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Asymptotic analysis of linearly elastic shells: III. A justification of W. T. Koiter’s model. (Analyse asymptotique des coques linéairement élastiques. III: Une justification du modèle de W. T. Koiter.) (French) Zbl 0837.73040

[For parts I, II see the authors, ibid. 318, No. 9, 863-868 (1994; Zbl 0823.73041) and 319, No. 1, 95-100 (1994; Zbl 0819.73043).]
We consider a family of linearly elastic shells with thickness \(2\varepsilon\) approaching zero, all having the same middle surface \(S\). For all \(\varepsilon>0\), let \(u_i^\varepsilon\) denote the covariant components of the displacement of the points of the shell, as given by three-dimensional elasticity; let \(\zeta_i^\varepsilon\) denote the covariant components of the displacement of the points of \(S\), as given by W. T. Koiter’s two-dimensional model. We show that the averages across the thickness \((1/2\varepsilon) \int^\varepsilon_{-\varepsilon} u_i^\varepsilon dx_3^\varepsilon\) and the functions \(\zeta_i^\varepsilon\) have the same principal part with respect to powers of \(\varepsilon\), whether it be in the “membrane-dominated”, or in the “bending-dominated” case. W. T. Koiter’s model is thus “at least as good” as the limit model given by the asymptotic analysis of the three- dimensional model.

MSC:

74K15 Membranes
74B05 Classical linear elasticity
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