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Methods Geomechanics, Vol. 2, 8793 Index A adiabatic 43–46, 49, 66, 67, 79, 256 advanced plasticity models 118 anisotropic elasticity 172 anisotropy 120, 163, 164, 169, 172, 210, 340 associated flow rule 18, 21, 28, 29, 32, 58, 69, 74, 83, 86, 96, 100, 108–114, 118, 130, 155, 176–179, 186, 209, 210, 261, 342 B back stress 27, 68, 89, 97, 108, 123, 130, 149, 153, 179, 180, 184, 207, 209 back stress function 144 backbone curve 192, 232, 233 bar structure 279 bending moment 299, 300 bending stiffness 298 body force 9, 245, 246 bounding surface plasicity 105–110, 118 bulk modulus 14, 44, 79, 162, 190 C canonical yield function 17, 58, 268– 271, 275, 330, 334, 335 Cauchy small strain tensor 7 Cauchy stress 8, 11, 286 classical thermodynamics 35, 36, 47, 59, 256, 320 classical thermodynamics of fluids 40, 43, 48 Clausius-Duhem inequality 38, 161 cohesionless soils 209 cohesive material 270, 340 compatibility 8–11, 278, 280, 282 complementary energy 48, 77, 164, 168, 170, 172, 318 compliance matrix 22, 23, 49, 66, 171 conjugate variables 57 conservation of energy 37 consistency condition 20–25, 63, 91, 97, 109, 111, 115, 124, 145, 179, 197, 239 constitutive behaviour 10, 21, 42, 59, 69, 74, 75, 88, 91–93, 96–98, 102, 124, 125, 148, 149, 155, 180, 190, 195, 197, 254, 258, 262, 266, 302, 313 constitutive models 1–4, 8, 11, 19, 62, 65, 74, 79, 115, 117, 156, 182, 210, 220, 255, 256, 259, 262, 273, 304, 340– 343 constraints 1, 71, 73, 84, 205, 264–266, 270, 278–282, 302 continuous field of yield surfaces 119, 151, 155, 340 continuous hyperplastic model 142, 177, 191 continuous hyperplasticity 133, 146, 155, 203, 210, 224, 300, 341 continuous material memory 119 continuum mechanics 6, 8–11, 48, 243, 253, 260, 318, 342 contraction 18, 87 convective derivative 9, 242 convex analysis 4, 17, 217, 263–266, 271, 275, 304, 306, 321, 325, 330, 331, 334 convex function 265, 328–331, 334 convex sets 327, 334 346 Index convexity 58, 321 coupled materials 32, 176 creep 211, 237, 238 creep rupture 238 critical state 28, 87, 186, 187, 191, 195, 203, 204, 209, 210, 340 cross-coupling 112, 180, 184 D damage mechanics 274 damage parameter 274, 277 Darcy’s law for fluid flow 254 decoupled materials 103 deformation gradient tensor 6 degenerate transform 57, 263 density 9, 47, 172, 242–244, 254, 286 deviatoric stress 78, 84, 94, 99, 129, 154, 159, 182, 200, 201 differential 10, 11, 20, 47, 57, 59, 68, 74, 88, 120, 138, 150, 181, 193, 216, 217, 225–227, 232, 233, 247, 250, 255, 264, 295, 296, 299, 300, 306–310, 313, 322– 324, 328 dilation 29, 73, 86, 205, 259, 261, 270, 271, 340 Dirac impulse function 306 displacement 6–11, 31, 114, 179, 257, 290–297 displacement gradient tensor 6 dissipation 3, 5, 30, 37, 40, 41, 50, 53– 62, 66–75, 82–103, 121–131, 136, 139, 143–145, 149, 153, 161, 176, 178, 183, 188–196, 204–217, 221, 223, 229, 241, 249, 251, 256, 257, 262, 268–271, 275, 278, 284, 285, 289, 293, 294, 297, 298, 301–304, 332–335, 339 dissipation functional 136, 138, 143, 144, 152, 178, 230 dissipative coupling 176 dissipative generalised stress 3, 54, 56, 75, 91, 94, 96, 99, 125, 129, 152, 178, 213, 261 dissipative generalised stress function 136, 143, 148, 191 dissipative materials 48, 54, 274, 304, 342 Drucker’s stability