On Lennard-Jones systems with finite range interactions and their asymptotic analysis

TL;DR

This paper analyzes the continuum limit of one-dimensional Lennard-Jones atom chains with finite-range interactions, deriving explicit formulas and extending fracture energy models to more general interaction ranges.

Contribution

It provides the first explicit continuum limit formula for finite-range Lennard-Jones interactions and extends fracture energy derivations to these cases.

Findings

Explicit continuum limit formula for finite-range Lennard-Jones interactions.

Homogenization via convexification of Cauchy-Born energy density.

Derivation of a 1D Griffith fracture energy model for finite-range interactions.

Abstract

The aim of this work is to provide further insight into the qualitative behavior of mechanical systems that are well described by Lennard-Jones type interactions on an atomistic scale. By means of $\Gamma$-convergence techniques, we study the continuum limit of one-dimensional chains of atoms with finite range interactions of Lennard-Jones type, including the classical Lennard-Jones potentials. So far, explicit formulae for the continuum limit were only available for the case of nearest and next-to-nearest neighbour interactions. In this work, we provide an explicit expression for the continuum limit in the case of finite range interactions. The obtained homogenization formula is given by the convexification of a Cauchy-Born energy density. Furthermore, we study suitably rescaled energies in which bulk and surface contributions scale in the same way. The related discrete-to-continuum limit yields a rigorous derivation of a one-dimensional version of Griffith' fracture energy and thus generalizes earlier derivations for nearest and next-to-nearest neighbors to the case of finite range interactions. A crucial ingredient to our proofs is a novel decomposition of the energy that allows for refined estimates.