#### Abstract

This article presents a finite element method (FEM) integrated with the nonlocal theory for analysis of the static bending and free vibration of the sandwich functionally graded (FG) nanoplates resting on the elastic foundation (EF). Material properties of nanoplates are assumed to vary through thickness following two types (Type A with homogeneous core and FG material for upper and lower layers and Type B with FG material core and homogeneous materials for upper and lower layers). In this study, the formulation of the four-node quadrilateral element based on the mixed interpolation of tensorial components (MITC4) is used to avoid “the shear-locking” problem. On the basis of Hamilton’s principle and the nonlocal theory, the governing equations for the sandwich FG nanoplates are derived. The results of the proposed model are compared with published works to verify the accuracy and reliability. Furthermore, the effects of geometric parameters and material properties on the static and free vibration behaviors of nanoplates are investigated in detail.

#### 1. Introduction

Nowadays, with the sophisticated development of technology, the investigation of nanostructures has been widely concerned by scientists in the world. However, the studies show that conventional computational theory for millimeter-sized structures is less accurate for nanometer-sized structures. So, in order to improve the accuracy for the analysis of nanostructures, many theories have been proposed such as the modified couple stress theory [1], the strain gradient theory [2], and the nonlocal theory [3, 4]. Among these theories, the nonlocal theory [3, 4] has been used popularly in the literature due to simplicity and high accuracy. Li et al. [5] developed a new nonlocal model to solve the static and dynamic problems for circular elastic nanosolids. Ansari et al. [6] used the nonlocal theory to analyze the free vibration of single-layered graphene plates. Arash and Wang [7] discussed about the nonlocal elastic theory in modeling carbon nanotubes and graphene. Asemi and Farajpour [8] studied thermomechanical vibration of graphene plates including surface effects by decoupling the nonlocal elasticity equations. Jalali et al. [9] used molecular dynamics combined with nonlocal elasticity approaches to investigate the effect of out-of-plane defects on vibration analysis of graphene. In addition, the nonlocal theory was also used in Refs. [10–20] to investigate the various performances of nanoplates. Recently, more attention has focused on the analysis of FG nanoplates considering the small-scale effects. Natarajan et al. [21] examined the vibration behavior of FG nanoplates by using the Mori-Tanaka homogenization scheme. Jung and Han [22] used the Navier solution to investigate the bending and free vibration responses of the Sigmoid FGM nanoplates. Nami et al. [23] studied the thermal buckling of FG nanoplates by using the third-order shear deformation theory (TSDT) combined with nonlocal elasticity. Hashemi et al. [24] analyzed the free vibration of moderately thick circular/annular FG Mindlin plates via nonlocal elasticity. Salehipour et al. [25] developed the analytical solution for free vibration analysis of the FG micro/nanoplates using the three-dimensional theory of elasticity accounting small-scale effect. Salehipour et al. [26] expanded the modified nonlocal elasticity for examining the natural frequency of the FG micro/nanoplates. Ansari et al. [27] analyzed the bending and free vibration responses of the FG nanoplates by using Eringen’s nonlocal theory.

In recent years, the application of FG materials has been widely spread in nanoscale devices such as thin films [28, 29] or fully released nanoelectromechanical systems (NEMS) [30] due to their excellent performance. Therefore, investigating various behaviors of the FG material nanostructures has attracted the attention of many researchers. Simsek [31] studied free longitudinal vibration of axially FG tapered nanorods using nonlocal effects. Simsek and Yurtcu [32] analyzed the bending and buckling of FG nanobeams based on the nonlocal Timoshenko beam theory. Natarajan et al. [33] considered the free flexural vibration of FG nanoplates. Nazemnezhad and Hashemi [34] analyzed the nonlinear free vibration of FG nanobeams, and Hashemi et al. [35, 36] investigated the nonlinear free vibration of piezoelectric FG nanobeams using nonlocal elasticity. In the above-mentioned studies, most of them used the analytical solutions to analyze the behavior of FG nanoplates. However, the analytical solutions are limited or even impossible when the geometry, boundary conditions, or types of load become more complicated. As an alternative, numerical methods have been proposed to fill in the gaps of these problems.

