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 ........................................... 9
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 ........................................ 15
1.4 Review of finite elasticity ............................ 18
1.5 Review of molecular dynamics ........................... 20
1.5.1 Lagrangian equations of motion .................. 20
1.5.2 Hamiltonian equations of motion ................. 22
1.5.3 Interatomic potentials .......................... 24
1.5.4 Two-body (pair) potentials ...................... 24
1.5.5 Embedded-Atom-Method (EAM) ...................... 27
1.6 Elements of lattice dynamics ........................... 29
1.6.1 Crystal lattice structures ...................... 29
1.6.2 Crystallographic system ......................... 32
1.6.3 Lattice dynamics ................................ 35
1.7 Exercises .............................................. 37
2 GREEN'S FUNCTION AND FOURIER TRANSFORM ...................... 39
2.1 Basics of Green's function ............................. 39
2.2 Fourier transform ...................................... 42
2.3 Examples of Green's functions .......................... 47
2.4 Static Green's function for 3D linear elasticity ....... 50
2.5 Green's function for Stokes equations .................. 56
2.6 Radon transform ........................................ 58
2.7 Green's function for elastodynamics .................... 62
2.8 Lattice statics Green's function (LSGF) ................ 65
2.8.1 Maradudin's solution for the screw dislocation ... 66
2.9 Exercises .............................................. 70
3 MICROMECHANICAL HOMOGENIZATION THEORY ....................... 73
3.1 Ergodicity principle and representative volume
element (RVE) .......................................... 74
3.1.1 Ergodic principle ............................... 75
3.1.2 Representative volume element ................... 78
3.2 Average field in an RVE ................................ 80
3.3 Definition of eigenstrain, eigenstress, and
inclusion .............................................. 86
3.4 Eshelby's equivalent eigenstrain method ................ 86
3.5 Fundamental equations of micro-elasticity .............. 89
3.5.1 Method of Fourier transform ..................... 90
3.5.2 Method of Green's function ...................... 91
3.6 Eshelby's solution to the inclusion problem in an
infinite space ......................................... 94
3.6.1 Interior solution of ellipsoidal inclusion ...... 95
3.6.2 Eshelby's conjectures ........................... 99
3.6.3 Exterior solution of ellipsoidal inclusion ..... 102
3.6.4 The second derivatives of Green's function
and the Eshelby tensors ........................ 106
3.7 Applications of eigenstrain theory .................... 108
3.7.1 Strain field in embedded quantum dots .......... 108
3.7.2 Dislocation problems ........................... 112
3.7.3 Stress intensity factor for a flat
ellipsoidal crack .............................. 115
3.8 John Douglas Eshelby (1916-1981) ...................... 121
3.9 Exercises ............................................. 125
4 EFFECTIVE ELASTIC MODULUS .................................. 131
4.1 Effective modulus for composites with dilute
suspension phases ..................................... 131
4.1.1 Basic equations for average stress and
strain ......................................... 131
4.1.2 Homogenization: equivalent stress/strain
conditions ..................................... 133
4.1.3 Example: elastic modulus of isotropic
composites ..................................... 136
4.2 Self-consistent method ................................ 138
4.3 Mori-Tanaka method .................................... 143
4.3.1 Tanaka-Mori lemma .............................. 143
4.3.2 Mori-Tanaka's mean field theory ................ 146
4.4 Rodney Hill ........................................... 151
4.5 Exercises ............................................. 156
5 COMPARISON VARIATIONAL PRINCIPLES .......................... 159
5.1 Review of variational calculus ........................ 159
5.2 Extremum variational principles in linear
elasticity ............................................ 163
5.2.1 Minimum potential energy principle ............. 163
5.2.2 Minimum complementary potential energy
principle ...................................... 166
5.2.3 Voigt bound and Ruess bound .................... 168
