Three-dimensional exact analysis of a simply supported functionally gradient piezoelectric plate

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Abstract

An exact three-dimensional analysis is presented for a functionally gradient piezoelectric material rectangular plate that is simply supported and grounded along its four edges. The state equations of the functionally gradient piezoelectric material are developed based on the state space approach. Assuming that the mechanical and electric properties of the material have the same exponent-law dependence on the thickness-coordinate, we obtain an exact three-dimensional solution of the coupling electroelastic fields in the plate under mechanical, and electric loading on the upper and lower surfaces of the plate. The influences of the different functionally gradient material properties on the structural response of the plate to the mechanical and electric stimuli are then studied through examples.

Introduction

Piezoelectric materials have been widely used as actuators and sensors in smart and adaptive systems due to their intrinsic coupling of mechanical and electric fields (Gandhi and Thompson, 1992; Rao and Sunar, 1994). In order to achieve large deformation, piezoelectric actuators are often constructed as bimorph or stacked form, by bonding together two piezoelectric ceramic sheets in strip or plate forms. While these designs can provide large displacement, they have great disadvantages. The bonding of two different piezoelectric materials or identical piezoelectric materials with different poling directions will cause severe interfacial stress concentration, and trigger the initiation and propagation of micro-cracks near the interface which may lead to failure of the devices. Such drawbacks reduce the reliability and life span of piezoelectric devices and limit their applications.

In order to overcome the drawbacks of conventional piezoelectric bimorphs, a new kind of piezoelectric materials, named functionally gradient piezoelectric materials (FGPMs), has been developed (Zhu and Meng, 1995; Wu et al., 1996; Shelley et al., 1999). FGPM is a kind of piezoelectric material with material composition and properties varying continuously along certain directions. The piezoelectric devices can be entirely made of FGPM or use FGPM as a transit interlayer between different piezoelectric materials. The advantage of this new kind of materials is that no discernible internal boundaries exist and failures from interfacial stress concentrations developed in conventional bimorphs can be avoided. FGPM actuators can thus produce large displacements while minimizing the internal stress concentrations, which will greatly improve the reliability and life of piezoelectric actuators. Nowadays, advancement of modern materials processing technology has enabled the fabrication of materials with arbitrary compositional gradient in a controlled fashion. The relationship between the material compositional gradient and the electromechanical responses of FGPM structures is very important in the design of FGPM devices. This research subject is so new that only a few results can be found in the literature. Most of the available results on the structural analysis of FGPM plate were based on a laminated structure scheme by which the FGPM plate was approximately modeled as a laminated structure. For example, this scheme was employed by Liu and Tani (1994) to study the wave propagation in FGPM plates, by Chen and Ding (2002) to analyze the free vibration of FGPM rectangular plates. Other related works include: Lim and He (2001) obtained an exact solution of a compositionally graded piezoelectric layer under uniform stretch, bending and twisting; Reddy and Cheng (2001) obtained a three-dimensional solution of smart functionally gradient plate; Li and Weng, 2002a, Li and Weng, 2002b, Hu et al. (2002), Jin and Zhong (2002) studied the problems of an antiplane crack in functionally gradient piezoelectric materials.

The objective of this work is to present an exact solution of a simply supported functionally gradient piezoelectric rectangular plate based on three-dimensional electroelasticity theory. The obtained exact solution could serve as a benchmark result to assess other approximate methodologies or as a basis for establishing simplified FGPM plate theories.

Section snippets

Formulation

Consider a FGPM rectangular plate of uniform thickness h, as shown in Fig. 1. Introduce a Cartesian coordinate system {xi} (i=1,2,3) such that the bottom and top surfaces of the undeformed plate lie in the plane x3=0 and x3=h. The lengths of the edges of the plate in x1- and x2-direction are respectively denoted by a and b. Throughout the paper, the Einsteinian summation convention over repeated indices of tensor components is used, with Latin indices ranging from 1 to 3 while Greek indices

Solution

The state variables that satisfy the boundary condition (14) can be assumed asΠ=u1u2σ33D3=∑m=1n=1Umn(x3)cosmπx1asinnπx2bVmn(x3)sinmπx1acosnπx2beα(x3/h)Zmn(x3)sinmπx1asinnπx2beα(x3/h)Dmn(x3)sinmπx1asinnπx2bΓ=σ13σ23u3φ=∑m=1n=1eα(x3/h)Xmn(x3)cosmπx1asinnπx2beα(x3/h)Ymn(x3)sinmπx1acosnπx2bWmn(x3)sinmπx1asinnπx2bΦmn(x3)sinmπx1asinnπx2b

Substituting , into (6), we obtain the following matrix equationMmnx3=KmnMmnwhereMmn=UmnVmnZmnDmnXmnYmnWmnΦmnTandKmn=K1mnK3mnK4mnK2mnwith K1mn, K2mn, K3mn and

Numerical examples

In this section numerical study of FGPM square plate (a=b=1 m, h=0.1 m), which is simply supported and grounded on its four lateral edges, will be made based on the above exact solution. The material chosen for the study is PZT-4 that has the material properties at x3=0, as follows (Cheng et al., 2000):c11110=c22220=139GPa,c33330=115GPa,c11220=77.8GPa,c11330=c22330=74.3GPa,c23230=c31310=25.6GPa,c12120=30.6GPa,e3110=e3220=−5.2C/m2,e3330=15.1C/m2,e1130=e2230=12.7C/m2,λ110220=1.306×104pF/m,λ330

Concluding remarks

An exact three-dimensional solution is obtained for a FGPM rectangular plate simply supported and grounded along its four edges by means of the state space approach. The mechanical and electric properties of the material were assumed to have the same exponent-law dependence on the thickness-coordinate of the plate. The obtained solution is valid for arbitrary mechanical and electric loads applied on the upper and lower surfaces of the plate and can play as a benchmark result when establishing

Acknowledgements

This work was supported by the National Natural Science Foundation of China (nos. 10072041 and 10125209) and the Teaching and Research Award Fund for Outstanding Young Teachers in High Education Institutions of MOE, PR China.

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