The stability of a rapid mode II shear crack with finite cohesive traction

Abstract

It has been found in numerical simulations by Andrews [1976] that a growing mode II shear crack may at first rapidly accelerate toward the Rayleigh wave speed vR and then after a short period of readjustment start running at a speed slightly greater than (2)½vS, where vS is the shear wave speed. In order to throw some light on this phenomenon we study in detail the steady motion of a semi‐infinite crack which is driven by a following point load which remains at constant distance d from the crack tip. A cohesive traction which is a function of relative displacement acts in the Dugdale type tip region [Dugdale, 1960] which is small in comparison with d. We consider crack speeds vC in the range vC < vP, the P wave speed, and find that for vC < vR and for vS < vC < (2)½vS the load is a decreasing function of vC, while for (2)½vS < vC < vP the load increases with vC. The former situation is interpreted as instability, and the latter as stability. The range vR < vC < vS appears to be forbidden. This is in agreement with, and partially explains, Andrews's results. It also agrees with an earlier analytic solution obtained by Craggs [1960] for vC < vR. We find that for vC < vR the displacement in the tip region has a shape independent of vC but suffers a contraction analogous to a Lorentz contraction so that the length goes to zero as vC ↑ vR.