The field equations of classical defect theory were re-written using an exterior calculus. Inelastic 1-forms were introduced whose exterior derivatives gave rise to the dislocation-density 2-forms. As an example, a problem with axial and anti-axial dislocation densities of compact support was considered. The problem was split into 2 regions; an inner part where the dislocation density was specified, and an outer part where the dislocation density was zero. Continuity of the mapping function, and a traction vector, were used to connect the 2 solutions. The results could be viewed as representing a continuum approach to the singular field formulation of dislocation theory; using distributed incompatibilities of finite strength.

Matching the Inner and Outer Solutions in the Continuum Theory of Dislocations. D.G.B.Edelen, D.C.Lagoudas: International Journal of Engineering Science, 1999, 37[1], 59-73