Interactions beyond nearest neighbours and rigidity of discrete energies: a compactness result and an application to dimension reduction

TL;DR

This paper investigates the role of interactions beyond nearest neighbours in discrete energies, showing they help enforce rigidity and can replace certain constraints, with applications to nanowire dimension reduction and dislocation nucleation.

Contribution

It provides a novel analysis of how extended interactions influence rigidity and energy bounds, applying these results to nanowire models and dislocation nucleation.

Findings

Interactions beyond nearest neighbours penalize orientation changes.

A compactness result for surface-scaled energies is established.

Bounds on the Gamma-limit of the energy are derived.

Abstract

We analyse the rigidity of 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 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.