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Trang 6Index
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
Trang 7346 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
Trang 8Index 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
Trang 9348 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
Trang 10Index 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