Thermoelastic analysis of FG-CNTRC cylindrical shells with various boundary conditions and temperature-dependent characteristics using quasi-3D higher-order shear deformation theory

Abstract

Abstract. This study focuses on performing static analysis of FG-CNTRC cylinder shells with various boundary restrictions, including thermomechanical responses. The governing equations are developed by taking into account the temperature-dependent material features, the quasi-3D high-order shear deformation hypothesis, and the normal transverse stress effect. The temperature gradient inside the thickness is expected to fluctuate, and the distribution pattern is derived by using the heat transfer equation and considering the temperature boundary limitations. A singular trigonometric series and the Laplace transform are used in an analytics solution to address basic equations. This study primarily examines the stress levels at the border region. The findings indicate that it is crucial to take into account the abrupt rise in stress at the boundary area, particularly when the shell’s relative length is small. The reciprocal impact of pressure and temperature load is also emphasized. Significant findings indicate that thermal load may either augment or diminish stress levels, contingent upon the orientation of the pressure and thermal load effect. The results of the study of this issue serve as the foundation for the calculation and design of relevant structures in practical applications. Furthermore, this serves as a foundation for the creation of more intricate issues in the forthcoming.

Introduction

Since their discovery by Iijima ¹ in 1991, CNTs (carbon nanotubes) have played a meaningful role in nanomaterial technology. The results of many studies have shown that CNTs have high strength and low density.^2–5 Besides, CNTs are also known as materials with outstanding electrical and thermal conductivity.^6–8 This is because they have special physical characteristics that make them excellent choices for reinforcement in nanocomposites. Numerous studies demonstrate that the physical and mechanical characteristics of nanocomposite materials are greatly enhanced by the implementation of a small amount of CNT into the polymer matrix.^9–14 For conventional nanocomposites (carbon nanotube-reinforced composites - CNTRC), the mechanical and physical properties do not change with location because the CNT is uniformly distributed in the matrix. To promote the efficiency of the structures, the concept of Functionally Graded Materials (FGMs) was introduced into nanocomposites to appear as a model of functionally graded carbon nanotube-reinforced composites (FG-CNTRC). The first FG-CNTRC models were introduced by Shen in 2009. ¹⁵ These models were then further developed by Shen in subsequent studies.^16–19 Later, many scientists used them to analyze the FG-CNTRC systems. These explorations have been evaluated by Liew and colleagues,^20,21 Soni and coworkers, ²² Zhang and his team, ²³ Kurmar and colleagues, ²⁴ Behera and coworkers, ²⁵ and Ebrahimi and his research team. ²⁶ The study of FG-CNTRC structures under thermal effects is an issue of great scholarly and practical value. In an effort to approach the actual model, scientists have continued to develop models of heat transfer and the relationships between temperature and the physical and mechanical properties of FG-CNTRC structures. Cuong et al. ²⁷ showcased finite element analysis is used to study the static bending behavior of rotating beams made of functionally graded-glass/polymer laminated reinforced composites (FG-GPLRC) in the presence of geometric flaws in heat environments. Boutaleb and his team ²⁸ the buckling behavior of imperfect beams made of functionally graded carbon nanotube-reinforced composite materials.

In many studies, the thermal gradient is ignored when the temperature of the whole structure is uniform. Shen ¹⁵ presented the nonlinear static analysis of the FG-CNTRC plates under the transverse uniformly and sinusoidally distributed loads in the thermal environment. The fundamental formulations were established according to Reddy’s higher-order shear deformation theory (HSDT) with the von Kármán nonlinear kinematic. The load-deflection and load-bending moment curves were obtained employing a two-step perturbation method. Shen ¹⁷ conducted the post-buckling analysis of the FG-CNTRC shell under axial compression. The buckling loads and post-buckling balance paths were defined using a singular perturbation method. Under various uniform temperature rises computational examples focus on the post-buckling analysis of the perfect and imperfect FG-CNTRC shells under the axial loading. Using the same theory and method, Shen ¹⁹ investigated the post-buckling of the FG-CNTRC shells under pressure load in thermal media. The nonlinear oscillation of FG-CNTRC plate structures placed on an elastic substrate subjected to thermal load was studied by Shen and Wang. ²⁹ They employed HSDT to establish the motion equation and an improved perturbation technique to solve these equations. Gawah and colleagues ³⁰ suggested an enhanced first-order shear deformation framework for analyzing waves propagating in FG-CNTRC beams supported by a viscoelastic substrate. Using first-order shear deformation plate theory (FSDT), Zhu and coworkers ³¹ presented static and free oscillation evaluation of FG-CNTRC plate structures. They employed the finite element method (FEM) to study the plates with various boundary constraints. Gholami and Ansari^32,33 studied how the initial geometric defect affects the nonlinear reverberation of rectangular plates made of functionally graded carbon nanotube-reinforced composite materials. Using Reddy’s HSDT and von Kármán-type equations, Shen and Wang ³⁴ investigated the nonlinear oscillation and static behavior of the multilayer plates with the CNTRC face sheet placed on an elastic substrate. The motion equations, which include the structure, elastic substrate association, and thermal impact, were resolved by a two-step perturbation method. Using the same theory, Shen ³⁵ employed a multi-scale approach to conduct buckling and post-buckling analysis of FG-CNTRC shells in the thermal media. It was considered that FG-CNTRCs' material characteristics were temperature-dependent. Xiang and Shen studied the nonlinear oscillation of FG-CNTRC shells at high temperatures. ¹⁹ The frequencies of the shells were determined by an improved perturbation technique. Using Reddy’s HSDT and nonlinear Von Karman theory, Shen và Xiang ³⁶ explored the post-buckling of FG-CNTRC under mixed radial and axial loads with uniform temperature rise. Zaitoun and Chikh^37,38 proposed a very effective computational model to analyze the vibration characteristics of a sandwich plate with functionally graded materials in a hygrothermal environment, taking into account the impacts of a viscoelastic foundation. Ninh and Bich ³⁹ used the conventional shell concept with Von Kármán theory to study the nonlinear vibration of piezoelectric FG-CNTRC structures under thermo-mechanical reactions. Galerkin’s approach was used to solve the motion equations of the structure in uniform temperature rise. Using the same theory and method, Van Tung and Trang ⁴⁰ investigated the sensitivities of imperfection on the nonlinear behavior of simply supported FG-CNTRC plates in thermal media. Postbuckling evaluation of CNTRC sandwich panels placed on elastic substrate was examined by Long and Tung. ⁴¹ The plate under uniaxial compressive and thermomechanical loads Authors employed the FSDT and von Kármán theory to establish motion equations. The Galerkin approach is employed to solve these equations of simply supported plates. Using FSDT and a semi-analytical solution, the nonlinear oscillation of stiffened FG-CNTRC plates was presented by Dong et al. ⁴² They used the Galerkin approach and the fourth-order Runge-Kutta approach to determine the vibration behaviors of plates and cylindrical panels. Using the classical shell theory (CST) and von Kármán theory, Dong et al. ⁴³ examined the nonlinear buckling and post-buckling of stiffened CNTRC shells by an elastic foundation with uniformly distributed temperature. These structures were reinforced with longitudinal or circumferential stiffeners made from CNTRC. The Galerkin method was used to solve the fundamental equations. Lafi et al. ⁴⁴ examined how the thermodynamic properties of FG sandwich plates, which are placed on different elastic foundations, is affected by the combination of variable distribution theories and boundary circumstances. Al-Osta et al. ⁴⁵ investigated the impact of porosity on the bending performance of AFG ceramic-metal plates under hygro-thermo-mechanical conditions using an integrated plate model. Based on the same theory and method, Tien et al. ⁴⁶ presented nonlinearity buckling of torsion-loaded CNTRC shells stiffened by CNTRC stiffeners in a thermal environment. Dat and colleagues ⁴⁷ explored the nonlinear oscillation of intelligent multilayer plates in hygrothermal media. A CNTRC core is merged in double electro-magneto-elastic face layers to create the sandwich plate. Using Reddy’s HSDT to develop the fundamental formulations of plates and the Galerkin approach to solve these expressions. Tounsi and his research ream^48–51 investigated how various boundary conditions and the hygro-thermal environment affect the free vibration responses of sandwiched plates made of functionally graded materials (FGM) and resting on a viscoelastic substrate. According to the FSDT and the Galerkin approach, Thanh and coworkers ⁵² examined the nonlinearity dynamical reaction and oscillation of imperfect FG-CNTRC shells resting on an elastic substrate. The combined load acting on the shell includes the dynamically distributed load on the outer surface and the axial load. The temperature in the shell is simulated to be unaltered throughout the whole structure. Mudhaffar and coworkers ⁵³ conducted a study on the influence of a viscoelastic substrate on the bending behavior of a functionally graded plate under hygro-thermo-mechanical stresses.

It can be seen that a few studies have considered the influence of thermal gradient loads on FG-CNTRC structures but used the expectations that the material possessions were temperature-independent. Liew and Alibeigloo ⁵⁴ used the three-dimensional (3D) hypothesis of elasticity to investigate the bending behavior of FG-CNTRC plates. Based on the same approach, Alibeigloo ⁵⁵ examined static responses of FG-CNTRC plates attached to thin piezoelectric components under the thermoelectric loads. They supposed that the temperature gradient has no impact on the thermo-elastic constants of the plates. Thermoelastic analysis of FG-CNTRC panels with the simply supported boundary condition was conducted by Alibeigloo. ⁵⁶ In this study, the 3D hypothesis of elasticity and state space approach are employed to examine the static behavior of panels. The author presupposed that the possessions of the FG-CNTRC material were temperature-independent. Hieu and Van Tung ⁵⁷ performed buckling and post-buckling analyses of FG-CNTRC shells under the pressure in various temperature conditions. The Galerkin method is taken into calculation to determine critical buckling pressures for the shells.

There are very few publications that simultaneously consider the influence of thermal gradient loadings and the influence of temperature on the material properties (temperature-dependent properties). Using the 3D elasticity theory and differential quadrature method, Pourasghar and coworkers ⁵⁸ studied the static behavior of the CNTRC shells under thermo-mechanical loads. The temperature distribution along the thickness direction and the material properties were temperature-dependent. To satisfy simple boundary conditions and simplify the solving process, suitable temperature and displacement functions were employed. Based on the same approach, Pourasghar and Chen ⁵⁹ conducted the static analysis of the CNTRC panels abutting by an elastic substrate. Moradi-Dastjerdi and colleagues ⁶⁰ used a mesh-free approach to examine the bending behavior of the FG cylinders in a thermal environment. The temperature distribution along the radial direction and the material characteristics were temperature-dependent.

To be able to accurately analyze the mechanical response of the FG-CNTRC structure, one of the important issues is determining the material properties. There are technological difficulties when producing and determining FG-CNT material parameters from experiments. When distributing CNTs into the matrix material, the CNTs can accumulate into clusters, and distributing the CNTs into the structure according to the rule of thickness variation is really a challenge. ⁶¹ Currently, there are several methods of manufacturing FGM materials being researched that have the potential to be put into commercial production.^62,63

This work makes use of the theoretical FG-CNTRC material model that Shen proposed and developed.^16–19 A number of methods have been researched to model FG-CNTRC materials, the most popular of which are the Eshelbly - Mori - Tanaka methods and the rule of mixture. Using the rule of mixture is simple and convenient. However, conventional mixing rules do not provide the level of accuracy needed for CNTRC materials. To achieve the necessary accuracy, Shen^16–19 used the extended rule of mixture by adding CNT effective coefficients to correct results. These coefficients are evaluated by matching the effective mechanical properties of FG-CNTRC obtained from the molecular dynamics simulations^64,65 with those from the rule of mixtures. Ansari et al.^66,67 conducted a thermal postbuckling study of FG-CNTRC plates with different forms and attributes that vary with temperature, using the VDQ-FEM approach. Van and Minh^68,69 studied the mechanical behavior of composite panels with dowel connections including sliding between layers. Some studies related to the content of this article can be referred to here.^70–82

The aforementioned papers indicate a scarcity of research that concurrently examines the impact of both the thermal gradient and the temperature-dependent characteristic. Furthermore, it is worth noting that, to the author’s understanding, no investigations have been conducted on the FG-CNTRC shells under varying boundary conditions inside a thermal environment. The thermal and displacement functions were often considered to be maintained just on all edges or the essential boundary. It is worth mentioning that earlier findings have shown that the stress constituents at the clamped border region exhibit significantly higher magnitudes compared to other locations, particularly in the case of a short shell including one clamped edge and one free edge.^83–85

This study examines the thermoelastic behavior of cylindrical shells made of FG-CNTRC under various boundary conditions in a thermal condition. The radial direction exhibits a distribution of temperature, whereas the material qualities are contingent upon the temperature. The main research content of this work is to focus on analyzing stress characteristics of a shell at the boundary area and the mutual interaction between mechanical load and thermal load that has not been considered in previous publications. To conduct this, using the HSDT that includes normal transerve stress, simultaneously consider parameters depending on temperature and thermal gradient loadings determined from the heat transfer equation. An analytical method using simple trigonometric series and the Laplace transform is employed to solve problems with different boundary conditions.

Fundamental equations

Consider the FG-CNTRC cylindrical shell with the geometric parameters including the length L, radius R, and thickness h described in Figure 1. The symbols $u,v,$ and $w$ represent respectively the displacement components of a point in the shell in the $\xi ,\theta ,$ and $z$ directions. The load acting on the shell includes pressure on the inner surface $q_{in}$ , pressure on the outer surface $q_{out}$ , and thermal load $q_T$ . The temperature distribution law in the structure is $T\left(\xi ,\theta ,z\right)$ , with the temperature on the outer surface being $T_{out}$ and the temperature on the inner surface being $T_{in}$ .

Figure 1.

The geometry of cylindrical shell.

The temperature-dependent material characteristics

The cylindrical shell is mixed of isotropic matrix reinforced with CNTs. Five types of CNT distribution configurations were investigated, including uniform distribution (UD) and four cases of linear distribution according to thickness: FG-∧, FG-V, FG-X, and FG-O.

The effective temperature-dependent material characteristics of FG-CNTRC are obtained from the extended rule of mixed as:^41,47

$E_{11}\left(T\right)={\eta }_1{V}_{CNT}{E}_{11}^{CNT}\left(T\right)+V_m{E}_m\left(T\right),$

(1.a)

$\frac{{\eta }_2}{E_{22}\left(T\right)}=\frac{V_{CNT}}{E_{22}^{CNT}\left(T\right)}+\frac{V_m}{E_m\left(T\right)},$

(1.b)

$\frac{{\eta }_3}{G_{12}\left(T\right)}=\frac{V_{CNT}}{G_{12}^{CNT}\left(T\right)}+\frac{V_m}{G_m\left(T\right)},$

(1.c)

${\upsilon }_{12}=V_{CNT}^{*}{\upsilon }_{12}^{CNT}+V_m{\upsilon }_m,$

(1.d)

$\rho =V_{CNT}^{*}{\rho }_{CNT}+V_m{\rho }_m,$

(1.e)

${\alpha }_{11}\left(T\right)=\frac{V_{CNT}{E}_{11}^{CNT}\left(T\right){\alpha }_{11}^{CNT}\left(T\right)+V_m{E}_m\left(T\right){\alpha }_m\left(T\right)}{V_{CNT}{E}_{11}^{CNT}\left(T\right)+V_m{E}_m\left(T\right)},$

(1.f)

${\alpha }_{22}\left(T\right)=\left(1+{\upsilon }_{12}^{CNT}\right)V_{CNT}{\alpha }_{22}^{CNT}\left(T\right)+\left(1+{\upsilon }_m\right)V_m{\alpha }_m\left(T\right)-{\upsilon }_{12}{\alpha }_{11}\left(T\right),$

(1.g)

$k_{11}=V_{CNT}{k}_{11}^{CNT}+V_m{k}_m,$

(1.h)

$\frac{1}{k_{22}}=\frac{V_{CNT}}{k_{22}^{CNT}}+\frac{V_m}{k_m},$

(1.i)

where ${\eta }_i\hspace{0.17em}\left(i=1,2,3\right)$ are CNT effectiveness coefficients. $E_{ii}^{CNT},G_{12}^{CNT},{\upsilon }_{12}^{CNT},{\rho }_{CNT},{\alpha }_{ii}^{CNT},$ and $k_{ii}^{CNT}$ are respectively Young’s modulus, shear modulus, ratio of Poisson, mass density, thermal expansion coefficients, and thermal conductivity in the longitudinal $\left(i=1\right)$ and transverse $\left(i=2\right)$ direction of the CNT. $E_m,$ $G_m,{\upsilon }_m,{\rho }_m,{\alpha }_m,$ and $k_m$ are Young’s modulus, shear modulus, ratio of Poisson, mass density, thermal expansion coefficients, and thermal conductivity, respectively, of the matrix. $V_{CNT}$ and $V_m$ are volume fraction of CNT and isotropic matrix, respectively.

