Thin-Walled Structures 135 (2019) 560–574 Contents lists available at ScienceDirect Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws Full length article Torsional postbuckling behavior of FG-GRC laminated cylindrical shells in thermal environments T Hui-Shen Shena,b, , Y. Xiangc ⁎ a School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China c School of Computing, Engineering and Mathematics, Western Sydney University, Locked Bag 1797, Penrith, NSW 2751, Australia b A R TICL E INFO A BSTR A CT Keywords: Cylindrical shell Torsion Postbuckling Nanocomposites Functionally graded materials Temperature dependent properties This paper presents the buckling and postbuckling behaviors of graphene-reinforced composite (GRC) laminated cylindrical shells subjected to torsion in thermal environments. The GRC layers of the shell are arranged in a piece-wise functionally graded (FG) distribution pattern in the thickness direction and each layer of the shell contains different volume fraction of graphene reinforcement. The extended Halpin-Tsai micromechanical model is employed to determine the temperature dependent material properties of GRC layers. The governing equations of the GRC laminated cylindrical shells under torsion are derived based on a higher-order shear deformation shell theory with the geometric nonlinearity being defined by the von Kármán strain-displacement relationship. A singular perturbation technique along with a two-step perturbation approach is employed to determine the buckling torques and the torsional postbuckling equilibrium paths of the FG-GRC laminated cylindrical shells in thermal environments. The numerical results obtained reveal that the piece-wise FG distribution of graphene volume fraction can enhance the buckling torque and the torsional postbuckling strength while the rise of temperature may lead to the reduction of the torsional buckling torques and torsional postbuckling strength of the GRC laminated cylindrical shell. 1. Introduction Composite laminated cylindrical shell is a common type of structural component which is used in a wide range of engineering applications. The shells may fail due to buckling when the shells are subjected to axial, radial or torsional loading. Most of the studies in the open literature have been made on the postbuckling behaviors of functionally graded (FG) metal/ceramic composite shell structures [1–10] and FG carbon nanotube reinforced composite (CNTRC) shell structures [11–14] subjected to axial compressive load or radial pressure. There are, however, a few works considering FGM cylindrical shells under torsion [15–18]. The buckling and postbuckling problems of cylindrical shells under torsion are different from those under loading of other types in both formulations and solution methods. The main difficulty involved in the torsional buckling is that firstly the single-wave buckling mode which is useful in the analysis of compressive buckling does not work and secondly the boundary conditions can hardly be satisfied beforehand by the assumed buckling mode. Among the available analytic solutions, for example, either the approximate solutions of Loo [19] and Nash [20], or the double series ⁎ solutions of Tabiei and Simitses [21] cannot satisfy both the boundary conditions and the equilibrium equations of cylindrical shells simultaneously. Recently, Shen [22] investigated the torsional buckling and postbuckling behaviors of FG-CNTRC cylindrical shells in thermal environments. In his study, the temperature dependent material properties of CNTRCs were estimated by the extended Voigt model (rule of mixture). It was confirmed that there exists a shear stress as well as an associated compressive stress when the CNTRC cylindrical shell is subjected to torsion [22]. Due to graphene's exceptional mechanical, thermal and electrical properties, graphene sheets may be considered as the ideal reinforcement materials for high performance structural composites. Many research results showed that the material properties of graphene sheets are anisotropic and temperature dependent [23–27]. Ni et al. [23] observed anisotropic mechanical properties for a graphene sheet along different load directions. Reddy et al. [24] confirmed that graphene behaves like an orthotropic material, in particular the shear modulus is much lower than that of a graphene when it is treated as an isotropic material. Shen et al. [25] found that single layer graphene sheets exhibit anisotropic, size-dependent and temperature-dependent Corresponding author at: School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China. E-mail address: hsshen@sjtu.edu.cn (H.-S. Shen). https://doi.org/10.1016/j.tws.2018.11.025 Received 17 September 2018; Received in revised form 9 November 2018; Accepted 20 November 2018 0263-8231/ © 2018 Elsevier Ltd. All rights reserved. Thin-Walled Structures 135 (2019) 560–574 H.-S. Shen, Y. Xiang properties. Graphene sheets can be aligned in polymer matrix to achieve better reinforcement effect as reported in [28,29]. However, the physical interactions between graphene and polymer matrix is weak and this has led to relatively low load transfer efficiency between graphene and polymer matrix in nanocomposites [30,31]. A better way to enhance the buckling behavior of graphene reinforced composite (GRC) structures is to arrange graphene reinforcement in a functionally graded manner to GRCs rather than to distribute graphene reinforcement uniformly in GRCs. Shen et al. [32] reported that the buckling load as well as the postbuckling strength of a GRC laminated plate can be increased as a result of a functionally graded piece-wise graphene reinforcement. In their analysis, the graphene sheet reinforcements are assumed to be aligned and oriented in the polymer matrix layer-bylayer and the anisotropic and temperature dependent material properties of the GRC are estimated through the extended Halpin-Tsai model which contains the graphene efficiency parameters. Song et al. [33] reported that by dispersing a very small amount of graphene platelets (GPLs) into the polymer matrix can significantly improve the critical buckling load and postbuckling load-carrying capacity of a plate. In their analysis, a transverse isotropic multilayered model was adopted. The GPLs are assumed to be randomly oriented and uniformly dispersed in the polymer matrix. The material properties of the graphene platelets reinforced composite (GPLRC) are assumed to be independent