Research Article | Open Access
Pseudo-Three-Dimensional Analysis for Functionally Graded Plate Integrated with a Piezoelectric Fiber Reinforced Composite Layer
In this paper, a pseudo-three-dimensional method is proposed to investigate static behavior analysis of functionally graded (FG) plate integrated with a piezoelectric fiber reinforced composite (PFRC) layer by the hyperbolic shear and normal deformation theory. The present method is a displacement-based theory which accounts for hyperbolic variation of in-plane displacement field and parabolic variation of transverse displacement field. The linear electrical potential function in the PFRC layer is modeled. The governing equations of present method are derived by the minimum potential energy principle and Navier’s procedure is used to solve the equations. Numerical results are presented to demonstrate the efficiency of the proposed method. The effects of some parameters including material composition, aspect ratios, and applied voltages on the deformations of the plate are investigated. Compared with the available data of numerical method and 3D method, the presented method is more suitable for the smart FG structure.
The functionally graded materials (FGMs) have become an important issue for advanced structural applications in many areas such as aerospace, automation, medicine, energy, and optoelectronic. Due to the important applications of the FGMs, many researchers affect improving material processing, fabrication processing, and studying the mechanics and mechanism of FGM structures. Various approaches of analytical, numerical, and experimental methods have been proposed to deal with the problems involving FGMs. Since the early 1990s, displacement-based theories had been used in the analysis of FG beam, plate, and shell structures. These theories were derived by making suitable assumptions concerning kinematics of deformation and stress state through the thickness direction, such as the classical plate theory (CLPT), the first-order shear deformation theory (FSDT) [1, 2], and the third-order shear deformation theory (TSDT) [3–5]. These assumptions allow the reduction of a 3D problem to a 2D problem and neglect normal deformation effects (). The CLPT is based on the Kirchhoff hypothesis and neglects the effect of transverse shear strain, which applies to thin plates, but not to medium and thick plates. The FSDT considers the shear deformation effect on linear variation of in-plane displacements through the thickness direction; this assumption leads to a constant shear stress state; thus, a shear correction factor is required to compensate for the difference between the actual and assumed stress state. Usually, the factor is dependent on many parameters including geometry, boundary conditions, and loading conditions and it is hard to find out consistently. To overcome the drawbacks, Reddy proposed the third-order shear deformation plate theory which considered the shear deformation effects based on a higher-order variation of in-plane displacement fields through the thickness of the plate and the shear correction factor is not required. The theory including normal deformation effects () can be firstly found in the literature by Kant and Manjunatha  using the finite element method, in which the cubic polynomial expansion were applied in both in-plane and transverse displacement fields. Some other methods considering the normal deformation effects have been proposed in the literatures [7–13]. The aforementioned theories can be called higher-order shear and normal deformation theory (HOSNT), which include higher-order variation transverse displacement field. Recently, the effect of thickness stretching has been investigated by Carrera et al.  using the finite element approximation. Neves et al.  presented bending and free vibration analysis of FG plates using radial basis function collocation method based on a hyperbolic shear deformation theory. Hebali and Thai [15, 16] developed quasi-3D solutions of bending and vibration analysis of FG plates using a hyperbolic shear deformation theory. Belabed et al.  obtained quasi-3D solutions for static and vibration analysis of the FG plate. Some other theories based on displacement assumptions can be found in the review article  in detail.
Three-dimensional exact solutions for FG plates are rather restricted in general and difficult to find unless some geometry and boundary conditions are relatively simple. Analytical 3D solutions for FG plates are very useful since they provided benchmark to assess the accuracy of various 2D theories, quasi-3D theories, and numerical methods. Fortunately, there are some studies about finding 3D exact solutions for FG plates. Cheng and Batra  presented 3D thermoelastic deformations of the FG elliptic plate. Reddy and Cheng  obtained 3D solutions of thermomechanical deformations of rectangular FG plates. An analytical solution of three-dimensional thermomechanical deformations of a simply supported FG rectangular plate subjected to time-dependent thermal load was presented by Vel and Batra . Anderson  analyzed 3D elasticity solutions for a sandwich composite plate with FGM core. Senthil  obtained 3D exact solutions for free and forced vibration of the FG plate and Kashtalyan  developed 3D elasticity solutions for the FG plate subjected to transverse loading. The bending analysis of an exponentially graded thick rectangular plate using both two-dimensional trigonometric shear deformation and three-dimensional elasticity theory was given by Zenkour .
