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Asymptotic analysis of linearly elastic shells. IV: “Sensitive membrane” shells. (Analyse asymptotique des coques linéairement élastiques. IV: Coques “membranaires sensitives”.) (French) Zbl 0843.73053

Summary: [For parts I, III see the authors, ibid. Sér. I 318, No. 9, 863-868 (1994; Zbl 0823.73041) and Sér. I 319, 299-304 (1994); for part II see the authors and B. Miara, ibid. Sér. I 319, No. 1, 95-100 (1994; Zbl 0819.73043).]
We consider a family of linearly elastic shells indexed by their half-thickness \(\varepsilon\), all having the same middle surface \(S=\varphi(\overline\omega)\), with \(\varphi: \overline\omega\subset \mathbb{R}^2\to \mathbb{R}^3\), and clamped along a portion of their lateral face whose trace on \(S\) is \(\varphi(\gamma_0)\), where \(\gamma_0\) is a fixed portion of \(\partial \omega\) with length \(\gamma_0 > 0\). Let \((\gamma_{\alpha\beta}(\eta))\) be the linearized strain tensor of \(S\). We make an essential geometric and kinematic assumption, according to which the semi-norm \(|\cdot|^M_\omega\) defined by \(|\eta|^M_\omega = \{\sum_{\alpha,\beta}|\gamma_{\alpha\beta}(\eta)|^2_{L^2(\omega)}\}^{1/2}\) is a norm over the space \(\mathbb{V}(\omega)=\{\eta\in \mathbb{H}^1(\omega); \eta=0\) on \(\gamma_0\}\), excluding however the already treated “membrane” case, where \(\gamma_0=\partial\omega\) and \(S\) is elliptic; this assumption is satisfied for instance if \(\gamma_0\neq \partial\omega\) and \(S\) is elliptic, or if \(S\) is a portion of a hyperboloid of revolution. We then show that, as \(\varepsilon\to 0\), the averages across the thickness of the covariant components of the displacement of the points of the shell, strongly converge in the completion of \(\mathbb{V}(\omega)\) for \(|\cdot|^M_\omega\), toward the solution of a “sensitive” membrane variational problem.

MSC:

74K15 Membranes
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