Vibration Characteristic Analysis of Sandwich Composite Plate Reinforced by Functionally Graded Carbon Nanotube-Reinforced Composite on Winkler/Pasternak Foundation

Abstract

This paper presents numerical investigations into the free vibration properties of a sandwich composite plate with two fiber-reinforced plastic (FRP) face sheets and a functionally graded carbon nanotube-reinforced composite (FG-CNTRC) core made of functionally graded carbon nanotube-reinforced composite resting on Winkler/Pasternak elastic foundation. The material properties of the FG-CNTRC core are gradient change along the thickness direction with four distinct carbon nanotubes reinforcement distribution patterns. The Hamilton energy concept is used to develop the equations of motion, which are based on the high-order shear deformation theory (HSDT). The Navier method is then used to obtain the free vibration solutions. By contrasting the acquired results with those using finite elements and with the previous literature, the accuracy of the present approach is confirmed. Moreover, the effects of the modulus of elasticity, the carbon nanotube (CNT) volume fractions, the CNT distribution patterns, the gradient index p, the geometric parameters and the dimensionless natural frequencies’ elastic basis characteristics are examined. The results show that the FG-CNTRC sandwich composite plate has higher dimensionless frequencies than the functionally graded material (FGM) plate or sandwich plate. And the volume fraction of carbon nanotubes and other geometric factors significantly affect the dimensionless frequency of the sandwich composite plate.

1. Introduction

Sandwich composite plates are widely used in many industrial applications such as the aerospace manufacturing, marine, and machinery fields. In the engineering environment, these structural elements are frequently exposed to erratic loads like ocean waves, wind, and noise. Prolonged vibration can lead to fatigue failure, which can jeopardize structural safety. Thus, from a practical design perspective, it is essential to understand their vibration properties [1,2,3,4,5,6,7,8,9,10]. A spring-supported structure is formed by a sequence of independent linear elastic springs in the Winkler–Pasternak model. When examining how structures like buildings, bridges, and machine foundations behave, the Winkler–Pasternak model is especially helpful. It makes it possible for engineers to thoroughly examine a structure’s performance and safety in actual soil situations. For instance, in marine structures, the bottom bases of subsea structures (such as the subsea foundations in Figure 1) can be represented as plate structures resting on a Winkler–Pasternak foundation, which can take into account the Winkler spring stiffness of the seabed in the vertical direction and the Pasternak model for resistance to horizontal shear deformation [11]. The Winkler–Pasternak foundation can more accurately evaluate the dynamic characteristics of marine subsea plate structures with soil–structure interactions.

Many researchers have examined the free vibration of sandwich plates using the Winkler–Pasternak (WP) model, based on the shear deformation theories [13,14,15,16,17,18]. Arani and Shariyat [19] assessed the vibrational responses of sandwich panels on WP foundation. Using first-order shear deformation theory (FSDT), Kumar and Harsha [20] studied the bending and diversion of FG panels on elastic foundations. Three-order shear deformation theory (TSDT) was used by Hadji et al. [21] to show the vibrational responses of a FG sandwich plate on an elastic base with an emphasis on geometrical parameters and material distribution. The nonlinear vibration of a sandwich FG plates on a resilient foundations, considering various material parameters was investigated by Li et al. [22]. Shahrany et al. [23] utilized Navier’s theoretical solution to study the dynamic response of a sandwich beam on an elastic basis. Selim and Liu [24] used a numerical technique based on HSDT for a parametric study of FG-GPLs on robust foundations. Zaitoun et al. [25] used HSDT to create a modeling strategy and obtain a precise answer for the buckling response of a FG sandwich supported on an elastic foundation. An analysis approach for examining the properties of a sandwich plate’s free vibration that is supported by a Pasternak foundation was introduced by Zenkour and El-Shahrany [26]. A novel method for examining the vibration characteristics of sandwich FGM plates resting on an elastic foundation in a heated environment was presented by Singh and Harsha [27]. Shahsavari et al. [28] used an elastic-foundation methodology to conduct multiple parametric investigations for the dynamic responses of FG imperfect plates. In addition, many researchers have also studied numerical methods, like the finite element method. Paik and Seo [29] used the nonlinear finite element method to study the application of ultimate strength to steel plate structures under combined forces. To forecast the steel model’s ultimate strength, Putranto et al. [30] carried out a ESL finite element analysis.

Fiber-reinforced plastics (FRPs) are popular and incredibly high-performing composite materials that combine the flexibility and malleability of a matrix resin with the high strength of fibers and are a hot research topic today. They are used extensively in many different industries due to their exceptional mechanical qualities, low weight, and resistance to corrosion. FRPs have many applications, ranging from sports equipment and aerospace to transportation and construction, and their use is growing as technology advances [31].

In recent years, FG-CNTRC has drawn more attention because it eliminates interfacial tensions by gradually changing its material composition as thickness increases [32,33,34,35,36,37]. Due to their exceptional temperature resistance, lightweight design, and extreme durability, FGM sandwich constructions can withstand high temperatures without losing structural strength or stiffness while also weighing less overall. FGMs are perfect for the nuclear reactor, mechanical engineering, aerospace, and marine industries because of these qualities. This soft, low-density, high-aspect ratio material has developed into a high-quality reinforcing phase in recent years due to the use of carbon nanotubes (CNTs) [38,39,40,41,42,43]. According to previous research, adding carbon nanotubes greatly enhances the characteristics of metals, ceramics, and polymers. In the research conducted by Mohebpour et al. [44], a numerical approach was presented to study dynamic responses of FG-CNTRC pipes. The vibrational responses of FG-CNTRC, indicating the impacts of CNT volume fractions and distribution patterns on vibration behavior were investigated by Tayeb et al. [45]. Cho [46] presented a numerical analysis based on element method to research the large deflection static problem of FG-CNTRC plate on elastic basis. Duong et al. [47] conducted a thorough investigation of the stress centralization phenomena and the static analysis of FG-CNTRC cylindrical shells. Duc and Minh [48] investigated the free vibration behavior of cracked FG-CNTRC plates using shear deformation theory and finite element method. Taheri and Memarzadeh [49] studied the dynamic behavior of cracked CNTRC plates under load and investigated the effect of various parameters on the vibrational behavior. Zeighami and Jafari [50] used a numerical solution to analyze thermal stresses of FG-CNTRC plate. Cong et al.’s study [51] examined the behavior of the laminated double-curved thin shell made of FG-CNTRC under different geometrical conditions. The vibration properties of a sandwich comprising FG-CNTRC panels and a metal core were examined by Watts et al. [52]. Duc and Minh [48] used TSDT to study the dynamic characteristics of FG-CNTRC plates with cracks. A high-order discretization was presented by Lin et al. [53] to study the FG-CNTRC beam’s random vibration behavior. The aforementioned research has demonstrated that carbon nanotubes can greatly enhance a material’s mechanical qualities. However, the application of CNTs in FRP-FGM-FRP sandwich plates has never been developed, and its kinetic properties are not yet clear.

Therefore, the novelty and importance of this work are found in the functional core layer that is created when CNTs are added to FGMs made of metals and ceramics to create the FG-CNTRC plate. This further enhances the sandwich composite plate’s dimensionless fundamental frequency. Furthermore, it is believed that the Winkler/Pasternak foundation is the location of this plate. Hamilton’s principle and HSDT were used to develop differential equations, and Navier’s method was used to obtain the dynamic results. Moreover, the impact of several aspects on the non-dimensional frequency of the sandwich plate are examined, including geometric parameters, gradient index, carbon nanotube volume percentage, carbon nanotube dispersal type, and elastic foundation parameters.

2. Theoretical Formulation

2.1. Description of the Model

In this paper, a rectangular sandwich composite plate in a Cartesian coordinate system (x-y-z) with two fiber-reinforced plastics (FRP) face sheets and a novel hybrid composite core is considered. The hybrid core consists of carbon nanotube-reinforced composites (CNTRCs) and metal–ceramic functionally graded (FG) matrix. Winkler and Pasternak foundations support the sandwich plate, which has dimensions of a, b, and h. Furthermore, the core and face sheets have thicknesses of h_c and h_f, respectively. The schematic diagram of the structure is shown in Figure 2.

