Postbuckling behavior of CNT-reinforced composite cylindrical shell surrounded by an elastic medium and subjected to combined mechanical loads in thermal environments

Abstract

Cylindrical shells are usually buckled under complex and combined loading conditions. This article presents an analytical approach to investigate the buckling and postbuckling behaviors of cylindrical shells reinforced by single-walled carbon nanotubes, surrounded by an elastic medium, exposed to thermal environments, and subjected to combined axial compression and lateral pressure loads. Carbon nanotubes (CNTs) are imbedded into matrix phase by uniform distribution or functionally graded distribution along the thickness direction. The properties of constituents are assumed to be temperature dependent, and effective properties of CNT-reinforced composite (CNTRC) are determined by an extended rule of mixture. Governing equations are based on the classical shell theory (CST) taking von Karman–Donnell nonlinearity and surrounding elastic foundations into consideration. Three-term form of deflection is assumed to satisfy simply supported boundary conditions, and Galerkin method is applied to obtain nonlinear load–deflection relations from which buckling loads and postbuckling equilibrium paths are determined. Numerical examples are carried out to show the effects of CNT volume fraction, distribution types, thermal environments, preexisting nondestabilizing lateral pressure and axial compression loads, and elastic medium on the buckling and postbuckling behaviors of CNTRC cylindrical shells.

Keywords

Carbon nanotube reinforced composite nanocomposite cylindrical shell buckling and postbuckling combined loads temperature effects

Introduction

Structural components in the form of circular cylindrical shells are widely used in engineering applications. Since cylindrical shells are usually subjected to complex loading conditions, their stability is an important problem and should be addressed for the sake of accurate prediction and safe design. Bagherizadeh et al.¹ used adjacent equilibrium criterion and analytical solution to study linear buckling behavior of functionally graded material (FGM) cylindrical shells surrounded by Pasternak elastic foundation. Huang and Han^2
–4 used an analytical approach based on Ritz energy method to investigate buckling and postbuckling behaviors of thin FGM cylindrical shells under external pressure, axial compression, and combined loading. Nonlinear static stability, vibration, and dynamic buckling of thin FGM cylindrical shells with stiffeners and surrounding elastic medium have been treated in works of Dung and coworkers.^5,6 Buckling, postbuckling, and geometrically nonlinear responses of FGM plates with various shapes have been addressed in works^7

–10 based on numerical approaches.

Carbon nanotubes (CNTs) have been attractive subject of many science fields since first work of Iijima in 1991.¹¹ No previous material has possessed extraordinary mechanical, thermal, and electrical properties attributed to CNTs. There are two main types of CNTs: single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs). The former is rolled seamlessly from a single sheet of graphene to form a cylinder with diameter of order of 1 nm and length of up to centimeters, and the latter consist of an array of such cylinders formed concentrically and separated about 0.35 nm.^12
–14 Due to superlative properties and very large aspect ratio, CNTs are used as ideal filler into polymer matrix to form carbon nanotube-reinforced composite (CNTRC), an advanced class of nanocomposite.

The concept of functionally graded carbon nanotube reinforced composite (FG-CNTRC) was first presented in work of Shen¹⁵ in which CNTs are reinforced into polymer matrix in such a way that their volume fraction is a linear function of thickness variable. This proposal of Shen has motivated subsequent studies on static and dynamic responses of structural components made of FG-CNTRC. Based on IMLS-Ritz method,^16,17 Zhang et al.^18,19 and Zhang²⁰ analyzed the nonlinear bending of thick FG-CNTRC plates resting on Pasternak elastic foundations and nonlinear response of FG-CNTRC plates with elastically restrained edges and internal supports. Geometrically nonlinear analyses of skew, quadrilateral, and triangular FG-CNTRC plates have been carried out by Zhang and Liew^21,22 and Zhang et al.²³ Lei et al.²⁴ and Zhang et al.²⁵ used meshless methods to study the free vibration of rectangular FG-CNTRC plates without and with elastic restraints of edges, respectively. Static response and free vibration of laminated FG-CNTRC plates have been analyzed in works of Lei et al.²⁶ and Zhang and Selim.²⁷ The effects of in-plane loads and piezoelectric patches on the vibration response of FG-CNTRC plates with various shapes have been examined in works of Zhang and coworkers.^28

–31 Linear buckling of FG-CNTRC rectangular and skew plates under mechanical loads has been investigated in works of Lei et al.³² and Zhang et al.³³ making use of first-order shear deformation theory and numerical methods. Using analytical and semi-analytical solutions, mechanical buckling behavior of FG-CNTRC rectangular plates has been studied in works of Mehrabadi et al.³⁴ and Wang et al.,³⁵ respectively. Zhang³⁶ made use of IMLS-Ritz method for buckling analysis of nanocomposite plates of polygonal planform. Zhang and Liew³⁷ and Zhang et al.^38,39 studied the postbuckling behavior of FG-CNTRC plates under compressive loads taking into account elastic foundations³⁷ and elastic constraints of boundary edges.³⁹ Shen and Zhang⁴⁰ presented a study on buckling and postbuckling of FG-CNTRC plates under two types of thermal loading. Kiani⁴¹ used Ritz method with shape functions as Chebyshev polynomials to examine postbuckling of FG-CNTRC plates under uniform temperature rise. Tung⁴² considered effects of elastic foundations and tangential edge constraints on the buckling and postbuckling behaviors of FG-CNTRC plates under two types of thermal load. Dynamic behavior of FG-CNTRC plates in supersonic airflow and subjected to dynamic loading has been considered in previous works.^43,44

Linear buckling of FG-CNTRC cylindrical panels with piezoelectric layers under axial compression has been treated in analytical work of Nasihatgozar et al.⁴⁵ Macias et al.⁴⁶ made use of numerical simulation based on shell finite element to analyze linear buckling behavior of FG-CNTRC cylindrical panels under axial compression and shear. Liew et al.⁴⁷ researched postbuckling behavior of FG-CNTRC cylindrical panels under axial compression using a meshless approach. Based on a semi-analytical approach with a higher order shear deformation theory and a perturbation technique, Shen⁴⁸ and Shen and Xiang^49
–51 presented investigations on postbuckling behavior of FG-CNTRC cylindrical panels subjected to external pressure, axial compression, uniform temperature rise, and combined loads. Recently, Tung and Trang^52,53 used Galerkin method to investigate the nonlinear postbuckling response of thin FG-CNTRC cylindrical panels under external pressure and axial compression taking effects of elastic foundations and tangential edge constraints into consideration. Lei et al.⁵⁴ used the element-free kp-Ritz method to analyze the dynamic stability of FG-CNTRC cylindrical panels.

