Thermal buckling and postbuckling of CNT-reinforced composite cylindrical shell surrounded by an elastic medium with tangentially restrained edges

Abstract

Buckling and postbuckling behavior of thin composite cylindrical shells reinforced by carbon nanotubes (CNTs), surrounded by elastic media and exposed to uniform temperature rise, are investigated in this article. CNTs are reinforced into isotropic matrix phase through uniform distribution or functionally graded distributions across the thickness direction. Material properties are assumed to be temperature dependent, and effective elastic moduli of CNT-reinforced composite are determined according to extended rule of mixture. Formulations are based on the classical thin shell theory taking Von Karman–Donnell nonlinearity, surrounding elastic media and elastic constraints of boundary edges into consideration. Multi-term solutions of deflection and stress function are assumed to satisfy simply supported boundary condition, and Galerkin method is applied to obtain nonlinear relation of thermal load and deflection. An iteration algorithm is used to determine buckling temperatures and postbuckling paths. Numerical examples are given to analyze the effects of volume fraction and distribution type of CNTs, geometrical parameters, degree of tangential edge constraint, buckling mode, and surrounding elastic media on the buckling temperatures and postbuckling strength of thermally loaded nanocomposite cylindrical shells.

Keywords

CNT-reinforced composite thermal load buckling and postbuckling cylindrical shell tangential edge constraint elastic foundation

Introduction

Due to superior mechanical, thermal and electrical properties and very large aspect ratio, carbon nanotubes (CNTs) are usually embedded into isotropic matrix to form nanocomposite known as CNT-reinforced composite (CNTRC).^1
–3 The concept of functionally graded CNT-reinforced composite (FG-CNTRC) is introduced to obtain optimal distribution of CNTs and desired response of nanocomposite structures.⁴ The generation of FG-CNTRC material has motivated investigations on the static and dynamic responses of structural components made of this new class of advanced materials.

Linear buckling behavior of FG-CNTRC plates under mechanical compressive loads has been studied in works^5

–9 based on numerical approaches and¹⁰ using a semi-analytical approach. Kiani^11,12 used Chebyshev–Ritz method to deal with linear buckling problem of FG-CNTRC rectangular plates subjected to shear and parabolic loadings. Kiani and Mirzaei¹³ also studied buckling behavior of FG-CNTRC skew plates under shear load. Based on a numerical approach, the nonlinear resonant dynamics of CNTRC beams and plates have been analyzed by Gholami et al.^14,15 Gholami and Ansari^16

–20 used numerical methods to deal with the nonlinear resonance and bending and stability problems of shear deformable nanocomposite annular and rectangular plates. Buckling behavior of FG-CNTRC cylindrical panels under axial and shear loads has been analyzed in the work of Macias et al.²¹ utilizing shell finite elements. Based on an analytical approach, Nasihatgozar et al.²² investigated the linear buckling behavior of FG-CNTRC cylindrical panels under axial compression. Using adjacent equilibrium criterion and generalized differential quadrature method, Jam and Kiani²³ presented a linear buckling analysis for FG-CNTRC conical shells subjected to lateral pressure.

Postbuckling behavior of FG-CNTRC plates under mechanical loads has been analyzed in the work of Zhang and Liew²⁴ using an element-free approach. Shen and Zhu²⁵ investigated postbuckling of sandwich plates with FG-CNTTRC face sheets under compressive and thermal loads. Making use of asymptotic solutions and a perturbation technique, Shen and Xiang^26
–28 presented investigations on the nonlinear stability of FG-CNTRC cylindrical panels resting on elastic foundations and subjected to mechanical loads in thermal environments. Similarly, postbuckling behavior of higher order shear deformable FG-CNTRC circular cylindrical shells under axial compression, lateral pressure, and combined mechanical loads in thermal environments has been addressed in the works of Shen and Xiang.^29
–31 Hieu and Tung³² used an analytical method to conduct the results of postbuckling analysis for thin FG-CNTRC cylindrical shells surrounded by elastic media under combined mechanical loads in thermal environments. Based on an analytical approach, Ansari et al.³³ considered the effects of piezoelectric layers on postbuckling response of FG-CNTRC cylindrical shells under mechanical loads. Recently, the postbuckling response of FG-CNTRC cylindrical shells with piezoelectric layers subjected to torsional load has been examined by Ninh³⁴ employing Galerkin method.

Thermal buckling and postbuckling behavior of nanocomposite plates and shells are problems of considerable importance. Shen and Zhang³⁵ used asymptotic solutions and a two-step perturbation technique to analyze postbuckling behavior of thick FG-CNTRC plates subjected to uniform temperature rise and in-plane temperature variation. Based on Ritz method and first-order shear deformation theory, Mirzaei and Kiani^36,37 treated the linear buckling problem of FG-CNTRC rectangular and skew plates subjected to uniform temperature rise. Mirzaei and Kiani³⁸ made use of singular discrete convolution method to solve linear stability equations for thermal buckling analysis of FG-CNTRC conical shells. Thermal postbuckling problem of FG-CNTRC plates and sandwich plates with FG-CNTRC face sheets has been considered in the works of Kiani^39,40 using Chebyshev–Ritz method. Shen and Xiang⁴¹ investigated the thermal postbuckling of FG-CNTRC cylindrical panels resting on elastic foundations. Postbuckling behavior of FG-CNTRC cylindrical shells subjected to uniform temperature rise has been analyzed by Shen⁴² using a higher order shear deformation theory and asymptotic solutions. Recently, thermal buckling and postbuckling behavior of FG graphene-reinforced composite laminated plates have been analyzed in works^43

–46 using different approaches.

