On the Role of Entropy Flux and Entropy Production in the Modeling of Shape Memory Alloys

A thermodynamically consistent model of shape memory alloys is developed for a body in a uniaxial setting under a tensile stress. The evolution properties are described using the temperature, the martensite fraction, and the stress as independent variables. The innovative approach is based on a general form of the Clausius-Duhem inequality (really, an equality) where the entropy flux and the entropy production rate are given by constitutive functions. Thermodynamic restrictions and a suitable splitting of the entropy and deformation functions transform the Clausius-Duhem inequality into an evolutionary partial differential equation. As a result, both temperature-induced and stress-induced phase transitions and their related hysteretic loops are carefully modelled by properly choosing the free energy, dynamic functions, and the entropy production rate. Furthermore, a region of equilibrium states follows from a stationary condition on the free energy. Next, a generalization is given by letting the constitutive function depend on appropriate gradients within a Lagrangian and an Eulerian formulation. Both formulations are allowed by the occurrence of the extra-entropy flux that turns out to be proportional to the pertinent rates of temperature, stress, and mass fraction.