Based on theory of elasticity, static analysis of FG-CNTRC cylindrical shells imbedded in piezoelectric layers under thermo–electro–mechanical load has been carried out by Alibeigloo and Pasha Zanoosi.⁵⁵ Shen and Xiang⁵⁶ investigated nonlinear vibration of FG-CNTRC cylindrical shells in thermal environments using an improved perturbation technique. Employing a numerical approach based on two different shear deformation theories, Song et al.⁵⁷ analyzed linear vibration of FG-CNTRC cylindrical shells with temperature effects. Zhang et al.⁵⁸ investigated the impact responses of FG-CNTRC cylindrical shells. Based on asymptotic solutions and a perturbation technique, Shen and his coworker have performed investigations on postbuckling behaviors of higher order shear deformable FG-CNTRC cylindrical shells subjected to axial compression,⁵⁹ external pressure,⁶⁰ combined mechanical loads,⁶¹ thermal load,⁶² and torsional load.⁶³ Utilizing a Ritz energy approach, Ansari et al.⁶⁴ presented an analytical study on postbuckling behavior of thin FG-CNTRC cylindrical shells under electromechanical loads. Making use of Galerkin method, Ninh⁶⁵ dealt with postbuckling problem of thin FG-CNTRC cylindrical shells under torsional load. Both works of Ansari et al.⁶⁴ and Ninh⁶⁵ used three-term form of deflection and considered the effects of piezoelectric layers on postbuckling behavior of FG-CNTRC cylindrical shells. Jam and Kiani⁶⁶ used adjacent equilibrium criterion and numerical method to examine the linear buckling of FG-CNTRC conical shells under external pressure.

The present article analytically investigates the buckling and postbuckling of CNTRC cylindrical shells with surrounding elastic foundations exposed to thermal environments and subjected to combined action of axial compression and external pressure. CNTs are reinforced into matrix phase by uniform distribution (UD) or functionally graded (FG) distribution. Material properties of constituents are assumed to be temperature dependent, and effective properties of CNTRC are estimated by an extended rule of mixture through a micromechanical approach. Formulations are based on the CST taking into account von Karman–Donnell nonlinear terms. Three-term solution of deflection and stress function are assumed to satisfy simply supported boundary conditions and Galerkin method is applied to obtain nonlinear load–deflection relations from which buckling loads and postbuckling paths are determined. Numerical examples are carried out to analyze the effects of CNT volume fraction, distribution patterns, thermal environments, preexisting nondestabilizing loads, and elastic foundations on the buckling and postbuckling behaviors of FG-CNTRC cylindrical shells.

CNTRC cylindrical shell surrounded by an elastic medium

Consider a CNTRC cylindrical shell of length L, mean radius R, and thickness h. The cylindrical shell is surrounded by an elastic medium and is referred to a coordinate system $x y z$ origin of which is located on the middle surface at an end; x and y are in axial and circumferential directions of the shell, respectively; and z is in the direction of the inward normal to the middle surface as shown in Figure 1. In this study, SWCNTs are reinforced into isotropic polymer matrix by UD or four FG distributions named as FG-V, FG-Λ, FG-O, and FG-X, and volume fraction $V_{CNT}$ of CNTs corresponding to these five types of CNT distribution is 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}^{*} = \frac{w_{CNT}}{w_{CNT} + (ρ_{CNT} / ρ_{m}) (1 - w_{CNT})}

in which $w_{CNT}$ is the mass fraction of CNTs in the CNTRC cylindrical shell, $ρ_{CNT}$ and $ρ_{m}$ are the densities of CNTs and matrix, respectively. It is described from equation (1) that the outer and inner surfaces of the cylindrical shell are CNT-rich for FG-V and FG-Λ types, respectively, the middle surface of the cylindrical shell is CNT-rich in case of FG-O type and both outer and inner surfaces are enriched by CNTs for FG-X type. Based on a micromechanical approach, effective properties of composite materials may be determined according to the Mori–Tanaka model or the rule of mixture. While the former is more complex, the latter is simple and effective for predicting overall properties of composites. An excellent agreement between Mori–Tanaka approach and rule of mixture for buckling analysis results of FG-CNTRC shell panels has been shown in the work of Macias et al.⁴⁶ The present study applies the extended rule of mixture to estimate the effective Young moduli and shear modulus of the CNTRC cylindrical shell as¹⁵

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 matrix phase, respectively. Due to imperfect interaction between two phases, coefficients $η_{j}$ $(j = 1, 2, 3)$ , known as CNT efficiency parameters, are inserted into equation (3) to account for the size-dependent material properties.

Effective Poisson ratio weakly depending on temperature and position is determined by

ν_{12} = V_{CNT}^{*} ν_{12}^{CNT} + V_{m} ν^{m}

where $ν_{12}^{CNT}$ and $ν^{m}$ are Poisson ratios of the CNT and matrix, respectively. The effective thermal expansion coefficients of CNTRC cylindrical shell in the longitudinal and transverse directions can be determined as^15,40,59

α_{11} = V_{CNT} α_{11}^{CNT} + V_{m} α^{m}

α_{22} = (1 + ν_{12}^{CNT}) V_{CNT} 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 of CNT and matrix, respectively.

Formulations

Based on the CST, strain components at a distance z from the middle surface of the cylindrical shell 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 strains $ε_{x 0}, ε_{y 0}, γ_{x y 0}$ of the middle surface and curvature changes $χ_{x}, χ_{y}, χ_{x y}$ are related to displacement components $u, v, w$ of the middle surface in coordinate directions $x, y, z$ , respectively, as follows

(\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})

where subscript prime indicates partial derivative and von Karman–Donnell nonlinear terms are maintained. Stress–strain relations of a CNTRC cylindrical shell in a thermal environment are expressed 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} \\ ε_{y} \\ γ_{x y} \end{matrix}} - {\begin{matrix} α_{11} \\ α_{22} \\ 0 \end{matrix}} Δ T)

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}

and $Δ T$ is uniform temperature rise in comparison with a thermal stress free reference state.

Force and moment intensities in the CNTRC cylindrical shell are related to the stresses by the equations

(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

and these intensities are expressed in specific form through equations (6) and (8) as

(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}

where coefficients $e_{i j} (i, j = 1 \div 3)$ and $e_{k l T}$ ( $k, l = 1, 2$ ) are defined as in works of Tung and Trang.^52,53

Based on the CST, nonlinear equilibrium equations for a geometrically perfect cylindrical shell surrounded by an elastic medium are

N_{x, x} + N_{x y, y} = 0

12a

N_{x y, x} + N_{y, y} = 0

12b

M_{x, x x} + 2 M_{x y, x y} + M_{y, y y} + N_{x} w_{, x x} + 2 N_{x y} w_{, x y} + N_{y} w_{, y y} + N_{y} / R + q - q_{f} = 0

12c

where q is the lateral pressure having positive and negative values in cases of external and internal pressures, respectively, and q_f is surrounding elastic medium-cylindrical shell interactive pressure represented by Pasternak model as

q_{f} = k_{1} w - k_{2} Δ w

in which k₁ is modulus of Winkler springs, k₂ is stiffness of shear layer in Pasternak model and $Δ = \partial^{2} / \partial x^{2} + \partial^{2} / \partial y^{2}$ is the Laplace operator.