2.2. Material Properties of CNT-Reinforced Plates

The material composition of the FG plate progressively changes from all-ceramic on the upper surface to all-metal on the bottom surface, with material characteristics varying in accordance with power law regulations [54]. The hybrid composite core’s matrix is made of metal–ceramic functionally graded materials. For instance, the mixing rule determines the power law distribution of the effective modulus of elasticity and density, which varies continuously along the thickness direction. The core matrix’s material properties are shown in Equation (1a,b). For the FGM core, Poisson’s coefficient is assumed to be constant (i.e., it is independent of the vertical coordinate z). Moreover, Young’s modulus E and mass density p are independent of temperature.

$$E_m(z)=(E_t-E_b){({\displaystyle \frac{z}{h_c}}+{\displaystyle \frac{1}{2}})}^p+E_b$$

(1a)

$${\rho }_m(z)=({\rho }_t-{\rho }_b){({\displaystyle \frac{z}{h_c}}+{\displaystyle \frac{1}{2}})}^p+{\rho }_b$$

(1b)

where $E_m$ and ${\rho }_m$ denote the elasticity modulus and density of the interlayer matrix; $E_t$ and $E_b$ denote the elasticity modulus of the ceramic and metal; ${\rho }_t$ and ${\rho }_b$ denote the density of the ceramic and metal, respectively; and p is the gradient index.

In this research, four types of CNT distributions (UD, FG-V, FG-O, and FG-X) along the thickness direction of the intermediate layer are investigated, as seen in Figure 3. The volume fractions of CNTs with different distribution types are shown in Equation (2a–d):

$$\mathrm{UD}:V_{CNT}(z)=V_{CNT}^{*}$$

(2a)

$$\mathrm{FG}\text{-}\mathrm{V}:V_{CNT}(z)=(1+{\displaystyle \frac{2z}{h_c}})V_{CNT}^{*}$$

(2b)

$$\mathrm{FG}\text{-}\mathrm{O}:V_{CNT}(z)=2(1-{\displaystyle \frac{2\left|z\right|}{h_c}})V_{CNT}^{*}$$

(2c)

$$\mathrm{FG}\text{-}\mathrm{X}:V_{CNT}(z)=2({\displaystyle \frac{2\left|z\right|}{h_c}})V_{CNT}^{*}$$

(2d)

where

$$V_{CNT}^{*}(z)={\displaystyle \frac{{\omega }_{CNT}}{{\omega }_{CNT}+\frac{{\rho }_{CNT}}{{\rho }_m}-\frac{{\rho }_{CNT}}{{\rho }_m}{\omega }_{CNT}}}$$

(3)

In which ${\omega }_{CNT}$ denotes the mass fraction of carbon nanotubes; ${\rho }_{CNT}$ and ${\rho }_m$ are the densities of the carbon nanotubes and matrix, respectively; and z is the interlayer thickness coordinate.

The effective elastic modulus, shear modulus, Poisson’s ratio, and mass density of the hybrid composite core in this work can be obtained from Equation (4a–f) by incorporating the efficiency parameters of CNTs:

$$E_{11}(z)={\eta }_1{V}_{CNT}(z)E_{11}^{CNT}+V_m{E}^m$$

(4a)

$${\displaystyle \frac{{\eta }_2}{E_{22}(z)}}={\displaystyle \frac{V_{CNT}(z)}{E_{22}^{CNT}}}+{\displaystyle \frac{V_m(z)}{E^m}}$$

(4b)

$${\displaystyle \frac{{\eta }_3}{G_{12}(z)}}={\displaystyle \frac{V_{CNT}(z)}{G_{12}^{CNT}}}+{\displaystyle \frac{V_m(z)}{G^m}}$$

(4c)

$${\mu }_{12}(z)=V_{CNT}(z){\mu }_{12}^{CNT}+V_m(z){\mu }^m$$

(4d)

$${\mu }_{21}(z)={\displaystyle \frac{{\mu }_{12}(z)}{E_{11}(z)}}{E}_{22}(z)$$

(4e)

$$\rho (z)=V_{CNT}(z){\rho }^{CNT}+V_m(z){\rho }^m$$

(4f)

where $E_{11}^{CNT}$, $E_{22}^{CNT}$, $G_{12}^{CNT}$, ${\mu }_{12}^{CNT}$, and ${\rho }^{CNT}$ denote the longitudinal and transverse elastic modulus, shear modulus, Poisson’s ratio, and density of carbon nanotubes, respectively. $E^m$, $G^m$, ${\mu }^m$, and ${\rho }^m$ denote the corresponding properties of the isotropic FGM matrix, respectively. ${\eta }_1$, ${\eta }_2$, and ${\eta }_3$ are the efficiency parameters considering the three directions of CNTs, and the values of the CNTs efficiency parameters used according to Shen [55] are shown in

Table 1. Three different nanotube volume fractions’ values for the efficiency parameters [55].

VCNT	η1	η2	η3
0.12	0.137	1.022	0.715
0.17	0.142	1.626	1.138
0.28	0.141	1.585	1.109

. $V_{CNT}$ and $V_m$ denote the volume fractions of the CNTs and FGM matrix materials, and we obtain $V_{CNT}$ + $V_m$ =1.

2.3. HSDT-Based Models

The HSDT-based displacements of the novel sandwich composite plate along x, y, and z can be given as

$$\left\{\begin{array}{l}u(x,y,z,t)\\ v(x,y,z,t)\\ w(x,y,z,t)\end{array}\right\}=\left\{\begin{array}{l}{u}_0(x,y,t)\\ v_0(x,y,t)\\ w_0(x,y,t)\end{array}\right\}-\left\{\begin{array}{c}z{\displaystyle \frac{\partial w_0}{\partial x}}\\ z{\displaystyle \frac{\partial w_0}{\partial y}}\\ 0\end{array}\right\}+\left\{\begin{array}{c}f(z){\theta }_x\\ f(z){\theta }_y\\ 0\end{array}\right\}$$

(5)

where the midplane surface is indicated by the subscript 0, and the displacements in the x, y, and z directions are indicated by u, v, and w. The transverse shear shape function is denoted by $f(z)$. Rotations of the midplane normal along the x and y axes are represented by the numbers ${\theta }_x$ and ${\theta }_y$. Equation (6) can be used to express the strain at a location in the plate when taking into account transverse and in-plane shear strains.

$$\epsilon ={\epsilon }^0+z{\epsilon }^b+f(z){\epsilon }^s;\gamma =f\prime (z){\gamma }^0$$

(6)

where

$$\begin{array}{l}{\epsilon }^0=\left\{\begin{array}{l}{\epsilon }_x^0\\ {\epsilon }_y^0\\ {\gamma }_{xy}^0\end{array}\right\}=\left\{\begin{array}{l}{\displaystyle \frac{\partial u_0}{\partial x}}\\ {\displaystyle \frac{\partial v_0}{\partial y}}\\ {\displaystyle \frac{\partial v_0}{\partial x}}+{\displaystyle \frac{\partial u_0}{\partial y}}\end{array}\right\},{\epsilon }^b=\left\{\begin{array}{l}{\epsilon }_x^b\\ {\epsilon }_y^b\\ {\gamma }_{xy}^b\end{array}\right\}=\left\{\begin{array}{l}-{\displaystyle \frac{{\partial }^2{w}_0}{\partial x^2}}\\ -{\displaystyle \frac{{\partial }^2{w}_0}{\partial y^2}}\\ -2{\displaystyle \frac{{\partial }^2{w}_0}{\partial x\partial y}}\end{array}\right\},{\epsilon }^s=\left\{\begin{array}{l}{\epsilon }_x^s\\ {\epsilon }_y^s\\ {\gamma }_{xy}^s\end{array}\right\}=\left\{\begin{array}{l}{\displaystyle \frac{\partial {\theta }_x}{\partial x}}\\ {\displaystyle \frac{\partial {\theta }_y}{\partial y}}\\ {\displaystyle \frac{\partial {\theta }_x}{\partial y}}+{\displaystyle \frac{\partial {\theta }_y}{\partial x}}\end{array}\right\},\\ {\gamma }^0=\left\{\begin{array}{l}{\gamma }_{xz}^0\\ {\gamma }_{yz}^0\end{array}\right\}=\left\{\begin{array}{l}{\theta }_x\\ {\theta }_y\end{array}\right\}\end{array}$$

