Homogenisation of Nonlinear Heterogeneous Thin Plate When the Plate Thickness and In-Plane Heterogeneities are of the Same Order of Magnitude

Abstract

In this work, we propose a new two-scale finite-strain thin plate theory for highly heterogeneous plates described by a repetitive periodic microstructure. For this type of theory, two scales exist, the macroscopic one is linked to the entire plate and the microscopic one is linked to the size of the heterogeneity. We consider the case when the plate thickness is comparable to in-plane heterogeneities. We assume that the nonlinear macroscopic part of the model is of Kirchhoff–Love type. We obtain the nonlinear homogenised model by performing simultaneously both the homogenisation and the reduction of the initial three-dimensional plate problem to a two-dimensional one. Since nonlinear equations are difficult to solve, we linearise this homogenised Kirchhoff–Love plate theory. Finally, we discuss the treatment of edge effects in the vicinity of the lateral boundary of the plate.