Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 55, 3, pp. 883-896, Warsaw 2017 DOI: 10.15632/jtam-pl.55.3.883 NONLOCAL ANALYSIS OF SINGLE AND DOUBLE-LAYERED GRAPHENE CYLINDRICAL PANELS AND NANO-TUBES UNDER INTERNAL AND EXTERNAL PRESSURES CONSIDERING THERMAL EFFECTS Shahriar Dastjerdi Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University, Mashhad, Iran e-mail: dastjerdi shahriar@yahoo.com Massoumeh Lotfi, Mehrdad Jabbarzadeh Department of Mechanical Engineering, Mashhad branch, Islamic Azad University, Mashhad, Iran Mechanical behavior of a bilayer graphene cylindrical panel and nano-tube is studied based on nonlocal continuum mechanics with regard to this aim, von-Karman assumptions and nonlocal theoryofEringenare considered.Then, the governing equations andboundary con- ditions have been derived applying energy method. While analyzing the bilayer cylindrical panel, the van der Waals interaction between the layers is considered in calculations. The constitutive equations are developed for nano-tubes under internal and external pressures. In order to solve the governing equations, the semi-analytical polynomial method (SAPM), whichwas presented by the authors before, is utilized and bending behavior of bilayer cylin- drical panels and nano-tubes is investigated. Finally, the effects of temperature, boundary conditions, elastic foundation, loading, van der Waals interaction between the layers and single layer to bilayer analyses are studied for graphene cylindrical panels and nano-tubes. Keywords: cylindrical graphene panels, nano-tube, nonlocal elasticity theory, first-order she- ar deformation theory 1. Introduction Shells and panels are the most frequently applied engineering structures and are categorized in several groups. The curvature of shells is one basis of the categorization which is most si- gnificant in geometric shapes (cylindrical, conical, and spherical, etc.) and related equations. Another basis of the categorization is thickness of the shell, which divides into two groups of low thickness (shell) andhigh thickness (plate), for the thickness-to-radius ratio of (h/R)max ¬ 1/20, it is assumed to be a low thickness shell. According to ever-increasing application of shells in industries, the significance of studying behavior of shell structures in different cases of loading includingbendingandbucklingbecomesmorepronounced.One interestingnano structurewhich scientifics are devoted to is the carbon nano-tube. The tubes were discovered in 1991 (Iijima, 1991). Single wall carbon nano-tubes can be assumed as 2D single layer graphene sheets which are rolled. The extraordinary attributes of graphene sheets, carbon nano-tubes and their exclu- sive specifications made them a reason to progress and renovation of difference researches and nano-science (Tans et al., 1997; Martel et al., 1998; Postma et al., 2001). Experimental tests in nano scale are difficult and expensive. Therefore, developing a suitable mathematical model for nano structures is important. Continuummechanics model is an appropriate and a low-cost solution. Disadvantages of the continuummethod is that it is not possible to estimate the exact small scale effects. When the size of structures is small, the effects of small scale on mechanical behavior are very important. To eliminate the disadvantage in the classic continuum model, Eringen and Edelen (1972) presented a nonlocal elasticity theory and modified the classical continuum mechanics to consider the effects of small size. By considering the effects of size in 884 S. Dastjerdi et al. related equations asmaterial properties, the nonlocal theory of Eringen is extremely effective in presenting an exact and appropriatemodel of nano structures. Hence to study behavior of nano plates in many presented studies, the nonlocal theory is applied. Murmu and Pradhan (2009) applied the theory of nonlocal elasticity to rectangular single layer graphenesheets in elastic environment.TheyappliedbothmodelsofWinkler andPasternak to simulate reactions of graphene sheets with their elastic environment. Reddy andPang (2008) studied bending and buckling of carbon nano-tubes based on nonlocal theory. Aghababaei and Reddy (2009) investigated bending and free vibration equations based on the third-order shear deformation theory/Thenonlocal effectswere then solved analytically for a isotropic rectangular sheet with simple boundary conditions. They concluded