Replacing ordinary derivatives by gauge derivatives in the continuum theory of dislocations to compensate the action of the symmetry group

Abstract

When the symmetry group of a body is continuous it plays a fundamental role on the nonlinear continuum theory of dislocations: it induces a non-uniqueness to the field that describes the defects – the uniform reference – and affects also other fundamental ingredients of the theory. The purpose of the present paper is to examine how certain important quantities of the dislocation theory are affected from symmetry's group action. Apart from the uniform reference we study how the deformation gradient, the first and second Piola–Kirchhoff stress tensors, the elasticities of the material and the momentum equation are affected from the action of the symmetry group. This action is inhomogeneous, namely, differs from point to point. A similar inhomogeneous action of a group may be found in gauge theories. Prompt by the gauge approach, we propose the use of the gauge covariant exterior derivative to compensate for the action of the symmetry group on the uniform reference. The main advantage of using this derivative is that the momentum equation for the static case retains its divergence form. It remains an open question how the Yang–Mills potentials may be determined for the present theory.