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Ciarlet P.G. — Mathematical elasticity. Volume II: Theory of plates
Ciarlet P.G. — Mathematical elasticity. Volume II: Theory of plates



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Название: Mathematical elasticity. Volume II: Theory of plates

Автор: Ciarlet P.G.

Аннотация:

The objective of Volume II is to show how asymptotic methods, with the thickness as the small parameter, indeed provide a powerful means of justifying two-dimensional plate theories. More specifically, without any recourse to any a priori assumptions of a geometrical or mechanical nature, it is shown that in the linear case, the three-dimensional displacements, once properly scaled, converge in H 1 towards a limit that satisfies the well-known two-dimensional equations of the linear Kirchhoff-Love theory; the convergence of stress is also established.

In the nonlinear case, again after ad hoc scalings have been performed, it is shown that the leading term of a formal asymptotic expansion of the three-dimensional solution satisfies well-known two-dimensional equations, such as those of the nonlinear Kirchhoff-Love theory, or the von K?rm?n equations. Special attention is also given to the first convergence result obtained in this case, which leads to two-dimensional large deformation, frame-indifferent, nonlinear membrane theories. It is also demonstrated that asymptotic methods can likewise be used for justifying other lower-dimensional equations of elastic shallow shells, and the coupled pluri-dimensional equations of elastic multi-structures, i.e., structures with junctions. In each case, the existence, uniqueness or multiplicity, and regularity of solutions to the limit equations obtained in this fashion are also studied.



Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1997

Количество страниц: 561

Добавлена в каталог: 09.02.2014

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Предметный указатель
Marguerre — von Karman shallow shell      439
Marguerre — von Karman shallow shell, scaled two-dimensional equations      443
Marguerre — von Karman shallow shell, three-dimensional equations      440
Marguerre — von Karman shallow shell, two-dimensional equations      446
mass density      104
Membrane (tension in a membrane)      342
Membrane energy      69 317
Membrane equation (linear membrane equation)      341 432
Membrane equations (of the Kirchhoff — Love theory)      67 76 112 402
Membrane equations (of the Kirchhoff — Love theory), existence of solutions      53 67
Membrane equations (of the Kirchhoff — Love theory), regularity of solutions      54 67
Membrane equations (of the Kirchhoff — Love theory), uniqueness of solutions      53 67
Membrane theory (large deformation, nonlinear membrane theory)      340 353 355 365
Membrane theory (large deformation, nonlinear membrane theory), existence of solutions      340
Method of formal asymptotic expansions      82 90 268 356 358 381 442
Metric (change of metric tensor)      319
Micro-vibrations      199
Middle surface of a plate      16
Middle surface of a shell      213
Min-max principle      107 109
Minimization of a functional      21 22 42 43 69 135 171 243 262 263 284 336 349 352 416
Minimizer of an energy      see "Minimization of a functional"
Minimum principle      107 109
Mixed finite element method      24
Modal synthesis by substructuring method      198
Model multi-structure      204
Moderately slanted plate      245 247
Moderately slanted plate theory      245 247
Moderately thin plate      86
Moderately thin plate theory      86
Monge — Ampere equation      416
Monge — Ampere form      391
Multi-dimensional problem      163 170 175 197
Multi-structure      see "Elastic multi-structure"
Multiplicity of solutions      see "Nonuniqueness of solutions"
Multiply connected domain      402 420 448
Naghdi plate theory      87
Naghdiplate theory      87
Nash — Moser implicit function theorem      324
Natural state      17 19 260
Nikodym set      388 395
Non-frame-indifferent plate theory      19 323 364 407
Nonconforming finite element method      24
Nonhomogeneous elastic material      85
Nonlinear elasticity (three-dimensional nonlinear elasticity)      263
Nonlinear elastodynamics      327
Nonlinear flexural theory (large deformation nonlinear flexural theory)      343
Nonlinear flexural theory (large deformation nonlinear flexural theory), existence of solutions      345
Nonlinear Kirchhoff — Love plate theory      317 321 326 344 356 364
Nonlinear Kirchhoff — Love plate theory as a small displacement theory      324 335
Nonlinear Kirchhoff — Love plate theory, existence of solutions      316 363
Nonlinear Kirchhoff — Love plate theory, regularity of solutions      316
Nonlinear Kirchhoff — Love theory of a plate      317 324 326 327 334 344 364
Nonlinear membrane theory (large deformation nonlinear membrane theory)      327 340 353 355 365
Nonlinear Reissner — Mindlin plate theory      328
Nonlinear small displacement theory      324 335 362
Nonlinearly elastic clamped plate problem      262
Nonlinearly elastic clamped plate, limit scaled three-dimensional equations      277
Nonlinearly elastic clamped plate, scaled three-dimensional equations      265 295
Nonlinearly elastic clamped plate, scaled two-dimensional energy      283
Nonlinearly elastic clamped plate, scaled two-dimensional equations      279
Nonlinearly elastic clamped plate, three-dimensional boundary conditons      262
Nonlinearly elastic clamped plate, three-dimensional constitutive equation      260
Nonlinearly elastic clamped plate, three-dimensional energy      263
Nonlinearly elastic clamped plate, three-dimensional equations      262 294
Nonlinearly elastic clamped plate, three-dimensional equations of equilibrium      259
Nonlinearly elastic clamped plate, three-dimensional principle of virtual work      259
Nonlinearly elastic clamped plate, time-dependent problem      327
Nonlinearly elastic clamped plate, two-dimensional boundary conditions      315 317
Nonlinearly elastic clamped plate, two-dimensional constitutive equations      317
Nonlinearly elastic clamped plate, two-dimensional energy      283 316
Nonlinearly elastic clamped plate, two-dimensional equations      279 315
Nonlinearly elastic clamped plate, two-dimensional equations of equilibrium      317
Nonlinearly elastic clamped plate, two-dimensional principle of virtual work      317
Nonlinearly elastic clamped shallow shell      356
Nonlinearly elastic clamped shallow shell, scaled two-dimensional equations      358
Nonlinearly elastic clamped shallow shell, three-dimensional equations      357
Nonlinearly elastic clamped shallow shell, two-dimensional equations      360
Nonlinearly elastic material      260
Nonlinearly elastic multi-structure      175
Nonlinearly elastic rod      325 328 335
Nonlinearly elastic shallow shell      356
Nonlinearly elastic shallow shell in Cartesian coordinates      361
Nonlinearly elastic von Karman plate      see "von Karman plate"
Nonlinearly elastic von Karman plate problem      374
Nonuniqueness of solutions      288 313 426
Numerical approximation      24 87 179 180 198 438
Obstacle (plate-lying on an obstacle)      53 85 124 438
One-dimensional rod theory      83 85 112