Tangential elasticity of boundary edges is inherent in practical situations and dramatically influences the behavior of composite structures as mentioned in previous works relating to FG material plates and shells.^47

–50 Zhang et al.^51,52 used element-free approach to analyze the large deformation and mechanical postbuckling responses of FG-CNTRC plates with elastically restrained edges. Based on an analytical approach, the nonlinear stability problems of FG-CNTRC plates and cylindrical panels with tangentially restrained edges under thermal, mechanical, and thermomechanical loadings have been dealt with in the works of Tung and Trang.^53

–58 Recently, Hieu and Tung⁵⁹ presented an analytical investigation on nonlinear stability of FG-CNTRC toroidal shell segments subjected to uniform external pressure taking effects of tangential restraints of boundary edges into consideration. More recently, thermal and thermomechanical postbuckling responses of CNTRC sandwich plates with elastically restrained edges have been studied by Long and Tung.^60,61 Shen’s study⁴² is the only work examining thermal postbuckling of CNTRC cylindrical shells with immovable edges and without surrounding elastic media. However, in practical situations, the cylindrical shells may be surrounded by elastic media and their edges are elastically restrained. Evident lack of investigations on thermal postbuckling of nanocomposite cylindrical shells with elastically restrained edges and surrounding elastic media motivates the present work.

The present article investigates the buckling and postbuckling behavior of thin CNTRC circular cylindrical shells surrounded by elastic media and subjected to uniform temperature rise. Material properties of constituents are assumed to be temperature dependent and effective elastic moduli of CNTRC are estimated according to extended rule of mixture. The classical thin shell theory is used for formulations in which Von Karman–Donnell nonlinearity, surrounding elastic medium–shell interaction, and elasticity of tangential edge constraints are taken into consideration. Multi-term solutions of deflection and stress function are assumed to satisfy simply supported boundary conditions and Galerkin method is adopted to obtain nonlinear relation between temperature and deflection. The effects of distribution types and volume fraction of CNTs, degree of tangential edge constraint, buckling mode, geometrical ratios, and elastic foundations on buckling temperatures and postbuckling temperature-deflection paths are analyzed through numerical examples. As shown in review articles of Du et al.⁶² and Gohardani et al.,⁶³ although there are some key problems of CNT/polymer composites such as orientation of CNTs, dispersion of CNTs within polymer matrix, imperfect bond between CNTs and matrix, and contacts between individual CNTs, nanocomposites have many practical applications in diverse fields, especially in aerospace sciences.

CNT-reinforced composite cylindrical shell surrounded by an elastic medium

Consider a circular cylindrical shell of length L, curvature radius R, and thickness h defined in a coordinate system $x y z$ origin of which is located on the middle surface at one end of the shell, x and y axes are in axial and circumferential directions, respectively, and z axis is in direction of inward normal to the middle surface of the shell as shown in Figure 1. The cylindrical shell is reinforced by CNTs and surrounded by an elastic medium modelled as a two-parameter elastic foundation. The CNTs are reinforced into isotropic matrix phase through uniform distribution (UD) or four different types of FG distributions named as FG- $Λ$ , FG-V, FG-O, and FG-X, and volume fractions $V_{CNT}$ of CNTs corresponding to these distribution patterns are defined as

Figure 1.

Configuration and coordinate system of a cylindrical shell surrounded by an elastic medium.

V_{CNT} = {\begin{matrix} V_{CNT}^{*} (UD) \\ (1 + \frac{2 z}{h}) V_{CNT}^{*} (FG-Λ) \\ (1 - \frac{2 z}{h}) V_{CNT}^{*} (FG-V) \\ 2 (1 - \frac{2 | z |}{h}) V_{CNT}^{*} (FG-O) \\ 2 (\frac{2 | z |}{h}) V_{CNT}^{*} (FG-X) \end{matrix}

where $V_{CNT}^{*}$ is total volume fraction of CNTs and its definition has been given in many previous works, for example.^4,35,42 The present study uses extended rule of mixture to estimate the effective elastic moduli and shear modulus of CNTRC as follows

E_{11} = η_{1} V_{CNT} E_{11}^{CNT} + V_{m} E^{m}

\frac{η_{2}}{E_{22}} = \frac{V_{CNT}}{E_{22}^{CNT}} + \frac{V_{m}}{E^{m}}

\frac{η_{3}}{G_{12}} = \frac{V_{CNT}}{G_{12}^{CNT}} + \frac{V_{m}}{G^{m}}

where $E_{11}^{CNT}$ , $E_{22}^{CNT}$ , and $G_{12}^{CNT}$ are the Young moduli and shear modulus, respectively, of the CNT, whereas E^m, G^m and $V_{m} = 1 - V_{CNT}$ denote Young modulus, shear modulus, and volume fraction of isotropic matrix material, respectively. In equations (2) to (4), coefficients $η_{j} (j = 1 \div 3)$ are called CNT efficiency parameters introduced to account for the size-dependent material properties.

Effective Poisson ratio of CNTRC is assumed to be independent of position and temperature and determined as

ν_{12} = V_{CNT}^{*} ν_{12}^{CNT} + (1 - V_{CNT}^{*}) ν^{m}

where $ν_{12}^{CNT}$ and $ν^{m}$ denote Poisson ratios of the CNT and matrix, respectively. Subsequently, effective coefficients of thermal expansion of CNTRC in the longitudinal and transverse directions are determined as^41,42

α_{11} = \frac{V_{CNT} E_{11}^{CNT} α_{11}^{CNT} + V_{m} E^{m} α^{m}}{V_{CNT} E_{11}^{CNT} + V_{m} E^{m}}

α_{22} = (1 + ν_{12}^{CNT}) V_{C N T} α_{22}^{CNT} + (1 + ν^{m}) V_{m} α^{m} - ν_{12} α_{11}

where $α_{11}^{CNT}$ , $α_{22}^{CNT}$ , and $α^{m}$ are thermal expansion coefficients, respectively, of the CNT and isotropic matrix.