The volume fraction of CNT and isotropic matrix are related as follows:

$V_{CNT}+V_m=1$

(2)

The CNT volume fraction distribution for five cases are expressed as: ⁵⁶

$\begin{array}{ll}{V}_{CNT}=V_{CNT}^{*}& \hspace{1em}\text{for}\hspace{0.17em}\text{UD}\end{array}$

(3.a)

$\begin{array}{ll}{V}_{CNT}=2\left(\frac{1}{2}-\frac{z}{h}\right)V_{CNT}^{*}& \hspace{1em}\text{for}\hspace{0.17em}\text{FG}-\mathrm{ʌ}\end{array}$

(3.b)

$\begin{array}{ll}{V}_{CNT}=2\left(\frac{1}{2}+\frac{z}{h}\right)V_{CNT}^{*}& \text{for}\hspace{0.17em}\text{FG}-\mathrm{V}\end{array}$

(3.c)

$\begin{array}{ll}{V}_{CNT}=2\left(1-\frac{2\left|z\right|}{h}\right)V_{CNT}^{*}& \text{for}\hspace{0.17em}\text{FG}-\mathrm{O}\end{array}$

(3.d)

$\begin{array}{ll}{V}_{CNT}=4\frac{\left|z\right|}{h}{V}_{CNT}^{*}& \text{for}\hspace{0.17em}\text{FG}-\mathrm{X}\end{array}$

(3.e)

in which

$V_{CNT}^{*}=\frac{{\rho }_m}{{\rho }_m+\frac{{\rho }_{CNT}}{{\mathrm{W}}_{CNT}}-{\rho }_{CNT}}$

(4)

where ${\mathrm{W}}_{CNT}$ is the mass fraction of CNT.

The other effective elastic constants are defined as follows:

$E_{22}=E_{33},G_{12}=G_{13}=G_{23},{\upsilon }_{21}=\frac{E_{22}}{E_{11}}{\upsilon }_{12},{\upsilon }_{13}={\upsilon }_{12},{\upsilon }_{31}={\upsilon }_{32}={\upsilon }_{23}={\upsilon }_{21},{\alpha }_{33}={\alpha }_{22},k_{33}=k_{22}$

(5)

Governing equations

The displacement field based on quasi-3D higher-order shear deformation concept, taking the normal transverse stress into calculation, is expressed by:^83,86

$\begin{array}{l}u\left(\xi ,\theta ,z\right)=u_0\left(\xi ,\theta \right)+u_1\left(\xi ,\theta \right)z+u_2\left(\xi ,\theta \right)\frac{z^2}{2!}+u_3\left(\xi ,\theta \right)\frac{z^3}{3!},\\ v\left(\xi ,\theta ,z\right)=v_0\left(\xi ,\theta \right)+v_1\left(\xi ,\theta \right)z+v_2\left(\xi ,\theta \right)\frac{z^2}{2!}+v_3\left(\xi ,\theta \right)\frac{z^3}{3!},\\ w\left(\xi ,\theta ,z\right)=w_0\left(\xi ,\theta \right)+w_1\left(\xi ,\theta \right)z+w_2\left(\xi ,\theta \right)\frac{z^2}{2!},\end{array}$

(6)

The transverse displacement w is developed as a quadratic polynomial in radial coordinates, not a constant like some other HSDTs. With this expansion, a the transverse stress component appears when performing calculations.

Strain and displacement according to the linear relationship are determined as:

$\begin{array}{l}{\epsilon }_{\xi }=\frac{1}{R}\frac{\partial u}{\partial \xi },{\epsilon }_{\theta }=\frac{1}{R+z}\frac{\partial v}{\partial \theta }+\frac{w}{R+z},{\epsilon }_z=\frac{\partial w}{\partial z},{\gamma }_{\xi z}=\frac{\partial u}{\partial z}+\frac{1}{R}\frac{\partial w}{\partial \xi },\\ {\gamma }_{\theta z}=\frac{\partial v}{\partial z}-\frac{v}{R+z}+\frac{1}{R+z}\frac{\partial w}{\partial \theta },{\gamma }_{\xi \theta }=\frac{1}{R}\frac{\partial v}{\partial \xi }+\frac{1}{R+z}\frac{\partial u}{\partial \theta }.\end{array}$

(7)

By substituting the displacement components in equation (6) into equation (7), one gets the strain expressions of the following form:

$\begin{array}{c}{\epsilon }_{\xi }=\frac{1}{R}\left[\frac{\partial u_0}{\partial \xi }+\frac{\partial u_1}{\partial \xi }z+\frac{\partial u_2}{\partial \xi }\frac{z^2}{2}+\frac{\partial u_3}{\partial \xi }\frac{z^3}{3!}\right],\\ {\epsilon }_{\theta }=\frac{1}{R+z}\left[\frac{\partial v_0}{\partial \theta }+w_0+\left(\frac{\partial v_1}{\partial \theta }+w_1\right)z+\left(\frac{\partial v_2}{\partial \theta }+w_2\right)\frac{z^2}{2}+\frac{\partial v_3}{\partial \theta }\frac{z^3}{3!}\right],\\ {\gamma }_{\xi \theta }=\frac{1}{R}\left(\frac{\partial v_0}{\partial \xi }+\frac{\partial v_1}{\partial \xi }z+\frac{\partial v_2}{\partial \xi }\frac{z^2}{2}+\frac{\partial v_3}{\partial \xi }\frac{z^3}{3!}\right)+\frac{1}{R+z}\left(\frac{\partial u_0}{\partial \theta }+\frac{\partial u_1}{\partial \theta }z+\frac{\partial u_2}{\partial \theta }\frac{z^2}{2}+\frac{\partial u_3}{\partial \theta }\frac{z^3}{3!}\right),\\ {\gamma }_{\xi z}=\frac{1}{R}\left[\frac{\partial w_0}{\partial \xi }+R{u}_1+\left(\frac{\partial w_1}{\partial \xi }+R{u}_2\right)z+\left(\frac{\partial w_2}{\partial \xi }+R{u}_3\right)\frac{z^2}{2}\right],{\epsilon }_z=\left(w_1+w_{2}z\right),\\ {\gamma }_{\theta z}=\frac{1}{R+z}\left[\frac{\partial w_0}{\partial \theta }+R{v}_1-v_0+\left(\frac{\partial w_1}{\partial \theta }+R{v}_2\right)z+\left(\frac{\partial w_2}{\partial \theta }+R{v}_3+v_2\right)\frac{z^2}{2}+2v_3\frac{z^3}{3!}\right].\end{array}$

(8)

When material characteristics depend on the temperature, the linear stress-strain relationship is determined as:

$\left\{\begin{array}{l}{\sigma }_{\xi }\\ {\sigma }_{\theta }\\ {\sigma }_z\\ {\tau }_{z\theta }\\ {\tau }_{\xi z}\\ {\tau }_{\xi \theta }\end{array}\right\}=\left[\begin{array}{cccccc}{C}_{11}\left(T,z\right)& C_{12}\left(T,z\right)& C_{13}\left(T,z\right)& 0& 0& 0\\ C_{21}\left(T,z\right)& C_{22}\left(T,z\right)& C_{23}\left(T,z\right)& 0& 0& 0\\ C_{31}\left(T,z\right)& C_{32}\left(T,z\right)& C_{33}\left(T,z\right)& 0& 0& 0\\ 0& 0& 0& C_{44}\left(T,z\right)& 0& 0\\ 0& 0& 0& 0& C_{55}\left(T,z\right)& 0\\ 0& 0& 0& 0& 0& C_{66}\left(T,z\right)\end{array}\right].\left\{\begin{array}{l}{\epsilon }_{\xi }\\ {\epsilon }_{\theta }\\ {\epsilon }_z\\ {\gamma }_{z\theta }\\ {\gamma }_{\xi z}\\ {\gamma }_{\xi \theta }\end{array}\right\}-\left\{\begin{array}{l}{\beta }_{\xi }\\ {\beta }_{\theta }\\ {\beta }_z\\ 0\\ 0\\ 0\end{array}\right\}\mathrm{\Delta }T,$

(9)

where $\mathrm{\Delta }T=T-T_{ref}$ and $T_{ref}$ is ambient temperature with no thermal strains. The stiffness elements, $C_{\text{ij}}$ are derived as follows:

$\begin{array}{l}{C}_{11}\left(T,z\right)=\frac{E_{11}\left(T,z\right)}{\mathrm{\Delta }}\left(1-{\upsilon }_{23}{\upsilon }_{32}\right),C_{22}\left(T,z\right)=\frac{E_{22}\left(T,z\right)}{\mathrm{\Delta }}\left(1-{\upsilon }_{31}{\upsilon }_{13}\right),\\ C_{33}\left(T,z\right)=\frac{E_{33}\left(T,z\right)}{\mathrm{\Delta }}\left(1-{\upsilon }_{21}{\upsilon }_{12}\right),C_{12}\left(T,z\right)=\frac{E_{11}\left(T,z\right)}{\mathrm{\Delta }}\left({\upsilon }_{21}+{\upsilon }_{23}{\upsilon }_{31}\right),\\ C_{13}\left(T,z\right)=\frac{E_{11}\left(T,z\right)}{\mathrm{\Delta }}\left({\upsilon }_{31}+{\upsilon }_{21}{\upsilon }_{32}\right),C_{23}\left(T,z\right)=\frac{E_{22}\left(T,z\right)}{\mathrm{\Delta }}\left({\upsilon }_{32}+{\upsilon }_{12}{\upsilon }_{31}\right),\\ C_{44}\left(T,z\right)=G_{23}\left(T,z\right),C_{55}\left(T,z\right)=G_{13}\left(T,z\right),C_{66}\left(T,z\right)=G_{12}\left(T,z\right),\\ \mathrm{\Delta }=1-{\upsilon }_{12}{\upsilon }_{21}-{\upsilon }_{23}{\upsilon }_{32}-{\upsilon }_{31}{\upsilon }_{13}-2{\upsilon }_{12}{\upsilon }_{23}{\upsilon }_{32}\end{array}$

(10)

${\beta }_{\xi },{\beta }_{\theta },{\beta }_z$ are the stress modules regarding to the thermal expansion coefficients ${\alpha }_{\text{ii}}$ according to the following formula:

$\left\{\begin{array}{l}{\beta }_{\xi }\\ {\beta }_{\theta }\\ {\beta }_z\end{array}\right\}=\left(\begin{array}{l}{C}_{11}\left(T,z\right){\alpha }_{11}+C_{12}\left(T,z\right){\alpha }_{22}+C_{13}\left(T,z\right){\alpha }_{33}\\ C_{21}\left(T,z\right){\alpha }_{11}+C_{22}\left(T,z\right){\alpha }_{22}+C_{23}\left(T,z\right){\alpha }_{33}\\ C_{31}\left(T,z\right){\alpha }_{11}+C_{32}\left(T,z\right){\alpha }_{22}+C_{33}\left(T,z\right){\alpha }_{33}\end{array}\right)$

(11)

Use the principle of virtual work to establish the fundamental equations. The virtual work principle for the shell, according to the virtual displacements, one obtains:

$\begin{array}{l}\underset{\xi \theta z}{\iiint }\left({\sigma }_{\xi }\delta {\epsilon }_{\xi }+{\sigma }_{\theta }\delta {\epsilon }_{\theta }+{\sigma }_z\delta {\epsilon }_z+{\tau }_{\xi \theta }\delta {\gamma }_{\xi \theta }+{\tau }_{\xi z}\delta {\gamma }_{\xi z}+{\tau }_{\theta z}\delta {\gamma }_{\theta z}\right)R^2\left(1+\frac{z}{R}\right)d\xi d\theta dz-\\ -\underset{\xi }{{\displaystyle \int }}\underset{\theta }{{\displaystyle \int }}\left[q_{out}\delta w^{out}\left(1+\frac{h}{2R}\right)+q_{in}\delta w^{in}\left(1-\frac{h}{2R}\right)\right]R^{2}d\xi d\theta =0,\end{array}$

(12)

in which $\delta w^{out}=\delta w_0+\frac{h}{2}\delta w_1+\frac{h^2}{8}\delta w_2$ , $\delta w^{in}=\delta w_0-\frac{h}{2}\delta w_1+\frac{h^2}{8}\delta w_2$ .

Substitute equations (6) -(8) and (10) into equation (12), then integrate the newly obtained equation through the thickness of the structure to derive the bellow equation:

$\begin{array}{l}\underset{\xi \theta }{\iint }(N_{\xi }\delta {\epsilon }_{\xi }^0+M_{\xi }\delta {\epsilon }_{\xi }^1+N_{\xi }^{*}\delta {\epsilon }_{\xi }^2+M_{\xi }^{*}\delta {\epsilon }_{\xi }^3+N_{\theta }\delta {\epsilon }_{\theta }^0+M_{\theta }\delta {\epsilon }_{\theta }^1+N_{\theta }^{*}\delta {\epsilon }_{\theta }^2+M_{\theta }^{*}\delta {\epsilon }_{\theta }^3+Q_z\delta {\epsilon }_z^0+\\ +S_z\delta {\epsilon }_z^1+N_{\xi \theta }\delta {\gamma }_{\xi \theta }^0+M_{\xi \theta }\delta {\gamma }_{\xi \theta }^1+N_{\xi \theta }^{*}\delta {\gamma }_{\xi \theta }^2+M_{\xi \theta }^{*}\delta {\gamma }_{\xi \theta }^3+N_{\theta \xi }\delta {\gamma }_{\theta \xi }^0+M_{\theta \xi }\delta {\gamma }_{\theta \xi }^1+N_{\theta \xi }^{*}\delta {\gamma }_{\theta \xi }^2+\\ +M_{\theta \xi }^{*}\delta {\gamma }_{\theta \xi }^3+Q_{\xi }\delta {\gamma }_{\xi z}^0+S_{\xi }\delta {\gamma }_{\xi z}^1+Q_{\xi }^{*}\delta {\gamma }_{\xi z}^2+Q_{\theta }\delta {\gamma }_{\theta z}^0+S_{\theta }\delta {\gamma }_{\theta z}^1+Q_{\theta }^{*}\delta {\gamma }_{\theta z}^2+S_{\theta }^{*}\delta {\gamma }_{\theta z}^3)Rd\xi d\theta -\\ -\underset{\xi }{{\displaystyle \int }}\underset{\theta }{{\displaystyle \int }}\left(F_0\delta w_0+F_1\delta w_1+F_2\delta w_2\right)R^{2}d\xi d\theta =0,\end{array}$

(13)

where the symbols are given as follows:

