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

Authors: Pham Van Vinh; Le Quang Huy
Journal: Shock and Vibration
 (2021)

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## Abstract

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.


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## Body

## 1. Introduction

The 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. Therefore, 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 [1–4], the nonlocal elasticity theory [5–7], and strain gradient theory and nonlocal strain gradient theory [8–13]. More details on these theories and their applications can be read from the literature review of Thai 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. The 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. [30–32] 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 two-variable 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 [42–46] 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. Thai 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]. These 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. Therefore, 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. The 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. [59–61] 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. The 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 stress-strain 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. This 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-to-thickness ratio, and the aspect ratio incorporating with variable nonlocal parameter are studied cautiously.


## 2. Material Properties of FGM Nanoplates with Porosity

A rectangular FGM nanoplate with porosity is considered. The dimensions of the plate area×b, and the thickness is h. The Cartesian coordinate xyz is placed at the middle surface of the FGM nanoplates as shown in Figure 1.Figure 1
The geometric model of FGM porous nanoplates.The distribution of the ceramic and metal components through the thickness of the plate is described via the volume fractions with the power-law function. The 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]:(1)pz=P0cosπzh,for pattern A,P01−cosπzh,for pattern B,P0cosπz2h+π4,for pattern C,P0cosπz2h−π4,for pattern D,where P0≤0.5 is the maximum porosity value [72]. The effective Young’s modulus, shear modulus, and mass density can be calculated as the following formula [58, 71]:(2)Ez=E˜z1−pz,ρz=ρ˜z1−pz,Gz=Ez21−νz,μz=μ˜z,where E˜z,ρ˜z,νz, 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]:(3)E˜z=Em+Ec−Emzh+12k,ρ˜z=ρm+ρc−ρmzh+12k,νz=νm+νc−νmzh+12k,μ˜z=μm+μc−μmzh+12k,where k is a nonnegative power-law index. In this study, the FGM nanoplates are made of mixture of several individual components. The 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/ZrO2 FGM nanoplates with k=3. Figure 2(a) displays the normalized distribution of porosity through the thickness of the plates. Figures 2(b) and 2(c) plot the variation of the effective Young’s modulus and mass density of the material through the thickness direction of the nanoplates. The variation of the effective nonlocal parameter of the material through the thickness of the nanoplates with μm=2 and different values of nonlocal parameter’s ratio μc/μm is presented in Figure 2(d).Table 1
Material properties of some individual metal/ceramic components.
MaterialsAlSUS304Al2O3Si3N4ZrO2Young’s modulus (GPa)70201.04380348.43200Mass densitykg/m327078166380023705700Poisson’s ratio0.30.30.30.240.3Nonlocal parameter22———Figure 2
The distribution of porosity and effective material properties ofAl/ZrO2 FGM nanoplates: (a) distribution of porosity; (b) k = 3, 0, P0 = 0.5; (c) k = 3, 0, P0 = 0.5; (d) k = 3.
(a)(b)(c)(d)


## 3. Theoretical Formulation

### 3.1. Simple First-Order Shear Deformation Theory

#### 3.1.1. Kinematic

In this study, a simple first-order shear deformation theory (S-FSDT) which was introduced by Thai and Choi [74, 75] is used to describe the kinematics of the nanoplates. The displacement field of this theory can be written as the following formula:(4)ux,y,z,t=ux,y,t−z∂wb∂x,vx,y,z,t=vx,y,t−z∂wb∂y,wx,y,z,t=wbx,y,t+wsx,y,t.The strain fields of the plate are obtained as(5)εxεyγxy=εx0εy0γxy0+zκxκyκxy,γxzγyz=γxz0γyz0,where(6)εx0εy0γxy0=∂u∂x∂v∂y∂u∂y+∂v∂x,κxκyκxy=−∂2wb∂x2∂2wb∂y22∂2wb∂x∂y,γxz0γyz0=∂ws∂x∂ws∂y.


#### 3.1.2. The Relations of Constitutive

The constitutive equations of the plate are expressed as(7)σxσyτxy=C11C120C12C22000C66εxεyγxy,(8)τxzτyz=C5500C44γxzγyz,where(9)C11=C22=Ez1−νz2,C12=νzC11,C44=C55=C66=Ez21+νz.


### 3.2. The Nonlocal Elasticity Theory

In Eringen’s nonlocal elasticity theory [5–7], the stress at any point in a continuum body depends on the strains at all neighbor points. The differential form of the classical nonlocal elasticity theory of Eringen is as the following formula [7]:(10)1−μ∇2σij=tij,where σijandtij are, respectively, the nonlocal and local stress tensors, ∇2=∂2/∂x2+∂2/∂y2 is the second Laplace operator, and μ=e0l2nm2 is the nonlocal parameter, in which e0 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 e0 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:(11)txtytxy=1−μz∇2σxσyτxy=C11C120C12C22000C66εxεyγxy,(12)tyztxz=1−μz∇2τyzτxz=C4400C55γyzγxz.


