Static and dynamic multipolar field expansions of dislocations and cracks in solids

This article provides a comprehensive analysis of the way multipolar field expansions, of common use in the study of point defects in solids, may be extended to study the long range fields of dislocations and cracks. The long range fields of such defects are of relevance in fields as disparate as dislocation dynamics, microcrack and fragmentation, or radiation damage studies. The article provides a general framework for the development of multipolar field expansions in the continuum; one that may be used for any generalised force distribution. The general framework is combined with the Burridge-Knopoff force representation of dislocations and cracks, both in the planar and in the three dimensional cases, to achieve their respective multipolar field expansions for generalised dislocation loop and crack geometries. It is shown that, despite its simplicity, the multipolar field expansions provide a very accurate measure of the far field of both planar and three dimensional dislocations and cracks, and that the accuracy increases as higher order terms (i.e., quadrupolar, octopolar, etc) are introduced into the expansion. The formulation is then extended to the elastodynamic case. Both a spatial-temporal multipolar field expansion and a spatial multipolar field expansion, are developed. The spatial-temporal multipolar field expansion is seen to capture only the leading terms of the elastodynamic fields, whilst the spatial multipolar expansions are seen to be very accurate at capturing the long range field behaviour so long as the characteristic speed of the dislocation or crack are a fraction of the longitudinal speed of sound.