Asymptotic analysis of linearly elastic shells. II: “Bending-dominated” shells. (Analyse asymptotique des coques linéairement élastiques. II: Coques “en flexion”.) (French) Zbl 0819.73043

A family of linearly elastic shells, clamped on a portion of their lateral surface, all having the same middle surface \(S = \varphi (\overline\omega)\), where \(\overline\omega\subset \mathbb {R}^ 2\) is a bounded, connected, open set with a Lipschitz-continuous boundary, and \(\overline\omega: \varpi \to \mathbb {R}^ 3\) is a mapping of class \(C^ 3\), has been considered. It is assumed that, as the shell thickness \(2\varepsilon\) approaches zero, the applied body force density is \(O (\varepsilon^ 2)\). According to geometric and kinematic assumptions, a certain space \(V_ f (\omega)\) of inextensional displacements, which is a closed subspace of \(H^ 1 (\omega) \times H^ 1 (\omega) \times H^ 2 (\omega)\), is not reducible to \(\{0\}\); the latter assumption is satisfied, if, in particular, \(S\) is a portion of a cylinder clamped along a generatrix. Using beforehand proved Korn inequality, the following statement has been demonstrated: As \(\varepsilon \to 0\), three covariant components of the displacement of the points of the shells, being defined on a fixed open set \(\Omega = \omega \times ]-1,1[\), converge in \(H^ 1 (\Omega)\) to limits \(u_ i\) independent of the transverse variable \(x_ 3\). The averages \({1\over2} \int^{+1} _{-1} u_ i dx_ 3\) solve then the two-dimensional problem for the “bending- dominated” shells, whose equations written on the whole space \(V_ f (\omega)\) are therefore justified.


74K15 Membranes
74B05 Classical linear elasticity