Exploiting symmetries in solid-to-shell homogenization, with application to periodic pin-reinforced sandwich structures

In this paper, a novel set of Periodic Boundary Conditions named Multiscale Periodic Boundary Conditions (MPBCs) that apply to reduced Unit Cells (rUCs) and enable the two-scale (solid-to-shell) numerical homogenization of periodic structures, including their bending and twisting response, is presented and implemented in an FE code. Reduced Unit Cells are domains smaller than the Unit Cells (UCs), obtained by exploiting the internal symmetries of the UCs. When applied to the solid-to-shell homogenization of a sandwich structure with unequal skins, the MPBCs enable the computation of all terms of the fully-populated ABD matrix with negligible error, of the order of machine precision. Furthermore, using the MPBCs it is possible to correctly simulate the mechanical response of periodic structures using rUCs (retrieving the same results as if conventional UCs were used), thus enabling a significant reduction of both modelling/meshing and analysis CPU times. The results of these analyses demonstrate the relevance of the proposed approach for an efficient multiscale modelling of periodic materials and structures.