Document Type : Research Paper
Authors
^{1} Department of Mechanical Engineering, Arak University, Arak, 3815688349, Iran  Institute of Nanosciences & Nanotechnolgy, Arak University, Arak, 3815688349, Iran
^{2} Department of Mechanical Engineering, Arak University, Arak, 3815688349, Iran
Abstract
Keywords
Sizedependent ElectroMechanical Vibration Analysis of FGPM Composite Plates using Modified Shear Deformation Theories
K. Khorshidi^{ a,b*}, M. Karimi ^{a}, A. Siahpush ^{a}
^{a} Department of Mechanical Engineering, Arak University, Arak, 3815688349, Iran
^{b} Institute of Nanosciences & Nanotechnolgy, Arak University, Arak, 3815688349, Iran
KEYWORDS 

ABSTRACT 
FGPM, Electromechanical loads, Free Vibration, Nonlocal, Modified shear deformation theory 
In the current study, the modified shear deformation theories are used for analyzing the electromechanical vibration of inhomogeneous piezoelectric nanoplates in conjunction with the nonlocal elasticity theory. These theories not only satisfy transverse shear tractionfree conditions on the top and bottom surfaces of the plate but also consider exponential and trigonometric distributions for the transverse shear deformations. Heterogeneity of the structure is supposed to be in the thickness direction of the nanoplate and it is assumed that the simply supported functionally graded piezoelectric nanoplate is subjected to a biaxial force and an external electric voltage. The governing equations of the vibrating functionally graded piezoelectric nanoplate are obtained by using Hamilton’s principle, which are then solved by using Navier’s method to achieve the vibrational behavior of the structure. The detailed discussion is presented to explain the influences of the various parameters on the natural frequencies of functionally graded piezoelectric nanoplates. It is concluded that increasing gradient index parameter, nonlocal parameter, thickness ratio and compressive forces lead to a decrease in natural frequencies while rising tensile forces and aspect ratio increase the natural frequencies.

