Material and spatial gauge theories of solids—II. Problems with given material dislocation densities

Abstract

Effects of material defects that arise from local action of the material translation symmetry group are analyzed. These problems are uncoupled by the assumptions that only the material translation group acts locally and that the material dislocation density 2-forms (material Cartan torsion) are both specified and 2-plane supported (elementary). All essential geometric quantities are evaluated. Since the material dislocation densities are specified, the only dynamical variables are the three displacement functions of the deformed body. Explicit forms of the dynamic equilibrium equations are obtained. These give new definitions of effective linear momentum and effective stress. Exact solutions of the field equations reproduce the stress field of straight screw dislocations, but the stress field for straight edge dislocations only obtain in the weak defect field limit. Both of these classes of problems arise through the local action of spatial material translations. The governing equations for problems with local action of the material time translation group are derived. They are shown to yield stress distributions that depend on the velocity components. Solutions of the field equations indicate that such models can represent relaxation phenomena.