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Statics and dynamics of thin shells. II: Case of inhibited flexion. Membrane approximation. (Statique et dynamique des coques minces. II: Cas de flexion pure inhibée. Approximation membranaire.) (French) Zbl 0712.73056
Summary: We continue the asymptotic study of thin shells of part I [ibid. 309, No. 6, 411–417 (1989; Zbl 0697.73051)]. We characterize the subspace \(G\) of pure flexion (for which the Riemannian metrics of the mean surface remains invariant). \(G\) reduces to the null displacement if there is a uniqueness theorem for a certain boundary value problem. In such a case, the membrane approximation (where the flexion energy is neglected) is justified for the evolution and spectral problems. According to the fact that the uniqueness theorem concerns a well-posed or an ill-posed problem, the origin does not belong or belongs to the essential spectrum. In the former case, the membrane approximation is justified for the static problem. In the later case the asymptotic problem remains open for the static problem.

74K25 Shells
74H45 Vibrations in dynamical problems in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
49R50 Variational methods for eigenvalues of operators (MSC2000)
35R25 Ill-posed problems for PDEs
35B25 Singular perturbations in context of PDEs