In the domain of material science, there are many smart or intelligent substances made by piezoelectric materials. Piezoelectric materials have been broadly utilized as components of sensors and actuators in control systems due to their electromechanical properties, allowing to convert electrical energy into mechanical energy and vice versa. With the trends toward device miniaturization, piezoelectric materials and their nanostructures have received much attention among researchers who are interested in both experimental and theoretical studies. Nanosized structures, where thickness is routinely based on microns and submicrons, have been substantially used in many devices and systems, such as microsensors, microactuators, nano/microelectromechanical systems (NEMS and MEMS). The active vibration control of plates is one of researchers’ favorite topics. Li et al. [1] investigated the vibration control of plates using piezoelectric patches. Khorshidi et al. [2] investigated the active vibration control of circular plates including piezoelectric layers under the plane sound wave excitation.
Conventional continuum models do not entail size effects due to the lack of material length scale parameters. Thus, there is a need for the development of sizedependent continuum models capturing size effects. In general, sizedependent models are based on nonclassical continuum theories such as the nonlocal elasticity theory [3], the couple stress theory [4], and the strain gradient theory [5]. In small scale structures, the influences of size are very evident and important, which has already been determined by many experiments and atomistic simulations [6, 7]. This is why the classical continuum model, which is a scaleindependent theory, does not succeed to meet the computational requisites of nanostructures. Therefore, extensive highorder theories such as the nonlocal elasticity theory, the strain gradient theory, the couple stress theory and the micropolar theory, are developed to modify the classical continuum model.
The nonlocal elasticity theory presented by Eringen [3] is usually implemented to evaluate the scale effect of nanostructures. This theory states that the stress of each point of the structure is dependent on the strains of all points in the body. This theory is applied to the nanoplates by imposing a length scale parameter into the constitutive equations as a material parameter. Meanwhile, different nonlocal and local models were developed to solve the buckling [8], linear and nonlinear vibrations of nanostructures such as graphene, carbon nanotubes, sheets, and etc. [913].
Additionally, the nonlocal theory has already been developed to examine the sizedependent mechanical behavior of the piezoelectric nanostructures [1420]. Many mathematical and theoretical models have been presented for various structures with piezoelectric sensors and actuators. The main disadvantage of these structures is that the high stress concentrations commonly grow at the interface of layers subjected to the electromechanical forces. This weakness limits the suitability of piezoelectric devices in the application which the devices need high dependability. With regards to the low performance of the usual layered piezoelectric structures, the idea of functionally graded materials (FGMs) has been developed to the piezoelectric materials by modern advances in the metallurgical science and production techniques. In contrast to laminated composite materials, there are no interlaminar stresses for these materials as they vary continuously throughout the structure. The functionally graded piezoelectric materials, because of their immeasurable employment in many engineering fields, are used in various systems such as micro/nano electromechanical systems. Due to the steady change in the material contribution and properties from the bottom to the surface, this category of advanced materials is called functionally graded piezoelectric materials (FGPMs). The electric field and stress components depend on each other in the functionally graded piezoelectric materials. Also, their material properties continuously change in one (or more) direction(s). These attributes expand the intricacy and laboriousness of analyzing structures made of these materials. In recent years, the behaviors of functionally graded materials of plates have been studied extensively [2129].
Plate deformation theories are separated into stress based and displacementbased theories. Displacement based theories can be divided into the classical plate theory and shear deformation plate theories. The use of shear deformation theories for composite and sandwich plates have been the subject of intense research. Transverse shear stress components in thickness direction are neglected in the classical theory of plates and this theory is used only for thin plates since it leads to inappropriate results for thick plates especially those made from advanced composites. Under the assumption of classical plate theory straight lines normal to the midplane before deformation, remain straight and normal to the midplane after deformation. The first order shear deformation theory is considered as an improvement over the classical plate theory. It is based on the supposition that straight lines normal to the midplane before deformation remain straight but, inevitably, not normal to the deformed midplane. Firstorder shear deformation theory incorporates the shear deformation effects with a constant transverse shear deformation distribution along the thickness of the plate. Thus, it violates stressfree conditions at the bottom and top of the plate and needs a shear correction factor to compensate this error. The limitations of the classical plate theory and the first order shear deformation theory resulted in the development of higher order shear deformation theories. Various distributions of transverse shear stress can be found in the literature for analyzing different structures [3043]. In the exponential and trigonometric shear deformation theories, exponential and trigonometric functions are applied distribution.
The intent of this study is to investigate the free electromechanical vibration of functionally graded piezoelectric nanoplates using the nonlocal elasticity theory. The mechanical and electrical properties of the assumed structure are considered to change according to the power law through the thickness of the nanoplate, and the structure is imposed to the different loadings. After obtaining the vibration characteristics of the nanoplate using Navier’s method, some comparison studies are provided to describe the reliability and efficacy of the present work. In the discussion section, the influences of the different parameters like nonlocal parameter, gradient index, biaxial forces, the external electrical force and the geometrical ratios on the electromechanical vibration characteristics of functionally graded piezoelectric nanoplates are shown.
Consider a finite rectangular nanoplate made of functionally graded piezoelectric materials of length , width and thickness , defined in the rectangular coordinate system . The nanoplate is subjected to a mechanical biaxial force P_{0} (compressive force or tensile force) and an external electric voltage V_{0} as shown in Fig. 1. The poling direction of the piezoelectric medium is parallel to the positive zaxis. The effective material properties of considered functionally graded piezoelectric nanoplate continuously vary through the thickness direction in accordance with the power law distribution as follows
, 
(1) 
where and v represent the material properties at the upper and lower surfaces, respectively, and and are the corresponding volume fractions defined as
, 
(2) 
And a gradient index implies the variation of material properties across the nanoplate thickness.
21. Nonlocal theory for FGPMs
In order to consider length scale on mechanical behavior of piezoelectric nanoplates, nonlocal theory has been commonly utilized. According to the Eringen's nonlocal elasticity theory, stress at a point X in a body depends on strains of the whole domain [3]. By introducing a nonlocal attenuation function to consider for the effect of longrange interatomic forces, the nonlocal elasticity theory explains satisfactorily some phenomena such as the high frequency vibration and wave dispersion. Mathematically, neglecting the effect of body forces, the basic equations for a functionally graded and nonlocal piezoelectric solid can be expressed as [2, 3].
, 
(3) 
, 
(4) 
(5) 

