Quantitative aspects of the rigidity of branching microstructures in shape memory alloys via H-measures

TL;DR

This paper uses H-measures to quantify the rigidity of branching microstructures in shape memory alloys, providing new estimates on interface clustering and energy density behavior near habit planes.

Contribution

It introduces a $B^{2/3}_{1, obreak ext{,} obreak ext{infinity}}$-estimate for twin characteristic functions, revealing the clustering scale of interfaces and energy bounds in shape memory alloys.

Findings

Larger-scale interfaces cluster on a set of Hausdorff-dimension $3-rac{2}{3}$.

The dimension of interface clustering is likely optimal.

Provides a local lower bound for energy density blow-up near habit planes.

Abstract

We quantify the rigidity of branching microstructures in shape memory alloys undergoing cubic-to-tetragonal transformations in the geometrically linearized theory by making use of Tartar's H-measures. The main result is a $B^{2/3}_{1,\infty}$-estimate for the characteristic functions of twins, which heuristically suggests that the larger-scale interfaces can cluster on a set of Hausdorff-dimension $3-\frac{2}{3}$. We provide evidence indicating that the dimension is optimal. Furthermore, we get an essentially local lower bound for the blow-up behavior of the limiting energy density close to a habit plane.