Homogenization of nonlinear Cosserat plate including growth theory when the thickness of the plate and the size of the in-plane heterogeneities are of the same order of magnitude

Abstract

In this work, we present a new two-scale finite-strain plate theory for highly heterogeneous plates described by a repetitive periodic microstructure. Two scales exist, the macroscopic scale is linked to the entire plate and the microscopic one is linked to the size of the heterogeneity. This work aims to propose such a theory for thick plates in a nonlinear setting when the thickness and the size of heterogeneities are of the same order of magnitude. The homogenization theory for large deformation with growth is suitable for the modelization of nearly incompressible plant tissue. This model is suitable for wavy leaves. For thick plates, the transverse normal stress and transverse shearing are modelized at both microscopic and macroscopic levels. At the macroscopic level, we consider a nonlinear Cosserat plate model. At the microscopic level, we impose that the average of contribution of the microscopic displacement to rotation angles is equal to zero. We also deal with the problem of boundary layer problem near the lateral boundary. The model recently proposed by Pruchnicki is valid for thin heterogeneous plates; we present an extension for thick plates that takes into account both transverse normal stress and shearing. This model is equivalent to the first model presented but it involves a second-order derivative of the macroscopic displacement field.