Operator of linearized elasticity (three-dimensional linearized elasticity)      18 105
Orientation-preserving condition      217 336
Partially clamped plate      17
Periodic boundary conditions      84 323
Perturbed bifurcation of solutions of the von Karman equations      435 436
Piezoelectric plate      86 118
Place (boundary condition of place)      16 258
Plate lying on an obstacle      53 85 124 438
Plate theory derived by a constraint method      88
Plate theory derived by integration across the thickness      81 328
Plate theory with rapidly varying thickness      85 327
Plate theory with varying thickness      85 123
Plate with holes      85 438
Plate with rapidly varying thickness      85 327
Plate with stiffeners      175
Plate with varying thickness      85 123
Plate with varying thickness of a plate      85 123
Poisson ratio      19
Pressure load      424
Properly invariant plate theory      335
Quasi-convex envelope      351 352 354
Quasi-convex function      351
Quasi-static problem      116
Quasilinear equations      322 340
Rapidly varying thickness of a plate      85 327
Rayleigh quotient      107 110
Reduced von Karman equation      411 426 429 434 435
Reference configuration      16
Regularity of solutions of partial differential equations      48 54 290 302 316 404 420 446
Reissner — Mindlin plate theory      84 86
Rigidity of an elastic material      17
Rotational inertia term      118
Rotor blade      177 246
Satellite      177
Scaled Airy stress function      390 444
Scaled deformation      337 350
Scaled displacement      25 83 89 125 351 441
Scaled eigenfunction      108 193
Scaled eigensolution      108 193
Scaled eigenvalue      108 193
Scaled energy      30 43 97 283 351
Scaled flexural equations (of the linear Kirchhoff — Love theory)      35 48
Scaled inverted constitutive equation      295 300 380
Scaled Kirchhoff — Love displacement field      47 110 115 142 195 276 278 333 344 384
Scaled linearized strain      32
Scaled linearized strains      32
Scaled Marguerre — von Karman equations      445
Scaled membrane equations (of the Kirchhoff — Love theory)      35 53 162
Scaled principle of virtual work      295 380
Scaled strains      32
Scaled stress      57 58 83 295
Scaled three-dimensional equations      218
Scaled three-dimensional equations for a linearly elastic clamped plate      28
Scaled three-dimensional equations for a linearly elastic multi-structure      140
Scaled three-dimensional equations for a linearly elastic shallow shell      237
Scaled three-dimensional equations for a nonlinearly elastic clamped plate      265 295
Scaled three-dimensional equations for a von Karman plate      380
Scaled two-dimensional equations      229
Scaled two-dimensional equations for a linearly elastic clamped plate      42 57
Scaled two-dimensional equations for a linearly elastic shallow shell      229
Scaled two-dimensional equations for a Marguerre — von Karman shallow shell      443
Scaled two-dimensional equations for a nonlinearly elastic clamped plate      279 289
Scaled two-dimensional equations for a nonlinearly elastic shallow shell      358
Scaled two-dimensional equations for a von Karman plate      384
Scaled vector field      27 295
Scaled von Karman equations      390
Scaling of stresses      57 83 295 378
Scalings of the deformations      337 350
Scalings of the displacements      25 83 89 125 137 183 215 264 295 329 357 378 441
Scalings of the stresses      57 83 295 378
Second Piola — Kirchhoff stress tensor      18 259 260 266 448
Semilinear equations      322 407
Shallow rod      85 246
Shallow shell      245
Shell      212
Simply connected domain      389
Simply connected domain in $R^{n}$      389
Simply supported plate      102 103 122
Singular perturbations      46
Singularities of solutions of partial differential equations      24 53 56 174
Sliding edges      323
Small displacement linear plate theory      73
Small displacement nonlinear plate theory      324
Small displacement theory      73 324
Sobolev space      8 58 261 291
Solar panel of a satellite      177
Spectral theory      106 427
St Venant — Kirchhoff elastic material      260
St Venant — Kirchhoff material      260 262 263 354 357 374
Stabilization of elastic multi-structures      203
Stabilization of elastic multi-structures (linearly elastic multi-structures)      203
Stationary point of a functional      262 283 361
Stationary solution      105 189
Stiff eigenvalue problem      198
Stiff elliptic problem      171
Stiff time-dependent problem      202
Stiffener (plate with stiffeners)      175
Stored energy function      336 349
Stored energy function of a St Venant — Kirchhoff material      336
strain      260
Strain tensor      319
Strain tensor of the middle surface of a plate      68 319
Stress      18 259
Stress couple      321
Stress function (Airy stress function)      see "Airy stress function"
Stress resultant      321 406
String      328 348
Substructure      133
Substructure of an elastic multi-structure (linearly elastic multi-structure)      133
Substructuring method (modal synthesis by substructuring method)      198
Surface force (applied surface force)      16
Tension in a membrane      342
Term of order p in a      269
Term of order p in a formal asymptotic expansion      269
Thermo-elastic material      85 420
Thermoelastic plate      85
Thickness of a plate      16
Thickness of a shell      213
Thin-walled rod      175
