Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

representation theorem for invariants.. 215 re.presentation theorem for isotropic tensor func-

tions, 201, 202, 217 residual strain, 298

residual stress. 208

response coefficients. 201-204. 217, 224, 234 response function, 197-200, 207, 210, 218, 223 retardation time, 280, 284, 287-289, 358 reversible process? l68,, "170, 172, 173, 348-

351. 359, 366

Reynolds' transport theorem, 138-140 rheological model, 286-288, 367-369

right Cauchy-Green tensor.; ue Cauchy-Ore-en tensor. right

right stretch tensor; see stretch tensor, right

rigid-body, 82, 99. 100

ri_gid-body motion, 82, 153

superimposed, 187-191, 198, 200, 209, 213,214

rigid-body rotation_, 86, 99., I00'.' 1.88, J95

.rigid-body translation, 62, 71. l 8"8

rigid transformation, time-independent, J84

Rivlin-Ericksen re.presentation theorem, 94, 20 J rotated rate of deformation tensor., I01, I58 rotation, 16, 20, 28-3-1, 87, 99, 100, 209, 267

pure., 85, 86

rigid-body, 86. 99, I00_, 188, 195

rotation -tensor, 86-88, 99-10 I , 1.28, 194, 209

Euc.lidcan transformation of, ·.J 89

rate or, 97

rmatio.na1 momentum, balance of, 14 I, 142

rubber, 309-311. 317-320, 346-349, 356, 357,

36"J

carbon-black lilied, 242, 243. 297, 298

ideal, 311, 333

natural. 311 real, 310

rubber balloon, 239-242 ~snap back' of, 242

·snap through' of, 242, 249

rubberband.309.310,326,348-351,355,356

cooling effect, 309, 351

heating -effect, 309, 351

rubber-like material, 235, 296. 311. 333, 337340, 357, 403

. compressible, 244-247 incompressible, 235-244

stress-strain-temperature response of,, 339,

343,362

Saint...Venant Kirchhoff model, 250, 251, 365 modified, 251

scalar, 1

scalar field, 45-48, 50, 66, 68, 69., 98, I06, 13·8-

·140, 149 objective, 185, 186

scaJar function, 40

scalar multip"lication, 2, 10 scaJar product, 2, 32

triple, .8

second law of thermodynamics, .l 66-l68, 170,

l72, 175,208

second Piola-Kirchho.ff stress tensor, ·127. 199,

210 effcctive,, 299

equilibrium part, 28.5, 362.. 363

Euclidean transformation of, 191 fictitious,230,231,234,289

isochoric contribution, 230, 234, 245, 246,

303,339

non-equi1ibrium part, 285, 362, 363 volumetric contribution., 230, 233, 245_, 303,

339

second-order tensor, 9-20, 36'.' 37

transpose, 13, 14 scJf-equilibratcd stress fl-cld, 145

separation of tie points, 307; see also -end-to-

end distance of tie points

shear

pure, 92. 93, 227, 250_, 304 simple, 93, 94, 227, 243, 366, 367 uniform, 93

shear direction, 93

shear modulus_, 227, 236, 247 lemperature dependent, 364

shear planes,, 93

shear rate, J05, 203, 366 shear stresses. I16, 117

maximium and minimum, 120-.122 Simo-Tay·lor-Piste.r variationaJ principle, 410, 411 simple material, 290

simple shear, 93, 94, 227, 243, 366, 367 simple tension, 226, 250, 294, 295, 343-352

singJe-field variational princi.p1e, 377-39 l, 3.95-

401

snap buckling, 242 softening parameler, 342

solenoidal vector field, 48, 50 solid, 205, 309-3 :l l

solid mechanics, 205 soluLion -technique

incremental/iterative, 252, 293, 392, 397, 405

staggered, 331, 332, 342 unconditionally stable, 33'1

source

enlropy. 167, 168 154, 155 heal, 162-1-64, 168

mass, 132 point, 51, 68

spatial mass density, 133..... J36, 141, 142, 148.

