Comparing the condition of strong ellipticity and the solvability for a purely elastic problem and the corresponding dislocated problem

Abstract

The aim of this paper is a comparison of a purely elastic problem and the corresponding dislocated one at two levels: (a) strong ellipticity; and (b) solvability. The corresponding dislocated problem is defined through Noll’s theory of materially uniform but inhomogeneous bodies, which is a theory of frozen dislocations. The framework for strong ellipticity is general enough; no specific assumption is made for the elastic field, the field of the defects and the elastic energy. Solvability (existence and uniqueness) in suitable spaces is examined only for a specific distribution of edge dislocations with a particular expression for the energy (neo-Hookean like) and the elastic field (anti-plane shear). The outcome is that strong ellipticity is not affected by the field of the defects. Namely, if one starts from a strongly elliptic elastic material, he will arrive at a dislocated one with the same property. For the dislocated problem we find bounds that the field of the defects has to satisfy in order for solvability to be retained when solvability of the corresponding elastic problem is known. The whole framework is designed for the finite hyperelastostatic case.