1. Introduction
Many natural porous materials have been widely used for thousands of years. Compared with continuous medium materials, porous materials have excellent impact resistance, electrical conductivity, energy absorption, and thermal management properties [
1,
2,
3,
4,
5]. Porous materials are often used in biological tissue, sound insulation materials, and new photoelectric elements [
6]. As a kind of porous material, metal foam has high strength and stiffness [
7]. At present, many scholars have probed into the mechanical behavior of porous materials and the influence of various factors on the materials.
Nguyen et al. [
8] investigated the bucking, bending, and vibration of functionally graded porous (FGP) beams by the Ritz method. Akbaş [
9,
10,
11] examined bucking and vibration by the finite element method. By the differential transform method (DTM), Ebrahimi and Mokhtari [
12] presented the vibration of rotating FGP beams. Rjoub and Hamad [
13] reported on the vibration of FGP beams by the Transfer Matrix Method. Wattanasakulpong and Chaikittiratana [
14] found that a uniform distribution of porosities has an obvious effect on natural frequencies. Chen et al. [
15,
16,
17] studied the buckling and bending of shear-deformable FGP beams by the Ritz method. Hoa et al. [
18] studied nonlinear buckling and post-buckling of cylindrical shells by the three-terms solution and Galerkin’s method. Under various boundary conditions, Chan et al. [
19] discussed nonlinear buckling and post-buckling of imperfect FG porous sandwich cylindrical panels by Galerkin’s method.
In order to meet the high efficiency of civil engineering structures and the high precision development of aerospace engineering devices, beams need to be thinner and thinner, and at the same time, it is necessary to improve the material strength of its structure so as to increase the effective space and load. Nanomaterials have good mechanical, thermal, optical, and electrical properties [
20], so carbon nanomaterials are often regarded as nanofillers to heighten matrix materials’ properties. These include graphene platelets [
21], carbon nanotubes [
22], and fullerenes. In 2004, British scientists were the first to peel graphene sheets from graphite with an extremely high tensile, Young modulus, and surface area [
23]. On account of the porous structure of metal foam, the stiffness and strength are weakened compared with that of dense metal. By filling the carbon nanomaterials into the matrix materials, the properties of porous materials are able to be efficiently improved. At present, there have been many studies on graphene-reinforced porous materials.
Many papers have been published on graphene-reinforced porous composite beams, plates, and shells. Kitipornchai et al. [
24,
25] investigated the static and dynamic mechanical behavior of graphene-reinforced FGP beams. Yas et al. [
26,
27] presented the buckling and vibration of graphene-reinforced FGP beams in thermal environments. Yang et al. [
28] studied the buckling and vibration of graphene-reinforced FGP plates. Teng and Wang [
29] explained the nonlinear forced vibration of simply supported graphene-reinforced FGP plates. Dong et al. [
30] researched the buckling of spinning graphene-reinforced FGP shells. By Galerkin’s method, Zhou et al. [
31] revealed the nonlinear buckling of graphene-reinforced FGP cylindrical shells. Under impulsive loading, Yang et al. [
32] studied nonlinear forced vibration and the dynamic buckling of graphene-reinforced FGP arches.
Graphene-reinforced composite porous beams and plates are easily affected by the thermal environment, resulting in a decrease in their structural stiffness. Therefore, it is of great significance to study their thermal buckling, free vibration, and dynamic instability for engineering practices. To the best of the authors’ knowledge, no relevant literature has studied the dynamic instability of graphene-enhanced porous materials based on elastic foundations, thermal environments, and axial forces. The present paper mainly investigates the dynamic instability, thermal buckling, and free vibration of functionally graded graphene platelet-reinforced porous beams on an elastic foundation under a thermal environment and axial forces. Three modes of GPL patterns and porosity distributions are considered. Based on the theory of the Timoshenko beam, the governing equation is obtained by the Hamilton principle. On the basis of the differential quadrature method (DQM), the governing equations are changed into Mathieu–Hill equations and the main unstable regions of porous composite beams are studied by the Bolotin method. Moreover, we also use the two-step perturbation method (TSPM) to calculate the thermal buckling and free vibration. The effects of porosity coefficients and GPL’s weight fraction, initial thermal loading, slenderness ratio, geometry and size, boundary conditions, foundation stiffness, and dispersion pattern are discussed.
