An Existence Theorem for Nonlinearly Elastic ‘Flexural’ Shells

Abstract

The two-dimensional equations of a nonlinearly elastic ‘flexural’ shell have been recently identified and justified by V. Lods and B. Miara, by means of the method of formal asymptotic expansions applied to the three-dimensional equations of nonlinear elasticity. These equations can be recast as a minimization problem for a ‘two-dimensional energy’ over a manifold of ‘admissible deformations’. The stored energy function is a quadratic expression in terms of the exact difference between the curvature tensor of the deformed middle surface and that of the undeformed one; the admissible deformations are those that preserve the metric of the undeformed middle surface and satisfy boundary conditions of clamping or of simple support. We establish here that this minimization problem has at least one solution.