High−frequency vibrations of crystal plates under initial stresses

Abstract

A system of six two−dimensional plate equations is derived for motions of small−amplitude waves or vibrations superimposed on finite, elastic deformations due to static, initial stresses. In the stress−strain relations, the nonliner terms associated to the third−order elastic stiffness coefficients are included. These equations accomodate the coupling of the six lowest modes of vibration, i.e., the flexure, extension, face−shear, thickness−shear, thickness−twist, and thickness−stretch modes, and all their anharmonic overtones. The new equations are applied to the rotated Y cuts of quartz in studying the thickness−shear and flexural vibrations. The changes in the resonance frequencies of the fundamental thickness−shear vibrations are computed as functions of the direction of initially applied force and of the angle of rotated Y cuts. The predicted results are compared with experimental data and with existing computed results. An explicit formula is obtained for the change of fundamental thickness−shear frequencies in terms of initial deformations and the second− and third−order elastic stiffness coefficients. Subject Classification: 40.24.