(7)

where ${\epsilon }_x$ and ${\epsilon }_y$ are the normal strains, ${\gamma }_{xy}$ is the in-plane shear strain, ${\gamma }_{xz}$ and ${\gamma }_{yz}$ are the transverse shear strains. In this paper, due to the anisotropic FRP face sheets and the isotropic core, their link between stress and strain can be stated as

$${\left\{\begin{array}{l}{\sigma }_x\\ {\sigma }_y\\ {\tau }_{xy}\\ {\tau }_{yz}\\ {\tau }_{xz}\end{array}\right\}}^{(k)}=\left[\begin{array}{ccccc}{\overline{Q}}_{11}^{(k)}& {\overline{Q}}_{12}^{(k)}& {\overline{Q}}_{16}^{(k)}& 0& 0\\ {\overline{Q}}_{12}^{(k)}& {\overline{Q}}_{22}^{(k)}& {\overline{Q}}_{26}^{(k)}& 0& 0\\ {\overline{Q}}_{16}^{(k)}& {\overline{Q}}_{26}^{(k)}& {\overline{Q}}_{66}^{(k)}& 0& 0\\ 0& 0& 0& {\overline{Q}}_{44}^{(k)}& {\overline{Q}}_{45}^{(k)}\\ 0& 0& 0& {\overline{Q}}_{45}^{(k)}& {\overline{Q}}_{55}^{(k)}\end{array}\right]{\left\{\begin{array}{l}{\epsilon }_x\\ {\epsilon }_y\\ {\gamma }_{xy}\\ {\gamma }_{yz}\\ {\gamma }_{xz}\end{array}\right\}}^{(k)}$$

(8a)

$$\begin{array}{l}\left\{\begin{array}{l}{\sigma }_x\\ {\sigma }_y\\ {\tau }_{yz}\\ {\tau }_{xz}\\ {\tau }_{xy}\end{array}\right\}=\left[\begin{array}{ccccc}{Q}_{11}& Q_{12}& 0& 0& 0\\ Q_{12}& Q_{22}& 0& 0& 0\\ 0& 0& Q_{44}& 0& 0\\ 0& 0& 0& Q_{55}& 0\\ 0& 0& 0& 0& Q_{66}\end{array}\right]\left\{\begin{array}{l}{\epsilon }_x\\ {\epsilon }_y\\ {\gamma }_{yz}\\ {\gamma }_{xz}\\ {\gamma }_{xy}\end{array}\right\}\end{array}$$

(8b)

where ${\sigma }_x$ and ${\sigma }_y$ are the in-plane normal stresses, ${\tau }_{xy}$ is the in-plane shear stress, and ${\tau }_{xz}$ and ${\tau }_{yz}$ are the transverse shear stresses. k denotes the kth layer of the FRP plate. The stiffness coefficients Q_ij for the FRP face sheets are expressed as

$$Q_{11}=Q_{22}={\displaystyle \frac{E(z)}{1-{\mu }^2}},Q_{12}={\displaystyle \frac{\mu E(z)}{1-{\mu }^2}},Q_{44}=Q_{55}=Q_{66}={\displaystyle \frac{E(z)}{2(1+\mu )}}$$

(9)

and

$$\begin{array}{l}{\overline{Q}}_{11}^{(k)}=Q_{11}{\cos }^4{\alpha }_k+2(Q_{12}+2Q_{66}){\sin }^2{\alpha }_k{\cos }^2{\alpha }_k+Q_{22}{\sin }^4{\alpha }_k\\ {\overline{Q}}_{12}^{(k)}=Q_{12}({\sin }^4{\alpha }_k+{\cos }^4{\alpha }_k)+(Q_{11}+Q_{22}-4Q_{66}){\sin }^2{\alpha }_k{\cos }^2{\alpha }_k\\ {\overline{Q}}_{22}^{(k)}=Q_{11}{\cos }^4{\alpha }_k+2(Q_{12}+2Q_{66}){\sin }^2{\alpha }_k{\cos }^2{\alpha }_k+Q_{11}{\sin }^4{\alpha }_k\\ {\overline{Q}}_{16}^{(k)}=(Q_{11}-Q_{12}-2Q_{66})\sin {\alpha }_k{\cos }^3{\alpha }_k+(Q_{12}-Q_{22}+2Q_{66}){\sin }^3{\alpha }_k\cos {\alpha }_k\\ {\overline{Q}}_{26}^{(k)}=(Q_{11}-Q_{22}+2Q_{66})\sin {\alpha }_k{\cos }^3{\alpha }_k+(Q_{11}-Q_{12}-2Q_{66}){\sin }^3{\alpha }_k\cos {\alpha }_k\\ {\overline{Q}}_{66}^{(k)}=Q_{66}({\sin }^4{\alpha }_k+{\cos }^4{\alpha }_k)+(Q_{11}+Q_{22}-2Q_{12}-2Q_{66}){\sin }^2{\alpha }_k{\cos }^2{\alpha }_k\\ {\overline{Q}}_{44}^{(k)}=Q_{44}{\cos }^2{\alpha }_k+Q_{55}{\sin }^2{\alpha }_k\\ {\overline{Q}}_{45}^{(k)}=(Q_{55}-Q_{44})\sin {\alpha }_k\cos {\alpha }_k\\ {\overline{Q}}_{55}^{(k)}=Q_{55}{\cos }^2{\alpha }_k+Q_{44}{\sin }^2{\alpha }_k\end{array}$$

(10)

where ${\alpha }_k$ denotes the angle of fiber formation in the kth layer. The stiffness coefficients Q_ij for the hybrid composite core are expressed as

$$Q_{11}={\displaystyle \frac{E_{11}(z)}{1-{\mu }_{12}{\mu }_{21}}},Q_{12}={\displaystyle \frac{{\mu }_{21}{E}_{11}(z)}{1-{\mu }_{12}{\mu }_{21}}},Q_{22}={\displaystyle \frac{E_{22}(z)}{1-{\mu }_{12}{\mu }_{21}}},Q_{44}=G_{23},Q_{55}=G_{13},Q_{66}=G_{12}$$

(11)

The forces and moments of the sandwich plate can be written in Equation (12).

