Topics in Microstructure: Compatibility and Convergence

In the design of shape-memory alloys for use in applications, significant attention is given to preventing functional fatigue, whereby the material begins to perform significantly worse after just a few deforming-heating-cooling cycles.

This functional fatigue arises due to imperfections introduced in the crystal lattice on cooling, by way of incompatibility between the lattices of the cooler martensite phase and hotter austenite phase. Consequently, research focuses on improving compatibility between the phases.

We discuss a property of the lattice known as the triplet condition, recently considered in an alloy developed by the group of Tomonari Inamura at the Tokyo Institute of Technology.

In particular, we use the framework of nonlinear elasticity to explain why materials satisfying the triplet condition enjoy greater compatibility than their contemporaries.

In studying this problem, we touch upon poorly understood problems in nonlinear elasticity and the calculus of variations, such as multiwell differential inclusion problems and weak notions of curvature.

Additionally, we investigate a dynamical problem in 1-dimensional nonlinear elasticity using techniques from gradient flows on probability spaces.