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(FS1S2,…,θ) as a function of a uniform deformation F, a set of internal atomic shift degrees of freedom Si , 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 B2 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 B2 (CsCl) and B19 (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