Handbook of micromechanics and nanomechanics

Front Cover......Page 1
Copyright......Page 5
Contents......Page 6
Preface......Page 26
1.1 Introduction......Page 28
1.2.1 Background......Page 29
1.2.2 Transfer Matrix Method......Page 31
1.3 Treatment of Damping......Page 35
1.3.1 Viscously DampedWaves in 1D Homogeneous Media......Page 36
1.3.2 Viscously Damped Waves in 1D Phononic Materials......Page 38
1.4 Treatment of Nonlinearity......Page 41
1.4.1 Finite-Strain Waves in 1D Homogenous Media......Page 42
1.4.1.1 Equation of motion......Page 43
1.4.1.2 Dispersion relation......Page 45
1.4.2 Finite-Strain Waves in 1D Phononic Materials......Page 48
References......Page 50
2.1 Introduction......Page 56
2.2.1 Negative Effective Mass......Page 59
2.2.2 Negative Effective Modulus......Page 65
2.3.1 Composites with Coated Sphere Inclusions......Page 67
2.3.2 Chiral Metamaterials......Page 75
2.4.1 Cloaking in Quasi-Static Approximation......Page 79
2.4.2 Transformation Acoustics and Elasticity......Page 84
2.4.3 Acoustic Imaging beyond the Diffraction Limit......Page 89
2.5 Conclusions......Page 93
References......Page 94
Phase Field Approach and Micromechanics in Ferroelectric Crystals......Page 100
3.1 Introduction......Page 101
3.2 The Fundamentals of Phase Field Approach......Page 102
3.2.1 Free Energy and the Constitutive Relations......Page 103
3.2.2 The Kinetics of Non-Equilibrium Process......Page 111
3.3 Applications of Phase Field Approach......Page 115
3.3.1 Applications in Ferroelectric Bulk Single Crystals......Page 116
3.3.2 Applications in Ferroelectric Bulk Polycrystals......Page 121
3.3.3 Applications in Ferroelectric Thin Films......Page 123
3.3.4 Applications in Ferroelectric Low Dimensional Structures......Page 128
3.4 The Fundamentals of Micromechanics Approach......Page 131
3.4.1 Constitutive Relations......Page 132
3.4.2 Gibbs Free Energy for Phase Transition and Domain Switch......Page 135
3.4.3 Thermodynamic Driving Force......Page 139
(i)......Page 142
(ii)......Page 143
(iii)......Page 144
3.5 Applications of Micromechanics Approach......Page 145
3.5.2 The Shift of Curie Temperature under Hydrostatic Pressure......Page 146
3.5.3 Hysteresis and Butterfly-Shaped Axial Strain vs. Electric Field Relations......Page 147
3.5.4 Double Hysteresis of a Ferroelectric Crystal Above TC......Page 148
C......Page 149
3.5.6 Development of Rank-1 and Rank-2 Domain Pattern, and Influence of a Compressive Stress in BaTiO3......Page 151
3.6 Concluding Remarks......Page 154
Appendix: The Coefficients in Equation (3.24)......Page 156
References......Page 158
4.1 Introduction......Page 168
DomainWalls......Page 175
4.2.2 Domain Walls with Point Defects......Page 176
4.2.3 Steps in DomainWalls......Page 177
4.3.1.1 Micro Canonical Ensemble Theory......Page 179
4.3.1.2 Canonical Ensemble Theory......Page 180
4.3.2 Shell Potentials......Page 181
4.3.3 Lattice Statics......Page 184
4.3.3.1 Defective Crystals and Symmetry Reduction......Page 185
4.3.3.2 Hessian Matrix for the Bulk Crystal......Page 187
4.3.4.1 Finite systems......Page 188
4.3.4.2 Perfect crystals......Page 191
4.3.4.3 Lattices with massless particles......Page 194
4.3.4.4 Defective Crystals......Page 197
4.3.5 Quasi-Newton Method......Page 200
4.4.1 Normal and Shear Strains......Page 202
4.4.1.1 Perfect domain walls......Page 203
4.4.1.2 Domain Walls with Oxygen Vacancies......Page 206
4.4.2 Steps in DomainWalls......Page 209
4.4.2.1 Pb–Pb steps......Page 210
4.4.2.3 Pb–Ti Steps......Page 214
4.4.3 External Electric Fields......Page 215
4.4.3.1 Perfect domain walls......Page 218
4.4.3.2 Defective domain walls......Page 221
4.4.4 Temperature......Page 223
References......Page 228
Micromechanics-Based Constitutive Modeling of Chain-Structured Ferromagnetic Particulate Composites......Page 238
5.1 Introduction......Page 239
