Magneto Axisymmetric Vibration of FG-GPLs Reinforced Annular Sandwich Plates with an FG Porous Core Using DQM and a New Shear Deformation Theory

Abstract

Based on the differential quadrature procedure (DQP), the vibrational response of functionally graded (FG) sandwich annular plates enhanced with graphene platelets (GPLs) and with an FG porous core is illustrated in this paper. The current annular plate is assumed to deform axisymmetrically and expose to a radial magnetic field. The Lorentz magnetic body force is deduced via Maxwell’s relations. The effective physical properties of the upper and lower layers of the sandwich plate are obtained by employing the Halpin–Tsai model. Our technique depends on a new four-unknown shear deformation theory to depict the displacements. In addition, the motion equations are established via Hamilton’s principle. The motion equations are solved by employing the DQP. In order to study the convergence of the DQ method, the minimum number of grid points needed for a converged solution is ascertained. In addition, the current theory’s outcomes are compared with those of previous higher-order theories. The effects of the porosity distribution type, porosity factor, GPLs distribution pattern, GPLs weight fraction, inner-to-outer radius ratio, outer radius-to-thickness ratio, magnetic field parameters, core thickness, and elastic substrate parameters on the nondimensional vibration frequencies are discussed.

1. Introduction

Reinforcements like graphene platelets (GPLs) are extensively used to improve the material’s mechanical characteristics. Because of their excellent toughness-to-weight ratio, reinforced composite structures have found extensive use in a variety of engineering fields, including aerospace, automotive, and marine engineering. Graphene represents the best reinforcing nanofillers due to its remarkable physical and mechanical characteristics, such as the highest strength and rigidity, the best thermal and electrical conductivity, and so on [1,2,3,4]. Polymers, metals, and ceramics can be enhanced with GPLs, which gives them high resistance, improved mechanical strength, and excellent thermal properties [1]. In this context, FG graphene-enhanced nanocomposite plates are considered to be one of the most important recent innovations in this field. These plates are characterized by their graded composition, which gives them unique mechanical and thermal properties. Song et al. [5] implemented the buckling and bending of graphene/polymer composite plates via the first-order theory (FPT). On the other hand, Abazid [6] explored the thermal buckling of GPLs metal foam nanoplates under the impact of 2D magnetic fields and humid conditions. Zargaripoora et al. [7] investigated the free vibration behavior of nanoplates strengthened by GPLs by employing the finite element method. On the basis of quasi-three-dimensional theory, wave propagation in porous nanocomposite plates immersed in an elastic media and subjected to mechanical and Lorentz forces was studied by Abazid et al. [8]. Zhang et al. [9] investigated the nonlinear analysis of rotating GPLs-reinforced metal foams annular plates resting on a viscoelastic foundation under pulse loads utilizing the FPT and von-Karman’s nonlinearity. The nonlinear vibration and bending responses of GPLs-reinforced piezoelectric porous curved beams were illustrated by Yang et al. [10], who employed Hamilton’s principle to deduce the nonlinear equations and used the DQP to solve them. The literature contains a large amount of other research on graphene-reinforced composites, such as that of Li et al. [11], Zhang et al. [12], Ding and She [13], Zhang et al. [9], Yee et al. [14], Sobhy [15], and Garg et al. [16].

Functionally graded materials (FGMs) exhibit progressive changes in their material structure and mechanical characteristics in a variety of directions [17]. This is a great choice for composite constructions that experience severe inconsistencies in structural and thermal properties along the boundaries of multiple materials [18]. Furthermore, the gradual alterations in composition may be customized to adapt for diverse operating situations while also meeting varying objectives. FGMs are often composed of several kinds of ceramics and metals or a variety of other materials. The ceramic material used in an FGM provides heat insulation benefits while also protecting the metal from damage caused by oxidation and corrosion. The metallic substance toughens and strengthens the FGM. FGMs are presently being developed for broad usage as construction components in severe high-temperature settings and other fields. In this particular circumstance, FGM structures have garnered substantial research efforts in recognition of their distinctive benefits and extensive prospective practical usage in the aviation industry, engineering, etc. Furthermore, recent studies that have examined the characteristics and behaviors of FG-GPLs-reinforced structures include [19,20,21,22,23].

A porous material is a type of material that contains small internal holes or pores [24,25]. These pores allow materials to absorb and pass fluids or gases through them. Porous materials such as metal foams in a graphene-FG microstructure are utilized as lightweight structures to ensure adequate rigidity stiffness relative to their weight. Correspondingly, numerous technical sectors, including civil construction, the automobile industry, and aerospace, frequently use porous structures [26]. Such materials’ internal porosities may be evenly distributed, unevenly distributed, or FG across the thickness of the structure. The strength of such materials is decreased because of the interior pores. Porous materials may be strengthened with harder elements, such as graphene nanoplates or carbon nanotubes [8,27], or they may be sandwiched between two strong sheets to minimize this disadvantage [28]. Chen et al. [29] explored the effect of porosity of sandwich beams with FG porous cores. In order to maximize critical force while minimizing beam mass, Magnucka-Blandzi and Magnucki [30] developed a sandwich beam with a porous core. In a different work, Chen et al. [29] examined the vibration response of porous beams using various porosity patterns. They found that a decrease in stiffness caused by different porosity distributions had a considerable impact on frequencies of the plate. Wattanasakulpong and Eiadtrong [31] investigated the vibration behavior of sandwich panels with an FG metal foam core exposed to a dynamic load utilizing the FPT. Pham et al. [32] illustrated the effects of a blast load on the vibration of sandwich panels with an FG porous core using the mixed interpolation of tensorial components triangular element and the Newmark-beta approach. Sobhy [33] studied the impacts of moisture and external voltage on the thermal buckling of sandwich panels composed of FG piezoelectric faces and a porous core. Meanwhile, Sobhy [34] used a nonlocal strain gradient model and a refined four-variable theory to elucidate the static bending behavior of annular and circular sandwich nanoplates with a metal foam core and piezoelectric faces resting on elastic foundations and exposed to an electric voltage and mechanical load. Based on various shear deformation theories, different behaviors of sandwich beams [35,36,37] and sandwich shells [38,39] with porous cores have been studied.

Since the differential quadrature method (DQM) is a useful numerical technique for solving ordinary and partial differential equations with initial or boundary conditions, it is used to solve the governing equations. Furthermore, in comparison to previous numerical approaches, the DQM predicts fairly accurate numerical outcomes with substantially fewer mech points [40]. A weighted linear sum of the function’s values throughout the whole domain is used to estimate the partial derivatives of a function at a specific discrete point in accordance with the DQM. Employing the DQ approach, linear free vibration and nonlinear vibration responses of FG-GPLs-reinforced annular and circular plates and cylindrical shells were studied by Nie and Zhong [41], Mercan et al. [42], Liu et al. [43], Liang et al. [44], Alsebai et al. [45] and Alali et al. [46].

As discussed in the above literature review, an axisymmetric vibrational analysis of FG sandwich annular plates strengthened with GPLs having an FG porous core and subjected to a radial magnetic field has not been considered in the available literature. Furthermore, a new improved four-unknown shear deformation plate theory that takes transverse shear strain into account to get the displacement field is used in this research. To discretize the equations of motion and convert them into an algebraic system of equations, the DQM is utilized. This system represents an eigenvalue problem that can be solved to obtain the eigenfrequencies of the sandwich annular plates. Next, the validity of the current theory is confirmed by comparing its results with the results of some other higher-order theories. In addition, the impacts of the graphene distribution type, GPLs volume fraction, magnetic field, and electric voltage on the eigenfrequencies of the FG porous sandwich annular plates are illustrated.

2. Mathematical Formulation

Figure 1 depicts a three-layered annular plate, two FG metal/GPLs face sheets, and a metal foam core placed on an elastic foundation. The face sheets are assumed to be perfectly bonded to the core. This annular plate has an inner radius of $r_1$ and an outer radius of $r_2$. Each of the top and bottom layers has a thickness $T_{GPL}$, while the middle layer has a thickness $T_{core}$; therefore, the total plate thickness is $T=2T_{GPL}+T_{core}$. In this investigation, the cylindrical coordinate system $(R,\theta ,Z)$ is used. The current approach takes into account both axial symmetry and geometry.

2.1. Refined Four-Variable Shear and Normal Deformations Theory

Shimpi’s theory [47] is utilized to model the present theory and includes the following assumptions:

• Since the displacements are tiny, the strains involved are also very tiny.
• The transverse normal stress ${\sigma }_{ZZ}$ is negligible in comparison to the in-plane stresses ${\sigma }_{RR}$ and ${\sigma }_{\theta \theta }$.
• The transverse displacement $u_Z$ contains two components: the bending component ${\varphi }^b$ and the shear component ${\varphi }^s$. Both components are functions of R and t only.
• The mid-plane radial displacement u is added to Shimpi’s two-variable plate theory [47].

Therefore, the displacement field can be written as:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_R(R,Z,t)\\ u_Z(R,Z,t)\end{array}\right\}=\left\{\begin{array}{c}u(R,t)\\ {\varphi }^b(R,t)+{\varphi }^s(R,t)\end{array}\right\}-Z\left\{\begin{array}{c}\frac{\partial {\varphi }^b}{\partial R}\\ 0\end{array}\right\}-G\left(Z\right)\left\{\begin{array}{c}\frac{\partial {\varphi }^s}{\partial R}\\ 0\end{array}\right\},u_{\theta }(R,Z,t)=0,\end{array}$$

(1)

where $G\left(Z\right)=Z-\overline{G}\left(Z\right)$, for which

$$\overline{G}\left(Z\right)=\frac{T}{2}{sinh}^{-1}\left(\frac{2Z}{T}\right)-\frac{2\sqrt{2}{Z}^3}{3T^2}.$$

(2)

The shape function $\overline{G}\left(Z\right)$ takes many forms such as the third-order plate theory (TPT) [48], the sinusoidal plate theory (SPT) [49], the hyperbolic plate theory (HPT) [50], and the exponential plate theory (EPT) [51] that can be respectively given as:

$$\begin{array}{c}\overline{G}\left(Z\right)=Z\left(1-\frac{4Z^2}{3T^2}\right),\hfill \\ \overline{G}\left(Z\right)=\frac{T}{\pi }sin\left(\frac{\pi Z}{T}\right),\hfill \\ \overline{G}\left(Z\right)=Tsinh\left(\frac{Z}{T}\right)-Zcosh\left(\frac{1}{2}\right),\hfill \\ \overline{G}\left(Z\right)=Zexp\left(\frac{-2Z^2}{T^2}\right).\hfill \end{array}$$

(3)

The general form of the strain–displacement relationship in cylindrical coordinates is now given as:

$$\begin{array}{cc}\hfill {\epsilon }_{RR}=& \frac{\partial u_R}{\partial R},{\epsilon }_{\theta \theta }=\frac{\partial u_{\theta }}{R\partial \theta }+\frac{u_R}{R},{\epsilon }_{ZZ}=\frac{\partial u_Z}{\partial Z},\hfill \\ \hfill {\epsilon }_{R\theta }=& \frac{\partial u_{\theta }}{\partial R}+\frac{\partial u_R}{R\partial \theta }-\frac{u_{\theta }}{R},{\epsilon }_{\theta Z}=\frac{\partial u_{\theta }}{\partial Z}+\frac{\partial u_Z}{R\partial \theta },{\epsilon }_{RZ}=\frac{\partial u_R}{\partial Z}+\frac{\partial u_Z}{\partial R}.\hfill \end{array}$$

(4)