(6) 

(7) 

(8) 
where is continuum occupying region, , , and are the components of the stress, strain, electric displacement and electric field, respectively. Also, C_{ijkl}, e_{ijk}, d_{ik} and ρ are the elastic, piezoelectric, dielectric coefficients and mass density, respectively, Φ is electric potential. represents the nonlocal attenuation function, considering size effects at the reference point. is a calibration constant appropriate to each material.
Fig. 1. Geometry and dimensions of the functionally graded piezoelectric nanoplate under eletromechenical loadings. 
The measure of is determined experimentally or estimated such that the relations of the nonlocal elasticity model could provide satisfactory approximations to the atomic dispersion curves of the plane waves with those obtained from the atomistic lattice dynamics [44]. a and l are the internal (e.g. C–C bond length, granular distance, lattice parameter) and external (e.g. crack length, wavelength) characteristic lengths of the nanostructures, respectively. When the internal characteristic length is zero, nonlocal elasticity reduces to the classical (local) elasticity [45]. The constitutive equations for the functionally graded piezoelectric solids may be demonstrated with an equivalent differential form as,
, 
(9) 
, 
(10) 
where is the Laplace operator; is the scale coefficient revealing the size effect on the response of nanostructures. For the functionally graded piezoelectric structures, where thickness is much smaller than its length and width, the behavior of the structure can be estimated based on twodimensional analysis. Therefore, for plates in the (x, y, z) coordinate system, ignoring the nonlocal behavior in the thickness direction, the nonlocal constitutive relations (9) and (10) can be approximated as,
, 
(11) 
, 
(12) 
where and are respectively the reduced elasticity constants, piezoelectric constants and dielectric constants for the functionally graded piezoelectric nanoplate under the plane stress state [18]. These constants are defined as,
(13) 

(14) 

(15) 

(16) 

(17) 

(18) 

(19) 

(20) 
where and are respectively the elasticity constants, piezoelectric constants and dielectric constants.
22. Free vibration analysis of the FGPMs
Based on the exponential and trigonometric shear deformation theories, the displacement field of the functionally graded piezoelectric nanoplate is given as below [39],
, 
(21) 
, 
(22) 
, 
(23) 
where and are the inplane displacements at any point , and denote displacement of the midplane and is deflection of the functionally graded piezoelectric nanoplate in the x, y and z directions, respectively. Also and are rotations of the normals to the midplane about the y and x axes, respectively, additionally, represents the distribution of the transverse shear strains and stresses along the thickness.
Time is expressed by t, and for exponential and trigonometric shear deformation theories can be described as follows [20],
, 
(24) 
. 
(25) 
The present nanoplate model should be assumed to satisfy the Maxwell equation [46]. The electric potential is approximately assumed as a combination of cosine and linear variation.
, 
(26) 
where , φ is the spatial and time variation of the electric potential in the midplane of the nanoplate, is the external electric voltage and Ω is the natural frequency of the functionally graded piezoelectric nanoplate. Hence the components of the linear strain tensor of the functionally graded piezoelectric nanoplate are
, , 
(27,28) 

, 
(29) 

, , . 
(3032) 