### 3.3. Equations of Motion

The equations of motion are achieved via Hamilton’s principle as follows:(13)0=∫0TδU−δKdt,where δU is the variation of the strain energy and δK is the variation of the kinematic energy of the plate. The variation of the strain energy is obtained as the following expression [33, 35]:(14)δU=∫A∫−h/2h/2σxδεx+σyδεy+τxyδεxy+τxzδεxz+τyzδεyzdzdA.The variation of the kinematic energy of the plate is expressed as [33, 35]:(15)δK=∫A∫−h/2h/2u˙δu˙+v˙δv˙+w˙δw˙ρzdzdA.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) as the following formulas:(16)δu:∂Nx∂x+∂Nxy∂y=I0u¨−I1∂w¨b∂x−∇2L0u¨−L1∂w¨b∂x,δv:∂Ny∂y+∂Nxy∂x=I0v¨−I1∂w¨b∂y−∇2L0v¨−L1∂w¨b∂y,δwb:∂2Mx∂x2+2∂2Mxy∂x∂y+∂2My∂y2=I0w¨b+w¨s+I1∂u¨∂x+∂v¨∂y−I2∇2w¨b−∇2L0w¨b+w¨s+L1∂u¨∂x+∂v¨∂y−L2∇2w¨b,δus:∂Qx∂x+∂Qy∂y=I0w¨b+w¨s−∇2L0w¨b+w¨s,where Ni,Mi,Li, and Ri are the local stress resultants which are calculated by(17)NxNyNxy=∫−h/2h/2txtytxydz,MxMyMxy=∫−h/2h/2txtytxyzdz,QxQy=∫−h/2h/2txztyzdz.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(18)NMQ=AB0BD000Hε0κγ0,where(19)Aij,Bij,Dij=∫−h/2h/2Cij1,z,z2dz,Hij=ks∫−h/2h/2Cijdz,where ks being the shear correction factor, which is usually taken as ks=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 I0,I1,I2 and L0,L1,L2 are calculated as the following formulas:(20)I0,I1,I2=∫−h/2h/2ρz1,z,z2dz,(21)L0,L1,L2=μ∫−h/2h/2ρz1,z,z2dz.When the nonlocal parameterμ is constant, one gets L0,L1,L2=μI0,I1,I2. 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. This is the novelty of the present study in comparison with other works. Such that the coefficients I0,I1,andI2 are calculated via equation (20), while the coefficients L0,L1,andL2 are calculated as follows:(22)L0,L1,L2=∫−h/2h/2μzρz1,z,z2dz.This is the novelty of the present study. The classical nonlocal elasticity theory with constant nonlocal parameter is achieved by settingμc/μm=1; thus, μc=μm=μ.


### 3.4. Analytical Solution

In this study, a fully simply supported nanoplate is considered. The simply supported edge is(23)Nx=v=wb=ws=Mx=0,atx=0,a,u=Ny=wb=ws=My=0,aty=0,b.By applying Navier’s solution, the displacement fields of the plates are assumed as(24)ux,y,t=∑m=1∞∑n=1∞Umneiωtcosαxsinβy,vx,y,t=∑m=1∞∑n=1∞Vmneiωtsinαxcosβy,wbx,y,t=∑m=1∞∑n=1∞Wbmneiωtsinαxsinβy,wsx,y,t=∑m=1∞∑n=1∞Wsmneiωtsinαxsinβy,where α=mπ/a,β=nπ/b, i2=−1, Δ=Umn,Vmn,Wbmn,Wsmn, and ω is the free frequency of the nanoplates. Substituting equation (24) into equation (4) and then equation (16), one gets(25)K−ω2MΔ=0,where KandM are the stiffness matrix and the mass matrix and their elements are KijandMij,i,j=1,4¯ which are calculated as the following formulas:(26)K11=A11α2+A66β2;K12=αA12+A66β;K13=−B11α3−B12+2B66β2α;K14=0;K22=A22β2+A66α2;K23=−B22β3−α2B12+2B66β;K24=0;K33=D11α4+2β2D12+2D66α2+D22β4;K34=0;K44=H44β2+H55α2,M11=α2+β2L0+I0;M12=0;M13=−αα2+β2L1+I1;M14=0;M22=α2+β2L0+I0;M23=−βα2+β2L1+I1;M24=0;M33=L2α4+2L2β2+I2+L0α2+L2β4+I2+L0β2+I0;M34=α2+β2L0+I0;M44=α2+β2L0+I0.


## 3.1. Simple First-Order Shear Deformation Theory

### 3.1.1. Kinematic

In this study, a simple first-order shear deformation theory (S-FSDT) which was introduced by Thai and Choi [74, 75] is used to describe the kinematics of the nanoplates. The displacement field of this theory can be written as the following formula:(4)ux,y,z,t=ux,y,t−z∂wb∂x,vx,y,z,t=vx,y,t−z∂wb∂y,wx,y,z,t=wbx,y,t+wsx,y,t.The strain fields of the plate are obtained as(5)εxεyγxy=εx0εy0γxy0+zκxκyκxy,γxzγyz=γxz0γyz0,where(6)εx0εy0γxy0=∂u∂x∂v∂y∂u∂y+∂v∂x,κxκyκxy=−∂2wb∂x2∂2wb∂y22∂2wb∂x∂y,γxz0γyz0=∂ws∂x∂ws∂y.


### 3.1.2. The Relations of Constitutive

The constitutive equations of the plate are expressed as(7)σxσyτxy=C11C120C12C22000C66εxεyγxy,(8)τxzτyz=C5500C44γxzγyz,where(9)C11=C22=Ez1−νz2,C12=νzC11,C44=C55=C66=Ez21+νz.


## 3.1.1. Kinematic

In this study, a simple first-order shear deformation theory (S-FSDT) which was introduced by Thai and Choi [74, 75] is used to describe the kinematics of the nanoplates. The displacement field of this theory can be written as the following formula:(4)ux,y,z,t=ux,y,t−z∂wb∂x,vx,y,z,t=vx,y,t−z∂wb∂y,wx,y,z,t=wbx,y,t+wsx,y,t.The strain fields of the plate are obtained as(5)εxεyγxy=εx0εy0γxy0+zκxκyκxy,γxzγyz=γxz0γyz0,where(6)εx0εy0γxy0=∂u∂x∂v∂y∂u∂y+∂v∂x,κxκyκxy=−∂2wb∂x2∂2wb∂y22∂2wb∂x∂y,γxz0γyz0=∂ws∂x∂ws∂y.


## 3.1.2. The Relations of Constitutive

The constitutive equations of the plate are expressed as(7)σxσyτxy=C11C120C12C22000C66εxεyγxy,(8)τxzτyz=C5500C44γxzγyz,where(9)C11=C22=Ez1−νz2,C12=νzC11,C44=C55=C66=Ez21+νz.


## 3.2. The Nonlocal Elasticity Theory

In Eringen’s nonlocal elasticity theory [5–7], the stress at any point in a continuum body depends on the strains at all neighbor points. The differential form of the classical nonlocal elasticity theory of Eringen is as the following formula [7]:(10)1−μ∇2σij=tij,where σijandtij are, respectively, the nonlocal and local stress tensors, ∇2=∂2/∂x2+∂2/∂y2 is the second Laplace operator, and μ=e0l2nm2 is the nonlocal parameter, in which e0 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 e0 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:(11)txtytxy=1−μz∇2σxσyτxy=C11C120C12C22000C66εxεyγxy,(12)tyztxz=1−μz∇2τyzτxz=C4400C55γyzγxz.