Substituting Equation (1) into Equation (4) leads to:

$$\begin{array}{cc}\hfill {\epsilon }_{RR}=& \frac{\partial u}{\partial R}-Z\frac{{\partial }^2{\varphi }^b}{\partial R^2}-G\frac{{\partial }^2{\varphi }^s}{\partial R^2},\hfill \\ \hfill {\epsilon }_{\theta \theta }=& \frac{1}{R}\left(u-Z\frac{\partial {\varphi }^b}{\partial R}-G\frac{\partial {\varphi }^s}{\partial R}\right),\hfill \\ \hfill {\epsilon }_{ZZ}=& 0,{\epsilon }_{R\theta }=0,{\epsilon }_{\theta Z}=0,\hfill \\ \hfill {\epsilon }_{RZ}=& {\overline{G}}^{\prime }\frac{\partial {\varphi }^s}{\partial R},{\overline{G}}^{\prime }=\frac{\mathrm{d}\overline{G}}{\mathrm{d}Z}.\hfill \end{array}$$

(5)

The stresses of each layer are given as [52]:

$$\begin{array}{cc}\hfill & {\sigma }_{RR}^{\left(i\right)}=C_{11}^{\left(i\right)}{\epsilon }_{RR}+C_{12}^{\left(i\right)}{\epsilon }_{\theta \theta },\hfill \\ & {\sigma }_{\theta \theta }^{\left(i\right)}=C_{12}^{\left(i\right)}{\epsilon }_{RR}+C_{11}^{\left(i\right)}{\epsilon }_{\theta \theta },\hfill \\ & {\sigma }_{RZ}^{\left(i\right)}=C_{55}^{\left(i\right)}{\epsilon }_{RZ},i=1,2,3,\hfill \end{array}$$

(6)

where

$$\begin{array}{c}\hfill C_{11}^{\left(i\right)}=\frac{E^{\left(i\right)}}{[1-{\left({\nu }^{\left(i\right)}\right)}^2]},C_{12}^{\left(i\right)}=\frac{{\nu }^{\left(i\right)}{E}^{\left(i\right)}}{[1-{\left({\nu }^{\left(i\right)}\right)}^2]},C_{55}^{\left(i\right)}=\frac{E^{\left(i\right)}}{2[1+{\nu }^{\left(i\right)}]}.\end{array}$$

(7)

2.2. The Magnetic Field

The relationships between the current density $\overrightarrow{J}$ and the distribution vector of the magnetic field $\overrightarrow{h}$ are given by Maxwell’s equations as [53,54]:

$$\begin{array}{cc}\hfill & \overrightarrow{J}=\nabla \times \overrightarrow{h},\hfill \\ & \overrightarrow{h}=\nabla \times (\overrightarrow{u}\times \overrightarrow{M}),\hfill \end{array}$$

(8)

where $\overrightarrow{M}(M_R,0,0)$ represents the in-plane magnetic field vector, and $\overrightarrow{u}(u_R,0,u_Z)$ represents the displacement vector. The vector of the distributed magnetic field $\overrightarrow{h}$ can be expanded as:

$$\begin{array}{c}\hfill \overrightarrow{h}=M_R\left(\frac{u_Z}{R}+\frac{\partial u_Z}{\partial R}\right)\widehat{Z}.\end{array}$$

(9)

By substituting Equation (9) into Equation (8), the current density $\overrightarrow{J}$ can be expressed as:

$$\begin{array}{c}\hfill \overrightarrow{J}=M_R\left(\frac{\partial u_Z}{R^2\partial \theta }+\frac{{\partial }^2{u}_Z}{R\partial R\partial \theta }\right)\widehat{R}-M_R\left(\frac{\partial u_Z}{R\partial R}+\frac{{\partial }^2{u}_Z}{\partial R^2}-\frac{u_Z}{R^2}\right)\widehat{\theta }.\end{array}$$

(10)

The Lorentz magnetic force vector per unit volume $\overrightarrow{F}$ is given by [53]:

$$\begin{array}{c}\hfill \overrightarrow{F}=\gamma (\overrightarrow{J}\times \overrightarrow{M}),\end{array}$$

(11)

where $\gamma$ is the permeability of the magnetic field. By substituting Equation (10) into Equation (11), we get:

$$F_R=0,F_{\theta }=0,F_Z=\gamma M_R^2\left(\frac{\partial u_Z}{R\partial R}+\frac{{\partial }^2{u}_Z}{\partial R^2}-\frac{u_Z}{R^2}\right).$$

(12)

2.3. Plate Properties

2.3.1. Nanocomposite Face Layers

The face layers are made of metal reinforced with GPLs that can be equally dispersed or smoothly modulated across the thickness of the layer. The Young’s modulus and Poisson’s ratio of the top and bottom layers have the following form in accordance with the modified Halpin–Tsai model and mixture law [55]:

$$\begin{array}{cc}\hfill & E^{\left(j\right)}\left(Z\right)=\frac{1}{8}[3E_1^{\left(j\right)}\left(Z\right)+5E_2^{\left(j\right)}\left(Z\right)],E_K^{\left(j\right)}\left(Z\right)=\frac{1+{\xi }_K^g{\eta }_K{F}_g^{\left(j\right)}}{1-{\eta }_K{F}_g^{\left(j\right)}}{E}_m,\hfill \\ & {\eta }_K=\frac{E_g-E_m}{E_g+{\xi }_K^g{E}_m},{\xi }_1^g=2\frac{L^g}{T^g},{\xi }_2^g=2\frac{W^g}{T^g},K=1,2,j=1,3,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & {\nu }^{\left(j\right)}\left(Z\right)=F_m^{\left(j\right)}\left(Z\right){\nu }_m+F_g^{\left(j\right)}\left(Z\right){\nu }_g,\hfill \\ \hfill & {\rho }^{\left(j\right)}=F_m^{\left(j\right)}\left(Z\right){\rho }_m+F_g^{\left(j\right)}\left(Z\right){\rho }_g,F_m^{\left(j\right)}\left(Z\right)=1-F_g^{\left(j\right)}\left(Z\right),j=1,3,\hfill \end{array}$$

(14)

where $E_g$ and ${\nu }_g$ are graphene’s Young’s modulus and Poisson’s ratio, respectively; $E_m$ and ${\nu }_m$ are metal’s Young’s modulus and Poisson’s ratio, respectively; $L^g$, $W^g$, and $T^g$ are the length, width, and thickness of the GPLs; $\rho$ is the mass density of the layers; ${\rho }_g$ and ${\rho }_m$ are graphene’s density and metal’s density, respectively. There are three types of GPLs distributions, as shown in Figure 2:

Type A:

In this type, the GPLs are evenly dispersed throughout the sandwich faces’ thickness; one may write [5,56]:

$$\begin{array}{c}\hfill F_g^{\left(j\right)}\left(Z\right)=V_g,T_0\le Z\le T_1\mathrm{and}{T}_2\le Z\le T_3,j=1,3,\mathrm{UD},\end{array}$$

(15)

where $T_0=-T/2$ and $T_3=T/2$; $T_1$ and $T_2$ are the layer interface coordinates (see Figure 1). Meanwhile, $V_g$ is provided as [5]:

$$\begin{array}{c}\hfill V_g=\frac{W_G}{W_G+\left(\frac{{\rho }_g}{{\rho }_m}\right)(1-W_G)},\end{array}$$

(16)

where $W_G$ represents the weight fraction of the GPLs.

Type B:

In this state, the interfaces ($T_1$ and $T_2$) are graphene-free, while the upper and lower surfaces of the plate are graphene-rich $\left(\frac{\nabla -FG}{\Delta -FG}\right)$; thus, we can type [57]:

$$\begin{array}{cc}\hfill & F_g^{\left(1\right)}\left(Z\right)=V_{g}cos(\frac{(2Z-T_0-T_1)\pi }{4(T_0-T_1)}-\frac{\pi }{4}),T_0\le Z\le T_1,\hfill \\ & F_g^{\left(3\right)}\left(Z\right)=V_{g}cos(\frac{(2Z-T_3-T_2)\pi }{4(T_3-T_2)}-\frac{\pi }{4}),T_2\le Z\le T_3.\hfill \end{array}$$

(17)

Type C:

In the final situation, the graphene distribution is the inverse of the preceding type: $\left(\frac{\Delta -FG}{\nabla -FG}\right)$; thus, $F_g^{\left(j\right)}\left(Z\right)$ is given as [57]:

$$\begin{array}{cc}\hfill & F_g^{\left(1\right)}\left(Z\right)=V_{g}cos(\frac{(2Z-T_0-T_1)\pi }{4(T_0-T_1)}+\frac{\pi }{4}),T_0\le Z\le T_1,\hfill \\ & F_g^{\left(3\right)}\left(Z\right)=V_{g}cos(\frac{(2Z-T_3-T_2)\pi }{4(T_3-T_2)}+\frac{\pi }{4}),T_2\le Z\le T_3.\hfill \end{array}$$

(18)

It is worth noting that the distribution types of GPLs play a crucial role in determining the response of the plates. In particular, Type A might improve the mechanical properties of the plates more than the other types. On the other hand, Type B is characterized by a strong bond between the two face layers and the core layer, where the interfaces between the layers are only metal. Therefore, it is expected that the third type will be less efficient than the other two types.

2.3.2. Porous Core Layer

The metal foam core is considered to be constructed of aluminum with evenly dispersed or FG porosities throughout the core thickness (see Figure 3). The current study takes into account three alternative porosity distributions:

Porous-I:

In this case, the pores are dispersed equally across the thickness of the core (UD) as follows [58]:

$$\begin{array}{cc}\hfill & P^{\left(2\right)}\left(Z\right)=P_m(1-{\beta }_0\lambda ),T_1\le Z\le T_2,\hfill \\ & {\rho }^{\left(2\right)}\left(Z\right)={\rho }_m\sqrt{1-{\beta }_0\lambda },T_1\le Z\le T_2,\hfill \end{array}$$

(19)

where P denotes the mechanical properties, like Young’s modulus E and shear modulus G, and ${\beta }_0$ denotes the porosity coefficient, where $0\le {\beta }_0<1$, and the parameter $\lambda$ is defined as [58]:

$$\begin{array}{c}\hfill \lambda =\frac{1}{{\beta }_0}-\frac{1}{{\beta }_0}(\frac{2}{\pi }\sqrt{1-{\beta }_0\lambda }-\frac{2}{\pi }+1{)}^2.\end{array}$$

(20)

Porous-II:

From the interfaces to the core’s mid-plane, the volume fraction of the porosities increases (O-FG). As a result, the mid-plane has the lowest mechanical property values. As a result, the properties are as follows [57]:

$$\begin{array}{cc}\hfill & P^{\left(2\right)}\left(Z\right)=P_m\left\{1-{\beta }_0\left[1-{\left(\frac{2Z-T_1-T_2}{T_1-T_2}\right)}^2\right]\right\},T_1\le Z\le T_2,\hfill \\ & {\rho }^{\left(2\right)}\left(Z\right)={\rho }_m\left\{1-\left(1-\sqrt{1-{\beta }_0}\right)\left[1-{\left(\frac{2Z-T_1-T_2}{T_1-T_2}\right)}^2\right]\right\},T_1\le Z\le T_2.\hfill \end{array}$$

(21)

Porous-III:

The mid-plane of the core is pore-free in this instance, but the interfaces are pore-rich (X-FG). As a result, we can write [57]:

$$\begin{array}{cc}\hfill & P^{\left(2\right)}\left(Z\right)=P_m\left[1-{\beta }_0{\left(\frac{2Z-T_1-T_2}{T_1-T_2}\right)}^2\right],T_1\le Z\le T_2,\hfill \\ & {\rho }^{\left(2\right)}\left(Z\right)={\rho }_m\left\{1-\left(1-\sqrt{1-{\beta }_0}\right){\left(\frac{2Z-T_1-T_2}{T_1-T_2}\right)}^2\right\},T_1\le Z\le T_2.\hfill \end{array}$$

(22)

The Poisson’s ratios of the cores in the preceding three categories are computed as follows [59]:

$$\begin{array}{c}\hfill {\nu }^{\left(2\right)}\left(Z\right)={\nu }_m[0.342{(1-\frac{{\rho }^2\left(Z\right)}{{\rho }_m})}^2-1.21(1-\frac{{\rho }^2\left(Z\right)}{{\rho }_m})+1]+0.221(1-\frac{{\rho }^2\left(Z\right)}{{\rho }_m}).\end{array}$$

(23)

Once again, the porosity distribution types significantly influence the reaction characteristics of the plates. Since the upper and lower surfaces of Type II are pore-free, the bond between the core and the face layers is enhanced. On the other hand, for porosity Type III, the thermal conductivity of the core at the interfaces might be substantially lower than that of the other types, reducing heat transfer to the core layer. Therefore, in high thermal environments, this type will be the best to use. The impact of distinct distribution types of GPLs and porosity on the vibrational response of FG sandwich annular plates will be discussed in more detail in Section 5.

