Towards uniformly $\Gamma$-equivalent theories for nonconvex discrete systems

Lucia Scardia; Anja Schlömerkemper; Chiara Zanini

doi:10.3934/dcdsb.2012.17.661

Article Contents

2012, Volume 17, Issue 2: 661-686. Doi: 10.3934/dcdsb.2012.17.661

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Towards uniformly $\Gamma$-equivalent theories for nonconvex discrete systems

1.
Eindhoven University of Technology, PO Box 513, WH 4.10, 5600 MB Eindhoven
2.
Institute for Mathematics, University of Würzburg, Emil-Fischer-Str. 40, 97074 Würzburg
3.
Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino

Received: August 2010

Revised: August 2011

Published: March 2012

Abstract / Introduction Related Papers Cited by

Abstract

In this paper we consider a one-dimensional chain of atoms which interact with their nearest and next-to-nearest neighbours via a Lennard-Jones type potential. We prescribe the positions in the deformed configuration of the first two and the last two atoms of the chain.
We are interested in a good approximation of the discrete energy of this system for a large number of atoms, i.e., in the continuum limit.
We show that the canonical expansion by $\Gamma$-convergence does not provide an accurate approximation of the discrete energy if the boundary conditions for the deformation are close to the threshold between elastic and fracture regimes. This suggests that a uniformly $\Gamma$-equivalent approximation of the energy should be made, as introduced by Braides and Truskinovsky, to overcome the drawback of the lack of accuracy of the standard $\Gamma$-expansion.
In this spirit we provide a uniformly $\Gamma$-equivalent approximation of the discrete energy at first order, which arises as the $\Gamma$-limit of a suitably scaled functional.

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Mathematics Subject Classification: 74R10, 74A45, 74G65, 82B21, 41A60.

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References

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