Correction Factors for Mindlin Higher-order Plate Theory with the Consideration of Electrodes

Mindlin plate theory has been the choice for the analysis of high frequency vibrations of piezoelectric quartz crystal resonators, which utilize electroded crystal plates vibrating at the thickness-shear and overtone modes to achieve frequency generation functions and applications. For this purpose, Mindlin plate equations have been simplified, modified, and corrected to accommodate the actual geometry, structural complications, practical boundary conditions, and various bias fields such as temperature change and initial stresses. Accordingly, the correction of Mindlin plate theory should also include complications through factors which adjust the thickness-shear frequencies to the exact three-dimensional solutions from earlier studies with the first-order plate equations. In recent studies, correction factors for equations up to the third-order theory have been introduced and they have the capability to make the cut-off frequencies for the thickness-shear modes accurate up to much higher-orders and improving the accuracy of the extensional mode groups also. These correction factors are derived based on the general principle of comparison and matching of exact frequencies of interested vibration modes in the coupled two-dimensional equations, and the inclusion of structural complications such as electrodes is an natural extension of the derivation with practical importance. In this study, we consider the effect of thin electrodes through their mass loading formulation by adding the mass ratio in the inertia terms as demonstrated by Mindlin and others. Consequently, a set of frequency equations for both thickness-shear and extensional modes are established with the inclusion of mass ratio of electrodes. These equations are solved numerically for mass ratios to obtain the corresponding correction factors which have been modified by the presence of electrodes. These correction factors can be utilized in the improvement of applications of the higher-order plate equation- s with the finite element implementation, which remains as one of the reliable tool in the precise analysis and design of quartz crystal resonators. The derivation of these correction factors can be further incorporated into nonlinear equations of piezoelectric solids.