Reducing the domain in the mechanical analysis of periodic structures, with application to woven composites

A theoretical framework is developed leading to a sound derivation of Periodic Boundary Conditions (PBCs) for the analysis of domains smaller then the Unit Cells (UCs), named reduced Unit Cells (rUCs), by exploiting non-orthogonal translations and symmetries. A particular type of UCs, Offset-reduced Unit Cells (OrUCs) are highlighted. These enable the reduction of the analysis domain of the traditionally defined UCs without any loading restriction. The relevance of the framework and its application is illustrated through practical examples: 2D woven laminates and 3D woven composites.The framework proposed is used to develop an algorithm that automatically: (i) selects the smallest rUC for a given loading, (ii) determines and (iii) applies the appropriate periodic boundary conditions to the Finite Element Model (FEM). Coupled with a modelling/meshing tool, it provides a strategy for the efficient automatic numerical modelling and analysis of periodic structures. The algorithm is applied and validated for 2D woven orthogonal weaves. The results suggest the relevance of the proposed algorithm towards the efficient multiscale modelling of this class of materials.