Elsevier

Composite Structures

Volume 104, October 2013, Pages 71-84
Composite Structures

An efficient finite element model for static and dynamic analyses of functionally graded piezoelectric beams

https://doi.org/10.1016/j.compstruct.2013.04.010Get rights and content

Abstract

This study deals with static, free vibration and dynamic response of functionally graded piezoelectric material (FGPM) beams using an efficient three-nodded beam element. This beam finite element is based on a refined sinus model. It does not require shear correction factor and ensures continuity conditions for displacements, transverse shear stresses as well as boundary conditions on the upper and lower surfaces of the FGPM beam. This conforming finite element does not suffer from shear locking. The number of the mechanical unknowns is independent of the number of layers. A high-order electrical potential field is considered through each graded piezoelectric layer. The proposed FE is validated through static, free vibration and dynamic tests for FGPM beams. For various electrical and mechanical boundary conditions, excellent agreement is found between the results obtained from the proposed formulation and reference results from open literature or 3D FEM.

Introduction

It is well known that piezoelectric materials have been widely used as sensors and actuators in control systems due to their excellent electro-mechanical properties, easy fabrication, design flexibility, and efficiency to convert electrical energy into mechanical energy. Extensive studies have been carried out, and many theoretical and mathematical models have been presented for laminated composite structures with piezoelectric sensors and actuators until now [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [37]. Traditional piezoelectric sensors and actuators are often made of several layers of different piezoelectric materials. In these sandwich devices, adhesive epoxy resin is usually used to bond the piezoelectric layers, which causes some problems to arise. The principal weakness of these structures is that the high stress concentrations are usually appeared at the layer interfaces under mechanical or electrical loading. These stress concentrations lead to the initiation and propagation of micro-cracks near the interfaces of two bonded piezoelectric layers. This drawback restricts the usefulness of piezoelectric devices in the areas where the devices require high reliability.

In order to overcome these disadvantages, FGPM sensors and actuators were introduced and fabricated by Zhu and Meng [11], and Wu et al. [12]. FGPMs are a kind of piezoelectric materials whose mechanical and electrical properties vary continuously in one or more directions. FGPM actuators can produce not only large displacements but also reduce the internal stress concentrations and consequently improve significantly the lifetime of piezoelectric actuators. Takagi et al [40] used a mixture system of PZT and Pt for the fabrication of FGPM bimorph actuators. To this end, they mixed the PZT and Pt powders together. The Pt contents of the powder mixtures were 0, 10, 20 and 30 vol% for the graded layers and the electrode layer. The mixtures were stacked layer by layer into a steel die according to the lamination scheme [PZT, PZT/10%Pt, PZT/20%Pt, PZT/30%Pt]S, then pressed. The formed specimens were cut to beams with dimensions 12×2×3 mm2. Nowadays, the FGPMs, as intelligent materials, have been used extensively in applications of sensors and actuators in the micro-electro-mechanical system and smart structures.

A considerable number of papers involving the behavior of FGPMs already exists. Lim and He [13] obtained an exact solution for a compositionally graded piezoelectric layer under uniform stretch, bending and twisting load. Reddy and Cheng [14] obtained a 3D solution for smart FGPM plates. Zhong and Shang [15] presented an exact 3D solution for FGPM rectangular plates, by means of the state space approach. Lu et al. [16] obtained an exact solution for simply supported FGPM laminates in cylindrical bending by Stroh-like formalism. Using this method, Lu et al. [17] also proposed exact solutions for simply supported FGPM plates. Liu and Shi [18], and Shi and Chen [19] obtained closed form solutions for the FGPM cantilever beams using the two-dimensional (2D) theory of elasticity and the Airy stress function. Xiang and Shi [20] investigated thermo-electro-elastic response of a FGPM sandwich cantilever beam. They also employed the Airy stress function in order to study the effect of parameters such as the electromechanical coupling, functionally graded index, temperature change and thickness ratio on the static behavior of actuators/sensors. Using the so called state-space based differential quadrature method (SSDQM), Yang and Zhifei [21] investigated the free vibration of a FGPM beam.

