Extracting a kinetic relation from the dynamics of a bistable chain

Abstract

We integrate Newton's second law for a chain of masses and bistable springs with a spinodal region with the goal of extracting a kinetic relation for propagating phase boundaries. Our numerical experiments correspond to the impact on a bar made of phase changing material. By reading off the spring extensions ahead and behind the phase boundaries in our numerical experiments, we compute a driving force and plot it as a function of the phase boundary velocity to get a kinetic relation. We then show that this kinetic relation results in solutions to Riemann problems in continuum bars that agree with the corresponding numerical experiments on the discrete mass–spring chain. We also integrate Langevin's equations of motion for the same chain of masses and springs to account for the presence of a heat bath at a fixed temperature. We find that the xt-plane looks similar to the purely mechanical numerical experiments at low temperatures but at high temperatures there is an increased incidence of random nucleation events. Using results from both impact and Riemann problems, we show that the kinetic relation is a function of the bath temperature.