Influence of Variable Nonlocal Parameter and Porosity on the Free Vibration Behavior of Functionally Graded Nanoplates

This paper studies the inﬂuence of the variable nonlocal parameter and porosity on the free vibration behavior of the functionally graded nanoplates with porosity. Four patterns of distribution of the porosity through the thickness direction are considered. The classical nonlocal elasticity theory is modiﬁed to take into account the variation of the nonlocal parameter through the thickness of the nanoplates. The governing equations of motion are established using simple ﬁrst-order shear deformation theory and Hamilton’s principle. The closed-form solution based on Navier’s technique is employed to solve the governing equations of motion of fully simply supported nanoplates. The accuracy of the present algorithm is proved via some comparison studies in some special cases. Then, the eﬀects of the porosity, the variation of the nonlocal parameter, the power-law index, aspect ratio, and the side-to-thickness ratio on the free vibration of nanoscale porous plates are investigated carefully. The numerical results show that the porosity and nonlocal parameter have strong eﬀects on the free vibration behavior of the nanoplates.


Introduction
e applications of micro/nanostructures in the micro/ nanoelectromechanical system (MEMS and NEMS) are increasing rapidly based on the development of micro/ nanotechnology, structures, and materials. At the micro/ nanodimension, the small-scale effects on the thermal and mechanical behaviors of micro/nanostructures cannot be neglected. To address this problem, molecular dynamics (MD) can be served as an excellent method to predict the exact behavior of the micro/nanostructures, but the MD usually costs expensive computation. erefore, many researchers have been focused on developing higher-order continuum theory to analyze small-scale structures, for instance, the couple stress theory and modified couple stress theory [1][2][3][4], the nonlocal elasticity theory [5][6][7], and strain gradient theory and nonlocal strain gradient theory [8][9][10][11][12][13]. More details on these theories and their applications can be read from the literature review of ai et al. [14]. Among them, the nonlocal elasticity theory which is first established by Eringen is usually used in combination with shear deformation theories by many scientists to analyze nanoplates, nanobeams, and nanoshells. Many remarkable works on the analysis of nanostructures can be reviewed herein. Zhang et al. [15] analyzed the buckling of multiwalled carbon nanotubes under compression with small-scale effects. Hu et al. [16] developed a nonlocal shell model to analyze wave propagation in single-and double-walled carbon nanotubes. e free vibration of multiwalled carbon nanotubes embedded in an elastic foundation has been analyzed by Li and Kardomateas [17] using a nonlocal elastic shell model. Wang and Varadan [18] employed nonlocal elastic shell theory to analyze the wave propagation of carbon nanotubes. Rouhi and Ansari [19] developed a nonlocal Flugge shell model to study the axial buckling of double-walled carbon nanotubes with different end conditions. Pradhan and Phadikar [20] investigated the small-scale effect on the vibration of multilayered graphene sheets resting on an elastic foundation using nonlocal continuum models. Aksencer and Aydogdu [21] analyzed the vibration and buckling of nanoplates using nonlocal elasticity theory and Levy type solution. Shen et al. [22] investigated the free vibration of a single-layered graphene sheet-based nanomechanical sensor using nonlocal Kirchhoff plate theory. Zhang et al. [23,24] applied nonlocal elasticity theory to analyze free vibration and buckling of single-layered graphene sheets based on the kp-Ritz method. Ansari et al. [25,26] analyzed single-and multilayered graphene sheets using nonlocal elasticity theory. Hosseini-Hashemi et al. [27] developed an exact solution for free vibration of functionally graded circular/annular Mindlin nanoplates using nonlocal elasticity theory. Anjomshoa and Tahani [28] analyzed the free vibration of orthotropic circular and elliptical nanoplates resting on elastic foundations. Fatima et al. [29] developed a nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoplates. Zenkour et al. [30][31][32] analyzed the thermal buckling and bending of orthotropic carbon nanotubes using nonlocal elasticity theory and mixed variation formula. Aghababaei and Reddy [33] developed a nonlocal third-order shear deformation theory to analyze bending and free vibration of nanoplates. Ansari and Sahmani [34] analyzed the buckling behavior of single-layered graphene sheets using a nonlocal plate model and MD simulations. Hosseini-Hashemi et al. [35] employed an analytical solution to study the buckling and free vibration of nanoplates via nonlocal third-order shear deformation theory. Daneshmehr et al. [36,37] analyzed stability and free vibration of functionally graded nanoplates via nonlocal elasticity theory and higher-order shear deformation