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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Archive for Rational Mechanics and Analysis, Springer Verlag, 2011, 200, pp.881-943. 〈10.1007/s00205-010-0378-7〉
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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, Springer Verlag, 2011, 200, pp.881-943. 〈10.1007/s00205-010-0378-7〉. 〈inria-00437765〉

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