Mathematical Elasticity, Vol III, Theory of Shells

Front Cover......Page 1
Mathematical Elasticity: Theory of Shells......Page 4
Copyright Page......Page 5
Table of Contents......Page 34
Mathematical Elasticity: General plan......Page 3
Mathematical Elasticity: General preface......Page 6
Preface to Volume I......Page 10
Preface to Volume II......Page 16
Preface to Volume III......Page 28
Differential geometry at a glance......Page 40
Three-dimensional elasticity in curvilinear coordinates at a glance......Page 48
Two-dimensional linear shell equations at a glance......Page 52
Two-dimensional nonlinear shell equations at a glance......Page 58
PART A: LINEAR SHELL THEORY......Page 63
Introduction......Page 65
1.1 Three-dimensional linearized elasticity in Cartesian coordinates......Page 67
1.2 Curvilinear coordinates and metric tensor in a three- dimensional domain......Page 74
1.3 The variational equations of three-dimensional linearized elasticity in curvilinear coordinates......Page 84
1.4 Covariant derivatives and Christoffel symbols in a three- dimensional domain......Page 94
1.5 Linearized change of metric tensor in curvilinear coordinates......Page 97
1.6 The boundary value problem of three-dimensional linearized elasticity in curvilinear coordinates......Page 99
1.7 A lemma of J. L. Lions; three-dimensional Korn's inequalities and infinitesimal rigid displacement lemma in curvilinear coordinates......Page 102
1.8 Existence and uniqueness theorem in curvilinear coordinates......Page 112
1.9 Complement: Recovery of a three-dimensional manifold from its metric tensor field......Page 116
Exercises......Page 118
Introduction......Page 123
2.1. Curvilinear coordinates and metric tensor on a surface......Page 125
2.2. Curvature tensor on a surface......Page 135
2.3. Covariant derivatives and Christoffel symbols on a surface......Page 147
2.4. Linearized change of metric tensor on a surface......Page 152
2.5. Linearized change of curvature tensor on a surface......Page 155
2.6. Inequalities of Korn's type and infinitesimal rigid displacement lemma on a general surface......Page 162
2.7. Inequality of Korn's type and infinitesimal rigid displacement lemma on an elliptic surface......Page 179
2.8 Complement: Recovery of a surface from its metric and curvature tensor fields......Page 192
Exercises......Page 194
Introduction......Page 199
3.1. The three-dimensional equations of a linearly elastic shell......Page 203
3.2. The three-dimensional equations over a domain independent of ε......Page 211
3.3. Geometrical and mechanical preliminaries......Page 216
3.4. The two-dimensional equations of linearly elastic "membrane" and "flexural" shells derived by means of a formal asymptotic analysis......Page 223
3.5. Summary of the convergence theorems......Page 245
Exercises......Page 252
Introduction......Page 255
4.1. Linearly elastic elliptic membrane shells: Definition, example, and assumptions on the data; the three- dimensional equations over a domain independent of ε......Page 258
4.2. Averages with respect to the transverse variable......Page 263
4.3. A three-dimensional inequality of Korn's type for a family of linearly elastic elliptic membrane shells......Page 267
4.4. Convergence of the scaled displacements as ε → 0......Page 271
4.5. The two-dimensional equations of a linearly elastic elliptic membrane shell existence, uniqueness, and regularity of solutions formulation as a boundary value problem......Page 285
4.6. Justification of the two-dimensional equations of a linearly elastic elliptic membrane shell; commentary and refinements......Page 292
Exercises......Page 297
Introduction......Page 303
5.1. Linearly elastic generalized membrane shells: Definition and assumptions on the data the three-dimensional equations over a domain independent of ε......Page 307
5.2. Analytical preliminaries......Page 310
5.3. A three-dimensional inequality of Korn's type for a family of linearly elastic shells......Page 320
5.4. Generalized membrane shells of the first and second kinds......Page 323
5.5. Admissible applied forces......Page 326
5.6. Convergence of the scaled displacements as ε → 0......Page 328
5.7. The two-dimensional equations of a linearly elastic generalized membrane shell; existence and uniqueness of solutions......Page 349
5.8. Justification of the two-dimensional equations of a linearly elastic generalized membrane shell; examples, commentary, and refinements......Page 353
Exercises......Page 359
Introduction......Page 361
6.1. Linearly elastic flexural shells: Definition, examples, and assumptions on the data; the three-dimensional equations over a domain independent of ε......Page 364
6.2. Convergence of the scaled displacements ase ε → 0......Page 370
6.3. The two-dimensional equations of a linearly elastic flexural shell; existence and uniqueness of solutions......Page 379
6.4. Justification of the two-dimensional equations of a linearly elastic flexural shell; commentary and refinements......Page 385
Exercises......Page 390
Introduction......Page 395
7.1. The two-dimensional Koiter equations for a linearly elastic shell: Existence, uniqueness, and regularity of solutions; formulation as a boundary value problem......Page 397
7.2. Justification of Koiter's equations for all types of linearly elastic shells......Page 407
7.3. Koiter's equations: Additional commentary and bibliographical notes......Page 422
7.4. The two-dimensional Naghdi equations for a linearly elastic shell; existence and uniqueness of solutions......Page 425
7.5. Other linear shell theories......Page 429
7.6. Linear shallow shell theories......Page 431
Exercises......Page 434
PART B: NONLINEAR SHELL THEORY......Page 441
Introduction......Page 443
8.1. Three-dimensional nonlinear elasticity in Cartesian coordinates......Page 448
8.2. Three-dimensional nonlinear elasticity in curvilinear coordinates......Page 454
8.3. The three-dimensional equations of a nonlinearly elastic shell......Page 465
8.4. The three-dimensional equations over a domain independent of ε......Page 469
8.5. Geometrical and mechanical preliminaries......Page 473
8.6. The method of formal asymptotic expansions......Page 475
8.7. The leading term is of order zero......Page 477
8.8. Identification of a two-dimensional variational problem satisfied by the leading term......Page 486
Exercises......Page 492
Introduction......Page 495
9.1 Nonlinearly elastic membrane shells: Definition, examples, and assumptions on the data......Page 498
9.2 The two-dimensional equations as a variational problem......Page 505
9.3 The two-dimensional equations as a minimization problem......Page 507
9.4 The two-dimensional equations of a nonlinearly elastic membrane shell derived by means of a formal asymptotic analysis commentary......Page 509
9.5 The two-dimensional equations of a nonlinearly elastic membrane shell derived by means of r-convergence theory commentary......Page 515
Exercises......Page 527
Introduction......Page 531
10.1. Identification of a two-dimensional variational problem satisfied by the leading term when there are nonzero admissible inextensional displacements......Page 534
10.2. Nonlinearly elastic flexural shells: Definition, examples, and assumptions on the data......Page 564
10.3. The two-dimensional equations as a variational problem......Page 569
10.4. The two-dimensional equations as a minimization problem......Page 581
10.5. The two-dimensional equations of a nonlinearly elastic flexural shell derived by means of a formal asymptotic analysis commentary......Page 583
10.6. Existence of solutions to the minimization problem Exercises......Page 588
11.1. The two-dimensional Koiter equations for a nonlinearly elastic shell......Page 607
11.2. Other nonlinear shell theories......Page 611
11.3. Nonlinear shallow shell theories......Page 614
References......Page 619
Index......Page 645