Using Eq. (26), the electric fields can be expressed as,
, 
(33) 
, 
(34) 
. 
(35) 
Then, the strain energy U of the piezoelectric nanoplate is given by
. 
(36) 
The first variation of strain energy can be written as,
, 
(37) 
where the resultant forces (N_{x}, N_{y}), (P_{x},P_{y},P_{xy}) and (Q_{xz},Q_{yz}), bending moment M_{x}, M_{y} and twisting moment M_{xy} can be written as follows,
, 
(38) 
, 
(39) 
, 
(40) 
. 
(41) 
The kinetic energy T can be calculated as,
, 
(42) 
where dotover script convention indicates the differentiation with respect to the time variable t. From Eqs. (1214) and (28), the first variation of the kinetic energy can be expressed as,
, 
(43) 
where inertia coefficients can be expressed as,
. 
(44) 
The work done by external forces is denoted by W_{f}
, 
(45) 
where A denotes the domain occupied by the midplane of the functionally graded piezoelectric nanoplate and (F_{Px}, F_{Py}) and (F_{Ex}, F_{Ey}) are the normal forces induced by the biaxial force P_{0}, external electric voltage V_{0} along the x and yaxis. Moreover, is average value of along the thickness which can be written as,
, 
(46) 
. 
(47) 
Herein, Hamilton’s principle is used to derive equations of motion based on the modified shear deformation theories. The principle in case of local form can be stated as follows,
, 
(48) 
By substituting Eqs. (37, 43 and 45) in the expression (48) of the principle of virtual works, integrating it by parts and setting the coefficients of , , , , and to zero, the governing equations of motion for a functionally graded piezoelectric nanoplate are derived as,
, 
(49) 
, 
(50) 
, 
(51) 
, 
(52) 
, 
(53) 
. 
(54) 
It is assumed that the functionally graded piezoelectric nanoplate has simply supported boundary conditions on all four edges and its edges are covered with electrodes and electrically grounded to zero potential. These mechanical and electrical boundary conditions can be written as follows,
, 
(55) 
, 
(56) 
, 
(57) 
, 
(58) 
, 
(59) 
, 
(60) 
, 
(61) 
, 
(62) 
where (n_{x}, n_{y}) denote the direction cosines of the outward unit normal to the boundary of the midplane. By employing Eqs. (21, 23) on Eqs. (4954) the nonlocal governing equations can be rewritten as,
, 
(63) 
, 
(64) 
, 
(65) 
, 
(66) 
, 
(67) 
. 
(68) 
Solution for the Eqs. (6368) with aforementioned boundary conditions are assumed as following double Fourier series forms
, 
(69) 
where , , and m and n denote the number of half waves in the x and y directions, respectively. , , , , and are Fourier coefficients to be determined for each (m,n). Clearly, the expansions in Eq. (69) exactly satisfy the boundary conditions in Eqs. (5562) for any , , , , and . Substituting Eqs. (69) into Eqs. (6368) yields
, 
(70) 
, 
(71) 
, 
(72) 
, 
(73) 
, 
(74) 
, 
(75) 
where A_{i} and B_{j} can be find in Appendix A. After simplification of the obtained equations, system of equations can be expressed as following matrix form,
, 
(76) 
where the vector is
, 
(77) 
Also and are stiffness and mass matrix respectively. This matrix eigenvalue problem should be solved to obtain the natural frequencies of functionally graded piezoelectric nanoplate [20].
3. Results and discussion
This section will present the numerical results for the electromechanical vibration of the functionally graded piezoelectric nanoplate under simply supported boundary conditions. The functionally graded piezoelectric nanoplate made of PZT4 and PZT5H piezoelectric materials are the subject of analysis. The top surface of the nanoplate is PZT4 rich, while the bottom one is PZT5H rich. Table 1 represents the electromechanical properties of these constituents [47]. In addition, it is assumed that the length of the functionally graded piezoelectric nanoplate is , the width of the nanoplate is and thickness is . The effects of various values of the dimensionless nonlocal parameter , gradient index λ, biaxial force , external electric potential , aspect ratio and sidetothickness ratio on the dimensionless natural frequencies of the functionally graded piezoelectric nanoplates were investigated and discussed in details.
To check the validity and accuracy of the present analysis, the results obtained from the present analysis are compared with those available in the literature for functionally graded piezoelectric square plates, and also for homogeneous piezoelectric nanoplates ( ). Bodaghi and Shakeri [47] analyzed the free vibration of functionally graded cylindrical panels based on the nonlocal theory of elasticity using the analytical method. And the free vibration analysis of nonlocal homogeneous piezoelectric nanoplates was presented by Ke et al. based on Mindlin’s theory using the Differential quadrature (DQ) method [18] and also Liu et al. analyzed the free vibration of piezoelectric nanoplates based on the Kirchhoff’s theory with the analytical method [17]. Therefore, this subsection will present two examples for the nonlocal elastic nanoplates to ensure the validity of the present model. In Table 2 the dimensionless fundamental frequencies of simply supported homogeneous piezoelectric nanoplates calculated by the present method are compared with those reported by Liu et al. [17] and Ke et al. [18].
It is observed that the results obtained on the basis of the present method have good agreement with the available data. The parameters used in this example are , , , and . It can be said that the effect of shear deformation is significant for small thicknesstolength ratios and it diminishes with an increase in lengthtothickness ratio. As a second example, Table 3 depicts the dimensionless natural frequencies of the functionally graded piezoelectric nanoplate for different gradient index values. In this reference, material properties of the functionally graded piezoelectric plate vary from PZT4 to PZT5H according to the power law function.
The input parameters are , , , and and the piezoelectric plate has been analyzed with the exponential shear deformation theory. And again, the present results agree well with the Mindlin’s plate theory.
The difference of the results is due to the difference between distributions of transverse shear deformation and rotation effects in modified shear deformation theories. Moreover, modified shear deformation theories do not require any shear correction factor and satisfy transverse shear tractionfree conditions on the top and bottom surfaces of the nanoplate. The effect of the nonlocal parameter μ on the first six dimensionless natural frequencies of the functionally graded piezoelectric nanoplate with , and is listed in the Table 4. Note that the nonlocal parameter corresponds to the classical piezoelectric nanoplate model without considering the nonlocal effect. It can be observed that the dimensionless natural frequencies predicted by the present model tend to be smaller than those of the classical model. The nonlocal parameter µ has an important influence on natural frequencies and increasing nonlocal parameter decreases the natural frequencies. This phenomenon is caused by nonlocal effect, which considers the conjoint effect of all points in the domain, and it decreases the stiffness of the nanostructures and as a result drops the values of the natural frequencies.
As seen in Table 5, the dimensionless natural frequency parameter has experienced a decline by increasing of the gradient index of functionally graded piezoelectric nanoplate which is due to the changing of material properties from PZT4 to PZT5H.
Table 1. Material properties of the PZT4 and the PZT 5H [47].


