| In order to facilitate the design of quartz resonators, the overall goal of the current work was to develop an accurate three-dimensional finite element model for the anisotropic frequency response of quartz. The model was based on the linear incremental equations for superimposed small vibrations onto nonlinear thermoelastic stressed media. The frequency response of the model was benchmarked to experimental data from quartz pressure sensors with temperature ranging from 50 °C to 200 °C and pressure from 14 psi to 20,000 psi. This direct finite element approach for frequency response at such high pressure had not been previously examined in literature.;The normalized frequency response to the change in external pressure from 14 psi to 20,000 psi matched very well with experimental data for lower temperatures, having a maximum deviation of only 7.5% at 20,000 psi when assuming constant 50 °C temperature. However, the same deviation grew to 25.7% assuming a higher 200 °C constant temperature. Similarly, the temperature-frequency response at constant pressure from 50 °C to 200 °C matched the experimental trend well for lower pressures, but this agreement deteriorated as pressure increased.;The nature of the observed frequency deviations suggests that changes in the third-order elastic constants with temperature, a quartz material definition that is not currently available in literature, could play a significant role in accurately modeling the frequency response at such conditions, and that the lack of such properties is the primary source of the error in both temperature and pressure response. This hypothesis was tested by using a novel method of giving the third-order elastic constants linear temperature dependence based on a single scalar parameter, for which a specific value was empirically derived for the AT-Cut quartz pressure sensor studied. Specifically, modeling a scalar decrease in magnitude of about 0.0775% per °C in the third-order elastic coefficients was shown to decrease the error of the simulation at the highest temperature and pressure from 25.7% to 4.0%, and this improvement was mirrored throughout the range of temperatures and pressures tested. Furthermore, knowing the expected benefit of implementing the third-order elastic coefficients as functions of temperature should aid future researchers in deciding if defining their full anisotropic temperature derivatives is practical, and relevant possibilities for such a study are given. |
