Feeding and dissipative waves in fracture and phase transition: II. Phase-transition waves

Abstract

Discrete and homogeneous models of a structured material are considered to resolve difficulties in the analysis of dynamic phase transition. The discrete model is a chain consisting of particles connected by massless bonds, while the continuous model is represented by a partial differential equation with higher than the second order of coordinate derivatives. The macrolevel constitutive law is represented by a bi-linear stress–strain relation, such that the transition from the first, stiffer phase to the second one is irreversible. Solutions of two types, macrolevel-associated and microlevel, are derived. The first type of solution is characterized by a macrolevel feeding wave (the wave delivering energy to the phase-transition front is of a zero wave number), while the microlevel solutions correspond to a nonzero feeding wave number. Subsonic, intersonic and supersonic phase-transition waves are described. For the homogeneous model it is shown that the contradiction between the limiting stress and energy criteria, inherent for the macrolevel formulation of the problem, is eliminated if and only if the phase transition does not concern the highest-order modulus. Total structure- and speed-dependent dissipation, as the energy carried by microlevel waves away from the phase-transition front, as well as parameters of each dissipative wave are determined. For the fourth-order partial differential equation, the existence of the Maxwell type, dissipation-free, subsonic phase-transition wave is shown. In this case, the microstructure plays the role of a catalyst. Common and distinctive properties of the discrete and homogeneous models are discussed.