Analysis of functionally graded carbon nanotube- reinforced composite structures: A review

Functionally graded carbonnanotube-reinforced composite (FG-CNTRC) is a novel nanomaterial; the mechanical behavior of FG-CNRC has become a hot topic in theMaterials Science and Engineering Science recently, thanks to its excellent mechanical and electrical properties after its fusion with matrix. In this paper, the review efforts for research progress on the modeling and analysis of FG-CNTRC structures are carried out. Firstly, the development background of FG-CNRC is presented, as well as some basic theories and main equations for mechanical analysis of FG-CNTRC structure. Then, the mechanical behaviors of FG-CNTRC beams, plates, and shells under loading conditions are comprehensively reviewed, with the emphasis on discussing the bending, buckling, and vibration behaviors of the structures. Finally, the future research orientation of the field is considered and prospected.


Introduction
Beams, plates, and shells, as fundamental components, have been applied extensively in various types of engineering structure [1]. They make up most of the products in many industrial fields, such as construction [2], aviation [3], naval architecture [4], etc. Under the practical circumstances, these structural elements are commonly subjected to various forms of dynamic loadings in their service life [5]. To avoid the damage caused by structural decay, it is significant to develop their mechanical analysis and make some judgments about whether they have the ability to work well [6]. At present, many advanced analysis theories and accurate computing methods have been proposed constantly in order to reflect the essential features of the structure and evaluate the mechanical property of structures with the kind of formats or materials [7].
Nanomaterials, due to their particular dimension effect, exhibit unusual physical, chemical, and biological behavior, so they have been applied to various fields [8].
As recognized, metal-matrix composites with ceramic fibers and particulates could prefer higher specific strength and elastic modulus than their monolithic alloys [9]. Using the nanomaterials as particulates and compositing with conventional materials, their mechanical properties in certain conditions would be further increased [10]. For instance, Mohan et al. [11] reported that the presence of the high-modulus graphene in a low modulus matrix can lead to significant reinforcement. Wu et al. [12] texted matrix composites reinforced by graphene nanoplatelets and found that the microhardness value of nanocomposite increases by 24.51% compared to the matrix. Liu et al. [13] focused on the Honeycomb structure and showed the result that with the filling circular tubes into a buckling area, a considerable mass efficiency improvement with respect to deflection resistance can be obtained. Scholars in growing number are exploring how to improve the mechanical properties of traditional structures and trying to find a new kind of material which can make the fundamental components own an excellent property.
Carbon nanotube (CNT), a type of carbonaceous nanomaterial, was first found by Iijima [14]. Its excellent mechanical and electrical properties make it a prevalent topic in nanotechnology [15]. Lau and Hui [16] have reported that the nanotubes own remarkable mechanical properties with theoretical Young's modulus and tensile strength as high as 1 TPa and 200 GPa, respectively. Lu and Hu [17] used the computational simulation for predicting mechanical properties of carbon nanotubes (CNTs) and get the result that the CNTs have a high elastic modulus of 0.349 TPa. So, the study of the carbon nanotube-reinforced composite structure is of great significance and has attracted the attention of scholars [18] (Figure 1).
This article first introduces a new type of nanocomposites, the functionally graded carbon nanotube-reinforced (FG-CNTRC), then reviews the mechanical analysis of the composites used in the beams, plates, and shells in recent years. It mainly focused on the theoretical method and solving results in the process of analysis. Finally, some consideration about study and the direction of future research was prospected.

Functionally graded carbon nanotube-reinforced composite structures
Evidence shows that the CNTs' content and arrangement in composite may affect the mechanical property of structures sensitively [20]. How to improve the macro mechanical properties of carbon nanotube-reinforced composite structures in the case of low content initiated the scholars' ponder. Chinese researcher Shen [21] first proposed the concept of functionally graded carbon nanotube-reinforced composite structures, which the volume content of CNTs in the material was distributed in gradient. This has made the composite acquire a satisfying reinforcing agent and designable characteristic, so that the research of FG-CNTRC has become popular in recent decades.

Background of functionally graded materials
As we know, composite materials have the outstanding material characteristics cause of the complementary and correlation of each component. However, when the composite structures encounter the extreme working conditions, the delamination failure or crack phenomena will be emerged [22]; then the composite materials are subjected to sharp transition of properties at the interface which can lead to component failure by delamination [23].
