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

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

Foundation, analysis, and numerical investigation of a variational network-based model for rubber

“…Yet this does not provide a pure continuum description of rubber elasticity. As proved in [3], the model "converges" to a continuum model as the meshsize goes to zero. However, the limiting model highly depends on the geometry of the mesh in terms of isotropy properties for instance.…”

“…In fact, the estimate from above is satisfied by W nn provided we consider any order of the Taylor expansion of the inverse of the Langevin function (for instance (3), in which case p = 10). For the volumetric term, there is a technical difficulty: the Helmholtz energy density (5) does not satisfy Hypothesis 1 since W Helm (ξ) blows up as det ξ → 0 (as it is desirable in nonlinear elasticity).…”

“…As for the Treloar model, whenW c is replaced by its Taylor expansion up to order p (see (3) for p = 5), this gives rise to a Taylor expansion of (6) that we denote byW p AB . In order to consider non isochoric deformations, one should add the Helmholtz volumetric energy as for the Treloar type energies.…”

“…It is a consequence of the integration over all the directions for (T). For (V), one has to assume in addition that the stochastic network is isotropic in the mean (the expectation of the stochastic network is isotropic), which implies the isotropy of the energy density, as proved in [3,Theorem 6].…”

Numerical homogenization: survey, new results, and perspectives

“…This analytical approximation should at least satisfy similar properties as the homogenized nonlinear map (say convexity, isotropy, etc.). This strategy has been used in [16] to approximate the homogenized energy density associated with a discrete model for rubber introduced in [33] and whose homogenization limit has been etablished in [3]. There, the homogenized energy density is known to be quasiconvex, isotropic, and minimal at identity.…”