Extraction of Nonlinear Thermo-Electroelastic Equations for High Frequency Vibrations of Piezoelectric Resonators with Initial Static Biases

In this paper, the general case of an anisotropic thermo-electro elastic body subjected to static biasing fields is considered. The biasing fields may be introduced by heat flux, body forces, external surface tractions, and electric fields. By introducing proper thermodynamic functions and employing variational principle for a thermo-electro elastic body, the nonlinear constitutive relations and the nonlinear equation of motion are extracted. The equations have the advantage of employing the Lagrangian strain and second Piola-Kirchhoff stress tensor with symmetric characteristics. These equations are used to analyze the high frequency vibrations of piezoelectric resonators under finite biasing fields. A system of three dimensional equations is derived for initial and incremental fields on the body. Capability of the equations in numerical modelling of temperature-frequency and force-frequency effects in quartz crystal is demonstrated. The numerical results compare well with the data from experiments. These equations may be used in accurate modelling of piezoelectric devices subjected to thermo electro mechanical loads.