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Index acceleration 94 additive decomposition 296 ALMANSI´s strain tensor 101 anisotropic material 189 anti(sym)metric tensor 13 antisymmetriser 40 area placement tensor 106 ARIANO and RIVLIN´s theorem

223 ARRUDA-BOYCE model 227 axial vector of a tensor 29 back strain 272 back stress 272 balance of linear momentum 131 balance of moment of momentum

139 balance of power 142 balance of work 142 bijective 4 bilinear form 27 BIOT strain tensor 101 BIOT stress tensor 144 BLATZ-KO model 226 body forces 136 BOLTZMANN´s axiom 141, 142 boundary values 231 BRAUER and NOLL´s theorem 191 caloro-dynamical state 160 CAUCHY´s 1st law of motion 139 CAUCHY´s 2nd law of motion 141 CAUCHY´s lemma 139 CAUCHY´s stress tensor 139 CAUCHY´s theorem 139, 156 CAUCHY-GREEN tensor 98 CAUCHY-STOKES decomposition

107 CAYLEY-HAMILTON theorem 24,

52 center of mass 125 centripetal acceleration 134 characteristic polynomial 21 CHRISTOFFEL symbol 77 CIARLET model 226

CLAUSIUS-DUHEM inequality 152, 173, 209, 289, 292

CLAUSIUS-PLANCK inequality 152

coaxial tensors 26 COLEMAN and NOLL´s theorem

210 conjugate 145 conservation of mass 121 consistency condition 277, 287 constraint 167, 172 contact forces 136 contravariant component 75 convected stress tensor 144 convective coordinates 98 coordinate system 72 COOS (coordinate system) 5, 72 CORIOLIS acceleration 134 COSSERAT medium 140 COTTER-RIVLIN rate 148, 276 covariant component 76 covariant derivative 79 critical resolved shear stress 308 cross product 7 curl 57, 82 cylindrical COOS 84 deformation gradient 96 deformation process 249 derivative 54 derivative of the principal invariants

60 determinant 14 deviator 14 deviatoriser 40 diagonalisable tensor 23 directional derivative 53 displacement 93 displacement gradient 100 dissipation 152 dissipation inequality 152 divergence 57, 87 divergence of a tensor field 82 DOYLE-ERICKSEN formula 215

INDEX

dual bases 8 dyad 17, 36 dyadic product 17 eigenbasis 22 eigenprojector 25 eigenprojector representation 25 eigenspace 22 eigenvalue 21 eigenvector 21 elastic material 178 elastic range 255 elastic reference law 268 energy balance 150, 151 entropy 151 equivalent stress 257 ERICKSEN problem 244 EUCLIDean metric 71 EUCLIDean shifter 71 EUCLIDean space 71 EUCLIDean transformation 132 EULER´s equations of motion 131 EULER´s velocity formula 109 EULERean description 94 EULER-RODRIGUES

representation 32 evolution function 252 extra stress 168 face-centred cubic crystal 307 field 75 FINGER tensor 99 FINGER´s theorem 220 flow rule 273 fluid 188 FOURIER´s theorem 149 frame of reference 131 FRECHET derivative 54 free energy 151 GALILEIan transformations 136 GATEAUX differential 53 GAUSS transformation 87 GAUSS´ theorem 87 general linear group 12, 16 general orthogonal group 35 general unimodular group 16 generalised strain tensor 103 generalised stress tensor 145 global balance equation 126

GOLDENBLAT´s theorem 222 gradient 54 gradient basis 74 GREEN´s strain tensor 101 group 12 HAIGH-WESTERGAARD plane

263 hardening 271 heat conduction inequality 210 heat flux 149 heat source 149 heat supply 149 HELMHOLTZ free energy 151 HENCKY´s strain tensor 101, 147 HILL model 226 HILL strain 103 HILL´s yield criterion 258 homogeneous body 185 homogenisation 232 HOOKE´s law 203 HUBER-v. MISES yield criterion

258, 262 hyperelastic 211 hyperelastic ranges 255 identity tensor 11 incompressibility 169 indefinite tensor 27 inextensibility 170 initial values 231 inner product of tensors 15, 20, 43 inner product of vectors 6 interface 127 intermediate placement 293 internal constraint 167, 172 internal energy 149 internal variable 251 intrinsic 165 invariant under change of observer

135 inverse tensor 12 isochoric motion 108 isoclinic placement 295 isomorphism 183, 217 isomorphy 183, 268, 269, 285, 304 isotropic hardening 272, 274 isotropic material 189 isotropic tensor function 46, 48

336

J2 -theory 262 JACOBI matrix 73 jump balance 130 kinematic hardening 272, 276 kinetic energy 142 KIRCHHOFF stress tensor 143 KRONECKER symbol 8 KUHN-TUCKER condition 278 LAGRANGEan description 94 LAMÉ constants 204 latent hardening 310 lattice 303 lattice basis 304 left CAUCHY-GREEN tensor 99 left stretch tensor 98 left subsymmetry 41 LEIBNIZ rule 55 LIE-derivative 147 linear 6 linear momentum 125 linear space 5 linear strain tensor 110 liquid crystal 195 loading condition 266, 286 local action 154 local balance equation 127 logarithmic rate 148 logarithmic strain tensor 101 MANDEL´s stress tensor 144 MANDEL´s stress tensor 295 mass 121 mass density 121 material COOS 94 material description 94 material functional 154 material stress tensor 144 matrix of tensor components 18 metric coefficient 77 moment of momentum 125 MOONEY-RIVLIN model 227 motion 92 multiplicative decomposition 293 MURNAGHAN-model 227 nabla operator 83 NANSON´s formula 106, 126 natural bases 75