that two first order and third order shear deformation theories had similar results while related results from nonlocal theory of thin shell were difference. Pradhan (2009) analyzed buckling of isotropic graphene rectangular plates based on the third-order shear deformation theory and the Navier method. He concluded that by increasing the small scale effects or decreasing length of the plate, the ratio of nonlocal buckling loads to the local case would be increased. The first solution of the buckling problem of cylindrical shells under uniform lateral pressurewas presented byBrush andAlmroth (1975). Then many extensive researches on buckling of cylindrical shells by different materials and different boundary conditions have been done. Hoff and Soong (1965) studied buckling of a cylindrical shell under axial load in different conditions. They showed that if the ratio of shell length to its thickness was large enough, the value of stress in buckling mode would always be independent from length of the shell and the maximum critical stress would occurr for simply supported boundary conditions. Linghai et al. (2008) considered buckling of a cylindrical panel under axial load by a differential quadrature method in which the load was perpendicular to the surface. Zhao and Liew (2009) analyzed a cylindrical panel made of a functionally graded material (FGM) under thermal and dynamic loads based on the first order nonlinear shear deformation theory by applying the free element method. Nguyen and Hoang (2010) studied stability of a cylindrical panel made of a functionally graded material under axial pressure and simple supported boundary conditions. It was found that by increasing the ratio of length to radius of the cylindrical panel, critical load always increased.Khazaeinejad et al. (2010) analyzed buckling of cylindrical shells made of a functionally gradedmaterial under both axial loads and internal pressure by using the first order shear deformation theory. They concluded that by decreasing the power factor of material properties, critical load increased. Lancaster et al. (2004) investigated theeffect of imperfectionson thebuckling load in the form of local initial stress,whichwereprobablymore typical in practice than inpurely geometric ones. Bisagni andCordisco (2003) experimentally studied buckling and post-buckling behavior of four unstiffened thin-walledCFRPcylindrical shells forwhich the test equipment allowed application of axial and torsional loadings. Their results identified the effect of laminate orientation, showed that the buckling loadswere essentially independent of the load sequence anddemonstrated that the shells were able to sustain load in the post-buckling field without any damage. Degenhardt et al. (2010) performed buckling tests and buckling simulations on CFRP cylindrical shells to investigate the imperfection sensitivity and to validate the applied simulation methodologies. Fazelzadeh andGhavanloo (2014) purused vibration characteristics of curved graphene ribbons (CGRs) embedded in an orthotropic elastic shell and investigated that there was significant dependence of natural frequencies on the curvature change. Biswas (2014) presented nonlinear analysis of plate and shell structures under mechanical and thermal loadings. Ansari et al. (2016) studied size-dependent nonlinear mechanical behavior of third-order shear deformable functionally graded microbeams using the variational differential quadrature method. They used the gradient elasticity theory in their study. Zhang et al. (2016) applied FSDT element free IMLS-Ritz method to analyze free vibration of triangular CNT-reinforced composite plates subjected to in-plane stresses. Nonlocal analysis of single and double-layered graphene cylindrical panels... 