The effect of viscosity on thickness-shear and flexural vibrations of crystal plates

The vibration analysis of crystal plates has been an essential part of quartz crystal resonator design with Mindlin plate theory for high frequency vibrations. As the vibration frequency of quartz crystal resonators continues climbing to meet new application requirements and resonator design to accommodate more restrictions on device parameters, additional considerations of material properties like the viscosity will provide new ways to study some known phenomena such as nonlinear effect and estimate resonator parameters that are important in circuit applications. With Mindlin plate theory, we consider the elastic constants of quartz crystal are viscous through adding the frequency dependent imaginary part with known experimental data. Then the vibrating plate has complex elastic constants and is frequency dependent with slight difference from linear elastic constants we are familiar with. These equations will give dispersion relations and frequency spectra with frequency dependence. We found that the inclusion of complex elastic constants does not change the dispersion relation, consequently the frequency spectra, significantly due to the relatively small viscosity in quartz crystal. However, these results, including the new equations, are essential in the modeling of crystal resonators working at high frequency and estimating basic circuit parameters such as the resistance and quality factor based on material property. These results are also useful in the resonator design and applications, and the new parameters of resonator performance are also important in the continuous efforts in the study of the nonlinear effect of crystal resonators that is believed to be related to the viscosity effect.