Effective elastic moduli of two-phase transversely isotropic composites with aligned clustered fibers
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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 bounds (appropriately modified for the clustered microstructure) and the classical Hashin-Shtrikman 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.