Formulations

In the present study, nanocomposite cylindrical shells are assumed to be thin and geometrically perfect, and the classical shell theory (CST) is used for formulations. Based on the CST, nonlinear strains in Von Karman–Donnell sense are expressed as

ε_{x} = ε_{x 0} + z χ_{x}, ε_{y} = ε_{y 0} + z χ_{y}, γ_{x y} = γ_{x y 0} + 2 z χ_{x y}

where

(\begin{matrix} ε_{x 0} \\ ε_{y 0} \\ γ_{x y 0} \end{matrix}) = (\begin{matrix} u_{, x} + w_{, x}^{2} / 2 \\ v_{, y} - w / R + w_{, y}^{2} / 2 \\ u_{, y} + v_{, x} + w_{, x} w_{, y} \end{matrix}), (\begin{matrix} χ_{x} \\ χ_{y} \\ χ_{x y} \end{matrix}) = (\begin{matrix} - w_{, x x} \\ - w_{, y y} \\ - w_{, x y} \end{matrix})

in which $u, v, w$ are displacement components of a point on the middle surface in coordinate directions $x, y, z$ , respectively, and subscript prime indicates partial derivative, for example $u_{, x} = \partial u / \partial x$ .

The CNTRC shell is assumed to be exposed to a heated environment temperature of which is uniformly raised from reference value $T_{0} = 300 K$ to final value T, and temperature change $Δ T = T - T_{0}$ is independent of coordinate variables. Stress components in the CNTRC cylindrical shell are determined as

(\begin{matrix} σ_{x} \\ σ_{y} \\ σ_{x y} \end{matrix}) = (\begin{matrix} Q_{11} & Q_{12} & 0 \\ Q_{12} & Q_{22} & 0 \\ 0 & 0 & Q_{66} \end{matrix}) (\begin{matrix} ε_{x} - α_{11} Δ T \\ ε_{y} - α_{22} Δ T \\ γ_{x y} \end{matrix})

where

Q_{11} = \frac{E_{11}}{1 - ν_{12} ν_{21}}, Q_{22} = \frac{E_{22}}{1 - ν_{12} ν_{21}}, Q_{12} = \frac{ν_{21} E_{11}}{1 - ν_{12} ν_{21}}, Q_{66} = G_{12}

Force and moment resultants in the CNTRC cylindrical shell are calculated through the stress components as

(N_{x}, N_{y}, N_{x y}) = \int_{- h / 2}^{h / 2} (σ_{x}, σ_{y}, σ_{x y}) d z, (M_{x}, M_{y}, M_{x y}) = \int_{- h / 2}^{h / 2} (σ_{x}, σ_{y}, σ_{x y}) z d z

From equations (8), (10), and (12), the force and moment resultants are expressed as follows

\begin{array}{l} (N_{x}, M_{x}) = (e_{11}, e_{12}) ε_{x 0} + ν_{21} (e_{11}, e_{12}) ε_{y 0} + (e_{12}, e_{13}) χ_{x} + ν_{21} (e_{12}, e_{13}) χ_{y} - (e_{11 T}, e_{12 T}) Δ T \\ (N_{y}, M_{y}) = ν_{12} (e_{21}, e_{22}) ε_{x 0} + (e_{21}, e_{22}) ε_{y 0} + ν_{12} (e_{22}, e_{23}) χ_{x} + (e_{22}, e_{23}) χ_{y} - (e_{21 T}, e_{22 T}) Δ T \\ (N_{x y}, M_{x y}) = (e_{31}, e_{32}) γ_{x y 0} + 2 (e_{32}, e_{33}) χ_{x y} \end{array}

where coefficients $e_{i j} (i, j = 1 \div 3)$ and $e_{k l T} (k, l = 1, 2)$ can be found in works.^55,56

Based on the CST, nonlinear equilibrium equation of the CNTRC cylindrical shell surrounded by an elastic medium modelled as a Winkler–Pasternak foundation has the form

\begin{array}{l} a_{11} w_{, x x x x} + a_{21} w_{, y y y y} + a_{31} w_{, x x y y} + a_{41} f_{, x x y y} - f_{, y y} w_{, x x} + 2 f_{, x y} w_{, x y} \\ - f_{, x x} w_{, y y} - \frac{f_{, x x}}{R} + k_{1} w - k_{2} (w_{, x x} + w_{, y y}) = 0 \end{array}

where $f (x, y)$ is a stress function defined such that $N_{x} = f_{, y y}$ , $N_{y} = f_{, x x}$ , $N_{x y} = - f_{, x y}$ , k₁ and k₂ are elastic modulus of Winkler springs and stiffness of shear layer of Pasternak model, respectively, and coefficients $a_{i 1} (i = 1 \div 4)$ have their definitions as in works^55,56

From equations (9) and (13), strain compatibility equation of the CNTRC cylindrical shell has the form

a_{12} f_{, x x x x} + a_{22} f_{, x x y y} + a_{32} f_{, y y y y} + a_{42} w_{, x x x x} + a_{52} w_{, x x y y} + a_{62} w_{, y y y y} - w_{, x y}^{2} + w_{, x x} w_{, y y} + \frac{w_{, x x}}{R} = 0

where coefficients $a_{j 2} (j = 1 \div 6)$ have been defined in the works.^55,56

In the present study, two boundary edges of the CNTRC cylindrical shell are assumed to be simply supported and tangentially restrained, and corresponding boundary conditions are expressed as

w = M_{x} = 0, N_{x} = N_{x 0} at x = 0, L

where $N_{x 0}$ is fictitious compressive force resultant due to elevated temperature at tangentially restrained edges and defined as

Δ_{x} c = N_{x 0}

in which c is average stiffness of tangential constraint and $Δ_{x}$ is average end-shortening displacement expressed as

Δ_{x} = - \frac{1}{2 π R L} \int_{0}^{2 π R} \int_{0}^{L} \frac{\partial u}{\partial x} d x d y

It is indicated from equation (17) that the values of $c = 0$ (i.e. $N_{x 0} = 0$ ), $c \to \infty$ (i.e. $Δ_{x} = 0$ ), and $0 < c < \infty$ represent movable, immovable, and partially movable edges of cylindrical shell, respectively.

