## Bifurcation and Stability of Multilattices with Applications to Martensitic Transformations in Shape Memory Alloys

#### Ryan S. Elliott, *
UMN
*

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 strong
thermomechanical (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. 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.