An energy method for buckling behavior analysis of functionally graded carbon nanotube-reinforced composite sandwich structures

The present work aims to investigate the buckling performance of sandwich structure of functionally graded carbon nanotube-reinforced composite (FG-CNTRC). Through first-order shear deformation theory, an analytical model for the sandwich structure of FG-CNTRC was established. The governing equation for the prediction of the buckling performance of the sandwich structure of FG-CNTRC was obtained through energy method. There was analytical solution that can satisfy both boundary conditions. The theoretical model and method were verified by literature analysis, and the influence of each parameter on the buckling performance was evaluated and performed on the basis of the corroborated model. The findings can lay a solid foundation of the design and application of the sandwich structure of FG-CNTRC.

The gradient distribution of CNTs in the matrix forms a new material, namely FG-CNTRC [12] that can improve the structural performance and reduce the consumption of carbon nanotubes. Various theoretical models have been applied to study the mechanical properties, including first shear deformation theory (FSDT) [13], second shear deformation theory (SSDT) [14], refined shear deformation theory (RSDT) [15], higher shear deformation theory (HSDT) [16], and others [17]. According to FSDT, Civalek et al [18] studied CNTreinforced cross-ply laminated composite plates on free vibration and buckling behaviors. Nguyen et al [19] adopted the FSDT for the postbuckling behaviors of FG-CNTRC. Daikh et al [20] studies the static buckling stability and defection of single-walled FG-CNTRC plates through the HSDT. Karami et al [21] adopted SSDT for studying the static and dynamic properties of FG-CNTRC panel, as well as the stability. Civalek et al [22] analyzed dynamic performance and stability of the FG-CNTR laminated non-rectangular sheets through the four-point straight-face transformation method. For the agglomerated CNT-reinforced composite nanoplates, Daghigh et al [23] analyzed the bending performance and stability via the Hamilton's principle. Cho et al [24] investigated the bending, free vibration, and buckling behaviors of the FG-CNTRC sheet applying the 2D meshfree natural element method. Adhikari et al [25] predicted the buckling behavior of FG-CNTRC laminates under different non-uniform in-plane edge loads with finite element solutions and elastic methods. By putting forward some Kantorovich-Galerkin methods, Wang et al [26] studied about thin-CNTRC sheets on buckling and free vibration behaviors through classical sheet theory. Dash et al [27] applied a semi-analytical method to study the free vibration and buckling performance of carbon nanotube and fiber reinforced composites in thermal environment.
Later, some researchers began to study the sandwich structures of FG-CNTRC. Foroutan et al [28] analyzed the postbuckling response of sandwich cylindrical panels through analytical and semi-analytical methods. Aref et al [29] used Ritz Method to study dynamic performance and stability of FG-GNPs reinforced composite sandwich beam. Mehar et al [30] explored the critical thermal buckling temperature of FG-CNTRC sandwich shells. Via the geometric transformation method, Civalek et al [31] explored the buckling behavior of the FG-CNTRC thick inclined plate under biaxial and uniaxial loading. Mohammadimehr et al [32] employed a generalized difference orthogonal method to study on the CNTreinforced sandwich beams about the free vibration and buckling behaviors. Maneengam et al [33] combined the HSDT and finite element method to investigate the dynamic performance of multi-walled carbon nanotube reinforced composite honeycomb shell structure. Shareef et al [34] investigated vibration characteristics of carbon nanotubes-reinforced magnetorheological elastomer composite sandwich plate under different speed and magnetic field. Ansari et al [35] explored free vibration of FG-CNTRC sandwich cylindrical shells via the HSDT and variational differential quadrature method. Comparing the results obtained by the HSDT with the results obtained by the FSDT, the validity of the derived differential equation was verified.
According to related survey, different theoretical methods are applied for investigating the static, stability, and dynamic properties of the Single-layer FG-CNTRC structure. The Galerkin method, generalized differential quadrature method, Hamilton's principle, and finite element method are adopted for studying the mechanical properties of FG-CNTRC sandwich structures. The minimum potential energy exhibits high accuracy and low computational cost when it is employed for the mechanical behavior of composite structures [36,37]. However, considering the effect of shear deformation and rotary inertias, there are scarcely any reports to explore the buckling behavior of the sandwich plates of FG-CNTRC via the minimum potential energy. To explore the buckling behavior of FG-CNTRC sandwich structures more precisely, an analysis model was established using the geometric equations of structure, constitutive relationship, and minimum potential energy. In addition, the shear correction coefficients have been added to improve the calculation accuracy. The theoretical model and method were verified by literature analysis. Furthermore, influence of different parameters (including CNT distribution, volume fraction of nanotubes, different boundary conditions, core-skin thickness ratio, lengthwidth ratio, and elasticity modulus ratio) on buckling performance (critical buckling load) has been studied and discussed.
2. Mechanical model of FG-CNTRC sandwich structure 2.1. Basic assumptions and model description For the derivation of the governing differential equations, the assumptions are given as below: (1) The displacement relationship between the layers conforms to first-order shear deformation model; (2) There is no no-slip between the skin layer and the core layer; and (3) Transverse displacement of all points in each layer, which is normal to the plate, is the same. Figure 1 displays the geometry and dimensions of the sandwich structures of FG-CNTRC. a and b represent the length and width of thin plate. h represents the total thickness of the sandwich structure of FG-CNTRC. The thickness of core layer and skin layer is h c and h f , respectively. Figure 2 shows the distribution of CNTs in the matrix. Among them, CNTs are uniformly distributed (UD) or functionally graded (FG) alongside the thickness direction. According to the different distribution of carbon nanotubes in the matrix material, carbon nanotube reinforced composites were named differently. Five different configurations are created and studied, namely UD CNTRC face-sheet, FG-X CNTRC face-sheet, FG-O CNTRC face-sheet, FG-Λ CNTRC face-sheet, and FG-V CNTRC face-sheet.