of temperature. The equivalent isotropic Young's modulus of the GPLRC is obtained by using the modified Halpin–Tsai model, where two homogenization weight coefficients 3/8 and 5/8 are used. Note that this equivalent isotropic model is only suitable for the case of the GPL having aspect ratio lGPL/wGPL = 5/3, otherwise the results may be incorrect [34–36]. The postbuckling behaviors of FG-GRC laminated cylindrical shells subjected to either axial compression or external pressure in thermal environments were studied by Shen and Xiang [37,38]. In the present study, the torsional postbuckling behaviors of GRC laminated cylindrical shells consisting of uniformly distributed (UD) and piece-wise FG volume fractions of graphene will be investigated. The anisotropic and temperature dependent material properties of GRC laminates are estimated by the extended Halpin-Tsai micromechanical model which contains the efficiency parameters. These graphene efficiency parameters are determined by matching the elastic moduli of GRCs obtained from the molecular dynamics (MD) simulations to those predicted from the Halpin–Tsai model. The nonlinear equations for the postbuckling of GRC laminated cylindrical shells under torsion are derived based on the Reddy's higher order shear deformation shell theory and the von Kármán-type kinematic assumptions. The initial geometric imperfections and the prebuckling deformations of the shell are both taken into account. A singular perturbation technique in conjunction with a twostep perturbation approach is applied to solve the postbuckling equations and to obtain the postbuckling equilibrium paths of the shell. The full nonlinear behaviors of torsional postbuckling for GRC laminated cylindrical shells under different thermal environmental conditions are discussed in detail. E22 = 2 G12 = 3 2. Multi-scale model for FG-GRC laminated cylindrical shells under torsion in which aG, bG and hG denote the length, width and the effective thickness of the graphene sheet, respectively, and Fig. 1. A GRC laminated cylindrical shell: (a) geometry and coordinate system; (b) configurations of GRC layers: (1) FG-V, (2) FG- , (3) FG-X, (4) FG-O. the shell in the middle surface. Several micromechanical models have been developed to predict the effective material properties of GRCs, for example, the Mori-Tanaka model [39], the Voigt model (rule of mixture) [40] and the Halpin-Tsai model [41]. The Mori-Tanaka model is applicable to microparticles and the Voigt model is suitable for the fiber fillers. The Halpin–Tsai model was developed for 2D aligned anisotropic fillers. It has also been reported that in nanoscale these three models cannot predict the effective material properties of GRCs accurately and should be modified [42,43]. Accordingly, the extended Halpin–Tsai model [32] is applied to evaluate the Young's and the shear moduli for a GRC layer as E11 = We consider a GRC laminated cylindrical shell with length L, mean radius R and thickness h to have N laminated plies, as shown in Fig. 1. The graphene reinforcement is assumed to be either armchair (referred to as 90-ply) or zigzag (referred to as 0-ply). Each ply consists of a mixture of the graphene reinforcement and the polymer matrix, and each ply may have different value of graphene volume fraction. The shell is in a piece-wise FG pattern when the volume fraction of graphene in each ply is different. The shell is located in the Cartesian coordinates (X, Y, Z) where X is in the axial direction, Y is in the circumferential direction and Z is in the direction of the inward normal to the middle surface of the shell. The origin of the coordinate system is at one end of 1 + 2(aG / hG ) 1 1 1 + 2(bG /hG ) 1 G 12 VG Em (1a) G 22 VG G 22 VG 1 1 G 11 VG G 11 VG Em (1b) Gm (1c) G 11 = G E11 /E m 1 G m E11/ E + 2aG / hG (2a) G 22 = G E22 /E m 1 G m E22/ E + 2bG /hG (2b) G m G G12/ G 12= G G12/ Gm m 1 (2c) m G where E and G are the Young's and shear moduli of the matrix, E11 , G G E22 and G12 are the Young's and shear moduli of the graphene sheet. The volume fractions of the graphene sheet and the matrix are denoted by 561 Thin-Walled Structures 135 (2019) 560–574 H.-S. Shen, Y. Xiang VG and Vm, respectively, and we have VG +Vm = 1. Due to the intermolecular effect, strain gradients effect and surface effect, in nanoscale the GRC mechanical properties cannot be directly determined by the conventional Halpin-Tsai model. Therefore, the efficiency parameters j (j = 1,2,3) are used in the conventional Halpin-Tsai model to account for these size-dependent effects. The values of 1, 2 and 3 may be determined by matching the elastic moduli of GRCs predicted by the extended Halpin–Tsai model against the ones obtained from the MD simulations [44]. In the current study, the material properties of both the polymer matrix and the graphene are considered to be temperature dependent. The longitudinal and transverse thermal expansion coefficients of the GRC layer are given by G G VG E11 11 + G + VG E11 11 = 22 = (1 + G 12 ) VG G 11 , G 22 Q11 = Q12 = G 22 m) V m m and are thermal expansion coefficients of the grawhere G phene and the matrix. The Poisson's ratios m and 12 are assumed to weakly depend on temperature variation and the Poisson's ratio 12 of the GRC layer is defined by 12 = VG G 12 m We assume the shell is geometrically imperfect and a torque is uniformly applied along the end edges of the shell. Based on the Reddy's third order shear deformation shell theory [45] and considering the thermal effects, the governing equations for a GRC laminated cylindrical shell under torsion can be expressed by ¯) L˜ 11 (W L˜ 12 ( L˜ 13 ( y ) + L˜14 (F ) x) L˜ 15 (N¯ T ) (5) L˜ 25 (N¯ T ) ¯ ) + L˜ 32 ( ¯x ) L˜ 31 (W L˜ 33 ( ¯y ) + L˜ 34 (F¯ ) L˜ 35 (N¯ T ) L˜ 36 (S¯T ) = 0 (7) L˜ 42 ( ¯x ) + L˜ 43 ( ¯y ) + L˜ 44 (F¯ ) L˜ 45 (N¯ T ) L˜ 46 (S¯T ) = 0 (8) ¯) L˜ 41 (W (6) T ¯ T ¯T Mxy Pxy N¯ xy N tk k=1 tk 1 Ax Ay Axy 2 R N¯ xy dY 0 T S¯xy = T ¯ xy M 4 3h2 2 R 0 (13a) * A22 2F¯ 2F¯ 4 * * * + A12 + B21 E21 X2 Y2 3h2 2W 2W ¯ ¯ ¯ 4 W * * + E22 + E21 3h2 X2 Y2 R ¯ W Y 1 2 ¯x X 2 * + B22 4 * E22 3h2 ¯ W ¯* W Y Y ¯y Y * N¯ xT + A22 * N¯ yT ) dY = 0 (A12 (13b) Note that both the periodicity condition of Eq. (13) and the in-plane boundary condition V̄ = 0 (at X = 0, L) achieve the same effect. Hence, V̄ = 0 in Eqs. (12a) and (12b) may be neglected when Eq. (13) is applied. For the postbuckling problem considered, we need to determine the postbuckling load-shortening