Effective elastic moduli of two-phase transversely isotropic composites with aligned clustered fibers

Abstract

In this paper, we address the issue of the effective elastic moduli of transversely isotropic composites reinforced with aligned clustered continuous fibers. “Clustering” implies that there are portions of the matrix with a dense reinforcement of fibers and other portions with a sparse reinforcement. The clustering effect is characterized by a probability density distribution in “local” fiber volume fractions, obtained from the Dirichlet tessellation of a microstructure. Using a combination of Christensen and Lo's solution of a 3-phase boundary value problem and Hill's self-consistent method, the effective moduli are derived in terms of the probability density distribution function. It is shown that a unimodal distribution (representative of a random microstructure) has a modest effect on the effective moduli whereas a bimodal distribution (representative of a clustered microstructure) has a significant effect over a wide range of inclusion/matrix properties. A parametric study demonstrates that clustering has a significant effect on the shear moduli and the plane strain bulk modulus of the transversely isotropic composite and has a negligible effect on the longitudinal Young's modulus and the major Poisson's ratio. The theory has been compared with the Hashin-Rosen [1] bounds (appropriately modified for the clustered microstructure) and the classical Hashin-Shtrikman [2] bounds, and the theoretical predictions have been found to be bracketed by both bounds. In addition, the plane strain bulk modulus of a sample clustered periodic microstructure is computed by the developed theory and also by the finite element analysis, and the modulus computed by both approaches demonstrates a sensitivity to clustering.