For circular cylindrical shell, the following circumferential closed condition must be satisfied

\int_{0}^{2 π R} \int_{0}^{L} \frac{\partial v}{\partial y} d x d y = 0

From equations (9), (13) and definition of stress function, the following expressions are obtained

\begin{array}{l} \frac{\partial u}{\partial x} = \frac{1}{(1 - ν_{12} ν_{21}) e_{11} e_{21}} [e_{21} f_{, y y} - ν_{21} e_{11} f_{, x x} - (ν_{12} ν_{21} e_{11} e_{22} - e_{12} e_{21}) w_{, x x} \\ - ν_{21} (e_{11} e_{22} - e_{12} e_{21}) w_{, y y}] + \frac{e_{21} e_{11 T} - ν_{21} e_{11} e_{21 T}}{(1 - ν_{12} ν_{21}) e_{11} e_{21}} Δ T - \frac{1}{2} w_{, x}^{2} \end{array}

\begin{array}{l} \frac{\partial v}{\partial y} = \frac{1}{(1 - ν_{12} ν_{21}) e_{11} e_{21}} [e_{11} f_{, x x} - ν_{12} e_{21} f_{, y y} - ν_{12} (e_{12} e_{21} - e_{11} e_{22}) w_{, x x} \\ - (ν_{12} ν_{21} e_{12} e_{21} - e_{11} e_{22}) w_{, y y}] + \frac{e_{11} e_{21 T} - ν_{12} e_{21} e_{11 T}}{(1 - ν_{12} ν_{21}) e_{11} e_{21}} Δ T - \frac{1}{2} w_{, y}^{2} + \frac{w}{R} \end{array}

To satisfy boundary conditions (16), multi-term solutions of deflection and stress function are assumed as

\begin{array}{l} w (x, y) = W_{0} + W_{1} sin β_{m} x sin δ_{n} y + W_{2} {sin}^{2} β_{m} x \\ f (x, y) = A_{1} cos 2 β_{m} x + A_{2} cos 2 δ_{n} y + A_{3} sin β_{m} x sin δ_{n} y \end{array}

+ A_{4} sin 3 β_{m} x sin δ_{n} y - σ_{0 y} h \frac{x^{2}}{2} + N_{x 0} \frac{y^{2}}{2}

where $β_{m} = m π / L$ and $δ_{n} = n / R$ with $m, n$ are numbers of half wave and full wave in the axial and circumferential directions, respectively, $W_{0}, W_{1}$ , and W₂ are unknown amplitudes corresponding to prebuckling, linear buckling, and nonlinear buckling states of the deflection, respectively, and $σ_{0 y}$ is average stress in the circumferential direction. By substituting solutions (22) and (23) into compatibility equation (15), coefficients $A_{i} (i = 1 \div 4)$ are determined and details of these coefficients can be found in the work of Hieu and Tung.³²

Next, introduction of solutions (22) and (23) into equation (21) and placing the resulting expression into equation (19) yield the average circumferential stress as follows

\begin{array}{l} σ_{0 y} = - ν_{12} \frac{{\bar{e}}_{21}}{{\bar{e}}_{11}} {\bar{N}}_{x 0} + \frac{1}{{\bar{e}}_{11}} ({\bar{e}}_{11} {\bar{e}}_{21 T} - ν_{12} {\bar{e}}_{21} {\bar{e}}_{11 T}) Δ T \\ + \frac{{\bar{e}}_{21}}{R_{h}} (1 - ν_{12} ν_{21}) ({\bar{W}}_{0} + \frac{{\bar{W}}_{2}}{2}) - \frac{n^{2} {\bar{e}}_{21}}{8 R_{h}^{2}} (1 - ν_{12} ν_{21}) {\bar{W}}_{1}^{2} \end{array}

where

R_{h} = R / h, ({\bar{N}}_{x 0}, {\bar{e}}_{11}, {\bar{e}}_{21}, {\bar{e}}_{11 T}, {\bar{e}}_{21 T}, {\bar{W}}_{0}, {\bar{W}}_{1}, {\bar{W}}_{2}) = \frac{1}{h} (N_{x 0}, e_{11}, e_{21}, e_{11 T}, e_{21 T}, W_{0}, W_{1}, W_{2}) .

Similarly, putting the solutions (22) and (23) into equation (20) and substituting the obtained expression into equations (17) and (18) give the following result of the fictitious compressive force resultant

{\bar{N}}_{x 0} = - λ {\bar{e}}_{11 T} Δ T - λ ν_{21} \frac{{\bar{e}}_{11}}{R_{h}} ({\bar{W}}_{0} + \frac{{\bar{W}}_{2}}{2}) + λ {\bar{e}}_{11} \frac{ν_{21} n^{2} L_{R}^{2} + m^{2} π^{2}}{8 R_{h}^{2} L_{R}^{2}} {\bar{W}}_{1}^{2} + λ {\bar{e}}_{11} \frac{m^{2} π^{2}}{4 R_{h}^{2} L_{R}^{2}} {\bar{W}}_{2}^{2}

where

λ = \frac{c}{c + e_{11}}, L_{R} = L / R .