2.4. Elastic Foundation

The flexible foundation is a layer beneath the plate that helps to distribute the weight and stress. Polymers, rubber, and flexible metals can be used to create the flexible foundation. In this context, the flexible foundation has a significant impact on the plate’s ability to absorb vibrations and shocks. Further, the flexible foundation better distributes stresses along the plate and absorbs vibration energy. The load is distributed across a broader area on the ground when employing a flexible foundation, which can minimize the pressure on the plate. This broader distribution can improve the overall performance of the plate by decreasing undesirable deformations or bending. For adding foundation stiffness, the Pasternak model has been defined as [8,60,61]:

$$\begin{array}{c}\hfill S=K_w{u}_Z-K_P(\frac{{\partial }^2{u}_Z}{\partial R^2}+\frac{1}{R}\frac{\partial u_Z}{\partial R}),\end{array}$$

(24)

where S, $K_w$, $u_Z$, and $K_P$ represent the density reaction of the foundations, the Winkler stiffness, the transverse displacement, and the shear (Pasternak) foundation stiffness, respectively.

3. Governing Equations

To find the equations of motion, Hamilton’s principle is used and is stated as:

$$\begin{array}{c}\hfill {\int }_{t_1}^{t_2}[\delta {\pi }_s-\delta {\pi }_f+\delta {\pi }_k]\mathrm{d}t=0,\end{array}$$

(25)

where $\delta {\pi }_s$ is the variation of the strain energy, $\delta {\pi }_f$ is the variation of the work done by an external force, and $\delta {\pi }_k$ is the variation of the kinetic energy; these are given as:

$$\begin{array}{c}\hfill \delta {\pi }_s=\sum _{i=1}^3{\int }_0^{2\pi }{\int }_{r_1}^{r_2}{\int }_{T_{i-1}}^{T_i}{\sigma }_{ij}^{\left(i\right)}\delta {\epsilon }_{ij}R\mathrm{d}Z\mathrm{d}R\mathrm{d}\theta ,\end{array}$$

(26)

$$\begin{array}{c}\hfill \delta {\pi }_f={\int }_0^{2\pi }{\int }_{r_1}^{r_2}{\int }_{-\frac{T}{2}}^{\frac{T}{2}}\overrightarrow{F}·\delta \overrightarrow{u}R\mathrm{d}Z\mathrm{d}R\mathrm{d}\theta -{\int }_0^{2\pi }{\int }_{r_1}^{r_2}S\delta u_{Z}R\mathrm{d}R\mathrm{d}\theta ,\end{array}$$

(27)

$$\begin{array}{c}\hfill \delta {\pi }_k=\sum _{i=1}^3{\int }_0^{2\pi }{\int }_{r_1}^{r_2}{\int }_{T_{i-1}}^{T_i}{\rho }^{\left(i\right)}{\ddot{u}}_i\delta u_{i}R\mathrm{d}Z\mathrm{d}R\mathrm{d}\theta .\end{array}$$

(28)

By inserting Equations (26)–(28) into Equation (25) and integrating by parts, we obtain

$$\begin{array}{cc}\hfill & \delta u:\frac{\partial N_{RR}}{\partial R}+\frac{N_{RR}}{R}-\frac{N_{\theta \theta }}{R}-I_1\ddot{u}+I_2\frac{\partial {\ddot{\varphi }}^b}{\partial R}+I_3\frac{\partial {\ddot{\varphi }}^s}{\partial R}=0,\hfill \\ & \delta {\varphi }^b:\frac{{\partial }^2{M}_{RR}}{\partial R^2}+\frac{2}{R}\frac{\partial M_{RR}}{\partial R}-\frac{1}{R}\frac{\partial M_{\theta \theta }}{\partial R}-I_2\frac{\partial \ddot{u}}{\partial R}-\frac{1}{R}{I}_2\ddot{u}+I_4\frac{{\partial }^2{\ddot{\varphi }}^b}{\partial R^2}+\frac{1}{R}{I}_4\frac{\partial {\ddot{\varphi }}^b}{\partial R}\hfill \\ & +I_5\frac{{\partial }^2{\ddot{\varphi }}^s}{\partial R^2}+\frac{1}{R}{I}_5\frac{\partial {\ddot{\varphi }}^s}{\partial R}-I_1{\ddot{\varphi }}^b-I_1{\ddot{\varphi }}^s-\gamma M_R^2\frac{u_Z}{R^2}T+\frac{1}{R}\gamma M_R^2\frac{\partial u_Z}{\partial R}T+\gamma M_R^2\frac{{\partial }^2{u}_Z}{\partial R^2}T\hfill \\ & -S=0,\hfill \\ & \delta {\varphi }^s:\frac{{\partial }^2{S}_{RR}}{\partial R^2}+\frac{2}{R}\frac{\partial S_{RR}}{\partial R}-\frac{1}{R}\frac{\partial S_{\theta \theta }}{\partial R}+\frac{\partial S_{RZ}}{\partial R}+\frac{S_{RZ}}{R}-I_3\frac{\partial \ddot{u}}{\partial R}-\frac{1}{R}{I}_3\ddot{u}+I_5\frac{{\partial }^2{\ddot{\varphi }}^b}{\partial R^2}\hfill \\ & +\frac{1}{R}{I}_5\frac{\partial {\ddot{\varphi }}^b}{\partial R}+I_6\frac{{\partial }^2{\ddot{\varphi }}^s}{\partial R^2}+\frac{1}{R}{I}_6\frac{\partial {\ddot{\varphi }}^s}{\partial R}-I_1{\ddot{\varphi }}^b-I_1{\ddot{\varphi }}^s-\gamma M_R^2\frac{u_Z}{R^2}T+\frac{1}{R}\gamma M_R^2\frac{\partial u_Z}{\partial R}T\hfill \\ & +\gamma M_R^2\frac{{\partial }^2{u}_Z}{\partial R^2}T-S=0,\hfill \end{array}$$

(29)

where

$$\begin{array}{cc}\hfill & N_{RR}=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{\sigma }_{RR}^{\left(i\right)}\mathrm{d}Z,M_{RR}=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{\sigma }_{RR}^{\left(i\right)}Z\mathrm{d}Z,S_{RR}=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{\sigma }_{RR}^{\left(i\right)}G\left(Z\right)\mathrm{d}Z,\hfill \\ & N_{\theta \theta }=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{\sigma }_{\theta \theta }^{\left(i\right)}\mathrm{d}Z,M_{\theta \theta }=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{\sigma }_{\theta \theta }^{\left(i\right)}Z\mathrm{d}Z,S_{\theta \theta }=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{\sigma }_{\theta \theta }^{\left(i\right)}G\left(Z\right)\mathrm{d}Z,\hfill \\ & S_{RZ}=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{\sigma }_{RZ}^{\left(i\right)}{\overline{G}}^{\prime }\left(Z\right)\mathrm{d}Z,\hfill \end{array}$$

(30)

and

$$\begin{array}{cc}\hfill & I_1=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{\rho }^{\left(i\right)}\mathrm{d}Z,I_2=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{\rho }^{\left(i\right)}Z\mathrm{d}Z,\hfill \\ & I_3=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{\rho }^{\left(i\right)}G\left(Z\right)\mathrm{d}Z,I_4=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{\rho }^{\left(i\right)}{Z}^2\mathrm{d}Z,\hfill \\ & I_5=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{\rho }^{\left(i\right)}ZG\left(Z\right)\mathrm{d}Z,I_6=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{\rho }^{\left(i\right)}{G}^2\left(Z\right)\mathrm{d}Z.\hfill \end{array}$$

(31)

By substituting Equations (5) and (6) into Equation (30) and then into Equation (29), the equations of motion are rewritten as:

$$\begin{array}{cc}\hfill & -A_2\frac{{\partial }^3{\varphi }^b}{\partial R^3}-A_3\frac{{\partial }^3{\varphi }^s}{\partial R^3}+A_1\frac{{\partial }^{2}u}{\partial R^2}-A_2\frac{{\partial }^2{\varphi }^b}{R\partial R^2}-A_3\frac{{\partial }^2{\varphi }^s}{R\partial R^2}+A_1\frac{\partial u}{R\partial R}\hfill \\ & +A_2\frac{\partial {\varphi }^b}{R^2\partial R}+A_3\frac{\partial {\varphi }^s}{R^2\partial R}-A_1\frac{u}{R^2}-I_1\ddot{u}+I_2\frac{\partial {\ddot{\varphi }}^b}{\partial R}+I_3\frac{\partial {\ddot{\varphi }}^s}{\partial R}=0,\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & -A_4\frac{{\partial }^4{\varphi }^b}{\partial R^4}-A_5\frac{{\partial }^4{\varphi }^s}{\partial R^4}+A_2\frac{{\partial }^{3}u}{\partial R^3}-A_4\frac{2{\partial }^3{\varphi }^b}{R\partial R^3}-A_5\frac{2{\partial }^3{\varphi }^s}{R\partial R^3}+A_2\frac{2{\partial }^{2}u}{R\partial R^2}+A_4\frac{{\partial }^2{\varphi }^b}{R^2\partial R^2}\hfill \\ & +A_5\frac{{\partial }^2{\varphi }^s}{R^2\partial R^2}-A_2\frac{\partial u}{R^2\partial R}-A_4\frac{\partial {\varphi }^b}{R^3\partial R}-A_5\frac{\partial {\varphi }^s}{R^3\partial R}+A_2\frac{u}{R^3}-I_2\frac{\partial \ddot{u}}{\partial R}-\frac{1}{R}{I}_2\ddot{u}\hfill \\ & +I_4\frac{{\partial }^2{\ddot{\varphi }}^b}{\partial R^2}+\frac{1}{R}{I}_4\frac{\partial {\ddot{\varphi }}^b}{\partial R}+I_5\frac{{\partial }^2{\ddot{\varphi }}^s}{\partial R^2}+\frac{1}{R}{I}_5\frac{\partial {\ddot{\varphi }}^s}{\partial R}-I_1{\ddot{\varphi }}^b-I_1{\ddot{\varphi }}^s-\gamma M_R^2\frac{u_Z}{R^2}T+\frac{1}{R}\gamma M_R^2\frac{\partial u_Z}{\partial R}T\hfill \\ & +\gamma M_R^2\frac{{\partial }^2{u}_Z}{\partial R^2}T-S=0,\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & -A_5\frac{{\partial }^4{\varphi }^b}{\partial R^4}-A_6\frac{{\partial }^4{\varphi }^s}{\partial R^4}+A_3\frac{{\partial }^{3}u}{\partial R^3}-A_5\frac{2{\partial }^3{\varphi }^b}{R\partial R^3}-A_6\frac{2{\partial }^3{\varphi }^s}{R\partial R^3}+A_3\frac{2{\partial }^{2}u}{R\partial R^2}+A_5\frac{{\partial }^2{\varphi }^b}{R^2\partial R^2}\hfill \\ & +\frac{{\partial }^2{\varphi }^s}{\partial R^2}[\frac{A_6}{R^2}+A_7]-A_3\frac{\partial u}{R^2\partial R}-A_5\frac{\partial {\varphi }^b}{R^3\partial R}+\frac{\partial {\varphi }^s}{R\partial R}[\frac{-A_6}{R^2}+A_7]+A_3\frac{u}{R^3}-I_3\frac{\partial \ddot{u}}{\partial R}\hfill \\ & -\frac{1}{R}{I}_3\ddot{u}+I_5\frac{{\partial }^2{\ddot{\varphi }}^b}{\partial R^2}+\frac{1}{R}{I}_5\frac{\partial {\ddot{\varphi }}^b}{\partial R}+I_6\frac{{\partial }^2{\ddot{\varphi }}^s}{\partial R^2}+\frac{1}{R}{I}_6\frac{\partial {\ddot{\varphi }}^s}{\partial R}-I_1{\ddot{\varphi }}^b-I_1{\ddot{\varphi }}^s-\gamma M_R^2\frac{u_Z}{R^2}T\hfill \\ & +\frac{1}{R}\gamma M_R^2\frac{\partial u_Z}{\partial R}T+\gamma M_R^2\frac{{\partial }^2{u}_Z}{\partial R^2}T-S=0,\hfill \end{array}$$

