Traction, 12
Transformation rules, 347
Tribo-fatigue entropy, 129, 130
True stress, 385, 386
True stress-strain curves, 384, 385
Two-phase composites
dispersed elastic spherical inhomogeneity,
277
effective elastic moduli, 279
elastic matrix, 277
inter-particle interactions, 278
non-interacting solution (see Noninteracting solution)
pairwise interacting solution (see Pairwise
interacting solution)
particle interaction, 290
plastic deformation, 290
viscoplastic matrix, 284
Two-phase interaction approximate solutions
conditional probability function, 294
ensemble integration, 294
expression, 294
mathematical manipulation, 295
non-interacting, 292, 293
particle configuration, 292
procedure, 293
spherical particle, 293
Two-point conditional probability function, 294
U
UMT Cosserat continuum implementation
coupling, 256
entropy generation rate, 266
radial return algorithm- rate dependent
model, 269–273
radial return algorithm- rate independent
model, 266–269
rate dependent material without degradation
constitutive equation, 262
TSI, 263, 264
yield surface, 262
rate independent material without
degradation
consistency parameter, 258, 260
constitutive model, 257, 261
constitutive tensor, 262
couple back-stress, 257
deviatoric component, 257
evolution equations, 259
flow rules, 258
flow theory, 261
generalized strain tensor, 257
hardening parameter, 257
Hooke’s law, 259–262
kinematic hardening modulus, 258
rate form, 256
yield surface, 257
size effects, 256
viscoplastic creep law, 265
UMT rate dependent model-strain gradientformulation, 267
Uniaxial stress-strain hysteresis loop, 339, 340
Uniaxial tension, 205
Unified mechanics theory, 100, 395, 411, 416
finite deformations (see Finite
deformations)
TSI (see Thermodynamic state index (TSI))
Uniform motion, 7
Universal law of degradation
additive constant, 185
electromagnetic radiation theory, 182
entropy, 183
molecular-kinetic considerations, 182
probability, 184
Stirling’s theorem, 185
thermodynamics, 182
V
Vacancy diffusion mechanism, 406
Vanishing internal resistance, 365
Vector quantity, 399
Velocity gradient, 343, 344, 348, 357
Virtual work
principal, 355–357
Virtual work rate, 353
Viscoplastic creep rate law, 207
Viscoplastic flow rule, 360, 389
Viscoplastic model, 365–367, 416
Viscoplastic strain, 290
Viscoplastic strain rate, 366
Viscoplasticity, 107
Viscoplasticity model
back stress evolution, 323
ensemble-volume averaged stress norm, 322
filler particles, 322
kinematic hardening behavior, 322
kinematic hardening function, 324
plastic strain increment equations, 322
UMT framework, 322
viscoplastic behavior, 322
viscoplastic strain rate, 323
von-Mises type, 322
yield function, 323
Viscous flow, 363
Index
441
Transformation rules, 347
Tribo-fatigue entropy, 129, 130
True stress, 385, 386
True stress-strain curves, 384, 385
Two-phase composites
dispersed elastic spherical inhomogeneity,
277
effective elastic moduli, 279
elastic matrix, 277
inter-particle interactions, 278
non-interacting solution (see Noninteracting solution)
pairwise interacting solution (see Pairwise
interacting solution)
particle interaction, 290
plastic deformation, 290
viscoplastic matrix, 284
Two-phase interaction approximate solutions
conditional probability function, 294
ensemble integration, 294
expression, 294
mathematical manipulation, 295
non-interacting, 292, 293
particle configuration, 292
procedure, 293
spherical particle, 293
Two-point conditional probability function, 294
U
UMT Cosserat continuum implementation
coupling, 256
entropy generation rate, 266
radial return algorithm- rate dependent
model, 269–273
radial return algorithm- rate independent
model, 266–269
rate dependent material without degradation
constitutive equation, 262
TSI, 263, 264
yield surface, 262
rate independent material without
degradation
consistency parameter, 258, 260
constitutive model, 257, 261
constitutive tensor, 262
couple back-stress, 257
deviatoric component, 257
evolution equations, 259
flow rules, 258
flow theory, 261
generalized strain tensor, 257
hardening parameter, 257
Hooke’s law, 259–262
kinematic hardening modulus, 258
rate form, 256
yield surface, 257
size effects, 256
viscoplastic creep law, 265
UMT rate dependent model-strain gradientformulation, 267
Uniaxial stress-strain hysteresis loop, 339, 340
Uniaxial tension, 205
Unified mechanics theory, 100, 395, 411, 416
finite deformations (see Finite
deformations)
TSI (see Thermodynamic state index (TSI))
Uniform motion, 7
Universal law of degradation
additive constant, 185
electromagnetic radiation theory, 182
entropy, 183
molecular-kinetic considerations, 182
probability, 184
Stirling’s theorem, 185
thermodynamics, 182
V
Vacancy diffusion mechanism, 406
Vanishing internal resistance, 365
Vector quantity, 399
Velocity gradient, 343, 344, 348, 357
Virtual work
principal, 355–357
Virtual work rate, 353
Viscoplastic creep rate law, 207
Viscoplastic flow rule, 360, 389
Viscoplastic model, 365–367, 416
Viscoplastic strain, 290
Viscoplastic strain rate, 366
Viscoplasticity, 107
Viscoplasticity model
back stress evolution, 323
ensemble-volume averaged stress norm, 322
filler particles, 322
kinematic hardening behavior, 322
kinematic hardening function, 324
plastic strain increment equations, 322
UMT framework, 322
viscoplastic behavior, 322
viscoplastic strain rate, 323
von-Mises type, 322
yield function, 323
Viscous flow, 363
Index
441