On the macroscopic free energy functions for shape memory alloys

Most of the models for the macroscopic behaviour of shape memory alloys (SMA) rely upon the assumption of equal material properties of the phases while, on the contrary, experiments show significant differences. On the basis of the variational formulation of the problem governing the behaviour of a linear elastic heterogeneous material with prescribed eigenstrains, macroscopic free energies for SMA are defined taking into account the phase heterogeneity. The general structure and the dependence on the macroscopic state variables of such functions are discussed and formal expressions in terms of proper concentration tensors given. In the case of an underlying two-phase microstructure exact connections between the quantities that determine the free energies (macroscopic transformation strain, interaction energy, effective thermal expansion tensor) and the effective elastic compliance are derived. Estimates of the SMA macroscopic free energies based on Reuss, Voigt and Hashin-Shtrikman bounds for the effective elastic moduli are explicitly calculated and compared in the specific case of a NiTi alloy.