Chapter 21 - Asymptotic homogenization approach applied to Cosserat heterogeneous media

Abstract

This work deals with the two-scale asymptotic homogenization method (AHM) applied to Cosserat heterogeneous media with periodic structure. The asymptotic homogenization process is understood as a solution method of a problem modeled by partial differential equations with rapidly oscillating properties in terms of a double scale asymptotic expansion. It turns on the original problem into an equivalent one in a media with homogeneous properties. In other words, the AHM extracts constant parameters from very fast oscillating coefficients defined in an heterogeneous media. Starting from the basic equations for a 3D heterogeneous micropolar material and based on microscopic-macroscopic description, the corresponding local problems and the effective coefficients obtained by AHM are explicitly described. As an example, layered Cosserat composites are also studied using AHM and focusing on the work for centro-symmetric materials. In this case, simple analytical expressions of the effective properties are reported for laminated composites with an arbitrary number of layers. The overall behavior of Cosserat bilaminate composites with cubic-symmetric constituents provides a homogenized material described by 18 independent moduli, 9 stiffness and 9 torque effective constants respectively, corresponding to the symmetry group referred as rotations by 90∘, which implies invariance of Cijpq and Dijpq under rotations of 90∘. Numerical results for the effective properties of bilaminate composite with cubic constituents are computed.