Free vibration characteristics of functionally graded Mindlin nanoplates resting on variable elastic foundations using the nonlocal elasticity theory

Abstract

In this research, free vibration behavior of thick functionally graded nanoplates is carried out using the Chebyshev spectral collocation method. It is assumed that the plates are resting on variable elastic foundations. Eringen’s nonlocal elasticity theory is used to capture the size effect, and Mindlin’s first-order shear deformation plate theory is employed to model the thick nanoplates. Hamilton’s principle along with the differential form of Eringen’s constitutive relations are utilized to obtain the governing partial differential equations of motion for the functionally graded nanoplates under consideration. A numerical solution is presented by applying the spectral collocation method and the natural frequencies are obtained. A parametric study is conducted to study the effects of several factors on the natural frequencies of the functionally graded nanoplates. It is found that the parameters of the variable elastic foundation (Winkler and shear moduli), thickness to length ratio, length to width ratio (aspect ratio), the nonlocal scale coefficient, the gradient index, the foundation type, and the boundary conditions have a remarkable influence on the free vibration characteristics of the functionally graded nanoplates.

Introduction

Micro and nano structures are popular in many advanced technological fields such as aerospace engineering, composites, and electronics for their novel properties comparing with other structures and mechanisms at the standard order of dimensions. These features allow them to be part of smart and reliable devices in various machines and mechanisms.

In order to enhance the specific strength (strength divided by density) of the structures used in micro/nano electromechanical systems (MEMS/NEMS), orthotropic, anisotropic, composite, and functionally graded (FG) materials are commonly used. To insure the smoothness variation of the material properties along the thickness of the structure, FG materials (which are manufactured from a combination of ceramics and metals) can be utilized, where the properties of these FG materials can be tailored for specific purposes. On the other hand, composite materials suffer from pre-mature failure as a result of the sudden change in the properties between the material’s plies. In addition, these materials may experience a decay in the elastic characteristics due to an unstable medium and sticky layers. ¹ Similarly, the properties of micro and nano materials such as carbon nanotubes and graphene sheets cannot be considered homogeneous due to the influence of lattice distance or grain size on their mechanical properties.

The impact of the size effect on the characteristics of the micro/nano structures can be predicted by experimental and atomistic simulations and models. Unfortunately, these techniques may be restricted by the cost and computational issues. As the theories of modeling different structures and systems are scale free, they are incapable of capturing the effect of the small-scale order on the performance of micro/nano structures. Hence, these theories and relations are insufficient in investigating the characteristics of these systems. ² Therefore, to apply the continuum mechanics theories in the analyses of different micro and nano structures, the scale effect should be taken into account. Various theories have been suggested to study the performance of micro/nano structures such as Cosserat elastic theory, Cosserat theory with constrained rotations, multipolar elastic theory, and the nonlocal elasticity theory. ³ The nonlocal elasticity theory will be employed in this research to perform the free vibrations of FG Mindlin plates embedded in variable elastic medium.

Eringen’s nonlocal elasticity theory has been extensively utilized to model and analyze the micro/nano structures. Aghababaei and Reddy ⁴ applied this theory, along with a modified version of a simple higher order theory of laminated plates, to study the deflection and the natural frequencies of nanoplates. Shen et al. ⁵ carried out the nonlinear vibrations of single layer graphene sheet in thermal medium. The sheet was modeled as a thin orthotropic nanoplate with von Kármán nonlinearity. The numerical results revealed that the nonlocal elasticity theory can successfully predict the natural frequencies of the graphene sheet. Moreover, Ansari et al. ⁶ applied Eringen’s theory to examine the free vibrations of single-layered graphene sheets. The first-order shear deformation theory was used to model these sheets, and the frequencies were computed via the generalized differential quadrature method.

In another study, Pouresmaeeli et al. ⁷ obtained closed form solutions for the natural frequencies of double-orthotropic nanoplates bonded by a set of identical springs and resting on Winkler-type layers. Eringen’s nonlocal theory was adopted to obtain the mathematical model, the impacts of the nonlocal index, the stiffness of the connecting springs and the Winkler foundation, and the length-to-width ratio on the free vibrations of the double nanoplates. Khorshidi and Fallah ⁸ implemented Eringen’s theory to investigate the buckling analysis of FG nanoplates. It was shown that the scale coefficient, power law indexes, and aspect and thickness ratios all have a remarkable influence on the buckling behavior of the FG rectangular nanoplates.

Shahidi et al. ⁹ examined the vibration analysis of orthotropic nonuniform Kirchhoff nanoplates. The effect of the thickness, nonlocal scale coefficient, boundary conditions, and the geometries on the frequencies was investigated. Namin and Pilafkan ¹⁰ investigated the vibration behavior of imperfect graphene sheets via Eringen’s theory. Mindlin’s plate theory was used to derive the equations of motion which were solved using the differential quadrature method. Zhang et al. ¹¹ applied the nonlocal elasticity theory to study the vibrational behavior of quadrilateral graphene sheets in a magnetic field. The governing equations of motion were solved by employing the kp-Ritz method. The influences of the skew angles, nonlocal scale coefficient, the magnetic field, and the type of boundary conditions on the fundamental natural frequencies of the single-layered graphene sheets were discussed. Furthermore, Li and Hu ¹² analyzed the torsional vibrations of tubes made of bidirectional FG materials. It was assumed that the material properties vary along the length and the radius directions of the nanotube. It was concluded that the torsional frequencies are increased by decreasing nonlocal parameter. However, it was observed that this parameter does not have an effect on the mode shapes of the nanotubes.

The mechanical response and characteristics of nonuniform bidirectional FG beam sensors were investigated by Khaniki and Rajasekaran. ¹³ The modified couple stress theory along with Hamilton’s principle were used to formulate the model of the microbeams with linear, exponential, and parabolic cross-sectional areas. A mixed finite element method, Gaussian quadrature method, and Wilson’s Lagrangian multiplier were used to solve the governing equations. Moreover, Hosseini-Hashemi and Khaniki ¹⁴ applied Eringen’s nonlocal elasticity theory to carry out the effect of a moving nanoparticle/load on the mechanical behavior of thick FG nanoplates lying on a Kelvin–Voigt viscoelastic layer. The Galerkin approach and the Runge–Kutta method were employed to solve the governing equations of motion. Rajasekaran and Khaniki ¹⁵ analyzed the static deformation, stability, and free vibration characteristics of nonhomogeneous nonuniform nanobeams in the framework of nonlocal strain gradient theory. The beam was made of axially FG materials and was assumed to have nonuniform cross-section. Finite element and Gaussian quadrature methods, in conjunction with Wilson’s Lagrangian multiplier, were applied to solve different problems.

Recently, Ebrahim and Barati ¹⁶ applied a Galerkin-based solution and the nonlocal strain gradient theory to investigate the vibration behavior of axially FG nanobeams resting on variable elastic foundation. It was assumed that the material properties vary axially following a power law distribution. The elastic foundation consists of a Winkler layer and a Pasternak layer with constant stiffness. Linear, parabolic, and sinusoidal variations of Winkler foundation were considered. It was shown that the layers have an increasing influence on the frequencies. Khaniki et al. ¹⁷ used Eringen’s two-phase integral model to study the vibration analysis of nanobeams resting on axially varying elastic foundation. A power law relation was considered to present the stiffness variation of the elastic foundation. The generalized differential quadrature was utilized to obtain the first four frequency terms of the nanobeams with different boundary conditions. Other scientists and researchers used Eringen’s nonlocal theory to study the behavior of different dynamical systems at the micro/nano scale.^6,18–33

Eringen’s theory has integral and differential forms. For some boundary and loading conditions (especially for cantilever beams), inconsistent and illogical findings may be resulted by the differential model.^34–37 However, for MEMS modeled as cantilever beams, using an integral-based nonlocal elastic model, a solution identical to the classical local stress beam model without any small-scale effect is obtained. ³⁸ On the other hand, Eringen ³ showed that the integral constitutive equation can be converted exactly into a corresponding differential form for some kernels. As Eringen’s two-phase local/nonlocal theory is well-posed and has the ability and flexibility to model various structures with several boundary conditions, it has been utilized to model the dynamics of structures at the nanoscale in the recent years.^39–41

In many applications, micro/nano structures may be resting on variable elastic foundations. Introducing a variable elastic foundation model must be considered in order to model any imperfect geometry and material changes in foundation. This situation may arise in the case of buried nano structures. Moreover, electrically actuated micro structures, which are used to model resonant sensors, are examples of microplates resting on variable elastic foundation.

