An analytical study of vibration in functionally graded piezoelectric nanoplates: nonlocal strain gradient theory

In this paper, we analytically study vibration of functionally graded piezoelectric (FGP) nanoplates based on the nonlocal strain gradient theory. The top and bottom surfaces of the nanoplate are made of PZT-5H and PZT-4, respectively. We employ Hamilton’s principle and derive the governing differential equations. Then, we use Navier’s solution to obtain the natural frequencies of the FGP nanoplate. In the first step, we compare our results with the obtained results for the piezoelectric nanoplates in the previous studies. In the second step, we neglect the piezoelectric effect and compare our results with those obtained for the functionally graded (FG) nanoplates. Finally, the effects of the FG power index, the nonlocal parameter, the aspect ratio, and the side-tothickness ratio, and the nanoplate shape on natural frequencies are investigated.


Introduction
In the past four decades, microelectromechanical systems have been widely used in engineering. Examples of the systems are beams, plates, shells, and gears. The problem of predicting mechanical properties of nano/micro structures is an important subject in physics and engineering due to their potential applications. With the fast development of technology, the nano/micro structures have received considerable attention of researchers during the last decade [1][2][3][4][5] . The structures have been employed into many areas like actuators, bio-engineering, nanocomposite and nanoelectromechanical devices.
With advances to nanotechnology, nanoelectromechanical systems (NEMSs) have been fabricated and employed in industry due to their superior features. Due to interesting features of NEMSs such as electrical, optical, and other properties, the systems have better applications compared with microelectromechanical systems. In the NEMSs, the size-dependent effect plays an important role [6][7][8] . The significant size-dependent effect has been authenticated in nanoscale structures. For example, Li et al. [9] used nonlocal strain gradient models in examining the sizedependent effects on the static and dynamical behaviors of micro/nano structures. Also, Zhu and Li [10] formulated the longitudinal dynamic problem of a size-dependent elasticity rod by utilizing an integral form of the nonlocal strain gradient theory.
There are several methods to study the size-dependent mechanical properties of micro/nanoscale structures. Examples of the methods are continuum mechanical theories, experimental methods, and molecular dynamic (MD) simulations. Hitherto, many authors have used MD simulations, atomistic models, and the classical continuum elasticity theory to determine the mechanical response in the NEMSs. Also, modified continuum models have been extensively used in the studies of nanomechanics [11][12] .
The inhomogeneous materials made of at least two constituent phases are called functionally graded materials (FGMs). Both the compositional profile and the material properties of FGMs vary smoothly and continuously [13][14][15] . FGMs have received tremendous amount of interest in the past few years due to their major potential in applications such as biomedical implants and heat exchanger tubes [14] . Hitherto, many works have been done on FGMs in the last decade. For example, Li et al. [16] studied the bending, buckling, and vibration behaviors of axially functionally graded (FG) nanobeams. The FG nanoplates have been employed in many applications like NEMSs [17][18][19] . The wave propagation of FG nanoplate under nonlinear thermal loading was studied by Ebrahimi et al. [20] . For more details, the readers can refer to Refs. [21]- [25].
Piezoelectric materials have their excellent properties. The materials can be employed in piezoelectric transducers, ultrasonic, and smart systems and structures [26][27][28][29] . The FGM is a kind of piezoelectric materials which is used for removing the stress concentrations and interfacial debonding [30][31][32] .
Our studies show that, despite some recent investigations on vibration behaviors of functionally graded piezoelectric (FGP) nanoplates, this problem based on the nonlocal strain gradient theory has not been studied so far. Thus, our motivation is to compare this theory with other models and the correctness of this theory.
In this paper, we study the free vibration of an FGP nanoplate based on the nonlocal strain gradient theory. We assume that the electric potential is zero along the edges of the surface electrodes. We use Hamilton's principle and derive the governing equations. Then, we solve analytically the equations and determine the natural frequency of the FGP nanoplate.