Parameters 
PZT4 
PZT5H 
139 
126 

74 
83.9 

115 
117 

77.8 
79.1 

74 
83.9 

25.6 
23 

30.6 
23.5 

5.2 
6.5 

15.1 
23.3 

12.7 
17 

6.46 
15.05 

5.62 
13.02 

7500 
7500 
Table 2. Comparison of dimensionless natural frequency for piezoelectric nanoplate under different distributions of transverse shear deformation. 

μ 

FSDT [18] 
CPT [17] 
Tri. 
Exp. 
FSDT [18] 
CPT [17] 
Tri. 
Exp. 

0.1570 
0.1574 
0.1566 
0.1568 
0.6068 
0.6290 
0.6059 
0.6064 
0 
0.1435 
0.1439 
0.1429 
0.1432 
0.5545 
0.5748 
0.5535 
0.5541 
0.1 
0.1174 
0.1177 
0.1168 
0.1171 
0.4536 
0.4702 
0.4529 
0.4533 
0.2 
0.0943 
0.0945 
0.0937 
0.0940 
0.3641 
0.3775 
0.3634 
0.3637 
0.3 
0.0770 
0.0772 
0.0764 
0.0768 
0.2976 
0.3085 
0.2967 
0.2972 
0.4 
0.0645 
0.0646 
0.0638 
0.0642 
0.2491 
0.2582 
0.2487 
0.2489 
0.5 
Table 3. Comparison of the dimensionless fundamental frequency for the simply supported functionally graded piezoelectric square plate using the exponential shear deformation theory. 


λ 






0 
1 
3 
5 
10 

20 
Exp. 
0.01378 
0.01300 
0.01277 
0.01269 
0.01257 
FSDT [23] 
0.01380 
0.01301 
0.01290 
0.01285 
0.01279 

10 
Exp. 
0.05363 
0.05063 
0.04979 
0.04946 
0.04901 
FSDT [23] 
0.05365 
0.05065 
0.05025 
0.05008 
0.04985 
Table 4. The effect of the nonlocal parameter μ on the dimensionless natural frequencies of functionally graded piezoelectric nanoplate. 