To date, no reports in the literature of the vibration behavior of thick FG nanoplates resting on variable elastic foundations are available. Motivated by these considerations and for the sake of improving the design of MEMS/NEMS, the objective of this research is to examine the vibrational behavior of thick FG nanoplates embedded in variable elastic foundations. The differential form of Eringen’s nonlocal theory will be utilized.

A nonlocal Mindlin plate model for FG nanoplates resting on variable elastic foundation

At the macro scale, the stress tensor at a given point in an elastic field is a function of the strains at the same point. However, according to Eringen, the stress field at a point in an elastic domain depends on the stress tensors at all other points. ³

In a nonlocal elastic field, the stress tensor t_ij is expressed as

$$t_{\mathrm{ij}}=\underset{V}{\int }\alpha \left(|x\prime -x|\right){\sigma }_{\mathrm{ij}}(x\prime )\mathrm{dV}(x\prime )$$

(1)

where x is a reference point in an elastic continuum, $\alpha (|x\prime -x|)$ is the nonlocal modulus, and $|x\prime -x|$ is the Euclidean distance.

Alternatively, Eringen proposed a differential operator form of equation (1) that reflects the material dispersions. This linear differential operator ς is defined by $\varsigma =1-(e_{0}l{)}^2{\nabla }^2$ , where $e_0$ is a material constant property that could be estimated either by experiments or simulations, l denotes the internal characteristic length, and ${\nabla }^2$ denotes the Laplace operator in the Cartesian coordinate system, and is given by

$${\nabla }^2=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}$$

(2)

Thus, equation (1) is reformulated as

$$(1-{\left(e_{0}l\right)}^2{\nabla }^2)t_{\mathrm{ij}}={\sigma }_{\mathrm{ij}}$$

(3)

Due to its convenience and simplicity, the differential form given in equation (3) has been widely used by many investigators in employing the nonlocal theory to carry out the vibration characteristics of micro/nano structures.

According to the first-order shear deformation theory, the displacement components of the middle surface of the plate in the x, y, and z directions are given as

$$\begin{array}{c}{u}_x=z{\psi }_x(x,y,t)\hfill \\ u_y=z{\psi }_y(x,y,t)\hfill \\ u_z=\overline{w}(x,y,t)\hfill \end{array}$$

(4)

where $\overline{w}$ is the displacement along the z-axis (out-plane displacement); ${\psi }_x$ and ${\psi }_y$ are the angular (rotational) displacements in the xz and yz planes due to bending; and t is the time.

The normal strains ${\epsilon }_{\mathrm{xx}}$ and ${ϵ}_{\mathrm{yy}}$ , and the shear strains ${\gamma }_{\mathrm{xy}}$ , ${\gamma }_{\mathrm{xz}}$ , and ${\gamma }_{\mathrm{yz}}$ , are given by

$$\begin{array}{c}{\epsilon }_{\mathrm{xx}}=z{\psi }_{x,x},{\epsilon }_{\mathrm{yy}}=z{\psi }_{y,y},{\epsilon }_{\mathrm{zz}}=0\hfill \\ {\gamma }_{\mathrm{xy}}=z\left(\frac{{\psi }_{x,y}+{\psi }_{y,x}}{2}\right),\hfill \\ {\gamma }_{\mathrm{xz}}=z\left(\frac{{\psi }_x+{\overline{w}}_{,x}}{2}\right),{\gamma }_{\mathrm{yz}}=z\left(\frac{{\psi }_y+{\overline{w}}_{,y}}{2}\right)\hfill \end{array}$$

(5)

Based on Hooke’s law, and in light of equation (3), the constitutive relations are expressed as

$${\sigma }_{\mathrm{xx}}-{\left(e_{0}l\right)}^2(\frac{{\partial }^2{\sigma }_{\mathrm{xx}}}{\partial x^2}+\frac{{\partial }^2{\sigma }_{\mathrm{xx}}}{\partial y^2})=\frac{E\left(z\right)}{1-{\nu }^2}({ϵ}_{\mathrm{xx}}+\nu {ϵ}_{\mathrm{yy}})$$

(6a)

$${\sigma }_{\mathrm{yy}}-{\left(e_{0}l\right)}^2(\frac{{\partial }^2{\sigma }_{\mathrm{yy}}}{\partial x^2}+\frac{{\partial }^2{\sigma }_{\mathrm{yy}}}{\partial y^2})=\frac{E\left(z\right)}{1-{\nu }^2}({ϵ}_{\mathrm{yy}}+\nu {ϵ}_{\mathrm{xx}})$$

(6b)

$${\tau }_{\mathrm{xy}}-{\left(e_{0}l\right)}^2(\frac{{\partial }^2{\tau }_{\mathrm{xy}}}{\partial x^2}+\frac{{\partial }^2{\tau }_{\mathrm{xy}}}{\partial y^2})=G\left(z\right){\gamma }_{\mathrm{xy}}$$

(6c)

$${\tau }_{\mathrm{xz}}-{\left(e_{0}l\right)}^2(\frac{{\partial }^2{\tau }_{\mathrm{xz}}}{\partial x^2}+\frac{{\partial }^2{\tau }_{\mathrm{xz}}}{\partial y^2})=G\left(z\right){\gamma }_{\mathrm{xz}}$$

(6d)

$${\tau }_{\mathrm{yz}}-{\left(e_{0}l\right)}^2(\frac{{\partial }^2{\tau }_{\mathrm{yz}}}{\partial x^2}+\frac{{\partial }^2{\tau }_{\mathrm{yz}}}{\partial y^2})=G\left(z\right){\gamma }_{\mathrm{yz}}$$

(6e)

where E(z) is the modulus of elasticity, $G(z)=E(z)/[2(1+\nu )]$ is the shear modulus, v is Poisson’s ratio, ${\sigma }_{\mathrm{xx}}$ and ${\sigma }_{\mathrm{yy}}$ are the normal stresses, and ${\tau }_{\mathrm{xy}}$ , ${\tau }_{\mathrm{xz}}$ , and ${\tau }_{\mathrm{yz}}$ are the shear stresses. The resultant moments and shear forces are expressed as

$$M_{\mathrm{ii}}=\underset{-h/2}{\overset{h/2}{\int }}{\sigma }_{\mathrm{ii}}\mathrm{zdz},i=x,y$$

(7a)

$$M_{\mathrm{xy}}=\underset{-h/2}{\overset{h/2}{\int }}{\tau }_{\mathrm{xy}}\mathrm{zdz}$$

(7b)

$$Q_j={\kappa }^2\underset{-h/2}{\overset{h/2}{\int }}{\tau }_{\mathrm{jz}}\mathrm{dz},j=x,y$$

