Calculus of Variations and Geometric Measure Theory

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


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


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.