Derivation of homogenized Euler-Lagrange equations for von Karman rod

Abstract

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 K\'arm\'an 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 K\'arm\'an rod model. The analogous result holds true for the von K\'arm\'an plate model.