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INDEX

341

Index acceleration 94 additive decomposition 299 ALMANSI´s strain tensor 101 anisotropic material 189 anti(sym)metric tensor 13 antisymmetriser 40 area placement tensor XII, 106 ARIANO and RIVLIN´s theorem

223 ARRUDA-BOYCE model 229 ATTARD model 227 axial vector of a tensor 29 back strain 275 back stress 275 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 233 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, 292, 295 CLAUSIUS-PLANCK inequality

152 coaxial tensors 26 COLEMAN and NOLL´s theorem

210 conjugate 145 conservation of mass 121 consistency condition 280, 290 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, 279 COUETTE flow 117 covariant component 76 covariant derivative 79 critical resolved shear stress 312 cross product 7 curl 57, 82 cylindrical COOS 84 deformation gradient 96 deformation process 251 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

342

divergence of a tensor field 82 DOYLE-ERICKSEN formula 215 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 257 elastic reference law 271 energy balance 150, 151 entropy 151 equivalent stress 259 ERICKSEN problem 246 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 254 extra stress 168 face-centred cubic crystal 310 field 75 FINGER tensor 99 FINGER´s theorem 220 flow rule 276 fluid 188 FOURIER´s theorem 149 frame of reference 131 FRECHET derivative 54 free energy 151 FUNG model 229 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 GENT model 229 global balance equation 126 GOLDENBLAT´s theorem 222 gradient 54 gradient basis 74 GREEN´s strain tensor 101 group 12 HAIGH-WESTERGAARD plane

266 hardening 275 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 260 homogeneous body 185 homogenisation 234 HOOKE´s law 203 HUBER-v. MISES yield criterion

261, 265 hyperelastic 211 hyperelastic ranges 257 identity tensor 11 incompressibility 169 indefinite tensor 27 inextensibility 170 initial values 233 inner product of tensors 15, 20, 43 inner product of vectors 6 interface 127 intermediate placement 296 internal constraint 167, 172 internal energy 149 internal variable 253 intrinsic 165 invariant under change of observer

135 inverse tensor 12 isochoric motion 108 isoclinic placement 298 isomorphism 183, 217

INDEX

INDEX

343

isomorphy 183, 271, 272, 288, 307 isotropic hardening 275, 277 isotropic material 189 isotropic tensor-function 46, 48 J2 -theory 265 JACOBI matrix 74 jump balance 130 kinematic hardening 275, 279 kinetic energy 142 KIRCHHOFF stress tensor 143 KNOWLES model 228 KRONECKER symbol 8 KUHN-TUCKER condition 281 LAGRANGEan description 94 LAMÉ constants 204 latent hardening 313 lattice 306 lattice basis 307 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 269, 288 local action 154 local balance equation 127 logarithmic rate 148 logarithmic strain tensor 101 MANDEL´s stress tensor 144 MANDEL´s stress tensor 298 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 296 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 270 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 228 OLDROYD rate 147, 201, 279 ONB (orthonormal basis) 10 orientation preserving tensor 16 orthogonal group 35 orthogonal tensor 31 orthonormal basis 10 overstress 294 partial derivative 56 perfect heat conduction 173 PETROSKI and CARLSSON´s

theorem 246 PFI 158 physical component 75 PIOLA identity 126 PIOLA-KIRCHHOFF stress tensor

143 PISM 157, 163 placement 91 plastic consistency-parameter 277 plastic potential 280 plastic spin 302 plastic stress tensor 274 plastic transformation 272 plasticity 256 PMO 155 polar decomposition 98, 119 polar decomposition theorem 33 polar media 140

344

position vector 72 positive semi-definite tensor 27 positive-definite tensor 27 power of stresses 142 power of the external forces 142 principal invariants 14 principal stresses 141 principle of determinisms 153, 168 principle of determinisms for

thermo-mechanical materials 160 principle of equivalence of work 236 principle of form invariance 158 principle of invariance under

superimposed rigid body motions 157

principle of local action 154 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 RAGHAVAN-VORP model 228 rate of deformation tensor 107 rate-independent 254, 276, 289 RAYLEIGH product 44, 45 reaction principle 139 reaction stress 168 reduced elastic forms 178 reduced form 164, 252 reduced thermo-elastic forms 179 reference placement 92, 180 REINER fluid 159 relative stress tensor 144 representative volume element 234 residual inequality 132, 139, 150 resolved shear stress 311 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 41 rigidity 171

RIVLIN-SAUNDERS model 228 rotation tensor 98 scalar product of tensors 15, 20, 43 scalar product of vectors 6 SCHMID tensor 311 SCHMID´s law 312 self-hardening 313 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 312 slip system 308 slip system theory 306 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 233 state 253 stiffness tetrad 200 strain energy 211 stress power 142 stress power in plasticity 274 stress principle of EULER and

CAUCHY 137 stress rate 147 stress vector 137 stretch tensor 98 stretching tensor 107 subsymmetry of a tetrad 41 superimposed rigid body motion 109 SYLVESTER´s formula 26 symmetric tensor 13 symmetriser 40 symmetry group 186, 253

INDEX

INDEX

345

symmetry transformation 46, 217, 253

symmetry-transformation 185 tangent basis 74 tangential stiffness tetrad 281 TAYLOR problem 315 temperature 151 tensor 11 tensor basis 18 tensor product 17 tensor surface 28 tensor-function 26 tetrad 38 texture 305 thermo-elastic material 179 thermo-elastic range 287 thermo-kinematical process 160 time derivative 94 trace 14 traction 137 transport equation 123, 124 transposed tensor 12 transposed tetrad 40 transposer 40 transversely isotropic 195 TRESCA´s yield criterion 265 triad 37

triclinic 188 triple product 7 TRUESDELL rate 148, 201, 279 undistorted reference placement 189,

218 undistorted state 189 uniform body 185 unimodular group 16 unimodular tensor 16 universal solution 238 VALANIS-LANDEL model 228 vector product 7 vector space 5 velocity 94 velocity gradient 107 versor 32 virtual power 142 viscoplasticity 294 viscous fluid 158 VOIGT representation 42 work conjugate 145 YEOH model 228 yield criterion 258, 288 yield surface 257, 288 ZAREMBA-JAUMANN rate 148,

279 zero tensor 11