From equations (7) and (11), equilibrium equation of a geometrically perfect CNTRC cylindrical shell is rewritten in the form as

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} - q + k_{1} w - k_{2} (w_{, x x} + w_{, y y}) = 0

where $f (x, y)$ is a stress function defined as

N_{x} = f_{, y y}, N_{y} = f_{, x x}, N_{x y} = - f_{, x y}

and coefficients $a_{11}, a_{21}, a_{31}, a_{41}$ are given in the previous works.^52,53

By virtue of equations (7), (11), and (15), the strain compatibility equation of a geometrically perfect CNTRC cylindrical shell is expressed as

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 $a_{i 2}$ $(i = 1 \div 6)$ have been defined in the works.^42,52,53

The present study considers simply supported CNTRC cylindrical shells with freely movable edges subjected to a combined action of uniform lateral pressure q and uniform axial compression pressure P at two ends. The associated boundary conditions are

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

where $N_{x 0} = - P h$ is prebuckling compressive force resultant. To satisfy boundary conditions (equation (17)) approximately, the following solutions of deflection and stress function are assumed ^2
–4

w (x, y) = W_{0} + W_{1} sin β_{m} x sin δ_{n} y + W_{2} {sin}^{2} β_{m} x

\begin{array}{l} 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 \\ + 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} \end{array}

where $β_{m} = m π / L$ , $δ_{n} = n / R$ with $m, n$ are numbers of half wave and full wave in the axial and circumferential directions, respectively, W₀, W₁, and W₂ are unknown amplitudes corresponding to prebuckling, linear buckling, and nonlinear buckling states of the deflection, respectively. In addition, in the equation (19), $σ_{0 y}$ is average stress in the circumferential direction, and $A_{i} (i = 1 \div 4)$ are coefficients to be determined. Next, placing solutions (18) and (19) into the strain compatibility equation (16) gives the coefficients $A_{i} (i = 1 \div 4)$ as follows

\begin{array}{l} A_{1} = \frac{1}{32 a_{12} β_{m}^{2}} (δ_{n}^{2} W_{1}^{2} + 16 a_{42} β_{m}^{2} W_{2} - \frac{4}{R} W_{2}) \\ A_{2} = \frac{β_{m}^{2} W_{1}^{2}}{32 a_{32} δ_{n}^{2}} \\ A_{3} = \frac{W_{1}}{a_{12} β_{m}^{4} + a_{22} β_{m}^{2} δ_{n}^{2} + a_{32} δ_{n}^{4}} (\frac{β_{m}^{2}}{R} - a_{42} β_{m}^{4} - a_{52} β_{m}^{2} δ_{n}^{2} - a_{62} δ_{n}^{4} - β_{m}^{2} δ_{n}^{2} W_{2}) \\ A_{4} = \frac{β_{m}^{2} δ_{n}^{2} W_{1} W_{2}}{81 a_{12} β_{m}^{4} + 9 a_{22} β_{m}^{2} δ_{n}^{2} + a_{32} δ_{n}^{4}} \end{array}

Subsequently, solutions (18) and (19) are introduced 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 yields the following expressions

σ_{0 y} \frac{h}{R} - q + k_{1} (W_{0} + \frac{W_{2}}{2}) = 0

21a

a_{13} - a_{23} W_{2} + a_{33} W_{1}^{2} + a_{43} W_{2}^{2} - σ_{0 y} h δ_{n}^{2} + N_{x 0} β_{m}^{2} = 0

21b

k_{1} W_{0} + a_{14} W_{2} - a_{24} W_{1}^{2} + a_{34} W_{1}^{2} W_{2} + β_{m}^{2} W_{2} N_{x 0} + σ_{0 y} \frac{h}{R} - q = 0

21c

where coefficients $a_{j 3} (j = 1 \div 4)$ and $a_{k 4} (k = 1 \div 3)$ are displayed in equation (1A) of Appendix 1.

Next, for the 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 (7), (11), and (15), $v_{, y}$ is expressed in terms of partial derivatives of w and f. Then, introducing the solutions (18) and (19) into the $v_{, y}$ and setting the result into equation (22) yield the following expression of the average circumferential stress

σ_{0 y} h = - ν_{12} \frac{e_{21}}{e_{11}} N_{x 0} + (e_{21 T} - \frac{e_{21}}{e_{11}} ν_{12} e_{11 T}) Δ T + (1 - ν_{12} ν_{21}) e_{21} [\frac{1}{R} (W_{0} + \frac{W_{2}}{2}) - \frac{δ_{n}^{2}}{8} W_{1}^{2}]

Now, introduction of $N_{x 0} = - P h$ and $σ_{0 y} h$ from equation (23) into equation (21) gives the following system of equations

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

f_{12} - f_{22} {\bar{W}}_{0} - f_{32} {\bar{W}}_{2} + f_{42} {\bar{W}}_{1}^{2} + f_{52} {\bar{W}}_{2}^{2} - f_{62} P - f_{72} Δ T = 0

f_{13} {\bar{W}}_{0} + f_{23} {\bar{W}}_{2} - f_{33} {\bar{W}}_{1}^{2} + f_{43} {\bar{W}}_{1}^{2} {\bar{W}}_{2} - (f_{53} {\bar{W}}_{2} - f_{63}) P + f_{73} Δ T - q = 0

where the details of coefficients $f_{i 1} (i = 1 \div 4)$ , $f_{j 2} (j = 1 \div 7)$ , and $f_{k 3} (k = 1 \div 7)$ can be found in equation (1B) in Appendix 1 in which

\begin{matrix} ({\bar{a}}_{11}, {\bar{a}}_{21}, {\bar{a}}_{31}, {\bar{e}}_{13}, {\bar{e}}_{23}, {\bar{e}}_{33}) = \frac{1}{h^{3}} (a_{11}, a_{21}, a_{31}, e_{13}, e_{23}, e_{33}), ({\bar{a}}_{12}, {\bar{a}}_{22}, {\bar{a}}_{32}) = (a_{12}, a_{22}, a_{32}) h \\ ({\bar{a}}_{41}, {\bar{a}}_{42}, {\bar{a}}_{52}, {\bar{a}}_{62}, {\bar{e}}_{11}, {\bar{e}}_{21}, {\bar{e}}_{31}, {\bar{e}}_{11 T}, {\bar{e}}_{21 T}, {\bar{W}}_{0}, {\bar{W}}_{1}, {\bar{W}}_{2}) \\ = \frac{1}{h} (a_{41}, a_{42}, a_{52}, a_{62}, e_{11}, e_{21}, e_{31}, e_{11 T}, e_{21 T}, W_{0}, W_{1}, W_{2}) \\ ({\bar{e}}_{12}, {\bar{e}}_{22}, {\bar{e}}_{32}) = \frac{1}{h^{2}} (e_{12}, e_{22}, e_{32}), R_{h} = R / h, L_{R} = L / R, (K_{1}, K_{2}) = (k_{1} R^{2}, k_{2}) \frac{R^{2}}{E_{0}^{m} h^{3}} \end{matrix}