(7c)

where κ² is the shear correction factor and h is the thickness of the plate. Substituting equations (7a–7c) into equations (6a–6e) yields the nonlocal moments and shear forces as

$$M_{\mathrm{xx}}-{\left(e_{0}l\right)}^2(\frac{{\partial }^2{M}_{\mathrm{xx}}}{\partial x^2}+\frac{{\partial }^2{M}_{\mathrm{xx}}}{\partial y^2})=A(\frac{\partial {\psi }_x}{\partial x}+v\frac{\partial {\psi }_y}{\partial y})$$

(8a)

$$M_{\mathrm{yy}}-{\left(e_{0}l\right)}^2(\frac{{\partial }^2{M}_{\mathrm{yy}}}{\partial x^2}+\frac{{\partial }^2{M}_{\mathrm{yy}}}{\partial y^2})=A(v\frac{\partial {\psi }_x}{\partial x}+\frac{\partial {\psi }_y}{\partial y})$$

(8b)

$$M_{\mathrm{xy}}-{\left(e_{o}l\right)}^2(\frac{{\partial }^2{M}_{\mathrm{xy}}}{\partial x^2}+\frac{{\partial }^2{M}_{\mathrm{xy}}}{\partial y^2})=\frac{(1+\upsilon )}{2}A(\frac{\partial {\psi }_y}{\partial x}+\frac{\partial {\psi }_x}{\partial y})$$

(8c)

$$Q_x-{\left(e_{0}l\right)}^2(\frac{{\partial }^2{Q}_x}{\partial x^2}+\frac{{\partial }^2{Q}_x}{\partial y^2})={\kappa }^2\mathrm{Bh}({\psi }_x+\frac{\partial \overline{w}}{\partial x})$$

(8d)

$$Q_y-{\left(e_{0}l\right)}^2(\frac{{\partial }^2{Q}_y}{\partial x^2}+\frac{{\partial }^2{Q}_y}{\partial y^2})={\kappa }^2\mathrm{Bh}({\psi }_y+\frac{\partial \overline{w}}{\partial y})$$

(8e)

Following the work of Hosseini-Hashemi et al., ⁴² the modulus of elasticity (Young’s modulus) and the density are suggested to be changed smoothly across the thickness of the plate as

$$E\left(z\right)=(E_c-E_m){(\frac{z}{h}+\frac{1}{2})}^{\alpha }+E_m$$

(9a)

$$\rho \left(z\right)=({\rho }_c-{\rho }_m){(\frac{z}{h}+\frac{1}{2})}^{\alpha }+{\rho }_m$$

(9b)

where the subscripts m and c represent the metallic- and ceramic-rich surfaces. The FG plate is formed from two different materials. The top surface (z = h/2) of the plate is ceramic-rich, whereas the bottom surface (z =−h/2) is metal-rich, and $\alpha$ is the volume fraction exponent (gradient index) and is in the range of (0, $\infty$ ). In equations (8a–8e), A and B are given by

$$A=\frac{1}{(1-{\nu }^2)}\left[\frac{\alpha E_m(8+3\alpha +{\alpha }^2)+3E_c(2+\alpha +{\alpha }^2)}{12(1+\alpha )(2+\alpha )(3+\alpha )}\right]$$

(10a)

$$B=\frac{1}{2(1+\nu )}\frac{E_c+\alpha E_m}{1+\alpha }$$

(10b)

The equations of motion of the free vibrations for a rectangular FG nonlocal plate embedded in a variable elastic foundation, shown in Figure 1, are derived using Mindlin’s plate theory and Hamilton’s principle. These equations are given as

$$\frac{\partial M_{\mathrm{xx}}}{\partial x}+\frac{\partial M_{\mathrm{xy}}}{\partial y}-Q_x=\frac{1}{12}C{h}^3\frac{{\partial }^2{\psi }_x}{\partial t^2}$$

(11a)

$$\frac{\partial M_{\mathrm{xy}}}{\partial x}+\frac{\partial M_{\mathrm{yy}}}{\partial y}-Q_y=\frac{1}{12}C{h}^3\frac{{\partial }^2{\psi }_y}{\partial t^2}$$

(11b)

$$\frac{\partial Q_x}{\partial x}+\frac{\partial Q_y}{\partial y}-K_{w}G\left(x\right)\overline{w}+K_s\frac{{\partial }^2\overline{w}}{\partial x^2}+K_s\frac{{\partial }^2\overline{w}}{\partial y^2}=\mathrm{Dh}\frac{{\partial }^2\overline{w}}{\partial t^2}$$

(11c)

where

$$C=\frac{\alpha {\rho }_m(8+3\alpha +{\alpha }^2)+3{\rho }_c(2+\alpha +{\alpha }^2)}{(1+\alpha )(2+\alpha )(3+\alpha )}$$

(12a)

$$D=\frac{{\rho }_c+\alpha {\rho }_m}{1+\alpha }$$

(12b)

Figure 1.

A nonlocal FG nanoplate resting on a variable elastic foundation.

Substituting the constitutive relations of Eringen (equations (8a–8e)) into equations (11a–11c) yields

$$\begin{array}{c}A(\frac{{\partial }^2{\psi }_x}{\partial x^2}+\frac{(1-\nu )}{2}\frac{{\partial }^2{\psi }_x}{\partial y^2}+\frac{(1+\nu )}{2}\frac{{\partial }^2{\psi }_y}{\partial x\partial y})\hfill \\ -{\kappa }^2\mathrm{Bh}({\psi }_x+\frac{\partial w}{\partial x})\hfill \\ =(1-{\left(e_{0}l\right)}^2(\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}))\frac{1}{12}C{h}^3\frac{{\partial }^2{\psi }_x}{\partial t^2}\hfill \end{array}$$

(13a)

$$\begin{array}{c}A(\frac{{\partial }^2{\psi }_y}{\partial y^2}+\frac{(1-\nu )}{2}\frac{{\partial }^2{\psi }_y}{\partial x^2}+\frac{(1+\nu )}{2}\frac{{\partial }^2{\psi }_x}{\partial x\partial y})\hfill \\ -{\kappa }^2\mathrm{Bh}({\psi }_y+\frac{\partial w}{\partial y})\hfill \\ =(1-{\left(e_{0}l\right)}^2(\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}))\frac{1}{12}C{h}^3\frac{{\partial }^2{\psi }_y}{\partial t^2}\hfill \end{array}$$

(13b)

$$\begin{array}{c}{\kappa }^2\mathrm{Bh}(\frac{\partial {\psi }_x}{\partial x}+\frac{\partial {\psi }_y}{\partial y}+\frac{{\partial }^{2}w}{\partial x^2}+\frac{{\partial }^{2}w}{\partial y^2})\hfill \\ -(1-{\left(e_{0}l\right)}^2(\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}))K_{w}G\left(x\right)w+K_s\hfill \\ (1-{\left(e_{0}l\right)}^2(\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}))(\frac{{\partial }^{2}w}{\partial x^2}+\frac{{\partial }^{2}w}{\partial y^2})\hfill \\ =(1-{\left(e_{0}l\right)}^2(\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}))\mathrm{Dh}\frac{{\partial }^{2}w}{\partial t^2}\hfill \end{array}$$

(13c)

where $K_s$ and $K_w$ are the coefficients of the shear and Winkler foundations, respectively.

In the present study, the variable Winkler elastic foundation is a function of the x coordinate only, and its variation with x may be either linear, parabolic, or sinusoidal (as Pradhan and Murmu ⁴³ and Sobhy ⁴⁴ )

$$G\left(x\right)=\{\begin{array}{c}(1+\zeta \frac{x}{a}),\mathrm{Linear}\hfill \\ \multicolumn{1}{c}{(1+\zeta {\left(\frac{x}{a}\right)}^2),\mathrm{Parabolic}}\\ \multicolumn{1}{c}{(1+\zeta \sin \left(\frac{\pi x}{a}\right)),\mathrm{Sinusoidal}}\end{array}$$

(14)

where $\zeta$ is a varied parameter.

