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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

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$x^{\varepsilon}_{3}$-dependent constitutive equation      321
$\Gamma$-convergence      83 95 101 348
$\Gamma$-limit      95 97
A priori assumption (of a geometrical or mechanical nature)      76 81 86 87 88 120 321
Adhesive (elastic adhesive)      136 177
Admissible deformation      336
Admissible deformations (set of admissible deformations)      336
Airy stress function      403 406 446
Airy stress function for the Marguerre — von Karman equations      446
Airy stress function for the von Karman equations      403
Anisotropic elastic material      85 327
Anisotropic elastic material (plate made with an anisotropic elastic material)      85 327
Anisotropic plate      85
Ansatz of the method of formal asymptotic expansions      90 269 297 338 358 381 442
Applied body force      16
Applied force      16
Applied surface force      16
Applied surface force along the lateral face      373 377 439
Applied surface forces along the lateral face of a plate      373
Applied surface forces along the lateral face of a shallow shell      439
Assumptions on applied body forces      27 94 113 126 139 184 216 265 329 350 358 379 441
Assumptions on applied surface forces      27 94 126 216 265 329 350 358 379 441
Assumptions on the data      27 77 83 94 108 113 139 184 192 198 216 244 265 325 350 358 379 441
Assumptions on the Lame constants      27 78 94 108 113 139 173 181 192 198 216 265 329 379 441
Assumptions on the mass density      108 192 198
Asymptotic analysis      24 33 83 137 264
Asymptotic analysis and $\Gamma$-convergence      97
Asymptotic expansions (method of formal asymptotic expansions)      82 90 268 297 356 358 381 442
Asymptotic method (formal asymptotic method)      82
Babuska's paradox      53
Beam inserted in an elastic foundation      176
Bending moment      321
Bifurcation of solutions of the MarGuerre — von Karman equations      446
Bifurcation of solutions of the von Karman equations      433 437
Bifurcation parameter      424
Bifurcation theory      433 438
Biharmonic operator      49
Body force (applied body force)      16
Boundary condition of clamping      16 188 317 327
Boundary condition of place      16 258
Boundary condition of pressure      409
Boundary condition of traction      259
Boundary conditions for a linearly elastic clamped plate      18 181 188
Boundary conditions for a Marguerre — von Karman shallow shell      439
Boundary conditions for a nonlinearly elastic clamped plate      315 327
Boundary conditions for a von Karman plate      374 376
Boundary conditions for the Airy stress function      403 408 446
Boundary conditions for the Marguerre — von Karman equations      446
Boundary conditions for the von Karman equations      403 407
Boundary layer      102 103
Branching of solutions      433
Brouwer fixed point theorem      416
Buckling of a von Karman plate      424 437
Canonical von Karman equations      409
Cartesian coordinates      218 242 361
Cauchy — Green strain tensor      326 337
Cauchy — Green strain tensor (right Cauchy — Green strain tensor)      326 337
Change of curvature tensor      68 319
Change of curvature tensor of the middle surface      68 319
Change of curvature tensor of the middle surface of a plate      68 319
Change of metric tensor      319
Change of metric tensor of the middle surface      319
Change of metric tensor of the middle surface of a plate      319
Circular von Karman plate      437
Clamped plate      15 16 181
Clamping "in average"      101
Classical plate theory      81 321
Classical two-dimensional plate theory      81 321
Coerciveness of a functional      285 347 419 449
Compatibility condition on applied forces      377 390 404 440
Completely clamped plate      16
Composite material      85
Conforming finite element method      24
Constitutive equation      76 260
Constitutive equation of a St Venant — Kirchhoff material      260
Constraint method      88
Controllability of a time-dependent problem      118 203
Controllability of elastic multi-structures      203
Controllability of elastic multi-structures (linearly elastic multi-structures)      203
Controllability of plates      118 420
Controllability of von Karman plates      420
Convergence of a formal asymptotic expansion      84
Convergence of a formal expansion      84
Convergence of scaled displacements      34 83 100 109 115 121 142 199 229 352
Convergence of scaled eigenfunctions      109 194
Convergence of scaled eigenvalues      109 194
Convergence of scaled stresses      58 83 84 121
Convergence of stresses      83 121 189
Convergence of the displacements      34 75 83 100 109 115 121 142 229 352
Corrector      102
Cosserat plate theory      86 328
Coupled multi-dimensional problem      170 175
Coupled, pluri-dimensional, time-dependent problem      202
Coupled, pluridimensional, eigenvalue problem      197
Curvature (change of curvature tensor)      319
Curvilinear coordinates      219
De-scalings of the displacements      64 116 167 240 360 386 403 404 405 445
De-scalings of the stresses      69 320 405
Dead load      258 408 424
Definition of a shallow shell      216 245 246 247 362 441
Deformation      258 260 335
Deformation gradient      336
Deformed configuration      258