5.2 Fundamental Solution to Magneto-Elastic Problems......Page 242
5.2.1 Modified Magnetic Green’s Functions......Page 243
5.2.2 Modified Elastic Green’s Functions Considering Body Force......Page 245
5.2.3 Modified Elastic Green’s Functions Considering Eigenstrain......Page 247
5.2.4.1 Magnetostatic field......Page 249
5.2.4.2 Magnetic force between particles......Page 252
5.2.4.3 Elastic field......Page 254
5.3.1 Magnetic Behavior of Magnetic Composites......Page 255
5.3.2 Magneto-Elastic Behavior of Ferromagnetic Composites......Page 259
5.4 Results and Discussion......Page 265
5.5 Conclusions......Page 274
References......Page 276
6.1 Introduction......Page 282
6.2.1 Basic Equations......Page 283
6.2.2 Domian Wall Motion......Page 285
6.2.3 Functional Grading......Page 288
6.2.4 Finite Element Model......Page 289
6.3.1 Material and Specimen Preparation......Page 290
6.3.2 Clamped-Free FGPM Plates......Page 291
6.3.3 Clamped-Clamped FGPM Plates......Page 292
6.4.1 Results of Clamped-Free Bimorphs......Page 293
6.4.2 Results of Clamped-Clamped Bimorphs......Page 295
6.5 Conclusions......Page 297
References......Page 298
7.1 Introduction......Page 300
7.2 Preliminary Definitions......Page 307
7.3 Mixture Kinematics......Page 309
7.4.1 Conservation of Mass......Page 311
7.4.3 Conservation of Momentum......Page 313
7.5 Entropy Principle and Constitutive Relations......Page 315
7.6 Deformation Dependent Solute flux Relations......Page 320
7.7 Summary......Page 322
References......Page 323
Micromechanics of Nanocomposites with Interface Energy Effect......Page 330
8.1 Introduction......Page 331
8.2.1 Geometry of a Deformable Interface/Surface......Page 334
8.2.2 Constitutive Relations of the Interface at Finite Deformation......Page 338
8.2.3 The “Three Configurations” Concept......Page 340
8.2.4 The Lagrangian and Eulerian Descriptions of Interface Equilibrium Equations......Page 345
8.3.1 The Infinitesimal Interface Strain......Page 350
8.3.2 Constitutive Relations of the Interface......Page 351
8.3.3 Linearization of Constitutive Relations in Bulk Solids with a Residual Elastic Field......Page 353
8.3.4 Lagrangian Description of the Young–Laplace Equation......Page 355
8.4.1 Extension of the Conventional Micromechanics by Taking into Account the Interface Energy Effect......Page 356
8.4.2 Effective Moduli of a Composite Filled with Spherical Particles......Page 358
8.4.3 Influence of Particle-Size Distribution......Page 364
8.4.4 Effective Properties of Thermoelastic Nanocomposite......Page 365
8.5 Summary......Page 371
References......Page 372
9.1 Introduction......Page 376
9.2 Interface Micromechanics Model......Page 378
9.3.1 The Gurtin–Murdoch Theory......Page 380
9.4.1 Dirichlet Boundary Value Problem......Page 381
9.4.2 Neumann Boundary Value Problem......Page 382
9.5.1 Composite Material Homogenization......Page 383
9.5.2 Correction to Wavelength for Quantum Dots......Page 386
References......Page 388
10.1 Introduction......Page 390
10.2 Overview of Nonlocal Gradient Plasticity Theory......Page 394
10.3 Thermodynamics of Nonlocal Gradient Plasticity Theory......Page 399
10.3.1 Principle of Virtual Power and Balance Laws......Page 401
10.3.2 Nonlocal Plasticity Yield Condition......Page 403
10.3.3 Nonlocal Clausius–Duhem Inequality......Page 404
10.3.4 Maximum Rate of Energy Dissipation Principle......Page 406
10.3.5 Assuming Function for the Free Energy and Plastic Dissipation......Page 407
10.3.6 Constitutive Equations......Page 408
10.4 Physical Interpretation of the Material Length Scales......Page 409
10.5 Applications to Size Effects in Metallic Systems......Page 411
References......Page 415
Strain Gradient Solutions of Eshelby-Type Inclusion Problems......Page 422
11.1 Introduction......Page 423
11.2.1 Simplified Strain Gradient Elasticity Theory (SSGET)......Page 424
