Calculus of Variations and Geometric Measure Theory
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R. Alicandro - M. Cicalese - A. Gloria

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

created by cicalese on 01 Dec 2009
modified by alicandr on 05 Jun 2011

[BibTeX]

Published Paper

Inserted: 1 dec 2009
Last Updated: 5 jun 2011

Journal: Archive Rational Mech. Anal.
Volume: 20
Number: 3
Pages: 881-943
Year: 2011

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 $F_{\varepsilon}$ stored in the deformation of an $\varepsilon$-scaling of a stochastic lattice $\Gamma$-converge to a continuous energy functional when $\varepsilon$ 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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