curves of the shell under torsion. The average end-shortening relationship may be expressed by x 1 2 RL = L (9a) * + B12 1 2 (9b) 2 R L 0 0 1 2 RL = P¯xT P¯yT T P¯xy (12c) which yields and S̄ is defined by ¯ xT M ¯ yT M MS = 0 V¯ dY = 0 Y T T S¯x T S¯y (11) (12b) 2 R (1, Z , Z 3) T dZ k , 12 21 ¯ = V¯ = ¯y = 0, M ¯ x = P¯x = 0 (simply supported) W 0 where W̄ is the initial geometric imperfection, W̄ is the additional deflection in the Z direction, ¯x and ¯y are the rotations of the normals to the middle surface with respect to the Y - and X - axes, and F̄ is the stress function defined by N¯x = 2F¯ / Y 2 , N¯ y = 2F¯ / X 2 and 2F ¯ / X Y . It is noted that, in Eqs. (5) and (6), the nonlinear N¯xy = operator L̃ ( ) denotes the geometric nonlinearity in the von Kármán sense, and the other linear operators L̃ij ( ) in Eqs. (5)–(8) are defined in Appendix A. The shell is assumed to be in a constant temperature field at an isothermal state. The thermal forces N̄ T , thermal moments M̄ T and higher order moments P̄ T of the shell associated with elevated temperature are defined by = 21 E11 1 where V̄ is the in-plane displacement in the Y-direction, MS = 2 R2h s and s is the shear stress, and M̄x is the bending moment and P̄x is the higher order moment, as defined in [45]. Also we have the periodicity condition * ¯ xT P¯xT N¯ xT M T ¯ yT P¯yT N¯ y M , 12 21 (12a) R ¯) L˜ 21 (F¯ ) + L˜ 22 ( ¯x ) + L˜ 23 ( ¯y ) L˜ 24 (W 1 ¯ 1˜ ¯ ¯ *, W ¯) L (W + 2W + W ,XX = 2 R E11 1 ¯ = V¯ = ¯x = ¯y = 0 (clamped) W ¯ T) L˜16 (M 1 ¯ ¯ +W ¯ *, F¯ ) F ,XX = L˜ (W R (10) in which E11, E22, G12 are the effective Young's and shear moduli, respectively, and 12 and 21 are the Poisson's ratios of the GRC layer. The two end edges of the shell are considered to be clamped or simply supported. The boundary conditions for such a shell can be expressed as X = 0, L: (4) m + Vm 11 22 E22 , 1 12 21 Q16 = Q26 = 0, Q44 = G23, Q55 = G13, Q66 = G12, (3b) 12 11 1 0 0 1 0 0 Q22 = (3a) + (1 + Q¯ 12 Q¯ 16 Q¯ 22 Q¯ 26 Q¯ 26 Q¯66 where 11 and 22 are, respectively, the longitudinal and transverse thermal expansion coefficients of the GRC layer. In the present analysis, the graphene reinforcement in GRC layer is assumed to be armchair or zigzag type, hence, the transformed elastic constants Q̄ij (ij = 1,2,6) is identical to Qij for a GRC layer, where Em m Vm Vm E m Q¯ 11 Q¯ 12 Q¯ 16 Ax Ay = Axy 2 R 0 4 * E12 3h2 ¯ W X 2 U¯ dXdY X L 0 ¯y Y * A11 2F ¯ Y2 * + A12 2F ¯ X2 * + B11 4 * E11 3h2 ¯x X 2W 2W ¯ ¯ 4 * * E11 + E12 X2 Y2 3h2 ¯ W ¯* W X X * N¯ xT + A12 * N¯ yT ) dXdY (A11 (14) where = T-T0 is the temperature rise from the reference temperature T0 at which there are no thermal strains in the shell, and where Ū is the in-plane displacement in the X-direction and 562 x is the Thin-Walled Structures 135 (2019) 560–574 H.-S. Shen, Y. Xiang end-shortening displacement of the shell. Also, we need to determine the postbuckling torque-rotation curves of the shell under torsion. The twist angle of the shell under the torsional load may be expressed by 1 L = 1 L L + 2F ¯ X Y * A66 0 ¯x 4 * E66 3h2 * B66 + Y ¯y + X 14 F ,xx = x) L13 ( y ) + 2L (W 14 1 2 2 24 x = 5/4 2W In order to solve this nonlinear torsional postbuckling problem, a singular perturbation method in associate with a two-step perturbation technique [46] is applied. Accordingly, the governing Eqs. (5)–(8) are first deduced to the dimensionless forms as L12 ( 220 2 24 24 611 x2 2 + ( 24 T1 2 0 24 L22 ( x ) 24 L31 (W ) + L32 ( x) 24 W ,xx = + L41 (W ) 2L (W + W *, F ) (16) s + 2W *, W ) + 244 2 2W L33 ( y ) + 14 L34 (F ) =0 (18) L42 ( x ) + L43 ( y ) + 14 L44 (F ) =0 (19) Pb 2 [D * D * ]1/2 11 22 24 * A22 * A11 * D22 * D11 = (AxT , A yT ) R * A11 * A22 = 1/2 , 5 ( x, s) = x L * A12 ,( * A22 = T 1, T 2 ) * E66 4 , * D22 * A11 * A22 * ]1/4 3h2 [D11 1/2 3/4 h [D * D * A * A * ]3/16 xL R 11 22 11 22 , * D22 * ]1/2 2 1/2 [D11 666 = in which N A xT A yT = k=1 and tk Ax dZ Ay tk 1 = 0 (clamped) x = W= y = 0, Mx = Px = 0 (simply supported) 2 0 2F x y dy + s W 2 x 24 y + 24 511 x + 233 x y W W* x x 2F x y 2 2W 666 + x y x 24 566 y + y x W W W W* W W* + + x y y x x y 24 dx (25) (26) j /4 + 1w j /4 + 1 (x , j /4 (x , y ), j =0 j /4 ( y, ) = j /4f y ), f (x , y, ) = x ) j /4 (x , y ), y (x , j /4 ( y, ) = y ) j /4 (x , y ), j =1 (27a) are defined by W= 1 2 x2 j=1 ˜ (x , , y , ) W 5/4 =0 j /4 + 1W ˜ j /4 + 1 (x , = (21) , y ), j=1 F˜ (x , , y , ) where Ax and Ay are given in detail in Eq. (10). The boundary conditions expressed by Eq. (12) become x = 0, : y 24 x (x , (20) A yT 2F 5 j =1 1/2 [D * D * A * A * ]5/16 11 22 11 22 AxT y2 (24) 266 0 w (x , y , ) = L1/2R3/4 , 2W where is a small perturbation parameter (provided Z̄ > 2.96) as defined in Eq. (20). In Eq. (26), w (x , y , ) , f (x , y, ) , x (x , y, ), ˜ (x , , y, ) , ) represent the regular solutions of the shell, W y (x , y , ˆ ˜ ˜ ˜ F (x , , y , ) , x (x , , y , ) , y (x , , y, ) and W (x , , y, ) , Fˆ (x , , y, ) , ˆx (x , , y , ) , ˆy (x , , y, ) are the boundary layer solutions near the x = 0 and x = edges, respectively. Moreover, the regular and boundary layer solutions can be expanded in the perturbation forms as , = 2 T dy = 0 ˆ (x , , y , ), ˜ (x , , y , ) + W W = w (x , y , ) + W F = f (x , y, ) + F˜ (x , , y, ) + Fˆ (x , , y , ), ˜ ˆ x = x (x , y , ) + x (x , , y , ) + x (x , , y , ), ˆ ˜ y = y (x , y, ) + y (x , , y , ) + y (x , , y , ), 1/4 1 4 B* E* , * D22 * A11 * A22 * ]1/4 66 3h2 66 [D11 1/2 3/4 h [D * D * A * A * ]3/16 sL R 11 22 11 22 = , p= 1/2 [D * D * ]1/2 11 22 566 s , 622 Usually, the shell structures have Z̄ 10 in practice [15,22]. From Eq. (20), we always have < < 1. Note that Eqs. (16)–(19) are the boundary layer type equations when < 1. Singular perturbation technique is a useful tool for solving the nonlinear equations of the boundary layer type. To this end, we assume that ¯,W ¯ *) (W L ¯ L2 = x= ,Z= , (W , W *) = , * D22 * A11 * A22 * ]1/4 R Rh [D11 ( ¯x , ¯y ) L F¯ , F= 2 ( x, y) = 2 , * D22 * A11 * A22 * ]1/4 * D22 * ]1/2 [D11 [D11 2 L2 1 ¯ x , 4 P¯x , = R [D11 * D22 * A11 * A22 * ]1/4 , M (Mx , Px ) = 2 2 * [D11 * D22 * A11 * A22 * ]1/4 D11 L2 3h2 4 y2 1 2 24 y2 5/4 (17) X Y , y= , L R = + x2 5 T 2) T ]dx dy 1 = 24 L 24 (W ) where the non-dimensional operators Lij() and L() are defined as in Shen [47]. In the above equations, the non-dimensional parameters are defined by x 5 T 1) T2 240 and the angle of twist may be re-written in dimensionless form as + 1 2 24 y 14 L14 (F ) 24 L 23 ( y ) + 522 W W* + ( y y 2F 2 2 24 0 24 L 21 (F ) + 2W y + x As has been shown in [47], we need to consider the boundary layer effect which is of the order 5/4 on the solution of a shell under torsion. Therefore, we can re-written the end-shortening equation of Eq. (14) in a dimensionless form as 3. Solution procedure 11 (W ) 24 2 W y 2 24 x + y2 (23) (15) 2L 1 2 24 W 2F 2 5 x2 2W ¯ 4 * 2E66 X Y 3h2 ¯ W ¯ ¯ W ¯* ¯ W ¯* W W W dX + + X Y X Y Y X + 2F 2 0 U V dX + Y X L 0 = and the condition for periodicity in Eq. (13a) becomes j /4 + 2F ˜j /4 + 2 (x , = , y ), j=1 ˜x (x , , y , ) = (22a) j /4 + 3/2 ( j= 0 ˜x ) j/4 + 3/2 (x , , y ), j =0 (22c) 563 ˜y ) j /4 + 2 (x , , y ), ˜y (x , , y, ) = (22b) j /4 + 2 ( (27b) Thin-Walled Structures 135 (2019) 560–574 H.