Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

lndex

spatial mass, 133-136, 14 I, :142, 148, ·149t

202 surface, 174-176 volume, 175

density in the motion, 133; see .a/so density, spatial mass

derivative

direction al; ,\·ee di recti on al derivative Gateaux, 46

Lie time, 106-108, 193-196, 376, 400 normal, 46

of a scalar field, 45

of a scalar function, 40 of a tc nsor field, 50

of a tensor .function scalar-valued, 4·1-42

.-tensor-valued, 42-44 of a vector field, 49

description

Eu.lcri an (spatial), 60, 61

Lagrangian (material; refercntia-l), 60, 61 determinant., Jacobian, 74; see also volume ratio determinant of a matrix, 7

determinant of a tensor, 15 derivative of, 4 I , 42

deviutor, :) 9

dev.iatoric operator, 19, 230, 233, 285 deviatoric tensor, 19

direct product, I0

direct.ional derivative, 46-48, ·65, 95, I0 I, I06, I07, 374-40 I, 404, 405

of a vector function, 37~376, 393 Dirichlet bounda~y condition, 378, 384, 409 displacement field, 61, 62

actual change of, 373 increment of. 393-395 prcscribed,379,382,385,409

virtual, 373-376, 382, 385, 394, 404 displacement gradient tensor, 73, 85 dissipation, internal, 170, 173, 28 l-288, 299,

358-363,369 dissipative material, 278-280, 332 distribution function

Gaussian, 308, 313, 314, 319 inverse Langevin. 244 Langevin.. 319., 356

divergence

material,, 65, 74

of a vector field, 48 divergence theorem., Gauss', 52. ·. dot product

of tensors, I2, 13 of vectors, 2, 3, 5

Duhamel-Neumann hypothesis, 344 Duhamel's law of heal conduction, ·110, ·111 ..

325,342

dummy "index, 4, 12-14, 35 dyad, 10

dyadic, lO

dynamical process, .148, 153, 159., I60,· l90,

198 dosed, 211, 212

eigenvalues or a tensor, 24-26 eigenvectors of a tensor, 24-26

·elas-tic fluid, 125, 126, 204

elastic mater"ial, 197; see also Cauchy-elastic ma1crial; .hyperclastic material; thcrrno(visco)ela.stic material

elastic ·potential; see model elastic solid_; see. material

e.lasticitics; see also elasticity tensor

.isentropic

rcforcntia·11cnsor of, 329, 330 spatial tensor of, 330

isotherm.al

referential tensor of, 328, 330 spatial tensor of, 328

referential tensor of, 25.2 spatial -tensor of, 253

elasticity; see hyperclasticity; thc.rmoclasticity; thcrmoviscoelaslici ty

c"lastic.ity .tensor, 252-265, 328-333, 40 I algorithmic, 293, 294

components of, 252, 253

decoupled representation of, 254-257, 265,

303,340-342,406 cffcc:tive, 302, 303 fictitious

in material description.. 255-257, 262, 272

in spatial description, 265 iscntropic_.. 329-331, 354

e:lasticity lensor (comd.) isothermal, 328, 330-333. 353

major symmetries,, 253, 328, 40 l in material description, 252, 253

minor symmetries, 253, 396, 398, 400, 40 I numeric al aspects, 33 l , 332

in spatial description, 253 spectral form of

in material description, 257-260, 263,

264

in spa.tia] description, 260 end-to-end.distance of tic poincs, 307, 308, 312-

318

mean value of, 312 end-to-end vector, 31.2

'energetic elastic' material., 311 energy, 131, 132

activation. 364

balance o.f; see balance of energy conserved, 160

free, 173; see also Helmholtz free-energy

.function

internal, 155., 157, 164, 173, 323-326, 329

changein,3J1,319~334

~netic, 153-157, 160,392 mechanical, 152-155, 166 non-recoverable,, 279, 296 potential; see potential energy

