# Pseudo-Three-Dimensional Analysis for Functionally Graded Plate Integrated with a Piezoelectric Fiber Reinforced Composite Layer

Authors: Wei Wang; Sen Li; Lin-Quan Yao; Shi-Chao Yi
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

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.


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## Body

## 1. Introduction

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 (εz=0). 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 (εz≠0) can be firstly found in the literature by Kant and Manjunatha [6] 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. [14] using the finite element approximation. Neves et al. [10] 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. [17] 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 [18] 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 [19] presented 3D thermoelastic deformations of the FG elliptic plate. Reddy and Cheng [20] 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 [21]. Anderson [22] analyzed 3D elasticity solutions for a sandwich composite plate with FGM core. Senthil [23] obtained 3D exact solutions for free and forced vibration of the FG plate and Kashtalyan [24] 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 [25].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 [26] gave 3D theoretical analysis of functionally graded rectangular plates bonded to a piezoelectric plate subjected to transient thermal loading. Reddy and Cheng [27] obtained 3D solutions of FG plates with an attached piezoelectric active material. He et al. [28] 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 [29]. Kargarnovin et al. [30] gave the vibration analysis of a simply FG rectangular plate with piezoelectric patches on its top or bottom surface. Brischetto and Carrera [31] 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 [32] 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 [33] 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 [34]. Shiyekar and Kant [35] 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 thicknessh, length a, and width b as shown in Figure 1. The top and bottom surfaces of the FG plate are at z=±h/2 and the edges of the plate are parallel to x and y axes, respectively. The top surface of the FG plate is integrated with a piezoelectric fiber reinforced composite (PFRC) layer with thickness hp, 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 E(z) was given by [25] (1)Ez=E0ep0.5+z/h,z∈-h2,h2,which dictated the material property through the thickness of the FG plate with a gradient parameter (0≤p≤+∞)p. Young’s modulus Eb=E0 and Et=E0ep 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 [36].Figure 1
Geometry of the rectangular FG plate integrated with a PFRC layer.


### 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 by(2)ux,y,z,t=u0x,y,t+zu1x,y,t+fzu2x,y,t,vx,y,z,t=v0x,y,t+zv1x,y,t+fzv2x,y,t,wx,y,z,t=w0x,y,t+zw1x,y,t+z2w2x,y,t,where u, v, and w are displacement components in the x, y, and z direction, respectively. The parameters (ui,vi,wi)(i=0,1,2) are unknown functions to be determined. f(z) 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 (3)fz=sinh⁡πzh.


### 2.3. Strain-Displacement Relations and Constitutive Equations

The linear strain-displacement relationship for FG piezoelectric materials is given by(4a)εx=∂u0∂x+z∂u1∂x+fz∂u2∂x,(4b)εy=∂v0∂y+z∂v1∂y+fz∂v2∂y,(4c)εz=w1+2zw2,(4d)γxy=∂u0∂y+∂v0∂x+z∂u0∂y+∂v0∂x+fz∂u0∂y+∂v0∂x,(4e)γyz=v1+f′zv2+∂w0∂y+z∂w1∂y+z2∂w2∂y,(4f)γxz=u1+f′zu2+∂w0∂x+z∂w1∂x+z2∂w2∂x.For piezoelectric materials, the constitutive equations can be expressed as(5a)σk=Ckε-eTE,(5b)D=eε+κE.and, for elastic functionally graded material, the constitutive equations are (6)σk=Ckε,where σ, ε, E, and D represent the stress vector, the strain vector, the electric field vector, and the electric displacement vector, respectively. C, e, and κ are the elastic stiffness matrix, the piezoelectric constants matrix, and the dielectric matrix, respectively. The superscript k 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 k 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(7a)σ=σxσyσzσyzσxzσxyT,(7b)ε=εxεyεzγyzγxzγxyT.The electric field vector and the electric displacement vector are given by(8a)D=DxDyDzT,(8b)E=ExEyEzT,where the electric field vector E related to electric potential ϕ is given by (9)Ex=-∂ϕx,y,z∂x,Ey=-∂ϕx,y,z∂y,Ez=-∂ϕx,y,z∂z.The elastic stiffness matrix, the piezoelectric matrix, and the dielectric matrix are given by(10a)C=C11C12C13000C12C22C23000C13C23C33000000C44000000C55000000C66,(10b)e=0000e150000e2400e31e32e33000,(10c)κ=κ11000κ22000κ33,where Cij are the elastic constants, for isotropic functionally graded material, defined by (11)C11=C22=C33=1-νEz1-2ν1+νC12=C13=C23=νEz1-2ν1+νC44=C55=C66=Ez21+ν.


### 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:(12)δU+δV=0.The variation of strain energy is given explicitly by (13)δU=∑k=12∫A∫hkhk+1σxkδεx+σykδεy+σzkδε+σxykδγxy+σxzkδγxz+σyzkδγyzdzdA=∫ANx∂δu0∂x+Nxy∂δu0∂y+Nxy∂δv0∂x+Ny∂δv0∂y+Qxz∂δw0∂x+Qyz∂δw0∂y+Mx∂δu1∂x+Mxy∂δu1∂y+Qxzδu1+Mxy∂δv1∂x+My∂δv1∂y+Qyzδv1+Mxz∂δw1∂x+Myz∂δw1∂y+Qzδw1+Px∂δu2∂x+Pxy∂δu2∂y+Pxzδu2+Pxy∂δv2∂x+Py∂δv2∂y+Pyzδv2+Rxz∂δw2∂x+Ryz∂δw2∂y+2Mzδw2dAwhere N, M, P, Q, and R are the stress resultants defined by (14a)Nx,Ny,Nxy=∑k=12∫hkhk+1σxk,σyk,σxykdz,(14b)Mx,My,Mxy,Mxz,Myz,Mz=∑k=12∫hkhk+1σxk,σyk,σxyk,σxzk,σyzk,σzkzdz,(14c)Px,Py,Pxy=∑k=12∫hkhk+1σxk,σyk,σxykfzdz,(14d)Qxz,Qyz,Qz=∑k=12∫hkhk+1σxzk,σyzk,σzkdz,(14e)Pxz,Pyz=∑k=12∫hkhk+1σxzk,σyzkf′zdz,(14f)Rxz,Ryz=∑k=12∫hkhk+1σxzk,σyzkz2dz,and hk(i=1,2,3) 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, h1, h2, h3 are -h/2, h/2, h/2+hp, respectively. Conversely, h1, h2, h3 are -h/2-hp, -h/2, h/2, respectively.The variation of potential energy of applied transverse loadq can be expressed as (15)δV=-∫Aqδw0+zw1+z2w2dA.Substitute the expressions ofδU and δV in (13) and (15) into (12); then integrate by parts and collect the coefficients of δui, δvi, and δwi(i=0,1,2). The following governing equations of present theory can be obtained: (16a)δu0:∂Nx∂x+∂Nxy∂y=0,(16b)δv0:∂Nxy∂x+∂Ny∂y=0,(16c)δw0:∂Qxz∂x+∂Qyz∂y+q=0,(16d)δu1:∂Mx∂x+∂Mxy∂y-Qxz=0,(16e)δv1:∂Mxy∂x+∂My∂y-Qyz=0,(16f)δw1:∂Mxz∂x+∂Myz∂y-Qz+zq=0,(16g)δu2:∂Px∂x+∂Pxy∂y-Pxz=0,(16h)δv2:∂Pxy∂x+∂Py∂y-Pyz=0,(16i)δw2:∂Rxz∂x+∂Ryz∂y-2Mz+z2q=0.


## 2.1. Geometrical Configuration

Consider a rectangular FG plate with total thicknessh, length a, and width b as shown in Figure 1. The top and bottom surfaces of the FG plate are at z=±h/2 and the edges of the plate are parallel to x and y axes, respectively. The top surface of the FG plate is integrated with a piezoelectric fiber reinforced composite (PFRC) layer with thickness hp, 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 E(z) was given by [25] (1)Ez=E0ep0.5+z/h,z∈-h2,h2,which dictated the material property through the thickness of the FG plate with a gradient parameter (0≤p≤+∞)p. Young’s modulus Eb=E0 and Et=E0ep 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 [36].Figure 1
Geometry of the rectangular FG plate integrated with a PFRC layer.


## 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 by(2)ux,y,z,t=u0x,y,t+zu1x,y,t+fzu2x,y,t,vx,y,z,t=v0x,y,t+zv1x,y,t+fzv2x,y,t,wx,y,z,t=w0x,y,t+zw1x,y,t+z2w2x,y,t,where u, v, and w are displacement components in the x, y, and z direction, respectively. The parameters (ui,vi,wi)(i=0,1,2) are unknown functions to be determined. f(z) 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 (3)fz=sinh⁡πzh.