(34)

where

$$\begin{array}{cc}\hfill & A_1=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{C}_{11}^{\left(i\right)}\mathrm{d}Z,A_2=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{C}_{11}^{\left(i\right)}Z\mathrm{d}Z,A_3=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{C}_{11}^{\left(i\right)}G\left(Z\right)\mathrm{d}Z,\hfill \\ & A_4=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{C}_{11}^{\left(i\right)}{Z}^2\mathrm{d}Z,A_5=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{C}_{11}^{\left(i\right)}ZG\left(Z\right)\mathrm{d}Z,\hfill \\ & A_6=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{C}_{11}^{\left(i\right)}{G}^2\left(Z\right)\mathrm{d}Z,A_7=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{C}_{55}^{\left(i\right)}({\overline{G}}^{\prime }\left(Z\right){)}^2\mathrm{d}Z.\hfill \end{array}$$

(35)

By separating the variables, the displacement components can be expressed as:

$$\begin{array}{c}\hfill u(R,t)=U\left(R\right)\mathcal{V}\left(t\right),{\varphi }^b(R,t)={\Phi }^b\left(R\right)\mathcal{V}\left(t\right),{\varphi }^s(R,t)={\Phi }^s\left(R\right)\mathcal{V}\left(t\right),\end{array}$$

(36)

where $\mathcal{V}\left(t\right)=e^{i\omega t}$, in which $i=\sqrt{-1}$, and $\omega$ is the eigenfrequency. By substituting Equation (36) into Equations (32)–(34), we get the following ordinary differential equations:

$$\begin{array}{cc}\hfill & -A_2\frac{{\mathrm{d}}^3{\Phi }^b}{\mathrm{d}{R}^3}-A_3\frac{{\mathrm{d}}^3{\Phi }^s}{\mathrm{d}{R}^3}+A_1\frac{{\mathrm{d}}^{2}U}{\mathrm{d}{R}^2}-A_2\frac{{\mathrm{d}}^2{\Phi }^b}{R\mathrm{d}{R}^2}-A_3\frac{{\mathrm{d}}^2{\Phi }^s}{R\mathrm{d}{R}^2}+A_1\frac{\mathrm{d}U}{R\mathrm{d}R}\hfill \\ & +\frac{\mathrm{d}{\Phi }^b}{\mathrm{d}R}\left(\frac{A_2}{R^2}-I_2{w}^2\right)+\frac{\mathrm{d}{\Phi }^s}{\mathrm{d}R}\left(\frac{A_3}{R^2}-I_3{w}^2\right)-U\left(\frac{A_1}{R^2}-I_1{w}^2\right)=0,\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill & -A_4\frac{{\mathrm{d}}^4{\Phi }^b}{\mathrm{d}{R}^4}-A_5\frac{{\mathrm{d}}^4{\Phi }^s}{\mathrm{d}{R}^4}+A_2\frac{{\mathrm{d}}^{3}U}{\mathrm{d}{R}^3}-A_4\frac{2{\mathrm{d}}^3{\Phi }^b}{R\mathrm{d}{R}^3}-A_5\frac{2{\mathrm{d}}^3{\Phi }^s}{R\mathrm{d}{R}^3}+A_2\frac{2{\mathrm{d}}^{2}U}{R\mathrm{d}{R}^2}\hfill \\ & +\frac{{\mathrm{d}}^2{\Phi }^b}{\mathrm{d}{R}^2}\left(\frac{A_4}{R^2}-I_4{w}^2+K_P+\gamma M_R^{2}T\right)+\frac{{\mathrm{d}}^2{\Phi }^s}{\mathrm{d}{R}^2}\left(\frac{A_5}{R^2}-I_5{w}^2+K_P+\gamma M_R^{2}T\right)\hfill \\ & -\frac{\mathrm{d}U}{\mathrm{d}R}\left(\frac{A_2}{R^2}-I_2{w}^2\right)-\frac{\mathrm{d}{\Phi }^b}{R\mathrm{d}R}\left(\frac{A_4}{R^2}+I_4{w}^2-K_P-\gamma M_R^{2}T\right)\hfill \\ & -\frac{\mathrm{d}{\Phi }^s}{R\mathrm{d}R}\left(\frac{A_5}{R^2}+I_5{w}^2-K_P-\gamma M_R^{2}T\right)+\frac{U}{R}\left(\frac{A_2}{R^2}+I_2{w}^2\right)\hfill \\ & +{\Phi }^b\left(I_1{w}^2-K_w-\frac{\gamma M_R^{2}T}{R^2}\right)+{\Phi }^s\left(I_1{w}^2-K_w-\frac{\gamma M_R^{2}T}{R^2}\right)=0,\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill & -A_5\frac{{\mathrm{d}}^4{\Phi }^b}{\mathrm{d}{R}^4}-A_6\frac{{\mathrm{d}}^4{\Phi }^s}{\mathrm{d}{R}^4}+A_3\frac{{\mathrm{d}}^{3}U}{\mathrm{d}{R}^3}-A_5\frac{2{\mathrm{d}}^3{\Phi }^b}{R\mathrm{d}{R}^3}-A_6\frac{2{\mathrm{d}}^3{\Phi }^s}{R\mathrm{d}{R}^3}+A_3\frac{2{\mathrm{d}}^{2}U}{R\mathrm{d}{R}^2}\hfill \\ & +\frac{{\mathrm{d}}^2{\Phi }^b}{\mathrm{d}{R}^2}\left(\frac{A_5}{R^2}-I_5{w}^2+K_P+\gamma M_R^{2}T\right)+\frac{{\mathrm{d}}^2{\Phi }^s}{\mathrm{d}{R}^2}\left(\frac{A_6}{R^2}+A_7-I_6{w}^2+K_P+\gamma M_R^{2}T\right)\hfill \\ & -\frac{\mathrm{d}U}{\mathrm{d}R}\left(\frac{A_3}{R^2}-I_3{w}^2\right)-\frac{\mathrm{d}{\Phi }^b}{R\mathrm{d}R}\left(\frac{A_5}{R^2}+I_5{w}^2-K_P-\gamma M_R^{2}T\right)\hfill \\ & -\frac{\mathrm{d}{\Phi }^s}{R\mathrm{d}R}\left(\frac{A_6}{R^2}-A_7+I_6{w}^2-K_P-\gamma M_R^{2}T\right)+\frac{U}{R}\left(\frac{A_3}{R^2}+I_3{w}^2\right)\hfill \\ & +{\Phi }^b\left(I_1{w}^2-K_w-\frac{\gamma M_R^{2}T}{R^2}\right)+{\Phi }^s\left(I_1{w}^2-K_w-\frac{\gamma M_R^{2}T}{R^2}\right)=0.\hfill \end{array}$$

(39)

In the current analysis, the following simply supported conditions are used at $R=r_1$ and $R=r_2$:

$$\begin{array}{c}\hfill {\Phi }^b={\Phi }^s=N_{RR}=0.\end{array}$$

(40)

4. Solution Procedure

In this part, the equilibrium Equations (37)–(39) are solved using the DQM in the radial direction. The current annular plate is discretized by n grid points in the domain $(r_1\le R\le r_2)$. According to the DQM, the displacement component differentiations are approximated as a weighted linear sum of function values at all discrete sites as follows [40]:

$$\begin{array}{cc}\hfill & {\left.\frac{{\mathrm{d}}^{p}U}{\mathrm{d}{R}^p}\right|}_{R=R_i}=\sum _{j=1}^n{C}_{ij}^{\left(p\right)}{U}_j,U_j=U\left(R_j\right),\hfill \\ & {\left.\frac{{\mathrm{d}}^p{\Phi }^b}{\mathrm{d}{R}^p}\right|}_{R=R_i}=\sum _{j=1}^n{C}_{ij}^{\left(p\right)}{\Phi }_j^b,{\Phi }_j^b={\Phi }^b\left(R_j\right),\hfill \\ & {\left.\frac{{\mathrm{d}}^p{\Phi }^s}{\mathrm{d}{R}^p}\right|}_{R=R_i}=\sum _{j=1}^n{C}_{ij}^{\left(p\right)}{\Phi }_j^s,{\Phi }_j^s={\Phi }^s\left(R_j\right),i=1,2,\cdots ,n,\hfill \end{array}$$

(41)

where $C_{ij}^{\left(p\right)}$ denotes the weighting coefficients for the pth-order derivative. They are provided as [40]:

$$\begin{array}{cc}\hfill & C_{ij}^{\left(1\right)}=\frac{K\left(R_i\right)}{(R_i-R_j)K\left(R_j\right)},i,j=1,2,\cdots ,n;i\ne j,\hfill \\ & C_{ii}^{\left(1\right)}=-\sum _{l=1}^n{C}_{li}^{\left(1\right)},i=1,2,\cdots ,n;l\ne i,\hfill \\ & K\left(R_i\right)=\prod _{j=1}^n(R_i-R_j),i\ne j.\hfill \end{array}$$

(42)

Furthermore, the weighting coefficients $C_{ij}^{\left(p\right)},(p>1),$ for higher-order derivatives are defined as [40]:

$$\begin{array}{c}\hfill C_{ij}^{\left(p\right)}=\sum _{l=1}^n{C}_{il}^{\left(1\right)}{C}_{lj}^{(p-1)},i,j=1,2,\cdots ,n.\end{array}$$

(43)

However, the Gauss–Chebyshev–Lobatto approach is applied to estimate the mesh points $R_i$ as [40]:

$$\begin{array}{c}\hfill R_i=r_1+\frac{r_2-r_1}{2}\left[1-cos\left(\pi \frac{i-1}{n-1}\right)\right].\end{array}$$

(44)

