Gamma-expansion for a 1D confined Lennard-Jones model with point defect

Thomas Hudson; Thomas Hudson

doi:10.3934/nhm.2013.8.501

Networks and Heterogeneous Media

2013, Volume 8, Issue 2: 501-527. doi: 10.3934/nhm.2013.8.501

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Gamma-expansion for a 1D confined Lennard-Jones model with point defect

Thomas Hudson ^{1
,}

1.
Mathematical Institute, 24-29 St Giles', Oxford, OX1 3LB

Received: 01 April 2012 Revised: 01 April 2013
Primary: 49J45; Secondary: 70C20, 74B20, 74G65, 74Q15.

We compute a rigorous asymptotic expansion of the energy of a point defect in a 1D chain of atoms with second neighbour interactions. We propose the Confined Lennard-Jones model for interatomic interactions, where it is assumed that nearest neighbour potentials are globally convex and second neighbour potentials are globally concave. We derive the $\Gamma$-limit for the energy functional as the number of atoms per period tends to infinity and derive an explicit form for the first order term in a $\Gamma$-expansion in terms of an infinite cell problem. We prove exponential decay properties for minimisers of the energy in the infinite cell problem, suggesting that the perturbation to the deformation introduced by the defect is confined to a thin boundary layer.
- Energy asymptotics,
- point defect,
- gamma convergence,
- discrete to continuum limits
Citation: Thomas Hudson. Gamma-expansion for a 1D confined Lennard-Jones model with point defect[J]. Networks and Heterogeneous Media, 2013, 8(2): 501-527. doi: 10.3934/nhm.2013.8.501

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Abstract

We compute a rigorous asymptotic expansion of the energy of a point defect in a 1D chain of atoms with second neighbour interactions. We propose the Confined Lennard-Jones model for interatomic interactions, where it is assumed that nearest neighbour potentials are globally convex and second neighbour potentials are globally concave. We derive the $\Gamma$-limit for the energy functional as the number of atoms per period tends to infinity and derive an explicit form for the first order term in a $\Gamma$-expansion in terms of an infinite cell problem. We prove exponential decay properties for minimisers of the energy in the infinite cell problem, suggesting that the perturbation to the deformation introduced by the defect is confined to a thin boundary layer.

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