Ultrasonic data and the thermodynamics of solids

This is primarily a tutorial paper on the use of ultrasonic velocity data, in conjunction with data on the specific heat and thermal expansion, to find the adiabatic and isothermal elastic coefficients and their first derivatives with respect to pressure and temperature. Introductory sections on the equations of continuum mechanics and on isentropic propagation of small-amplitude waves establish the basis for subsequent derivations. Two sections on established loss mechanisms illustrate some limitations of the assumption of isentropic propagation used in the remainder of the paper, and tend to justify this assumption for moderate frequencies. One of these sections derives the classical solution for the loss arising from heat conduction in a thermoelastic but nonviscous anisotropic solid. In this case, instead of assuming isentropic changes of state, the energy equation is used to determine the nature of the changes which take place. Another brief section discusses losses of a viscous type. Except for a final section which reviews some applications of third-order elastic coefficients, the remainder of the paper deals with the interpretation of data on the transit times for ultrasonic waves ("velocity data"). It is proved that the bulk modulus at any pressure is related to the effective elastic coefficients for wave propagation at that pressure by the same formula as at zero pressure. Hence, the correct derivative of the bulk modulus is obtained by differentiating the formula, interpreting the derivatives as derivatives of the effective elastic coefficients. Formulas and numerical examples are given for the conversion from the adiabatic to the isothermal bulk modulus and the conversion from pressure and temperature derivatives of the adiabatic bulk modulus to pressure and temperature derivatives of the isothermal bulk modulus. General formulas for the processing of ultrasonic data are given and specialized to cubic crystals. It is emphasized that all the first pressure derivatives of effective elastic coefficients, evaluated at zero pressure, can be calculated from the data without calculating the path length and density at any elevated pressure.