It was recalled that it had been pointed out that asymptotic stress fields could be constructed, around the tip of a general mixed mode-I and mode-II crack in elastic perfectly plastic solids, by the use of elastic sectors whose stress magnitude was below the yield stress. When elastic sectors which were below the yield stress existed, the rotation pseudo-stress within an elastic sector was logarithmic singular with distance, r, from the crack tip. It also had to be logarithmic singular within a plastic fan sector that was adjacent to an elastic sector. The areal dislocation density field within a plastic fan adjacent was therefore singular: (r0/r)ln(r0/r). In order to justify the existence of below yield stress elastic sectors, it was necessary to show that the logarithmic singular stress field that was produced by the crack plane surface dislocation distribution was cancelled out by another logarithmic singular stress field; produced by the areal dislocation distributions within the fan plastic sectors. That was carried out explicitly here.

Asymptotic Crack Tip Stress, Rotation Pseudo-Stress and Dislocation Fields for Mixed Mode-I and II Cracks in Elastic Perfectly Plastic Solids. J.Weertman: Mechanics of Materials, 2003, 35[3-6], 433-52