It is clear from equation (26) that ${\bar{N}}_{x 0}$ is zero-valued in the case of $λ = 0$ (i.e. $c = 0$ ) implying that boundary edges are freely movable.

Now, solutions (22) and (23) are substituted into the equilibrium equation (14), then applying Galerkin method on whole region of the cylindrical shell ( $0 \leq x \leq L$ , $0 \leq y \leq 2 π R$ ) to the resulting equation, with attention to equations (24) and (26), leads to the following equations

g_{11} (2 {\bar{W}}_{0} + {\bar{W}}_{2}) - g_{21} {\bar{W}}_{1}^{2} - g_{31} {\bar{W}}_{2}^{2} + g_{41} Δ T = 0

g_{12} - g_{22} {\bar{W}}_{0} - g_{32} {\bar{W}}_{2} + g_{42} {\bar{W}}_{1}^{2} + g_{52} {\bar{W}}_{2}^{2} - g_{62} Δ T = 0

g_{13} {\bar{W}}_{0} + g_{23} {\bar{W}}_{2} - g_{33} {\bar{W}}_{1}^{2} - g_{43} {\bar{W}}_{2}^{2} - g_{53} {\bar{W}}_{0} {\bar{W}}_{2} + g_{63} {\bar{W}}_{1}^{2} {\bar{W}}_{2} + (g_{73} - g_{83} {\bar{W}}_{2}) Δ T + g_{93} {\bar{W}}_{2}^{3} = 0

where the details of coefficients $g_{i 1} (i = 1 \div 4)$ , $g_{j 2} (j = 1 \div 6)$ , and $g_{k 3} (k = 1 \div 9)$ are displayed in equation (1A) of Appendix 1.

From equations (28) and (29), it is deduced that

{\bar{W}}_{0} = g_{14} {\bar{W}}_{2} + g_{24} {\bar{W}}_{2}^{2} + g_{34} Δ T - g_{44}

{\bar{W}}_{1}^{2} = g_{15} {\bar{W}}_{2} + g_{25} {\bar{W}}_{2}^{2} + g_{35} Δ T - g_{45}

where coefficients $g_{i 4}$ and $g_{i 5}$ ( $i = 1 \div 4$ ) can be found in equation (2A) of Appendix 2.

Introduction of equations (31) and (32) into equation (30) leads to the following nonlinear relation

Δ T = \frac{1}{g_{16} + g_{26} {\bar{W}}_{2}} (g_{36} + g_{46} {\bar{W}}_{2} + g_{56} {\bar{W}}_{2}^{2} + g_{66} {\bar{W}}_{2}^{3})

in which coefficients $g_{i 6}$ ( $i = 1 \div 6$ ) are given in equation (2B) of Appendix 2.

By setting ${\bar{W}}_{2} = 0$ , buckling thermal loads are determined as

Δ T_{b} = \frac{g_{36}}{g_{16}}

and critical buckling thermal load $Δ T_{c r}$ is the smallest value among buckling thermal loads and is determined by minimizing $Δ T_{b}$ with respect to numbers $m, n$ . It is noticed that $g_{16}$ approaches zero and $Δ T_{b}$ approaches infinity when $λ$ (i.e. c) tends to zero. This fact implies that the CNTRC circular cylindrical shells are not buckled as boundary edges are freely movable.

It is clear from equation (22) that maximum deflection of the CNTRC cylindrical shell is

W_{max} = W_{0} + W_{1} + W_{2}

From equations (31), (32), and (35), the non-dimensional value of maximum deflection is determined as

\begin{array}{l} {\bar{W}}_{max} = W_{max} / h = (g_{14} + 1) {\bar{W}}_{2} + g_{24} {\bar{W}}_{2}^{2} + g_{34} Δ T - g_{44} \\ + {(g_{15} {\bar{W}}_{2} + g_{25} {\bar{W}}_{2}^{2} + g_{35} Δ T - g_{45})}^{1 / 2} \end{array} .

Equations (33) and (36) are used to analyze postbuckling behavior of CNTRC cylindrical shells under uniform temperature rise. Due to temperature dependence of material properties, critical buckling thermal loads and postbuckling temperature-deflection paths are determined through an iteration process detailed steps of which have been described in the work⁴⁷ and are omitted here for the sake of brevity.

Results and discussion

Numerical results are presented in this section for circular cylindrical shells made of poly(methyl methacrylate), referred to as PMMA, as matrix material and reinforced by (10,10) single-walled CNTs (SWCNTs). The temperature-dependent properties of the PMMA are $E^{m} = (3.52 - 0.0034 T)$ GPa, $α^{m} = 45 (1 + 0.0005 Δ T) \times 10^{- 6} K^{- 1}$ , $ν^{m} = 0.34$ in which $T = T_{0} + Δ T$ and $T_{0} = 300 K$ (room temperature). The temperature-dependent properties of the (10,10) SWCNTs have been given in the works^{26,29,35,36,40,42} and are omitted here for the sake of brevity. The CNT efficiency parameters $η_{j} (j = 1 \div 3)$ used in this study are the same as those given in the works,^29,35,42 specifically, $(η_{1}, η_{2}, η_{3}) = (0.137, 1.022, 0.715)$ for the case of $V_{C N T}^{*} = 0.12$ , $(η_{1}, η_{2}, η_{3}) = (0.142, 1.626, 1.138)$ for the case of $V_{C N T}^{*} = 0.17$ , and $(η_{1}, η_{2}, η_{3}) = (0.141, 1.585, 1.109)$ for the case of $V_{CNT}^{*} = 0.28$ .