885 Dastjerdi et al. (2016a) derived constitutive equations for graphene plates in Cartesian and cylindrical coordinate systems based on the nonlocal first and higher order shear deformation theories. Zhang et al. (2015) presentedanonlocal continuummodel for vibration of single-layered graphene sheets based on the element-free kp-Ritz method. All the mentioned solving methods are popular in analyzing the small scale effects of nano structures. Continuing, Dastjerdi et al. (2016b,c) worked on static analysis of single layer annular/circular graphene sheets (Dastjerdi et al., 2016b) and the effect of vacant defects on bending analysis of circular graphene sheets (Dastjerdi et al., 2016c) based on the nonlocal theory of Eringen. They presented a new semi- analytical polynomial method (SAPM) for solving ordinary and partial differential equations. In this paper, for the first time, the nonlinear bending analysis of bilayer graphene cylindrical panels and nano-tubes has been studied. Nonlocal constitutive equations have been derived and solved applying SAPM (Dastjerdi et al., 2016c). It has been tried to consider all effective parameters on study such as boundary conditions, nonlocal parameter, temperature, loading, elastic foundation, etc. 2. Nonlocal theory Different non-classical theories have been presented to analyze structures in nano and micro scales such as strain-gradient theories, couples stress and the nonlocal theory of Eringen.With regard to considering the simplicity of the equations and satisfactory results of the nonlocal theory, this theory has been applied by many researchers to analysis of mechanical behavior of nano structures. In other words, this theory has been applied in the present article, considering its advantages. According to the nonlocal theory of Eringen, stress in a specified point of surface is a function of strain in all points of the surface. As below in the integral equation, it is shown as a function of local stress (Eringen and Edelen, 1972) σij = ∫ V α(|χ′−χ|)σLij(χ ′) dV (χ′) (2.1) The integral must be calculated on the total surface. χ is a point whose stress can be calculated according to stress of all χ′ points of the structure.The indices i, j are the same for x, y, z of the shell coordinate components. The distance between χ and χ′ can be presented by |χ′−χ| and α(|χ′ −χ|), which are nonlocal weight functions. σLij and σ NL ij are local and nonlocal stresses, respectively. Finally, Eq. (2.1) can be expressed for the 2D case as (1−µ∇2)σNLij = σ L ij µ =(e0a) 2 ∇2 = ∂2 ∂x2 + ∂2 ∂y2 (2.2) In theabove,a is related to internal dimensionsand small size, e0 is thematerial coefficientwhich results from experimental studies. The value of the nonlocal parameter µ is not specified exactly and is dependent on boundary conditions, mode number, layer quantity and type of movement (Eringen and Edelen, 1972). Wang and Wang (2007) illustrated that the value of the nonlocal parameter (e0a) is between 0 and 2nm. Consequently, many researchers have considered this range of the nonlocal parameter in their investigations. As a result, in this paper, the nonlocal parameter is assumed between 0 and 2nm, too. 3. Governing equations A bilayer graphene cylindrical panel is considered as shown in Fig. 1, resting on a Winkler- -Pasternak elastic mediumunder uniform transverse loading q. TheWinkler-Pasternak stiffness coefficients are kw andkp, respectively.Winkler considers the elastic foundation to be completely 886 S. Dastjerdi et al. Fig. 1. Schematic view of a bilayer graphene cylindrical panel rested onWinkler-Pasternak elastic foundation made of linear springs (P = kww). If the mentioned foundation is affected by a transverse loading q, the springs would not be influenced beyond the loaded domain. In the Pasternak elastic matrix, the effect of shear force among the spring elements is consi- dered,which is completed by connecting the ends of the springs to the plate that only undergoes transverse shear deformation. The relationship between the load and deflection is obtained by assuming the vertical equilibrium of a shear layer (P = kww+kp∇ 2w). The amount of van der Waals interaction depends on the distance between the layers. ko and k ∗ o can be introduced as a stiffness of spring force (van der Waals interaction) as ko(w2−w1)+k ∗ o(w2−w1) 3 in which w1 represents defection of the upper layer and w2 – the bottom layer. In practical investiga- tions, the values of w1 and w2 are close to each other. So, the term (w2−w1)3 is extremely insignificant. As a result, the non-linear term can be neglected for simplicity of calculations, and the van derWaals interaction can be considered linear as ko(w2−w1). In this study, the constitutive equations arederivedbasedonthefirst-order sheardeformation theory (FSDT) in which the neglected assumptions in the classical plate theory are considered with more accuracy. According to the first-order shear deformation theory, the displacement field can be written as follows Ui = ui+zϕi V i = vi+zψi Wi = wi (3.1) where ui, vi and wi are the displacement components of the mid-plane along the x, y and z directions, respectively. ϕi and ψi refer to the rotation functions of the transverse normal to y and x directions. The index i = 1,2 refers to the upper and bottom layers, respectively. The strain filed is then as below Nonlocal analysis of single and double-layered graphene cylindrical panels... 