## 2.3. Strain-Displacement Relations and Constitutive Equations

The linear strain-displacement relationship for FG piezoelectric materials is given by(4a)εx=∂u0∂x+z∂u1∂x+fz∂u2∂x,(4b)εy=∂v0∂y+z∂v1∂y+fz∂v2∂y,(4c)εz=w1+2zw2,(4d)γxy=∂u0∂y+∂v0∂x+z∂u0∂y+∂v0∂x+fz∂u0∂y+∂v0∂x,(4e)γyz=v1+f′zv2+∂w0∂y+z∂w1∂y+z2∂w2∂y,(4f)γxz=u1+f′zu2+∂w0∂x+z∂w1∂x+z2∂w2∂x.For piezoelectric materials, the constitutive equations can be expressed as(5a)σk=Ckε-eTE,(5b)D=eε+κE.and, for elastic functionally graded material, the constitutive equations are (6)σk=Ckε,where σ, ε, E, and D represent the stress vector, the strain vector, the electric field vector, and the electric displacement vector, respectively. C, e, and κ are the elastic stiffness matrix, the piezoelectric constants matrix, and the dielectric matrix, respectively. The superscript k 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 k 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(7a)σ=σxσyσzσyzσxzσxyT,(7b)ε=εxεyεzγyzγxzγxyT.The electric field vector and the electric displacement vector are given by(8a)D=DxDyDzT,(8b)E=ExEyEzT,where the electric field vector E related to electric potential ϕ is given by (9)Ex=-∂ϕx,y,z∂x,Ey=-∂ϕx,y,z∂y,Ez=-∂ϕx,y,z∂z.The elastic stiffness matrix, the piezoelectric matrix, and the dielectric matrix are given by(10a)C=C11C12C13000C12C22C23000C13C23C33000000C44000000C55000000C66,(10b)e=0000e150000e2400e31e32e33000,(10c)κ=κ11000κ22000κ33,where Cij are the elastic constants, for isotropic functionally graded material, defined by (11)C11=C22=C33=1-νEz1-2ν1+νC12=C13=C23=νEz1-2ν1+νC44=C55=C66=Ez21+ν.


## 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:(12)δU+δV=0.The variation of strain energy is given explicitly by (13)δU=∑k=12∫A∫hkhk+1σxkδεx+σykδεy+σzkδε+σxykδγxy+σxzkδγxz+σyzkδγyzdzdA=∫ANx∂δu0∂x+Nxy∂δu0∂y+Nxy∂δv0∂x+Ny∂δv0∂y+Qxz∂δw0∂x+Qyz∂δw0∂y+Mx∂δu1∂x+Mxy∂δu1∂y+Qxzδu1+Mxy∂δv1∂x+My∂δv1∂y+Qyzδv1+Mxz∂δw1∂x+Myz∂δw1∂y+Qzδw1+Px∂δu2∂x+Pxy∂δu2∂y+Pxzδu2+Pxy∂δv2∂x+Py∂δv2∂y+Pyzδv2+Rxz∂δw2∂x+Ryz∂δw2∂y+2Mzδw2dAwhere N, M, P, Q, and R are the stress resultants defined by (14a)Nx,Ny,Nxy=∑k=12∫hkhk+1σxk,σyk,σxykdz,(14b)Mx,My,Mxy,Mxz,Myz,Mz=∑k=12∫hkhk+1σxk,σyk,σxyk,σxzk,σyzk,σzkzdz,(14c)Px,Py,Pxy=∑k=12∫hkhk+1σxk,σyk,σxykfzdz,(14d)Qxz,Qyz,Qz=∑k=12∫hkhk+1σxzk,σyzk,σzkdz,(14e)Pxz,Pyz=∑k=12∫hkhk+1σxzk,σyzkf′zdz,(14f)Rxz,Ryz=∑k=12∫hkhk+1σxzk,σyzkz2dz,and hk(i=1,2,3) 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, h1, h2, h3 are -h/2, h/2, h/2+hp, respectively. Conversely, h1, h2, h3 are -h/2-hp, -h/2, h/2, respectively.The variation of potential energy of applied transverse loadq can be expressed as (15)δV=-∫Aqδw0+zw1+z2w2dA.Substitute the expressions ofδU and δV in (13) and (15) into (12); then integrate by parts and collect the coefficients of δui, δvi, and δwi(i=0,1,2). The following governing equations of present theory can be obtained: (16a)δu0:∂Nx∂x+∂Nxy∂y=0,(16b)δv0:∂Nxy∂x+∂Ny∂y=0,(16c)δw0:∂Qxz∂x+∂Qyz∂y+q=0,(16d)δu1:∂Mx∂x+∂Mxy∂y-Qxz=0,(16e)δv1:∂Mxy∂x+∂My∂y-Qyz=0,(16f)δw1:∂Mxz∂x+∂Myz∂y-Qz+zq=0,(16g)δu2:∂Px∂x+∂Pxy∂y-Pxz=0,(16h)δv2:∂Pxy∂x+∂Py∂y-Pyz=0,(16i)δw2:∂Rxz∂x+∂Ryz∂y-2Mz+z2q=0.


## 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(17a)ϕx,y,z=1hpz-h2ϕ0x,y,h2≤z≤h2+hp,(17b)orϕx,y,z=-1hpz+h2ϕ0x,y,-h2+hp≤z≤-h2,according to the top surface or the bottom surface of the FG plate attached with the PFRC layer [34]. ϕ0(x,y) 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 (ui,vi,wi)(i=0,1,2) and electric potential ϕ0. 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 q0=-40N/m2 is applied on the top surface of the smart FG plate. For convenience, the following nondimensional parameters are used for presenting the numerical results:(18)u∗=E0uq0s3h,w∗=100E0wq0s4h,σx∗,σy∗,σxy∗=σx,σy,σxyq0s2σxz∗,σyz∗=σxz,σyzq0s,σz∗=σzq0,s=ah.


### 3.2. Boundary Conditions

The following boundary conditions of present theory for simply supported FG plates are imposed:(19a)v0=w0=v1=w1=v2=w2=Nx=Mx=Px=0atx=0,a,(19b)u0=w0=u1=w1=u2=w2=Ny=My=Py=0aty=0,b.Based on Navier’s procedure, the following displacement components(ui,vi,wi) are assumed: (20)uix,y,t=∑m=1∞∑n=1∞Uimncos⁡αxsin⁡βy,vix,y,t=∑m=1∞∑n=1∞Vimnsin⁡αxcos⁡βy,wix,y,t=∑m=1∞∑n=1∞Wimnsin⁡αxsin⁡βy,where α=mπ/a, β=nπ/b, and Uimn,Vimn,Wimn(i=0,1,2) are unknown coefficients to be determined.The transverse loadq and electric potential ϕ0 are also expanded in double-Fourier series as (21)qx,y=∑m=1∞∑n=1∞qmnsin⁡αxsin⁡βy,ϕ0x,y=∑m=1∞∑n=1∞ϕ0mnsin⁡αxsin⁡βy.Substituting (20) and (21) into (16a)-(16i), we can obtain the algebraic equations: (22)KU=f,where(23)U=U0mn,V0mn,W0mn,U1mn,V1mn,W1mn,U2mn,V2mn,W2mnT.and the symmetric matrix K and vector f are given in Appendix B.


### 3.3. Mechanical Properties

In this section, a simply supported rectangular FG plate attached with a PFRC layer under transverse loadq is taken into consideration. The thickness of the FG plate is considered as 3mm and the PFRC layer is taken as 250um. The following material properties of the bottom surface of the FG plate are given by (24)E0=200GPa,ν=0.3.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 [33]:(25)C11=32.6GPa,C12=4.3GPa,C13=4.76GPa,C22=C33=7.2GPa,C23=3.85GPa,C44=1.05GPa,C55=C66=1.29GPa,e31=-6.76C/m2,κ11=κ22=0.037×10-9C/Vm,κ33=10.64×10-9C/Vm.The other coefficientse32, e33, e15, and e24 of the PFRC layer are smaller compared to the effective piezoelectric constant e31 and therefore the piezoelectric coefficients e32, e33, e15, and e24 are not considered for deriving the system equations.