149,202

spatial strain tensor, 79-82

spatial slrcss-entropy tensor, 330, 332

spatial stress-temperature tensor, 328, 329, 332,

333

spatial time derivative, 66-68.. 98 spatial velocity gradient,, 95-97, 99

Eud idean transformation oC 190 specific heat capacity, 325-327

constant, 335, 347-349

at constant defonnation, 325-327, 329, 330, 334,360

at constant stress, 327

spectral decomposition of a tensor, 25, 26 spherical tensor, 19, 3 I

spin, 98

spjn angular mome.ntum, .143, 152

spi.n tensor,, 97-IOO, 154, 156, 192-195

Euclidean transformation oC 191 ·physical .imerprelation .of., 105

spring, 279, 280, 286-288, 309, 310, 351, 368 spring-and-dashpol model, 286, 367

square-root theorem, 86 staggered mclhod, 331

staggered solution technique, 33 ·1, 332, 342 state function, thermodynamic, 161

slate of plane strain, 84., 92

state of plane stress, 126, 225, 227, 237 stale of stress, 123-126; ~ftee al.w stress state

stale variables, thermodynamic, 161, 278, 305,

3.21

static condcnsation9 411

static problem9 380, 395

stationary potential energy, principle of, 386-

392

statistical concept, 306-309

statistical theory, 238, 239. 308

Gaussian, 312, 318., 319, 339

non-Gaussian, 244, 3 I9, 356

statistical thermodynamics, 305 steady flow9 "149

steady motion, 68

stiffness matrix, 253, 397, 406 geometric.al (initial slress), 397 ill-conditioned, 391, 402, 406 material, 397

Stokes' heat Hux theorem, 162-·l 64 Stokes·~ theorem, 53

stored energy, 207; .me t1l.w strain-energy func-

Lion

stored-energy .function, 207; see al.w strain-ener- gy function

strain

p.lain, 84

.principal directions of, 89, 90, 201, 2 I9, 258, .259

residual, 298

strain energy, 160, 207-2 ·1 I, 214-2.22, 233~238,

242-247,252-254,283,320-339;see ttl.m strain-energy function

total, 159

strain ..cncrgy factor, 290 strain-energy function, 160, 207

complemc~tary, 408

decoupled representation of, 229, 231, 233,

.244, 245, 337' 389 effective, 298-300

isochoric, 303

forms of, 209, 2 I0., 235-251 global .minimum of. 208

lime derivative of, 208, 210, 218, 229 strain-equivalence, principle of, 300

strain invariants, 215, 216, 223, 224, 233, 238,

243

strain ratc9 2:87, 358, 359, 362, 369 strain rate tensor., material, I0 I, ·107, 158 strain space plasticity, 301

strain tensor, 76-85 Biol, .88

Cauchy-Green; .\'ee Cauchy-Green t.cnsor

eigenvalues of, 89-92 eigenvectors of, 89-92

EuJer-Almansi;-see Euler-Almansi strain tensor

Green-Lagrange; see Green-Lagrange strain tensor

He.ncky, ·ss

material, 76-79 rate of. 97

spatial, 79-82 stress; se.e also stresses

algorithmic, 291. 293, 294

Cauchy, 111

P.io1a. l 11 residual, 208 state of, 123-126

stress-entropy tensor referential, 330, 354 spatial,. 330, 332

stress-free configuration, 208, 247, 362'.' 363, 379,390

stress function, 322, 323, 347 stress matrix, 112, 117, 120 stress ·PIancs, 121

stress power, 153-l 59, 164, :170, 173, 218, 230,

280,319

effective. 299 stress rate~ 107

Grccn-Naghdi, 194_, 195 Jaumann-Zaremba, 194, 195 objective, 19.3-196

Oldroyd, 193-196,253,332,400 Truesdell, 195

stress relation, 197-202; see also constilulivc equations

stress softening, 296-298, 303, 304

stress state, 123-126

.biaxial, 124, J26

cquibiax"ial~ .124

ho1nogcneous, 123 hydrostatic, 125, -126, 23"1 plane,126,225,227,237 pure nonnal, 123, 124