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

It is deduced from equations (24) and (25) that

{\bar{W}}_{0} = f_{14} {\bar{W}}_{2} + f_{24} {\bar{W}}_{2}^{2} + f_{34} P + f_{44} Δ T + f_{54} q - f_{64}

28a

{\bar{W}}_{1}^{2} = f_{15} {\bar{W}}_{2} + f_{25} {\bar{W}}_{2}^{2} + f_{35} P + f_{45} Δ T + f_{55} q - f_{65}

28b

where coefficients $f_{i 4}, f_{i 5} (i = 1 \div 6)$ are given in detail in equation (2A) in Appendix 2.

In what follows, two cases of combined mechanical loads will be considered.

Case 1: CNTRC cylindrical shell is subjected to a preexisting nondestabilizing (internal or external) lateral pressure and then under uniform axial compression.

In this case, it is assumed that

q = γ P

where $γ$ is non-dimensional parameter indicating load ratio. Introduction of equations (28) and (29) into equation (26) leads to the following nonlinear relation

P = \frac{1}{f_{16} + f_{26} {\bar{W}}_{2}} [f_{36} + f_{46} {\bar{W}}_{2} + f_{56} {\bar{W}}_{2}^{2} + f_{66} {\bar{W}}_{2}^{3} - (f_{76} + f_{86} {\bar{W}}_{2}) Δ T]

where coefficients $f_{j 6} (j = 1 \div 8)$ are displayed in equation (2C) in Appendix 2. By setting ${\bar{W}}_{2} = 0$ , buckling axial compression loads of laterally pre-pressurized CNTRC cylindrical shells are determined as

P_{b} = \frac{1}{f_{16}} (f_{36} - f_{76} Δ T)

It is obvious from equation (18) that maximum deflection of the cylindrical shell is

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

A combination of equations (28), (29), and (32) gives non-dimensional maximum deflection as

\begin{array}{l} {\bar{W}}_{max} = W_{max} / h = (f_{14} + 1) {\bar{W}}_{2} + f_{24} {\bar{W}}_{2}^{2} + (f_{34} + γ f_{54}) P + f_{44} Δ T - f_{64} \\ + {(f_{15} {\bar{W}}_{2} + f_{25} {\bar{W}}_{2}^{2} + f_{35} P + γ f_{55} P + f_{45} Δ T - f_{65})}^{1 / 2} \end{array}

Equations (30) and (33) are used for postbuckling analysis of CNTRC cylindrical shells under high values of axial compression combined with relatively low lateral pressure.

Case 2: CNTRC cylindrical shell is subjected to a preexisting nondestabilizing axial compression and then under uniform lateral pressure.

In this case, it is assumed that

P = η q

where $η = 1 / γ$ is a load proportional parameter. Substituting equations (28) and (34) into equation (26) yields the following nonlinear expression

q = \frac{1}{{f^{'}}_{16} + {f^{'}}_{26} {\bar{W}}_{2}} [f_{36} + f_{46} {\bar{W}}_{2} + f_{56} {\bar{W}}_{2}^{2} + f_{66} {\bar{W}}_{2}^{3} - (f_{76} + f_{86} {\bar{W}}_{2}) Δ T]

where

\begin{array}{l} {f^{'}}_{16} = η (f_{33} f_{35} - f_{13} f_{34} - f_{63}) + 1 + f_{33} f_{55} - f_{13} f_{54} \\ {f^{'}}_{26} = η (f_{53} - f_{43} f_{35}) - f_{43} f_{55} \end{array}

Buckling pressures of axially pre-compressed CNTRC cylindrical shells are obtained from equation (35) by setting ${\bar{W}}_{2} = 0$ as

q_{b} = \frac{1}{{f^{'}}_{16}} (f_{36} - f_{76} Δ T)

Non-dimensional maximum deflection of the cylindrical shell in this case of loading is deduced from equations (28), (32), and (34) as

\begin{array}{l} {\bar{W}}_{max} = W_{max} / h = (f_{14} + 1) {\bar{W}}_{2} + f_{24} {\bar{W}}_{2}^{2} + (η f_{34} + f_{54}) q + f_{44} Δ T - f_{64} \\ + {(f_{15} {\bar{W}}_{2} + f_{25} {\bar{W}}_{2}^{2} + η f_{35} q + f_{55} q + f_{45} Δ T - f_{65})}^{1 / 2} \end{array}

Equations ( 35) and (38) are used for postbuckling analysis of CNTRC cylindrical shells under high values of lateral pressure combined with relatively low axial compression load.

Results and discussion

This section presents numerical examples of buckling and postbuckling analyses for cylindrical shells made of poly(methyl methacrylate), referred to as PMMA, as matrix material and reinforced by (10,10) SWCNTs. The temperature dependent properties of the PMMA are $E^{m} = (3.52 - 0.0034 T)$ GPa, $α^{m} = 45 (1 + 0.0005 Δ T) \times 10^{- 6} K^{- 1}$ , and $ν^{m} = 0.34$ in which $T = T_{0} + Δ T$ and $T_{0} = 300 K$ (room temperature). The temperature dependent properties of the (10,10) SWCNTs are given at discrete values of temperature in the works of Shen and coworkers^40,
62 and as continuous functions of temperature in the work of Shen⁴⁸ and are omitted here for the sake of brevity. The CNT efficiency parameters $η_{j} (j = 1 \div 3)$ are estimated by matching the Young moduli $E_{11}$ , $E_{22}$ and shear modulus $G_{12}$ of the CNTRC determined from the extended rule of mixture to those from the molecular dynamics simulations and given in the works of Shen and coworkers^40,
59,
62 as $(η_{1}, η_{2}, η_{3}) = (0.137, 1.022, 0.715)$ for the case of $V_{CNT}^{*} = 0.12$ , $(η_{1}, η_{2}, η_{3}) = (0.142, 1.626, 1.138)$ for the case of $V_{CNT}^{*} = 0.17$ and $(η_{1}, η_{2}, η_{3}) = (0.141, 1.585, 1.109)$ for the case of $V_{CNT}^{*} = 0.28$ .

