Document Type : Research Article
Authors
Department of Mechanical Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran
Abstract
Keywords
Buckling Analysis of Nano Composite Plates Based on Combination of the Incremental Load Technique and Dynamic Relaxation Method
V. Zeighami ^{a}, M.E. Golmakani ^{a*}
^{a} Department of Mechanical Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran
KEYWORDS 

ABSTRACT 
CNTRC plate Buckling Incremental loading DR method 
In this paper, a different method, incremental load technique in conjunction with dynamic relaxation (DR) method, is employed to analyze the buckling behavior of composite plates reinforced with functionally graded (FG) distributions of singlewalled carbon nanotubes (SWCNTs) along the thickness direction. The properties of carbonnanotubes reinforced composite (CNTRC) plate were determined through modified rule of mixture. The nonlinear governing relations are obtained incrementally in the form of partial differential equations (PDEs) based on firstorder shear deformation theory (FSDT) and Von Karman nonlinear strain. In the proposed method, for finding the critical buckling load, the mechanical loads are applied to the CNTRC plate incrementally so that in each load step the incremental form of PDEs are solved by the DR method combined with the finite difference (FD) discretization technique. Finally, the critical buckling load is determined from the loaddisplacement curve. In order to verify the accuracy of the present method, the results are compared with those available in the literatures. Finally, a detailed parametric study is carried out and results demonstrate that the change of carbon nanotube volume fraction, plate widthtothickness ratio, plate aspect ratio, boundary condition and loading condition have pronounced effects on the buckling strength of CNTRC plates. It is seen that for all types of loading, boundary conditions and both cases of with and without presence of elastic foundation the FGX and FGO have the highest and lowest values of buckling loads. 
Carbon nanotubes (CNTs) have been widely accepted owing to their remarkable mechanical, electrical and thermal properties and the applications of CNTs are thus drawing much attention currently. Conventional fiberreinforced composite materials are normally made of stiff and strong fillers with microscale diameters embedded into various matrix phases.
The discovery of CNTs may lead to a new way to improve the properties of resulting composites by changing reinforcement phases to nanoscaled fillers [1]. Carbon nanotubes are considered as a potential candidate for the reinforcement of polymer composites, provided that good interfacial bonding between CNTs and polymer and proper dispersion of the individual CNTs in the polymeric matrix can be guaranteed [2]. Since the load transfer between the nanotube and the matrix is less than perfect, several micromechanical models have been developed to predict properties of CNTreinforced nanocomposites. Fidelus et al. [3] examined thermomechanical properties of epoxybased nanocomposites with low weight fractions of randomly oriented single and multiwalled carbon nanotubes with a ruleofmixture type prediction of the modulus. Based on the rule of mixture, Anumandla and Gibson [4] presented a comprehensive closed form micromechanics model for estimating the elastic modulus of nanotubereinforced composites. Han and Elliot [5] presented classical molecular dynamics (MD) simulations to model polymer/CNT composites constructed by embedding a single wall (10, 10) CNT into two different amorphous polymer matrices. The CNT–polymer interfacial shear strength was determined according to a series of pullout tests of individual carbon nanotubes embedded within polymer matrix by Wagner et al. [6,7], which demonstrated that carbon nanotubes are effective in reinforcing a polymer due to remarkably high separation stress. Molecular dynamics (MD) simulations were performed to predict the interfacial bonding by considering threedimensional crosslinks and stronger interfacial adhesion can be achieved through functionalization of the nanotube surface to form chemical bonding to the chains of polymer matrix [8,9]. Another fundament issue is the dispersion of carbon nanotubes in the matrix, since CNTs are tend to agglomerate and entangle because of their enormous surface area and high aspect ratio. Aminofunctionalized CNTs are therefore developed to improve their dispersion in polymer resins [10]. The constitutive models and mechanical properties of carbon nanotube polymer composites have been studied analytically, experimentally, and numerically. A review and comparisons of mechanical properties of single and multiwalled carbon nanotube reinforced composites fabricated by various processes were given by Coleman et al. [11]. In actual structural applications, carbon nanotubereinforced composites (CNTRC), as a type of advanced material, may be incorporated in the form of beams, plates or shells as structural components. It is thus of importance to explore mechanical