11.2.2 Green’s Functions......Page 426
11.3.1 3-D Inclusion Problems......Page 427
11.3.1.1 Spherical inclusion......Page 430
11.3.1.2 Cylindrical inclusion......Page 433
11.3.1.3 Ellipsoidal inclusion......Page 437
11.3.1.4 Polyhedral inclusion......Page 442
11.3.2.1 Circular cylindrical inclusion......Page 453
References......Page 458
12.1 Introduction......Page 462
12.2.1 Couple-Stress Elasticity......Page 466
12.2.2 Dipolar Gradient Elasticity......Page 468
12.3.1 Problem Statement and Plane-Strain States......Page 470
12.3.2 Outline of Analysis and Results......Page 472
12.4.1 Problem Statement and Plane-Strain States......Page 479
12.4.2 Outline of Analysis and Results......Page 481
12.5 Conclusions......Page 486
References......Page 487
13.1 Introduction......Page 492
13.2 Representation of the Solutions by Fourier Series......Page 500
13.3.1 Cauchy Integrals, Plemelj Formula and Elliptic Functions......Page 503
13.3.2 Solutions to Problem......Page 507
13.3.2.1 Specification to a Vigdergauz structure......Page 511
13.3.3 Solutions to Problem......Page 515
13.3.3.1 Specification to a Vigdergauz structure......Page 522
13.4 Summary and Discussion......Page 526
Appendix: Verification of Equation (13.3.72)......Page 528
References......Page 531
14.1 Introduction......Page 532
14.2.1 Stochastic Energy Variational Principle......Page 534
14.2.2 Stochastic Hashin–Shtrikman Variational Principle......Page 538
14.3.1 Correlation-Based Variational Bounds of Elastic Moduli......Page 540
14.3.2 Correlation-Based Variational Bounds of Transport Properties......Page 544
14.4.1 Formulation of Ellipsoidal Bounds......Page 545
14.4.2 Estimates of Ellipsoidal Bounds of Elastic Moduli......Page 548
14.4.3 Estimates of Ellipsoidal Bounds of Transport Properties......Page 551
14.5.1 Optimal Percolation Thresholds of 3D Composites......Page 554
14.5.2 Optimal Percolation Thresholds of 2D Composites......Page 557
14.5.3 The Dimensional Effect......Page 559
Appendix A. Eshelby’s Tensors of Ellipsoids A.1 Fourth-Rank Eshelby’s Tensors in Elasticity......Page 561
A.2 Second-rank Eshelby’s Tensors in Transport......Page 562
References......Page 563
15.1 Introduction......Page 566
15.2.1 ABC Kinematics......Page 570
15.2.2 ABC Variational Statement......Page 573
15.2.2.1 Kinetic power......Page 574
15.3.1 General Concept......Page 575
15.3.2 Four Blending Laws: Effective Moduli for Sampling Points......Page 578
15.3.2.2 Self-consistent model......Page 579
15.3.2.3 Two energy-based models......Page 581
15.4.1 The ABC Model......Page 583
15.4.2 Results and Discussion......Page 586
15.5 Conclusions......Page 591
References......Page 592
Microstructural Characterization of Metals Using Nanoindentation......Page 596
16.1 Introduction......Page 597
16.2 Physical Interpretation of Length Scales......Page 602
16.3 Model for Temperature and Rate Indentation Size Effect (TRISE)......Page 604
16.4.1.1 Nanoindentation experiments and sample preparation......Page 612
16.4.1.2 Comparison of experimental results with the developed model......Page 614
16.4.2 Iron......Page 618
16.4.3 Nickel......Page 621
16.4.4 Niobium (Nb)......Page 623
16.4.5 Tungsten......Page 627
16.4.6 Gold Thin Film......Page 629
16.4.7 Single-Crystal Platinum......Page 630
16.5 Conclusion......Page 633
References......Page 636
A Multiscale Modeling of Multiple Physics......Page 646
17.1 Introduction......Page 647
17.2.1 Lattice Dynamics......Page 652
17.2.2 Kinematic Constraints......Page 655
17.2.3 Summation Rules on Force Calculations......Page 659
17.3.1 Non-Equilibrium Molecular Dynamics Simulation......Page 661
17.3.2 Coarse-Grained Non-equilibrium Molecular Dynamics Simulation......Page 666
17.3.3 Electromagnetic Effects......Page 668
17.4 Summary......Page 670
References......Page 672
18.1 Introduction......Page 676
