A Rigidity Result for a Reduced Model of a Cubic-to-Orthorhombic Phase Transition in the Geometrically Linear Theory of Elasticity

TL;DR

This paper analyzes a simplified 2D model of a cubic-to-orthorhombic phase transition in shape-memory alloys, demonstrating the rigidity of complex microstructures involving crossing twins under small energy perturbations.

Contribution

It introduces a variational model that proves the rigidity of crossing twin microstructures and determines their optimal scaling behavior in a simplified phase transition model.

Findings

Crossing twin structures are rigid under small energy perturbations.

The model establishes the optimal scaling behavior of microstructures.

Results relate to and extend previous work by Capella, Otto, Dolzmann, and Müller.

Abstract

We study a simplified two-dimensional model for a cubic-to-orthorhombic phase transition occuring in certain shape-memory-alloys. In the low temperature regime the linear theory of elasticity predicts various possible patterns of martensite arrangements: Apart from the well known laminates this phase transition displays additional structures involving four martensitic variants -- so called crossing twins. Introducing a variational model including surface energy, we show that these structures are rigid under small energy perturbations. Combined with an upper bound construction this gives the optimal scaling behavior of incompatible microstructures. These results are related to papers by Capella and Otto as well as to a paper by Dolzmann and M\"uller.