responses of the structures made of CNTRC. Wuite and Adali [12] firstly used the classical laminated beam theory to analyze symmetric crossply and angleply laminated beams stacked with multiple transversely isotropic layers reinforced by CNTs based on micromechanical constitutive models developed according to the Mori–Tanaka method. Vodenitcharova and Zhang [13] developed a continuum model for pure bending of a straight nanocomposite beam with a circular cross section reinforced by a singlewalled carbon nanotube. Formica et al. [14] studied vibration behaviors of cantilevered CNTRC plates by employing an equivalent continuum model according to the Mori–Tanaka scheme. The buckling behaviors of laminated composite plates reinforced by SWCNTs were investigated analytically and numerically based on the classical laminated plate theory and thirdorder shear deformation theory, respectively, to consider the optimized orientation of CNTs for obtaining the highest critical load and corresponding mode shape were calculated for different kinds of boundary conditions as well as aspect ratios of the plates [15]. Motivated by the concept of functionally graded materials (FGM), Shen [16] suggested that for CNTbased composite structures the distributions of CNTs within an isotropic matrix were designed purposefully to grade with certain rules along desired directions for the improvement of the mechanical properties of the structures and the nonlinear bending behaviors of the resulting functionally graded CNTreinforced composite (FGCNTRC) plates in thermal environments were presented. Zhu et al. [17] discovered that CNT reinforcements distributed close to top and bottom are more efficient than those distributed near the midplane for increasing the stiffness of CNTRC plates. Based on the Timoshenko beam theory, nonlinear free vibrations of functionally graded CNTRC beams were analyzed with the Ritz method and direct iterative technique by Ke et al. [18]. They found linear and nonlinear frequencies of FGCNTRC beam with symmetrical distribution of CNTs are higher than those of beams with uniform or asymmetrical distribution of CNTs. By using a multiscale approach, Shen and Zhang [19] discussed thermal buckling and post buckling behaviors of functionally graded nanocomposite plates reinforced by SWCNTs subjected to inplane temperature variation. Aref et al [20, 21] investigated HigherOrder ThermoElastic Analysis of FGCNTRC Cylindrical Vessels Surrounded by a Pasternak Foundation and TwoDimensional ElectroElastic Analysis of FGCNTRC Cylindrical Laminated Pressure Vessels with Piezoelectric Layers Based on ThirdOrder Shear Deformation Theory. Based on a higher order shear deformation plate theory, nonlinear vibration of FGSWCNT plates rested on elastic foundation in thermal environments was investigated by an improved perturbation technique [22]. Aref et al [23] investigated Effect of characteristics and distribution of porosity on electroelastic analysis of laminated vessels with piezoelectric facesheets based on higherorder modeling.
In the present study, a different method, incremental load technique in conjunction with dynamic relaxation (DR) method, is used to study the buckling of singlewalled carbon nanotube reinforced composite plates resting on an elastic foundation and subjected to different tensilecompressive loads. A uniform and three kinds of functionally graded distributions of CNTs along the thickness direction of plate are considered. The properties of composite material in each point were determined by modified rule of mixture. All governing equations were obtained incrementally based on first order shear deformation theory (FSTD) and Von Karman nonlinear strains. Using the principle of minimum potential energy, the set of coupled nonlinear equilibrium equations was obtained in incremental form for different boundary conditions. The dynamic relaxation (DR) method combined with the finite difference discretization technique is employed to find the critical buckling load for simply supported and clamped boundary conditions. To verify the present results and formulations, some comparison studies are carried out between the obtained results and the available solutions in the literature. Excellent agreement between the obtained and available results is observed. A detailed parametric study is carried out to investigate the effects of volume fraction of nanotubes, arrangement of nanotubes, widthtothickness ratio, elastic foundation and aspect ratio on the buckling load of nanocomposite plates with clamped and simply supported boundary conditions.
Fig. 1 shows the coordinate system, geometry and loadings of nanocomposite plate with different distributions of CNTs. The reinforced composite plates have four different distributions of nanotubes with definite length a, width b and thickness h.
In the present work, based on the modified rule of mixture and introducing the CNT efficiency parameters the effective material properties (elastic modules and Poisson ratios) of CNTRC plate can be expressed as follows [16]:
(1) 