## 3.3. Equations of Motion

The equations of motion are achieved via Hamilton’s principle as follows:(13)0=∫0TδU−δKdt,where δU is the variation of the strain energy and δK is the variation of the kinematic energy of the plate. The variation of the strain energy is obtained as the following expression [33, 35]:(14)δU=∫A∫−h/2h/2σxδεx+σyδεy+τxyδεxy+τxzδεxz+τyzδεyzdzdA.The variation of the kinematic energy of the plate is expressed as [33, 35]:(15)δK=∫A∫−h/2h/2u˙δu˙+v˙δv˙+w˙δw˙ρzdzdA.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) as the following formulas:(16)δu:∂Nx∂x+∂Nxy∂y=I0u¨−I1∂w¨b∂x−∇2L0u¨−L1∂w¨b∂x,δv:∂Ny∂y+∂Nxy∂x=I0v¨−I1∂w¨b∂y−∇2L0v¨−L1∂w¨b∂y,δwb:∂2Mx∂x2+2∂2Mxy∂x∂y+∂2My∂y2=I0w¨b+w¨s+I1∂u¨∂x+∂v¨∂y−I2∇2w¨b−∇2L0w¨b+w¨s+L1∂u¨∂x+∂v¨∂y−L2∇2w¨b,δus:∂Qx∂x+∂Qy∂y=I0w¨b+w¨s−∇2L0w¨b+w¨s,where Ni,Mi,Li, and Ri are the local stress resultants which are calculated by(17)NxNyNxy=∫−h/2h/2txtytxydz,MxMyMxy=∫−h/2h/2txtytxyzdz,QxQy=∫−h/2h/2txztyzdz.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(18)NMQ=AB0BD000Hε0κγ0,where(19)Aij,Bij,Dij=∫−h/2h/2Cij1,z,z2dz,Hij=ks∫−h/2h/2Cijdz,where ks being the shear correction factor, which is usually taken as ks=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 I0,I1,I2 and L0,L1,L2 are calculated as the following formulas:(20)I0,I1,I2=∫−h/2h/2ρz1,z,z2dz,(21)L0,L1,L2=μ∫−h/2h/2ρz1,z,z2dz.When the nonlocal parameterμ is constant, one gets L0,L1,L2=μI0,I1,I2. 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. This is the novelty of the present study in comparison with other works. Such that the coefficients I0,I1,andI2 are calculated via equation (20), while the coefficients L0,L1,andL2 are calculated as follows:(22)L0,L1,L2=∫−h/2h/2μzρz1,z,z2dz.This is the novelty of the present study. The classical nonlocal elasticity theory with constant nonlocal parameter is achieved by settingμc/μm=1; thus, μc=μm=μ.


## 3.4. Analytical Solution

In this study, a fully simply supported nanoplate is considered. The simply supported edge is(23)Nx=v=wb=ws=Mx=0,atx=0,a,u=Ny=wb=ws=My=0,aty=0,b.By applying Navier’s solution, the displacement fields of the plates are assumed as(24)ux,y,t=∑m=1∞∑n=1∞Umneiωtcosαxsinβy,vx,y,t=∑m=1∞∑n=1∞Vmneiωtsinαxcosβy,wbx,y,t=∑m=1∞∑n=1∞Wbmneiωtsinαxsinβy,wsx,y,t=∑m=1∞∑n=1∞Wsmneiωtsinαxsinβy,where α=mπ/a,β=nπ/b, i2=−1, Δ=Umn,Vmn,Wbmn,Wsmn, and ω is the free frequency of the nanoplates. Substituting equation (24) into equation (4) and then equation (16), one gets(25)K−ω2MΔ=0,where KandM are the stiffness matrix and the mass matrix and their elements are KijandMij,i,j=1,4¯ which are calculated as the following formulas:(26)K11=A11α2+A66β2;K12=αA12+A66β;K13=−B11α3−B12+2B66β2α;K14=0;K22=A22β2+A66α2;K23=−B22β3−α2B12+2B66β;K24=0;K33=D11α4+2β2D12+2D66α2+D22β4;K34=0;K44=H44β2+H55α2,M11=α2+β2L0+I0;M12=0;M13=−αα2+β2L1+I1;M14=0;M22=α2+β2L0+I0;M23=−βα2+β2L1+I1;M24=0;M33=L2α4+2L2β2+I2+L0α2+L2β4+I2+L0β2+I0;M34=α2+β2L0+I0;M44=α2+β2L0+I0.