Validation

To validate the proposed approach, the buckling behavior of simply supported thin CNTRC cylindrical shells under uniform temperature rise is considered. Shen’s work⁴² is the only study on the subject of thermal buckling and postbuckling of CNTRC cylindrical shells in the literature. The critical buckling temperatures of thin CNTRC cylindrical shells with immovable edges and without surrounding elastic foundations are shown in Table 1 in comparison with results reported by Shen⁴² using asymptotic solutions and a perturbation technique. As can be seen, a good agreement is achieved in this comparison.

Table 1.

Comparisons of critical buckling temperatures $T_{c r} = T_{0} + Δ T_{c r}$ (K) of thin CNTRC cylindrical shells with immovable edges and without elastic foundation under uniform temperature rise $(R / h = 100, V_{CNT}^{*} = 0.17)$ .

$L / R$	Reference	UD	FG-X	FG-V	FG- $Λ$
1	Shen⁴²	383.9038	397.2143	383.0082	373.9069
	Present	379.2385	390.6587	378.7498	370.3782
$\sqrt{3}$	Shen⁴²	387.6421	394.7082	385.5435	376.2598
	Present	385.5992^a	392.1993^a	378.8005^b	370.7300^b
$\sqrt{5}$	Shen⁴²	383.4142	394.9890	383.7665	374.7634
	Present	379.1865	389.3015	379.8709	371.4522

UD: uniform distribution; CNTRC: carbon nanotube-reinforced composite.

Buckling mode $(m, n) = (1, 7)$ for $L / R = 1$ ; $(m, n) = (2, 7)$ for $L / R = \sqrt{5}$ ; ^a $(m, n) = (1, 6)$ , ^b $(m, n) = (2, 7)$ .

In what follows, buckling and postbuckling behavior of thin CNTRC cylindrical shells with tangentially restrained edges surrounded by elastic media and subjected to uniform temperature rise will be analyzed in tabular and graphical forms. To measure degree of tangential constraints of boundary edges in a more convenient way, non-dimensional tangential stiffness parameter $λ$ is introduced as equation (27). Evidently, values of $λ = 0$ (i.e. $c = 0$ ) and $λ = 1$ (i.e. $c \to \infty$ ) represent movable and immovable edges, respectively, and intermediate degrees of tangential edge constraint are described by $0 < λ < 1$ (i.e. $0 < c < \infty$ ). For the sake of brief expression, temperature dependent and independent properties are mentioned as T-D and T-ID properties, respectively, and T-ID properties are those calculated at room temperature (T₀ = 300 K). In the following illustrations, the effects of surrounding elastic foundation are characterized by non-dimensional stiffness parameters $K_{1}, K_{2}$ defined as in equation (1B) of Appendix 1. In addition, CNTRC cylindrical shells are assumed to be without foundation interaction and with immovable edges, unless otherwise specified.

Thermal buckling analysis

The effects of CNT volume fraction $V_{CNT}^{*}$ and distribution pattern, and various degrees of in-plane constraint of boundary edges on the critical buckling temperature of CNTRC cylindrical shells are examined in Table 2. It is evident that critical buckling temperatures are considerably decreased as parameter $λ$ is increased from 0.5 (partially movable edges) to 1.0 (immovable edges). This means that compressive thermal stress on the edges is higher and cylindrical shell is buckled sooner as tangential edge constraints become more rigorous. It is interesting to note that critical buckling temperatures corresponding to values of $V_{CNT}^{*} = 0.17$ and $V_{CNT}^{*} = 0.28$ are the highest and lowest, respectively. In other words, an intermediate value of CNT volume fraction can give the best capacity of thermal buckling resistance of CNTRC cylindrical shells. Table 2 also indicates that, among five types of CNT distribution, FG-X and FG-O patterns of CNT distribution lead to the highest and lowest values of critical buckling temperatures, respectively, and UD cylindrical shells have intermediate critical temperatures.

Table 2.

Comparisons of critical buckling temperatures $T_{c r} = T_{0} + Δ T_{c r}$ (K) of CNTRC cylindrical shells under uniform temperature rise ( $R / h = 100$ , $L / R = 1$ , $(K_{1}, K_{2}) = (0, 0)$ , $(m, n) = (1, 7)$ ).

$V_{CNT}^{*}$	$λ$	UD	FG-X	FG-O	FG-V	FG- $Λ$
0.12	0.5	427.11 (11.3%)^a	444.48 (13.2%)	410.80 (9.6%)	424.36 (11.5%)	411.55 (9.4%)
	0.75	391.53 (6.5%)	404.18 (7.8%)	379.67 (5.4%)	389.82 (6.6%)	380.06 (5.3%)
	1.0	371.92 (4.3%)	382.01 (5.2%)	362.50 (3.5%)	370.67 (4.3%)	362.76 (3.5%)
0.17	0.5	438.71 (13.4%)	458.02 (15.9%)	422.34 (11.7%)	437.09 (13.9%)	423.82 (11.6%)
	0.75	400.46 (7.9%)	414.68 (9.5%)	388.50 (6.7%)	399.65 (8.1%)	389.48 (6.7%)
	1.0	379.25 (5.2%)	390.68 (6.4%)	369.74 (4.4%)	378.75 (5.4%)	370.44 (4.4%)
0.28	0.5	414.14 (9.5%)	437.81 (12.5%)	397.45 (7.8%)	411.08 (9.6%)	403.11 (8.4%)
	0.75	381.79 (5.4%)	399.24 (7.3%)	369.69 (4.3%)	379.81 (5.5%)	373.79 (4.7%)
	1.0	364.06 (3.5%)	378.03 (4.8%)	354.49 (2.8%)	362.60 (3.5%)	357.72 (3.0%)

UD: uniform distribution; CNTRC: carbon nanotube-reinforced composite.

