Governing equations for a piezoelectric plate with graded properties across the thickness

Abstract

Two-dimensional first-order governing equations for electroded piezoelectric crystal plates with general symmetry and thickness-graded material properties are deduced from the three-dimensional equations of linear piezoelectricity by Mindlin's general procedure of series expansion. Mechanical displacements and thickness-graded material properties, i.e., the elastic stiffnesses, piezoelectric coefficients, dielectric permittivities, and mass density, are expanded in powers of the thickness coordinate, while electric potential is expanded in a special series in order to accommodate the specified electric potentials at electroded faces of the plate. The effects of graded material properties on the piezoelectrically induced stresses or deformations by the applied surface potentials are clearly exhibited in these newly derived equations which reduce to Mindlin's first-order equations of elastic anisotropic plates when the material properties are homogeneous. Closed form solutions are obtained from the three-dimensional equations of piezoelectricity and from the present two-dimensional equations for both homogeneous plates and bimorphs of piezoelectric ceramics. Dispersion curves for homogeneous plates and bimorphs and resonance frequencies for bimorph strips with finite width are computed from the solutions of three-dimensional and two-dimensional equations. Comparison of the results shows that predictions from the two-dimensional equations are very close to those from the three-dimensional equations.

A comprehensive analysis of functionally graded sandwich plates: Part 1—Deflection and stresses

“…Reiter et al (1997) and Reiter and Dvorak (1998) have performed detailed finite element studies of discrete models containing simulated particulate and skeletal microstructures and compared results with those computed from homogenized models in which effective properties were derived by the MoriTanaka and the self-consistent methods. Lee and Yu (1998) and Lee et al (1999) have expanded the mechanical displacements, electric potential and the material moduli as power series in the thickness coordinate and derived plate equations of different orders for FG piezoelectric disks, infinite plates and strips. The response of FG ceramic-metal plates has been investigated by Praveen and Reddy (1998) using a plate finite element that accounts for the transverse shear strains, rotatory inertia and moderately large rotations in von Kármán sense.…”

The fabrication and deformation capabilities of two dimensionally graded DEPP FGP

“…Other FGP modeling techniques typically represent the electric field as a constant across the actuator or layer thickness [27][28][29] or represent material property gradients as a collection of discrete material layers. [27][28][29][30][31] Both of these approaches can result in significant errors in the predicted performance 18 . The variations must be accurately captured to produce a robust modeling methodology.…”

Impedance and admittance matrices of symmetric piezoelectric annular bimorphs and their applications

“…Hui et al 19 studied the electromechanical static response of the circular multilayered piezoelectric plates. Lee et al 20,21 derived two-dimensional first-order governing a͒ Electronic mail: sungha@email.hanyang.ac.kr equations for the piezoelectric crystal plates with the thickness-graded material properties and obtained the RF of flexural and thickness-shear vibration. However, the impedance and admittance matrices of a piezoelectric annular bimorph ͑PAB͒ and a piezoelectric circular bimorph ͑PCB͒ have not been reported yet.…”