NAVIER-STOKES fluid 159, 170 neo-HOOKE model 227 NEUMANN´s potential 212 neutral loading 267 NOLL´s rule 187 NOLL´s theorem on hyperelastic

materials 212 non-polar medium 141 norm of a tensor 15 objective 135 objectivity 155 observer 131 OGDEN model 227 OLDROYD rate 147, 201, 276 ONB (orthonormal basis) 10 orientation preserving tensor 16 orthogonal group 35 orthogonal tensor 31 orthonormal basis 10 overstress 291 partial derivative 56 perfect heat conduction 173 PETROSKI and CARLSSON´s

theorem 244 PFI 158 physical component 75 PIOLA identity 126 PIOLA-KIRCHHOFF stress tensor

143 PISM 157, 163 placement 91 plastic consistency-parameter 274 plastic potential 277 plastic spin 298 plastic stress tensor 271 plastic transformation 269 plasticity 254 PMO 155 polar decomposition 98, 119 polar decomposition theorem 33 polar media 140 position vector 72 positive semi-definite tensor 27 positive-definite tensor 27 power of stresses 142 power of the external forces 142

INDEX 337

INDEX

principal invariants 14 principal stresses 141 principle of determinisms 153, 168 principle of determinisms for

thermo-mechanical materials 160 principle of equivalence of work 234 principle of form invariance 158 principle of invariance under

superimposed rigid body motions 157

principle of local action 154 principle of local action for simple

thermo-mechanic materials 160 principle of local action for thermo-

mechanical materials 160 principle of material objectivity 155 principle of virtual power 142 product rule 55 projector 25 proper orthogonal tensor 32 quadratic form 27 rate of deformation tensor 107 rate-independent 252, 273, 286 RAYLEIGH product 44 reaction principle 139 reaction stress 168 reduced elastic forms 178 reduced form 164, 250 reduced thermo-elastic forms 179 reference placement 92, 180 REINER fluid 159 relative stress tensor 144 representative volume element 232 residual inequality 132, 139, 150 resolved shear stress 307 REYNOLDS´ transport equation

123, 124 RICHTER representation 199, 220 RIEMANN´s curvature tensor 78 right CAUCHY-GREEN tensor 98 right stretch tensor 98 right subsymmetry 40 rigidity 171 RIVLIN-SAUNDERS model 227 rotation tensor 98 scalar product of tensors 15, 20, 43 scalar product of vectors 6

SCHMID tensor 308 SCHMID´s law 308 self-hardening 310 SETH family of strain tensors 104 shear locking 171 similarity transformation 22, 45 simple material 154 simple shear 113, 119 skew tensor 13, 29 slip rule 309 slip system 305 slip system theory 303 small deformations 110 solid 189 spatial COOS 94 spatial description 94 special linear group 16 special orthogonal group 35 special unimodular group 16 spectral form 23 spherical COOS 86 spherical tensor 11 spin tensor 107 St. VENANT-KIRCHHOFF law 202 stability 231 state 251 stiffness tetrad 200 strain energy 211 stress power 142 stress power in plasticity 271 stress principle of EULER and

CAUCHY 137 stress rate 147 stress vector 137 stretch tensor 98 stretching tensor 107 subsymmetry of a tetrad 40 superimposed rigid body motion 109 SYLVESTER´s formula 26 symmetric tensor 13 symmetriser 39 symmetry group 186, 251 symmetry transformation 46, 217,

251 symmetry-transformation 185 tangent basis 74 tangential stiffness tetrad 278

338

TAYLOR problem 312 temperature 151 tensor 10 tensor basis 18 tensor function 26 tensor product 17 tensor surface 28 tetrad 38 texture 302 thermo-elastic material 179 thermo-elastic range 284 thermo-kinematical process 160 time derivative 94 trace 14 transport equation 123, 124 transposed tensor 12 transposed tetrad 40 transposer 39 transversely isotropic 195 TRELOAR´s model 227 TRESCA´s yield criterion 262 triad 37 triclinic 188 triple product 7

TRUESDELL rate 148, 201, 276 undistorted reference placement 189,

218 undistorted state 189 uniform body 185 unimodular group 16 unimodular tensor 16 universal solution 236 VALANIS-LANDEL model 227 vector product 7 vector space 5 velocity 94 velocity gradient 106 versor 32 virtual power 142 viscoplasticity 291 viscous fluid 158 VOIGT representation 42 work conjugate 145 yield criterion 256, 285 yield surface 255, 285 ZAREMBA-JAUMANN rate 148,

276 zero tensor 11

INDEX 339