New bounds on effective elastic moduli of two-component materials

Abstract

Based on the perturbation solution, we derive new bounds on the effective moduli of a two-component composite material which are exact up to fourth order in $\delta\mu$ = $\mu_1$ - $\mu_2$ and $\delta\kappa$ = $\kappa_1$ - $\kappa_2$, where $\mu_i$ and $\kappa_i$, i = 1, 2, are the shear and bulk modulus, respectively, of the phases. The bounds on the effective bulk modulus involve three microstructural parameters whereas eight parameters are needed in the bounds on the effective shear modulus. For engineering calculations, we recommend the third-order bounds on the effective shear modulus which require only two geometrical parameters. We show in detail how Hashin-Shtrikman's bounds can be extended and how Walpole's bounds can be improved using two inequalities on the two geometrical parameters that appear in the third-order bounds on the effective shear modulus. The third- and fourth-order bounds on the effective moduli are shown to be more restrictive than, or at worst, coincident with, existing bounds due to Hashin and Shtrikman, McCoy, Beran and Molyneux and Walpole.