Ryan S. Elliott

Department of Aerospace Engineering

The

*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.