target="_blank" rel="nofollow" href="#fb3_img_img_ca450434-cf9c-59e8-a000-2dbefd7d0e17.png" alt="alpha overbar plus StartFraction left-parenthesis k Subscript p Baseline minus k Subscript m Baseline right-parenthesis left-parenthesis alpha Subscript p Baseline minus alpha Subscript m Baseline right-parenthesis upper V Subscript p Baseline upper V Subscript m Baseline Over upper V Subscript p Baseline k Subscript p Baseline plus upper V Subscript m Baseline k Subscript m Baseline plus k Subscript m Baseline k Subscript p Baseline slash left-parenthesis four-thirds mu Subscript p Baseline right-parenthesis EndFraction less-than-or-equal-to alpha Subscript eff Baseline less-than-or-equal-to alpha overbar plus StartFraction left-parenthesis k Subscript p Baseline minus k Subscript m Baseline right-parenthesis left-parenthesis alpha Subscript p Baseline minus alpha Subscript m Baseline right-parenthesis upper V Subscript p Baseline upper V Subscript m Baseline Over upper V Subscript p Baseline k Subscript p Baseline plus upper V Subscript m Baseline k Subscript m Baseline plus k Subscript m Baseline k Subscript p Baseline slash left-parenthesis four-thirds mu Subscript m Baseline right-parenthesis EndFraction period"/>(3.66)
Walpole [7, Equation (26)] has derived rigorous bounds for the effective shear modulus, which can, for an isotropic two-phase composite, be expressed in the following form having the same structure as the result (3.59)
where μmin* and μmax* are defined by (3.49). The structure of (3.67) is identical to that given by Torquato [2, Equations (21.73)–(21.75)].
To conclude this section summarising results, it is useful to provide the relationships between the bulk and shear moduli and the elastic constants which are more frequently encountered in applications. It follows from (2.208) that the effective Young’s modulus Eeff and effective Poisson’s ratio νeff for an isotropic composite are given by
3.7 Comparison of Predictions with Known Results
When assessing the validity of undamaged particulate composites, it is particularly valuable to compare predictions using formulae for the relevant effective properties with those obtained from the use of alternative methods. This procedure can provide the confidence for use of the formulae in practical situations, and this approach to validation is now followed. When considering the effective bulk modulus, thermal expansion coefficient and thermal conductivity for two-phase composites having spherical particles of the same size, the results obtained using Maxwell’s methodology are identical to the realistic bounds. They are also identical to estimates for effective properties obtained by applying the composite spheres assemblage model (see the review by Hashin [1]) for a particulate composite to a representative volume element comprising just one particle and a matrix region that is consistent with the volume fractions of the composite. In the case of shear modulus, the result (3.58) obtained using Maxwell’s methodology, corresponds exactly to one of the variational bounds whenever (kp−km)(μp−μm)≥0. For the case (kp−km)(μp−μm)≤0, it can be shown that μmin*≤μm*≤μmax* and a comparison of (3.59) and (3.67) then indicates that the result (3.58) for the effective shear modulus derived using Maxwell’s methodology must lie between the bounds (3.67). In addition, the results (3.55–3.58), for two-phase composites arising from the use of Maxwell’s methodology, are such that κeff→κp, keff→kp, μeff→μp and αeff→αp, respectively, when Vp→1, limits requiring Vp values attained only for a range of particle sizes, as for the composite spheres assembly model.
Effective properties of two-phase composites, derived using Maxwell’s methodology, may be expressed as a mixtures estimate plus a correction term, as seen from (3.55)–(3.58). The correction is always proportional to the product VpVm, and it involves the square of property differences for the case of conductivity, bulk and shear moduli, and the product of differences of the bulk compressibility and expansion coefficient for the case of thermal expansion. These results are the preferred common form for effective properties, having the advantage that conditions governing whether an extreme value is an upper or lower bound are then easily determined. In addition, such conditions determine when both upper and lower bounds coincide with each other, and with predictions based on Maxwell’s methodology, leading to exact nontrivial predictions for all volume fractions. For example, when μp=μm the bounds for bulk modulus given by (3.62) are equal to the exact solution for any values of kp, km and the volume fractions, and they are equal to the result (3.56) indicating that Maxwell’s methodology leads, in this special nontrivial case, to an exact result for all volume fractions for which the composite is isotropic. For the case of thermal expansion, it follows from (3.65) and (3.66) that exact results are also obtained for any values of kp, km, αp, αm and Vp, and they are equal to (3.57) indicating that Maxwell’s methodology again leads, in a special nontrivial case, to an exact result for all volume fractions.
Results for effective properties of two-phase composites, are such that Maxwell’s methodology, the composite spheres assemblage model when it can generate exact results, and the realistic variational bound, all lead to the same result. This suggests very strongly that the realistic bound is a much better estimate of effective properties for spherical particles than the other bound, which can be obtained simply by interchanging particle and matrix properties and the volume fractions. Further evidence of this phenomenon is provided in Figures 3.2–3.4 and Tables 3.1 and 3.2, where use has been made of the conductivity results of Sangani and Acrivos [10], based on the use of spherical harmonic expansions, and the results of Arridge [11] who used harmonics up to 11th order to estimate accurate values of bulk modulus and thermal
