Index

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8-node brick element, 68, 87, 324

20-node brick element, 69, 70, 90, 141,

167, 324

4-node tetrahedral element, 71

10-node tetrahedral element, 72–74, 141,

324

6-node wedge element, 73

15-node wedge element, 73, 141

absolute temperature, 37, 313

acceleration, 20, 24, 120–122, 126, 138

accuracy, 76, 120, 126, 129, 173

active slip planes, 249, 258, 259

additive decomposition, 226

alternating symbols, 15, 32, 55

angular momentum, 26, 31, 37

angular velocity, 27, 103

anisotropic material, 216, 225, 244, 260

annihilation of material, 184

area change, 8, 218

Arruda–Boyce model, 193, 212

arteries, 219

associative, 230, 246, 263, 265

asymptotic expansion, 132

automatic meshing routines, 73

axiom of

material invariance, 46

objectivity, 46

axisymmetric elements, 166

backward Euler, 240, 248, 293

Baker–Ericksen inequality, 191

Bauschinger effect, 230

beam elements, 162, 165

beam subject to hydrostatic pressure, 154

The Finite Element Method for Three-dimensional Thermomechanical Applications Guido Dhondt

2004 John Wiley & Sons, Ltd ISBN: 0-470-85752-8

bilinear mapping, 181

biological tissue, 225

blackbody, 319, 322, 323

Boltzmann constant, 319

boundary conditions, 38, 39, 41–43, 91,

97, 101, 116, 118, 155, 305,

312

brick elements, 64, 70, 75, 90, 104, 162,

167, 293

buckling, 113, 114, 154

bulk modulus, 54, 279

Cailletaud model, 246–248, 252, 256, 262

cantilever beam subject to torsion, 170

Cauchy stress tensor, 32

Cauchy’s theorem, 30, 40

Cauchy–Green tensor, 6, 7, 18

centrifugal loading, 103, 150, 312

characteristic equation, 128, 178, 185, 201,

218, 301

Christoffel symbols, 11, 35

Clausius–Duhem inequality, 27, 227, 276,

305

coerciveness inequality, 180, 189

cofactor, 8, 36, 58, 152, 173, 174, 188

commutative, 265

complex conjugate, 14, 93, 106

compressive stress, 50, 104

conduction coefficient, 50, 53, 310

conjugate pairs, 37

conservation of

angular momentum, 26

energy, 26, 306

mass, 25, 315

momentum, 25, 29

336 INDEX

consistency, 120, 126, 129, 229, 251, 268,

270

consistent mass matrix, 78, 138, 140

consistent tangent stiffness matrix, 233

constitutive equations, 28, 37, 38, 43–46,

154

constitutive theory, 44

constrained minimization problem, 277

continuum tangent, 242, 244

contravariant tensor, 3

convective coefficient, 308

convergence, 120, 123, 143, 144, 242,

253, 270, 300

convex hull, 181

convexity, 180

Coriolis force, 24

couple, 26

covariant derivative, 10, 35

covariant tensor, 3

creep, 232, 234, 236, 247, 256–258, 286,

292

critical solution, 110

cross section, 162

cyclic symmetry, 91, 93, 116

deformation gradient, 5, 7, 59, 180, 273,

274, 278

deformation rate tensor, 21, 24, 273

density, 25

deviatoric stress, 236, 237, 262, 278, 284–

286, 295

diffuse emitter, 319

dihedral angle, 73

displacement vector, 67

displacements, 1, 11, 38, 64, 67, 68, 176,

309

dissipative stress, 61, 62

dot product, 3, 8

double contraction, 17, 265

double roots, 206

driving flux, 310, 312

dual basis, 15

dyadic product, 4

eigenvalue problem, 14, 18, 33, 106, 114,

116, 118, 128

eigenvalues of

the Cauchy stress tensor, 33

the Lagrangian strain tensor, 14

eigenvectors, 14–16, 18, 19, 33, 59

elastic materials, 47, 49, 51, 58, 59

elastic predictor, 257, 270

elasticity tensor, 248, 262

elastomeric foam, 195, 208

element capacity matrix, 310

element conduction matrix, 310

element force vector, 67

element mass matrix, 67, 140

element stiffness matrix, 67

elliptic system, 187, 188

emissivity, 308, 319, 321

energy density, 27

entropy, 27, 34, 37, 38, 44, 48–50, 60,

227, 307

entropy density, 27

environmental temperature, 308

equivalent bulk modulus, 210, 212

equivalent environmental temperature, 323

equivalent Poisson coefficient, 212

Eulerian strain tensor, 10

evolution equations, 228–230, 246, 252,

258, 289

existence theorems, 189

expansion coefficient, 50, 53, 102, 104

extreme strains, 177–179

fiber-reinforced material, 219

finite difference equations, 128

finite elements, 120

first law of thermodynamics, 27

flow rule, 230, 231, 258, 269, 270, 277,

280, 281, 284, 288

fluid temperature, 313

force jump, 126, 138

forging process, 225

Fourier components, 119

free energy, 48, 53, 177, 227, 275–277,

305

frequency calculations, 112

friction coefficient, 109, 127

full integration, 76, 87, 89, 105, 166, 167

