Crack-tip process zone as a bifurcation problem

Abstract

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.

Phase transformations, fracture, and other structural changes in inelastic materials

“…In this case, the PT conditions are the same as for elastic materials, which does not explain the strong influence of plastic strain on PT. In this case, the plastic dissipation at the interface is absent, and our driving force (35) for isothermal processes coincides with the Eshelby driving force [93], i.e., like for elastic materials.…”

“…After some transformations that include universal conditions (2)-(4) for a coherent interface, one arrives at Eqs. (35) and (36) that determine the thermodynamic driving force for a phase interface propagation and the corresponding dissipation rate. Note that for evaluation of the integrals in Eq.…”

“…Besides, stresses produced in the course of a PT may cause fracture, in particular during cyclic loading of SMAs [156]. PFA has been widely used for modeling fracture (see Section 19.5) and PTs (see Sections 11.4,11.6 and 16), and their interaction [34,35,48,73,329,397,473]. However, only several papers [73,177,338,397,473] study both fracture and PT within the PFA.…”

Shape transformation of a wake following the process zone at the tip of a propagating crack

“…This often generates a process zone, a nano-to micro-scale region possessing specific properties. Such a zone can often be regarded as a phase of the solid, different from the one in the bulk [1,2]. It will be referred to here as the transfromational process zone (TPZ).…”

“…We develop an alternative approach within which the zone is described not only by the strain field, ε, but also by an internal variable, referred to as the order parameter, η, describing the difference between the zone and the matrix [1,2,24,30,31]. So far, the formation and dynamics of the wake have not been addressed.…”