Free Vibration and Static Bending Analysis of Piezoelectric Functionally Graded Material Plates Resting on One Area of Two-Parameter Elastic Foundation

Free vibration and static bending analysis of piezoelectric functionally graded material plates resting on one area of the two-parameter elastic foundation is ﬁrstly investigated in this paper. The third-order shear deformation theory of Reddy and 8-node plate elements are employed to derive the ﬁnite element formulations of the structures; this theory does not need any shear correction factors; however, the mechanical response of the structure is described exactly. Veriﬁcation problems are performed to evaluate the accuracy of the proposed theory and mathematical model. A wide range of parameter study is investigated to ﬁgure out the eﬀect of geometrical, physical, and material properties such as the plate dimension, volume fraction index, piezoelectric eﬀect, elastic foundation coeﬃcients, and the square size of the area of the foundation on the free vibration and static bending of piezoelectric functionally graded material plates. These numerical results of this work aim to contribute to scientiﬁc knowledge of these smart structures in engineering practice.


Introduction
Nowadays, due to the development of the science of technology, a huge range of smart materials such as functionally graded materials, shape memory alloys, shape memory polymers, magnetostrictive materials, and piezoelectric materials has been considered and applied to engineering practice, in which piezoelectric materials with mainly twocomponent actuators and sensors are employed and performed in a wide range of engineering structures [1][2][3][4][5]. Normally, one smart structure is commonly made of laminated composites from ceramic proportions. erefore, at the surfaces among such layers, due to discontinuities of different materials, the stress concentrations usually occur; thus, this is one of the main reasons deriving the failures of the laminated composite structures. However, this issue can be easily overcome by taking advantage of functionally graded piezoelectric materials (FGPM). In the last few decades, considering the mechanical behavior of piezoelectric FGM structures has become more irresistible. To confirm this knowledge, numerous published scientific papers relating to piezoelectric analyses for FGPM plates can be found in [6][7][8][9][10][11][12][13][14][15][16]. As a result, in this work, we do not need to talk about the basic information of FGM structures as several types of FGM materials such as power-law functionally graded materials (P-FGM), sigmoid law functionally graded materials (S-FGM), and exponential law functionally graded materials (E-FGM) were presented in detail in [17,18]. e mechanical performances of these FGM structures are also investigated in the following papers. Tinh et al. [19] studied mechanical behaviors of heated functionally graded plates in the high-temperature environment by using the finite element method and a new third-order shear deformation plate theory. Nguyen and his coworkers [20] investigated the mechanical buckling of stiffened functionally graded material (FGM) plates. A wide range of studied parameters was carried out such as the effects of material distribution, the thickness-to-width ratio, and stiffener parameters on the buckling characteristics of the stiffened FGM plates. Nam et al. [21] used the finite element method and the phase-field model to consider the buckling behavior of stiffened functionally graded (FG) plate with cracks. Besides, the interactions of the mechanical structures with the elastic foundations have been taken into account in many investigations of scientists worldwide. For example, to describe the behavior between the plate and foundation, different types of mathematical elastic foundation models have been introduced [22,23]. e simplest one is Winkler, usually called one-parameter elastic foundations, which assumes the foundation as a series of dependent springs without coupling impacts among each other [24]. en, this model is developed by Pasternak taking into account a shear spring component to model the interactions among the separated springs in Winkler's elastic foundation [25][26][27][28][29][30]. Besides, substantial investigations depended on classical plate theory [31,32], first-order shear deformation theory [24,[33][34][35], higher-order shear deformation [36], and three-dimensional elasticity theory (3D theory) [32,[37][38][39]. Zenkour and coworkers [40] introduced the bending static analysis of fiber-reinforced viscoelastic composite plate resting on the two-parameter elastic foundation of Pasternak. Several explorations concerning different mechanical responses of FG plates using HSDT and resting on elastic foundations have been arisen [41][42][43]. Abderrahmane et al. [44] used the analytical method to study the dynamic behaviors of the FG Sandwich plate resting on an elastic foundation based on refined shear deformation theory. Abdallah et al. [45] employed a cubic shear deformation theory and the exact solution to examine the bending response of FG porous plates. Mohamed et al. [46] used a new innovative 3-unknown HSDT to research buckling and free vibration of FG Sandwich plates resting on elastic foundations. Mokhtar et al. [47] employed a four-variable quasi-3D HSDT to give out the exact solution for the static and vibration problems of thick advanced composite plates. Sara and her coworkers [48] studied the buckling response of FG Sandwich plates resting on elastic foundation based on a novel four-unknown integral model. Salah et al. [49] also studied buckling of FG Sandwich plates resting on elastic foundation subjected to hygro-thermo-mechanical loads. Mohammed et al. [50] investigated the effects of the boundary conditions on the bending and free vibration behavior of FGM Sandwich plates using a four-unknown refined integral plate theory.
ere are many approaches to calculating the static bending, free vibration, and dynamic problems of FGM plates such as the classical plate theory, first-order shear deformation theory (FSDT), and higher-order shear deformation theories. e classical plate theory [51] is suitable for thin plates, and it does not take into account the effect of the shear force. Although the FSDT takes into account the effect of shear force, it cannot describe the exact distribution of shear stress according to the plate thickness; besides, the calculation needs to use the shear correction factor. e third-order shear deformation theory [8,52] presents exactly the distribution of shear stress in the thickness direction and does not use any shear correction factors. Notwithstanding, although the calculation is more complicated than the classical plate theory and the FSDT, Reddy's plate theory is suitable for solving the complicated structures such as piezoelectric homogeneous and heterogeneous structures. Moreover, some piezoelectric plates are only partially leaning on other structures, possibly modeled as one area elastic foundation, which has not been specified in the existing studies. It is also because the plate only rests on one area elastic foundation, so the mechanical behavior of this plate will be different from the case of a fully elastic foundation. e modeling and calculation of these structures using the higher-order shear deformation theory of Reddy are interesting problems on how to use them most effectively in engineering practice. is is also the main purpose of this work. Besides, the 8-node plate element is used to present high confidence through comparison with previously published literature. e rest of this paper is as follows. Section 2 introduces shortly piezoelectric functionally graded plates. Finite element formulations for free vibration and static bending of piezoelectric FGM plates are presented in detail in Section 3. Section 4 performs verification problems, numerical results, and discussions. Some bold conclusions are summed up in Section 5.

