Table of Contents
Preface vii
1 Introduction 1
1.1 What are micromechanics and nanomechanics? 1
1.2 Vectors and tensors 4
1.2.1 Vector algebra 4
1.2.2 Tensor algebra 6
1.2.3 Inversion formula for fourth-order isotropic tensor 10
1.2.4 Tensor analysis 11
1.3 Review of linear elasticity theory 13
1.3.1 Governing equations 13
1.3.2 Betti's reciprocal theorem and the Somigliana identity 16
1.4 Review of finite elasticity 19
1.5 Review of molecular dynamics 21
1.5.1 Lagrangian equations of motion 21
1.5.2 Hamiltonian equations of motion 23
1.5.3 Interatomic potentials 25
1.5.4 Two-body (pair) potentials 26
1.5.5 Embedded-atom method 28
1.6 Elements of lattice dynamics 31
1.6.1 Crystal lattice structures 31
1.6.2 Crystallographic system 35
1.6.3 Lattice dynamics 37
1.7 Exercises 39
2 Green's Function and Fourier Transform 41
2.1 Basics of Green's function 41
2.2 Fourier transform 44
2.3 Examples of Green's functions 50
2.4 Static Green's function for 3D linear elasticity 53
2.5 Green's function for Stokes equations 60
2.6 Radon transform 62
2.7 Green's function for elastodynamics 67
2.8 Lattice statics Green's function 70
2.8.1 Maradudin's solution for the screw dislocation 72
2.9 Exercises 75
3 Micromechanical Homogenization Theory 77
3.1 Ergodicity principle and representative volume element 78
3.1.1 Ergodic principle 79
3.1.2 Representative volume element 82
3.2 Average field in an RVE 84
3.3 Hill-Mandel lemma in finite deformation 92
3.4 Jean Mandel (1907-1982) 95
3.5 Definition of eigenstrain, eigenstress, and inclusion 96
3.6 Eshelby's equivalent eigenstrain method 97
3.7 Fundamental equations of microelasticity 100
3.7.1 Method of Fourier transform 100
3.7.2 Method of Green's function 102
3.8 Eshelby's solution to the inclusion problem in an infinite space 106
3.8.1 Interior solution of ellipsoidal inclusion 107
3.8.2 Eshelby's conjectures 112
3.8.3 Exterior solution of ellipsoidal inclusion 115
3.8.4 The second derivatives of Green's function and the Eshelby tensors 121
3.9 Applications of eigenstrain theory 122
3.9.1 Strain field in embedded quantum dots 122
3.9.2 Dislocation problems 126
3.9.3 Stress intensity factor for a flat ellipsoidal crack 130
3.10 John Douglas Eshelby (1916-1981) 137
3.11 Exercises 141
4 Effective Elastic Modulus 147
4.1 Effective modulus for composites with dilute suspension phases 147
4.1.1 Basic equations for average stress and strain 147
4.1.2 Homogenization: equivalent stress-strain conditions 149
4.1.3 Example: elastic modulus of isotropic composites 153
4.2 Self-consistent method 155
4.3 Mori-Tanaka method 161
4.3.1 Tanaka-Mori lemma 161
4.3.2 Mori-Tanaka's mean field theory 164
4.4 Rodney Hill (1921-2011) 170
4.5 Exercises 174
5 Variational Principles and Computational Homogenization 177
5.1 Review of variational calculus 177
5.2 Extremum variational principles in linear clasticity 182
5.2.1 Minimum potential energy principle 182
5.2.2 Minimum complementary potential energy principle 185
5.2.3 Voigt bound and Reuss bound 187
5.3 Hashin-Shtrikman variational principles 191
5.4 Hashin-Shtrikman bounds 197
5.5 Review of functional analysis and convex analysis 208
5.5.1 Concept of convexity 213
5.5.2 Gâteaux variation and convex functional 215
5.5.3 Primal variational problems 217
5.6 Legendre transformation and duality 218
5.7 Legendra Fenchel transformation in linear elasticity 225
5.8 Talbot-Willis variational principles 231