Degeneracy of the Marguerre — von Karman equations into the membrane equation (linear membrane equation)      446
Degeneracy of the von Karman equations into the membrane equation (linear membrane equation)      432
Displacement      16
Displacement approach      27 33 34 268 300 383
Displacement gradient      260
Displacement of the middle surface of a plate      64 313 403 404
Displacement-stress approach      33 83 120 268 294 295 300 327 380 381
Displacement-traction problem      262
Domain decomposition method      180
Domain in $R^{n}$      7
Domain in $R^{n}$ as a Nikodym set      388
Eigenfunction      105 190
Eigensolution      105
Eigenvalue      105 190
Eigenvalue coupled multi-dimensional problem      197
Eigenvalue problem for an elastic multi-structure      189 199
Eigenvalue problem for an elastic multi-structure (linearly elastic multi-structure)      194 199
Eigenvalue problem for folded plates      199
Eigenvalue problem for rods      112
Eigenvalue problem for the flexural equations (of the linear Kirchhoff — Love theory)      111
Eigenvalue problem for the scaled flexural equations      111
Eigenvalue problem in three-dimensional elasticity      105
Eigenvalue problems for folded plates      199
Elastic adhesive      136 177
Elastic material      17 260
Elastic multi-structure (linearly elastic multi-structure)      133
Elastic multi-structure (nonlinearly elastic multi-structure)      175
Elastic multi-structure with a shallow shell      175 178 246
Elastoplastic plate      85 101
Ellipticity of a bilinear form      21 22 23 50 55 165 240
Energy, coerciveness      285 347 419 449
Energy, existence of a minimum      69 287
Equations of linearized elasticity (three-dimensional linearized elasticity)      18
Error estimates      88
Error estimates for stresses      102
Error estimates for the displacements      83 102 103 104
Error estimates for the stresses      102
Example of a nonlinearly elastic material      260
Example of an elastic multi-structure (linearly elastic multi-structure)      134 176 177 178 182
Existence of solutions      21 48 53 164 238 247 263 282 287 316 323 328 340 345 363 404 416 420 446
Existence theory for eigenvalue problems      106
Existence theory for time-dependent problems      113
Fibers (materials with fibers)      85
Finite element of class $C^{0}$      87
Finite element of class $C^{1}$      87
First Piola — Kirchhoff stress tensor      327
Flexural energy      69 317
Flexural equations (of the linear Kirchhoff — Love theory)      67 75
Flexural equations (of the linear Kirchhoff — Love theory), existence of solutions      48 67
Flexural equations (of the linear Kirchhoff — Love theory), regularity of solutions      48 67
Flexural equations (of the linear Kirchhoff — Love theory), uniqueness of solutions      48 67
Flexural rigidity of a plate      67 319 386 404
Flexural theory (inextensional, large deformation, nonlinear flexural theory)      343 345
Folded plates      175 180 199
Formal asymptotic method      82
Frame-indifference      19 323 328 337 364 365 407
Frame-indifferent flexural theory      343
Frame-indifferent membrane theory      340 353 355 365
Frame-indifferent plate theory      335 340
Functional      see "Energy" "Minimization
General constitutive equation      325
General elastic material      325 407
General nonlinearly elastic material      325
Generalized Korn's inequality      126 144 224 231 236 239
Generic character of the nonlinear Kirchhoff — Love plate theory      327
Generic character of the von Karman equations      407
Green — St Venant strain tensor      260 267 326
Hierarchic plate theory      87
Hilbert uniqueness method (HUM)      118 203
Holes (plates with holes)      85 438
Homogeneous elastic material      17 260 325
Homogenization theory      85 437
Hooke's law      19 76 88
Hybrid finite element method      24
Hyper-elastic material      262
Hyperelastic material      262
Implicit function theorem      263 323 364 450
In-plane displacement      64 313 404
In-plane displacement of the middle surface      64
Inextensional deformation      343
Inextensional flexural theory      343
Infimizing sequence of an energy      287
Infinite energy      176
Integration across the thickness      81
Integration across the thickness of a plate      81
Inverted constitutive equation      294 375
Isotropic elastic material      17 260 325
Junction between a plate and a rod      175
Junction between a plate and a shallow shell      245
Junction between a plate and a three-dimensional structure      163 168
Junction between a shallow shell and a plate      245
Junction between a three-dimensional structure and a plate      163 168
Junction between a three-dimensional structure and a rod      176
Junction between a three-dimensional structure and a shallow shell      175 178
Junction between nonlinearly elastic substructures      175
Junction between plates      175
Junction between plates and rods      175
Junction between rods      175 199
Junction between shells      180
Junction conditions for a coupled, multi-dimensional problem      148 170 172 175 197
Junction conditions for a three-dimensional problem      139
Justification of assumptions on the data      94 329
Justification of the displacement-stress approach      313
Justification of the linear Kirchhoff — Love plate theory      72
