Three-Dimensional Modeling of Piezoelectric Materials

This paper deals with 3-D modeling of piezoelectric materials. The model is based on an exact description of the potential and electric field inside a material. Moreover, coherent piezoelectric equations are used. Modeling has been applied to rectangular and cylindrical elements. In each case, the exact equations of the displacements along the three coordinate axes and the corresponding electric impedance are calculated. The general resonance conditions are stated for these two geometries. It is shown that, contrary to the 1-D models, a unique equation describes lateral and thickness vibrations, or radial and thickness vibrations. These properties enable us to analytically calculate the frequency spectrum of rectangular elements, thick disks, or cylinders and also thick rings or hollow cylinders versus the width to thickness ratio. It is then very easy to determine the corresponding dispersion diagram related to each geometry sample. These resonance conditions are similar to those deduced from the 1-D model described in the IEEE standard but are more general and necessitate no cancelling out assumptions. In addition, contrary to 1-D models, the wave velocities and the permittivity are independent of the element geometry (parallelepiped or cylindrical). The wave velocities are equal to those stated for the wave propagation in infinite medium and measured with pulse-echo techniques. It is the coupling inside the material which modifies the resonance conditions and not the geometrical dimensions of the vibrating element. 3-D modeling and 1-D radial mode of the admittance of a thick disk are calculated and compared with experimental measurements. Theoretical and measured admittances are compared and discussed.