An elastic theory of dislocations

Abstract

A theory is presented with the intention of describing the internal stresses set up by slip processes in crystalline materials. It is based on the concept of dislocations as they occur in the mathematical theory of elasticity, and follows on the work of Volterra. The present theory starts from the assumption that only the three stress components which act across a surface in an elastic body need be continuous at that surface. This less restrictive condition on the stresses makes possible a more general type of dislocation, one with arbitrary elastic discontinuities of displacement. The theory is stated in cylindrical polar co-ordinates, and general formulae are derived for calculating the stresses corresponding to given dislocations. Four examples are solved completely involving plane and screw dislocations in circular cylinders. The relations between the mathematical dislocations and some simple types of slip in crystals are described, and certain equivalent systems of dislocations explained. It is shown that if the slip bands in a deformed crystal are replaced by the appropriate dislocations, the stresses in the remaining elastic region may be calculated by means of this theory.