$$\left\{\begin{array}{l}{N}_x\\ M_x\\ S_x\\ N_y\\ M_y\\ S_y\\ N_{xy}\\ M_{xy}\\ S_{xy}\end{array}\right\}=\left[\begin{array}{ccccccccc}{A}_{11}& A_{12}& A_{16}^k& B_{11}& B_{12}& B_{16}^k& C_{11}& C_{12}& C_{16}^k\\ B_{11}& B_{12}& B_{16}^k& D_{11}& D_{12}& D_{16}^k& E_{11}& E_{12}& E_{16}^k\\ C_{11}& C_{12}& C_{16}^k& E_{11}& E_{12}& E_{16}^k& F_{11}& F_{12}& F_{16}^k\\ A_{12}& A_{22}& A_{26}^k& B_{12}& B_{22}& B_{26}^k& C_{12}& C_{22}& C_{26}^k\\ B_{12}& B_{22}& B_{26}^k& D_{12}& D_{22}& D_{26}^k& E_{12}& E_{22}& E_{26}^k\\ C_{12}& C_{22}& C_{26}^k& E_{12}& E_{22}& E_{26}^k& F_{12}& F_{22}& F_{26}^k\\ A_{16}^k& A_{26}^k& A_{66}& B_{16}^k& B_{26}^k& B_{66}& C_{16}^k& C_{26}^k& C_{66}\\ B_{16}^k& B_{26}^k& B_{66}& D_{16}^k& D_{26}^k& D_{66}& E_{16}^k& E_{26}^k& E_{66}\\ C_{16}^k& C_{26}^k& C_{66}& E_{16}^k& E_{26}^k& E_{66}& F_{16}^k& F_{26}^k& F_{66}\end{array}\right]\left\{\begin{array}{l}{\epsilon }_x^0\\ {\epsilon }_y^0\\ {\gamma }_{xy}^0\\ {\epsilon }_x^b\\ {\epsilon }_y^b\\ {\gamma }_{xy}^b\\ {\epsilon }_x^s\\ {\epsilon }_y^s\\ {\gamma }_{xy}^s\end{array}\right\},\left\{\begin{array}{l}{N}_{xz}\\ N_{yz}\end{array}\right\}=\left[\begin{array}{cc}{G}_{55}& G_{45}^k\\ G_{45}^k& G_{44}\end{array}\right]\left\{\begin{array}{l}{\gamma }_{xz}^0\\ {\gamma }_{yz}^0\end{array}\right\}$$

(12)

where $P_{ij}^k$ (P = A, B, C, D, E, F and G) = 0 when k = CNT, and P ≠ 0 when p = FRP. These coefficients can be specified as

$$\begin{array}{l}(A_{ij},B_{ij},C_{ij},D_{ij},E_{ij},F_{ij},G_{ij})={\eta }_{ij}={\eta }_{ij}^t+{\eta }_{ij}^c+{\eta }_{ij}^b={\displaystyle {\int }_{-\frac{h}{2}}^{-\frac{h_c}{2}}{\overline{Q}}_{ij}^{(k)}\{1,z,f(z),z^2,zf(z),f^2(z),f\prime (z)\}}dz\\ +{\displaystyle {\int }_{-\frac{h_c}{2}}^{\frac{h_c}{2}}{Q}_{ij}\{1,z,f(z),z^2,zf(z),f^2(z),f\prime (z)\}}dz+{\displaystyle {\int }_{\frac{h_c}{2}}^{\frac{h}{2}}{\overline{Q}}_{ij}^{(k)}\{1,z,f(z),z^2,zf(z),f^2(z),f\prime (z)\}}dz\end{array}$$

(13)

where $\eta$ = A, B, C, D, E, F, and G; $i,j=1,2,4,5\mathrm{and}6$; and the superscript “t”, “c”, and “b” represent the sandwich plate’s top, middle, and bottom layers, respectively. The governing equations of motion for the sandwich plate can be determined by using Hamilton’s principle, which is given by

$${\displaystyle {\int }_0^t(\delta U+\delta V_e-\delta T)}dt=0$$

(14)

where the variations in strain energy are denoted by $\delta U$, the elastic basis’s potential energy is represented by $\delta V_e$, and the kinetic energy is represented by $\delta T$. The plate’s strain energy can be described as

$$\begin{array}{l}\delta U={\displaystyle {\int }_A{\displaystyle \underset{-h/2}{\overset{h/2}{\int }}(}}{\sigma }_x\delta {\epsilon }_x+{\sigma }_y\delta {\epsilon }_y+{\sigma }_{xy}\delta {\gamma }_{xy}+{\sigma }_{xz}\delta {\gamma }_{xz}+{\sigma }_{yz}\delta {\gamma }_{yz})dAdz\\ ={\displaystyle {\int }_A}[N_x{\displaystyle \frac{\partial \delta u_0}{\partial x}}-M_x{\displaystyle \frac{{\partial }^2\delta w_0}{\partial x^2}}+S_x{\displaystyle \frac{\partial \delta {\theta }_x}{\partial x}}+N_y{\displaystyle \frac{\partial \delta v_0}{\partial y}}-M_y{\displaystyle \frac{{\partial }^2\delta w_0}{\partial y^2}}+S_y{\displaystyle \frac{\partial \delta {\theta }_y}{\partial y}}+N_{xy}({\displaystyle \frac{\partial \delta u_0}{\partial y}}+{\displaystyle \frac{\partial \delta v_0}{\partial x}})\\ -2M_{xy}{\displaystyle \frac{{\partial }^2\delta w_0}{\partial x\partial y}}+S_{xy}({\displaystyle \frac{\partial \delta {\theta }_x}{\partial y}}+{\displaystyle \frac{\partial \delta {\theta }_y}{\partial x}})+N_{xz}\delta {\theta }_x+N_{yz}\delta {\theta }_y]dA\end{array}$$

(15)

For the variation in elastic potential energy, it can be expressed as

$$\delta V_e={{\displaystyle {\int }_{A}f}}_e\delta w_{0}dA$$

(16)

where $f_e$ is specifically expressed in the form [56]

$$\begin{array}{l}{f}_e(Winkler)=q_{Winkler}=K_w{w}_0,\\ f_e(Pasternak)=q_{Pasternak}=K_w{w}_0-G_p{\nabla }^2{w}_0\end{array}$$

(17)

where ${\nabla }^2={\partial }^2/\partial x^2+{\partial }^2/\partial y^2$ is the Laplace operator in a rectangular Cartesian coordinate system [57]. The kinetic energy of the plate is specified as

$$\begin{array}{l}T={\displaystyle {\int }_A{\displaystyle {\int }_{-h/2}^{h/2}}(\dot{u}\delta \dot{u}+\dot{v}\delta \dot{v}+\dot{w}\delta \dot{w})\rho (z)dzdA}\\ {\displaystyle {\int }_A}\{I_0({\dot{u}}_0\delta {\dot{u}}_0+{\dot{v}}_0\delta {\dot{v}}_0+{\dot{w}}_0\delta {\dot{w}}_0)-I_1({\dot{u}}_0{\displaystyle \frac{\partial \delta {\dot{w}}_0}{\partial x}}+{\displaystyle \frac{\partial {\dot{w}}_0}{\partial x}}\delta {\dot{u}}_0+{\dot{v}}_0{\displaystyle \frac{\partial \delta {\dot{w}}_0}{\partial x}}+{\displaystyle \frac{\partial {\dot{w}}_0}{\partial x}}\delta {\dot{v}}_0)\\ +I_2({\displaystyle \frac{\partial {\dot{w}}_0}{\partial x}}{\displaystyle \frac{\partial \delta {\dot{w}}_0}{\partial x}}+{\displaystyle \frac{\partial {\dot{w}}_0}{\partial y}}{\displaystyle \frac{\partial \delta {\dot{w}}_0}{\partial y}})+I_3({\dot{u}}_0\delta {\dot{\theta }}_x+{\dot{\theta }}_x\delta {\dot{u}}_0+{\dot{v}}_0\delta {\dot{\theta }}_y+{\dot{\theta }}_y\delta {\dot{v}}_0)+I_4({\dot{\theta }}_x\delta {\dot{\theta }}_x+{\dot{\theta }}_y\delta {\dot{\theta }}_y)\\ -I_5({\displaystyle \frac{\partial {\dot{w}}_0}{\partial x}}\delta {\dot{\theta }}_x+{\dot{\theta }}_x{\displaystyle \frac{\partial \delta {\dot{w}}_0}{\partial x}}+{\displaystyle \frac{\partial {\dot{w}}_0}{\partial y}}\delta {\dot{\theta }}_y+{\dot{\theta }}_y{\displaystyle \frac{\partial \delta {\dot{w}}_0}{\partial y}})\}dA\end{array}$$