To validate the proposed approach, the buckling behavior of simply supported CNTRC cylindrical shells without elastic foundations under combined mechanical loads are considered. Critical buckling loads of CNTRC cylindrical shells with different values of load proportional parameter $R γ / 2 h$ (which is denoted by b₂ in the work of Shen and Xiang⁶¹) are given in Table 1 in comparison with results reported by Shen and Xiang⁶¹ using asymptotic solutions and a perturbation technique. As can be seen, a very good agreement is achieved in this comparison.

Table 1.

Comparisons of critical buckling loads $(P_{c r}, q_{c r})$ (in MPa) of CNTRC cylindrical shells subjected to combined axial and external pressure loads ( $R / h = 30$ , $L^{2} / R h = 300$ , $T = 300 K$ ).

$\frac{R γ}{2 h}$	Source	$V_{CNT}^{*} = 0.12$		$V_{CNT}^{*} = 0.17$
$\frac{R γ}{2 h}$	Source	UD	FG-X	UD	FG-X
0	Shen and Xiang⁶¹	(100.620, 0)	(118.848, 0)	(162.180, 0)	(196.376, 0)
	Present	(102.994, 0)^a	(118.790, 0)^b	(165.306, 0)^a	(197.158, 0)^a
0.02	Shen and Xiang⁶¹	(76.903, 0.103)	(83.935, 0.112)	(127.790, 0.170)	(142.763, 0.190)
	Present	(77.328, 0.103)	(83.965, 0.112)	(128.444, 0.171)	(142.905, 0.190)
0.1	Shen and Xiang⁶¹	(29.891, 0.199)	(32.625, 0.218)	(49.669, 0.331)	(55.490, 0.370)
	Present	(30.048, 0.200)	(32.627, 0.218)	(49.911, 0.333)	(55.529, 0.370)

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

^a Buckling mode $(m, n) = (2, 4)$ .

^b $(m, n) = (1, 3)$ , otherwise $(m, n) = (1, 4)$ .

In what follows, buckling and postbuckling behaviors of CNTRC cylindrical shells under combined mechanical loads in thermal environments will be analyzed. For the sake of brief expression, CNTRC cylindrical shells are assumed to be placed at room temperature ( $T = 300 K$ ) and without surrounding elastic foundations ( $K_{1} = K_{2} = 0$ ) unless otherwise specified.

Buckling analysis

Table 2 considers the effects of CNT volume fraction $V_{C N T}^{*}$ , distribution types, and load proportional parameter $γ$ on the buckling behavior of CNTRC cylindrical shells under combined axial compression and external pressure. As can be seen, buckling loads are considerably increased and decreased as $V_{CNT}^{*}$ and $γ$ are increased, respectively. This table also indicates that FG-X and FG-O cylindrical shells have the highest and lowest values of buckling combined loads, respectively. Buckling load of UD shell is slightly higher than that of FG-V shell under axial compression only (i.e. $γ = 0$ ) at small value of CNT volume fraction ( $V_{CNT}^{*} = 0.12$ ) and, inversely, buckling load of UD shell is slightly lower than that of FG-V shell under combined loads ( $γ \neq 0$ ) and/or for CNT-richer shells.

Table 2.

Critical buckling loads $P_{c r}$ (in MPa) of CNTRC cylindrical shells subjected to combined axial compression and external pressure ( $q_{c r} = γ P_{c r}$ , $R / h = 100$ , $L / R = 1.5$ , $T = 300 K$ ).^a

$V_{CNT}^{*}$	$γ \times 10^{3}$	UD	FG-X	FG-O	FG-V	FG-Λ
0.12	0	31.41 (1,6)	33.99 (1,6)	25.59 (2,8)	30.55 (2,8)	26.76 (2,8)
	1	15.47 (1,7)	16.93 (1,7)	14.18 (1,7)	15.94 (1,7)	14.05 (1,7)
	2	10.02 (1,8)	11.01 (1,8)	9.15 (1,8)	10.28 (1,8)	9.20 (1,7)
0.17	0	51.87 (1,6)	56.74 (1,6)	41.13 (2,8)	49.11 (2,8)	43.16 (2,8)
	1	25.56 (1,7)	28.48 (1,7)	23.34 (1,7)	26.50 (1,7)	23.55 (1,7)
	2	16.62 (1,8)	18.64 (1,7)	15.00 (1,8)	17.14 (1,8)	15.41 (1,7)
0.28	0	61.45 (1,6)	72.15 (1,6)	51.28 (2,8)	62.48 (2,8)	57.22 (2,8)
	1	30.20 (1,7)	36.67 (1,7)	26.78 (1,8)	31.41 (1,7)	28.80 (1,7)
	2	19.45 (1,8)	24.01 (1,7)	16.81 (1,8)	20.21 (1,8)	18.80 (1,8)

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

^a Number in brackets indicates buckling mode $(m, n)$

Next, effects of CNT volume fraction, distribution type, thermal environment, and elastic foundation on the interaction buckling curves of CNTRC cylindrical shells subjected to combined axial compression and external pressure are shown in Figures 2 to 5. Figure 2 demonstrates that the interaction buckling curve of FG-V CNTRC shell is slightly higher than that of UD CNTRC shell and that the stable regions of FG-X and FG-Λ shells are the largest and smallest, respectively, among four distribution types of CNTRC cylindrical shell. Figure 3 shows that the stable region of CNTRC cylindrical shell become smaller when environment temperature is increased. Subsequently, Figure 4 indicates that the interaction buckling curve of CNTRC cylindrical shell is considerably enhanced as CNT volume fraction is increased. As final illustration for buckling analysis, Figure 5 considers effects of surrounding elastic foundation on the interaction buckling curve of CNTRC cylindrical shells under combined loads. It is clear that the stable region of the shell is significantly broadened when the shell is surrounded by elastic foundation, especially Pasternak type foundation.

Figure 2.

Effects of CNT distribution types on the interaction buckling curves of CNTRC cylindrical shells under combined axial compression and external pressure.

Figure 3.

Effects of thermal environments on the interaction buckling curves of CNTRC cylindrical shells under combined axial compression and external pressure.

Figure 4.

Effects of CNT volume fraction on the interaction buckling curves of CNTRC cylindrical shells under combined axial compression and external pressure.

Figure 5.

Effects of surrounding elastic foundations on the interaction buckling curves of CNTRC cylindrical shells under combined axial compression and external pressure.

