Bounds for Effective Bulk Modulus of Heterogeneous Materials

Bounds for the effective bulk modulus for statistically isotropic and homogeneous materials have been developed in terms of statistical information, i.e., one‐point and three‐point correlation function from variational principles. Aside from the one‐point correlation function, i.e., the volume fraction, this statistical information is difficult or impossible to obtain for real materials. For a broad class of heterogeneous materials which we shall call cell materials, the functions of the three‐point correlation function that appear in the bounds of effective bulk modulus are simply a number for each phase. Furthermore, this number has a range of values $\frac{1}{9}$ to ⅓ and a simple geometric significance. The number $\frac{1}{9}$ implies a cell of spherical shape, the number ⅓ a cell of plate‐like shape, and all other cell shapes, no matter how irregular, have a corresponding number between. Each value of this number determines a new set of bounds which are substantially narrower and always within the best bounds for two‐phase media in terms of volume fraction alone (i.e., Hashin‐Shtrikman bounds). For dilute suspensions the new bounds are so narrow in most cases as to be essentially an exact solution. There is a substantial improvement over previous bounds for a finite suspension and yet greater improvement for multiphase materials where the geometric characteristics of each phase are known. The shape factor G is found to have exactly the same range of numerical values and the same geometric significance as was found in the determination of effective dielectric constant bounds. It was found further that under certain conditions the bounds on effective bulk modulus and dielectric constant become numerically identical.