Average stress in the matrix and effective moduli of randomly oriented composites

By considering the variation of average stress in the matrix, some elastic properties of randomly oriented composites are established as a function of aspect ratio. Both three- and two-dimensional random orientations, resulting, respectively, in a complete and transverse isotropy, are considered. As the shape of inclusions changes, the isotropic bulk and shear moduli are shown to vary within the Hashin-Shtrikman bounds. The aspect ratio dependence of the five in-plane and out-of-plane moduli with planar orientations are also explicitly given; these results suggest that the in-plane properties are most effectively reinforced by fibrous inclusions, whereas the out-of-plane ones are more responsive to the disc type. The accompanying variations of average stress in the matrix are seen to be closely related to the corresponding variations of the moduli. Comparisons with some limited experimental data also show a reasonable agreement.