The motion Equations (37)–(39) can be discretized using Equation (41), resulting in $(n-2)$ linear algebraic equations as follows:

$$\begin{array}{cc}\hfill & -A_2\sum _{j=1}^n{C}_{ij}^{\left(3\right)}{\Phi }_j^b-A_3\sum _{j=1}^n{C}_{ij}^{\left(3\right)}{\Phi }_j^s+A_1\sum _{j=1}^n{C}_{ij}^{\left(2\right)}{U}_j-A_2\frac{1}{R_i}\sum _{j=1}^n{C}_{ij}^{\left(2\right)}{\Phi }_j^b\hfill \\ & -A_3\frac{1}{R_i}\sum _{j=1}^n{C}_{ij}^{\left(2\right)}{\Phi }_j^s+A_1\frac{1}{R_i}\sum _{j=1}^n{C}_{ij}^{\left(1\right)}{U}_j+\sum _{j=1}^n{C}_{ij}^{\left(1\right)}{\Phi }_j^b\left(\frac{A_2}{R_i^2}-I_2{w}^2\right)\hfill \\ & +\sum _{j=1}^n{C}_{ij}^{\left(1\right)}{\Phi }_j^s\left(\frac{A_3}{R_i^2}-I_3{w}^2\right)+U_i\left(\frac{-A_1}{R_i^2}+I_1{w}^2\right)=0,\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill & -A_4\sum _{j=1}^n{C}_{ij}^{\left(4\right)}{\Phi }_j^b-A_5\sum _{j=1}^n{C}_{ij}^{\left(4\right)}{\Phi }_j^s+A_2\sum _{j=1}^n{C}_{ij}^{\left(3\right)}{U}_j-A_4\frac{2}{R_i}\sum _{j=1}^n{C}_{ij}^{\left(3\right)}{\Phi }_j^b\hfill \\ & -A_5\frac{2}{R_i}\sum _{j=1}^n{C}_{ij}^{\left(3\right)}{\Phi }_j^s+A_2\frac{2}{R_i}\sum _{j=1}^n{C}_{ij}^{\left(2\right)}{U}_j+\sum _{j=1}^n{C}_{ij}^{\left(2\right)}{\Phi }_j^b\left(\frac{A_4}{R_i^2}-I_4{w}^2+K_P+\gamma M_R^{2}T\right)\hfill \\ & +\sum _{j=1}^n{C}_{ij}^{\left(2\right)}{\Phi }_j^s\left(\frac{A_5}{R_i^2}-I_5{w}^2+K_P+\gamma M_R^{2}T\right)+\sum _{j=1}^n{C}_{ij}^{\left(1\right)}{U}_j\left(\frac{-A_2}{R_i^2}+I_2{w}^2\right)\hfill \\ & -\frac{1}{R_i}\sum _{j=1}^n{C}_{ij}^{\left(1\right)}{\Phi }_j^b\left(\frac{A_4}{R_i^2}+I_4{w}^2-K_P-\gamma M_R^{2}T\right)\hfill \\ & -\frac{1}{R_i}\sum _{j=1}^n{C}_{ij}^{\left(1\right)}{\Phi }_j^s\left(\frac{A_5}{R_i^2}+I_5{w}^2-K_P-\gamma M_R^{2}T\right)+\frac{U_i}{R_i}\left(\frac{A_2}{R_i^2}+I_2{w}^2\right)\hfill \\ & +{\Phi }_i^b\left(I_1{w}^2-K_w-\frac{\gamma M_R^{2}T}{R_i^2}\right)+{\Phi }_i^s\left(I_1{w}^2-K_w-\frac{\gamma M_R^{2}T}{R_i^2}\right)=0,\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill & -A_5\sum _{j=1}^n{C}_{ij}^{\left(4\right)}{\Phi }_j^b-A_6\sum _{j=1}^n{C}_{ij}^{\left(4\right)}{\Phi }_j^s+A_3\sum _{j=1}^n{C}_{ij}^{\left(3\right)}{U}_j-A_5\frac{2}{R_i}\sum _{j=1}^n{C}_{ij}^{\left(3\right)}{\Phi }_j^b\hfill \\ & -A_6\frac{2}{R_i}\sum _{j=1}^n{C}_{ij}^{\left(3\right)}{\Phi }_j^s+A_3\frac{2}{R_i}\sum _{j=1}^n{C}_{ij}^{\left(2\right)}{U}_j+\sum _{j=1}^n{C}_{ij}^{\left(2\right)}{\Phi }_j^b\left(\frac{A_5}{R_i^2}-I_5{w}^2+K_P+\gamma M_R^{2}T\right)\hfill \\ & +\sum _{j=1}^n{C}_{ij}^{\left(2\right)}{\Phi }_j^s\left(\frac{A_6}{R_i^2}+A_7-I_6{w}^2+K_P+\gamma M_R^{2}T\right)+\sum _{j=1}^n{C}_{ij}^{\left(1\right)}{U}_j\left(\frac{-A_3}{R_i^2}+I_3{w}^2\right)\hfill \\ & -\frac{1}{R_i}\sum _{j=1}^n{C}_{ij}^{\left(1\right)}{\Phi }_j^b\left(\frac{A_5}{R_i^2}+I_5{w}^2-K_P-\gamma M_R^{2}T\right)\hfill \\ & +\frac{1}{R_i}\sum _{j=1}^n{C}_{ij}^{\left(1\right)}{\Phi }_j^s\left(\frac{-A_6}{R_i^2}+A_7-I_6{w}^2+K_P+\gamma M_R^{2}T\right)+\frac{U_i}{R_i}\left(\frac{A_3}{R_i^2}+I_3{w}^2\right)\hfill \\ & +{\Phi }_i^b\left(I_1{w}^2-K_w-\frac{\gamma M_R^{2}T}{R_i^2}\right)+{\Phi }_i^s\left(I_1{w}^2-K_w-\frac{\gamma M_R^{2}T}{R_i^2}\right)=0,i=2,3,\cdots ,(n-1).\hfill \end{array}$$

(47)

Furthermore, the boundary conditions can be represented in a discretized form as follows:

Simply supported (S):

$$\begin{array}{cc}\hfill & {\Phi }_i^b={\Phi }_i^s=A_1\sum _{j=1}^n{C}_{ij}^{\left(1\right)}{U}_j-A_2\sum _{j=1}^n{C}_{ij}^{\left(2\right)}{\Phi }_j^b-A_3\sum _{j=1}^n{C}_{ij}^{\left(2\right)}{\Phi }_j^s+B_1\frac{U_i}{R_i}-B_2\frac{1}{R_i}\sum _{j=1}^n{C}_{ij}^{\left(1\right)}{\Phi }_j^b\hfill \\ & -B_3\frac{1}{R_i}\sum _{j=1}^n{C}_{ij}^{\left(1\right)}{\Phi }_j^s=A_2\sum _{j=1}^n{C}_{ij}^{\left(1\right)}{U}_j-A_4\sum _{j=1}^n{C}_{ij}^{\left(2\right)}{\Phi }_j^b-A_5\sum _{j=1}^n{C}_{ij}^{\left(2\right)}{\Phi }_j^s+B_2\frac{U_i}{R_i}\hfill \\ & -B_4\frac{1}{R_i}\sum _{j=1}^n{C}_{ij}^{\left(1\right)}{\Phi }_j^b-B_5\frac{1}{R_i}\sum _{j=1}^n{C}_{ij}^{\left(1\right)}{\Phi }_j^s=A_3\sum _{j=1}^n{C}_{ij}^{\left(1\right)}{U}_j-A_5\sum _{j=1}^n{C}_{ij}^{\left(2\right)}{\Phi }_j^b\hfill \\ & -A_6\sum _{j=1}^n{C}_{ij}^{\left(2\right)}{\Phi }_j^s+B_3\frac{U_i}{R_i}-B_5\frac{1}{R_i}\sum _{j=1}^n{C}_{ij}^{\left(1\right)}{\Phi }_j^b-B_6\frac{1}{R_i}\sum _{j=1}^n{C}_{ij}^{\left(1\right)}{\Phi }_j^s=0,i=1,n,\hfill \end{array}$$

(48)

where

$$\begin{array}{cc}\hfill & B_1=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{C}_{12}^{\left(i\right)}\mathrm{d}Z,B_2=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{C}_{12}^{\left(i\right)}Z\mathrm{d}Z,\hfill \\ & B_3=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{C}_{12}^{\left(i\right)}G\left(Z\right)\mathrm{d}Z,B_4=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{C}_{12}^{\left(i\right)}{Z}^2\mathrm{d}Z,\hfill \\ & B_5=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{C}_{12}^{\left(i\right)}ZG\left(Z\right)\mathrm{d}Z,B_6=\sum _{i=1}^3{\int }_{T_{i-1}}^{T_i}{C}_{12}^{\left(i\right)}{G}^2\left(Z\right)\mathrm{d}Z.\hfill \end{array}$$

(49)

System (45)–(47) represents an eigenvalue problem that can be solved with the boundary conditions (48) to obtain the eigenfrequency.

5. Numerical Results

In the current section, various numerical examples are presented to demonstrate the effects of various parameters on the frequency of a GPLs-reinforced annular plate with an FG porous core, including the porosity factor ${\beta }_0$, the porosity distribution types, the GPLs weight fraction $W_G$, the GPLs distribution type, the outer radius-to-thickness ratio $r_2/T$, the magnetic field parameter ${\overline{M}}_R$, the core thickness $T_c/T$, and the elastic substrate parameters $g_1$ and $g_2$. The dimensionless formulations are defined by:

$$\begin{array}{cc}\hfill & \Omega =\omega r_2^2\sqrt{\frac{T{\rho }_m}{E_m}},{\overline{M}}_R=\frac{\gamma T{r}_2^2{M}_R^2}{D_m}{g}_1=\frac{K_w{r}_2^4}{D_m},g_2=\frac{K_P{r}_2^2}{D_m},D_m=\frac{E_m{T}^3}{12(1-{\nu }_m^2)}.\hfill \end{array}$$

(50)

The material properties of the constituent materials are taken as: $E_m=70$ GPa, ${\nu }_m=0.3$, ${\rho }_m=2700$ Kg/m³, $E_g=1.01$ TPa, ${\nu }_g=0.186$, and ${\rho }_g=1060$ Kg/m³.

The numerical examples use the following fixed data (unless otherwise declared): $W_G=0.1$, ${\overline{M}}_R=50$, $T_c/T=0.1$, $g_1=100$, $g_2=10$, ${\beta }_0=0.2$, $r_1/r_2=0.1$, $r_2/T=10$, $T=0.2$ m, $L^g=15$ nm, $W^g=9$ nm, and $T^g=0.188$ nm.

Firstly,

Table 1. Convergence of the results of FG porous nanocomposite annular plates ($r_2/T=8,T_R=T_c/T$).

n	${\mathit{T}}_{\mathit{R}}=0.1$	Error	${\mathit{T}}_{\mathit{R}}=0.2$	Error	${\mathit{T}}_{\mathit{R}}=0.3$	Error	${\mathit{T}}_{\mathit{R}}=0.4$	Error
7	$1.2917$	-	$1.2795$	-	$1.2637$	-	$1.2424$	-
9	$2.9013$	$1.6096$	$2.7686$	$1.4891$	$2.6314$	$1.3677$	$2.4888$	$1.2454$
11	$2.8621$	$0.0393$	$2.7674$	$0.0012$	$2.6302$	$0.0011$	$2.4877$	$0.0011$
13	$2.8996$	$0.0375$	$2.7669$	$0.0005$	$2.6297$	$0.0005$	$2.4872$	$0.0005$
15	$2.8994$	$0.0002$	$2.7667$	$0.0002$	$2.6296$	$0.0002$	$2.4871$	$0.0002$
17	$2.8993$	$0.0001$	$2.7667$	$0.0001$	$2.6295$	$0.0001$	$2.4870$	$0.0001$

${\mathrm{Error}}_i={\Omega }_i-{\Omega }_{i-1}$.

shows a convergence analysis for the current outcomes of the FG porous annular plate enhanced by GPLs for various values of core thickness $T_c/T$. The minimal number of grid points needed for a converged solution in the DQM is calculated in this table. It is worth noting that 17 grid points are adequate to produce a converged solution.

