Lemm J.M. — Mathematical elasticity. Theory of shells :: Электронная библиотека попечительского совета мехмата МГУ

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Lemm J.M. — Mathematical elasticity. Theory of shells

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Название: Mathematical elasticity. Theory of shells

Автор: Lemm J.M.

Аннотация:

The objective of Volume III is to lay down the proper mathematical foundations of the two-dimensional theory of shells. To this end, it provides, without any recourse to any a priori assumptions of a geometrical or mechanical nature, a mathematical justification of two-dimensional nonlinear and linear shell theories, by means of asymptotic methods, with the thickness as the "small" parameter.

Язык:

Рубрика: Физика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2000

Количество страниц: 659

Добавлена в каталог: 06.02.2014

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Предметный указатель

$\Gamma$ -convergence      233 326 459
$\Gamma$ -limit      460
A priori assumption of a geometrical or mechanical nature      336 363 546
Admissible applied force      265 292 293 298 351
Admissible applied forces for a linearly elastic generalized membrane shell      265 292 293 298
Admissible applied forces for Koiter's linear shell equations      351
Admissible displacement      303 439 503
Admissible inextensional deformation      530
Admissible inextensional displacement      439 503
Admissible linearized inextensional displacement      303
Analytic functions      121 125
Anisotropic elastic material      162 178 234 326 360 366 550
Ansatz of the method of formal asymptotic expansions      164 413
Applied body force      7 25 386
Applied force (admissible applied force)      265 292 293 298 351
Applied surface force      7 26 386
Area element      19
Area element on a surface      70
Assumption on the middle surface of a shallow shell      369 371 552
Assumptions on applied body forces      152 162 164 179 180 199 229 232 233 247 265 305 322 326 414 442 443 458 507
Assumptions on applied surface forces      152 162 164 179 180 199 229 247 265 305 322 414 442 443 507
Assumptions on the data      see "Applied body force" "Applied "Lame "Shallow
Assumptions on the Lame constants      152 162 199 229 247 303 322 442 443 507
Asymptotic expansions (method of formal asymptotic expansions)      164 413
Asymptotic line      83 295
Average with respect to the transverse variable      201
Badly-inhibited bending      292
Benchmark problem      361
Bending in shells      see "Flexural-dominated behavior" "Flexural "Linearly "Nonlinearly
Bending moment      344
Body force      see "Applied body force"
Boundary condition of place      8 32 387
Boundary condition of pressure      455
Boundary condition of traction      388
Boundary layer      351 362 368 371 374
Boundary layer in a shell      351 362
Budiansky — Sanders change of curvature tensor on a surface (linearized change of curvature tensor on a surface)      367
Budiansky — Sanders linearized change of curvature tensor      367
Canonical extension      144
Cartesian components      6
Cartesian components of the displacement      7 387 459
Cartesian components of the displacement field      7 387 459 463
Cartesian coordinates      6 67
Cartesian frame      6
Cauchy — Green strain tensor      9
Cauchy — Green strain tensor (right Cauchy — Green strain tensor)      9
Cauchy — Green strain tensor (right)      9
Center of curvature      74 77
Change of curvature tensor      508
Change of curvature tensor on a surface      508 519 524 548
Change of curvature tensor on a surface (linearized change of curvature tensor on a surface)      93 174 181 308 322 338 344
Change of metric tensor      388 398
Change of metric tensor in a three-dimensional domain (linearized change of metric tensor in a three-dimensional domain) in Cartesian coordinates      9
Change of metric tensor in a three-dimensional domain (linearized change of metric tensor in a three-dimensional domain) in curvilinear coordinates      36
Change of metric tensor in a three-dimensional domain in Cartesian coordinates      388
Change of metric tensor in a three-dimensional domain in curvilinear coordinates      398
Change of metric tensor on a surface      437 450 524 548
Change of metric tensor on a surface (linearized change of metric tensor on a surface)      91 170 181 210 228 308 321 338 344 366
Christoffel symbols      29 33
Christoffel symbols of the first kind      55 60
Christoffel symbols of the second kind      55 60
Christoffel symbols on a surface      88 136
Christoffel symbols on a surface of the first kind      130
