New third-order bounds on the effective moduli of $N$-phase composites

We develop some new bounds on the effective moduli of $N$-phase composites. These new bounds are accurate up to and including terms of third order in $O\left ( {\left | {{K_i} - {K_j}} \right |,\left | {{\mu _i} - {\mu _j}} \right |} \right )$, where ${K_i}$ and ${\mu _i}$ 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 $\left | {\delta \mu } \right | = O\left ( {\left | {{\mu _i} - {\mu _j}} \right |} \right )$ be small.