Simple formulae for the effective moduli of unidirectional aligned composites

Simple closed form expressions for the effective tensile moduli of an unidirectionally aligned two phase composite reinforced by fiber-like or flake-like inclusions of aspect ratio a are obtained. The case of alpha much greater than 1 corresponds to fiber-like inclusions whereas the case of alpha much less than 1 corresponds to flake-like inclusions. These expressions are based on the exact results obtained by Tandon and Weng, who used the Mori-Tanaka method to obtain the effective moduli of such composites. These expressions are then compared with various empirically based formulae such as the Halpin-Tsai equation and the Modified Rule of Mixtures equation. It is found that very large discrepancies result when the modulus ratio E-0/E-1 is sufficiently small compared with the aspect ratio of the flake-like inclusions, where E-1 and E-0 are the Young's modulus of the inclusions and the matrix, respectively. Simple criteria for the validity of the rule of mixtures relation are also given. Also, the predictions of the simple closed form expressions are compared to available experimental results and favorable agreements are obtained.