## 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.

### MSC:

 74K15 Membranes 74B05 Classical linear elasticity