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

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 $SB{V}^p$ functions whose jump sets have a prescribed orientation.

References in corpus (1)

• Energy Bounds for a Compressed Elastic Film on a Substrate

Cited by in corpus (4)

• On the Energy Scaling Behaviour of a Singularly Perturbed Tartar Square
• On Scaling Laws for Multi-Well Nucleation Problems without Gauge Invariances
• On the Energy Scaling Behaviour of Singular Perturbation Models with Prescribed Dirichlet Data Involving Higher Order Laminates
• On the Scaling of the Cubic-to-Tetragonal Phase Transformation with Displacement Boundary Conditions