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 for the effective isotropic properties ϕ of the composite
The inequalities (3.12), (3.30) and (3.48), valid for all volume fractions, lead to rigorous bounds valid for any phase geometries that are statistically isotropic. They have the following common structure that is strongly related to the structure defined by (3.51) and (3.52) for effective properties determined using Maxwell’s methodology
By comparing the bounds (3.53) with (3.51) when J = 1, the result for thermal conductivity obtained using Maxwell’s methodology is exactly the lower bound for κeff when κmin=κm, and the upper bound when κmax=κm. When κmin<κm<κmax, the result for effective thermal conductivity obtained using Maxwell’s methodology lies between the bounds for all volume fractions. A comparison of (3.53) with (3.51) when J = 2 shows that the result obtained for bulk modulus using Maxwell’s methodology leads exactly to the lower bound for keff when μmin=μm, and the upper bound when μmax=μm. When μmin<μm<μmax, the result for effective bulk modulus obtained using Maxwell’s methodology lies between the bounds for all volume fractions. A comparison of (3.53) with (3.51) when J = 3 shows that the result (3.51) obtained for shear modulus using Maxwell’s methodology leads exactly to the lower bound for μeff when μmin*=μm*, and the upper bound when μmax*=μm*. When μmin*<μm*<μmax*, the result for effective shear modulus obtained using Maxwell’s methodology lies between the bounds for all volume fractions. Thus, it has been shown that effective thermoelastic properties, obtained above using Maxwell’s methodology, do not lie beyond rigorous bounds for properties for all volume fractions consistent with isotropic effective properties. This characteristic of Maxwell’s methodology provides significant evidence that its validity is not confined to small volume fractions.
3.5.2 Two-phase Composites
When N = 1, it follows from (3.2) and (3.11) that the result first derived by Maxwell [3] for the analogous case of electrical conductivity is obtained, which may be expressed in the form of a mixtures estimate plus a correction term so that
It follows from (3.28), (3.29) and (3.47) derived using