To overcome this problem, the concept of functionally graded materials (FGM) was firstly proposed by Niino [24]. These materials replace sharp interface with gradient interface which results in smooth transition of properties from one material to the other; then they could eliminate interface problems like stress concentrations and poor adhesion [25]. Therefore, this composite structure can adapt to the special environment and be widely used in aerospace, defense, and so forth [26]. Several kinds of analysis methods have been developed in order to design and evaluate the FG structure better. For instance, Li and Pang [27][28][29][30][31] provide a new semi-analytical method to analyze the free vibration of functionally graded circular cylindrical shells under complex boundary conditions. In addition, their team extended the research object to porous cylindrical shell [32] and axisymmetric doubly curved shells [33]. Karami et al. [34] investigated FG nanoplates made of anisotropic material (beryllium crystal as a hexagonal material) and used Galerkin's approach to solve the buckling problem for different boundary conditions. reinforced composites [36]. Due to limitation of manufacturing, the distribution forms of carbon nanotubes are presented in the following types: UD, FG-V, FG-Λ, FG-X, and FG-O [37], which are depicted in Figure 2.
Type UD represents that the CNTs volume fraction was uniformly distributed in Z direction. In type FG-V and FG-Λ, the CNT fiber concentration increases and decreases through the selected direction, respectively. In type X, the CNT's volume fraction decreases gradually from the top surface to the middle and then increases symmetrically until reaching the other side. Type O materials have opposite CNT distribution method by having the maximum CNT concentration in the middle of the composite and the lowest at the surfaces [39]. The respective representation of CNTs' volume fraction can be expressed as follows [40]: where z are the coordinate value along the thickness direction, h are the structure thickness, and V tcnt are the total volume fraction of CNTs which can be written by: where w cnt is the mass fraction of CNTs, ρ cnt , ρ m are the mass density of CNTs, and isotropic matrix, V m , is the total volume fraction of matrix. Based on the above, there are mainly two typical micromechanical models that have been approved by most scholars; one is the extended rule of mixture [41] and the other is the Eshelby-Mori-Tanaka [42], both in order to estimate the effective constitutive law of the elastic isotropic matrix with dispersed elastic inhomogeneities (CNTs).
The rule of mixture is based on Krenchel's model for three-dimensional randomly dispersed short-fiber composites. It introduces the CNTs' efficiency parameters to indicate the size-dependent material properties [43]. After finishing and correction, the effective Young's modulus and shear modulus can be written as [44]: where ( = ) η i 1, 2, 3 i represents the CNTs' efficiency parameters, which are determined by the results of the molecular dynamics simulation.
The Eshelby-Mori-Tanaka model was established by the theory of Eshelby in micromechanics; it combined the concept of average stress in the matrix from Mori-Tanaka [45]. It can be used to the CNTs as clustered particles randomly distributed in the matrix. Kamarian et al. [46] suggests that this model considered some important parameters such as agglomeration effect of CNTs, while the rule of mixture cannot. The effective bulk modulus K and shear modulus G are given by: r r m r m r r (11) where the subscript r and m, respectively, represent the parameter of particle and matrix. α, β, and η can be determined by the Hill's elastic moduli of CNT. For SWCNT, the effective Young's modulus E and Poisson's ratio v can be expressed as

The fundamental theory of FG-CNTRC
Essentially, FG-CNTRC is a kind of nonhomogeneous composite materials, so that the conventional linear theory will not be suitable for them. Thanks to the efforts of numerous researchers, the analysis computation on material nonlinear has gradually been developed and improved. There are four approaches used in the previous studies in order to conduct the nonlinear model of FG-CNTRC in the governing equations.

Von-Kármán strain-displacement relation
It has been proposed by Von-Kármán and developed in the 1960s by Stansky and Whitney. It aims to analyze the heterogeneous anisotropic plates and nonlinear structure [47]. As the external load acting on the composite, the ratio of out-of-plane deformation to thickness is no longer a small quantity and the membrane force plays a great role in bending equilibrium. According to these, the strain-displacement relation can be written by: (16) where ε x , ε y , and ε xy are the strain on the middle surface; u, v, and w denote the displacements in x, y, and z directions.