-S. Shen, Y. Xiang ˆ (x , , y , ) W s j /4 + 1W ˆ j /4 + 1 (x , = , y ), j=1 j /4 + 2F ˆj /4 + 2 (x , , y ), ˆx (x , , y , ) = ˆx ) j/4 + 3/2 j /4 + 3/2 ( j= 0 (x , , y ), j /4 + 2 ( ˆy ) j /4 + 2 where the boundary layer variables and Wm = (27c) j =0 =( are defined by sin mx sin ny , As discussed in [47], the solution w2 (x , y ) = which was a suitable solution for the shell subjected to radial pressure [38], is not applicable for the shell subjected to torsion. The admissible solution for the shell under torsion may be chosen as (2) (2) + A11 w2 (x , y )= A00 sin mx sin(kx + ny ) (2) (2) + a11 cos mx cos(kx + ny ) + A02 cos 2(kx + ny ) (29) However, for the torsional postbuckling analysis of a cylindrical shell, the solution of Eq. (29) is not a case when the nonlinear operator L () is under consideration. Following Shen's works [15,22], for the case of torsional postbuckling of a cylindrical shell, we may take the initial buckling mode to have the form (2) (2) w2 (x , y ) = A00 sin(mx + A11 + 2µ (2) ky )sin ny + a11 cos(mx ky )cos ny (30) cos 2ny (2) [A11 sin(mx (2) ky )sin ny + a11 cos(mx ky )cos ny] (31) = 2 (0) s (2) s (2) (A11 ) + (0) x (T ) x + (4) s 4 (2) (A11 ) + (32) and x = (2) x 4 (2) (A11 ) + (34) 2 (2) (A11 ) + (4) x 4 (2) (A11 ) + 2 6 Wm (35a) + ¯ h W 1 + 1/4 * * * * C3 [D11D22A11 A22 ] h 5 (35b) In this section, the buckling and postbuckling behaviors of perfect and imperfect, GRC laminated cylindrical shells under torsion in a thermal environmental condition are considered. In numerical calculations, Poly (methyl methacrylate), referred to as PMMA, is selected for the matrix, and the material properties of PMMA matrix are adopted as m m = 0.34, = 45(1 +0.0005 T) × 10−6/K and m E = (3.52–0.0034 T) GPa, where T = T0 + T and T0 = 300 K (room temperature). Also, we select graphene sheets of zigzag type (referred to as 0-ply) as reinforcements. The anisotropic and temperature dependent material properties of a monolayer graphene with effective thickness G hG = 0.188 nm are listed in Table 1 [44]. As discussed previously, the Halpin–Tsai model needs to be modified to account for the small scale effect and other effects on the GRC material properties through the introduction of the graphene efficiency parameters j (j = 1,2,3). The values of the efficiency parameters 1, 2 and 3 are determined by matching the Young's moduli E11 and E22 and shear modulus G12 of GRCs predicted from the Halpin–Tsai model to those from the MD simulations [37,38]. These temperature dependent graphene efficiency parameters are listed in Table 2. Also, we assume that G13 = G23 = 0.5 G12. Since this is an unsolved problem, there are no relevant works available for direct comparison in the open literature. The correctness and accuracy of the boundary layer theory and the two-step perturbation method solutions for many examples of isotropic and composite laminated cylindrical shells subjected to torsion were verified by Shen [15,22,48]. In addition, the buckling torques for clamped, FG-CNTRC cylindrical shells under an environmental temperature of T = 300, 400 or 500 K are calculated and compared in Table 3 with the Galerkin method solutions of Ninh [49] based on the classical thin shell theory. In Table 3, the shell has R/h = 30 and h= 2 mm, the extended Voigt model (rule of mixture) is adopted to determine the temperature dependent material properties of CNTRCs and the CNT efficiency parameters are taken to be 1= 0.141, 2 = 1.585 and 3 = 1.109 for the * = 0.28. Moreover, the buckling loads for (0/90/0)S lamicase of VCN nated cylindrical shells under torsion are calculated and compared in Table 4 with the FEM results of Park et al. [50]. The computing data adopted are: R = 190.5 mm, R/h = 100, E11 = 149.62 GPa, E22 = 9.92 GPa, G12 =G13 = 4.48 GPa, G23 = 2.55 GPa and 12 = 0.28. These two additional comparison studies confirm that the present solutions agree well with the existing results. A parametric study has been carried out and typical results are in which µ is defined as the imperfection parameter. Substituting the regular and boundary layer displacements and forces in Eqs. (26)–(27) into the governing Eqs. (16)–(19), we obtain three sets of perturbation equations by collecting terms with the same order of . These regular and boundary layer perturbation equations may be solved step-by-step. Then we use Eqs. (30) and (31) to solve these perturbation equations of each order and to match the regular solutions with the boundary layer solutions at each end edge of the shell. Thereafter, we obtain the asymptotic solutions W, F, x and y in the postbuckling region (as shown in Appendix B). It is evident that, from Eq. (B.1), the shell has nonlinear prebuckling deformations. Also, from Eq. (B.2) we observed that there exists a shear stress in conjunction with an associate compressive stress when the GRC laminated cylindrical shell is subjected to torsion. The torsional buckling and postbuckling behaviors of GRC laminated cylindrical shells will be influenced by this compressive stress. Unfortunately, this compressive stress was neglected in many previous works for torsional buckling of isotropic/anisotropic cylindrical shells [16–18,20,21]. Substituting solution F into the boundary condition (22c), and solutions W, F, x and y into Eqs. (24) and (25), the postbuckling loaddeflection, load-shortening and load-rotation relationships can be obtained as s (4) s 4. Numerical results and discussion where the continuous variable k needs to be determined firstly through Eq. (C.3). The initial geometric imperfection is assumed to have the form W * (x , y , ) = 2 (2) (A11 ) + In Eq. (32)–(35) all symbols are described in detail in Appendix C. It is noted that s(i) , x(i ) and s(i ) (i = 0,2,4,…) are also temperature dependent for GRC laminated shells. By minimizing the buckling load in Eq. (32) with respect to m and n, which denote the buckling mode half-wave and full wave numbers in the X and Y directions respectively, the initial buckling load along with the associate buckling mode (m, n) of a perfect GRC shell can be obtained simultaneously. In the next section, nonlinear curves of postbuckling load-deflection, load-shortening, and load-rotation relationships will be presented for FG-GRC laminated cylindrical shells subjected to torsion and in a thermal environmental condition. (28) x )/ (2) A11 (2) A02 (2) s in which (x , , y ), ˆ y (x , , y , ) = , + (2) = Wm A11 j=1 = x/ (0) s (2) In Eqs. (32)–(34), we take ( A11 ) as the second perturbation parameter which relates to the dimensionless maximum deflection Wm. By taking (x, y) = ( (1 + k/n)/2 m, /2 n) in Eq. (B.1), we have Fˆ (x , , y, ) = = (33) 564 Thin-Walled Structures 135 (2019) 560–574 H.-S. Shen, Y. Xiang Table 1 Temperature-dependent material properties for monolayer graphene (aG = 14.76 nm, bG = 14.77 nm, thickness hG = 0.188 nm, Temperature(K) G E11 (GPa) G E22 (GPa) G G12 (GPa) 300 400 500 1812 1769 1748 1807 1763 1735 683 691 700 Table 2 Temperature dependent efficiency parameters of graphene/PMMA nanocomposites. T (K) VG 300 0.03 0.05 0.07 0.09 0.11 0.03 0.05 0.07 0.09 0.11 0.03 0.05 0.07 0.09 0.11 400 500 2 1 2.929 3.068 3.013 2.647 2.311 2.977 3.128 3.060 2.701 2.405 3.388 3.544 3.462 3.058 2.736 T (K) 11.842 15.944 23.575 32.816 33.125 13.928 15.229 22.588 28.869 29.527 16.712 16.018 23.428 29.754 30.773 Source UD FG-X FG- 300 Present and Shen [22] Ninh [49] Present and Shen [22] Ninh [49] Present and Shen [22] Ninh [49] 4.03 4.16 3.51 3.64 2.63 2.87 5.17 5.40 4.56 4.76 3.68 3.85 3.56 3.67 3.11 3.30 2.41 2.47 400 500 Park et al. [50] Present 1 5 0.1714 0.0806 0.1706 (1,3)a 0.0936 (1,2) a (× 10−6/K) G 11 ) [44]. (× 10−6/K) − 0.95 − 0.40 − 0.08 Lay-up UD FG-X FG-V FG- FG-O 18.98b 18.99b 19.04b 17.07b 17.26b 17.31b 16.55b 16.81b 16.88b 20.58b 20.48b 20.62b 18.64b 18.81b 18.83b 18.31b 18.58b 18.57b (+8.4%) (+7.8%) (+8.3%) (+9.2%) (+9.0%) (+8.8%) (+10.6%) (+10.5%) (+10.0%) 14.52b 14.61b 14.66b 13.24b 13.37b 13.38b 13.27b 13.37b 13.34b 14.98b 15.04b 15.10b 13.50b 13.62b 13.71b 13.33b 13.41b 13.55b 12.97b 13.09b 13.03b 11.62b 11.81b 11.77b 11.40b 11.62b 11.61b 20.73b 20.71b 20.89b 18.68b 18.83b 18.91b 18.43b 18.28b 18.58b 22.40b (+8.1%) 22.65b (+9.4%) 22.84b (+9.3%) 20.65b (+10.5%) 20.80b (+10.5%) 20.84b (+10.2%) 19.95c (+8.2%) 20.46c (+11.9%) 20.53c (+10.5%) 15.45b 15.70b 15.68b 14.28b 14.37b 14.53b 14.33b 14.56b 14.51b 16.01b 16.18b 16.18b 14.42b 14.64b 14.66b 14.66b 14.66b 14.60b 13.58b 13.66b 13.78b 12.40b 12.55b 12.57b 12.31b 12.48b 12.51b 20.00c 20.05c 20.06c 16.91c 17.42c 17.53c 14.73c 16.18c 16.47c 21.09c 21.08c 21.19c 18.14c 18.58c 18.72c 16.49c 17.66c 17.89c 16.22c 16.25c 16.28c 13.50c 14.13c 14.26c 11.47c 13.09c 13.46c 16.49c 16.51c 16.57c 13.54c 14.14c 14.32c 11.50c 13.40c 13.47c 14.80b 15.01c 14.98c 11.97c 12.73c 12.87c 9.49c 11.56c 11.84c (+5.5%) (+5.1%) (+5.6%) (+7.3%) (+6.7%) (+6.9%) (+11.9%) (+9.1%) (+8.6%) a Difference = 100%[(Ms)cr(FG) - (Ms)cr(UD)]/(Ms)cr(UD). b Buckling mode (m, n) = (1, 2). c Buckling mode (m, n) = (2, 2). Table 4 Comparisons of buckling loads (Nxy)cr (N/m × 106) for (0/90/0)S cylindrical shells subjected to torsion (R = 190.5 mm, R/h = 100, E11 = 149.62 GPa, E22 = 9.92 GPa, G12 = G13 = 4.48 GPa, G23 = 2.55 GPa, 12 = 0.28). L/R 3 G = 4118 kg/m − 0.90 − 0.35 − 0.08 Z̄ = 400 300 (0)10 (0/90/0/90/0)S (0/90)5T 400 (0)10 (0/90/0/90/0)S (0/90)5T 500 (0)10 (0/90/0/90/0)S (0/90)5T Z̄ = 600 300 (0)10 (0/90/0/90/0)S (0/90)5T 400 (0)10 (0/90/0/90/0)S (0/90)5T 500 (0)10 (0/90/0/90/0)S (0/90)5T Z̄ = 800 300 (0)10 (0/90/0/90/0)S (0/90)5T 400 (0)10 (0/90/0/90/0)S (0/90)5T 500 (0)10 (0/90/0/90/0)S (0/90)5T Table 3 Comparisons of torque (Ms)cr (in kN*m) for perfect, CNTRC cylindrical shells * = 0.28, (m, n)= (3, subjected to torsion [Z̄ = 800, R/h = 30, h= 2 mm, VCN 2)]. T(K) = 0.177, Table 5 Buckling torque (Ms)cr (in kN*m) for perfect, GRC laminated cylindrical shells subjected to torsion (R/h = 30, h = 2 mm). 3 2.855 2.962 2.966 2.609 2.260 2.896 3.023 3.027 2.603 2.337 3.382 3.414 3.339 2.936 2.665 G 11 G 12 distribution of graphene volume fractions is inversed, i.e. [(0.03)2/ (0.05)2/(0.07)2/(0.09)2/(0.11)2], referred to as FG- . And for Type X, a mid-plane symmetric graded distribution of graphene volume fractions is achieved, i.e. [0.11/0.09/0.07/0.05/0.03]S, while for type O, the graphene volume fractions are assumed to have symmetric graded distribution [0.03/0.05/0.07/0.09/0.11]S, referred to as FG-X and FGO, respectively. In such a way, the shells of UD and FG types will have the same value of total volume fraction of graphene. It is noted that, in ¯ */h represents the dimensionless maximum initial geoFigs. 2–5, W ¯ / h represents the additional metric imperfection of the shell and W deflection of the shell, respectively. Table 5 presents the buckling torque (MS)cr (in kN*m) for perfect, (0)10, (0/90/0/90/0)S and (0/90)5T GRC shells subjected to torsion with different environmental temperatures. Four types of FG-GRC shells are considered in Table 5 that are referred to as FG-V, FG- , FG-X and FG-O. The results for a UD-GRC laminated cylindrical shell are also listed in Table 5 for direct comparisons. It has been reported that the glass transition temperature of PMMA will have a significant increase when graphene sheets are added in PMMA [52]. Hence, in the present study, the environmental temperatures are taken to be T = 300, 400 and 500 K. Table 5 shows that the buckling torques of the FG-X shell are greater than those of the UD-GRC laminated cylindrical shell. As shown Buckling mode (m, n). shown in Table 5 and Figs. 2–5. For all cases discussed below, the GRC shell consists of ten plies with each ply having identical thickness of 0.2 mm, so that the total thickness of the shell is h = 2 mm. The geometric parameter of the shell Z̄ ( = L2/Rh) is set to be 400, 600 and 800, and R/h is of the value 30. The shell is clamped on both ends. Apart from the UD GRC laminated cylindrical shell where each ply has the same value of the graphene volume fraction, i.e. VG = 0.07, four types of FG-GRC laminated cylindrical shells, namely FG-V, FG- , FG-X and FG-O, are considered. It has been reported that GRCs may contain the volume fraction of graphene reinforcement by up to 21% [51]. Accordingly, for Type V, the graphene volume fractions of ten plies are assumed to have piece-wise graded distribution [(0.11)2/(0.09)2/ (0.07)2/(0.05)2/(0.03)2], referred to as FG-V, while for Type , the 565 Thin-Walled Structures 135 (2019) 560–574 H.