Two Variant Modeling of Shape Memory Materials: Unfolding a Phase Diagram Triple Point

For shape memory materials, the combined effects of hysteresis and phase mixing are described in the absence of stress by the four transformation temperatures: Mf, Ms, As, Af. These temperatures can be viewed as the four-way splitting of a single transformation temperature that applies when both effects are absent. Treating the combined effect of hysteresis and phase mixing in the presence of stress have led to internal variable models in which transformation zones replace abrupt transition boundaries in traditional stress vs. temperature thermodynamic phase diagrams. We discuss how these descriptions emerge from a hysteresis and phase mixing mediated unfolding of an underlying phase diagram for a three species system involving austenite and two variants of martensite. The unfolding concept dictates certain general features of the resulting phase diagram zonal geometry, correlates this geometry with particular transformations and transformation combinations, and provides insight into the evolutionary kinetics that governs these transformations. A complete evolutionary model for arbitrary temperature and single stress component histories can be extracted from knowledge of the purely thermal behavior coupled with material data specification in terms of: transformation strain, austenite and martensite elastic moduli, transformation latent heat, and the start and finish stresses for martensite detwinning. The accompanying evolutionary kinetics is given an incremental formulation that captures shape memory and superelasticity, and also allows for the identification of attracting states associated with periodic stressing. The kinetics consistently permits an algebraic description of transformation progression for phase species undergoing depletion, but not for those phase species that are in flux or are increasing under multiple concurrent transformation. This latter result is established by an analysis of the path independent integrability of the evolution kinetics which serves to clarify the resulting description with respect to other models.