First-order shear deformation theory
Since the volume content of reinforcement in FG-CNTRC was distributed in gradient, the equivalent transverse shear modulus of structure is much less than the equivalent elastic modulus in the fiber direction [48], so that the FG-CNTRC is sensitive to the transverse shear deformation and the transverse shearing distortion must be considered. First-order shear deformation theory (FSDT) [49,50] takes account of the effect of transverse shearing distortion and the displacement function can be expressed as: where u, v, and w are the displacements of a generic point in the FG-CNTRC plates and u 0 , v 0 , and w 0 represent the displacements projections on the mid-plane. ψ x and ψ y are the transverse normal which can be given by: First-order shear deformation (FSDT) is one of the most popular approaches to establish the governing equations of FG-CNTRC. Wang et al. [51] used it to analyze the free and transient vibration behavior of composite laminated open cylindrical shells. Lee and Hwang [52] studied the nonlinear transient behaviors of FG-CNTRC spherical shells based on the FSDT. In the analysis of beam, FSDT is also known as Timoshenko beam theory. It is worthy to note that researchers when applying this theory tend to introduce a shear correction factor in order to balance the shear strain energy changed by the FSDT [53]. In some study [54], the shear correction factor is taken as 5/6 referring to the isotropic materials, but most researchers [55] preferred the other formula which was proposed by Efraim and Eisenberger [56]; this formula considered the change of Poisson's ratio in thickness direction and can be written as where V m and V cnt are the total volume fraction of matrix and CNTs, v m and v cnt are the Poisson's ratio of them.

Higher order shear deformation theory
Actually, the shear correction factor is closely linked not only with the Poisson's ratio, but also with material properties, laying mode, etc. [57]. So, in order to overcome the bad effects caused by it, several high-order shear deformation theories have been developed, and the most commonly used one was suggested by Reddy [58]. The displacement expressions of it can be expressed as: Analysis of FG-CNTRC structures: a review  1411 where, u, v, w, u 0 , v 0 , w 0 , z, ψ x , and φ y exist in similar significance with FSDT, h is the thickness of structure. After arranging, the shear deformation function ( ) f z is defined as Based on the Higher order shear deformation theory (HSDT), Shen and Xiang [59] derived the motion equations of a CNTRC beam on an elastic foundation and considered the influence caused by thermo. Phungvan et al. [60] considered the effects of carbon nanotube volume fraction and plate width-to-thickness ratio on natural frequencies and deflections of FG-CNTRC. To further simulate the continuity conditions of transverse shear stress at layer interfaces, different higher order polynomial and trigonometric functions already have been tried, as shown in the Table 1.

Three-dimensional elasticity theory
Although using higher order shear deformation theory can get a relatively accurate result about the composite materials structure, it is still a two-dimensional equivalent theory. With the increase of the thickness ratio and the difference of the materials in each layer, the theoretical error increases sharply [67]. Therefore, three-dimensional elasticity theory has attracted the researchers' attention [68], as it abandons the assumption of displacement or stress and consider all the stress and displacement components as well as the interlaminar continuity condition.
However, during the process of application, a large number of equilibrium equations may lead to the difficulty of solution [69]. So, some exact and more targeted calculation methods have been proposed to analyze specific case in different geometries, boundary conditions, and layers, such as Pagano's classical approach [70], state space approach [71], series expansion [72], perturbation methods [73], and 3-D finite element method [74]. Because of space limitations, these approaches could not be introduced in detail. For FG-CNTRC, Alibeigloo [75] used the state space technique across the thickness direction to study the bending behavior of cylindrical panel. Thomas calculated the dynamic responses of FG-CNTRC shell structure subjected to impulse load by 3-D finite element method [76] (Figure 3).

Mechanical analysis of FG-CNTRC
As mentioned above, beams, plates, and shells are the most fundamental components in engineering structure, so that any kind of new materials to be used have to be analyzed for the mechanical response of components under the actions of environmental loads or external load. In this section, the review of mechanical analysis of FG-CNTRC is discussed from four aspects (beams, plates, shell, and component), mainly covering the bending, buckling, and vibration of FG-CNTRC.