Theory and model
The nonlocal stress field and strain gradient effects are related by two scale parameters [46] . Thus, the stress field is given by where σ (0) ij and σ (1) ij are the zero order (classical stress) and first order stresses, respectively. The stresses correspond to the strain ε ij and the strain gradient ∇ε ij as Here, ζ 0 = e 0 a and ζ 1 = e 1 a denote the nonlocal stress effects, C ijkl is the elastic coefficient, and l is the strain gradient parameter. When the nonlocal functions α 0 (x, x ′ , e 0 a) and α 1 (x, x ′ , e 1 a) satisfy the developed conditions by Eringen, the constitutive relation of the nonlocal strain gradient theory has the following form: Assuming e 0 = e 1 = e, we can write the general constitutive relation in Eq. (4) as In the nonlocal piezoelasticity, the stress tensor and the electric displacement at a point x depend on not only the strain components and electric-field components at the same point but also another point x ′ of the body. The basic constitutive equations for piezoelectric materials based on the nonlocal strain gradient theory are given by where E i and D i are the electric field and the electric displacement, respectively. Also, e kij and Ξ ijk are piezoelectric constants and dielectric constants, respectively, α(|x − x ′ |, τ ) represents the nonlocal modulus, |x − x ′ | is the distance, and τ is the scale coefficient that includes the small-scale factor. Based on the Eringen model, the constitutive equations (6) and (7) can be simplified to the equivalent differential constitutive equations as follows: Since we employ the Cartesian coordinates in our calculations, the stress and displacement due to the electric field use the same gradient. It is noted that the integral constitutive equations in this work are simplified as an approximate differential formulation. Recently, Zhu and Li [10] showed that the nonlocal differential and integral elasticity based models may not be equivalent to each other.

FGMs
Consider a flat FGP nanoscale plate with the length, the width, and the uniform thickness equal to l a , l b , and h, respectively (see Fig. 1). ϕ Fig. 1 The geometry of an FGP nanoplate Consider that the FGP plate is made by the combination of two kinds of piezoelectric materials. Based on the rule of mixture, the effective material property of the FGP plate (P eff ) is given by [44] P eff (z) = P 2 + ( where P 1 and P 2 are the bottom and the upper surface properties of the FGP plate, respectively. Also, g is the FG power index.
It should be noted that for g = 0, P eff = P 1 , and for g = ∞, P eff = P 2 .

Governing equations
Based on the Kirchhoff plate theory, one can write the displacement field as where t is the time, and u 0 , v 0 , and w 0 are the displacement components in three directions (x, y, and z), respectively. Using the above equations, we can obtain the strains as Moreover, we can obtain [45][46][47] φ(x, y, where h is the thickness of the piezoelectric plate, ϕ(x, y, t) is the electric potential in the midplane, V 0 denotes the amplitude, Ω is the circular excitation frequency, and β = π h . Applying Eq. (16), one can obtain the electric fields as Based on the plane-stress assumption, we have σ zz = 0, and our system is a two-dimensional structure. Therefore, the constitutive relations are given by where C 44 = 1 2 (C 11 − C 12 ). Now, using σ 33 in Eq. (18), we can obtain the following relation: where Using the above relations and Eqs. (5) and (6), we can determine the strain constitutive relations for the piezoelectric materials based on the strain gradient theory. The relations are given by where The constitutive relations for electric displacements are given by Employing previous relations, we now rewrite the following relations: where C33 . The strain energy of the FGP plate using the nonlocal strain gradient theory is given by Here, it is noted that we have not taken the higher-order stresses σ (1) xz and σ (1) yz in the strain gradient of the above equations because we have assumed that the size-dependent effects in the thickness direction of the plate are small and thereby we have neglected them. In this regard, recently, Li et al. [9] and Tang et al. [48][49] studied the size-dependent effects in the thickness direction of beams and plates.
Inserting Eqs. (10)- (17) into Eq. (30), we obtain Now, we define the forces and bending moments as Employing the above equations, the strain energy of the FGP plate can be written as The kinetic energy of the FGP plate is given by where Using Hamilton's principle, i.e., (δU − δK) dt = 0, we obtain the following relations: ∂D x ∂x cos(βz) + ∂D y ∂y cos(βz) + D z β sin(βz) dz = 0.
Now, we can write the constitutive relations for forces as Using the following relations: we have where The constitutive relations for moments are given by The constitutive relations for electric displacements are given by Employing the above equations, one can obtain the governing equations of an FGP nanoplate as

Solution procedures
To solve the above equations, the solutions can be considered as where ω mn is the circular frequency, and m and n are the half wave numbers.

