Dislocation pile-ups, slip-bands, ellipsoids, and cracks

The classic theories of dislocation pile-ups, initiated by Eshelby, Frank and Nabarro, and by Leibfried, can be greatly simplified if it is recognised that the dislocations in the pile-up will experience uniform stress if they are lodged in an ellipsoidal interface. Elementary algebra then produces the familiar results from continuum theory. It seems possible that the ellipsoid construction may represent physical reality if it is taken to represent a three-dimensional slip-band. If so, there are concentrated forces spreading the band perpendicular to the slip band as well as parallel to it. Such ellipsoids also represent Mode II and Mode III cracks, and give a method for dealing with the more complicated Mode I cracks.