Bending response of functionally graded shallow shells under the action of non-linear thermomechanical loading considering the effects of transverse normal strain

Abstract

In this article, the bending response of single-layer functionally graded shells of double curvature is investigated under non-linear thermomechanical loadings with simply supported boundary conditions. A higher-order trigonometric shear and normal deformation theory is applied to the present study. The theory satisfies the traction-free boundary condition at the shell’s extreme top and bottom surfaces and gives a cosine distribution of transverse shear stresses through the thickness. The principle of virtual work is employed to obtain the governing differential equations. The Navier solution technique is used further to solve governing equations for the simply supported boundary conditions of the shell. The present study mainly focuses on the study of effects of transverse normal strain, shear deformation, radii of curvature, and volume fraction distributions on the bending response of shells of double curvature, such as cylindrical, spherical, hyperbolic, and elliptical. Since very little or no literatures are available on thermomechanical analysis of functionally graded shells in the open literature, the authors have formulated parabolic shear and normal deformation theory and first-order shear deformation theory to compare the present results. The numerical results of hyperbolic and elliptical shells will be a benchmark for future researchers.

Introduction

The functionally graded material (FGM) is one of the advanced composite materials used for the first time in Japan in 1983. In FGMs, the material’s elastic properties are continuously graded through the thickness of the structural element under consideration. The physical characteristics of the material vary along the thickness direction (i.e., z-direction) according to the power-law of the material gradation. Notable FGM applications include shell structures widely used in many engineering structures, like nuclear reactors, spacecraft, missiles, ships, etc. Koizumi, ¹ Kawasaki and Watanabe, ² Watari et al., ³ Pompe et al., ⁴ Muller et al. ⁵ carry out a detailed discussion on FGM manufacuring and its applications. Castanie et al. ⁶ reviewed the issues concerning sandwich structures for aeronautical applications. Therefore, investigating the static response of the FGM shells under thermomechanical loads is necessary for their safe design and to avoid failure.

The classical shell theory (CST) ⁷ developed by Kirchhoff in 1850, neglects the effect of transverse shear deformation. Therefore, it provides an inaccurate result of displacements and stresses for thick plates and shells wherein shear deformation effects are significant. Therefore, Mindlin ⁸ in 1951 developed a first-order shear deformation theory (FSDT) that considers the effects of transverse shear stresses for both thin as well as moderately thick plates and shells. Still, it fails to give the realistic shear stress condition at the top and bottom of the plate and shell element. However, this theory gives constant values of transverse shear stresses across the thickness of the structural element, which is practically unacceptable.

This necessitated the development of higher-order shear and normal deformation theory (HSNDT), which will consider the effects of both transverse shear and normal deformations and give the realistic boundary conditions at the top and bottom of the element under consideration. The review articles elaborating various modelling techniques of FGM and its integrity in the analysis of FGM structures are discussed by Thai and Kim, ⁹ Swaminathan and Sangeetha, ¹⁰ Ghatage et al., ¹¹ Boggarapu et al., ¹² Swaminathan et al., ¹³ Abrate and Sciuva, ¹⁴ and Garg and Chalak. ¹⁵

Punera and Kant ¹⁶ studied the elastostatics of laminated and functionally graded sandwich open cylindrical shells using a refined higher-order shear and normal deformation theory. Ghugal et al. ¹⁷ developed trigonometric shear and normal deformation theory to analyse laminated composite spherical shells subjected to sinusoidal mechanical/thermal loads with simply supported boundary conditions. Shinde et al. ¹⁸ analysed the thermal response of a thick isotropic rectangular plate using hyperbolic shear deformation theory. Punera et al. ¹⁹ presented analytical solutions for laminated and FG sandwich open cylindrical shells under mechanical and thermal loads using a refined higher-order shear and normal deformation theory. Kugler et al. ²⁰ derived an efficient low-order finite shell element for the thermo-elastic analysis of shell structures made of FGM or multilayer composites. Kushnir et al. ²¹ performed an analytic investigation of the stress-strain state of a FG cylindrical shell of finite length using equations of the refined theory of shells. Dai and Dai ²² studied the thermoelastic bending behavior of an FGM cylindrical shell subjected to a uniform transverse mechanical load and non-uniform thermal loads using the classical linear shell theory. Li et al. ²³ developed a four-variable refined plate theory to investigate the thermomechanical bending response of FGM sandwich plates. Zenkour and Alghamdi ²⁴ studied the bending analysis of FG ceramic-metal sandwich plates under the effect of mechanical and thermal loads using a refined higher-order shear and normal deformation theory. Sayyad and Ghumare ²⁵ studied the thermomechanical bending analysis of FG sandwich plates using a quasi-three-dimensional theory.

Filippi et al. ²⁶ used 1D Carrera’s Unified Formulation (CUF) to perform static analyses of FG structures. Cinefra et al. ²⁷ used refined shell theories to analyse simply supported FG shell thermomechanically. Cinefra et al. ²⁸ employed CUF-based refined models for the static analysis of plates and shells made of FGM under the action of mechanical loads. Ramos et al. ²⁹ presented an analytical solution for the thermoelastic analysis of simply supported FG sandwich plates using CUF. Cinefra et al. ³⁰ performed the linear static thermal stress analysis of composite structures using a shell finite element method. Carrera et al. ³¹ investigated the static response of multilayered plates and shells embedded with functionally graded material layers. Wu et al. ³² developed a unified formulation of geometrically non-linear refined shell theory to analyse highly flexible shell structures. Sayyad and Ghugal ³³ presented the static and free vibration analysis of doubly-curved functionally graded material shells using equivalent single-layer shell theories. Shinde and Sayyad^34–37 presented a fifth-order shear and normal deformation theory for the static and thermoelastic analysis of laminated composite, sandwich, and orthotropic shells. Shinde et al. ³⁸ assessed a new higher-order shear and normal deformation theory for the static response of functionally graded shallow shells. Armendariz et al. ³⁹ developed a finite element model to analyse functionally graded shells. Czekanski and Zozulya ⁴⁰ developed a higher-order theory for analysing functionally graded shells. Aghdam et al. ⁴¹ developed a semi-analytical solution for the static response of moderately thick doubly curved functionally graded panels. Nath and Das ⁴² presented a zigzag theory for the static and free vibration analysis of multilayered functionally graded material cylindrical shells and rectangular plates. Shahsiah et al. ⁴³ employed first-order shell theory and the Sanders kinematics relations to study the thermal instability of functionally graded shallow spherical shells. Sheng and Wang ⁴⁴ employed von Karman’s non-linear theory to analyze functionally graded cylindrical shells under mechanical and thermal loads. Sladek et al. ⁴⁵ used first-order shear deformation theory for the static and dynamic analysis of shallow shells with functionally graded and orthotropic material properties. Reddy ⁴⁶ developed a higher-order shear deformation theory to analyse laminated composite plates. Reddy ⁴⁷ extended the Sanders shell theory to analyse doubly curved laminated shells. Krishna Murty ⁴⁸ developed a consistent plate theory that considers the cubic variation of the direct stress and parabolic variation of the shear stress across the depth of the beam. Daouadji et al. ⁴⁹ employed a modified first-order shear deformation theory to investigate the bending behavior of an advanced porous functionally graded plate subjected to the combined effect of mechanical, thermal, and moisture loads while resting on a viscoelastic foundation. Bentabet et al. ⁵⁰ modeled and applied a refined integral plate theory incorporated into Eringen’s nonlocal elasticity theory to study the flexural response and buckling analysis of functionally graded nanoscale plates under thermomechanical loadings. Draiche et al. ⁵¹ developed an efficient integral higher-order shear and normal deformation theory based on a unified and enriched kinematic model to investigate the static bending response of functionally graded sandwich curved beams under uniform mechanical loading. Belbachir et al. ⁵² addressed a refined plate theory to describe the bending response of anti-symmetric cross-ply laminated plates under uniformly distributed non-linear thermal and mechanical loadings. Mazari et al. ⁵³ employed a quasi-3D hyperbolic shear deformation theory to investigate the bending analysis of simply supported functionally graded thick plates with in-plane stiffness variation. Benounas et al. ⁵⁴ used an improved finite element shear model to precisely analyze the static bending response of functionally graded doubly curved shallow shells under various loading conditions. Vinh and Tounsi ⁵⁵ studied the free vibration analysis of the functionally graded doubly curved nanoshells using nonlocal first-order shear deformation theory with variable nonlocal parameters. Vinh et al. ⁵⁶ modified and applied the classical Eringen’s nonlocal elasticity theory to investigate the free vibration behavior of functionally graded porous doubly curved shallow nanoshells with variable nonlocal parameters. Benounas et al. ⁵⁷ proposed a novel finite element model based on the improved first-order shear deformation theory to investigate the vibration behavior of functionally graded doubly curved shallow shells. Belarbi et al. ⁵⁸ analyzed the free vibration behavior of functionally graded doubly curved shallow nanoshells using an improved first-order shear deformation theory model. Daikh et al. ⁵⁹ introduced a new hyperbolic shear deformation theory for the free vibration analysis of cosine functionally graded doubly curved shells under various boundary conditions. Vu et al. ⁶⁰ combined a new refined quasi-3D logarithmic shear deformation theory and an advanced moving Kriging interpolation-based meshless method to study the static bending, free vibration, and compressive buckling analysis of isotropic and sandwich functionally graded plates resting on the elastic foundations. Vu ⁶¹ employed an advanced element-free Galerkin method based on moving Kriging interpolation in conjunction with a new simple quasi-3D hyperbolic shear deformation theory for the mechanical behavior analysis of functionally graded porous plates resting on elastic foundations. Vu et al. ⁶² presented a numerical method for analyzing the compressive buckling of porous power-law functionally graded plates resting on Pasternak foundations based on a novel refined quasi-3D hyperbolic shear deformation theory. Vu and Cao ⁶³ investigated the static bending responses and natural frequency characteristics of functionally graded porous plates resting on elastic foundations using a four-variable hyperbolic quasi-3D shear deformation theory. Cao and Vu^64,65 explored the free vibration, flexural, and buckling behavior of functionally graded porous plates resting on a Kerr-type elastic foundation. This investigation employed an innovative trigonometric shear deformation theory with five variables. Cao and Vu ⁶⁶ presented a quasi-3D trigonometric shear deformation theory to analyze the free vibration behavior of sandwich plates with two porous functionally graded skins and an auxetic honeycomb core resting on a Kerr-type elastic foundation. Kolapkar and Sayyad ⁶⁷ examined the bending behavior of functionally graded material shells under linear and non-linear hygrothermomechanical loadings using higher-order trigonometric shear and normal deformation theory.

Studies on the effects of transverse normal strain

Zenkour ⁶⁸ used a refined higher-order plate theory for the bending analysis of functionally graded plates. Zenkour and Alghamdi ⁶⁹ studied the thermoelastic bending response of FG ceramic-metal sandwich plates using a refined higher-order shear and normal deformation theory. Mantari et al. ⁷⁰ presented a static analysis of FG single and sandwich plates using CUF with five new displacement fields of the non-polynomial form. Ghumare and Sayyad ⁷¹ analysed a functionally graded beam to investigate the bending behaviour under the hygrothermomechanical loading using a new fifth-order shear and normal deformation theory. Ghumare and Sayyad ⁷² developed a new fifth-order shear and normal deformation theory for the bending analysis of functionally graded plates resting on a Winkler-Pasternak elastic foundation subjected to non-linear hygrothermomechanical loading. Naik and Sayyad ⁷³ used a fifth-order shear and normal deformation theory for the mechanical, thermal, and hygrothermal stress analysis of composite layered and sandwich plates on simply supported edge conditions. Carrera et al. ⁷⁴ evaluated the effect of thickness stretching in plate/shell structures made up of FGM. Recently, a few more articles have been found assessing the effects of transverse normal strain on the static and free vibration analysis of laminated composite and FGM shallow shells.^75–81 Chakraborty et al. ⁸² employed first-order shear deformation theory to study thermoelastic behavior of functionally graded beam structures.