Beams
In the series-study of bending, Wuite and Adali [78] performed a benchmark research to study static responses of CNTRC beams according to the classical beam theory. They found that the different stacking sequences in components would influence material properties. Wattanasakulpong and Ungbhakorn's research [79] proved that this feature also applies to FG-CNTRC. They presented linear bending of FG-CNTRC beams resting on the Pasternak elastic foundation and got the result that, with the distribution of FG-X, the beam can be the strongest with the smallest transverse displacement, followed by the FG-UD, FG-V, and FG-O beams. Shen and Xiang [59] further considered the nonlinear bending behavior of simply supported FG-CNTRC beams resting on the Pasternak elastic foundation as a result of thermal effects. Based Touratier Akavci and Tanrikulu Mantari on the Reddy higher order shear deformation theory and applied the two-step perturbation technique, they obtained the effect of temperature variation on the nonlinear bending behavior of CNTRC beams, which the deflections are increased with increase in temperature. Kumar and Srinivas [80] perform a numerical analysis on the static and dynamic behaviors of beams made up of functionally graded carbon nanotube reinforced polymer and hybrid laminated composite containing the layers of carbon-reinforced polymer with CNTs. The hybrid laminated composite beam was considered to have a combination of FG-CNTRC and FRC layers, whose material modeling and mathematical formulation for multilayer beam are described by the Timoshenko beam theory. Unlike in ref. [80] where each CNTRC layer is assumed to be linear functionally graded, in Yang study [81] the CNTRC layers are arranged in a piecewise functionally graded (FG) pattern in the thickness direction of the beam. The novelty of their study can be reflected by the identification of the negative Poisson's ratio of CNTRC laminated beams with the functionally graded configurations by performing the nonlinear bending analysis ( Figure 4).
Based on the CNTs employing rule of mixture and Timoshenko beam theory, Yas and Samadi [82] investigate the buckling of FG-CNTRC beams on elastic foundations. The governing equations are derived through using Hamilton's principle and then solved by using the generalized differential quadrature method (GDQM). They respectively considered the beam with the different distributions under the four boundary conditions, including hinged-hinged (H-H), clamped-hinged (C-H), clampedclamped (C-C), and clamped-free (C-F) and obtained the results that FG-X distribution has higher critical buckling load in comparison with other distributions. However, Shen and Xiang [59] proposed that CNTRC beam with intermediate CNT volume fraction does not necessarily have intermediate nonlinear frequencies, buckling temperatures, and thermal postbuckling strengths; it means that the buckling analysis of FG-Λ by Yas and Samadi may be reconsidered.
Rafiee et al. [83] discussed buckling behavior of FG-CNTRC beams with surface-bonded piezoelectric layers by multiple scales method. According to the Euler-Bernoulli beam theory and von-Kármán geometric nonlinearity, this paper found that thermal buckling phenomenon may be delayed by applying the appropriate voltage to the actuator piezoelectric layers, but the solution method is valid only for beams which are clamped on both ends and ignored the shear deformation of FG-CNTRC beams. In fact, the defects of carbon nanotubes may affect their mechanical properties whether in macroscopic or in microscopic case [84]. Wu et al. [85] first extended the research to the composite with various geometric imperfections; they suggested that the mechanical behavior of beam structures is sensitive to the presence of a small imperfection, especially the postbuckling behavior. Based on the first-order shear deformation beam theory and modified Newton-Raphson iterative technique, they drew the postbuckling equilibrium paths of imperfect and perfect CNTRC beams, respectively.
In addition, the sandwich beam was studied by Kiani and his coauthors. The research [86] showed that due to the antipathetic lateral loading, there is a snap-through phenomenon when the temperature elevation is large enough and the snap-through intensity will be influenced by the temperature, the thickness ratio, and the volume fraction of CNTs. Based on the first-order shear deformation theory and von Kármán type of geometrical nonlinearity, thermal postbuckling response of a sandwich beam made of a stiff host core and carbon nanotube (CNT)-reinforced face sheets is analyzed though Ritz method [87]. It is shown that graded profile of CNTs, length to thickness ratio, host thickness to face thickness ratio, volume fraction of CNTs, boundary conditions, and temperature dependency are important factors on critical buckling temperature and postbuckling equilibrium path of sandwich beams. Recently, Khosravi et al. [88] innovatively change the study object from stationary structures to rotating FG-CNTRC structures, with the beam acquiring a constant angular rotating speed. Prebuckling deformations of the beam are studied carefully to discuss the conditions for thermal and inertial buckling under the simultaneous actions of rotation and heating.
The vibration characteristics of FG-CNTRC beam also have been a concern for scholars. Asadi and his coauthors [89] set the scope of the study in aerospace applications; they analyze the nonlinear dynamic responses of functionally graded carbon nanotube-reinforced composite beams exposed to axial supersonic airflow in thermal environments. With regard to the first-order shear deformation theory and harmonic differential quadrature method, they applied a direct iterative procedure to determine thermal bifurcation points and critical aerodynamic pressure.