µ 

0 
0.1 
0.2 
0.3 
0.4 
0.5 

Exp. 
0.07336 
0.06434 
0.04945 
0.03813 
0.03046 
0.02516 

Tri. 
0.07335 
0.06433 
0.04944 
0.03811 
0.03045 
0.02515 

Exp. 
0.13944 
0.11037 
0.07579 
0.05527 
0.04295 
0.03496 

Tri. 
0.13940 
0.11033 
0.07577 
0.05525 
0.04294 
0.03495 

Exp. 
0.20308 
0.14716 
0.09451 
0.06718 
0.05163 
0.04180 

Tri. 
0.20300 
0.14710 
0.09447 
0.06715 
0.05161 
0.04178 

Exp. 
0.23930 
0.16554 
0.10337 
0.07279 
0.05573 
0.04503 

Tri. 
0.23918 
0.16546 
0.10332 
0.07276 
0.05571 
0.04501 

Exp. 
0.26031 
0.17548 
0.10807 
0.07576 
0.05791 
0.04674 

Tri. 
0.26017 
0.17539 
0.10801 
0.07572 
0.05788 
0.04672 

Exp. 
0.34881 
0.21252 
0.12509 
0.08654 
0.06580 
0.05298 

Tri. 
0.34857 
0.21237 
0.12500 
0.08648 
0.06575 
0.05294 
In Fig. 2, the effects of the aspect ratio and the nonlocal parameter with , and on the nondimensional fundamental frequency of the piezoelectric nanoplates using the exponential shear deformation theory are shown. Thickness value of the piezoelectric nanoplate is taken to be h=5 nm. The depicted results reveal that the dimensionless fundamental natural frequency first increases mildly when the aspect ratio varies from 0.2 to1.0 and then swiftly increases when the aspect ratio changes from 1.0 to 3.0. Also, the size effect on the piezoelectric nanoplate has an increasingly strong trend with the increase of the aspect ratio.
The effect of the lengthtothickness ratio on the first dimensionless natural frequency ω_{1} of piezoelectric nanoplates is shown in Fig. 3 with , , and . It is observed that the dimensionless fundamental natural frequency reduces quickly with the raise of the lengthtothickness ratio. That is to say, the fundamental frequency for thick and relatively thick nanoplates is higher than thin nanoplates. Furthermore, Fig. 3 illustrates that when lengthtothickness ratio increases, the importance of the size effect reduces.
Figs. 45 examine the effect of different amounts of the biaxial force on the first six dimensionless natural frequencies of the piezoelectric nanoplate with , and . The positive and negative values of the biaxial forces signify the compressive force and tensile force, respectively.
Table 5. The effect of the gradient index λ on the dimensionless natural frequencies of functionally graded piezoelectric nanoplates. 



