Integral Representation Results for Energies Defined on Stochastic Lattices and Application to Nonlinear Elasticity

Roberto Alicandro, Marco Cicalese, Antoine Gloria

Research output: Contribution to journalArticlepeer-review

46 Scopus citations

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 Fε stored in the deformation of an ε scaling of a stochastic lattice Γ-Converge to a continuous energy functional when ε 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.

Original languageEnglish
Pages (from-to)881-943
Number of pages63
JournalArchive for Rational Mechanics and Analysis
Volume200
Issue number3
DOIs
StatePublished - Jun 2011
Externally publishedYes

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