strain (stored), 160; see also strain-energy function

thermal, 161, 172

total, 155, 157, 164, 165 energy flux vector, 165

energy functional, 159, 386, 389, 408 penalty form of, 389

energy-momentum tensor, 21 l enthalpy, 324

~entropic elastic' material, 311, 333-336 simple tension of, 343-356

entropic e'lasticity concept of, 333

for a stretched .Piece of rubber, 346-351 entropic theory

modified, 333-335, 33'8, 346, 348 purely {strictly), 333-336, 339, 352

entropy, 166-168, 315-319. 328, 329, 358-363

changein,310,319,329,334,353

equiHbrium part.. 362, 363

non-equHibrium part, 362, 363 of a single chain, 315, 3 16

total production of, I 67 entropy flux

Cauchy (true), 167 Piola-Kirchhoff (nominal), 167

entropy function, 322., 347, 359 entropy inequality principle, 167, 361 entropy input, rate of, J67,, 168, 172

entropy produclion, 167, J68., 282, 358; see also dissipation, internal

by .conduction of heal, 169 entropy source. 167, l 68 equation

global (integral) form of, 134 local (differential) form of, l 34

equation of equHihrium, Cauchy~s. 145.. ·191, 405

equation of evolution; see evolution ·equation equation of motion, 144-146

Cauchy's first, ·144-146. 148, 176. 342,

378~381

Cauchy's ·second, 147 weak form of.. 380

equation of state, 161 T 197 caloric, 323

thennal, 322-324, 347

equilibrium, thermodynamic (thermal). 161, 208, 282,285,359,363,365,366

equilibrium state, 16 I, 28:?, 283, 359, 362., 363 ·equilibrium thermodynamics, 168

Eshelby tensor, 21.1, 212

Euclidean space'.t 4, 57, 59, I80, 181 Euclidean transfonnation, 1. 81

of stress tensors. 19.0, 191

of various kincmatical quantities, 189., 190 Euler-Almansi strain tensor, 82, 88

Euclidean transformation of, 191 first variation of, 376, 377, 382

linearization of, 395, 398 mated al time dcrivative of, 400

Lie lime derivative of, l07 material lime derivative or, 102

Euler-Lagrange equation, 381, 405 .. 408, 409, 411, 412

Eulerian (spatial) description, 60, 6.1 Eulerian form, 60

.event, 180-182, 187

.evolution equation, 281, 282, 286-290, 302, 359,

363,366,369

expansion coefficient, linear, 339, 346, 349, 352 extension

strip..biaxial, 93

uniform (uni.axial), 92, 124 external constraints

holonomic. 379 no.nholonomic, 379

external variables, 278, 282, 287, 289., '364, 368

fiber, 77-79,267,273

direction, ·84, 96, 266-269, 273-276,

extensible, 270 inextensibl-e, 270, 275-277

fiber-reinforced composite, 265, 266, 272, 274 fibers, families of, 265,

field

harmonic, 50, 51, 68, 69, .137 material, 64, 65

mechanical, 306 scalar; see scalar field spatial, 64-67

tensor; see tensor fie'ld

themml, 306

vector; see vector field Finger deformation tensor, 8'J finite element method

displaccmcnl·..based, 402, 403, 411 hybrid, 389

Jacobian-pressure fonnulation. 410

n~xed,389,391,403.404

first law of the.nnodynamics, 164-166.,] 75, 319 first Piola-Kirchhoffstrcss tensor, l U-114, 127,

128, 199, 207

Euclidean .transformation of, 190, ·191 first Piofa..Kirchhoff traction veclor, I 11, J 13,