Functionally graded CNTRC
There is a linear correlation between the volume fraction of matrix and CNTs and the thickness of the structure. By considering five distributions of CNTs alongside the thickness direction, the volume fraction can be mathematically described as below [38]: where, the value V * stands for the given volume fraction of CNTs. The value V CNT represent the volume fraction of CNTs along z direction, and the value V m represent the volume fraction of the matrix along z direction. In addition, the volume fraction of the matrix can be calculated as V V 1. The effective material properties can be expressed as follows [38]. and m u stand for the Poisson's ratios of the nanocomposite, the CNTs, and the matrix. Table 1 shows the efficiency parameters , ,

Stress-strain relation
Based on FSDT and basic assumptions, displacement at any point in each layer of the sandwich structures of FG-CNTRC is shown as follows [40]:  rotation angles of the normal to center plane along the x and y axes; h i represents the thickness of the skin layers and the core layer. Based on the continuous relationship of interlayer displacement, the calculation formulas of the core layer displacement of the sandwich structure of FG-CNTRC can be expressed as follows [40]: As for the displacement-strain relationship of the FG-CNTRC layers, the shear correction coefficients have been added, which improves the accuracy of the calculation. The calculation formula is: e stands for the in-plane strain vector of the FG-CNTRC layers, and g stands for the out-of-plane strain vector of the FG-CNTRC layers, and The value g f is the shear correction coefficient, and The displacement and strain relationship of the core layer is as follows e stands for the in-plane strain vector of the core layers, and . c c y z c x z The value g c is the shear correction factor, and g 5 6.
c / = As for the stress-strain relation of the FG-CNTRC layers, the calculation formulas are [40]:

Derivation of governing equations
As for the FG-CNTRC layer, the calculation formula of strain energy is expressed as: where, U f means the strain energy of FG-CNTRC layer.
As for the core layer, the calculation formula of strain energy is as follows: c V cx cx cy cy cxy cxy cyz cyz cxz cxz where, U c means the strain energy of the core layer.
The internal force is expressed as follows: where N x and N y represent the axial pressure and N xy indicates the shear load. The work done by the longitudinal load in the buckling process of the sheet can be expressed by: The calculation formula of the total potential energy can be written as: Two boundary conditions (simply supported and clamped) for the sandwich structures of FG-CNTRC are considered.
The boundary condition for the four edges simply supported (SSSS) can be expressed [41]: According to the minimum potential energy principle [42], the governing equation of the sandwich structures of FG-CNTRC can be originated from the following relation for solving the buckling behavior.

Verifcation and comparison
The material and dimensional parameters are tabulated in [39]. The geometric parameters of FG-CNTRC are given as below: a b h 500 mm, 5 mm. In table 2, it is observed that the agreement is very good, although there are slight differences between the results. Further, the sandwich structure is used to prove the validity of the theoretical model.
Based on a natural element hierarchical model, Cho et al [43] analyzed the stability of the sandwich plates with FG-CNTRC layers. The material parameters in [43] are applied to verify the theoretical model in this paper. These parameters of UD nanocomposite face-sheets can be solved according to equations (1)- (5 This geometric dimension is as below: a = b = 0.1 m, h = 10 mm. The four sides of the sandwich structures are simply supported. When the core-face thickness ratio (CFTR) takes different values, the dimensionless critical buckling loads obtained by the natural element method and the minimum potential energy principle are shown in table 3.
It can be seen from table 3 that the error of the results obtained by the two theoretical models is within 4%, which proves the validity of the theoretical model in this paper.