For convenience, the following dimensionless quantities are defined as ⁴²

$$\begin{array}{c}X=\frac{x}{a},Y=\frac{y}{b},{\overline{\Psi }}_X={\psi }_x,{\overline{\Psi }}_Y={\psi }_y,\hfill \\ \overline{W}=\frac{\overline{w}}{h},\delta =\frac{h}{a},\gamma =\frac{h}{b},\eta =\frac{a}{b},\hfill \\ {\overline{K}}_s=\frac{K_s{a}^2}{A{h}^3},{\overline{K}}_w=\frac{K_w{a}^4}{A{h}^3},\mu =\frac{e_{0}l}{a},\overline{\beta }=\omega a^2\sqrt{\frac{C}{A{h}^2}}\hfill \end{array}$$

(15)

Assuming sinusoidal variation in time as

$$({\overline{\Psi }}_X,{\overline{\Psi }}_Y,\overline{W})(X,Y)=({\Psi }_X,{\Psi }_Y,W)(X,Y)e^{i\omega t}$$

(16)

Substituting equations (15) and (16) into equations (13a) to (13c) yields the following dimensionless equations

$$\begin{array}{c}\frac{{\partial }^2{\Psi }_X}{\partial X^2}+\left(\frac{1-\nu }{2}\right){\eta }^2\frac{{\partial }^2{\Psi }_X}{\partial Y^2}\hfill \\ +\left(\frac{1+\nu }{2}\right)\eta \frac{{\partial }^2{\Psi }_Y}{\partial X\partial Y}-\frac{F}{{\delta }^2}({\Psi }_X+\delta \frac{\partial W}{\partial X})\hfill \\ =\frac{{\delta }^2{\overline{\beta }}^2}{12}(-{\Psi }_X+{\mu }^2\frac{{\partial }^2{\Psi }_X}{\partial X^2}+{\mu }^2{\eta }^2\frac{{\partial }^2{\Psi }_X}{\partial Y^2})\hfill \end{array}$$

(17a)

$$\begin{array}{c}{\eta }^2\frac{{\partial }^2{\Psi }_Y}{\partial Y^2}+\left(\frac{1-\nu }{2}\right)\frac{{\partial }^2{\Psi }_Y}{\partial X^2}\hfill \\ +\left(\frac{1+\nu }{2}\right)\eta \frac{{\partial }^2{\Psi }_X}{\partial X\partial Y}-\frac{F}{{\delta }^2}({\Psi }_Y+\gamma \frac{\partial W}{\partial Y})\hfill \\ =\frac{{\delta }^2{\overline{\beta }}^2}{12}(-{\Psi }_Y+{\mu }^2\frac{{\partial }^2{\Psi }_Y}{\partial X^2}+{\mu }^2{\eta }^2\frac{{\partial }^2{\Psi }_Y}{\partial Y^2})\hfill \end{array}$$

(17b)

$$\begin{array}{c}\delta \frac{\partial {\Psi }_X}{\partial X}+\gamma \frac{\partial {\Psi }_Y}{\partial Y}+{\delta }^2\frac{{\partial }^{2}W}{\partial X^2}+{\gamma }^2\frac{{\partial }^{2}W}{\partial Y^2}\hfill \\ -\frac{{\overline{K}}_W{\delta }^4}{F}G\left(X\right)W+\frac{{\overline{K}}_W{\delta }^4{\mu }^2}{F}\frac{{\partial }^{2}G\left(X\right)}{\partial X^2}W\hfill \\ +\frac{2{\overline{K}}_W{\delta }^4{\mu }^2}{F}\frac{\partial G\left(X\right)}{\partial X}\frac{\partial W}{\partial X}+\frac{{\overline{K}}_W{\delta }^4{\mu }^2}{F}G\left(X\right)\frac{{\partial }^{2}W}{\partial X^2}\hfill \\ +\frac{{\overline{K}}_W{\delta }^4{\eta }^2{\mu }^2}{F}G\left(X\right)\frac{{\partial }^{2}W}{\partial Y^2}\hfill \\ +\frac{{\overline{K}}_S{\delta }^4}{F}\frac{{\partial }^{2}W}{\partial X^2}+\frac{{\overline{K}}_S{\delta }^2{\gamma }^2}{F}\frac{{\partial }^{2}W}{\partial Y^2}\hfill \\ -\frac{{\overline{K}}_S{\delta }^4{\mu }^2}{F}\frac{{\partial }^{4}W}{\partial X^4}-\frac{2{\overline{K}}_S{\delta }^2{\gamma }^2{\mu }^2}{F}\frac{{\partial }^{4}W}{\partial X^2\partial Y^2}\hfill \\ -\frac{{\overline{K}}_S{\delta }^2{\gamma }^2{\eta }^2{\mu }^2}{F}\frac{{\partial }^{4}W}{\partial Y^4}\hfill \\ =\frac{D{\delta }^4{\overline{\beta }}^2}{\mathrm{FC}}(-W+{\mu }^2\frac{{\partial }^{2}W}{\partial X^2}+{\mu }^2{\eta }^2\frac{{\partial }^{2}W}{\partial Y^2})\hfill \end{array}$$

(17c)

where

$$F=\frac{{\kappa }^{2}B}{A}$$

The shear correction factor is given as

$$\begin{array}{c}\multicolumn{1}{c}{{\kappa }^2(\alpha ,\delta )=\frac{5}{6}+C_1(e^{-C_2\alpha }-e^{-C_3\alpha })(10\delta -2)}\\ -C_4(e^{-C_5\alpha }-e^{-C_6\alpha })(10\delta -1)\hfill \end{array}$$

(18)

where the values of the constants C_i $(i=1,2,\dots ,6)$ depend on the FG material and are given in Hosseini-Hashemi et al. ⁴² In the present study, the values of the properties for the metal and the ceramics used in the nonlocal FG nanoplate are given in Table 1. Herein, Poisson’s ratio is assumed to be constant through the plate and is taken as 0.3.

Table 1.

Material properties used in the FG plate. ⁴²

Properties	Metal	Ceramic
	Aluminum (Al)	Zirconia (ZrO2)	Alumina (Al2O3)
E (GPa)	70	200	380
ρ (kg/m3)	2702	5700	3800

FG: functionally graded.

Solution procedure: Chebyshev collocation method

As the governing partial differential equations of motion of the FG nanoplate under consideration are coupled and have several varying coefficients, it is clear that finding exact solutions for the displacements is challenging. Hence, these equations are discretized by applying the Chebyshev collocation method (CCM). The details of this method including the Chebyshev points and the Chebyshev differentiation matrix are given in Trefethen ⁴⁵ and are omitted in the present article for brevity.

Due to its accuracy, fast convergence, and simplicity in implementation, the CCM has been utilized to carry out the vibrations and buckling characteristics of one, two, and three dimensional continuous systems (at the macro and micro levels).^46–52 When discretizing ordinary and partial differential equations by the CCM, the nth derivative of a function is given by Dn = (D_N)ⁿ.