114, .144

in material description, 1·65 in spatial description, l 63

first variation of a function

in material description, 374,, 375 in spatial descriplion, 375-377

flow

irrotalional, 149 steady, 149

flow behavior index., 367

ftuid, 205

·elastic, I25, l 26, 204 Reiner-Rivlin, 202. 204 v.iscous, 202, 203

Newtonian. 203,.204, 286, 287, 367, 369 fluid mechanics, .205

flux 52, 136, 139, 176, 150 entropy; see entropy ·Hux

heat; see heat flux

force

body; see body force contact, ll 'I

external, 110, 387

inertia, 142,378,382,384,386

interna·t~ I I0

resultant, 110., 1.28, "142 retractive, 310, 311

thermodynamic; see the.rmodynamic force forces, system of, .147-149, 15.3

Fourier's Jaw of heat conductio.n, 171, 342, 348, 354

fourth-order tensor, 22-24 transpose of, 23 fractional-step method, 33 l

frame-indifferent spatial fields, 182, 185, 186.; see also objectivity

free energy, 173, .206, 267-270, 273-275, 280,

298,321-326~328-335,347,357,361,

364; ;ree also Helmho'hz frce-.encrgy

function

configuration~,284,285,304~362,364

Gibbs,323y324

free-energy factor, 364 free index. 4~ 1.s. 34, 43

free vibration, 154

freely jointed chain, 312-315, 319 friction, 160, J66, 31 I

function

convex, strictly, 229, 244, 30.3, 390, 407 linear.. 32, 41, J62

nonlinear. 41, 351, 380, 393 penalty, 390, 391

scalar, 40, 43 tensor, 40-43 test, 380-382 vector, 40 weighting, 380

funclional., 119, 290, 391, 392 energy, 159, 386, 389, 408 Hcllinger-Rcissner, 408, 409 Lyapunov,332

.perturbed, 406

stalionary position of, I J9

Galcrkin method, 402 Galilean transformation, .185 gas ·constant, 364

Gateaux derivative, 46; see al.w directional deri- vative

GfiteauX. operator, 47, 374, 393 Gauss' divergence Lheorcm, 52

Gauss·i an chain, 3 12-316, 3 l"9; see a/so chain Gaussian distribution runcLion, 308, 3 I3, 314,

319

-Gaussian statistical theory, 312, 318, 319, 339 Gibbs free energy, 323, 324

Gibbs function, 323

Gibbs relation, 322-324, 358 glass trnnsi.tion temperature, 309

-global (integral) fonn of.an equation, 134 Gough..Joule effect, 309, 31 l, 326,, 327, 349 Gradient

material, 65, 66, 74 or a scalar field., 45

spatial, 66, 74

of a (.second-order) tensor field, 50 of a tensor function

scalar-valued, 41-42 tensor-valued, 42-44

transposed., 49

of u vector field, 49

Green deformation tensor, 78

Grcc.n··elastic material, 206; see t1lso :hyperclas.. tic mate.rial

Grccn-Gauss-Ostrogradskii theorem, 53 Green-Lagrange strain tensor, 79, 82, 83, 88,

209,252,253,353,365,366 Euclidean transformalion of, 1·91 first variation of, 375 linearization of, 394

time derivative of, WO, IOl, 107, 158 Grcen-Naghdi stress rate. 194, 195 ground ·subslance, 265

g.rowth conditions,, 208, 251

Hamilton,,s variational pri.ncip1c, 391, 392

harmonic field, 50, 51,, 68, 69, 137

heat, 162

heat capacity, specific; see specific heat. capacity heat conduction

Duhamcl's law of, 170, 171,, 325, 342 Fourier's law of, 171, 342, 348, 354

heat conducti-on inequality, 170, 324, 342

heal Hux

Cauchy (true), 162. 163, 170 Piola-Kirchhoff (nominal),, 162, 163, 324,

342,354,359

heat tlux theorem, SLokes~'t 16.2-1·64

heating (cooling)., structura.l; see inelastic :heat- ing; thermoelastic heating; thermovis-

cocla~tic .heating

heat source, 162-164, 168 heat transfer; see heat

Hcllingcr-Reissner functional, 409 Helling.er-Reissner variational principle, 408, 409

Helmhollz free-energy function, l 73, 206; see al.m free energy

decoupled representation of, 27 J, 277, 283,,

303,304

Hcncky strain tensor~ "88 Hessian operator, 50 heterogeneous material, 207

hidden variable~, 278; see al.w internal variables higher-order lensor, 20-24, 37

hist.ory term, 293

history variables, 278, 300; see also .internal variables

homogeneous material, .207, 32 l Hooke's law, 286

Hu-Washizu variational principle, 412 hydrostatic pressure, 125, 222, 245, 390, 391,

