Nonlinear Continua Part 14 pot

acceleration, 10convected, 104 material, 11 admissible displacements, 184 admissible transformation, 213 Almansi deformation tensor, 35, 49, 58 pull-back, 43 associated plastic flow, see

acceleration, 10

convected, 104

material, 11

admissible displacements, 184

admissible transformation, 213

Almansi deformation tensor, 35, 49, 58

pull-back, 43

associated plastic flow, see associative

plastic flow

associative plastic flow, 144

back-stress tensor, 142, 161

balance of energy, 109

Eulerian (spatial) formulation, 109

Lagrangian (material) formulation,

112

localized material form, 112

localized spatial form, 111

balance of moment of momentum, 105

Eulerial (spatial) formulation, 105

localized spatial form, 107

symmetry of stress measures, 107

balance of momentum

Eulerian (spatial) formulation, 96

Lagrangian (material) formulation,

103

localized material form, 104

localized spatial form, 97

balance of momentum principle, 95

balance principles, 85

base vectors, 216

contravariant, 218

covariant, 216

Bernstein formula, 157

Bianchi identity, 243 body-attached loads, 187 Bridgman experimental observations,

140, 144 buckling, 188, 199

Cartesian coordinates, 5, 213 Cauchy elastic material, see elastic material model

Cauchy stress tensor, 69—72, 78 symmetry, 107

Cauchy Theorem, 71 chemical energy, 109 classical objectivity, see objectivity Clausius-Duhem inequality, 162 compatibility, 61

configuration reference, 8, 10, 11, 22, 33—35 spatial, 10, 11, 25, 33, 35 configurations, 7

conjugate stress/strain rate measures, 72

consecutive rotations, 52 conservative loads, 195, 197 consistency equation, 150, 170 constant direction load, 187 constitutive relations, 115 continuity equation, 93 continuous body, 7 continuous media hypothesis, 7 kinematics, 7 contravariant transformation rule, 214 control surface, 88

control volume, 88

convex yield surface, 142

coordinates

convected, 13, 34

Eulerian, 52

fixed Cartesian, 11

Lagrangian, 52

material, 8, 14

spatial, 8, 11, 14

coordinates transformation, 213

corotational stress rate, 84

covariance, 44, 49

covariant derivative, 233

of a general tensor, 236

of a vector, 233

covariant rates, 58

covariant transformation rule, 215

cross product between vectors, 229

curvilinear coordinates, 5, 213

damage mechanics, 115

decomposition

left polar, 25

multiplicative, 23, 25

polar, 21

numerical algorithm, 28

right polar, 22

deformation gradient tensor, 13, 47

inverse, 14

multiplicative decomposition, 158

transpose, 14

density, 9

deviatoric components

Cauchy stress tensor, 141

Green-Lagrange strain tensor, 126

displacement vector, 9

distribuited torques, 69

divergence of a tensor, 238

Doyle-Ericksen formula, 122

Drucker’s postulate, 142, 144, 145, 147

dyads, 224

eigenvalues, 28—33

eigenvectors, 28—33, 230

elastic energy, 109, 120

elastic material model, 120

elasticity tensor, 123

spatial, 123

symmetries, 124

elastoplastic material model 1D case, 135

finite strains, 155 general formulation, 140 infinitesimal strains, 135 thermal eects, 170 energy conjugate, 73, 75, 76, 79 energy dissipation, 162

entropy, 167 equilibrium, see balance of momentum principle

equipresence, 116 equivalent plastic strain, 153 equivalent plastic strain rate, 152 equivalent stress, 152

essential boundary conditions, 205 Euclidean space, 62, 213

event, 45

fiber, 14 Finger deformation tensor, 25, 33, 58 first law of Thermodynamics, see balance of energy

flow rule, 140, 142, 144, 177 follower load, 187

forces concentrated, 70 external, 67 internal, 67 per unit mass, 68 surface, 68 Fourier’s law, 169 free energy, 162, 165 frictional material, 148

generalized Gauss’ theorem, 88 gradient of a tensor, 237 Green deformation tensor, 21, 33 Green elastic material, see hyperelastic material model

Green-Lagrange strain tensor, 34, 48 Green-Naghdi stress rate, 84

hardening law, 135, 140, 151, 162, 172, 175

heat flux, 110 heat source, 110 Helmholtz’s free energy, 167 Hencky strain tensor, 35, 56, 161 time derivative, 79

homeomorphism, 8

hydrostatic component

Cauchy stress tensor, 141

hyperelastic material model, 120

hyperelasticity, 115, 120

hypoelastic material model, 121, 155

incompressible flow, 180

incremental formulation, 164, 189

infinitesimal strain tensor, 65

instant, 7

internal energy, 109

isocoric deformation, 21

isometric transformation, 45

isotropic hardening, 151, 161, 172

isotropic materials, 125

J2-yield function, see von Mises yield

function

Jacobian, 213

of the transformation, 19

time rate, 86

Jaumann stress rate, 84

jump discontinuity, 90

condition, 93

kinematic constraints, 207, 209

kinematic evolution, 50

kinematic hardening, 151, 154, 161, 170,

175

kinetic energy, 73

Kirchho stress tensor, 74, 82, 84

Kotchine’s theorem, 93

Lagrange criterion, 88

Lagrange multipliers, 207

physical interpretation, 208

Lagrangian system, 51

Laplacian of a tensor, 239

Lee’s multiplicative decomposition, 158

Left Cauchy-Green deformation tensor,

see Finger deformation tensor

left stretch tensor, 25, 28, 48

physical interpretation, 26

pull-back, 43

Levi-Civita tensor, 229

Lie derivative, 56, 58, 82, 84

local action, 117

Logarithmic strain tensor, see Hencky

strain tensor

mapping, 8 mass, 9 mass-conservation principle, 74, 93 Eulerian (spatial) formulation, 93 Lagrangian (material) formulation, 95

