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## 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: $$||{\bf u}(\epsilon)-\bfzeta ||_ {H^1(\Omega)\times H^1(\Omega)\times L^2(\Omega)}\le C\epsilon^a,\quad a={1\over6},\tag1$$where $\Omega=\omega\times(-1,\,1)\subset{\bf R}^3$. 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 $\bf f$

Topics: Shells, elasticity, asymptotics, error estimates, three-dimensional model, [ MATH ] Mathematics [math], [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], [ MATH.MATH-FA ] Mathematics [math]/Functional Analysis [math.FA], [ MATH.MATH-MP ] Mathematics [math]/Mathematical Physics [math-ph], [ PHYS.MECA.SOLID ] Physics [physics]/Mechanics [physics]/Mechanics of the solides [physics.class-ph]
Publisher: IOS Press
Year: 1998
OAI identifier: oai:HAL:hal-01478594v1
Provided by: Hal-Diderot