Subscript m Baseline left-parenthesis b slash r right-parenthesis Superscript 5 Baseline zero width space zero width space plus 2 left-parenthesis 1 plus nu Subscript m Baseline right-parenthesis upper D overbar Subscript m Baseline left-parenthesis b slash r right-parenthesis cubed right-bracket sine 2 theta sine 2 phi comma 3rd Row sigma Subscript r phi Superscript m Baseline equals 2 mu Subscript m Baseline left-bracket gamma minus left-parenthesis 24 slash 5 right-parenthesis upper D overbar Subscript m Baseline left-parenthesis b slash r right-parenthesis Superscript 5 Baseline zero width space zero width space plus 2 left-parenthesis 1 plus nu Subscript m Baseline right-parenthesis upper D overbar Subscript m Baseline left-parenthesis b slash r right-parenthesis cubed right-bracket sine theta cosine 2 phi comma EndLayout right-brace b less-than r less-than infinity period"/>(3.44)
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
it follows that μmax*≥μmin* for all values of the bulk and shear moduli, indicating that the ‘max’ and ‘min’ subscripts are used in an appropriate sense. It should be noted that kmin and μmin may be associated with different phases, and similarly for kmax and μmax.
3.5 Summary of Results
3.5.1 Multiphase Composites
Key results, (3.10) for thermal conductivity, (3.27) for bulk modulus and (3.46) for shear modulus, derived using Maxwell’s methodology have the following simple common structure, involving ‘mixtures’ formulae