(18)

where the differentiation with regard to the time variable is indicated by the superscripted dot, and $\rho (z)$ denotes the density along the thickness direction z. $I_0,I_1,I_2,I_3,I_4,I_5$ are the inertia terms, which are expressed as

$$\begin{array}{l}(I_0,I_1,I_2,I_3,I_4,I_5,)={\displaystyle {\int }_{-h/2}^{-h_c/2}[1,z,z^2,f(z),f(z)z,f^2(z)]}\rho (z)dz\\ +{\displaystyle {\int }_{-h_c/2}^{h_c/2}[1,z,z^2,f(z),f(z)z,f^2(z)]}\rho (z)dz+{\displaystyle {\int }_{h_c/2}^{h/2}[1,z,z^2,f(z),f(z)z,f^2(z)]}\rho (z)dz\end{array}$$

(19)

Substituting Equations (15), (16), and (18) into Equation (14), the governing equations for the plate can be derived using the Hamiltonian energy principle and collected $\delta u_0$,$\delta v_0$, $\delta w_0$, $\delta {\theta }_x$ and $\delta {\theta }_y$, respectively denoted as

$$\delta u_0:{\displaystyle \frac{\partial N_x}{\partial x}}+{\displaystyle \frac{\partial N_{xy}}{\partial y}}=I_0{\ddot{u}}_0-I_1{\displaystyle \frac{\partial \ddot{w}}{\partial x}}-I_3{\displaystyle \frac{\partial \ddot{\theta }}{\partial x}}$$

(20a)

$$\delta v_0:{\displaystyle \frac{\partial N_y}{\partial y}}+{\displaystyle \frac{\partial N_{xy}}{\partial x}}=I_0{\ddot{v}}_0-I_1{\displaystyle \frac{\partial \ddot{w}}{\partial y}}-I_3{\displaystyle \frac{\partial \ddot{\theta }}{\partial y}}$$

(20b)

$$\delta w_0:{\displaystyle \frac{{\partial }^2{M}_x}{\partial x^2}}+{\displaystyle \frac{{\partial }^2{M}_y}{\partial y^2}}+2{\displaystyle \frac{{\partial }^2{M}_{xy}}{\partial x\partial y}}-f_e=I_0{\ddot{w}}_0+I_1({\displaystyle \frac{\partial \ddot{u}}{\partial x}}+{\displaystyle \frac{\partial \ddot{v}}{\partial y}})-I_4({\displaystyle \frac{{\partial }^2{\ddot{\theta }}_x}{\partial x^2}}+{\displaystyle \frac{{\partial }^2{\ddot{\theta }}_y}{\partial y^2}})+I_0{\ddot{w}}_0$$

(20c)

$$\delta {\theta }_x:{\displaystyle \frac{\partial s_x}{\partial x}}+{\displaystyle \frac{\partial s_{xy}}{\partial y}}-N_{xz}=I_2{\ddot{u}}_0-I_4{\displaystyle \frac{\partial {\ddot{w}}_0}{\partial x}}+I_5{\ddot{\theta }}_x$$

(20d)

$$\delta {\theta }_y:{\displaystyle \frac{\partial s_y}{\partial y}}+{\displaystyle \frac{\partial s_{xy}}{\partial x}}-N_{yz}=I_2{\ddot{v}}_0-I_4{\displaystyle \frac{\partial {\ddot{w}}_0}{\partial y}}+I_5{\ddot{\theta }}_y$$

(20e)

Equation (12) is substituted into Equation (20a)–(20e), and by replacing the coefficients in the equation, expansions of the equation $\delta u_0$, $\delta v_0$, $\delta w_0$, $\delta {\theta }_x$, and $\delta {\theta }_y$ can be derived, and these expansions are listed in Appendix A.

2.4. Solutions for Simply Supported Sandwich Plates

The displacement variables for the sandwich plate to satisfy the boundary conditions are given as follows:

$$\begin{array}{l}{u}_0=U_{ij}\cos (\lambda x)\sin (\mu y)e^{iwt},v_0=V_{ij}\sin (\lambda x)\cos (\mu y)e^{iwt},w_0=W_{ij}\sin (\lambda x)\sin (\mu y)e^{iwt}\\ {\theta }_x=X_{ij}\cos (\lambda x)\sin (\mu y)e^{iwt},{\theta }_y=Y_{ij}\sin (\lambda x)\cos (\mu y)e^{iwt}\end{array}$$

(21)

where $U_{ij}$, $V_{ij}$, $W_{ij}$, $X_{ij}$, and $Y_{ij}$ are the five unknown parameters, w represents the natural frequency. The variables λ = iπ/a, μ = jπ/b, where i and j are truncated numbers. For sinusoidally distributed loads, the following equation is obtained by substituting Equation (21) into the expansions of the equation, i.e., $\delta u_0$, $\delta v_0$, $\delta w_0$, $\delta {\theta }_x$, and $\delta {\theta }_y$:

$$\left(\left[\begin{array}{ccccc}{P}_{11}& P_{12}& P_{13}& P_{14}& P_{15}\\ P_{21}& P_{22}& P_{23}& P_{24}& P_{25}\\ P_{31}& P_{32}& P_{33}& P_{34}& P_{35}\\ P_{41}& P_{42}& P_{43}& P_{44}& P_{45}\\ P_{51}& P_{52}& P_{53}& P_{54}& P_{55}\end{array}\right]-w^2\left[\begin{array}{ccccc}{Q}_{11}& Q_{12}& Q_{13}& Q_{14}& Q_{15}\\ Q_{21}& Q_{22}& Q_{23}& Q_{24}& Q_{25}\\ Q_{31}& Q_{32}& Q_{33}& Q_{34}& Q_{35}\\ Q_{41}& Q_{42}& Q_{43}& Q_{44}& Q_{45}\\ Q_{51}& Q_{52}& Q_{53}& Q_{54}& Q_{55}\end{array}\right]\right)\left\{\begin{array}{l}{U}_{ij}\\ V_{ij}\\ W_{ij}\\ X_{ij}\\ Y_{ij}\end{array}\right\}=\left\{\begin{array}{l}0\\ 0\\ 0\\ 0\\ 0\end{array}\right\}$$

(22)

where

$$\begin{array}{l}{P}_{11}=-A_{11}{\lambda }^2-2A_{16}\lambda \mu -A_{66}{\mu }^2,P_{12}=-A_{12}\lambda \mu -A_{16}{\lambda }^2-A_{26}{\mu }^2-A_{66}\lambda \mu ,P_{13}=B_{11}{\lambda }^3+B_{12}\lambda {\mu }^2+3B_{16}{\lambda }^2\mu \\ +B_{26}{\mu }^3+2B_{66}\lambda {\mu }^2,P_{14}=-C_{11}{\lambda }^2-2C_{16}\lambda \mu -C_{66}{\mu }^2,P_{15}=-C_{12}\lambda \mu -C_{16}{\lambda }^2-C_{26}{\mu }^2-C_{66}\lambda \mu ,P_{22}=-A_{22}{\mu }^2\\ -2A_{26}\lambda \mu -A_{66}{\lambda }^2,P_{23}=B_{12}{\lambda }^2\mu +3{{\rm B}}_{26}\lambda {\mu }^2+B_{16}{\lambda }^3+2{{\rm B}}_{66}{\lambda }^2\mu +{{\rm B}}_{22}{\mu }^3,P_{24}=-C_{12}\lambda \mu -C_{26}{\mu }^2-C_{16}{\lambda }^2\\ -C_{66}\lambda \mu ,P_{25}=-C_{22}{\mu }^2-2C_{26}\lambda \mu -C_{66}{\lambda }^2,P_{33}=-D_{11}{\lambda }^4-2D_{12}{\lambda }^2{\mu }^2+4D_{16}{\lambda }^3\mu +4D_{26}\lambda {\mu }^3-4D_{66}{\lambda }^2{\mu }^2\\ -D_{22}{\mu }^4-\underset{q_{Pasternak}}{\underbrace{\underset{q_{Winkler}}{\underbrace{k_w}}-G_p({\lambda }^2+{\mu }^2)}},P_{34}=E_{11}{\lambda }^3-3E_{16}{\lambda }^2\mu -E_{26}{\mu }^3+E_{12}\lambda {\mu }^2+2E_{66}\lambda {\mu }^2,P_{35}=E_{12}\lambda {\mu }^2\\ -E_{16}{\lambda }^3-3E_{26}\lambda {\mu }^2+E_{22}{\mu }^3+2E_{66}{\lambda }^2\mu ,P_{44}=-F_{11}{\lambda }^2-2F_{16}\lambda \mu -F_{66}{\mu }^2-G_{55},P_{45}=-F_{12}\lambda \mu -F_{16}{\lambda }^2\\ -F_{26}{\mu }^2-F_{66}\lambda \mu -G_{45},P_{55}=-G_{44}-F_{22}{\mu }^2-F_{66}{\lambda }^2-2F_{26}\lambda \mu \end{array}$$