5.9 Ponte Castañeda variational principle 231
5.9.1 Effective property and nonlinear potential 231
5.9.2 Variational method based on a linear comparison solid 233
5.10 Zvi Hashin 236
5.11 Computational homogenization (I) 236
5.11.1 Linear heterogeneous composites 237
5.11.2 The Hill-Mandel conditions 240
5.11.3 Nonlinear case I: Hyoperelastic materials under small deformation 243
5.11.4 Nonlinear case II: Hyperelastic materials with finite deformation 248
5.12 Exercises 252
6 Eshelby Tensors in a Finite Volume and Their Applications 255
6.1 Introduction 255
6.2 The inclusion problem of a finite RVE 256
6.3 Properties of the radially isotropic tensor 259
6.4 Eshelby tensors for finite domains 263
6.4.1 Dirichlet-Eshelby tensor 263
6.4.2 Neumann-Eshelby tensor 267
6.5 Average Eshelby tensors and average disturbance fields 273
6.5.1 Average Eshelby tensors 273
6.5.2 Average disturbance fields 275
6.6 Improvements of classical homogenization methods 277
6.6.1 Dilute suspension model 277
6.6.2 A refined Mori-Tanaka model 279
6.6.3 Multiphase variational bounds 281
6.7 Application to multiscale finite element methods 291
6.7.1 Variational multiscale eigenstrain formulation 292
6.7.2 Modal analysis of the modified smart element 299
6.7.3 Numerical examples 301
6.8 Exercises 306
7 Micromechanics-Based Damage Theory 307
7.1 Spherical void growth in linear viscous solids 307
7.2 McClintock solution to cylindrical void growth problem 310
7.3 Gurson model 315
7.4 Gurson-Tvergaar-Needleman model 321
7.5 A cohesive microcrack damage model 323
7.5.1 Average theorem for a cohesive RVE 325
7.5.2 Penny-shaped cohesive crack under uniform triaxial tension 326
7.5.3 Effective elastic material propertics of an RVE 330
7.5.4 Microcohesive crack damage models 334
7.6 Frank A. McClintock (1921-2011) 335
7.7 Exercises 337
8 Introduction of Dislocation Theory 341
8.1 Screw dislocation 341
8.1.1 The solution of a screw dislocation 343
8.1.2 Image stress of a screw dislocation in a half-space 346
8.1.3 Eshelby's twist: screw dislocation in a finite whisker 347
8.2 Edge dislocation 348
8.2.1 Image stress for an edge dislocation 350
8.3 Peach-Koehler force 353
8.4 Point defects 358
8.4.1 Displacement field induced by a point defect 359
8.4.2 Formation volume tensor 361
8.5 Continuum theory of dislocation 362
8.5.1 Volterra and Mura's formulas 363
8.5.2 The Burgers formula 366
8.5.3 The Peach-Koehler stress formula for dislocation loop 369
8.6 Discrete dislocation dynamics 372
8.6.1 Galerkin weak form formulation 372
8.6.2 Finite element implementation 374
8.7 Peierls-Nabarro model 377
8.7.1 Hilbert transform 377
8.7.2 Peirls-Nabarro dislocation model 378
8.7.3 Misfit energy and the Peierls force 383
8.7.4 Variable core model 389
8.7.5 Story of the Peierls-Nabarro model 393
8.8 Dislocations in epitaxial thin films 394
8.8.1 Frenkel & Kontorova and Frank & van der Merwe models 395
8.8.2 Matthews and Blackeslee's equilibrium theory 402
8.8.3 Mobility of screw dislocations in a thin film 404
8.9 Exercises 410
9 Configurational Mechanics of Defects 413
9.1 Configurational force: Eshelby's energy-momentum tensor 413
9.1.1 Eshelby's thought experiment 414
9.1.2 Lessons from J. D. Eshelby 421
9.1.3 J-integral and energy release rate G 423
9.2 James R. Rice 426
9.3 Continuum defect theory 426
9.3.1 Defect and compatibility condition 427
9.3.2 Kröner's continuum defect theory 428
