## Asymptotic analysis of linearly elastic shells. I: “Membrne-dominated” shells. (Analyse asymptotique des coques linéairement élastiques. I: Coques “membranaires”.)(French)Zbl 0823.73041

Summary: We consider a family of linearly elastic shells, clamped along their entire lateral surface, all having the same middle surface $$S = \varphi (\overline \omega)$$, where $$\omega \subset \mathbb{R}^ 2$$ is a bounded, connected, open set with a smooth boundary and $$\varphi : \overline \omega \to \mathbb{R}^ 3$$ is a smooth mapping. We assume that, as the thickness $$2 \varepsilon$$ approaches zero, the applied body force density is $$O(1)$$ with respect to $$\varepsilon$$, and finally, we make an essential geometrical assumption on the middle surface $$S$$, which is satisfied if $$S$$ is uniformly elliptic. We then show that, as $$\varepsilon \to 0$$, the three covariant components of the displacements of the points of the shell, once defined over the fixed open set $$\Omega = \omega \times ]-1,1[$$, converge in $$L^ 2 (\Omega)$$ to limits $$u_ i$$ that are independent of the transverse variable $$x_ 3$$, the averages over the thickness of the tangential components converging in addition in $$H^ 1 (\omega)$$. Finally, we show that the averages $${1\over 2} \int^ 1_{-1} u_ i dx_ 3$$ solve the classical bi-dimensional problem of a “membrane-dominated” shell, whose equations, posed over the space $$H^ 1_ 0 (\omega) \times H^ 1_ 0 (\omega) \times L^ 2 (\omega)$$, are therefore justified.

### MSC:

 74K15 Membranes 35Q72 Other PDE from mechanics (MSC2000) 35B40 Asymptotic behavior of solutions to PDEs