theory. Malezadeh and Shojaee [38] developed a nonlocal twovariable refined plate theory to analyze the free vibration of nanoplates. Narendar and Gopalakrishnan [39,40] applied a combination of two-variable shear deformation theory and nonlocal elasticity theory to analyze buckling of micro/ nanoplates. Hoa et al. [41] established a novel nonlocal higher-order with only one unknown variable to analyze the bending and free vibration of functionally graded nanoplates. Sobhy and Zenkour [42][43][44][45][46] and Zenkour and Sobhy [47] developed several nonlocal shear deformation theories to analyze single-, double-, and multilayered graphene sheets as well as orthotropic nanoplates with and without elastic foundations. ai et al. [48] developed a nonlocal sinusoidal plate theory to study micro/nanoplates. Fattahi et al. [49] applied nonlocal elasticity theory to determine vibrational behavior of FGM nanoplates. Sahmani and Safaei [50] studied the large-amplitude oscillations of composite conical nanoshells considering surface stress effect. To take into account the thickness stretching effects, Sobhy and Radwan [51] developed a new quasi-3D nonlocal plate theory to investigate the vibration and buckling behavior of functionally graded nanoplates. Bessaim et al. [52] established a nonlocal quasi-3D trigonometric plate theory for free vibration analysis of micro/nanoplates.
Functionally graded materials (FGMs) were first introduced by some Japanese's scientists in the 1950s. Due to their high performance, they have been used widely in many files of engineering, industry, and other special areas of engineering [53,54]. ese materials also applied to produce micro/nanostructures. It is noticed that the porosity usually occurs during the process of manufacture and/or they can be created intentionally to reduce the mass of the structures and control the vibration of the structures. In the case of attended creation, the print-3D technology can be used to create porous structures with desired distribution of porosity. erefore, numerous works have been done on the investigation of free and forced vibration of FGM macro/ micro/nanoplates with and without porosity. Rezaei et al. [55] analyzed the free vibration of FGM plates with porosity using a simple four-variable plate theory. Riadh et al. [56] used a higher-order and normal shear deformation theory to analyze the free vibration response of FGM porous plates. e vibration analysis of temperature-dependent FGM beams with porosity was investigated by Ebrahimi and Jafari [57]. Gao et al. [58] studied wave propagation in FGM porous plates reinforced with graphene platelets in which Young's modulus and mass density depend on the porosity distribution through the thickness of the plates, while Poisson's ratio does not depend on the porosity. Moradi-Dastjerdi et al. [59][60][61] analyzed thermo-electro-mechanical, free vibration, and buckling behaviors of advanced smart sandwich plates including the effects of porosity. In these studies, Young's modulus, mass density, and Poisson's ratio are assumed to depend on the individual material properties and porosity. Akbaş et al. [62] examined the bending and vibration of FGM nanoplates with porosity. Wattanasakulpong and Ungbhakorn [63] studied the linear and nonlinear vibration of elastically restrained ends of FGM beams with porosity. Allahkarami et al. [64] examined the dynamic stability of bidirectional FGM cylindrical shells with porosity. Fan et al. [65] used a modified couple stress theory to analyze geometrically nonlinear vibration of FGM microplates with porosity. e effects of porosity on the free vibration of FGM nanoplates resting on Winkler-Pasternak elastic foundation were investigated by Mechab and his coworkers [66]. Shahverdi and Barati [67] analyzed the free vibration of porous FGM nanoplates. Barati and Shahverdi [68] analyzed forced vibration of FGM nanoplates with porosity under dynamic load using a general nonlocal stressstrain gradient theory. Shahsavari et al. [69] analyzed the shear buckling behavior of nanoplates with porosity using a new size-dependent quasi-3D theory. Daikh et al. [70] investigated the buckling behavior of FGM sandwich nanoplates with porosity with the effects of heat conduction via nonlocal strain gradient theory.
According to the literature review, most studies on the mechanical behavior of FGM nanoplates with and without porosity are carried out based on the nonlocal elasticity theory with a constant nonlocal parameter. On the other hand, it is obvious that the nonlocal parameter depends on the material components. So, it is necessary to consider the variation of the nonlocal parameter on the mechanical analysis of FGM nanoplates. is is the main aim of the present study, in which the classical Eringen's elasticity theory is modified to consider the variable nonlocal parameter for free vibration analysis of FGM porous nanoplates. In addition, the effects of the porosity, the side-tothickness ratio, and the aspect ratio incorporating with variable nonlocal parameter are studied cautiously.

