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 expansion for body-centred cubic (b.c.c.) and face-centred cubic (f.c.c.) arrays of spherical particles having the same size. Whereas Sangani and Acrivos considered simple cubic, b.c.c. and f.c.c. arrays of spheres, only the f.c.c. results are shown in Figure 3.2, for the values κp/κm=0.01,10,∞ of the phase contrast, as a larger range of volume fractions can be considered. It is seen that there is excellent agreement between predictions based on Maxwell’s result (3.55) and the results of Sangani and Acrivos for a wide range of particulate volume fractions. Results (not shown) indicate that the agreement is less good at large volume fractions of particulate, when comparing Maxwell’s result with the simple-cubic and b.c.c. results of Sangani and Acrivos.
Figure 3.2 Dependence of ratio of effective and matrix thermal conductivities for a two-phase composite on particulate volume fraction for a face-centred cubic array of spherical particles, at various phase contrasts.
Figure 3.4 Dependence of the effective thermal expansion coefficient for a two-phase composite on particulate volume fraction (see Table 3.2 for numerical values).
Table 3.1 Estimates of effective bulk modulus (GPa) for a two-phase particulate composite.
| Vp | Maxwell’s Methodology | Arridge [11] (f.c.c.) | Arridge [11] (b.c.c.) | Torquato [13] Lower bound |
|---|---|---|---|---|
| 0 | 80.56 | 80.56 | 80.56 | 80.56 |
| 0.1 | 88.06 | 88.01 | 88.09 | 88.09 |
| 0.2 | 96.61 | 96.47 | 96.67 | 96.71 |
| 0.3 | 106.42 | 106.28 | 106.56 | 106.66 |
| 0.4 | 117.80 | 117.76 | 118.12 | 118.23 |
| 0.5 | 131.16 | 131.56 | 131.90 | 131.83 |
| 0.6 | 147.07 | 148.79 | 148.76 | — |
| 0.6802 (max.) | 162.20 | — | 165.39 | — |
| 0.7 | 166.33 | 171.67 | — | — |
| 0.7405 (max.) | 175.33 | 183.54 | — | — |
| 0.8 | 190.12 | — | — | — |
| 0.9 | 220.26 | — | — | — |
Table 3.2 Estimates of thermal expansion coefficient (×106 K –1) of a two-phase particulate composite.
| Vp | Maxwell’s Methodology | Arridge [11] (f.c.c.) | Arridge [11] (b.c.c.) | Torquato [13] Upper bound |
|---|---|---|---|---|
| 0 | 22.5 | 22.5 | 22.5 | 22.5 |
| 0.1 | 20.13 | 20.1 | 20.1 | 20.12 |
| 0.2 | 17.87 | 17.9 | 17.9 | 17.85 |
| 0.3 | 15.73 | 15.8 | 15.7 | 15.69 |
| 0.4 | 13.70 | 13.7 | 13.7 | 13.63 |
| 0.5 | 11.76 | 11.7 | 11.7 | 11.67 |
| 0.6 | 9.91 | 9.7 | 9.7 | — |
| 0.6802 (max.) | 8.49 | — | 8.1 | — |
| 0.7 | 8.15 | 7.7 | — | — |
| 0.7405 (max.) | 7.45 | 6.9 | — | — |
| 0.8 | 6.46 | — |