-S. Shen, Y. Xiang Fig. 3. Effect of shell parameter Z̄ on the postbuckling behavior of (0/90/0/90/ 0)S GRC laminated cylindrical shells subjected to torsion: (a) load-shortening; (b) load-rotation. Fig. 2. Postbuckling behavior of (0/90/0/90/0)S GRC laminated cylindrical shells with different types of graphene reinforcements subjected to torsion: (a) load-shortening; (b) load-rotation. curves of FG-X and UD (0/90/0/90/0)S GRC laminated shells with Z̄ = 400, 600 and 800 subjected to torsion at T = 300 K. Unlike in the case of FG-CNTRC shells [22], in the present case the shell with Z̄ = 600 has a higher postbuckling strength and a lower slope of the postbuckling load-shortening and load-rotation curves than the shells with Z̄ = 400 and 800. The postbuckling equilibrium paths of (0/90/0/ 90/0)S GRC laminated shells with Z̄ = 400 and 600 are weakly unstable, while the postbuckling equilibrium paths of (0/90/0/90/0)S GRC laminated shells with Z̄ = 800 is unstable. This is because the GRC laminated shell with Z̄ = 800 has the buckling mode (m, n) = (2, 2), while the GRC laminated shells with Z̄ = 400 and Z̄ = 600 have the buckling mode (m, n) = (1, 2). Fig. 4 shows the effect of temperature changes on the torsional postbuckling behavior of FG-X and UD (0/90/0/90/0)S GRC laminated shells with Z̄ = 400 in a thermal environment of T = 300, 400 and 500 K. As expected, the rise of temperature decreases the elastic moduli and the strength of the GRCs due to the temperature dependent material properties of both graphene sheets and PMMA. Like in the case of FGCNTRC shells [22], an initial extension occurs in the postbuckling loadshortening curves when temperature rises. We observe that both buckling torque and torsional postbuckling strength are decreased as in the brackets, the buckling torques are increased by about 8–11% for the shell with Z̄ = 400 and about 5–12% for the shell with Z̄ = 800. Hence, we consider only the FG-X GRC laminated cylindrical shell in the following examples, except for Fig. 2. We observe that the buckling torque is decreased when temperature rises. The percentage of the decrease is about 11.5% for the UD (0/90/0/90/0)S GRC laminated shell and is about 9.3% for the FG-X (0/90/0/90/0)S GRC laminated shell with Z̄ = 400 when temperature rises from T = 300 K to T = 500 K. Fig. 2 illustrates the postbuckling load-shortening and load-rotation curves for (0/90/0/90/0)S GRC shells with four FG patters of types V, , O and X and subjected to torsion at T = 300 K. Fig. 2 also includes the case for UD-GRC laminated shell for comparison purpose. All considered shells have the shell geometric parameter Z̄ = 400. We observe that the FG-X GRC laminated cylindrical shell has the highest, while the FG-O GRC laminated cylindrical shell has the lowest postbuckling strength among the five shells. Unlike in the case of FG-CNTRC shells [22], the postbuckling load-shortening and load-rotation curves of both FG- and FG-V GRC laminated shells are lower than those of the UD GRC shell. Fig. 3 presents the postbuckling load-shortening and load-rotation 566 Thin-Walled Structures 135 (2019) 560–574 H.-S. Shen, Y. Xiang Fig. 4. Effect of temperature variation on the postbuckling behavior of (0/90/ 0/90/0)S GRC laminated cylindrical shells subjected to torsion: (a) loadshortening; (b) load-rotation. Fig. 5. Postbuckling behavior of three GRC laminated cylindrical shells subjected to torsion: (a) load-shortening; (b) load-rotation. temperature increases for FG-X and UD (0/90/0/90/0)S GRC laminated shells. Fig. 5 presents the postbuckling load-shortening and load-rotation curves of FG-X and UD (0)10, (0/90/0/90/0)S and (0/90)5T GRC laminated shells with Z̄ = 400 subjected to torsion at T = 300 K. It is observed that the postbuckling load-shortening curves and load-rotation curves of shells with three different lamination arrangements have negligible difference for both UD and FG-X GRC laminated cylindrical shells. used to determine the anisotropic and temperature dependent material properties of GRCs. The postbuckling equations are solved by applying a singular perturbation method along with a two-step perturbation approach. Parametric studies for UD and FG-GRC laminated cylindrical shells with low graphene volume fractions have been carried out. The numerical results reveal that the FG-X shell has a higher buckling torque and postbuckling strength than the other types of FG-GRC laminated shells. The results confirm that the postbuckling equilibrium path is weakly stable for GRC laminated cylindrical shells under torsion and the shell structure is virtually imperfection-insensitive. 5. Concluding remarks Acknowledgments New findings on the torsional postbuckling behaviors of FG-GRC laminated cylindrical shells under torsion and in a thermal environmental condition have been presented. Each ply of the shell may have different volume fraction of graphene which results in a piece-wise functionally graded pattern. The extended Halpin-Tsai micromechanical model is The supports for this work, provided by the National Natural Science Foundation of China under Grant 51779138, and the Australian Research Council grant DP140104156 are gratefully acknowledged. 567 Thin-Walled Structures 135 (2019) 560–574 H.