$\begin{array}{l}{\epsilon }_{\xi }^0=\frac{\partial u_0}{\partial \xi },{\epsilon }_{\xi }^1=\frac{\partial u_1}{\partial \xi },{\epsilon }_{\xi }^2=\frac{\partial u_2}{\partial \xi },{\epsilon }_{\xi }^3=\frac{\partial u_3}{\partial \xi },{\epsilon }_{\theta }^0=\frac{\partial v_0}{\partial \theta }+w_0,{\epsilon }_{\theta }^1=\frac{\partial v_1}{\partial \theta }+w_1,\\ {\epsilon }_{\theta }^2=\frac{\partial v_2}{\partial \theta }+w_2,{\epsilon }_{\theta }^3=\frac{\partial v_3}{\partial \theta },{\epsilon }_z^0=w_1,{\epsilon }_z^1=w_2,{\gamma }_{\xi \theta }^0=\frac{\partial v_0}{\partial \xi },{\gamma }_{\xi \theta }^1=\frac{\partial v_1}{\partial \xi },\\ {\gamma }_{\xi \theta }^2=\frac{\partial v_2}{\partial \xi },{\gamma }_{\xi \theta }^3=\frac{\partial v_3}{\partial \xi },{\gamma }_{\theta \xi }^0=\frac{\partial u_0}{\partial \theta },{\gamma }_{\theta \xi }^1=\frac{\partial u_1}{\partial \theta },{\gamma }_{\theta \xi }^2=\frac{\partial u_2}{\partial \theta },{\gamma }_{\theta \xi }^3=\frac{\partial u_3}{\partial \theta },\\ {\gamma }_{\xi z}^0=\frac{\partial w_0}{\partial \xi }+R{u}_1,{\gamma }_{\xi z}^1=\frac{\partial w_1}{\partial \xi }+R{u}_2,{\gamma }_{\xi z}^2=\frac{\partial w_2}{\partial \xi }+R{u}_3,{\gamma }_{\theta z}^0=\frac{\partial w_0}{\partial \theta }+R{v}_1-v_0,\\ {\gamma }_{\theta z}^1=\frac{\partial w_1}{\partial \theta }+R{v}_2,{\gamma }_{\theta z}^2=\frac{\partial w_2}{\partial \theta }+R{v}_3+v_2,{\gamma }_{\theta z}^3=2v_3.\end{array}$

(14)

The force resultants in equation (13) are determined as follows:

$\left[\begin{array}{l}{N}_{\xi }\\ M_{\xi }\\ N_{\xi }^{*}\\ M_{\xi }^{*}\end{array}\right]=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}{\sigma }_{\xi }\left(1+\frac{z}{R}\right)\left[\begin{array}{l}1\\ z\\ z^2/2\\ z^3/6\end{array}\right]dz,\left[\begin{array}{l}{Q}_z\\ S_z\end{array}\right]=\underset{-h}{\overset{+h}{{\displaystyle \int }}}{\sigma }_z\left(1+\frac{z}{R}\right)\left[\begin{array}{l}1\\ z\end{array}\right]dz,$

$\left[\begin{array}{l}{N}_{\xi \theta }\\ M_{\xi \theta }\\ N_{\xi \theta }^{*}\\ M_{\xi \theta }^{*}\end{array}\right]=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}{\tau }_{\xi \theta }\left(1+\frac{z}{R}\right)\left[\begin{array}{l}1\\ z\\ z^2/2\\ z^3/6\end{array}\right]dz,\left[\begin{array}{l}{Q}_{\xi }\\ S_{\xi }\\ Q_{\xi }^{*}\end{array}\right]=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}{\tau }_{\xi z}\left(1+\frac{z}{R}\right)\left[\begin{array}{l}1\\ z\\ z^2/2\end{array}\right]dz,$

(15)

$\left[\begin{array}{l}{N}_{\theta }\\ M_{\theta }\\ N_{\theta }^{*}\\ M_{\theta }^{*}\end{array}\right]=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}{\sigma }_{\theta }\left[\begin{array}{l}1\\ z\\ z^2/2\\ z^3/6\end{array}\right]dz,\left[\begin{array}{l}{N}_{\theta \xi }\\ M_{\theta \xi }\\ N_{\theta \xi }^{*}\\ M_{\theta \xi }^{*}\end{array}\right]=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}{\tau }_{\xi \theta }\left[\begin{array}{l}1\\ z\\ z^2/2\\ z^3/6\end{array}\right]dz,\left[\begin{array}{l}{Q}_{\theta }\\ S_{\theta }\\ Q_{\theta }^{*}\\ S_{\theta }^{*}\end{array}\right]=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}{\tau }_{\theta z}\left[\begin{array}{l}1\\ z\\ z^2/2\\ z^3/6\end{array}\right]dz,$

$F_i=\frac{q_{in}}{R}\left(R-\frac{h}{2}\right)\frac{{\left(-h/2\right)}^i}{i!}+\frac{q_{out}}{R}\left(R+\frac{h}{2}\right)\frac{{\left(h/2\right)}^i}{i!},i=0,1,2.$

The force resultants are derived from the elastic force resultants and the thermal force resultants as follows:

$\left[\begin{array}{l}{N}_{\xi }\\ M_{\xi }\\ N_{\xi }^{*}\\ M_{\xi }^{*}\end{array}\right]=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}\left(C_{11}{\epsilon }_{\xi }+C_{12}{\epsilon }_{\theta }+C_{13}{\epsilon }_z\right)\left(1+\frac{z}{R}\right)\left[\begin{array}{l}1\\ z\\ z^2/2\\ z^3/6\end{array}\right]dz-\left[\begin{array}{l}{N}_{\xi }^T\\ M_{\xi }^T\\ N_{\xi }^{T*}\\ M_{\xi }^{T*}\end{array}\right],$

$\left[\begin{array}{l}{N}_{\theta }\\ M_{\theta }\\ N_{\theta }^{*}\\ M_{\theta }^{*}\end{array}\right]=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}\left(C_{21}{\epsilon }_{\xi }+C_{22}{\epsilon }_{\theta }+C_{23}{\epsilon }_z\right)\left[\begin{array}{l}1\\ z\\ z^2/2\\ z^3/6\end{array}\right]dz-\left[\begin{array}{l}{N}_{\theta }^T\\ M_{\theta }^T\\ N_{\theta }^{T*}\\ M_{\theta }^{T*}\end{array}\right],$

$\left[\begin{array}{l}{Q}_z\\ S_z\end{array}\right]=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}\left(C_{31}{\epsilon }_{\xi }+C_{32}{\epsilon }_{\theta }+C_{33}{\epsilon }_z\right)\left(1+\frac{z}{R}\right)\left[\begin{array}{l}1\\ z\end{array}\right]dz-\left[\begin{array}{l}{Q}_z^T\\ S_z^T\end{array}\right],$

(16)

$\left[\begin{array}{l}{N}_{\theta \xi }\\ M_{\theta \xi }\\ N_{\theta \xi }^{*}\\ M_{\theta \xi }^{*}\end{array}\right]=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}{C}_{44}{\gamma }_{\xi \theta }\left[\begin{array}{l}1\\ z\\ z^2/2\\ z^3/6\end{array}\right]dz,\left[\begin{array}{l}{N}_{\xi \theta }\\ M_{\xi \theta }\\ N_{\xi \theta }^{*}\\ M_{\xi \theta }^{*}\end{array}\right]=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}{C}_{44}{\gamma }_{\xi \theta }\left(1+\frac{z}{R}\right)\left[\begin{array}{l}1\\ z\\ z^2/2\\ z^3/6\end{array}\right]dz,$

$\left[\begin{array}{l}{Q}_{\xi }\\ S_{\xi }\\ Q_{\xi }^{*}\end{array}\right]=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}{C}_{55}{\gamma }_{\xi z}\left(1+\frac{z}{R}\right)\left[\begin{array}{l}1\\ z\\ z^2/2\end{array}\right]dz,\left[\begin{array}{l}{Q}_{\theta }\\ S_{\theta }\\ Q_{\theta }^{*}\\ S_{\theta }^{*}\end{array}\right]=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}{C}_{66}{\gamma }_{\theta z}\left[\begin{array}{l}1\\ z\\ z^2/2\\ z^3/6\end{array}\right]dz\hspace{0.17em}.$

The thermal force resultants are determined as follows:

$\left[\begin{array}{l}{N}_{\xi }^T\\ M_{\xi }^T\\ N_{\xi }^{T*}\\ M_{\xi }^{T*}\end{array}\right]=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}\left(C_{11}{\alpha }_{11}+C_{12}{\alpha }_{22}+C_{13}{\alpha }_{33}\right)\mathrm{\Delta }T\left(1+\frac{z}{R}\right)\left[\begin{array}{l}1\\ z\\ z^2/2\\ z^3/6\end{array}\right]dz,\left[\begin{array}{l}{N}_{\theta }^T\\ M_{\theta }^T\\ N_{\theta }^{T*}\\ M_{\theta }^{T*}\end{array}\right]=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}\left(C_{21}{\alpha }_{11}+C_{22}{\alpha }_{22}+C_{23}{\alpha }_{33}\right)\mathrm{\Delta }T\left[\begin{array}{l}1\\ z\\ z^2/2\\ z^3/6\end{array}\right]dz,$

(17)

$\left[\begin{array}{l}{Q}_z^T\\ S_z^T\end{array}\right]=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}\left(C_{31}{\alpha }_{11}+C_{32}{\alpha }_{11}+C_{33}{\alpha }_{11}\right)\mathrm{\Delta }T\left(1+\frac{z}{R}\right)\left[\begin{array}{l}1\\ z\end{array}\right]dz.$

The governing equations of shell are determined from equations (13), and (15)–(17) by integrating the displacement gradients by parts and setting the components $\delta u_k$ , $\delta v_k$ and $\delta w_k$ to zero separately. The system of equations determined according to HOSNT type quasi-3D includes 11 equations corresponding to 11 unknowns, as follows:

$\delta u_0:\frac{\partial N_{\xi }}{\partial \xi }+\frac{\partial N_{\theta \xi }}{\partial \theta }=0,$

$\delta v_0:\frac{\partial N_{\theta }}{\partial \theta }+\frac{\partial N_{\xi \theta }}{\partial \xi }+Q_{\theta }=0,$

$\delta w_0:\frac{\partial Q_{\xi }}{\partial \xi }+\frac{\partial Q_{\theta }}{\partial \theta }-N_{\theta }-R{F}_0=0,$

$\delta u_1:\frac{\partial M_{\xi }}{\partial \xi }+\frac{\partial M_{\theta \xi }}{\partial \theta }-R{Q}_{\xi }=0,$

$\delta v_1:\frac{\partial M_{\xi \theta }}{\partial \xi }+\frac{\partial M_{\theta }}{\partial \theta }-R{Q}_{\theta }=0,$

$\delta w_1:\frac{\partial S_{\xi }}{\partial \xi }+\frac{\partial S_{\theta }}{\partial \theta }-M_{\theta }-R{Q}_z-R{F}_1=0,$

(18)

$\delta u_2:\frac{\partial N_{\xi }^{*}}{\partial \xi }+\frac{\partial N_{\theta \xi }^{*}}{\partial \theta }-R{S}_{\xi }=0,$

$\delta v_2:\frac{\partial N_{\xi \theta }^{*}}{\partial \xi }+\frac{\partial N_{\theta }^{*}}{\partial \theta }-R{S}_{\theta }-Q_{\theta }^{*}=0,$

$\delta w_2:\frac{\partial Q_{\xi }^{*}}{\partial \xi }+\frac{\partial Q_{\theta }^{*}}{\partial \theta }-N_{\theta }^{*}-R{S}_z-R{F}_2=0,$

$\delta u_3:\frac{\partial M_{\xi }^{*}}{\partial \xi }+\frac{\partial M_{\theta \xi }^{*}}{\partial \theta }-R{Q}_{\xi }^{*}=0,$

$\delta v_3:\frac{\partial M_{\xi \theta }^{*}}{\partial \xi }+\frac{\partial M_{\theta }^{*}}{\partial \theta }-R{Q}_{\theta }^{*}-2S_{\theta }^{*}=0.$

The boundary condition of equation (18) are determined as:

- For the boundary $\xi ={\xi }_1,{\xi }_2$

$\begin{array}{l}{N}_{\xi }={\overline{N}}_{\xi }\hspace{0.17em}\cup \hspace{0.17em}{u}_0={\overline{u}}_0,\hspace{0.17em}{N}_{\xi \theta }={\overline{N}}_{\xi \theta }\hspace{0.17em}\cup \hspace{0.17em}{v}_0={\overline{v}}_0,\hspace{0.17em}{Q}_{\xi }={\overline{Q}}_{\xi }\hspace{0.17em}\cup \hspace{0.17em}{w}_0={\overline{w}}_0,\\ M_{\xi }={\overline{M}}_{\xi }\hspace{0.17em}\cup \hspace{0.17em}{u}_1={\overline{u}}_1,\hspace{0.17em}{M}_{\xi \theta }={\overline{M}}_{\xi \theta }\hspace{0.17em}\cup \hspace{0.17em}{v}_1={\overline{v}}_1,\hspace{0.17em}{S}_{\xi }={\overline{S}}_{\xi }\hspace{0.17em}\cup \hspace{0.17em}{w}_1={\overline{w}}_1,\\ N_{\xi }^{*}={\overline{N}}_{\xi }^{*}\hspace{0.17em}\cup \hspace{0.17em}{u}_2={\overline{u}}_2,\hspace{0.17em}{N}_{\xi \theta }^{*}={\overline{N}}_{\xi \theta }^{*}\hspace{0.17em}\cup \hspace{0.17em}{v}_2={\overline{v}}_2,\hspace{0.17em}{Q}_{\xi }^{*}={\overline{Q}}_{\xi }^{*}\hspace{0.17em}\cup \hspace{0.17em}{w}_2={\overline{w}}_2,\\ M_{\xi }^{*}={\overline{M}}_{\xi }^{*}\hspace{0.17em}\cup \hspace{0.17em}{u}_3={\overline{u}}_3,\hspace{0.17em}{M}_{\xi \theta }^{*}={\overline{M}}_{\xi \theta }^{*}\hspace{0.17em}\cup \hspace{0.17em}{v}_3={\overline{v}}_3,\end{array}$

(19)

• - For the boundary $\theta ={\theta }_1,{\theta }_2$

$\begin{array}{l}{N}_{\theta \xi }={\overline{N}}_{\theta \xi }\hspace{0.17em}\cup \hspace{0.17em}{u}_0={\overline{u}}_0,\hspace{0.17em}{N}_{\theta }={\overline{N}}_{\theta }\hspace{0.17em}\cup \hspace{0.17em}{v}_0={\overline{v}}_0,\hspace{0.17em}{Q}_{\theta }={\overline{Q}}_{\theta }\hspace{0.17em}\cup \hspace{0.17em}{w}_0={\overline{w}}_0,\\ M_{\theta \xi }={\overline{M}}_{\theta \xi }\hspace{0.17em}\cup \hspace{0.17em}{u}_0={\overline{u}}_0,\hspace{0.17em}{M}_{\theta }={\overline{M}}_{\theta }\hspace{0.17em}\cup \hspace{0.17em}{v}_1={\overline{v}}_1,\hspace{0.17em}{S}_{\theta }={\overline{S}}_{\theta }\hspace{0.17em}\cup \hspace{0.17em}{w}_1={\overline{w}}_1,\\ N_{\theta \xi }^{*}={\overline{N}}_{\theta \xi }^{*}\hspace{0.17em}\cup \hspace{0.17em}{u}_2={\overline{u}}_2,\hspace{0.17em}{N}_{\theta }^{*}={\overline{N}}_{\theta }^{*}\hspace{0.17em}\cup \hspace{0.17em}{v}_2={\overline{v}}_2,\hspace{0.17em}{Q}_{\theta }^{*}={\overline{Q}}_{\theta }^{*}\hspace{0.17em}\cup \hspace{0.17em}{w}_2={\overline{w}}_2,\\ M_{\theta \xi }^{*}={\overline{M}}_{\theta \xi }^{*}\hspace{0.17em}\cup \hspace{0.17em}{u}_3={\overline{u}}_3,\hspace{0.17em}{M}_{\theta }^{*}={\overline{M}}_{\theta }^{*}\hspace{0.17em}\cup \hspace{0.17em}{v}_3={\overline{v}}_3.\end{array}$

(20)

The determination of all forms of boundary circumstances may be achieved by the use of equations (19) and (20). The number of boundaries is found to be equivalent to the number of degrees of freedom in the system of differentiation equations.