Equations (17a) to (17c) are discretized by utilizing the CCM as

$$\left[\begin{array}{c}\mathrm{GE}1\hfill \\ \multicolumn{1}{c}{\mathrm{GE}1}\\ \multicolumn{1}{c}{\mathrm{GE}3}\end{array}\right]\left\{U\right\}={\overline{\beta }}^2\left[\begin{array}{c}\mathrm{RH}1\hfill \\ \multicolumn{1}{c}{\mathrm{RH}1}\\ \multicolumn{1}{c}{\mathrm{RH}3}\end{array}\right]\left\{U\right\}$$

(19)

where {U} is the displacement vector for the plate and

$$\begin{array}{c}\multicolumn{1}{c}{E1=(\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \end{array}\right]\otimes (D2\otimes I))}\\ +(\left(\frac{1-\nu }{2}\right){\eta }^2\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \end{array}\right]\otimes (I\otimes D2))\hfill \\ +(\left(\frac{1+\nu }{2}\right)\eta \left[\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \end{array}\right]\otimes (D1\otimes D1))\hfill \\ -(\frac{F}{{\delta }^2}\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \end{array}\right]\otimes (I\otimes I))\hfill \\ -(\frac{F}{\delta }\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]\otimes (D1\otimes I))\hfill \end{array}$$

(20a)

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{GE}2=({\eta }^2\left[\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \end{array}\right]\otimes (I\otimes D2))}\\ +(\left(\frac{1-\nu }{2}\right)\left[\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \end{array}\right]\otimes (D2\otimes I))\hfill \\ +(\left(\frac{1+\nu }{2}\right)\eta \left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \end{array}\right]\otimes (D1\otimes D1))\hfill \\ -(\frac{F}{{\delta }^2}\left[\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \end{array}\right]\otimes (I\otimes I))\hfill \\ -(\frac{F\gamma }{{\delta }^2}\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]\otimes (I\otimes D1))\hfill \end{array}$$

(20b)

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{GE}3=(\delta \left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \end{array}\right]\otimes (D1\otimes I))}\\ +(\gamma \left[\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \end{array}\right]\otimes (I\otimes D1))\hfill \\ +({\delta }^2\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]\otimes (D2\otimes I))\hfill \\ +({\gamma }^2\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]\otimes (I\otimes D2))\hfill \\ -(\frac{{\overline{K}}_W{\delta }^4}{F}\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]\otimes (G\left(X\right)\otimes I))\hfill \\ +(\frac{{\overline{K}}_W{\delta }^4{\mu }^2}{F}\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]\otimes (\frac{{\partial }^{2}G\left(X\right)}{\partial X^2}\otimes I))\hfill \\ +(\frac{2{\overline{K}}_W{\delta }^4{\mu }^2}{F}\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]\otimes (\frac{\partial G\left(X\right)}{\partial X}*D1\otimes I))\hfill \\ +(\frac{{\overline{K}}_W{\delta }^4{\mu }^2}{F}\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]\otimes (G\left(X\right)*D2\otimes I))\hfill \\ +(\frac{{\overline{K}}_W{\delta }^4{\eta }^2{\mu }^2}{F}\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]\otimes (G\left(X\right)\otimes D2))\hfill \\ +(\frac{{\overline{K}}_S{\delta }^4}{F}\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]\otimes (D2\otimes I))\hfill \\ +(\frac{{\overline{K}}_S{\delta }^2{\gamma }^2}{F}\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]\otimes (I\otimes D2))\hfill \\ -(\frac{{\overline{K}}_S{\delta }^4{\mu }^2}{F}\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]\otimes (D4\otimes I))\hfill \\ -(\frac{2{\overline{K}}_S{\delta }^2{\gamma }^2{\mu }^2}{F}\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]\otimes (D1\otimes D1))\hfill \\ -(\frac{{\overline{K}}_S{\delta }^2{\gamma }^2{\eta }^2{\mu }^2}{F}\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]\otimes (I\otimes D4))\hfill \end{array}$$

(20c)

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{RH}1=(\frac{-{\delta }^2}{12}\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \end{array}\right]\otimes (I\otimes I))}\\ +(\frac{{\delta }^2{\mu }^2}{12}\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \end{array}\right]\otimes (D2\otimes I))\hfill \\ +(\frac{{\delta }^2{\mu }^2{\eta }^2}{12}\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \end{array}\right]\otimes (I\otimes D2))\hfill \end{array}$$

(20d)

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{RH}2=(\frac{-{\delta }^2}{12}\left[\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \end{array}\right]\otimes (I\otimes I))}\\ +(\frac{{\delta }^2{\mu }^2}{12}\left[\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \end{array}\right]\otimes (D2\otimes I))\hfill \\ +(\frac{{\delta }^2{\mu }^2{\eta }^2}{12}\left[\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \end{array}\right]\otimes (I\otimes D2))\hfill \end{array}$$

(20e)

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{RH}3=(\frac{-D{\delta }^4}{\mathrm{FC}}\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]\otimes (I\otimes I))}\\ +(\frac{D{\delta }^4{\mu }^2}{\mathrm{FC}}\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]\otimes (D2\otimes I))\hfill \\ +(\frac{D{\delta }^4{\mu }^2{\eta }^2}{\mathrm{FC}}\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]\otimes (I\otimes D2))\hfill \end{array}$$

(20f)

where ⊗ is the Kronecker product. $G(X)$ , $\partial G(X)/\partial X$ , and ${\partial }^{2}G(X)/\partial X^2$ are diagonal M × M matrices, and M = N + 1 represents the number of the Chebyshev points in the x and y directions. Each of GE1, GE2, GE3, RH1, RH2, and RH3 has a size of M² × 3M². After rewriting the equations of motion according to the CCM, the eigenvalue problem for the FG Mindlin nanoplates with clamped (C) and simply supported (S) boundary conditions is solved. It should be noted that notation SCSC denotes that edges X = 0 and X = 1 are simply supported (S), and edges Y = 0 and Y = 1 are clamped (C).

Boundary conditions

In the present study, two types of boundary conditions, simply supported (S) and clamped (C), are taken into consideration. For a clamped edge, the displacements W, Ψ_X, and Ψ_Y equal zero, whereas for a simply supported edge at X = 0 or 1, the displacements W, Ψ_Y, and the moment M_XX equal zero. The boundary conditions (in dimensionless form) at the edge X = 0 are applied using the CCM and the Kronecker product as

(C):

$$(\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \end{array}\right]\otimes (\left[\begin{array}{ccc}1\hfill & 0\hfill & \cdots 0\hfill \end{array}\right]\otimes I))\left\{U^T\right\}=0$$

(21a)

$$(\left[\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \end{array}\right]\otimes (\left[\begin{array}{ccc}1\hfill & 0\hfill & \cdots 0\hfill \end{array}\right]\otimes I))\left\{U^T\right\}=0$$

(21b)

$$(\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]\otimes (\left[\begin{array}{ccc}1\hfill & 0\hfill & \cdots 0\hfill \end{array}\right]\otimes I))\left\{U^T\right\}=0$$

(21c)

(S):

$$(\left[\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \end{array}\right]\otimes (\left[\begin{array}{ccc}1\hfill & 0\hfill & \cdots 0\hfill \end{array}\right]\otimes I))\left\{U^T\right\}=0$$

(22a)

$$(\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]\otimes (\left[\begin{array}{ccc}1\hfill & 0\hfill & \cdots 0\hfill \end{array}\right]\otimes I))\left\{U^T\right\}=0$$

(22b)

$$((\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \end{array}\right]\otimes (D1(1,:)\otimes I))+v(\left[\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \end{array}\right]\otimes (\left[\begin{array}{ccc}1\hfill & 0\hfill & \cdots 0\hfill \end{array}\right]\otimes D1)))\left\{U^T\right\}=0$$

(22c)