## 4. Numerical Results and Discussion

### 4.1. Comparison Study

To present the validity of the current algorithm, the free vibration of fully simply supported square porous FGM plates is considered. The plates are made ofAl/Al2O3, the dimensions of the plate are a=b=1, and the thickness is h. The 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]:(27)Ez=Ec−Emzh+12k+Em−Ec+EmP02,ρz=ρc−ρmzh+12k+ρm−ρc+ρmP02.The nondimensional frequencies of the plates are calculated asω¯=ωhρm/Em. The 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 higher-order shear deformation theory (HSDT).Table 2
The comparison of the nondimensional frequencies of the FGM plates with porosity.
b/aP0Methodk=0k=0.1k=0.5k=1a/h=50.50Rezaei et al. [55]0.2719460.2620170.2315730.208997Present0.2706640.2608100.2305910.2081580.2Rezaei et al. [55]0.2804780.2689600.2308750.198452Present0.2791550.2677270.2299330.1977350.4Rezaei et al. [55]0.2929770.2792940.2284530.172779Present0.2915960.2780200.2275850.17230810Rezaei et al. [55]0.4182100.4032210.3570540.322491Present0.4153410.4005150.3548400.3205980.2Rezaei et al. [55]0.4313300.4139560.3562900.306810Present0.4283710.4111900.3541610.3051850.4Rezaei et al. [55]0.4505520.4299360.3530950.268252Present0.4474610.4270790.3511270.267173a/h=100.50Rezaei et al. [55]0.0719090.0692080.0609870.054979Present0.0718120.0691180.0609140.0549160.2Rezaei et al. [55]0.0741650.0710290.0607210.052052Present0.0740650.0709360.0606510.0519990.4Rezaei et al. [55]0.0774700.0737370.0599430.045029Present0.0773660.0736410.0598790.04499510Rezaei et al. [55]0.1136930.1094490.0965100.087024Present0.1134540.1092250.0963290.0868690.2Rezaei et al. [55]0.1172600.1123330.0961190.082444Present0.1170130.1121040.0959450.0823130.4Rezaei et al. [55]0.1224850.1166240.0949360.071421Present0.1222280.1163870.0947770.071336Secondly, 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/Si3N4 with several values of side-to-thickness ratio a/h, and nonlocal parameters μ are considered. The dimensions of the plates are a=b=10nm and the thickness is h. The effective material properties are calculated via a power-law function. The comparison between the nondimensional fundamental frequencies ω^=10ωhρm/Em of the FGM nanoplates of SUS304/Si3N4 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].Table 3
The comparison of the nondimensional frequencies of the FGM plates with porosity.
a/hkMethodμ=02μ=0.52μ=12μ=1.52μ=2250Sobhy and Radwan [51] (quais-3D)5.123775.001844.682434.263703.83015Sobhy and Radwan [51] (HSDT)5.107024.985494.667134.249763.81763Present5.105874.984374.666074.248803.816771Sobhy and Radwan [51] (quais-3D)3.044452.972002.782212.533402.27580Sobhy and Radwan [51] (HSDT)3.018602.946772.758592.511902.25648Present3.018732.946892.758712.512002.256575Sobhy and Radwan [51] (quais-3D)2.441722.383622.231402.031851.82525Sobhy and Radwan [51] (HSDT)2.424432.366742.215602.017471.81232Present2.433562.375652.223952.025071.81915MetalSobhy and Radwan [51] (quais-3D)2.122312.071801.939501.766061.58648Sobhy and Radwan [51] (HSDT)2.112612.062341.930641.757991.57923Present2.112082.061821.930151.757551.57883100Sobhy and Radwan [51] (quais-3D)1.390151.357071.270411.156801.03917Sobhy and Radwan [51] (HSDT)1.388291.355251.268711.155251.03778Present1.388241.355211.268671.155211.037751Sobhy and Radwan [51] (quais-3D)0.827820.808120.756520.688860.61882Sobhy and Radwan [51] (HSDT)0.822500.802920.751650.684430.61484Present0.822960.803380.752080.684820.615185Sobhy and Radwan [51] (quais-3D)0.668900.652990.611290.556620.50002Sobhy and Radwan [51] (HSDT)0.664850.649030.607580.553250.49699Present0.665940.650090.608570.554150.49780MetalSobhy and Radwan [51] (quais-3D)0.578170.564410.528370.481120.43220Sobhy and Radwan [51] (HSDT)0.576950.563220.527250.480100.43128Present0.576930.563200.527240.480090.43127500Sobhy and Radwan [51] (quais-3D)0.057330.055960.052390.047700.04285Sobhy and Radwan [51] (HSDT)0.057300.055930.052360.047680.04283Present0.057300.055930.052360.047680.042831Sobhy and Radwan [51] (quais-3D)0.034170.033350.031220.028430.02554Sobhy and Radwan [51] (HSDT)0.033980.033170.031050.028270.02540Present0.034010.033200.031080.028300.025425Sobhy and Radwan [51] (quais-3D)0.027690.027030.025310.023040.02070Sobhy and Radwan [51] (HSDT)0.027540.026880.025170.022910.02058Present0.027560.026900.025180.022930.02060MetalSobhy and Radwan [51] (quais-3D)0.023880.023310.021820.019870.01785Sobhy and Radwan [51] (HSDT)0.023850.023290.021800.019850.01783Present0.023860.023290.021800.019850.01784


