1. Introduction
The physical understanding and numerical simulation of the buckling responses of laminated composite plates have been the focus of intense efforts due to the extended use of fibrous composites in automotive, aerospace, naval architecture, and other fields of modern engineering technology. It has been deemed necessary to establish the practical limits of the load-carrying capability of structures made from fiber-reinforced composite materials. Such structures of plates/panels that could buckle are common in everyday engineering practice—for instance, flanges, and webs of rolled or built-up beams and columns, aircraft wing, tail and rudder panels, aircraft fuselage panels, and vehicle body panels.
Over the last years, a great number of theoretical theories have been developed to study buckling or free-vibration behavior and mode shapes of rectangular composite laminates [
1,
2,
3,
4,
5,
6,
7,
8]. The approximate analytical techniques that have been applied to the buckling and postbuckling of flat composite plates usually have simple boundary conditions, modeled by the classical lamination theory, and subjected to simple loading states [
9,
10,
11,
12,
13,
14]. The finite element method (FEM) can approach the linear instability [
15,
16,
17] of composite plates with complex geometry, mixed boundary conditions, variable thickness, and temperature-dependent material properties. The FEM has enabled nonlinear postbuckling analysis and sensitivity analysis, which can be used to study the sensitivity of the buckling and postbuckling responses to variations in the different material and lamination parameters of the plate [
18].
Meanwhile, experimental studies have been performed by many researchers to compare experimental results with those of previous analytical methods. Ashton et al. [
19] carried out an experimental study on the uniaxial compressive stability of rectangular boron–epoxy laminated plates. The buckling loads were determined for different boundaries utilizing Southwell plots. Gopalan et al. [
20] carried out an experimental study on the buckling characteristics of a woven flax/bio epoxy laminated composite plate subjected to axial compressive loads.
Fiber-reinforced polymer matrix composites have emerged as a major class of structural materials and are being considered for use as a replacement of many conventional materials in an overwhelming number of weight-critical components in the aerospace, automotive, and other industries on the grounds that their strength/weight ratio and stiffness/weight ratio are high [
21,
22,
23]. The simultaneous development of nanotechnology and the corresponding discovery of novel nanomaterials and nanostructures have given a new perspective on the use of composite materials since the enhanced characteristics of nanostructures can importantly increase the already-improved mechanical properties of typical composite materials [
24].
Nanocomposites have the potential of becoming the future structural material owing to their greater mechanical properties and superior thermal, electrical, optical, and other properties [
25,
26,
27,
28]. The mechanical properties of the nanoreinforcement are considerably high, and the ratio of their surface area to volume is high as well, which means that a great interfacial interaction with the matrix can be provided [
29]. These are the leading reasons for such highly improved properties in nanocomposites [
30]. The usage of polymer nanocomposites as a material for primary structures is in its early stage, but their potential in future aerospace applications has been realized [
31]. Recently, a joint lab was established by the Airbus Beijing Engineering Centre (ABEC) and the National Center for Nanoscience and Technology of China (NCNST) to explore the application of nanoscience in the aeronautic industry [
32]. Vidya et al. [
33] reviewed the physical attributes and mechanical behavior of polymer nanocomposites using different types of nanofillers and various techniques to fabricate and characterize nanocomposites with a focus on ballistic and aerospace applications. Their review has shown the importance of nanoparticle inclusion into polymers in both mechanical and physical properties. Yip et al. [
34] experimentally investigated the interlaminar shear strength and flexural strength of nanocomposites with different proportions of carbon nanotubes (CNTs). Their results have shown 15.7% and 9.2% improvements with 0.75% hundred resin CNT content into the polymer in both interlaminar shear and flexural strength.
Nanographite–polymer composites are intensively studied worldwide by many researchers to find ways to properly exploit their advantages for establishing novel polymer-based nanocomposites. Research efforts are guided by the very high values of modulus of elasticity, mechanical strength, and electrical conductivity exhibited by the nanocarbon filler. Embedding a carbonaceous nanofiller in a polymer resin has been proved to be conducive to the mechanical and physical properties of carbon–polymer nanocomposites even at a low content. Graphitic nanofillers, such as CNTs [
35], graphene [
36], and graphite nanoplatelets, when appropriately dispersed in a polymer matrix, provide a strong impetus for both thermomechanical [
37,
38,
39] and electrical performance [
40].