In order to develop the structures with self-controlling and self-monitoring capabilities, piezoelectric material is an ideal material which can be used as distributed actuators and sensors. Piezoelectric material is such that when it is subjected to a mechanical load, it generates an electrical charge which is usually called direct piezoelectric effect. Conversely, mechanical stress or strain occurs when piezoelectric material is subjected to an applied voltage; this phenomenon is known as the converse piezoelectric effect. The essential feature of piezoelectric materials is the capability of energy transformation between electrical energy and mechanical energy. Hence, piezoelectric materials can be tailored for developing smart structures and have been widely used in various applications such as automotive sensors, actuators, ultrasonic transducers, microelectromechanical system (MEMS) technology, and active damping devices. The functionally graded plates integrated with piezoelectric materials have also been studied by many researchers. Ootao and Tanigawa  gave 3D theoretical analysis of functionally graded rectangular plates bonded to a piezoelectric plate subjected to transient thermal loading. Reddy and Cheng  obtained 3D solutions of FG plates with an attached piezoelectric active material. He et al.  presented the active control of piezoelectric FG plate using the finite element formulation based on the classical plate theory. The nonlinear vibration and dynamic response of FG plates bonded with a piezoelectric layer in thermal environments are developed by Huang and Sheng . Kargarnovin et al.  gave the vibration analysis of a simply FG rectangular plate with piezoelectric patches on its top or bottom surface. Brischetto and Carrera  investigated the piezoelectric FG plate using a refined plate theory.
Although the piezoelectric materials have been widely used as intelligent control devices, due to the small stress-strain coefficients of the monolithic piezoelectric material, the control authority of monolithic piezoelectric material is very low. Mallik and Ray  proposed the concept of piezoelectric fiber reinforced composite (PFRC) and obtained the effective coefficients of piezoelectric fiber reinforced composite through micromechanical analysis. It has been found that, when the fiber volume fraction exceeds a critical fiber volume fraction, the effective piezoelectric coefficient becomes significantly larger than the corresponding coefficient of the piezoelectric material. Ray and Sachade  dealt with 3D exact solutions for the static analysis of the FG plate integrated with a PFRC layer subjected to an electromechanical loading. They subsequently studied the effects of the variation of the piezoelectric angle in the PFRC layer using the finite element model . Shiyekar and Kant  used a higher-order shear and normal deformation theory to analyze the FG plate integrated with a piezoelectric reinforced composite layer.
In present work, a pseudo-three-dimensional method is presented to study a simply supported piezoelectric FG plate by the hyperbolic shear and normal deformation theory. The effects of some parameters including material composition, aspect ratios, and applied voltages on the deformations of the plate are investigated. The obtained results are compared with numerical solutions and 3D exact solutions to verify the accuracy and the efficiency of the present method.
2. Mathematical Modeling and Formulation
2.1. Geometrical Configuration
Consider a rectangular FG plate with total thickness , length , and width as shown in Figure 1. The top and bottom surfaces of the FG plate are at and the edges of the plate are parallel to and axes, respectively. The top surface of the FG plate is integrated with a piezoelectric fiber reinforced composite (PFRC) layer with thickness , acting as a distributed actuator of the FG plate. In the FG plate, due to gradually changing of volume fraction, the material properties are assumed to obey an exponential function through the thickness direction. The effective Young’s modulus was given by  which dictated the material property through the thickness of the FG plate with a gradient parameter . Young’s modulus and represent the material properties of the bottom surface and the top surface of the FG plate, respectively. Poisson’s ratio of the FG plate was assumed to be constant for that the effect of Poisson’s ratio on the deformation is much less than that of Young’s modulus .