pure shear, 124 pure tangential, 124 tria-xial, 124 uniform shear, J24

stress-temperature lensor

referential, 328-330, 339-342., 353, 354 decoupled representation of, 340-342

spatial, 328, 329, 332,, 333 uniaxial tension, 124

stress tensor, 1ll-115, 11-9, 120, ·123-129 alternative, 127-129

Biol., 128, 158

Cauchy; .we Cauchy stress tensor -corotatcd, 128

eigenvalues -of, 119, 120 eigenvectors of, 11.9

first Piola-Kirchhoff; see first Piola-Kirch- hoff stress tensor

Kirchhoff; see Kirchhoff stress tensor Mandel, 128,, 158, 386

nominal, 111

second Piola-Kirchho.ff; see second PiolaKirchhoff s-tress tensor

true_, 1.J I

stress theorem, Cauchy's, I ll-114, 147, .-]48, ·150, 154, 175

stress vector, t 11 ·stresses; see also stress

compressive, 117

maximum and minimum, 126 non-equilibrium, 285, 287-290, 359, 363,

365,366,368 normal, 116, .117

maximium and minimum, 119, 120 reaction, 202, 271,, 276

shear; see shear stresses sign convention for, I 15 tangential, I "17

-tensile, 117 stretch, 78

principal; see principal ·stretches pure, 85-87

stretch ratio, 78; see also stretch stretch tensor, 85

Euclidean transformation or, J 89, 190 left (spatial).. 85-88, 218~ 2.19

·right (material), 85-88, 12-8, "I 98, 199. 209 ti me derj vative o.f, 99, 158

spectral decomposition of,, 90 stretch vcctur, 78.• 8l, 87

inverse, 81 structural tensors, 274

subscript comma, 45 summation convention, 4 surface

boundary; see boundary surface dosed, 52, 53

control, 132, 136, 149, 150 open.,53

surface density, 174-176

surface element, 52, 74, 110-117.. 119, 162, 383 material (undeformed).. 74

spatial (deformed), 74 first variation of.. 377 time derivative of, I04

surface traction, 109-111, 160 surroundings, 131, 161. 162, 175 ~ymbolic notation, 3, 5 symmetries

major, 253, 328, 401

minor, 253, 396, J98 .. 400, 401 symmetry of Cauchy stress lensor, 147 system,, 13·1, 132

boundary of, 131

closed, 131-133, 141., 319 condition of, l 61

conservative., 159, 160, 319, 386 isolated, 132, 319

macroscopic, 55, 56 microscopic, 55 open, 132, 136, 149 orthonormal, 3 right.-handec.I, 3 wall of, 13 I

system of forces, ·147-149, 153

tangent moduli; see also elasticity tensor algorithmic, 257

consistent .linearized, .257, 293

tangent stiffness matrix; see stiffness matrix tangent vector, 53, 70,, 7 l, 74

.materia·I, 70, Tl spatial, 70t 71

Taylor's ·expansion., 41, 42, 45, 77, 80, 244, 393 temperature, 168-173, 3.19-369

absolute, 1.68

.change in, 334 Celsius, 168, 319 Fahrenheit, ·I ·68

Kelvin, 168, 319

reference, 333-339. 342, 344-348 temperature function, 323

temperature gradient, l 70. 171, 324, 354, 359 tensile stresses, 117

tension

pure, 124

simple~ 226, 250, 294, 295., 343-352 uni.form (uniaxial), 124

tensor, 9 antisymmetric, 16

Cartesia.n, I 0

of chemical potential, 211, 212 contravariant, 36, 83, ·121., 193, 194, 253 ·covariant, 36, 83

deviato.ric, 19 eigenvalues .of, 24-26

=eigenvectors of, 24--26 fourth-order, 22-24 hjgher-order, 20-24, 37 inertia, l60

-inverse of, 15, I6 isotropic, 30-32 metric, 37, 39 mixed., 36

negative definite, 1I negative semi-definite, l I nonsingular, 15

norm of, 157 365 order (rank) of, 20 orthogonal, ".J 6

positive definite, 11, 25, 78, 81, 85, 87 positive semi·dcfinile, 11, 170, 17 l projection; see projection lcnsor second-order, 9-20, 36, 37

singular, l 5

skew, 16., .17., 98-1 00

spectral decomposition of, 25, 26 spherical, t9, 31

spin; see spin tensor symmetric, ·16,17 third-order, 20....;22 trace of, 14

two-point, 71, 82, 86, 90, l 11, I 89-191 tensor field, 45, 49...,.52

convected rate uf, 193, 196 co-rotational rate of, 192~ 196 frame-.indifferem, .185 objective, 185.. 1.86, l 90-194