In the case of the nanostructures resting on the elastic foundation (EF), some typical works can be mentioned here as follows. Wang and Li [37] investigated the static bending behavior of the nanoplates embedded in an EF. Narendar and Gopalakrishnan [38] studied the wave dispersion of a single-layered graphene sheet embedded in an elastic polymer matrix. Pouresmaeeli et al. [39] investigated the vibration behaviors of nanoplates embedded in a viscoelastic medium. Zenkour and Sobhy [40] considered thermal buckling of nanoplates on the EF by using the sinusoidal shear deformation theory. Panyatong et al. [41] combined the nonlocal theory and the surface stress to investigate the bending behavior of nanoplates embedded in an elastic medium.

It is well known that the classical plate theories are limited only for thin plates. Therefore, many studies in recent years have developed Reissner-Mindlin plate theories to investigate both thick and thin plates. However, the shear-locking phenomenon in these theories can arise when the thickness of plates becomes very small. To eliminate the shear-locking phenomenon, many improved techniques have been introduced such as the reduced integration method [42], discrete shear gap method (DSG) [43], assumed natural strains technique (ANS) [44], and mixed interpolation of tensorial components (MITC) [45–48]. Among these methods, the four-node quadrilateral element using MITC technique, which gives the so-called MITC4 element, has attracted the interest of many scientists around the world due to the simplicity of formulation and reasonable computational cost.

Based on the best of authors’ knowledge, the static and free vibration analysis of sandwich FG nanoplates resting on the elastic foundation using the MITC4 element integrated with the nonlocal theory has not been published yet. This motivates us to develop the MITC4 element integrated with the nonlocal theory to accurately describe the stress-deformation and displacement of the FG nanoplates resting on the elastic foundation in this work. The accuracy and reliability of the proposed method are verified by comparing its numerical results with those available in the literature. Furthermore, the effects of geometric parameters and material properties on the bending and free vibration responses of sandwich FG nanoplates are also investigated in detail.

#### 2. Theoretical Formulation

##### 2.1. FG Sandwich Nanoplates

In this study, we consider a sandwich FG nanoplate has length , width , and thickness which rests on the Winkler foundation. The properties of FG materials of the nanoplate are given by power-law exponent function as follows: where mentions the effective material property such as the modulus of elasticity , mass density , and Poisson’s ratio ; the characters and are the metal and ceramic constituents, respectively; is the power-law index; and is the volume fraction of the material. In this study, two types of the FG sandwich nanoplates are considered as shown in Figure 1.

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###### 2.1.1. Type A: Sandwich Nanoplates with FG Skins—Homogeneous Core

The bottom and top surfaces are composed of the FG materials, while the core is ceramic (Figure 1(a)), and is given by the following formula:

###### 2.1.2. Type B: Sandwich Nanoplates with Homogeneous Skins—FG Core

In this type, the bottom and top surfaces are metal and ceramic, respectively, while the core is composed of the FG materials as shown in Figure 1(b). is given by the following formula:

##### 2.2. Plate Equations Based on Nonlocal Elasticity

Based on the nonlocal theory [1], the constitutive relation of a Hookean solid is determined by where ; is the small-scale effect factor or nonlocal factor; is an internal characteristic length, and is a constant; and is the stress tensor at a point which is calculated by the local theory. When equals zero, the nonlocal continuum theory degenerates into classical elasticity theories. Stress tensor is determined by where with

According to the first-order shear deformation theory (FSDT), the displacement field of the nanoplates is expressed as where are the displacements at any point (, , ); are the displacements at the mid-plane; and are the rotation angles of the cross-section around the -axis and -axis, respectively.

