Geometric theory of smooth and singular defects

• We review the theory of de Rham currents as a generalization of differential forms.
• We present continuous dislocations in terms of decomposable differential forms.
• The presence of defects is expressed in terms of the non-vanishing of a form.
• Singular defects are viewed as the non-vanishing of the boundary of a current.
• Frank׳s rules are obtained from the nilpotence of the boundary operator.

A unified theory of material defects, incorporating both the smooth and the singular descriptions, is presented based upon the theory of currents of Georges de Rham. The fundamental geometric entity of discourse is assumed to be represented by a single differential form or current, whose boundary is identified with the defect itself. The possibility of defining a less restrictive dislocation structure is explored in terms of a plausible weak formulation of the theorem of Frobenius. Several examples are presented and discussed.