404, 410., 411

hydrostatic stress, state of, 125, 126, 23 I hypcrclastic .ma·terial, 205-304,

anisotropic, 214 compressible, 227-235

constitutive equations for, 206-208 mcomprcss1. . .blc,.. ???..........- ??7

isochoric (distortional) elastic response of,

229, 245, 246, 283. 290, 294, 303,

337.338,390,403

isotropic, 212-222

Jocally or.thotropic, 276, 277

ortholropic., 275, 276 transversely ·isotropic, .265-272

volumetric (dilatio.nal) elastic -response o·C

229, 244., 245, 283, 290., 303, 337 t

338,390

with isotropic damage, 295-304 with two families or fibers, 273~275

work done on, 21.l, 212

hyperelasticity, 206

compressible, 228-234

hypoclastic material, 254 hysteresis,, 279, 296, 309

ideal rubber., 311, 318, 333

identity tensor, lO; see also unit tensor impenetrability of matter, 74

incomprcssibiH-ty constraints,, ·.103, 2.22, 223, 225, 237,247,270,403,405,407

incompressible ·hypcrclasticity, 222, 223 -incompressible .isotropic hyperelastidty, 223-226

incompressible material, 202, 203, 222-227, 333,

389, 391, 402

mechankaJly, 345-348,, 355

near1y, 228, 389, 402, 403, 409

incrcmcntallitcrative solution techniques, 252,

293.,392,397,405

incremental objectivity, 292

index

dummy, 4, J2-14, 35

free, 4, 18, 34, 43

live, 4 summation, 4

index notation, 3-5

inelastic (plastic) heating (cooling), structural,

360

inertia forc·c, 142, 378, 382, 384, 386 inertia tcnsort l 60

initial boundary-value problem, 377-382 strong (classical) form of, 379 weak (variational) form of, 381

initial conditions, 289., 302, 363, 379-382

-integral theorems, 52-54

integration method, (sclcctivc-)reduccd, 391 integrity bases, 268

internal constraints, I03, 202., 222.. 223, 270..

389, 41 ·1

internal -energy, 155, 157, 164, 17.3, 323-326,

329

.internal-energy funccion, 3.23t 329, 331, 354 internal variable model, 281, 282

internal variables, 278, 280-284, 287-290, 292,

293,298,299,358

concept of, 278, 279, .283., 361 interpo1ation .functions, 406, 411

.jntr1nsic .angular moment·um, "J 43

invarjants, 25, 89, 20.1-203., 268-270, 274-276

modified, 233

principalt 25 pseudo-, 268, 276

representat-ion theorem for, 215

strain, 215 .. 216, 223, 224, 233, 238, 243

stress, 120., J22 theory of, 268

inverse Langevin func-tion, 244 inverse square law, 51

inverse ·stretch ratio, 81

irreversible process, 168, 170, 279, 281, 282

irrotatiomll flow, 149

irrotational motion, 6.9, 98 :irrotational vector field, 4'8., 50

iscntro.pic elasticity .tensor

in material description_, 329-331, 354

in spatial description, 330

isentrop"ic operator split, 332

isentropic process, 172, 173,. 323

reversible, 172

isochoric motion!' 75, 103, ·136

isothermal elastichy tensor

in material description, 328, 330, 331, 353 in spatial descriplion, 328, 332, 333

isothcnnal operator split, 331

isothermal process, 172,. 322, 323, 346, 356 isotropic material, 120, 201, 214

Lhermal1y, .J 71, 342, 345, 346, 348

Lransvcniely, .265-272

·isotropic ·tensor, 30-32

isolrop ic tensor funetion, 20 l , 2"13-21 8_, 220,

268,274

first representation theorem for, 20 I. 217 second representation theorem for, 202, 217