Parametric study
The influence of different parameters (including CNT distribution, volume fraction of nanotubes, different boundary conditions, elasticity modulus ratio, core-skin thickness ratio, and length-width ratio) on the buckling properties of the sandwich structures of FG-CNTRC was investigated applying validated theoretical model. The core material parameters are as below: E G 0.038 GPa, 0.0144 GPa, 100 kg m c c c 3 r = = = - [44]. It is necessary to specify that the 'Λ-V' sandwich configuration indicates that the bottom and top CNT distributions of the sandwich structures of FG-CNTRC are 'Λ' and 'V', respectively.

Effects of elasticity modulus ratio on the structural critical buckling load
The effect of the elasticity modulus ratio (E m /E c ) of the sandwich structures of FG-CNTRC on the structural critical buckling load can be firstly explored. Figures 3(a)-(d) shows the effect of elasticity modulus ratio (E m /E c ) on structural CBL. From figures 3(a)-(d), when the elasticity modulus ratio (E m /E c ) increases, the CBL of the sandwich structures of FG-CNTRC with the increase of various configurations. It is due to the fact that the overall stiffness and the ability to resist deformation of the structure increase with the increase of elasticity modulus ratio (E m /E c ).

Impacts of thickness ratio on the structural critical buckling load
Then, the effect of thickness ratio (h c /h f ) of the sandwich structures of FG-CNTRC on the structural critical buckling load was investigated. Figure 4 reveals the impact of volume fraction on relationships between thickness ratio (h c /h f ) and structural CBL of the sandwich structures of FG-CNTRC with various configurations.
From figures 4(a)-(f), the structural CBL are found to linearly increase with the increase of thickness ratio (h c /h f ). The reason refers to that the stiffness of the sandwich structures of FG-CNTRC is influenced by the distance between the upper and lower panels. When the thickness ratio (h c /h f ) increases, that is, the core thickness increases, the distance between the upper and lower panels increases, resulting in the strengthening of the rigidity of the structure. So the ability to resist instability is enhanced. When the four sides of the sandwich structures of FG-CNTRC are simply supported, the 'Λ-V' sandwich configuration exhibits the largest CBL while the 'V-Λ' sandwich configuration shows the minimum CBL. We can further observe that for the sandwich structures of FG-CNTRC with clamped-support boundary (V 0.17 = * ). When the smaller value of thickness ratio (h c /h f ) is taken, the CBL of 'Λ-V' sandwich configuration is the largest, while that of 'V-Λ' sandwich configuration is the smallest. When the bigger value of thickness ratio (h c /h f ) is taken, the CBL of 'X-X' sandwich configuration is the largest, while that of 'V-Λ' sandwich configuration is the smallest. Table 4 demonstrates the effects of two boundary conditions as well as volume fraction on structural CBL of the sandwich structures of FG-CNTRC.
According to table 4, the sandwich structures of FG-CNTRC with higher given volume fraction V * of CNTs have higher critical buckling loads for any given configuration types. Compared with the four-side simply supported constraints, the sandwich structures of FG-CNTRC with four-side clamped support have better stability. When the value V * is 0.11, for the simply supported boundary condition, the critical buckling loads of 'O-O' type structure and 'V-Λ' type structure are reduced by 3.32% and 19.30% respectively compared with the 'UD-UD' type structure, and the critical buckling loads of 'X-X' type structure and 'Λ-V' type structure are increased by 3.20% and 16.82% respectively. When the value V * is 0.11, for the for clamped boundary conditions, the critical buckling loads of 'O-O' type structure and 'V-Λ' type structure are reduced by 7.29% and 19.74% respectively compared with the 'UD-UD' type structure, and the critical buckling loads of 'X-X' type structure and 'Λ-V' type structure are increased by 7.23% and 11.74% respectively. The distribution of carbon nanotubes in sandwich structure has influence on the stiffness and stability of the structure. In the 'X-X' and 'Λ-V' sandwich configuration, carbon nanotubes are more distributed on the upper and lower surfaces of the sandwich structure, which gives full play to the ability of FG-CNTRCDS to resist deformation, thus greatly improving the stiffness and stability of the sandwich structure. In the 'V-Λ' and 'O-O' sandwich configuration, the carbon nanotubes are less filled on the upper and lower surfaces, which make the sandwich structure unable to exert its high bearing capacity, thus greatly reducing the stability of the sandwich structure. So, the 'X-X' sandwich configuration and 'Λ-V' sandwich configuration have better stability, whereas 'V-Λ' type structures exhibit the worst stability.

5.
3. Impacts of length-to-width ratio on the structural critical buckling load Then, the effect of length-to-width ratio (a/b) on the CBL of five sandwich configurations is explored.