In case the boundary conditions are applied at the edge X = 1, the vector [1 0 … 0] is replaced by the vector [0 0 … 1], and D1(1,:) is replaced by D1(M,:), where D1(1,:) is defined as

$$\begin{array}{c}D1(1,:)=\hfill \\ \left[\begin{array}{ccccc}D1(1,1)\hfill & D1(1,2)\hfill & D1(1,3)\hfill & \cdots \hfill & D1(1,M)\hfill \end{array}\right]\hfill \end{array}$$

and D1(M,:) is defined as

$$\begin{array}{c}D1(M,:)=\hfill \\ \left[\begin{array}{ccccc}D1(M,1)\hfill & D1(M,2)\hfill & D1(M,3)\hfill & \cdots \hfill & D1(M,M)\hfill \end{array}\right]\hfill \end{array}$$

More details can be found in Sari and Butcher. ⁴⁷

Comparison studies

In order to check the validity and accuracy of the present model of the FG Mindlin nanoplates, a comparison study is performed. Initially, the results of the FG local plate (by setting the nonlocal parameter to zero) resting on an elastic foundation are compared to those computed by Hashemi et al. ⁴² Next, the outcomes of a nanoplate with constant properties (isotropic) are compared with the results of Hashemi et al. ⁵³ In Tables 2 and 3, the fundamental nondimensional frequencies for SSSS, SSSC, and SCSC Mindlin FG beams are shown and are defined as $\hat{\beta }=\omega h\sqrt{{\rho }_c/E_c}$ . Moreover, in Table 4, the frequencies of the suggested model for nonlocal Mindlin plates are compared with those obtained by Hashemi et al. ⁵³ In this case, the dimensionless frequencies are defined as ${\beta }^{*}=\omega a^2\sqrt{\rho h/D}$ , where D is the flexural rigidity of the nanoplate. As seen in these tables, the frequencies of the present solution can attain the same accuracy of those calculated by Hosseini-Hashemi et al. ⁴² and Hosseini-Hashemi et al. ⁵³ These tables reflect the reliability of the numerical technique. In Table 3, the frequency ratio (FR) is given as

$$\mathrm{FR}=\frac{{\beta }^{*}\mathrm{computed}\mathrm{by}\mathrm{Eringen}\prime s\mathrm{theory}}{{\beta }^{*}\mathrm{computed}\mathrm{by}\mathrm{classical}\mathrm{theory}}$$

Table 2.

Fundamental frequency parameter $\hat{\beta }=\omega h\sqrt{{\rho }_c/E_c}$ of the Al/Al₂O₃ rectangular FG Mindlin plate with δ = 0.15, η = 0.5, and $\zeta$  = 0.

$({\overline{K}}_W,{\overline{K}}_s)$	$\alpha$	Method	Boundary conditions
			SSSS	SSSC	SCSC
(0,0)	0	Present study	0.08006	0.08325	0.08729
		Hosseini-Hashemi et al. 42	0.08006	0.08325	0.08729
	0.25	Present study	0.07320	0.07600	0.07950
		Hosseini-Hashemi et al. 42	0.07320	0.07600	0.07950
	1	Present study	0.06335	0.06541	0.06790
		Hosseini-Hashemi et al. 42	0.06335	0.06541	0.06790
	5	Present study	0.05379	0.05524	0.05695
		Hosseini-Hashemi et al. 42	0.05379	0.05524	0.05695
	$\infty$	Present study	0.04076	0.04263	0.04443
		Hosseini-Hashemi et al. 42	0.04100	0.04263	0.04443
(100, 10)	0	Present study	0.12870	0.13097	0.13376
		Hosseini-Hashemi et al. 42	0.12870	0.13097	0.13376
	0.25	Present study	0.11842	0.12040	0.12280
		Hosseini-Hashemi et al. 42	0.11842	0.12040	0.12280
	1	Present study	0.10519	0.10659	0.10824
		Hosseini-Hashemi et al. 42	0.10519	0.10659	0.10824
	5	Present study	0.09223	0.09318	0.09426
		Hosseini-Hashemi et al. 42	0.09223	0.09318	0.09426
	$\infty$	Present study	0.06551	0.06708	0.06809
		Hosseini-Hashemi et al. 42	0.06591	0.06708	0.06808

FG: functionally graded; SSSS: simply supported at the edge X=0, simply supported at the edge Y=0, simply supported at the edge X=1, and simply supported at the edge Y=1; SSSC: simply supported at the edge X=0, simply supported at the edge Y=0, simply supported at the edge X=1, and clamped at the edge Y=1; SCSC: simply supported at the edge X=0, clamped at the edge Y=0, simply supported at the edge X=1, and clamped at the edge Y=1.

Table 3.

Fundamental frequency parameter $\hat{\beta }=\omega h\sqrt{{\rho }_c/E_c}$ of the Al/ZrO₂ FG Mindlin plate with a/b = 1.5 and $\zeta$  = 0.

$({\overline{K}}_W,{\overline{K}}_s)$	$\delta$	$\alpha$	Method	Boundary conditions
				SSSS	SSSC	SCSC
(0,0)	0.05	0	Present study	0.02392	0.03130	0.04076
			Hosseini-Hashemi et al. 42	0.02392	0.03129	0.04076
		0.25	Present study	0.02269	0.02899	0.03664
			Hosseini-Hashemi et al. 42	0.02269	0.02899	0.03664
		1	Present study	0.02156	0.02667	0.03250
			Hosseini-Hashemi et al. 42	0.02156	0.02667	0.03250
		5	Present study	0.02180	0.02677	0.03239
			Hosseini-Hashemi et al. 42	0.02180	0.02677	0.03239
		$\infty$	Present study	0.02046	0.02689	0.03502
			Hosseini-Hashemi et al. 42	0.02046	0.02689	0.03502
(250, 25)		0	Present study	0.03421	0.04022	0.04815
			Hosseini-Hashemi et al. 42	0.03421	0.04021	0.04815
		0.25	Present study	0.03285	0.03786	0.04412
			Hosseini-Hashemi et al. 42	0.03285	0.03786	0.04412
		1	Present study	0.03184	0.03577	0.04035
			Hosseini-Hashemi et al. 42	0.03184	0.03577	0.04035
		5	Present study	0.03235	0.03615	0.04053
			Hosseini-Hashemi et al. 42	0.03235	0.03615	0.04053
		$\infty$	Present study	0.02936	0.03456	0.04138
			Hosseini-Hashemi et al. 42	0.02937	0.03455	0.04138

FG: functionally graded; SSSS: simply supported at the edge X=0, simply supported at the edge Y=0, simply supported at the edge X=1, and simply supported at the edge Y=1; SSSC: simply supported at the edge X=0, simply supported at the edge Y=0, simply supported at the edge X=1, and clamped at the edge Y=1; SCSC: simply supported at the edge X=0, clamped at the edge Y=0, simply supported at the edge X=1, and clamped at the edge Y=1.

Table 4.

Fundamental frequency parameter ${\beta }^{*}=\omega a^2\sqrt{\rho h/D}$ for the isotropic SSSS nonlocal Mindlin plates $({\overline{K}}_W={\overline{K}}_S=0,{\kappa }^2=0.86667,\delta =0.1,\alpha =\zeta =0)$ .

µ		0	0.2	0.4	0.6	0.8
	ω NL	FR	FR	FR	FR	FR
η = 1.0/0.6
Present study	35.0643	1.0000	0.6335	0.3789	0.2633	0.2006
Hosseini-Hashemi et al. 53	35.0643	1.0000	0.6335	0.3789	0.2633	0.2005
η = 1.0
Present study	19.0840	1.0000	0.7475	0.4904	0.3512	0.2708
Hosseini-Hashemi et al. 53	19.0840	1.0000	0.7475	0.4904	0.3512	0.2708

FR: frequency ratio.

