Abstract:
We analyse the rigidity of discrete energies where at least nearest and nextto-
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 surface-scaled energy and we give bounds
on its possible Gamma-limit.
In the second part of the paper we follow the approach developed in the first part
to study a discrete model for (possibly heterogeneous) nanowires. In the heterogeneous
case, by applying the compactness result shown in the first part of the paper, we obtain
an estimate on the minimal energy spent to match different equilibria. This gives insight
into the nucleation of dislocations in epitaxially grown heterostructured nanowires.
