Dislocations in the elastic fields of randomly distributed defects

In recent years, the behavior of dislocations in random solid solutions has received renewed interest, and several models have been discussed where random alloys are treated as effective media containing random distributions of dilatation and compression centers. More generally speaking, the arrangement of defects in metals and alloys is always characterized by statistical disorder, and the same is true for the fluctuating fields that arise from the superposition of the stress and strain fields of many defects. In order to develop models for the dynamics of dislocations interacting with such fields, a statistical description of the dislocation energy landscape and the associated configurational forces is needed, as well as methods for coarse graining these forces over dislocation segments of varying length and shape. In this context the problem arises how to regularize the highly singular stress fields associated with dislocations and other defects. Here we formulate an approach which is based upon evaluating the interaction energies and interaction forces between singular dislocations and other defects including solutes, modelled as point-like dilatation centers, and other dislocations. We characterize the interactions in terms of the probability densities of interaction energies and interaction forces, and the corresponding spatial correlation functions. We also consider the effects of dislocation core regularization, either in terms of continuously distributed Burgers vectors or by using the formalism of gradient elasticity of Helmholtz type to formulate a regularized energy functional. We demonstrate that the stress fields arising from randomly distributed defects obey in general non-Gaussian statistics, and discuss implications for the motion of dislocations.