Regarding the modeling of advanced composites reinforced by nanomaterials such as CNTs or graphene nanoplatelets, many studies have been conducted. Civalek et al. [
41] recently investigated free-vibration and buckling behaviors of CNT-reinforced cross-ply laminated composite plates adopting a first-order shear deformation theory and using the method of discrete singular convolution for the numerical solution of the problems. They also studied the free-vibration behavior of CNT-reinforced composite microbeams [
42] deriving microstructure-dependent governing differential equations by applying Hamilton’s principle on the basis of couple stress theory and several beam theories and solving them by using Navier’s solution method. Jalaei and Civalek [
43] examined the dynamic instability of viscoelastic porous functionally graded nanobeam embedded on visco-Pasternak medium subjected to an axially oscillating loading as well as the magnetic field. Porosity-dependent material properties of the porous nanobeam were described via a modified power-law function. The viscoelasticity of the nanostructure was considered according to the Kelvin–Voigt model. Employing Eringen’s differential law in conjunction with the Timoshenko beam theory, the motion equations were derived via Hamilton’s variational principle. Akbaş et al. [
44] studied the dynamic responses of a fiber-reinforced composite beam under a moving load using the Timoshenko beam theory.
In the last two decades, the state of the achieved dispersion of nanoinclusions and the interfacial effects between matrix and reinforcing phase have been the interest of study both analytically and experimentally, as factors such as waviness, agglomeration, and orientation of nanofillers have a crucial role in the overall performance of the nanocomposites. Craveiro and Loja [
45] theoretically estimated the agglomeration effect of CNT-reinforced composite thin plates. CNT-based material properties were determined using the two-parameter model of agglomeration based on the Eshelby–Mori–Tanaka approach, while FEM was conducted for behavioral analysis through the higher-order shear deformation theory based on the displacement field of Kant. From their results, it can be concluded that the agglomeration effect deteriorates the mechanical behavior of the composite plates. Rafiee and Eskandariyun [
46] developed a novel multiscale modeling approach to predict Young’s modulus of graphene/polymer composites using deterministic modeling in preference to stochastic modeling. The dispersion of graphene in the matrix was captured at the mesoscale considering the formation of local aggregates, while the orientation of graphene was captured at the macroscale. Comparison between the results has shown that the fully agglomerated model presents lower values.
Many researchers have carried out theoretical and numerical studies to determine the effect of nanotubes on the material properties of composites. Georgantzinos et al. [
47] investigated a laminated composite drive shaft reinforced by MWCNTs for modal and linear buckling analysis using an analytical approach as well as FEM. The Halpin–Tsai model was employed to calculate the elastic modulus of composites having randomly oriented nanotubes. In another recent study, Taş and Soykok [
48] theoretically determined the engineering constants of CNT-based composite lamina. Bending analysis was performed on a composite plate under concentrated and distributed load. Lei et al. [
49] successfully applied the element-free kp-Ritz method to the buckling analysis of functionally graded CNT-reinforced composite plates under different in-plane loading conditions in a thermal environment. CNT-based material properties were determined through a micromechanical model based on either the Eshelby–Mori–Tanaka approach or the extended rule of mixture. The results showed that the changes of carbon nanotube volume fraction, plate width-to-thickness ratio, plate boundary condition and aspect ratio, loading condition, and temperature had a distinct effect on the buckling response of CNT-reinforced plates.
Rectangular plates are frequently used in structural design problems and are subjected to mechanical or thermal loads. While the stability problems of rectangular plates with neat composite materials are well discussed in the literature, when the polymeric matrix of the plate material is embedded with nanoparticles, such as graphene and nanotubes [
50,
51,
52], the stability problem requires further investigation. In this study, the development of suitable computational procedures based on finite element analysis for the prediction of the mechanical behavior of rectangular plates manufactured from laminated composite materials reinforced with MWCNTs and considering factors such as agglomeration, orientation, and waviness is presented.