(2) 
(3) 
Where , , and refer to elasticity modulus and shear modulus of carbon nanotubes, respectively. Also, and indicate the elasticity modulus and shear modulus of the substrate and is the CNT efficiency parameters which is introduced by Shen [16] for accounting the smallscale effect and other effects on the material properties of CNTRCs. The values of is determined by matching the elastic moduli of CNTRCs predicted by the molecular dynamics (MD) simulations with those obtained from the modified rule of mixture in Eqs. (1) – (3). Furthermore, and are volume fractions of carbon nanotubes and substrate, respectively, and expressed as follows:
(4) 
(a) Uniform distribution 
(b) FGV distribution 
(c) FGV distribution 
(d) FGX distribution 
Fig. 1. Nanocomposite plates subjected to various types of loading with (a) uniform, (b) FGV, (c) FGO and (d) FGX distributions of CNTs. 
The volume fraction of the uniform and three types of functionally graded distributions of the carbon nanotubes is determined through following expressions [17].
(UDCNTRC) 
(5) 

(FGV CNTRC) 

(FGO CNTRC) 

(FGX CNTRC) 
So that
(6) 
Where is the mass fraction of carbon nanotubes in composite plate. In above expressions, and refer to densities of substrate and carbon nanotubes, respectively. It is noticed that the four types of CNTRC plates possess the same mass fraction ( ) and volume of CNTs. According to the following relation, Poisson’s ratio, , is assumed to be uniformly distributed over the thickness of the functionally graded CNTRC plates.
(7) 
The firstorder shear deformation theory (FSDT) is employed to predict the displacement fieldof CNTRC plate. Based on FSDT, the displacement field is expressed as follows [24].
(8) 
Where , and denote displacement components of the mid–plane in the , and directions, respectively. Moreover, and represent rotations of a transverse normal about and axes, respectively. In order to predict the buckling load by the DR method the equilibrium equations should be derived in the incremental form. Thus, all of the following governing equations are derived in the incremental form of variables. By assuming small strains and moderate rotations, based on the incremental nonlinear von Karman strain–displacement relations, the strain components associated with the displacement field of Eq. (8) are expressed as follows [24]:
(9) 
In which


(10) 
According to the Hooke’s law, the incremental constitutive relations are defined as follows:
(11) 
In which the components are defined by
(12) 

In above relations, and refer to elasticity moduli of CNTRC plate along x and y directions, respectively. In addition, , and denote shear moduli and and refer to Poisson ratios. The incremental forces, moments and shear stress resultants can be expressed by the following expressions.
(13) 

By substituting Eqs. (9) to (12) in Eq. (13), the incremental forces, moments and shearstress resultants is obtained as follows
(14) 

(15) 

(16) 
where A, B, D and A^{s}are the extensional, coupling, bending, and shear stiffness, respectively, which are obtained by the following expressions:
(17) 


(18) 

Where and is transverse shear correction coefficient that assumed to be along the thickness direction [17]. By substituting Eq. (10) in Eqs. (14) to (16), the incremental form of stress resultants can be obtained based on displacement field as follows:


(19) 
Using the principle of minimum total potential energy, the equilibrium equations are obtained based on FSDT as follows:


(20) 
In which, and are Winkler and shear coefficients of foundation parameters, respectively. It is noticed that incremental load is the transverse mechanical load and must be removed from the third relation of Eq. (20) for the buckling analysis. By substituting Eq. (19) in Eq. (20), the equilibrium equations are obtained based on displacement field as follows
(21) 
For the buckling of CNTRC plates, the following boundary conditions are employed:
AClamped Support at and
(22) 

BSimply Support at and
(23) 

Because of the effectiveness and efficiency of dynamic relaxation (DR) method to solve highly nonlinear problems, the DR technique in conjunction with finite difference discretization scheme has been employed in this study to analyze the nonlinear differential equations of the CNTRC plate. The DR is an explicit iterative procedure which is employed to transfer a boundary value problem into timestepping initial value problem. This aim can be achieved by adding fictitious inertia and damping terms to the equilibrium equations (for more details see [2627]):
(24) 
Where , are fictitious mass and damping matrices, respectively. Also, , and refer to vectors of virtual speed, acceleration and external forces at n^{th} iteration, respectively. In addition, is displacement vector and denotes stiffness matrix. Dynamic relaxation iteration method is generally unstable. So, the mass and damping matrices should be defined to guarantee the stability and convergence of the iterative procedure. In order to obtain explicit solution, matrix should be diametrical. Based on Gershgorin theorem, mass matrix is obtained through following expressions [27].
(25) 

(26) 
In above expressions, is internal forces (the lefthand side of the equilibrium equations (Eqs. (21)). Based on Zhang’s suggestion, the damping coefficient can be determined according to the following expression [27]:
(27) 
In addition, damping matrix should be dependent to the mass matrix according to the following expression:
(28) 
To complete the transformation process, the velocity and acceleration terms must be replaced with the following equivalent central finitedifference expressions:
(29) 

