Analysis and Mathematical Physics

Remarkable martensitic microstructures are observed in the alloy  $Ti_{76}Nb_{22}Al_{2}$ , which undergoes a cubic to orthorhombic transformation with six martensitic variants $\mathbf U_i=\mathbf U_i^T>0$ having middle eigenvalue $\lambda_2(\mathbf U_i)$ very close to 1. Assuming that $\lambda_2(\mathbf U_i)=1$ there are exactly 12 matrices in the set of energy wells $\bigcup_{i=1}^6SO(3)\mathbf U_i$ that are rank-one connected to $\mathbf 1$. This set of 12 matrices has no rank-one connections.  We attempt to understand the observed microstructures by studying gradient Young measures, exact gradients and $T_N$-configurations supported on these 12 matrices. This is joint work with Tomonari Inamura and Francesco Della Porta.