Intrinsic Formulation of the Kirchhoff-Love Theory of Nonlinearly Elastic Shallow Shells

Abstract

The classical formulation of the Kirchhoff-Love theory of nonlinearly elastic shallow shells consists of a system of nonlinear partial differential equations and boundary conditions whose unknowns are the Cartesian components of the displacement field of the middle surface of the shell subjected to applied forces. We show that this system is equivalent to a system whose sole unknowns are the bending moments and stress resultants inside the middle surface of the shell. This system thus provides a direct method for computing the stresses appearing in such a shell, without any recourse to the displacement field. To this end, we first establish specific compatibility conditions of Saint-Venant type for the bending moments and stress resultants; we then identify the boundary conditions that these fields must satisfy.