Homogenization and dimensional reduction for nonlinear multilayered plates including biological growth when the plate thickness and the size of the heterogeneities are not of the same order of magnitude

Abstract

In this work, we address the important problem of the homogenization and the dimensional reduction for nonlinear plates including biological growth effect when the plate thickness and the size of the heterogeneities are not of the same order of magnitude. The theory when the thickness of the plate and the in-plane heterogeneities are of the same order of magnitude has been previously addressed by the first author. In our reduction method, the thickness of the plate is small but does not go to zero; however, the homogenization method adopted is a standard asymptotic analysis since the size of the heterogenities goes to zero. For the sake of simplicity, the distribution of material heterogeneities is assumed to be repetitive periodic. For the case when the period is very much smaller than the thickness of the plate, we first have to consider the limit when the size of the heterogeneities goes to zero. We obtain a multilayered plate with homogeneous layers, for which we propose a two-dimensional plate model. Then, we consider the case when the period is very much bigger than the thickness of the plate. We first have to consider homogenization in the plane parallel to the mid-plane of the plate. This method is only meaningful when the geometry of the heterogeneities does not depend on the thickness direction of the plate. Then, we can obtain a plate model from multilayered plate in which each layer is homogeneous. Possible applications for future numerical works are given through work references and photographs.