λ 


0 
0.3 
1 
5 
10 

Exp. 
0.03768 
0.03073 
0.04046 
0.03020 
0.03009 
0.02987 

Tri. 
0.03767 
0.03070 
0.03044 
0.03019 
0.03007 
0.02985 

Exp. 
0.05306 
0.04331 
0.04295 
0.04263 
0.04249 
0.04219 

Tri. 
0.05304 
0.04330 
0.04294 
0.04262 
0.04248 
0.04218 

Exp. 
0.06371 
0.05203 
0.05163 
0.05128 
0.05113 
0.05079 

Tri. 
0.06368 
0.05201 
0.05161 
0.05126 
0.05111 
0.05077 

Exp. 
0.06872 
0.05614 
0.05573 
0.05538 
0.05522 
0.05486 

Tri. 
0.06869 
0.05611 
0.05571 
0.05535 
0.05519 
0.05484 

Exp. 
0.07138 
0.05832 
0.05791 
0.05755 
0.05739 
0.05702 

Tri. 
0.07134 
0.05829 
0.05788 
0.05752 
0.05736 
0.05700 

Exp. 
0.08101 
0.06622 
0.06580 
0.06546 
0.06528 
0.06489 

Tri. 
0.08095 
0.06617 
0.06575 
0.06541 
0.06524 
0.06485 

Clearly, when the stronger biaxial tensile force is applied, the natural frequencies increase to higher values, and the reverse trend will occur for the biaxial compressive force . The explanation is that compressive force weakens but tensile force strengthens the stiffness of piezoelectric nanoplates.
Fig. 2. The effects of the aspect ratio and nonlocal parameter µ on the dimensionless fundamental frequency ω_{1} of functionally graded piezoelectric nanoplate with , and . 
Fig. 3. The effect of the lengthtothickness ratio on the dimensionless fundamental frequency ω_{1} of functionally graded piezoelectric nanoplate. 
The effect of the external electric voltage on the first six dimensionless natural frequencies of piezoelectric nanoplates with , and under exponential and trigonometric distributions of transverse shear deformation is described in Figs. 67, respectively.
It is found that the natural frequencies of the piezoelectric nanoplates are utterly dependent on the variation of the external electric voltage. So that the negative voltage heightens the natural frequencies of the piezoelectric nanoplates, while the positive voltage causes the contrary effect.
This is due to the fact that the axial compressive and tensile forces are generated in the nanostructure nanoplate by applying the positive and negative voltages, respectively. The tensile forces raise the stiffness value of the piezoelectric nanoplate structure, and therefore lead to the greater value of the natural frequencies while the compressive ones have the opposite reaction.
Fig. 4. The effect of the biaxial force P_{0}(N) on the dimensionless natural frequencies of the functionally graded piezoelectric nanoplate using exponential shear deformation theory with , and . 
Fig 5. The effect of the biaxial force P_{0}(N) on the dimensionless natural frequencies of the functionally graded piezoelectric nanoplate using trigonometric shear deformation theory with , and . 
Fig. 6. The effect of the biaxial force V_{0}(V) on the dimensionless natural frequencies of the functionally graded piezoelectric nanoplate under exponential distribution of transverse shear deformation with , and . 
Fig. 7. The effect of the biaxial force V_{0}(V) on the dimensionless natural frequencies of the functionally graded piezoelectric nanoplate under trigonometric distribution of transverse shear deformation with , and . 
4. Conclusions
In this paper, free vibration of the functionally graded piezoelectric nanoplates based on exponential and trigonometric shear deformation theories using the nonlocal elasticity theory is investigated. Navier solution is applied to analysis the free vibration of the functionally graded piezoelectric nanoplates. Hamilton’s principle is used to derive the governing equations of motion and associated boundary conditions. The material properties of functionally graded piezoelectric nanoplate are assumed to change through the thickness with a power law distribution. What presented herein illustrates the influences of the variations of the nonlocal parameter, gradient index, aspect ratio, lengthtothickness ratio, biaxial force and external electric voltage on the dimensionless natural frequencies values of a functionally graded piezoelectric nanoplate. From the comparison and discussion of numerical results, following conclusions can be deduced:
Acknowledgments
The authors gratefully acknowledge the funding by Arak University, under Grant No. 92/9834.
Nomenclature
Length of the plate 

Width of the plate 

Thickness of the plate 

Young’s modulus 

Poisson’s ratio 

Mass density 

Displacement in the direction 

Displacement in the direction 

Displacement in the direction 

Normal stresses 

Gradient index parameter 

Nabla operator 

Electric field 

Electric displacement 

Shear modulus 

Material length scale parameter 

Natural frequency of the nanoplate 

P_{0} 
Biaxial force 
Normal strains 

External electric voltage 

Electric potential 

Plate inertias 
Appendix A
, , , 
(A13) 

, , , 
(A46) 

, , , 
(A78) 

, , , 
(A1012) 

, , , 
(A1215) 

, , , 
(A1518) 

, , , 
(A1821) 

, , , 
(A2224) 

, , , 
(A2527) 

, , 
(A28, 29) 

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