Buckling analysis of functionally graded carbon nanotube-reinforced curved panels under axial compression and shear

Buckling analysis of functionally graded carbon nanotube-reinforced curved panels under axial compression and shear. Enrique Garc´ ıa-Mac´ ıasa,∗, Luis Rodriguez-Temblequea, Rafael Castro-Triguerob, Andr´ es S´ aeza aDepartment of Continuum Mechanics and Structural Analysis, School of Engineering, University of Seville , Camino de los Descubrimientos s/n, E-41092-Seville, Spain bDepartment of Mechanics, University of Cordoba, Campus de Rabanales, Cordoba, CP 14071, Spain Abstract The need for high-performance lightweight materials for the design of aerospace structures predicts a remarkable future role of carbon nanotubes (CNTs). In particular, thin curved fuselage panels are widely employed. However, the theoretical difficulties in the underlying differential problem induced by the curvature, as well as a still under development technology of CNT-based composites, the number of studies dealing with buckling behavior of this structural typology is limited. The present study aims to provide some insight into the linear buckling analysis of functionally graded carbon-nanotube reinforced (FG-CNTRC) cylindrical curved panels under compressive and shear loading. Effective properties of materials of the panels reinforced by single-walled carbon nanotubes (SWCNTs) are estimated through a micromechanical model based on either the Eshelby-MoriTanaka approach or the extended rule of mixtures. A series of numerical simulations have been carried out to inspect the influence of curvature, panel aspect ratio, the distribution profile of reinforcements (uniform and three non-uniform distributions) and CNTs orientation angle on the buckling critical load under compressive and shear loading in uniform thermal environments. Results demonstrate that the change of fiber orientation, CNTs distribution, panel aspect ratio, loading condition and temperature have noticeable effects on the buckling strength and buckling modes of FG-CNTRC curved panels. Keywords: Functionally graded Carbon nanotube, Cylindrical panel, Buckling, Shell finite elements, Temperature-dependent properties 1. Introduction The interest in composite materials for the design of aerospace structures has been steadily growing over the last decades. The upsurge in the need for lightweight structures able to give high level performances has led to a more intensive use of advanced materials, such as fiber reinforced composites, laminates, sandwiches, foams and nanostructures [1]. In particular, carbon nanotubes (CNTs) exhibit manifold interesting features as reinforcing fibers for high strength materials [2] and smart materials with self-sensing capabilities [3, 4]. The introduction of composite materials to a larger extent on commercial aircrafts such as Boeing 787 and Airbus A380, predicts a remarkable role of carbon nanotube-reinforced composites (CNTRC) for both aeronautical and aerospace industry [5]. A promising direction is the application of CNTs as reinforcements in Functionally Graded Materials (FGM). This branch of advanced materials is characterized by spatially continuous varying properties [6]. These materials are inhomogeneous composites characterized by smooth and continuous variations in both compositional profile and material properties, feature that eludes characteristic issues of laminates such as delamination and debonding [7–9]. An interesting application of these new materials is on curved fuselage panels. However, due to the mathematical difficulties involved in their formulation, works on prediction of the critical buckling loads of CNTRC cylindrical unstiffened curved panels subjected to uniform axial compression and shear are scarce in the literature. Papers related to the buckling theory of curved panels are not so numerous in comparison to the literature devoted to flat shell or cylindrical shell buckling. In the case of isotropic curved panels under axial compressive loading, some analytical solutions were proposed under the assumption of critical loads equal to the case of full revolution cylinders. This is the case of the reference expressions developed by Redshaw [10] and Timoshenko [11]. Stowell [12] proposed a modified form of Redshaw’s expression taking