### 3.4. Numerical Validation

Numerical results of the displacementu∗, w∗ and the stress σx∗, σy∗, σxy∗ are presented in Tables 1–4. The results for different aspect ratios (s=10,20,100), applied voltages (V=0,100,-100), and different values of the functionally graded material parameter are obtained. The results are compared with those provided by 3D theory [33], the finite element method [34], and the higher-order shear and normal deformation theory (HOSNT) [35]. 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.Table 1
Results of the FG plate(Et/Eb=10) attached to the PFRC layer on the top surface with and without applied voltage.
Aspect ratio s=10 s=20 s=100 Quantity Theory V=0 V=100 V=-100 V=0 V=100 V=-100 V=0 V=100 V=-100 w∗ HOSNT [35] -0.9575 187.8140 -189.7290 -0.9251 45.5336 -47.3839 -0.9147 0.9342 -2.7635 3D [33] -0.9553 186.8222 -188.7329 -0.9252 45.5393 -47.3897 -0.9155 0.9368 -2.7678 FEM [34] -0.9485 183.9178 -185.8148 -0.9233 45.2938 -47.1403 -0.9145 0.9328 -2.7619 Present -0.9544 187.6270 -189.5358 -0.9249 45.5708 -47.4207 -0.9155 0.9360 -2.7669  u∗ HOSNT [35] -0.0194 -0.0233 -0.0156 -0.0195 -0.0340 -0.0049 -0.0195 -0.0202 -0.0187 0.0093 -5.9832 6.0018 0.0093 -1.4685 1.4870 0.0092 -0.0496 0.0681 3D [33] -0.0194 -0.0307 -0.0081 -0.0195 -0.0337 -0.0053 -0.0195 -0.0202 -0.0188 0.0093 -5.9373 5.9558 0.0093 -1.4670 1.4855 0.0093 -0.0497 0.0682 FEM [34] -0.0195 0.0459 -0.0849 -0.0195 -0.0297 -0.0093 -0.0195 -0.0202 -0.0187 0.0093 -5.9402 5.9588 0.0093 -1.4655 1.4840 0.0093 -0.0496 0.0681 Present -0.0196 0.0672 -0.1063 -0.0195 -0.0276 -0.0114 -0.0195 -0.0202 -0.0188 0.0092 -5.9753 5.9938 0.0093 -1.4688 1.4874 0.0093 -0.0496 0.0682  σ x ∗ HOSNT [35] 0.0873 -5.9143 6.0891 0.0873 -1.3682 1.5427 0.0872 0.0296 0.1448 -0.4247 206.3480 -207.1980 -0.4191 50.5808 -51.4189 -0.4172 1.6138 -2.4483 3D [33] 0.0871 -5.8052 5.9794 0.0873 -1.3742 1.5489 0.0874 0.0291 0.1457 -0.4201 203.9840 -204.8249 -0.4171 50.3911 -51.2253 -0.4161 1.6124 -2.4446 FEM [34] 0.0893 -6.1985 6.3772 0.0893 -1.4170 1.5427 0.0893 0.0299 0.1486 -0.4250 208.1964 -209.0447 -0.4250 51.4183 -51.4189 -0.4247 1.6447 -2.4941 Present 0.0881 -6.2363 6.4124 0.0876 -1.3995 1.5747 0.0874 0.0291 0.1457 -0.4252 206.0779 -206.9283 -0.4182 50.4945 -51.3309 -0.4160 1.6113 -2.4434  σy∗ HOSNT [35] 0.0873 -19.8310 20.0051 0.0872 -4.9293 5.1036 0.0871 -0.1140 0.2882 -0.4256 59.2543 -60.1056 -0.4199 14.4701 -15.3100 -0.4181 0.1771 -1.0134 3D [33] 0.0870 -19.6720 19.8460 0.0872 -4.9333 5.1078 0.0873 -0.1145 0.2892 -0.4213 57.9752 -58.8187 -0.4180 14.3338 -15.1698 -0.4170 0.1751 -1.0090 FEM [34] 0.0893 -20.1638 20.3424 0.0893 -5.0400 5.2185 0.0892 -0.1167 0.2951 -0.4260 58.2432 -59.0951 -0.4259 14.5560 -15.4079 -0.4256 0.1772 -1.0283 Present 0.0880 -19.9654 20.1414 0.0875 -4.9489 5.1239 0.0873 -0.1144 0.2891 -0.4261 59.1653 -60.0174 -0.4191 14.3937 -15.2319 -0.4169 0.1746 -1.0083  σxy∗ HOSNT [35] -0.0469 6.9021 -6.9960 -0.0470 1.6983 -1.7923 -0.0470 0.0229 -0.1169 0.2251 -71.0439 71.4941 0.2242 -17.4332 17.8816 0.2239 -0.4803 0.9281 3D [33] -0.0469 6.8593 -6.9530 -0.0470 1.6982 -1.7922 -0.0470 0.0230 -0.1171 0.2242 -70.4771 70.9256 0.2243 -17.4232 17.8717 0.2243 -0.4813 0.9298 FEM [34] -0.0481 7.1027 -7.1989 -0.0481 1.7394 -1.8356 0.0480 0.0233 -0.1194 0.2290 -71.7882 72.7882 0.2290 -17.7746 18.2326 0.2289 -0.4907 0.9486 Present -0.0472 7.0270 -7.1215 -0.0471 1.7079 -1.8021 -0.0470 0.0230 -0.1171 0.2239 -70.9436 71.3913 0.2242 -17.4457 17.8940 0.2242 -0.4810 0.9295 (i)   w ∗ = w ∗ ( a / 2 , b / 2,0 ), u∗=u∗(0,b/2,∓h/2), (σx∗,σy∗)=(σx∗,σy∗)(a/2,b/2,∓h/2), and σxy∗=σxy∗(0,0,∓h/2).Table 2
Results of the FG plate(Et/Eb=0.1) attached to the PFRC layer on the top surface with and without applied voltage.
Aspect ratio s=10 s=20 s=100 Quantity Theory V=0 V=100 V=-100 V=0 V=100 V=-100 V=0 V=100 V=-100 w∗ HOSNT [35] -8.6140 3164.12 -3181.35 -8.3577 790.3310 -807.0460 -8.2750 23.7481 -40.2981 3D [33] -8.7397 3253.30 -3270.80 -8.5437 813.7105 -830.7979 -8.4806 23.9160 -40.5745 FEM [34] -8.6673 3212.90 -3230.30 -8.4154 797.9455 -814.7764 -8.3292 24.4974 -41.4587 Present -8.7085 3299.60 -3317.01 -8.5275 815.3451 -832.4000 -8.4698 24.4438 -41.3835  u∗ HOSNT [35] -0.0905 -3.8486 3.6674 -0.0899 -1.1482 0.9883 -0.0897 -0.1336 -0.0458 0.1653 -112.9800 113.3110 0.1689 -26.5921 26.9299 0.1701 -0.8807 1.2209 3D [33] -0.0914 -3.1845 3.0017 -0.0912 -0.9839 0.8015 -0.0912 -0.1285 -0.0539 0.1686 -114.0117 114.3489 0.1735 -27.0971 27.4441 0.1750 -0.8991 1.2493 FEM [34] -0.0901 -3.0418 2.8616 -0.0902 -1.0641 0.8838 -0.0901 -0.1322 -0.0480 0.1714 -107.7880 108.1308 0.1716 -26.3653 26.7085 0.1715 -0.8840 1.2270 Present -0.0910 -2.3375 2.1555 -0.0911 -0.9344 0.7522 -0.0911 -0.1285 -0.0537 0.1703 -109.6641 110.0046 0.1737 -26.7836 27.1311 0.1748 -0.8969 1.2466  σx∗ HOSNT [35] 0.3908 -33.7320 34.5138 0.3848 -7.7446 8.5744 0.3829 0.0628 0.7031 -0.0848 47.3070 -47.4767 -0.0846 11.2600 -11.4293 -0.0845 0.3627 -0.5319 3D [33] 0.4042 -41.0915 41.8999 0.4031 -9.8557 10.6619 0.4027 -0.0061 0.8115 -0.0807 44.3075 -44.4689 -0.0796 10.6491 -10.8083 -0.0793 0.3451 -0.5036 FEM [34] 0.4066 -42.1225 42.9357 0.4068 -9.6579 10.4715 0.4066 0.0199 0.8013 -0.0793 43.0275 -43.1862 -0.0794 10.5905 -10.7493 -0.0793 0.3460 -0.5046 Present 0.4083 -44.3395 45.1561 0.4027 -9.8812 10.6866 0.4009 0.0013 0.8006 -0.0804 42.7266 -42.8874 -0.0791 10.4844 -10.6425 -0.0786 0.3421 -0.4994  σy∗ HOSNT [35] 0.3763 -163.0770 163.8300 0.3697 -41.0169 41.7565 0.3676 -1.2950 2.0302 -0.0866 30.8163 -30.9899 -0.0862 7.7083 -7.8808 -0.0861 0.2267 -0.3989 3D [33] 0.3900 -170.3199 171.0999 0.3883 -43.1562 43.9327 0.3877 -1.3654 2.1408 -0.0825 28.1032 -28.2681 -0.0812 7.1105 -7.2729 -0.0808 0.2088 -0.3704 FEM [34] 0.3916 -172.9020 173.6851 0.3913 -43.5671 44.3496 0.3909 -1.3748 2.1565 -0.0810 28.0609 -28.2229 -0.0810 7.0834 -7.2453 -0.0809 0.2072 -0.3689 Present 0.3940 -172.3681 173.1561 0.3877 -43.0789 43.8543 0.3857 -1.3566 2.1280 -0.0820 28.0188 -28.1828 -0.0806 7.0440 -7.2052 -0.0802 0.2061 -0.3664  σ x y ∗ HOSNT [35] -0.2116 55.3721 -55.7955 -0.2098 13.8615 -14.2811 -0.2091 0.3560 -0.7743 0.0408 -19.0577 19.1394 0.0416 -4.6504 4.7336 0.0418 -0.1448 0.2285 3D [33] -0.2138 56.9184 -57.3461 -0.2131 14.2724 -14.6986 -0.2128 0.3692 -0.7948 0.0416 -19.4500 19.5333 0.0427 -4.7790 4.8645 0.0431 -0.1491 0.2353 FEM [34] -0.2148 57.9332 -58.3627 -0.2148 14.3377 -14.7673 -0.2147 0.3667 -0.7962 0.0431 -19.1536 19.2399 0.0431 -4.7614 4.8477 0.0431 -0.1489 0.2352 Present -0.2127 58.3656 -58.7911 -0.2126 14.3408 -14.7661 -0.2126 0.3684 -0.7936 0.0420 -19.1476 19.2315 0.0428 -4.7524 4.8379 0.0430 -0.1488 0.2348 (i)   w ∗ = w ∗ ( a / 2 , b / 2,0 ), u∗=u∗(0,b/2,∓h/2), (σx∗,σy∗)=(σx∗,σy∗)(a/2,b/2,∓h/2), and σxy∗=σxy∗(0,0,∓h/2).Table 3
Results of the FG plate(Et/Eb=10) attached to the PFRC layer on the bottom surface with and without applied voltage.
Aspect ratio s=10 s=20 s=100 Quantity Theory V=0 V=100 V=-100 V=0 V=100 V=-100 V=0 V=100 V=-100 w∗ HOSNT [35] -0.9472 359.1410 -361.0400 -0.9172 89.2463 -91.0807 -0.9067 2.7015 -4.1494 3D [33] -0.9428 359.0874 -360.9938 -0.9164 89.5468 -91.3877 -0.9076 2.7147 -4.5346 FEM [34] -0.9416 360.3108 -362.1940 -0.9205 89.4064 -91.2392 -0.9100 2.7028 -4.5180 Present -0.9486 363.9614 -365.8587 -0.9192 89.8131 -91.6515 -0.9098 2.7131 -4.5327  u∗ HOSNT [35] -0.0191 12.6577 -12.6960 -0.0192 2.9598 -2.9982 -0.0192 0.0975 -0.1359 0.0093 0.2688 -0.2502 0.0093 0.0861 -0.0675 0.0093 0.0125 0.0060 3D [33] -0.0192 12.4664 -12.5049 -0.0193 2.9557 -2.9942 -0.0193 0.0978 -0.1364 0.0093 0.2578 -0.2391 0.0093 0.0835 -0.0649 0.0093 0.0124 0.0062 FEM [34] -0.0193 11.9237 -11.9622 -0.0193 2.9147 -2.9532 -0.0192 0.0974 -0.1359 0.0093 0.1967 -0.1781 0.0093 0.0811 -0.0625 0.0093 0.0125 0.0061 Present -0.0193 11.9869 -12.0256 -0.0193 2.9241 -2.9627 -0.0193 0.0977 -0.1363 0.0093 0.1609 -0.1424 0.0093 0.0775 -0.0589 0.0093 0.0124 0.0062  σx∗ HOSNT [35] 0.0874 -50.2609 50.4364 0.0877 -11.8397 12.0150 0.0876 -0.3808 0.5561 -0.4202 47.7941 -48.6351 -0.4147 11.1666 -11.9961 -0.4129 0.0441 -0.5698 3D [33] 0.0865 -48.6233 48.7963 0.0866 -11.6559 11.8291 0.0866 -0.3775 0.5507 -0.4228 47.6247 -48.4703 -0.4179 11.4491 -12.2848 -0.4163 0.0564 -0.8890 FEM [34] 0.0883 -47.8463 48.0230 0.0883 -11.7681 11.9448 0.0883 -0.3842 0.5608 -0.4250 50.8126 -51.6626 -0.4250 11.7704 -12.6204 -0.4246 0.0546 -0.9039 Present 