Piezoelectric Functionally Graded Plates
Considering that a piezoelectric FGM plate with the core layer is FGM, the other outer ones attacked the piezoelectric components, called actuator and sensor layers. One area of the plate rests on the elastic foundation with the length d e and two stiffness coefficients k w and k s (Figure 1).
Assuming that the core layer is the FGM made of ceramic and metal materials, the material properties of the material at the point with z coordinate have the following expressions [52]: Poisson's ratio, mass density, and the volume fraction of i-th material, x and y are lines of in-plane of midsurface, z defines the normal axis, c denotes ceramic, and m denotes metal. e volume fractions of materials in this paper are given by [52] where n is the material parameter with a nonnegative value.

Finite Element Formulations for Free Vibration of Piezoelectric FGM Plates
Many plate theories can be used to explore the mechanical behavior of the plate structures. In this work, the high-order shear deformation plate theory of Reddy is employed; therefore, the displacement field at any points within the plate is given as follows [52]: where u 0 ; v 0 ; w 0 are the displacements at a point (x, y, z � 0); φ x , φ y are the transverse normal rotations in xz-and yzsurfaces, respectively. e strains can be written in detail as Equation (4) can be written in a short form as follows:  Mathematical Problems in Engineering e linear piezoelectric coupling between the elastic field and the electric field can be expressed by the direct and the converse piezoelectric equations; the relationship between stresses and strains are written by [53,54] (i) E v is the electric field, which can be defined through the variation of voltage as follows [55]: where the effect of the electric field is considered only in the thickness direction of the plate; it is symbolized as E z and E z is calculated through the voltage and the thickness as (9) in which h p is the thickness of actuator and sensor layers, h p � h a and h p � h s . (ii) D is the electric displacement vector. (iii) e denotes the piezoelectric stress coefficients with the following particular expression: (iv) p is the dielectric tensor at constant mechanical strain with the following expression: is work uses an 8-node element, where each node has 5 degrees of freedom: where N j is the Lagrangian interpolation function of an 8node element and ξ, η are the natural coordinates.
So the strains relate to element displacement vector as follows: 4 Mathematical Problems in Engineering At this time, the electric vector of actuator and sensor layers is calculated through two degrees of freedom of actuator and sensor layers as follows: in which where h a and h s are the thicknesses of the actuator and sensor layers, respectively. After integrating from − h/2 to h/2, the normal force, bending moments, high-order bending moments, shear force, and high-order shear force can be expressed as where e total strain energy of the piezoelectric FGM plate can be given by