.tensor function, 40-43 tensor product, 10-12

test function, 380-382

LhcrmaJ.conductivi.ty, coefficient of, 171, 342 therma.1 conductivity tensor

material, 171

spatial, 170

Lhcnnal energy, 161, 172 thennal ·energy, balance of, 164

thermal equation of slate, 322-324, 347 thermal equilibrium, 161; see .a/so thermody-

namic equilibrium

thermal expansion, 344, 347, 348" 351, 352 thermal field, 306

thermal power, 162, 164, 165, 172, 175, 3.19

Lherma:J work, rate of.. 1·62; see al.w thermal power

thermul.ly isolropic mater.iaJ, 171, 342, 345, 346,

348

thermodynamic continuum, 16.l

359,363,365.366 thermodynamic force, 208, 28 l ~ 299, 303

maximum, 300, 302

lhcrmodynamic Maxwell relation, 325.. 352, 353 thermodynamic potentials, 321-325, 338., 33'9,

359

general structure of, 334, 335 thermodynamic process, 161, 164., 166-.168,

172,319,330 irreversibility or, 167

the.nnodynamic reciprocal relation, 325

thermodynamics

classical, 305

·continuum, 131, 16 l, 305, 325

equilibrium, 168

irreversible, 168

of materials, 305-369 non-equilibrium, 168, 280, 282, 285 notati·on used .in, 161, I62

reversible, 168

statistical, 305

with int.emal variables., 357-369

.thermodynamic ·slate, 16 ·J, 278, 280-282, 321,

322.,357,362

thermodynamic state function, I61 thermodynamic state variables, .l 61, 278, 305~

32.l

thermoelastic heating (cooling), structural, 326,

327,336,348,349,361 decoupled representation of, 343

thermoelastic inversion point, 350-352

thennoclastic material, 323

thermoelasticity, finite" 306, 327, 33.2 of macroscopic networks, 3 'll-321 one-dimensional, 352-354

Lhcrmome~hankaJ coupling effects., 3277 360,

363 lhermomcchanical device, 367

lhermomcchani·cal problem, coupled, 305, 331,

332,342,344,348

thermostatics.. 161, 168

Lhermoviscoefastic heating (cooling), structural, 360,.361

thermoviscoelastic material, 358 thermoviscoelastic mode.I, 3.61-364 Lhermoviscoelasticity, finite,, 306., 357-360

third-order tensor, 20-22

thre.e.:fi.eld var.iational principle, 409-413 tie point. 307. 308, 312-314, 317

time

fina·1,319

initial, 58, 289,, 363, 379

instant of. 57-59,, ·1 ·so, 188, 372, 385

'modi tied', .289

reference, 58., 110, 266, 344

relaxation (retardation), 280, 284, 287-289,

358 time derivative

Lie, 106-108,193-196,376,400

local, ·66

material; see material time derivative

sputial, 66-68, 98

substantial, 64

total, 64

time increment, 291.

time integration algorithm, 290-293

time interval, 180, 18 .t

dosed, 211, 212, 283, 291, 300, 3.19, 320,

361.,392

scmi..open, 289, 363, 366

time-shift, 181, 187, .188

torque

body, 143

pure, I "10 resultant, 142

tOlal differential, 41. 42, 45, 9 l, 252, 328 total-Lagrangian formulation, 399

trace of a tensor, 14 traction vector, I09-11 "8

B-iot, I 2-8

Cauchy(true), .111, 113-"120., 142, 147, 148 coup"led, 143

first Piola-Kirchhoff (nominal), 111, J 13, 114, 144

prescribed, 379, 385, 412 trajectory, 59

transformation law

for basis vectorst 28, 29 tensorial, 29, 30 vectorial, 29

lrnnsfonnation .matrix, 29 translation, 28, 62, 209

rigid-body, 62, 71, 188

transport theorem, Reynolds', 138-·l 40 transversely isotropic material., 265-272 triadic .product of vectors, 21

