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

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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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Authors and Affiliations

  1. DAEIMI, Università di Cassino, via Di Biasio 43, 03043, Cassino (FR), Italy

    Roberto Alicandro

  2. Dipartimento di Matematica e Applicazioni ‘R. Caccioppoli’, Università di Napoli, via Cintia, 80126, Napoli, Italy

    Marco Cicalese

  3. Projet SIMPAF, INRIA Lille-Nord Europe Parc Scientifique de la Haute Borne, 40, avenue Halley Bat.A, Park Plaza, 59650, Villeneuve d’Ascq, France

    Antoine Gloria

Authors
  1. Roberto Alicandro
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  2. Marco Cicalese
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  3. Antoine Gloria
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Correspondence to Roberto Alicandro.

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Communicated by S. Müller

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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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