postulate 4 Drucker-Prager model 87, 261 dry density 243, 250 dummy subscripts 21 E effective angle of friction 259 effective stresses 159–162, 187 elastic material 13, 15, 133, 288, 339 elastic strains 16, 19, 20, 176 elasticity 11–16, 20, 77–80, 111, 162– 164, 209, 265, 277, 286, 288, 339, 340 elastic-viscoplastic model 216 elliptical yield surfaces 178, 186 end bearing 293–296 endochronic theory 117, 118, 343 energy functional 134, 193, 230 energy function 42, 45, 47, 49, 68, 80, 120, 137, 190, 215, 256, 298 entropy 36, 38, 40–46, 55, 65–70, 74, 75, 80, 248–251, 302, 320 entropy flux 54, 249, 301 equation of state 36, 45, 46 equations of motion 248, 254 equilibrium 8–11, 32, 36, 41, 51, 248, 278, 280, 295, 296 Euclidian distance 326 Euler’s theorem 56, 136, 212, 323 Euler-Almansi tensor 7 Eulerian formulation 7, 242, 250 evolution equations 19, 50, 53, 65, 142, 230 extensive quantities 40, 47, 242, 251 extremum principles 2, 303, 321 F Fenchel dual 265, 266, 275, 304, 321, 331–335 fibre-reinforced material 288 finite element 2, 62, 112, 138, 172, 230, 343 First Law 36, 38, 246 flexible pile 294, 298 flow potential 213, 215, 218, 219, 222, 224, 230, 233, 270, 277, 303 flow potential functional 230, 234 flow rule 17, 22, 24, 29, 57, 63, 83–86, 89, 94, 99, 101, 103, 111, 122, 143– 146, 179, 210, 261, 335 fluid 40, 242–248, 253–262, 315 fluxes 241–243, 250–254, 262 force potential 213, 215, 218, 219, 223, 225, 251–256, 259, 261, 268, 275, 277, 303 force potential functional 229, 230 Index 347 Fourier heat conduction law 261 Frechet derivative/differential 144, 193, 195, 295, 298, 302, 308–310 free energy functional 134, 143, 144, 196, 234, 296 friction 28, 29, 32, 37, 74, 84–86, 159, 205, 209, 210, 261, 271, 340 frictional material 28–32, 69, 90, 103, 204, 210 G gas constant 45 Gateaux derivative/differential 307, 308 gauge function 264, 268, 269, 304, 330– 335 Gauss’s divergence theorem 243 generalised fluxes 242 generalised forces 242, 296 generalised failure criterion 207 generalised signum function 82 generalised stress 53–58, 65, 68, 70, 73, 75, 82, 85–93, 96, 97, 123, 124, 130, 135, 138, 144, 150, 192, 195, 205, 207, 213, 229, 261, 268, 270, 278, 289, 303, 332 generalised stress function 135, 137, 144, 145, 194, 229 generalised tensorial signum function 83, 306 generalised thermodynamics 1, 3, 54, 133, 155, 341 geotechnical materials 2, 28, 74, 142, 160, 205, 210, 222, 271, 339 gravitational acceleration vector 9 Green-Lagrange strain tensor 7 H hardening laws 28 hardening modulus 22, 24, 109, 110, 148 hardening parameters 18, 19 hardening plasticity 19, 22, 24, 342 heat capacity 259 heat engine 39, 40 heat flow/flux 36, 39, 41, 44, 50, 66, 74, 161, 245, 249, 262, 301 heat supply 37, 40, 41, 54 Heaviside step function 306, 336 Hessian 70, 71, 316 hierarchy of models 15, 80, 102, 220 homogeneous first-order function 56, 58, 70, 73, 75, 121, 188, 229, 269, 303, 333 homogeneous function 88, 212, 214, 269, 318, 331 Hooke’s law 95, 100, 130, 155 hyperbolic stress-strain law 191, 192 hyperelastic material 14, 15, 48 hyperelasticity 15, 20, 253, 273 hypoelastic material 13, 15 hypoelasticity 15, 20 hypoplasticity 117 hysteretic behaviour 28, 107, 110, 111, 162, 200, 233 I Il'iushin's postulate of plasticity 32 image point 106–110 incompressibility