^aDifference = $100 % (T_{c r}^{T - I D} - T_{c r}^{T - D}) / T_{c r}^{T - D}$ .

Subsequently, the effects of geometrical ratios and surrounding elastic media on the critical buckling temperatures of FG-CNTRC cylindrical shells with immovable edges are considered in Table 3. As shown, although buckling mode $(m, n)$ are changed due to change of $L / R$ ratio, this ratio has very slight effect on critical buckling temperatures. Meanwhile, critical buckling temperature is significantly reduced and number of full wave in circumferential direction n is increased when $R / h$ ratio is higher (i.e. thinner shells). Furthermore, surrounding elastic foundations have beneficial influence on the thermal buckling resistance capability of FG-CNTRC cylindrical shells. In addition, it is recognized from Tables 2 and 3 that critical temperatures in the case of T-ID properties are significantly higher than those corresponding to T-D properties (e.g. difference is over 10% in the case of $R / h = 100$ , $L / R = 1$ , $V_{CNT}^{*} = 0.17$ , $λ = 0.5$ ). Nevertheless, deteriorative effects of T-D properties on critical temperatures are negligibly small for thin CNTRC cylindrical shells with immovable edges.

Table 3.

Effects of geometrical ratios and elastic media on the critical buckling temperatures $T_{c r} = T_{0} + Δ T_{c r}$ (K) of FG-CNTRC cylindrical shells with immovable edges (FG-X, $V_{CNT}^{*} = 0.17$ , $λ = 1$ ).

$L / R$	$(K_{1}, K_{2})$	$R / h$
$L / R$	$(K_{1}, K_{2})$	100	200	300
1	(0,0)	390.68 (6.4%)^a (1,7)^b	349.43 (2.3%) (1,9)	335.85 (1.3%) (1,10)
	(500,0)	403.94 (7.1%) (1,7)	353.41 (2.5%) (1,9)	336.86 (1.2%) (2,12)
	(500,10)	424.14 (8.8%) (1,7)	362.46 (3.1%) (1,9)	339.09 (1.3%) (2,12)
1.5	(0,0)	389.78 (6.8%) (1,6)	349.65 (2.2%) (2,10)	334.09 (1.1%) (2,11)
	(500,0)	409.53 (7.4%) (2,8)	351.76 (2.3%) (2,10)	335.15 (1.2%) (2,11)
	(500,10)	423.16 (8.6%) (2,7)	357.74 (2.7%) (2,9)	338.90 (1.4%) (2,11)
2.0	(0,0)	390.68 (6.4%) (2,7)	349.43 (2.3%) (2,9)	334.19 (1.1%) (3,11)
	(500,0)	403.94 (7.1%) (2,7)	353.41 (2.5%) (2,9)	335.03 (1.2%) (3,12)
	(500,10)	424.14 (8.8%) (2,7)	362.46 (3.1%) (2,9)	338.11 (1.3%) (3,11)

^aDifference = $100 % (T_{c r}^{T - I D} - T_{c r}^{T - D}) / T_{c r}^{T - D}$ .

^bBuckling mode $(m, n)$ .

The effects of CNT volume fraction and degree of tangential edge constraint on the critical buckling temperatures of FG-CNTRC cylindrical shells with T-D and T-ID properties are graphically displayed in Figure 2. As mentioned, the critical temperature is the lowest as CNT volume fraction is the highest, and the critical temperatures are rapidly decreased as edges are more rigorously restrained in tangential displacement. Furthermore, difference between critical temperatures corresponding to cases of T-D and T-ID properties become smaller for higher values of $λ$ parameter.

Figure 2.

Effects of tangential edge constraint and CNT volume fraction on the critical buckling temperature of FG-CNTRC cylindrical shells with T-D and T-ID properties.

Thermal postbuckling analysis

First example for thermal postbuckling analysis is shown in Figure 3 considering the effects of CNT distribution types on the postbuckling behavior of CNTRC cylindrical shells with immovable edges under uniform temperature rise. It is evident that the FG-X distribution gives the strongest postbuckling response of thermally loaded CNTRC cylindrical shells and postbuckling path corresponding to FG-V pattern of CNT distribution is higher than postbuckling paths corresponding to four the remaining types of CNT distribution in small region of deflection. It is realized that, unlike case of mechanically loaded CNTRC cylindrical shells,³² the postbuckling equilibrium paths of thermally loaded CNTRC cylindrical shells are relatively stable. Next, the effects of buckling mode $(m, n)$ on the thermal postbuckling behavior of FG-CNTRC cylindrical shells are examined in Figure 4. In this example, with a definite geometry ( $L / R = 1.8$ and $R / h = 150$ ), the probability of buckling at mode shape $(m, n) = (2, 8)$ is the highest because critical buckling temperature is reached at this mode shape, and postbuckling path corresponding to mode shape $(m, n) = (2, 8)$ is very stable. Nevertheless, the CNTRC cylindrical shell also may be buckled at mode shape $(m, n) = (1, 8)$ and postbuckling equilibrium path corresponding to this mode shape is unstable because the cylindrical shell experiences a snap-through response. Because postbuckling path in deep region of deflection corresponding to buckling mode $(m, n) = (1, 8)$ is considerably lower, on design point of view, this mode shape is more practical choice.

Figure 3.

Effects of CNT distribution types on the thermal postbuckling behavior of CNTRC cylindrical shells with immovable edges.

Figure 4.

Effects of buckling mode (m, n) on the thermal postbuckling behavior of FG-CNTRC cylindrical shells with immovable edges.

