  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].