(30) 
By substituting Eqs. (29) and (30) in Eq. (24), the equilibrium equations can be rearranged into an initial value format as follows:
(31) 
By integrating the velocities at the end of each load step, the incremental displacements can be obtained as:
(32) 
Therefore, the displacement equilibrium equations and Eqs. (31)(32) together with the appropriate boundary conditions in their finite difference forms, constitute the set of equations for the sequential DR method. For the sake of brevity, the DR algorithm which is clearly explained in [26, 27] is omitted.
In order to find the critical buckling load from the loaddisplacement curve the total displacements of each load must be computed. For this purpose, the obtained incremental displacements in each load step should be added to the displacements determined from the previous load steps as follows:
(33) 
This process continues till the code diverges, and this is a sign of buckling event. Clearly, critical buckling load is a specified load in which a large amount of displacement is occurred compared to the previous load steps.
In Table 1, the present solutions for buckling loads of CNTRC plates with clamped and simply supported boundary conditions and different arrangements of CNTs and loadings are compared with the results reported by Lei et al. [28]. It is noted that loadings include uniaxial compression along axis x , biaxial compression and biaxial compressive and tensile loading . As shown in Table 1, for different load conditions, the current solutions are in good consistency with those of Lei et al. [28] and the reliability and accuracy of the present formulation and results are verified.
In this section, the effects of various parameters on the buckling behavior of FGCNTRC plate are presented. The material properties of polymeric phase are taken from [16]. Also, the elastic properties of SWCNTs in armchair state (10 and 10) are considered as , , which are reported in [19].
Table 2 shows the Young modulus of CNTRC plate in x and y directions as well as CNT efficiency parameters which are taken from [5, 19] based on MD and modified rule of mixture, respectively, for various values of CNTs volume fractions. Furthermore, as reported by Shen and Zhang [19], , and .
In the following results, the effects of nanotubes volume fraction, distribution of nanotubes, widthtothickness ratio of plates and plate geometry are investigated on the critical buckling load of nanocomposite plates with clamped and simply supported boundary conditions. In the results, the boundary conditions are considered with all edges clamped and simply supported which named by CCCC and SSSS, respectively. The dimensionless buckling load is defined by and the plate thickness is assumed with .
Fig. 2 and Fig. 3 show the dimensionless buckling load of simply supported nanocomposite plate in terms of volume fractions of CNTs for different distributions of CNTs, loading states and two widthtothickness ratios and , respectively. As seen, the FGX and FGO distributions have the highest and lowest values of buckling loads, respectively, for all loading states, CNTs distributions and widthtothickness ratios. Furthermore, increase of CNTs volume fraction from 0.12 to 0.17 causes a significant rise of buckling load for different CNTs distributions and loading states. While, increasing the CNTs volume fraction from 0.17 to 0.28 does not have significant effect on the buckling load.
Considering the results for two different widthtothickness ratios of and shows that increasing CNTs volume fraction has the greater effect on the buckling strength of nanocomposite plate. Also, it is seen that increasing the thickness of nanocomposite plate has the more effect on the buckling load for FG distribution of CNTs compared to uniform ones.
Table 1 Comparison of dimensionless buckling load between the present study and those of Lei [28] for different boundary conditions, loading and CNTs distribution. 



Present study 
Lei [28] 
Present study 
Lei [28] 



C 
C 
SS 
SS 

ɤ_{1}= 1_{, }ɤ_{2}=0 
UD 
25.19 
25.73 
13.75 
14.11 

FGX 
26.67 
27.89 
16.15 
17.06 

FGO 
22.38 
21.12 
10.55 
9.83 

FGV 
24.54 
 
13.43 
 








ɤ_{1}= 1,_{, }ɤ_{2}=1 
UD 
30.95 
31.10 
27.82 
28.47 

FGX 
31.43 
31.39 
28.98 
29.18 

FGO 
27.14 
27.93 
25.26 
24.05 

FGV 
31.43 
 
28.98 
 








ɤ_{1,} ɤ_{2}= 1 
UD 
9.0476 
9.28 
5.71 
5.88 

FGX 
9.54 
9.65 
6.19 
6.44 

FGO 
8.57 
8.60 
5.00 
4.90 

FGV 
8.57 
 
5.00 
 

Table 2 Young’s moduli for PMMA/CNT composites reinforced by (10, 10) tube in room temperature reported by [5] and [19]. 

