Temperature derivatives of elastic stiffness derived from the frequency-temperature behavior of quartz plates

Abstract

Linear field equations for small vibrations superposed on thermally-induced deformations by steady and uniform temperature changes are derived from the nonlinear field equations of thermoelasticity in Lagrangian formulation. From the solutions of these equations for the thickness vibrations, the temperature derivatives of elastic stiffness are related analytically to the known or measured properties such as the second- and third-order elastic stiffnesses, thermal expansion coefficients, and temperature coefficients of frequency of quartz plates. Six values of the first temperature derivative C(1)pq and six values of the effective second temperature derivative C̃(2)pq are calculated from the temperature coefficients of frequency measured by Bechmann, Ballato, and Lukaszek for various doubly-rotated quartz plates. The presently calculated values are compared with the first temperature derivatives obtained by Sinha and Tiersten. In the incremental stress-strain-temperature relations, certain expressions involving the elastic stiffnesses, temperature derivatives, and thermal expansion coefficients can be identified as having similar significance as the temperature coefficients of Cpq defined by Bechmann. Values of these expressions are calculated and compared with the existing values. The loci of the zeros of the first and second order temperature coefficients of frequency for thickness shear (B and C) modes and the frequency-temperature characteristics of the LC cut are studied and compared with experimental values.