Crack-tip process zone as a bifurcation problem

Stress concentration at a crack tip generates a solid structural transformation in its vicinity, the process zone. We argue that its formation represents a local phase transition described by a multicomponent order parameter. We derive a system of equations describing the dynamics of the order parameter driven by an inhomogeneous, time-dependent stress field in the solid and show that it exhibits a bifurcation. The latter corresponds to the emergence of a process zone characterized by the distribution of the order parameter localized in the vicinity of the crack tip. The emergence temperature T_{*} considerably differs from the temperature of the bulk phase transformation T_{c}. We demonstrate that T_{*} exhibits a universal behavior T_{*}-T_{c}∼K_{I}^{4/3}, in terms of the stress intensity factor K_{I}, and that the zone universally vanishes upon achieving a critical velocity. These facts together give rise to a universal dynamic phase diagram.