MD (Ref. [5]) 
Rule of mixture [19] 

0.12 
94.6 
2.9 
94.78 
0.137 
2.9 
1.022 

0.17 
138.9 
4.9 
138.68 
0.142 
4.9 
1.626 

0.28 
224.2 
5.5 
224.5 
0.141 
5.5 
1.585 

(a) 

(b) 

(c) 

Fig. 2 Effect of CNTs volume fraction on the buckling load of FGCNTRC plate ( ) with simply supported boundary conditions for (a) , 

(a) 

(b) 

(c) 

Fig. 3 Effect of CNTs volume fraction on the buckling load of FGCNTRC plate ( ) with simply supported boundary conditions for (a) , 

Fig. 4 and Fig. 5 consider the effect of widthtolength ratio of FGCNTRC plate ( and ) on the buckling load with different loading types for clamped and simply supported boundary conditions, respectively. As seen, for two loading modes of and the buckling load decreases by increasing the widthtolength ratio in both CCCC and SSSS boundary conditions. Furthermore, the decrease of buckling load is more significant for biaxial compressivetensile load compared to uniaxial compressive load . Furthermore, it is depicted that effect of widthtolength ratio on the buckling load of CCCC boundary condition is much less than the SSSS one (50% decrease versus 30% decrease by 2.5 times increase of widthtolength ratio for different CNTs distributions). However, for loading type of , highest and lowest decline of dimensionless buckling load are associated with FGO (i.e., 42 percent) and FGX (i.e., 21 percent) distributions, respectively. It can be rephrased that FGX distribution has the highest resistance against reduction of critical buckling load for both of loading modes. Finally, as shown in Fig. 4 (c) in the case of biaxial compressive loading and clamped boundary condition, by raising the widthtolength ratio the variation of buckling load be completely different compared to other loading types. For example, in clamped boundary condition and loading type of biaxial compression increasing the widthtolength ratio to a specified value goes up the buckling load significantly and more increase of this ratio does not have considerable effect on the buckling load.
However, as illustrated in Fig. 5 (c), for SSSS boundary condition and UD, FGX and FGV distributions with growing up the widthtolength ratio a little increase of buckling load can be observed. Although, in SSSS case and FGO distribution increasing this ratio decreases the buckling load smoothly.
In the following results, the buckling behavior of CNT reinforced composite plate on elastic foundation has been studied for different volume fraction of CNTs, widthtothickness ratios, widthtolength ratios and two types of CCCC and SSSS boundary conditions.
Tables 3 to 5 show the effects of elastic foundation coefficient on dimensionless buckling load of nanocomposite plate with thickness of h=2mm, the widthtothickness ratio ( ) and volume fraction ( ) with SSSS and CCCC boundary conditions for uniaxial compressive loading , compressivetensile loading and biaxial compressive loading , respectively.
(a) 
(b) 
(c) 
Fig. 4 Effect of widthtolength ratio on the buckling load of FGCNTRC plate ( ) with clamped boundary conditions for (a) , 
(a) 
(b) 
(c) 
Fig. 5 Effect of widthtolength ratio on the buckling load of FGCNTRC plate ( ) with simply supported boundary conditions for 
Table 3 Effects of Elastic Substrate (Ks and Kw) on Dimensionless Buckling Load of Nanocomposite Plate 



UD 
FGV 
FGX 
FGO 
K_{w} 
K_{s} 
Simply Support 

0 
0 
18.40 
17.73 
19.07 
17.19 
20 
18.67 
17.99 
19.34 
17.46 

50 
19.07 
18.4 
19.74 
17.76 







20 
0 
20.01 
19.20 
20.55 
18.80 
20 
20.28 
19.47 
20.81 
19.07 

50 
20.68 
19.87 
21.22 
19.47 







50 
0 
22.29 
21.49 
22.83 
21.08 
20 
22.56 
21.75 
23.10 
21.35 

50 
22.96 
22.16 
23.50 
21.75 



Clamped Support 

0 
0 
34.11 
34.38 
35.05 
31.42 
20 
34.51 
34.78 
35.59 
31.83 

50 
35.05 
35.45 
36.12 
32.36 







20 
0 
36.39 
36.53 
37.33 
32.77 
20 
36.93 
36.93 
37.60 
33.17 

50 
37.33 
37.60 
38.27 
33.71 







50 
0 
39.21 
39.48 
30.02 
34.51 
20 
39.61 
39.75 
40.42 
34.78 

50 
40.15 
40.42 
40.96 
35.32 
Table. 4 Effects of Elastic Substrate (Ks and Kw) on Dimensionless Buckling Load of Nanocomposite Plate ( , and ) and biaxial compressivetensile load . 