into account the influence of the ∗Corresponding author. Email address: [email protected] (Enrique Garc´ ıa-Mac´ ıas) boundary conditions. However, gross differences have been found with these simplifications in comparison to some experimental results in the literature [13–15]. Further investigations were conducted by Yamura et al. [16] who studied numerically the buckling plastic collapse and by Park et al. [17, 18] who provided a simplified method to estimate the ultimate strength based on Faulkner’s formule for a single plate with a newly defined slenderness parameter including curvature effect. Le Tran et al. [19] proposed semi-empirical formulae for predicting the elastic buckling and ultimate strength based on finite element simulations. Similar analyses were also conducted for the case of buckling of curved shells under shear loading. The initial analysis were conducted by Legget [20] for long, slightly curved plates under uniform shear stresses. There are some other analytical results in the literature (see e.g. [21, 22]) although due to the nonexistence of simple trigonometric shape functions for shear buckling modes of curved panels, numerical simulations by finite element analysis has proven to be more feasible [23, 24]. With regard to the analysis of CNTRC and FG-CNTRC structural elements, the number of publications has considerably increased in recent years with plenty of newly results. One of the first works concerning non-uniform distributions of CNTs within an isotropic matrix was published by Shen [25]. In this work, a nonlinear vibration analysis of FG-CNTRC plates in thermal environments was presented. Shen and Zhang [26] analyzed the thermal buckling and postbuckling behavior of uniform and symmetric FG-CNTRC plates under in-plane temperature variation. The results showed that functionally graded CNT distributions can rise the buckling temperature as well as the thermal postbuckling strength of plates. However, it is also observed that in some cases the plate with intermediate nanotube concentration may not present intermediate buckling temperature and initial thermal postbuckling strength. Based on the homogenization framework of Eshelby-Mori-Tanaka, Arani et al. [27] investigated analytically and numerically the buckling behavior of moderately thick CNTRC rectangular plates subjected to uniaxial compression. Adopting classical laminate plate theory and third-order shear deformation theory, the authors optimized the orientation of CNTs to achieve the highest critical load. A similar contribution by Mehrabadi et al. [28] analyzed the mechanical buckling of FG-CNT reinforced composite Mindlin plates subjected to both uniaxial and biaxial in-plane loadings. The buckling analysis of FG-CNTRC plates under in-plane mechanical loads was also carried out by Lei et al. [29] using the element-free kp-Ritz method. In that work, a micromechanical model based on either the Eshelby-Mori-Tanaka approach or the extended rule of mixture showed good synergism between both methodologies for the case of uniaxially CNT reinforced composites. Zhu et al. [30] proposed a meshless local Petrov-Galerkin approach based on the moving Kriging interpolation technique for the geometrically nonlinear thermoelastic analysis of FG plates in thermal environments. Later work of the same group dealt with the mechanical and thermal buckling analysis of ceramic-metal FG plates [31]. Lei et al. [32] employed the IMLS-Ritz method for the buckling analysis of thick FG-CNT skew plates resting on Pasternak foundations. Similar results can be found in the literature concerning the buckling analysis of FG-CNTRC thick plates resting on Winkler [33] and Pasternak foundations [34], the buckling analysis of thick FG-CNTRC skew plates under biaxial compression [35], the postbuckling behavior of FG-CNTRC plates with elastically restrained edges under axial compression [36], as well as of biaxial compressed arbitrarily straight-sided quadrilateral ceramic-metal FGM plates [37]. This methodology has been found useful for other manifold applications such as the free vibration of moderately thick laminated CNTRC plates [38] and triangular plates [39], large deformation of FG-CNT reinforced composite quadrilateral plates [40], and FG-CNTRC plates with elastically restrained edges [41]. Those