On a Gamma-convergence analysis of a quasicontinuum method

TL;DR

This paper uses Gamma-convergence to analyze a quasicontinuum method, showing that it accurately reproduces atomistic behavior in elastic regimes and identifying conditions for accurate fracture modeling.

Contribution

It provides a rigorous Gamma-convergence analysis of a quasicontinuum method, highlighting conditions for matching atomistic energies and fracture behavior.

Findings

Minimizers and energies match in elastic limit as atoms increase.

Fracture behavior depends on the choice of representative atoms.

Proper selection of representative atoms ensures accurate fracture energy and location.

Abstract

We investigate a quasicontinuum method by means of analytical tools. More precisely, we compare a discrete-to-continuum analysis of an atomistic one-dimensional model problem with a corresponding quasicontinuum model. We consider next and next-to-nearest neighbour interactions of Lennard-Jones type and focus on the so-called quasinonlocal quasicontinuum approximation. Our analysis, which applies $\Gamma$-convergence techniques, shows that, in an elastic setting, minimizers and the minimal energies of the fully atomistic problem and its related quasicontinuum approximation have the same limiting behaviour as the number of atoms tends to infinity. In case of fracture this is in general not true. It turns out that the choice of representative atoms in the quasicontinuum approximation has an impact on the fracture energy and on the location of fracture. We give sufficient conditions for the choice of representative atoms such that, also in case of fracture, the minimal energies of the fully atomistic energy and its quasicontinuum approximation coincide in the limit and such that the crack is located in the atomistic region of the quasicontinuum model as desired.