A new homogenization scheme for beam and plate structures without a priori requirements on boundary conditions

Abstract

This contribution picks up on a novel approach for a first order homogenization procedure based on the Irving-Kirkwood theory and provides a finite element implementation as well as applications to beam and plate structures. It does not have the fundamental problems of dependency from representative volume element (RVE) size in determining the shear and torsional stiffness for beams and plates, that is present in classic Hill-Mandel methods. Due to the possibility of using minimal boundary conditions whilst simultaneously reusing existing homogenization algorithms, creation of models and numerical implementation are much more straight forward. The presented theory and FE formulation are limited to materially and geometrically linear problems. The approach to determining shear stiffness is based on the assumption of a quadratic shear stress distribution over the height (and width in case of the beam), which causes warping of the cross-section under transverse shear loading. Results for the homogenization scheme are shown for various beam and plate configurations and compared to values from well known analytical solutions or computed full scale models.