Theoretical aspects of selecting repeated unit cell model in micromechanical analysis using displacement-based finite element method

Abstract

Repeated Unit Cell (RUC) is a useful tool in micromechanical analysis of composites using Displacement-based Finite Element (DFE) method, and merely applying Periodic Displacement Boundary Conditions (PDBCs) to RUC is almost a standard practice to conduct such analysis. Two basic questions arising from this practice are whether Periodic Traction Boundary Conditions (PTBCs, also known as traction continuity conditions) are guaranteed and whether the solution is independent of selection of RUCs. This paper presents the theoretical aspects to tackle these questions, which unify the strong form, weak form and DFE method of the micromechanical problem together. Specifically, the solution’s independence of selection of RUCs is dealt with on the strong form side, PTBCs are derived from the weak form as natural boundary conditions, and the validity of merely applying PDBCs in micromechanical Finite Element (FE) analysis is proved by referring to its intrinsic connection to the strong form and weak form. Key points in the theoretical aspects are demonstrated by illustrative examples, and the merits of setting micromechanical FE analysis under the background of a clear theoretical framework are highlighted in the efficient selection of RUCs for UniDirectional (UD) fiber-reinforced composites.