A Study of the 1 + 2 Partitioning Scheme of Fibrous Unitcell under Reduced-Order Homogenization Method with Analytical Influence Functions

The multiscale computational method with asymptotic analysis and reduced-order homogenization (ROH) gives a practical numerical solution for engineering problems, especially composite materials. Under the ROH framework, a partition-based unitcell structure at the mesoscale is utilized to give a mechanical state at the macro-scale quadrature point with pre-evaluated influence functions. In the past, the “1-phase, 1-partition” rule was usually adopted in numerical analysis, where one constituent phase at the mesoscale formed one partition. The numerical cost then is significantly reduced by introducing an assumption that the mechanical responses are the same all the time at the same constituent, while it also introduces numerical inaccuracy. This study proposes a new partitioning method for fibrous unitcells under a reduced-order homogenization methodology. In this method, the fiber phase remains 1 partition, but the matrix phase is divided into 2 partitions, which refers to the “1 + 2” partitioning scheme. Analytical elastic influence functions are derived by introducing the elastic strain energy equivalence (Hill-Mandel condition). This research also obtains the analytical eigenstrain influence functions by alleviating the so-called “inclusion-locking” phenomenon. In addition, a numerical approach to minimize the error of strain energy density is introduced to determine the partitioning of the matrix phase. Several numerical examples are presented to compare the differences among direct numerical simulation (DNS), “1 + 1”, and “1 + 2” partitioning schemes. The numerical simulations show improved numerical accuracy by the “1 + 2” partitioning scheme.