Need for future research

Based on the aforementioned literature review, the observations that need further study are

(1) Analysis of laminated composite and FGM plates and shells under mechanical loading have been extensively studied in the literature. However, there are limited studies available on the thermomechanical analysis of FGM shells. To the best of author’s knowledge, very few or no papers have been found in the literature on the thermomechanical analysis of FGM shells.

(2) It is also noted from the literature published by Carrera ⁷⁴ that the analysis of advanced composite shells is meaningless unless the effects of transverse normal strain, i.e., thickness stretching, are considered in the kinematics of shell theories. With reference to the section mentioned above, in the open literature, plenty of research articles are available on the analysis of laminated composite and FGM plates considering the effects of transverse normal strain. However, articles available on the analysis of FGM shells ignore the effects of transverse normal strain, which plays an important role in the accurate structural analysis of FGM shells under the action of thermomechanical loading conditions, especially when the temperature varies across the thickness of the shells.

(3) The review above also shows that the literature on analyzing FGM cylindrical and spherical shells is found in good amounts. However, the literature on hyperbolic and elliptical paraboloid shells is very limited.

To fulfil the research gap mentioned in the above section, the authors have applied trigonometric shear and normal deformation theory for the bending analysis of doubly curved FGM shells, considering the effects of transverse normal strain under thermomechanical loading. The principle of virtual work is applied to obtain the governing differential equations, and further, these equations are solved using the Navier solution technique. Since the literature on thermomechanical analysis of functionally graded shells is not available in the open literature, the authors have formulated parabolic shear deformation theory and first-order shear deformation theory to compare the present results obtained using trigonometric shear and normal deformation theory. The present study focuses on the effects of transverse normal strain, shear deformation, radii of curvature, and volume fraction distributions on the bending of shells of double curvature, such as cylindrical, spherical, hyperbolic, and elliptical. The results of hyperbolic and elliptical shells will serve as a benchmark for future researchers.

Geometry of the FGM shell

A simply supported single-layer FGM shell element on a rectangular planform, as shown in Figure 1, is considered for the mathematical formulation. A shell has dimensions ‘a’ and ‘b’ along the ‘x’ and ‘y’ directions, respectively, and ‘h’ along the z-direction. R₁ and R₂ are the principal radii of curvature of the mid-plane along the ‘x’ and ‘y’ directions, respectively. The top surface of the shell is located at (z = -h/2), a mid-plane is located at (z = 0), and the bottom surface is located at (z = +h/2). A transverse load of intensity q (x, y) is applied on the shell’s top surface. The FGM used to construct the shell has gradations of elastic properties such as modulus of elasticity and coefficient of thermal expansion in the direction of thickness, i.e., z-direction, according to the power-law. Following is the simple form of the power-law, which is used for the gradation of material properties across the thickness of the shell. ⁸²

$\begin{array}{l}E\left(z\right)=E_m+\left(E_c-E_m\right)V_f\hspace{0.17em}\\ \alpha \left(z\right)={\alpha }_m+\left({\alpha }_c-{\alpha }_m\right)V_f\end{array}$

(1)

where E(z) and α(z) are the values of modulus of elasticity and coefficient of thermal expansion at any point on the shell element in the z-direction; E _m and E _c are the modulus of elasticity of metal and ceramic, respectively, whereas, α _m and α _c are the coefficients of thermal expansion of metal and ceramic respectively; V _f is the volume fraction which is given as

$V_f{=\left(0.5+\frac{z}{h}\right)}^p$

(2)

where ‘p’ is the power-law index. When p = 0, the shell is fully made up of ceramic, and when p = ∞, the shell is fully made up of metal. Figure 2 shows the variation of material properties according to the power-law for various values of the power-law index.

Figure 1.

Geometry and coordinate systems of the FGM shell.

Figure 2.

Gradation of material properties for various values of power-law index.

Mathematical formulation

About the present theory

The present theory uses the trigonometric function in the displacement field of the shell element. However, this function was originally suggested by Levy ⁸³ in 1877 and then by Touratier ⁸⁴ in 1991. Later on, the trigonometric function is used by Sayyad and Ghugal^85,86 for the static, buckling, and free vibration analysis of laminated composite plates and shells. Recently, Tamnar and Sayyad ⁷⁵ as well as Kolapkar and Sayyad ⁸¹ used the trigonometric theory for the mechanical analysis of FGM shells. This study used trigonometric shear and normal deformation theory for the thermomechanical analysis of FGM shells of double curvature under various geometric conditions.

Displacement field

The displacement field of the present theory for the shell element is written as follows. ⁷⁵

$\begin{array}{l}u\left(x,y,z\right)={\left[1+\left(z/R_1\right)\right]u}_0\left(x,y\right)-z\frac{\partial w_0}{\partial x}+f\left(z\right)\hspace{0.17em}{\theta }_x\\ v\left(x,y,z\right)=\left[1+\left(z/R_2\right)\right]v_0\left(x,y\right)-z\frac{\partial w_0}{\partial y}+f\left(z\right)\hspace{0.17em}{\theta }_y\\ w\left(x,y,z\right)=w_0\left(x,y\right)+C_1{f}^{\prime }\left(z\right)\hspace{0.17em}{\theta }_z\end{array}$

(3)

where u, v, w are the displacement of any point on the shell domain in x-, y-, and z-directions, respectively; $u_0\left(x,y\right),v_0\left(x,y\right)\hspace{0.17em}\mathrm{and}{\hspace{0.17em}w}_0\left(x,y\right)$ are the displacements of a point on mid-plane in x-, y-, and z-directions respectively; ${\theta }_x\left(x,y\right),{\theta }_y\left(x,y\right)\hspace{0.17em}\mathrm{and}\hspace{0.17em}{\theta }_{\mathrm{z}}\left(x,y\right)$ are the shear rotations. ‘f (z)’ is the function associated with the transverse shear strain distribution and C₁ is the arbitrary constant account for the consideration of effects of transverse normal strain. When C₁ = 0, ε _z = 0, and when C₁ = 1, ε _z ≠ 0, where ‘ε _z ’ is the transverse normal strain component.

Strain-displacement relationships

The linear relationships between strain quantities and displacement components for the doubly-curved shells are obtained from the theory of elasticity equations. ⁷⁵

$\begin{array}{l}{\epsilon }_x=\frac{\partial u_0}{\partial x}-z\frac{{\partial }^2{w}_0}{\partial x^2}+f\left(z\right)\frac{\partial {\theta }_x}{\partial x}+\frac{w_0}{R_1}+C_1{f}^{\prime }\left(z\right)\frac{{\theta }_z}{R_1}\hspace{0.17em}\\ {\epsilon }_y=\frac{\partial v_0}{\partial y}-z\frac{{\partial }^2{w}_0}{\partial y^2}+f\left(z\right)\frac{\partial {\theta }_y}{\partial y}+\frac{w_0}{R_2}+C_1{f}^{\prime }\left(z\right)\frac{{\theta }_z}{R_2}\hspace{0.5em}\\ {\epsilon }_z=C_1{f}^{″}\left(z\right){\theta }_z\hspace{0.5em}\\ {\gamma }_{xy}=\frac{\partial u_0}{\partial y}+\frac{\partial v_0}{\partial x}-2z\frac{{\partial }^2{w}_0}{\partial x\partial y}+f\left(z\right)\frac{\partial {\theta }_x}{\partial y}+f\left(z\right)\frac{\partial {\theta }_y}{\partial x}\hspace{0.17em}\\ {\gamma }_{xz}=f^{\prime }\left(z\right){\theta }_x+C_1{f}^{\prime }\left(z\right)\frac{\partial {\theta }_z}{\partial x}\hspace{1em}\\ {\gamma }_{yz}=f^{\prime }\left(z\right){\theta }_y+C_1{f}^{\prime }\left(z\right)\frac{\partial {\theta }_z}{\partial y}\hspace{1em}\end{array}$

(4)

where ( ${\epsilon }_x,{\epsilon }_y,{\epsilon }_z,{\gamma }_{xy},{\gamma }_{xz},{\gamma }_{yz})$ are the normal and shear strain components

$f\left(z\right)=\frac{h}{\pi }\left(\sin \frac{\pi z}{h}\right),f^{\prime }\left(z\right)=\cos \left(\frac{\pi z}{h}\right),f^{″}\left(z\right)=-\sin \left(\frac{\pi z}{h}\right)\frac{\pi }{h}$

(5)

Stress-strain relationship

Thermomechanical stress components associated with the non-zero strain components mentioned in equation (4) are obtained using the following thermal stress-strain relationship. ²⁵

$\left\{\begin{array}{l}{\sigma }_x\\ {\sigma }_y\\ {\sigma }_z\\ {\tau }_{xy}\\ {\tau }_{xz}\\ {\tau }_{yz}\end{array}\right\}=\left[\begin{array}{cccccc}{Q}_{11}\left(z\right)& Q_{12}\left(z\right)& Q_{13}\left(z\right)& 0& 0& 0\\ Q_{12}\left(z\right)& Q_{22}\left(z\right)& Q_{23}\left(z\right)& 0& 0& 0\\ Q_{13}\left(z\right)& Q_{23}\left(z\right)& Q_{33}\left(z\right)& 0& 0& 0\\ 0& 0& 0& Q_{66}\left(z\right)& 0& 0\\ 0& 0& 0& 0& Q_{55}\left(z\right)& 0\\ 0& 0& 0& 0& 0& Q_{44}\left(z\right)\end{array}\right]\left\{\begin{array}{c}{\epsilon }_x-\alpha \left(z\right)\mathrm{\Delta }T\\ {\epsilon }_y-\alpha \left(z\right)\mathrm{\Delta }T\\ {\epsilon }_z-\alpha \left(z\right)\mathrm{\Delta }T\\ {\gamma }_{xy}\\ {\gamma }_{xz}\\ {\gamma }_{yz}\end{array}\right\}$

(6)

where $\left({\sigma }_x,{\sigma }_y,{\sigma }_z,{\tau }_{xy},{\tau }_{xz},{\tau }_{yz}\right)$ are the normal and shear stress components; $\alpha \left(z\right)$ is the coefficient of thermal expansion, and $Q_{ij}\left(z\right)$ are the stiffness coefficients given as follows-

for, ε _z = 0,

$\begin{array}{l}{Q}_{11}\left(z\right)=Q_{22}\left(z\right)=\frac{E\left(z\right)}{\left(1-{\mu }^2\right)}\\ Q_{12}\left(z\right)=\frac{E\left(z\right)\mu }{\left(1-{\mu }^2\right)}\\ Q_{44}\left(z\right)=Q_{55}\left(z\right)=Q_{66}\left(z\right)=\frac{E\left(z\right)}{2\left(1+\mu \right)}\end{array}$

(7)

for, ε _z ≠ 0

$\begin{array}{l}{Q}_{11}\left(z\right)=Q_{22}\left(z\right)=Q_{33}\left(z\right)=\frac{E\left(z\right)\left(1-\mu \right)}{\left(1+\mu \right)\left(1-2\mu \right)}\\ Q_{12}\left(z\right)=Q_{13}\left(z\right)=Q_{23}\left(z\right)=\frac{E\left(z\right)\mu }{\left(1+\mu \right)\left(1-2\mu \right)}\\ Q_{44}\left(z\right)=Q_{55}\left(z\right)=Q_{66}\left(z\right)=\frac{E\left(z\right)}{2\left(1+\mu \right)}\end{array}$

(8)

where μ is the Poisson’s ratio, which is taken as constant in the formulation for simplicity. The temperature profile adopted for the thermomechanical analysis is defined as follows ²⁵ :

$\mathrm{\Delta }T\left(x,y,z\right)=T_0\left(x,y\right)+\frac{z}{h}{T}_1\left(x,y\right)+\frac{f\left(z\right)}{h}{T}_2\left(x,y\right)$

(9)

where T₀ represents the constant variation of temperature, T₁ represents the linear variation, and T₂ represents the non-linear variation through the thickness of the shell, as shown in Figure 3.