For common boundary circumstances, the expressions in equation (19) can be expressed as:

- For fully clamped boundary constraint:

$u_i=v_i=w_j=0,i=0,1,2,3,\hspace{0.17em}j=0,1,2$

(21.a)

• - For fully simply supported boundary constraint:

$N_{\xi }=M_{\xi }=N_{\xi }^{*}=M_{\xi }^{*}=0,v_i=w_j=0,i=0,1,2,3,\hspace{0.17em}j=0,1,2$

(21.b)

• - For fully free boundary constraint:

$N_{\xi }=M_{\xi }=N_{\xi }^{*}=M_{\xi }^{*}=0,Q_{\xi }=S_{\xi }=Q_{\xi }^{*}=0,N_{\xi \theta }=M_{\xi \theta }=N_{\xi \theta }^{*}=M_{\xi \theta }^{*}=0.$

(21.c)

The boundary constraint in equation (20) will have a periodic form with respect to θ when considered for a closed cylindrical shell.

Temperature distribution

For the FG-CNTRC cylindrical structure with distribution of CNT along the radial course, the steady state three-dimensional heat conduction equation is can be obtained as the following:^56,59,60

$\frac{1}{R+z}\frac{\partial }{\partial z}\left(k_{33}r\frac{\partial T}{\partial z}\right)+\frac{k_{22}}{{\left(R+z\right)}^2}\frac{{\partial }^{2}T}{\partial {\theta }^2}+\frac{k_{11}}{R^2}\frac{{\partial }^{2}T}{\partial {\xi }^2}=0$

(22)

In the event that the temperature varies in the radial course of the cylindrical shell, the temperature distributed function is determined according to the following equation:

$\frac{1}{R+z}\frac{\partial }{\partial z}\left(k_{33}r\frac{\partial T}{\partial z}\right)=0$

(23)

The corresponding temperature boundary conditions on the outer surface and inner surface of the cylindrical shell are determined as follows:

$T=T_{in}\hspace{0.17em}\text{at}\hspace{0.17em}\hspace{0.17em}z=-h/2$

(24.a)

$T=T_{out}\hspace{0.17em}\text{at}\hspace{0.17em}\hspace{0.17em}z=h/2$

(24.b)

The expression of $k_{33}$ in terms of z can be defined from equation (1). Substituting expression just dertermined with note $r=R+z$ , we can be obtained the heat transfer equation as follows:

$\frac{1}{R+z}\frac{\partial }{\partial z}\left(\left(R+z\right)k_{33}\left(z\right)\frac{\partial T}{\partial z}\right)=0$

(25)

By solving (24), the temperature distributed law along the thickness is defined as:

- For case UD:

$T\left(z\right)=\frac{A_1}{k_{33U}}\ln \left(R+z\right)+A_2$

(26)

where $A_1,A_2$ are the integral constants can be derived from temperature boundary circumstances (24):

$A_1=\frac{k_{33U}\left(T_{in}-T_{out}\right)}{\ln \left(\frac{R-h/2}{R+h/2}\right)},A_2=\frac{T_{out}\hspace{0.17em}\ln \left(R-h/2\right)-T_{in}\hspace{0.17em}\ln \left(R+h/2\right)}{\ln \left(\frac{R-h/2}{R+h/2}\right)}$

• - For case FG-Λ:

$T\left(z\right)=A_3.G_{1\mathrm{\Lambda }}.\hspace{0.17em}\ln \left(R+z\right)+A_3.G_{2\mathrm{\Lambda }}.z+A_4$

(27)

where:

$G_{1\mathrm{\Lambda }}=\left(1+2R/h\right).k_m.V_{CNT}^{*}+\left[1-\left(1+2R/h\right).V_{CNT}^{*}\right].k_{33}^{CNT},G_{2\mathrm{\Lambda }}=\frac{2}{h}.V_{CNT}^{*}.\left(k_{33}^{CNT}-k_m\right)$

$A_3=\frac{T_{in}-T_{out}}{G_{1\mathrm{\Lambda }}.\hspace{0.17em}\ln \left(\frac{R-h/2}{R+h/2}\right)-G_{2\mathrm{\Lambda }}.h},$

$A_4=\frac{\left[G_{1\mathrm{\Lambda }}.\hspace{0.17em}\ln \left(R-h/2\right)-G_{2\mathrm{\Lambda }}.h/2\right].T_{out}-\left[G_{1\mathrm{\Lambda }}.\hspace{0.17em}\ln \left(R+h/2\right)+G_{2\mathrm{\Lambda }}.h/2\right].T_{in}}{G_{1\mathrm{\Lambda }}.\hspace{0.17em}\ln \left(\frac{R-h/2}{R+h/2}\right)-G_{2\mathrm{\Lambda }}.h}$

• - For case FG-V:

$T\left(z\right)=A_5.G_{1V}.\hspace{0.17em}\ln \left(R+z\right)+A_5.G_{2V}.z+A_6$

(28)

where:

$G_{1V}=\left(1-2R/h\right).k_m.V_{CNT}^{*}+\left[1-\left(1-2R/h\right).V_{CNT}^{*}\right].k_{33}^{CNT},G_{2V}=\frac{2}{h}.V_{CNT}^{*}.\left(k_m-k_{33}^{CNT}\right)$

$A_5=\frac{T_{in}-T_{out}}{G_{1V}.\hspace{0.17em}\ln \left(\frac{R-h/2}{R+h/2}\right)-G_{2V}.h}$

$A_6=\frac{\left[G_{1V}.\hspace{0.17em}\ln \left(R-h/2\right)-G_{2V}.h/2\right].T_{out}-\left[G_{1V}.\hspace{0.17em}\ln \left(R+h/2\right)+G_{2V}.h/2\right].T_{in}}{G_{1V}.\hspace{0.17em}\ln \left(\frac{R-h/2}{R+h/2}\right)-G_{2V}.h}$

• - For case FG-O:

$T\left(z\right)=\{\begin{array}{l}{A}_7.G_{1O}.\hspace{0.17em}\ln \left(R+z\right)+A_7.G_{2O}.z+A_8\left(-h/2\le z<0\right)\\ A_7.G_{3O}.\hspace{0.17em}\ln \left(R+z\right)+A_7.G_{4O}.z+A_9\left(0\le z\le h/2\right)\end{array}$

(29)

where

$G_{1O}=\left(2-4R/h\right).k_m.V_{CNT}^{*}+\left[1-\left(2-4R/h\right).V_{CNT}^{*}\right].k_{33}^{CNT},G_{2O}=\frac{4}{h}.V_{CNT}^{*}.\left(k_m-k_{33}^{CNT}\right),$

$G_{3O}=\left(2+4R/h\right).k_m.V_{CNT}^{*}+\left[1-\left(2+4R/h\right).V_{CNT}^{*}\right].k_{33}^{CNT},G_{4O}=\frac{4}{h}.V_{CNT}^{*}.\left(k_{33}^{CNT}-k_m\right)$

$A_7=\frac{T_{out}-T_{in}}{\mathrm{\Delta }},A_8=\frac{1}{{\mathrm{\Delta }}_O}.\left[\begin{array}{l}\left[-G_{1O}.\hspace{0.17em}\ln \left(R-h/2\right)+G_{2O}.h/2\right].T_{out}+\\ +\left[G_{1O}.\hspace{0.17em}\ln \hspace{0.17em}R-G_{3O}.\hspace{0.17em}\ln \hspace{0.17em}R+G_{3O}.\hspace{0.17em}\ln \left(R+h/2\right)+G_{4O}.h/2\right].T_{in}\end{array}\right]$

$A_9=\frac{1}{{\mathrm{\Delta }}_O}.\left[\begin{array}{l}\left[G_{1O}.\hspace{0.17em}\ln \hspace{0.17em}R-G_{3O}.\hspace{0.17em}\ln \hspace{0.17em}R-G_{1O}.\hspace{0.17em}\ln \left(R-h/2\right)+G_{2O}.h/2\right].T_{out}+\\ +\left[G_{3O}.\hspace{0.17em}\ln \left(R+h/2\right)+G_{4O}.h/2\right].T_{in}\end{array}\right]$

$\begin{array}{l}{\mathrm{\Delta }}_O=-G_{1O}.\hspace{0.17em}\ln \left(R-h/2\right)+G_{2O}.h/2-G_{3O}.\hspace{0.17em}\ln \left(R+h/2\right)-G_{4O}.h/2+\\ +G_{1O}.\hspace{0.17em}\ln \hspace{0.17em}R-G_{3O}.\hspace{0.17em}\ln \hspace{0.17em}R\end{array}$

• - For case FG-X:

$T\left(z\right)=\{\begin{array}{l}{A}_{10}.G_{1X}.\hspace{0.17em}\ln \left(R+z\right)+A_{10}.G_{2X}.z+A_{11}\left(-h/2\le z<0\right)\\ A_{10}.G_{3X}.\hspace{0.17em}\ln \left(R+z\right)+A_{10}.G_{4X}.z+A_{12}\left(0\le z\le h/2\right)\end{array}$

(30)

where:

$G_{1X}=\left(4R/h\right).k_m.V_{CNT}^{*}+\left[1-\left(4R/h\right).V_{CNT}^{*}\right].k_{33}^{CNT},G_{2X}=\frac{4}{h}.V_{CNT}^{*}.\left(k_{33}^{CNT}-k_m\right),$

$G_{3X}=\left[1-\left(4R/h\right).V_{CNT}^{*}\right].k_{33}^{CNT}-\left(4R/h\right).k_m.V_{CNT}^{*},G_{4X}=\frac{4}{h}.V_{CNT}^{*}.\left(k_m-k_{33}^{CNT}\right)$

$A_{10}=\frac{T_{out}-T_{in}}{{\mathrm{\Delta }}_X},A_{11}=\frac{1}{{\mathrm{\Delta }}_X}\left[\begin{array}{l}\left[-G_{1X}.\hspace{0.17em}\ln \left(R-h/2\right)+G_{2X}.h/2\right].T_{out}+\\ +\left[G_{1X}.\hspace{0.17em}\ln \hspace{0.17em}R-G_{3X}.\hspace{0.17em}\ln \hspace{0.17em}R+G_{3X}.\hspace{0.17em}\ln \left(R+h/2\right)+G_{4X}.h/2\right].T_{in}\end{array}\right]$

$A_{12}=\frac{1}{{\mathrm{\Delta }}_X}.\left[\begin{array}{l}\left[G_{1X}.\hspace{0.17em}\ln \hspace{0.17em}R-G_{3X}.\hspace{0.17em}\ln \hspace{0.17em}R-G_{1X}.\hspace{0.17em}\ln \left(R-h/2\right)+G_{2X}.h/2\right].T_{out}+\\ +\left[G_{3X}.\hspace{0.17em}\ln \left(R+h/2\right)+G_{4X}.h/2\right].T_{in}\end{array}\right]\hspace{0.17em}\begin{array}{c}{\mathrm{\Delta }}_X=-G_{1X}.\hspace{0.17em}\ln \left(R-h/2\right)+G_{2X}.h/2-G_{3X}.\hspace{0.17em}\ln \left(R+h/2\right)\\ -G_{4X}.h/2+G_{1X}.\hspace{0.17em}\ln \hspace{0.17em}R-G_{3X}.\hspace{0.17em}\ln \hspace{0.17em}R\end{array}$

Solving procedure

The equation system (18) is calculated as:

$\begin{array}{c}{\displaystyle \sum }_{i=0}^3\left(H_{1i}^l+H_{1i,11}^l\frac{{\partial }^2}{\partial {\xi }^2}+H_{1i,22}^l\frac{{\partial }^2}{\partial {\theta }^2}\right)u_i+{\displaystyle \sum }_{i=0}^3{H}_{2i,12}^l\frac{{\partial }^2}{\partial \xi \partial \theta }{v}_i+\\ +{\displaystyle \sum }_{i=0}^2{H}_{3i,1}^l\frac{\partial }{\partial \xi }{w}_i=0,l=1,...,4,\\ {\displaystyle \sum }_{i=0}^3{H}_{1i,12}^k\frac{{\partial }^2}{\partial \xi \partial \theta }{u}_i+{\displaystyle \sum }_{i=0}^3\left(H_{2i}^k+H_{2i,11}^k\frac{{\partial }^2}{\partial {\xi }^2}+H_{2i,22}^k\frac{{\partial }^2}{\partial {\theta }^2}\right)v_i+\\ +{\displaystyle \sum }_{i=0}^2{H}_{3i,2}^k\frac{\partial }{\partial \theta }{w}_i=0,k=5,...,8,\\ {\displaystyle \sum }_{i=0}^3{H}_{1i,1}^n\frac{\partial }{\partial \xi }{u}_i+{\displaystyle \sum }_{i=0}^2\left(H_{3i}^n+H_{3i,11}^n\frac{{\partial }^2}{\partial {\xi }^2}+H_{3i,22}^n\frac{{\partial }^2}{\partial {\theta }^2}\right)w_i+{\displaystyle \sum }_{i=0}^3{H}_{2i,2}^n\frac{\partial }{\partial \theta }{v}_i\\ =H_{qout}^n{q}_{out}+H_{qin}^n{q}_{in}+H_{Tout}^n\mathrm{\Delta }{T}_{out}+\mathrm{\Delta }{H}_{Tin}^n\mathrm{\Delta }{T}_{in},n=9,...,11\end{array}$

(31)

here, the parameters $H_b^a$ are constant, whose values depend on the geometric parameters of the shell, including radius R, thickness ratio h/R, material properties, including Poisson’s coefficient μ, and Young’s modulus E. $H_b^a$ is determined by calculating the coefficients according to expressions (6), (7), (9), (16), and (17), then grouping them according to the derivative terms of u, v, w. The specific expressions for the coefficients $H_b^a$ are presented in the Appendix.