The beam and plate/shell theories have been presented for the structural analysis of FGPM structures in the literature. But, in most of the available beam and plate/shell models, it is assumed that the functionally graded piezoelectric layer consists of a number of laminates, where the material properties within each laminate are invariant. Liu and Tani [22] used this method to study the wave propagation in FGPM plates. Chen and Ding [23] analyzed the free vibration of FGPM rectangular plates using the aforementioned method. Lee [24] used a layerwise finite element formulation to investigate the displacement and stress response of a FGPM bimorph actuator. Such models are sufficiently accurate but it needs a high computational cost. By using the Timoshenko beam theory, Yang and Xiang [25] investigated the static and dynamic response of FGPM actuators under thermo-electro-mechanical loadings. In their work, the numerical results were obtained by using the differential quadrature method (DQM). A comprehensive study on the static, dynamic and free vibration response of FGPM panels under different sets of mechanical, thermal and electrical loadings using the finite element method was presented by Behjat et al. [26]. Behjat et al. [27] investigated also the static bending, free vibration and dynamic response of FGPM plates under mechanical and electrical loading using the first order shear deformation theory. Doroushi et al. [28] studied free and forced vibration characteristics of a FGPM beam under thermo-electro-mechanical loads using third-order shear deformation beam theory. Wu et al. [29] derived a high-order theory for FGPM shells based on the generalized Hamilton’s principle.

In this paper, a refined Sinus beam finite element is evaluated for static bending, free vibration and transient dynamic response of FGPM beams. This element is based on the Sinus model introduced in Touratier [30]. Then, it has been extended to take into account the interlaminar continuity of the transverse shear stresses in Polit and Touratier [31] for plates, and in Dau et al. [32] for shells. The coupling with the piezoelectric effect is carried out in Ossadzow-David and Touratier [33] and Fernandes and Pouget [34] using an analytical approach. The original Sinus model has been enriched in Vidal and Polit [35] by introducing a layer refinement in the kinematics, and then extended to thermal effects (Vidal and Polit [36]). In the present study, it is intended to extend these last works to static and dynamic analyses of FGPM beams.

The proposed beam element is a three-nodded element which satisfies the continuity conditions between layers of laminates for displacements, transverse shear stress and the free conditions at the top and bottom surfaces of the beam. The element is totally free of shear locking. It has four independent generalized displacements. Concerning the electrical part, a high-order electrical potential field is considered. In contrast with many of the available studies on FGPMs, which consider one of the graded piezoelectric properties variable in the thickness direction, here it is assumed that all material properties are variable in the thickness of the beam. The results obtained from the present finite element model are in good agreement with previous published and coupled finite element (ABAQUS) reference solutions. Moreover, the computational cost of the present element is very low in comparison with the available layer-wise beam and plate/shell theories.

Section snippets

Geometry, coordinate system

The laminated beam has a rectangular uniform cross section of length L, width b, height h, and is made of Nl layers either completely or in part constituted of FGPMs. The geometric parameters and the chosen Cartesian coordinate system (x1, x2, x3) are shown in Fig. 1.

Constitutive equations

The 3D linear constitutive equations of the kth layer, polarized along its thickness direction in its global material coordinate system can be expressed as:σ11σ22σ33σ23σ13σ12(k)=c11c12c1300c16c12c22c2300c26c13c23c3300c36000c44c450000c

Finite element formulation

In this section, finite element approximations are defined for the mechanical and electrical variables of Eqs. (6), (8), (11). As the highest derivative of w in the expression of the strain energy is of second-order, this variable is interpolated using the C1-continuous Hermite cubic shape functions. As far as the rotation ω3 is concerned, the quadratic Lagrangian shape functions are chosen. Furthermore, if an identical order is adopted for the shape functions of both w0,1 and ω3 in the

Numerical examples

In this section, several static and dynamic tests are presented to validate the proposed finite element. The mechanical and electrical material properties which are employed in this section are given in Table 1. Present results are issued from a MATLAB program written by the authors. It is worthy to note that the exact integration procedure is employed for the calculation of the stiffness matrix of beam elements.

Conclusions

A conforming three-nodded beam element is presented for the static, modal and transient response of FGPM beams. The kinematics is based on a refined sinus model. All displacement and transverse shear stress continuities are ensured at layer interfaces as well as the free boundary conditions on the top and bottom of the beam. The number of the mechanical unknowns remains low and is independent of the number of layers. A LayerWise high-order approximation is used for the electrical potential.

In

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