Planar distributions of dislocations III. The exact positions of important dislocations when the number in an array is large

Abstract

It is shown that if large numbers of complete dislocations are piled-up against a coplanar barrier dislocation, the equilibrium positions occupied by those discrete dislocations in the immediate vicinity of the barrier can be found directly from the characteristics, in the barrier’s neighbourhood, of the function representing the situation where all the dislocations are smeared into a continuous distribution, assuming that they have the same Burgers vector. Since such characteristics are readily obtained by examining the relevant singular integral equation for the model, determination of the important dislocation positions becomes a simple procedure. The approach is exact and refers to the limiting situation where the number of dislocations is very large; on the other hand, the smeared-discrete compromise approach (described in earlier papers in this series), in which the important dislocations remain discrete while the remainder are smeared into a continuous distribution, is approximate but is applicable to the more general situation where the number of dislocations is sufficiently large for the distance between the important dislocations to be small compared with the array length.