New Third-Order Bounds on the Effective Moduli of n-Phase Composites (original) (raw)

We develop some new bounds on the effective moduli of TV-phase composites. These new bounds are accurate up to and including terms of third order in 0(|-Kj\, |^-Hj\), where Kt and n, are the bulk and shear modulus, respectively, of phase i. These bounds use the same statistical information as McCoy's and Beran-Molyneux's bounds but are tighter than, or at worst coincident with, the latter bounds. We also present in the appendix a new perturbation solution for the effective moduli which only requires that | <5|i | = 0(|-fij |) be small. 1. Introduction. We consider the theoretical determination of the effective moduli of a composite material. The composite material in question is comprised of N phases distributed in such a way that the overall material is homogeneous in a statistical sense. Each phase is assumed isotropically elastic and its Lame moduli are assumed known. The problem has a long history and has been reviewed by Hashin [1], Hale [2], Watt et al. [3] and McCoy [4]. In particular, we are concerned with the problem of determining bounds on the effective shear modulus ne and the effective bulk modulus Ke of the composite. These bounds may be conveniently classified by their width. That is, if the upper and lower bounds on Ke, say Ku and Kt, respectively, differ by a term of the order 0(<5v"+'), where