Secondly, to ensure the accuracy of the current theory, the frequency obtained via the present theory is compared with that estimated using the third-order plate theory (TPT) [48], the sinusoidal plate theory (SPT) [49], the hyperbolic plate theory (HPT) [50], and the exponential plate theory (EPT) [51] for various plate geometry parameter values as shown in

Table 2. Comparison of the results of the present theory for the frequency $\Omega$ of a GPLs-reinforced annular plate with an FG porous core with the results of other theories.

${\mathit{r}}_2/\mathit{T}$	${\mathit{r}}_1/{\mathit{r}}_2$	TPT [48]	SPT [49]	HPT [50]	EPT [51]	Present
10	$0.1$	$3.62419$	$3.62419$	$3.62419$	$3.62419$	$3.62419$
	$0.2$	$3.31653$	$3.31653$	$3.31653$	$3.31653$	$3.31653$
	$0.3$	$2.68767$	$2.68631$	$2.68781$	$2.68524$	$2.68662$
	$0.4$	$2.17241$	$2.17182$	$2.17247$	$2.17135$	$2.17195$
	$0.5$	$2.05191$	$2.05112$	$2.05191$	$2.05049$	$2.05130$
20	$0.1$	$7.24838$	$7.24838$	$7.24838$	$7.24838$	$7.24838$
	$0.2$	$6.63306$	$6.63306$	$6.63306$	$6.63306$	$6.63306$
	$0.3$	$2.80871$	$2.80831$	$2.80874$	$2.80815$	$2.80847$
	$0.4$	$2.23078$	$2.23064$	$2.23071$	$2.23053$	$2.23067$
	$0.5$	$2.17024$	$2.16992$	$2.17027$	$2.16967$	$2.16999$
30	$0.1$	$10.87256$	$10.87256$	$10.87256$	$10.87256$	$10.87256$
	$0.2$	$9.94959$	$9.94959$	$9.94959$	$9.94959$	$9.94959$
	$0.3$	$2.82940$	$2.82927$	$2.82942$	$2.82916$	$2.82930$
	$0.4$	$2.24134$	$2.24128$	$2.24135$	$2.24123$	$2.24129$
	$0.5$	$2.19826$	$2.19810$	$2.19827$	$2.19798$	$2.19814$
40	$0.1$	$12.62866$	$12.62634$	$12.62890$	$12.62451$	$12.62687$
	$0.2$	$12.76462$	$12.76230$	$12.76485$	$12.76049$	$12.76283$
	$0.3$	$2.83644$	$2.83637$	$2.83645$	$2.83631$	$2.83638$
	$0.4$	$2.24501$	$2.24498$	$2.24501$	$2.24495$	$2.24498$
	$0.5$	$2.20858$	$2.20849$	$2.20859$	$2.20842$	$2.20851$
50	$0.1$	$12.74444$	$12.74289$	$12.74460$	$12.74167$	$12.74324$
	$0.2$	$12.89102$	$12.88947$	$12.89118$	$12.88825$	$12.88982$
	$0.3$	$2.83966$	$2.83961$	$2.83966$	$2.83957$	$2.83962$
	$0.4$	$2.24671$	$2.24668$	$2.24671$	$2.24667$	$2.24669$
	$0.5$	$2.21345$	$2.21339$	$2.21345$	$2.21334$	$2.21340$

. It should be observed that the frequencies offered by our theory are extremely consistent with those obtained by the other, higher-order theories. Furthermore, this table shows the results for various values of the outer radius-to-thickness ratio $r_2/T$ and the inner-to-outer radius ratio $r_1/r_2$. As shown in this table, the eigenfrequency increases as the ratio $r_2/T$ increases and the ratio $r_1/r_2$ decreases because the increase in the inner-to-outer radius ratio $r_1/r_2$ leads to a decrease in the plate width; hence, the annular plate becomes weaker.

Table 3. The frequency $\Omega$ of a GPLs-reinforced annular plate with an FG porous core for different values of inner-to-outer radius ratio $r_1/r_2$, outer radius-to-thickness ratio $r_2/T$, and weight fraction of GPLs $W_G$ (type A) ($T_c/T=0.2$).

${\mathit{W}}_{\mathit{G}}$	${\mathit{r}}_1/{\mathit{r}}_2$	${\mathit{r}}_2/\mathit{T}=10$	${\mathit{r}}_2/\mathit{T}=20$	${\mathit{r}}_2/\mathit{T}=30$	${\mathit{r}}_2/\mathit{T}=40$	${\mathit{r}}_2/\mathit{T}=50$
$0.1$	$0.1$	$3.45840$	$6.91671$	$10.37520$	$12.50054$	$12.63196$
	$0.2$	$3.16457$	$6.32914$	$9.49371$	$12.63682$	$12.77847$
	$0.3$	$2.64327$	$2.78527$	$2.80948$	$2.81768$	$2.82143$
	$0.4$	$2.14791$	$2.21516$	$2.22724$	$2.23144$	$2.23337$
	$0.5$	$2.02688$	$2.15425$	$2.18558$	$2.19721$	$2.20272$
$0.3$	$0.1$	$5.25929$	$10.51858$	$15.77788$	$19.06438$	$19.27266$
	$0.2$	$4.83757$	$9.67515$	$14.51272$	$19.16868$	$19.38888$
	$0.3$	$3.78301$	$4.05868$	$4.11357$	$4.13300$	$4.14203$
	$0.4$	$2.98221$	$3.11984$	$3.14735$	$3.15713$	$3.16167$
	$0.5$	$2.57020$	$2.64910$	$2.66782$	$2.67471$	$2.67811$
$0.5$	$0.1$	$6.46262$	$12.92523$	$19.38785$	$23.54438$	$23.79938$
	$0.2$	$5.96345$	$11.92689$	$17.89034$	$23.64136$	$23.90945$
	$0.3$	$4.58008$	$4.92612$	$4.99636$	$5.02136$	$5.03301$
	$0.4$	$3.56945$	$3.74259$	$3.77768$	$3.79011$	$3.79603$
	$0.5$	$3.00713$	$3.08113$	$3.09749$	$3.10348$	$3.10630$

,

Table 4. The frequency $\Omega$ of a GPLs-reinforced annular plate with an FG porous core for different values of inner-to-outer radius ratio $r_1/r_2$, outer radius-to-thickness ratio $r_2/T$, and weight fraction of GPLs $W_G$ (type B) ($T_c/T=0.2$).

${\mathit{W}}_{\mathit{G}}$	${\mathit{r}}_1/{\mathit{r}}_2$	${\mathit{r}}_2/\mathit{T}=10$	${\mathit{r}}_2/\mathit{T}=20$	${\mathit{r}}_2/\mathit{T}=30$	${\mathit{r}}_2/\mathit{T}=40$	${\mathit{r}}_2/\mathit{T}=50$
$0.1$	$0.1$	$2.93543$	$5.87086$	$8.80621$	$11.41827$	$11.57085$
	$0.2$	$2.68318$	$5.36635$	$8.04953$	$10.73270$	$11.72338$
	$0.3$	$2.40694$	$2.56998$	$2.59556$	$2.60392$	$2.60768$
	$0.4$	$1.99251$	$2.06498$	$2.07729$	$2.08141$	$2.08342$
	$0.5$	$1.91637$	$2.06641$	$2.10484$	$2.11912$	$2.12589$
$0.3$	$0.1$	$4.13848$	$8.27695$	$12.41543$	$16.39444$	$16.65699$
	$0.2$	$3.79834$	$7.59667$	$11.39501$	$15.19334$	$16.77831$
	$0.3$	$3.19026$	$3.53593$	$3.60621$	$3.63108$	$3.64263$
	$0.4$	$2.58143$	$2.75033$	$2.78485$	$2.79713$	$2.80286$
	$0.5$	$2.29262$	$2.39291$	$2.42024$	$2.43087$	$2.43591$
$0.5$	$0.1$	$4.87654$	$9.75307$	$14.62961$	$19.47708$	$19.80334$
	$0.2$	$4.48737$	$8.97474$	$13.46211$	$17.94948$	$19.91742$
	$0.3$	$3.68636$	$4.12317$	$4.21542$	$4.24846$	$4.26387$
	$0.4$	$2.95577$	$3.17110$	$3.21653$	$3.23284$	$3.24046$
	$0.5$	$2.57556$	$2.66715$	$2.68980$	$2.69841$	$2.70252$

and

Table 5. The frequency $\Omega$ of a GPLs-reinforced annular plate with an FG porous core for different values of inner-to-outer radius ratio $r_1/r_2$, outer radius-to-thickness ratio $r_2/T$, and weight fraction of GPLs $W_G$ (type C) ($T_c/T=0.2$).

${\mathit{W}}_{\mathit{G}}$	${\mathit{r}}_1/{\mathit{r}}_2$	${\mathit{r}}_2/\mathit{T}=10$	${\mathit{r}}_2/\mathit{T}=20$	${\mathit{r}}_2/\mathit{T}=30$	${\mathit{r}}_2/\mathit{T}=40$	${\mathit{r}}_2/\mathit{T}=50$
$0.1$	$0.1$	$2.93543$	$5.87086$	$8.80621$	$9.65646$	$9.73211$
	$0.2$	$2.68318$	$5.36635$	$8.04953$	$9.83057$	$9.91635$
	$0.3$	$2.13381$	$2.14971$	$2.14907$	$2.14859$	$2.14832$
	$0.4$	$1.77801$	$1.79123$	$1.79341$	$1.79416$	$1.79451$
	$0.5$	$1.89657$	$1.99483$	$2.01435$	$2.02127$	$2.02441$
$0.3$	$0.1$	$4.13848$	$8.27695$	$12.41543$	$13.07747$	$13.17954$
	$0.2$	$3.79834$	$7.59667$	$11.39501$	$13.21485$	$13.32873$
	$0.3$	$2.80084$	$2.90316$	$2.92066$	$2.92662$	$2.92935$
	$0.4$	$2.25143$	$2.30241$	$2.31166$	$2.31488$	$2.31637$
	$0.5$	$2.12881$	$2.24002$	$2.26534$	$2.27458$	$2.27893$
$0.5$	$0.1$	$4.87654$	$9.75307$	$14.62961$	$15.25590$	$15.37391$
	$0.2$	$4.48737$	$8.97474$	$13.46211$	$15.37919$	$15.51052$
	$0.3$	$3.20118$	$3.33752$	$3.36254$	$3.37123$	$3.37524$
	$0.4$	$2.54227$	$2.61137$	$2.62437$	$2.62893$	$2.63105$
	$0.5$	$2.29633$	$2.39271$	$2.41537$	$2.42374$	$2.42761$

depict the effects of the outer radius-to-thickness ratio $r_2/T$, the inner-to-outer radius ratio $r_1/r_2$, and the GPLs weight fraction $W_G$ on the frequency of a GPLs-reinforced annular plate with an FG porous core. It can be noted that, irrespective of the graphene distribution type, the frequency gradually increases with an increasing outer radius-to-thickness ratio $r_2/T$ and a decreasing inner-to-outer radius ratio $r_1/r_2$. On the other hand, the frequency increases as the graphene weight fraction $W_G$ increases because the plate becomes stiffer by increasing the graphene component.

Table 6. The frequency $\Omega$ of a GPLs-reinforced annular plate with an FG porous core for different values of the inner-to-outer radius ratio $r_1/r_2$ and the outer radius-to-thickness ratio $r_2/T$ (Porous-I) ($T_c/T=0.6$).