887 εix = ∂ui ∂x + 1 2 (∂wi ∂x )2 +z ∂ϕi ∂x −α∆T εiy = wi R + ∂vi ∂y + 1 2 (∂wi ∂y )2 +z ∂ψi ∂y −α∆T γixy = ∂vi ∂x + ∂ui ∂y + (∂wi ∂x ∂wi ∂y ) +z (∂ϕi ∂y + ∂ψi ∂x ) γixz = ϕi+ ∂wi ∂x γiyz = ψi+ ∂wi ∂y − vi R (3.2) Now to obtain equilibriumequations, an energymethod is applied inwhich variation of potential energy for external loads and strain energy must be zero δπ = δU +δV =0 δV = ∫∫ A (q −kww0) δw0dA (3.3) and δU = ∫∫∫ A (σijδεij) dv (3.4) Equation (3.4) describes the strain energy which can be expanded as follows δU = ∫∫∫ A (σixδεix+σiyδεiy +σixyδεixy +σiyzδγiyz +σixzδγixz) dv i =1,2 (3.5) The stress andmoment resultants are specified as follows [Nix,Niy,Nixy] NL = h/2 ∫ −h/2 [σix,σiy,σixy] NL dz [Mix,Miy,Mixy] NL = h/2 ∫ −h/2 [σix,σiy,σixy] NLz dz [Qiy,Qix] NL = ks h/2 ∫ −h/2 [σiyz,σixz] NL dz (3.6) where ks is the shear factor correction coefficient for the first-order shear deformation theory, which is taken 5/6. Now by substituting Eqs. (3.6) into Eq. (3.3) and integrating with respect to z, Eq. (3.3) can be developed. The obtained total energy equation must be equal to zero. The equilibrium equations are based on the nonlocal theory of Eringen. Applying Eq. (2.2) into the resultants (Eqs. (3.3)) and then substituting into the nonlocal governing equations, the constitutive equations can be obtained in the local form of the resultants considering the effects of the nonlocal parameter. Dastjerdi et al. (2016a) explained the process of obtaining the local form of governing equations from nonlocal ones. Finally, the nonlocal constitutive equations for the bilayer graphene cylindrical panel can be expressed as follows (some inconsiderable terms are neglected) ∂NiLx ∂x + ∂NiLxy ∂y =0 ∂NiLy ∂y + QiLy R + ∂NiLxy ∂x =0 ∂MiLx ∂x + ∂MiLxy ∂y −QiLx =0 ∂MiLy ∂y + ∂MiLxy ∂x −QiLy =0 (3.7) 888 S. Dastjerdi et al. —upper layer − N1Ly R + ∂Q1Ly ∂y + ∂Q1Lx ∂x +(1−µ∇2) [ N1Lx ∂2w1 ∂x2 +2N1Lxy ∂2w1 ∂x∂y +N1Ly ∂2w1 ∂y2 +q −ko(w2−w1) ] =0 (3.8) — bottom layer − N2Ly R + ∂Q2Ly ∂y + ∂Q2Lx ∂x +(1−µ∇2) [ N2Lx ∂2w2 ∂x2 +2N2Lxy ∂2w2 ∂x∂y +N2Ly ∂2w2 ∂y2 +ko(w2−w1)+(−kww2+kp∇ 2w2) ] =0 (3.9) 4. Boundary conditions All possible boundary conditions have been assumed as simply supported (S), clamped (C) and free (F) edges, which can be written as follows (for the first line 0 < x < Lx and for the second line 0 < y < Ly) S : ui = vi =wi = ψi = Mix =0 ui = vi =wi = ϕi = Miy =0 C : ui = vi = wi = ϕi = ψi =0 ui = vi = wi = ϕi = ψi =0 F : Nix = Nixy = Qix = Mix = Mixy =0 Nix = Nixy = Qiy = Miy = Mixy =0 (4.1) 5. Solution Taking a look on the governing equations, it is clear that a system of nonlinear partial differen- tial equations has been obtained whose solution is not possible to be determined by common analytical methods. In the paper, a new semi-analytical polynomialmethod (SAPM) is applied, which was presented before by the authors (Dastjerdi et al., 2016c). The SAPM formulation is extremely simple and its accuracy was proved before (Dastjerdi et al., 2016c). According to the mentioned explanations for SAPM, to achieve a solution to the governing equations, the below displacements and rotations can be introduced (Dastjerdi et al., 2016c) (i = 1,2 for the upper and bottom layers) ui = N ∑ k=1 M ∑ t=1 ai{k+t−[1−(k−1)(M−1)]}x (k−1)y(t−1) vi = N ∑ k=1 M ∑ t=1 ai{k+t−[1−(k−1)(M−1)]+MN}x (k−1)y(t−1) wi = N ∑ k=1 M ∑ t=1 ai{k+t−[1−(k−1)(M−1)]+2MN}x (k−1)y(t−1) ϕi = N ∑ k=1 M ∑ t=1 ai{k+t−[1−(k−1)(M−1)]+3MN}x (k−1)y(t−1) ψi = N ∑ k=1 M ∑ t=1 ai{k+t−[1−(k−1)(M−1)]+4MN}x (k−1)y(t−1) (5.1) Nonlocal analysis of single and double-layered graphene cylindrical panels... 889 By considering Eqs. (5.1) in constitutive equations (3.7)-(3.9), the partial differential equations will be transformed intoalgebraic equations.Theobtainedequations canbesolvedbyconsidering boundary conditions. 