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 Γ-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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Alicandro, R., Cicalese, M. & Gloria, A. Integral Representation Results for Energies Defined on Stochastic Lattices and Application to Nonlinear Elasticity. Arch Rational Mech Anal 200, 881–943 (2011). https://doi.org/10.1007/s00205-010-0378-7
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DOI: https://doi.org/10.1007/s00205-010-0378-7