DERIVATION OF A HOMOGENIZED VON-KÁRMÁN PLATE THEORY FROM 3D NONLINEAR ELASTICITY

STEFAN NEUKAMM, IGOR VELČIĆ
2013 Mathematical Models and Methods in Applied Sciences  
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
more » ... 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 outof-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.
doi:10.1142/s0218202513500449 fatcat:y6in36yaeravnaxuetmigybzj4