Mathematical Problems in Engineering
where N w � 8 j�1 [0, 0, N j , 0, 0], F is the surface loading, V is the voltage applied to actuator and sensor layers, q e is the nodal displacement vector, and k w and k s are the elastic foundation parameters. Equation (19) is expressed in matrix form as where the stiffness matrices are calculated as follows: 6 Mathematical Problems in Engineering So for the plate element resting on a two-parameter elastic foundation, two components k w N T w N w + k s ((zN w / zx) T (zN w /zx) + (zN w /zy) T (zN w /zy)) will be added to the element stiffness matrix as shown in equation (21). For the plate element with the foundation, these two components are neglected. For the convenience of calculation, the element mesh will be divided so that the element node points will be on the boundary of the elastic domain, which means that an element will lie entirely or not in the domain of the elastic foundation.
For the static problem, minimizing the total potential energy function by variables, we have the following equation: Equation (29) can be shortened in the form with only displacement variable q e as erefore, the static equilibrium equation of the whole structure has the following form: e kinetic energy of the plate element is determined by formula Mathematical Problems in Engineering in which ρ p , ρ a , and ρ s are the densities of the plate, actuator layer, and sensor layer, respectively; N is the shape function matrix; L and the element mass matrix M e are calculated as follows: For the free vibration problem, to find frequencies of the plate, the following equation is solved: Hence, the voltage applied to the structure V will generate an electric force vector as shown in equation (29), thereby affecting the static displacement of the structure. is paper has not considered the effect of this potential on the vibration response of the structure.
rough the above calculation formulas, using Reddy's higher-order shear deformation theory, the element stiffness matrix and the element mass matrix show more components, leading to more complicated calculations than the classical plate theory and the FSDT, and no shear correction factor is required. By using the 8-node plate element and the finite element method, the equilibrium equation of the structure is derived. To verify the reliability of the proposed theory and mathematical model, the following section will present a specific comparison to determine this problem. e plate is fully and simply supported (SSSS), and the first natural frequencies of the plate obtained from this numerical work, FEM and Kirchoff-love plate theory [51], and the analytical solution [56] are presented in Table 1. Herein, Table 1 also presents the numerical results with different meshes. It can be seen clearly that an 8 × 8 mesh can return accurate data. For this mesh, in Table 2, pre-Kirchoff-love plate sends the first ten natural frequencies of the piezoelectric FGM plate. From Tables 1 and 2, the proposed method is verified. For all the following investigations, an 8 × 8 mesh is used.

Numerical Results and Discussion
Next, the effect of the elastic foundation (EF) is considered; the plate is placed on the elastic foundation with two parameters.
e FGM (Al/ZrO 2 ) plate has a � 0.2 m and b � a/1.5, the plate thickness of h � a/10 and h � a/20, and the material properties E c � 200 GPa, ] c � 0.3, ρ c � 5700 kg/m 3 , E m � 70 GPa, ] m � 0.3, and ρ m � 2702 kg/m 3 . e plate is fully and simply supported. Two nondimensional parameters of the EF are given as follows: e nondimensional natural frequency of the plate is defined as Table 3 presents the nondimensional natural frequencies of the plate with different thicknesses and volume fraction indexes obtained from this work and the analytical solution of Hosseini-Hashemi et al. [57]. It can be easily seen that the results have a good agreement.