Material Properties of FGM Nanoplates with Porosity
A rectangular FGM nanoplate with porosity is considered. e dimensions of the plate are a × b, and the thickness is h. e Cartesian coordinate xyz is placed at the middle surface of the FGM nanoplates as shown in Figure 1.
e distribution of the ceramic and metal components through the thickness of the plate is described via the volume fractions with the power-law function. e porosity is distributed through the thickness of the plate with four different patterns, namely, patterns A, B, C, and D, and they are described by the following formula [58,71]: for pattern A, for pattern C, for pattern D, where P 0 ≤ 0.5 is the maximum porosity value [72]. e effective Young's modulus, shear modulus, and mass density can be calculated as the following formula [58,71]: where E(z), ρ(z), ](z), and μ(z) are the effective Young's modulus, mass density, Poisson's ratio, and nonlocal parameter of the perfect FGM nanoplates; they are calculated via the power-law function as follows [58,71,73]: where k is a nonnegative power-law index. In this study, the FGM nanoplates are made of mixture of several individual components. e material properties of these individual components are given in Table 1. Figure 2 presents the dispersion of porosity and effective material properties of the Al/ZrO 2 FGM nanoplates with k � 3. Figure 2

Simple First-Order Shear Deformation eory
3.1.1. Kinematic. In this study, a simple first-order shear deformation theory (S-FSDT) which was introduced by ai and Choi [74,75] is used to describe the kinematics of the nanoplates. e displacement field of this theory can be written as the following formula: e strain fields of the plate are obtained as

e Relations of Constitutive.
e constitutive equations of the plate are expressed as where 3.2. e Nonlocal Elasticity eory. In Eringen's nonlocal elasticity theory [5][6][7], the stress at any point in a continuum body depends on the strains at all neighbor points. e differential form of the classical nonlocal elasticity theory of Eringen is as the following formula [7]: where σ ij and t ij are, respectively, the nonlocal and local stress tensors, ∇ 2 � z 2 /zx 2 + z 2 /zy 2 is the second Laplace operator, and μ � (e 0 l) 2 (nm 2 ) is the nonlocal parameter, in which e 0 is a material constant which is determined via experimental or atomistic dynamics and l is an internal characteristic length. It is obvious that the nonlocal parameter of classical Eringen's theory is assumed to be constant. However, it can be seen that e 0 and l are two material-dependent parameters, so the nonlocal parameter μ should be a material-dependent parameter. Consequently, it is necessary to consider the variation of the nonlocal parameter through the thickness of FGM nanoplates. In this study, the nonlocal parameter is assumed to vary through the thickness direction of the FGM nanoplates. Hence, the nonlocal constitutive relations of the plates can be modified as follows:

