Convex Integration Arising in the Modelling of Shape-Memory Alloys: Some Remarks on Rigidity, Flexibility and Some Numerical Implementations

TL;DR

This paper explores convex integration solutions in shape-memory alloy modeling, linking their regularity to variational models with surface energy and presenting the first numerical implementations for phase transformation problems.

Contribution

It establishes a connection between convex integration regularity and variational surface energy models, and introduces novel numerical schemes for phase transformation simulations.

Findings

Convex integration solutions relate to lower bounds in variational models.

Numerical schemes successfully implemented for phase transformation.

Comparison of two convex integration algorithms.

Abstract

We study convex integration solutions in the context of the modelling of shape-memory alloys. The purpose of the article is two-fold, treating both rigidity and flexibility properties: Firstly, we relate the maximal regularity of convex integration solutions to the presence of lower bounds in variational models with surface energy. Hence, variational models with surface energy could be viewed as a selection mechanism allowing for or excluding convex integration solutions. Secondly, we present the first numerical implementations of convex integration schemes for the model problem of the geometrically linearised two-dimensional hexagonal-to-rhombic phase transformation. We discuss and compare the two algorithms from [RZZ16] and [RZZ17].