Jacobian determinant, 74; .~·ee also volume rafio

Jaumann-Zaremba rate, l93T 196

Jaumann-Zarcmba stress rate, 194, 195

Kelvin-Voigt model, 279, 280 kinetic energy~ J53-157, 160, 392

Kirchhoff stress tensor, 127, 147, 158, 159

Lie time derivative of, 194; .we also QJ.. droyd stress rate

linearization of, 398

material time derivative of, 400 Kronecker de:] ta, 5, 7

.mixed, 33, 35

L,agran.ge muI··tl.pl..1er, 119., ..----,.,..,, n4~- , ""5-- , "70- , 275,403,407,410

Lagrange-multiplier method, 119, 403-407 augmented, 406

linearization of, 405, 406 perturbcd,406,407

Lagrangian (material, referentia:J) description, 60,

61

Lagrangian form~ 60 Lame constants, 250

Landau order symbol, 41, 78, 393

Langevin distribulion function, 319, 356

Lapface's equation, 50 Laplacian operator, 50 latent heat, 326, 329

left Cauchy-Green tensor; see Cauchy-Gree.n tensor, left

left stretch tensor; see stretch tensor, tefl Legendre transformation, 173, 31.8, 323, 324,

326,334,408

length of a vector~ 2, 5, 35, 188 :]evel surface, 46, 50 Levi-Civita symbol, 6 l'HopitaJ's rule, 260, 264

.Lie Lime derivative, I06-108, 193-196, 376, 400

Jinear approximation of a function, 3.93, 394

.linear momentum., 14.1., 142

·balance of, 141-144, 149,, ·1so,175 linear operator, 9, 152, 175

lincar.transfonnation, 9, 12, 16, 7l, IOO

1inearization, concept of, 393-395 linearization of a function

in material descri.ption.. 393

in spatial description, 394t 395

linearization operator, 393, 3.94

lincarization process, consistent, 257, 339-342,

393

Hne element compressed, 78

·extended, 78

material (undeformed), 7.1-73., 75-78, 86 spatial (deformed), 7.1-73. 86

first variation of, 377 time derivative ofl I02

unstretched, 78

loads, 1.1l,379, 386-388, 397, 409, 412

'dead', 387, 395 pressure, 242, 395, 402

.local (differential) form of an equalfon., 134 local time derivative, 66

locking phenomena, 402 volumetric, 403, 406, 41 l

Lyapunov functional, 332

macroscopic approach, 56

macroscopic quantities, 57, 16 I, 278., 280 macroscopi·c system, 55, 56

Mandel stress tensor, 128, l 58, 38-6 mass, 56, 131-.136, 139, 140, 320

concentrated, 131, 151. conservation .of, 132-134, 116

Lime derivative of, 133. 134, .l 36 lotal, 134

mass center, 150., 15 l motion of, 151

mass density; see density

mass element, infinitesimal. 133, 140 mass sink, 132

mass source, 132

master balance princ.ip.Je, .J 74-177 in global form, 174-176

in local form, 1"76 master fic·ld equation, 176

master inequality principle, 175, .176 matcrjal

adiabatic, 325

Cauchy-e.lastic; see Cauchy-elastic material composite, 272-277

constrained, 222

dissipative~ 278-280, 332

elastic, 1.97; ,r;ee also Cauchy-elastic material

Index

'energetic clastkt, 311

4 Cntropic -elastict, 31 l, 333-336 heterogeneous, 207 homogeneous, 207, 321

hyperelastic; see hypcrelastic materia:J hypoclastic, .254

incompressible; see incompressible material inelastic, 278-281

isotropic; .vee isotropic material matrix; see matrix material

orLhotropic~ 275-277

perfccLly elastic, .208

ruhber-Hkc, 235; see also .rubber-like material

simple, 290 thermoelastic, 323 thermoviscoelastic. 358 transversely isotropic, 265 viscoe1astic, 283

material frame-indifference, principle of, 198200, 267, 292

material function, 20 l

material mode), 207; see alw model material o~jectivity. principle of, 198 material point, 56