Figure 3.

Variations of temperature through the thickness of the shell.

In FGM the gradation of materials takes in one of the preferred directions particularly along thickness direction. Therefore, $T_0\left(x,y\right),T_1\left(x,y\right),T_2\left(x,y\right)$ along with the function f(z) are chosen to reflect the realistic boundary condition. These terms ensure that the model aligns with physical constraints like continuity of temperature and heat flux. The proposed temperature distribution $\mathrm{\Delta }T\left(x,y,z\right)$ in the heat conduction equation is typically validated against realistic scenarios to ensure it accurately represents the thermal behavior of the material under given conditions.

Governing equations

The principle of virtual displacement is used in this study to obtain the governing equations associated with the present trigonometric shear and normal deformation theory ⁷⁵

${\int }_0^a{\int }_0^b{\int }_{-h/2}^{+h/2}\left({\sigma }_x\delta {\epsilon }_x+{\sigma }_y\delta {\epsilon }_y+{\sigma }_z\delta {\epsilon }_z+{\tau }_{xy}\delta {\gamma }_{xy}+{\tau }_{xz}\delta {\gamma }_{xz}+{\tau }_{yz}\delta {\gamma }_{yz}\right)dxdydz=\underset{0}{\overset{a}{{\displaystyle \int }}}\underset{0}{\overset{b}{{\displaystyle \int }}}q\left(x,y\right)\delta wdxdy$

(10)

Equation (11) is derived after the substitution of equation (4) through (9) into equation (10).

${\int }_0^a{\int }_0^b\left(N_x\frac{\partial \delta u_0}{\partial x}+N_{xy}\frac{\partial \delta u_0}{\partial y}+N_y\frac{\partial \delta v_0}{\partial y}+N_{xy}\frac{\partial \delta v_0}{\partial x}-M_x^b\frac{{\partial }^2\delta w_0}{\partial x^2}-M_y^b\frac{{\partial }^2\delta w_0}{\partial y^2}-2M_{xy}^b\frac{{\partial }^2\delta w_0}{\partial x\partial y}+M_x^s\frac{\partial \delta {\theta }_x}{\partial x}+M_{xy}^s\frac{\partial \delta {\theta }_x}{\partial y}{+M}_y^s\frac{\partial \delta {\theta }_y}{\partial y}+Q_x^s\delta {\theta }_x+Q_y^s\delta {\theta }_y+S^S\delta {\theta }_z+S_x\delta {\theta }_z+S_y\delta {\theta }_z\right)dxdy=\underset{0}{\overset{a}{{\displaystyle \int }}}\underset{0}{\overset{b}{{\displaystyle \int }}}q\left(x,y\right)\delta wdxdy$

(11)

where stress resultants involved in equation (11) are defined as follows.

$\begin{array}{l}\left(N_x,N_y,N_{xy},M_x^b,M_y^b,M_{xy}^b\right)=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}\left({\sigma }_x,{\sigma }_y,{\tau }_{xy},z{\sigma }_x,z{\sigma }_y,z{\tau }_{xy}\right)dz\hspace{0.17em}\\ \left(M_x^s,M_y^s,M_{xy}^s\right)=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}f\left(z\right)\left({\sigma }_x,{\sigma }_y,{\tau }_{xy}\right)dz\hspace{0.17em}\\ \left({S_{x,}{S}_{y,}Q}_x^S,Q_y^S\right)=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}{f}^{\prime }\left(z\right)\left({{\sigma }_x,{\sigma }_y,\tau }_{xz},{\tau }_{yz}\right)dz\hspace{0.17em}\\ S^S=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}{f}^{″}\left(z\right){\sigma }_{z}dz\hspace{0.17em}\end{array}$

(12)

where $\left(N_x,{\hspace{0.17em}N}_y,{\hspace{0.17em}N}_{xy}\right)$ are the in-plane force resultant, $\left(M_x^b,{\hspace{0.17em}M}_y^b,{\hspace{0.17em}M}_{xy}^b\right)$ are the resultant bending moments, $\left(M_x^s,{\hspace{0.17em}M}_y^s,{\hspace{0.17em}M}_{xy}^s\right)$ are the resultant shear moments, and $\left({S_{x,}{\hspace{0.17em}S}_{y,}\hspace{0.17em}Q}_x^S,{\hspace{0.17em}Q}_y^S,{\hspace{0.17em}S}^S\right)$ are the transverse shear and normal force resultants. By integrating equation (11) by parts and collecting the coefficients of the six unknown variables, one can derive the following six governing differential equations of stress resultants.

$\begin{array}{l}\delta u_0:\frac{\partial N_x}{\partial x}+\frac{\partial N_{xy}}{\partial y}=0\\ \delta v_0:\frac{\partial N_y}{\partial y}+\frac{\partial N_{xy}}{\partial x}=0\\ \delta w_0:\frac{{\partial }^2{M}_x^b}{\partial x^2}+\frac{{\partial }^2{M}_y^b}{\partial y^2}+2\frac{{\partial }^2{M}_{xy}^b}{\partial x\partial y}+q=0\\ \delta {\theta }_x:\frac{\partial M_x^s}{\partial x}+\frac{\partial M_{xy}^s}{\partial y}-Q_x^S=0\\ \delta {\theta }_y:\frac{\partial M_y^s}{\partial y}+\frac{\partial M_{xy}^s}{\partial x}-Q_y^S=0\\ \delta {\theta }_z:\frac{\partial Q_x^S}{\partial x}+\frac{\partial Q_y^S}{\partial y}-S^S=0\end{array}$

(13)

After substituting stress resultant expressions from equation (12) into the six governing equations stated in equation (13), one can derive governing equations in terms of unknown variables $\left(u_0,v_0,{\hspace{0.17em}w}_0,{\theta }_x,{\theta }_y,{\theta }_{\mathrm{z}}\right)$ .

Collecting the terms associated with the unknown $u_0$ leads to equation (14)

$\begin{array}{l}\delta u_0:A_{11}\left(\frac{{\partial }^2{u}_0}{\partial x^2}+\frac{1}{R_1}\frac{\partial w_0}{\partial x}\right)-B_{11}\frac{{\partial }^3{w}_0}{\partial x^3}+C_{11}\frac{{\partial }^2{\theta }_x}{\partial x^2}+A_{12}\left(\frac{{\partial }^2{v}_0}{\partial x\partial y}+\frac{1}{R_2}\frac{\partial w_0}{\partial x}\right)-B_{12}\frac{{\partial }^3{w}_0}{\partial x\partial y^2}+C_{12}\frac{{\partial }^2{\theta }_y}{\partial x\partial y}\\ +\left(\frac{F_{11}}{R_1}+\frac{F_{12}}{R_2}\right)\frac{\partial {\theta }_z}{\partial x}{C}_1+D_{13}\frac{\partial {\theta }_z}{\partial x}{C}_1+A_{66}\left(\frac{{\partial }^2{u}_0}{\partial y^2}+\frac{{\partial }^2{v}_0}{\partial x\partial y}\right)-2B_{66}\frac{{\partial }^3{w}_0}{\partial x\partial y^2}+C_{66}\left(\frac{{\partial }^2{\theta }_x}{\partial y^2}+\frac{{\partial }^2{\theta }_y}{\partial x\partial y}\right)\\ -\left(A_{11}^T+A_{12}^T+A_{13}^T\right)\frac{{\partial T}_0}{\partial x}-\left(\frac{B_{11}^T}{h}+\frac{B_{12}^T}{h}+\frac{B_{13}^T}{h}\right)\frac{{\partial T}_1}{\partial x}-\left(\frac{C_{11}^T}{h}+\frac{C_{12}^T}{h}+\frac{C_{13}^T}{h}\right)\frac{{\partial T}_2}{\partial x}=0\end{array}$

(14)

Collecting the terms associated with the unknown $v_0$ leads to equation (15)

$\begin{array}{l}\delta v_0:A_{12}\left(\frac{{\partial }^2{u}_0}{\partial x\partial y}+\frac{1}{R_1}\frac{\partial w_0}{\partial y}\right)-B_{12}\frac{{\partial }^3{w}_0}{\partial x^2\partial y}+C_{12}\frac{{\partial }^2{\theta }_x}{\partial x\partial y}+A_{22}\left(\frac{{\partial }^2{v}_0}{\partial y^2}+\frac{1}{R_2}\frac{\partial w_0}{\partial y}\right)+\left(\frac{F_{12}}{R_1}+\frac{F_{22}}{R_2}\right)\frac{\partial {\theta }_z}{\partial y}{C}_1\\ -B_{22}\frac{{\partial }^3{w}_0}{\partial y^3}+C_{22}\frac{{\partial }^2{\theta }_y}{\partial y^2}+D_{23}\frac{\partial {\theta }_z}{\partial y}{C}_1+A_{66}\left(\frac{{\partial }^2{u}_0}{\partial x\partial y}+\frac{{\partial }^2{v}_0}{\partial x^2}\right)-2B_{66}\frac{{\partial }^3{w}_0}{\partial x^2\partial y}+C_{66}\left(\frac{{\partial }^2{\theta }_x}{\partial x\partial y}+\frac{{\partial }^2{\theta }_y}{\partial x^2}\right)\\ -\left(A_{12}^T+A_{22}^T+A_{23}^T\right)\frac{{\partial T}_0}{\partial y}-\left(\frac{B_{12}^T}{h}+\frac{B_{22}^T}{h}+\frac{B_{23}^T}{h}\right)\frac{{\partial T}_1}{\partial y}-\left(\frac{C_{12}^T}{h}+\frac{C_{22}^T}{h}+\frac{C_{23}^T}{h}\right)\frac{{\partial T}_2}{\partial y}=0\end{array}$

(15)

Collecting the terms associated with the unknown $w_0$ leads to equation (16)