Results and discussion

The variations of the fundamental dimensionless frequencies with the parameter $\zeta$ for FG CCCC plates with η = 0.5, α = 1, µ = 0.0, δ = 0.1, ${\overline{K}}_S=5,{\overline{K}}_W=25$ at different types of elastic foundations are shown in Figure 2(a) and (b). From these figures, it can be depicted that the dimensionless frequencies rise as the parameter $\zeta$ grows, and this growth is more significant for the sinusoidal elastic foundation. For example, the frequencies of the Al/Al₂O₃ plates resting on a sinusoidal foundation increase by 21.23% as $\zeta$ increases from 0 to 10, compared to an increase by 12.49% and 7.02% for the linear and parabolic foundations, respectively. Furthermore, the frequencies of the Al/ZrO₂ plates increase by 19.68% for the sinusoidal foundation, 11.53% for the linear foundation, and 6.46% for the parabolic foundation as $\zeta$ increases from 0 to 10.

Figure 2.

The fundamental natural dimensionless frequency versus the variable parameter $\zeta$ for a CCCC FG Mindlin nanoplate, $\eta =0.5,\alpha =1,\mu =0,\delta =0.1,{\overline{K}}_S=5,{\overline{K}}_W=25$ , for different foundation types: (a) Al/Al₂O₃ and (b) Al/ZrO_2.

Figure 3(a) and (b) displays the influence of the nonlocal scale parameter µ on the fundamental frequencies of FG CCCC nanoplates with η = 0.5, α = 1, $\zeta$  = 3.0, δ = 0.1, ${\overline{K}}_S={\overline{K}}_W=10$ at different types of elastic foundations. These figures show that increasing the nonlocal scale parameter leads to a decrease in the frequencies due to the reduction in the stiffness of the plate. The dimensionless frequencies of the FG nanoplates resting on sinusoidal foundations are higher than those for the FG nanoplates resting on linear and parabolic foundations (when the nonlocal scale parameter µ is in the ranges of [0,0.48] and [0,0.44] for the Al/Al₂O₃ and Al/ZrO₂ plates, respectively). In contrary, these frequencies are smaller when µ is in the ranges of [0.52,1] for the Al/Al₂O₃ and Al/ZrO₂ plates. The figure shows that the nonlocal scale coefficient has a more remarkable influence on the dimensionless frequencies of the FG nanoplates resting on sinusoidal foundations. The frequencies of the plates resting on linear and parabolic foundations tend to have the same values as µ > 0.67. In Figure 4, it is revealed that the fundamental frequencies of FG SSSS plates with η = 1.0, α = 0.1, $\zeta$  = 5.0, δ = 0.05, and ${\overline{K}}_S=5,{\overline{K}}_W=25$ are higher for the plates resting on sinusoidal foundations, and the nonlocal scale coefficient has a greater impact on the dimensionless frequencies of the FG nanoplates resting on parabolic foundations.

Figure 3.

The fundamental natural dimensionless frequency versus the nonlocal parameter for a CCCC FG Mindlin nanoplate, $\eta =0.5,\alpha =1,\zeta =3.0,\delta =0.1,{\overline{K}}_S={\overline{K}}_W=10$ for different foundation types: (a) Al/Al₂O₃ and (b) Al/ZrO₂.

Figure 4.

Variation of the fundamental natural frequency with the nonlocal scale coefficient of an SSSS FG Mindlin plate, $\eta =1.0,\alpha =0.1,\zeta =5.0,\delta =0.05,{\overline{K}}_S=5,{\overline{K}}_W=25$ , for different foundation types: (a) Al/Al₂O₃ and (b) Al/ZrO₂.

Figure 5 shows the influence of the nonlocal scale coefficient on the dimensionless frequencies of FG SSSS nanoplates with η = 1.5, α = 4.0, $\zeta$  = 2.0, δ = 0.1, ${\overline{K}}_S=5,{\overline{K}}_W=25$ . It is noticed that the frequencies almost have the same values regardless of the type of the elastic foundation. This leads us to state that for certain boundary conditions and at some values of the parameters (η,α, $\zeta$ , δ, ${\overline{K}}_S$ , and ${\overline{K}}_W$ ), the frequencies do not depend on the type of the elastic foundation. Figure 6 depicts the relation between the dimensionless fundamental frequencies and the gradient index α of FG SSSS plates with η = 0.5, µ = 0.1, $\zeta$  = 1.0, δ = 0.1, and ${\overline{K}}_S={\overline{K}}_W=10$ . The results show that the rise of the gradient index α, in the range of [0,20], leads to a decrease in the dimensionless frequencies for the Al/Al₂O₃ plates. The figure reveals that the dimensionless frequencies of the FG nanoplates lying on sinusoidal medium are slightly larger than those resting on linear and parabolic foundations. For the Al/ZrO₂ plates, the dimensionless frequencies decrease as α varies from 0.01 to 0.81, and as α varies from 0.81 to 3.808, the dimensionless frequencies increase. Then, they decrease as α varies from 3.808 to 20. Similar observations can be made for Figure 7.

Figure 5.

The fundamental natural frequency versus the nonlocal scale coefficient for an SSSS FG Mindlin nanoplate, $\eta =1.5,\alpha =4.0,\zeta =2.0,\delta =0.1,{\overline{K}}_S=5,{\overline{K}}_W=25$ , for different foundation types: (a) Al/Al₂O₃ and (b) Al/ZrO₂.

Figure 6.

Effect of the fundamental natural dimensionless frequency on the gradient index for an SSSS FG Mindlin nanoplate, $\eta =0.5,\mu =0.1,\zeta =1.0,\delta =0.1,{\overline{K}}_S={\overline{K}}_W=10$ , for different foundation types: (a) Al/Al₂O₃ and (b) Al/ZrO₂.

Figure 7.

The fundamental natural dimensionless frequency versus the gradient index for an SCSC FG Mindlin nanoplate, $\eta =1.5,\mu =0.05,\zeta =5.0,\delta =0.2,{\overline{K}}_S=25,{\overline{K}}_W=100$ , for different foundation types: (a) Al/Al₂O₃ and (b) Al/ZrO₂.

The effect of the slenderness ratio δ (the thickness of the plate divided by its length in the x-axis) on the dimensionless frequencies is examined for the FG SSSS nanoplates (η = 1.0, µ = 0.1, α = 0.2, $\zeta$  = 1.0, and ${\overline{K}}_S=10,{\overline{K}}_W=100$ ) with various types of elastic foundations are displayed in Figure 8. The figure reveals that increasing the slenderness ratio causes a drop in the dimensionless frequencies. The dimensionless frequencies for the plates resting on sinusoidal foundations are higher than those resting on linear and parabolic foundations. Furthermore, the dimensionless frequencies of the Al/ZrO₂ plates are larger than those of the Al/Al₂O₃ plates. In Figures 9 and 10, it is shown that the dimensionless frequencies rise as the Winkler medium stiffness parameter ${\overline{K}}_W$ and the shear foundation stiffness parameter ${\overline{K}}_S$ increase.

Figure 8.

The fundamental natural dimensionless frequency versus the thickness ratio of an SSSS FG Mindlin nanoplate, $\eta =1.0,\mu =0.1,\zeta =1.0,\alpha =0.2,{\overline{K}}_S=10,{\overline{K}}_W=100$ , for different foundation types: (a) Al/Al₂O₃ and (b) Al/ZrO₂.

Figure 9.

Variation of the fundamental dimensionless natural frequency with the Winkler foundation stiffness parameter of an SSSS FG Mindlin nanoplate, $\eta =0.5,\delta =0.1,\mu =0,\zeta =1.0,\alpha =1.0,{\overline{K}}_S=25$ , for different foundation types: (a) Al/Al₂O₃ and (b) Al/ZrO₂.

Figure 10.

