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

• Published in: Archive for rational mechanics and analysis 2011-06, Vol.200 (3), p.881-943
• Main Authors: Alicandro, Roberto, Cicalese, Marco, Gloria, Antoine
• Format: Article
• Language: English
• Subjects: Classical Mechanics; Complex Systems; Exact sciences and technology; Fluid- and Aerodynamics; Fundamental areas of phenomenology (including applications); Mathematical and Computational Physics; Mathematics; Numerical Analysis; Physics; Physics and Astronomy; Solid mechanics; Static elasticity (thermoelasticity...); Structural and continuum mechanics; Theoretical
• ISSN: 0003-9527; 1432-0673
• DOI: 10.1007/s00205-010-0378-7

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