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Index

a priori stability estimates, 280, 286, 288, 289, 296, 325, 329-331

in/-sup condition for nonconform­ing formulations, 404-416

algorithmic symmetrization of frictional stiffness, 174-179,

226 assembly

of contact contributions, 149, 150, 152

of element contributions, 62, 68

augmented Lagrangian methods, 91-94,99-101,133-134, 169-179, 226, 315, 336, 356

algorithmic performance, 196-209

Bernstein polynomial, 382, 384, 389

Cauchy-Green tensor, 19 inelastic, 36

Clausius-Duhem inequality, 281 consistent algorithmic linearization,

61-63 of frictional tractions, 161, 177,

178,224-225 of neo-Hookean hyperelastic­

ity, 65 of small strain elastoplastic­

ity, 63 of thermomechanical friction,

259-261 contact detection, see contact search­

ing contact force vector, 146, 162-168,

178-179,226,265-269 contact pressure, 72, 77, 298 contact searching, 152-158 contact segments, 151-152,424 contact stiffness, 146, 162-168, 178-

179,226, 263,265-269 contact virtual work, 79-81, 102,

105, 137-139, 141, 148, 256,420

linearization of, 141-144 control points

452 Index

of a smooth three dimensional patch, 383-389

convected kinematic frame for frictional response, 120-

121, 131, 137, 159, 161, 166, 168, 169, 220-222

convected relative velocity, 118, 119 Courant stability limit, 53 current configuration, see spatial

configuration

deformation gradient, 18 directional derivative, 65, 66, 81,

84,101,140-143,150,161 discrete derivative, 311

elasticity finite strain, 33 linear, 10, 74

energy consistent algorithm, 288, 325-347

energy stability, 297-304 energy-momentum algorithm, 279,

280,286,288,289,294 using discontinuous velocity

updates, 347-365 energy-momentum methods

for elastodynamics, 304-311 entropy inequality, 246, 283, 285 Eulerian description, 21 explicit time stepping, 5, 52 external force vector, 45, 146

FEAP, 180 finite element discretization, 41-

43,47 of contact interaction, 145-

168, 316-318 first law of thermodynamics, 245,

279,280 flow operator, 123, 126 friction

Coulomb, 96-101, 325-328 in large deformations, 130-

134

in mortar descriptions, 420 nonlocal description, 106-108 rate and state dependent, 212-

238 in thermomechanical formu­

lations, 251-255 thermomechanically coupled,

238-294 frictional heating, 238, 244, 270,

275-278 frictional kinematics

in large deformations, 116-122

gap function, 72, 76, 81, 298 in large deformations, 114, 115

Green strain tensor, 20 Gregory patch, 383-389

Hermite cubic interpolation, 374, 378-382

Hertzian contact problem, 427 HHT integrator, see Hilber-Hughes­

Taylor integrator Hilber-Hughes-Taylor integrator,

50, 158, 160, 299 Newmark family, 50, 159, 189,

194, 295, 299 central differences, 51 trapezoidal rule, 51,58, 159,

300 hyperelasticity, 33

neo-Hookean, 34, 39, 135

mvp, see initial/boundary value problem

impact mechanics energy-momentum treatment

of, 295-365 implicit time stepping, 5, 51, 57 incremental load approach, 49 inelastic hardening

isotropic, 13 kinematic, 13

initial/boundary value problem

finite strain, 17, 38 thermomechanically coupled,

241-244 with frictional contact, 110-

137 kinematically linear, 8, 14

with frictional contact, 94-101

with frictionless contact, 70-79

strong form, 9, 31, 134-137 weak form, 16, 39

frictional contact, 101-102, 137-144

frictionless contact, 79-81 intermediate contact surface, 373,

416-422,424,425 internal force vector, 45, 146 isoparametric interpolation, 147

Kuhn-Tucker conditions for contact interaction, 73, 77,

86, 130, 131, 420 in inelasticity, 13, 38 .

Lagrange multiplier methods, 85-89,298,314,333,356

Lagrangian description, 21 Lie derivative, 31, 36

with respect to relative ve­locity, 125-129

with respect to surface veloc­ity, 122-125

line search, 59-61 loading/unloading conditions, see

K uhn-Tucker conditions, inelastic

mass matrix, 45, 146 material frame indifference, 27-

31 of contact rate variables, 121-

129, 160 material time derivative, 21, 118

in terms of the directional deriva­tive, 141

Index 453

midpoint rule, 304, 332 mortar methods

for frictional contact, 416-432 for tied contact, 404-416 in contact description, 404-

432

Nanson's formula, 26 Newton-Raphson solution proce­

dure, 57-58, 159, 261 numerical integration, see quadra­

ture

objectivity, see material frame in­difference

patch test for contact, 369, 370, 373, 417,

425 penalty methods, 89-91, 99, 131-

133, 159-168, 221, 315, 335, 356

persistency condition in impact, 309, 313, 330, 331,

333, 335, 347 Piola traction, 26, 119 Piola transformation, see Nanson's

formula plasticity

finite strain multiplicative, 35, 39, 136

small strain rate independent, 12,54,75

polar decomposition, 19, 24

quadrature of contact integral, 148-152,

262 quasistatic problem, 48, 56, 146,

159, 261

radial return algorithm, 55 rate dependence

of friction, 212-2141 217, 235 rate of deformation tensor

material, 23

454 Index

spatial, 22 reference configuration, 17, 110 rotated configuration, 24

second law of thermodynamics, 244, 246, 247, 279

self contact, 156-157,181 semidiscrete formulation, 48, 146 Signorini problem, 70 simultaneous iteration

in augmented Lagrangian meth­ods, 174

single surface contact, see self con­tact

slip advected bases, 117, 128, 130, 131, 136, 137

small strain tensor, 10 spatial configuration, 17 spatial kinematic frame

for frictional response, 119-120, 130, 136, 159, 161, 164, 168

spatial relative velocity, 118, 119 spline interpolation, 374-378 staggered solution scheme

for coupled problems, 268, 269, 271,280

stick-slip motion, 212, 228-230 stress definition

Kirchhoff stress, 27 Cauchy stress, 9, 24 first Piola-Kirchhoff stress, 26 rotated stress, 27 second Piola-Kirchhoff stress,

26 strong form, see initial/boundary

value problem, strong form

Taylor impact problem, 188, 295 thermal softening

bulk, 275 frictional, 238, 240, 244, 249,

251, 279, 285, 293, 294 thermodynamical consistency

algorithmic, 279-294

in frictional model formula­tion, 244-251

two body contact problem, 75, 110 two pass contact algorithms, 157-

158

Uzawa's method, 170-174

variational inequalities, 81, 102-106

viscoplastic regularization in frictional model formula­

tion, 218-220

weak form, see initial/boundary value problem, weak form