$\begin{array}{l}\delta w_0:B_{11}\left(\frac{{\partial }^3{u}_0}{\partial x^3}+\frac{1}{R_1}\frac{{\partial }^2{w}_0}{\partial x^2}\right)-H_{11}\frac{{\partial }^4{w}_0}{\partial x^4}+I_{11}\frac{{\partial }^3{\theta }_x}{\partial x^3}+\left(\frac{J_{11}}{R_1}+\frac{J_{12}}{R_2}\right)C_1\frac{{\partial }^2{\theta }_z}{\partial x^2}+B_{12}\left(\frac{{\partial }^3{v}_0}{\partial x^2\partial y}+\frac{1}{R_2}\frac{{\partial }^2{w}_0}{\partial x^2}\right)\\ -H_{12}\frac{{\partial }^4{w}_0}{\partial x^2\partial y^2}+I_{12}\frac{{\partial }^3{\theta }_y}{\partial x^2\partial y}+K_{13}\frac{{\partial }^2{\theta }_z}{\partial x^2}{C}_1+B_{12}\left(\frac{{\partial }^3{u}_0}{\partial x\partial y^2}+\frac{1}{R_1}\frac{{\partial }^2{w}_0}{\partial y^2}\right)-H_{12}\frac{{\partial }^4{w}_0}{\partial x^2\partial y^2}+I_{12}\frac{{\partial }^3{\theta }_x}{\partial x\partial y^2}\\ +\left(\frac{J_{12}}{R_1}+\frac{J_{22}}{R_2}\right)C_1\frac{{\partial }^2{\theta }_z}{\partial y^2}+B_{22}\left(\frac{{\partial }^3{v}_0}{\partial y^3}+\frac{1}{R_2}\frac{{\partial }^2{w}_0}{\partial y^2}\right)-H_{22}\frac{{\partial }^4{w}_0}{\partial y^4}+I_{22}\frac{{\partial }^3{\theta }_y}{\partial y^3}+K_{23}\frac{{\partial }^2{\theta }_z}{\partial y^2}{C}_1\\ +2B_{66}\left(\frac{{\partial }^3{u}_0}{\partial x\partial y^2}+\frac{{\partial }^3{v}_0}{\partial x^2\partial y}\right)-4H_{66}\frac{{\partial }^4{w}_0}{\partial x^2\partial y^2}+2I_{66}\left(\frac{{\partial }^3{\theta }_x}{\partial x\partial y^2}+\frac{{\partial }^3{\theta }_y}{\partial x^2\partial y}\right)-\frac{A_{11}}{R_1}\left(\frac{\partial u_0}{\partial x}+\frac{w_0}{R_1}\right)+\frac{B_{11}}{R_1}\frac{{\partial }^2{w}_0}{\partial x^2}\\ -\frac{C_{11}}{R_1}\frac{\partial {\theta }_x}{\partial x}-\frac{1}{R_1}\left(\frac{F_{11}}{R_1}+\frac{F_{12}}{R_2}\right)C_1{\theta }_z-\frac{A_{12}}{R_1}\left(\frac{\partial v_0}{\partial y}+\frac{w_0}{R_2}\right)+\frac{B_{12}}{R_1}\frac{{\partial }^2{w}_0}{\partial y^2}-\frac{C_{12}}{R_1}\frac{\partial {\theta }_y}{\partial y}-\frac{D_{13}}{R_1}{C}_1{\theta }_z-\frac{A_{12}}{R_2}\left(\frac{\partial u_0}{\partial x}+\frac{w_0}{R_1}\right)\\ +\frac{B_{12}}{R_2}\frac{{\partial }^2{w}_0}{\partial x^2}-\frac{C_{12}}{R_2}\frac{\partial {\theta }_x}{\partial x}-\frac{1}{R_2}\left(\frac{F_{12}}{R_1}+\frac{F_{22}}{R_2}\right)C_1{\theta }_z-\frac{A_{22}}{R_2}\left(\frac{\partial v_0}{\partial y}+\frac{w_0}{R_2}\right)+\frac{B_{22}}{R_2}\frac{{\partial }^2{w}_0}{\partial y^2}-\frac{C_{22}}{R_2}\frac{\partial {\theta }_y}{\partial y}-\frac{D_{23}}{R_2}{C}_1{\theta }_z\\ -\left(B_{11}^T+B_{12}^T+B_{13}^T\right)\frac{{\partial }^2{T}_0}{{\partial x}^2}+\left(A_{11}^T+A_{12}^T+A_{13}^T\right)\frac{T_0}{R_1}-\left(\frac{H_{11}^T}{h}+\frac{H_{12}^T}{h}+\frac{H_{13}^T}{h}\right)\frac{{\partial }^2{T}_1}{{\partial x}^2}\\ +\left(B_{11}^T+B_{12}^T+B_{13}^T\right)\frac{T_1}{{hR}_1}-\left(B_{12}^T+B_{22}^T+B_{23}^T\right)\frac{{\partial }^2{T}_0}{{\partial y}^2}+\left(A_{12}^T+A_{22}^T+A_{23}^T\right)\frac{T_0}{R_2}\\ +\left(B_{12}^T+B_{22}^T+B_{23}^T\right)\frac{T_1}{{hR}_2}-\left(\frac{H_{12}^T}{h}+\frac{H_{22}^T}{h}+\frac{H_{23}^T}{h}\right)\frac{{\partial }^2{T}_1}{{\partial y}^2}-\left(\frac{I_{11}^T}{h}+\frac{I_{12}^T}{h}+\frac{I_{13}^T}{h}\right)\frac{{\partial }^2{T}_2}{{\partial x}^2}\\ +\left(C_{11}^T+C_{12}^T+C_{13}^T\right)\frac{T_2}{{hR}_1}-\left(\frac{I_{12}^T}{h}+\frac{I_{22}^T}{h}+\frac{I_{23}^T}{h}\right)\frac{{\partial }^2{T}_2}{{\partial Y}^2}+\left(C_{12}^T+C_{22}^T+C_{23}^T\right)\frac{T_2}{{hR}_2}=-q\end{array}$

(16)

Collecting the terms associated with the unknown ${\theta }_x$ leads to equation (17)

$\begin{array}{l}\delta {\theta }_x:C_{11}\left(\frac{{\partial }^2{u}_0}{\partial x^2}+\frac{1}{R_1}\frac{\partial w_0}{\partial x}\right)-I_{11}\frac{{\partial }^3{w}_0}{\partial x^3}+L_{11}\frac{{\partial }^2{\theta }_x}{\partial x^2}+C_{12}\left(\frac{{\partial }^2{v}_0}{\partial x\partial y}+\frac{1}{R_2}\frac{\partial w_0}{\partial x}\right)-I_{12}\frac{{\partial }^3{w}_0}{\partial x\partial y^2}\\ +\left(\frac{M_{11}}{R_1}+\frac{M_{12}}{R_2}\right)\frac{\partial {\theta }_z}{\partial x}{C}_1+L_{12}\frac{{\partial }^2{\theta }_y}{\partial x\partial y}+N_{13}\frac{\partial {\theta }_z}{\partial x}{C}_1+C_{66}\left(\frac{{\partial }^2{u}_0}{\partial y^2}+\frac{{\partial }^2{v}_0}{\partial x\partial y}\right)-2I_{66}\frac{{\partial }^3{w}_0}{\partial x\partial y^2}+L_{66}\left(\frac{{\partial }^2{\theta }_x}{\partial y^2}+\frac{{\partial }^2{\theta }_y}{\partial x\partial y}\right)\\ -O_{55}{\theta }_x-O_{55}\frac{\partial {\theta }_z}{\partial x}{C}_1-\left(C_{11}^T+C_{12}^T+C_{13}^T\right)\frac{{\partial T}_0}{\partial x}-\left(\frac{I_{11}^T}{h}+\frac{I_{12}^T}{h}+\frac{I_{13}^T}{h}\right)\frac{{\partial T}_1}{\partial x}-\left(\frac{L_{11}^T}{h}+\frac{L_{12}^T}{h}+\frac{L_{13}^T}{h}\right)\frac{{\partial T}_2}{\partial x}=0\end{array}$

(17)

Collecting the terms associated with the unknown ${\theta }_y$ leads to equation (18)

$\begin{array}{l}\delta {\theta }_y:C_{12}\left(\frac{{\partial }^2{u}_0}{\partial x\partial y}+\frac{1}{R_1}\frac{\partial w_0}{\partial y}\right)-I_{12}\frac{{\partial }^3{w}_0}{\partial x^2\partial y}+L_{12}\frac{{\partial }^2{\theta }_x}{\partial x\partial y}+C_{22}\left(\frac{{\partial }^2{v}_0}{\partial y^2}+\frac{1}{R_2}\frac{\partial w_0}{\partial y}\right)-I_{22}\frac{{\partial }^3{w}_0}{\partial y^3}\\ +\left(\frac{M_{12}}{R_1}+\frac{M_{22}}{R_2}\right)\frac{\partial {\theta }_z}{\partial y}{C}_1+L_{22}\frac{{\partial }^2{\theta }_y}{\partial y^2}+N_{23}\frac{\partial {\theta }_z}{\partial y}{C}_1+C_{66}\left(\frac{{\partial }^2{u}_0}{\partial x\partial y}+\frac{{\partial }^2{v}_0}{\partial x^2}\right)-2I_{66}\frac{{\partial }^3{w}_0}{\partial x^2\partial y}+L_{66}\left(\frac{{\partial }^2{\theta }_x}{\partial x\partial y}+\frac{{\partial }^2{\theta }_y}{\partial x^2}\right)\\ -O_{44}{\theta }_y-O_{44}\frac{\partial {\theta }_z}{\partial y}{C}_1-\left(C_{12}^T+C_{22}^T+C_{23}^T\right)\frac{{\partial T}_0}{\partial y}-\left(\frac{I_{12}^T}{h}+\frac{I_{22}^T}{h}+\frac{I_{23}^T}{h}\right)\frac{{\partial T}_1}{\partial y}-\left(\frac{L_{12}^T}{h}+\frac{L_{22}^T}{h}+\frac{L_{23}^T}{h}\right)\frac{{\partial T}_2}{\partial y}=0\end{array}$

(18)

Collecting the terms associated with the unknown ${\theta }_z$ leads to equation (19)

$\begin{array}{l}\delta {\theta }_z:O_{55}\left(\frac{\partial {\theta }_x}{\partial x}{C}_1+\frac{{\partial }^2{\theta }_z}{\partial x^2}{C}_1^2\right)+O_{44}\left(\frac{\partial {\theta }_y}{\partial y}{C}_1+\frac{{\partial }^2{\theta }_z}{\partial y^2}{C}_1^2\right)-\frac{F_{11}}{R_1}{C}_1\left(\frac{\partial u_0}{\partial x}+\frac{w_0}{R_1}\right)+\frac{J_{11}}{R_1}{C}_1\frac{{\partial }^2{w}_0}{\partial x^2}\hspace{0.17em}\\ -\frac{M_{11}}{R_1}\frac{\partial {\theta }_x}{\partial x}{C}_1-\frac{F_{12}}{R_1}{C}_1\left(\frac{\partial v_0}{\partial y}+\frac{w_0}{R_2}\right)+\frac{J_{12}}{R_1}{C}_1\frac{{\partial }^2{w}_0}{\partial y^2}-\frac{1}{R_1}\left(\frac{O_{11}}{R_1}+\frac{O_{12}}{R_2}\right)C_1^2{\theta }_z-\frac{M_{12}}{R_1}\frac{\partial {\theta }_y}{\partial y}{C}_1\\ -\frac{P_{13}}{R_1}{C}_1^2{\theta }_z-\frac{F_{12}}{R_2}{C}_1\left(\frac{\partial u_0}{\partial x}+\frac{w_0}{R_1}\right)+\frac{J_{12}}{R_2}{C}_1\frac{{\partial }^2{w}_0}{\partial x^2}-\frac{M_{12}}{R_2}\frac{\partial {\theta }_x}{\partial x}{C}_1-\frac{F_{22}}{R_2}{C}_1\left(\frac{\partial v_0}{\partial y}+\frac{w_0}{R_2}\right)+\frac{J_{22}}{R_2}{C}_1\frac{{\partial }^2{w}_0}{\partial y^2}\\ -\frac{1}{R_2}\left(\frac{O_{12}}{R_1}+\frac{O_{22}}{R_2}\right)C_1^2{\theta }_z-\frac{M_{22}}{R_2}\frac{\partial {\theta }_y}{\partial y}{C}_1-\frac{P_{23}}{R_2}{C}_1^2{\theta }_z-D_{13}{C}_1\left(\frac{\partial u_0}{\partial x}+\frac{w_0}{R_1}\right)+K_{13}{C}_1\frac{{\partial }^2{w}_0}{\partial x^2}-N_{13}\frac{\partial {\theta }_x}{\partial x}{C}_1\\ -D_{23}{C}_1\left(\frac{\partial v_0}{\partial y}+\frac{w_0}{R_2}\right)+K_{23}{C}_1\frac{{\partial }^2{w}_0}{\partial y^2}-\left(\frac{P_{13}}{R_1}+\frac{P_{23}}{R_2}\right)C_1^2{\theta }_z-N_{23}\frac{\partial {\theta }_y}{\partial y}{C}_1-S_{33}{C}_1^2{\theta }_z\\ -\left(F_{11}^T+F_{12}^T+F_{13}^T\right)\frac{T_0{C}_1}{R_1}-\left(J_{11}^T+J_{12}^T+J_{13}^T\right)\frac{T_1{C}_1}{{hR}_1}-\left(F_{12}^T+F_{22}^T+F_{23}^T\right)\frac{T_0{C}_1}{R_2}\\ -\left(J_{12}^T+J_{22}^T+J_{23}^T\right)\frac{T_1{C}_1}{{hR}_2}-\left(D_{13}^T+D_{23}^T+D_{33}^T\right)T_0{C}_1-\left(K_{13}^T+K_{23}^T+K_{33}^T\right)\frac{T_1{C}_1}{h}\\ -\left(M_{11}^T+M_{12}^T+M_{13}^T\right)\frac{T_2{C}_1}{h{R}_1}-\left(M_{12}^T+M_{22}^T+M_{23}^T\right)\frac{T_2{C}_1}{h{R}_2}-\left(N_{13}^T+N_{23}^T+N_{33}^T\right)\frac{T_2{C}_1}{h}=0\end{array}$

(19)

The mechanical and thermal stiffness coefficients of the governing differential equations are defined as follows.

