Anti-plane shear cracks in ideally plastic crystals

Cracks in ductile single crystals are analyzed here for geometries and orientations such that two-dimensional states of anti-plane shear constitute possible deformation fields. The crystals are modelled as ideally plastic and yield according to critical resolved shear stresses on their slip systems. Restrictions on the asymptotic forms of stress and deformation fields at crack tips are established for anti-plane loading of stationary and quasistatically growing cracks, and solutions are presented for several specific orientations in f.c.c. and b.c.c. crystals. The asymptotic solutions are complemented by complete elastic-plastic solutions for stationary and growing cracks under small scale yielding, based on previous work by Rice (1967, 1984) and Freund (1979). Remarkably, the plastic zone at a stationary crack tip collapses into discrete planes of displacement and stress discontinuity emanating from the tip; plastic flow consists of concentrated shear on the displacement discontinuities. For the growing crack these same planes, if not coincident with the crack plane, constitute collapsed plastic zones in which velocity and plastic strain discontinuities occur, but across which the stresses and anti-plane displacement are fully continuous. The planes of discontinuity are in several cases coincident with crystal slip planes but it is shown that this need not be the case, e.g., for orientations in which anti-plane yielding occurs by multi-slip, or for special orientations in which the crack tip and the discontinuity planes are perpendicular to the activated slip plane.