### 4.2. Parameter Study

In the current investigation, a functionally graded nanoplate ofAl/ZrO2 with porosity is considered. The material properties of Al and ZrO2 are given in Table 1. The nonlocal parameter of Al is assumed to be constant and given as μm=2nm2 and plays as the reference nonlocal value. For convenience, the following nondimensional frequencies are used:(28)Ω=10ωhρmEm.Tables4–8 show the nondimensional fundamental frequency of the FGM nanoplates with the dimension of a=10nm2 and the thickness of h=a/10. The 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 μc/μm 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. The 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 P0 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 P0 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.Table 4
The nondimensional fundamental frequency of the porous FGM nanoplates of pattern A.
a/hP0μc/μmk=0k=0.5k=1k=2k=5k=∞50.10.51.13701.05031.01530.99780.99680.904411.05350.99710.97450.96880.98030.90441.50.98600.95120.93830.94210.96450.904420.93010.91120.90580.91750.94950.90440.20.51.15141.06301.02721.00951.00870.915811.06681.00920.98590.97980.99150.91581.50.99850.96280.94910.95250.97510.915820.94180.92220.91620.92740.95950.91580.50.51.20701.11191.07311.05411.05360.960011.11831.05571.02941.02161.03350.96001.51.04671.00730.99060.99201.01460.960020.98730.96500.95590.96480.99660.9600100.10.50.62250.57240.55300.54520.54890.495110.57680.54330.53070.52930.53980.49511.50.53980.51820.51080.51460.53110.495120.50920.49630.49310.50110.52290.49510.20.50.63190.58060.56070.55280.55660.502610.58550.55100.53800.53640.54710.50261.50.54800.52560.51780.52140.53810.502620.51690.50340.49980.50760.52950.50260.50.50.66880.61270.59070.58190.58620.531910.61960.58160.56650.56380.57500.53191.50.58000.55480.54490.54730.56440.531920.54700.53140.52570.53220.55440.5319Table 5
The nondimensional fundamental frequency of the porous FGM nanoplates of pattern B.
a/hP0μc/μmk=0k=0.5k=1k=2k=5k=∞50.10.51.11141.02770.99380.97670.97530.884011.02970.97560.95410.94880.96000.88401.50.96380.93070.91880.92320.94530.884020.90910.89140.88720.89960.93130.88400.20.51.09731.01530.98200.96500.96330.872811.01670.96370.94280.93770.94860.87281.50.95150.91940.90810.91270.93450.872820.89750.88060.87690.88950.92110.87280.50.51.04420.96860.93710.92030.91720.830610.96750.91930.90020.89530.90480.83061.50.90560.87690.86730.87230.89280.830620.85420.83990.83770.85100.88130.8306100.10.50.60590.55790.53930.53180.53500.481910.56140.52950.51770.51650.52660.48191.50.52540.50500.49840.50250.51860.481920.49560.48370.48120.48950.51090.48190.20.50.59690.55000.53180.52430.52730.474710.55300.52200.51050.50940.51930.47471.50.51760.49790.49160.49570.51160.474720.48820.47680.47460.48310.50430.47470.50.50.56340.52070.50380.49630.49810.448110.52200.49420.48380.48280.49140.44811.50.48860.47130.46610.47030.48490.448120.46080.45140.45020.45880.47870.4481Table 6
The nondimensional fundamental frequency of the porous FGM nanoplates of pattern C.
a/hP0μc/μmk=0k=0.5k=1k=2k=5k=∞50.10.51.12631.03941.00390.98590.98540.895911.04360.98620.96310.95670.96880.89591.50.97680.94040.92690.93000.95310.895920.92130.90050.89440.90550.93810.89590.20.51.12771.03881.00220.98320.98340.897011.04490.98500.96070.95340.96620.89701.50.97800.93890.92400.92610.94990.897020.92240.89860.89110.90100.93430.89700.50.51.12061.02490.98450.96160.96350.891311.03830.96960.94080.92930.94400.89131.50.97180.92240.90250.90000.92560.891320.91660.88150.86850.87330.90820.8913100.10.50.61560.56520.54560.53740.54130.489610.57030.53620.52330.52140.53220.48961.50.53380.51120.50350.50680.52350.489620.50350.48940.48580.49330.51530.48960.20.50.61640.56480.54440.53550.53980.490310.57120.53540.52170.51910.53030.49031.50.53460.51020.50160.50420.52130.490320.50420.48830.48370.49040.51280.49030.50.50.61190.55560.53260.52110.52610.486710.56690.52540.50880.50330.51530.48671.50.53060.49970.48790.48730.50520.486720.50050.47750.46940.47280.49560.4867Table 7
The nondimensional fundamental frequency of the porous FGM nanoplates of pattern D.
a/hP0μc/μmk=0k=0.5k=1k=2k=5k=∞50.10.51.12631.04241.00880.99220.99040.895911.04360.99010.96900.96420.97480.89591.50.97680.94490.93360.93850.96000.895920.92130.90540.90180.91470.94580.89590.20.51.12771.04541.01280.99700.99430.897011.04490.99360.97350.96960.97920.89701.50.97800.94870.93850.94430.96480.897020.92240.90940.90710.92090.95100.89700.50.51.12061.04711.01901.00630.99920.891311.03830.99770.98260.98170.98630.89131.50.97180.95470.94980.95880.97380.891320.91660.91680.92010.93750.96190.8913100.10.50.61560.56740.54900.54180.54490.489610.57030.53880.52720.52650.53640.48961.50.53380.51420.50790.51230.52820.489620.50350.49260.49050.49930.52050.48960.20.50.61640.56940.55170.54510.54770.490310.57120.54110.53020.53010.53940.49031.50.53460.51660.51110.51620.53150.490320.50420.49520.49390.50340.52400.49030.50.50.61190.57100.55640.55200.55170.486710.56690.54400.53650.53850.54460.48671.50.53060.52050.51860.52600.53790.486720.50050.49980.50230.51430.53140.4867Table 8
The nondimensional fundamental frequency of the perfect FGM nanoplates.
a/hP0μc/μmk=0k=0.5k=1k=2k=5k=∞500.51.12421.03901.00450.98730.98610.894211.04160.98630.96430.95880.97010.89421.50.97490.94090.92860.93270.95500.894220.91960.90130.89650.90860.94050.89421000.50.61420.56510.54610.53850.54190.488510.56900.53630.52410.52290.53320.48851.50.53260.51160.50460.50850.52490.488520.50240.48990.48710.49530.51690.4885Continuously, the influence of the porosity on the fundamental frequency of the square FGM nanoplate is considered. The nondimensional fundamental frequency of perfect FGM nanoplates is given in Table8. In the cases of a/h=10 and μc/μm=2, Figure 3 presents the variation of the nondimensional fundamental frequency of the FGM nanoplates as functions of maximum porosity value P0. It can be seen that the effects of the porosity on four types of porosity dispersion are different. When P0=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 P0 increases, the fundamental frequency increases rapidly for both k=0.5 and k=2. On the opposite side, when P0 increases, the fundamental frequency of pattern B decreases rapidly for both k=0.5 and k=2. The 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.Figure 3
The effects of porosity on the fundamental frequencies of the porous FGM nanoplates: (a)μc/μm=2 and k=0.5; (b) μc/μm=2 and k=2.
(a)(b)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. Figure4 presents the ratio between the frequencies of the FGM nanoplates with porosity and those of the perfect FGM nanoplates Ωm,nporous/Ωm,nperfect. 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.Figure 4
The effects of porosity on the high frequencies of the porous FGM nanoplates: (a) Mode (5, 5); (b) Mode (10, 10); (c) Mode (20, 20); (d) Mode (100, 100).
(a)(b)(c)(d)The role of the variable nonlocal parameter on the free vibration of the square porous FGM nanoplates witha/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 μc/μm; however, the effects of the nonlocal parameters are different from homogeneous perfect nanoplates. Figures 5(a) and 5(b) demonstrate the effects of the increase in the nonlocal parameters ratio on the nondimensional fundamental frequencies of the homogeneous nanoplates (k=0). It can be seen that the effects of the nonlocal parameters ratio on the fundamental frequencies of the nanoplates depend on the patterns and coefficients of porosity. Besides, the effects of the porosity of patterns A and B are significant than those of patterns C and D of porosity. In the cases of FGM nanoplates with k=2, the effects of the nonlocal parameters ratio μc/μm on the nondimensional fundamental frequencies of the FGM nanoplates are more different from those of the homogeneous ones as shown in Figures 5(c) and 5(d). When the porosity parameters are small P0=0.2, the fundamental frequencies of the FGM nanoplates decrease at similar rate with the increase in the nonlocal parameters ratio μc/μm. When P0=0.5 and k=2, the decrease rates of the FGM porous nanoplates of patterns A and B are similar to the perfect ones. The 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. Therefore, the effects of the nonlocal parameters ratio on the fundamental frequencies of the FGM porous nanoplates depend on power-law index k, porosity parameter P0, and the distribution of porosity.Figure 5
The effects of the variation of the nonlocal parameter on fundamental frequencies of the porous FGM nanoplates: (a)k=0 and P0=0.2; (b) k=0 and P0=0.5; (c) k=2 and P0=0.2; (d) k=2 and P0=0.5.
(a)(b)(c)(d)Figure6 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 P0=0.5. When μc/μm=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 power-law index k increases from 2 to 4, and the fundamental frequencies decrease gently when k > 4. When μc/μm=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. The 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.Figure 6
The influence of the power-law index on the fundamental frequencies of the porous FGM nanoplates: (a)μc/μm=0.5 and P0=0.5; (b) μc/μm=2 and P0=0.5.
(a)(b)The influences of the porosity on the fundamental frequencies of the rectangular porous FGM nanoplates witha/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.Figure 7
The effects of the aspect ratio on the fundamental frequencies of the FGM nanoplates: (a)μc/μm=0.5 and P0=0.5; (b) μc/μm=2 and P0=0.5.
(a)(b)Lastly, the dependence of the fundamental frequencies of the square FGM porous nanoplates withk=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-to-thickness ratio increases. The 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. The fundamental frequencies of the square porous FGM nanoplates are increasing in order of pattern B, C, D, and A.Figure 8
The influence of the side-to-thickness ratio on the fundamental frequencies of the porous FGM nanoplates: (a)μc/μm=0.5 and P0=0.5; (b) μc/μm=2 and P0=0.5.
(a)(b)


