The effect of spatial distribution on the effective behavior of composite materials and cracked media

Abstract

Estimates of the Hashin-Shtrikman type are developed for the overall moduli of composites consisting of a matrix containing one or more populations of inclusions, when the spatial correlations of inclusion locations take particular "ellipsoidal" forms. Inclusion shapes can be selected independently of the shapes adopted for the spatial correlations. The formulae that result are completely explicit and easy to use. They are likely to be useful, in particular, for composites that have undergone a prior macroscopically uniform large deformation. To the extent that the statistics that are assumed may not be realized exactly, the formulae provide approximations. Since, however, they are derived as variational approximations for composites with some explicit statistics that are realizable, they are free from some of the drawbacks of competitor approximations such as that of Mori and Tanaka (1973 Acta Metall. 21, 571-574), which can generate tensors of effective moduli which fail to satisfy a necessary symmetry requirement. The new formulae are also the only ones known that take explicit account, at least approximately, of inclusion shape and spatial distribution independently. © 1995.