As previously mentioned, applying the first-order beam theories may cause errors because of the shear correction factor. Lin and Xiang [90] performed a comparative analysis on free vibrations of an FG-CNTRC beam between the first-order and third-order shear deformable beam theories. The result showed a substantial difference between these two theories as shown in Figure 5, especially for beams with both edges clamped. Jam and Kiani [91] examined the low velocity impact response of FG-CNTRC beams in thermal environment. The solution of the resulting equations is traced in time using the wellknown Runge-Kutta method. In fact, the vibrational resonance can also occur at excitation frequencies Ω which are multiple of the natural frequency ω [92]. As the excitation frequency Ω is near to ω/n or nω, the superharmonic and subharmonic resonances would occur, respectively, which bring some effect to the structure. Wu et al. [93] investigated the nonlinear primary and super-harmonic resonances of FG-CNTRC beams and their paper first used the incremental harmonic balance (IHB) method to solve the discretized equations. The numerical results showed that super-harmonic resonance exhibits only in the case of FGΛ-CNTRC beam, while does not occur in the case of other three beams reinforced by symmetrically or uniformly distributed CNTs. In addition, they extended the research to subharmonic resonance [94] and obtained a similar conclusion about the effects of material, geometry, and excitation parameters on the responses. Heidari and Arvin [95] discussed the free vibration of rotating FG-CNTRC Timoshenko beam and suggested the fundamental natural frequency of the considered hinged-clamped beam by the augmentation in the rotation speed initially, but after a threshold value it shows a opposite effect because of the induced compressive force by the centrifugal force which tends to destabilize the beam.
Besides the single-layer beams, the sandwich beam and FG-CNTRC beams received much attention in the recent years. Mirzaei and Kiani [96] studied the large amplitude free vibration of temperature-dependent sandwich beams with carbon nanotube-reinforced face sheets, applying the polynomial-based Ritz method into the Hamilton principle.
Kamarian et al. [46] used the Eshelby-Mori-Tanaka approach to assume the material properties in order to consider the agglomeration effect of CNTs and the results presented the fact that the free vibrations of sandwich beams are seriously affected by CNTs agglomeration. Vo-Duy and Hohuu [97] established a laminated composite beam analysis model and considered the effect of the numbers of layers, using the finite element method to solve the model under various boundary conditions. As the matrix of the laminated beams is cracked, the dynamic behavior of the structure may be changed evidently. Fan and Wang [98] focused on the matrix-cracked shear deformable laminated beams on elastic foundations in thermal environments and established two kinds of damage models for matrix cracking, namely self-consistent model and elasticity theory model.

Plates
Shen [21] first investigated the bending behavior of functionally graded carbon nanotube-reinforced composite plates, based on the Von-Kármán strain-displacement relation and Reddy type of higher order shear deformation plate theory. Applying the two-step perturbation technique, the effect of the thermal was considered. Since then, static, dynamic, and buckling behaviors of FG-CNTRC structures have been studied and reported in the literature.
Zhu et al. [99] carried out bending analysis of FG-CNTRC plates by FEM based on first-order shear deformation plate theory (FSDT) with some similar conclusions. Phungvan et al. [60] proposed an effective formulation to investigate the static behavior of FG-CNTRC plates, using isogeometric elements based on Nonuniform Rational B-Spline (NURBS) basis functions. Compared with the traditional FEM, the IGA easily fulfills the continuity requirements for plate elements stemming from the HSDT, which is the key of this study. Zhang and his coauthors [55] further proposed an element-free IMLS-Ritz method, which used a set of scattered nodes to replace the meshing in the problem domain.
Recently, Sobhy [100] employed a new shape function, based on the higher order shear deformation theory, to analyze the bending response of FG-CNT rectangular plate under the double-layered elastic foundations in thermal environments. In addition, Aakash and his coauthors [101] applied the inverse hyperbolic shear deformation theory to study the similar problem; actually his method also provided a new shear deformation function which used the inverse hyperbolic function to satisfy the zero transverse shear stress conditions at the extreme surfaces of the plate. Keleshteri et al. [102] selected the FG-CNTRC annular plates as the research object and investigated the effect of thickness profile in detail. The generalized differential quadrature method is adopted and the nonlinear system of equations is solved via the Newton-Raphson iterative method. For laminated plates, Natarajan et al. [103] investigated the bending of sandwich plates with CNTRC face sheets using QUAD-8 shear flexible element, which accounts for the realistic variation of the displacements through the thickness and the possible discontinuity in slope at the interface. Chavan and Lal [104] developed a 9-node isoparametric element with seven degrees of freedom per node to acquire precise computation of the deflection and stresses of reinforced composite plate. Based on the Reddy's third-order shear deformation plate and two-step perturbation approach, Shen and his coauthors [105] presented the findings on the nonlinear bending behaviors of FG-CNTRC laminated plates with negative Poisson's ratios. They found that the (±22)3T CNT/PmPV and (±70)3T CNT/PmPV laminated plate correspond to the maximum NPR. In fact, besides the theories we mentioned above, there are also some other approaches to derive the strain-displacement relation. Sciuva and Sorrenti [106] presents an application of the extended Refined Zigzag Theory (eRZT) in conjunction with the Ritz method to analyze the mechanical property of FG-CNTRC sandwich plates. He got the numerical results of bending under transverse uniform which were contrasted with FSDT and Reddy's TSDT. It confirms the superior predictive capabilities of the eRZT over the traditional FSDT and TSDT, also for FG-CNTRC sandwich plates.
