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

We develop some new bounds on the effective moduli of N N -phase composites. These new bounds are accurate up to and including terms of third order in O ( | K i − K j | , | μ i − μ j | ) O ( { | {{K_i} - {K_j}} |, | {{μ _i} - {μ _j}} |} ) , where K i {K_i} and μ i {μ _i} are the bulk and shear modulus, respectively, of phase i 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 | δ μ | = O ( | μ i − μ j | ) | {δ μ } | = O ( { | {{μ _i} - {μ _j}} |} ) be small.