condition 72, 94, 98– 102, 128, 130, 152, 155 incompressibility constraint 79, 287 incompressible elasticity 78, 81 incremental response 48, 62, 68, 69, 74, 75, 90, 92, 96, 123, 124, 138, 145, 150, 215, 230, 234, 239, 303, 308 incremental strain vector 107, 108 incremental stress vector 106, 107, 116 incremental stress-strain relationship 2, 19–21, 64, 112, 142, 239 indicator function 264–266, 270, 271, 329–335 inertial effects 261, 262 initial and boundary conditions 8, 255 initial stiffness 147, 162, 192, 234, 276 intensive quantities 40, 253 internal coordinate 134, 137, 228, 285, 290 internal energy 36–46, 49, 54, 55, 66, 78, 246–255, 279–282, 303, 320 internal function 103, 121, 134–137, 155, 179, 198, 228–234, 342 internal variables 1, 10, 33, 49, 53, 54, 71, 74, 84, 103, 120–125, 131–135, 142, 155, 173, 198, 224, 225, 228–230, 241, 242, 251, 264, 278, 280, 289, 301, 330 intrinsic time 117 invariants of the tensor 311 irrecoverable behaviour 15 irreversible behaviour 50, 51, 117, 274 348 Index isentropic 43–46, 67 isothermal 14, 43, 46–49, 66, 74, 79, 102, 258, 259, 287 isotropic elasticity 78, 83 isotropic hardening 25–28, 92–95, 101, 103, 210, 341 isotropic thermoelasticity 49, 79 Iwan model 125–127, 149, 150 K kinematic hardening 27, 28, 97–103, 112–115, 119, 121, 123, 130, 142, 147, 151, 155, 156, 185, 186, 196, 207, 209, 228, 231, 233, 342 kinematic internal variable 53, 103, 120, 175, 225, 257 kinetic energy 245, 246, 255 L Lagrangian formulation 7, 242, 250 Lagrangian multiplier 72, 87, 206, 261, 278, 280, 287 large displacement theory 9 large strain analysis 5, 242 Laws of Thermodynamics 15, 162, 210 Legendre transform 4, 42, 43, 46–49, 56, 57, 68, 69, 72, 73, 82, 88, 89, 122, 123, 137, 143, 144, 167, 205, 212, 213, 255, 263, 273, 309, 315–324, 331, 333 Legendre-Fenchel transformation 82, 230, 256, 261, 321, 331 limiting strain 182 linear elastic region 100, 119, 179, 181 linear elasticity 13, 14, 78, 265, 318 linear hardening 27, 96–98, 127, 128, 341 link to conventional plasticity 102, 121 loading history 110, 159, 172 loading surface 106, 107, 118 logarithmic stress-strain curve 180, 191 M Macaulay brackets 92, 116, 179, 217 mapping rule 106, 118 Masing rules 28, 147, 151, 185 mass balance equations 244, 246, 254 mass flux 243 material derivative 242, 246, 250 Maxwell’s relations 43 mean stress 84, 159, 162, 169, 172, 202 mechanical dissipation 50, 55, 75, 86, 136, 161, 262, 302 mechanical power 36, 37 memory of stress reversals 120 micromechanical energy 209 Minkowski function 330 mixed invariants 313 Modified Cam-Clay model 162 modulus coupling 176 modulus of subgrade reaction 297 multiple internal variables 53, 120, 131, 135, 224–228, 231 multiple stress reversals 177 multiple surface models 111, 118, 125, 142 multisurface hyperplasticity 119 N nested surface models 111, 118 non-associated plastic flow 2, 32, 204 non-dilative plasticity 271 non-dissipative materials 48 non-intersection condition 112–117 non-linear elasticity 1, 165 non-linear viscous behaviour 219 non-uniqueness 190 normal cone 329, 330, 333–335 normality 18, 31, 103, 123, 144 notation 5 O one-dimensional elastoplasticity 81 Onsager reciprocity relationships 254 orthogonality condition 53, 56, 63, 226, 232, 296 overconsolidated clays 177, 187, 