functional analysis density theorem, 41

INDEX 337

gas equation of state, 314

gas temperature, 313

Gauss integration schemes, 76

generalized complex eigenvalue problem,

116

generalized eigenvalue problem, 106, 114

geometric boundary conditions, 39, 41,

91

global coordinates, 64, 81, 87, 102

global mass matrix, 67

global stiffness matrix, 67, 92, 93

gradient, 36

gray surfaces, 321

Gurtin’s experiment, 180

harmonic excitation, 112

heat equation, 51, 308, 314

heat flux, 27, 50, 91, 155, 276, 306–308

heat sources, 27

heat transfer, 305

hemispherical shell, 166

Hermitian matrix, 93, 96

high-frequency dissipation, 120, 122, 134

hinge, 171

hourglassing, 69, 78, 87, 90, 162

Hu–Washizu weak form, 90

hybrid elements, 90, 176

hyperelastic materials, 59, 177, 190

hypoelastic formulation, 273

I-cross section, 162

ideal gas, 313, 314

implicit summation, 16, 168

in-phase amplitude, 112

incompressibility constraint, 174

incremental stability, 191

infinitesimal rotation, 11, 54, 55

infinitesimal strain, 12, 13, 54, 58, 143,

190, 214, 226, 283

inhomogeneous multiple point constraint,

102

initial conditions, 109

initial tangent stiffness, 209

inner product, 2, 181, 275

instability, 130, 148

integral equation, 67

integration schemes, 76, 78, 138

interaction coefficients, 247

intermediate configuration, 275

internal dynamic variable, 227, 228

internal energy, 26, 27, 31

internal kinematic variable, 227

invariants, 15, 58, 128

isochoric motion, 8

isochoric term, 185, 190

isoparametric formulation, 64

isotropic hardening, 230, 231, 234, 237,

285

isotropic linear elastic materials, 52

Jacobian determinant, 6, 36, 47, 65, 75,

215

Jacobian matrix, 5

kinematic constraints, 171

kinematic hardening, 237, 238, 241, 270,

285, 286

kinetic energy, 26, 139

Kirchhoff stress, 34, 281

knot, 162, 167

Lagrangian multiplier, 13, 91

Lagrangian strain tensor, 10, 13

Lamґe’s constants, 53

large deformation stiffness, 113, 147, 154

large rotations, 159, 190

left Cauchy–Green tensor, 7, 18, 274

length change, 6, 10, 13, 21, 218

linear constraints, 96

linear dynamic analysis, 108

linear elastic materials, 51, 104

local coordinates, 64, 69, 71, 73, 76, 78,

81, 102, 140, 175

local cylindrical coordinate system, 99

local rectangular coordinate system, 98

localization matrix, 67

localization of

the conservation of angular mass, 31

the conservation of energy, 31

the conservation of mass, 28

the conservation of momentum, 29

the entropy inequality, 31

localization principle, 28

338 INDEX

lowest eigenfrequency, 113, 154

lumped mass matrix, 140

material coordinate system, 226

material form of

the conservation of angular

momentum, 37

the conservation of energy, 37

the conservation of mass, 35

the conservation of momentum, 35

the entropy inequality, 37

material orientation, 101, 102

material symmetry, 45, 216

maximum plastic dissipation, 276

mean rotation, 167, 170

mean value theorem, 29, 179

mechanical grade, 46, 47, 59, 144, 275

mechanical work, 314

memory, 45, 47, 177, 225

metals, 225

metric tensor, 2–4, 14–16, 18, 33, 275

minor determinant, 8

modal analysis, 116

modal dynamics, 108, 120

Mooney–Rivlin model, 58, 192

multicorrector step, 125

multiple point constraint, 91–94, 96, 97,

102, 118, 155, 161, 174, 317

multiplicative decomposition, 273

multisurface viscoplasticity, 246

natural boundary conditions, 91

neo-Hooke model, 58, 192

network, 313

Newmark algorithm, 120, 121

Newton’s second law, 25

Newton’s third law, 29, 39

Newton–Raphson method, 143, 144, 241,

242, 249, 290, 293

nodal diameter, 114–116

nodal temperature, 309

nonlocal effects, 45

nonpolar theories, 26

Norton law, 232, 263, 286

numerical integration, 75, 82,

139

objectivity, 22

observable quantities, 44

Ogden model, 58, 191, 194, 195, 211

one-form, 3–5, 8, 15, 16, 20, 30, 33, 39,

82

opaque materials, 321

orientation tensor, 244

orthogonal basis, 108

orthogonal matrix, 18, 23, 101

orthogonal vectors, 18

orthogonality of eigenvectors, 14

orthotropy, 45

out-of-phase amplitude, 112

overdetermined system, 84, 96

path length, 148

permutation symbols, 36

perturbation of

the displacement field, 39

the temperature field, 307

piecewise-linear loading, 112

Piola–Kirchhoff stress tensor of the first

kind, 33

Piola–Kirchhoff stress tensor of the second

kind, 34

Planck constant, 319