UD 
FGV 
FGX 
FGO 


K_{w} 
K_{s} 
Simply Support 

0 
0 
37.8704 
38.6761 
38.5418 
34.5131 

20 
38.2732 
39.079 
38.9447 
34.7816 

50 
38.8104 
39.7504 
39.6161 
35.4531 








20 
0 
38.8104 
39.7504 
39.4818 
35.1845 

20 
39.2133 
40.1533 
39.8847 
35.5874 

50 
39.m7504 
40.6905 
40.5562 
36.1245 








50 
0 
40.1533 
41.0934 
40.8248 
36.1245 

20 
40.4219 
41.4962 
41.2276 
36.3931 

50 
41.0934 
42.0334 
41.7648 
37.0646 



Clamped Support 

0 
0 
42.302 
43.6449 
43.2421 
37.796 

20 
42.7049 
44.0478 
42.6449 
37.1989 

50 
43.5106 
44.7192 
44.3164 
37.7361 








20 
0 
43.9135 
45.2564 
44.8535 
37.7361 

20 
44.364 
45.6593 
45.2564 
38.1389 

50 
44.9878 
46.3307 
45.9279 
38.6761 








50 
0 
46.1965 
47.5394 
47.0022 
39.2133 

20 
46.5993 
48.0765 
47.5394 
39.6161 

50 
47.2708 
48.748 
48.765 
40.1533 

For all types of loading, CNTs distributions and boundary conditions the effect of Winkler coefficient on increase of critical dimensionless buckling load is significantly higher than shear coefficient of the substrate . As seen, for all types of loading and both cases of with and without presence of elastic foundation the FGX and FGO have the highest and lowest values of buckling loads, respectively, for both SSSS and CCCC boundary conditions.
Fig. 6 represent the effect of CNTs volume fraction on the critical buckling load of SSSS nanocomposite plate with UD and FG distributions for different coefficients of elastic substrate and various types of loadings.
As seen, for both types of UD and FG distributions of nanocomposite plate with and without presence of elastic foundation increasing the volume fraction of CNTs from 0.12 to 0.17 causes a significant increase of buckling load. However, by increasing the volume fraction of CNTs from 0.17 to 0.28 the raising rate of buckling load decreases for both types of with and without elastic foundation and CNTs distributions. Obviously, presence of elastic foundation increases the buckling load and the effect of Winkler foundation on the buckling load is much greater than the shear coefficient of Pasternak foundation.
Fig. 7 show the effect of widthtothickness ratio on the buckling load of SSSS nanocomposite plate with and without elastic foundation and two types of UD and FGX distributions for loading types of (a) , (b) and (c) . As seen, with increase of widthtothickness ratio the influence of elastic foundation on the buckling load goes up significantly. Furthermore, the effect of CNTs distribution on the buckling load is much greater when the elastic foundation does not exist.
Table 5 Effects of Elastic Substrate (Ks and Kw) on Dimensionless Buckling Load of Nanocomposite Plate 



UD 
FGV 
FGX 
FGO 
K_{w} 
K_{s} 
Simply Support 

0 
0 
7.3266 
6.3801 
7.7473 
6.1038 
20 
7.4919 
6.4573 
7.8449 
6.1905 

50 
7.7122 
6.6708 
8.0583 
6.4040 







20 
0 
8.2080 
7.0977 
8.592 
6.8843 
20 
8.3733 
7.2578 
8.6987 
6.9910 

50 
8.5936 
7.4713 
8.9656 
7.2578 







50 
0 
9.5526 
8.3252 
9.8728 
8.1117 
20 
9.7661 
8.4319 
10.0863 
8.2184 

50 
9.9262 
8.6987 
10.2997 
8.4853 



Clamped Support 

0 
0 
13.0263 
12.0863 
13.4292 
11.9520 
20 
13.2949 
12.3549 
13.6978 
12.0863 

50 
13.5635 
12.6234 
13.9664 
12.3549 







20 
0 
13.9664 
12.892 
14.3692 
12.7579 
20 
14.1007 
13.1606 
14.5035 
12.8921 

50 
14.3692 
13.4292 
14.9064 
13.2949 







50 
0 
15.1750 
14.1007 
15.5779 
13.8321 
20 
15.3093 
14.2350 
15.7122 
14.1007 

