To model these changes in lattice stability, a continuum-level thermoelastic energy density for a bi-atomic multilattice is derived from a set of temperature-dependent atomic potentials. The Cauchy-Born kinematic assumption is employed to ensure, by the introduction of internal atomic shifts, that each atom is in equilibrium with its neighbors. Stress-free equilibrium paths as a function of temperature are numerically investigated, and an asymptotic analysis is used to identify the paths emerging from "multiple bifurcation" points that are encountered. The stability of each path against all possible bounded perturbations is determined by calculating the phonon spectra of the crystal. The advantage of this approach is that the stability criterion includes perturbations of all wavelengths instead of only the long wavelength information that is available from the stability investigation of homogenized continuum models. The above methods will be reviewed, and results corresponding to both reconstructive and proper martensitic transformations will be presented. Of particular interest is the prediction of a transformation that has been experimentally observed in CuAlNi, AuCd, and other shape memory alloys.