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Limit process from the three-dimensional elasticity to the asymptotic theory of thin shells. (Passage à la limite de l’élasticité tridimensionnelle à la théorie asymptotique des coques minces.) (French) Zbl 0701.73080
Summary: We perform an asymptotic expansion of the two-scale kind analogous to that of homogenization theory involving a large scale associated with the medium surface and a small scale associated with the thickness $$2\epsilon$$ of the shell. There are two very different asymptotic schemes according to the medium surface admitting whether or not “pure flexions” (i.e. “inextensional” displacements keeping invariant the first fundamental form). In the first case the limit behavior is given by a “pure flexion”, and in the second, by the membrane theory. Consequently, we find again, starting from the three-dimensional elasticity, the results obtained by the author [ibid. 309, No. 6, 411–417 (1989; Zbl 0697.73051); and No. 7, 531–537 (1989)] from the Koiter theory of shells.

MSC:
 74S30 Other numerical methods in solid mechanics (MSC2010) 74P10 Optimization of other properties in solid mechanics 74E05 Inhomogeneity in solid mechanics 74K15 Membranes 74K25 Shells 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)