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.


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