The buckling behavior of FG-CNTRC plates also has attracted the most attention from scholars. Wang [107] applied a semi-analytical solution and discussed the buckling of FG-CNTRC plates based on the classical plate theory and Galerkin technique. The transverse displacement is expressed in a sum of products of characteristic beam functions in one direction and unknown functions in the other and it can be suitable for arbitrary combinations of boundary conditions. Shen and Zhang [108] adopted a two-step perturbation technique for the buckling and postbuckling of rectangular plates subjected to uniform or in-plane parabolic temperature loading, but all edges of the plate are considered to be simply supported in this research. Torabi et al. [109] developed a unified numerical formulation for the thermal buckling of FG-CNTRC plates in the variational framework. They employed the variational differential quadrature (VDQ) approach to present the governing equations, in which the total potential energy of the structure can be instantly discretized applying twodimensional GDQ-based differential and integral operators. Further, they [110] improved the solution process and proposed a VDQ-FEM technique in which the space domains of plate are transformed into a number of finite elements in order to overcome the deficiency of VDQ used in structural with concave domains, as shown in Figure 6.
Kiani [111] examined the shear buckling of FG-CNTRC rectangular plates and proposed a Ritz-based solution with Chebyshev polynomials to construct an eigenvalue problem associated with the onset of buckling. The result showed that increasing the volume fraction of CNT can increase the buckling capacity of the plate. On the basis of it, their research group [112] expanded the research object to skew plate through the coordinate changes. Civalek and Jalaei [113] take the FG-CNTRC skew plates as the research subjects. With the help of discrete singular convolution method in geometric transformation, the related governing equations of skew plate buckling and boundary conditions are transformed from the physical domain into a square computational domain and the accuracy has been validated by comparing with other existing literature. Based on the Eshelby-Mori-Tanaka model, Safaei and his coauthors [114] investigated the bucking behaviors of sandwich plates with the consideration of porosity and the formation of CNT clusters. They used amesh-free method to obtain results that porosity considerably improved thermal buckling behavior, but reduced the critical mechanical loads of sandwich plates. Jiao [115] investigates the buckling behaviors of thin rectangular FG-CNTRC plate subjected to the arbitrarily nonuniform loaddistributed partial edge compression loads. They transformed the arbitrarily distributed partial edge compression load at one plate edge to the load acting on all internal points of this edge, then obtained the accurate critical buckling loads and buckling modes.
In the series-study of vibration, Zhang and his coauthors [116] employed the element-free IMLS-Ritz method to investigate the free vibration of FG-CNT plates with elastically restrained edges, based on the FSDT. They provided a set of vibration frequencies of structural and found that the FG-X types furnish the highest frequency values of all the cases. Further, using the same method, they analyzed the thick plates which were rested on elastic foundations and compared their solutions with the existing results [117].
Based on the FSDT and variational principle, Zhong et al. [118] adopted the modified Fourier series to replace the traditional admissible functions of the Ritz method in order to remove the potential discontinuous of the original rectangular plates unknown and their derivatives at the edges. Thai and his coauthors [119] presented a moving Kriging (MK) mesh-free method combined with HSDT and nonlocal theory for small size effect analysis of FG CNTRC nanoplates. Unlike the other mesh-free method, this method does not require to use the Lagrange multiplier or penalty methods to impose essential boundary condition. García-Macías and his coauthors [120] considered the vary of mechanical properties and the uncertainties inherent in the fabrication technique and focused on the Structural Health Monitoring. This study on the basis of stochastic representation of the grading profiles used the Kriging and High-Dimensional Model Representations (RS-HDMR) to surrogate the finite element model. Based on the Reddy's high-order shear deformation theory, Di et al. [121] established the calculation model for the impact system with the use of the weak form quadrature element method. In addition, they employed the nonlinear Hertzian contact law to describe the impact process between a rigid impactor and the rectangular FGCNTRC plate.