342, 343 overconsolidation ratio 172, 175, 200, 341 P partial derivative/differential 307 partial Legendre transformations 319 passive variables 59, 122, 319 perfect gas 35, 36, 41, 44–46 perfect plasticity 18–23, 32, 81, 103, 233 permeability coefficient 259 pile capacity 290 Index 349 pin-jointed structures 277 Piola-Kirchhoff stress tensor 250 plastic moduli 148 plastic modulus function 151, 178, 192 plastic multiplier 18, 20, 23, 65, 108, 111, 207 plastic potential 2, 17–22, 29, 32, 33, 58, 86, 111, 122, 210, 261 plastic strain 16–29, 32, 33, 49, 57, 67, 73, 82–86, 89, 90, 93–99, 105–114, 117, 121–123, 126, 129, 131, 138, 144, 149, 150, 154, 156, 172–177, 180, 188, 189, 192, 207, 234, 261, 271, 274, 275, 291, 335 plastic strain increments 16, 18, 69, 173 plastic strain rate tensor 88, 121 plastic work 19, 24, 29, 30, 90, 209, 210, 335 plasticity theory 1–5, 16, 18, 28, 33, 35, 57, 58, 62, 89, 107, 117, 143, 177, 263, 264, 300, 304, 321, 323, 334, 335 Poisson's ratio 162 polar function 269, 270, 304, 333–335 pore fluid 160, 161, 243–249, 253, 254, 257 pore water pressure 160 porosity 243, 256 porous continua 241, 339 porous medium 241–243, 248, 253 potential functionals 142, 148, 151, 177, 210 potential functions 2, 74, 88, 89, 93, 98, 102, 121, 125, 128, 156, 209, 238, 241, 254, 258, 262, 303, 341 potentials 2, 59, 74, 122, 173, 176, 213, 217, 232, 262, 264, 302, 303, 313, 315, 340 power input 37, 160, 161 Prager’s translation rule 28, 114 preconsolidation pressure 162, 172– 175, 196, 198, 235 pressure 36, 40–47, 74, 160–163, 166, 172, 175, 196, 202, 204, 210, 245, 247, 250, 253, 258, 259, 320, 341 principal stretches 287 prismatic beams 284 property 36, 38, 42, 46, 54, 57, 74, 88, 229, 253, 301, 331 proportional loading 131, 155, 180–183 Q quadratic functions 78, 318 quasi-homogeneous dissipation function 254 R rate effects 211, 239 rate process theory 221, 223, 233, 236 rate-dependent materials 212, 215, 221, 228, 230, 239, 273 rate-dependent models 224 rate-independent materials 1, 3, 51, 117, 136, 230, 303 rates of the plastic strains 87 rational mechanics 49, 133, 155 rational thermodynamics 2, 3 redundant structure 281 reservoir 38–40 reversibility 40, 117, 191 reversible materials 40 reversible processes 41, 49, 341 rigid pile 290, 296, 297 rigid-plastic materials 84 rubber elasticity 286, 287 S saturated granular materials 160 secant shear stiffness 177, 191 Second Law 3, 38, 54, 161, 248 shear modulus 14, 79, 162, 188, 290, 341 sign convention 8 simple shear 16, 26, 94, 99, 102 singular transformation 58, 71, 73, 138, 230, 320, 321, 324 skin friction 293 sliding element 97, 98, 126–128, 134 slip stress 97, 98, 126, 134, 149 small deformations 6–8 small displacement 7, 8 small strain analysis 5–8, 47 small strain region 179 small strain stiffness 162 small strains 6–8, 50, 162, 183, 203, 257, 286 soil skeleton 161, 242–250, 253–261 soils 2, 28, 32, 33, 74, 107, 112, 118, 119, 159, 162, 163, 172, 174, 183, 186, 191, 195, 198, 204, 221, 339–343 [...]... 331, 333 thermodynamics 1, 2, 4, 5, 15, 18, 31, 35, 36, 40, 42, 47–51, 54, 66, 133, 137, 162, 242, 256, 304, 341 thermodynamics of fluids 40, 47 thermodynamics with internal variables 1, 49 thermoelasticity 48, 79, 80, 261, 340 thermomechanics of continua 47 Third Law 36 tortuosity 245, 255, 256 total differential 307 tractions 10, 245, 247 triaxial test 159 , 160, 183, 339 true stress space 68, 69, 86,... 