Equivalent transformation strain and its relation with martensite volume fraction for isotropic and anisotropic shape memory alloys

The present paper deals with the superelastic behavior of both isotropic and anisotropic shape memory alloys (SMA). Recently, a macroscopic model, which permits to simulate the superelasticity of SMA under complex multi-axial loading, has been proposed by Bouvet et al. [Bouvet, C., Calloch, S., Lexcellent, C., 2004. A phenomenological model for pseudoelasticity of shape memory alloys under multi-axial proportional and non-proportional loadings. Eur. J. Mech. A Solids 23, 37–61]. In this model, a conjecture concerning the proportionality of the equivalent transformation strain with the martensite volume fraction has been adopted. The main goal of this study is to show the validity of this conjecture when the stress state is multi-axial. In a first part, the case of isotropic SMA is considered. An equivalent stress and an equivalent transformation strain are introduced. In the second section, the case of anisotropic SMA is considered. The previous equivalent stress and equivalent transformation strain are generalized to take into account the anisotropy of the material. The relation between the equivalent transformation strain and the martensite volume fraction is discussed by using, on one hand, experimental results under proportional tension-torsion loadings and, on the other hand, a polycrystalline model.