Derivation of Wachtman's Equation for the Temperature Dependence of Elastic Moduli of Oxide Compounds

Abstract

Recently, Wachtman and his colleagues showed that a good description of the temperature dependence of the Young's modulus of several oxides is $Y=Y_0-b_{1}T\exp (-\frac{T_0}{T})$ where $Y_0$ is the modulus at absolute zero and $b$ and $T_0$ are constants. Wachtman and his colleagues pointed out that they had no theoretical basis for this equation, but demonstrated that it worked very well over a large temperature range (1000°K), and they discussed the physical basis for the parameters $b_1$ and $T_0$. In this paper we show that the form of their equation for $Y$ is equivalent to an equation for the bulk modulus derived from the Mie-Grüneisen equation of state. The Wachtman empirical equation for Young's modulus apparently is valid only when the variation of Poisson's ratio with temperature is small, for in that case $\frac{d\ln E}{\mathrm{dT}}$ is equal to $\frac{d\ln B}{\mathrm{dT}}$, where $B$ is the bulk modulus.