0.0872 -47.1332 47.3076 0.0867 -11.5509 11.7243 0.0865 -0.3768 0.5499 -0.4252 52.1247 -52.9752 -0.4183 11.7060 -12.5427 -0.4161 0.0558 -0.8880  σy∗ HOSNT [35] 0.0877 -32.4460 32.6220 0.0879 -8.0425 8.2183 0.0878 -0.2373 0.4130 -0.4175 186.4510 -187.2890 -0.4130 46.6489 -47.4769 -0.4111 1.4757 -2.2979 3D [33] 0.0867 -31.3030 31.4763 0.0868 -7.8872 8.0607 0.0868 -0.2338 0.4074 -0.4212 185.7531 -186.5955 -0.4162 46.9151 -47.7476 -0.4146 1.4889 -2.3181 FEM [34] 0.0885 -31.8707 32.0478 0.0885 -8.0314 8.2085 0.0885 -0.2378 0.4147 -0.4233 190.3620 -191.2086 -0.4232 47.8967 -48.7431 -0.4229 1.5168 -2.3626 Present 0.0874 -31.4381 31.6129 0.0869 -7.8865 8.0602 0.0867 -0.2334 0.4068 -0.4236 188.7205 -189.5677 -0.4166 47.0638 -47.8971 -0.4144 1.4874 -2.3162  σxy∗ HOSNT [35] -0.0462 21.2813 21.7741 -0.0465 5.2540 -5.3470 -0.0465 0.1638 -0.2568 0.2239 -62.8314 63.2801 0.2234 -15.6605 16.1073 0.2231 -0.4137 0.8600 3D [33] -0.0466 21.4662 -21.5594 -0.0467 5.2583 -5.3516 -0.0467 0.1646 -0.2579 0.2249 -62.8348 63.2847 0.2240 -15.7140 16.1620 0.2237 -0.4161 0.8634 FEM [34] -0.0476 21.4743 -21.5695 -0.0476 5.3330 -5.4282 -0.0476 0.1674 -0.2626 0.2282 -64.9662 65.4226 0.2282 -16.0699 16.5264 0.2282 -0.4228 0.8791 Present -0.0468 21.1201 -21.2138 -0.0467 5.2341 -5.3275 -0.0467 0.1644 -0.2578 0.2232 -64.4092 64.8557 0.2236 -15.8057 16.2528 0.2237 -0.4159 0.8632 (i)   w ∗ = w ∗ ( a / 2 , b / 2,0 ), u∗=u∗(0,b/2,∓h/2), (σx∗,σy∗)=(σx∗,σy∗)(a/2,b/2,∓h/2), and σxy∗=σxy∗(0,0,∓h/2).Table 4
Results of the FG plate(Et/Eb=0.1) attached to the PFRC layer on the bottom surface with and without applied voltage.
Aspect ratio s=10 s=20 s=100 Quantity Theory V=0 V=100 V=-100 V=0 V=100 V=-100 V=0 V=100 V=-100 w∗ HOSNT [35] -9.1979 1776.2 -1794.6 -8.9792 430.7050 -448.6630 -8.9093 8.5929 -26.4115 3D [33] -9.2748 1748.2 -1807.5 -9.0549 436.0142 -454.1240 -8.9338 8.7587 -26.7281 FEM [34] -9.2761 1789.0 -1766.7 -9.0233 430.4678 -448.5145 -8.9364 8.6323 -26.5051 Present -9.2090 1794.1 -1812.5 -9.0354 435.9295 -454.0002 -8.9801 8.7430 -26.7031  u∗ HOSNT [35] -0.0877 57.1545 -57.3300 -0.0874 14.0393 -14.2140 -0.0872 0.4755 -0.6500 0.1878 0.7828 -0.4073 0.1913 0.4707 -0.0880 0.1925 0.2053 0.1797 3D [33] -0.0890 57.1020 -57.2801 -0.0886 14.1124 -14.2896 -0.0885 0.4783 -0.6552 0.1890 0.5288 -0.1507 0.1925 0.3882 -0.0032 0.1936 0.2029 0.1844 FEM [34] -0.0877 57.1020 -57.0792 -0.0878 14.0431 -14.2186 -0.0877 0.4760 -0.6514 0.1931 -0.0201 0.4063 0.1932 0.3963 -0.0100 0.1930 0.2046 0.1815 Present -0.0883 57.3954 -57.5720 -0.0884 14.1204 -14.2972 -0.0884 0.4779 -0.6546 0.1890 -0.3857 0.7637 0.1925 0.3335 0.0515 0.1936 0.2030 0.1843  σ x ∗ HOSNT [35] 0.4122 -200.2660 201.0910 0.4101 -49.1317 49.9519 0.4094 -1.5643 2.3830 -0.0870 5.1151 -5.2891 -0.0853 1.1740 -1.3446 -0.0848 -0.0349 -0.1347 3D [33] 0.4023 -195.9356 196.7419 0.4011 -48.4127 49.2150 0.4005 -1.5486 2.3495 -0.0882 5.4977 -5.6753 -0.0871 1.2984 -1.4726 -0.0866 -0.0313 -0.1419 FEM [34] 0.4060 -198.9894 199.8015 0.4061 -49.1587 49.9708 0.4058 -1.5729 2.3844 -0.0882 5.7883 -5.9647 -0.0882 1.3181 -1.4945 -0.0881 -0.0328 -0.1435 Present 0.4072 -197.2765 198.0908 0.4012 -48.3838 49.1861 0.3992 -1.5440 2.3424 -0.0883 5.8572 -6.0338 -0.0869 1.3117 -1.4856 -0.0865 -0.0317 -0.1413  σy∗ HOSNT [35] 0.4209 -58.3826 59.2244 0.4185 -14.2690 15.1060 0.4177 -0.1697 1.0052 -0.0862 18.5407 -18.7130 -0.0845 4.6123 -4.7813 -0.0840 0.1044 -0.2724 3D [33] 0.4115 -54.9286 55.7515 0.4093 -13.5769 14.3954 0.4085 -0.1529 0.9700 -0.0880 18.8895 -19.0655 -0.0863 4.7369 -4.9096 -0.0858 0.1082 -0.2798 FEM [34] 0.4148 -54.2976 55.1272 0.4147 -13.5690 14.3984 0.4143 -0.1484 0.9771 -0.0874 19.2664 -19.4412 -0.0874 4.8161 -4.9909 -0.0873 0.1096 -0.2842 Present 0.4156 -55.5255 56.3568 0.4094 -13.5211 14.3400 0.4075 -0.1491 0.9640 -0.0875 19.1071 -19.2822 -0.0861 4.7396 -4.9118 -0.0857 0.1077 -0.2791  σxy∗ HOSNT [35] -0.2164 67.1785 -67.6112 -0.2153 16.4960 -16.9266 -0.2150 0.4517 -0.8817 0.0450 -6.5236 6.6136 0.0458 -1.6054 1.6971 0.0461 -0.0201 0.1123 3D [33] -0.2193 67.4898 -67.9285 -0.2182 16.6863 -17.1227 -0.2178 0.4581 -0.8937 0.0453 -6.5681 6.6587 0.0461 -1.6255 1.7177 0.0464 -0.0207 0.1135 FEM [34] -0.2208 68.2317 -68.6734 -0.2209 16.8966 -17.3383 -0.2208 0.4635 -0.9051 0.0472 -6.7491 6.8436 0.0472 -1.6521 1.7466 0.0472 -0.0207 0.1151 Present -0.2177 67.8268 -68.2621 -0.2177 16.6922 -17.1276 -0.2177 0.4574 -0.8928 0.0453 -6.7182 6.8087 0.0461 -1.6333 1.7256 0.0464 -0.0207 0.1134 (i)   w ∗ = w ∗ ( a / 2 , b / 2,0 ), u∗=u∗(0,b/2,∓h/2), (σx∗,σy∗)=(σx∗,σy∗)(a/2,b/2,∓h/2), and σxy∗=σxy∗(0,0,∓h/2).Tables1 and 2 show the results of the FG plate attached with the PFRC layer on the top surface for Et/Eb=10 and 0.1, 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 Et/Eb=0.1 are much larger than the FG plates with Et/Eb=10.In order to illustrate the accuracy of present method, the transverse stressesσxz∗, σyz∗, σz∗ obtained by present method and 3D theory [33] are presented in Tables 5 and 6. The results are computed for FG plates attached to the PFRC layer on the top surface with Et/Eb=10 and 0.1, respectively. It can be seen that the present results for both thick and thin FG plates are in good agreement with 3D exact solutions.Table 5
The transverse stress results of the FG plate(Et/Eb=10) attached to the PFRC layer on the top surface with and without applied voltage.
Aspect ratio s=10 s=20 s=100 Quantity Theory V=0 V=100 V=-100 V=0 V=100 V=-100 V=0 V=100 V=-100 σxz∗ 3D[33] -0.2218 2.7211 -3.1646 -0.2225 0.4628 -0.9079 -0.2228 -0.1960 -0.2495 Present -0.2218 2.6559 -3.1001 -0.2226 0.4572 -0.9024 -0.2228 -0.1961 -0.2495  σyz∗ 3D [33] -0.2215 43.9606 -44.4035 -0.2223 10.9839 -11.4284 -0.2225 0.2278 -0.6729 Present -0.2221 44.1326 -44.5763 -0.2224 10.9897 -11.4345 -0.2225 0.2276 -0.6726  σz∗ 3D [33] -0.3537 43.9487 -44.6562 -0.3554 10.8462 -11.5570 -0.3559 0.0938 -0.8056 Present -0.3552 44.3729 -45.0834 -0.3557 10.8666 -11.5781 -0.3559 0.0935 -0.8053 (i)   σ x z ∗ = σ x z ∗ ( 0 , b / 2 , h / 2 ), σyz∗=σyz∗(a/2,0,h/2), and σz∗=σz∗(a/2,b/2,h/2).Table 6
The transverse stress results of the FG plate(Et/Eb=0.1) attached to the PFRC layer on the top surface with and without applied voltage.
Aspect ratio s=10 s=20 s=100 Quantity Theory V=0 V=100 V=-100 V=0 V=100 V=-100 V=0 V=100 V=-100 σxz∗ 3D [33] -0.2248 -41.5579 41.1083 -0.2255 -10.8859 10.4348 -0.2258 -0.6565 0.2049 Present -0.2248 -42.3466 41.8971 -0.2254 -10.9476 10.4967 -0.2256 -0.6570 0.2057  σyz∗ 3D [33] -0.2114 79.9749 -80.3977 -0.2117 20.2434 -20.6668 -0.2118 0.6118 -1.0353 Present -0.2110 80.4406 -80.8627 -0.2113 20.2345 -20.6571 -0.2114 0.6101 -1.0329  σz∗ 3D [33] -0.6177 115.6052 -116.8407 -0.6184 28.8665 -30.1034 -0.6186 0.5663 -1.8036 Present -0.6177 117.0028 -118.2382 -0.6173 28.8020 -30.0366 -0.6171 0.5598 -1.7941 (i)   σ x z ∗ = σ x z ∗ ( 0 , b / 2 , h / 2 ), σyz∗=σyz∗(a/2,0,h/2), and σz∗=σz∗(a/2,b/2,h/2).In Figures2(a) and 2(b), the variation of the in-plane displacement u∗ and the transverse displacement w∗ through the thickness of thin FG plates (s=100) attached to the PFRC layer on the top surface is shown. The in-plane displacement u∗ is linear variation, while the transverse displacement is kept at a constant. The in-plane stresses σx∗, σy∗, σxy∗ and the transverse stresses σxz∗, σyz∗, σz∗ 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 e31, the variation of the normal stress σx∗ near the top surface is more than the normal stress σy∗ in the FG plate.Figure 2
The variation of displacement results through the thickness of the FG plate(Et/Eb=10,s=100) attached with the PFRC layer on the top surface with or without applied voltage. (a) The in-plane displacement u∗. (b) The transverse displacement w∗.
(a) (b)Figure 3
The variation of in-plane stress results through the thickness of the FG plate(Et/Eb=10,s=100) attached with the PFRC layer on the top surface with or without applied voltage. (a) The in-plane normal stress σx∗. (b) The in-plane normal stress σy∗. (c) The in-plane shear stress σxy∗.
(a) (b) (c)Figure 4
The variation of transverse stress results through the thickness of the FG plate(Et/Eb=10,s=100) attached with the PFRC layer on the top surface with or without applied voltage. (a) The transverse shear stress σxz∗. (b) The transverse shear stress σyz∗. (c) The transverse normal stress σz∗.
(a) (b) (c)