Variation of the fundamental dimensionless natural frequency with the shear foundation stiffness parameter of an SCSC FG Mindlin nanoplate, $\eta =1.0,\delta =0.05,\mu =0.1,\zeta =1.0,\alpha =0.5,{\overline{K}}_W=100$ , for different foundation types: (a) Al/Al₂O₃ and (b) Al/ZrO₂.

Figure 11 displays the impact of the boundary conditions on the fundamental dimensionless frequencies of an FG plate (η = 1.0, δ = 0.05, α = 1.0, $\zeta$  = 4.0, ${\overline{K}}_S=5,{\overline{K}}_W=25$ ) lying on a linear elastic foundation. The figure reveals that the fundamental dimensionless frequencies of the FG CCCC nanoplates are larger compared to those of the FG SCSC and SSSS nanoplates. The frequencies of the SCSC plates decrease by 42%, and the CCCC by 40%, whereas the SSSS decreases by 39.3% as the nonlocal scale coefficient µ varies over the range 0 to 1. The effect of the nonlocal scale coefficient µ on the dimensionless frequencies for FG plates (η = 1.5, δ = 0.1, α = 1, $\zeta$  = 2.0, ${\overline{K}}_S={\overline{K}}_W=10$ ) with various boundary conditions and lying on parabolic elastic mediums is presented in Figure 12. For the Al/Al₂O₃ and Al/ZrO₂ materials, the SSSS plates have higher dimensionless frequencies for µ > 0.7. It is noticed that for the Al/ZrO₂ CCCC and SCSC plates, the frequencies have the same values for µ > 0.47. The influence of the nonlocal scale coefficient on the fundamental dimensionless frequencies of FG nanoplates (η = 0.5, δ = 0.1, α = 3.0, $\zeta$  = 1.0, ${\overline{K}}_S=5,{\overline{K}}_W=10$ ), with different boundary conditions and resting on sinusoidal foundations, is demonstrated in Figure 13.

Figure 11.

Variation of the fundamental dimensionless natural frequency with the nonlocal scale coefficient of an FG Mindlin nanoplate lying on a linear elastic layer, $\eta =1.0,\delta =0.05,\zeta =4.0,\alpha =1,{\overline{K}}_S=5,{\overline{K}}_W=25$ , for different boundary conditions: (a) Al/Al₂O₃ and (b) Al/ZrO₂.

Figure 12.

Variation of the fundamental dimensionless natural frequency with the nonlocal scale coefficient of an FG Mindlin nanoplate resting on a parabolic elastic foundation, $\eta =1.5,\delta =0.1,\zeta =2.0,\alpha =1,{\overline{K}}_S={\overline{K}}_W=10$ , for different boundary conditions: (a) Al/Al₂O₃ and (b) Al/ZrO₂.

Figure 13.

Variation of the fundamental dimensionless natural frequency with the nonlocal scale parameter of an FG Mindlin nanoplate resting on a sinusoidal elastic foundation, $\eta =0.5,\delta =0.1,\zeta =1.0,\alpha =3,{\overline{K}}_S=5,{\overline{K}}_W=10$ , for different boundary conditions: (a) Al/Al₂O₃ and (b) Al/ZrO_2.

In Figures 14 to 16, the first three dimensionless frequencies versus the nonlocal scale parameter $\mu$ are illustrated for FG nanoplates resting on linear, parabolic, and sinusoidal elastic foundations, respectively. As predicted, the frequencies are reduced due to the growth of the nonlocal scale parameter μ, and the reduction is more noticeable for the second and the third modes. For the Al/Al₂O₃ SCSC plate (η = 1.5, δ = 0.15, α = 10.0, $\zeta$  = 2.0, ${\overline{K}}_S=5,{\overline{K}}_W=20$ ), the first dimensionless frequency decreases by 73.5%, the second dimensionless frequency by 77.98%, and the third dimensionless frequency by 84% as the parameter µ varies over the range 0 to 1, as shown in Figure 14(a). Moreover, it is noticed that for larger values of µ, these frequencies almost approach the same value. In Figure 15(b), the frequencies tend to have the same value for µ > 0.5. In contrast, Figure 16 reveals that the frequencies of the SSSS FG nanoplate (η = 0.5, δ = 0.1, α = 4.0, $\zeta$  = 1.0, ${\overline{K}}_S={\overline{K}}_W=10$ ) resting on a sinusoidal foundation are deviated from each other as the nonlocal scale coefficient rises. The frequencies of the Al/Al₂O₃ plates are more sensitive to the rise of the µ parameter than those of the Al/ZrO₂ plates. For instance, the first dimensionless frequency of the Al/Al₂O₃ plates decreases by 38.46%, the second frequency by 35.29%, and the third by 50.7%. However, the first dimensionless frequency of the Al/ZrO₂ plates decreases by 26%, the second frequency by 34.8%, and the third by 42.36%. It is worthwhile to mention that the dimensionless frequencies in Figures 2 to 16 are given as

$$\beta =\omega \frac{a^2}{h}\sqrt{{\rho }_c/E_c}$$

Figure 14.

Variation of the first three dimensionless natural frequencies with the nonlocal scale parameter of an SCSC FG Mindlin nanoplate lying on a linear elastic layer, $\eta =1.5,\delta =0.15,\zeta =2.0,\alpha =10,{\overline{K}}_S=5,{\overline{K}}_W=20$ : (a) Al/Al₂O₃ and (b) Al/ZrO₂.

Figure 15.

Variation of the first three dimensionless natural frequencies with the nonlocal scale parameter of an SCSC FG Mindlin nanoplate resting on a parabolic elastic medium, $\eta =1.5,\delta =0.10,\zeta =2.0,\alpha =10,{\overline{K}}_S=25,{\overline{K}}_W=100$ : (a) Al/Al₂O₃ and (b) Al/ZrO₂.

Figure 16.

Variation of the first three dimensionless natural frequencies with the nonlocal scale parameter of an SSSS FG Mindlin nanoplate resting on a sinusoidal elastic foundation, $\eta =0.5,\delta =0.10,\zeta =1.0,\alpha =4,{\overline{K}}_S={\overline{K}}_W=10$ : (a) Al/Al₂O₃ and (b) Al/ZrO₂.

Conclusion

The natural vibration behavior of thick nonlocal FG nanoplates lying on variable elastic foundations was investigated. Eringen’s nonlocal theory was used to account for the small-scale effects of the FG material, and Hamilton’s principle was utilized to obtain the partial differential equations that govern the behavior of the FG Mindlin nanoplates lying on variable elastic mediums. The CCM was applied to calculate the dimensionless natural frequencies. The partial differential equations with variable coefficients were discretized in an efficient and convenient scheme. The influences of different parameters such as the nonlocal scale coefficient, the slenderness ratio, Winkler and shear foundation stiffness parameters, boundary conditions, gradient indices, and the type of the elastic medium on the dimensionless natural frequencies were performed.

The findings showed that the dimensionless natural frequencies decline by the rise of the nonlocal scale coefficient, the thickness ratio, and the gradient index. Moreover, the dimensionless natural frequencies grow as the values of the variable parameter $(\zeta )$ , and the Winkler and the shear foundation stiffness parameters get larger. It is observed that the type of the elastic medium has a remarkable impact on the natural dimensionless frequencies. Due to its significant influence on the natural vibration behavior, it is concluded that the nonlocal scale coefficient should be considered when analyzing, examining, and designing nano thick structures lying on elastic mediums. Authors hope that the results and the findings obtained herein may be important for engineers modeling thick micro/nano FG structures embedded in variable elastic foundations. For future work, it is recommended to perform the vibration behavior of nanoplates with surface effects subjected to magnetic and electroelastic forces.