Mathematics

We analyse the rigidity of non-convex discrete energies where at least nearest
and next-to-nearest neighbour interactions are taken into account. Our purpose is to
show that interactions beyond nearest neighbours have the role of penalising changes of
orientation and, to some extent, they may replace the positive-determinant constraint
that is usually required when only nearest neighbours are accounted for. In a discrete to
continuum setting, we prove a compactness result for a family of surface-scaled energies
and we give bounds on its possible Gamma-limit in terms of interfacial energies that
penalise changes of orientation.