2. Model Construction
Under axial force and uniform temperature change , we consider a FGP multilayer beam that rests on a two-parameter elastic foundation in an initial stress-free state at the reference temperature .
As seen in
Figure 1,
L,
b and
h, respectively, represent the length, width, and thickness of the beam, and
and
are the Winkler stiffness and shearing layer stiffness. Among others, the thickness
h is divided into n layers, each of which is
.
Figure 2 considers three porosity distributions and GPL patterns. Because of disparate dispersion patterns, GPL patterns can be divided as A, B, and C, and the GPLs volume content
is smoothly on the
-axis. According to different porosity distributions,
’s peak values can be denoted as
. Assuming three GPL patterns have the same total amount of GPLs will result in
.
and are denoted as the maximum and minimum elastic moduli of the non-uniform porous beams without GPLs, respectively. In addition, represents the elastic moduli of the uniform porosity distribution beams.
The relationships of elastic moduli
, mass density
, and thermal expansion coefficient
of FGP beams for three porosity distributions are given by the following formulas [
25,
33]
where
in which
. Further,
,
, and
are the maximum values of
,
, and
, respectively. The porosity coefficient
is referred to as
Based on the Gaussian Random Field (GRF) scheme, the mechanical property of closed-cell cellular solids is denoted by [
34]
Using Equation (
4), the coefficient of mass density
is given by the following formula
Similarly, using the closed-cell GRF scheme, Poisson’s ratio
is defined as [
35]
where
is Poisson’s ratio of pure non-porous matrix materials and
Due to the total masses being the same for the three porosity distributions,
in Equation (
2) can be defined as
in which
M represents all porosity distributions, as shown in the following equation
According to the distribution patterns, the volume fraction of GPLs
is denoted by
in which
.
The relationship between the weight fraction of GPLs
and the volume fraction of GPLs
is given by
Based on Halpin–Tsai micromechanics model [
36,
37,
38,
39], the elastic moduli
of the nanocomposites is defined as
where
where
,
, and
are the average length, width, and thickness of GPLs.
and
represent the elastic moduli of the metal and GPLs.
By the following mixture rule, the mass density
, Poisson’s ratio
, and thermal expansion coefficient
of the metal matrix reinforced by GPLs can be obtained as
in which
,
,
, and
are the mass density, Poisson’s ratio, thermal expansion coefficient, and volume fraction of the metals. Furthermore,
,
,
, and
are the corresponding properties of GPLs.
5. Conclusions
The effect of GPL nanofillers on FGP composite beams under thermal environments, thermal buckling, thermo-mechanical vibration, and dynamic instability are investigated. Among them, weight fraction, normalized static axial force, porosity coefficient, dispersion pattern, boundary conditions, initial thermal loading, geometry and size, foundation stiffness, and slenderness ratio are studied. The following conclusions are obtained:
- •
Porosity 1 reinforced by GPL A of the beam has the biggest value of critical buckling, temperature rise, dimensionless fundamental frequency, and the origin of dynamic instability. The non-uniform, symmetric porosity distribution and GPL pattern have the strongest enhancement.
- •
The porosity coefficient has an important influence on thermal buckling, thermo-mechanical vibration, and dynamic instability. When the porosity coefficient grows, the origin of the dynamic instability shows a decreasing trend, but the dimensionless fundamental frequency and critical buckling temperature rise both increase.
- •
The addition of GPL nanofillers can enhance the beam stiffness significantly, and the mechanical performance is enhanced with increases.
- •
The values of thermo-mechanical vibration and dynamic instability decrease with normalized static axial force and initial thermal loading increase.
- •
Winkler and Pasternak foundations both strengthen the stiffness of the beam. It is noted that shearing layer stiffness has a better enhancement effect than Winkler foundation stiffness.