Resistors, 131
Return mapping algorithm
classical theory rate dependent model, 214
deviatoric strain increment vector, 209
explicit/implicit method, 209
flow rule, 210
generalized midpoint rule, 209
integration parameter, 210
linearization (see Linearization (Consistent
Jacobian))
Newton method, 211
rate independent case, 211
relative stress tensor, 210
yield surface, 210
Rice, 118, 119, 121, 122
Richeton’s model, 366
Rotation tensor, 345, 347
Rubber-elasticity model, 367
Rubbery modulus, 375
Rubbery shear modulus, 375
S
Saturation value, 376
Second law of thermodynamics, 119, 351, 372,
417
Clausius-Duhem inequality, 80
definition, 80, 81
entropy (see Entropy)
pendulum, 80
Second law of unified mechanics theory, 133,
134
Second Piola-Kirchhoff stress, 61–63, 368
Second-order tensor, 14, 18
Self-diffusion, 399
Self-organizing systems, 117
Shear strain, 30, 59
Shear stresses, 19, 20, 28, 65
Single ellipsoidal inclusion, 285
Skew symmetric tensor, 220
Small rotation formulation, 31, 32
Small strain
definition, 29
local coordinates, 33, 36–38
material coordinates, 32–35
pure rigid body motion, 31
pure uniaxial strain, 29
shear strain, 30, 31
local coordinates, 39
spatial vs. local coordinates, 40
small rotation formulation, 31, 32
Small strain theory, 419
Sorbolle’s electromigration theory, 401
Spatial description, 32, 49
Spherical stress tensor, 26
Spin tensor, 43, 344, 345, 348, 350, 420
State distribution, 140, 141
State functions, 85, 94, 95
State variables, 85, 94, 95, 98, 99, 101, 102,
105–108
Statistical mechanics, 135, 136, 191
Stiffness matrix, 416
Strain
definition, 28
formulation, 28
samll (see Small strain)
Strain equivalence principle, 208, 322
Strain rate, 363–367, 369, 371, 373, 377–379,
381, 383–386
Strain rate histories, 385, 386
Strain rate measures, 371
Stress
arbitrary plane, 15–18
definition, 12–14
Stress components, 371
Stress gradient, 395–397, 400
diffusion driving force, 404–406
induced diffusion, 406
Stress invariants
characteristic equation, 24
deviatoric stresses, 27, 28
hydrostatic stress, 26
principal axes, 25
scalar quantity, 25
spherical stress tensor, 26
Stress measures, 354, 357
Stress modulus, 380
Stress-strain curves, 376, 380, 382
Stress-strain relationship, 231, 336
Stress tensor, 19, 347, 355
arbitrary tetrahedron, 22
Cartesian coordinate system, 21
direction cosines, 23, 24
principal axes, 22
principal planes, 22
principal stresses, 24
symmetric, 22, 24
3-dimensional stress-tensor, 22
Stretching rate tensor, 344, 345
Stretching tensor, 43
Stretch-like internal variable, 362
Sudden brittle failure, 373
Surface stress vector, 123
Surface tractions, 355
438
Index
Return mapping algorithm
classical theory rate dependent model, 214
deviatoric strain increment vector, 209
explicit/implicit method, 209
flow rule, 210
generalized midpoint rule, 209
integration parameter, 210
linearization (see Linearization (Consistent
Jacobian))
Newton method, 211
rate independent case, 211
relative stress tensor, 210
yield surface, 210
Rice, 118, 119, 121, 122
Richeton’s model, 366
Rotation tensor, 345, 347
Rubber-elasticity model, 367
Rubbery modulus, 375
Rubbery shear modulus, 375
S
Saturation value, 376
Second law of thermodynamics, 119, 351, 372,
417
Clausius-Duhem inequality, 80
definition, 80, 81
entropy (see Entropy)
pendulum, 80
Second law of unified mechanics theory, 133,
134
Second Piola-Kirchhoff stress, 61–63, 368
Second-order tensor, 14, 18
Self-diffusion, 399
Self-organizing systems, 117
Shear strain, 30, 59
Shear stresses, 19, 20, 28, 65
Single ellipsoidal inclusion, 285
Skew symmetric tensor, 220
Small rotation formulation, 31, 32
Small strain
definition, 29
local coordinates, 33, 36–38
material coordinates, 32–35
pure rigid body motion, 31
pure uniaxial strain, 29
shear strain, 30, 31
local coordinates, 39
spatial vs. local coordinates, 40
small rotation formulation, 31, 32
Small strain theory, 419
Sorbolle’s electromigration theory, 401
Spatial description, 32, 49
Spherical stress tensor, 26
Spin tensor, 43, 344, 345, 348, 350, 420
State distribution, 140, 141
State functions, 85, 94, 95
State variables, 85, 94, 95, 98, 99, 101, 102,
105–108
Statistical mechanics, 135, 136, 191
Stiffness matrix, 416
Strain
definition, 28
formulation, 28
samll (see Small strain)
Strain equivalence principle, 208, 322
Strain rate, 363–367, 369, 371, 373, 377–379,
381, 383–386
Strain rate histories, 385, 386
Strain rate measures, 371
Stress
arbitrary plane, 15–18
definition, 12–14
Stress components, 371
Stress gradient, 395–397, 400
diffusion driving force, 404–406
induced diffusion, 406
Stress invariants
characteristic equation, 24
deviatoric stresses, 27, 28
hydrostatic stress, 26
principal axes, 25
scalar quantity, 25
spherical stress tensor, 26
Stress measures, 354, 357
Stress modulus, 380
Stress-strain curves, 376, 380, 382
Stress-strain relationship, 231, 336
Stress tensor, 19, 347, 355
arbitrary tetrahedron, 22
Cartesian coordinate system, 21
direction cosines, 23, 24
principal axes, 22
principal planes, 22
principal stresses, 24
symmetric, 22, 24
3-dimensional stress-tensor, 22
Stretching rate tensor, 344, 345
Stretching tensor, 43
Stretch-like internal variable, 362
Sudden brittle failure, 373
Surface stress vector, 123
Surface tractions, 355
438
Index