The continuity of dislocations

Abstract

Dislocation continuity is derived from the Bilby–Kondo theory of dislocations using exterior calculus. Dislocation density is represented by the torsion vector-valued two-form. Burgers vectors are associated with the vector part of the torsion while dislocation lines are associated with the two-form part. The exterior derivative of the torsion is shown to vanish when the crystal curvature vanishes. This implies two simultaneous continuity conditions: Burgers vector conservation and continuity of dislocation lines. On the other hand, dislocation continuity is violated when the curvature does not vanish. Since this can occur on grain boundaries it is inferred that grain boundaries are regions where crystal curvature is concentrated.