Use simple trigonometric series to express the load and displacement components to convert the system of partial differential equations (31) into a system of ordinary differential equations. When considering a closed cylindrical shell, the loads and displacement components are expressed as follows:

$u_i\left(\xi ,\theta \right)=U_{i0}\left(\xi \right)+{\displaystyle \sum }_{m=1}^{\infty }\left[U_{im}^1\left(\xi \right)\cos \hspace{0.17em}m\theta +U_{im}^2\left(\xi \right)\sin \hspace{0.17em}m\theta \right],$

$v_i\left(\xi ,\theta \right)=V_{i0}\left(\xi \right)+{\displaystyle \sum }_{m=1}^{\infty }\left[V_{im}^1\left(\xi \right)\sin \hspace{0.17em}m\theta -V_{im}^2\left(\xi \right)\cos \hspace{0.17em}m\theta \right],$

$w_j\left(\xi ,\theta \right)=W_{j0}\left(\xi \right)+{\displaystyle \sum }_{m=1}^{\infty }\left[W_{jm}^1\left(\xi \right)\cos \hspace{0.17em}m\theta +W_{jm}^2\left(\xi \right)\sin \hspace{0.17em}m\theta \right],$

(32)

$q_{in}\left(\xi ,\theta \right)=Q_{in0}\left(\xi \right)+{\displaystyle \sum }_{m=1}^{\infty }\left[Q_{inm}^1\left(\xi \right)\cos \hspace{0.17em}m\theta +Q_{inm}^2\left(\xi \right)\sin \hspace{0.17em}m\theta \right],$

$q_{out}\left(\xi ,\theta \right)=Q_{out0}\left(\xi \right)+{\displaystyle \sum }_{m=1}^{\infty }\left[Q_{outm}^1\left(\xi \right)\cos \hspace{0.17em}m\theta +Q_{outm}^2\left(\xi \right)\sin \hspace{0.17em}m\theta \right],$

$T_{out}=T_{out0}\left(\xi \right)+{\displaystyle \sum }_{m=1}^{\infty }\left[T_{outm}^1\left(\xi \right)\cos \hspace{0.17em}m\theta +T_{outm}^2\left(\xi \right)\sin \hspace{0.17em}m\theta \right],$

$T_{in}=T_{in0}\left(\xi \right)+{\displaystyle \sum }_{m=1}^{\infty }\left[T_{inm}^1\left(\xi \right)\cos \hspace{0.17em}m\theta +T_{inm}^2\left(\xi \right)\sin \hspace{0.17em}m\theta \right],$

$i=0,...,3;j=0,...,2.$

The type of displacement solution depends on the type of load acting on the shell, so the value of m depends on the type of load. The value m here characterizes the periodicity of the load according to the circular direction coordinate θ on a cross-section of the shell. In the general case, any form of load on a cross-section can be approximated by trigonometric series in θ. The higher the m value, the higher the level of accuracy, but the complexity and calculation time also increase.

Some common special cases: when the load on a cross-section is constant according to coordinates θ, the load only depends on ξ then m = 0; or when the load is trigonometric $q=-Q_0\hspace{0.17em}\sin \left(\pi x/L\right)\sin \left(\theta \right)$ , then m = 1.

Substituting the expressions in equation (32) into equation (31), we can derive the differential equations to define $U_{i0}\left(\xi \right),V_{i0}\left(\xi \right),W_{j0}\left(\xi \right)$ as follows:

$\begin{array}{l}{\displaystyle \sum }_{i=0}^3\left(H_{1i}^l+H_{1i,11}^l\frac{d^2}{d{\xi }^2}\right)U_{i0}+{\displaystyle \sum }_{i=0}^2{H}_{3i,1}^l\frac{d{W}_{j0}}{d\xi }=0,l=1,...,3,\\ {\displaystyle \sum }_{i=0}^3\left(H_{2n}^k+H_{2n,11}^k\frac{d^2}{d{\xi }^2}\right)V_{i0}=0,k=5,...,8,\\ {\displaystyle \sum }_{i=0}^3{H}_{1i,1}^n\frac{d{U}_{i0}}{d\xi }+{\displaystyle \sum }_{i=0}^2\left(H_{3i}^n+H_{3i,11}^n\frac{d^2}{d{\xi }^2}\right)W_{i0}\\ =H_{qout}^n{Q}_{out0}+H_{qin}^n{Q}_{in0}+H_{Tout}^n{T}_{out0}+H_{Tin}^n{T}_{in0},n=9,...,11.\end{array}$

(33)

When the load is axisymmetric, the components $V_{i0}\left(\xi \right)=0$ .

Components $U_{im}^1\left(\xi \right)$ , $V_{im}^1\left(\xi \right)$ , $W_{jm}^1\left(\xi \right)$ and $U_{im}^2\left(\xi \right)$ , $V_{im}^2\left(\xi \right)$ , $W_{jm}^2\left(\xi \right)$ are calculated from the following system of differential equations:

$\begin{array}{c}{\displaystyle \sum }_{i=0}^3\left(H_{1i}^l+H_{1i,11}^l\frac{d^2}{d{\xi }^2}-m^2{H}_{1i,22}^l\right)U_{im}^{\left(s\right)}+m{\displaystyle \sum }_{i=0}^3{H}_{2i,12}^l\frac{d}{d\xi }{V}_{im}^{\left(s\right)}+\\ +{\displaystyle \sum }_{i=0}^2{H}_{3i,1}^l\frac{d}{d\xi }{W}_{im}^{\left(s\right)}=0,l=1,..,4,\\ -m{\displaystyle \sum }_{i=0}^3{H}_{1i,12}^k\frac{d}{d\xi }{U}_{im}^{\left(s\right)}+{\displaystyle \sum }_{i=0}^3\left(H_{2i}^k+H_{2i,11}^k\frac{d^2}{d{\xi }^2}-m^2{H}_{2i,22}^k\right)V_{im}^{\left(s\right)}-\\ -m{\displaystyle \sum }_{i=0}^2{H}_{3i,2}^k{W}_{im}^{\left(s\right)}=0,k=5,..,8,\\ {\displaystyle \sum }_{i=0}^3{H}_{1i,1}^n\frac{d}{d\xi }{U}_{im}^{\left(s\right)}+{\displaystyle \sum }_{i=0}^2\left(H_{3i}^n+H_{3i,11}^n\frac{d^2}{d{\xi }^2}-m^2{H}_{3i,22}^n\right)W_{im}^{\left(s\right)}+m{\displaystyle \sum }_{i=0}^3{H}_{2i,2}^n{V}_{im}^{\left(s\right)}\\ =H_{qout}^n{Q}_{outm}^{\left(s\right)}+H_{qin}^n{Q}_{inm}^{\left(s\right)}+H_{Tout}^n\mathrm{\Delta }{T}_{outm}^{\left(s\right)}+H_{Tin}^n\mathrm{\Delta }{T}_{inm}^{\left(s\right)},n=9,..,11,s=1,2.\end{array}$

(34)

Equation (33) is a special case of equation (34) when m = 0.

Let ${\stackrel{\sim }{U}}_{im}\left(p\right)$ , ${\stackrel{\sim }{V}}_{im}\left(p\right)$ , ${\stackrel{\sim }{W}}_{jm}\left(p\right)$ , ${\stackrel{\sim }{Q}}_{outm}\left(p\right)$ , ${\stackrel{\sim }{Q}}_{inm}\left(p\right)$ , ${\stackrel{\sim }{T}}_{outm}\left(p\right)$ and ${\stackrel{\sim }{T}}_{inm}\left(p\right)$ be the image functions of $U_{im}\left(\xi \right)$ , $V_{im}\left(\xi \right)$ , $W_{jm}\left(\xi \right)$ , $Q_{outm}\left(\xi \right)$ , $Q_{inm}\left(\xi \right)$ , $T_{outm}\left(p\right)$ and $T_{inm}\left(p\right)$ respectively. Apply the Laplace transform to equation (34), we obtain the equations to determine ${\stackrel{\sim }{U}}_{im}\left(p\right)$ , ${\stackrel{\sim }{V}}_{im}\left(p\right),$ and ${\stackrel{\sim }{W}}_{jm}\left(p\right)$ in the form of algebraic equations as:

$\begin{array}{c}{\displaystyle \sum }_{i=0}^3\left(H_{1i}^l+H_{1i,11}^l{p}^2-m^2{H}_{1i,22}^l\right){\stackrel{\sim }{U}}_{im}\left(p\right)+m{\displaystyle \sum }_{i=0}^3{H}_{2i,12}^{l}p{\stackrel{\sim }{V}}_{im}\left(p\right)+{\displaystyle \sum }_{i=0}^2{H}_{3i,1}^{l}p{\stackrel{\sim }{W}}_{im}\left(p\right)=\\ ={\displaystyle \sum }_{i=0}^3{H}_{1i,11}^l\left(p{C}_{im}^{10}+C_{im}^{11}\right)+m{\displaystyle \sum }_{i=0}^3{H}_{2i,12}^l{C}_{im}^{20}+{\displaystyle \sum }_{i=0}^2{H}_{3i,1}^l{C}_{jm}^{30},l=1,...,4,\end{array}$

$\begin{array}{l}-m{\displaystyle \sum }_{i=0}^3{H}_{1i,12}^{k}p{\stackrel{\sim }{U}}_{im}\left(p\right)+{\displaystyle \sum }_{i=0}^3\left(H_{2i}^k+H_{2i,11}^k{p}^2-m^2{H}_{2i,22}^k\right){\stackrel{\sim }{V}}_{im}\left(p\right)-m{\displaystyle \sum }_{i=0}^2{H}_{3i,2}^k{\stackrel{\sim }{W}}_{im}\left(p\right)=\\ =-m{\displaystyle \sum }_{i=0}^2{H}_{1i,12}^k{C}_{im}^{10}+{\displaystyle \sum }_{i=0}^2{H}_{2i,11}^k\left(p{C}_{im}^{20}+C_{im}^{21}\right),k=5,...,8,\end{array}$

(35)

$\begin{array}{l}{\displaystyle \sum }_{i=0}^3{H}_{1i,1}^{n}p{\stackrel{\sim }{U}}_{im}\left(p\right)+{\displaystyle \sum }_{i=0}^2\left(H_{3i}^n+H_{3i,11}^n{p}^2-m^2{H}_{3i,22}^n\right){\stackrel{\sim }{W}}_{im}\left(p\right)+m{\displaystyle \sum }_{i=0}^3{H}_{2i,2}^n{\stackrel{\sim }{V}}_{im}\left(p\right)=\\ =H_{qout}^n{\stackrel{\sim }{Q}}_{outm}\left(p\right)+H_{qin}^n{\stackrel{\sim }{Q}}_{inm}\left(p\right)+H_{Tout}^n\mathrm{\Delta }{\stackrel{\sim }{T}}_{outm}\left(p\right)+H_{Tin}^n\mathrm{\Delta }{\stackrel{\sim }{T}}_{inm}\left(p\right)+\\ +{\displaystyle \sum }_{i=0}^3{H}_{1i,1}^n{C}_{im}^{10}+{\displaystyle \sum }_{i=0}^2{H}_{3i,11}^n\left(p{C}_{im}^{30}+C_{im}^{31}\right),n=9,...,11.\end{array}$

For a closed cylindrical shell, the corresponding boundary conditions in equation (21) at position $\xi =0$ are satisfied by the integration constants $C_{im}^{10}$ , $C_{im}^{20}$ , $C_{jm}^{30}$ , $C_{im}^{11}$ , $C_{im}^{21}$ , $C_{jm}^{31}$ , $i=0,1,2,3;j=0,1,2$ as:

- The clamped condition in equation (21.a):

$C_{im}^{10}=0,C_{im}^{20}=0,C_{jm}^{30}=0,i=0,1,2,3;j=0,1,2.$

(36)

• - The simply supported condition in equation (21.b):

$C_{im}^{11}=0,C_{im}^{20}=0,C_{jm}^{30}=0,i=0,1,2,3;j=0,1,2.$

(37)

• - The free condition in equation (21.c):

$N_{\xi }=0:{\displaystyle \sum }_{i=0}^K{G}_{1i}^1{C}_{im}^{11}+m{\displaystyle \sum }_{i=0}^K{G}_{2i}^1{C}_{im}^{20}+{\displaystyle \sum }_{j=0}^{K-1}{G}_{3j}^1{C}_{jm}^{30}=0,$

$M_{\xi }=0:{\displaystyle \sum }_{i=0}^K{G}_{1i}^2{C}_{im}^{11}+m{\displaystyle \sum }_{i=0}^K{G}_{2i}^2{C}_{im}^{20}+{\displaystyle \sum }_{j=0}^{K-1}{G}_{3j}^2{C}_{jm}^{30}=0,$

$N_{\xi }^{*}=0:{\displaystyle \sum }_{i=0}^K{G}_{1i}^3{C}_{im}^{11}+m{\displaystyle \sum }_{i=0}^K{G}_{2i}^3{C}_{im}^{20}+{\displaystyle \sum }_{j=0}^{K-1}{G}_{3j}^3{C}_{jm}^{30}=0,$

$M_{\xi }^{*}=0:{\displaystyle \sum }_{i=0}^K{G}_{1i}^4{C}_{im}^{11}+m{\displaystyle \sum }_{i=0}^K{G}_{2i}^4{C}_{im}^{20}+{\displaystyle \sum }_{j=0}^{K-1}{G}_{3j}^4{C}_{jm}^{30}=0,$

$N_{\xi \theta }=0:-m{\displaystyle \sum }_{i=0}^K{G}_{1i}^5{C}_{im}^{10}+{\displaystyle \sum }_{i=0}^K{G}_{2i}^5{C}_{im}^{21}=0,$

$M_{\xi \theta }=0:-m{\displaystyle \sum }_{i=0}^K{G}_{1i}^6{C}_{im}^{10}+{\displaystyle \sum }_{i=0}^K{G}_{2i}^6{C}_{im}^{21}=0,$

(38)

$N_{\xi \theta }^{*}=0:-m{\displaystyle \sum }_{i=0}^K{G}_{1i}^7{C}_{im}^{10}+{\displaystyle \sum }_{i=0}^K{G}_{2i}^7{C}_{im}^{21}=0,$

$M_{\xi \theta }^{*}=0:-m{\displaystyle \sum }_{i=0}^K{G}_{1i}^8{C}_{im}^{10}+{\displaystyle \sum }_{i=0}^K{G}_{2i}^8{C}_{im}^{21}=0,$

$Q_{\xi }=0:{\displaystyle \sum }_{i=0}^K{G}_{1i}^9{C}_{im}^{11}+m{\displaystyle \sum }_{i=0}^K{G}_{2i}^9{C}_{im}^{20}+{\displaystyle \sum }_{j=0}^{K-1}{G}_{3j}^9{C}_{jm}^{30}=0,$

$S_{\xi }=0:{\displaystyle \sum }_{i=0}^K{G}_{1i}^{10}{C}_{im}^{11}+m{\displaystyle \sum }_{i=0}^K{G}_{2i}^{10}{C}_{im}^{20}+{\displaystyle \sum }_{j=0}^{K-1}{G}_{3j}^{10}{C}_{jm}^{30}=0,$

$Q_{\xi }^{*}=0:{\displaystyle \sum }_{i=0}^K{G}_{1i}^{11}{C}_{im}^{11}+m{\displaystyle \sum }_{i=0}^K{G}_{2i}^{11}{C}_{im}^{20}+{\displaystyle \sum }_{j=0}^{K-1}{G}_{3j}^{11}{C}_{jm}^{30}=0.$

$i=0,1,2,3;\hspace{0.17em}j=0,1,2.$

Coefficients $G_{1i}^n$ , $G_{2i}^n$ , $G_{3j}^n$ in equation (38) are determined from the expressions of the force resultants in equations (15)–(17).

From the approach to the boundary problem above, through equations (36)–(38), the boundary conditions at an end of shell are self-satisfied as we assign appropriate values to the coefficients $C_{im}^{10}$ , $C_{im}^{20}$ , $C_{jm}^{30}$ , $C_{im}^{11}$ , $C_{im}^{21}$ , $C_{jm}^{31}$ . Simultaneously, half of the integration constants are self-satisfied, which simplifies the process of solving equation (33).

Using the boundary condition at $\xi =0$ for ${\stackrel{\sim }{U}}_{im}\left(p\right)$ , ${\stackrel{\sim }{V}}_{im}\left(p\right)$ and ${\stackrel{\sim }{W}}_{jm}\left(p\right)$ and solving equations (35), one obtains the components of the image functions. Applying the inverse Laplace transform to these image function solution expressions, we obtain the expressions $U_{im}\left(\xi \right)$ , $V_{im}\left(\xi \right)$ and $W_{jm}\left(\xi \right)$ . Using the boundary constraint at $\xi =L/R$ , we can obtain the value of the remaining half of the integration coefficient. Details about this procedure can be referred to in the articles:^83,85

Substituting the determined expressions $U_{im}\left(\xi \right)$ , $V_{im}\left(\xi \right),$ and $W_{jm}\left(\xi \right)$ into equation (32), we can derive the expressions of the displacement components $u_i\left(\xi ,\theta \right)$ , $v_i\left(\xi ,\theta \right),$ and $w_j\left(\xi ,\theta \right).$ From equations (6)–(10), the expressions of the ${\sigma }_{\xi },{\sigma }_{\theta }$ and ${\tau }_{\xi \theta }$ can be obtained.

