Available theories for the variation of elastic moduli in two-phase materials are reviewed. It appears that only three rigorous independent solutions currently exist: the upper Voigt limit, the lower Reuss limit, and the interdependent boundaries obtained by Hashin and Shtrikman. The lower Hashin-Shrikman boundary appears to best describe the behavior of elastic properties over the complete range between the end-members. This boundary was previously derived by Kerner for the case of spherical particles. Plots of moduli against volume fraction always have positive (concave-upward) curvature, with the exception of the linear behavior of the Voigt model. In terms of weight percent, the curvatures may be positive, negative, or zero, according to the relative values of the densities and moduli of the end-members. Sonic velocities and adiabatic moduli are reported for PbOB2O3, Li2OSiO2, and PbOSiO2 glasses. The Hashin-Shtrikman/Kerner relations describe the moduli fairly well over known or suspected two-phase composition regions. Maxima and minima previously reported for certain properties in the LiOSiO2 and the PbOSiO2 systems were not verified in this investigation. The Hashin-Shtrikman/Kerner relations fit the experimental data fairly well over the two-phase regions in the sodium silicate system and in the alkali borate glass systems, with the fit being poorest for the potassium borates. No immiscibility is expected near 12 wt % (24 mole %) Na2O in the Na2OP2O2O5 system, or over the range 0-10 wt% Na2O(O-16 mole % Na2O) in the Na2OGeO2 system.
ASJC Scopus subject areas
- Electronic, Optical and Magnetic Materials
- Ceramics and Composites
- Condensed Matter Physics
- Materials Chemistry