuration with VMD......Page 457
10.11.2 Preparation of the NAMD Input File......Page 459
10.11.4 Error Messages and Recommended Action......Page 460
10.12.1 Preliminary Steps......Page 461
10.12.2 Preparation of the NAMD Input Files......Page 463
10.12.4 Run NAMD Simulation and Data Processing......Page 465
10.13.1 General Features of LAMMPS......Page 467
10.13.3 Building LAMMPS and Run......Page 468
10.13.4 Examples......Page 470
10.14.1 Multiscale Model Development......Page 477
10.14.3 Running mpi Simulation for Multiscale Analysis and Data Processing......Page 480
10.A.2 Using the KNOPPIX CD to Install the GNU/Linux System......Page 482
10.A.3 ssh and scp......Page 483
10.A.4 Fortran and C Compiler......Page 484
10.A.5 Visual Molecular Dynamics (VMD)......Page 486
10.B.1 Program Structure, Write to Terminal and Write to File......Page 487
10.B.2 Do Cycle, Formatted Output......Page 489
10.B.3 Arrays and Allocation......Page 490
10.C.1 Introduction......Page 491
10.C.2 Simple Commands......Page 492
10.D.1 Force Calculation in Atomistic Simulation......Page 493
10.E.1 General Information......Page 494
10.E.2 Atom Decomposition......Page 495
10.E.4 Domain Decomposition......Page 496
Appendix 10.F Supplemental Materials and Software for Geometric Model Development in Atomistic Simulation......Page 497
10.F.1 Model Development for Model Coordinates Coincident with Main Crystal Axes......Page 498
10.F.2 Model Development for Model Coordinates not Coincident with Crystal Axes......Page 501
References......Page 503
Postface......Page 505
Index......Page 507