Governing equations of piezoelectric plates with graded properties across the thickness

Two-dimensional governing equations for electroded piezoelectric plates with graded material properties across the thickness are derived from the three-dimensional equations of linear piezoelectricity by extending Mindlin´s power series expansion to the mechanical displacement, electric potential and the thickness-graded properties, i.e., the elastic stiffnesses, piezoelectric coefficients, dielectric permittivity, and mass density. The couplings of the extensional and flexural motions due to the graded material properties, in addition to those due to the anisotropy of material properties, are clearly disclosed in these newly derived equations which reduce to Mindlin´s equations of homogeneous nonpiezoelectric plates when the graded material properties contain only the zeroth-order terms in the power series expansion. Dispersion curves for both the homogeneous piezoelectric plates and piezoelectric bimorphs are calculated from the 3-D equations and from the present 2-D first-order equations, and the comparison shows that the agreement is very close for frequencies up to and including the fundamental thickness-shear frequencies. An electrode-plated piezoelectric ceramic cantilever plate with thickness-graded piezoelectric coefficients is studied in detail. Resonance frequencies, modes of vibrations and static responses are computed and discussed