The deformation field of the FG nanoplate is defined as follows:

Equation (11) may be written as with

From Equations (4)–(13), the nonlocal force and moment resultants are determined by

Equations (14) and (15) can be rewritten as where which leads to

By using Hamilton’s principle, the equilibrium equations of the FG nanoplates are written by the following formula [49]: in which is the active force which is the combination between the external force in the -direction and the reaction force of the elastic foundation as follows

The mass inertia moment components are determined by the following formula:

Substituting Equations (18)–(20) into Equations (21)–(25), we obtain equilibrium equations of the sandwich FG nanoplates according to the nonlocal theory as follows:

Finally, we perform a few simple calculations by multiplying the equations from Equations (28)–(32), respectively, with the variables and integrating on the domain and adding together the sides of each equation, to give the final equation as follows:

##### 2.3. Finite Element Formulation

###### 2.3.1. Basic Formulation

In this study, we use the four-node plate element, each node has 5 degrees of freedom (dof) to discretize the nanoplates. Then the nodal displacement vector can be defined as follows: where is the displacement vector at the node of element and is expressed as

The displacement field in the plate element is interpolated through the node displacement vector as where,,,, are the shape functions in the forms of

The matrices in Equation (37) have the size () where are the Lagrange interpolations, and it is shown in Appendix.

Substituting Equation (36) into Equation (33), the FEM of the typical element can be expressed as where is the element stiffness matrix defined as where , , are the bending, shear element stiffness matrices, and the foundation stiffness matrix, respectively, and are defined as where

###### 2.3.2. Transverse Shear Strains of the Element Based on MITC

The shear strains are approximated by interpolation in the natural coordinate system by [45]: where is the Jacobian matrix and are the midpoints of the edges as shown in Figure 2; are the shear deformation of the points , respectively.

By transforming the shear deformation according to the dof of the element displacement vector , we obtain where with in which

From here, the element stiffness matrix of the nanoplate can be recalculated as follows:

#### 3. Numerical Results

In this section, three numerical examples are performed to demonstrate the reliability and accuracy of the proposed method. In numerical analysis, the material properties are given in Table 1. For convenience, the following dimensionless displacement, stress, and frequencies of the plate are defined as

##### 3.1. Convergence and Accuracy Study

Firstly, the FG nanoplate using the material #2 but with the nonlocal factor is considered. The numerical results with other mesh sizes are presented in Tables 2 and 3 and compared with the analytical method in Refs. [50, 51]. It can be seen that the results using the MITC4 element converge faster than those using the Q4 element in both static and free vibration problems. In the static bending problem, the results of proposed method using mesh size are as accurate as those in Ref. [50], while in the free vibration problem, we only need to use mesh size to ensure the same accuracy as in Ref. [51]. Hence, to ensure the accuracy of the proposed method, the paper will use the mesh element to analyze both static bending and free vibration responses of nanoplates.

Next, Table 4 presents the results of the static bending analysis of the square nanoplates on the EF including the dimensionless displacement , dimensionless stress , and . It is seen that the obtained results of the present method match well with the analytical method in Ref. [52]. These results hence demonstrate the reliability and accuracy of the proposed method.

##### 3.2. Static Bending Problem

We now investigate the static bending response of the fully simple-supported (SSSS) sandwich (Type A) FG nanoplate resting on the Winkler foundation under the sinusoidal load . The dimensions of the nanoplate are given by , , and the material property is chosen by (material #3) as shown in Table 1. The power-law index varies from 0 to 10, and the nonlocal factor varies from 1 to 4. Five cases of the ratio of layer thickness, 2-1-2, 2-1-1, 2-2-1, 1-2-1, and 1-8-1 together with the foundation stiffness , are considered.

Tables 5 and 6 present the numerical results of the dimensionless displacement and stresses. Figures 3 and 4 illustrate the variation of the dimensionless displacement of the nanoplate center and the dimensionless stresses when varies from 0 to 10 corresponding to . Figures 5 and 6 present the variation of the dimensionless displacement and stresses when varies from 0 to 4 corresponding to . It can be seen that the increase of power-law index leads to the increase of the metal content in the FG structures and makes the FG nanoplates become softer. This result hence leads to the increase of the displacement and stress of the nanoplates. With the same power-law index , the increase of the geometric parameter and of the nonlocal factor leads to the increase of the dimensionless displacement and stress of the FG nanoplates. Furthermore, the dimensionless displacement and stress of the sandwich FG nanoplate obtain the minimum values in the case of the ratio of layers thickness 1-8-1 and obtain the maximum values in the case of the ratio of layers thickness 2-1-2. These results show that the sandwich FG nanoplate with a constant thickness will have the higher stiffness corresponding to the increase of the thickness of the core layer (ceramic).

