The complex behavior of shape memory alloys (e.g. NiTi or AgCd) has attracted a lot of attention over the past two decades, partially due to the applicability of these alloys in active structures. When subjected to applied mechanical loading, these materials undergo a martensitic phase transformation from a high-strain parent phase, called austenite, to a low-strain martensite phase. The solid-solid martensitic phase transition is characterized by a lattice deformation that occurs through the correlated migration of phase boundaries. The energy dissipation associated with the motion of phase boundaries leads to a hysteretic behavior called pseudoelasticity. The dynamics of phase boundaries plays an important role in the origin and size of pseudoelastic hysteresis.
Following the pioneering work of Ericksen , it has become common to use a non-monotone, up-down-up stress-strain relation to model the materials undergoing solid-solid phase transitions. The corresponding dynamic initial-value problem possesses a family of solutions and thus is ill-posed (James ). One way to regularize the problem in the dynamic setting is to postulate an additional kinetic relation that expresses the jump in entropy across a phase boundary, called a driving force, as a function of the interface velocity and temperature (Truskinovsky , Abeyaratne and Knowles ). Such a relation and an additional nucleation criterion are sufficient to obtain a unique solution of the dynamic problem.
In this talk I will discuss the derivation of an explicit formula for a kinetic relation governing the motion of a phase boundary in a bilinear thermoelastic material. The relation was obtained by studying traveling wave solutions of a regularized problem that includes viscosity, heat conduction and convection. Both inertia and latent heat of transformation are taken into account. By investigating the effect of material parameters on the kinetic relation, one can calculate the range of parameters for which the relation becomes non-monotone, in agreement with recent numerical and asymptotic calculations (Ngan and Truskinovsky , Turteltaub ). The non-monotone kinetic relations lead to an irregular propagation of phase boundaries, as shown in the work of Rosakis and Knowles  where such non-monotonicity was postulated. The model predicts a nonzero resistance to phase boundary motion and captures local self-heating, in qualitative agreement with recent experiments on shape-memory-alloy wires (Leo, Shield and Bruno , Shaw and Kyriakides ). If time permits, I will also discuss the recent numerical simulations of phase transitions in a finite thermoelastic bar undergoing time-dependent displacement-controlled loading.