Subsequently, the effects of tangential constraint of boundary edges on the thermal postbuckling behavior of FG-CNTRC cylindrical shells are illustrated in Figure 5. As can be observed, buckling temperatures and postbuckling curves of the shells are considerably dropped as tangential stiffness parameter $λ$ is increased from 0.4 to 1.0. Snap-through instability can occur for nanocomposite cylindrical shells with partially movable edges ( $λ \neq 1$ ) and may be vanished for the shells with immovable edges ( $λ = 1$ ). In addition, it is recognized from Figure 5 that prebuckling deflection of the shell is decreased as edges are more rigorously restrained. Specifically, CNTRC cylindrical shell is deflected outwards (i.e. negative deflection) prior to buckling and, in this example, non-dimensional prebuckling deflections are $W_{0} / h = - 1.66, - 1.11, - 0.84, - 0.67$ for $λ = 0.4, 0.6, 0.8, 1.0$ , respectively. Consequently, this example reflects a fact that thermally loaded CNTRC cylindrical shells with partially movable edges have large prebuckling deformation and unstable postbuckling response.

Figure 5.

Effects of tangential constraint of boundary edges on the thermal postbuckling behavior of FG-CNTRC cylindrical shells under uniform temperature rise.

Next, thermal postbuckling paths of FG-CNTRC cylindrical shells corresponding to different values of CNT volume fraction $V_{CNT}^{*}$ are plotted in Figure 6. It is interesting that, among three values of $V_{CNT}^{*}$ , the paths corresponding to $V_{CNT}^{*} = 0.17$ and $V_{CNT}^{*} = 0.28$ are the highest and lowest, respectively. This fact suggests that an average volume percentage of CNTs can result in the best postbuckling response of CNTRC cylindrical shell under thermal loading. In other words, unlike case of mechanically loaded CNTRC cylindrical shells,^29

–32 high percentage of CNTs has no positive effect on thermally loaded CNTRC cylindrical shell. As a subsequent illustration, Figure 7 assesses the effects of length-to-radius $L / R$ ratio on the thermal postbuckling behavior of FG-CNTRC cylindrical shells with immovable edges. As can be seen, there is no a definite tendency of variation in buckling temperature versus the $L / R$ ratio. For longer cylindrical shell and smaller numbers of buckling mode, snap-through response can occur in the postbuckling region.

Figure 6.

Effects of CNT volume fraction on the thermal postbuckling behavior of FG-CNTRC cylindrical shells with various degrees of tangential edge constraint.

Figure 7.

Effects of L/R ratio on the thermal postbuckling behavior of FG-CNTRC cylindrical shells with immovable edges under uniform temperature rise.

Subsequently, Figure 8 considers the effects of radius-to-thickness $R / h$ ratio on the thermal postbuckling behavior of FG-CNTRC cylindrical shells with immovable edges. Obviously, buckling temperatures and postbuckling paths of the shells are substantially lowered as the $R / h$ ratio is increased. In addition, prebuckling negative deflection is larger and snap-through phenomenon in postbuckling response is more pronounced for thinner CNTRC cylindrical shells. Finally, the effects of surrounding elastic media on the thermal postbuckling behavior of FG-CNTRC cylindrical shells with partially movable edges $(λ = 0.8)$ are analyzed in Figure 9. It is realized that buckling temperatures and postbuckling equilibrium paths of CNTRC cylindrical shells are significantly enhanced due to the embrace of surrounding elastic foundations, especially Pasternak type foundation model. Furthermore, the intensity of snap-through response (measured by difference between bifurcation and lowest points on temperature-deflection path) is not reduced and prebuckling negative deflection is remarkably increased because of surrounding elastic foundations. Specifically, in this example, negative deflections in prebuckling state of thermally loaded CNTRC cylindrical shells are predicted as $W_{0} / h = - 0.90, - 1.06, - 1.23, - 1.39$ for non-dimensional foundation stiffness parameters $(K_{1}, K_{2}) = (0, 0), (500, 0), (1000, 1), (500, 10)$ , respectively.

Figure 8.

Effects of R/h ratio on the thermal postbuckling behavior of FG-CNTRC cylindrical shells with immovable edges under uniform temperature rise.

Figure 9.

Effects of surrounding elastic media on the thermal postbuckling behavior of FG-CNTRC cylindrical shells with partially movable edges under uniform temperature rise.

Concluding remarks

The buckling and postbuckling behavior of thin CNTRC cylindrical shells with simply supported and tangentially restrained edges subjected to uniform temperature rise have been presented. Multi-term solutions and Galerkin method are used to solve nonlinear governing equations, and an iteration procedure is adopted to determine critical buckling temperatures and postbuckling temperature-deflection paths. The results indicate that an average value of CNT volume percentage can lead to the best capacities of buckling resistance and postbuckling load carrying of thermally loaded CNTRC cylindrical shells. In addition, FG-X type distribution of CNTs gives the highest buckling temperatures and postbuckling paths. The study also reveals that buckling temperature, postbuckling strength, intensity of snap-through response, and prebuckling negative deflection are considerably reduced as boundary edges are more rigorously restrained in tangential displacement. Furthermore, surrounding elastic foundations effectively improves buckling loads and postbuckling strength of thermally loaded CNTRC cylindrical shells.

Footnotes

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Appendix 1

The details of coefficients $g_{i 1} (i = 1 \div 4)$ , $g_{j 2} (j = 1 \div 6)$ , and $g_{k 3} (k = 1 \div 9)$ in the equations (28) $\div$ (30) are defined as

in which

and $E_{0}^{m}$ is value of E^m calculated at room temperature (T₀ = 300 K).

Appendix 2

Specific definitions of coefficients $g_{i 4}$ and $g_{i 5} (i = 1 \div 4)$ in the equations (31) and (32) are

The details of coefficients $g_{i 6} (i = 1 \div 6)$ in the equation (33) are defined as

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