ellipsoidal shape and orientation, 283
Eshelby’s equivalence principle, 280
fictitious equivalent eigenstrain, 281
Green’s function, 281
inhomogeneities, 280
integral equation, 282–284
inter-particle interaction effects, 283
linear elastic matrix material, 282
micromechanical, 283
perturbed strain field, 281
phase particles, 280, 283
recast, 282
renormalization procedure, 282
RVE, 280, 281
Enthalpy, 101
Entropic fundamental relation, 87
Entropy, 81, 82, 98
balance equations, 418–421
Clausius-Duhem Inequality, 91–94
concept, 80
definition, 80, 85, 118, 126
degradation metric, 129
density, 419
disorder, 96–99, 183, 192
Euler equation, 87–90
extensive parameter, 82
flat surface, 82
flux to heat flux, 359
fundamental governing equations, 84
fundamental relation, 83
generation mechanisms, 191, 389
generation rate, 132, 133, 135, 416
Gibbs-Duhem Relation, 85–87
hypothetical reversible engine, 85
inductive-reasoning, 81
intensive variable, 82
internal energy, 83
irreversible, 359
ISO-TEST simulations, 389
mathematical quantity, 81
measurements, 131
Newtonian (continuum) mechanics, 94–96
Newtonian mechanics, 120
Newtonian mechanics formulation, 118
non-equilibrium states, 83
nonnegative, 373
permutability measure, 137, 176, 180
plastic deformation energy, 125
production, 82, 116, 127, 351, 352, 371, 390
production in irreversible process, 90–91
production rate, 126, 420
production values, 389
property, 83
resonator, 185–188
source strength or internal, 417
stable equilibrium, 84
state functions, 85
states of energy, 82
statistical mechanics, 136, 367
and stiffness matrices, 411
temperature means, 81
thermodynamic potential
(see Thermodynamic potential)
third law of thermodynamics, 84
TSI, 194
zero Kelvin temperature, 84
Entropy balance equation, 216
Entropy generation mechanisms, 324
Entropy generation rate, 216
Entropy law and balance
Clausius-Duhem inequality, 223, 224, 226,
228
conservation of energy, 226
continuum mechanics, 225
direction, 225
elastic deformation, 229
elastic energy, 226
entropy, 228
equation, 227
flux quantity, 227
Gibbs relation, 226
heat conduction, 228
heat transfer, 223
Helmholtz free energy, 226
imaginary reversible isothermal path, 223
irreversible process, 224
irreversible thermodynamics, 224
local equilibrium assumption, 224
macroscopic system, 223
mathematical expressions, 225
production, 227
RVE, 224
temperatures, 228
thermal boundary conditions, 228
thermo-mechanical problems, 228
Entropy production, 351
Equivalent stretch rate, 386
Eshelby tensor, 283
Eshelby theory, 285, 302
Eshelby’s equivalence principle, 280
Euler equation, 87–90
Euler form, 87
Euler relations, 88
Eulerian description, 32, 42
Eulerian strain tensor, 52, 54
Exponential-mapping, 384
Index
431
Eshelby’s equivalence principle, 280
fictitious equivalent eigenstrain, 281
Green’s function, 281
inhomogeneities, 280
integral equation, 282–284
inter-particle interaction effects, 283
linear elastic matrix material, 282
micromechanical, 283
perturbed strain field, 281
phase particles, 280, 283
recast, 282
renormalization procedure, 282
RVE, 280, 281
Enthalpy, 101
Entropic fundamental relation, 87
Entropy, 81, 82, 98
balance equations, 418–421
Clausius-Duhem Inequality, 91–94
concept, 80
definition, 80, 85, 118, 126
degradation metric, 129
density, 419
disorder, 96–99, 183, 192
Euler equation, 87–90
extensive parameter, 82
flat surface, 82
flux to heat flux, 359
fundamental governing equations, 84
fundamental relation, 83
generation mechanisms, 191, 389
generation rate, 132, 133, 135, 416
Gibbs-Duhem Relation, 85–87
hypothetical reversible engine, 85
inductive-reasoning, 81
intensive variable, 82
internal energy, 83
irreversible, 359
ISO-TEST simulations, 389
mathematical quantity, 81
measurements, 131
Newtonian (continuum) mechanics, 94–96
Newtonian mechanics, 120
Newtonian mechanics formulation, 118
non-equilibrium states, 83
nonnegative, 373
permutability measure, 137, 176, 180
plastic deformation energy, 125
production, 82, 116, 127, 351, 352, 371, 390
production in irreversible process, 90–91
production rate, 126, 420
production values, 389
property, 83
resonator, 185–188
source strength or internal, 417
stable equilibrium, 84
state functions, 85
states of energy, 82
statistical mechanics, 136, 367
and stiffness matrices, 411
temperature means, 81
thermodynamic potential
(see Thermodynamic potential)
third law of thermodynamics, 84
TSI, 194
zero Kelvin temperature, 84
Entropy balance equation, 216
Entropy generation mechanisms, 324
Entropy generation rate, 216
Entropy law and balance
Clausius-Duhem inequality, 223, 224, 226,
228
conservation of energy, 226
continuum mechanics, 225
direction, 225
elastic deformation, 229
elastic energy, 226
entropy, 228
equation, 227
flux quantity, 227
Gibbs relation, 226
heat conduction, 228
heat transfer, 223
Helmholtz free energy, 226
imaginary reversible isothermal path, 223
irreversible process, 224
irreversible thermodynamics, 224
local equilibrium assumption, 224
macroscopic system, 223
mathematical expressions, 225
production, 227
RVE, 224
temperatures, 228
thermal boundary conditions, 228
thermo-mechanical problems, 228
Entropy production, 351
Equivalent stretch rate, 386
Eshelby tensor, 283
Eshelby theory, 285, 302
Eshelby’s equivalence principle, 280
Euler equation, 87–90
Euler form, 87
Euler relations, 88
Eulerian description, 32, 42
Eulerian strain tensor, 52, 54
Exponential-mapping, 384
Index
431