Postbuckling analysis

As first example for postbuckling analysis, Figure 6 analyzes the effects of preexisting nondestabilizing external pressure (i.e. four different values of $γ$ ) on the postbuckling behavior of CNTRC cylindrical shell subjected to axial compression. It is evident that postbuckling equilibrium path is lower and intensity of snap-through response is more benign when load-proportional parameter $γ$ is increased. In addition, number of circumferential waves n is increased at larger values of preexisting external pressure. Figure 7 considers the effects of three CNT distribution types FG-X, FG-V, and UD on the postbuckling behavior of CNTRC cylindrical shell subjected to axial compression alone ( $γ = 0$ ) or combined with external pressure ( $γ = 5 \times 10^{- 4}$ ). Obviously, the equilibrium path of FG-X shell is the highest and the most stable. In comparison with UD shell, the postbuckling equilibrium path of FG-V shell is higher in initial postbuckling region and the snap-through response of FG-V shell is more severe. In addition, the equilibrium paths of FG-V and UD shells under combined loads are closer to each other.

Figure 6.

Illustrative shape of buckling mode $(m, n) = (1, 6)$ (a) and postbuckling behavior of CNTRC cylindrical shells subjected to axial compression combined with preexisting external lateral pressure (b).

Figure 7.

Effects of CNT distribution types on postbuckling behavior of CNTRC cylindrical shells subjected to axial compression combined with preexisting external lateral pressure.

Next, Figure 8 plotted with three different values of CNT volume fraction shows that the postbuckling strength of CNTRC cylindrical shell is considerably higher and difference between equilibrium paths corresponding to values $γ = 10^{- 4}$ and $γ = 5 \times 10^{- 4}$ becomes bigger at higher value of CNT volume percentage. In other words, detrimental effect of preexisting external pressure on the postbuckling behavior of axially loaded CNTRC cylindrical shell is more pronounced for CNT-richer shell. A comparison of postbuckling behavior of axially loaded CNTRC cylindrical shells with varying temperatures ( $T = 300 K$ and $500 K$ ) and various values of preexisting lateral pressure is carried out in Figure 9, in which negative and positive values of $γ$ represent internal and external pressures, respectively. As can be observed, load–deflection curves are strongly reduced as values of T and/or $γ$ are increased. Furthermore, the severity of snap-through response is increased as preexisting pressure is internal and is decreased as environment temperature is elevated. Subsequently, the effects of elastic foundations on the postbuckling behavior of CNTRC cylindrical shell exposed to a thermal environment ( $T = 400 K$ ), subjected to axial compression combined with internal or external pressure are displayed in Figure 10. In contrast to preexisting external pressure, internal pressure and surrounding elastic foundation have beneficial influences on the postbuckling behavior of CNTRC cylindrical shells under axial compression. Specifically, axial load–deflection curves are significantly enhanced as CNTRC cylindrical shells are surrounded by elastic foundations, especially Pasternak type foundations.

Figure 8.

Effects of CNT volume fraction on postbuckling behavior of CNTRC cylindrical shells subjected to axial compression combined with preexisting external lateral pressure.

Figure 9.

Effects of thermal environments on postbuckling behavior of CNTRC cylindrical shells subjected to axial compression combined with internal and external lateral pressures.

Figure 10.

Effects of elastic foundations on postbuckling behavior of CNTRC cylindrical shells under combined loads in a thermal environment ( $(m, n) = (1, 6)$ for $γ = - 2.5 \times 10^{- 4}$ and $γ = 0$ , $(m, n) = (1, 7)$ for $γ = 2.5 \times 10^{- 4}$ ).

The remainder of this section presents numerical examples of postbuckling analysis for CNTRC cylindrical shells subjected to external pressure combined with preexisting nondestabilizing axial compression. Figure 11, plotted with four different values of load-proportional parameter $η$ , shows that preexisting nondestabilizing axial compression load has detrimental influences on the postbuckling behavior of CNTRC cylindrical shell subjected to external pressure. More specifically, external pressure–deflection equilibrium paths are dropped and intensity of snap-through response is increased as $η$ is increased. Next, Figure 12 shows that difference between external pressure–deflection equilibrium paths corresponding to cases of $η = 0$ and $η = 500$ become bigger as CNT volume fraction is increased. This means the deteriorative influences of preexisting axial load on the postbuckling behavior of externally pressurized CNTRC cylindrical shells are more remarked for CNT-richer shells.

Figure 11.

Postbuckling behavior of CNTRC cylindrical shells subjected to external lateral pressure combined with preexisting axial compression.

Figure 12.

Effects of CNT volume fraction on the postbuckling behavior of CNTRC cylindrical shells subjected to external lateral pressure combined with preexisting axial compression.

The interactive effects of thermal environments and preexisting axial load on the postbuckling behavior of CNTRC cylindrical shells under external pressure are examined in Figure 13. Obviously, increases in parameter $η$ and temperature T cause a remarked decrease in pressure load bearing capability of the shells. In addition, the intensity of snap-through response is increased and decreased at higher values of $η$ and T, respectively. In this example, prebuckling deflections predicted are $W_{0} / h = 0.063, 0.055, 0.0475$ for $η = 0, 300, 600$ at $T = 300 K$ , respectively, and $W_{0} / h = - 0.7973, - 0.8045, - 0.8117$ for $η = 0, 300, 600$ at $T = 500 K$ , respectively. These demonstrate a fact that pressure-loaded cylindrical shell is deflected outward prior to buckling at elevated temperature and the amplitude of prebuckling deflection is increased due to increase in preexisting axial load. Finally, Figure 14 proves that surrounding elastic foundations have pronouncedly beneficial influences on the postbuckling response of CNTRC cylindrical shells under external pressure combined with axial compression. Specifically, postbuckling equilibrium paths are higher and more stable (i.e. more benign snap-through phenomenon) through embrace of surrounding elastic media.

Figure 13.

Effects of thermal environments on the postbuckling behavior of CNTRC cylindrical shells subjected to external lateral pressure combined with axial compression.

Figure 14.

Effects of elastic foundations on postbuckling behavior of CNTRC cylindrical shells under external pressure combined with axial compression in a thermal environment.

Concluding remarks

An analytical investigation on the buckling and postbuckling behaviors of thin CNTRC cylindrical shells surrounded by elastic media, exposed to thermal environments, and subjected to combined action of axial compression and lateral pressure has been presented. Above analysis results suggest the following remarks:

The interaction buckling curves are higher (i.e. stable regions are larger) for CNT-richer shells and/or due to surrounding elastic media, especially for FG-X CNTRC shells, and are lower as temperature is elevated.

Postbuckling equilibrium paths of axially compressed CNTRC cylindrical shells are considerably lower and more stable through increase in preexisting nondestabilizing external pressure. In contrast, preexisting internal pressure leads to higher axial load–deflection curves and more intense snap-through response.