2.2. Displacement Field
The present higher-order shear and normal deformation theory have been given by Neves et al. [9, 10, 37–39]. They have given the bending, vibration, and buckling analysis of FG plates using meshless radial basis function collocation method based on polynomial, hyperbolic, and sinusoidal higher-order displacement field. Navier’s solutions for a simply supported FG plate with a PFRC layer are not given for present higher-order shear and normal deformation theory which is given bywhere , , and are displacement components in the , , and direction, respectively. The parameters are unknown functions to be determined. is the shape function representing the distribution of transverse strain and shear stress through the thickness direction. The shape function can be polynomial function, hyperbolic function, or sinusoidal function. In this paper, a hyperbolic sinusoidal function is used for the bending analysis of the smart FG plate such that
2.3. Strain-Displacement Relations and Constitutive Equations
The linear strain-displacement relationship for FG piezoelectric materials is given by
For piezoelectric materials, the constitutive equations can be expressed as and, for elastic functionally graded material, the constitutive equations are where , , , and represent the stress vector, the strain vector, the electric field vector, and the electric displacement vector, respectively. , , and are the elastic stiffness matrix, the piezoelectric constants matrix, and the dielectric matrix, respectively. The superscript represents either the PFRC layer or the FG layer according to the location of the PFRC layer. For example, if the top surface of plate is integrated with the PFRC layer then the values of as 1 and 2 denote the FG plate and the PFRC layer, respectively, and vice versa.
The components of stress and strain vectors are given by
The electric field vector and the electric displacement vector are given by where the electric field vector related to electric potential is given by
The elastic stiffness matrix, the piezoelectric matrix, and the dielectric matrix are given by where are the elastic constants, for isotropic functionally graded material, defined by
2.4. Strain Energy due to Governing Equations
The minimum potential energy principle is used herein to derive governing equations and the principle can be determined by a summation of the variation of strain energy and work done by external force as follows: The variation of strain energy is given explicitly by where , , , , and are the stress resultants defined by and are the thickness coordinates from the bottom to the top of the smart FG plate. For example, if the PFRC layer is attached on the top of the FG plate, , , are , , , respectively. Conversely, , , are , , , respectively.
The variation of potential energy of applied transverse load can be expressed as
3. Results and Discussion
3.1. Numerical Illustration
It is assumed that the surface of the PFRC layer being in contact with the FG plate is suitably grounded. Since the thickness of the PFRC layer is very small, the electric potential can be considered as linear variation through the thickness of the PFRC layer. The electric potential function can be expressed as either according to the top surface or the bottom surface of the FG plate attached with the PFRC layer . is a generalized electric potential function at any point in the PFRC layer.
Substituting (2)-(6) and (17a)-(17b) into (14a)-(14f), the stress resultants can be eventually expressed in terms of generalized displacements and electric potential . The stress resultants and the corresponding coefficients are defined as relations in Appendix A. The FG plate attached to a piezoelectric layer either on the top or on the bottom is taken here. The sinusoidal mechanical loading and electric potential are applied to the smart FG plate. The mechanical load is applied on the top surface of the smart FG plate. For convenience, the following nondimensional parameters are used for presenting the numerical results:
3.2. Boundary Conditions
The following boundary conditions of present theory for simply supported FG plates are imposed:
Based on Navier’s procedure, the following displacement components are assumed: where , , and are unknown coefficients to be determined.
The transverse load and electric potential are also expanded in double-Fourier series as
3.3. Mechanical Properties
In this section, a simply supported rectangular FG plate attached with a PFRC layer under transverse load is taken into consideration. The thickness of the FG plate is considered as and the PFRC layer is taken as . The following material properties of the bottom surface of the FG plate are given by
The piezoelectric fiber and matrix of the PFRC material are made of PZT5H and epoxy, respectively. Considering 40% fiber volume fraction, the following elastic and piezoelectric constants of the PFRC layer are obtained by using micromechanics models :
The other coefficients , , , and of the PFRC layer are smaller compared to the effective piezoelectric constant and therefore the piezoelectric coefficients , , , and are not considered for deriving the system equations.
