Analysis of the quasi-nonlocal approximation of linear and circular chains in the plane

TL;DR

This paper analyzes the stability and error of quasi-nonlocal and Cauchy-Born approximations for atomistic chains in the plane, revealing their limitations and providing sharp estimates for lattice stability and modeling errors.

Contribution

It offers a detailed stability analysis of these approximations for planar chains, including circular configurations, and derives optimal error bounds for atomistic strain predictions.

Findings

Cauchy-Born and quasi-nonlocal approximations overestimate lattice stability under general perturbations.

Sharp estimates of the critical strain for lattice stability are provided.

Optimal order a priori error estimates for atomistic strain approximation are established.

Abstract

We give an analysis of the stability and displacement error for linear and circular atomistic chains in the plane when the atomistic energy is approximated by the Cauchy-Born continuum energy and by the quasi-nonlocal atomistic-to-continuum coupling energy. We consider atomistic energies that include Lennard-Jones type nearest neighbor and next nearest neighbor pair-potential interactions. Previous analyses for linear chains have shown that the Cauchy-Born and quasi-nonlocal approximations reproduce (up to the order of the lattice spacing) the atomistic lattice stability for perturbations that are constrained to the line of the chain. However, we show that the Cauchy-Born and quasi-nonlocal approximations give a finite increase for the lattice stability of a linear or circular chain under compression when general perturbations in the plane are allowed. We also analyze the increase of the lattice stability under compression when pair-potential energies are augmented by bond-angle energies. Our estimates of the largest strain for lattice stability (the critical strain) are sharp (exact up to the order of the lattice scale). We then use these stability estimates and modeling error estimates for the linearized Cauchy-Born and quasi-nonlocal energies to give an optimal order (in the lattice scale) {\em a priori} error analysis for the approximation of the atomistic strain in $\ell^2_\epsilon$ due to an external force.