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

Abstract

We study convex integration solutions in the context of the modelling of shape-memory alloys. The purpose of the article is twofold, treating both rigidity and flexibility prop- erties: 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 geomet- rically linearised two-dimensional hexagonal-to-rhombic phase transformation. We discuss and compare the two algorithms from Rüland et al. (J Elast. 2019. https://doi. org/10.1007/s10659-018-09719-3; SIAM J Math Anal 50(4):3791–3841, 2018) and give a numerical estimate of the regularity attained.