Justification of the nonlinear Kirchhoff — Love plate theory      321 334
Justification of the scalings of the displacements      93 329
Justification of the scalings of the stresses      58 120
Kirchhoff — Love displacement field      65 67 120 173 188 243 314 322 406
Kirchhoff — Love hypothesis      76 81 120
Korn's inequality      13
Korn's inequality with boundary conditions      10 22 36 54 119 135 144 214 224
Korn's inequality without boundary conditions      10 54 119
Kuhn — Tucker relations      125486
Lagrange multiplier      77 125 172
Lame constants      17 19 261
Laminated plate      85 88
Large deformation nonlinear flexural theory      343
Large deformation nonlinear membrane theory      340 353
Lateral face of a plate      15
Lateral face of a shallow shell      439
Lax — Milgram lemma      21 23 39 50 55
Leading term in a formal asymptotic expansion      269
Limit displacement inside the plate      64 76 313 404
Limit displacement of the middle surface      64 313 403 404
Limit displacements inside the plate      64 76 313 404
Limit displacements of the middle surface      64 313 386
Limit energy      69 171
Limit energy of an elastic multi-structure (linearly elastic multi-structure)      171
Limit energy of an elastic multistructure      171
Limit equations of an elastic multi-structure (linearly elastic multi-structure)      163 168
Limit scaled energy      44
Limit scaled stress      58 62 302 387 444
Limit scaled stresses      58 62 302 387 444
Limit scaled three-dimensional equations      34
Limit scaled three-dimensional equations for a linearly elastic clamped plate      34
Limit scaled three-dimensional equations for a nonlinearly elastic clamped plate      277
Limit scaled three-dimensional equations for a von Karman plate      382
Limit stress      69 76 174 319 322 328 361 404
Linear elastodynamics      104 112 189 199
Linear Kirchhoff — Love plate theory      67 72 75 79 86 94 342
Linear Kirchhoff — Love plate theory as a small displacement theory      73 335
Linear Kirchhoff — Love plate theory, eigenvalue problem      111
Linear Kirchhoff — Love plate theory, existence of solutions      67
Linear Kirchhoff — Love plate theory, regularity of solutions      67
Linear Kirchhoff — Love plate theory, uniqueness of solutions      67
Linear Kirchhoff — Love theory of a plate      67 72 75 79 86 94 342
Linear membrane equation      341 432 446
Linear small displacement theory      73 335
Linearized bending moment      70
Linearized boundary condition of traction      18
Linearized displacement-traction problem      17
Linearized elasticity (three-dimensional linearized elasticity)      13 24
Linearized elasticity (three-dimensional linearized elasticity), eigenvalue problem      105
Linearized elasticity (three-dimensional linearized elasticity), time-dependent problem      104 112
Linearized elastodynamics      104 112 189 199
Linearized strain      19
Linearized strain tensor      19
Linearized strains      19
Linearized stress couple      70 76
Linearized stress resultant      70 76
Linearized stress tensor      18
Linearly elastic clamped plate problem      20
Linearly elastic clamped plate, eigenvalue problem      105
Linearly elastic clamped plate, scaled three-dimensional equations      28
Linearly elastic clamped plate, scaled two-dimensional energy      43
Linearly elastic clamped plate, scaled two-dimensional equations      42 57
Linearly elastic clamped plate, three-dimensional boundary conditions      18
Linearly elastic clamped plate, three-dimensional energy      23
Linearly elastic clamped plate, three-dimensional equations      20
Linearly elastic clamped plate, three-dimensional Hellinger — Reissner variational principle      23 82 120
Linearly elastic clamped plate, three-dimensional principle of virtual work      23
Linearly elastic clamped plate, time-dependent problem      112
Linearly elastic clamped plate, two-dimensional boundary conditions      66 181
Linearly elastic clamped plate, two-dimensional energy      69
Linearly elastic clamped plate, two-dimensional equations      67
Linearly elastic clamped plate, two-dimensional flexural equations      67 111
Linearly elastic clamped plate, two-dimensional Hellinger — Reissner variational principle      71 76
Linearly elastic clamped plate, two-dimensional membrane equations      67 169
Linearly elastic clamped plate, two-dimensional principle of virtual work      71
Linearly elastic material      19
Linearly elastic multi-structures      see "Elastic multi-structure"
Linearly elastic rod      83 85 112
Linearly elastic shallow shell      242
Linearly elastic shallow shell in Cartesian coordinates      242
Linearly elastic shallow shell in curvilinear coordinates      219
Linearly elastic shell      213
Linearly elastic von Karman plate      122
Lions (lemma of J.L. Lions)      9 11 226
Live load      327 408 424 448
Locking phenomenon      87
Lower face of a plate      15
Lower face of a shallow shell      214
Marguerre — von Karman equations      446 447
Marguerre — von Karman equations, bifurcation of solutions      446
Marguerre — von Karman equations, degeneracy into the linear membrane equation      446
Marguerre — von Karman equations, existence of solutions      446
Marguerre — von Karman equations, regularity of solutions      446
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