## 4.1. Comparison Study

To present the validity of the current algorithm, the free vibration of fully simply supported square porous FGM plates is considered. The plates are made ofAl/Al2O3, the dimensions of the plate are a=b=1, and the thickness is h. The 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]:(27)Ez=Ec−Emzh+12k+Em−Ec+EmP02,ρz=ρc−ρmzh+12k+ρm−ρc+ρmP02.The nondimensional frequencies of the plates are calculated asω¯=ωhρm/Em. The 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 higher-order shear deformation theory (HSDT).Table 2
The comparison of the nondimensional frequencies of the FGM plates with porosity.
b/aP0Methodk=0k=0.1k=0.5k=1a/h=50.50Rezaei et al. [55]0.2719460.2620170.2315730.208997Present0.2706640.2608100.2305910.2081580.2Rezaei et al. [55]0.2804780.2689600.2308750.198452Present0.2791550.2677270.2299330.1977350.4Rezaei et al. [55]0.2929770.2792940.2284530.172779Present0.2915960.2780200.2275850.17230810Rezaei et al. [55]0.4182100.4032210.3570540.322491Present0.4153410.4005150.3548400.3205980.2Rezaei et al. [55]0.4313300.4139560.3562900.306810Present0.4283710.4111900.3541610.3051850.4Rezaei et al. [55]0.4505520.4299360.3530950.268252Present0.4474610.4270790.3511270.267173a/h=100.50Rezaei et al. [55]0.0719090.0692080.0609870.054979Present0.0718120.0691180.0609140.0549160.2Rezaei et al. [55]0.0741650.0710290.0607210.052052Present0.0740650.0709360.0606510.0519990.4Rezaei et al. [55]0.0774700.0737370.0599430.045029Present0.0773660.0736410.0598790.04499510Rezaei et al. [55]0.1136930.1094490.0965100.087024Present0.1134540.1092250.0963290.0868690.2Rezaei et al. [55]0.1172600.1123330.0961190.082444Present0.1170130.1121040.0959450.0823130.4Rezaei et al. [55]0.1224850.1166240.0949360.071421Present0.1222280.1163870.0947770.071336Secondly, 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/Si3N4 with several values of side-to-thickness ratio a/h, and nonlocal parameters μ are considered. The dimensions of the plates are a=b=10nm and the thickness is h. The effective material properties are calculated via a power-law function. The comparison between the nondimensional fundamental frequencies ω^=10ωhρm/Em of the FGM nanoplates of SUS304/Si3N4 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].Table 3
The comparison of the nondimensional frequencies of the FGM plates with porosity.
a/hkMethodμ=02μ=0.52μ=12μ=1.52μ=2250Sobhy and Radwan [51] (quais-3D)5.123775.001844.682434.263703.83015Sobhy and Radwan [51] (HSDT)5.107024.985494.667134.249763.81763Present5.105874.984374.666074.248803.816771Sobhy and Radwan [51] (quais-3D)3.044452.972002.782212.533402.27580Sobhy and Radwan [51] (HSDT)3.018602.946772.758592.511902.25648Present3.018732.946892.758712.512002.256575Sobhy and Radwan [51] (quais-3D)2.441722.383622.231402.031851.82525Sobhy and Radwan [51] (HSDT)2.424432.366742.215602.017471.81232Present2.433562.375652.223952.025071.81915MetalSobhy and Radwan [51] (quais-3D)2.122312.071801.939501.766061.58648Sobhy and Radwan [51] (HSDT)2.112612.062341.930641.757991.57923Present2.112082.061821.930151.757551.57883100Sobhy and Radwan [51] (quais-3D)1.390151.357071.270411.156801.03917Sobhy and Radwan [51] (HSDT)1.388291.355251.268711.155251.03778Present1.388241.355211.268671.155211.037751Sobhy and Radwan [51] (quais-3D)0.827820.808120.756520.688860.61882Sobhy and Radwan [51] (HSDT)0.822500.802920.751650.684430.61484Present0.822960.803380.752080.684820.615185Sobhy and Radwan [51] (quais-3D)0.668900.652990.611290.556620.50002Sobhy and Radwan [51] (HSDT)0.664850.649030.607580.553250.49699Present0.665940.650090.608570.554150.49780MetalSobhy and Radwan [51] (quais-3D)0.578170.564410.528370.481120.43220Sobhy and Radwan [51] (HSDT)0.576950.563220.527250.480100.43128Present0.576930.563200.527240.480090.43127500Sobhy and Radwan [51] (quais-3D)0.057330.055960.052390.047700.04285Sobhy and Radwan [51] (HSDT)0.057300.055930.052360.047680.04283Present0.057300.055930.052360.047680.042831Sobhy and Radwan [51] (quais-3D)0.034170.033350.031220.028430.02554Sobhy and Radwan [51] (HSDT)0.033980.033170.031050.028270.02540Present0.034010.033200.031080.028300.025425Sobhy and Radwan [51] (quais-3D)0.027690.027030.025310.023040.02070Sobhy and Radwan [51] (HSDT)0.027540.026880.025170.022910.02058Present0.027560.026900.025180.022930.02060MetalSobhy and Radwan [51] (quais-3D)0.023880.023310.021820.019870.01785Sobhy and Radwan [51] (HSDT)0.023850.023290.021800.019850.01783Present0.023860.023290.021800.019850.01784


