A derivation of linear elastic energies from pair-interaction atomistic systems

Andrea Braides; Margherita Solci; Enrico Vitali

doi:10.3934/nhm.2007.2.551

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2007, Volume 2, Issue 3: 551-567. Doi: 10.3934/nhm.2007.2.551

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A derivation of linear elastic energies from pair-interaction atomistic systems

1.
Dipartimento di Matematica, Università di Roma 'Tor Vergata', via della Ricerca Scientifica, 00133 Roma
2.
DAP, Università di Sassari, piazza Duomo, 6-07041, Alghero
3.
Dipartimento di Matematica 'F. Casorati', Università di Pavia, via Ferrata, 1-27100 Pavia

Received: September 30, 2006

Revised: May 31, 2007

Published: August 2007

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Abstract

Pair-interaction atomistic energies may give rise, in the framework of the passage from discrete systems to continuous variational problems, to nonlinear energies with genuinely quasiconvex integrands. This phenomenon takes place even for simple harmonic interactions as shown by an example by Friesecke and Theil [19]. On the other hand, a rigorous derivation of linearly elastic energies from energies with quasiconvex integrands can be obtained by $\Gamma$-convergence following the method by Dal Maso, Negri and Percivale [14]. We show that the derivation of linear theories by $\Gamma$-convergence can be obtained directly from lattice interactions in the regime of small deformations. Our proof relies on a lower bound by comparison with the continuous result, and on a direct Taylor expansion for the upper bound. The computation is carried over for a family of lattice energies comprising interactions on the triangular lattice in dimension two.

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Mathematics Subject Classification: Primary: 74B05, 49J45; Secondary: 46E39, 74B20.

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