Nonlinear theory of shallow shells (Applied Mathematical Sciences)

Contents......Page 14
Preface to the English Edition......Page 6
Preface to the Russian Edition......Page 7
From the Editor......Page 11
1. Results from the Theory of Surfaces......Page 18
2. S-Coordinates in Space. Formation of a Shell. Components of Finite Deformation in S-Coordinates and Their Simplification......Page 31
3. The Kirchhoff–Love Hypotheses. Their Mathematical and Mechanical Content. Computation of Deformations of a Shallow Shell Using the Kirchhoff–Love Hypotheses......Page 36
4. Potential Energy of Deformation of a Shallow Shell......Page 43
5. Independent Displacements, Generalized Stresses and the Work of External Forces Under the Kirchhoff–Love Hypotheses......Page 50
6. Boundary Value Problems in Displacements of the Moderate Bending Theory for Shallow Shells......Page 53
7. Boundary Value Problems with Airy Stress Function in the Moderate Bending Theory for Shallow Shells......Page 61
8. Some Remarks on Nonlinear Shallow Shell Theory. A Historical Survey......Page 73
9. Some General Mathematical Results......Page 77
10. General Mathematical Results (Continued)......Page 90
11. The Function Spaces H[sub(t)], t = 5, 6, 7, 8. Properties of Their Elements......Page 99
12. The Function Spaces H[sub(k)], k = 1, 2, 3, 4. Properties of Their Elements......Page 113
13. The Generalized Formulation of Boundary Value Problems in Displacements. Reduction to Operator Equations. The Physical Meaning of Generalized Solutions......Page 124
14. Some Properties of the Operators K[sub(tk)], G[sub(kk)]......Page 133
15. Computation of the Winding Number of the Vector Field w – G[sub(kk)](w) on Spheres of Large Radius in H[sub(k)]: Preliminary Lemmas......Page 140
16. Computation of the Winding Number of the Vector Field w – G[sub(kk)](w) on Spheres of Large Radius in H[sub(k)]. Solvability of the Main Boundary Value Problems in…......Page 147
17. The Generalized Formulation of the Boundary Value Problems of Shallow Shells with an Airy Stress Function. Reduction to Operator Equations. Physical Interpretation…......Page 162
18. Main Properties of the Operators K[sub(9k)](w), G[sub(k)](w)......Page 170
19. Computation of the Winding Number of the Vector Field w – G[sub(k)](w) on Spheres of Large Radius in H[sub(k)]. Solvability of the Main Boundary Value Problems of…......Page 176
20. Differentiability Properties of Generalized Solutions of the Problems tk and 9k. Conditions for the Existence of Classical Solutions......Page 186
21. The Variational Approach to the Problem of Solvability of Boundary Value Problems of Nonlinear Shallow Shell Theory in Displacements......Page 198
22. The Variational Approach to the Problem of Solvability of Boundary Value Problems of Nonlinear Shallow Shell Theory with an Airy Stress Function......Page 212
23. Expansion in Powers of a Small-parameter (Nonsingular Solutions)......Page 223
24. Expansion in Powers of a Small-parameter (Singular Solutions). The Liapunov–Schmidt Method......Page 229
25. The Newton–Kantorovich Method......Page 236
26. Variational Methods for Approximate Solutions of Problems tk (k = 1, 2, 3, 4; t = 5, 6, 7, 8). The Version of Papkovich......Page 246
27. The Bubnov–Galerkin–Ritz Method for Approximate Solution of Problems tk (k = 1, 2, 3, 4; t = 5, 6, 7, 8). The Versions of Mushtari and Vlasov......Page 257
28. Error Estimates for the Bubnov–Galerkin–Ritz (BGR) Method in Some Problems of the Nonlinear Theory of Shallow Shells......Page 268
29. Formulation of the Problem of Stability in the Nonlinear Theory of Shallow Shells. Local Uniqueness of Solutions. Conditions for Global Uniqueness......Page 280
30. Physical Stiffness of Shells. Connection with Geometrical Stiffness of the Middle Surface......Page 290
31. Well-Posedness of Problems of the Nonlinear Theory of Shallow Shells: Its Relation to Physical Stability......Page 298
32. Momentless State of Shells. Passage to the Linearized Problem. Spectral Properties of the Linearized Problem......Page 313
33. Global Stability of Shells in Problems tk. Existence of Lower Critical Numbers. Some Estimates for U-Decompositions......Page 323
34. Global Stability of Shells in Problems 9k. Existence of Lower Critical Values. Some Estimates for U-Decompositions......Page 337
35. Bifurcation of Solutions in a Neighborhood of the Momentless State......Page 342
36. Variational Methods in Global Stability of Shallow Shells......Page 347
37. Some Problems of Global Stability of Plates......Page 353
38. A Probabilistic Model of Operations of a Shallow Shell Under Moderate Bending......Page 360
Some Unsolved Problems of the Mathematical Theory of Shells......Page 370
References......Page 372
List of Symbols......Page 392
B......Page 398
D......Page 399
H......Page 400
N......Page 401
P......Page 402
S......Page 403
T......Page 404
W......Page 405