9.3.3 Continuum disclination theory 431
9.3.4 Conservation law in continuum defect theory 432
9.4 Multiscale energy-momentum tensor 438
9.5 Ekkehart Kröner (1919-2000) 445
9.6 Exercises 445
10 Nanomechanics: Small-Scale Coarse-Grained Models 447
10.1 Gurtin-Murdoch surface clasticity model 447
10.1.1 Projection operator 447
10.1.2 Gurtin-Murdoch theory 450
10.1.3 Spherical inclusion problem 452
10.2 Atomistic mechanical stress tensor 455
10.2.1 Virial stress 456
10.2.2 Hardy stress 459
10.3 Cauchy-Born rule 462
10.4 Atomistically informed constitutive relations 464
10.4.1 Multiscale constitutive relations 466
10.4.2 CB rule for non-Bravais lattice 468
10.5 Higher order CB rule and multiscale cohesive zone model 470
10.5.1 Multiscale cohesive zone method 471
10.5.2 The first-order CB rule in the zeroth-order process zone (bulk element) 472
10.5.3 The second-order CB rule in the first-order process zone 473
10.5.4 The third-order CB rule in the second-order process zone 474
10.5.5 EAM potential-based multiscale constitutive model 475
10.5.6 Bubble mode and benchmark test 476
10.5.7 Multiscale FEM formulation 478
10.5.8 A numerical example 480
10.6 Higher order CB rule-based MCZM for silicaon crystals 485
10.6.1 Tersoff potential based constitutive equation 488
10.6.2 Second-order stress tensor 492
10.6.3 Lennard-Jones potential 495
10.6.4 Simulation of fracture toughness 496
10.7 Exercises 500
11 Periodic Microstructure and Asymptotic Homogenization 503
11.1 Unit cell and Fourier series 503
11.1.1 Fourier transform of displacement field and strain field 505
11.1.2 Fourier series transform of stress field 507
11.2 Eigenstrain homogenization 508
11.3 Introduction to asymptotic homogenization 516
11.3.1 1D model problem 516
11.3.2 A multiple dimension example 522
11.4 Variational characterization 530
11.5 Multiscale finite element method 534
11.5.1 Asymptotic homogenization of linear elasticity 534
11.5.2 Finite element formulation 539
11.6 A numerical example: Multiscale formulation for slab structure 541
11.7 Dynamic homogenization - asymptotic homogenization of wave equation 547
11.7.1 Homogenized wave equation to scale ε0 552
11.7.2 Homogenized wave equation at the scle ε1 554
11.7.3 Homogenized wave equation of scale ε2 557
11.7.4 Dispersion relation 560
11.8 G-, H-, and Δ-convergence 560
11.8.1 Strong convergence and week convergence 561
11.8.2 G-convergence 565
11.8.3 H-convergence 569
11.8.4 Δ-convergence 570
11.9 Toshia Mura (1925-2009) 570
11.10 Exercises 571
12 Introduction to Crystal Plasticity 575
12.1 Micromechanics of crystallographic slip 575
12.2 Schmid's law 578
12.3 Finite deformation crystal plasticity 580
12.3.1 Constitutive model 584
12.4 Homogenization of plastic behaviors of polycrystalline composites 586
12.4.1 Determining active slip systems 587
12.4.2 The Taylor-Bishop-Hill model 591
12.4.3 Self-consistent model (I) 594
12.4.4 Self-consistent model (II) 596
12.5 Geometrically necessary dislocation density 598
12.6 G. I. Taylor (1886-1975) 600
12.7 Exercise 601
Appendix A Appendix of Chapter 6 603
A.1 Integration formulas 603
A.2 Table of Eshelby tensor coefficients for three-layer shell model 606
A.3 Finite Eshelby tensors for circular inclusion 608
Appendix B Noether's theorems 611
B.1 Noether's theorem for a vector filed 611
B.2 Noether's theorem for a tensorial field 612
Bibliography 617
Author Index 635
Subject Index 639