$\begin{array}{l}\left(A_{ij},B_{ij},C_{ij},D_{ij},F_{ij}\right)=Q_{ij}\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}\left[1,z,f\left(z\right),f^{″}\left(z\right),f^{\prime }\left(z\right)\right]dz\hspace{0.17em}\\ \left(H_{ij},I_{ij},J_{ij},K_{ij},L_{ij}\right)=Q_{ij}\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}\left[z^2,f\left(z\right)z,f^{\prime }\left(z\right)z,f^{″}\left(z\right)z,f{\left(z\right)}^2\right]dz\hspace{0.17em}\\ \left(M_{ij},N_{ij},O_{ij},P_{ij},S_{ij}\right)=Q_{ij}\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}\left[f^{\prime }\left(z\right)f\left(z\right),f^{″}\left(z\right)f\left(z\right),f^{\prime }{\left(z\right)}^2,f^{″}\left(z\right)f^{\prime }\left(z\right),f^{″}{\left(z\right)}^2\right]dz\end{array}$

(20)

and

$\begin{array}{l}\left(A_{ij}^T,B_{ij}^T,C_{ij}^T,D_{ij}^T,F_{ij}^T\right)=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}{Q}_{ij}\left[\alpha \left(z\right),z\alpha \left(z\right),f\left(z\right)\alpha \left(z\right),f^{″}\left(z\right)\alpha \left(z\right),f^{\prime }\left(z\right)\alpha \left(z\right)\right]dz\\ \left(H_{ij}^T,I_{ij}^T,J_{ij}^T,K_{ij}^T\right)=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}{Q}_{ij}\left[z^2\alpha \left(z\right),f\left(z\right)z\alpha \left(z\right),f^{\prime }\left(z\right)z\alpha \left(z\right),f^{″}\left(z\right)z\alpha \left(z\right)\right]dz\\ \left(L_{ij}^T,M_{ij}^T{,N}_{ij}^T\right)=\underset{-h/2}{\overset{+h/2}{{\displaystyle \int }}}{Q}_{ij}\left[{f{\left(z\right)}^2\alpha \left(z\right),f}^{\prime }\left(z\right)f\left(z\right)\alpha \left(z\right),f^{″}\left(z\right)f\left(z\right)\alpha \left(z\right)\right]dz\end{array}$

(21)

The general boundary conditions of the present theory are of the following form.

$\begin{array}{l}\mathbf{At}\hspace{0.17em}\mathbf{the}\hspace{0.17em}\mathbf{edge},\mathit{x}=\mathbf{0}\hspace{0.17em}\mathbf{and}\hspace{0.17em}\mathit{x}=\mathit{a}\\ \mathrm{Either}\hspace{0.17em}{N}_{x\hspace{0.17em}}=0\hspace{0.17em}\mathrm{or}\hspace{0.17em}{u}_{0\hspace{0.17em}}\mathrm{is}\hspace{0.17em}\mathrm{prescribed}\\ \begin{array}{l}\mathrm{Either}\hspace{0.17em}{N}_{xy\hspace{0.17em}}=0\hspace{0.17em}\mathrm{or}\hspace{0.17em}{v}_{0\hspace{0.17em}}\mathrm{is}\hspace{0.17em}\mathrm{prescribed}\\ \mathrm{Either}\hspace{0.17em}{M}_x^b=0\hspace{0.17em}\mathrm{or}\hspace{0.17em}{w}_{0\hspace{0.17em}}\mathrm{is}\hspace{0.17em}\mathrm{prescribed}\\ \begin{array}{l}\mathrm{Either}\hspace{0.17em}{M}_{xy}^b=0\hspace{0.17em}\mathrm{or}\hspace{0.17em}{\partial w}_0/\partial x\hspace{0.17em}\mathrm{is}\hspace{0.17em}\mathrm{prescribed}\\ \mathrm{Either}\hspace{0.17em}{M}_x^S=0\hspace{0.17em}\mathrm{or}\hspace{0.17em}{\theta }_{x\hspace{0.17em}}\mathrm{is}\hspace{0.17em}\mathrm{prescribed}\\ \begin{array}{l}\mathrm{Either}\hspace{0.17em}{M}_{xy}^S=0\hspace{0.17em}\mathrm{or}\hspace{0.17em}{\theta }_{y\hspace{0.17em}}\mathrm{is}\hspace{0.17em}\mathrm{prescribed}\\ \mathrm{Either}\hspace{0.17em}{Q}_{xz\hspace{0.17em}}=0\hspace{0.17em}\mathrm{or}\hspace{0.17em}{\theta }_{z\hspace{0.17em}}\mathrm{is}\hspace{0.17em}\mathrm{prescribed}\end{array}\end{array}\end{array}\end{array}$

(22)

$\begin{array}{l}\mathbf{At}\hspace{0.17em}\mathbf{the}\hspace{0.17em}\mathbf{edge},\mathit{y}=\mathbf{0}\hspace{0.17em}\mathbf{and}\hspace{0.17em}\mathit{y}=\mathit{b}\\ \mathrm{Either}\hspace{0.17em}{N}_{y\hspace{0.17em}}=0\hspace{0.17em}\mathrm{or}\hspace{0.17em}{v}_{0\hspace{0.17em}}\mathrm{is}\hspace{0.17em}\mathrm{prescribed}\\ \begin{array}{l}\mathrm{Either}\hspace{0.17em}{N}_{xy\hspace{0.17em}}=0\hspace{0.17em}\mathrm{or}\hspace{0.17em}{u}_{0\hspace{0.17em}}\mathrm{is}\hspace{0.17em}\mathrm{prescribed}\\ \mathrm{Either}\hspace{0.17em}{M}_y^b=0\hspace{0.17em}\mathrm{or}\hspace{0.17em}{w}_{0\hspace{0.17em}}\mathrm{is}\hspace{0.17em}\mathrm{prescribed}\\ \begin{array}{l}\mathrm{Either}\hspace{0.17em}{M}_{xy}^b=0\hspace{0.17em}\mathrm{or}\hspace{0.17em}{\partial w}_0/\partial y\hspace{0.17em}\mathrm{is}\hspace{0.17em}\mathrm{prescribed}\\ \mathrm{Either}\hspace{0.17em}{M}_y^S=0\hspace{0.17em}\mathrm{or}\hspace{0.17em}{\theta }_{y\hspace{0.17em}}\mathrm{is}\hspace{0.17em}\mathrm{prescribed}\\ \begin{array}{l}\mathrm{Either}\hspace{0.17em}{M}_{xy}^S=0\hspace{0.17em}\mathrm{or}\hspace{0.17em}{\theta }_{x\hspace{0.17em}}\mathrm{is}\hspace{0.17em}\mathrm{prescribed}\\ \mathrm{Either}\hspace{0.17em}{Q}_{yz\hspace{0.17em}}=0\hspace{0.17em}\mathrm{or}\hspace{0.17em}{\theta }_{z\hspace{0.17em}}\mathrm{is}\hspace{0.17em}\mathrm{prescribed}\end{array}\end{array}\end{array}\end{array}$

(23)

The navier solution

The Navier solution technique is employed in the present study to obtain the closed-form solution for the static analysis of FGM shells with simply-supported boundary conditions. ⁸¹ The Navier solution technique has some limitations, such as it is strictly applicable only to simply supported plates/shells; this technique is not directly applicable to plates/shells with clamped, free, or mixed boundary conditions without modifications; to use this technique for the analysis of composite structures, the geometry of the structures must be either rectangular or on a rectangular planform. Therefore, it is applied to the shell’s simply-supported boundary conditions. Some of the typical examples of the simply-supported shells in engineering problems are reinforced concrete cylindrical roof shell resting on beams, curved bottom slab of box girder resting on bearings, cylindrical shell panel between stiffeners, cylindrical shell between two rigid rings, etc. After this validation, the present work can be extended in the future for the study of restrained boundary conditions of the plate/shell. Some of the typical examples of the simply-supported shells in engineering problems are shell roof monolithically cast with edge beam and column, shell-type pier head integrated with columns, fuselage or rocket stage welded to bulkhead, end welded to thick end plate, etc.

A semi-analytical solution for the thermomechanical problem of simply-supported FGM shells is obtained using the Navier solution technique. The following are the simply supported boundary conditions associated with the present theory, which are exactly satisfied using the Navier solution technique. ⁷⁵

$\mathbf{At}\hspace{0.17em}\mathbf{the}\hspace{0.17em}\mathbf{edge}\hspace{0.17em}\mathit{x}=\mathbf{0}\hspace{0.17em}\mathbf{and}\hspace{0.17em}\mathit{x}=\mathit{a}$

$v_0=w_0=M_x^b=M_x^S=N_x={\theta }_y={\theta }_z=0$

(24)

$\mathbf{At}\hspace{0.17em}\mathit{y}=\mathbf{0}\hspace{0.17em}\mathbf{and}\hspace{0.17em}\mathit{y}=\mathit{b}$

$u_0=w_0=M_y^b=M_y^S=N_y={\theta }_x={\theta }_z=0$

(25)

According to Navier’s solution technique, the unknown variables $\left(u_0,{\hspace{0.17em}v}_{0,}\hspace{0.17em}{w}_0,{\hspace{0.17em}\theta }_x,{\hspace{0.17em}\theta }_y,{\hspace{0.17em}\theta }_z\right)$ involved in the six governing equations are assumed in the form of double trigonometric series to satisfy the boundary conditions stated in equations (26) and (27) exactly.

$\begin{array}{l}{u}_0={\displaystyle \sum }_{m=1}^{\infty }{\displaystyle \sum }_{n=1}^{\infty }{u}_{mn}\hspace{0.17em}\cos \hspace{0.17em}\alpha x\hspace{0.17em}\sin \hspace{0.17em}\beta y,{\theta }_x={\displaystyle \sum }_{m=1}^{\infty }{\displaystyle \sum }_{n=1}^{\infty }{\theta }_{xmn}\hspace{0.17em}\cos \hspace{0.17em}\alpha x\hspace{0.17em}\sin \hspace{0.17em}\beta y,\\ v_0={\displaystyle \sum }_{m=1}^{\infty }{\displaystyle \sum }_{n=1}^{\infty }{v}_{mn}\hspace{0.17em}\sin \hspace{0.17em}\alpha x\hspace{0.17em}\cos \hspace{0.17em}\beta y,{\theta }_y={\displaystyle \sum }_{m=1}^{\infty }{\displaystyle \sum }_{n=1}^{\infty }{\theta }_{ymn}\hspace{0.17em}\sin \hspace{0.17em}\alpha x\hspace{0.17em}\cos \hspace{0.17em}\beta y,\\ w_0={\displaystyle \sum }_{m=1}^{\infty }{\displaystyle \sum }_{n=1}^{\infty }{w}_{mn}\hspace{0.17em}\sin \hspace{0.17em}\alpha x\hspace{0.17em}\sin \hspace{0.17em}\beta y,{\theta }_z={\displaystyle \sum }_{m=1}^{\infty }{\displaystyle \sum }_{n=1}^{\infty }{\theta }_{zmn}\hspace{0.17em}\sin \hspace{0.17em}\alpha x\hspace{0.17em}\sin \hspace{0.17em}\beta y\end{array}$

(26)

where α = mπ/a, β = nπ/b; m and n are the positive odd integers varying from 1 to ∞; $\left(u_{mn},{\hspace{0.17em}v}_{mn},w_{mn},{\hspace{0.17em}\theta }_{xmn},{\hspace{0.17em}\theta }_{ymn},{\hspace{0.17em}\theta }_{zmn}\right)$ are the unknown coefficients to be determined. The expressions for the transverse mechanical and thermal loads acting on the top surface of the shells are also expressed in the double trigonometric series form as follows.