-S. Shen, Y. Xiang Appendix A In Eqs. (5)–(8) the linear and nonlinear operators are defined by 4 4 4 4 * * + F21 * + 4F66 *) * F11 + (F12 + F22 3h2 X4 X2 Y 2 Y4 L˜11 ( )= L˜12 ( * ) = D11 L˜13 ( * + 2D66 *) ) = (D12 4 * L˜14 ( ) = B21 X2 2 ¯ T) = L˜16 (M 4 X4 2 L˜15 (N¯ T ) = 3 3 4 * + 2D66 *) * + 2F66 *) (F12 + (D12 3h2 X3 X Y2 3 3 4 4 * + 2F66 *) * * (F21 F22 + D22 X2 Y Y3 3h2 3h2 4 * F11 3h2 X2 * + B22 * + (B11 *) 2B66 2 * N¯ xT + B21 * N¯ Ty ) + 2 (B11 2 ¯ xT ) + 2 (M 4 L˜ 21 ( * ) = A22 L˜ 24 ( )= * + B12 T * N¯ xy (B66 )+ X Y 2 T ¯ xy (M )+ X Y 4 X2 Y 2 Y2 * + A66 *) + (2A12 2 Y2 * N¯ xT + B22 * N¯ Ty ) (B12 ¯ yT ) (M 4 4 * + A11 Y4 3 3 4 4 ˜ * * * B66 *) * E66 *) L22 ( ) = B21 E21 (E11 + (B11 X3 X Y2 3h2 3h2 3 3 4 4 * B66 *) * E66 *) * * L˜ 23 ( ) = (B22 E12 + B12 (E22 X2 Y Y3 3h2 3h2 2 X2 X Y 8 16 D55 + 4 F55 h2 h 4 * * F11 H11 3h2 4 *) 2E66 2 * N¯ xT + A22 * N¯ yT ) (A12 ) = A55 4 + 2 3h X2 Y 2 4 4 * * + E22 * E21 + (E11 X4 3h2 L˜ 25 (N¯ T ) = L˜ 31 ( X4 Y4 T * N¯ xy (A66 )+ 2 Y2 Y4 * N¯ xT + A12 * N¯ Ty ) (A11 X 3 X3 4 * + E12 X2 Y 2 3 4 * + 2H66 *) (H12 3h2 * + 2F66 *) + (F21 X Y2 8 16 D55 + 4 F55 h2 h 2 2 8 16 8 16 * * * * * + * D11 F H D66 F66 H66 + 11 11 X2 Y2 3h2 9h4 3h2 9h4 4 16 * + D66 *) * + F21 * + 2F66 *) + * + H66 *) L˜ 33 ( ) = (D12 (F12 (H12 3h2 9h4 L˜ 34 ( ) = L˜ 22 ( ) L˜ 32 ( ) = A55 L˜ 35 (N¯ T ) = 4 * N¯ xT + B21 * E11 3h2 T L˜ 36 (S¯T ) = L˜ 41 ( * B11 X X (S¯x ) + X 2 (S¯xy) + 2 X2 Y 2 3 X2 Y * + F22 8 16 * + * F22 H22 3h2 9h4 * B12 3 4 * H22 3h2 Y3 2 Y2 4 * N¯ xT + B22 * E12 3h2 4 * N¯ yT E22 3h2 T Y (S¯y ) 2 2 4 T * N¯ xy E66 3h2 Y 4 T * N¯ xy + E66 3h2 Y T L˜ 46 (S¯T ) = L˜ ( ) = * B66 X Y (S¯xy ) 4 4 * + 2F66 *) * + 2H66 *) (H12 + 2 (F12 3h 3h2 L˜ 42 ( ) = L˜ 33 ( ) 8 16 L˜ 43 ( ) = A 44 D44 + 4 F44 h2 h 2 8 16 * * * * D66 F H D22 + 66 66 3h2 9h 4 X2 L˜ 44 ( ) = L˜ 23 ( ) L˜ 45 (N¯ T ) = * B66 + T Y 8 16 D44 + 4 F44 h2 h ) = A 44 4 * N¯ Ty E21 3h2 2 X Y 2 2 + X Y X Y 2 (A.1) Y 2 X2 In the above equations, [ Aij* ], [Bij*], [Dij*], [Eij*], [Fij* ] and [Hij*] are the reduced stiffness matrices determined through relationships [53] A* = A 1, B* = A 1B , D* = D–BA 1B , E* = A 1E, F * = F–EA 1B , H* = H–EA 1E (A.2) where Aij, Bij, Dij, etc., are the shell panel stiffnesses that are defined by N (Aij , Bij , Dij , Eij, Fij , Hij ) = k=1 N (Aij , Dij , Fij ) = k=1 tk tk 1 tk tk 1 (Q¯ ij )k (1, Z , Z 2, Z 3, Z 4, Z 6) (Q¯ ij ) k (1, Z 2, Z 4 ) dZ (i, j = 1, 2, 6) (i , j = 4, 5) (A.3a) (A.3b) 568 Thin-Walled Structures 135 (2019) 560–574 H.-S. Shen, Y. Xiang Appendix B For a shell under torsion, the solutions W, F, (5/4) A 00 5/4 W= x (5/4) (5/4) A 00 a01 cos 2 [A (2) 00 + x (5/4) (5/4) A00 a01 cos (2) + A11 sin(mx and x x (5/4) + a10 sin x (5/4) + a10 sin in the postbuckling region may be expressed as y exp x x exp (2) ky )sin ny + a11 cos(mx (2) ky )cos ny + A02 cos 2ny (2) (2) (2) (2) (2) (A00 A11 sin ky sin ny + a11 cos ky cos ny + A02 cos 2ny ) a01 cos x x (2) sin +a10 exp x (2) (2) (2) (2) (2) (A00 + ( 1)m 1A11 sin ky sin ny + ( 1)ma11 cos ky cos ny + A02 cos 2ny ) a01 cos (3) (3) 3 [A (3) + A (3) sin(mx ky )sin ny + a11 cos(mx ky )cos ny + A02 00 11 (4) (4) 4 [A (4) + A (4) sin(mx ky )sin ny + a11 cos(mx ky )cos ny + A20 00 11 (4) (4) (4) cos 2ny + A13 sin(mx ky )sin 3ny + a13 cos(mx ky )cos 3ny A02 5 O( ) + + + + (0) B00 F= + 9/4 (2) B00 2 y2 2 (0) C00 xy + y2 2 ( ( x x (5/4) (9/4) + A00 b01 cos (2) + (A 00 (3) B00 3 y2 2 y2 2 x exp x cos 2ny] cos 2(mx ky ) (4) + A04 cos 4ny] (B.1) ky )sin ny x (9/4) + b10 sin x (9/4) + b10 sin (2) +a10 sin (1) C00 xy (2) (2) C00 xy + B11 sin(mx (5/4) (9/4) A00 b01 cos + (1) B00 x ) exp ( ) x ) exp ( x ) (3) (3) C00 xy + B02 cos 2ny (2) (2) (2) A11 sin ky sin ny + a11 cos ky cos ny + A 02 cos 2ny ) (b (3) 01 x cos x (3) sin + b10 ) exp ( x ) (2) (2) (2) + (A00 + ( 1)m 1A11 sin ky sin ny + ( 1)ma11 cos ky cos ny (b (2) cos 2ny + A02 + 4 (4) B00 y2 2 (4) B13 sin(mx x = 7/4 + 2 [C (2) 11 + 5/2 (3) 01 x cos (4) (4) C00 xy + B20 cos 2(mx (4) ky )sin 3ny + b13 cos(mx (5/4) (7/4) A00 c10 sin cos( mx (2) A00 x (3) sin + b10 x ky )sin ) exp ( x ) (4) ky ) + B02 cos 2ny ky )cos 3ny ] + O ( 5) x exp (B.2) x (5/4) (7/4) + A00 c10 sin (2) ny + c11 sin( mx ky )cos x exp ny ] (2) (2) (2) (5/2) A11 sin ky sin ny + a11 cos ky cos ny + A 02 cos 2ny c10 sin x exp x (2) (2) (2) + (A00 + ( 1)m 1A11 sin ky sin ny + ( 1)ma11 cos ky cos ny (2) (5/2) + A02 cos 2ny c10 sin + + + y + + + 3 [C (3) cos( 11 4 [C (4) cos( 11 (4) cos(mx C13 = 2 [D (2) 11 mx mx ky )sin ky )sin ky )sin 3ny + sin(mx x x exp (3) ny + c11 sin( mx ny + (4) c11 sin( mx (4) c13 sim (mx ny] ky )cos (4) sin 2( mx ny + C20 ky ) ky )cos 3ny] + O ( 5) (2) ky )cos ny + d11 cos(mx (3) 3 [D (3) sin(mx cos(mx ky )cos ny + d11 11 (4) 4 [D (4) sin(mx cos(mx ky )cos ny + d11 11 (4) (4) sin(mx ky )cos 3ny + d13 cos(mx D13 ky )cos (B.3) ky )sin ny ] (3) sin 2ny] ky )sin ny + D02 (4) sin 2ny ky )sin ny + D02 ky )sin 3ny] + O ( 5) (B.4) In the above equations, all coefficients are related and can be expressed in terms of by substituting Eqs. (B.1)-(B.4) into periodicity condition (23). 569 (2) A11 , except for (j ) A00 (j = 5/4,2,3,4) which may be determined Thin-Walled Structures 135 (2019) 560–574 H.-S. Shen, Y. Xiang Appendix C In Eqs. (32)–(34) m 2n (0) s = 2 (g210 2 24 m g220 2 )(1 + µ) g220 5/4 + 24 2 (1 + µ)2 (g210 2 ) g220 1+µ g210 g32 + g220 g31) + (g220 g310 + g210 g320)] 1/4 g31 (g220 g310 + g210 g320) + g32 (g220 g320 + g210 g310) g120 1 24 + 2 + 2 2 (1 + µ) m (1 + µ) g210 g220 14 24 µ (1 + µ)2 2 + g2 ) 2g210 g310 g320 + g220 (g310 320 2 2 g210 g220 g110 g320 + g120 g310 µ m4 (1 + µ)2 + 3/4 2 2 24 g31 2g210 g310 g320 + g220 (g310 + g320 ) 2 2 g210 g220 (1 + µ) 14 2 2 24 g32 g210 (g310 + g320 ) + 2g220 g310 g320 2 2 g210 g220 2 + 3g 2 ) + g 2 2 g g g ( µ 220 310 210 g320 (3g310 + g320 ) 310 320 24 2 2 (1 + µ)2 g210 g220 2 + g 2 ) + 2g g120 (g310 µ2 110 g310 g320 320 + 6 m (1 + µ)3 14 + (1 + µ) 7/4 2 2 2 2 24 g31 g220 g310 (g310 + 3g320 ) + g210 g320 (3g310 + g320 ) 2 2 g210 g220 2 2 2 + g2 ) + g g g220 g320 (3g310 210 g310 (g310 + 3g320 ) 320 + 24 32 2 2 (1 + µ) g210 g220 4 2 2 4 2 2 24 µ 4g210 g310 g320 (g310 + g320 ) + g220 (g310 + 6g310 g320 + g320 ) 2 2 (1 + µ)2 g210 g220 + (1 + µ) m 2n (2) s = 2 6 14 24 m g210 g220 2 )2 g220 5/4 2 2g09 (g210 2 4 14 24 m 2 8g09 g210 (g210 2 ) g220 2 + g2 g 3g210 220 220 g310 2 2 + 3g 2 g + g210 220 210 g320 g32 1/4 + 2 4 14 24 m g220 [g (1 + 2µ) 2 )(1 + µ ) 14 g220 + g31 2 4 2 2 14 24 m g320 (g210 + 3g220 ) 2 )2 (1 + µ ) g220 1/4 2 2g09 (g210 g302 (1 + µ)2] 2 4g09 (g210 + 11/4 , 1/4 µ 14 24 m2 2 + g 2 ) + 2g g120 (g210 110 g210 g220 220 2 2 g220 8g09 g210 (g210 ) 14 2 + g 2 )(3g 2 + g 2 ) + 2g 2 2 g220 (g310 210 g310 g320 (g210 + 3g220 ) 320 210 220 2 2 1+µ g210 g220 24 24 (1 + µ) 2 + 3g 2 )(g g 2 2 g210 (g210 220 32 310 + g31 g320 ) + g220 (3g210 + g220 )(g31 g310 + g32 g320 ) 2 2 g210 g220 3/4 2 2 )[2(1 + µ)2 + 3(1 + 2µ)] + g g220 2 m2n4 4g220 (g210 210 g200 (1 + µ) 3/4 g 1 + 2µ 3/4 + 2 24 2 2 2 g210 13 g220 (g210 g220)(1 + µ) 4(g210 )(1 + µ) g210 g200 2 24 m g220 8 2 2 14 24 m g220 2 2 ) g220 8g09 (g210 2 2 2 2 14 24 m g320 g302 (g210 + g220 ) 3/4 4g310 g14] 3/4 + 2 2 ) g220 4g09 g210 (g210 [g210 g302 g11 2 2 14 24 m g14 (1 + 2µ) 2 2 )(1 + µ)2 8g09 g210 (g210 g220 2 + g 2 )g 3/4 [(g210 220 320 + 2g210 g220 g310] 2 2 2 + g 2 )g (3g210 220 220 g320 + (g210 + 3g220 ) g210 g310 2 2 14 24 m g320 2 2 2g09 g210 (g210 )(1 + µ) g220 2 g210 3/4 2 g220 2 2 2 2 14 24 m (3g210 + g220 ) g220 g11 (g31 g310 ) 3/4 2 2 )2 (1 + µ) 16g09 (g210 g220 2 2 2 2 14 24 m (g210 + 3g220 ) g210 g11 (g32 g320 ) 3/4 , 2 2 )2 (1 + µ) 16g09 (g210 g220 (4) s = + 2 3 10 2 + 3g 2 ) g220 (g210 m 14 24 m (1 + µ) 220 4 2 (g 2 2 ) 2 2 2 2n 64g09 g g ( 210 220 210 g220 ) (1 + µ) 2 2 )(g 2 g 2 ) g 2 g220 (g210 23 24 210 g220 2 g210 2 2 2 2 2 g24 (g210 g220 + g220 ) + 2g210 g220 g23 3g210 2 2 2 g24 (g210 2g210 + 3g220 )] + 2 (1 + µ) g220 g23 (3g210 2 g210 2 g23 2 (1 + µ) g220 g24 (3g210 2 ) g220 2 g23 2 g210 g g g 2 220 220 23 2 g210 g23 2 ) g220 g210 g24 2 g24 2 + g2 ) g210 g24 (g210 220 2 g24 g g g + 2 220 210 23 2 g210 g23 g220 g24 2 g24 g + 6 220 R1 g210 2 + g2 ) g210 g23 (g210 220 2 g24 g2 + g2 + 210 2 220 R2 g210 5/4 , (C.1) 570 Thin-Walled Structures 135 (2019) 560–574 H.