${\mathit{\beta }}_0$	${\mathit{r}}_1/{\mathit{r}}_2$	${\mathit{r}}_2/\mathit{T}=10$	${\mathit{r}}_2/\mathit{T}=20$	${\mathit{r}}_2/\mathit{T}=30$	${\mathit{r}}_2/\mathit{T}=40$	${\mathit{r}}_2/\mathit{T}=50$
$0.1$	$0.1$	$2.72497$	$5.44993$	$8.17490$	$10.89987$	$11.26156$
	$0.2$	$2.49151$	$4.98303$	$7.47454$	$9.96605$	$11.41602$
	$0.3$	$2.23661$	$2.50761$	$2.53736$	$2.54687$	$2.55101$
	$0.4$	$1.93981$	$2.02395$	$2.03781$	$2.04249$	$2.04462$
	$0.5$	$1.84099$	$2.03665$	$2.08098$	$2.09765$	$2.10556$
$0.3$	$0.1$	$2.73089$	$5.46178$	$8.19267$	$10.92356$	$11.45363$
	$0.2$	$2.49809$	$4.99618$	$7.49428$	$9.99237$	$11.61191$
	$0.3$	$2.24354$	$2.55254$	$2.58533$	$2.59572$	$2.60032$
	$0.4$	$1.97081$	$2.06336$	$2.07847$	$2.08354$	$2.08585$
	$0.5$	$1.84745$	$2.07738$	$2.12569$	$2.14393$	$2.15261$
$0.6$	$0.1$	$2.75916$	$5.51833$	$8.27749$	$11.03665$	$11.84989$
	$0.2$	$2.52529$	$5.05059$	$7.57588$	$10.10118$	$12.01566$
	$0.3$	$2.26905$	$2.64491$	$2.68498$	$2.69748$	$2.70296$
	$0.4$	$2.03245$	$2.14456$	$2.16263$	$2.16863$	$2.17135$
	$0.5$	$1.86941$	$2.15875$	$2.21583$	$2.23765$	$2.24806$

,

Table 7. The frequency $\Omega$ of a GPLs-reinforced annular plate with an FG porous core for different values of the inner-to-outer radius ratio $r_1/r_2$ and the outer radius-to-thickness ratio $r_2/T$ (porous-II) ($T_c/T=0.6$).

${\mathit{\beta }}_0$	${\mathit{r}}_1/{\mathit{r}}_2$	${\mathit{r}}_2/\mathit{T}=10$	${\mathit{r}}_2/\mathit{T}=20$	${\mathit{r}}_2/\mathit{T}=30$	${\mathit{r}}_2/\mathit{T}=40$	${\mathit{r}}_2/\mathit{T}=50$
$0.1$	$0.1$	$2.72531$	$5.45062$	$8.17592$	$10.90123$	$11.27346$
	$0.2$	$2.49184$	$4.98369$	$7.47553$	$9.96737$	$11.42781$
	$0.3$	$2.23701$	$2.51017$	$2.54036$	$2.55002$	$2.55430$
	$0.4$	$1.94074$	$2.02598$	$2.04003$	$2.04477$	$2.04693$
	$0.5$	$1.84125$	$2.03739$	$2.08208$	$2.09889$	$2.10688$
$0.3$	$0.1$	$2.73431$	$5.46879$	$8.20318$	$10.93758$	$11.49275$
	$0.2$	$2.50127$	$5.00254$	$7.50380$	$10.00507$	$11.65068$
	$0.3$	$2.24637$	$2.56091$	$2.59522$	$2.60610$	$2.61092$
	$0.4$	$1.97382$	$2.07005$	$2.08581$	$2.09101$	$2.09351$
	$0.5$	$1.84975$	$2.07992$	$2.12939$	$2.14814$	$2.15707$
$0.6$	$0.1$	$2.77761$	$5.55521$	$8.33282$	$11.11042$	$11.94643$
	$0.2$	$2.54181$	$5.08371$	$7.62569$	$10.16759$	$12.11144$
	$0.3$	$2.28374$	$2.66534$	$2.70911$	$2.72280$	$2.72881$
	$0.4$	$2.04035$	$2.16095$	$2.18060$	$2.18713$	$2.19009$
	$0.5$	$1.88132$	$2.16586$	$2.22549$	$2.24845$	$2.25944$

and

Table 8. The frequency $\Omega$ of a GPLs-reinforced annular plate with an FG porous core for different values of the inner-to-outer radius ratio $r_1/r_2$ and the outer radius-to-thickness ratio $r_2/T$ (porous-III) ($T_c/T=0.6$).

${\mathit{\beta }}_0$	${\mathit{r}}_1/{\mathit{r}}_2$	${\mathit{r}}_2/\mathit{T}=10$	${\mathit{r}}_2/\mathit{T}=20$	${\mathit{r}}_2/\mathit{T}=30$	${\mathit{r}}_2/\mathit{T}=40$	${\mathit{r}}_2/\mathit{T}=50$
$0.1$	$0.1$	$2.72479$	$5.44957$	$8.17436$	$10.89915$	$11.21445$
	$0.2$	$2.49104$	$4.98207$	$7.47311$	$9.96414$	$11.36829$
	$0.3$	$2.23602$	$2.49678$	$2.52556$	$2.53478$	$2.53888$
	$0.4$	$1.93307$	$2.01465$	$2.02810$	$2.03264$	$2.03472$
	$0.5$	$1.84021$	$2.02824$	$2.07149$	$2.08772$	$2.09542$
$0.3$	$0.1$	$2.72891$	$5.45783$	$8.18674$	$10.91566$	$11.29909$
	$0.2$	$2.49541$	$4.99081$	$7.48622$	$9.98163$	$11.45526$
	$0.3$	$2.24042$	$2.51690$	$2.54624$	$2.55559$	$2.55974$
	$0.4$	$1.94907$	$2.03251$	$2.04618$	$2.05079$	$2.05289$
	$0.5$	$1.84426$	$2.04960$	$2.09410$	$2.11081$	$2.11873$
$0.6$	$0.1$	$2.74774$	$5.49548$	$8.24322$	$10.99096$	$11.48015$
	$0.2$	$2.51333$	$5.02666$	$7.53999$	$10.05332$	$11.64057$
	$0.3$	$2.25708$	$2.55922$	$2.58967$	$2.59930$	$2.60356$
	$0.4$	$1.98222$	$2.06927$	$2.08339$	$2.08813$	$2.09029$
	$0.5$	$1.85847$	$2.09156$	$2.13834$	$2.15591$	$2.16426$

depict the effects of the porosity factor ${\beta }_0$ on the frequency of a GPLs-reinforced circular/annular plate with an FG porous core for various values of the outer radius-to-thickness ratio $r_2/T$ and the inner-to-outer radius ratio $r_1/r_2$. It is found that, regardless of the variations of the ratios $r_2/T$ and $r_1/r_2$, the frequency gradually increases with an increasing porosity factor ${\beta }_0$ since the plate becomes lighter. It is also noted that the difference between the results of the three porosity types is slight.

The influence of the magnetic field parameter ${\overline{M}}_R$ on the frequency of a GPLs-reinforced annular plate with or without an FG porous core is depicted in

Table 9. The frequency $\Omega$ of a GPLs-reinforced annular plate with an FG porous core for different values of the magnetic field parameter ${\overline{M}}_R$, the inner-to-outer radius ratio $r_1/r_2$, and the outer radius-to-thickness ratio $r_2/T$ ($T_c/T=0$).

${\overline{\mathit{M}}}_{\mathit{R}}$	${\mathit{r}}_1/{\mathit{r}}_2$	${\mathit{r}}_2/\mathit{T}=10$	${\mathit{r}}_2/\mathit{T}=20$	${\mathit{r}}_2/\mathit{T}=30$	${\mathit{r}}_2/\mathit{T}=40$	${\mathit{r}}_2/\mathit{T}=50$
0	$0.1$	$3.78511$	$7.57022$	$11.35533$	$12.32216$	$12.42809$
	$0.2$	$3.46402$	$6.92804$	$10.39205$	$12.34207$	$12.45619$
	$0.3$	$2.34392$	$2.48982$	$2.51952$	$2.53012$	$2.53507$
	$0.4$	$1.75904$	$1.83196$	$1.84672$	$1.85199$	$1.85444$
	$0.5$	$1.39447$	$1.41808$	$1.42271$	$1.42448$	$1.42526$
100	$0.1$	$3.78511$	$7.57022$	$11.35533$	$13.10209$	$13.20503$
	$0.2$	$3.46402$	$6.92804$	$10.39205$	$13.34898$	$13.46559$
	$0.3$	$2.88701$	$2.89661$	$2.89333$	$2.89184$	$2.89101$
	$0.4$	$2.42077$	$2.43442$	$2.43667$	$2.43746$	$2.43782$
	$0.5$	$2.56396$	$2.74667$	$2.77239$	$2.78151$	$2.78575$
200	$0.1$	$3.78511$	$7.57022$	$11.35533$	$13.84635$	$13.94750$
	$0.2$	$3.46402$	$6.92804$	$10.39205$	$13.85607$	$14.42084$
	$0.3$	$2.79490$	$2.69504$	$2.67677$	$2.67045$	$2.66755$
	$0.4$	$2.73538$	$2.74577$	$2.74929$	$2.75065$	$2.75121$
	$0.5$	$2.56396$	$3.39930$	$3.41867$	$3.42558$	$3.42880$

,

Table 10. The frequency $\Omega$ of a GPLs-reinforced annular plate with an FG porous core for different values of the magnetic field parameter ${\overline{M}}_R$, the inner-to-outer radius ratio $r_1/r_2$, and the outer radius-to-thickness ratio $r_2/T$ ($T_c/T=3$).

${\overline{\mathit{M}}}_{\mathit{R}}$	${\mathit{r}}_1/{\mathit{r}}_2$	${\mathit{r}}_2/\mathit{T}=10$	${\mathit{r}}_2/\mathit{T}=20$	${\mathit{r}}_2/\mathit{T}=30$	${\mathit{r}}_2/\mathit{T}=40$	${\mathit{r}}_2/\mathit{T}=50$
0	$0.1$	$3.29501$	$6.59002$	$9.88503$	$11.97895$	$12.13395$
	$0.2$	$3.01502$	$6.03004$	$9.04506$	$12.00375$	$12.16475$
	$0.3$	$2.21325$	$2.42378$	$2.46854$	$2.48467$	$2.49221$
	$0.4$	$1.68716$	$1.79144$	$1.81341$	$1.82132$	$1.82502$
	$0.5$	$1.36481$	$1.39534$	$1.40156$	$1.40379$	$1.40483$
100	$0.1$	$3.29501$	$6.59002$	$9.88503$	$12.77211$	$12.92070$
	$0.2$	$3.01502$	$6.03004$	$9.04506$	$12.06008$	$13.18007$
	$0.3$	$2.70851$	$2.84856$	$2.84548$	$2.84349$	$2.84241$
	$0.4$	$2.37977$	$2.40221$	$2.40479$	$2.40565$	$2.40604$
	$0.5$	$2.23097$	$2.70291$	$2.73938$	$2.75235$	$2.75840$
200	$0.1$	$3.29501$	$6.59002$	$9.88503$	$13.18003$	$13.67082$
	$0.2$	$3.01502$	$6.03004$	$9.04506$	$12.06008$	$14.13854$
	$0.3$	$2.70851$	$2.66906$	$2.64006$	$2.62994$	$2.62527$
	$0.4$	$2.44472$	$2.71928$	$2.72263$	$2.72406$	$2.72476$
	$0.5$	$2.23097$	$3.35626$	$3.38256$	$3.39199$	$3.39641$

and

Table 11. The frequency $\Omega$ of a GPLs-reinforced annular plate with an FG porous core for different values of the magnetic field parameter ${\overline{M}}_R$, the inner-to-outer radius ratio $r_1/r_2$, and the outer radius-to-thickness ratio $r_2/T$ ($T_c/T=0.6$).

