Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry

Abstract When describing a crystal containing an arbitrary distribution of dislocation lines it is often convenient to treat the distribution as continuous, and to specify the state of dislocation as a function of position. Formally, however, there is then no ‘good crystal’ anywhere, and difficulties arise in defining Burgers circuits and the dislocation tensor. The dislocated state may be defined precisely by relating the local basis at each point to that of a reference lattice. The dislocation density may then be defined; it is important to distinguish this from the local dislocation density. The geometry of the continuously dislocated crystal is most conveniently analyzed by treating the manifold of lattice points in the final state as a non-Riemannian one with a single asymmetric connexion. The coefficients of connexion may be expressed in terms of the generating deformations relating the dislocated crystal to the reference lattice. The tensor defining the local dislocation density is then the torsion tensor associated with the asymmetric connexion. Some properties of the connexion are briefly discussed and it is shown that it possesses that of distant parallelism, in conformity with the requirement that the dislocated lattice be everywhere unique.