$\begin{array}{l}q={\displaystyle \sum }_{m=1}^{\infty }{\displaystyle \sum }_{n=1}^{\infty }{q}_{mn}\hspace{0.17em}\sin \hspace{0.17em}\alpha x\hspace{0.17em}\sin \hspace{0.17em}\beta y\\ \left(T_{0,}{T}_1,T_2\right)={\displaystyle \sum }_{m=1}^{\infty }{\displaystyle \sum }_{n=1}^{\infty }\left(T_{0mn},T_{1mn},T_{2mn}\right)\sin \hspace{0.17em}\alpha x\hspace{0.17em}\sin \hspace{0.17em}\beta y\end{array}$

(27)

where $\left(q_{mn},{\hspace{0.17em}T}_{0mn},{\hspace{0.17em}T}_{1mn},T_{2mn}\right)$ are the Fourier coefficients for the mechanical and thermal loads, respectively. For sinusoidal load, $\left(q_{mn}=q_0,{\hspace{0.17em}T}_{0mn}=t_0,{\hspace{0.17em}T}_{1mn}=t_1,T_{2mn}=t_2\right)$ , where $\left(q_0,{\hspace{0.17em}t}_0,{\hspace{0.17em}t}_1,{\hspace{0.17em}t}_2\right)$ are the maximum intensities of the mechanical and thermal loads with m = 1 and n = 1. Substituting equations (26) and (27) into governing equations (14)–(19) leads to the six simultaneous equations, which can be written in the following matrix form.

$\left[K\right]\left\{\Delta \right\}=\left\{F\right\}$

(28)

where $\left[K\right],\left\{\mathrm{\Delta }\right\},\left\{F\right\}$ are the stiffness matrix, vector of unknowns, and force vector, respectively; the elements of $\left[K\right],\left\{\mathrm{\Delta }\right\},\left\{F\right\}$ are mentioned in the Appendix.

Numerical results and discussions

To illustrate the accuracy and efficiency of the present theory, the FG shell made up of titanium (top surface metal-rich) and zirconia (bottom surface ceramic-rich) is considered for the analysis. The following numerical example is solved in the present study to predict the thermomechanical response of FG shells using the present theory.

Example: In this example, thermomechanical bending analysis of FGM shells is performed, and the corresponding numerical results are presented in Tables 2–13. The numerical analysis and discussions are carried out on four types of shells: spherical, cylindrical, hyperbolic, and elliptical. These shells are analysed using non-linear thermomechanical loading.

The various transverse shear strain functions used in the present study are as follows.

(1) Trigonometric shear deformation theory (TSDT) of Levy ⁸³ and Touratier ⁸⁴

$f\hspace{0.17em}\left(z\right)=\frac{h}{\pi }\left(\sin \frac{\pi z}{h}\right),f^{\prime }\left(z\right)=\cos \left(\frac{\pi z}{h}\right)$

• (2) Parabolic shear deformation theory (PSDT) of Reddy^46,47 and Krishna Murty ⁴⁸

$f\hspace{0.17em}\left(z\right)=z\left[1-\frac{4z^2}{3h^2}\right],f^{\prime }\left(z\right)=\left[1-4\frac{z^2}{h^2}\right]$

• (3) First-order shear deformation theory (FSDT) by Mindlin ⁸

$f\hspace{0.17em}\left(z\right)=z,f^{\prime }\left(z\right)=1$

The typical mechanical properties of the metal and ceramic used in the FGM shells are shown in Table 1.

Table 1.

Material properties used in the FGM shells.

Properties	References	Metal: (Titanium)	Ceramic: (Zirconia)
Modulus of Elasticity (E)	69	66.2 GPa	117.0 GPa
Poisson’s ratio (µ)	69	0.3	0.3
Coefficient of thermal expansion ( ${\alpha }_0$ )	69	10.3 (10−6/K)	7.11 (10−6/K)

The following non-dimensional forms present the numerical values of displacements and stresses for the thermomechanical analysis of shells. The value of E₀ = 1.0 in equation (29).

$\begin{array}{l}\overline{w}\left(\frac{a}{2},\frac{b}{2},0\right)=\frac{{10}^3\times w}{\left(q_0{a}^4/E_0{h}^3\right)+\left({{10}^3\alpha }_0{T}_{1mn}{a}^2/h\right)},{\overline{\sigma }}_x\left(\frac{a}{2},\frac{b}{2},-\frac{h}{2}\right)=\frac{10\times {\sigma }_x}{\left(q_0{a}^2/h^2\right)+\left(10E_0{\alpha }_0{T}_{1mn}{a}^2/h^2\right)}\\ {\overline{\tau }}_{xz}\left(0,\frac{b}{2},0\right)=\frac{{\tau }_{xz}}{\left(q_{0}a/h\right)+\left(E_0{\alpha }_0{T}_{1mn}a/10h\right)},{\overline{\sigma }}_z\left(\frac{a}{2},\frac{b}{2},-\frac{h}{2}\right)=\frac{10\times {\sigma }_x}{\left(q_0{a}^2/h^2\right)+\left(10E_0{\alpha }_0{T}_{1mn}{a}^2/h^2\right)}\end{array}$

(29)

Discussion on the Numerical Results

The results of displacement and stress obtained by Navier’s method depend on the number of terms taken in the trigonometric series expansion. However, the number of terms for sinusoidal loading conditions are one i.e. m = 1 and n = 1. Therefore does not required convergence.

Tables 2–5 show the results of the non-dimensional transverse displacements for FGM shells subjected to a combination of mechanical load and non-linear thermal (thermomechanical) load. Tables consider the effects of the power-law index, radii of curvature, and transverse normal strain on the non-dimensional transverse displacement for FGM shallow shells. Tables reveal an increase in the transverse displacements with an increase in the power-law indices, irrespective of the inclusion of transverse normal strain. In the case of spherical, cylindrical, and elliptical shells, the transverse displacement increases with an increase in the values of the radii of curvature. In contrast, the hyperbolic shell yields the same value of transverse displacement irrespective of an increase in the radii of curvature attributed to the sagging and hogging nature of the shell.

Table 2.

Effects of power-law indices and radii of curvature on the non-dimensional transverse displacement $\left(\overline{w}\right)$ of the FG spherical shell (R₁ = R₂ = R) subjected to non-linear thermomechanical load (a = 10h, a = b, q₀ = 100, t₀ = 0, t₁ = 100, t₂ = 100).

				R/a
p	Theory	Reference	ε z	1	5	10	50	100
0	TSNDT	Present	εz ≠ 0	0.197175	0.770825	0.844365	0.870908	0.871764
	TSDT	Present	εz = 0	0.223873	0.783342	0.849699	0.873374	0.874135
	HSDT	Krishna Murty 48	εz ≠ 0	0.199681	0.779548	0.854160	0.881107	0.881976
	PSDT	Reddy46,47	εz = 0	0.227056	0.794524	0.861835	0.885850	0.886622
	FSDT	Mindlin 8	εz = 0	0.253336	0.881242	0.955229	0.981601	0.982449
1	TSNDT	Present	εz ≠ 0	0.232043	0.945049	1.033285	1.062300	1.062828
	TSDT	Present	εz = 0	0.284303	0.959026	1.035225	1.059024	1.059249
	HSDT	Krishna Murty 48	εz ≠ 0	0.235322	0.957338	1.046967	1.076419	1.076950
	PSDT	Reddy46,47	εz = 0	0.288331	0.972639	1.049923	1.074059	1.074287
	FSDT	Mindlin 8	εz = 0	0.321289	1.077841	1.162714	1.189181	1.189429
2	TSNDT	Present	εz ≠ 0	0.250646	0.984581	1.071226	1.098312	1.098588
	TSDT	Present	εz = 0	0.306694	1.000293	1.075241	1.097448	1.097456
	HSDT	Krishna Murty 48	εz ≠ 0	0.255243	1.000254	1.088414	1.115906	1.116172
	PSDT	Reddy46,47	εz = 0	0.311063	1.014576	1.090599	1.113125	1.113133
	FSDT	Mindlin 8	εz = 0	0.347035	1.125244	1.208738	1.233468	1.233481
4	TSNDT	Present	εz ≠ 0	0.266654	1.012932	1.098878	1.125715	1.125993
	TSDT	Present	εz = 0	0.322402	1.032521	1.107648	1.130120	1.130184
	HSDT	Krishna Murty 48	εz ≠ 0	0.271861	1.031166	1.118836	1.146172	1.146447
	PSDT	Reddy46,47	εz = 0	0.327041	1.047349	1.123550	1.146343	1.146407
	FSDT	Mindlin 8	εz = 0	0.365638	1.162682	1.246278	1.271246	1.271315
8	TSNDT	Present	εz ≠ 0	0.276468	1.038710	1.127465	1.156291	1.156779
	TSDT	Present	εz = 0	0.328902	1.061628	1.140241	1.165054	1.165371
	HSDT	Krishna Murty 48	εz ≠ 0	0.281447	1.057946	1.148688	1.178185	1.178688
	PSDT	Reddy46,47	εz = 0	0.333655	1.076927	1.156666	1.181829	1.182149
	FSDT	Mindlin 8	εz = 0	0.373582	1.196360	1.283796	1.311291	1.311630
10	TSNDT	Present	εz ≠ 0	0.278496	1.047206	1.137467	1.167212	1.167784
	TSDT	Present	εz = 0	0.329559	1.070668	1.150911	1.176703	1.177113
	HSDT	Krishna Murty 48	εz ≠ 0	0.283318	1.066375	1.158683	1.189144	1.189736
	PSDT	Reddy46,47	εz = 0	0.334318	1.086097	1.167491	1.193646	1.194061
	FSDT	Mindlin 8	εz = 0	0.374370	1.206689	1.295955	1.324532	1.324972

Table 3.

Effects of power-law indices and radii of curvature on the non-dimensional transverse displacement $\left(\overline{w}\right)$ of the FG cylindrical shell (R₁ = R and R₂ = ∞) subjected to non-linear thermomechanical load (a = 10h, a = b, q₀ = 100, t₀ = 0, t₁ = 100, t₂ = 100).