6. Numerical results and discussion Thecurvature radiusof thegraphenepanelhasbeenassumedto tendto infinity inorder toobtain a rectangular plate. Then, the results are compared with those by Dastjerdi and Jabbarzadeh (2016). According to Fig. 2, it can be seen that the results are close to each other. The small differences occurred because of tending the radius of the cylindrical panel to infinity.Also, it can be concluded that the differences increase for more flexible boundary conditions. However, the maximum difference is about 2.5%, which is insignificant. Consequently, the obtained results of DQM (Dastjerdi and Jabbarzadeh, 2016) are in compliance with the results of SAPM in this paper. Fig. 2. Comparison between the results of the present paper andDastjerdi and Jabbarzadeh (2016) Table 1 indicates the effect of temperature on the results for different types of boundary conditions. It is observed that in higher temperature, the maximum deflection increases. The variations are approximately linear and there is not a considerable differencebetweenCCCCand SSSS boundary conditions. The variations are the same. Also, it is observed that by increasing the nonlocal coefficient, the deflection decreases and the rate of variations remains constant. In CCCCboundaryconditions, by increasing thenonlocal effect, thedeflection decreases faster, but in SSSS boundary conditions, the variations are small. For example, deflections for e0a =1nm and e0a = 2nm are not considerably different, but in CCCC boundary condition, they are different. Itmay result that in SSSSboundaryconditions, due to itsmoreflexibility, temperature raise versus an increase in the small-scale effects is more than in CCCC boundary conditions, and by increasing e0a, the deflection has a less descending rate. The properties of the panel are presented below Lx =5nm θ = π h =0.34nm R =5nm Ex = Ey =1.06TPa νxy = νyx =0.3 q =0.5GPa kw =1.13GPa/nm kp =1.13Pam α =2.02 ·10 −6Co−1 (6.1) 890 S. Dastjerdi et al. Table 1.Effect of temperature on nonlocal analysis ∆T w [nm] [C◦] e0a =0nm e0a =1nm e0a =2nm Rm = w∗ e0a=1nm w∗ e0a=0nm Rm = w∗ e0a=2nm w∗ e0a=0nm 0 0.0307 0.0294 0.0249 0.9557 0.8086 CCCC 100 0.0321 0.0307 0.0262 0.9576 0.8156 500 0.0374 0.0362 0.0316 0.9653 0.8441 1000 0.0443 0.0432 0.0390 0.9746 0.8800 0 0.0326 0.0318 0.0311 0.9769 0.9560 SSSS 100 0.0340 0.0332 0.0325 0.9782 0.9572 500 0.0396 0.0389 0.0381 0.9822 0.9616 1000 0.0466 0.0461 0.0451 0.9873 0.9657 To survey the effect ofWinkler-Pasternak elastic foundation,Tables 2 and 3present different types of boundary conditions. In both tables, higher values of elastic the foundation stiffness lead to smaller deflections. As mentioned before, SSSS boundary conditions have more effects on the results, as seen in Table 1. For example, in Table 2, it is considered that by increasing e0a in CCCC boundary conditions, the normal falling rate is available, but in SSSS boundary conditions, the falling rate is smaller. Thementioned variations can be observed inTable 3, too, but it seems that the effect of Pasternak elastic foundation is more evident. Basically, Winkler and Pasternak elastic foundations have a fairly important role in the analysis. It is due to the direct effect of the nonlocal coefficient on the elasticity ofWinkler andPasternak foundations in Eq. (3.9). Consequently, the availability and their values are important in the small scale effect when the nonlocal elasticity theory is applied. Table 2.Effect ofWinkler elastic foundation on the results kw w [nm] [GPa/nm] e0a =0nm e0a =1nm e0a =2nm Rm = w∗ e0a=1nm w∗ e0a=0nm Rm = w∗ e0a=2nm w∗ e0a=0nm 0 0.0335 0.0336 0.0330 1.0020 0.9850 CCCC 1 0.0316 0.0311 0.0288 0.9841 0.9114 5 0.0259 0.0238 0.0188 0.9189 0.7259 10 0.0210 0.0183 0.0131 0.8714 0.6238 0 0.0354 0.0363 0.0377 1.0254 1.0649 SSSS 1 0.0333 0.0335 0.0325 1.0060 0.9759 5 0.0268 0.0255 0.0209 0.9514 0.7798 10 0.0214 0.0195 0.0144 0.9112 0.6729 The van derWaals force is a kind of force which depends on the distance between the layers (Dastjerdi and Jabbarzadeh, 2016). As the distance between two layers decreases, the van der Waals force increases. Figure 3a is drawn for CCCCboundary conditions, and Fig. 3b for SSSS boundaryconditions.According to thefigures, it is observed thatby increasingk0, thedeflections of two layers approach each other, and after a specified value, an increase in k0 has no impact on the results. In both Figs. 3a and 3b, also it is observed that by increasing the nonlocal parameter, the deflections of both layers approach each other faster than before. Therefore, increasing the nonlocal effect results in an increase in van der Waals interaction between the layers. In Fig. 3b, it can be concluded that by increasing small-scale effects, more variation results rather than in CCCC boundary conditions. However, the variations are approximately similar and no considerable change is observed. In this case, the elastic foundation is available and the variations in both cases of CCCC and SSSS boundary conditions are approximately similar. Nonlocal analysis of single and double-layered graphene cylindrical panels... 