In this paper we study the effects of simultaneous homogenization and dimension reduction in the context of convergence of stationary points for thin nonhomogeneous rods under the assumption of the von Kármán scaling. Assuming stationarity condition for a sequence of deformations close to a rigid body motion, we prove that the corresponding sequences of scaled displacements and twist functions converge to a limit point, which is the stationary point of the homogenized von Kármán rod model. The analogous result holds true for the von Kármán plate model.

We use the notion of stochastic two-scale convergence introduced in [ZP06] to solve the problem of stochastic homogenization of the elastic plate in the bending regime.

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We prove that, in the limit of vanishing thickness, equilibrium configurations of inhomogeneous, three-dimensional non-linearly elastic rods converge to equilibrium configurations of the variational limit theory. More precisely, we show that, as h ց 0, stationary points of the energy E h , for a rod Ω h ⊂ R 3 with cross-sectional diameter h, subconverge to stationary points of the Γ-limit of E h , provided that the bending energy of the sequence scales appropriately. This generalizes earlier results for homogeneous materials to the case of materials with (not necessarily periodic) inhomogeneities.

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