2. Theoretical Approach
In this section, the determination of the mechanical properties of a CNT-reinforced nanocomposite is presented using a combination of theoretical models, such as the Halpin–Tsai (H–T) equations and rule of mixtures. Elastic constants such as Young’s modulus, Poisson’s ratio, and shear modulus quantify the relationship between stress and strain and lead us to estimate the mechanical behavior of nanocomposites under certain loads.
According to previous studies, current processing techniques in the use of CNTs for enhancing the thermomechanical properties of nanocomposites lead to an agglomerated state for the CNTs. Developing effective ways of CNT dispersion in polymers is one of the critical steps involved in the preparation of hybrid CNT-reinforced nanocomposites. Another critical factor that influences the behavior of nanocomposites is CNT waviness. According to images taken from scanning electron microscopy (SEM) and transmission electron microscopy, CNTs remain to a large extent curved during embedding in the polymer matrix [
53].
In the present work, three critical factors, including random dispersion, nonstraight shape, and agglomerated state of MWCNTs, are incorporated into the H–T model. A unidirectional carbon-fiber-reinforced composite lamina with a hybrid MWCNT–polymer matrix is shown in
Figure 1.
2.1. Hybrid CNT–Polymer Matrix Elastic Constants
The elastic modulus of an aligned straight CNT-reinforced polymer nanocomposite can be predicted by using the well-established H–T micromechanical model as follows [
53]:
in which
and
where
Em and
Ecnt are Young’s modulus of the polymer matrix and CNT, respectively. Additionally,
Vcnt is the volume fraction of CNTs, while
dcnt and
Lcnt are the nanotube’s average outer diameter and length, respectively.
The H–T model in Equation (1) estimates the elastic modulus of straight-aligned CNT-reinforced nanocomposites with the assumption of uniform dispersion of CNTs into the polymer matrix. In most real applications, CNTs are not straight aligned, while a perfect and uniform dispersion of CNTs into the polymer matrix is very difficult to be achieved. In such cases, Equation (3) is no longer applicable. In the present study, we take into account three critical factors (orientation, waviness, and agglomeration) in our mechanical property estimations, and therefore, these factors should be incorporated into the modified H–T model.
First, an orientation factor is included in Equation (3) to calculate the random orientation of CNTs into the nanocomposite. It is assumed that a CNT is randomly oriented in two dimensions when its length is greater than the specimen thickness, resulting in
fR = 1/3, and that it is randomly oriented in three dimensions when its length is much smaller than the specimen thickness, resulting in
fR = 1/6. In this study, the orientation factor is considered to be
fR = 1/6, and Equation (3) is rewritten as:
Second, the waviness factor is included in Equation (4) to determine the waviness of either CNTs or MWCNTs into the nanocomposite:
where
A is the amplitude of a wavy CNT and
W is a half wavelength, as illustrated in
Figure 2.
Here, the waviness factor is considered to be equal to
fW = 0.6, and thus, Equation (4) takes the following form:
The H–T micromechanical model is further altered to include the CNT agglomerated state in the polymer matrix. Therefore, an agglomeration efficiency factor,
fA, is added in Equation (6). In the present study, the values of
α and
β are considered to be 10 and 0.9, and Equation (6) may be rewritten as:
where
in which parameters
α and
β are related to the degree of CNT agglomeration.
The efficiency factors (i.e., orientation, waviness, and agglomeration of CNTs) may be adjusted by using characterization techniques. The characterization methods can be used to provide the representative average values for the efficiency factors according to the production process adopted in the manufacture of the nanocomposites.
The shear modulus of the CNT–polymer matrix, as it exhibits quasi-isotropic behavior, can be calculated by the following equation [
39]:
Similarly, regarding Poisson’s ratio for the pure and CNT–polymer matrix, it may approximately be written as [
39]:
The density of the hybrid matrix can also be calculated from the following equation [
39]:
where
ρcnt and
ρm are the densities of the nanotube and matrix, respectively, and
Vcnt and
Vm are the volume fractions of the nanotube and matrix, respectively.
The density of the nanocomposite lamina can be calculated from the following equation [
39]:
where
ρf is the density of the fibers, and
Vf and
Vm-cnt are the volume fractions of the fibers and hybrid (CNT-reinforced) matrix, respectively.
