Nonlinear Mindlin plate equations for the thickness-shear vibrations of crystal plates

Mindlin plate equations have wide applications in engineering fields with piezoelectric crystal resonators in particular due to its accurate prediction of high frequency vibrations of plates in the vicinity of thickness-shear mode. As a linear theory based on the assumption of infinitesimal deformation, its applications have been limited mainly to vibration frequency analysis. Many important properties of quartz crystal resonators such as the electrical circuit parameters, and nonlinear phenomena such as the drive-level dependence (DLD), have to be studied with the consideration of higher-order material constants and subsequent inclusion of nonlinear strain components. This, in turn, implies that the consideration of large deformation related to the driving electrical field. The need of nonlinear theory for the analysis of high frequency vibrations of piezoelectric crystal plates have been noticed before, and research work concerning large electrical fields and deformation have been initiated for the calculation of electrical properties and DLD of quartz crystal resonators by many investigators recently. In this study, we start with higher-order constitutive relation which includes higher-order elastic constants. Then, nonlinear terms of strain components are considered in a compatible and systematic manner. By following Mindlin's procedure in deriving the two-dimensional equations, nonlinear Mindlin plate equations for large deformation are obtained. The equations take the familiar form of Mindlin plate theory except the inclusion of nonlinear terms in the strain tensor.