Christoffel symbols on a surface of the second kind      130
Circular cylinder      66 191
Clamped shell      101 116 146 339
Codazzi — Mainardi identities      131 133
Coerciveness of a functional      431 452 466 538
Complementing boundary condition      123 236 237 373
Composite material (shell made with a composite material)      368 371
Cone      296 305 329 505 540
Continuation theorem (unique continuation theorem)      125 126 294
Contravariant basis      21
Contravariant basis on a surface      86
Contravariant basis on a surface of the tangent plane      72
Contravariant components of a vector      56
Contravariant components of the applied body force density      25
Contravariant components of the applied surface force density      26
Contravariant components of the first Piola — Kirchhoff stress tensor      403
Contravariant components of the linearized stress tensor      38
Contravariant components of the metric tensor      19 20 57
Contravariant components of the metric tensor of a surface      72
Contravariant components of the metric tensor on a surface      72
Contravariant components of the resultant stress tensor      224
Contravariant components of the second Piola — Kirchhoff stress tensor      402
Contravariant components of the three-dimensional elasticity tensor      32
Contravariant components of the two-dimensional elasticity tensor of a shell      101
Controllability of shells      234 361 371
Convergence of displacements      183 209 231 233 268 291 308 324 370 461
Convergence of stresses      234 237 326 331
Convergence of the displacements      183 209 231 233 268 291 308 324 370 461
Coordinate line      17
Coordinate line on a surface      68
coordinates      see also "Cartesian coordinates" "Curvilinear
Cosserat shell theory      363 549
Cosserat surface (one director Cosserat surface)      363 549
Covariant basis      17
Covariant basis of the tangent plane      69
Covariant basis on a surface      86
Covariant components of a vector      25 56
Covariant components of the change of metric tensor      398
Covariant components of the curvature tensor      81
Covariant components of the curvature tensor on a surface      81
Covariant components of the displacement      25
Covariant components of the displacement field      25
Covariant components of the displacement field of a surface      86
Covariant components of the displacement of a surface      86
Covariant components of the Green — St Venant strain tensor      397
Covariant components of the linearized change of curvature tensor      93
Covariant components of the linearized change of metric tensor      36
Covariant components of the linearized change of metric tensor on a surface      91
Covariant components of the linearized strain tensor      36
Covariant components of the metric tensor      18 57
Covariant components of the metric tensor of a surface      69
Covariant components of the metric tensor on a surface      69
Covariant derivative of a tensor field      38 94 133 224 402
Covariant derivative of a vector field      30 33 451
Covariant derivative of a vector field on a surface      88
Curvature      see also "Curvature tensor on a surface"
Curvature of a planar curve      75 76
Curvature tensor      81
Curvilinear coordinates      15
Curvilinear coordinates on a surface      65
Cylinder      296 328 440 465 504 539
Cylindrical coordinates      15
Definition of a linearly elastic elliptic membrane shell      196
Definition of a linearly elastic flexural shell      199 302
Definition of a linearly elastic generalized membrane shell      245
Definition of a linearly elastic shallow shell      369 371
Definition of a nonlinearly elastic flexural shell      502
Definition of a nonlinearly elastic membrane shell (theory derived by formal analysis)      439
Definition of a nonlinearly elastic shallow shell      552 553 555
Deformation      9 387 456
Deformation gradient      10
Deformed configuration      9 287
Developable surface      83
Developpee      75
Director field on a surface      363
Displacement field in three-dimensional elasticity      7
Displacement field of a surface      86
Displacement gradient      10 389
Displacement-traction problem      390
Domain in $\mathbb{R}^{n}$       6 40
Donnel — Mushtari — Vlasov nonlinear shallow shell theory      555
Donnell — Mushtari — Vlasov nonlinear shallow shell theory      555
EDGE      see "Fold"
Eigenvalue problem for Koiter's linear shell equations      358
Eigenvalue problem for shells      234 326 358 371 555
Eigenvalue problem for the two-dimensional Koiter equations      358
Eigenvalue problems for shells      234 326 358 371