891 Table 3.Effect of Pasternak elastic foundation on the results kp w [nm] [Pam] e0a =0nm e0a =1nm e0a =2nm Rm = w∗ e0a=1nm w∗ e0a=0nm Rm = w∗ e0a=2nm w∗ e0a=0nm 0 0.0335 0.0336 0.0330 1.0029 0.9851 CCCC 1 0.0328 0.0321 0.0289 0.9786 0.8811 5 0.0302 0.0266 0.0182 0.8808 0.6026 10 0.0273 0.0216 0.0122 0.7912 0.4469 0 0.0354 0.0363 0.0377 1.0254 1.0650 SSSS 1 0.0349 0.0360 0.0367 1.0315 1.0516 2 0.0345 0.0355 0.0347 1.0289 1.0058 10 0.0302 0.0299 0.0251 0.9900 0.8311 Fig. 3. Variation of deflection versus van derWaals interaction between the layers for (a) CCCC, (b) SSSS boundary conditions Fig. 4. The effect of van derWaals interaction on the ratio Rb 892 S. Dastjerdi et al. For better perception, Fig. 4 shows variation of the ratio between the deflections of the bottom layer to the upper layer Rb versus an increase in the van der Waals force. It is clearly observed that at the beginningof risingk0, the rate of variations is elevated, and in the following, it decreases.Totally, variations inSSSSboundaryconditions are greater than inCCCCboundary conditions. This observation results from Figs. 3a and 3b too. Also, by increasing the nonlocal effects,Rbbecomesgreater. Inone specifiedk0,which in this case is about10GPa/nm,Rbhas the maximumdistance fromother ones in four cases of boundaryconditions andnonlocal coefficients. An increase in radiusof thenanographeneshell (R)makes curvature lower andthecylindrical shell transforms into a flat rectangular sheet. Figure 5 shows the effects of increasing R on the results. By increasing R, the deflection grows and the rate of variation is high. Continuing, the rate becomes less and remains constant. An infinite value of R means that the cylindrical shell is transformed into a flat rectangular plate. Therefore, by bending the rectangular plate, its deflection subsequently decreases. In SSSS boundary conditions, due to its flexibility, the deflection variations due to R are slow. Also the effect of nonlocal analysis in CCCC boundary conditions is greater than in SSSS boundary conditions. According to Fig. 5, it is observed that when R rises, the value of Rm in CCCC boundary conditions decreases more. In this case, the nonlocal effects on the results are more evident (the specifications of panel are similar to Eq. (6.1), only Ly =5nm and ∆T =0). Fig. 5. Deflection due to the increase of radius R of the cylindrical panel As told before, the nonlocal analysis is more effective in small scales. In this regard, Fig. 6a and 6b are drawn. It is observed that by increasing length of the graphene shell, the deflection increases too, and then, the rate of ascending becomes less and remains fixed at one specified valuewhose further increase in size has no any considerable impact on the results. It is observed that the increasing length results in smaller nonlocal effects, and the effects become greater by decreasing the size. Increasing the size leads to a rise in the ascending rate of reduction of the nonlocal effect, and this conclusion is clearly obvious for higher values of e0a. Generally, in small sizes, the nonlocal effects are considerable and nonlocal analysis must be applied. But in larger sizes, the classical theory gives acceptable results by considering a suitable approximation. So, the classical theory is applied for the sizes greater than the nano scale. Now a nano-tube under internal pressure is studied. This case could be assumed as a shell with radius R whose curvature between both ending edges is 2π in θ direction, so that the Nonlocal analysis of single and double-layered graphene cylindrical panels... 