studies concluded that the FG-X profile results in the stiffest CNT configuration, unlike the FG-O profile which generates the least stiffconfiguration. Similar conclusions were reached by Zhang et al. [42] who investigated the large deflection behavior of FG-CNTRC plates resting on elastic foundations under transversely distributed loads. It is also worth mentioning the proposal of these authors of a state-space Levy method for the vibration analysis of FG-CNTRC plates subjected to in-plane loads based on higher-order shear deformation theory [43]. Nevertheless, the number of publications dealing with cylindrical panels is less numerous. It is noteworthy the work of Lei et al. [44] which presented parametric studies of the dynamic stability of CNTRC-FG cylindrical panels under static and periodic axial forces using the mesh-free-kp-Ritz method and the Eshelby-Mori-Tanaka homogenization scheme. Shen and Xiang [45] employed the higher-order shear deformation theory with a von K´ arm´ an-type of kinematic nonlinearity to study the postbuckling of axially compressed FG-CNTRC cylindrical panels resting on elastic foundations in thermal environment. Regarding the state of the art of mechanical analysis of FG-CNTRC composites [46], only a limited number of works reported on the buckling analysis of FG-CNTRC cylindrical panels under axial compressive and tangential forces. The influence of the different variables involved in the micromechanics of functionally graded materials, such as volume fraction, CNT distributions, fiber orientation angle and temperature-dependent material properties, on the buckling behavior of curved panels is still unexplored. The objective of the present study is to investigate the effect of the main design variables which influence the linear buckling behavior of FG-CNTRC unstiffened curved panels. The panels have been modeled with the commercial FEA software ANSYS v15.0 with effective properties of materials estimated by means of a micromecamechanical model based on either the Eshelby-Mori2 Tanaka approach or the extended rule of mixtures. Detailed parametric studies have been carried out to inspect the influence of curvature, panel aspect ratio, the distribution profile of reinforcements (uniform and three non-uniform distributions) and CNTs orientation angle on the buckling critical load under compressive and shear loading in uniform thermal environments. The results show substantial influence of these variables on the buckling strength and buckling modes of FG-CNTRC curved panels. The paper is organized as follows. Section 2 introduces the terminology used for the parametrization of FG-CNTRC cylindrical curved panels as well as the homogenization schemes. Section 3 sets up the basis of the eigenvalue buckling analysis of FG-CNTRC cylindrical curved panels. Section 4 presents some comparison analyses with the existing literature, buckling analysis of FG-CNTRC curved panels under axial compressive and tangential forces. Finally, Section 5 presents the conclusions derived from this work. 2. FG-CNTRC cylindrical curved panels The FG-CNTRC cylindrical curved panel and the corresponding local coordinate system {u,v,w}considered in this paper are shown in Fig. 1a. This panel is assumed to be thin and of projected width b, straight length a, radius R, span angle βand uniform constant thickness t. In order to characterize the curvature of the panels, it is defined a non-dimensional parameter Z=s2/R·t(modified Bartdorf’s parameter deprived of the effects of Poisson’s ratio), being sthe curved length of the panel. From a physical point of view, this parameter is proportional to the sagitta of the curved edges h(given by b2/8·R) divided by its thickness t. In order to visualize the curvature parameter Z, the schematic curved shapes of selected curvature parameters are shown in Fig. 1b. The CNTs are assumed to be uniaxially aligned at αdegrees with respect to the axial direction (x-axis) and functionally graded along the thickness of the panel by four different distributions, namely UD, FG-V, FG-O and FG-X. UD-CNTRC represents the uniform distribution whilst FG-V, FG-O and FG-X CNTRC are linear distributions of carbon nanotubes along the radial direction. According to these distributions (Fig. 2), the CNT volume fractions VCNT (z) are