18.2 Atomistic Formulation of Microscopic Balance Equations......Page 679
18.3 Numerical Implementation by Finite Element Method......Page 683
18.4 Simulation Results of Dislocation Dynamics......Page 685
18.4.1 Dislocation Nucleation and Migration......Page 686
18.4.2 Dislocation–Dislocation Interactions......Page 691
18.4.3 Dislocation-Stacking Fault Interactions......Page 696
18.4.4 Dislocations in a Submicron Thin Sheet Specimen......Page 699
18.5 Simulation of Brittle Fracture......Page 702
18.5.1 Comparison of FE and MD Simulations......Page 703
18.5.2 Large-Scale FE Simulations......Page 709
18.6 Discussions......Page 717
References......Page 720
Timescaling in Multiscale Mechanics of Nanowires and Nanocrystalline Materials......Page 724
19.1 Introduction......Page 725
19.2 Method and Framework......Page 731
19.2.1 Interatomic Potential......Page 732
19.2.2 Satisfaction of Dynamical Requirements......Page 734
19.2.3 Simulation Details......Page 736
19.3 Results and Discussions......Page 738
19.3.1 Analyses of the Si Nanowire Deformation......Page 740
19.3.2 Analyses of the Thin Film and Bulk Polycrystalline Si Deformation......Page 744
19.4 Conclusions......Page 751
References......Page 753
Modeling and Simulation of Carbon Nanotube Based Composites and Devices......Page 756
20.1 Introduction......Page 757
20.2.1 Molecular Dynamics......Page 762
20.2.2 Continuum Approximation......Page 764
20.2.3 Temperature-Related Homogenization......Page 766
20.2.4 Bridging Domain Coupling Method......Page 767
20.3 Carbon Nanotube......Page 769
20.3.1 Molecular Dynamics Simulation......Page 770
20.3.2 Fracture of SWNTs with One Vacancy Defect......Page 771
20.3.3 Fracture of SWNTs with Randomly Located Vacancy Defects......Page 773
20.4.1 Molecular Modeling and Simulations......Page 775
20.4.2 Multiscale Modeling and Simulations......Page 779
20.5.1 Molecular Modeling and Simulations......Page 783
20.5.2 Continuum Modeling and Simulations......Page 787
20.5.3 CNT-Based Memory Cells......Page 789
20.6.1 Multiscale Modeling......Page 794
20.6.2 Simulations and Results......Page 797
20.7 Conclusions......Page 799
References......Page 800
21.1 Introduction......Page 812
21.2.1 Overview......Page 815
21.2.2 Basic Notations and Configurations......Page 816
21.2.3.1 Governing equations of molecular dynamics......Page 818
21.2.3.2 Coarse grained model based on virtual atom cluster model......Page 819
21.2.3.3 The space-time formulation......Page 821
21.2.4 Space-Time Approximation Based on Enrichment......Page 825
21.3.1 Multiscale Spatial Discretization......Page 828
21.3.2 Multiscale Temporal Discretization......Page 830
21.3.3 Interface Treatment Based on Bridging Scale Method......Page 831
21.4.1 Selection of Enrichment Function......Page 834
21.4.2 Wave Speed and Dispersion Relation......Page 836
21.5 Example Problems......Page 837
21.5.1 Space-Time FEM with Direct Hand-Shake......Page 838
21.5.2 Space-Time FEM with Bridge Scale Method......Page 840
21.5.3 Extended Space-Time FEM with Bridging Scale Method......Page 843
21.5.4 Application to Nonlinear Potential......Page 846
21.6 Summary......Page 847
References......Page 848
22.1 Introduction......Page 854
22.2 Theoretical and Semi-Empirical Model for Macroporous Foams......Page 857
22.3 Experimental Studies on Nanoporous Foams......Page 859
22.3.1 Elastic Modulus of Nanoporous Metals......Page 861
22.3.2 Yield Strength of Nanoporous Metals......Page 862
22.3.3 Tensile Response of Nanoporous Metals......Page 871
22.4 Molecular Dynamics Study of Nanoporous Metals......Page 873
22.4.1 Model Generation......Page 875
22.4.2 Simulation Methods......Page 879
22.4.3 Results and Discussion......Page 880
22.4.3.1 The influence of ligament and joint size on the softening behavior......Page 883
22.5 Conclusion......Page 889
References......Page 890
Numerical Characterization of Nanowires......Page 896