material strain rate tensor, 10-t, 107, 158 materia] strain tensor, 76-79

material time derivative, 64-68, 95, 96, 99-108, 133-141, 192-196, 258, 259, 399--40 l

of a material field. 64._, 65 -of a spatial fieJd_, 66-68 material v.e.locity gradient, 95, 96

matrix

column, 12 diagonal, 26 inverse of, 34 orthogonal, 29 row, 12 ·square, 12

stress, ll 2, I 17. 120

matrix mat·eriaL 265-267, 272-275

incompressible isotropic,, 270, 275-277 matrix notali on, 1l , 12

matrix product, 10

Maxwell element, 286, 287, 368 Maxwell model, 279

generalized, 286, 287, 368

Maxwell relation, thermodynamic, 325, 352, 353

mean dilatation method, 4·11 .. measurable quantities,, 278 mechanical device, 286 mechanical energy, 152-·t60

balance of, in material description, 155-.

157

balance of, in spatial descripLjon, 153-155 mechanical field, 306

mechanical power, external, J53-.156, 159, 160r

164, 165

mechanical theory, _purely, 173, 206·. mechanical work, rate of, 153; see also ex=tcrnal

mechanical power; stress power metric coefficients, 33, 36-38

metric tensor, 37, 39

microscopic approach, 55, 56, 295 microscopic system, 55

mid...point rule, i92 mixed components, 36-39 mixed tensor, 36

model

Arruda and Boyce, 244, 248, 249_, 263 B-latz and Ko, 247., 248, 261, ·262

damage, 298-30 I :I 303, 304

MaxwcU; see Maxwell model Mooney-Rivlin; see Mooney-Rivlin model neo..Hookcnn; see neo-Hookean model Ogden; see Ogden model

rheological, 286-288, 367-369 Saint-Venant Kirchhoff, 250, 25 .1, 365

spring-and-dashpot, 286, 367 thermoviscoelaslic, 361-3.64

Vurga,238-242,239,356 viscoe:Jastic, 283-290

with internal variables, 278....;282 Yeoh,243

molecular network; see network molecule, 55

long-chain, 307, 311

moment uf momentum, balanc.e of, l 4.1, 142 moment, resultant, 142

momentum

angular (moment of; rotational), 141, 142 linear (translational), 141, 142

spin angufar, 143, 152

momentum balance principles, 141-152 for a closed system, I4.1-.149

for an open sys.Lem, 149-150

Mooney-Rivlin model, 203, 339

for compressible materia1s, 247

for incompressible materials, 238-243 motion., 59., ·60

affine, 71 , 3 I6 equation of, .144-.146 inverse, 59, 71, 80, ·163

irrolalional, 69, 98 isochoric, 75, 103, I 36 plane,69, 106., 137

rigid-body; see .rigid...body motion steady,-6.8

uniform, 68 vo1umc-prcscrving, 75

Mullins effect, 296-298

Nnbla operator, 45

Nanson's formula, 75. 104, l l3, 114., .146, :163 nco-Hooke.an model,, 203, 318., 339

for compressible .matcr"ia.ls, 247

for .incompress.iblc ma.tcriaJs, 23"8-243

network,238,244,307,311,312~361

elasticity of, 316-320

entropy change of, 317-320 network density, 312

.neutron scattering, 307

Newtonian nuid, 203

Newtonian shear :thinning phenomenon.. 366 Newtonian viscous fluid., 203, 204.. 286.. 287,

367,369

Newton's law of action and ·reaction., I 12 Newton's method, 257, 29], 295, 393

nomjnal stress tensor, 11.l; ,fee al.w first Piola-

Kirchhoff s.tress tensor

nominal traclion ve.ctor., ·I J.1; see al.m first PiolaKirchhoff traction vector

non-equilibrium state, 161, 279, 284, 362 non-equilibrium stresses, 285 .. 287-290, 35.9,

363.,365.,3-66,368

non-equilibrium thermodynamics, 'l68, 280, 282,

285

non-Gaussian statisticaJ theory.. 244, 3 19, 356 non-pl1lar continuum, 144

norm of a tensor, 15, 365 norm of a vector, 2 normal derivative, 46 normal stresses, .1 ·J 6, 117

maximium and minimum, 119, 120

normalizalion condition, 208., 216, 219.. 229..

