  We study the geometry of needle-shaped domains in shape-memory alloys.
Needle-shaped domains are ubiquitously found in martensites around macroscopic
interfaces between regions which are laminated in different directions, or
close to macroscopic austenite/twinned-martensite interfaces. Their geometry
results from the interplay of the local nonconvexity of the effective energy
density with long-range (linear) interactions mediated by the elastic strain
field, and is up to now poorly understood. We present a two-dimensional shape
optimization model based on finite elasticity and discuss its numerical
solution. Our results indicate that the tapering profile of the needles can be
understood within finite elasticity, but not with linearized elasticity. The
resulting tapering and bending reproduce the main features of experimental
observations on NiAl.
