TY - JOUR
T1 - Integral Representation Results for Energies Defined on Stochastic Lattices and Application to Nonlinear Elasticity
AU - Alicandro, Roberto
AU - Cicalese, Marco
AU - Gloria, Antoine
PY - 2011/6
Y1 - 2011/6
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=79956141973&partnerID=8YFLogxK
U2 - 10.1007/s00205-010-0378-7
DO - 10.1007/s00205-010-0378-7
M3 - Article
AN - SCOPUS:79956141973
SN - 0003-9527
VL - 200
SP - 881
EP - 943
JO - Archive for Rational Mechanics and Analysis
JF - Archive for Rational Mechanics and Analysis
IS - 3
ER -