Calculus of Variations and Geometric Measure Theory

C. I. Zeppieri

Stochastic homogenisation of singularly-perturbed integral functionals

created by zeppieri on 25 May 2015
modified by zeppieri1 on 20 Oct 2016


Published Paper

Inserted: 25 may 2015
Last Updated: 20 oct 2016

Journal: Ann. Mat. Pura Appl.
Volume: 195
Number: 6
Pages: 2183-2208
Year: 2016


We study the relative impact of small-scale random inhomogeneities and singular perturbations in nonlinear elasticity. More precisely, we analyse the asymptotic behaviour of the energy functionals \[ F_\varepsilon(\omega)(u)=\int_A \Big(f\Big(\omega,\frac{x}{\varepsilon}, Du\Big) +\varepsilon^2
\Delta u
^2\Big) \,dx, \] where $\omega$ is a random parameter and $\varepsilon>0$ denotes a typical length-scale associated with the variations in the elastic properties of the body. For $f$ stationary and ergodic, we show that when $\varepsilon\to 0$ the randomly inhomogeneous material described by $F_\varepsilon(\omega)$ behaves (almost surely) like a homogeneous deterministic material. The limit stored energy density is given in terms of an asymptotic cell formula in which the Laplacian perturbation explicitly appears.

Keywords: Gamma-convergence, nonlinear elasticity, stochastic homogenisation