A thermodynamical formulation for the constitutive modeling of a shape memory alloy with two martensite phases

This paper presents a thermodynamical formulation for the one-dimensional constitutive model for shape memory alloys (SMAs) proposed by the authors in Marfia and Rizzoni (Eur J Mech A Solids 40:166–185, 2013) and able to describe the pseudo-elastic and shape memory effects and the martensite detwinning. The model takes into account the asymmetric behavior in tension and compression and the different elastic properties of the three phases considered for the SMA material: austenite, tensile and compressive martensite. A new formulation based on two specific energy potentials, the Helmholtz and the Gibbs free energies, is proposed. For the two potentials an expression is given, depending on the martensite volume fractions taken as internal variables, and incorporating a mixing energy of the three phases as proposed in Fre´mond (C R Acad Sci Paris 304:239–244, 1987). An original analysis of the non dissipative and dissipative processes is carried out in the general framework of tension-compression asymmetry and different elastic properties of the three phases; in particular, in the dissipative case the nonnegativity of the dissipation is used to restrict evolutive processes. The numerical procedure developed in Marfia and Rizzoni (Eur J Mech A Solids 40:166–185, 2013) is applied to time integrate the evolutive equations of the internal variables. Applications are carried out in order to verify the effectiveness of the proposed model and to compare the numerical results of the model with the experimental results, available in the literature.