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Journal Articles Archive for Rational Mechanics and Analysis Year : 2011

Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity

Abstract

This article is devoted to the study of the asymptotic behavior of a class of energies defined on stochastic lattices. Under polynomial growth assumptions, we prove that the energy functionals $F_\e$ stored in the deformation of an $\e$-scaling of a stochastic lattice $\Gamma$-converge to a continuous energy functional when $\e$ goes to zero. In particular, the limiting energy functional is of integral type, and deterministic if the lattice is ergodic. We also generalize to systems and nonlinear settings well-known results on stochastic homogenization of discrete elliptic equations. As an application of the main result, we prove the convergence of a discrete model for rubber towards the nonlinear theory of continuum mechanics. We finally address some mechanical properties of the limiting models, such as frame-invariance, isotropy and natural states.
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Dates and versions

inria-00437765 , version 1 (01-12-2009)

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Roberto Alicandro, Marco Cicalese, Antoine Gloria. Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity. Archive for Rational Mechanics and Analysis, 2011, 200, pp.881-943. ⟨10.1007/s00205-010-0378-7⟩. ⟨inria-00437765⟩
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