Statics and dynamics of thin shells. I: Case of non-inhibited flexion. (Statique et dynamique des coques minces. I: Cas de flexion pure non inhibée.) (French) Zbl 0697.73051

Sanchez-Palencia, Évariste

C. R. Acad. Sci., Paris, Sér. I 309, No. 6, 411-417 (1989).

Summary: We consider thin shells in the context of the linearized theory of Koiter and the variational formulation of M. Bernadou and P. G. Ciarlet [Lect. Notes Econ. Math. Syst. 134, 89–136 (1976; Zbl 0356.73066)]. We consider the subspace \(G\) of the displacements for which the Riemannian metrix of the mean surface remains invariant (pure flexions). We study the asymptotic behavior for thickness tending to zero in the case when \(G\) does not reduce the null element (called of non-inhibited pure flexion). The asymptotic behavior is described by a variational problem in \(G\), where only the flexion energy is involved. The membrane energy appears only indirectly by the Lagrange multiplier associated with the subspace \(G\). We consider the static, dynamic and spectral problems (for small and medium frequencies).

MSC:

74K25	Shells
74H45	Vibrations in dynamical problems in solid mechanics
74S99	Numerical and other methods in solid mechanics
49R05	Variational methods for eigenvalues of operators (should also be assigned at least one other classification number in Section 49-XX)
49J27	Existence theories for problems in abstract spaces

Keywords:

minimization of elastic energy; space of kinematically admissible displacements; linearized theory of Koiter; pure flexions; asymptotic behavior; thickness tending to zero; spectral problems