To improve the accuracy of the transverse stress components, we use the method applied in previous studies.^83,85 In particular, the transverse stress components are calculated based on the equilibrium equation of 3D elasticity theory as:

${\tau }_{\xi z}=-\frac{1}{R+z}\underset{-h/2}{\overset{z}{{\displaystyle \int }}}\left[\left(1+\frac{z}{R}\right)\frac{\partial {\sigma }_{\xi }}{\partial \xi }+\frac{\partial {\tau }_{\xi \theta }}{\partial \theta }\right]dz,$

${\tau }_{\theta z}=-\frac{R}{{\left(R+z\right)}^2}\underset{-h/2}{\overset{z}{{\displaystyle \int }}}\left[\left(1+\frac{z}{R}\right)\frac{\partial {\sigma }_{\theta }}{\partial \theta }+{\left(1+\frac{z}{R}\right)}^2\frac{\partial {\tau }_{\xi \theta }}{\partial \xi }\right]dz,$

(39)

${\sigma }_z=-\frac{1}{R+z}\underset{-h/2}{\overset{z}{{\displaystyle \int }}}\left[\left(1+\frac{z}{R}\right)\frac{\partial {\tau }_{\xi z}}{\partial \xi }+\frac{\partial {\tau }_{\theta z}}{\partial \theta }-{\sigma }_{\theta }\right]dz+\frac{R-h/2}{R+h/2}{q}_{in}.$

Numerical results

Accuracy evaluation

In the first assement, to evaluate the accuracy and effectiveness of the quasi-3D high-order shear deformation theory, consider an laminated orthotropic cylindrical shell with the relative length L/R = 4, the relative thickness S = R/h. The shell structure under the sinusoidal distributed load on the inner surface $q=-Q_0\hspace{0.17em}\sin \frac{m\pi \xi R}{L}\cos \left(n\theta \right)$ with m = 1 and n = 4, and the simply supported boundary conditions are applied in the both ends.. The non-dimensional material properties are $E_1=25E_2,G_{23}=0.2E_2,G_{13}=G_{12}=0.5E_2$ , Poisson’s ratio ${\upsilon }_{12}=0.25$ .

Results are presented for these cases for different $R/h$ values in terms of the following non-dimensional parameters.

$\begin{array}{l}\overline{\mathrm{w}}=\frac{10E_1\mathrm{w}}{h{S}^4{Q}_0\hspace{0.17em}\sin \left(m\pi \xi R/L\right)\cos \left(n\theta \right)},{\overline{\sigma }}_z=\frac{10{\sigma }_z}{h{S}^2{Q}_0\hspace{0.17em}\sin \left(m\pi \xi R/L\right)\cos \left(n\theta \right)}\\ \left({\overline{\sigma }}_{\xi },{\overline{\sigma }}_{\theta }\right)=\frac{10\left({\sigma }_{\xi },{\sigma }_{\theta }\right)}{h{S}^2{Q}_0\hspace{0.17em}\sin \left(m\pi \xi R/L\right)\cos \left(n\theta \right)},{\overline{\tau }}_{\theta \xi }=\frac{10{\tau }_{\theta \xi }}{S^2{Q}_0\hspace{0.17em}\sin \left(m\pi \xi R/L\right)\sin \left(n\theta \right)}\\ {\overline{\tau }}_{z\xi }=\frac{10{\tau }_{z\xi }}{S{Q}_0\hspace{0.17em}\sin \left(m\pi \xi R/L\right)\cos \left(n\theta \right)},{\overline{\tau }}_{z\theta }=\frac{10{\tau }_{z\theta }}{S{Q}_0\hspace{0.17em}\sin \left(m\pi \xi R/L\right)\sin \left(n\theta \right)}.\end{array}$

(40)

The comparison studies have also been given in Table 1 for the two-layer (0°/90°) composite laminated cylindrical shell with S = 4,10,50, and 100. Identical thickness of each layer for all cases is taken into consideration. The presented example has been investigated by many researchers. The proposed present numerical results are compared with those obtained by the 3D analytical approach, ⁸⁷ the scaled boundary finite element method (SBFEM) based on 3D theory, ⁸⁸ the exact closed-form technique, ⁸⁹ as well as the first-order shear deformation theory (FSDT) and the third-order shear deformation theory (TSDT). ⁹⁰

Table 1.

Comparison of non-dimensional displacement at the middle for two layer (0/90^o) composite laminated cylindrical shell.

z	$\overline{\mathrm{w}}$	${\overline{\sigma }}_{\xi }$	${\overline{\sigma }}_{\xi }$	${\overline{\sigma }}_{\theta }$	${\overline{\sigma }}_{\theta }$	${\overline{\tau }}_{\theta \xi }$	${\overline{\tau }}_{\theta \xi }$	${\overline{\tau }}_{z\xi }$	${\overline{\tau }}_{z\theta }$	${\overline{\sigma }}_z$
	$0$	$\raisebox{1ex}{$h$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$	$-\raisebox{1ex}{$h$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$	$\raisebox{1ex}{$h$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$	$\raisebox{1ex}{$-h$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$	$\raisebox{1ex}{$h$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$	$-\raisebox{1ex}{$h$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$	$-\raisebox{1ex}{$h$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$	$\raisebox{1ex}{$h$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$	$\raisebox{1ex}{$h$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$
$R/h=4$
3D 87	6.100	0.2120	−0.9610	10.31	−1.789	−0.2007	0.2812	0.2758	4.440	−0.70
Ref 89	5.09696	0.20710	−0.71888	12.07122	−1.11616	−0.16858	0.22653	0.21269	5.48863	N
TSDT 90	6.66980	0.24272	−0.93037	12.57100	−1.68620	−0.20324	0.29249	N	N	N
FSDT 90	7.32821	0.23042	−0.90572	10.38730	−1.79515	−0.21694	0.32172	N	N	N
SBFE 88	6.18912	0.21442	−0.97076	10.29838	−1.80043	−0.22171	0.30240	N	N	N
Present	5.85673	0.14673	−0.99034	10.10688	−1.74027	−0.19461	0.26988	0.26985	4.64414	−0.728
$R/h=10$
3D 87	3.330	0.1930	−0.1689	10.59	−1.343	−0.12470	0.23250	0.1591	5.457	−1.68
Ref 89	3.16576	0.19089	−0.15665	10.95205	−1.20498	−0.11819	0.22150	0.15053	5.68891	N
TSDT 90	3.44978	0.20242	−0.17017	11.32900	−1.32470	−0.12606	0.24039	N	N	N
FSDT 90	3.67150	0.20389	−0.17541	10.92150	−1.42100	−0.13019	0.25724	N	N	N
SBFE 88	3.33316	0.19303	−0.17069	10.58798	−1.34344	−0.12670	0.23520	N	N	N
Present	3.28659	0.162825	−0.18097	10.47100	−1.33195	−0.12323	0.229571	0.157680	5.520317	−1.698
$R/h=50$
3D 87	2.242	0.21890	1.610	8.937	−0.9670	0.0784	0.3449	−0.0448	4.785	−6.29
Ref 89	2.23717	0.21866	1.60510	8.95433	−0.96152	0.07842	0.34440	−0.04424	4.79505	N
TSDT 90	2.28648	0.22122	1.62290	9.06240	−0.97207	0.07938	0.34835	N	N	N
FSDT 90	2.26272	0.22223	1.63873	9.02585	−0.98588	0.08080	0.35231	N	N	N
SBFE 88	2.24223	0.21889	1.60986	8.93654	−0.96713	0.078065	0.345486	N	N	N
Present	2.24087	0.21744	1.60965	8.93211	−0.96623	0.078337	0.344708	−0.04491	4.788837	−6.299
$R/h=100$
3D 87	1.367	0.18710	2.300	5.560	−0.57590	0.1819	0.3452	−0.1512	2.972	−7.71
Ref 89	1.36665	0.18708	2.29788	5.56430	−0.57495	0.18187	0.34514	−0.15097	2.97454	N
TSDT 90	1.37380	0.18838	2.30920	5.62630	−0.57771	0.18279	0.34690	N	N	N
FSDT 90	1.37812	0.18860	2.31605	5.60604	−0.58073	0.18354	0.34809	N	N	N
SBFE 88	1.36746	0.18715	2.29959	5.56017	−0.57596	0.18096	0.34531	N	N	N
Present	1.36720	0.18820	2.29956	5.56128	−0.57565	0.18187	0.34519	−0.15128	2.97395	−7.711

Note that *N: Unavailable.

The comparison shown in Table 1 demonstrates that the quasi-3D high-order shear deformation theory used in this study achieves a degree of accuracy that is comparable to the 3D elastic theory for both thick and thin shell scenarios. Significantly, when dealing with thick shells, the inaccuracies in findings obtained from alternative theories like FSDT and TSDT, when compared to results derived from 3D elastic theory, are much greater than the mistakes associated with the 3D quasi-high-order shear deformation theory used in this research. Moreover, the use of equation (39) has facilitated the computation of the normal stress component, yielding a result that closely aligns with the principles of 3D elasticity theory.

In the second assesment, the validation of the temperature distrubution is conducted by comparing present results with those of Moradi-Dastjerdi et al. ⁶⁰ Parameter of shell: inner radius $r_{in}=0.4m,$ outer radius $r_{out}=0.5m,$ the fraction volume CNT $V_{CNT}^{*}=0.17,$ temperature in inner and outer surface are $T_{in}=300K$ and $T_{out}=700K$ , respectively. From Figure 2, it can be seen that the results are in good agreement.

Figure 2.

The comparison of temperature distribution along thickness of FG-CNTRC cylindrical shell.

The third assessment evaluates the CNTRC cylindrical shell under thermal load and compared with published data of Pourasghar and coworkers. ⁵⁸ They assumed that temperature varied along radial directions and temperature-dependent material properties. The cylindrical shell has parameter as length $L=1m,$ middle radius $R=0.2m,$ ratio of radius to thickness $R/h=10$ . The material characteristics of PMMA, is chosen for the matrix, are ${\upsilon }_m=0.34,$ $E_m=\left(3.52-34\times {10}^{-4}T\right)GPa,$ ${\alpha }_m=45\left[1+5\times {10}^{-4}\left(T-T_{\text{ref}}\right)\right]·{10}^{-6}/K,$ $k_m=0.247W/mK.$ The reinforcing material is (3,3) SWCNTs, with the dynamic Young’s modulus defined as follows:

$E_{CNT}\left(T,\dot{\epsilon }\right)=\left(E_0+AT+B{T}^2+C{T}^3\right){\left(1+\dot{\epsilon }·{10}^{-6}\right)}^D$

(41)

where $\dot{\epsilon }$ is strain rate; A, B,C and D are constant which are not depended on loading rate and thermal environment. For zigzag SWCNT (3,3), these coefficients have the following values: $E_0=1.15,$ $\dot{\epsilon }=100,A=-2.8·{10}^{-4},$ $B=-3.98·{10}^{-8},$ $C=3.74·{10}^{-11},$ $D=0.27.$

The other material characteristics of SWCNT (3,3) are ${\alpha }_{11}^{CNT}=3.4584·{10}^{-6}/K,$ ${\alpha }_{22}^{CNT}={\alpha }_{33}^{CNT}=5.1682·{10}^{-6}/K,$ $k_{11}^{CNT}=3000W/mK$ , $k_{22}^{CNT}=k_{33}^{CNT}=100W/mK,{\upsilon }_{CNT}=0.175.$ CNT effective parameters are determined according to the CNT volume fraction as: $V_{CNT}^{*}=0.12,{\eta }_1=0.6696,$ ${\eta }_2=1.0211;{\eta }_3={\eta }_2.$ The temperature at the inner surface and the outer surface are $T_{in}$ and $T_{out}$ , respectively.

The temperature distribution was obtained by Pourasghar et al. ⁵⁸ as follows:

$T\left(r\right)=T_{in}+\frac{T_{out}-T_{in}}{C}\left[\begin{array}{l}\left(\frac{r-r_{in}}{h}\right)-\frac{K_{oin}}{\left(p+1\right)K_i}{\left(\frac{r-r_{in}}{h}\right)}^{p+1}+\frac{K_{oin}^2}{\left(2p+1\right)K_i^2}{\left(\frac{r-r_{in}}{h}\right)}^{2p+1}\\ -\frac{K_{oin}^3}{\left(3p+1\right)K_{in}^3}{\left(\frac{r-r_{in}}{h}\right)}^{3p+1}+\frac{K_{oin}^4}{\left(4p+1\right)K_{in}^4}{\left(\frac{r-r_{in}}{h}\right)}^{4p+1}\\ -\frac{K_{oin}^5}{\left(5p+1\right)K_{in}^5}{\left(\frac{r-r_{in}}{h}\right)}^{5p+1}\end{array}\right]$

(42)

where $K_{oin}=K_{out}-K_{in}$ and

$C=1-\frac{K_{oin}}{\left(p+1\right)K_{in}}+\frac{K_{oin}^2}{\left(2p+1\right)K_{in}^2}-\frac{K_{oin}^3}{\left(3p+1\right)K_{in}^3}+\frac{K_{oin}^4}{\left(4p+1\right)K_{in}^4}-\frac{K_{oin}^5}{\left(5p+1\right)K_{in}^5}$

In the case of UD cylindrical shell: $p=0,K_{out}=K_{in}$ .

Based on the comparative analysis shown in Figure 3, it is evident that a strong concurrence exists between the findings pertaining to the distribution of temperatures in the current study and the research conducted by Pourasghar and coworkers

Figure 3.

Comparison of the temperature distribution $\overline{\mathit{T}}=T/273$ along the radial direction with ${\mathit{T}}_{\mathit{o}\mathit{u}\mathit{t}}=100K$ .

To evaluate the need for consideration of the effect of temperature on material properties, perform calculations with two models: the former with temperature-dependent properties the latter with temperature-independent properties.

The comparison of non-dimensioness displacement and circumferential stress is shown in Figure 4. In this Figure, the symbols Tin300_Ref, Tin = 300_Present_Depend, and Tin = 300_Present_Independ are the results of Pourasghar et al., ⁵⁸ present work with temperature-dependent material characteristics and present work with temperature-independent material characteristics, respectively, in the case T _in . = 300K. Similar to Tin500_Ref, Tin = 500_Present_Depend, Tin = 500_Present_Independ in case T _in = 500K.

Figure 4.

Comparison of non-dimensional radial displacement $\overline{\mathit{w}}=w/\left({\alpha }_m{T}_o\right)$ and circumferential stress ${\overline{\mathit{\sigma }}}_{\mathit{\theta }}={\sigma }_{\theta }/\left({\alpha }_m{E}_m{T}_o\right)$ along the radial direction with ${\mathit{T}}_{\mathit{o}\mathit{u}\mathit{t}}=100K$ .

The comparison shown in Figure 4 demonstrates a strong concurrence between the outcomes obtained using temperature-dependent material characteristics and the findings reported by Pourasghar and collaborators. ⁵⁸ However, a notable discrepancy arises when contrasting outcomes obtained from the analysis of temperature-dependent material characteristics with those presented by Pourasghar and colleagues. ⁵⁸

Based on the aforementioned comparison and analysis findings, it can be ascertained that the current technique is deemed legitimate. Furthermore, it is important to take into account the temperature-dependent material characteristics while analyzing the cylinder shells made of functionally graded carbon nanotube reinforced composites (FG-CNTRC) subjected to thermal stresses.