50 
15.5779 
14.5035 
16.1151 
14.3692 
(a) 
(b) 
(c) 
Fig. 6 Effect of CNTs volume fraction on the dimensionless buckling load of simply supported FGCNTRC plate ( ) with different elastic foundation coefficients for (a) , (b) and (c) . 
(a) 
(b) 
(c) 
Fig. 7 Effect of widthtothickness ratio on the dimensionless buckling load of simply supported FGCNTRC plate ( ) with different elastic foundation coefficients for (a) , (b) and (c) . 
(a) 
(b) 
(c) 
Fig. 8 Effect of widthtolength ratio on the dimensionless buckling load of simply supported FGCNTRC plate
Fig. 8 illustrate the effect of widthtolength ratio on the buckling load of UD and FGX nanocomposite plate with and without presence of elastic foundation for loading types of (a) , (b) and (c) . Obviously, with increase of aspect ratio the buckling load decreases for both loading types of uniaxial compressive load and compressivetensile load and the decrease rate is greater in compressivetensile loading compared to uniaxial compressive case. But, against the mentioned loading types, increase of aspect ratio raises the buckling load gradually. It is also observed that the effect of Winkler module on the buckling load is much greater than the shear elastic foundation. 
Based on incremental load technique a new method is presented for buckling analysis of singlewalled carbon nanotube reinforced composite plates resting on an elastic foundation and subjected to different tensilecompressive loads. A uniform and three kinds of functionally graded distributions of CNTs along the thickness direction of plate are considered.
The properties of composite material in each point were determined by modified rule of mixture. All governing equations were obtained incrementally based on first order shear deformation theory (FSTD) and Von Karman nonlinear strains. Using the principle of minimum potential energy, the set of coupled nonlinear equilibrium equations was obtained in incremental form for different boundary conditions. The dynamic relaxation (DR) method combined with the finite difference discretization technique is employed to find the critical buckling load for simply supported and clamped boundary conditions. Some major inferences are as follows:
It is seen that increasing the thickness of nanocomposite plate has the more effect on the buckling load for FG distribution of CNTs compared to uniform ones.
For all types of loading and both cases of with and without presence of elastic foundation the FGX and FGO have the highest and lowest values of buckling loads, respectively, for both SSSS and CCCC boundary conditions.
For both types of with and without elastic foundation, CNTs distributions and boundary conditions, increasing the volume fraction of CNTs from 0.12 to 0.17 causes a significant increase of buckling load. However, by increasing the volume fraction of CNTs from 0.17 to 0.28 the raising rate of buckling load decreases.
Nomenclature
a, b, h 
length, width and thickness of CNTRC plate 
elasticity modulus and shear modulus of CNTRC plate 

