Homogenization assumptions for the two-scale analysis of first-order shear deformable shells

This contribution presents a multiscale approach for the analysis of shell structures using Reissner–Mindlin kinematics. A distinctive feature is that the thickness of the representative volume element (RVE) corresponds to the shell thickness. The main focus of this paper is on the choice of correct boundary conditions for the RVE. Three different types of boundary conditions, which fulfil the Hill–Mandel condition, are presented to bridge the two scales. A common feature is the application of zero-traction boundary conditions at the top and bottom surfaces of the RVE. Furthermore, an internal constraint is used to reduce the dependency of the stiffness components on the RVE size. The introduced boundary conditions differ mainly in the application of shear strains and their symmetry requirements on the RVE. The characteristic features are compared by means of linear-elastic benchmark tests. It is shown that the stress resultants and tangent stiffness components are obtained correctly. Moreover, the presented approach is verified using different macroscopic shell structures and different mesostructures. Both, linear and nonlinear small strain examples are compared to analytical values or full-scale solutions and demonstrate a wide applicability of the present formulation.