Theory of Plates, Volume II

Mathematical Elasticity: Theory of Plates......Page 4
Copyright Page......Page 5
Mathematical Elasticity: General plan......Page 6
Mathematical Elasticity: General Preface......Page 8
Preface to Volume I......Page 14
Preface to Volume II......Page 20
TABLE OF CONTENTS......Page 32
Main notations and definitions......Page 38
Plate equations at a glance......Page 54
Shallow shell equations at a glance......Page 58
PART A: LINEAR PLATE THEORY......Page 64
Introduction......Page 66
1.1. A lemma of J.L. Lions and the classical Korn inequal- ities......Page 70
1.2. The three-dimensional equations of a linearly elastic clamped plate......Page 77
1.3. Transformation into a problem posed over a domain independent of ε; the fundamental scalings of the unknowns and assumptions on the data; the displacement approach......Page 87
1.4. Convergence of the scaled displacements as ε → 0......Page 95
1.5. The limit scaled two-dimensional flexural and mem- brane equations: Existence, uniqueness, and regularity of solutions; formulation as boundary value problems......Page 110
1.6. Convergence of the scaled stresses as ε → 0; explicit forms of the limit scaled stresses......Page 120
1.7. The two-dimensional equations of a linearly elastic clamped plate; linear Kirchhoff-Love theory......Page 127
1.8. Justification of the linear Kirchhoff-Love theory......Page 135
1.9. Linear plate theories: Historical notes and commen- tary......Page 144
1.10. Justifications of the scalings and assumptions in the linear case......Page 152
1.11. Asymptotic analysis and F-convergence......Page 158
1.12. Error estimates......Page 164
1.13. Eigenvalue problems......Page 167
1.14. Time-dependent problems......Page 175
Exercises......Page 181
Introduction......Page 192
2.1. The three-dimensional equations of a linearly elastic multi-structure......Page 196
2.2. Transformation into a problem posed over two domains independent of e; the fundamental scalings of the unknowns and assumptions on the data......Page 200
2.3. Convergence of the scaled displacements as ε → 0......Page 204
2.4. The limit scaled problem: Existence and uniqueness of a solution; formulation as a boundary value problem......Page 225
2.5. Mathematical modeling of an elastic multi-structure by a coupled, multi-dimensional boundary value problem; junction conditions......Page 230
2.6. Commentary; refinements and generalizations......Page 234
2.7. Justification of the boundary conditions of a clamped plate......Page 243
2.8. Eigenvalue problems......Page 252
2.9. Time-dependent problems......Page 262
Exercises......Page 266
Introduction......Page 270
3.1. The three-dimensional equations of a linearly elastic clamped shell in Cartesian coordinates......Page 274
3.2. Transformation into a problem posed over a domain independent of ε; the fundamental scalings of the unknowns and assumptions on the data......Page 278
3.3. Technical preliminaries......Page 282
3.4. A generalized Korn inequality......Page 286
3.5. Convergence of the scaled displacements as ε → 0......Page 292
3.6. The limit scaled two-dimensional problem: Existence and uniqueness of a solution; formulation as a boundary value problem......Page 301
3.7. Justification of the two-dimensional equations of a linearly elastic shallow shell in Cartesian coordinates......Page 303
3.8. Definition of a “shallow” shell; commentary......Page 307
Exercises......Page 309
PART B: NONLINEAR PLATE THEORY......Page 312
Introduction......Page 314
4.1. The three-dimensional equations of a nonlinearly elastic clamped plate......Page 320
4.2. Transformation into a problem posed over a domain independent of ε; the fundamental scalings of the unknowns and assumptions on the data......Page 327
4.3. The method of formal asymptotic expansions: The displscement approach......Page 331
4.4. Cancellation of the factors of εq,-4 < q < q 5 0, in the scaled three-dimensional problem......Page 333
4.5. Identification of the leading term uo in the displacement approach......Page 339
4.6. The limit scaled two-dimensional problem: Existence and regularity of solutions, formulation as a boundary value problem......Page 345
4.7. The method of formal asymptotic expansions: The displacement-stress approach......Page 356
4.8. Identification of the leading term Σ° in the displace ment-stress approach; explicit forms of the limit scaled stresses......Page 364
4.9. The two-dimensional equations of a nonlinearly elastic clamped plate; nonlinear Kirchhoff-Love theory......Page 376
4.10. Justification of the nonlinear Kirchhoff-Love theory; commentary, refinements and generalizations......Page 384
4.11. Justification of the scalings and assumptions in the nonlinear case......Page 392
4.12. Frame-indifferent nonlinear membrane and flexural the- ories......Page 398
4.13. Frame-indifferent nonlinear membrane theory and F- convergence......Page 411
4.14. Nonlinearly elastic shallow shells in Cartesian coordi- nates......Page 419
Exercises......Page 425
Introduction......Page 430
5.1. The three-dimensional equations of a nonlinearly elastic von Kármán plate......Page 436
5.2. Transformation into a problem posed over a domain independent of ε; the fundamental scalings of the unknowns and assumptions on the data......Page 441
5.3. The method of formal asymptotic expansions: The displacement-stress approach......Page 444
5.4. Identification of the leading term u° the limit scaled "displacement" two-dimensional problem......Page 445
5.5. Identification of the leading term Σ°; explicit forms of the limit scaled stresses......Page 450
5.6. Equivalence of the limit scaled "displacement" problem with the scaled von Kármán equations......Page 451
5.7. Justification of the von Kármán equations of a non- linearly elastic plate; commentary and bibliographical notes......Page 466
5.8. The von Kármán equations: Existence and regularity of solutions......Page 472
5.9. The von Kármán equations: Uniqueness or nonuniqueness of solutions......Page 486
5.10. The von Kármán equations: Degeneracy into the linear membrane equation......Page 491
5.11. The von Kármáns equations: Bifurcation of solutions......Page 496
5.12. The Marguerre-von Kármán equations of a nonlinearly elastic shallow shell......Page 501
Exerciscs......Page 510
References......Page 514
Index......Page 542