Propagation stresses in phase transitions of an SMA wire: New analytical formulas based on an internal-variable model

In this paper, we study the stress-induced isothermal phase transitions of a shape memory alloy wire, with a focus on homogeneous or piecewise homogeneous deformations. Based on the constitutive model in the literature with the specific Helmholz free energy and rate of mechanical dissipation, the three-dimensional model is formulated. Identifying the characteristic axial strain as the small parameter, we arrive at the asymptotic one-dimensional equation which involves the stress, the axial strain and the phase state variable. By considering the evolution law of the phase state variable, the stress–strain relations corresponding to austenite, martensite and phase transition (phase mixture) regions are obtained. Although we take the dissipation into consideration, we find that each of the pure loading and unloading processes is equivalent to the Ericksen’s bar problem with three branches in the stress–strain curve. As a result, we successfully deduce the analytical formulas for the nucleation stresses and propagation stresses. The analytical results reveal explicitly how such important quantities depend on the material constants and the temperature. We also demonstrate that they capture a number of features observed in experiments. As an important application, we show that these formulas can be used for calibration of the material constants (such as the difference of the thermal free energies of two phases) by comparing with the measured nominal stress–strain curves. Three examples are also provided.