## 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(17a)ϕx,y,z=1hpz-h2ϕ0x,y,h2≤z≤h2+hp,(17b)orϕx,y,z=-1hpz+h2ϕ0x,y,-h2+hp≤z≤-h2,according to the top surface or the bottom surface of the FG plate attached with the PFRC layer [34]. ϕ0(x,y) 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 (ui,vi,wi)(i=0,1,2) and electric potential ϕ0. 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 q0=-40N/m2 is applied on the top surface of the smart FG plate. For convenience, the following nondimensional parameters are used for presenting the numerical results:(18)u∗=E0uq0s3h,w∗=100E0wq0s4h,σx∗,σy∗,σxy∗=σx,σy,σxyq0s2σxz∗,σyz∗=σxz,σyzq0s,σz∗=σzq0,s=ah.


## 3.2. Boundary Conditions

The following boundary conditions of present theory for simply supported FG plates are imposed:(19a)v0=w0=v1=w1=v2=w2=Nx=Mx=Px=0atx=0,a,(19b)u0=w0=u1=w1=u2=w2=Ny=My=Py=0aty=0,b.Based on Navier’s procedure, the following displacement components(ui,vi,wi) are assumed: (20)uix,y,t=∑m=1∞∑n=1∞Uimncos⁡αxsin⁡βy,vix,y,t=∑m=1∞∑n=1∞Vimnsin⁡αxcos⁡βy,wix,y,t=∑m=1∞∑n=1∞Wimnsin⁡αxsin⁡βy,where α=mπ/a, β=nπ/b, and Uimn,Vimn,Wimn(i=0,1,2) are unknown coefficients to be determined.The transverse loadq and electric potential ϕ0 are also expanded in double-Fourier series as (21)qx,y=∑m=1∞∑n=1∞qmnsin⁡αxsin⁡βy,ϕ0x,y=∑m=1∞∑n=1∞ϕ0mnsin⁡αxsin⁡βy.Substituting (20) and (21) into (16a)-(16i), we can obtain the algebraic equations: (22)KU=f,where(23)U=U0mn,V0mn,W0mn,U1mn,V1mn,W1mn,U2mn,V2mn,W2mnT.and the symmetric matrix K and vector f are given in Appendix B.