plane strain elements, 166

plane stress elements, 166

plane surface condition, 173

plastic correction, 295, 298

plastic deformation, 226, 227, 229, 230,

232, 238, 278, 289, 293

Poisson coefficient, 54

polar decomposition, 18

polyconvexity, 184, 189

polynomial form, 192

polynomials, 68, 69

positive definiteness, 6, 14

potential function, 193, 247, 284

predictor values, 122, 123, 125

pressure, 33, 54, 57, 59, 61, 81, 90, 152,

176, 213, 214, 236, 238, 293,

294, 314

pressure waves, 141

principal directions, 15, 18, 19

principal strains, 13, 15

INDEX 339

principal stresses, 52, 203

principle of

objectivity, 45, 47, 60, 62

virtual power, 38

virtual work, 38, 40, 42

proper transformations, 47

pull-back, 7, 22, 37, 274

push-forward, 6, 7, 22, 37, 274

radial return method, 241, 291

radiation power, 318, 320, 321

radiosity, 321

raising the index, 15

reciprocity rule, 322, 323

rectangular grid, 326

reduced integration, 76, 78, 85, 87, 90,

104, 105, 166, 167, 293

reduced invariants, 199

reduced polynomial model, 192

reference state, 1, 4, 6

reference temperature, 306, 308

reflections, 23

residual stress, 50, 57, 104

right Cauchy–Green tensor, 6, 274

rigid body motion, 13, 23, 24, 45, 91,

155, 167, 170

Riks method, 148

rotating blade, 113

rotating rod, 12

rotation, 22, 45

rotational degrees of freedom, 161, 162,

166, 167

rubber, 58, 76, 174, 187, 190, 193, 195,

273

second law of thermodynamics, 27, 50

second Piola–Kirchhoff stress, 37, 59

shape functions, 64, 68, 69, 72, 83, 84,

86, 104, 153, 309

shear deformation, 162

shear experiment, 196

shear locking, 69, 87, 88, 90

shear modulus, 54

shear stress, 256

shell elements, 162, 166, 167

shifters, 39

single crystals, 45, 244

single degree of freedom system, 127

slip direction, 244

slip plane, 244, 248, 256, 260

slip system, 244, 246, 251, 261, 262

snapping-through plate, 148

soils, 225, 236

solid angle, 318

spatial state, 4–6, 32

specific gas constant, 314

specific heat, 51, 53, 313

spectral intensity, 318, 319

spectral radius, 136

spectral, directional emissivity, 319

spectral, directional irradiation intensity,

320

spectral, hemispherical emissive power,

318

speed of light in vacuum, 319

spin tensor, 21

St Venant–Kirchhoff material, 53, 190,

212, 214, 216

stability, 120, 126, 130, 138, 191, 192

Stefan–Boltzmann constant, 308

Stefan–Boltzmann law, 319

Stokesian fluids, 62

stored-energy function, 49, 179, 180, 184,

186, 189–191, 194

straight-line condition, 171

strain measures, 9

stress power, 31

stress stiffness, 147

stress tensor, 31, 32, 34, 236, 285

stretch, 6, 19

strong form, 38, 39, 41–43

structural tensor, 17, 203, 216

subcritical solution, 110

summation rule for enclosures, 322

supercritical solution, 110

symmetric entry, 96

symmetric matrix, 183, 268, 280

Taylor series, 46, 129

temperature gradient, 50, 308

temperature loading, 103

tensorial basis, 15

340 INDEX

theory of invariants, 47

thermal grade, 47, 59, 144

thermal stress, 50, 53, 64, 104, 105, 307

thermomechanical processes, 27, 38

time derivative of the Jacobian determinant,

22

time shift, 23, 45

total derivative, 19, 21

total emissive power, 319–321

total hemispherical irradiation power, 320

total viscous strain rate, 256

total, hemispherical absorptivity, 321

total, hemispherical reflectivity, 321

trace, 15, 17, 53, 280

traction continuity condition, 39

traction forces, 151, 154, 312

transformations, 46, 47, 216

translation, 22, 45

transversely isotropic materials, 216

trial and error, 233

trial state, 289

trial stress, 295

triple roots, 202, 206

true stress, 32, 52

tubes with internal flow, 313

twisted surface, 167

two-point tensor, 5, 18, 34, 36

underscore, 16, 168

uniqueness, 180

vector product, 26, 81, 82, 99

velocity, 19, 20, 22, 24–26, 42, 108, 120,

124, 126

velocity gradient, 21

view factor, 322, 328

virtual displacement, 67

viscous shear strain rate, 256

visible triangles, 325, 326, 328

volume change, 8

volumetric locking, 69, 76, 78, 87,

176

volumetric stress, 295

volumetric term, 185, 190, 295

von Mises stress, 236–238, 293

von Mises yield surface, 286

warping, 162

weak form, 38, 39, 41, 43,

307

weak form of

the balance of momentum, 38,

63

the energy balance, 42

weight, 76, 139

work rate, 26

Yeoh model, 192

yield condition, 236, 241, 248, 275–277,

287, 289, 290

Young’s modulus, 54__