The stresses in the plastic region near the crack tip in a composite material of two different bonded materials, with an interface debonding zone being free of tractions, have been studied. The constituent materials were assumed to be elastic-perfectly plastic materials in this study. It was shown that, generally, this problem had a unique solution; however, for a particular homogeneous case, when there is one material only instead of two different materials, this problem was found to have two solutions. The first (continuous) solution coincides with that found earlier by Prandtl and later modified by Hill and Sokolovsky. Additionally, a second (discontinuous) solution, unnoticed by Prandtl, Hill, and Sokolovsky, was unveiled. Some reasons are suggested for prefering the second solution, since evidently both solutions cannot be valid simultaneously.