## 3.3. Mechanical Properties

In this section, a simply supported rectangular FG plate attached with a PFRC layer under transverse loadq is taken into consideration. The thickness of the FG plate is considered as 3mm and the PFRC layer is taken as 250um. The following material properties of the bottom surface of the FG plate are given by (24)E0=200GPa,ν=0.3.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 [33]:(25)C11=32.6GPa,C12=4.3GPa,C13=4.76GPa,C22=C33=7.2GPa,C23=3.85GPa,C44=1.05GPa,C55=C66=1.29GPa,e31=-6.76C/m2,κ11=κ22=0.037×10-9C/Vm,κ33=10.64×10-9C/Vm.The other coefficientse32, e33, e15, and e24 of the PFRC layer are smaller compared to the effective piezoelectric constant e31 and therefore the piezoelectric coefficients e32, e33, e15, and e24 are not considered for deriving the system equations.


## 3.4. Numerical Validation

Numerical results of the displacementu∗, w∗ and the stress σx∗, σy∗, σxy∗ are presented in Tables 1–4. The results for different aspect ratios (s=10,20,100), applied voltages (V=0,100,-100), and different values of the functionally graded material parameter are obtained. The results are compared with those provided by 3D theory [33], the finite element method [34], and the higher-order shear and normal deformation theory (HOSNT) [35]. 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.Table 1
Results of the FG plate(Et/Eb=10) attached to the PFRC layer on the top surface with and without applied voltage.
Aspect ratio s=10 s=20 s=100 Quantity Theory V=0 V=100 V=-100 V=0 V=100 V=-100 V=0 V=100 V=-100 w∗ HOSNT [35] -0.9575 187.8140 -189.7290 -0.9251 45.5336 -47.3839 -0.9147 0.9342 -2.7635 3D [33] -0.9553 186.8222 -188.7329 -0.9252 45.5393 -47.3897 -0.9155 0.9368 -2.7678 FEM [34] -0.9485 183.9178 -185.8148 -0.9233 45.2938 -47.1403 -0.9145 0.9328 -2.7619 Present -0.9544 187.6270 -189.5358 -0.9249 45.5708 -47.4207 -0.9155 0.9360 -2.7669  u∗ HOSNT [35] -0.0194 -0.0233 -0.0156 -0.0195 -0.0340 -0.0049 -0.0195 -0.0202 -0.0187 0.0093 -5.9832 6.0018 0.0093 -1.4685 1.4870 0.0092 -0.0496 0.0681 3D [33] -0.0194 -0.0307 -0.0081 -0.0195 -0.0337 -0.0053 -0.0195 -0.0202 -0.0188 0.0093 -5.9373 5.9558 0.0093 -1.4670 1.4855 0.0093 -0.0497 0.0682 FEM [34] -0.0195 0.0459 -0.0849 -0.0195 -0.0297 -0.0093 -0.0195 -0.0202 -0.0187 0.0093 -5.9402 5.9588 0.0093 -1.4655 1.4840 0.0093 -0.0496 0.0681 Present -0.0196 0.0672 -0.1063 -0.0195 -0.0276 -0.0114 -0.0195 -0.0202 -0.0188 0.0092 -5.9753 5.9938 0.0093 -1.4688 1.4874 0.0093 -0.0496 0.0682  σ x ∗ HOSNT [35] 0.0873 -5.9143 6.0891 0.0873 -1.3682 1.5427 0.0872 0.0296 0.1448 -0.4247 206.3480 -207.1980 -0.4191 50.5808 -51.4189 -0.4172 1.6138 -2.4483 3D [33] 0.0871 -5.8052 5.9794 0.0873 -1.3742 1.5489 0.0874 0.0291 0.1457 -0.4201 203.9840 -204.8249 -0.4171 50.3911 -51.2253 -0.4161 1.6124 -2.4446 FEM [34] 0.0893 -6.1985 6.3772 0.0893 -1.4170 1.5427 0.0893 0.0299 0.1486 -0.4250 208.1964 -209.0447 -0.4250 51.4183 -51.4189 -0.4247 1.6447 -2.4941 Present 0.0881 -6.2363 6.4124 0.0876 -1.3995 1.5747 0.0874 0.0291 0.1457 -0.4252 206.0779 -206.9283 -0.4182 50.4945 -51.3309 -0.4160 1.6113 -2.4434  σy∗ HOSNT [35] 0.0873 -19.8310 20.0051 0.0872 -4.9293 5.1036 0.0871 -0.1140 0.2882 -0.4256 59.2543 -60.1056 -0.4199 14.4701 -15.3100 -0.4181 0.1771 -1.0134 3D [33] 0.0870 -19.6720 19.8460 0.0872 -4.9333 5.1078 0.0873 -0.1145 0.2892 -0.4213 57.9752 -58.8187 -0.4180 14.3338 -15.1698 -0.4170 0.1751 -1.0090 FEM [34] 0.0893 -20.1638 20.3424 0.0893 -5.0400 5.2185 0.0892 -0.1167 0.2951 -0.4260 58.2432 -59.0951 -0.4259 14.5560 -15.4079 -0.4256 0.1772 -1.0283 Present 0.0880 -19.9654 20.1414 0.0875 -4.9489 5.1239 0.0873 -0.1144 0.2891 -0.4261 59.1653 -60.0174 -0.4191 14.3937 -15.2319 -0.4169 0.1746 -1.0083  σxy∗ HOSNT [35] -0.0469 6.9021 -6.9960 -0.0470 1.6983 -1.7923 -0.0470 0.0229 -0.1169 0.2251 -71.0439 71.4941 0.2242 -17.4332 17.8816 0.2239 -0.4803 0.9281 3D [33] -0.0469 6.8593 -6.9530 -0.0470 1.6982 -1.7922 -0.0470 0.0230 -0.1171 0.2242 -70.4771 70.9256 0.2243 -17.4232 17.8717 0.2243 -0.4813 0.9298 FEM [34] -0.0481 7.1027 -7.1989 -0.0481 1.7394 -1.8356 0.0480 0.0233 -0.1194 0.2290 -71.7882 72.7882 0.2290 -17.7746 18.2326 0.2289 -0.4907 0.9486 Present -0.0472 7.0270 -7.1215 -0.0471 1.7079 -1.8021 -0.0470 0.0230 -0.1171 0.2239 -70.9436 71.3913 0.2242 -17.4457 17.8940 0.2242 -0.4810 0.9295 (i)   w ∗ = w ∗ ( a / 2 , b / 2,0 ), u∗=u∗(0,b/2,∓h/2), (σx∗,σy∗)=(σx∗,σy∗)(a/2,b/2,∓h/2), and σxy∗=σxy∗(0,0,∓h/2).Table 2
Results of the FG plate(Et/Eb=0.1) attached to the PFRC layer on the top surface with and without applied voltage.
Aspect ratio s=10 s=20 s=100 Quantity Theory V=0 V=100 V=-100 V=0 V=100 V=-100 V=0 V=100 V=-100 w∗ HOSNT [35] -8.6140 3164.12 -3181.35 -8.3577 790.3310 -807.0460 -8.2750 23.7481 -40.2981 3D [33] -8.7397 3253.30 -3270.80 -8.5437 813.7105 -830.7979 -8.4806 23.9160 -40.5745 FEM [34] -8.6673 3212.90 -3230.30 -8.4154 797.9455 -814.7764 -8.3292 24.4974 -41.4587 Present -8.7085 3299.60 -3317.01 -8.5275 815.3451 -832.4000 -8.4698 24.4438 -41.3835  u∗ HOSNT [35] -0.0905 -3.8486 3.6674 -0.0899 -1.1482 0.9883 -0.0897 -0.1336 -0.0458 0.1653 -112.9800 113.3110 0.1689 -26.5921 26.9299 0.1701 -0.8807 1.2209 3D [33] -0.0914 -3.1845 3.0017 -0.0912 -0.9839 0.8015 -0.0912 -0.1285 -0.0539 0.1686 -114.0117 114.3489 0.1735 -27.0971 27.4441 0.1750 -0.8991 1.2493 FEM [34] -0.0901 -3.0418 2.8616 -0.0902 -1.0641 0.8838 -0.0901 -0.1322 -0.0480 0.1714 -107.7880 108.1308 0.1716 -26.3653 26.7085 0.1715 -0.8840 1.2270 Present -0.0910 -2.3375 2.1555 -0.0911 -0.9344 0.7522 -0.0911 -0.1285 -0.0537 0.1703 -109.6641 110.0046 0.1737 -26.7836 27.1311 0.1748 -0.8969 1.2466  σx∗ HOSNT [35] 0.3908 -33.7320 34.5138 0.3848 -7.7446 8.5744 0.3829 0.0628 0.7031 -0.0848 47.3070 -47.4767 -0.0846 11.2600 -11.4293 -0.0845 0.3627 -0.5319 3D [33] 0.4042 -41.0915 41.8999 0.4031 -9.8557 10.6619 0.4027 -0.0061 0.8115 -0.0807 44.3075 -44.4689 -0.0796 10.6491 -10.8083 -0.0793 0.3451 -0.5036 FEM [34] 0.4066 -42.1225 42.9357 0.4068 -9.6579 10.4715 0.4066 0.0199 0.8013 -0.0793 43.0275 -43.1862 -0.0794 10.5905 -10.7493 -0.0793 0.3460 -0.5046 Present 0.4083 -44.3395 45.1561 0.4027 -9.8812 10.6866 0.4009 0.0013 0.8006 -0.0804 42.7266 -42.8874 -0.0791 10.4844 -10.6425 -0.0786 0.3421 -0.4994  σy∗ HOSNT [35] 0.3763 -163.0770 163.8300 0.3697 -41.0169 41.7565 0.3676 -1.2950 2.0302 -0.0866 30.8163 -30.9899 -0.0862 7.7083 -7.8808 -0.0861 0.2267 -0.3989 3D [33] 0.3900 -170.3199 171.0999 0.3883 -43.1562 43.9327 0.3877 -1.3654 2.1408 -0.0825 28.1032 -28.2681 -0.0812 7.1105 -7.2729 -0.0808 0.2088 -0.3704 FEM [34] 0.3916 -172.9020 173.6851 0.3913 -43.5671 44.3496 0.3909 -1.3748 2.1565 -0.0810 28.0609 -28.2229 -0.0810 7.0834 -7.2453 -0.0809 0.2072 -0.3689 Present 0.3940 -172.3681 173.1561 0.3877 -43.0789 43.8543 0.3857 -1.3566 2.1280 -0.0820 28.0188 -28.1828 -0.0806 7.0440 -7.2052 -0.0802 0.2061 -0.3664  σ x y ∗ HOSNT [35] -0.2116 55.3721 -55.7955 -0.2098 13.8615 -14.2811 -0.2091 0.3560 -0.7743 0.0408 -19.0577 19.1394 0.0416 -4.6504 4.7336 0.0418 -0.1448 0.2285 3D [33] -0.2138 56.9184 -57.3461 -0.2131 14.2724 -14.6986 -0.2128 0.3692 -0.7948 0.0416 -19.4500 19.5333 0.0427 -4.7790 4.8645 0.0431 -0.1491 0.2353 FEM [34] -0.2148 57.9332 -58.3627 -0.2148 14.3377 -14.7673 -0.2147 0.3667 -0.7962 0.0431 -19.1536 19.2399 0.0431 -4.7614 4.8477 0.0431 -0.1489 0.2352 Present -0.2127 58.3656 -58.7911 -0.2126 14.3408 -14.7661 -0.2126 0.3684 -0.7936 0.0420 -19.1476 19.2315 0.0428 -4.7524 4.8379 0.0430 -0.1488 0.2348 (i)   w ∗ = w ∗ ( a / 2 , b / 2,0 ), u∗=u∗(0,b/2,∓h/2), (σx∗,σy∗)=(σx∗,σy∗)(a/2,b/2,∓h/2), and σxy∗=σxy∗(0,0,∓h/2).Table 3
Results of the FG plate(Et/Eb=10) attached to the PFRC layer on the bottom surface with and without applied voltage.
Aspect ratio s=10 s=20 s=100 Quantity Theory V=0 V=100 V=-100 V=0 V=100 V=-100 V=0 V=100 V=-100 w∗ HOSNT [35] -0.9472 359.1410 -361.0400 -0.9172 89.2463 -91.0807 -0.9067 2.7015 -4.1494 3D [33] -0.9428 359.0874 -360.9938 -0.9164 89.5468 -91.3877 -0.9076 2.7147 -4.5346 FEM [34] -0.9416 360.3108 -362.1940 -0.9205 89.4064 -91.2392 -0.9100 2.7028 -4.5180 Present -0.9486 363.9614 -365.8587 -0.9192 89.8131 -91.6515 -0.9098 2.7131 -4.5327  u∗ HOSNT [35] -0.0191 12.6577 -12.6960 -0.0192 2.9598 -2.9982 -0.0192 0.0975 -0.1359 0.0093 0.2688 -0.2502 0.0093 0.0861 -0.0675 0.0093 0.0125 0.0060 3D [33] -0.0192 12.4664 -12.5049 -0.0193 2.9557 -2.9942 -0.0193 0.0978 -0.1364 0.0093 0.2578 -0.2391 0.0093 0.0835 -0.0649 0.0093 0.0124 0.0062 FEM [34] -0.0193 11.9237 -11.9622 -0.0193 2.9147 -2.9532 -0.0192 0.0974 -0.1359 0.0093 0.1967 -0.1781 0.0093 0.0811 -0.0625 0.0093 0.0125 0.0061 Present -0.0193 11.9869 -12.0256 -0.0193 2.9241 -2.9627 -0.0193 0.0977 -0.1363 0.0093 0.1609 -0.1424 0.0093 0.0775 -0.0589 0.0093 0.0124 0.0062  σx∗ HOSNT [35] 0.0874 -50.2609 50.4364 0.0877 -11.8397 12.0150 0.0876 -0.3808 0.5561 -0.4202 47.7941 -48.6351 -0.4147 11.1666 -11.9961 -0.4129 0.0441 -0.5698 3D [33] 0.0865 -48.6233 48.7963 0.0866 -11.6559 11.8291 0.0866 -0.3775 0.5507 -0.4228 47.6247 -48.4703 -0.4179 11.4491 -12.2848 -0.4163 0.0564 -0.8890 FEM [34] 0.0883 -47.8463 48.0230 0.0883 -11.7681 11.9448 0.0883 -0.3842 0.5608 -0.4250 50.8126 -51.6626 -0.4250 11.7704 -12.6204 -0.4246 0.0546 -0.9039 Present 0.0872 -47.1332 47.3076 0.0867 -11.5509 11.7243 0.0865 -0.3768 0.5499 -0.4252 52.1247 -52.9752 -0.4183 11.7060 -12.5427 -0.4161 0.0558 -0.8880  σy∗ HOSNT [35] 0.0877 -32.4460 32.6220 0.0879 -8.0425 8.2183 0.0878 -0.2373 0.4130 -0.4175 186.4510 -187.2890 -0.4130 46.6489 -47.4769 -0.4111 1.4757 -2.2979 3D [33] 0.0867 -31.3030 31.4763 0.0868 -7.8872 8.0607 0.0868 -0.2338 0.4074 -0.4212 185.7531 -186.5955 -0.4162 46.9151 -47.7476 -0.4146 1.4889 -2.3181 FEM [34] 0.0885 -31.8707 32.0478 0.0885 -8.0314 8.2085 0.0885 -0.2378 0.4147 -0.4233 190.3620 -191.2086 -0.4232 47.8967 -48.7431 -0.4229 1.5168 -2.3626 Present 0.0874 -31.4381 31.6129 0.0869 -7.8865 8.0602 0.0867 -0.2334 0.4068 -0.4236 188.7205 -189.5677 -0.4166 47.0638 -47.8971 -0.4144 1.4874 -2.3162  σxy∗ HOSNT [35] -0.0462 21.2813 21.7741 -0.0465 5.2540 -5.3470 -0.0465 0.1638 -0.2568 0.2239 -62.8314 63.2801 0.2234 -15.6605 16.1073 0.2231 -0.4137 0.8600 3D [33] -0.0466 21.4662 -21.5594 -0.0467 5.2583 -5.3516 -0.0467 0.1646 -0.2579 0.2249 -62.8348 63.2847 0.2240 -15.7140 16.1620 0.2237 -0.4161 0.8634 FEM [34] -0.0476 21.4743 -21.5695 -0.0476 5.3330 -5.4282 -0.0476 0.1674 -0.2626 0.2282 -64.9662 65.4226 0.2282 -16.0699 16.5264 0.2282 -0.4228 0.8791 Present -0.0468 21.1201 -21.2138 -0.0467 5.2341 -5.3275 -0.0467 0.1644 -0.2578 0.2232 -64.4092 64.8557 0.2236 -15.8057 16.2528 0.2237 -0.4159 0.8632 (i)   w ∗ = w ∗ ( a / 2 , b / 2,0 ), u∗=u∗(0,b/2,∓h/2), (σx∗,σy∗)=(σx∗,σy∗)(a/2,b/2,∓h/2), and σxy∗=σxy∗(0,0,∓h/2).Table 4
Results of the FG plate(Et/Eb=0.1) attached to the PFRC layer on the bottom surface with and without applied voltage.
Aspect ratio s=10 s=20 s=100 Quantity Theory V=0 V=100 V=-100 V=0 V=100 V=-100 V=0 V=100 V=-100 w∗ HOSNT [35] -9.1979 1776.2 -1794.6 -8.9792 430.7050 -448.6630 -8.9093 8.5929 -26.4115 3D [33] -9.2748 1748.2 -1807.5 -9.0549 436.0142 -454.1240 -8.9338 8.7587 -26.7281 FEM [34] -9.2761 1789.0 -1766.7 -9.0233 430.4678 -448.5145 -8.9364 8.6323 -26.5051 Present -9.2090 1794.1 -1812.5 -9.0354 435.9295 -454.0002 -8.9801 8.7430 -26.7031  u∗ HOSNT [35] -0.0877 57.1545 -57.3300 -0.0874 14.0393 -14.2140 -0.0872 0.4755 -0.6500 0.1878 0.7828 -0.4073 0.1913 0.4707 -0.0880 0.1925 0.2053 0.1797 3D [33] -0.0890 57.1020 -57.2801 -0.0886 14.1124 -14.2896 -0.0885 0.4783 -0.6552 0.1890 0.5288 -0.1507 0.1925 0.3882 -0.0032 0.1936 0.2029 0.1844 FEM [34] -0.0877 57.1020 -57.0792 -0.0878 14.0431 -14.2186 -0.0877 0.4760 -0.6514 0.1931 -0.0201 0.4063 0.1932 0.3963 -0.0100 0.1930 0.2046 0.1815 Present -0.0883 57.3954 -57.5720 -0.0884 14.1204 -14.2972 -0.0884 0.4779 -0.6546 0.1890 -0.3857 0.7637 0.1925 0.3335 0.0515 