elasticity modulus and shear modulus of matrix 

elasticity modulus and shear modulus of carbon nanotubes 

volume fractions of substrate and carbon nanotubes 

CNT efficiency parameters 

mass fraction of carbon nanotubes in composite plate 

densities of substrate and carbon nanotubes 

Poisson’s ratio of CNTRC plate, substrate and CNT 

displacement components in x, y and z directions, respectively 

displacement components of the mid–plane 

rotations of transverse normal about and axes 

incremental strain components 

incremental stress components 

Plane stressreduced stiffnesses 

incremental forces, moments and shear stress resultants 

extensional, coupling, bending, and shear stiffness 

transverse shear correction coefficient 

Winkler and shear coefficients of foundation parameters 
References
[1] Thostenson, E. T., Ren, Z., Chou, T.W., 2001. Advances in the science and technology of carbon nanotubes and their composites: a review. Composites Science and Technology, 61(13), pp.18991912
[2] Fiedler, B., Gojny, F. H., Wichmann, M. H., Nolte, M. C., Schulte, K., 2006. Fundamental aspects of nanoreinforced composites. Composites Science and Technology, 66(16), pp.31153125.
[3] Fidelus, J., Wiesel, E., Gojny, F., Schulte, K., Wagner, H., 2005. Thermomechanical properties of randomly oriented carbon/epoxy nanocomposites. Composites Part A: Applied Science and Manufacturing, 36(11), pp.15551561.
[4] Anumandla, V., Gibson, R. F., 2006. A comprehensive closed form micromechanics model for estimating the elastic modulus of nanotubereinforced composites. Composites Part A: Applied Science and Manufacturing, 37(12), pp.21782185.
[5] Han, Y., Elliott, J, 2007. Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites. Computational Materials Science, 39(2), pp.315323.
[6] Cooper, C. A., Cohen, S. R., Barber, A. H., Wagner, H. D., 2002. Detachment of nanotubes from a polymer matrix. Applied Physics Letters, 81(20), pp.38733875.
[7] Barber, A. H., Cohen, S. R., Wagner, H. D., 2003. Measurement of carbon nanotube–polymer interfacial strength. Applied Physics Letters, 82(23), pp.41404142.
[8] Gou, J., Minaie, B., Wang, B., Liang, Z., Zhang, C., 2004. Computational and experimental study of interfacial bonding of singlewalled nanotube reinforced composites. Computational Materials Science, 31(34), pp.225236.
[9] Frankland, S., Caglar, A., Brenner, D., Griebel, M., 2002. Molecular simulation of the influence of chemical crosslinks on the shear strength of carbon nanotubepolymer interfaces. The Journal of Physical Chemistry B, 106(12), pp.30463048.
[10] Ma, P.C., Mo, S.Y., Tang, B.Z., Kim, J.K., 2010. Dispersion, interfacial interaction and reagglomeration of functionalized carbon nanotubes in epoxy composites. Carbon, 48(6), pp.18241834.
[11] Coleman, J. N., Khan, U., Blau, W. J., Gun’ko, Y. K., 2006. Small but strong: a review of the mechanical properties of carbon nanotubepolymer composites. Carbon, 44(9), pp.16241652.
[12] Wuite, J., Adali, S., 2005. Deflection and stress behaviour of nanocomposite reinforced beams using a multiscale analysis. Composite Structures, 71(34), pp.388396.
[13] Vodenitcharova, T., Zhang, L., 2006. Bending and local buckling of a nanocomposite beam reinforced by a singlewalled carbon nanotube. International journal of solids and structures, 43(10), pp.30063024.
[14] Formica, G., Lacarbonara, W., Alessi, R., 2010. Vibrations of carbon nanotubereinforced composites. Journal of Sound and Vibration, 329(10), pp.18751889.
[15] Arani, A. G., Maghamikia, S., Mohammadimehr, M., Arefmanesh, A., 2011. Buckling analysis of laminated composite rectangular plates reinforced by SWCNTs using analytical and finite element methods. Journal of Mechanical Science and Technology, 25(3), pp.809820.
[16] Shen, H.S., 2009. Nonlinear bending of functionally graded carbon nanotubereinforced composite plates in thermal environments. Composite Structures, 91(1), pp.919.
[17] Zhu, P., Lei, Z., Liew, K. M., 2012. Static and free vibration analyses of carbon nanotubereinforced composite plates using finite element method with first order shear deformation plate theory. Composite Structures, 94(4), pp.14501460.
[18] Ke, L.L., Yang, J., Kitipornchai, S., 2010. Nonlinear free vibration of functionally graded carbon nanotubereinforced composite beams. Composite Structures, 92(3), pp.676683.
[19] Shen, H.S., Zhang, C.L., 2010. Thermal buckling and postbuckling behavior of functionally graded carbon nanotubereinforced composite plates. Materials and Design, 31(7), pp.34033411.
[20] Mohammadi, M., Arefi, M., Dimitri, R., Tornabene, F., 2019. Higherorder thermoelastic analysis of FGCNTRC cylindrical vessels surrounded by a Pasternak foundation. Nanomaterials, 9(1), 79.
[21] Mohammadi, M., Arefi, M., Ahmadi, S. A., 2020. Twodimensional electroelastic analysis of FGCNTRC cylindrical laminated pressure vessels with piezoelectric layers based on thirdorder shear deformation theory. Journal of Pressure Vessel Technology, 142(2), 021304.
[22] Wang, Z.X., Shen, H.S.,2011. Nonlinear vibration of nanotubereinforced composite plates in thermal environments. Computational Materials Science, 50(8), pp.23192330.
[23] Arefi, M., Mohammadi, M., Rabczuk, T., 2019. Effect of characteristics and distribution of porosity on electroelastic analysis of laminated vessels with piezoelectric facesheets based on higherorder modeling. Composite Structures, 225, 111085.
[24] Reddy, J. N., 2004. Mechanics of laminated composite plates and shells: theory and analysis. CRC press.
[25] Kadkhodayan, M., Zhang, L., Sowerby, R., 1997. Analyses of wrinkling and buckling of elastic plates by DXDR method. Computers and Structures, 65(4), pp.561574.
[26] Zhang, L., Kadkhodayan, M., Mai, Y.W., 1994. Development of the maDR method. Computers and Structures, 52(1), pp.18.
[27] Underwood, P., 1983. Dynamic relaxation in computational method for transient analysis. Amsterdam: Elsevier, 245265.
[28] Lei, Z., Liew, K., Yu, J., 2013. Buckling analysis of functionally graded carbon nanotubereinforced composite plates using the elementfree kpRitz method. Composite Structures, 98, pp.160168.