Slip bands, ellipsoids, pile-ups and cracks

Abstract

The theory of dislocation pile-ups in a plane was the subject of an epoch-making paper by Eshelby, Frank and Nabarro (1951). The ideas they introduced continue to be of central importance in all discussions of fracture-mechanics and work-hardening, which are themselves the subject of many reviews and text-books. In this paper, we present a version of the theory based on Eshelby's (1957) theory of elastic inclusions. This treatment has the merit of greatly simplifying the essential mathematical steps—or rather, allowing them to occur off-stage, as it were—while enabling the basic formulae and physical ideas to be expounded. In addition to being a possible method of teaching the subject, the theory can form the basis of a realistic theory of slip bands and shows how the results for planar pile-ups follow from a treatment of three-dimensional packets of slip in the limit of confinement to a single plane.