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Figure 7 shows the variation of the dimensionless stress of sandwich FG square nanoplate along with the thickness corresponding to two different proportions of material layer, 1-2-1 and 2-2-1, and corresponding to the variation of power-law indexes . Note that the nonlocal factor is fixed. It is seen that when the power-law index equals zero (), the sandwich FG nanoplate becomes an isotropic nanoplate, and hence the stress distributions through the thickness of FG nanoplates are linear. When the power-law indexes are different from zero (), the stress distributions through the thickness of FG nanoplates will vary according to the folding curve types. When the core layer is homogeneous ceramic, the stress will have the jumping step at the contact surface of the middle layer with the top and bottom layers.

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The effects of the foundation stiffness on the static bending response of the sandwich FG nanoplates are presented in Table 7, and the graphs of dimensionless displacements and stresses are shown in Figure 8. It can be seen that the increase of (from 0 to 200) leads to the decrease of the dimensionless displacement and stress for all cases of the nanoplate thickness.

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##### 3.3. Free Vibration Problem

In this example, we consider the free vibration response of a sandwich FG nanoplate (Type A) with geometric dimensions , , resting on the Winkler foundation. Figure 9 shows the numerical results of the dimensionless frequencies of the FG nanoplates when the volume fraction exponent varies from 0 to 10 with . It can be seen that the increase of the volume fraction exponents leads to the decrease of the dimensionless frequencies. Table 8 and Figure 10 present the numerical results of the dimensionless frequencies of the FG nanoplates with the nonlocal factor varying from 0 to 4 and the volume fraction exponent corresponding to five cases of the ratio of layers thickness. It is noted that the increase of the nonlocal factor leads to the decrease of the dimensionless frequencies. Furthermore, Figure 11 presents the variation of the dimensionless frequencies of the FG nanoplates versus the variations of the volume fraction exponent and the nonlocal factor . It can be observed that the increase of these values leads to the decrease of the dimensionless frequencies.

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Table 9 and Figure 12 demonstrate the effects of the foundation stiffness (varying from 0 to 200) on the free vibration of the sandwich FG nanoplates. It is found that the increase of the foundation stiffness leads to the increase of the dimensionless frequencies of the FG nanoplates. In addition, the natural frequencies of the CCCC nanoplates are larger than those of the SSSS sandwich FG nanoplate. The results are quite reasonable because the SSSS boundary condition inherently offers more flexible boundary conditions than the CCCC boundary condition. Figure 13 shows the first six mode shapes of the SSSS sandwich FG square nanoplate with . It can be seen that the second and third mode shapes are similar to each other (the second fundamental frequency is approximated the third fundamental frequency). This phenomenon is suitable for the actual symmetrical plates under the same boundary conditions.

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#### 4. Conclusions

In this article, a new finite element method using the MITC4 element integrated with the nonlocal theory is proposed for the analysis of the static bending and free vibration of the sandwich functionally graded (FG) nanoplates resting on the Winkler foundation. Material properties of nanoplates are assumed to vary through thickness following two types (Type A with homogeneous core and FG material for upper and lower layers and Type B with FG material core and homogeneous materials for upper and lower layers). The obtained results of the proposed method agree excellently with others published works in the literature. The proposal method is free shear-locking and converges faster than the Q4 element and hence is very promising for the analysis of more complicated problems of the sandwich functionally graded (FG) nanoplates resting on the elastic foundation.

#### Appendix

#### The Shape Function of the Quadrilateral Element

The Lagrange interpolations in Equation (38):

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2019.330.