Computation of dynamical phase transitions in solids

Abstract

This paper deals with the dynamics of phase boundaries in a nonlinear elastic two-phase material. We consider the elasticity system in 1D and the equations of anti-plane shear motion in 2D, where effects of viscosity and capillarity are neglected. These first-order conservation laws allow to represent phase boundaries as shock-like sharp interfaces. However, in contrast to what is known for homogeneous materials, the entropy inequality does not select a unique solution, and an additional criterion, the so-called kinetic relation, is required. Based on a scheme introduced by Hou, Rosakis and LeFloch [T. Hou, Ph. Rosakis, P.G. LeFloch, A level-set approach to the computation of twinning and phase-transition dynamics, J. Comput. Phys. 150 (1999) 302–331] we focus on the numerical solution of a specific model system. Using a level-set technique to enforce the kinetic relation on the discrete level leads to a reformulation of the original system in the form of a system of conservation laws coupled to a Hamilton–Jacobi equation for each phase boundary. The numerical method for the reformulated system is constructed for unstructured meshes (in 2D), and a self-adaptive algorithm is introduced. In the 1D-case we show that the reformulated system has a solution that corresponds to exact dynamical phase boundaries of the elasticity system which obey the kinetic relation. To validate the method in 2D, we present computations on the interaction of a plane wave with a phase boundary. The efficiency of the adaptation mechanism is demonstrated by an example showing the development of microstructures by twinning.