Two-dimensional equations for electroelastic plates under biasing fields

Two-dimensional equations for piezoelectric plates have been very effective in modeling piezoelectric resonators. To predict the behavior of resonators under environmental effects like temperature change or acceleration, the theory of incremental motions in an electroelastic body under biasing fields is necessary. Existing two-dimensional equations for electroelastic plates under biasing fields employ various simplifying assumptions. For example, electroelastic couplings are often neglected for materials like quartz with weak piezoelectric effect. Spatially uniform and time-independent biasing fields are usually assumed so that the resulting equations have constant coefficients. The study of resonators made from new materials with strong piezoelectric coupling and the treatment of, e.g., resonator vibration sensitivity require plate equations with full electroelastic coupling and time-dependent or spatially varying biasing fields. We develop two-dimensional equations for an electroelastic plate under general biasing fields. No assumptions on the biasing fields are made. Full electroelastic coupling is taken into account. A set of two-dimensional equations for coupled extension and flexure with shear deformations are obtained. The application of the equations in resonator vibration sensitivity is shown by an example