2.2. Unidirectional Composite Lamina Elastic Constants
Young’s modulus of a unidirectional lamina in the longitudinal direction,
E1, and Poisson’s ratio,
υ12, can be calculated by utilizing the rule of mixtures [
39]:
where
Ef, Vf, and
vf are Young’s modulus of the fiber, the volume fraction of the fiber, and Poisson’s ratio of the fiber, respectively. In this study, the volume fractions of the fibers and matrix were assumed to be 60% and 40%, respectively.
According to experimental results, the values obtained for transverse Young’s modulus,
E2; Poisson’s ratio,
υ23; and shear modulus,
G12 and
G23, are not well matched with the values calculated using the rule of mixtures. Due to these incompatibilities, semiempirical models have been developed for design purposes, such as Halpin–Tsai, as they can be used over a wide range of elastic properties and fiber volume fractions. The H–T equation for calculating these material properties is given by [
39]:
where
Pm-cnt means the related properties of the CNT–polymer matrix, and
P can be considered transverse Young’s modulus;
E2, transverse Poisson’s ratio,
υ23; in-plane shear modulus,
G12; and out-of-plane shear modulus,
G23. The parameter
η is an experimental factor computed by using the next expression [
39]:
where
Pf means the related properties of the fiber. The term
ξ is called reinforcing factor and depends on fiber geometry, packing geometry, and loading conditions. For circular fibers in a square array,
ξ = 2 for
E2, and
ξ = 1 for
υ23,
G12, and
G23, as referred in [
48].
5. Conclusions
In this work, the critical buckling load of laminated composite plates reinforced by CNTs was investigated using FEM. The modified polymer matrix with different CNT volume fractions was theoretically evaluated using the modified H–T micromechanical model considering the effects of the widely observed waviness, agglomeration, and orientation of CNTs. The results obtained from FEM were in good agreement with those from the open literature. The predictions of the modified H–T model were compared with the corresponding available experimental and analytical results found in the open literature to verify the applicability of the approach. The comparisons showed a very good agreement.
The CNT nanoparticles considerably influenced the engineering constants that were used to determine material properties such as Young’s modulus, shear modulus, and Poisson’s ratio of the composite lamina. The comparison between the results regarding the pure and reinforced matrix indicates that the CNTs enhance the matrix material properties. However, it is established that two critical factors, such as the waviness and agglomeration of the CNTs, may considerably reduce the stiffening effect of CNTs. According to the numerical outcome, the following conclusions can be drawn:
Young’s modulus of the CNT/polymer matrix appears to be considerably affected by the waviness and agglomeration state of the CNTs.
Young’s modulus of the CNT/polymer matrix can be improved by up to 49.18% with the addition of 10% by volume CNTs, considering varying factors such as orientation, waviness, and agglomeration of the CNTs.
The mechanical properties E1, E2, G12, and G23 of the composite lamina can be increased by up to 0.60%, 18.38%, 31.93%, and 24.21%, respectively, with the addition of 10% by volume CNTs in the matrix, considering orientation, waviness, and agglomeration effects in the calculations.
The critical buckling load rises exponentially regarding the increase of fiber orientation angle (θο) for an eight-layer symmetric cross-ply laminated rectangular plate stacked as (θο/−θο).
In consequence of adding 10% by volume CNTs into the conventional composite, the critical buckling load of the laminated composite plate showed great improvements. These enhancements measured from 0.97% to 19.11% regarding the critical buckling load, taking into account critical factors, such as the waviness and agglomeration of the CNTs.
The critical buckling load of the laminated nanocomposite plate seems to be significantly affected by the waviness and agglomeration state of CNTs. The presence of CNT waviness and agglomeration in the polymer matrix results in a negative effect on its reinforcing role.
Polymer nanocomposites represent a promising class of engineering materials. Besides nanofiller properties and geometrical characteristics, the major factors for the performance of CNT/reinforced nanocomposites are the state of the agglomeration, waviness, orientation and the interface between the polymer matrix and nanofiller. These key factors appear to have a crucial role in the overall behavior of nanocomposites and, thus, are the focus of the scientific research community. This work could be a guide concerning the efficient design and development of composite structures and devices with carbon nanotube inclusions.