Governing equations for a piezoelectric plate with gradedproperties across the thickness

摘要

Two-dimensional first-order governing equations for electrodednpiezoelectric crystal plates with general symmetry and thickness-gradednmaterial properties are deduced from the three-dimensional equations ofnlinear piezoelectricity by Mindlin's general procedure of seriesnexpansion. Mechanical displacements and thickness-graded materialnproperties, i.e., the elastic stiffnesses, piezoelectric coefficients,ndielectric permittivities, and mass density, are expanded in powers ofnthe thickness coordinate, while electric potential is expanded in anspecial series in order to accommodate the specified electric potentialsnat electroded faces of the plate. The effects of graded materialnproperties on the piezoelectrically induced stresses or deformations bynthe applied surface potentials are clearly exhibited in these newlynderived equations which reduce to Mindlin's first-order equations ofnelastic anisotropic plates when the material properties are homogeneous.nClosed form solutions are obtained from the three-dimensional equationsnof piezoelectricity and from the present two-dimensional equations fornboth homogeneous plates and bimorphs of piezoelectric ceramics.nDispersion curves for homogeneous plates and bimorphs and resonancenfrequencies for bimorph strips with finite width are computed from thensolutions of three-dimensional and two-dimensional equations. Comparisonnof the results shows that predictions from the two-dimensional equationsnare very close to those from the three-dimensional equations