${\overline{\mathit{M}}}_{\mathit{R}}$	${\mathit{r}}_1/{\mathit{r}}_2$	${\mathit{r}}_2/\mathit{T}=10$	${\mathit{r}}_2/\mathit{T}=20$	${\mathit{r}}_2/\mathit{T}=30$	${\mathit{r}}_2/\mathit{T}=40$	${\mathit{r}}_2/\mathit{T}=50$
0	$0.1$	$2.73089$	$5.46178$	$8.19267$	$10.82260$	$11.02036$
	$0.2$	$2.49809$	$4.99618$	$7.49428$	$9.99237$	$11.05634$
	$0.3$	$1.94258$	$2.20352$	$2.26182$	$2.28306$	$2.29303$
	$0.4$	$1.51377$	$1.64233$	$1.67076$	$1.68111$	$1.68597$
	$0.5$	$1.25866$	$1.29468$	$1.30224$	$1.30499$	$1.30627$
100	$0.1$	$2.73089$	$5.46178$	$8.19267$	$10.92356$	$11.87347$
	$0.2$	$2.49809$	$4.99618$	$7.49428$	$9.99237$	$12.14720$
	$0.3$	$2.24354$	$2.59419$	$2.57964$	$2.57274$	$2.56925$
	$0.4$	$2.02467$	$2.25650$	$2.25682$	$2.25693$	$2.25698$
	$0.5$	$1.84745$	$2.60073$	$2.64365$	$2.65898$	$2.66613$
200	$0.1$	$2.73089$	$5.46178$	$8.19267$	$10.92356$	$12.67992$
	$0.2$	$2.49809$	$4.99618$	$7.49428$	$3.55904$	$3.45135$
	$0.3$	$2.24354$	$2.46485$	$2.42912$	$2.41691$	$2.41148$
	$0.4$	$2.02467$	$2.62179$	$2.62684$	$2.62906$	$2.63016$
	$0.5$	$1.84745$	$3.24021$	$3.27054$	$3.28156$	$3.28675$

for different values of the outer radius-to-thickness ratio $r_2/T$ and the inner-to-outer radius ratio $r_1/r_2$. It can be seen that the frequency monotonically increases with an increase in the magnetic field parameter ${\overline{M}}_R$. One can conduct that the presence of the magnetic field enhances the plate strength.

Table 12. The frequency $\Omega$ of a GPLs-reinforced annular plate with an FG porous core for different values of the shear elastic substrate stiffness $g_2$, the inner-to-outer radius ratio $r_1/r_2$, and the outer radius-to-thickness ratio $r_2/T$ ($g_1=0$).

${\mathit{g}}_2$	${\mathit{r}}_1/{\mathit{r}}_2$	${\mathit{r}}_2/\mathit{T}=10$	${\mathit{r}}_2/\mathit{T}=20$	${\mathit{r}}_2/\mathit{T}=30$	${\mathit{r}}_2/\mathit{T}=40$	${\mathit{r}}_2/\mathit{T}=50$
20	$0.1$	$3.45840$	$6.91671$	$10.37520$	$12.57104$	$12.70205$
	$0.2$	$3.16457$	$6.32914$	$9.49371$	$12.65827$	$12.87656$
	$0.3$	$2.67273$	$2.79571$	$2.81476$	$2.82099$	$2.82371$
	$0.4$	$2.17469$	$2.23203$	$2.24195$	$2.24536$	$2.24693$
	$0.5$	$2.11098$	$2.26067$	$2.29559$	$2.30835$	$2.31436$
40	$0.1$	$3.45840$	$6.91671$	$10.37520$	$12.71755$	$12.84778$
	$0.2$	$3.16457$	$6.32914$	$9.49371$	$12.65827$	$13.07717$
	$0.3$	$2.74371$	$2.81842$	$2.82535$	$2.82719$	$2.82794$
	$0.4$	$2.25592$	$2.29301$	$2.29907$	$2.30112$	$2.30206$
	$0.5$	$2.31699$	$2.48732$	$2.52325$	$2.53611$	$2.54211$
60	$0.1$	$3.45840$	$6.91671$	$10.37520$	$12.86270$	$12.99219$
	$0.2$	$3.16457$	$6.32914$	$9.49371$	$12.65827$	$13.27517$
	$0.3$	$2.78841$	$2.80649$	$2.80131$	$2.79891$	$2.79770$
	$0.4$	$2.32685$	$2.34765$	$2.35101$	$2.35219$	$2.35274$
	$0.5$	$2.34169$	$2.67632$	$2.71006$	$2.72202$	$2.72759$

,

Table 13. The frequency $\Omega$ of a GPLs-reinforced annular plate with an FG porous core for different values of the shear elastic substrate stiffness $g_2$, the inner-to-outer radius ratio $r_1/r_2$, and the outer radius-to-thickness ratio $r_2/T$ ($g_1=200$).

${\mathit{g}}_2$	${\mathit{r}}_1/{\mathit{r}}_2$	${\mathit{r}}_2/\mathit{T}=10$	${\mathit{r}}_2/\mathit{T}=20$	${\mathit{r}}_2/\mathit{T}=30$	${\mathit{r}}_2/\mathit{T}=40$	${\mathit{r}}_2/\mathit{T}=50$
20	$0.1$	$3.45840$	$6.91671$	$10.37520$	$12.57759$	$12.70857$
	$0.2$	$3.16457$	$6.32914$	$9.49371$	$12.65827$	$12.88296$
	$0.3$	$2.70335$	$2.82538$	$2.84418$	$2.85036$	$2.85314$
	$0.4$	$2.21218$	$2.26893$	$2.27875$	$2.28213$	$2.28369$
	$0.5$	$2.14844$	$2.29674$	$2.33137$	$2.34404$	$2.34991$
40	$0.1$	$3.45840$	$6.91671$	$10.37520$	$12.72404$	$12.85422$
	$0.2$	$3.16457$	$6.32914$	$9.49371$	$12.65827$	$13.08347$
	$0.3$	$2.77378$	$2.84785$	$2.85461$	$2.85652$	$2.85726$
	$0.4$	$2.29218$	$2.32905$	$2.33498$	$2.33702$	$2.33795$
	$0.5$	$2.34169$	$2.52018$	$2.55587$	$2.56864$	$2.57461$
60	$0.1$	$3.45840$	$6.91671$	$10.37520$	$12.86911$	$12.99856$
	$0.2$	$3.16457$	$6.32914$	$9.49371$	$12.65827$	$13.28139$
	$0.3$	$2.81827$	$2.83610$	$2.83094$	$2.82854$	$2.82734$
	$0.4$	$2.36208$	$2.38278$	$2.38614$	$2.38732$	$2.38787$
	$0.5$	$2.34169$	$2.70693$	$2.74048$	$2.75237$	$2.75791$

and

Table 14. The frequency $\Omega$ of a GPLs-reinforced annular plate with an FG porous core for different values of the shear elastic substrate stiffness $g_2$, the inner-to-outer radius ratio $r_1/r_2$, and the outer radius-to-thickness ratio $r_2/T$ ($g_1=400$).

${\mathit{g}}_2$	${\mathit{r}}_1/{\mathit{r}}_2$	${\mathit{r}}_2/\mathit{T}=10$	${\mathit{r}}_2/\mathit{T}=20$	${\mathit{r}}_2/\mathit{T}=30$	${\mathit{r}}_2/\mathit{T}=40$	${\mathit{r}}_2/\mathit{T}=50$
20	$0.1$	$3.45840$	$6.91671$	$10.37520$	$12.58415$	$12.71508$
	$0.2$	$3.16457$	$6.32914$	$9.49371$	$12.65827$	$12.88936$
	$0.3$	$2.73362$	$2.85465$	$2.87330$	$2.87943$	$2.88219$
	$0.4$	$2.24905$	$2.30524$	$2.31497$	$2.31832$	$2.31986$
	$0.5$	$2.18526$	$2.33225$	$2.36662$	$2.37919$	$2.38511$
40	$0.1$	$3.45840$	$6.91671$	$10.37520$	$12.73052$	$12.86065$
	$0.2$	$3.16457$	$6.32914$	$9.49371$	$12.65827$	$13.08978$
	$0.3$	$2.80352$	$2.87697$	$2.88375$	$2.88554$	$2.88627$
	$0.4$	$2.32787$	$2.36445$	$2.37035$	$2.37237$	$2.37321$
	$0.5$	$2.34169$	$2.55261$	$2.58808$	$2.60077$	$2.60661$
60	$0.1$	$3.45840$	$6.91671$	$10.37520$	$12.87552$	$13.00492$
	$0.2$	$3.16457$	$6.32914$	$9.49371$	$12.65827$	$13.28760$
	$0.3$	$2.84289$	$2.86541$	$2.86026$	$2.85787$	$2.85667$
	$0.4$	$2.39671$	$2.41740$	$2.42076$	$2.42194$	$2.42249$
	$0.5$	$2.34169$	$2.73719$	$2.77056$	$2.78239$	$2.78781$

display the effects of the Winkler $g_1$ and shear elastic substrate $g_2$ stiffnesses on the frequency of a GPLs-reinforced annular plate with an FG porous core for various values of the outer radius-to-thickness ratio $r_2/T$ and the inner-to-outer radius ratio $r_1/r_2$. As expected, the presence of the elastic foundation enhances the plate stiffness, so the frequency gradually increases with an increase in the elastic substrate parameters $g_1$ and $g_2$.

To explain the effects of various parameters on the frequency of a GPLs-reinforced (type A) annular plate with a porous core (porous-I) in graphical form, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 are presented. As shown in Figure 4, the eigenfrequency $\Omega$ of the GPLs-reinforced annular plate with a porous core increases linearly when $r_2/T\in [5,34]$. After that, it increases slowly as the outer radius-to-thickness ratio $r_2/T$ increases. As expected and as mentioned above in Table 3, Table 4 and Table 5, a noticeable increment in the frequency $\Omega$ occurs with an increasing in the GPLs weight fraction $W_G$.

The influences of the porosity factor ${\beta }_0$, Winkler $g_1$ and shear elastic substrate $g_2$ stiffnesses, magnetic field parameter ${\overline{M}}_R$, and core thickness ratio $T_c/T$ on the eigenfrequency $\Omega$ of a GPLs-reinforced annular plate with a porous core are illustrated in Figure 5, Figure 6, Figure 7 and Figure 8, respectively. It can be noted that the eigenfrequency $\Omega$ increases with increasing ${\beta }_0$, $g_1$, $g_2$, and ${\overline{M}}_R$. However, it decreases as the ratio $T_c/T$ increases.

6. Conclusions

The current article discusses the vibrational behavior of FG-GPLs-strengthened sandwich annular plates with an FG porous core lying on an elastic medium and exposed to a radial magnetic field. Based on three graphene distribution types, the volume fraction of the GPLs is varied through the thickness. Further, the porosities are graded across the core thickness according to three patterns of distribution. The Lorentz magnetic body force is deduced via Maxwell’s relations. The displacements are modeled based on a novel shear deformation theory, while the motion equations are depicted from Hamilton’s principle. The DQM is employed to convert the motion equations into a linear algebraic system that can be solved to obtain the minimum eigenfrequency. Convergence analysis is done to find the minimum mesh points that are needed to obtain reliable findings. The results of the current theory are in a good agreement with those obtained by other theories. Moreover, numerous parametric examples have been carried out to show the impacts of different parameters on the eigenfrequency of the composite annular plate. Accordingly, the ensuing items can be concluded:

• The presence of the elastic foundation and the magnetic field boosts the strength of the plate, so the eigenfrequency increases by increasing the elastic foundation parameters and the magnetic field parameter.
• A noticeable decrement in the eigenfrequency occurs as the inner-to-outer radius ratio increases and the outer radius-to-thickness ratio decreases.
• It is known that the addition of graphene components raises the plate stiffness, so the eigenfrequency grows as the GPLs weight fraction increases.
• Since increasing the porosities in the core layer makes the plate lighter, increasing the porosity factor leads to an increase in frequency.