				R/a
p	Theory	Reference	ε z	1	5	10	50	100
0	TSNDT	Present	εz ≠ 0	0.477032	0.844366	0.864962	0.871764	0.871978
	TSDT	Present	εz = 0	0.506471	0.849699	0.868083	0.874135	0.874326
	HSDT	Krishna Murty 48	εz ≠ 0	0.482092	0.854161	0.875070	0.881976	0.882193
	PSDT	Reddy46,47	εz = 0	0.513687	0.861835	0.880483	0.886622	0.886816
	FSDT	Mindlin 8	εz = 0	0.571425	0.955229	0.975709	0.982449	0.982661
1	TSNDT	Present	εz ≠ 0	0.579541	1.033287	1.056491	1.062828	1.062827
	TSDT	Present	εz = 0	0.627630	1.035225	1.054597	1.059249	1.059125
	HSDT	Krishna Murty 48	εz ≠ 0	0.586630	1.046968	1.070531	1.076950	1.076945
	PSDT	Reddy46,47	εz = 0	0.636529	1.049923	1.069570	1.074287	1.074162
	FSDT	Mindlin 8	εz = 0	0.707344	1.162714	1.184264	1.189429	1.189289
2	TSNDT	Present	εz ≠ 0	0.614134	1.071228	1.093255	1.098588	1.098460
	TSDT	Present	εz = 0	0.664745	1.075241	1.093654	1.097456	1.097223
	HSDT	Krishna Murty 48	εz ≠ 0	0.623690	1.088415	1.110795	1.116172	1.116034
	PSDT	Reddy46,47	εz = 0	0.674227	1.090599	1.109276	1.113133	1.112897
	FSDT	Mindlin 8	εz = 0	0.750012	1.208738	1.229239	1.233481	1.233223
4	TSNDT	Present	εz ≠ 0	0.640876	1.098879	1.120697	1.125993	1.125869
	TSDT	Present	εz = 0	0.692743	1.107648	1.126192	1.130184	1.129981
	HSDT	Krishna Murty 48	εz ≠ 0	0.652047	1.118837	1.141075	1.146447	1.146316
	PSDT	Reddy46,47	εz = 0	0.702699	1.123550	1.142359	1.146407	1.146201
	FSDT	Mindlin 8	εz = 0	0.782822	1.246278	1.266886	1.271315	1.271088
8	TSNDT	Present	εz ≠ 0	0.659078	1.127466	1.150581	1.156779	1.156756
	TSDT	Present	εz = 0	0.710751	1.140241	1.160309	1.165371	1.165290
	HSDT	Krishna Murty 48	εz ≠ 0	0.670502	1.148689	1.172337	1.178688	1.178667
	PSDT	Reddy46,47	εz = 0	0.721002	1.156666	1.177019	1.182149	1.182067
	FSDT	Mindlin 8	εz = 0	0.803999	1.283796	1.306056	1.311630	1.311533
10	TSNDT	Present	εz ≠ 0	0.663707	1.137469	1.161205	1.167784	1.167800
	TSDT	Present	εz = 0	0.714964	1.150911	1.171637	1.177113	1.177077
	HSDT	Krishna Murty 48	εz ≠ 0	0.674962	1.158684	1.182982	1.189736	1.189757
	PSDT	Reddy46,47	εz = 0	0.725271	1.167491	1.188510	1.194061	1.194023
	FSDT	Mindlin 8	εz = 0	0.808840	1.295955	1.318944	1.324972	1.324924

Table 4.

Effects of power-law indices and radii of curvature on the non-dimensional transverse displacement $\left(\overline{w}\right)$ of the FG hyperbolic shell (R₁ = R and R₂ = -R) subjected to non-linear thermomechanical load (a = 10h, a = b, q₀ = 100, t₀ = 0, t₁ = 100, t₂ = 100).

				R/a
p	Theory	Reference	ε z	1	5	10	50	100
0	TSNDT	Present	εz ≠ 0	0.872156	0.872054	0.872051	0.872050	0.872050
	TSDT	Present	εz = 0	0.874389	0.874389	0.874389	0.874389	0.874389
	HSDT	Krishna Murty 48	εz ≠ 0	0.882360	0.882270	0.882267	0.882266	0.882266
	PSDT	Reddy46,47	εz = 0	0.886880	0.886880	0.886880	0.886880	0.886880
	FSDT	Mindlin 8	εz = 0	0.982732	0.982732	0.982732	0.982732	0.982732
1	TSNDT	Present	εz ≠ 0	1.062775	1.062653	1.062650	1.062648	1.062648
	TSDT	Present	εz = 0	1.058844	1.058844	1.058844	1.058844	1.058844
	HSDT	Krishna Murty 48	εz ≠ 0	1.076872	1.076765	1.076761	1.076760	1.076760
	PSDT	Reddy46,47	εz = 0	1.073876	1.073876	1.073876	1.073876	1.073876
	FSDT	Mindlin 8	εz = 0	1.188974	1.188974	1.188974	1.188974	1.188974
2	TSNDT	Present	εz ≠ 0	1.098286	1.098161	1.098157	1.098156	1.098156
	TSDT	Present	εz = 0	1.096831	1.096831	1.096831	1.096831	1.096831
	HSDT	Krishna Murty 48	εz ≠ 0	1.115830	1.115720	1.115717	1.115716	1.115716
	PSDT	Reddy46,47	εz = 0	1.112500	1.112500	1.112500	1.112500	1.112500
	FSDT	Mindlin 8	εz = 0	1.232789	1.232789	1.232789	1.232789	1.232789
4	TSNDT	Present	εz ≠ 0	1.125705	1.125576	1.125572	1.125570	1.125570
	TSDT	Present	εz = 0	1.129620	1.129620	1.129620	1.129620	1.129620
	HSDT	Krishna Murty 48	εz ≠ 0	1.146123	1.146011	1.146007	1.146006	1.146006
	PSDT	Reddy46,47	εz = 0	1.145835	1.145835	1.145835	1.145835	1.145835
	FSDT	Mindlin 8	εz = 0	1.270687	1.270687	1.270687	1.270687	1.270687
8	TSNDT	Present	εz ≠ 0	1.156697	1.156561	1.156556	1.156555	1.156555
	TSDT	Present	εz = 0	1.165050	1.165050	1.165050	1.165050	1.165050
	HSDT	Krishna Murty 48	εz ≠ 0	1.178587	1.178469	1.178465	1.178464	1.178464
	PSDT	Reddy46,47	εz = 0	1.181823	1.181823	1.181823	1.181823	1.181823
	FSDT	Mindlin 8	εz = 0	1.311259	1.311259	1.311259	1.311259	1.311259
10	TSNDT	Present	εz ≠ 0	1.167781	1.167642	1.167637	1.167636	1.167636
	TSDT	Present	εz = 0	1.176879	1.176879	1.176879	1.176879	1.176879
	HSDT	Krishna Murty 48	εz ≠ 0	1.189718	1.189598	1.189594	1.189593	1.189593
	PSDT	Reddy46,47	εz = 0	1.193822	1.193822	1.193822	1.193822	1.193822
	FSDT	Mindlin 8	εz = 0	1.324695	1.324695	1.324695	1.324695	1.324695

Table 5.

Effects of power-law indices and radii of curvature on the non-dimensional transverse displacement $\left(\overline{w}\right)$ of the FG elliptical shell (R₁ = R and R₂ = 1.5 R) subjected to non-linear thermomechanical load (a = 10h, a = b, q₀ = 100, t₀ = 0, t₁ = 100, t₂ = 100).

				R/a
p	Theory	Reference	ε z	1	5	10	50	100
0	TSNDT	Present	εz ≠ 0	0.260694	0.799205	0.852638	0.871256	0.871851
	TSDT	Present	εz = 0	0.289737	0.809084	0.857094	0.873684	0.874213
	HSDT	Krishna Murty 48	εz ≠ 0	0.263701	0.808334	0.862559	0.881461	0.882064
	PSDT	Reddy46,47	εz = 0	0.293858	0.820636	0.869336	0.886165	0.886701
	FSDT	Mindlin 8	εz = 0	0.327641	0.909956	0.963468	0.981946	0.982535
1	TSNDT	Present	εz ≠ 0	0.310180	0.979541	1.042817	1.062555	1.062848
	TSDT	Present	εz = 0	0.364706	0.989045	1.043277	1.059169	1.059225
	HSDT	Krishna Murty 48	εz ≠ 0	0.314211	0.992372	1.056648	1.076676	1.076968
	PSDT	Reddy46,47	εz = 0	0.369874	1.003086	1.058089	1.074206	1.074263
	FSDT	Mindlin 8	εz = 0	0.411897	1.111294	1.171673	1.189342	1.189402
2	TSNDT	Present	εz ≠ 0	0.333299	1.018700	1.080382	1.098483	1.098565
	TSDT	Present	εz = 0	0.391466	1.030024	1.082990	1.097521	1.097396
	HSDT	Krishna Murty 48	εz ≠ 0	0.338926	1.034973	1.097722	1.116074	1.116146
	PSDT	Reddy46,47	εz = 0	0.397045	1.044733	1.098459	1.113199	1.113072
	FSDT	Mindlin 8	εz = 0	0.442679	1.158378	1.217365	1.233551	1.233414
4	TSNDT	Present	εz ≠ 0	0.352674	1.046812	1.107945	1.125886	1.125971
	TSDT	Present	εz = 0	0.410811	1.062330	1.115430	1.130211	1.130134
	HSDT	Krishna Murty 48	εz ≠ 0	0.359136	1.065727	1.128082	1.146343	1.146423
	PSDT	Reddy46,47	εz = 0	0.416720	1.077584	1.131444	1.146435	1.146356
	FSDT	Mindlin 8	εz = 0	0.465522	1.195869	1.254929	1.271347	1.271259
8	TSNDT	Present	εz ≠ 0	0.364704	1.073533	1.136980	1.156533	1.156791
	TSDT	Present	εz = 0	0.420050	1.092647	1.148552	1.165230	1.165362
	HSDT	Krishna Murty 48	εz ≠ 0	0.371016	1.093542	1.158421	1.178433	1.178701
	PSDT	Reddy46,47	εz = 0	0.426116	1.108392	1.165096	1.182007	1.182140
	FSDT	Mindlin 8	εz = 0	0.476640	1.230889	1.293021	1.311483	1.311617
10	TSNDT	Present	εz ≠ 0	0.367288	1.082550	1.147205	1.167482	1.167809
	TSDT	Present	εz = 0	0.421414	1.102263	1.159456	1.176911	1.177119
	HSDT	Krishna Murty 48	εz ≠ 0	0.373430	1.102512	1.168647	1.189423	1.189764
	PSDT	Reddy46,47	εz = 0	0.427495	1.118146	1.176158	1.193857	1.194067
	FSDT	Mindlin 8	εz = 0	0.478234	1.241866	1.305442	1.324758	1.324976

Tables 6–9 show the effects of the power-law index and radii of curvature on the non-dimensional in-plane normal stresses for FGM shallow shells under thermomechanical loading with and without considering the transverse normal strain. The maximum value of in-plane normal stresses is observed in the spherical shells, whereas the minimum values are observed in the hyperbolic shells. It also reveals that the in-plane normal stress decreases with an increase in the value of the power-law index in all four shells.

Table 6.

Effects of power-law indices and radii of curvature on the non-dimensional in-plane normal stress $\left({\overline{\sigma }}_x\right)$ of the FG spherical shell (R₁ = R₂ = R) subjected to non-linear thermomechanical load (a = 10h, a = b, q₀ = 100, t₀ = 0, t₁ = 100, t₂ = 100).

