Recent advances in computational speed have resulted in the ability to model composite materials using larger representative volume elements (RVEs) with greater numbers of inclusions than have been previously studied. It is often necessary to assume periodicity for the effective evaluation of material properties, failure analysis, or constitutive law development for composite materials. Imposing periodic boundary conditions on very large RVEs can mean enforcing thousands of constraint equations. In addition, most commercial finite element codes incorporate these constraints on a global level, thereby drastically reducing computational speed. The present study investigates a method that uses a local implementation of the constraints that does not adversely affect the computational speed. As a step toward a three-dimensional formulation, the present study utilizes a two-dimensional triangular RVE of a periodically-spaced regular hexagonal array of composite material containing fibers of equal radii. In the present study, the finite element method is employed to obtain the response of the RVE. To impose the boundary conditions along the edges, this study utilizes a cubic interpolant to model the displacement field along the matrix edges and a linear interpolant to model the field along the fiber edges. The method eliminates the need for the conventional node-coupling scheme for imposing periodic boundary conditions, consequently reducing the number of unknowns to the interior degrees of freedom of the RVE along with a finite number of global parameters. The method results in a valuable computational savings that greatly simplifies the pre-processing stage of the analysis.
Classical and enriched finite element formulations for Bloch-periodic boundary conditions