A kinematically-enhanced relaxation scheme for the modeling of displacive phase transformations

Abstract

In this contribution, a micro-mechanically motivated, energy relaxation-based constitutive model for phase transformation, martensite reorientation and twin formation in shape memory alloys is proposed. The formulation builds on an idealized parametrization of the austenite-twinned martensite microstructure through first- and second-order laminates. To estimate the effective rank-one convex energy density of the phase mixture, the concept of laminate-based energy relaxation is applied. In this context, the evolution of the energetic and dissipative internal state variables, that describe characteristic microstructural features, is computed via constrained incremental energy minimization. This work also suggests a first step towards the continuous modeling of twin formation within the framework of energy relaxation and can be viewed as a generalization of earlier models suggested by Bartel and Hackl (2009) and Bartel et al. (2011). More specifically, in the current model the orientation of martensitic variants in space is not pre-assigned. Variants are rather left free to arrange themselves relative to the martensite-martensite interface in an energy-minimizing fashion, where, however, it is assumed that they form crystallographically-twinned pairs. The formulation also eliminates the need to introduce specific expressions for the Bain strains in each of the martensitic variants, by relating them to a master variant and utilizing the information about their absolute orientation. The predictive capabilities of the proposed modeling framework are demonstrated in several representative numerical examples. In the first part of the results section, the focus is placed on purely energetic analysis, and the particular influence of the different microstructural degrees of freedom on the relaxed energy densities and the corresponding stress-strain responses is investigated in detail. In the second part, macro-homogeneous uniaxial strain and shear loading cases are analyzed for the dissipative case. It is shown, that the proposed model, which, compared to purely phenomenological macro-scale models, has the advantage of strong micro-mechanical motivation, is capable of qualitatively predicting central features of single crystal shape memory alloy behavior, such as the phase diagram in stress-temperature space, and pseudo-elastic and pseudo-plastic responses, while simultaneously providing valuable insight into the underlying micro-scale mechanisms.