## 4.2. Parameter Study

In the current investigation, a functionally graded nanoplate ofAl/ZrO2 with porosity is considered. The material properties of Al and ZrO2 are given in Table 1. The nonlocal parameter of Al is assumed to be constant and given as μm=2nm2 and plays as the reference nonlocal value. For convenience, the following nondimensional frequencies are used:(28)Ω=10ωhρmEm.Tables4–8 show the nondimensional fundamental frequency of the FGM nanoplates with the dimension of a=10nm2 and the thickness of h=a/10. The 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 μc/μm 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. The 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 P0 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 P0 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.Table 4
The nondimensional fundamental frequency of the porous FGM nanoplates of pattern A.
a/hP0μc/μmk=0k=0.5k=1k=2k=5k=∞50.10.51.13701.05031.01530.99780.99680.904411.05350.99710.97450.96880.98030.90441.50.98600.95120.93830.94210.96450.904420.93010.91120.90580.91750.94950.90440.20.51.15141.06301.02721.00951.00870.915811.06681.00920.98590.97980.99150.91581.50.99850.96280.94910.95250.97510.915820.94180.92220.91620.92740.95950.91580.50.51.20701.11191.07311.05411.05360.960011.11831.05571.02941.02161.03350.96001.51.04671.00730.99060.99201.01460.960020.98730.96500.95590.96480.99660.9600100.10.50.62250.57240.55300.54520.54890.495110.57680.54330.53070.52930.53980.49511.50.53980.51820.51080.51460.53110.495120.50920.49630.49310.50110.52290.49510.20.50.63190.58060.56070.55280.55660.502610.58550.55100.53800.53640.54710.50261.50.54800.52560.51780.52140.53810.502620.51690.50340.49980.50760.52950.50260.50.50.66880.61270.59070.58190.58620.531910.61960.58160.56650.56380.57500.53191.50.58000.55480.54490.54730.56440.531920.54700.53140.52570.53220.55440.5319Table 5
The nondimensional fundamental frequency of the porous FGM nanoplates of pattern B.
a/hP0μc/μmk=0k=0.5k=1k=2k=5k=∞50.10.51.11141.02770.99380.97670.97530.884011.02970.97560.95410.94880.96000.88401.50.96380.93070.91880.92320.94530.884020.90910.89140.88720.89960.93130.88400.20.51.09731.01530.98200.96500.96330.872811.01670.96370.94280.93770.94860.87281.50.95150.91940.90810.91270.93450.872820.89750.88060.87690.88950.92110.87280.50.51.04420.96860.93710.92030.91720.830610.96750.91930.90020.89530.90480.83061.50.90560.87690.86730.87230.89280.830620.85420.83990.83770.85100.88130.8306100.10.50.60590.55790.53930.53180.53500.481910.56140.52950.51770.51650.52660.48191.50.52540.50500.49840.50250.51860.481920.49560.48370.48120.48950.51090.48190.20.50.59690.55000.53180.52430.52730.474710.55300.52200.51050.50940.51930.47471.50.51760.49790.49160.49570.51160.474720.48820.47680.47460.48310.50430.47470.50.50.56340.52070.50380.49630.49810.448110.52200.49420.48380.48280.49140.44811.50.48860.47130.46610.47030.48490.448120.46080.45140.45020.45880.47870.4481Table 6
The nondimensional fundamental frequency of the porous FGM nanoplates of pattern C.
a/hP0μc/μmk=0k=0.5k=1k=2k=5k=∞50.10.51.12631.03941.00390.98590.98540.895911.04360.98620.96310.95670.96880.89591.50.97680.94040.92690.93000.95310.895920.92130.90050.89440.90550.93810.89590.20.51.12771.03881.00220.98320.98340.897011.04490.98500.96070.95340.96620.89701.50.97800.93890.92400.92610.94990.897020.92240.89860.89110.90100.93430.89700.50.51.12061.02490.98450.96160.96350.891311.03830.96960.94080.92930.94400.89131.50.97180.92240.90250.90000.92560.891320.91660.88150.86850.87330.90820.8913100.10.50.61560.56520.54560.53740.54130.489610.57030.53620.52330.52140.53220.48961.50.53380.51120.50350.50680.52350.489620.50350.48940.48580.49330.51530.48960.20.50.61640.56480.54440.53550.53980.490310.57120.53540.52170.51910.53030.49031.50.53460.51020.50160.50420.52130.490320.50420.48830.48370.49040.51280.49030.50.50.61190.55560.53260.52110.52610.486710.56690.52540.50880.50330.51530.48671.50.53060.49970.48790.48730.50520.486720.50050.47750.46940.47280.49560.4867Table 7
The nondimensional fundamental frequency of the porous FGM nanoplates of pattern D.
a/hP0μc/μmk=0k=0.5k=1k=2k=5k=∞50.10.51.12631.04241.00880.99220.99040.895911.04360.99010.96900.96420.97480.89591.50.97680.94490.93360.93850.96000.895920.92130.90540.90180.91470.94580.89590.20.51.12771.04541.01280.99700.99430.897011.04490.99360.97350.96960.97920.89701.50.97800.94870.93850.94430.96480.897020.92240.90940.90710.92090.95100.89700.50.51.12061.04711.01901.00630.99920.891311.03830.99770.98260.98170.98630.89131.50.97180.95470.94980.95880.97380.891320.91660.91680.92010.93750.96190.8913100.10.50.61560.56740.54900.54180.54490.489610.57030.53880.52720.52650.53640.48961.50.53380.51420.50790.51230.52820.489620.50350.49260.49050.49930.52050.48960.20.50.61640.56940.55170.54510.54770.490310.57120.54110.53020.53010.53940.49031.50.53460.51660.51110.51620.53150.490320.50420.49520.49390.50340.52400.49030.50.50.61190.57100.55640.55200.55170.486710.56690.54400.53650.53850.54460.48671.50.53060.52050.51860.52600.53790.486720.50050.49980.50230.51430.53140.4867Table 8
The nondimensional fundamental frequency of the perfect FGM nanoplates.
