Deformation concentration for martensitic microstructures in the limit of low volume fraction

TL;DR

This paper analyzes the transition in microstructure patterns of shape memory alloys under low volume fractions, deriving a reduced model that captures the energy scaling and pattern formation using $ ext{Gamma}$-convergence.

Contribution

It introduces a new reduced model for microstructure energy in shape memory alloys, connecting pattern formation with Mumford-Shah type functionals under specific constraints.

Findings

Derived a $ ext{Gamma}$-limit functional similar to Mumford-Shah with constraints.

Identified the energy transition from uniform to patterned microstructures.

Developed an approximation result for $SBV^p$ functions with prescribed jump orientations.

Abstract

We consider a singularly-perturbed nonconvex energy functional which arises in the study of microstructures in shape memory alloys. The scaling law for the minimal energy predicts a transition from a parameter regime in which uniform structures are favored, to a regime in which the formation of fine patterns is expected. We focus on the transition regime and derive the reduced model in the sense of $\Gamma$-convergence. The limit functional turns out to be similar to the Mumford-Shah functional with additional constraints on the jump set of admissible functions. One key ingredient in the proof is an approximation result for $SBV^p$ functions whose jump sets have a prescribed orientation.