893 Fig. 6. Variations of deflection versus of the length shell Lx for (a) CCCC, (b) SSSS boundary conditions edges are connected to each other. Nano-tubes can be usedwidely. For example, to transfer fluid through nano-tubes, the relatedmechanical properties must be specified. These nano-tubes can be applied in medical purposes such as nano vessels. In this study, the value of expansion and contraction of a nano graphene tube under internal and external pressure are surveyed. At first, a bi-layer nano-tube is studied, whose layers are connected to each other by van derWaals interaction. The ratio of deflection of thebottom layer to the upper layer is definedasRb. Now, a bilayer nano-tube is assumedwith the specifications given below (Rb = wbottom layer/wupper layer) Lx =15nm h =0.34nm Ex = Ey =1.06TPa νxy = νyx =0.3 q =0.5GPa kw =1.13GPa/nm p =1.13Pam ∆T =0 e0a =1nm (6.2) Fig. 7. Variation of nano-tube radius on Rb for different van derWaals interactions Figure 7 shows the effect of increasing radius of the nano-tube on its expansion Rb for different values of van derWaals forces. It is observed that by increasing the value of R, it leads to an increase in Rb too, and in the following, its raising rate lowers and, finally, an increase 894 S. Dastjerdi et al. in R, does not have any considerable effect on variation of Rb any longer. The obtained results are similar with the related results for the shell type studied in Fig. 5. As it is expected, an increase in the van derWaals force has direct effect on the deflection of two bottom and upper layers. By increasing the van derWaals force, these deflections approach each other faster. To reinforce single layer nano-tubes, bilayer nano-tubes canbeapplied.To survey the effect of beingbilayer nano-tubes on the results, Fig. 8 is presented (Rbs = wbilayer/wsingle layer). Figure 8 shows variation of Rbs versus nano-tube radius. As it is observed, by increasing radius R, the deflections of both bilayer and single layer nano-tubes approach each other, but for small values of radius R, the deflection of the bilayer sheet is considerably less than the single layer one. In this case, strength of the bilayer sheet is greater than the single layer one, but for a larger radius, the strength becomes smaller and the deflection of the bilayer sheet approaches the single layer sheet. Therefore, a single layer nano-tube can be used instead of nano-tubewith R radius. Figure 8 is very useful for selection a nano-tube based on the single layer or bilayer one. The figure could be drawn for different conditions, and according to the requirements of the problem, the design must be chosen (ko =45GPa/nm) Fig. 8. Single layer to bilayer Rbs versus variations of nano-tube radius 7. Conclusions In thispaper, for thefirst time, themechanical behavior of agraphenecylindricalpanel andnano- -tube under internal and external pressure is studied. The system is embedded in an elastic matrix and the effect of thermal conduction has been taken into account too. Remarkable conc- lusions can be listed as follows: • An increase in the nonlocal effect results in an increase in van der Waals interaction between the layers. • The small scale effect decreases along the increase of temperature. • By increasing nonlocal effects, the deflections of the upper and bottom layers approach each other. • Whatever radius R of the shell is smaller, the nonlocal decrease effects. So, the maximum nonlocal effects occur for rectangular plates (when radius R tends to infinity). Nonlocal analysis of single and double-layered graphene cylindrical panels... 895 • An increase in size leads to a rise in the ascending rate of reduction of the nonlocal effect. • Whatever the radius of nano-tubeR rises, the deflection of two layers approach each other. • The strength of the bilayer nano-tubedecreases due to an increase in radius.Consequently, for a high range of radius R, a single layer nano-tube can be applied instead of a bilayer one. Acknowledgement I want to dedicate this paper to dr. Jabbarzadeh for his help andmy father Reza Dastjerdi who has supportedme in every second of my life. References 1. Aghababaei R., Reddy J.N., 2009, Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates, Journal of Sound and Vibration, 326, 277-289 2. Ansari R., Faghih Shojaei M., Gholami R., 2016, Size-dependent nonlinear mechanical be- havior of third-order shear deformable functionally gradedmicrobeams using the variational diffe- rential quadraturemethod,Composite Structures, 136, 669-683 3. Bisagni Ch., Cordisco P., 2003, An experimental investigation into the buckling and post- -buckling of CFRP shells under combined axial and torsion loading, Composite Structures, 60, 391-402 4. Biswas P., 2014, Thermal stresses, deformations and vibrations of plates and shells – a nonlinear approach,Procedia Engineering, 144, 