Three-dimensional constitutive equation      260
Three-dimensional eigenvalue problem for a plate      105
Three-dimensional elasticity tensor      20 24 261 318
Three-dimensional energy      23 336
Three-dimensional equations      213
Three-dimensional equations for a linearly elastic clamped plate      20
Three-dimensional equations for a linearly elastic multi-structure      135
Three-dimensional equations for a linearly elastic shallow shell      213
Three-dimensional equations for a Marguerre — von Karman shallow shell      440
Three-dimensional equations for a nonlinearly elastic clamped plate      262
Three-dimensional equations for a nonlinearly elastic shallow shell      357
Three-dimensional equations for a von Karman plate      376
Three-dimensional equations of an elastic multi-structure (linearly elastic multi-structure)      135
Three-dimensional equations of equilibrium      259
Three-dimensional Hellinger — Reissner variational principle      23 82 120
Three-dimensional principle of virtual work      23 259 294 375
Three-dimensional response function      261
Three-dimensional set with a two-dimentional slit      163
Time-dependent coupled multi-dimensional problem      199
Time-dependent flexural equations (of the linear Kirchhoff — Love theory)      115 117
Time-dependent linear Kirchhoff — Love plate theory      115
Time-dependent membrane equations (of the Kirchhoff — Love theory)      116
Time-dependent nonlinear Kirchhoff — Love plate theory      327
Time-dependent problem for a von Karman plate      420
Time-dependent problem for an elastic multi-structure      199
Time-dependent problem for an elastic multi-structure (linearly elastic multi-structure)      199
Time-dependent problem for folded plates      203
Time-dependent problem in three-dimensional linearized elasticity      112
Time-dependent problems for folded plates      203
Time-dependent von Karman equations      420
Traction (boundary condition of traction)      259
Transmission conditions      136
Transmission problem      136
Transverse displacement      64 313 403
Transverse displacement of the middle surface      64
Tranverse displacement      64 313 403
Two-dimensional boundary conditions of a linearly elastic clamped plate      66
Two-dimensional boundary conditions of a nonlinearly elastic clamped plate      315 317
Two-dimensional boundary conditions of a von Karman plate      403 407
Two-dimensional boundary conditions of clamping      317
Two-dimensional constitutive equation      317 322
Two-dimensional constitutive equations of a plate      317 322
Two-dimensional eigenvalue problem for a plate      111
Two-dimensional elasticity tensor      68 319
Two-dimensional energy      69 283 316 361
Two-dimensional equations      242
Two-dimensional equations for a linearly elastic clamped plate      67
Two-dimensional equations for a linearly elastic shallow shell      242
Two-dimensional equations for a Marguerre — von Karman shallow shell      446
Two-dimensional equations for a nonlinearly elastic clamped plate      315
Two-dimensional equations for a nonlinearly elastic shallow shell      360
Two-dimensional equations for a von Karman plate      386
Two-dimensional equations of equilibrium      317 322 364
Two-dimensional flexural equations      67 75 172
Two-dimensional Hellinger — Reissner variational principle      71 76
Two-dimensional membrane equations      67 76 112
Two-dimensional plate theory      79 81
Two-dimensional principle of virtual work      71 317
Two-dimensional response function      318
Two-dimensional slit in a three-dimensional open set      163
Two-dimensional strain tensor      318 322
Two-dimensional time-dependent problem for a linearly elastic plate      115 117
Two-dimensional time-dependent problem for a nonlinearly elastic plate      327
Two-scale convergence      85
Uniqueness of solutions      21 48 53 164 238 247 288 328 426 450
Unperturbed bifurcation of solutions of the von Karman equations      433 434
Upper face of a plate      15
Upper face of a shallow shell      214
Vekua plate theory      87
Viscoplastic plate      85
von Karman equation (reduced von Karman equation)      411 426 429 434 435
Von Karman equations      403 404
von Karman equations, bifurcation of solutions      433 434 435 436 437
von Karman equations, degeneracy into the linear membrane equation      432
von Karman equations, equivalence with a two-dimensional displacement problem      389
von Karman equations, existence of solutions      404 416 420 448
von Karman equations, nonuniqueness of solutions      426
von Karman equations, regularity of solutions      404 420
von Karman equations, uniqueness of solutions      426 450
von Karman plate      375 376 425 447
von Karman plate lying on an obstacle      438
von Karman plate with holes      438
von Karman plate, scaled three-dimensional equations      380
von Karman plate, scaled two-dimensional equations      384
von Karman plate, thermoelastic scaled two-dimensional equations      420
von Karman plate, three-dimensional boundary conditions      374 376
von Karman plate, three-dimensional constitutive equations      374
von Karman plate, three-dimensional equations      374
von Karman plate, two-dimensional boundary conditions      386
von Karman plate, two-dimensional equations      386 403
Weak lower semi-continuity      284 347 418 449
Weak lower semi-continuity of a functional      284 347 418 449
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