Multipole representation of the elastic field of dislocation ensembles

Abstract

A multipole expansion method is developed to determine the elastic field of dislocation loop ensembles of arbitrary geometric complexity. The method results in reduction of the severe computational requirements in large-scale dislocation dynamics (DD) computer simulations without an artificial cutoff on the interaction range. Order of N, $O(N),$ algorithms for DD simulations is immediately accessible on the basis of the developed procedure. Examples of dislocation interaction with large dislocation arrays representing a tilt boundary and a dislocation wall show that the method results in speeding up the calculation of Peach-Kohler interaction forces by a factor of 100, with an error of less than 0.4%. The multipole expansion reveals a physical connection to Kr\oner's continuum theory of dislocations, with the zeroth order moment being Nye's dislocation density tensor. Higher-order tensors in the expansion correspond to moments of a basic tensor comprised of the tangent and Burgers vectors, and can be used to characterize the spatial distribution of dislocation loop ensembles.