Stress at the boundary area with different boundary conditions

The boundary area is a location that needs to be thoroughly investigated, but there are very few publications that carry out this content. In this section, the stresses at the clamped edge area of the shells with different boundary conditions are discussed. Consider the FG-CNTRC cylindrical shell under internal uniform pressure Q ₀ = 10 ⁷  Pa with different boundary conditions, including C-C, C-S, and C-F. The shell’s geometric parameters are radius R = 0.5 m, relative length L/R, radius and thickness ratio R/h. The temperature at inner and outer surface are T _in and T _out .

The dimensionless displacement and stresses are expressed as follows:

$\overline{w}=w/h;\hspace{0.17em}\left({\overline{\sigma }}_{\xi \xi },{\overline{\sigma }}_{\theta \theta },{\overline{\sigma }}_{zz},{\overline{\tau }}_{\xi \theta },{\overline{\tau }}_{\xi z},{\overline{\tau }}_{z\theta }\right)=\left({\sigma }_{\xi },{\sigma }_{\theta },{\sigma }_z,{\tau }_{\xi \theta },{\tau }_{\xi z},{\tau }_{z\theta }\right)/Q_0$

(43)

In present study, FG-CNTRC is made up of the fibers, which are the (10,10) SWCNTs, and the isotropic matrix material, which is PMMA.

The elasticity modulus and thermal expansion coefficient of PMMA as: ¹⁷

$E_m=\left(3.52-34\times {10}^{-4}T\right)GPa,{\alpha }_m=45\left[1+5\times {10}^{-4}\left(T-T_{\text{ref}}\right)\right]·{10}^{-6}/K$

(44)

where the ambient temperature $T_{ref}=300K$ .

The Poisson ratio ${\upsilon }_m=0.34$ and the thermal conductivity $k_m=5W/mK$ .

The properties of (10,10) SWCNTs can be obtained by the functions of temperature, ⁶⁰ as follow:

$\begin{array}{l}{E}_{11}^{CNT}=6.3998\times {10}^{12}-4.3384\times {10}^{9}T+7.4300\times {10}^6{T}^2-4.4583\times {10}^3{T}^3\\ E_{22}^{CNT}=8.0216\times {10}^{12}-5.4204\times {10}^{9}T+9.2750\times {10}^6{T}^2-5.5625\times {10}^3{T}^3\\ G_{12}^{CNT}=1.4076\times {10}^{12}+3.4762\times {10}^{9}T-6.9650\times {10}^6{T}^2+4.4792\times {10}^3{T}^3\\ {\alpha }_{11}^{CNT}=-1.1252\times {10}^{-6}+2.2917\times {10}^{-8}T-2.8870\times {10}^{-11}{T}^2+1.1363\times {10}^{-14}{T}^3\\ {\alpha }_{22}^{CNT}=5.43715\times {10}^{-6}-9.84625\times {10}^{-10}T+2.900\times {10}^{-13}{T}^2+1.2500\times {10}^{-17}{T}^3\end{array}$

(45)

The efficiency characteristics of CNT (carbon nanotube) are derived by aligning the effective properties of FG-CNTRC (functionally graded carbon nanotube reinforced composite) as indicated by the law of mixing, with those acquired via molecular dynamics (MD) simulation. The values of the volume fractions of CNT are shown below:

$V_{CNT}^{*}=0.12;\hspace{0.17em}{\eta }_1=0.137;\hspace{0.17em}{\eta }_2=1.022;\hspace{0.17em}{\eta }_3=0.7{\eta }_2;$

$V_{CNT}^{*}=0.17;\hspace{0.17em}{\eta }_1=0.142;\hspace{0.17em}{\eta }_2=1.626;\hspace{0.17em}{\eta }_3=0.7{\eta }_2;$

$V_{CNT}^{*}=0.28;\hspace{0.17em}{\eta }_1=0.141;\hspace{0.17em}{\eta }_2=1.585;\hspace{0.17em}{\eta }_3=0.7{\eta }_2;$

Figure 5 shows the results of the stresses of the FG-Λ shell at the edge position $\xi =0$ and the middle position $\xi =L/2R$ . The shell has different boundary conditions: C-C, C-S, C-F, and S-S, with V _CNT = 0.17, L/R = 1, R/h = 10, T _in = 400K, and T _out = 300K. From the stress graphs, it can be seen that the stress at the edge position has a much larger value than the value of the stress at the middle position of the shell. For example, in the case of C-C boundary conditions, the maximum axial stress at the boundary edge is about 240% of the maximum axial stress at the middle of the shell (graphs a and b in Figure 5), while the maximum radial stress at the edge is about 320% of the maximum radial stress at the middle of the shell (graphs g and h in Figure 5).

Figure 5.

Effect of the boundary conditions on stress components of C-F FG-Λ cylindrical shell with ${\mathit{V}}_{\mathit{C}\mathit{N}\mathit{T}}^{*}=0.17,\mathit{L}/R=1,R/h=10,T_{in}=400K,T_{out}=300K$ . (a) axial stress at $\xi =0$ , (b) axial stress at $\xi =L/2R$ , (c) circumferential stress at $\xi =0$ , (d) circumferential stress at $\xi =L/2R$ , (e) transverse shear stress at $\xi =0$ , (f) transverse shear stress at $\xi =L/2R$ , (g) radial stress at $\xi =0$ and (h) radial stress at $\xi =L/2R$ .

For more details on the variation of stress at the boundary area, examine the shell stress components in the region a distance x from the left edge $\xi =0$ , taking values from 0 to 5 times the shell thickness (5h). The investigation results are shown in Figures 6 and 7. In which:

- Figure 6 shows the stress results in the clamped boundary area of the shell with distribution CNT type UD and C-F boundary condition. The geometric dimensions include L/R = 2, R/h = 20, and the shell is subjected to internal pressure p = 10 MPa. The temperature on the inner surface is T _in = 400K, and the temperature on the outer surface is T _out = 300K.

- Figure 7 shows the stress result in the clamped boundary area of the shell with distribution CNT type FG-Λ and boundary condition C-C. The geometric dimensions include L/R = 5, R/h = 10, and the shell is subjected to internal pressure p = 10 MPa. The temperature on the inner surface is T _in = 500K, and the temperature on the outer surface is T _out = 300K.

Figure 6.

Variation of non-dimensional stress components at the clamped boundary area of C-F UD cylindrical shell with ${\mathit{V}}_{\mathit{C}\mathit{N}\mathit{T}}^{*}=0.28,\mathit{L}/R=2,R/h=20,T_{in}=400K,T_{out}=300K$ . (a) axial stress, (b) circumferential stress, (c) radial stress and (d) transverse shear stress.

Figure 7.

Variation of non-dimensional stress components at the clamped boundary area of C-C FG-Λ cylindrical shell with ${\mathit{V}}_{\mathit{C}\mathit{N}\mathit{T}}^{*}=0.28,\mathit{L}/R=5,R/h=10,T_{in}=500K,T_{out}=300K$ . (a) axial stress, (b) circumferential stress, (c) radial stress and (d) transverse shear stress.

From the investigation results, it can be seen that the stress components have a sudden jump in value at the clamped edge area. This sudden-change region is limited to an area about 1.5 times the shell thickness from the clamped edge. Outside this area, the variation of stress components is relatively smooth. The level of change in the value of axial stress, radial stress, and transverse shear stress is greater than that of circumferential stress.

Besides, the stress value at the boundary edge is greatly affected by the boundary conditions. To better understand this influence, perform calculations for the stress at the clamped boundary area (left edge) of shells with different boundary conditions, including C-C, C-S, and C-F.

Table 2 shows the results of the stress at the clamped boundary position of the FG-V shell with different boundary conditions, subjected to a uniform internal pressure load of 10 MPa, an internal temperature T _in = 300K, and T _out = 500K. The shell is considered to have different relative lengths L/R and relative thicknesses R/h. Similarly, the stress results of the FG-A shell subjected to similar pressure with internal temperature T _in = 400, T _out = 300K are shown in Table 3.

Table 2.

Stresses at clamped boundary area of FG-V with different boundary conditions, subjected to a uniform internal pressure Q = 10 MPa and $V_{CNT}^{*}=0.28$ , T _in = 300, T _out = 500.

	${\overline{\sigma }}_{\xi \xi }$	${\overline{\sigma }}_{\xi \xi }$	${\overline{\sigma }}_{\theta \theta }$	${\overline{\sigma }}_{\theta \theta }$	${\overline{\tau }}_{z\xi }$	${\overline{\tau }}_{z\xi }$	${\overline{\sigma }}_{zz}$	${\overline{\sigma }}_{zz}$
$\left(\xi ,\theta \right)$	$\left(0,0\right)$ $\left(\frac{h}{2R},0\right)$ $\left(\frac{h}{R},0\right)$	$\left(0,0\right)$ $\left(\frac{h}{2R},0\right)$ $\left(\frac{h}{R},0\right)$	$\left(0,0\right)$ $\left(\frac{h}{2R},0\right)$ $\left(\frac{h}{R},0\right)$	$\left(0,0\right)$ $\left(\frac{h}{2R},0\right)$ $\left(\frac{h}{R},0\right)$	$\left(0,0\right)$ $\left(\frac{h}{2R},0\right)$ $\left(\frac{h}{R},0\right)$	$\left(0,0\right)$ $\left(\frac{h}{2R},0\right)$ $\left(\frac{h}{R},0\right)$	$\left(0,0\right)$ $\left(\frac{h}{2R},0\right)$ $\left(\frac{h}{R},0\right)$	$\left(0,0\right)$ $\left(\frac{h}{2R},0\right)$ $\left(\frac{h}{R},0\right)$
z	$-h/2$	$h/2$	$-h/2$	$h/2$	$-3h/8$	$3h/8$	$-h/4$	$h/4$
$L/R=5,R/h=10$
C-C	7.017	−259.452	3.299	−4.945	4.024	11.528	1.561	−6.840
	4.111	−182.524	2.600	−3.360	2.272	6.543	−0.455	−0.589
	2.835	−134.761	2.734	−1.976	1.385	5.043	−0.629	−0.400
C-S	7.017	−259.452	3.299	−4.945	4.024	11.528	1.561	−6.840
	4.111	−182.524	2.600	−3.360	2.272	6.543	−0.455	−0.589
	2.835	−134.761	2.600	−1.976	1.385	5.043	−0.629	−0.400
C-F	6.959	−266.578	3.271	−4.966	4.018	11.558	1.539	−6.861
	4.062	−189.499	2.577	−3.378	2.276	6.556	−0.455	−0.591
	2.784	−141.647	2.711	−1.990	1.387	5.053	−0.629	−0.400
$L/R=0.5,R/h=10$
C-C	2.596	−113.732	1.197	−4.499	2.215	7.960	0.552	−6.116
	0.999	−66.391	0.790	−3.541	1.334	3.714	−0.683	−0.483
	0.199	−39.509	0.738	−2.906	0.775	2.496	−0.764	−0.338
C-S	2.596	−113.732	1.197	−4.499	2.215	7.960	0.552	−6.116
	0.999	−66.391	0.790	−3.541	1.334	3.714	−0.683	−0.483
	0.199	−39.509	0.790	−2.906	0.775	2.496	−0.764	−0.338
C-F	5.942	−230.992	2.787	−4.857	3.619	10.779	1.299	−6.684
	3.335	−160.334	2.158	−3.407	2.075	5.940	−0.506	−0.570
	2.156	−117.032	2.247	−2.187	1.259	4.494	−0.660	−0.389
$L/R=0.5,R/h=50$
C-C	22.595	−819.500	10.710	−6.654	10.770	24.788	5.215	−9.618
	13.963	−631.618	6.983	−5.361	5.897	17.052	0.224	−1.027
	9.820	−504.793	5.450	−4.198	3.860	14.366	−0.311	−0.675
C-S	27.785	−1030.437	13.185	−7.296	12.122	29.113	6.608	−10.662
	18.139	−812.023	9.042	−5.779	6.836	19.855	1.389	−0.171
	13.348	−664.405	9.042	−4.397	4.499	16.811	0.788	0.259
C-F	29.301	−1154.055	13.933	−7.668	11.806	25.224	7.704	−11.341
	19.547	−965.460	9.691	−6.324	5.902	17.035	0.255	−0.996
	15.430	−838.708	8.230	−5.053	3.852	14.373	−0.300	−0.667

Table 3.

Stresses at clamped boundary area of FG-∧ with different boundary conditions, subjected to a uniform internal pressure Q = 10 MPa and $V_{CNT}^{*}=0.28$ , T _in = 400, T _out = 300.

	${\overline{\sigma }}_{\xi \xi }$	${\overline{\sigma }}_{\xi \xi }$	${\overline{\sigma }}_{\theta \theta }$	${\overline{\sigma }}_{\theta \theta }$	${\overline{\tau }}_{z\xi }$	${\overline{\tau }}_{z\xi }$	${\overline{\sigma }}_{zz}$	${\overline{\sigma }}_{zz}$
$\left(\xi ,\theta \right)$	$\left(0,0\right)$ $\left(\frac{h}{2R},0\right)$ $\left(\frac{h}{R},0\right)$	$\left(0,0\right)$ $\left(\frac{h}{2R},0\right)$ $\left(\frac{h}{R},0\right)$	$\left(0,0\right)$ $\left(\frac{h}{2R},0\right)$ $\left(\frac{h}{R},0\right)$	$\left(0,0\right)$ $\left(\frac{h}{2R},0\right)$ $\left(\frac{h}{R},0\right)$	$\left(0,0\right)$ $\left(\frac{h}{2R},0\right)$ $\left(\frac{h}{R},0\right)$	$\left(0,0\right)$ $\left(\frac{h}{2R},0\right)$ $\left(\frac{h}{R},0\right)$	$\left(0,0\right)$ $\left(\frac{h}{2R},0\right)$ $\left(\frac{h}{R},0\right)$	$\left(0,0\right)$ $\left(\frac{h}{2R},0\right)$ $\left(\frac{h}{R},0\right)$
z	$-h/2$	$h/2$	$-h/2$	$h/2$	$-3h/8$	$3h/8$	$-h/4$	$h/4$
$L/R=4,R/h=10$
C-C	151.069	−8.171	−1.499	−4.365	6.659	4.305	−0.796	−2.037
	93.698	−3.268	−0.855	−1.038	5.077	2.005	−0.539	−0.560
	58.764	−0.926	0.023	0.867	3.892	1.146	−0.627	−0.287
C-S	145.599	−8.296	−1.526	−4.432	6.662	4.339	−0.816	−2.081
	88.080	−3.354	−0.882	−1.078	5.092	2.011	−0.539	−0.561
	53.045	−1.005	−0.882	0.832	3.903	1.149	−0.627	−0.287
C-F	145.637	−8.298	−1.526	−4.433	6.663	4.340	−0.816	−2.081
	88.105	−3.355	−0.881	−1.079	5.093	2.011	−0.539	−0.561
	53.062	−1.005	−0.001	0.832	3.905	1.150	−0.627	−0.287
$L/R=1,R/h=10$
C-C	110.293	−6.908	−1.709	−3.685	6.273	3.993	−1.137	−1.605
	55.462	−2.386	−1.115	−0.635	4.856	1.919	−0.552	−0.543
	22.022	−0.166	−0.349	1.130	3.698	1.092	−0.639	−0.281
C-S	118.868	−7.603	−1.663	−4.055	6.882	4.323	−1.046	−1.735
	59.063	−2.699	−1.009	−0.744	5.330	2.093	−0.532	−0.583
	22.328	−0.272	−1.009	1.194	4.123	1.204	−0.629	−0.301
C-F	146.798	−8.186	−1.521	−4.376	6.382	4.220	−0.808	−2.107
	91.645	−3.379	−0.902	−1.110	4.862	1.928	−0.548	−0.541
	58.213	−1.125	−0.047	0.729	3.697	1.095	−0.631	−0.276
$L/R=1,R/h=50$
C-C	895.483	−38.434	2.352	−20.581	21.023	13.800	5.569	−10.791
	728.690	−22.931	2.034	−11.371	15.331	5.812	−0.177	−1.440
	621.318	−16.688	2.310	−7.507	13.022	3.594	−0.421	−0.635
C-S	873.501	−38.482	2.242	−20.611	20.697	13.730	5.456	−10.867
	708.972	−23.054	1.927	−11.435	15.118	5.737	−0.189	−1.425
	603.085	−16.891	1.927	−7.619	12.834	3.543	−0.425	−0.626
C-F	850.258	−37.625	2.122	−20.150	20.445	13.518	5.254	−10.578
	687.481	−22.445	1.810	−11.130	14.960	5.677	−0.194	−1.411
	582.704	−16.349	2.077	−7.359	12.693	3.505	−0.429	−0.621

Stress values are taken at the edge positions ( $\xi =0$ ), half the shell thickness away from the edge ( $\xi =h/2R$ ), and half the shell thickness away from the edge ( $\xi =h/R$ ). From the results table, it can be seen that for shells with average relative lengths (L/R = 4; 5), the stress values are approximately equal with different boundary conditions. However, for shells with a relatively short length (L/R = 0.5; 1), the stress value at the attachment boundary of the C-F shell is much larger than the stress of the C-C and C-S shells. This difference is relatively large even for short shells with relatively thin thickness (R/h = 50) and increases in the case of both short and thick shells. For example, the axial stress value at the boundary area of the C-F FG-V shell with L/R = 0.5 and R/h = 50 is 140% compared to the C-C and C-S shells; this ratio increases to 203% when the shell has L/R = 0.5 and R/h = 10.

