External forces, 161
Eyring cooperative model, 367
F
Failure in electronics, 395, 398, 403, 418, 421
Far-field stress, 285
Fatigue fracture entropy (FFE), 128, 129, 371
Fick's first law, 399
Fick's second law, 399, 400
Finite deformations
Cauchy and Almansi tensors, 345
continuous one-to-one mapping, 343, 344
dual-mechanism viscoplastic model
(see Dual-mechanism viscoplastic
model)
elastic and plastic, 345, 346
elastic part, 344
finite strain (see Finite strain)
formulation, 343
frame indifferent model, 346
frame of reference indifference, 347–351
gradient, 343
incompressible plastic flow, 346
irrotational plastic flow, 346
material properties, 343, 377–381
mechanics, 343
multiplicative split, 346
plastic, 344
polymer, 343
rotation tensor, 345
spin tensor, 345
stretch tensors, 345
stretching rate tensor, 344, 345
TSI, 371–373
velocity gradient, 343, 344
Finite element analysis, 386
Finite element method, 396, 408
Finite element method implementation
anti-symmetric Cauchy stress tensor, 248
general couple stress theory, 250, 252
prime superscript notation, 249
reduced couple stress theory (see Reduced
couple stress theory)
reduced theory continuum, 249
symmetric bilinear forms, 249
symmetric Cauchy stress tensor, 248
translational degrees of freedom, 249
translational/rotational degrees, 248
Finite strain, 29
Clausius-Duhem inequality, 351
conservation of energy law, 351
constitutive relations (see Constitutive
relations)
definitions, 46
Eulerian description, 48
green deformation tensor, 49, 50
Helmholtz free energy, 351
Lagrangian formulation, 49
local coordinate system, 46, 47
Malvern formulation, 47
rate of deformation, 54, 55
referential coordinate system, 47
rotation tensors, 57
second law of thermodynamics, 351
spatial description, 49
spatial gradient, velocity vs. deformation
gradient tensor, 56
strain rate, 54
stretch tensors, 57
thermodynamic restrictions, 352–359
thermo-mechanical effects, 351
Finite strain formulation, 3
First law of thermodynamics, 351
continuum mechanics, 75
heat input, 78–80
potential function, 76
work done on the system (power input),
76–78
First Piola-Kirchhoff stress tensor, 60, 61
Flow theory
elastic constitutive relationship, 203, 204
flow rule, 204
isotropic hardening, 205
NLK rule, 205, 206
viscoplastic creep rate law, 207
Fluctuation theorem, 127
Flux quantity, 227
Force equilibrium, 411, 412
Fourier heat conduction equation, 228
Fourier’s law, 398
Frame indifferent model, 346
Free-volume theory, 373
Fully coupled thermo-mechanical equations
dissipative phenomena, 229
entropy generation rate, 232
Fourier’s Law, 231
fundamental equations, 232
gradient functions, 230
Helmholtz free energy, 230
inter-convertibility, 232
internal state, 229
isotropic thermal expansion coefficient, 231
Laplacian operator, 231
432
Index
Eyring cooperative model, 367
F
Failure in electronics, 395, 398, 403, 418, 421
Far-field stress, 285
Fatigue fracture entropy (FFE), 128, 129, 371
Fick's first law, 399
Fick's second law, 399, 400
Finite deformations
Cauchy and Almansi tensors, 345
continuous one-to-one mapping, 343, 344
dual-mechanism viscoplastic model
(see Dual-mechanism viscoplastic
model)
elastic and plastic, 345, 346
elastic part, 344
finite strain (see Finite strain)
formulation, 343
frame indifferent model, 346
frame of reference indifference, 347–351
gradient, 343
incompressible plastic flow, 346
irrotational plastic flow, 346
material properties, 343, 377–381
mechanics, 343
multiplicative split, 346
plastic, 344
polymer, 343
rotation tensor, 345
spin tensor, 345
stretch tensors, 345
stretching rate tensor, 344, 345
TSI, 371–373
velocity gradient, 343, 344
Finite element analysis, 386
Finite element method, 396, 408
Finite element method implementation
anti-symmetric Cauchy stress tensor, 248
general couple stress theory, 250, 252
prime superscript notation, 249
reduced couple stress theory (see Reduced
couple stress theory)
reduced theory continuum, 249
symmetric bilinear forms, 249
symmetric Cauchy stress tensor, 248
translational degrees of freedom, 249
translational/rotational degrees, 248
Finite strain, 29
Clausius-Duhem inequality, 351
conservation of energy law, 351
constitutive relations (see Constitutive
relations)
definitions, 46
Eulerian description, 48
green deformation tensor, 49, 50
Helmholtz free energy, 351
Lagrangian formulation, 49
local coordinate system, 46, 47
Malvern formulation, 47
rate of deformation, 54, 55
referential coordinate system, 47
rotation tensors, 57
second law of thermodynamics, 351
spatial description, 49
spatial gradient, velocity vs. deformation
gradient tensor, 56
strain rate, 54
stretch tensors, 57
thermodynamic restrictions, 352–359
thermo-mechanical effects, 351
Finite strain formulation, 3
First law of thermodynamics, 351
continuum mechanics, 75
heat input, 78–80
potential function, 76
work done on the system (power input),
76–78
First Piola-Kirchhoff stress tensor, 60, 61
Flow theory
elastic constitutive relationship, 203, 204
flow rule, 204
isotropic hardening, 205
NLK rule, 205, 206
viscoplastic creep rate law, 207
Fluctuation theorem, 127
Flux quantity, 227
Force equilibrium, 411, 412
Fourier heat conduction equation, 228
Fourier’s law, 398
Frame indifferent model, 346
Free-volume theory, 373
Fully coupled thermo-mechanical equations
dissipative phenomena, 229
entropy generation rate, 232
Fourier’s Law, 231
fundamental equations, 232
gradient functions, 230
Helmholtz free energy, 230
inter-convertibility, 232
internal state, 229
isotropic thermal expansion coefficient, 231
Laplacian operator, 231
432
Index