Numerical Results Study for Free Vibration of Piezoelectric FGM Plates.
is section investigates the effect of two parameters of the EF on natural frequencies of the piezoelectric FGM plate (Al/ZrO 2 ) with a � b � 0.  (36) and (38), respectively. For boundary conditions, some descriptions are as follows. If one edge of the plate is simply supported, it will be denoted as S. Similarly, C and F denote, respectively, clamp support and free support. erefore, the plate is fully and simply supported, called SSSS, the plate is clamped at one edge, and the other three edges are free, called CFFF.
(1) Effect of the Stiffness of the Elastic Foundation. Consider an elastic foundation with two parameters, K * s varies from 0 8 Mathematical Problems in Engineering to 500, and K * w changes in the range from 0 to 10000, and the volume fraction index n � 1. e plate is rested on one area of the elastic foundation with d e � a/3. e boundary condition is SSSS. e first five natural frequencies are shown in Figure 2. From these results, it can be seen that when increasing the values of two elastic foundation parameters, natural frequencies increase. Particularly, when K * s gets the small value, the curve of the natural frequency tends to depend on K * w much.
(2) Effect of the Length of the Elastic Foundation Area d. e . In this investigation, K * w varies from 0 to 10000 while K * s � 50, the volume fraction index n � 1, and the plate is fully and simply supported (SSSS). e plate is rested on one area of the elastic foundation with d e varying from a/12 to a/2. e nondimensional natural frequencies of the plate are presented in Figure 3. One can see that when increasing d e , which means increasing the area of the elastic foundation, the plate becomes stronger; therefore, the natural frequency of the plate increases. At the same time, when increasing the stiffness parameter K * w , the plate also becomes harder, and the first natural frequency of the plate increases.
Five natural vibration mode shapes of the plate in cases of d e � a/12 and d e � a/2 are shown in Figure 4. It can be seen  clearly that the higher the length d e is, the stronger the nondimensional natural frequencies as well as vibration mode shapes are affected, especially for the first natural frequency of the structure. Herein, to obtain the higher quality of vibration mode shapes, 12 × 12 mesh is used instead of 8 × 8 mesh as mentioned.
(3) Effect of Volume Fraction Index n. e plate is rested on one area of the elastic foundation with the length d e � a/3. In this exploration, when K * s � 50, four cases of K * w are considered, which are 0, 100, 500, and 1000, and in the case of K * w � 1000, K * s gets values in a range of 0, 10, 50, and 100. e plate is SSSS. Figures 5 and 6 present the nondimensional natural frequencies of the plate changes as a function of the volume fraction index n from 0 to 10. At the same time, the value of the volume fraction index in a range of 0.6 to 1.8 is found so that the first natural frequency of the plate is the smallest.
(4) Effect of Boundary Condition. In this section, the effect of boundary condition on the nondimensional natural frequencies of the plate is investigated. Four cases of boundary condition are considered: SSSS, CCCC, CFFF, and SCSC. Figure 7 presents the dependence of the nondimensional natural frequencies on the volume fraction index n and boundary condition. It can be observed that when n increases, the nondimensional natural frequencies go down sharply and then go up gradually. ey reach the minimum values around n � 1.5. Figure 8 shows the dependence of the nondimensional natural frequencies on K * w with four boundary conditions (K * s � 50, n � 1). It can be seen that when K * w increases, ω * does not change much.    deflections at y � b/2 obtained from this work and [51] are presented in Figures 9 and 10. Note that the upper layer of the plate is metal Ti-6Al-4V, and the lower one of the plate is ceramic Al 2 O 3 . From these comparisons, it can be concluded that the proposed method is verified. Next, the plate resting on an elastic foundation is considered. Consider a square plate with a � b � 0.2 m, h � a/10, and a/ 200, the material properties E c � E m � 320.24 GPa, and Poisson's ratio 0.26. e plate is SSSS and under a uniformly distributed load q 0 . e nondimensional elastic foundation parameters are calculated as  e nondimensional deflection at the center point of the plate is defined by the following formula: e comparative results obtained by this work, the differential quadrature method [58], and the analytical method [28] are presented in Table 4. From here, it can conclude that the proposed method is verified. e maximum nondimensional deflection is standardization as follows:

