Phase transitions in shape memory alloys: A non-isothermal Ginzburg-Landau mode

We propose a model to describe non-isothermal transitions from the austenite to the martensite phase occurring in shape memory materials. The phenomenon is set in the context of the Ginzburg-Landau theory of phase transitions, postulating a free energy depending on the temperature, the stress and the order parameter. In the one-dimensional case, when only two martensitic variants are involved and stress and deformation have a fixed direction, our choice of free energy allows us to deduce a phase diagram describing the main features of a typical SMA. The Ginzburg-Landau equation ruling the evolution of the order parameter is coupled with the equations of thermoelasticity by assuming a constitutive equation relating stress, strain and order parameter. The consistency of the model with the second law of Thermodynamics in the form of the Clausius-Duhem inequality is proved. Finally a possible generalization to a three-dimensional model is proposed, by introducing a tensor-valued order parameter.