Equations of Motion.
e equations of motion are achieved via Hamilton's principle as follows: where δU is the variation of the strain energy and δK is the variation of the kinematic energy of the plate. e variation of the strain energy is obtained as the following expression [33,35]: e variation of the kinematic energy of the plate is expressed as [33,35]: By substituting equations (7) and (8) into equation (14), inserting equation (4) into equation (15), and considering the nonlocal relations of equations (11) and (12), after integrating through the thickness of the plates, the governing equations of motion of the sandwich plates are derived from equation (13) where N i , M i , L i , and R i are the local stress resultants which are calculated by By substituting equations (11) and (12) into equation (17) and integrating through the thickness of the plates and then reordering these equations into matrix form, one gets where where k s being the shear correction factor, which is usually taken as k s � 5/6. In the case of conventional nonlocal elasticity theory, the nonlocal parameter μ is assumed to be constant through the thickness of the plates and the coefficients I 0 , I 1 , I 2 and L 0 , L 1 , L 2 are calculated as the following formulas: When the nonlocal parameter μ is constant, one gets (L 0 , L 1 , L 2 ) � μ(I 0 , I 1 , I 2 ). Consequently, the governing equations of motion equation (16) become the conventional governing equations of motion of the nanoplates with the constant nonlocal parameter. In this study, the nonlocal parameter is assumed to vary through the thickness of the plates as other material properties. is is the novelty of the present study in comparison with other works. Such that the coefficients I 0 , I 1 , and I 2 are calculated via equation (20), while the coefficients L 0 , L 1 , and L 2 are calculated as follows: is is the novelty of the present study. e classical nonlocal elasticity theory with constant nonlocal parameter is achieved by setting μ c /μ m � 1; thus, μ c � μ m � μ.

Analytical Solution.
In this study, a fully simply supported nanoplate is considered. e simply supported edge is By applying Navier's solution, the displacement fields of the plates are assumed as V mn e iωt sin αx cos βy, Wb mn e iωt sin αx sin βy, Ws mn e iωt sin αx sin βy, (24) where α � mπ/a, β � nπ/b, i 2 � −1, Δ � U mn , V mn , Wb mn , Ws mn }, and ω is the free frequency of the nanoplates. Substituting equation (24) into equation (4) and then equation (16), one gets where K and M are the stiffness matrix and the mass matrix

Comparison Study.
To present the validity of the current algorithm, the free vibration of fully simply supported square porous FGM plates is considered. e plates are made of Al/Al 2 O 3 , the dimensions of the plate are a � b � 1, and the thickness is h. e material properties of the perfect plates are calculated via power-law function, and the porosity is normally distributed through the thickness direction. Hence, the effective Young's modulus and effective mass density are computed as follows [55]: e nondimensional frequencies of the plates are calculated as ω � ωh ����� � ρ m /E m . e present results are compared with those of Rezaei et al. [55] in Table 2. According to this table, it can be concluded that the proposed results are in good agreement with those of Rezaei et al. [55] using higherorder shear deformation theory (HSDT).
Secondly, the numerical results of free vibration of FGM nanoplates using the proposed theory are compared with those of Sobhy and Radwan [51]. In this subsection, the FGM nanoplates are made of SUS304/Si 3 N 4 with several values of side-to-thickness ratio a/h, and nonlocal parameters μ are considered. e dimensions of the plates are a � b � 10 nm and the thickness is h. e effective material properties are calculated via a power-law function. e comparison between the nondimensional fundamental frequencies ω � 10ωh ����� � ρ m /E m of the FGM nanoplates of SUS304/Si 3 N 4 using the present theory and those of Sobhy using simple HSDT and quasi-3D theory is shown in Table 3. According to this table, it can be concluded that the present results are in good agreement with those of Sobhy and Radwan [51].

