Consideration of stiffness and mass effects of relatively thicker electrodes with Mindlin plate theory

Mindlin plate theory has been widely used in the high-frequency vibrations of piezoelectric crystal plates with emphasis on its applications in crystal resonator analysis and design. The plate equations were derived without considering the effect of electrodes from the beginning. But continuing efforts have been made to include the mechanical effect, or the mass loading, through the consideration of the mass ratio of the electrodes and crystal blank. Such a consideration has been effective for relatively thin electrodes before, but the ever-increasing mass ratio has been pressing further improvement to take into account relatively thicker electrodes. To extend Mindlin plate equations for these applications, we derive the plate equations systematically with the approximation of displacements in electrodes with those in the crystal blank. As a result, both mass and stiffness effects of electrodes are considered through ratios of the thickness, density, and elastic constants of the electrodes to those of the crystal blank, respectively, and the plate equations are modified accordingly. A practical design of the electrodes and crystal blank are analyzed to demonstrate the necessity of such modifications to Mindlin plate equations.