Uniqueness condition for dynamical phase transitions in a shape memory alloy bar

We consider dynamical phase transitions in a shape memory alloy bar. A known difficulty is to determine the solution for a propagating phase boundary uniquely, which has not been completely overcome yet. By adopting a strain-gradient continuum model with an internal variable, we formulate the one-dimensional equations. By considering the travelling-wave solution and using matched asymptotics, we derive a uniqueness condition to resolve this difficulty. We also obtain the stress–strain relations with different loading and unloading paths. We then apply this uniqueness condition to study phase boundary propagation in a shape memory alloy bar/wire subjected to a displacement-controlled tensile loading. We find that at a sufficiently slow loading rate, the stress along the bar is approximately the Maxwell stress, with some modification by the loading rate. We also obtain the relation between the propagation speed of the phase boundary and the loading rate, which is, remarkably, the same as the one in the literature obtained by fitting experimental data and validates the obtained uniqueness condition. Finally, we determine the nucleation time for phase transition analytically, which agrees well with the experimental data.