It is worth noting from Bonnecaze and Brady [12, Tables 2–4], who used a multipole method to estimate the conductivity of cubic arrays of spherical particles, that their results for the case that retains only dipole–dipole interactions correspond almost exactly (to three significant figures) with the results shown in Figure 3.2 obtained using Maxwell’s result (3.55). They did not make this comparison, considering only the results of Sangani and Acrivos [10]. This result suggests that Maxwell’s result, which was derived assuming particles do not interact, is in fact valid also for the case when particle interactions are represented by dipole–dipole interactions, and this might explain why Maxwell’s result is found to be a good approximation for a wide range of volume fractions. Further discussion of this issue is beyond the scope of this chapter. We note that for composites used in practice, the difference in the values of the thermomechanical properties (e.g. bulk modulus, thermal expansion coefficient) of the reinforcement and matrix, seldom lead to values of phase contrast that are greater than 10 or so. The phase contrast of the transport properties (such as electrical or thermal conductivity) can be very much greater. It follows that in practical situations, greater confidence may be placed in the Maxwell formulation being accurate at relatively large volume fractions for the thermomechanical properties when compared with the case of transport properties.
The results of Arridge [11] are based on the following properties for silicon carbide spheres in an aluminium matrix:
The corresponding values of bulk and shear moduli are kp=259.68GPa, km=80.56GPa, μp=202.94GPa, μm=26.85GPa, and are such that kp>km and μp>μm. It should be noted that an array of spheres in a b.c.c. or in an f.c.c. arrangement possesses cubic symmetry. The thermal expansion coefficient of such an array is, therefore, isotropic. Arridge’s results are given as mean values implying that the expansion coefficients differ slightly in various directions, a situation that could arise because an insufficient number of harmonics has been included in the representation.
Additional evidence concerning the accuracy of realistic bounds is given by Torquato [13] who considered the effect on bounds of geometrical factors relating to the reinforcement, and developed three-point bounds that are more restrictive than the conventional two-point bounds (equivalent to the Hashin–Shtrikman [6] bounds) for the case of bulk modulus and thermal expansion of suspensions of spheres. The definition, (Torquato [13], Equation (25)), has in fact been replaced by ζ2=(512−316ln3)Vp. Torquato’s [13] three-point bounds are compared in Figures 3.3 and 3.4 with the almost exact results of Arridge [11], and results obtained from Maxwell’s methodology and the two-point variational bounds. From (3.56) and (3.62), bulk modulus results using the Hashin–Shtrikman [6] lower bound and Maxwell’s methodology are identical as μp>μmfor Arridge’s properties. These results are very close to those obtained using the Arridge model for both f.c.c. and b.c.c. particle arrangements, and to the three-point lower bound estimate of Torquato [13]. The results of Arridge are shown for all volume fractions up to the closest packing value for f.c.c. and b.c.c. configurations of spherical particles. The f.c.c. and b.c.c. packing configurations lead to bulk moduli that are very close together for particulate volume fractions in the range 0 < Vp < 0.6. Furthermore, the results obtained using Maxwell’s methodology lie between the f.c.c. and b.c.c. estimates for volume fractions in the range 0 < Vp < 0.4. For a significant range of volume fractions, the Hashin–Shtrikman upper bound is seen in Figure 3.3 to be significantly different to the corresponding lower bound, and to the three-point upper bound of Torquato.
Figure 3.3 Dependence of effective bulk modulus for a two-phase composite on particulate volume fraction (see Table 3.1 for numerical values).
For the case of thermal expansion, the Hashin–Shtrikman [6] upper bound and Maxwell’s methodology result are identical as seen from (3.57) and (3.66), because for Arridge’s properties (kp−km)(μp−μm)(αp−αm)≤0. These results are seen in Figure 3.4 to be very close to those obtained using the Arridge model for both f.c.c. and b.c.c. particle arrangements, and to the three-point upper bound estimate of Torquato. The results of Arridge are again shown for all volume fractions up to the closest packing value for f.c.c. and b.c.c. configurations of spherical particles. The f.c.c. and b.c.c. packing configurations lead to expansion coefficients that are very close together, and very close to results obtained using Maxwell’s methodology, for particulate volume fractions in the range 0 < Vp < 0.5. For a significant range of volume fractions, the Hashin–Shtrikman lower bound is seen in Figure 3.4 to be significantly different to the corresponding upper bound, and to the three-point lower bound of Torquato. In view of the almost exact results of Arridge, and the observation that the three-point bounds for bulk modulus and thermal expansion derived by Torquato are reasonably close, it is deduced that Maxwell’s methodology provides accurate estimates of bulk modulus and thermal expansion coefficient for a wide range of volume fractions.
For the case of a simple cubic array of spherical particles with volume fractions in the range 0 < Vp < 0.4, Cohen and Bergman [14] (see Figure 3.4) have shown that bounds for shear modulus, obtained using a Fourier representation of an integrodifferential equation for the displacement field, are very close to the Hashin–Shtrikman lower bound when using properties for a glass–epoxy composite. The results of this chapter indicate that their bounds will also be very close to the result obtained using Maxwell’s methodology, showing again that its validity is not restricted to low volume fractions, as might be expected from the approximations made.
For the cases of bulk modulus and thermal expansion, Maxwell’s methodology is based on a stress distribution (3.24) in the matrix outside the sphere having radius b of effective medium, which is exact everywhere in the matrix (i.e. b<r<∞) and involves an r-dependence only through terms proportional to r−3. It follows from (3.23) that, for the discrete particle model (see Figure 3.1(a)), the asymptotic form for the stress field in the matrix as r→∞ has the same form as the exact solution for the equivalent effective medium model (see Figure 3.1(b)). The matching of the discrete and effective medium models at large distances, leading to an exact solution in the matrix (b<r<∞) of the effective medium model, is thought to be one reason why estimates for bulk modulus and thermal expansion coefficient of two-phase composites are accurate for a wide range of volume fractions. When estimating thermal conductivity using Maxwell’s