Effect of CNT distribution type

Consider the C-F boundary shell under internal uniform pressure Q ₀ = 10 ⁷  Pa with five configurations of the CNT distribution: FG-Ʌ, FG-V, UD, FG-O, FG-V. Geometric parameters of the cylindrical shell: R = 0.5 m; L/R = 2; R/h = 20. The temperature at inner and outer surface are T _in = 500 K and T _out = 300 K.

The results of temperature distribution, nondimensional displacement, and nondimensional stresses along the radial direction at middle position $\xi =L/2R$ and at boundary position $\xi =0$ with CNT volume fractions $V_{CNT}^{*}=0.12,0.17,0.28$ are presented in Figures 8–10, respectively. From these figures, one has:

- The CNT distribution types affect the temperature distribution in the shell. The temperature graphs of FG-V and FG-Λ, FG-X and FG-O, one by one, have opposite variation; the temperature distribution graph of the UD case varies linearly and is between the other cases (show plots a in Figures 8–10). For the case under consideration (T _in = 500 K and T _out = 300 K), the temperature of the FG-V configuration has the lowest value, while FG-Λ has the highest value. This can be explained based on the very large thermal conductivity of CNT compared to the matrix material. In the FG-V situation, a significant portion of carbon nanotubes (CNTs) exhibit a larger concentration towards the external surface of the shell, which corresponds to a relatively lower temperature. Conversely, in the FG-Λ scenario, the majority of CNTs are concentrated towards the internal surface of the shell, where the temperature is comparatively higher. The relationship between the distribution type of carbon nanotubes (CNTs) and the temperature distribution becomes more pronounced with an increase in the volume percentage of CNTs.

- The displacement at the middle position of the shell is highly dependent on the CNT distribution type. For shells subjected to internal pressure and internal thermal load, displacements in FG-X and FG-V types have the smallest value (show plots b in Figures 8–10).

- The CNT distribution type greatly affects the stresses at the boundary position ( $\xi =0$ ) and the middle position ( $\xi =L/2R$ ). The locations showing the greatest influence of the CNT distribution types were at the inner $z=-h/2$ , outer $z=h/2$ , or middle $z=0$ surfaces. This phenomenon may be explained by the reality that at these positions, the CNT volume fraction reaches its maximum or minimum value. Regarding the trend of variation, the stress values FG-V and FG-Λ, FG-O and FG-X tend to vary oppositely around the stress value of the UD type (show plots c, e, g, and i in Figures 8–10). For the case of internal pressure and thermal load on the inner surface, the maximum value of ${\overline{\sigma }}_{\xi \xi },{\overline{\sigma }}_{zz},{\overline{\tau }}_{\xi z}$ in the type of FG-Λ is the largest (show plots c, g, and i in Figures 8, 9, and 10), while the maximum value of ${\overline{\sigma }}_{\theta \theta }$ is smaller than the other distributions (show plots f in Figures 8–10).

Figure 8.

Effect of CNT distribution types on temperature distribution, non-dimensional displacement and stress components for C-F cylindrical shell with ${\mathit{V}}_{\mathit{C}\mathit{N}\mathit{T}}^{*}=0.12,\mathit{L}/R=2,\mathit{R}/h=20,{\mathit{T}}_{\mathit{i}\mathit{n}}=500K,{\mathit{T}}_{\mathit{o}\mathit{u}\mathit{t}}=300K$ . (a) Temperature distribution, (b) radial displacement at $\xi =L/2R$ , (c) axial stress at $\xi =0$ , (d) axial stress at $\xi =L/2R$ , (e) circumferential stress at $\xi =0$ , (f) circumferential stress at $\xi =L/2R$ , (g) transverse shear stress at $\xi =0$ , (h) transverse shear stress at $\xi =L/2R$ , (i) radial stress at $\xi =0$ , (k) radial stress at $\xi =L/2R$ .

Figure 9.

Effect of CNT distribution types on temperature distribution, non-dimensional displacement and stress components for C-F cylindrical shell with ${\mathit{V}}_{\mathit{C}\mathit{N}\mathit{T}}^{*}=0.17,\mathit{L}/R=2,\mathit{R}/h=20,{\mathit{T}}_{\mathit{i}\mathit{n}}=500K,{\mathit{T}}_{\mathit{o}\mathit{u}\mathit{t}}=300K$ . (a) Temperature distribution, (b) radial displacement at $\xi =L/2R$ , (c) axial stress at $\xi =0$ , (d) axial stress at $\xi =L/2R$ , (e) circumferential stress at $\xi =0$ , (f) circumferential stress at $\xi =L/2R$ , (g) transverse shear stress at $\xi =0$ , (h) transverse shear stress at $\xi =L/2R$ , (i) radial stress at $\xi =0$ , (k) radial stress at $\xi =L/2R$ .

Figure 10.

Effect of CNT distribution types on temperature distribution, non-dimensional displacement and stress components for C-F cylindrical shell with ${\mathit{V}}_{\mathit{C}\mathit{N}\mathit{T}}^{*}=0.28,\mathit{L}/R=2,\mathit{R}/h=20,{\mathit{T}}_{\mathit{i}\mathit{n}}=500K,{\mathit{T}}_{\mathit{o}\mathit{u}\mathit{t}}=300K$ . (a) Temperature distribution, (b) radial displacement at $\xi =L/2R$ , (c) axial stress at $\xi =0$ , (d) axial stress at $\xi =L/2R$ , (e) circumferential stress at $\xi =0$ , (f) circumferential stress at $\xi =L/2R$ , (g) transverse shear stress at $\xi =0$ , (h) transverse shear stress at $\xi =L/2R$ , (i) radial stress at $\xi =0$ , (k) radial stress at $\xi =L/2R$ .

Interaction effect between pressure load and themal load

To investigate the interaction between pressure load and thermal load, consider six cases, as shown in Figure 11, including:

- Only subject to an external pressure load (I)

- Withstands external pressure loads and external thermal loads (II)

- Withstand external pressure load and internal heat load (III)

- Only subject to an internal pressure load (IV);

- Withstands internal pressure load and external heat load (V)

- Withstands internal pressure load and internal heat load (VI)

Figure 11.

Six cases to investigate the interactive effects of pressure load and heat load on the stress of FG-CNTRC cylindrical shell.

Figure 12 shows the calculation results of the stress components of the FG-V shell subjected to external pressure load in the following cases: without thermal load (Tin = Tout = Tref = 300K), with internal thermal load (Tin = 400K, 500K, Tout = Tref = 300K), with external thermal load (Tin = Tref = 300K, Tout = 400K, 500K).

Figure 12.

The effect of the interaction of external pressure and thermal load on the stress of the C-S FG-V cylindrical shell with ${\mathit{V}}_{\mathit{C}\mathit{N}\mathit{T}}^{*}=0.28,\mathit{L}/R=3,\mathit{R}/h=10$ . (a) axial stress at $\xi =0$ , (b) axial stress at $\xi =L/2R$ , (c) circumferential stress at $\xi =0$ , (d) circumferential stress at $\xi =L/2R$ , (e) transverse shear stress $\xi =0$ , (f) transverse shear stress $\xi =L/2R$ , (g) radial stress at $\xi =0$ , h) radial stress at $\xi =L/2R$ .

Similarly, Figure 13 shows the calculation results of the stress components of the FG-Λ shell subjected to internal pressure load in the following cases: without thermal load (Tin = Tout = Tref = 300K), with internal thermal load (Tin = 400K, 500K; Tout = Tref = 300K), with external thermal load (Tin = Tref = 300K, Tout = 400K, 500K).

Figure 13.

The effect of the interaction of internal pressure and thermal load on the stress of the C-S FG-Λ cylindrical shell with ${\mathit{V}}_{\mathit{C}\mathit{N}\mathit{T}}^{*}=0.28,\mathit{L}/R=3,\mathit{R}/h=10$ . (a) axial stress at $\xi =0$ , (b) axial stress at $\xi =L/2R$ , (c) circumferential stress at $\xi =0$ , (d) circumferential stress at $\xi =L/2R$ , (e) transverse shear stress $\xi =0$ , (f) transverse shear stress $\xi =L/2R$ , (g) radial stress at $\xi =0$ , (h) radial stress at $\xi =L/2R$ .

From the graphs, there are some interesting results, as follows, about the mutual influence of pressure and thermal load.

For the case of pressure loading on the outside of the shell (I, II, III) shown in Figure 12

- The axial stress is proportional to the magnitude of the thermal load, in which the increase in the thermal load on the outer surface is greater than the thermal load on the inner surface (graph a and b in Figure 12).

- Circumferential stress is strongly influenced by the location of the thermal load on the inside or outside surface. The heat source on the inner surface will have a greater impact on the circumferential stress on the inside than on the outside of the shell. Conversely, the heat load source on the outside surface will have a greater influence on the stress on the outside surface compared to the inside surface. Comparing the level of influence, for the shell subjected to extenal pressure, for circumferential stress, the thermal load on the inside surface has a greater influence than the thermal load on the outside surface (graphs c and d in Figure 12). Notably from the graph d in Figure 12, internal thermal load reduces the circumferential stress value at the inner half of the thickness ( $-h/2\le z\le 0$ ) and increases its value at the outer half of the thickness $\left(0<z\le h/2\right)$ , whereas external thermal load increases the circumferential stress value at the inner half of the thickness ( $-h/2\le z\le 0$ ) and reduces its value the outer half of the thickness $\left(0<z\le h/2\right)$ .

- The transverse shear stress is proportional to the value of the thermal load. The larger the thermal load, the larger the maximum value of the transverse shear stress. The heat load on the outer surface has a greater influence than the heat load on the inner surface (graph e and f in Figure 12).

- The impact of the external pressure load is amplified by the rise in external heat load, as shown in the case of radial stress. Conversely, the internal temperature load mitigates the impact of the external pressure load. To clarify, the impact of the external pressure load and the heat load from the outside exhibit similar patterns, but the impact of the external pressure load and the heat load from the interior show contrasting patterns (graph g and h Figure 12).

For the case of pressure loading on the inside of the shell (IV, V, VI) shown in Figure 13

- The axial stress is directly proportional to the size of the thermal load, as shown in graphs a and b in Figure 13. The maximum axial stress occurs at the boundary edge when the thermal load on the outer surface exceeds the thermal load on the inner surface (graph an in Figure 13).

- Circumferential stress is strongly influenced by the location of the heat load on the inside or outside surface. The heat source on the inside will have a greater impact on the stress on the inside than on the outside of the shell. Conversely, the heat load source on the outside surface will have a greater influence on the stress on the outside surface compared to the inside surface. The interesting result is that, similar to the case of a shell subjected to external pressure, the thermal load on the inner surface will give a larger maximum value of circumferential stress than the case of thermal load on the outer surface (graph d in Figure 13). Similar to the case of a shell subjected to external pressure, internal thermal load reduces the circumferential stress value at the inner half of the thickness ( $-h/2\le z\le 0$ ) and increases its value at the outer half of the thickness $\left(0<z\le h/2\right)$ , whereas external thermal load increases the circumferential stress value at the inner half of the thickness ( $-h/2\le z\le 0$ ) and reduces its value the outer half of the thickness $\left(0<z\le h/2\right)$ .

- Just as when the shell is influenced by external pressure, when the shell is exposed to internal pressure, the transverse shear stress is directly proportional to the size of the heat load. The thermal load on the outer surface has a more significant impact than the thermal load on the inner surface (graphs e and f in Figure 13).

- Radial stress is greatly influenced by the location and magnitude of the thermal load. Considering the stress value at the boundary edge, it can be seen that at about half the thickness of the inner surface of the shell ( $-h/2\le z\le 0$ ), the external heat load increases the effect of the internal pressure load, while the heat load inside reduces the effect of internal pressure loads. In other words, the effects of internal pressure load and external heat load are in the same trend, while the effects of internal pressure load and internal heat load are opposite trends (graphs g and h in Figure 13). However, at half the outer shell thickness $\left(0<z\le h/2\right)$ , the influence of both internal and external thermal loads increases the influence of internal pressure loads.

Conclusion

This paper presents the static analysis of cylindrical shells made of FG-CNTRC subjected to thermo-mechanical stresses. The temperature distribution over the thickness is defined by the heat transfer equation and the boundary conditions for temperature. The characteristics of materials are influenced by temperature and are characterized in accordance with the expanded rule of mixing. Several conclusions may be derived from the numerical calculations conducted.

- Boundary conditions greatly affect the stress in the shell. The stress components at the clamped boundary location exhibit much greater magnitudes compared to those seen at the middle position. Notably, there is a sudden increase in stress values at the boundary area; the limit of this area is about 1.5 times the shell thickness from the boundary edge.

- The temperature distribution, displacement, and stress components of shells are significantly influenced by the CNT distribution type due to the substantial disparity in thermal conductivity and elastic modulus values between CNTs and the matrix material.

- Pressure load and thermal load interact with each other. An increase in the value of pressure load and thermal load can increase the magnitude of the axial stress value, in which the influence of the heat load from the outside surface is greater than the heat load on the inside surface.

The novelties and main contributions of the present paper include the following: First of all, using the high-order shear deformation theory, which includes normal stress, can show the phenomenon of sudden increases in stress at the boundary. Secondly, simultaneously considered temperature-dependent materials and thermal gradient loads determined from the heat transfer equation. Thirdly, the method using trigonometric series and Laplace transform can be solve to different boundary conditions. Finally, the article showed the interaction between pressure and temperature. The findings of this work have significant implications for the computation and design of cylindrical shell concepts composed of advanced materials, while also considering the influence of temperature variables.

From the above results, it can be seen that the advantage of HSDT, which includes normal stress combined with the solution method used in this work, is to point out the phenomenon of sudden stress increase at the boundary area of the shell. At the same time, this approach also shows that the interaction of heat load and pressure can increase or decrease the stress in the shell. The findings of this study have significant implications for engineers engaged in the calculation, design, and implementation of actual engineering projects.