given by R a 2·β t y x α h O b y z Z=200 Z=150 Z=100 Z=50 Z=0.0 s 1 2 3 4 E1 E2 E3 E4 w u v 12 (a) (b) Figure 1: a) Geometry and coordinate system of carbon nanotube nanoreinforced cylindrical panel. b) Schematic views of panels with different curvature parameters Z. VCNT =V∗ CNT (UD CNTRC) VCNT =4|z| tV∗ CNT (FG-X CNTRC) VCNT =(1 +2z t)V∗ CNT (FG-V CNTRC) VCNT =2(1 −2|z| t)V∗ CNT (FG-O CNTRC) (1) with V∗ CNT the volume fraction of CNTs that can be calculated from the mass fraction of nanotubes, wCNT , as V∗ CNT =wCNT wCNT +(ρCNT /ρm)−(ρCNT /ρm)wCNT (2) 3 where ρmand ρCNT are the densities of the matrix and CNTs, respectively. Hence, the cases of uniformly distributed (UD), i.e. VCNT =V∗ CNT , and functionally graded (FG) CNTRCs will have the same value of mass fraction of CNTs. r -t/2 0 t/2 V CNT 0 VCNT * 2 V CNT * UD FG-V FG-O FG-X · Figure 2: Variation of nanotube volume fraction (VCNT ) along the radial direction for types of FG-V, FG-O, FG-X and UD. The effective material properties of the two-phase nanocomposites mixture of uniaxially aligned CNTs reinforcements and a polymeric matrix, can be estimated according to the Mori-Tanaka scheme [47] or the rule of mixtures [48, 49]. The accuracy of the extended rule of mixtures (EROM) has been widely discussed and a remarkable synergism with the Mori-Tanaka scheme for functionally graded ceramic-metal beams is reported in [50]. 2.1. Extended Rule of Mixtures According to the extended rule of mixtures, the effective material properties of CNTRC panels can be expressed as [25] E11 =η1VCNT ECNT 11 +VmEm(3a) η2 E22 =VCNT ECNT 22 +Vm Em(3b) η3 G12 =VCNT GCNT 12 +Vm Gm(3c) where ECNT 11 ,ECNT 22 and GCNT 12 indicate the Young’s moduli and shear modulus of SWCNTs, respectively, and Em and Gmrepresent the corresponding properties of the isotropic matrix. In order to account for the scale-dependent material properties, the CNT efficiency parameters, ηj(j=1,2,3), are introduced and can be calculated by matching the effective properties of the CNTRC obtained from a molecular dynamics (MD) or multi-scale simulation with those from the rule of mixtures. VCNT and Vmare respectively the volume fractions of the carbon nanotubes and matrix, whose sum equals unity. Similarly, the thermal expansion coefficients in the longitudinal and transverse directions, α11 and α22, Poisson’s ratio ν12 and the density ρof the nanocomposites can be determined in the same way as ν12 =VCNT νCNT 12 +Vmνm(4a) ρ=VCNT ρCNT +Vmρm(4b) α11 =VCNT αCNT 11 +Vmαm(4c) α22 =(1 +νCNT 12 )VCNT αCNT 22 +(1 +νm)Vmαm−ν12α11 (4d) where νCNT 12 and νmare Poisson’s ratios, and αCNT 11 ,αCNT 22 and αmare the thermal expansion coefficients of the CNT and matrix, respectively. Note that ν12 is considered to be constant over the thickness of the functionally graded 4 CNTRC plates. In addition, we assume that G23 =G13 =G12.The rest of the other effective mechanical properties are defined as follows E33 =E22,G13 =G12, ν23 =E22 2G23 −1, ν13 =ν12, ν31 =ν21, ν32 =ν23, ν21 =ν12 E22 E11 (5) 2.2. Eshelby-Mori Tanaka approach The Mori-Tanaka method [51] allows to extension of the theory of Eshelby [52, 53], restricted to one single inclusion in a semi-infinite elastic, homogeneous and isotropic medium, to the case of multiple inhomogeneities embedded into a finite domain. The Eshelby-Mori-Tanaka approach, known as the equivalent inclusion-average stress method, is based on the equivalent elastic inclusion idea of Eshelby and the concept of average stress in the matrix due to Mori-Tanaka. In this paper, we consider a linear elastic polymer matrix reinforced by a large number of dispersed aligned and straight CNTs. According to Benveniste’s revision [54], the following expression of the effective elastic tensor is obtained: C=Cm+VCNT h(Cr−Cm)Ari(VmI+VCNT hAri)−1(6) where Iis the identity tensor, Cmis the stiffness tensor of the equivalent fiber, Cris the stiffness tensor of the equivalent fiber, and Aris the dilute mechanical strain concentration tensor for the fiber: Ar=hI+S(Cm)−1(Cr−Cm)i−1(7) The tensor Sis the Eshelby’s tensor, as given by Eshelby [52] and Mura [55]. The terms enclosed with angle brackets in Eq. 6 represent the average value of the term over all orientations defined by transformation from the local fiber coordinates (o−x0 1x0 2x0 3) to the global