Results and discussion
Here, we have carried out our analytical calculations for the dimensionless natural frequency Ω = ωl a l0 A11 PZT-4 of an FGP nanoplate by considering various parameters. For this purpose, we have considered an FGP nanoplate consisting of two-phase graded piezoelectric materials such as PZT-4 and PZT-5H. The top and bottom surfaces are PZT-5H and PZT-4, respectively. The material properties are listed in Table 1. Table 1 Material properties of PZT-4 and PZT-5H PZT In the first step, to check the present results, we have presented our results for the piezoelectric nanoplate by setting g = 0. The first four dimensionless frequencies of the piezoelectric nanoplate for different values of the nonlocal parameter µ = e 0 a/l a are listed in Table 2. In the table, we can compare our results with those of Refs. [44] and [50]. It is seen from the table that our results are in good agreement with those of Refs. [44] and [50]. It is observed that the natural frequencies are reduced with enhancing the nonlocal parameter for four dimensionless frequencies. The reason for this behavior is the decrease in the stiffness of nanostructures by increasing the nonlocal parameter. Table 2 Comparisons of the dimensionless natural frequencies of the FGP nanoscale plate for g = 0 In the second step, we ignore the piezoelectric effect and compare our results with those obtained by Natarajan et al. [50] for the FG nanoplate in Table 3. For this goal, we have considered an FG nanoplate including two-phase graded materials as silicon nitride (Si 3 N 4 ) and stainless steel (SUS304). The used parameters for Si 3 N 4 are ρ c = 2 370 kg/m 3 , E c = 348.43 × 10 9 N/m 2 and for SUS304 are ρ m = 8 166 kg/m 3 , E m = 201.04 × 10 9 N/m 2 . In Table 3, we have reported the obtained dimensionless fundamental frequency Ω = ωh ρc Gc and compared it with Refs. [44] and [50]. It is clear from Table 3 that our results are in agreement with those of Refs. [44] and [50]. Table 3 Comparisons of the dimensionless natural frequencies of the FG nanoplate for g = 0 la/l b la/h (e 0 a) 2 Ω 11 Ω 12 Ω 22 Present Ref. [50] Ref. [44] Present Ref. [50] Ref. [44] Present Ref. [50] Ref.  Figure 2 shows the dimensionless natural frequencies of an FGP nanoplate as a function of the length l a for three different values of the FG power index with e 0 a = 1. It is observed from the figure that the natural frequency is reduced by enhancing the length of the nanoplate for all modes. Also, the natural frequency is increased with enhancing the FG power index for a fixed length. , we have plotted the dimensionless natural frequency of the FGP nanoplate as a function of the aspect ratio l a /l b for different values of FG power index with e 0 a = 3. The natural frequency is increased with the aspect ratio. In addition, the natural frequency is increased with the FG power index for an aspect ratio.
Ω Ω Ω Ω Fig. 3 The dimensionless natural frequency versus the aspect ratio la/l b for different values of the power index (color online) In Figs. 4(a) and 4(b), we have presented the dimensionless natural frequency of the FGP nanoplate versus the length-to-thickness ratio (l a /h) for three different values of the FG power index and two values of the nonlocal parameters. In the figure, we have selected the length and the width equal to l a = l b = 50 nm while the thickness varies. It is seen from the figures that the dimensionless natural frequency is reduced with l a /h for two nonlocal parameters. It is noted that, with enhancing l a /h, the nanoplate becomes thinner and thus it has a lesser stiffness.
In Figs. 5(a) and 5(b), the dimensionless natural frequency (f 44 = Ω 44 ) is plotted as a function of the FG power index for different values of the strain gradient effect l and the nonlocal parameter µ. It is seen that the dimensionless natural frequency is increased with the strain gradient effect for the fixed nonlocal parameter. Also, by increasing the nonlocal parameter, the values of the dimensionless natural frequency are decreased for the fixed strain gradient effect.
Ω Ω µ µ µ µ Ω Ω Fig. 4 The dimensionless natural frequency versus the length-to-thickness ratio la/h for different values of the power index (color online) µ µ Fig. 5 The dimensionless natural frequency (f44 = Ω44) versus the FG power index (color online) Figure 6 shows the parameters A 11 , A 12 , and A 13 as a function of the FG power index. It is clear that the parameters have maximum values for the FG power index g, and the parameters decrease with the increasing values of g. Fig. 6 Variations of the parameters A11, A12, and A13 as a function of the FG power index (color online) Figure 7 shows the dimensionless natural frequency of the FGP nanoplate versus the surface area (l a l b ) for two different shapes of square and rectangle with h = 5 nm and µ = 1. It is obvious that the nanoplate shape has a significant effect on the natural frequency. It is seen that the natural frequency for the square nanoplate has larger values than the rectangular nanoplate. Ω Fig. 7 The dimensionless natural frequency versus the surface area with h = 5 nm and µ = 1 (color online)

Conclusions
In this work, the vibration behaviors of FGP nanoplates based on the nonlocal strain gradient theory are studied. Using Hamilton's principle, we derive the governing differential equations and then we solve the equations to determine the natural frequencies. We study the effect of various parameters on the natural frequency of FGP nanoplates. The conclusions are summarized as follows.
(i) The natural frequency is a decreasing function of the nonlocal parameter.
(ii) The natural frequency is a decreasing function of the length of the nanoplate for all modes.
(iii) The natural frequency is increased with the FG power index for a fixed length.
(iv) The natural frequency is increased with the aspect ratio l a /l b .
(v) The natural frequency is increased with the FG power index for an aspect ratio.
(vi) With enhancing l a /h, the nanoplate becomes thinner and thus it has a lesser stiffness.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.