We derive, via simultaneous homogenization and dimension reduction, the Γ-limit for thin elastic plates whose energy density oscillates on a scale that is either comparable to, or much smaller than, the film thickness. We consider the energy scaling that corresponds to Kirchhoff's nonlinear bending theory of plates.
We rigorously derive a homogenized von-Kármán plate theory as a Γ-limit from nonlinear three-dimensional elasticity by combining homogenization and dimension reduction. Our starting point is an energy functional that describes a nonlinear elastic, three-dimensional plate with spatially periodic material properties. The functional features two small length scales: the period ε of the elastic composite material, and the thickness h of the slender plate. We study the behavior as ε and h simultaneously converge to zero in the von-Kármán scaling regime. The obtained limit is a homogenized von-Kármán plate model. Its effective material properties are determined by a relaxation formula that exposes a non-trivial coupling of the behavior of the out-of-plane displacement with the oscillatory behavior in the in-plane directions. In particular, the homogenized coefficients depend on the relative scaling between h and ε, and different values arise for h ≪ ε, ε ~ h and ε ≪ h.
Starting from 3D elasticity equations we derive the model of the homogenized von Kármán plate by means of Γ-convergence. This generalizes the recent results, where the material oscillations were assumed to be periodic.
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