Parameter Study.
In the current investigation, a functionally graded nanoplate of Al/ZrO 2 with porosity is considered. e material properties of Al and ZrO 2 are given in Table 1. e nonlocal parameter of Al is assumed to be constant and given as μ m � 2(nm 2 ) and plays as the reference nonlocal value. For convenience, the following nondimensional frequencies are used: Tables 4-8 show the nondimensional fundamental frequency of the FGM nanoplates with the dimension of a � 10(nm 2 ) and the thickness of h � a/10. e nondimensional fundamental frequencies of four patterns of the porous FGM nanoplates are given in Tables 4-7. According to these tables, it can be seen that when the power-law index increases, the frequency of the FGM nanoplates decreases. Besides, the increase in the nonlocal parameter's ratio μ c /μ m leads to the decrease in the frequency of the FGM nanoplates. When the power-law index is infinity, k � ∞, the nondimensional fundamental frequency of the FGM nanoplates is independent of the variation of the nonlocal parameter's ratio. e reason is that when k � ∞, the FGM nanoplate becomes a fully metallic one; therefore, the nonlocal parameter is constant and equals the nonlocal parameter of the metal component through the thickness of the plate. Moreover, when the maximum porosity value P 0 increases, the nondimensional fundamental frequencies of the plates of type B and C decrease but those of the plates of types A and D increase because the increase in the maximum porosity coefficient P 0 leads to the decrease in both the stiffness and the mass of the plates. In the case of type A, most of the porosities are distributed around the middle plane of the plate, so the effect of the porosity to the mass matrix is stronger than that to the stiffness matrix. On the opposite side, the porosities of type B of the plate are distributed near two surfaces of the plate, so the effect of the porosity to the stiffness matrix is stronger than that to the mass matrix. In the case of type C, the porosity is distributed near the metal-rich surface, so the effect of the porosity to the stiffness is stronger than that to the mass. In the cases of type D, the porosity is distributed near the ceramic-rich surface, so the effect of the porosity on the mass is stronger than that to the stiffness.  Continuously, the influence of the porosity on the fundamental frequency of the square FGM nanoplate is considered. e nondimensional fundamental frequency of perfect FGM nanoplates is given in Table 8. In the cases of a/h � 10 and μ c /μ m � 2, Figure 3 presents the variation of the nondimensional fundamental frequency of the FGM nanoplates as functions of maximum porosity value P 0 . It can be seen that the effects of the porosity on four types of porosity dispersion are different. When P 0 � 0, the fundamental frequencies of the FGM nanoplates of four patterns A, B, C, and D are identical because these porous FGM nanoplates become perfect ones. For pattern A, when P 0 increases, the fundamental frequency increases rapidly for both k � 0.5 and k � 2. On the opposite side, when P 0 increases, the fundamental frequency of pattern B decreases rapidly for both k � 0.5 and k � 2. e fundamental frequency for pattern C decreases slower than that of pattern B, and the fundamental frequency of pattern D increases slower than that of pattern A.

