Derivation of a Homogenized Von-Kármán Plate Theory from 3d Nonlinear Elasticity
Stefan Neukamm and Igor Velčić
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Submission date: 02. Oct. 2012
published in: Mathematical models and methods in applied sciences, 23 (2013) 14, p. 2701-2748
DOI number (of the published article): 10.1142/S0218202513500449
MSC-Numbers: 35B27, 49J45, 74E30, 74Q05
Keywords and phrases: elasticity, dimension reduction, homogenization, von-Kármán plate theory, two-scale convergence
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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.