1023-1030 5. Brush D.O., Almroth B.O., 1975,Buckling of Bars, Plate and Shells, McGraw hill, NewYork 6. Dastjerdi Sh., Aliabadi Sh., Jabbarzadeh M., 2016a, Decoupling of constitutive equations formulti-layered nano-plates embedded in elasticmatrix based on non-local elasticity theory using first and higher order shear deformation theories, Journal of Mechanical Science and Technology, 30, 1253-1264 7. DastjerdiSh., JabbarzadehM., 2016,Nonlinearbendinganalysisofbilayerorthotropicgraphe- ne sheet resting onWinkler-Pasternak elastic foundation based on non-local continuummechanics, Composites Part B, 87, 161-175 8. Dastjerdi Sh., JabbarzadehM.,Aliabadi Sh., 2016b,Nonlinear static analysis of single layer annular/circular graphene sheets embedded inWinkler-Pasternakelasticmatrix based onnon-local theory of Eringen,Ain Shams Engineering Journal, 7, 873-884 9. Dastjerdi Sh., Lotfi M., Jabbarzadeh M., 2016c, The effect of vacant defect on bending analysis of graphene sheets based on the Mindlin nonlocal elasticity theory, Composites Part B, 98, 78-87 10. Degenhardt R., Kling A., Bethge A., Orf J., Karger L., Rohwer K., Calvi A., 2010, RolfZimmermannInvestigationson imperfection sensitivityanddeductionof improvedknock-down factors for unstiffened CFRP cylindrical shells,Composite Structures, 92, 1939-1946 11. Eringen A.C., Edelen D.G.B., 1972, On nonlocal elasticity, International Journal of Engine- ering Science, 10, 233-248 12. Fazelzadeh S.A., Ghavanloo E., 2014, Vibration analysis of curved graphene ribbons based on an elastic shell model,Mechanics Research Communications, 56, 61-68 13. Hoff N.J., Tsai-Chen S., 1965, Buckling or circular cylindrical shells in axial compression, International Journal of Mechanical Science, 7, 489-520 14. Iijima S., 1991, Helical microtubules of graphitic carbon,Nature, 354, 56-58 896 S. Dastjerdi et al. 15. Khazaeinejad P., Najafizade M.M., Jenabi J., 2010,On the buckling of functionally graded cylindrical shells under combined external pressure and axial compression, Journal of Pressure Vessel Technology ASME, 132-136 16. Lancaster E.R., Calladine C.R., Palmer S.C., 2004, Paradoxical buckling behaviour of a thin cylindrical shell under axial compression, International Journal of Mechanical Science, 42, 843-865 17. Linghai J., YongliangW., XinweiW., 2008, Buckling analysis of stiffened circular cylindrical panels using DQmethod,Thin Walled Structures, 46, 390-398 18. Martel R., Schmidt T., Shea H.R., Hertel T., Avouris P., 1998, Single and multi-wall carbon nanotube field-effect transistors,Applied Physics Letter, 73, 2447 19. Murmu T., Pradhan S.C., 2009, Vibration analysis of nano-single-layered graphene sheets em- bedded in elastic medium based on nonlocal elasticity theory, Journal of Applied Physics, 105, 064319 20. Nguyen D., Hoang Van T., 2010, Nonlinear analysis of stability for functionally graded cylin- drical panels under axial compression,Computational Materials Science, 49, S313-S316 21. PostmaH.W.C.,TeepenT.,YaoZ.,GrifoniM.,DekkerC., 2001,Carbonnanotube single- electron transistors at room temperature, Science, 293, 76 22. Pradhan S.C., 2009, Buckling of single layer graphene sheet based on nonlocal elasticity and higher order shear deformation theory,Physics Letters A, 373, 4182-4188 23. Reddy J.N., Pang S.D., 2008, Nonlocal continuum theories of beams for the analysis of carbon nanotubes, Journal of Applied Physics, 103, 023511 24. Tans S.J., Devoret M.H., Dai H., Thess A., Smalley R.E., Geerligs L.J., Dekker C., 1997, Individual single-wall carbon nanotubes as quantumwires,Nature, 386, 474-477 25. Wang Q., Wang C.M., 2007, The constitutive relation and small scale parameter of nonlocal continuummechanics for modeling carbon nanotube,Nanotechnology, 18, 1-10 26. Zhang L.W., Zhang Y., Zou G.L., Liew K.M., 2016, Free vibration analysis of triangular CNT-reinforced composite plates subjected to in-plane stresses using FSDT element-free method, Composite structures, 149, 247-260 27. Zhang Y., Lei Z.X., Zhang L.W., Liew K.M., Yu J.L., 2015, Nonlocal continuummodel for vibration of single-layered graphene sheets based on the element-free kp-Ritzmethod,Engineering Analysis with Boundary Elements, 56, 90-97 28. Zhao X., Liew K.M., 2009,Geometrically nonlinear analysis of functionally graded shells, Inter- national Journal of Mechanical Science, 51, 131-144 Manuscript received August 30, 2016; accepted for print March 21, 2017