1. Introduction
In the analysis and design of all civil engineering structures, the buckling response of the CNT-reinforced FGM plate caught the attention of many researchers in recent years. Currently, critical buckling loads are obtained using the Corr and Jennings [
1] simultaneous iteration technique. The critical buckling load is the maximum load in the elastic range of the material above which plates start to deflect laterally. If the material is stressed beyond the elastic range and into the non-linear (plastic) range, the buckling strength of a plate is smaller than the elastic buckling strength of a plate. When the load approaches the critical buckling load, the plate will bend significantly, and the material’s stress–strain behavior will diverge from linear. In FGM-type composite material, properties of material constituents are varied according to the required performance. In this paper, the material constituents were a metal matrix, CNT reinforcement, and fiber. The final material was made in two phases. Here, we calculated the minimum edge compressive load in the form of the non-dimensional critical buckling load, which is required to initiate the instability of the plate structure. FGM is widely employed in many areas such as machine, construction, defense, electronic, chemical, pharma, energy sources, nuclear, automotive, and shipbuilding industries. Because of the expanding use of FGMs in a range of structural applications, detailed theoretical models are required to anticipate their behavior.
Abrate [
2] used a classical plate theory, FSDT, and HSDT to study the dynamic, static, and buckling behaviors of thick and thin FGM plates. The significance of their study is that the response of the FGM plate can be analyzed without performing a direct analysis. Zenkour [
3] adopted a generalized shear deformation model to study the stress and displacement of FGM plates under uniform loading. They observed that the gradient material properties play a vital role in the response of the FGM plates. Zhang [
4] carried out a geometric non-linear analysis of CNT-reinforced FGM plates with column support. For modeling the structure, they used FSDT mathematical model with the von Kármán nonlinearity equation. Based on HSDT theory the Levy-type solution has been presented by Bodaghi andSaidi [
5] for buckling analysis of simply supported FGM plate to observe the effect of the various parameter such as volume fraction index, aspect ratio, side-thickness ratio, loading condition, and various boundary condition. Thai and Choi [
6] developed a refined displacement theory without considering the shear correction factor for calculating the critical buckling load of the FGM plates. Various numerical studies have been presented for dynamic, buckling, and post-buckling analysis of FGM plate, laminated, and shell structure [
7,
8,
9,
10,
11,
12,
13].
Kiani [
14] studied the buckling response of a CNT-based FGM plate subjected to mechanical load. The distribution of load is obtained using the 2D formulation. Feldman and Aboudi [
15] studied the buckling behavior of uniaxially loaded FGM plates. A combination of micromechanical and structural approaches is used to predict the effective material properties of non-homogeneous FGM plates. Zghal et al. [
16] carried out the buckling response of FGM- and CNT-reinforced FGM plates and cylindrical panels. The final material properties of these plates and cylindrical panels were achieved by the power law and the extended rule of a mixture. A simple power-law equation for calculating the effective material properties was used by Ramu and Mohanty [
17] for buckling analysis of FGM plates using the FEM method and noting that the critical buckling load in non-axial compression was greater than that in biaxial compression. Arani et al. [
18] used an analytical and a finite element approach to determine the critical buckling load of the CNT-reinforced composite plate, and the overall elastic properties of the material were calculated by the Mori–Tanaka approach. By adopting the simple rule of a mixture, the effective elastic properties of the FGM sandwich were calculated by Yaghoobi and Yaghoobi [
19] to calculate the critical buckling load under mechanical, thermal, and thermo–mechanical loading. A micromechanics model based on Halpin–Tsai and the extended mixture rule has been used by Hanifehlou and Mohammadimehr [
20] to predict the effective elastic properties of graphene platelets and CNT-reinforced FGM plates. Lei et al. [
21] and Wang et al. [
22] considered an extended rule of mixture approach for predicting the effective material properties of CNT-reinforced FGM for buckling analysis. By assuming the power law composition of the volume fraction of the constituent material, the effective material properties were calculated to investigate the buckling analysis of the FGM plate structure [
23]. Aragh et al. [
24] employed the Eshelby–Mori–Tanaka method to calculate the effective elastic properties of the material for vibration response of a continuous-grade CNT-reinforced cylindrical panel.
