zbMATH — the first resource for mathematics

An existence theorem for nonlinearly elastic ‘flexural’ shells. (English) Zbl 0929.74067
Summary: We apply two-dimensional equations of a nonlinearly elastic ‘flexural’ shell, by means of the method of formal asymptotic expansions, to the three-dimensional equations of the 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.

74K25 Shells
74B20 Nonlinear elasticity
35Q72 Other PDE from mechanics (MSC2000)
Full Text: DOI