Piezoelectric ceramic disks with thickness-graded material properties

A system of two-dimensional, first-order equations for electroded piezoelectric crystal plates with general symmetry and thickness-graded material properties recently was deduced from the three-dimensional equations of linear piezoelectricity. These equations are simplified for the two limiting cases of thickness-graded piezoelectric properties, i.e., the homogeneous plate and bimorph of piezoelectric ceramics. Closed-form solutions are obtained from these reduced equations for the flexural and thickness-shear vibrations and static response of bimorph disks as well as for the extensional and thickness-stretch vibrations of homogeneous disks. Frequency spectra and modes are computed and examined. Resonance frequencies for both homogeneous and bimorph disks of PZT-857 are computed and measured. The comparison of the results shows that the agreement is close.