Graphene is a material with one of the highest known strengths. The theoretical Young’s modulus of graphene is up to 1.0 TPa and it has an inherent tensile strength of 130 GPa. The morphology of graphene is similar to the lamellar structure of thin paper and the thickness of a single layer is only 0.335 nm. It is the thinnest two-dimensional material presently known. Furthermore, it has good toughness, reproducibility, high dispersibility, and good chemical and biocompatibility [
1]; thus, it has become an ideal reinforcement for composite materials. Additionally, a composite plate is made from a metal matrix plate, as a continuous phase, and has a sized graphene modifier as a dispersed phase. The modifier is dispersed in the matrix material through appropriate preparation methods in order to form a composite system that contains sized materials. Graphene reinforced porous composite plates have a lighter weight, higher stiffness, higher strength, and multi-functional properties that can meet the lightweight requirements of the aerospace field and inject new vitality into the development of aerospace materials.
Scholars have been paying more and more attention to the study of free vibration graphene reinforced porous composite plates. Rahman et al. [
2] used molecular mechanics and molecular dynamic simulations to study graphene epoxy resin-based composites. The results showed that adding graphene platelet (GPL) to the epoxy resin matrix could significantly improve the Young’s modulus and shear modulus of the matrix materials. King et al. [
3] found that carbon-filled epoxy composites have high specific stiffness, which can be used for the structural components of fixed-wing aircrafts. Feng et al. [
4] studied the vibration analysis of multi-layer graphene sheets using a continuum model. This paper confirms the use of the GNP aspect ratio and the two-dimensional randomly oriented filler Halpin–Tsai model, adjusted for platelet filler shapes, to predict the tensile modulus well for the GNP/epoxy composites, which provides a basic theory for our subsequent research. Alibeigloo [
5] used non-local continuum mechanics to conduct a three-dimensional vibration analysis of multi-layer graphene platelets embedded in a polymer matrix. The numerical results show that the bending properties can be significantly improved by adding a small amount of GPLs into the polymer matrix. Yang et al. [
6] studied the nonlinear bending and buckling behavior of graphene reinforced nanocomposite beams, based on the Timoshenko beam theory, and the results showed that composite beams doped with graphene platelets could exhibit better mechanical properties. Eringen [
7,
8,
9], by the use of nonlocal thermodynamics and invariance under rigid motions, obtained constitutive equations for the nonlinear micromorphic elastic solids and differential equations of nonlocal elasticity and solutions for screw dislocation and surface waves. Polit et al. [
10] analyzed the bending and elastic stability of thick beams with a hole based on the high-order shear deformation theory of the transverse tensile effect. The formula in this paper provides an important reference for the writing of this paper. Rafiee et al. [
11] experimentally found that GPLs has the distinct advantage of reinforcing nanofillers over carbon nanotubes at a very low content, which has also been theoretically confirmed [
12,
13,
14,
15]. Zhao et al. [
16] studied the rubbing of the mistuned bladed disk system with blades of variable thicknesses, and elastically supported shaft-variable thickness blades coupled with the finite element model was established. Sharma et al. [
17,
18] studied the reduction of sound pressure at the receiver location with a lumped mass at the optimal location, which was shown to be much more than what is achievable by a uniform distribution of the point mass over the plate. A novel concept of the local mitigation of the transmitted noise at a target receiver location is presented by controlling the directivity of the transmitted noise through a point mass attachment on the barrier surface. Zhao et al. [
19,
20,
21] findings shed an important light on the design of the novel graphene reinforced blade-shaft system and remarkably improved its dynamic performance. Sharma et al. [
22] studied the effect of uncertainties in material and geometric parameters on the acoustic performance of a viscoelastic coating. Zhao et al. [
23,
24,
25] investigated the free vibration behaviors of a functionally graded (FG) disk-shaft rotor system, which was reinforced with a graphene nanoplatelet (GPL) that rested on elastic supports. Shafiei et al. [
26] studied the size-dependent nonlinear vibration behavior of FG porous microbeams using the improved coupling of the stress and Euler–Bernoulli theory, and evaluated the effects of uniform and non-uniform porosity. Sharma et al. [
27] studied the effect of strong and weak coupling of void resonances on the transmission characteristics and drew the conclusion that strong coupling of the resonance of voids results in broadband attenuation of sound. Davletshin et al. [
28] found that an interlayer distance change leads to significant band gap size modulations and direct–indirect band gap transitions in the phosphorene–BN heterostructure. Babicheva et al. [
29] found that a less dense structure may actually be stronger due to the fact that all the interatomic bonds in it are loaded more uniformly. Savin et al. [
30] found that the thermal conductivity coefficient of the nanoribbon increases monotonically up to 10%, with an increasing twist angle; the regime of uniform twisting and twist deformation of nanoribbons can improve their mechanical and physical properties. Savin et al. [
31] revealed that layered materials can support surface ripplocations that are highly mobile, topologically solitary waves that efficiently transport mass and energy. Chen et al. [
32] conducted a numerical study on the crushing process of an FG porous structure, and pointed out that, under high-speed impact, a certain type of non-uniform asymmetric pore can significantly promote the energy absorption of foam metal. The above research results are very important and provide the basis for subsequent research. Before the porous composites can be applied to engineering applications, a large amount of corresponding research work is still needed in order to reveal the structural properties of porous composites.
In this paper, graphene platelets were mixed into copper-based, porous square plates to enhance the material properties of square copper-based plates. The study on the combination of a porous structure and graphene reinforced composites will better improve the performance of the copper substrate, and provide a reference for the preparation of a high performance graphene copper matrix composite material that is light weight, has a high modulus and high strength in the future. The Young’s modulus and shear modulus of the porous graphene composite plate were calculated using the Halpin–Tsai micromechanical model, and the vibration equation of the plate was established. The free vibration frequencies under four boundary conditions were solved by the calculation. The effects of porous distribution, the distribution mode, content and geometric size of the free vibration frequency of the porous graphene composite plate were analyzed.