Numerical Study for Static
(1) Effect of the Stiffness of the Foundation. Herein, let K * s increase gradually from 0 to 500 while K * w changes in the range from 0 to 10000. e volume fraction index n � 1. e plate is rested on one area of the elastic foundation with d e � a/3. e nondimensional defections w * are shown in Figures 11 and 12. ese figures indicate that when increasing the stiffness of the elastic foundation, the maximum deflections of the plate decrease; that is due to the increased total energy of the plate due to the addition of the stiffness component of the foundation, so the global stiffness matrix of the structure increases, reducing the deflection of the plate. Particularly, the smaller the value K * s gets, the higher the slope of deflection curve w * depending on K * s is. Besides, when the plate is subjected to both mechanical and electric loads, the maximum deflection of the plate decreases. e reason is that the piezoelectric effect reduces the maximum deflection. is can be explained as follows: when the plate is subjected to both mechanical and electrical loads, the force direction of the electrical loads is in the opposite direction of the mechanical load; therefore, the total force acting on the plate will be reduced, and the deflection of the plate decreases.
(2) Effect of Volume Fraction Index n. Parameters of the elastic foundation are K * s � 50, K * w � 0-1000, and d e � a/3. By changing the volume fraction index n in a range of 0 to 10, the deflection curve w * depending on n in the cases of V � 0 and V � 40 V is shown in Figures 13 and 14. From these figures, it can be seen that when the volume fraction index n increases, the maximum deflection w * increases; this can explain that when the volume fraction index n increases, the portion of the metal increases, the plate becomes softer, and the deflection of the plate w * increases. However, when the  piezoelectric effect is applied to the plate, the maximum deflection w * decreases.
(3) Effect of Boundary Condition and the Area of the Elastic Foundation. In this subsection, the plate is rested on the elastic foundation with K * s � 10 and K * w � 100. ree cases of boundary condition SSSS, CSCS, and SSCC are considered. e volume fraction index n � 1. By increasing gradually the length of the area of the elastic foundation d e in a range of 0 to a/2, nondimensional deflections w * with V � 0 V and V � 40 V are plotted in Figures 15 and 16. It can be seen easily that when the area of the elastic foundation increases, the maximum deflection of the plate decreases; the reason is that when d e increases, the area of the elastic foundation increases, the total energy of the structure increases due to the additional energy of the foundation, then the plate becomes stronger, and the deflection of the plate decreases. Besides, when applying the piezoelectric effect to the plate, the maximum deflection of the plate decreases, too.

Conclusions
is paper carries out the free vibration and static bending analysis of piezoelectric FGM plates resting on one area of the two-parameter elastic foundation. Finite element formulations are based on the third-order shear deformation theory of Reddy by using an 8-node plate element. e higher-order shear deformation theory of Reddy presents the advantages over the classical plate theory and the FSDT, especially the distribution of stresses in the thickness direction without the need for shear correction factors, which can solve free vibration, buckling, static, and dynamic behaviors of the plate made of various materials. e proposed theory and mathematical model are verified by comparing the numerical results of this work with those of exact solutions. e numerical comparisons show that, for the thirdorder shear deformation theory and the 8-node plate element used in this paper, the proposed approach has very good reliability; it can reasonably solve the problem of free oscillation and static bending of FGM plates with piezoelectric layers. Some key conclusions are summed up as follows: (i) Increasing the stiffness of the elastic foundation, the nondimensional natural frequencies increase while the maximum deflections of the plate decrease. is can explain that the total energy of the plate is added by the energy of the foundation; therefore, the stiffness of the structure increases, the plate becomes stronger, the natural frequency of the plate increases, and the deflection of the plate decreases. (ii) e larger the area of the elastic foundation (d e increases), the higher the nondimensional natural frequencies, and the smaller the maximum deflections. is parameter also affects the free vibration mode shapes of the plate. (iii) For each case of the elastic foundation, the corresponding volume fraction index n can be found so that the nondimensional natural frequencies reach the smallest values. (iv) When applying the piezoelectric effect to the plate, the maximum deflections of the structure decrease. is is the effect of the piezoelectric layers in the case of the plate under static load. is is due to the reason that the electrical force and mechanical force are in the opposite directions; therefore, the electrical force reduces the effect of the mechanical force, and the deflection of the plate decreases.
e numerical results in this work contribute to the knowledge of scientists when considering these types of piezoelectric structures in engineering practice.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e author declares that there are no conflicts of interest regarding the publication of this paper.