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MiS Preprint

Derivation of a Homogenized Von-Kármán Plate Theory from 3d Nonlinear Elasticity

Stefan Neukamm and Igor Velčić


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.

Oct 2, 2012
Oct 8, 2012
MSC Codes:
35B27, 49J45, 74E30, 74Q05
elasticity, dimension reduction, homogenization, von-Kármán plate theory, two-scale convergence

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2013 Repository Open Access
Stefan Neukamm and Igor Velčić

Derivation of a homogenized Von-Karman plate theory from 3d nonlinear elasticity

In: Mathematical models and methods in applied sciences, 23 (2013) 14, pp. 2701-2748