3.4. Numerical Validation
Numerical results of the displacement , and the stress , , are presented in Tables 1–4. The results for different aspect ratios , applied voltages , and different values of the functionally graded material parameter are obtained. The results are compared with those provided by 3D theory , the finite element method , and the higher-order shear and normal deformation theory (HOSNT) . In the HOSNT, the cubic polynomial expansion is applied in both in-plane and transverse displacement fields and it contains 12 unknown parameters. The finite element method analyzed the smart FG plate using eight node isoparametric quadrilateral element based on the first-order shear deformation theory. For the present method, it only has 9 unknowns to be determined and it is very efficient in predicting bending analysis of the smart FG plate for different aspect ratios and applied electric voltages. The aforementioned methods are derived by making displacement assumptions concerning the kinematics of deformations or the stress state through the thickness of the plate.
, , , and .
, , , and .
, , , and .
, , , and .
Tables 1 and 2 show the results of the FG plate attached with the PFRC layer on the top surface for and , respectively. The results of the FG plate attached with the PFRC layer on the bottom surface are presented in Tables 3 and 4, respectively. It can be observed that the results developed by the present method; the HOSNT and the FEM are very close to each other and the results are in good accordance with 3D exact solutions. These results demonstrate that when the PFRC layer is subjected to a positive voltage, it counteracts the deformation caused by the vertically applied downward mechanical load. Conversely, while for applying a negative voltage, the PFRC layer increases the deformation, the deformations are also significantly influenced by the sign of applied voltages and therefore the PFRC layer can be acting as a distributed actuator for functionally graded plates. Moreover, the deformations of the FG plates with are much larger than the FG plates with .
In order to illustrate the accuracy of present method, the transverse stresses , , obtained by present method and 3D theory  are presented in Tables 5 and 6. The results are computed for FG plates attached to the PFRC layer on the top surface with and , respectively. It can be seen that the present results for both thick and thin FG plates are in good agreement with 3D exact solutions.
, , and .
, , and .
In Figures 2(a) and 2(b), the variation of the in-plane displacement and the transverse displacement through the thickness of thin FG plates attached to the PFRC layer on the top surface is shown. The in-plane displacement is linear variation, while the transverse displacement is kept at a constant. The in-plane stresses , , and the transverse stresses , , are plotted in Figures 3 and 4, respectively. All the stress results are nonlinear variation through the thickness of the FG plate. The stress results change in the opposite trend when the PFRC layer is subjected to different signs of applied voltages. The effects of the sign of the applied voltage on numerical results are illustrated again from these figures. Because of the nonzero piezoelectric constant , the variation of the normal stress near the top surface is more than the normal stress in the FG plate.
This paper presented a pseudo-three-dimensional method to analyze a simply supported FG plate attached to a piezoelectric reinforced composite layer by the higher-order shear and normal deformation theory. The piezoelectric fibers in the PFRC layer are oriented longitudinally along the length of the FG plate. The present higher-order theory considers hyperbolic in-plane displacement field and parabolic transverse displacement field. The electrical potential in the PFRC layer is modeled as linear distribution. The governing equations are derived by the principle of the minimum potential energy and Navier’s procedure is applied to solve the equations. The static analysis of the smart FG plate subjected electromechanical load is developed to illustrate the efficiency of the proposed method. The variation of displacement and stress is significantly influenced by the sign of applied voltage on the PFRC layer. In the numerical results, the good agreement was found between the present results and 3D exact solutions for different aspect ratios, applied voltages, and functionally graded material parameters. The present theory is very accurate and efficient in predicting the bending deformations of smart FG plates.