On the homogenization of a nonlinear shell

In this paper we propose a multiscale finite-strain shell theory for simulating the mechanical response of a highly heterogeneous shell of varying thickness. To resolve this issue, a higher-order stress-resultant shell formulation based on multiscale homogenization is considered. At the macroscopic scale level, we approximate the displacement field by a fifth-order Taylor–Young expansion in thickness. We take account of the microscale fluctuations by introducing a boundary value problem over the domain of a three-dimensional representative volume element (RVE). The geometrical form and the dimensions of the RVE are determined by the representative microstructure of the heterogeneity. In this way, an in-plane homogenization is directly combined with a through thickness stress integration. As a result, the macroscopic stress resultants are the volume averages through RVE of microscopic stress. All microstructural constituents are modeled as first-order continua and three-dimensional continuum, described by the standard equilibrium and the constitutive equations. This type of theory is anxiously awaited.