Elastodynamics for a shell      see "Time-dependent equations"
Ellipsoid      121 197 297
Elliptic middle surface (shells with elliptic middle surface)      120 196 293
Elliptic point      83
Elliptic point on a surface      83
Elliptic surface      120 122 126 130 196
Energy      see also "Three-dimensional energy" "Two-dimensional "Two-dimensional
Equations of equilibrium in Cartesian coordinates      387
Equations of equilibrium in curvilinear coordinates      403
Error estimates for displacements      233 350 351 358 360
Euler characteristic of a surface      83
Eversion of shells      442 542 554
Example of a linearly elastic elliptic membrane shell      197
Example of a linearly elastic flexural shell      303 304 305 328 329 330
Example of a linearly elastic generalized membrane shell      293 294 295 296 297
Example of a nonlinearly elastic flexural shell      504 505 506 539 540
Example of a nonlinearly elastic membrane shell (theory derived by formal analysis)      440 441 465
Existence of a minimum energy      224 228 290 321 339 340 391 462 530 551
Existence of solutions      52 101 111 115 126 223 317 338 340 366 367 376 392 452 453 529 551 554
First fundamental form of a surface      69 72
First Piola — Kirchhoff stress tensor      388
First-order shell theory      550
Flexural theory      see "Shell theory"
Flexural-dominated behavior      361
fold      76 295 296 297
force      see "Applied body force" "Applied "Applied
Formal asymptotic expansion      164 414
Formula of Gauss      88
Frame-indifferent flexural theory      525 541
Frame-indifferent membrane theory      452 463 466
Frame-indifferent shell theory      452 463 466 525 541
Functional      see also "Energy" "Minimization
Functional as a $\Gamma$ -limit      460
Gauss — Bonnet theorem      83 133
Gaussian curvature      82 83 84 121 133
Genus of a surface      83
Geometrically exact shell theory      549
Green — St Venant strain tensor      388
Green — St Venant strain tensor in Cartesian coordinates      388
Green — St Venant strain tensor in curvilinear coordinates      397
Hierarchic shell theory      368
Higher-order shell theory      550
Hilbert uniqueness method (HUM)      361
Holmgren's uniqueness theorem      124
Homogeneous elastic material      6 388
Homogenization of shell equations      234 360
Hyperbolic middle surface (shells with hyperbolic middle surface)      295 362
Hyperbolic paraboloid      132
Hyperbolic point      83
Hyperbolic point on a surface      83
Hyperbolic surface      295
Hyperboloid of revolution      294
Hyperelastic material      391 392 454
Implicit function theorem      144 392 453 555
Inextensional deformation      530
Inextensional displacement      439 503
Infinitesimal rigid displacement      46
Infinitesimal rigid displacement lemma      49
Infinitesimal rigid displacement lemma on a general surface      109 366 377
Infinitesimal rigid displacement lemma on a general surface with little regularity      114
Infinitesimal rigid displacement lemma on an elliptic surface      122 295
Infinitesimal rigid displacement on a surface      109 115
Infinitesimal rigidity of a surface      297
Infinitesimal rotation field      133
Interior layer      362
Internal constraints (method of internal constraints)      368 550
Intrinsic linear shell theory      368
Intrinsic nonlinear shell theory      551
Isometric surfaces      85 439 441 531
Isotropic elastic material      6 388
Junctions between shells      360 454
Justification of assumptions on the data      162 425 443 476 507
Justification of scalings of displacements      162 415
Kirchhoff — Love assumption      336 372 546 550
Kirchhoff — Love plate theory      227 320 328
Koiter energy      344 548
Koiter strain energy      344 548
Koiter's equations for a linearly elastic elliptic membrane shell      347
Koiter's equations for a linearly elastic flexural shell      355
Koiter's equations for a linearly elastic generalized membrane shell      351
Koiter's linear shell equations      100 111 115 337 339 341 344 360 373 375