(23)

in which

$$\begin{array}{l}{Q}_{11}=-I_0,Q_{12}=0,Q_{13}=I_1\lambda ,Q_{14}=-I_3,Q_{15}=0,Q_{22}=-I_0,Q_{23}=I_1\mu ,Q_{24}=0,Q_{25}=-I_3,\\ Q_{33}=-I_0-I_2{\lambda }^2-I_2{\mu }^2,Q_{34}=I_4\lambda ,Q_{35}=I_4\mu ,Q_{44}=-I_5,Q_{45}=0,Q_{55}=-I_5\end{array}$$

(24)

3. Results and Discussion

This first section is to confirm accuracy of the present method before the impact of a number of parameters on the free vibration intrinsic frequency of sandwich plates is conducted, including the modulus of elasticity, gradient index, kw, Gp, aspect ratio, volume fraction, core thickness ratio (h_c/h), and type of distribution of carbon nanotubes. Regarding the transverse shear function f(z) of the displacement field, the shear shape function proposed by Li [15] is used in this study, which is expressed in the form of f(z) = $8.169z-7.169h\log [\sec ({\displaystyle \frac{z}{h}})+\tan ({\displaystyle \frac{z}{h}})]$, and its derivative g(z) is denoted as g(z) = $8.169-7.169\sec ({\displaystyle \frac{z}{h}})$. The geometric parameters used in this analysis are a = b = 1 m, h_f = 0.01 m, h_c = 0.1 m, kw = Gp = 100; the fiber layer is [0/90/90/0]; and the CNT type is UD.

Table 2. Material properties of FRP sheet plate.

Material	${\mathit{E}}_{11}^{\mathit{F}\mathit{R}\mathit{P}}$ (GPa)	${\mathit{E}}_{22}^{\mathit{F}\mathit{R}\mathit{P}}$ (GPa)	${\mathit{\mu }}_{12}^{\mathit{F}\mathit{R}\mathit{P}}$	${\mathit{G}}_{12}^{\mathit{F}\mathit{R}\mathit{P}}$ (GPa)	${\mathit{G}}_{13}^{\mathit{F}\mathit{R}\mathit{P}}$ (GPa)	${\mathit{G}}_{12}^{\mathit{F}\mathit{R}\mathit{P}}$ (GPa)	${\mathit{G}}_{23}^{\mathit{F}\mathit{R}\mathit{P}}$ (GPa)
FRP	40	1	0.25	0.6	0.6	0.6	0.5

[58],

Table 3. Material properties of FGM plate.

Material	Eb (GPa)	ρb (kg/m3)	Et (GPa)	ρt (kg/m3)	${\mathit{\mu }}_{\mathit{b}}$	${\mathit{\mu }}_{\mathit{t}}$
FGM	70	2702	210	2400	0.3	0.3

[15], and

Table 4. Material properties of the CNTs.

Material	${\mathit{E}}_{11}^{\mathit{c}\mathit{n}\mathit{t}}$ (GPa)	${\mathit{E}}_{22}^{\mathit{c}\mathit{n}\mathit{t}}$ (GPa)	${\mathit{G}}_{12}^{\mathit{c}\mathit{n}\mathit{t}}$ (GPa)	${\mathit{\mu }}_{12}^{\mathit{c}\mathit{n}\mathit{t}}$	${\mathit{\rho }}^{\mathit{c}\mathit{n}\mathit{t}}$ (kg/m3)	VCNT
CNT	5.6466	70.8	1.9445	0.175	1150	0.12

[38] show the material properties.

3.1. Comparison and Validation

To demonstrate the accuracy and practicality of the proposed approach, this section provides several situations that contrast the obtained results with finite element and the literature results. For convenience, the following intrinsic frequencies are in a dimensionless form: $\tilde{\omega }=\omega h\sqrt{{\rho }_b/E_b}$. The formula for calculating the error is $Diff.=|(\tilde{\omega }-{\tilde{\omega }}_{\mathrm{Re}f})/{\tilde{\omega }}_{\mathrm{Re}f}|\times 100$.

Table 5. Dimensionless fundamental frequencies $\tilde{\omega }=\omega h\sqrt{{\rho }_b/E_b}$ of the FG plates.

kw	Gp	h/a	Model	p
				0	0.5	1	2	5
0	0	0.1	TSDT [59]	0.1134	0.0975	0.0891	0.0819	0.0767
			Quasi-3D [28]	0.1135	0.0970	0.0882	0.0806	0.0755
			Li et al. [15]	0.1135	0.0970	0.0882	0.0806	0.0756
			Present	0.1134	0.0963	0.0868	0.0788	0.0740
		0.2	TSDT [59]	0.4154	0.3606	0.3299	0.30160	0.2765
			Quasi-3D [28]	0.4168	0.3586	0.3260	0.29610	0.2722
			Li et al. [15]	0.4168	0.3586	0.3260	0.29610	0.2723
			Present	0.4150	0.3551	0.3205	0.28920	0.2667
100	0	0.1	TSDT [59]	0.1162	0.1012	0.0933	0.0867	0.0821
			Quasi-3D [28]	0.1163	0.1006	0.0923	0.0853	0.0809
			Li et al. [15]	0.1163	0.1006	0.0923	0.0853	0.0809
			Present	0.1161	0.0999	0.0910	0.0836	0.0795
		0.2	TSDT [59]	0.4273	0.3758	0.3476	0.3219	0.2999
			Quasi-3D [28]	0.4284	0.3734	0.3431	0.3159	0.2950
			Li et al. [15]	0.4281	0.373	0.3427	0.3154	0.2946
			Present	0.4269	0.3702	0.3381	0.3097	0.2901
100	100	0.1	TSDT [59]	0.1619	0.1563	0.1542	0.1535	0.1543
			Quasi-3D [28]	0.1616	0.1551	0.1525	0.1512	0.1521
			Li et al. [15]	0.1613	0.1548	0.1521	0.1509	0.1517
			Present	0.1617	0.1549	0.1519	0.1505	0.1515
		0.2	TSDT [59]	0.6162	0.6026	0.5978	0.5970	0.5993
			Quasi-3D [28]	0.6137	0.5940	0.5856	0.5815	0.5843
			Li et al. [15]	0.6085	0.5878	0.5792	0.5750	0.5720
			Present	0.6156	0.5950	0.5852	0.5810	0.5834

compares the dimensionless fundamental frequencies of a simply supported FG plate on a Winkler–Pasternak foundation with gradient index (0, 0.5, 1, 2, and 5) and thickness-to-width ratios (h/a = 0.05, 0.1, 0.15, and 0.2) for various elastic foundation parameters. Noting that the following material attributes are employed in this computation: E_b = 70 GPa, ρ_b = 2702 kg/m³, E_t = 380 GPa, ρ_t = 3800 kg/m³, and ${\mu }_b={\mu }_t=0.3$. As seen in Table 5, the present results are consistent with the quasi-3D HSDT results reported by Hasani et al. [59], Shahsavari et al. [28], and Li et al. [15]. Furthermore, there is greater agreement between the current numerical results and the quasi-3D results in ref. [15]. The dimensionless intrinsic frequency of simply supported metal–ceramic FGM plates on the Winkler–Pasternak foundation may be accurately calculated using the current algorithm.

