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 \(F_\varepsilon (\omega )(u)=\int _A \Big (f\Big (\omega ,\frac{x}{\varepsilon }, Du\Big ) +\varepsilon ^2 |\Delta u|^2\Big ) \,\hbox {d}x,\) 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 \(\varepsilon \rightarrow 0\) 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.
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