trial solution, 291 triple scalar produc.t, 8

triple vector product, .8, 9, .19 Truesdell stress rate, t 95

true stress tensor, 111; .we a/.w Cauchy stress tensor

lrue traction vector, I l l; see also Cauchy traction vector

two~ficld variational principle, 402-409 two-point tensor, Tl, 82, 86, 90, 111, 189-·191

uni.form compression, 92, I24 uniform deformation, .92, 251 ll 352 unifonn extension, 92, 124 uniform motion, 68

uniform tension, 124 unit tensor

of fourth-order, 23, 31, 4.2

of second-order, l0, l .1, 30, 3.I, 37 unit vector, 3

update algorithm, 290, 291

.updated-Lagrangian formulation, 399

Va1anis-Landel hypothesis,, 237lt 338

Varga model, 23.8-242, 239, 356

variational equation, 382. 385, 38T, 389, 397,

405, 41] variational operator, 374

variational princi.p1e,, 37"1-413 HamHton 's, 391, 392

·Hcllingcr-Reissner, 408, 409

1-Iu-Washizu, 412

muJti-field,, 389, 403, 407, 4 =11

Simo-Taylor-Pistcr, 410, 41 J single-field, 377-391, 395-40"1 three-field, 409--413

two-field.. 402-409 variat-ional problem, 381

var.iation of a fuoctiont first, 374-377 veclort 1; .~ee al.w vectors

axial, 17ll 20, 48, 98, I00, I05 basis; see basis vectors contravariant, 35, 83, l63~ 186 covariant, 35, 83

dual, 17

length of, 2, 5, 35, 188 magnitude of, 2

norm of, 2 projection nf, 3

vector field, 45, 48-54, 61. 68. 83, 98., 106 conservative, 48

convected rate of, 193, 196 co-rotational rate of, 192 curl-free, 48 divergence-free, 48 frame-indifferent, I 82 irrotationa1, 48, 50 objective, 182-186, 188, 192 solenoidal., 48, 50

vector function, 40

vector operator, 45, 48-50 vector product, 5

.triple, .8, 9, 19 vectors, 1; see al.m vector

angle between, 2, 5, 16, 28, 33, 18"8,, 258 equal, 2

orthogonality of, 3 parallel, 6

triadic product·of, 21 velocity, angular, 59

velocity field. 62-65., 67, 68, 1.39, 14.1, 149~

378~379

under changes of observers, J 8.3-.185

Index

velocity gradient

material, 95, 96

spatial, 95-97, 99

Euclidean transformation of, 190 velocity potential., 69, 98

vibration, free, 154

virtua] displacement field,, 373-376, 382, 385,

394,,404

virtual pressure field,, 404 virtual work

external {mechanica] ), 382, 38.5,, 388, 391,

404

done by constant pressure, 383, 392 linearization of. 395., 402

internal (mechanical).. 382, 385, 386, 388,

391,395-397,405

linearization of, 396-40 I

principle of; see principle of vi.rtual work viscoelastic factor, 294

v.iscoelastic material, 283

viscoelastic model, 283-290 viscosity,, 203, 286, 365-367 viscosity index,, 367

viscous fluid, 202. 203.; see also fluid vo1ume, 56

control, 132,, 136., 149, 150

volume change, 227, 236, 310, 311,, 3·16, 317,

345-347 virtual, 41·0

volume-changing deformation, 228, 232, 4.10 volume density9 175

volume element, 52, 74, 313 incremental, 133

material (undeformed), 74 spatial (deformed),, 74

first variation of, 377 time derivative of, .104

volume-preserving dcfo.nnatlon, 228, 232 volume-preserving motion, 75

volume ratio, 74

Euclidean transformation of., 189 first variation oft 377

time dedvative of,, 103

volumetric .locking phenomena, 403, 406, 41.l von Neumann boundary condhion, 378., 384,

387,409 ·vorticity tensor, 97 vo.rticily vector, 98

455

·wo.rkconjugate, 159~ 170,·208,2.17, 252

work C·Onjugate pair, 159; see also work conju-

gate

work rate; see.external mechanical power; stress power; thermal power

Yeo.h model, 243

Young's moduli, 286, 368

-zero lcnsor, I0, 43

zero vector, 2