The results of the first four orders of non-dimensional free vibration responses based on different CNTs distribution types (FG-V, FG-O, and FG-X) and aspect ratios (a/ℎ = 10, 20 and 50) for the FG plates are given in

Table 6. Validation study of non-dimensional first four free vibration frequencies of SSSS CNTRC plate based on different CNTs configurations and aspect ratios (V_CNT = 0.11).

Type	a/ℎ	Mode	Method
			TSDT [60]	RPT [61]	Present
FG-V	10	(1,1)	12.461	12.755	12.4138
		(1,2)	17.077	17.128	16.9426
		(2,1)	27.401	27.158	31.2429
		(2,2)	31.943	33.227	33.558
	20	(1,1)	15.09	15.127	15.040
		(1,2)	19.883	19.606	19.715
		(2,1)	38.936	38.855	47.316
		(2,2)	47.858	48.298	49.655
	50	(1,1)	16.230	16.093	16.1644
		(1,2)	21.114	20.683	20.9135
		(2,1)	60.094	59.872	59.4529
		(2,2)	62.682	62.118	61.9633
FG-O	10	(1,1)	11.319	11.773	9.568
		(1,2)	16.137	16.469	14.906
		(2,1)	26.555	26.827	26.553
		(2,2)	29.408	31.858	29.507
	20	(1,1)	13.405	13.50	10.567
		(1,2)	18.398	18.371	16.309
		(2,1)	38.938	38.855	34.945
		(2,2)	42.997	44.759	38.272
	50	(1,1)	14.265	14.153	10.914
		(1,2)	19.338	19.154	16.803
		(2,1)	52.653	52.616	39.268
		(2,2)	55.451	55.123	42.846
FG-X	10	(1,1)	14.683	15.254	14.586
		(1,2)	18.687	18.825	18.551
		(2,1)	28.524	28.004	33.447
		(2,2)	34.442	35.980	35.488
	20	(1,1)	19.94	20.241	19.866
		(1,2)	23.771	23.573	23.611
		(2,1)	38.938	38.855	56.591
		(2,2)	52.336	49.581	58.344
	50	(1,1)	22.954	22.880	22.873
		(1,2)	26.741	26.183	26.547
		(2,1)	83.153	83.604	82.218
		(2,2)	84.956	83.703	83.992

. And it should be mentioned that the matrix is pure metal in this case. Comparing the obtained result with Selim et al. [60] and Pasha et al. [61], it can be observed that the present modeling results are in good agreement with the existing ones of Selim and Pasha. The material properties used are as follows: E_m = 2.1 GPa; ρ_m = 1150 kg/m³; and ${\mu }_m=0.34$. The efficiency parameters used are similar to those assumed by Shen [55]: η₁ = 0.149, η₂ = 0.939, and η₃ = η₂ for V_CNT = 0.11.

Since there are no available literature results on the novel hybrid composite plates for comparison, the present results with different CNT volume fractions are compared with finite element ones. Moreover, Reddy shear shape function is also used for comparison. Results based on the aforementioned methodologies are shown in

Table 7. Comparison of the first eight natural frequencies (Hz) of three-dimensional sandwich plates under different CNT volume fractions with simple support conditions.

VCNT	Method	Mode
		1	2	3	4	5	6	7	8	Diff. (%)
0.12	ABAQUS	668.7	1552.9	1698.1	2387.5	2890.9	3188.7	3552.7	3725.4	–
	Present—Reddy [62]	669.0	1554.0	1700.0	2391.9	2895.3	3195.9	3560.4	3734.8	0.16
	Present	668.7	1555.0	1701.6	2391.8	2899.6	3201.1	3561.8	3737.1	0.22
0.17	ABAQUS	835.5	2012.6	2020.8	3004.6	3765.2	3781.6	4577.8	4589.9	–
	Present—Reddy [62]	834.8	2012.6	2018.8	3006.1	3767.9	3777.6	4582.5	4590.2	0.06
	Present	832.8	2011.6	2017.6	3001.4	3771.0	3780.6	4580.0	4587.5	0.11
0.28	ABAQUS	922.9	2193.7	2269.2	3308.1	4095.9	4254.8	5001.3	5089.4	–
	Present—Reddy [62]	916.3	2178.6	2256.7	3291.4	4067.0	4235.7	4973.2	5067.9	0.57
	Present	914.7	2181.3	2260.6	3290.8	4081.4	4252.5	4979.5	5076.5	0.43

(comparisons of the mode shapes are shown in Figure 4). The correctness and viability of the current model are demonstrated by the good agreement between the frequencies and other data.

3.2. Parametric Studies

This section examines how the free vibration frequencies of sandwich plates set on Winkler–Pasternak foundations are affected by geometric parameters, material characteristics, and elastic stiffnesses. Figure 5 demonstrates the effect of the thickness ratio h_c/h, the elasticity modulus ratios E₁/E₂, and gradient index p on the dimensionless fundamental frequency of the sandwich plate. The thickness of sandwich plate is h = 0.1 m. Figure 5 illustrates that when the elastic modulus ratio rises, so does the dimensionless fundamental frequency. However, as Figure 5 illustrates, when the h_c/h increases from 0.2 to 0.5, the dimensionless fundamental frequencies somewhat drop for p = 1, 5, and 10 and E₁/E₂ = 5. Furthermore, when the gradient index rises, the dimensionless fundamental frequency falls, with the decline being more rapid for large h_c/h. Figure 5 similarly shows a sharp rise in the dimensionless fundamental frequency as thickness ratios grow.

Figure 6 examines the impact of the gradient index (p = 0.5, 1, 5, and 10), core thickness ratio, and CNT volume fraction (V_CNT = 0, 0.12, 0.17, and 0.28) on the dimensionless fundamental frequency of sandwich plates. It is evident that the sandwich plate’s dimensionless fundamental frequency grows dramatically as the core thickness ratio rises from 0.2 to 1, while it remains nearly constant for core thickness ratios between 0 and 0.2. Furthermore, as the volume percentage of CNTs grows, so does the dimensionless fundamental frequency. It has been demonstrated that the structural rigidity of sandwich plates can be considerably increased by increasing their thickness and CNT volume fraction.

Figure 7 shows how the dimensionless fundamental frequency of sandwich plates varies with changes in the gradient index, core thickness ratio, and elastic stiffness parameters. The sandwich plate is 0.1 m thick. This image illustrates how the fundamental frequency rapidly decreases for small graded indices and decreases with an increase in the gradient index p. The reason behind this phenomenon is that the elastic modulus decease with the increase in graded indexes, and this leads to the decrease in bending stiffness of the whole sandwich plates. In addition, the dimensionless fundamental frequency significantly increases with core thicknesses, indicating that the hybrid composite core can significantly improve the structural stiffness. Additionally, Figure 7 shows that the dimensionless fundamental frequency of the sandwich plate rises as kw and Gp grow, but that kw has little effect on the sandwich plate’s vibration frequency. And the dimensionless fundamental frequency of the sandwich plate is greatly impacted by the increase in Gp. The sandwich plate’s dimensionless fundamental frequency rises noticeably with increasing Gp. Furthermore, thinner core plates exhibit larger dimensionless fundamental frequencies at Gp = 100 compared to Gp = 0 for lower gradient indexes.

Figure 8 examines the effects of gradient index, elastic parameters, and thickness-to-width ratio on the sandwich plate’s dimensionless fundamental frequency. The ratio of thickness is h_c/h = 0.8. Because of the thin plates, the bending stiffness and fundamental frequencies rise with overall thickness, as shown in Figure 8, which shows that the dimensionless fundamental frequency steadily increases as the thickness to width ratio rises from 0.05 to 0.2. Additionally, Figure 8 shows that as kw and Gp grow, the sandwich plate’s vibration frequency rises, but kw has less of an impact on the sandwich plate’s dimensionless fundamental frequency. Additionally, as Gp increases, the dimensionless fundamental frequency will rise noticeably.

