Matching the inner and outer solutions in the continuum theory of dislocations

Abstract

The field equations of the classical defect theory are written using exterior calculus, whereby antiexact inelastic 1-forms are introduced whose exterior derivatives give rise to the dislocation density 2-forms. As an example of the formulation, a problem with axial and antiaxial dislocation densities of compact support is considered. The problem is split into two regions, the inner problem where the dislocation density is specified and the outer problem where the dislocation density is zero. Continuity of the mapping functions and traction vector is used to connect the two solutions. The results presented could be viewed as a continuum approach to the singular field formulation of the dislocation theory using distributed incompatibilities of finite strength.