Modelling of localization and propagation of phase transformation in superelastic SMA by a gradient nonlocal approach

In this work, a nonlocal phenomenological behavior model is proposed in order to describe the localization and propagation of stress-induced martensite transformation in shape memory alloy (SMA) wires and thin films. It is a nonlocal extension of an existing local model that was derived from a micromechanical-inspired Gibbs free energy expression. The proposed model uses, besides the local field of the internal variable, namely the martensite volume fraction, a nonlocal counterpart. This latter acts as an additional degree of freedom, which is determined by solving an additional partial differential equation (PDE), derived so as to be equivalent to the integral definition of a nonlocal quantity. This PDE involves an internal length parameter, dictating the global scale at which the nonlocal interactions of the underlying micromechanisms are manifested during phase transformation. Moreover, to account for the unstable softening behavior, the transformation yield force parameter is considered as a gradually decreasing function of the martensite fraction. Possible material and geometric imperfections that are responsible for localization initiation are also considered in this analysis. The obtained constitutive equations are implemented in the Abaqus® finite element code in one and two dimensions. This requires the development of specific finite elements having the nonlocal volume fraction variable as an additional degree of freedom. This implementation is achieved through the UEL user’s subroutine. The effect of martensitic localization on the superelastic global behavior of SMA wire and holed thin plate, subjected to tension loading, is analyzed. Numerical results show that the developed tool correctly captures the commonly observed unstable superelastic behavior characterized by nucleation and propagation of martensitic phase. In particular, they show the influence of the internal length parameter, appearing in the nonlocal model, on the size of the localization area and the stress nucleation peak.