23.1 Introduction......Page 897
23.2 Mechanical Properties of Perfect Nanowires......Page 899
23.2.1.1 Numerical techniques......Page 900
23.2.1.2 Tensile properties......Page 902
23.2.2 Nanowires under Compression......Page 908
23.2.3 Nanowires under Torsion......Page 912
23.2.4.1 Theoretical and numerical techniques for bending......Page 914
23.2.4.2 Bending properties......Page 918
23.2.5 Nanowires under Vibration......Page 922
23.3 Mechanical Properties of Defected Nanowires......Page 923
23.3.1 Effect by Grain Boundary and Twin Boundary......Page 924
23.3.2 Effect by Surface and Internal Defects......Page 925
23.3.2.1 Defect effect under tension......Page 926
23.3.2.2 Defect effect under other loading conditions......Page 930
23.4 Conclusions and Future Directions......Page 931
References......Page 932
Molecular Modeling of the Microstructure of Soft Materials: Healing, Memory, and Toughness Mechanisms......Page 944
24.1 Introduction......Page 945
24.2.1 Chemistry Meets Mechanics at the Molecular Scale......Page 948
24.2.2 Mechanics of Chemical vs. Physical Cross-links......Page 950
24.2.3 Nanodynamics of Soft Interfaces: Molecular Simulation Methods......Page 952
24.2.4 Thermomechanics of Molecular Interactions and Interfaces......Page 956
24.2.5 Exploring Free Energy Landscapes of Molecular Interfaces......Page 958
24.2.6 Nanodynamics of Interfaces: Interplay of Formation and Fracture......Page 961
24.3 Mechanical Behavior of Molecular Assemblies......Page 964
24.3.1.2 Approaches for detecting glass-transition in materials......Page 965
24.3.1.3 Comparison of approaches for detecting glass-transition in polymers......Page 968
24.3.1.4 Glass transition in stimuli responsive polymers: shape-memory effect......Page 970
24.3.2 Reversible Molecular Links and Implications for Functional Materials......Page 974
24.3.2.1 Challenges in investigating reversible chemical cross-links......Page 975
24.3.2.2 Hierarchical modeling frameworks for reversible bond networks......Page 977
24.3.2.3 Implications of reversible interfaces for self-healing, toughness and flaw-tolerance......Page 981
24.4 Conclusion and Outlook......Page 986
References......Page 987
25.1 Introduction......Page 1002
25.2 The Hierarchical Structure of Fish Scales......Page 1004
25.3 Tensile Testing of Individual Scales......Page 1006
25.4 Resistance to Sharp Penetration......Page 1009
25.5 Analytical Model......Page 1012
25.6.1 Analytical Model......Page 1014
25.6.2 Results......Page 1019
25.7 Conclusions......Page 1021
References......Page 1023
26.1 Introduction......Page 1026
26.2 Random Network Structure......Page 1029
26.3 Constitutive Behavior of Individual Fibers......Page 1034
26.4.1 Flexible Filaments......Page 1037
26.4.2.1 Affine versus non-affine deformation......Page 1038
26.4.2.2 Network elasticity......Page 1040
26.4.2.3 Field fluctuations and spatial correlations......Page 1042
26.4.2.4 Size effects......Page 1047
26.4.2.5 Mapping to continuum representations of network domains......Page 1048
26.4.2.6 Large deformations......Page 1050
26.5 Entangled Networks......Page 1051
26.6 Conclusion......Page 1054
References......Page 1055
Size-Dependent Probabilistic Damage Micromechanics and Toughening Behavior of Particle/Fiber Reinforced Composites......Page 1060
27.1 Introduction......Page 1061
27.2.1 Manufacturing Process Induced Damage......Page 1063
27.2.2 Thermal Residual Stress and Relaxation......Page 1065
27.2.3 Concept of Equivalent Inclusion Method......Page 1069
27.2.4 Volume Fraction Evolution of Debonded Particles......Page 1071
27.3.1 Effective Elastic-Damage Moduli of 4-Phase Composites......Page 1072
27.3.2 Effective Yield Function for Multi-Phase Elastoplastic Composites with Damage and Residual Stress......Page 1073
27.3.3 Dislocation Strengthening......Page 1075
