Time-dependent congruences of Volterra-type dislocations were investigated on the basis of generalized Frenet formulae in a Riemannian space. The analysis was applied to congruences of edge and mixed dislocations which were consistent with a continuous distribution of dislocations in an equidistant Riemannian space. In particular, the principal congruences of dislocations and the kinematics of congruences of mixed dislocations with clearly defined local slip planes were considered. It was shown that the geometry of such dislocation congruences led to a class of non-linear evolution equations which described the curvature and torsion of a congruence of curves in a Riemannian space. Additional conditions were imposed on this system of equations in order to describe the evolution of the curvature and torsion of congruences of edge dislocations. In the static case, an expression was obtained for the shear stresses which were required to bend prismatic edge dislocations, having zero torsion, that were located on the totally geodesic crystal surfaces. It was concluded that the congruences of the dislocations were associated with finite self-energy functions.

Congruences of Dislocations in Continuously Dislocated Crystals. A.Trzesowski: International Journal of Theoretical Physics, 2001, 40[3], 727-53