Elastic thin shells: Asymptotic theory in the anisotropic and heterogeneous cases. (English) Zbl 0844.73043

Asymptotic two-scale methods are used to develop a thin shell theory directly from linear three-dimensional elasticity. The asymptotic behavior of elastic heterogeneous shells as the thickness tends to zero is discussed using the variational formulation of the problem. The method worked out gives in the non-inhibited case the leading term of the displacement field which is contained in the subspace \(G\) of the inextensible displacements (pure bendings), and simultaneously an asymptotically smaller term belonging to the space orthogonal to \(G\). Both terms have contributions of the same order to the stress field. In the inhibited case \(G = \{0\}\), and the asymptotic behavior of the stress field can be given by the membrane theory. As in the case of anisotropic heterogeneous plates which exhibit coupling between flexion and traction, heterogeneous shells exhibit coupling between terms in \(G\) and in its orthogonal space.
The paper is organized as follows: sections 2 to 5 are only concerned with laminated shells; in section 6 the general heterogeneous case is considered in the framework of homogenization theory. The main features of the method and the asymptotic procedure are outlined in section 2. Section 3 is devoted to a local study of the stress field and of the corresponding membrane stresses. The description of the limit problem is given in section 4. Section 5 deals with some generalizations.


74K15 Membranes
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