-S. Shen, Y. Xiang and (0) p k m = (0) s (0) xp , + (2) p k m = (2) s (2) xp , + (4) p = k m (4) s + (4) xp , (C.2) in which k may be determined through the equation (0) xp (2) xp 2 (2) (A11 ) + (4) xp 4 (2) (A11 ) =0 (C.3) where (0) xp 1 2 = 2 (g210 2 24 m g210 2 )(1 g220 5/4 1 m2 (1 + µ) g110 2 g210 (1 + µ) 14 2 2 2g220 g310 g320 + g210 (g310 ) + g320 24 µ 2 2 (1 + µ)2 g210 g220 µ m4 (1 + µ)2 2 ) g220 1+µ g210 g31 g31 (g220 g320 + g210 g310) + g32 (g220 g310 + g210 g320 ) 24 + 2 (1 + µ)2 (g210 1/4 + g220 g32) + (g220 g320 + g210 g310 )] + 24 + + µ) g110 g310 + g120 g320 + 2 g220 3/4 24 g31 2 2 2g220 g310 g320 + g210 (g310 ) + g320 2 g210 (1 + µ) 14 2 ) + 2g210 g310 g320 + g320 24 g32 2 2 (1 + µ) g210 g220 2 2 2 g210 g310 (g310 + 3g320 ) + g220 g320 (3g310 24 µ 2 2 2 (1 + µ) g210 g220 2 g220 2 g220 (g310 + µ2 + µ )3 + 2 ) + g320 7/4 2 2 ) + 2g120 g310 g320 g110 (g310 + g320 m6 (1 14 2 2 2 2 24 g31 g220 g320 (3g310 + g320 ) + g210 g310 (g310 + 3g320 ) + 2 g210 (1 + µ) 2 g220 g310 (g310 24 g32 + (1 + µ) 2 4g220 g310 g320 (g310 24 µ (1 + (2) xp = + 2 3g320 ) 2 g210 + µ )2 2 2 6 14 24 m (g210 2 4g09 (g210 1 2 2 4 14 24 m 2 8g09 g210 (g210 4 2 2 4 + 6g310 + g320 g210 (g310 g320 ) 11/4 2 g220 5/4 2 2 g220 ) 2 2 g220 g320 3g210 + g220 1/4 g31 µ) + g32 2 2 2 4 14 24 m g220 g320 (3g210 + g220 ) 2 2 2 2g09 g210 (g210 g220) (1 + µ) [g14 (1 + 2µ) g302 (1 + µ)2] 1/4 1/4 2 2 g110 (g210 ) + 2g120 g210 g220 + g220 2 g220) 14 2 2 2 2 2 )(g210 ) + 2g220 g310 g320 (3g210 ) + g320 + 3g220 + g220 2 2 g210 g220 2 2 2 2 + 3g220 + g220 )(g31 g310 + g32 g320 ) + g220 (3g210 )(g32 g310 g210 (g210 2 2 g210 g220 2 14 24 m 2 8g09 g210 (g210 2 g210 (g310 24 µ + 2 g220 2 g320 )+ 2 g210 2 2 g220 ) 2 2 2 4 14 24 m (g210 + g220 ) 2 2 8g09 g210 (g210 g220)(1 + + 2 2 + g210 g320 (3g310 + g320 ) 2 + g220 ) 2 2 + g210 + 3g220 g210 g310 + 2 g220 1+µ 24 (1 + µ ) 2 24 m 2 2 + g220 g210 16 2 g210 g13 1 + 2µ + g31 g320) 3/4 2 2 4 4 (g 2 + g 2 ) (g 2 g220 )[2(1 + µ)2 + 3(1 + 2µ)] + g210 g200 (1 + µ) 3/4 24 m n 210 210 220 2 2 2 2 g210 (g210 g220)(1 + µ) g220 4(g210 )(1 + µ) g210 g200 2 2 2 2 2 2 ( ) m g + g 14 24 14 24 m g320 g302 g220 3/4 210 220 [g210 g302 g11 4g310 g14] 3/4 + 2 2 2 2 g220 16g09 g210 (g210 g220) 2g09 (g210 ) 2 2 14 24 m g14 (1 + 2µ ) 2 [(g 2 + g220 ) g310 + 2g210 g220 g320] 3/4 2 2 8g09 g210 (g210 )(1 + µ)2 210 g220 2 2 2 2 2 2 (3g210 ) g220 g310 + (g210 ) g210 g320 3/4 + g220 + 3g220 14 24 m g320 2 2 2 2 2g09 g210 (g210 )(1 + µ) g220 g210 g220 571 3/4 Thin-Walled Structures 135 (2019) 560–574 H.-S. Shen, Y. Xiang 2 2 2 14 24 m (3g210 2 + g220 ) g220 g11 (g32 2 16g09 (g210 2 2 2 14 24 m (g210 2 ) g210 g11 (g31 + 3g220 2 16g09 (g210 (4) xp + 2 2 g210 + g220 g210 g23 = 3/4 1 g220 g24 2 g23 g210 2 g24 2 2 + g220 g210 g220 g23 1+µ (0) x g310 ) 2 2 ) (1 + µ) g220 2 24 2 g210 g24 2 g23 g210 4 2 g24 (9/4) b01 2 + = 1 2 2 3g210 + g220 + 2g210 + 2g210 2 5 1/2 (9/4) b10 24 (2) x , (1 + µ ) 2 g g 2 + g220 2 2 2 )(g23 ) g210 23 210 g220 g24 1+µ + 3/4 2 2 2 2 3 10 + g220 (3g210 ) g220 1 14 24 m (1 + µ) 4 2 2 2 2 2 2 2 64g09 ) g210 (g210 g220 (g210 g220) = 2 (g210 + g320) 2 2 g220 ) (1 + µ) g210 g23 g220 g24 2 g23 g210 g24 2 g23 (T ) x = 2 2 ) g310 (g210 + g220 2 3 (g210 + 2 m 1/4 2 ) + g320 = 1 32 + 2m2n4 1 [( (0) s = 266 2 24 T 1 5 T 2) 1/4 T] 2 2 g210 + g220 3/4 2 g210 7/4 4g210 g220 g310 g320 11/4 2 g210 b11 64 4 (1 5/4, (C.4) 2 2 2 2 2 2 2 g310 (g210 + g220)(g310 + 3g320) 2g320 g210 g220 (3g310 + g320) 2 4 m g210 (4) x R1 g210 24 + m2 1 + 2µ 2g320 g210 g220 2 g210 2 2 g220)(g310 2 2 g210 + g220 2g220 R2 2 g24 24 2 2 1 b11 g210 + g220 2 8 2 g210 +3 2 g24 g220 g23 p, 2 2 2g24 g210 g220 2g210 + g220 2 2 8 2 14 24 m (1 + µ ) 2 2 4 4 n g09 g210 + µ) 2 2 g210 2 g220 2 7/4 14 24 + 2 g210 2 g220 2 g210 g09 g210 15/4 , (1 + µ) 2m4 m2 2 2(g210 2 )(1 + 2µ ) + g210 g200 (1 + g220 2 2 4(g210 g220 )(1 + µ ) g210 g200 2 g210 µ) 3/4 g310 2 11/4 , (C.5) s, 24 (2) s = 1 4 2 2 n 2g220 k g210 + g220 m2 1 + 2µ + 2 m m g210 g210 3/4 2 2 2 2 n g320 (g210 + g220) 2g310 g210 g220 k g310 (g210 + g220) 2g320 g210 g220 + 2 2 m g210 m g210 +2 +2 2 2 2 2 k (g210 + g220)(g310 + g320) 2 m m2g210 7/4 4g210 g220 g310 g320 2 2 2 2 n (g210 + g220 ) g310 g320 g210 g220 (g310 + g320 ) 2 m m2g210 11/4 2 2 2 2 2 2 k g310 (g210 + g220)(g310 + 3g320) 2g320 g210 g220 (3g310 + g320) 2 4 m m g210 + 2 2 2 2 2 2 n g320 (g210 + g220)(3g310 + g320) 2g310 g210 g220 (g310 + 3g320) 2 4 m m g210 (4) s + = 2 2 1 k m6 (1 + µ) 2 (g210 g220) 2 32 m g09 g210 g210 k 4m2n4 m 4 (1 + µ) 2 2 g210 2 g220 2 g210 2 14 24 3/4 g09 g210 2 2(g210 2 g210 2 g220 2 g210 15/4 , k n g + g m 310 m 320 (1 + µ )2m4 2 g220 )(1 + 2µ ) + g210 g200 (1 + 2 2 4(g210 g220 )(1 + µ ) g210 g200 µ) in which 572 7/4 2 11/4 , (C.6) Thin-Walled Structures 135 (2019) 560–574 H.-S. Shen, Y. Xiang g15 = g16 = + 14 c= 1 2 = 430 [ 220 ( 310 + 120 ) 320 ( 140 2 2 [( + )( 320 110 320 14 24 220 430 24 ( 320 140 120 g15 14 24 320 430 b c 2 g16 , = g17 3 b2 (5/4) (5/4) a01 = 1, a10 = b11 = 320 430 b+c 2 310 120 ) 310 220 )], 1/2 14 24 320 240 , b= 1/2 2 g19 = 220 )( 240 )], + g16 1/2 240 320 , g17 = 320 2 g18, g20 = g17 + (9/4) (3) , b01 = b01 = 1 (5/4) 2 2 (5/4) (a10 ) b + a10 2 b , 3 2 + 24 g19 , 4 320 g18 = 320 + 2 14 24 220 , 2 g18, b2 c+ 2 220 310 , 2 14 24 220 (9/4) (3) b10 = b10 = 2 2 + 4 24 g20 , (C.7) and other symbols are defined as in Shen [47]. properties of finite graphene, Nanotechnology 17 (2006) 864–870. [25] L. Shen, H.-S. Shen, C.-L. Zhang, Temperature-dependent elastic properties of single layer graphene sheets, Mater. Des. 31 (2010) 4445–4449. [26] Z. Xu, Graphene nano-ribbons under tension, J. Comput. Theor. Nanosci. 6 (2009) 625–628. [27] Georgios I. Giannopoulos, Ilias, G. 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