coordinates (o−x1x2x3)(Fig. 3). The matrix is assumed to be elastic and isotropic, with Young’s modulus Emand Poisson’s ratio νm. Each straight CNT is modeled as a long fiber with transversely isotropic elastic properties. Therefore, the composite is also transversely isotropic. The substitution of the non-vanishing components of the Eshelby’s tensor Sfor a straight, long fiber along the x2-direction in Eq. 7 gives the dilute mechanical strain concentration tensor. Then the substitution of Ar(Eq. 7) into Eq. 6 gives the tensor of effective elastic moduli of the composite reinforced by aligned and straight CNTs. In particular, the Hill’s elastic moduli are found as [47]: Micro scale Macro scale x3 x2 x1 x02 x01 x03 β α Figure 3: Representative volume element (RVE) including straight CNTs. k=Em{EmVm+2kr(1 +vm)[1 +VCNT (1 −2vm)]} 2(1 +vm)[Em(1 +VCNT −2vm)+2Vmkr(1 −vm−2v2 m)] (8) l=Em{vmVm[Em+2kr(1 +vm)] +2VCNT kr(1 −v2 m)} (1 +vm)[Em(1 +VCNT −2vm)+2Vmkr(1 −vm−2v2 m)] (9) n=E2 mVm(1 +VCNT −Vmvm)+2VmVCNT (krnr−l2 r)(1 +vm)2(1 −2vm) (1 +vm)[Em(1 +VCNT −2vm)+2Vmkr(1 −vm−2v2 m)] +Em[2V2 mkr(1 −vm)+VCNT nr(1 +VCNT −2vm)−4Vmlrvm] Em(1 +VCNT −2vm)+2Vmkr(1 −vm−2v2 m) (10) 5 p=Em[EmVm+2pr(1 +vm)(1 +VCNT )] 2(1 +vm)[Em(1 +VCNT )+2Vmpr(1 +vm)] (11) k=Em[EmVm+2mr(1 +vm)(3 +VCNT −4vm)] 2(1 +vm){Em[Vm+4VCNT (1 −vm)] +2Vmmr(3 −vm−4v2 m)}(12) where k,l,m,nand pare Hill’s elastic moduli of the composite; kis the plane-strain bulk modulus normal to the fiber direction, nis the uniaxial tension modulus in the fiber direction, lis the associated cross modulus, mand p is the shear moduli in planes normal and parallel to the fiber direction, respectively. kr,lr,mr,nrand prare the Hill’s elastic moduli for the reinforcing phase (CNTs). The material properties of the composite can be expressed in terms of engineering constants as [56]: E11 =n−l2 k(13) E22 =4m(k−l2) kn −l2+mn (14) ν12 =ν13 =l 2k(15) ν23 =n(k−m)−l2 n(k+m)−l2(16) G12 =G13 =p(17) 3. Formulation of eigenvalue buckling analysis of FG-CNTRC cylindrical curved panels The numerical studies of eigenvalue buckling analysis are conducted with the coomercial software ANSYS v15.0. The panel is modeled with the standard structural shell element, Shell 181 [57]. This element type is a quadrilateral 4-nodes element involving both bending and membrane properties with three rotational and three translational degrees of freedom per node. This element is suitable for thin to moderately-thick shell structures. The aim of this paper is focused on the study of the linear buckling of simply supported cylindrical curved panels subjected to axial pressure and shear forces. Hence, the boundary conditions considered in the modeling procedure are summarized in Table 1. Note that the boundary conditions are defined correspondingly in the local cylindrical coordinates {u,v,w}as represented in Fig. 1a. Table 1: Boundary conditions for buckling under axial compression and tangential forces Compression Shear u v w u v w Corner 1 cpl 0 1 1 1 1 Corner 2 cpl 0 1 1 0 1 Corner 3 1 0 1 1 0 1 Corner 4 1 1 1 1 0 1 Edge 1 cpl 0 1 0 0 1 Edge 2 0 0 1 0 0 1 Edge 3 1 0 1 0 0 1 Edge 4 0 0 1 0 0 1 (cpl coupled; 1 restrained; 0 free). The theoretical background of this procedure consists on the application of an arbitrary reference level of external load {R}re f , and perform a standard linear analysis to determine element stresses. The effects of membrane stresses on the lateral deflection are accounted for by the stress stiffness matrix [Kσ]re f which augments the conventional stiffness matrix [K]. For a generic load level, obtained by multiplying the reference load by the scalar λ, the stress stiffness matrix can be written [Kσ]=λ[Kσ]re f ;{R}=λ{R}re f (18) Eqs. 18 imply that multiplication of all loads Riin {R}by λalso multiplies the intensity of the stress field by λbut does not alter the distribution of stresses. Because the problem is assumed linear, the conventional stiffness 6 matrix [K]does not depend on loading. Let buckling displacements {δD}take place relative to displacements {D}re f of the reference configuration. Then because the external loads do not change at a bifurcation point, we have        ([K]+λcr [Kσ]){D}re f =λcr {R}re f ([K]+λcr [Kσ]){D}re f +{δD}=λcr {R}re f (19) Subtraction of the first equation from the second yields ([K]+λcr [Kσ]){δD}=0 (20) Eq. 20 is an eigenvalue problem whose smallest root λcr defines the smallest level of external load for which the bifurcation takes place, namely {R}cr =λcr {R}re f (21) The eigenvector {δD}associated with λcr is the buckling mode. Eq. 20 can be rewritten in a more compact way as [K]net {δD}=0. Because forces of this new problem are zero, it can be said that the stresses of critical intensity lead the net stiffness to be singular with respect to the buckling mode {δD}. Mathematically, [K]net has a zero determinant thus the linear bifurcation problem reduces to the following eigenvalue problem [K]+λcr [Kσ]re f =0 (22) Finally, in order to implement the constitutive equations of the resulting CNTRCs into the modeling, the preintegrated sections module of ANSYS is employed. This module permits the specification of the membrane, bending and coupling stiffness matrices of the shells. For this purpose, let the constitutive equations of the CNTRCs written in Voigt’s notation as follows                    s11 s22 s12 s23 s13                    =                    Q11(z)Q12(z) 0 0 0 Q12(z)Q22(z) 0 0 0 0 0 Q66(z) 0 0 000Q44(z) 0 0000Q55(z)                    ·                    γ11 γ22 γ12 γ23 γ13                    (23) Q11 =E11 1−ν12ν21 ,Q22 =E22 1−ν12ν21 ,Q12 =ν21 E11 1−ν12ν21 , Q66 =G12,Q44 =G23,Q55 =G13 (24) where E11 and E22 are effective Young’s moduli of CNTRC panels in the local coordinates; G12,G13 and G23 are the shear moduli; ν12 and ν12 are Poisson’s ratios obtained by the extended rule of mixtures (EROM) or the Eshelby-Mori-Tanaka approach. Note that Qi j varies with zaccording to the grading profile of the CNTRC along the thickness. Thus, the components of the extensional stiffness CE, bending extensional coupling stiffness CC, bending stiffness CB, and transverse shear stiffness CSare defined by the following integrals (Ci j E,Ci j C,Ci j B)=Rt/2 −t/2Qi j(z)·(1,z,z2)dz (i,j=1,2,6), Ci j S=1 ks Rt/2 −t/2Qi j(z)dz (i,j=4,5) (25) where ks denotes the transverse shear correction factor for FGM, given by [58] ks =6−(νCNT 12 VCNT +νmVm) 5(26) 4. Results and discussion In this section, several numerical examples are presented to study the buckling behavior of FG-CNTRC curved panels under uniform axial compression and shear in uniform thermal environment. Poly{(m-phenylenevinylene) -co-[(2,5-dioctoxy-p-phenylene)vinylene]}(PmPV) is considered as the matrix and the mean material properties are assumed to be νm=0.34, ρm=1.15g/m3and Em=(3.52 −0.0047T)GPa, where T=T0+ ∆Tand room temperature T0=300K. The armchair (10,10) SWCNTs are selected as reinforcements. For the Eshelby-MoriTanaka approach, we use the the analytical results of Popov et al. [59]. In the case of the extended rule of mixtures, the properties of SWCNTs are taken from the MD simulation carried out by Shen and Zhang [26] and summarized in Table 2. The key issue for the successful application of the extended rule of mixtures to CNTRCs 7 relies on the CNT efficiency parameters ηj(j=1,2,3). In the present study, the CNT efficiency parameters are determined by matching the Young’s moduli E11 and E22 from the MD simulations given by Han and Elliot [60] with the counterparts computed by the rule of mixtures. For example, η1=0.149 and η2=0.934 for the case of V∗ CNT =0.11, and η1=0.150 and η2=0.941 for the case of V∗ CNT =0.14, and η1=0.149 and η2=1.381 for the case of V∗ CNT =0.17. In addition, we assume that η3=η2. Unless otherwise specified, the Eshelby-Mori-Tanaka approach is applied to predict effective material properties of CNTRCs. For the subsequent analyses, it is set up a constant length of b=100cm and thickness t=b/50. In order to define the mesh density, convergence studies of panels with two different aspect ratios, namely a/b=1 and a/b=2, have first been performed as represented in Fig. 4. Different uniform mesh divisions of (a/b)8×8, (a/b)16×16, (a/b)32×32, (a/b)64×64 and (a/b)128×128 have been used, wherein (a/b)64×64 was found to provide accurate results and has been used for all the following results. Table 2: Temperature-dependent material properties for (10,10) SWCNT (L=9.26 nm, R=0.68 nm, h=0.067 nm, νCNT 12 =0.175) Temperature (K) ECNT 11 (TPa) ECNT 22 (TPa) GCNT 12 (TPa) αCNT 11 (×10−6/K) αCNT 22 (×10−6/K) 300 5.6466 7.0800 1.9445 3.4584 5.1682 400 5.5679 6.9814 1.9703 4.1496 5.0905 500 5.5308 6.9348 1.9643 4.5361 5.0189 700 5.4744 6.8641 1.9644 4.6677 4.8943 1000 5.2814 6.6220 1.9451 4.2800 4.7532 8×816 ×16 32 ×32 64 ×64 128 ×128 0 100 200 300 400 500 Log Number of Elements kσ a/b=1 h/R=0 h/R=0.5 h/R=1 16 ×8 32 ×16 64 ×32 128 ×64 256 ×128 0 100 200 300 400 500 Log Number of Elements kσ a/b=2 h/R=0 h/R=0.5 h/R=1 Figure 4: Convergence analysis of the buckling load intensity factor kσ=σcr σEof UD-CNTRC panels subjected to axial compression (SSSS, b=100cm, t=b/50, R=b/2, V∗ CNT =0.11) for different panel aspect ratios: a) a/b=1 and b) a/b=2. 