Micro-macromechanical analysis of heterogeneous materials: Macroscopically homogeneous vs periodic microstructures

A complete set of engineering moduli for two types of unidirectional composites with large fiber/matrix property contrasts was generated using the finite-element approach based on three sets of boundary conditions employed to calculate macroscopic moduli of statistically homogeneous and periodic heterogeneous materials. The boundary condition-dependent differences in the generated moduli highlight the differences between representative volume element and repeating unit cell concepts, which continue to be used interchangeably in the composite mechanics community. Homogeneous boundary conditions, which underpin the concept of a representative volume element, produce apparent engineering moduli that are known to converge asymptotically to effective moduli of a periodic composite from below and above with increasing number of uniformly-spaced inclusions at a rate that depends on the inclusion/matrix property contrast and on the particular modulus. Herein, new results are presented which demonstrate that the quality of the effective moduli estimates depends both on the type of homogeneous boundary conditions and on the particular engineering modulus regardless of the inclusion/matrix property mismatch. Further, not all effective engineering moduli are bounded by the apparent moduli obtained under homogeneous displacement and traction boundary conditions. The generated effective moduli are subsequently employed in a macrostructural problem simulating a simple tension test in order to estimate the accuracy of the effective modulus concept in a multiscale localization analysis setting. For the considered problem, homogenization-based localization analysis of microlevel stress fields produces accurate values relative to the fully heterogeneous results outside of a very small layer spanning one row/column of single-inclusion boundary unit cells.