Bouguenina et al. [
25] presented a solution to investigate the thermal buckling analysis of FGM plates. The presented solution was based on an analytical approach for constant thickness and a finite element approach for variable thickness. Mirzaei and Kiani [
26] studied the thermal buckling analysis of CNT-reinforced FGM plates, where CNT and the matrix material were assumed to be temperature-dependent. Singh et al. [
27] studied the buckling and vibration analysis of isotropic and sandwich FGM plates resting on an elastic foundation. They adopted a new sigmoid law to predict the effective elastic properties of the FGM plate. The buckling response and post-buckling response of pristine composite plates reinforced with graphene sheets were investigated by Zeverdejani et al. [
28]. The stability equations were solved using the eigenvalue problem, and the critical buckling loads were calculated for various boundary conditions. Fekrar et al. [
29] studied the buckling analysis of a ceramic-based FGM plate using only four-variable refined theory and demonstrated the accuracy and effectiveness of mathematical theory in analyzing the buckling behavior. A refined plate theory based on the secant function was used by Abdulrazzaq et al. [
30] to study the thermal buckling stability of clamped nano-size FGM plates. From their study, it can be observed that the buckling behavior of clamped FGM nanoplates was very sensitive to various parameters such as aspect and side-to-thickness ratios, material graduation, thermal condition, etc. The study of the influence of small-scale parameters on the vibration and buckling behavior of CNT-reinforced FGM plates was done by Shahraki et al. [
31]. The CNT-based FGM nanoplate was considered to rest on a Kerr elastic foundation. Costa and Loja [
32] represented the static analysis of a dual-phase moderately thick FGM plate. The CNT reinforcements were assumed to be added to the matrix material in the first phase.
Even though various studies on the buckling of FGM plates have been conducted based on a range of plate theories, no studies on the buckling of multiscale FGM plates based on the MTSDT theory were found. The present MTSDT mathematical theory has been modified to represent the kinematics field that captures normal and transverse cross-section deformation modes. The assumed in-plane fields incorporate the cubic degree of thickness terms and quadratic degree of thickness terms for the transverse component. The C1 continuity requirement associated with third-order shear deformation theory is avoided by developing a C0 FE formulation by replacing the out-of-plane derivatives with independent field variables. The present study can be used for the design and analysis of various types of hybrid composite curve panels, which are used in various engineering fields. The design charts can be obtained by the present model, which may be useful for the designer. Material properties, such as Young’s modulus, are supposed to change with plate thickness according to a power-law distribution of the volume percentage of the constituents. To the best of the authors’ knowledge, no experimental results on the present work are available in the literature; hence, present model results were validated with the closed-form elasticity solution and numerical analysis results available in the literature. To study the influence of various parameters, the non-dimensional critical buckling load was calculated for numerical analysis.
2. Geometrical Configuration and Effective Material Properties
A multiscale FGM plate of length a, width b, and thickness h, as shown in
Figure 1 was considered. In the buckling response of the plate, the rectangular Cartesian platform coordinates × and y were used. The co-ordinate planes × = 0, a and y = 0, b define the boundaries of the plate. The reference surface is the middle surface of the plate, defined by z = 0, where z is the thickness co-ordinate measured from the un-deformed middle surface of the plate.
The performance of these FGM plates might be improved by using a multiscale hierarchical FGM as shown in
Figure 2, which is made possible by combining the continuous fiber phase, the metal matrix, and CNT reinforcement. In such circumstances, the overall homogenization process can be divided into two phases: in the first phase, the dispersion of CNT in the metallic matrix yields a nanocomposite, and in the second phase, this nanocomposite receives ceramic inclusions in a graded manner, resulting in a CNT-reinforced multiscale composite. Since the CNTs are expected to be evenly distributed and randomly oriented throughout the matrix, the final mixture is considered an isotropic mixture. It is also expected that the bonding between CNT and matrix and dispersion of CNT in the matrix are perfect. Each CNT is assumed to be straight and has the same aspect ratio and mechanical properties. The matrix material is considered void-free, and the bonding between the matrix and fiber is excellent.
To evaluate the effective elastic properties of the material, a suitable approach should be adopted. A combination of the Halpin–Tsai equation [
33] and homogenization scheme can be adapted to predict the effective material properties of a three-phase multiscale FGM plate. The Halpin–Tsai equation is an empirical formula, known to be fit for calculating effective material properties of the mixture of the matrix and low fraction of the CNT reinforcement. The elastic properties of an anisotropic mixture of CNT and the matrix can be expressed as follows:
The volume fraction of carbon nanotube
and Poisson’s ratio of the nanocomposite
are calculated as [
34].
The volume fraction of dispersed fiber constituents is expressed as follows:
where
h and
Z are the respective total thickness and thickness coordinate in the transverse direction, having an origin on the middle surface of the plate. The exponential power
n permits the ceramic fiber to fluctuate in the thickness direction. The effective material characteristics of the final material fluctuate continuously according to Equation (5). In this paper, effective elastic material properties are calculated using a homogenization approach based on the Voigt rule of the mixture. as shown below:
Because of the dispersion of carbon nanotubes in the metal matrix, the effective Young’s modulus of the nanocomposite phase may be used instead of Young’s modulus of the matrix phase in the preceding equation. In this work, we assume the dispersion of carbon nanotubes in metal; therefore, we must first compute the effective material properties of the nanocomposite.