APPENDIX B: Bounds of Dederichs and Zeller for Multiphase Media

One of the bounds of Dederichs and Zeller (1973) is based on the assumption that, inside each grain of a multiphase material, the distribution of different phases is independent of the shape of the grain, and also independent of the phases of contiguous grains. Grains are therefore assumed to be completely uncorrelated, both internally and externally. The results obtained for bulk modulus are:  

$$K_{DZ}^\pm = \left[\sum_{n=1}^N \frac{f_n}{K_n +4g_\pm/3}\right]^{-1} - 4g_\pm/3,$$ (26)

where  

$$g_- = c_{44} \qquad\hbox{and}\qquad g_+ = c_{66}$$ (27)

in our present notation [see eq. (7)]. Similarly, for shear modulus, we have  

$$\mu_{DZ}^\pm = \left[\sum_{n=1}^N \frac{f_n}{\mu_n +\zeta_\pm}\right]^{-1} - \zeta_\pm,$$ (28)

where  

$$\zeta_- = \frac{c_{44}}{6} \frac{\left<9/\mu + 8/K\right\gt... ...right\gt^{-1} + 8g_-} {\left<1/K\right\gt^{-1} + 2g_-}\right)$$ (29)

and  

$$\zeta_+ = \frac{c_{66}}{6} \frac{\left<9K + 8\mu\right\gt}{\... ...ac{9\left<K\right\gt + 8g_+}{\left<K\right\gt + 2g_+}\right).$$ (30)

These bounds on bulk modulus are the same as those of Beran and Molyneux (1966) and Miller (1969). The upper bound on shear modulus is the same as that of McCoy (1970) and Silnutzer (1972). Because of the simple functional form of both sets of bounds, it is easy to show (Berryman, 1982) that they are always at least as restrictive as -- and, for nonnegligible volume fractions of inclusions, normally a significant improvement upon -- the Hashin-Shtrikman bounds (Hashin and Shtrikman, 1962; 1963).

We chose to consider these bounds here because they depend only on simple volume averages of the constituent elastic constants, and also because they show -- by way of contrast to the other bounds (see Figures 1 and 2) -- that it does indeed matter what assumptions are made about the microstructure of the composite.