Besides the single-layer, the laminated plates, like sandwich plates, also attracted the attention of researchers. Beni [122] first extended the Carrera Unified Formulation for analysis of the free vibration of asymmetric annular sector plates, where all three displacement components of layer k in a layered structure are expressed as a set of thickness functions. The governing equations were obtained employing the Principle of Virtual Displacements and solved with GDQ method (Figure 7). Moradi-Dastjerdi and Momeni-Khabisi [123] investigated the vibrational behavior of sandwich plates resting on Pasternak elastic foundation and subjected to periodic loads. It is worth noting that the researches consider the effects of CNTs aspect ratio and waviness, as the CNT curvature (waviness index) dramatically decreases modulus of elasticity. In order to overcome the limitation that continuum micromechanics equations cannot capture the scale difference between the nano-and micro-levels, Tahouneh Vahid [124] used the Eshelby-Mori-Tanaka approach to determine the effect of CNT agglomeration on the elastic properties of randomly oriented CNTRCs and provide a 3D elasticity solution for sandwich sectorial plates. Fu et al. [125] investigated the acoustic radiation behavior of laminated FG-CNTRC plates in thermal environments. The sound pressure and radiation efficiency are calculated through Rayleigh integral, and the results show that the peaks and dips of the structural and acoustic response will increase as the values of CNT volume fraction increase.
Some scholars also pay their attention to FG-CNTRC plates with the piezoelectric layer. For the CNT-reinforced composite plates with piezoelectric layers, Selim et al. [126] focused on the active vibration control of FG-CNTRC plate and adopted constant velocity feedback approach to determine two positions of piezoelectric sensor and actuator layers. Keleshteri et al. [127] dealt with large amplitude vibration analysis of FG-CNTRC annular sector plates with surface-bonded piezoelectric layers. The nonlinear frequencies of the FG-CNTRC annular sector plate were obtained through the generalized differential quadrature method along with direct iterative method.
Nguyen-Quang et al. [128] investigated the dynamic response of PFG-CNTRC plates by using the extended isogeometric method with nonuniform rational B-spline and the HSDT.

Shells
Because of the high load-carrying ability, the shell structures are largely employed for practical applications. To further enhance the physical properties of it, the research of FG-CNTRC shells has been increasingly developed by scholars. In static analysis, Zghal et al. [129] used a discrete double-directors shell element to derive the governing equations of FG-CNTRC cylindrical panel and investigate the bending behavior under the uniformly distributed load. The obtained results were compared with those given by Zhang et al. [130], which employed the mesh-free kp-Ritz and validated the effectivity of this model. In addition, the Kirchhoff shell theory was used to analyze large displacements and rotations of thin FG-CNTRC shell structures by Zghal et al. [131], which can assure notably the compromise between good accuracy and low computational costs. They carried out using four node and three node finite elements, overcoming the locking problems during the analysis. Ansari and Kumar [132] developed a finite element (FE) coding for the functionally graded CNT-reinforced doubly curved singly ruled truncated rhombic cone by using a C 0 nine noded element.
Alibeigloo [133] assumed that the effective thermoelastic constants are independent of temperature and obtained the temperature distribution in three dimensions by solving heat conduction differential equation with variable coefficient. Then the paper applied Fourier series expansion to acquire the stress and displacement fields along the axial and circumferential direction and state space technique along the radial direction. In addition, Alibeigloo and Zanoosi [134] considered thermoelectro-elastic deformations of FG-CNTRC cylindrical shell integrated with piezoelectric layers and come to a conclusion that the sign of stresses and displacements distribution in mechanical loading is opposite to the electric voltage loading so that adjusting the load voltage can be used to control bending behavior of FG-CNTRC layer.
The researches on buckling of FG-CNTRC shells mainly focus on different structural forms, numerical solution, and loading they are subjected. Mehri et al. [135] analyzed the bifurcation of a composite truncated conical shell with embedded single-walled carbon nanotubes subjected to combined axial compressive load and hydrostatic pressure. They proposed a semi-analytical solution on the basis of the trigonometric expansion through the circumferential direction along with the harmonic differential quadrature discretization in the meridional direction.
