Some closed-form solutions for effective moduli of composites containing randomly oriented short fibers

This investigation presents an approach to obtain the effective moduli of both two- and three-dimensional, randomly oriented composites in terms of the shape and volume fraction of fibers. The composite fiber is treated as an spheroidal inclusion that enables its geometry ranging from short fiber to continuous fiber. To simulate spatial fiber orientation, a probability density function controlled by two Euler angles is introduced. Furthermore, based upon the Mori-Tanaka mean-field theory to account for the interaction between the fibers and matrix, an analytical approach is developed to assess the effective moduli of composites containing randomly oriented short fibers. In particular, when the fibers are uniformly distributed over a given region, closed-form solutions for the effective moduli of a two-phase composite are obtained for four special distributions of fiber orientations. Both two- and three-dimensional random orientations, resulting respectively in a transversely isotropic and a fully isotropic composite, are analyzed explicitly. Numerical examples have been given for an E-Glass/Epoxy composite. Analysis results indicate that the effective moduli are strongly affected by the volume fraction, the aspect ratio, and the orientation of fibers.