245,283,287,298

notation absolute,, 3 direct, 3 index, 3-5

matrix, I :J, 12

subscript, 4

suffix, 4 symbolic, 3, 5

used in thermodynamics, I·61, l 62 numerical solution, 331, 392, 403

.numerical stabilily, 331

objective rate, "I 92, 193

objective spatial field, .1.82- I 94 objecr1vc stress rate, 193-1.96

objectivi~y

incremental., 292

.principle of material, .198

of a scalar .field, 185, 186

of a tensor field, 185., .186, 190-194 of a two-point t.cnsor field., 189-191 of a vector field, 182-.186, 188, 192

objectivity, requirement of, l 86, ·J 89-192, 214,

217t 2.91

observer, 180-18.9, .198, 200 change ·Of, 181-183

octahedral plane., 123

Ogden model

for compressible .materials.. 244-246 for incompressible malerials,, 235-242

thermodynamic .ex-tension of, 337-343..

347-351

Oldroyd stress rate., 193-196, 253, 3327 400

open system, 132, 136, 149

ope rator spIit adiabatic, 332 isentropic, 332 ·isothermal, 331

origin~28,57,58.344

orthogonal matrixt 29 orthogona·J tensor~ 16 orthogonality condition., 29 orthogonality of vectors, 3

orthotropic material, 275-.277

Index

parallelogram law,, 2 particle, 56

path line, 59

.penalty function, 390.. 391

penalty method, 389, 391, 404, 407

.for incompressibility, 389-391 penalty parameter, 390, 391, 407 perfectly elastic material, 208 permutation, even; odd, 6 permutation symbol, 6~ 7, 17, 21

permutation tensor, 21, 22, 31, 54, 147 perturbed functional, 406

perturbed Lagrange-multiplier method~ 406, 407 phenomenological approach, 205, 283, 295 phenomenological variable, 27.8, 300

Piola .dcfonnalion tensor, 79

Piola identi.ty, .146, 151 Piola-K.irchhoff stress tensor

first, '111-114, 127, 128, 199,, 207 Euclidean transformation of, 190, 191

second, 127, 199, 21 O; see.also second Pio- la-Kirchhoff stress tensor

Piola stress, 111

Piola transformation, 83, 84, 113, 127, 163, 190,

195,253

place, 58

plane motion, 69, l06, 137 plane strain, stale of, 84, 92

plane stress, state of., 126., 225, 227 ~ 237

point SOUJC·C, 51, 68

points, 58, 74 material, 56

neighboring, 76, 77

distance between, 76, 78, 8.2 relative position of, 180, 188

Poisson's cqualion, 50

Poisson's ratio, 247

polar continuum, 144, l47, 152 polar decomposition, 85-88., 90, 128

left, ·8'6 right, 86

polyconvexity, 207, 251 polymer chain; see chain polymer network; .'iee network position, current, 58

position, referential, 58

position vector, 58, 5.9, 141, ·1s1. I.82~ 266~ 344 potential, 'dissipative', 284, 362

.potential energy

external, 159, 387, 389, 404" · · "

of external loading., 159 "· · internal, 159, 387- .

total, 159. 160., 387-389, 392, 4.12· first vari.ati on of, 387

second variation of, 387.

stalionary position or, 187-389 . potential flow, 149

potential of a veclor field, 4.8

.potentials, thermodynamic, 32 l-325:t 338, 339.

