### Computing Bifurcation and Stability Properties of Crystals

**
Ryan Elliott, University of Minnesota
**

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. To study these transformations, a set of phenomenological temperature-dependent atomic pair-potentials is used to derive the crystal’s energy density *W*(**F**, **S**_{1}, **S**_{2},…,*θ*) as a function of a uniform deformation **F**, a set of internal atomic shift degrees of freedom **S**_{i }, and temperature *θ*. Special attention is paid to the evaluation of crystal structure stability. Using a specific set of temperature-dependent pair-potentials a stress-free bifurcation diagram is generated for the *B*2 binary crystal structure (with temperature serving as the loading parameter). A hysteretic transformation is suggested by the existence of certain stable equilibrium branches corresponding to *B*2 (CsCl) and *B*19 (orthorhombic) crystal structures. These results indicate the ability of temperature-dependent atomic potential models to provide valuable insight into the behavior of shape memory alloys such as NiTi, AuCd, and CuAlNi.

In this talk, I will provide an overview of the above formulation and numerical results. I will then present a description of the numerical techniques which proved to be invaluable for completing this work. These include fast methods for evaluating the stability of a given equilibrium crystal structure, so-called pseudo-arc-length methods for efficiently following equilibrium paths, and the use of projection operators for identifying new equilibrium paths which emerge from bifurcation points.

**Abstract Author(s): **Ryan Elliott<br />2005 Howes Scholar