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

Mathias Schäffner; Anja Schlömerkemper

doi:10.3934/nhm.2018005

Article Contents

2018, Volume 13, Issue 1: 95-118. Doi: 10.3934/nhm.2018005

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

Mathias Schäffner^1, and
Anja Schlömerkemper^2,

1.
TU Dresden, Department of Mathematics, Zellescher Weg 12-14, 01069 Dresden, Germany
2.
University of Würzburg, Institute of Mathematics, Emil-Fischer-Straẞe 40, 97074 Würzburg, Germany

Received: June 2017

Revised: October 2017

Published: March 2018

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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 $Γ$-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 formula 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 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.

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Mathematics Subject Classification: Primary: 49J45, 74R10, 74G10, 74G65.

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