Shock and Vibration
To investigate the effects of the porosity on the high frequencies of the square FGM nanoplates, the frequencies of four modes (5,5), (10,10), (20,20), and (100, 100) are considered. Figure 4 presents the ratio between the frequencies of the FGM nanoplates with porosity and those of the perfect FGM nanoplates Ω porous m,n /Ω perfect m,n . It can be seen clearly that the effects of the porosity on the high frequencies are different from those on the low frequencies. For mode (5,5), the effects of the porosity of patterns A and B are stronger than those of patterns C and D. For mode (10,10), the strong effects of porosity of four patterns are similar with two different trends and the frequencies of patterns A and D increase while those of patterns B and C decrease. For mode (20,20), the influences of the porosity of patterns C and D on the   frequencies of porous FGM nanoplates are stronger than those of patterns A and B. For mode (100, 100), the porosity of patterns C and D affects strongly the free vibration of porous FGM nanoplates but the frequencies of the FGM nanoplates of patterns A and B are almost unchanged.
e role of the variable nonlocal parameter on the free vibration of the square porous FGM nanoplates with a/h � 10 is investigated and presented in Figure 5. Generally, the fundamental frequency of the nanoplates decreases with the increase in the nonlocal parameter ratio μ c /μ m ; however, the effects of the nonlocal parameters are different from   Table 6: e nondimensional fundamental frequency of the porous FGM nanoplates of pattern C.       When the porosity parameters are small P 0 � 0.2, the fundamental frequencies of the FGM nanoplates decrease at similar rate with the increase in the nonlocal parameters ratio μ c /μ m . When P 0 � 0.5 and k � 2, the decrease rates of the FGM porous nanoplates of patterns A and B are similar to the perfect ones. e rate of reduction of the fundamental frequencies of the FGM porous nanoplates of pattern C is higher than the perfect ones, while the rate of reduction of the fundamental frequencies of the FGM porous nanoplates of pattern D is slower than the perfect ones. erefore, the effects of the nonlocal parameters ratio on the fundamental frequencies of the FGM porous nanoplates depend on power-law index k, porosity parameter P 0 , and the distribution of porosity. Figure 6 demonstrates the influence of the power-law index on the nondimensional fundamental frequencies of the square FGM nanoplates with porosity and a/h � 10 and P 0 � 0.5. When μ c /μ m � 0.5, the fundamental frequency of the porous FGM nanoplates decreases rapidly when the power-law index k increases from 0 to 2. When k > 2, the fundamental frequencies of the porous FGM nanoplates of patterns A and C increase slowly. For patterns B and D, the fundamental frequencies increase slowly when the powerlaw index k increases from 2 to 4, and the fundamental frequencies decrease gently when k > 4. When μ c /μ m � 2, the fundamental frequencies of four patterns decrease with the increase in the power-law index from 0 to 1. When k > 1, the fundamental frequencies increase with the increase in the power-law index. Especially, a minimum frequency exists when the power-law index increases. In the practical application, this phenomenon should be noticed to avoid the resonance behavior of the nanoplates. Besides, when the power-law index is equal to zero k � 0, the fundamental frequencies of patterns C and D are similar. e reason is that when k � 0, the FGM porous nanoplates become the homogeneous isotropic ones, so the effects of the porosity of patterns C and D are similar. e influences of the porosity on the fundamental frequencies of the rectangular porous FGM nanoplates with a/h � 10 and k � 2 are investigated in this subsection. According to Figure 7, the fundamental frequencies of the porous FGM nanoplates decrease when the aspect ratio of b/a increases. In general, the frequencies of the FGM nanoplates with the aspect ratio of b/a � 5 are two times smaller than those of square ones, and the frequencies of the square ones are the highest. When the aspect ratio increases from 1 to 2, the fundamental frequency of the FGM porous nanoplates decreases dramatically. When the aspect ratio greater than 2, the fundamental frequency of the FGM porous nanoplates decrease slowly.
Lastly, the dependence of the fundamental frequencies of the square FGM porous nanoplates with k � 2 on the variation of the side-to-thickness ratio a/h is examined. Figure 8 shows that the fundamental frequencies of the square FGM nanoplates with decrease in porosity rapidly as the side-tothickness ratio increases. e fundamental frequencies of the square porous FGM nanoplates with a/h � 20 are approximately five times smaller than those of the plates with a/h � 5. e fundamental frequencies of the square porous FGM nanoplates are increasing in order of pattern B, C, D, and A.

Conclusions
A comprehensive study on the effects of porosity and variable nonlocal parameter on the free vibration of the porous FGM nanoplates has been presented. A modified nonlocal elasticity theory with the variation of the nonlocal parameter has been established to analyze the nonlocal free vibration of FGM nanoplates. e numerical results presented the significant effects of the porosity and variable nonlocal parameter on the free vibration behavior of the FGM nanoplates. Depending on the dispersion and coefficient of the porosity, the frequencies of the nanoplates can increase or decrease. Several remarkable conclusions can be read as follows: (i) e variation of the nonlocal parameter should be considered due to its important role in the free vibration of FGM nanoplates (ii) e inclusion of the nonlocal parameter leads to the decrease in the frequency of the nonlocal porous FGM plates (iii) e porosity plays significant effects on the free vibration characteristics of the porous FGM nanoplates (iv) e influence of the porosity on the high frequencies is different from that of the low frequencies of the FGM nanoplates e outcomes of this study can serve as benchmark results for future works on the vibration analysis of micro/ nanostructures considering the variable of small-scale parameters as well as the effects of the porosity.

Data Availability
No data were used to support this study.

Conflicts of Interest
On behalf of all authors, the corresponding author states that there are no conflicts of interest.