5.3 Hashin-Shtrikman variational principles ............... 170
5.4 Hashin-Shtrikman bounds ............................... 176
5.5 Review of functional analysis and convex analysis ..... 186
5.5.1 Concept of convexity ........................... 191
5.5.2 Gateaux variation and convex functional ........ 193
5.5.3 Primal variational problems .................... 195
5.6 Legendre transformation and duality ................... 196
5.7 Legendre-Fenchel transformation in linear
elasticity ............................................ 203
5.8 Talbot-Willis variational principles .................. 204
5.9 Ponte Castaneda variational principle ................. 207
5.9.1 Effective property and nonlinear potential ..... 207
5.9.2 Variational method based on a linear
comparison solid ............................... 210
5.10 Zvi Hashin ............................................ 212
5.11 Exercises ............................................. 212
6 ESHELBY TENSORS IN A FINITE VOLUME AND THEIR
APPLICATIONS ............................................... 217
6.1 Introduction .......................................... 217
6.2 The inclusion problem of a finite RVE ................. 218
6.3 Properties of the radially isotropic tensor ........... 221
6.4 Eshelby tensors for finite domains .................... 224
6.4.1 Dirichlet-Eshelby tensor ....................... 225
6.4.2 Neumann-Eshelby tensor ......................... 229
6.5 Average Eshelby tensors and average disturbance
fields ................................................ 233
6.5.1 Average Eshelby tensors ........................ 234
6.5.2 Average disturbance fields ..................... 236
6.6 Improvements of classical homogenization methods ...... 237
6.6.1 Dilute suspension model ........................ 237
6.6.2 A refined Mori-Tanaka model .................... 238
6.6.3 Multiphase variational bounds .................. 241
6.7 Application to multiscale finite element methods ...... 251
6.7.1 Variational multiscale eigenstrain
formulation .................................... 251
6.7.2 Modal analysis of the modified smart element ... 258
6.7.3 Numerical examples ............................. 260
6.8 Exercises ............................................. 264
7 INTRODUCTION TO MICROMECHANICS AND NANOMECHANICS
MICROMECHANICAL DAMAGE THEORY .............................. 265
7.1 Spherical void growth in linear viscous solids ........ 265
7.2 McClintock solution to cylindrical void growth
problem ............................................... 267
7.3 Gurson model .......................................... 273
7.4 Gurson-Tvergaard-Needleman (GTN) model ................ 278
7.5 A cohesive micro-crack damage model ................... 281
7.5.1 Average theorem for a cohesive RVE ............. 282
7.5.2 Penny-shaped cohesive crack under uniform
triaxial tension ............................... 283
7.5.3 Effective elastic material properties of an
RVE ............................................ 287
7.5.4 Micro-cohesive-crack damage models ............. 291
7.6 Frank A. McClintock ................................... 293
7.7 Exercises ............................................. 293
8 INTRODUCTION OF DISLOCATION THEORY ......................... 297
8.1 Screw dislocation ..................................... 297
8.1.1 The solution of a screw dislocation ............ 298
8.1.2 Image stress of a screw dislocation in a half
space .......................................... 301
8.1.3 Eshelby's twist: screw dislocation in
a finite whisker ............................... 302
8.2 Edge dislocation ...................................... 303
8.2.1 Image stress for an edge dislocation ........... 306
8.3 Peach-Koehler force ................................... 308
8.4 Point defects ......................................... 313
8.4.1 Displacement field induced by a point defect ... 314
8.4.2 Formation volume tensor ........................ 315
8.5 Continuum theory of dislocation ....................... 317
8.5.1 Volterra and Mura's formulas ................... 317
8.5.2 The Burgers formula ............................ 320
8.5.3 Peach-Koehler stress formula for dislocation
loop ........................................... 323
8.6 Discrete dislocation dynamics (DDD) ................... 325
8.6.1 Galerkin weak form formulation ................. 325
8.6.2 Finite element implementation .................. 327
8.7 Peierls-Nabarro model ................................. 329
