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.
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
. 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]:
where
denotes the weighting coefficients for the
pth-order derivative. They are provided as [
40]:
Furthermore, the weighting coefficients
for higher-order derivatives are defined as [
40]:
However, the Gauss–Chebyshev–Lobatto approach is applied to estimate the mesh points
as [
40]:
The motion Equations (
37)–(
39) can be discretized using Equation (
41), resulting in
linear algebraic equations as follows:
Furthermore, the boundary conditions can be represented in a discretized form as follows:
Simply supported (S):
where
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
, the porosity distribution types, the GPLs weight fraction
, the GPLs distribution type, the outer radius-to-thickness ratio
, the magnetic field parameter
, the core thickness
, and the elastic substrate parameters
and
. The dimensionless formulations are defined by:
The material properties of the constituent materials are taken as:
GPa,
,
Kg/m
3,
TPa,
, and
Kg/m
3.
The numerical examples use the following fixed data (unless otherwise declared): , , , , , , , , m, nm, nm, and nm.
Firstly,
Table 1 shows a convergence analysis for the current outcomes of the FG porous annular plate enhanced by GPLs for various values of core thickness
. 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. 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
and the inner-to-outer radius ratio
. As shown in this table, the eigenfrequency increases as the ratio
increases and the ratio
decreases because the increase in the inner-to-outer radius ratio
leads to a decrease in the plate width; hence, the annular plate becomes weaker.
Table 3,
Table 4 and
Table 5 depict the effects of the outer radius-to-thickness ratio
, the inner-to-outer radius ratio
, and the GPLs weight fraction
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
and a decreasing inner-to-outer radius ratio
. On the other hand, the frequency increases as the graphene weight fraction
increases because the plate becomes stiffer by increasing the graphene component.
Table 6,
Table 7 and
Table 8 depict the effects of the porosity factor
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
and the inner-to-outer radius ratio
. It is found that, regardless of the variations of the ratios
and
, the frequency gradually increases with an increasing porosity factor
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
on the frequency of a GPLs-reinforced annular plate with or without an FG porous core is depicted in
Table 9,
Table 10 and
Table 11 for different values of the outer radius-to-thickness ratio
and the inner-to-outer radius ratio
. It can be seen that the frequency monotonically increases with an increase in the magnetic field parameter
. One can conduct that the presence of the magnetic field enhances the plate strength.
Table 12,
Table 13 and
Table 14 display the effects of the Winkler
and shear elastic substrate
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
and the inner-to-outer radius ratio
. 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
and
.
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
of the GPLs-reinforced annular plate with a porous core increases linearly when
. After that, it increases slowly as the outer radius-to-thickness ratio
increases. As expected and as mentioned above in
Table 3,
Table 4 and
Table 5, a noticeable increment in the frequency
occurs with an increasing in the GPLs weight fraction
.
The influences of the porosity factor
, Winkler
and shear elastic substrate
stiffnesses, magnetic field parameter
, and core thickness ratio
on the eigenfrequency
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
increases with increasing
,
,
, and
. However, it decreases as the ratio
increases.