10 nu Subscript m Baseline right-parenthesis mu Subscript eff Baseline plus left-parenthesis 7 minus 5 nu Subscript m Baseline right-parenthesis mu Subscript m Baseline EndFraction right-bracket comma"/>(3.43)
where μeff is the effective shear modulus of the isotropic particulate composite. It then follows from (3.38) and (3.40) that the exact matrix stress distribution, for a given value of μeff, is
As the stress distribution given by (3.41) must be identical at large distances from the cluster with that specified by (3.44) it follows, from a consideration of terms proportional to r−3, that
where use has been made of (3.1). On substituting (3.42) and (3.43) into (3.45), it can be shown using (3.1) that the following ‘mixtures’ result is obtained for the function 1/(μ+μm*)
On using (3.1), the effective shear modulus may be estimated using the following relation
It can be shown that the bounds for the effective shear modulus derived by Hashin and Shtrikman [6, Equations (3.44)–(3.50)] and the bounds derived by Walpole [7, Equation (26)] are identical and may be expressed in the following form that has the same structure as the result (3.46) derived using Maxwell’s methodology
The parameters kmin and μmin are the lowest values of the bulk and shear moduli of all phases in the composite, respectively, whereas kmax and μmax are the highest values. On writing