Asymptotic analysis of linearly elastic shells: error estimates in the membrane case

Abstract

International audienceThe linear static problem for a thin shell made of a homogeneous and isotropic elastic material is analyzed. It is assumed that the shell thickness $2\epsilon$ is constant, the neutral shell surface S\coloneq\bftheta(\omega) is bounded connected and elliptic and the shell edge $\Gamma$ is clamped. The solution ${\bf u}(\epsilon)$ of the three-dimensional problem is compared with the solution \bfzeta of the corresponding two-dimensional membrane problem. The following estimate is proved for $\epsilon$ small enough: ||{\bf u}(\epsilon)-\bfzeta ||_ {H^1(\Omega)\times H^1(\Omega)\times L^2(\Omega)}\le C\epsilon^a,\quad a={1\over6},\tag1where $\Omega=\omega\times(-1,\,1)\subset{\bf R}^3$. Here \bftheta\colon\ \omega\to{\bf R}^3 is a mapping of class $C^3$, the external body forces $f^i\in L^2(\Omega)$ with $\partial_{\alpha}f^{\alpha}\in L^2(\Omega)$, and \bfzeta(y,x_3)=\bfzeta(y)\in{\bf H}^2(\Omega). It is also shown that inequality (1) cannot hold with $a>{5\over6}$ even for smooth enough functions $\bf f$