Hysteresis and phase transitions for 1D and 3D models in shape memory alloys

By means of the Ginzburg-Landau theory of phase transitions, we study a non-isothermal model to characterize the austenite-martensite transition in shape memory alloys. In the first part of this paper, the onedimensional model proposed in [3] is modified by varying the expression of the free energy. In this way, the description of the phenomenon of hysteresis, typical of these materials, is improved and the related stressstrain curves are recovered. Then, a generalization of this model to the three dimensional case is proposed and its consistency with the principles of thermodynamics is proved. Unlike other three dimensional models, the transition is characterized by a scalar valued order parameter φ and the Ginzburg-Landau equation, ruling the evolution of φ, allows us to prove a maximum principle, ensuring the boundedness of φ itself.