359

power

mechanical, external9 153-156, 159, 160,

164, 165

stress; see stress power

therm.al, 162, 164, .165, 172, 175, 3 J9 power expended, .theorem of. .153

power law model, .367 pressure

hydrostatic; see hydrostatic pressure mean, 126

pressure boundary loading, 383, 384, 402 primary loading path, 29·6, 301

prindpaJ axes of a tensor, 24

principal directions, 24 referenlia·J, 89, 90

spatial, 90

of strain, 89, .90, 201, 219, 258, 259 of stress, 120-126., 201, 219, 220

prmc1pa. . · I .mvan,mts,· .· · · 25 , 'J 5·' "16, 'J?3-.-.. --,7?5....

principal planes, 120-122

principal stresses, 120-·:J 22, 21·9, 220, 225, 226, 237,246,258-260,341

principal stretches, 89-94., 21.9-222, 225-227, 236-242t257-260,316-319

modified, 228, 245, 246, 337-340 principa1 values or a tensor, 24, 25

principle of material frame-indifference, 198-

200, 267, 292

principle of material objectivity, '198

principle of stationary potential energy, 386-

392

principle of strain-equivalence, 300

principle of virtual displacement. 382; see a/so principle of virtual work

principle of virtual work.. 377-386, 388-392

principle or virtual work (comd.) linearization of, 392-402

in material description, 395-397 in spatial description., 397--40 I

in material description~ 384-386 in spatia·I description, 380-3'82

probability, 308, 313-315

probabilily density, 308, 3 t3, 315

process, 148

adiabatic, 172, 173,336,356

reversible, 172, 173, 348-351

dynamic~,148, 153,159, 160, 190~198

closed, 2 I ] , 212

.irreversible, 168, 170, 279, 281, 282 isentropic, 172, 173, 323

reversible, 172

isothermal, 172, 322, 323, 346, 356 quasi-equilibrium, 162

quasi-static, :l 62

reversible, 168,, 170, 172, J 73, 348-35 .I,

359,366

thermodynamic, 161., 164,. 166-168, 172,

319,330

production of entropy, local, 170; see also dissipation, intema:J

projection tensor

fourth-order, 24, 229-234, 255, 256, 285,

290

modified, 255

second-orde~., 1.8., 26, 117

projection of a vector, 3 pseudo-elasLidty, 30 I

pull-back operation., 82-84, 106, 107~ 127, 163,

375-377,385,395,397,399 pure rotation, 85, .86

pure shear, 92, 93, 227, 250. 304 with isotropic damaget 304

pure stretch, 85-87 pure tension, 124

push...forward -operation, 82-84, 106, l07. 127,

195,253,375-377,395,397~399

quasi-cquH ibrium process, 162

quasi-static problem, 154, 180 quasi ..static process, 162

rate

convected, 193, 196 co-rotational, 192, 196 objective, .l 92, 193

rate of

deformation tenso~, 97., 99, :101-107, 194,

202,2.18

Euclidean transformation of, I91 rotated, 10-.1, 158

physical interpretation of, I04 spectral decomposition of, l05

entropy i:npul, 167, 168, 172

external mechanical work, 153; see also external mechanical power

internal mcchani.cal work, 153; see also stress power

rotation tensor, 97

strain tensor, 97; see also .rate of deformation tensor

thennal work, l 62; see also thermal power transport, 139

reaction stresses, 202, 27·1, 276

reciprocal basis, 32, 35. 37 reciprocal basis vector, 32 recovery, 279

recurrence update formula, 293, 295, 393

reduction .factor, 298 reference body force, 144, 384 reference frame, 57

of observers, .18 l, 182, 1.84 spin of, 184

reference mass density, 133-135, 197 reference temperature, 333-339, 342, 344-348 reference time, 58, 1l0, 266, 344

referential stress-entropy tensor, 330, 354 refcrentiu-1 strcss-t·emperature tensor, 328-330.,

339-342,353,354

decoupled representation of, 340-342 referential thermal coefficients .of stress, 328 reflection, 16. 20, 28., 31

region., 52, 5.8-60

Reiner-Rivlin fluid, 202. 204 relaxation, 279, 280, 286, 36 l, 3-64 relaxation time, 280, 284, 287-.289, 358

re:Jaxation {retardation) process, 288, 358, 364,

365,368 relaxation tesl, 294 replacement operator, 5