a/hP0μc/μmk=0k=0.5k=1k=2k=5k=∞500.51.12421.03901.00450.98730.98610.894211.04160.98630.96430.95880.97010.89421.50.97490.94090.92860.93270.95500.894220.91960.90130.89650.90860.94050.89421000.50.61420.56510.54610.53850.54190.488510.56900.53630.52410.52290.53320.48851.50.53260.51160.50460.50850.52490.488520.50240.48990.48710.49530.51690.4885Continuously, the influence of the porosity on the fundamental frequency of the square FGM nanoplate is considered. The nondimensional fundamental frequency of perfect FGM nanoplates is given in Table8. In the cases of a/h=10 and μc/μm=2, Figure 3 presents the variation of the nondimensional fundamental frequency of the FGM nanoplates as functions of maximum porosity value P0. It can be seen that the effects of the porosity on four types of porosity dispersion are different. When P0=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 P0 increases, the fundamental frequency increases rapidly for both k=0.5 and k=2. On the opposite side, when P0 increases, the fundamental frequency of pattern B decreases rapidly for both k=0.5 and k=2. The 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.Figure 3
The effects of porosity on the fundamental frequencies of the porous FGM nanoplates: (a)μc/μm=2 and k=0.5; (b) μc/μm=2 and k=2.
(a)(b)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. Figure4 presents the ratio between the frequencies of the FGM nanoplates with porosity and those of the perfect FGM nanoplates Ωm,nporous/Ωm,nperfect. 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.Figure 4
The effects of porosity on the high frequencies of the porous FGM nanoplates: (a) Mode (5, 5); (b) Mode (10, 10); (c) Mode (20, 20); (d) Mode (100, 100).
(a)(b)(c)(d)The role of the variable nonlocal parameter on the free vibration of the square porous FGM nanoplates witha/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 μc/μm; however, the effects of the nonlocal parameters are different from homogeneous perfect nanoplates. Figures 5(a) and 5(b) demonstrate the effects of the increase in the nonlocal parameters ratio on the nondimensional fundamental frequencies of the homogeneous nanoplates (k=0). It can be seen that the effects of the nonlocal parameters ratio on the fundamental frequencies of the nanoplates depend on the patterns and coefficients of porosity. Besides, the effects of the porosity of patterns A and B are significant than those of patterns C and D of porosity. In the cases of FGM nanoplates with k=2, the effects of the nonlocal parameters ratio μc/μm on the nondimensional fundamental frequencies of the FGM nanoplates are more different from those of the homogeneous ones as shown in Figures 5(c) and 5(d). When the porosity parameters are small P0=0.2, the fundamental frequencies of the FGM nanoplates decrease at similar rate with the increase in the nonlocal parameters ratio μc/μm. When P0=0.5 and k=2, the decrease rates of the FGM porous nanoplates of patterns A and B are similar to the perfect ones. The 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. Therefore, the effects of the nonlocal parameters ratio on the fundamental frequencies of the FGM porous nanoplates depend on power-law index k, porosity parameter P0, and the distribution of porosity.Figure 5
The effects of the variation of the nonlocal parameter on fundamental frequencies of the porous FGM nanoplates: (a)k=0 and P0=0.2; (b) k=0 and P0=0.5; (c) k=2 and P0=0.2; (d) k=2 and P0=0.5.
(a)(b)(c)(d)Figure6 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 P0=0.5. When μc/μm=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 power-law index k increases from 2 to 4, and the fundamental frequencies decrease gently when k > 4. When μc/μm=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. The 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.Figure 6
The influence of the power-law index on the fundamental frequencies of the porous FGM nanoplates: (a)μc/μm=0.5 and P0=0.5; (b) μc/μm=2 and P0=0.5.
(a)(b)The influences of the porosity on the fundamental frequencies of the rectangular porous FGM nanoplates witha/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.Figure 7
The effects of the aspect ratio on the fundamental frequencies of the FGM nanoplates: (a)μc/μm=0.5 and P0=0.5; (b) μc/μm=2 and P0=0.5.
(a)(b)Lastly, the dependence of the fundamental frequencies of the square FGM porous nanoplates withk=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-to-thickness ratio increases. The 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. The fundamental frequencies of the square porous FGM nanoplates are increasing in order of pattern B, C, D, and A.Figure 8
The influence of the side-to-thickness ratio on the fundamental frequencies of the porous FGM nanoplates: (a)μc/μm=0.5 and P0=0.5; (b) μc/μm=2 and P0=0.5.
(a)(b)


## 5. 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. The 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)
The variation of the nonlocal parameter should be considered due to its important role in the free vibration of FGM nanoplates(ii)
The inclusion of the nonlocal parameter leads to the decrease in the frequency of the nonlocal porous FGM plates(iii)
The porosity plays significant effects on the free vibration characteristics of the porous FGM nanoplates(iv)
The influence of the porosity on the high frequencies is different from that of the low frequencies of the FGM nanoplatesThe 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.



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*Source: 1219429-2021-07-15.xml*