221, 253, 301, 320 temperature gradient 41, 55 Terzaghi’s principle of effective stress 253 thermal conductivity coefficient 259 thermal dissipation 41, 50, 54, 75, 301 thermal expansion 44, 46, 49, 79, 80 thermal expansion coefficient 259 thermally activated processes 221, 233 thermodynamic closed system 35 thermodynamic efficiency 39 thermodynamic equilibrium 36, 51 thermodynamic process 6, 7, 20,... 123, 130, 144, 179 U unchanged system 37–39 uncoupled materials 32 undamaged Helmholtz free energy 277 unified soil models 191 uniqueness 58, 190, 303, 321 unsaturated granular material 161, 340 Index V velocity 9, 160, 243–245, 255, 258 virgin consolidation line 198 visco-hyperplastic model 233, 234 viscous materials 212 voids ratio 162 volumetric behaviour 29, 78, 100, 129, 153 , 165–172, 182, 209,... 286, 320 S-shaped curve 177 standard material 5, 303 state variables 35–37, 42, 49, 251 stiffness 44, 46, 67, 109–112, 147, 151 , 156 , 162, 163, 166, 168–174, 177, 178, 182, 183, 192, 193, 196, 198, 201, 202, 210, 234, 266, 274–277, 280, 287, 291, 340, 342 stiffness matrix 20–25, 32, 48, 66, 170– 172, 178 strain contours 169, 170 strain decomposition 20 strain energy potential 14 strain hardening 19,... 289 strain-hardening hyperplasticity 88 strains 7–16, 23, 28, 31, 32, 66, 71, 72, 79, 85, 87, 117, 159 , 160, 163, 164, 167–170, 177, 186, 187, 191, 195, 203, 205, 226, 233, 243, 251, 262, 266, 277, 287–289 strain-softening behaviour 31 strength parameters 148 stress history 111, 112, 118, 200, 202 stress reversal 98, 114, 127, 151 , 184– 186, 201 stress tensor 9, 47, 154 , 163, 175, 301 stress–dilatancy... 175, 301 stress–dilatancy relation 210 stress-induced anisotropy 169, 172 stretch 286, 287 structural analysis 277, 300 structural anisotropy 172 St-Venant model 97, 98 subdifferential 82, 264–266, 321, 328– 335, 337 subgradient 275, 328 subscript notation 5, 301 support function 268, 269, 304, 332– 335 surroundings 36–38, 245 T tangent modulus 98, 127, 128, 150 temperature 32, 36–46, 54, 55, 65, 66,...350 Index source of heat 37 specific enthalpy 42, 256 specific entropy 40, 45, 248 specific Gibbs free energy 42, 93, 96, 98, 101, 102, 121, 142, 175, 177, 256 specific Gibbs free energy functional 179 specific heat 44–46 specific heat at constant pressure 45 specific heat at constant volume 45, 46 specific Helmholtz free energy 42, 175, 256 specific internal energy 40, 74, 246, 256, 301... 165–172, 182, 209, 271, 289 volumetric thermal expansion coefficients 258 von Mises yield surface 16, 26, 27, 83, 95, 100, 101, 130, 155 W weighting function 143, 148 151 , 177, 228 Winkler method 290 351 work conjugacy 10, 160 work hardening 19, 24, 341 Y yield stress 16, 26, 211, 284 yield surface 2, 16–33, 57–59, 68–70, 83–91, 95, 96, 100, 103–131, 142–147, 150 155 , 177–180, 184, 188–192, 195– 201, 204–210,... 68–70, 83–91, 95, 96, 100, 103–131, 142–147, 150 155 , 177–180, 184, 188–192, 195– 201, 204–210, 217, 225, 231, 233, 239, 261, 268–270, 289, 304, 330 Z Zeroth Law 36 Ziegler’s orthogonality condition 3, 75, 225, 254, 257, 261, 302 Ziegler’s translation rule 28, 114, 186 . 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