*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

**Abstract:**

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

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