Continuum dynamics of solid-solid phase transitions

Abstract

This work focuses on the applications in dynamics of recently developed continuum-mechanical models of solid-solid phase transitions. The dynamical problems considered here involve only one space coordinate, and attention is limited to hyperelastic materials that involve two phases. This investigation has two purposes. The first is to determine the predictions of the models in complicated situations. Secondly, the present study attempts to develop analytical and numerical approaches to problems that may be relevant to the interpretation and understanding of experiments involving phase transitions under dynamical conditions. The first problem studied involves the study of a semi-infinite bar initially in an equilibrium state that involves two material phases separated by a phase boundary at a given location. The end of the bar is suddenly subject to a constant impact velocity that persists for a finite time and is then removed. Interaction between the phase boundary and the elastic waves generated by the impact and subsequent reflections are studied in detail, and the trajectory of the phase boundary is determined exactly. The second task addressed involves the development of a Riemann solver to be applied to the numerical solution of Riemann problems for two-phase elastic materials. Riemann problems for such materials involve complications not present in the corresponding problems that arise, for example, in classical gas dynamics. Finally, a finite-difference method of Godunov type is developed for the numerical treatment of boundary-initial-value problems arising in the model of Abeyaratne and Knowles. The method is applied to specific problems.