Asymptotic analysis of heterogeneous Cosserat media

Abstract

The present work deals with the development of homogenization procedures for periodic heterogeneous linear elastic Cosserat media. It is resorted to asymptotic methods classically used in periodic homogenization. It is shown that the nature of the homogeneous equivalent medium depends on the hierarchy of three characteristic lengths: the size l of the heterogeneities, the Cosserat intrinsic lengths l c of the constituents and the typical size L of the considered structure. When l and l c are comparable and much smaller than L, the effective medium is proved to be a Cauchy continuum with volume couples, whereas the case l c∼ L leads to a Cosserat effective medium. Finite element simulations are provided in the case of a fiber-matrix composite for a large range of characteristic lengths l c and for two different volume fractions. Reference calculations involving every heterogeneity are compared to the response obtained using a homogeneous equivalent medium. The results confirm the predicted hierarchy of models and also show that a Cosserat effective medium still provide a good estimation when all characteristic lengths have the same order of magnitude.