On the dual formulation of boundary value problems in non-linear elastostatics

Abstract

Boundary value problems of non-linear elastostatics with a dual description in terms of the displacement gradient and the first Piola-Kirchhoff stress tensor are investigated. Methods of functional analysis are applied in order to construct both a variational functional in terms of geometrically admissible displacements, and, for nondegenerate critical points, a dual functional in terms of statically admissible Piola-Kirchhoff stresses. Elastic states are introduced. Their distance is used as an error measure. For the case of a locally convex displacement energy the energy method leads to an equivalent norm. For a locally nonconvex displacement energy only a seminorm is obtained. In this case the energy method fails. Therefore a new bilinear form is introduced, which leads to an equivalent norm. In both cases energy-normlike estimates are obtained.