8.7.1 Hilbert transform .............................. 330
8.7.2 Peierls-Nabarro dislocation model .............. 331
8.7.3 Misfit energy and the Peierls force ............ 335
8.7.4 Variable core model ............................ 341
8.7.5 Story of the Peierls-Nabarro model ............. 344
8.8 Dislocations in epitaxial thin films .................. 346
8.8.1 Frenkel & Kontorova and Frank & van der Merwe
models ......................................... 346
8.8.2 Matthews & Blackeslee's equilibrium theory ..... 353
8.8.3 Mobility of screw dislocations in a thin
film ........................................... 355
8.9 Exercises ............................................. 360
9 INTRODUCTION TO CONFIGURATIONAL MECHANICS .................. 363
9.1 Configurational force: Eshelby's energy-momentum
tensor ................................................ 363
9.1.1 Eshelby's thought experiment ................... 364
9.1.2 Lessons from J.D. Eshelby ...................... 371
9.1.3 J-integral and energy release rate G ........... 373
9.2 James R. Rice ......................................... 375
9.3 Configurational compatibility ......................... 376
9.3.1 Continuum dislocation theory ................... 376
9.3.2 Continuum disclination theory .................. 379
9.3.3 Re-combination I: The generalized Nye theory ... 380
9.3.4 Re-combination II: The Kröner-deWit theory ..... 381
9.3.5 Compatibility conservation laws ................ 381
9.4 Multiscale energy-momentum tensor ..................... 384
9.5 Ekkehart Kroner (1919-2000) ........................... 391
9.6 Exercises ............................................. 391
10 SMALL SCALE COARSE-GRAINED MODELS .......................... 393
10.1 Gurtin-Murdoch surface elasticity model ............... 393
10.1.1 Projection operator ........................... 393
10.1.2 Gurtin-Murdoch theory ......................... 396
10.1.3 Spherical inclusion problem ................... 397
10.2 Cohesive quasi-continuum finite element method ........ 399
10.2.1 Atomistic modeling ............................ 400
10.2.2 Quasi-continuum method ........................ 401
10.2.3 Interface Cauchy-Born rule .................... 404
10.3 Microscale or mesoscale stress ........................ 412
10.3.1 Virial stress ................................. 412
10.3.2 Hardy stress .................................. 415
10.4 Exercises ............................................. 418
11 PERIODIC MICROSTRUCTURE AND ASYMPTOTIC HOMOGENIZATION ...... 419
11.1 Unit cell and Fourier series .......................... 419
11.1.1 Fourier transform of displacement field and
strain field ................................... 421
11.1.2 Fourier series transform of stress field ....... 422
11.2 Eigenstrain homogenization ............................ 423
11.3 Introduction to asymptotic homogenization ............. 431
11.3.1 One-dimensional model problem .................. 431
11.3.2 A multiple dimension example ................... 436
11.4 Variational characterization .......................... 444
11.5 Multiscale finite element method ...................... 447
11.5.1 Asymptotic homogenization of linear
elasticity ..................................... 448
11.5.2 Finite element formulation ..................... 451
11.6 G-, H-, and Г- Convergence ............................ 454
11.6.1 Strong convergence and weak convergence ........ 454
11.6.2 G-convergence .................................. 457
11.6.3 H-convergence .................................. 462
11.6.4 Г-convergence .................................. 462
11.7 Toshia Mura ........................................... 463
11.8 Exercises ............................................. 464
Appendix A Appendix of Chapter 6 .............................. 467
A.l Integration formulas ................................. 467
A.2 Table of Eshelby tensor coefficients for
three-layer shell model .............................. 470
A.3 Finite Eshelby tensors for circular inclusion ........ 472
Appendix В Noether's Theorems ................................. 475
B.l Noether's theorem for a vector field ................. 475
B.2 Noether's theorem for a tensorial field .............. 476
Bibliography .................................................. 481
SUBJECT INDEX ................................................. 495
AUTHORS INDEX ................................................. 499
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