The geometric equation of dislocation dynamics

Abstract

Recent advances in the continuum theory of dislocations have been achieved mainly in two directions: (1) the differential geometric (non-linear) theory of stationary dislocations, and (2) the formal linear dislocation dynamics. These two are unified here to form a differential geometric dynamical theory of continuous distributions of dislocations. To begin with, the basic concepts of the geometric theory are briefly summed up. The fundamental geometric equation of time-dependent distortions is derived first in a physically instructive elementary way and afterwards by means of exact differential geometry. A symmetrized form of the equation is given in terms of deformations or the metric tensor. The physical meaning of a previously introduced dislocation current tensor is discussed. A general form of the continuity equation for the dislocation current is then given. Thereafter the forces acting on dislocations are dealt with in connection with energy dissipation during plastic deformation and Ohm's law for dislocations, which has been introduced recently. The dislocation conductivity in simple cubic crystals is discussed. Finally, an invariant partition of the torsion (or dislocation) tensor is introduced. The semi-symmetrical part of this tensor corresponds to volume deformations, while the remainder is associated with shape deformations only. The main unsolved problems are enumerated, and some concluding remarks, concerned with the correspondence between dislocation theory on the one hand and general relativity and differential geometry on the other, are added.