4.1. Comparison studies First, we carry out several numerical experiments to validate the present computational approach. However, since no results are available in the literature for FG-CNTRC curved panels, the comparison studies focus on the cases of simply supported (SSSS) orthotropic flat shells and isotropic curved panels, configurations which do have analytical solutions and are provided in the literature. In addition, the results obtained from the present computational approach are also compared to results available in the literature on the buckling behavior of FGCNTRC flat panels. 4.1.1. Orthotropic flat shells The case of orthotropic rectangular shells (Z=0, a×b) subjected to compressive edge forces was studied by Hwang and Lee [61]. In the case of buckling under uniaxial compression of simply supported shells, closed solutions are provided as follows: σcr σo =m2 R2+χ2 m2+2η(27) 8 where σcr is the critical buckling stress and mis the number of half-waves in the loading direction. The terms σo, R,λand ηare defined as: σo=π2√D11D22 b2h;χ=λ1/4(a/b) (28) λ=D22 D11 ;η=D12 +2D66 √D11D22 (29) with Di j the flexural rigidities of the shell. The width band thickness tof the shell are set to 1m and t=0.021m, respectively, and the Young’s modulus in x-direction Ex=1.81 ×1011N/m2, Young’s modulus in y-direction Ey=1.03 ×1010N/m2, shear modulus Gxy=7.53 ×109N/m2and Poisson’s ratio νxy =0.28. Buckling load parameters for the first mparameters (m=1,...,5) are depicted in Fig. 5. Here, it is observed that the number of buckle waves increases as the plate aspect ratio increases. Note that each mode shape takes place in a range of aspect ratios, and the buckling mode changes when there is a change in the energy state of the plate. In the case of the present computational approach, the first critical buckling load is represented which corresponds to the buckling mode with the lowest energy state, associated with the buckling eigenvalue problem, and thus it provides the minimum envelope of the analytical curves. 0 1 2 3 4 100 101 102 103 R Log(sigmacr /sigmao) Present Hwang and Lee [50], m=1 Hwang and Lee [50], m=2 Hwang and Lee [50], m=3 Hwang and Lee [50], m=4 Hwang and Lee [50], m=5 Figure 5: Comparison analysis of the critical buckling stress σcr/σoas a function of the adjusted plate aspect ratio Rfor the first five buckling modes (m=1,...,5) of a simply supported rectangular orthotropic plate. 4.1.2. Isotropic curved panels In this second test, the results of the present approach for unstiffened curved isotropic panels are compared to the theoretical formulae found in the literature. In particular, the numerical values of the critical buckling load are compared to the reference analytical expressions proposed by Redshaw [10], Timoshenko [11], Stowell [12] and Domb and Leigh [62], whose expressions are collected in [19]. In order to simplify the comparison, all the results are expressed by the so-called load intensity factor kσdefined by kσ=σcr σE ;σE=π2E 12(1 −ν2)t s2 (30) The results of this comparison are presented in Fig. 6 where the load intensity factor kσis plotted as a function of the curvature parameter Z. For values of Ztending towards zero, all formulae naturally converge to kσ=4 corresponding to the case of flat square plate. It is also remarkable that, for all curves, the buckling coefficients increase with Z. This tendency is due to the fact that for increasing values of Z, larger curvature values lead to higher circumferential membrane stresses in the radial direction. Nonetheless, some differences can be found in the asymptotic behavior of these methodologies. In the case of the formulae of Redshaw and Timoshenko, the buckling coefficients asymptotically converge