Graded piezoelectric cylinders subjected to high electric fields and comparison of their frequency response with piezoelectric plates

Abstract

In this paper we investigated a radially polarized piezoceramic cylinder with graded piezoelectric properties, and used a nonlinear model for piezoceramics subjected to high electric fields. We investigated the nonlinear behavior of this material by examining changes in its electric-field-dependent dielectric and piezoelectric coefficients caused by domain wall motion. The Galerkin finite-element method was used to solve the governing equations of the axisymmetrically loaded heterogeneous piezoceramic medium subjected to harmonic electrical loading. Stress, displacement, resonance, and frequency responses were compared for homogeneous and graded cylinders; additionally, we compared the results of linear and nonlinear studies. We showed that the effective stress was higher within the graded cylinder than within the homogeneous cylinder, and that the nonlinearity caused by domain wall motion was less pronounced for the graded cylinder than for the homogeneous cylinder. The frequency responses of homogeneous and heterogeneous piezocylinders were also compared with those of piezoelectric plates. We concluded that—unlike for graded plates, which have a more desirable frequency response than homogeneous plates—graded cylinders are not superior to homogeneous cylinders. The finite-element solution in this paper is verified by simulations using COMSOL Multiphysics software.

Appendix

The constitutive relations needed for thickness-vibration analysis of a one-dimensional piezoelectric plate polarized and graded in the thickness direction and constrained from in-plane motion are given by the following relations:

$$ \begin{gathered} \sigma_{zz} = c_{33}^{E} u_{,z} + e_{33} \phi_{,z} \hfill \\ D_{z} = e_{33} u_{,z} - \in_{33}^{S} \phi_{,z} \hfill \\ \end{gathered} $$

(19)

where $z$ is the thickness coordinate and $u$ is the displacement in the thickness direction.

Substituting the previous equation in (2) and considering the lack of shear stress in this problem, the governing equations are:

$$ \begin{gathered} c_{33}^{E} u_{,zz} + c_{33,z}^{E} u_{,z} + e_{33} \phi_{,zz} + e_{33,z} \phi_{,z} = \rho \textit{\"{u}} \hfill \\ e_{33} u_{,zz} + e_{33,z} u_{,z} - \in_{33}^{S} \phi_{,zz} - \in_{33,z}^{S} \phi_{,z} = 0 \hfill \\ \end{gathered} $$

(20)

The boundary conditions of the problem are similar to those of the cylinder:

$$ \begin{gathered} \phi \left( {z = 0} \right) = - \phi \left( {z = h} \right) = \phi_{0} e^{{j\omega_{c} t}} \hfill \\ \sigma_{rr} \left( {z = 0,t} \right) = \sigma_{rr} \left( {z = h,t} \right) = 0 \hfill \\ \end{gathered} $$

(21)

where $h$ is the plate thickness.