Koiter's linear shell equations for shells whose middle surface has little regularity      115 339 362
Koiter's nonlinear shallow shell equations      553
Koiter's nonlinear shell equations      547
Korn's inequality in Cartesian coordinates      49 113 207 217 253 257 see
Korn's inequality in curvilinear coordinates      44 48 53 59
Korn's type (inequality of Korn's type) on a general surface      103 111 118 134 318 339 356 377 551 554
Korn's type (inequality of Korn's type) on a general surface with little regularity      115 340
Korn's type (inequality of Korn's type) on an elliptic surface      126 197 208 216 225 261 349
Lagrange multiplier      319 539 550
Lame constants      7 389
Laminated shell      368 550
Large deformation nonlinear flexural shell theory      525
Large deformation nonlinear membrane shell theory      452 463
Large deformation, or large displacement, flexural theory      525
Large deformation, or large displacement, membrane theory      452 463
Large deformation, or large displacement, shell theory      452 463 525
Lateral face of a shell      146
Lax — Milgram lemma      54 101 117 192 225 278 285 288 319 339
Leading term in a formal asymptotic expansion      164 414
Length element      20
Length element on a surface      70
Limit displacements      226 232 289 319 447 462 522
Limit stresses      234 237 326 331 452 546
Line of curvature      82
Linear Budiansky — Sanders shell theory      367
Linear flexural shell theory      101 162 171 180 182 199 526
Linear intrinsic shell theory      368
Linear Koiter's shell theory      100 115 336 346 548
Linear membrane shell theory      117 126 162 170 178 182 196 199 232 246 452 466
Linear Naghdi's shell theory      363 365
Linear shallow shell theory      369
Linear shell theory      see "Shell theory"
Linear shell theory based on the method of internal constraints      368
Linearization trick      415
Linearized change of curvature tensor      93 174 181 308 322 338 344 526
Linearized change of metric tensor      9 36
Linearized change of metric tensor in a three-dimensional domain in Cartesian coordinates      9 36
Linearized change of metric tensor in a three-dimensional domain in curvilinear coordinates      36
Linearized change of metric tensor in Cartesian coordinates      9
Linearized change of metric tensor in curvilinear coordinates      36
Linearized change of metric tensor on a surface      91 170 181 210 228 308 321 338 344 366 452 526
Linearized displacement-traction problem      10
Linearized elasticity (three-dimensional linearized elasticity) in Cartesian coordinates      8 10 455
Linearized elasticity (three-dimensional linearized elasticity) in curvilinear coordinates      32 38 52
Linearized elasticity in Cartesian coordinates      10
Linearized elasticity in curvilinear coordinates      37
Linearized equations of equilibrium in Cartesian coordinates      12
Linearized equations of equilibrium in curvilinear coordinates      40
Linearized inextensional displacement      162 303
Linearized Kirchhoff — Love assumption      336 360 368 372
Linearized principle of virtual work in Cartesian coordinates      11
Linearized principle of virtual work in curvilinear coordinates      40
Linearized rotation field      364 376
Linearized strain      see also "Change of metric"
Linearized strain in Cartesian coordinates      9
Linearized strain in curvilinear coordinates      36
Linearized strain tensor      9 36
Linearized strain tensor in Cartesian coordinates      9
Linearized strain tensor in curvilinear coordinates      36
Linearized strains in Cartesian coordinates      9 36
Linearized strains in curvilinear coordinates      36
Linearized stress in Cartesian coordinates      11
Linearized stress in curvilinear coordinates      38
Linearized stress tensor      38
Linearized stresses in Cartesian coordinates      11
Linearized stresses in curvilinear coordinates      38 237
Linearized transverse shear strain tensor      366
Linearized transverse shear tensor      366
Linearly elastic flexural shell      111 541

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