towards a straight line with slope equal to the buckling coefficient of a revolution cylinder. On the contrary, the formulae of Domb and Leigh, Stowell and the present numerical approach converge towards half this value. These differences can be attributed to the different boundary conditions considered in these two sets of approaches. The assumption of a full revolution cylinder of the first set of formulae 9 +Nxy -Nxy +Nxy -Nxy h/R=0 h/R=0.45 α=0º α=30º α=90º α=60º Figure 15: First elastic buckling modes of simply supported UD-CNTRC flat panel (h/R=0) and curved panel (h/R=0.9) under shear forces with different fiber orientation angles α(b=100cm, a=b, t=b/50, V∗ CNT =0.11). +Nxy -Nxy +Nxy -Nxy h/R=0 h/R=0.45 Figure 16: First elastic buckling modes for UD-CNTRC flat panels (h/R=0) and curved panels (h/R=0.45) under shear forces with fiber orientation angle (α) of 30◦under shear forces (SSSS, b=100cm, a=b, t=b/50, V∗ CNT =0.11). 16 5. Conclusions In this paper, the critical buckling of FG-CNTRC cylindrical curved panels under compression and shear forces have been investigated by finite elements analyses. The plates are reinforced by SWCNTs and effective material properties of CNTRC plates are estimated either by the Eshelby-Mori-Tanaka approach or the extended rule of mixture. Comparison studies were performed to verify the accuracy and efficiency of the present approach and the results were found to be in good agreement with previously published works. Detailed case studies illustrate the influence of fiber angle, aspect ratio, temperature, SWCNTs grading profile and curvature . The key contributions of this study can be summarized as follows: •The curvature increases the buckling load values with respect to the flat configuration. This effect is explained by higher circumferential membrane stresses in the radial direction induced by an increasing curvature. •The buckling load intensity factor kσof flat panels subjected to compressive loading presents positive curvature with respect to the fiber orientation angle α. One clear maximum peak is found for fiber orientations aligned with the diagonal of the panel. On the contrary, the effects of the fiber orientation angle αon the buckling load factor become more complex as the curvature increases. It is shown a change from curves with one single maximum peak to curves with one minimum and two maximum peaks. •FG-X distribution leads to the largest buckling loads in all the studied cases whilst the FG-O distribution conducts to the minimum values. It is concluded that reinforcements distributed close to the top and bottom induce higher stiffness values. •In the case of curved panels, it is found some fiber orientation angle ranges in which there is not a big difference among the different CNT distributions. In particular, for curved panels with ratio h/R=0.9, in the range of fiber angles between 70◦and 85◦, all the fiber distributions present very similar results. •In all cases, the buckling load intensity factors have higher values when the volume fraction of CNT is larger as it raises the stiffness of the panels. •In the case of curved configurations under compressive loading, slight differences on the buckling loads are found for FG-V and UD distributions with certain values of fiber orientation angle and aspect ratio. This fact even results in the inversion of the relative order of these two distributions. •The buckling load intensity factors decrease as temperature increases in FG-CNTRC curved panels subjected to compressive and tangential forces as a result of the temperature-dependent material properties. •The critical loads of the FG-CNTRC panels under the negative shear are higher than those of positive shear for all fiber angles, exhibiting a maximum value for fibers approximately aligned with the diagonal direction (α=45◦). Acknowledgement This work was supported by the Ministerio de Econom´ ıa y Competitividad of Spain and the Consejer´ ıa de Econom´ ıa, Innovaci´ on, Ciencia y Empleo of Andaluc´ ıa (Spain) under projects DPI2014-53947-R and P12-TEP2546. E. G-M was also supported by a FPU contract-fellowship from the Spanish Ministry of Education Ref: FPU13/04892. The financial support is gratefully acknowledged. References [1] A. A. B. 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