Ryan S. Elliott
Department of Aerospace Engineering
Bifurcation and Stability of Multilattices
with Applications to Martensitic Transformations in
Shape Memory Alloys
Some of the most interesting and technologically important solid--solid transformations are the first order diffusionless transformations that occur in certain ordered multi-atomic crystals. These include the reconstructive martensitic transformations (where no group--subgroup symmetry relationship exists between the phases) found in steel and ionic compounds such as CsCl, as well as the thermally-induced, reversible, proper (group--subgroup relationships exist) martensitic transformations that occur in shape memory alloys such as NiTi. Shape memory alloys are especially interesting, for engineering applications, due to their strongthermomechanical (multi-physics) coupling. The mechanism responsible for these temperature-induced transformations is a change in stability of the crystal's lattice structure as
the temperature is varied.
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 (Bloch-wave method). 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.