Derivation of homogenized Euler-Lagrange equations for von Karman rod

TL;DR

This paper investigates how homogenization and dimension reduction affect the stationary points of thin nonhomogeneous rods under von Kármán scaling, deriving the limit equations for the homogenized model.

Contribution

It establishes the convergence of stationary points for nonhomogeneous rods to those of the homogenized von Kármán model under simultaneous homogenization and dimension reduction.

Findings

Sequences of scaled displacements converge to a stationary point of the homogenized model.

The results extend to the von Kármán plate model.

Provides a rigorous derivation of homogenized Euler-Lagrange equations.

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.