The dimensionless fundamental frequency of sandwich plates with varying aspect ratios is examined in

Table 8. Simple supported hybrid sandwich plates with different aspect ratios (h_c = 0.1; h_f = 0.01) for the variation in dimensionless natural frequencies (p = 1).

(kw,Gp)	VCNT		a/b
			1	1.2	1.4	1.6	1.8	2	3	4
(0,0)	0.12	UD	0.0158	0.0133	0.0118	0.0109	0.0103	0.0099	0.0089	0.0086
		FG-V	0.0155	0.0131	0.0117	0.0108	0.0103	0.0099	0.0089	0.0086
		FG-X	0.0164	0.0138	0.0122	0.0112	0.0106	0.0101	0.0092	0.0088
		FG-O	0.0145	0.0124	0.0111	0.0103	0.0097	0.0094	0.0085	0.0082
	0.17	UD	0.0196	0.0168	0.0151	0.0140	0.0133	0.0128	0.0116	0.0112
		FG-V	0.0193	0.0166	0.0150	0.0140	0.0133	0.0129	0.0117	0.0113
		FG-X	0.0207	0.0176	0.0149	0.0147	0.0139	0.0134	0.0121	0.0117
		FG-O	0.0177	0.0153	0.0139	0.0130	0.0124	0.0119	0.0109	0.0105
	0.28	UD	0.0216	0.0183	0.0164	0.0152	0.0144	0.0138	0.0125	0.0121
		FG-V	0.0212	0.0182	0.0164	0.0153	0.0146	0.014	0.0128	0.0123
		FG-X	0.0237	0.0201	0.018	0.0167	0.0158	0.0152	0.0138	0.0133
		FG-O	0.0182	0.0157	0.0142	0.0132	0.0125	0.0121	0.0110	0.0107
(100,100)	0.12	UD	0.0372	0.029	0.0239	0.0206	0.0182	0.0165	0.0122	0.0105
		FG-V	0.0370	0.0289	0.0239	0.0205	0.0182	0.0164	0.0122	0.0105
		FG-X	0.0375	0.0292	0.0241	0.0208	0.0184	0.0166	0.0123	0.0107
		FG-O	0.0363	0.0283	0.0233	0.0200	0.0177	0.0160	0.0118	0.0101
	0.17	UD	0.0394	0.0310	0.0259	0.0226	0.0202	0.0185	0.0143	0.0128
		FG-V	0.0392	0.0309	0.0259	0.0225	0.0202	0.0185	0.0144	0.0129
		FG-X	0.0399	0.0315	0.0264	0.0230	0.0206	0.0189	0.0147	0.0132
		FG-O	0.0378	0.0298	0.0249	0.0216	0.0193	0.0177	0.0136	0.0121
	0.28	UD	0.0412	0.0325	0.0272	0.0237	0.0213	0.0195	0.0152	0.0136
		FG-V	0.0409	0.0324	0.0272	0.0237	0.0213	0.0196	0.0154	0.0139
		FG-X	0.0424	0.0336	0.0282	0.0247	0.0223	0.0205	0.0163	0.0147
		FG-O	0.0384	0.0303	0.0253	0.0219	0.0196	0.0179	0.0138	0.0123

. Additionally, the impact of the volume percentage and distribution type of carbon nanotubes is taken into account. The dimensionless fundamental frequency is found to drop as the aspect ratio rises, with the tendency for the reduction to slow down as the aspect ratio approaches reasonably large values. For example, for the FG-V distribution with kw = Gp = 100, the dimensionless frequency decreases by 55.68% when the aspect ratio is increased from a/b = 1 to a/b = 2, and the dimensionless fundamental frequency decreases by 25.61% when a/b is increased from 2 to 3. This trend indicates that the increase in the volume fraction of carbon nanotubes can enhance the structural stiffness. Furthermore, the distribution of CNTs along the thickness direction significantly influences the free vibration of the sandwich plate because the FG-X has the highest dimensionless fundamental frequency compared to other distribution types.

Figure 9 examines the effects of CNT distribution types, volume fraction, and gradient index on the sandwich plate’s dimensionless fundamental frequency. Figure 9 shows that when CNTs have distinct distribution types in the core, FG-O has a relatively modest dimensionless fundamental frequency and FG-X is slightly greater than the other types. With the exception of the FG-O distribution type, it can be seen that the three CNT distribution types—UD, FG-V, and FG-X—have no discernible impact on the dimensionless fundamental frequency. Furthermore, Figure 9 demonstrates that as the volume proportion of CNTs grows, so does the dimensionless fundamental frequency. However, when the gradient index progressively rises, the dimensionless fundamental frequency falls.

The effect of elastic foundation parameters on the sandwich plate’s dimensionless fundamental frequency is investigated in Figure 10. The distribution types and volume proportion of CNTs are taken into account. Generally speaking, when the volume percentage of CNTs rises, the dimensionless fundamental frequency rises noticeably, demonstrating that CNTs increase the material’s rigidity. It is also observed that the dimensionless fundamental frequency increases with an increase in kw and Gp. However, the impact of kw on the dimensionless fundamental frequency is less, and the increase in Gp significantly increases the dimensionless fundamental frequency. Moreover, in good agreement with expectation, among the four distributions, FG-O has the relatively smallest dimensionless fundamental frequency and FG-X has the relatively largest dimensionless fundamental frequency.

Figure 11 investigates the effect of fiber’s various lamination schemes and elastic foundation parameters of the plate on the first four dimensionless free vibration frequencies of the sandwich plate. It should be noted that the influence of different lamination schemes has little effect on the first two dimensionless frequencies, but as the order increases, the differences in the dimensionless frequencies of the different lamination schemes become progressively more pronounced. In the last two free vibration frequencies, the dimensionless frequency is highest when the lamination scheme of the fiber is [0°/90°]_4s and is lowest when the lamination scheme of the fiber is [0°]_4s. In addition, in agreement with the previous cases, the dimensionless frequencies significantly increase with the increase in the elastic foundation parameters kw and Gp.

4. Conclusions

The free vibration behavior of a sandwich composite plate on the Winkler/Pasternak elastic basis is investigated analytically in this work. The sheets are a fiber-reinforced composite, whereas the core is a CNT-reinforced composite with a metal–ceramic FGM matrix. In the FGM plate, the CNTs are arranged in four different configurations. The HSDT-based motion equations are derived through the Hamilton energy principle. The free vibration solutions are then obtained by applying the Navier method. Furthermore, the accuracy of this research approach is confirmed by comparing the results of this paper’s method with those of finite elements and previous studies using several examples. The structure studied in this paper can not only promote the application of composites in the field of vibration control but also provide an important theoretical basis and technical support for improving the performance, reliability, and multifunctionality of engineering structures. In addition, the effect of characteristics parameters on the dimensionless frequency of sandwich plates is investigated. And some results of the study are observed as follows:

(1)

As the volume fraction of ceramics and CNTs increases, it leads to an increase in the stiffness of the plate and produces larger dimensionless free vibration frequencies.

(2)

For different distributions of CNT (UD, FG-V, FG-X, FG-O), the FG-O distribution significantly reduces the dimensionless frequency, while the remaining three distributions have no significant effect on the dimensionless frequency.

(3)

Both the elastic foundation parameters kw and Gp have positively correlated effects on the free vibration behavior of the FRP_FG-CNTRC_FRP plate. However, Gp affects the dimensionless fundamental frequency to a much greater extent than kw.

(4)

It can be clearly observed that the dimensionless fundamental frequency of the sandwich composite plate increases significantly with the increase in h_c/h, indicating that the FG-CNTRC core layer mentioned in this paper can enhance the stiffness of the plate well.