Planar distributions of dislocations

Abstract

Many problems in metal physics can be adequately described by planar arrays of dislocations, and in analysing them there are two extreme types of approach: either all the dislocations can be regarded as discrete entities, when a numerical procedure must in general be adopted, or they may be smeared into a continuous distribution, when a simple analytical solution is usually available. The present paper introduces a procedure whereby the majority of the dislocations in an array are smeared into a continuous distribution, while those that are intimately concerned with the particular physical phenomenon under consideration are allowed to remain discrete. The main advantage of this approach is that it allows for ease of solution, while preserving a high degree of physical reality. Particular types of dislocation array are considered, the accuracy of the method being demonstrated by comparing, where possible, the results with those obtained assuming that all the dislocations are discrete. Both singular integral equation and potential theory techniques are employed, and the results are applied in a brief discussion of the problem of cleavage crack nucleation in crystalline solids.