Stability of dispersive bi-atomic crystals

Understanding thermoelastic martensitic transformations is a fundamental component in the study of shape memory alloys. These transformations involve a hysteretic change in stability of the crystal lattice between an austenite (high symmetry) phase and a martensite (low symmetry) phase within a small temperature range. In previous work, a continuum energy density W(U; θ) (as a function of the right stretch tensor U and temperature θ) for a perfect bi-atomic crystal was derived based on temperature-dependent atomic pair-potentials. For this model, only high symmetry cubic configurations were found to be stable (local energy minimizers). The present work derives an energy density W(U,P(1),P(2), θ) that explicitly accounts for a set of internal atomic shifts P(i). In addition, the model permits the calculation of the crystal's dispersion relations which determine the stability of the crystal with respect to bounded perturbations of all wavelengths (Blochwaves). Using a specific model of a bi-atomic crystal with the temperature serving as the loading parameter, a stress-free bifurcation diagram is generated. Stable equilibrium branches corresponding to the B2 (cubic) and B19 (orthorhombic) crystal structures are found to exist and overlap for certain temperatures. The group-subgroup relationship between these two crystal structures is necessary for the shape memory effect. Thus, our results are consistent with the transformations that occur in shape memory alloys such as AuCd and NiTi.