0.1936 0.2030 0.1843  σ x ∗ HOSNT [35] 0.4122 -200.2660 201.0910 0.4101 -49.1317 49.9519 0.4094 -1.5643 2.3830 -0.0870 5.1151 -5.2891 -0.0853 1.1740 -1.3446 -0.0848 -0.0349 -0.1347 3D [33] 0.4023 -195.9356 196.7419 0.4011 -48.4127 49.2150 0.4005 -1.5486 2.3495 -0.0882 5.4977 -5.6753 -0.0871 1.2984 -1.4726 -0.0866 -0.0313 -0.1419 FEM [34] 0.4060 -198.9894 199.8015 0.4061 -49.1587 49.9708 0.4058 -1.5729 2.3844 -0.0882 5.7883 -5.9647 -0.0882 1.3181 -1.4945 -0.0881 -0.0328 -0.1435 Present 0.4072 -197.2765 198.0908 0.4012 -48.3838 49.1861 0.3992 -1.5440 2.3424 -0.0883 5.8572 -6.0338 -0.0869 1.3117 -1.4856 -0.0865 -0.0317 -0.1413  σy∗ HOSNT [35] 0.4209 -58.3826 59.2244 0.4185 -14.2690 15.1060 0.4177 -0.1697 1.0052 -0.0862 18.5407 -18.7130 -0.0845 4.6123 -4.7813 -0.0840 0.1044 -0.2724 3D [33] 0.4115 -54.9286 55.7515 0.4093 -13.5769 14.3954 0.4085 -0.1529 0.9700 -0.0880 18.8895 -19.0655 -0.0863 4.7369 -4.9096 -0.0858 0.1082 -0.2798 FEM [34] 0.4148 -54.2976 55.1272 0.4147 -13.5690 14.3984 0.4143 -0.1484 0.9771 -0.0874 19.2664 -19.4412 -0.0874 4.8161 -4.9909 -0.0873 0.1096 -0.2842 Present 0.4156 -55.5255 56.3568 0.4094 -13.5211 14.3400 0.4075 -0.1491 0.9640 -0.0875 19.1071 -19.2822 -0.0861 4.7396 -4.9118 -0.0857 0.1077 -0.2791  σxy∗ HOSNT [35] -0.2164 67.1785 -67.6112 -0.2153 16.4960 -16.9266 -0.2150 0.4517 -0.8817 0.0450 -6.5236 6.6136 0.0458 -1.6054 1.6971 0.0461 -0.0201 0.1123 3D [33] -0.2193 67.4898 -67.9285 -0.2182 16.6863 -17.1227 -0.2178 0.4581 -0.8937 0.0453 -6.5681 6.6587 0.0461 -1.6255 1.7177 0.0464 -0.0207 0.1135 FEM [34] -0.2208 68.2317 -68.6734 -0.2209 16.8966 -17.3383 -0.2208 0.4635 -0.9051 0.0472 -6.7491 6.8436 0.0472 -1.6521 1.7466 0.0472 -0.0207 0.1151 Present -0.2177 67.8268 -68.2621 -0.2177 16.6922 -17.1276 -0.2177 0.4574 -0.8928 0.0453 -6.7182 6.8087 0.0461 -1.6333 1.7256 0.0464 -0.0207 0.1134 (i)   w ∗ = w ∗ ( a / 2 , b / 2,0 ), u∗=u∗(0,b/2,∓h/2), (σx∗,σy∗)=(σx∗,σy∗)(a/2,b/2,∓h/2), and σxy∗=σxy∗(0,0,∓h/2).Tables1 and 2 show the results of the FG plate attached with the PFRC layer on the top surface for Et/Eb=10 and 0.1, 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 Et/Eb=0.1 are much larger than the FG plates with Et/Eb=10.In order to illustrate the accuracy of present method, the transverse stressesσxz∗, σyz∗, σz∗ obtained by present method and 3D theory [33] are presented in Tables 5 and 6. The results are computed for FG plates attached to the PFRC layer on the top surface with Et/Eb=10 and 0.1, respectively. It can be seen that the present results for both thick and thin FG plates are in good agreement with 3D exact solutions.Table 5
The transverse stress results of the FG plate(Et/Eb=10) attached to the PFRC layer on the top surface with and without applied voltage.
Aspect ratio s=10 s=20 s=100 Quantity Theory V=0 V=100 V=-100 V=0 V=100 V=-100 V=0 V=100 V=-100 σxz∗ 3D[33] -0.2218 2.7211 -3.1646 -0.2225 0.4628 -0.9079 -0.2228 -0.1960 -0.2495 Present -0.2218 2.6559 -3.1001 -0.2226 0.4572 -0.9024 -0.2228 -0.1961 -0.2495  σyz∗ 3D [33] -0.2215 43.9606 -44.4035 -0.2223 10.9839 -11.4284 -0.2225 0.2278 -0.6729 Present -0.2221 44.1326 -44.5763 -0.2224 10.9897 -11.4345 -0.2225 0.2276 -0.6726  σz∗ 3D [33] -0.3537 43.9487 -44.6562 -0.3554 10.8462 -11.5570 -0.3559 0.0938 -0.8056 Present -0.3552 44.3729 -45.0834 -0.3557 10.8666 -11.5781 -0.3559 0.0935 -0.8053 (i)   σ x z ∗ = σ x z ∗ ( 0 , b / 2 , h / 2 ), σyz∗=σyz∗(a/2,0,h/2), and σz∗=σz∗(a/2,b/2,h/2).Table 6
The transverse stress results of the FG plate(Et/Eb=0.1) attached to the PFRC layer on the top surface with and without applied voltage.
Aspect ratio s=10 s=20 s=100 Quantity Theory V=0 V=100 V=-100 V=0 V=100 V=-100 V=0 V=100 V=-100 σxz∗ 3D [33] -0.2248 -41.5579 41.1083 -0.2255 -10.8859 10.4348 -0.2258 -0.6565 0.2049 Present -0.2248 -42.3466 41.8971 -0.2254 -10.9476 10.4967 -0.2256 -0.6570 0.2057  σyz∗ 3D [33] -0.2114 79.9749 -80.3977 -0.2117 20.2434 -20.6668 -0.2118 0.6118 -1.0353 Present -0.2110 80.4406 -80.8627 -0.2113 20.2345 -20.6571 -0.2114 0.6101 -1.0329  σz∗ 3D [33] -0.6177 115.6052 -116.8407 -0.6184 28.8665 -30.1034 -0.6186 0.5663 -1.8036 Present -0.6177 117.0028 -118.2382 -0.6173 28.8020 -30.0366 -0.6171 0.5598 -1.7941 (i)   σ x z ∗ = σ x z ∗ ( 0 , b / 2 , h / 2 ), σyz∗=σyz∗(a/2,0,h/2), and σz∗=σz∗(a/2,b/2,h/2).In Figures2(a) and 2(b), the variation of the in-plane displacement u∗ and the transverse displacement w∗ through the thickness of thin FG plates (s=100) attached to the PFRC layer on the top surface is shown. The in-plane displacement u∗ is linear variation, while the transverse displacement is kept at a constant. The in-plane stresses σx∗, σy∗, σxy∗ and the transverse stresses σxz∗, σyz∗, σz∗ 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 e31, the variation of the normal stress σx∗ near the top surface is more than the normal stress σy∗ in the FG plate.Figure 2
The variation of displacement results through the thickness of the FG plate(Et/Eb=10,s=100) attached with the PFRC layer on the top surface with or without applied voltage. (a) The in-plane displacement u∗. (b) The transverse displacement w∗.
(a) (b)Figure 3
The variation of in-plane stress results through the thickness of the FG plate(Et/Eb=10,s=100) attached with the PFRC layer on the top surface with or without applied voltage. (a) The in-plane normal stress σx∗. (b) The in-plane normal stress σy∗. (c) The in-plane shear stress σxy∗.
(a) (b) (c)Figure 4
The variation of transverse stress results through the thickness of the FG plate(Et/Eb=10,s=100) attached with the PFRC layer on the top surface with or without applied voltage. (a) The transverse shear stress σxz∗. (b) The transverse shear stress σyz∗. (c) The transverse normal stress σz∗.
(a) (b) (c)


## 4. Conclusions

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.



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*Source: 8586310-2019-04-24.xml*