localized material form, 95 localized spatial form, 93 material

isotropic, 79, 81 material derivative, 12 material particle, 7, 14 material surface, 88 material time derivative, 11, 12 material-frame indierence, 116 mathematical model, 2

linear, 2 nonlinear, 2 metric, 219 Cartesian coordinates, 220 curvilinear coordinates, 220 metric tensor, 15, 57, 220, 228 contravariant components, 220 covariant components, 220 mixed components, 221 pull-backs of the spatial, 43 push-forward of the reference, 44 minimum potential energy principle, 197

momentum conservation principle, see balance of momentum principle Mooney-Rivlin material model, 131 motion, 8, 14

continuous body, 9, 10 Eulerian description, 11, 12 Lagrangian description, 10—12 regular, 8

moving control volume energy conservation, 111 mass conservation, 94 momentum conservation, 99 multiplicative decomposition of the deformation gradient, 158

n-poliad, 227 Nanson formula, 100 natural boundary conditions, 203, 205, 207

neo-Hookean material model, 131—133

Newtonian fluids, 180

no-slip condition, 181

nonassociated plasticity, 144, 149, 151

nonconservative loads, 198

nonconvex yield surface, 147

nonpolar media, 70

notation, 5

numerical model, 2

objective

physical law, 50

rates, 58

stress rate, 61, 81

objectivity, 44

classical, 47

criteria, 47

observation frame, see reference frame

Ogden hyperelastic material model, 129

Oldroyd stress rate, 81, 82

orthotropic material, 125

perfect fluid, 98

Euler equation, 98

perfectly plastic material, 147

permanent deformations, see plastic

deformations

permutation tensor, see Levi-Civita

tensor

physical components, 244

physical phenomena

observation, 1

quantification, 1

Piola identity, 102

Piola Kirchho stress tensor

first

symmetry, 107

second

symmetry, 107

Piola-Kirchho stress tensor

first, 74

second, 76, 83

plastic deformation, 135

plastic dissipation, 143, 164

maximization, 143, 164

Kuhn-Tucker conditions, 143, 164

plasticity, 115

point, see material particle

polar decomposition, see decomposition

polar media, 70 postbuckling, 201 potential energy, 195 power, 72

principle of maximum plastic dissipa-tion, 143

principle of stationary potential energy, 195

principle of virtual power, 194 principle of virtual work, 183 geometrically nonlinear problems, 186 projection theorem, see reciprocal theorem of Cauchy

proper transformation, 214 pull-back, 36, 75, 79 strain measures, 43 tensor components, 40 vector components, 36 push-forward, 42 strain measures, 43

quotient rule, 232

Rayleigh-Ritz method, 205 reciprocal theorem of Cauchy, 72 reference frame, 45

Reynolds’ transport theorem, 85 discontinuity surface, 90 generalized, 88

Riemann-Christoel tensor, 62, 240 right Cauchy-Green deformation tensor, see Green deformation tensor right stretch tensor, 22, 28, 48 physical interpretation, 26 rigid boundary conditions , see essential bounday conditions

rigid rotation, 15 rigid translation, 15 rotation tensor, 23, 26, 48 physical interpretation, 26 rotor of a tensor, 240

Serrin representation, 32 shear modulus, 126 softening material, 137, 147 space-attached loads, 187 spatial derivative, 12 spin tensor, see vorticity tensor stable materials, 144

strain measures, 33

strain rate eect, 176

strain rate tensor, 51

strain rates, 50

stress tensor, 79

stresses, 67

stresses power, 73

symmetry of stress measures, 107

tangential constitutive tensor, 150

tensor analysis, 213

tensors

covariant, 59

Eulerian, 47, 58

isotropic, 125, 156

Lagrangian, 47, 50

n-order, 227

orthogonal, 23

physical components, 245

second-order, 223

eigenvalues and eigenvectors, 225

symmetric, 22

two-point, 14, 23, 47, 50, 74

thermal energy, 109

thermo-elastoplastic constitutive model,

170

thermoelastic constitutive model, 167

time, 7

time rates, 50

total Lagrangian formulation, 190

total-Lagrangian Hencky material

model, 166

traction, 70

transformation

isometric, 45

Truesdell stress rate, 82

updated Lagrangian formulation, 190, 192

variational calculus, 183 variational consistency, 210 variational methods, 183 variations, 184

vector analysis, 213 vector components, 216 vectors, 215

velocity, 10 material, 10 velocity gradient tensor, 50 Veubeke-Hu-Washizu variational principles, 209

constitutives constraints, 211 kinematic constraints, 209 virtual displacements, 184 virtual strains, 185 virtual work, 185 viscoelasticity, 115 viscoplasticity, 115, 176 volumetric component Green-Lagrange strain tensor, 126 volumetric modulus, 126

von Mises yield function, 140, 141 vorticity tensor, 51

work hardening, 147

yield criterion, 135, 144, 161, 170 yield surface, 140, 144, 148, 171, 177 Young’s modulus, 126