Based on the first-order shear deformation shell theory, Nguyen and his coauthors [136] first used nonuniform rational B-Spline basis functions to perform postbuckling analysis of FG-CNTRC shells. It is an advanced numerical method integrated idea between CAD and FEM, in which the geometric data from computer-aided design (CAD) can be used directly for numerical simulation. Based on the three-dimensional elasticity theory, Liew and Alibeigloo [137] studied the buckling behavior of FG-CNTRC simply supported cylindrical panel with initial normal axial and circumferential stresses. Hajlaoui and his coauthors [138] innovatively used the enhanced solid-shell elements with a parabolic shear strain distribution imposed on the compatible strain part to investigate the buckling behavior of cylindrical shell under external pressure and axial compression. They employed the assumed natural strain (ANS) method and the enhanced assumed strain (EAS) method to overcome the locking phenomena in traditional threedimensional finite element modeling. Safarpour et al. [139] considered the effects of critical voltage and CNT reinforcement on the piezoelectric rotating cylindrical CNTRC shell. The results show that FG-O has the lowest critical voltage and the FG-V pattern has high stability area in comparison with other patterns, as shown in Figure 7. Mehar et al. [140] focused on the graded CNTreinforced composite sandwich shell structure; they derived the eigenvalue type of buckling equation by the variational technique, considering the geometrical distortion due to temperature loading, and solved numerically via isoparametric displacement controlled FEM (Figure 8).
For the free vibration analysis of carbon nanotubereinforced functionally graded composite shell, Qin and his coauthors [141] take the rotating FG-CNTRC cylindrical shells as the research subjects. They considered the Coriolis and centrifugal effects on the strain and kinetic energy of shell due to rotating and applied the Ritz method to derive the motion equations where the displacement fields of the shell are expressed by Chebyshev polynomials.
Kiani [142] carried out the research deals with the free vibration response of FG-CNTRC spherical panel. The solution method is based on the Ritz method whose shape functions are estimated according to the Gram-Schmidt process. With the similar method, Kiania and his coauthors [143] focused on the FG-CNTRC conical panels. They concluded that boundary conditions and angles of embrace of the conical shell play an important role on the fundamental frequencies of the structure. Based on the FSDT and Ritz-variational energy method, Wang et al. [144] used a semi-analytical method for vibration analysis of FG-CNTRC doubly curved panels and shells of revolution in which the translation and rotation displacements are expressed as the superposition of a standard cosine Fourier series and several auxiliary functions; some selected mode shapes of the shells are given in Figure 9. Hamilton's principle and the assumed mode method are used to formulate the equation of motion of the CNT-reinforced functionally graded closed cylindrical shell by Song and his coauthors [145]. They investigated the effects of natural frequency on the parameter. On the basis of free vibration analysis, Kiani [146] utilized the Newmark time marching scheme to trace the resulting dynamic equations in time and explore the influences of load velocity. Frikha and his coauthors [147] also adopted the same method in their prior research which used a double-directors finite shell element and temporal responses were drawn using the Newmark time scheme.
For the structures with an initial imperfection, based on the classical thin shell theory, Foroutan [148] studied the nonlinear vibration analysis of imperfect FG-CNTRC cylindrical panels subjected to the external pressure in the thermal environment. They presented the influence of parameters on results, like initial imperfection, temperature, and CNTs distributions. What's more, Nguyen et al. [149] used the Reddy's TSDT to analyze the imperfect thick FG-CNTRC double curved shallow shells subjected to the combination of blast load and temperature. The shell was assumed to rest on elastic foundations and solved through the Galerkin method and fourth-order Runge-Kutta method.

Conclusion
A comprehensive review is given in the present paper, in which the mechanical analysis of functionally graded carbon nanotube-reinforced composite beams, plates, and shells has been discussed. The review of various investigations on the bending, buckling, and vibration behavior of nanostructures including beams, plates, and nanoshells has been carried out. Based on the literature research, some considerations about study and the direction of future research are as follows: (1) It can be confirmed that the volume rate change of CNT has a significant effect on the bending, buckling and vibration characteristics of the functionally graded carbon nanotube-reinforced composite beams plates and shells structure. (2) Currently, the main solution methods for mechanical research of FG-CNTRC structure mainly include variational method, finite element method, generalized differential quadrature, two perturbation method, etc.
(3) There are few evaluation methods for the material parameters and CNT performance parameters of FG-CNTRC, and the two existing mainstream analysis methods have their own limitations. (4) The research on the monomer structure of functionally graded carbon nanotube-reinforced composites has been relatively complete, but there are relatively few studies on combined structures, such as spherical-cylindrical combined structures. can be more extensive research on the structure of FG-CNTRC, such as the investigation of vibroacoustic response.