Preexisting nondestabilizing axial compression load detrimentally influences on the postbuckling behavior of pressure-loaded CNTRC cylindrical shells because pressure–deflection curves are significantly dropped and snap-through response is more severe through increase in axial compression.

Postbuckling equilibrium paths are lower and snap-through response is more benign when CNTRC cylindrical shells under combined mechanical loads exposed to an elevated temperature.

Detrimental effects of preexisting external pressure (axial load) on the postbuckling behavior of CNTRC cylindrical shells subjected to axial load (external pressure) are more remarked for CNT-richer shells.

Postbuckling response of CNTRC cylindrical shells under combined loads is pronouncedly improved through surrounding elastic foundations, especially Pasternak type foundations.

Footnotes

Funding

The author(s) disclosed receipt of following financial support for the research, authorship, and/or publication of this article: This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2017.11.

Appendix 1

The coefficients $a_{j 3} (j = 1 \div 4)$ and $a_{k 4} (k = 1 \div 3)$ in the equations (21b) and (21c) are 1A

\begin{matrix} a_{13} = a_{11} β_{m}^{4} + a_{21} δ_{n}^{4} + a_{31} β_{m}^{2} δ_{n}^{2} + k_{1} + k_{2} (β_{m}^{2} + δ_{n}^{2}) \\ + \frac{1}{a_{12} β_{m}^{4} + a_{22} β_{m}^{2} δ_{n}^{2} + a_{32} δ_{n}^{4}} (\frac{β_{m}^{2}}{R} - a_{42} β_{m}^{4} - a_{52} β_{m}^{2} δ_{n}^{2} - a_{62} δ_{n}^{4}) (a_{41} β_{m}^{2} δ_{n}^{2} + \frac{β_{m}^{2}}{R}) \\ a_{23} = \frac{β_{m}^{2} δ_{n}^{2}}{a_{12} β_{m}^{4} + a_{22} β_{m}^{2} δ_{n}^{2} + a_{32} δ_{n}^{4}} [2 \frac{β_{m}^{2}}{R} + (a_{41} - a_{52}) β_{m}^{2} δ_{n}^{2} - a_{42} β_{m}^{4} - a_{62} δ_{n}^{4}] \\ + \frac{δ_{n}^{2}}{16 a_{12}} (\frac{4}{R} - 16 a_{42} β_{m}^{2}) \\ a_{33} = \frac{β_{m}^{4}}{16 a_{32}} + \frac{δ_{n}^{4}}{16 a_{12}} \\ a_{43} = \frac{β_{m}^{4} δ_{n}^{4}}{a_{12} β_{m}^{4} + a_{22} β_{m}^{2} δ_{n}^{2} + a_{32} δ_{n}^{4}} + \frac{β_{m}^{4} δ_{n}^{4}}{81 a_{12} β_{m}^{4} + 9 a_{22} β_{m}^{2} δ_{n}^{2} + a_{32} δ_{n}^{4}} \\ a_{14} = 4 a_{11} β_{m}^{4} + \frac{3}{4} k_{1} + β_{m}^{2} k_{2} - \frac{1}{4 a_{12} R} (4 a_{42} β_{m}^{2} - \frac{1}{R}) \\ a_{24} = \frac{β_{m}^{2} δ_{n}^{2}}{2 (a_{12} β_{m}^{4} + a_{22} β_{m}^{2} δ_{n}^{2} + a_{32} δ_{n}^{4})} (\frac{β_{m}^{2}}{R} - a_{42} β_{m}^{4} - a_{52} β_{m}^{2} δ_{n}^{2} - a_{62} δ_{n}^{4}) + \frac{δ_{n}^{2}}{16 a_{12} R} \\ a_{34} = \frac{β_{m}^{4} δ_{n}^{4}}{2 (a_{12} β_{m}^{4} + a_{22} β_{m}^{2} δ_{n}^{2} + a_{32} δ_{n}^{4})} + \frac{β_{m}^{4} δ_{n}^{4}}{2 (81 a_{12} β_{m}^{4} + 9 a_{22} β_{m}^{2} δ_{n}^{2} + a_{32} δ_{n}^{4})} \end{matrix}

The details of coefficients $f_{i 1} (i = 1 \div 4)$ , $f_{j 2} (j = 1 \div 7)$ and $f_{k 3} (k = 1 \div 7)$ in the equations (24) to (26) are defined as follows 1B

\begin{matrix} f_{11} = \frac{1}{2 R_{h}^{2}} (1 - ν_{12} ν_{21}) {\bar{e}}_{21} + \frac{K_{1} E_{0}^{m}}{2 R_{h}^{4}}; f_{21} = \frac{n^{2}}{8 R_{h}^{3}} (1 - ν_{12} ν_{21}) {\bar{e}}_{21} \end{matrix}

\begin{matrix} f_{31} = \frac{ν_{12} {\bar{e}}_{21}}{R_{h} {\bar{e}}_{11}}; f_{41} = \frac{1}{R_{h} {\bar{e}}_{11}} ({\bar{e}}_{11} {\bar{e}}_{21 T} - ν_{12} {\bar{e}}_{21} {\bar{e}}_{11 T}) \end{matrix}

\begin{matrix} f_{12} = {\bar{a}}_{11} \frac{m^{4} π^{4}}{R_{h}^{4} L_{R}^{4}} + {\bar{a}}_{21} \frac{n^{4}}{R_{h}^{4}} + {\bar{a}}_{31} \frac{m^{2} n^{2} π^{2}}{R_{h}^{4} L_{R}^{2}} + K_{1} \frac{E_{0}^{m}}{R_{h}^{4}} + K_{2} \frac{E_{0}^{m}}{R_{h}^{2}} (\frac{m^{2} π^{2}}{R_{h}^{2} L_{R}^{2}} + \frac{n^{2}}{R_{h}^{2}}) \end{matrix}

\begin{matrix} + (\frac{m^{2} π^{2}}{R_{h}^{3} L_{R}^{2}} - {\bar{a}}_{42} \frac{m^{4} π^{4}}{R_{h}^{4} L_{R}^{4}} - {\bar{a}}_{52} \frac{m^{2} n^{2} π^{2}}{R_{h}^{4} L_{R}^{2}} - {\bar{a}}_{62} \frac{n^{4}}{R_{h}^{4}}) (\frac{{\bar{a}}_{41} m^{2} n^{2} π^{2} L_{R}^{2} + m^{2} π^{2} R_{h} L_{R}^{2}}{{\bar{a}}_{12} m^{4} π^{4} + {\bar{a}}_{22} m^{2} n^{2} π^{2} L_{R}^{2} + {\bar{a}}_{32} n^{4} L_{R}^{4}}) \end{matrix}