27.4 Overall Elastoplastic-Damage Stress–Strain Responses with a Hybrid Effective Yield Function......Page 1076
27.5 Numerical Simulations......Page 1078
27.6.1 Fiber Bridging Stress......Page 1082
27.6.2 Crack Mouth Opening Displacement......Page 1086
References......Page 1087
Multiscale Asymptotic Homogenization of Heterogeneous Slab and Column Structures with Three-Dimensional Microstructures......Page 1094
28.1 Introduction......Page 1095
28.2.1 Governing Equations......Page 1098
28.2.2 Finite Element Discretization of Unit Cell Problem......Page 1101
(I) One-dimensional heterogeneous bar problem......Page 1104
(II) Two-dimensional plane strain problem......Page 1107
28.3.1 Definition of Slab Unit Cell......Page 1109
28.3.2 Consistent Asymptotic Expansion of Slab Displacement Field......Page 1111
28.3.3 Construction of Local and Global Problems for Slab Structure......Page 1112
28.3.4 Algorithm Verification for Homogeneous Slab......Page 1114
28.3.5 Homogenization Analysis for Heterogeneous Slab......Page 1116
28.3.6 Multiscale Analysis of Cantilever Beam Problem......Page 1118
28.4.1 Definition of Column Unit Cell......Page 1122
28.4.2 Consistent Asymptotic Expansion of Column Displacement Field......Page 1123
28.4.3 Construction of Local and Global Problems for Column......Page 1125
28.4.4 Algorithm Verification for Homogeneous Column......Page 1126
28.4.6 Multiscale Analysis of Heterogeneous Column......Page 1128
28.5 Conclusions......Page 1132
References......Page 1133
Computational Homogenization and Partial Overlap Coupling Between Micropolar Elastic Continuum Finite Elements and Elastic Spher......Page 1138
29.1 Introduction......Page 1139
29.3.1 Three-Dimensional (3D) Micropolar Linear Isotropic Elasticity and Balance Equations......Page 1142
29.3.2 1D Timoshenko Beam Kinematics with Axial Stretch and Resulting 1D Micropolar Linear Elasticity......Page 1143
29.3.3 Finite Element (FE) Implementation of 1D Micropolar Linear Elasticity......Page 1147
29.3.4 Convergence of 1D Micropolar Linear Elastic FE......Page 1152
29.4 1D String of Hertzian Nonlinear Elastic Discrete Element (DE) Spheres......Page 1153
29.5 1D String of Linear Elastic Discrete Element (DE) Spheres......Page 1155
29.6 Overlap Coupling between 1D Micropolar FEs and a String of Spherical DEs......Page 1156
29.6.1 3D Kinematics......Page 1157
29.6.2 3D Kinetic and Potential Energy Partitioning and Coupling......Page 1162
29.6.3 1D Full Overlap Coupling......Page 1168
29.6.4 1D Partial Overlap Coupling with Partial Overlay 1D Micropolar FE......Page 1173
Acknowledgments......Page 1180
References......Page 1181
Non-Concurrent Computational Homogenization of Nonlinear, Stochastic and Viscoelastic Materials......Page 1184
30.1 Introduction......Page 1185
30.2 Homogenization of Nonlinear Materials......Page 1188
30.2.1 Nonlinear Elasticity at Small Strains......Page 1189
30.2.2 Nonlinear Elasticity at Finite Strains......Page 1191
30.3 Viscoelasticity......Page 1194
30.4.1 Basic Ideas......Page 1196
30.4.2.1 Construction of the strain domain......Page 1198
30.4.3.1 Direct multidimensional interpolation......Page 1199
30.4.3.2 Separated variables......Page 1200
30.4.5 Issues Related to Finite Strains......Page 1202
30.5.1 Probabilistic Modeling of the Microstructure......Page 1203
30.5.2 Extension of NEXP for Stochastic Problems (S-NEXP Method)......Page 1204
30.6.1 Numerical Mapping, Discrete Scheme and Interpolation......Page 1206
30.6.2 Macroscopic Algorithm......Page 1207
30.7.1 Nonlinear Composite at Small Strains......Page 1209
30.7.2 Nonlinear Composite at Finite Strains......Page 1211
30.7.3 Stochastic Nonlinear Composite at Finite Strains......Page 1213
30.7.4 Linear Viscoelastic Composite......Page 1214
30.8 Conclusion and Prospects......Page 1217
References......Page 1218
Color Insert......Page 1224
Back Cover......Page 1256