\begin{matrix} f_{22} = \frac{n^{2}}{R_{h}^{3}} {\bar{e}}_{21} (1 - ν_{12} ν_{21}) \end{matrix}

\begin{matrix} f_{32} = \frac{1}{R_{h}^{4} ({\bar{a}}_{12} m^{4} π^{4} + {\bar{a}}_{22} m^{2} n^{2} π^{2} L_{R}^{2} + {\bar{a}}_{32} n^{4} L_{R}^{4})} [{\bar{a}}_{41} m^{4} n^{4} π^{4} + m^{4} n^{2} π^{4} R_{h} \end{matrix}

\begin{matrix} + m^{2} n^{2} π^{2} R_{h}^{4} L_{R}^{2} (\frac{m^{2} π^{2}}{R_{h}^{3} L_{R}^{2}} - {\bar{a}}_{42} \frac{m^{4} π^{4}}{R_{h}^{4} L_{R}^{4}} - {\bar{a}}_{52} \frac{m^{2} n^{2} π^{2}}{R_{h}^{4} L_{R}^{2}} - {\bar{a}}_{62} \frac{n^{4}}{R_{h}^{4}})] \end{matrix}

\begin{matrix} + \frac{n^{2}}{16 {\bar{a}}_{12} R_{h}^{2}} (\frac{4}{R_{h}} - 16 {\bar{a}}_{42} \frac{m^{2} π^{2}}{R_{h}^{2} L_{R}^{2}}) + \frac{n^{2} {\bar{e}}_{21}}{2 R_{h}^{3}} (1 - ν_{12} ν_{21}) \end{matrix}

\begin{matrix} f_{42} = \frac{m^{4} π^{4}}{16 {\bar{a}}_{32} R_{h}^{4} L_{R}^{4}} + \frac{n^{4}}{16 {\bar{a}}_{12} R_{h}^{4}} + \frac{n^{4} {\bar{e}}_{21}}{8 R_{h}^{4}} (1 - ν_{12} ν_{21}) \end{matrix}

\begin{matrix} f_{52} = \frac{m^{4} n^{4} π^{4}}{R_{h}^{4} ({\bar{a}}_{12} m^{4} π^{4} + {\bar{a}}_{22} m^{2} n^{2} π^{2} L_{R}^{2} + {\bar{a}}_{32} n^{4} L_{R}^{4})} \end{matrix}

\begin{matrix} + \frac{m^{4} n^{4} π^{4}}{R_{h}^{4} (81 {\bar{a}}_{12} m^{4} π^{4} + 9 {\bar{a}}_{22} m^{2} n^{2} π^{2} L_{R}^{2} + {\bar{a}}_{32} n^{4} L_{R}^{4})} \end{matrix}

\begin{matrix} f_{62} = \frac{m^{2} π^{2}}{R_{h}^{2} L_{R}^{2}} + ν_{12} \frac{n^{2} {\bar{e}}_{21}}{R_{h}^{2} {\bar{e}}_{11}}; f_{72} = \frac{n^{2}}{R_{h}^{2} {\bar{e}}_{11}} ({\bar{e}}_{11} {\bar{e}}_{21 T} - ν_{12} {\bar{e}}_{21} {\bar{e}}_{11 T}) \end{matrix}

\begin{matrix} f_{13} = K_{1} \frac{E_{0}^{m}}{R_{h}^{4}} + \frac{{\bar{e}}_{21}}{R_{h}^{2}} (1 - ν_{12} ν_{21}) \end{matrix}

\begin{matrix} f_{23} = \frac{{\bar{e}}_{21}}{2 R_{h}^{2}} (1 - ν_{12} ν_{21}) + 4 {\bar{a}}_{11} \frac{m^{4} π^{4}}{R_{h}^{4} L_{R}^{4}} + K_{1} \frac{3 E_{0}^{m}}{4 R_{h}^{4}} + K_{2} \frac{m^{2} π^{2} E_{0}^{m}}{R_{h}^{4} L_{R}^{2}} - \frac{1}{4 R_{h} {\bar{a}}_{12}} (4 {\bar{a}}_{42} \frac{m^{2} π^{2}}{R_{h}^{2} L_{R}^{2}} - \frac{1}{R_{h}}) \end{matrix}

\begin{matrix} f_{33} = \frac{n^{2} {\bar{e}}_{21}}{8 R_{h}^{3}} (1 - ν_{12} ν_{21}) + \frac{n^{2}}{16 {\bar{a}}_{12} R_{h}^{3}} \end{matrix}

\begin{matrix} + \frac{m^{2} n^{2} π^{2} L_{R}^{2}}{2 ({\bar{a}}_{12} m^{4} π^{4} + {\bar{a}}_{22} m^{2} n^{2} π^{2} L_{R}^{2} + {\bar{a}}_{32} n^{4} L_{R}^{4})} (\frac{m^{2} π^{2}}{R_{h}^{3} L_{R}^{2}} - {\bar{a}}_{42} \frac{m^{4} π^{4}}{R_{h}^{4} L_{R}^{4}} - {\bar{a}}_{52} \frac{m^{2} n^{2} π^{2}}{R_{h}^{4} L_{R}^{2}} - {\bar{a}}_{62} \frac{n^{4}}{R_{h}^{4}}) \end{matrix}

\begin{matrix} f_{43} = \frac{m^{4} n^{4} π^{4}}{2 R_{h}^{4} ({\bar{a}}_{12} m^{4} π^{4} + {\bar{a}}_{22} m^{2} n^{2} π^{2} L_{R}^{2} + {\bar{a}}_{32} n^{4} L_{R}^{4})} \end{matrix}

\begin{matrix} + \frac{m^{4} n^{4} π^{4}}{2 R_{h}^{4} (81 {\bar{a}}_{12} m^{4} π^{4} + 9 {\bar{a}}_{22} m^{2} n^{2} π^{2} L_{R}^{2} + {\bar{a}}_{32} n^{4} L_{R}^{4})} \end{matrix}

\begin{matrix} f_{53} = \frac{m^{2} π^{2}}{R_{h}^{2} L_{R}^{2}}; f_{63} = \frac{ν_{12} {\bar{e}}_{21}}{R_{h} {\bar{e}}_{11}}; f_{73} = \frac{1}{R_{h} {\bar{e}}_{11}} ({\bar{e}}_{11} {\bar{e}}_{21 T} - ν_{12} {\bar{e}}_{21} {\bar{e}}_{11 T}) \end{matrix}

Appendix 2

The coefficients $f_{i 4}, f_{i 5} (i = 1 \div 6)$ in the equation (28) are defined as

in which

The details of coefficients $f_{j 6} (j = 1 \div 8)$ in equation (30) are given as

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