				R/a
p	Theory	Reference	ε z	1	5	10	50	100
0	TSNDT	Present	εz ≠ 0	−6.581010	−2.576800	−2.432850	−2.646160	−2.694790
	TSDT	Present	εz = 0	−6.066133	−2.240764	−2.125959	−2.307948	−2.348664
	HSDT	Krishna Murty 48	εz ≠ 0	−6.706280	−2.599690	−2.444100	−2.656130	−2.705190
	PSDT	Reddy46,47	εz = 0	−6.028363	−2.141928	−2.024691	−2.208996	−2.250285
	FSDT	Mindlin 8	εz = 0	−5.740004	−1.377474	−1.244437	−1.447593	−1.493309
1	TSNDT	Present	εz ≠ 0	−4.939120	−0.357650	−0.223320	−0.498830	−0.558310
	TSDT	Present	εz = 0	−5.214850	−1.118565	−1.069906	−1.345979	−1.400757
	HSDT	Krishna Murty 48	εz ≠ 0	−5.110000	−0.394160	−0.247600	−0.524150	−0.584510
	PSDT	Reddy46,47	εz = 0	−5.166313	−1.004345	−0.954133	−1.233867	−1.289422
	FSDT	Mindlin 8	εz = 0	−4.792660	−0.123100	−0.064257	−0.373092	−0.434612
2	TSNDT	Present	εz ≠ 0	−4.209040	0.381339	0.472665	0.163501	0.100161
	TSDT	Present	εz = 0	−4.853734	−0.765535	−0.754612	−1.060668	−1.118977
	HSDT	Krishna Murty 48	εz ≠ 0	−4.434770	0.315826	0.419249	0.107256	0.042710
	PSDT	Reddy46,47	εz = 0	−4.801058	−0.645944	−0.633924	−0.944068	−1.003209
	FSDT	Mindlin 8	εz = 0	−4.394475	0.278442	0.295380	−0.047136	−0.112649
4	TSNDT	Present	εz ≠ 0	−3.434260	1.216199	1.283693	0.955112	0.889413
	TSDT	Present	εz = 0	−4.609730	−0.406506	−0.405785	−0.722125	−0.781791
	HSDT	Krishna Murty 48	εz ≠ 0	−3.731740	1.111351	1.193554	0.862798	0.795879
	PSDT	Reddy46,47	εz = 0	−4.554119	−0.281132	−0.279423	−0.600022	−0.660546
	FSDT	Mindlin 8	εz = 0	−4.122170	0.692098	0.697394	0.342800	0.275674
8	TSNDT	Present	εz ≠ 0	−2.719680	2.132671	2.211854	1.879744	1.812621
	TSDT	Present	εz = 0	−4.524935	−0.036298	−0.010985	−0.320469	−0.380203
	HSDT	Krishna Murty 48	εz ≠ 0	−3.080470	1.992672	2.091128	1.759415	1.691305
	PSDT	Reddy46,47	εz = 0	−4.467836	0.095082	0.121757	−0.191911	−0.252508
	FSDT	Mindlin 8	εz = 0	−4.022261	1.119735	1.152328	0.804760	0.737450
10	TSNDT	Present	εz ≠ 0	−2.518910	2.421930	2.510560	2.179375	2.111858
	TSDT	Present	εz = 0	−4.521489	0.079817	0.117759	−0.187663	−0.247332
	HSDT	Krishna Murty 48	εz ≠ 0	−2.890300	2.275997	2.385161	2.055212	1.986796
	PSDT	Reddy46,47	εz = 0	−4.464239	0.212918	0.252412	−0.057132	−0.117664
	FSDT	Mindlin 8	εz = 0	−4.017169	1.252548	1.299341	0.956233	0.888977

Table 7.

Effects of power-law indices and radii of curvature on the non-dimensional in-plane normal stress $\left({\overline{\sigma }}_x\right)$ of the FG cylindrical shell (R₁ = R and R₂ = ∞) subjected to non-linear thermomechanical load (a = 10h, a = b, q₀ = 100, t₀ = 0, t₁ = 100, t₂ = 100).

				R/a
p	Theory	Reference	ε z	1	5	10	50	100
0	TSNDT	Present	εz ≠ 0	−4.260260	−2.421720	−2.523910	−2.693690	−2.720340
	TSDT	Present	εz = 0	−3.797181	−2.125959	−2.209568	−2.348664	−2.370479
	HSDT	Krishna Murty 48	εz ≠ 0	−4.336340	−2.432730	−2.533270	−2.704070	−2.731000
	PSDT	Reddy46,47	εz = 0	−3.723823	−2.024691	−2.109275	−2.250285	−2.272410
	FSDT	Mindlin 8	εz = 0	−3.150134	−1.244437	−1.337338	−1.493309	−1.517816
1	TSNDT	Present	εz ≠ 0	−2.275090	−0.210210	−0.346870	−0.557010	−0.589410
	TSDT	Present	εz = 0	−2.723228	−1.069906	−1.208297	−1.400757	−1.429770
	HSDT	Krishna Murty 48	εz ≠ 0	−2.379000	−0.234260	−0.370440	−0.583190	−0.616100
	PSDT	Reddy46,47	εz = 0	−2.635539	−0.954133	−1.094275	−1.289422	−1.318849
	FSDT	Mindlin 8	εz = 0	−1.950744	−0.064257	−0.218677	−0.434612	−0.467208
2	TSNDT	Present	εz ≠ 0	−1.480200	0.486023	0.327708	0.101491	0.067183
	TSDT	Present	εz = 0	−2.313997	−0.754612	−0.912050	−1.118977	−1.149731
	HSDT	Krishna Murty 48	εz ≠ 0	−1.622940	0.432732	0.274008	0.044052	0.009065
	PSDT	Reddy46,47	εz = 0	−2.220685	−0.633924	−0.793377	−1.003209	−1.034405
	FSDT	Mindlin 8	εz = 0	−1.490027	0.295380	0.119609	−0.112649	−0.147218
4	TSNDT	Present	εz ≠ 0	−0.622260	1.297368	1.126649	0.890774	0.855276
	TSDT	Present	εz = 0	−1.968751	−0.405785	−0.569525	−0.781791	−0.813230
	HSDT	Krishna Murty 48	εz ≠ 0	−0.818980	1.207267	1.036751	0.797242	0.761055
	PSDT	Reddy46,47	εz = 0	−1.870283	−0.279423	−0.445278	−0.660546	−0.692439
	FSDT	Mindlin 8	εz = 0	−1.094750	0.697394	0.514252	0.275674	0.240290
8	TSNDT	Present	εz ≠ 0	0.216110	2.226101	2.054468	1.814034	1.777711
	TSDT	Present	εz = 0	−1.721867	−0.010985	−0.168930	−0.380203	−0.411755
	HSDT	Krishna Murty 48	εz ≠ 0	−0.031100	2.105358	1.935704	1.692718	1.655816
	PSDT	Reddy46,47	εz = 0	−1.619368	0.121757	−0.038230	−0.252508	−0.284520
	FSDT	Mindlin 8	εz = 0	−0.807557	1.152328	0.975313	0.737450	0.701883
10	TSNDT	Present	εz ≠ 0	0.462372	2.525024	2.354686	2.113292	2.076717
	TSDT	Present	εz = 0	−1.660906	0.117759	−0.036921	−0.247332	−0.278890
	HSDT	Krishna Murty 48	εz ≠ 0	0.206102	2.399604	2.231799	1.988228	1.951117
	PSDT	Reddy46,47	εz = 0	−1.557489	0.252412	0.095740	−0.117664	−0.149681
	FSDT	Mindlin 8	εz = 0	−0.737256	1.299341	1.125941	0.888977	0.853393

Table 8.

Effects of power-law indices and radii of curvature on the non-dimensional in-plane normal stress $\left({\overline{\sigma }}_x\right)$ of the FG hyperbolic shell (R₁ = R and R₂ = -R) subjected to non-linear thermomechanical load (a = 10h, a = b, q₀ = 100, t₀ = 0, t₁ = 100, t₂ = 100).

				R/a
p	Theory	Reference	ε z	1	5	10	50	100
0	TSNDT	Present	εz ≠ 0	−2.638690	−2.726250	−2.737210	−2.745980	−2.747080
	TSDT	Present	εz = 0	−2.393253	−2.393253	−2.393253	−2.393253	−2.393253
	HSDT	Krishna Murty 48	εz ≠ 0	−2.647270	−2.736750	−2.747940	−2.756900	−2.758020
	PSDT	Reddy46,47	εz = 0	−2.295508	−2.295508	−2.295508	−2.295508	−2.295508
	FSDT	Mindlin 8	εz = 0	−1.543408	−1.543408	−1.543408	−1.543408	−1.543408
1	TSNDT	Present	εz ≠ 0	−0.492990	−0.597070	−0.610100	−0.620530	−0.621830
	TSDT	Present	εz = 0	−1.459848	−1.459848	−1.459848	−1.459848	−1.459848
	HSDT	Krishna Murty 48	εz ≠ 0	−0.517830	−0.623850	−0.637120	−0.647730	−0.649060
	PSDT	Reddy46,47	εz = 0	−1.349359	−1.349359	−1.349359	−1.349359	−1.349359
	FSDT	Mindlin 8	εz = 0	−0.501011	−0.501011	−0.501011	−0.501011	−0.501011
2	TSNDT	Present	εz ≠ 0	0.164491	0.058194	0.044888	0.03424	0.032909
	TSDT	Present	εz = 0	−1.181534	−1.181534	−1.181534	−1.181534	−1.181534
	HSDT	Krishna Murty 48	εz ≠ 0	0.106960	−0.000400	−0.013830	−0.024580	−0.025920
	PSDT	Reddy46,47	εz = 0	−1.066667	−1.066667	−1.066667	−1.066667	−1.066667
	FSDT	Mindlin 8	εz = 0	−0.182977	−0.182977	−0.182977	−0.182977	−0.182977
4	TSNDT	Present	εz ≠ 0	0.954422	0.845708	0.832099	0.821208	0.819846
	TSDT	Present	εz = 0	−0.845723	−0.845723	−0.845723	−0.845723	−0.845723
	HSDT	Krishna Murty 48	εz ≠ 0	0.859922	0.750822	0.737174	0.726254	0.724889
	PSDT	Reddy46,47	εz = 0	−0.725404	−0.725404	−0.725404	−0.725404	−0.725404
	FSDT	Mindlin 8	εz = 0	0.203709	0.203709	0.203709	0.203709	0.203709
8	TSNDT	Present	εz ≠ 0	1.881263	1.768321	1.754180	1.742865	1.741451
	TSDT	Present	εz = 0	−0.444415	−0.444415	−0.444415	−0.444415	−0.444415
	HSDT	Krishna Murty 48	εz ≠ 0	1.758679	1.645762	1.631636	1.620334	1.618921
	PSDT	Reddy46,47	εz = 0	−0.317658	−0.317658	−0.317658	−0.317658	−0.317658
	FSDT	Mindlin 8	εz = 0	0.665059	0.665059	0.665059	0.665059	0.665059
10	TSNDT	Present	εz ≠ 0	2.182001	2.067449	2.053107	2.041630	2.040196
	TSDT	Present	εz = 0	−0.311581	−0.311581	−0.311581	−0.311581	−0.311581
	HSDT	Krishna Murty 48	εz ≠ 0	2.055708	1.941216	1.926893	1.915433	1.914001
	PSDT	Reddy46,47	εz = 0	−0.182850	−0.182850	−0.182850	−0.182850	−0.182850
	FSDT	Mindlin 8	εz = 0	0.816523	0.816523	0.816523	0.816523	0.816523

Table 9.

Effects of power-law indices and radii of curvature on the non-dimensional in-plane normal stress $\left({\overline{\sigma }}_x\right)$ of the FG elliptical shell (R₁ = R and R₂ = 1.5 R) subjected to non-linear thermomechanical load (a = 10h, a = b, q₀ = 100, t₀ = 0, t₁ = 100, t₂ = 100).

				R/a
p	Theory	Reference	ε z	1	5	10	50	100
0	TSNDT	Present	εz ≠ 0	−6.011690	−2.480300	−2.449700	−2.661470	−2.703170
	TSDT	Present	εz = 0	−5.480250	−2.164355	−2.142709	−2.321086	−2.355829
	HSDT	Krishna Murty 48	εz ≠ 0	−6.126580	−2.498240	−2.459970	−2.671560	−2.713650
	PSDT	Reddy46,47	εz = 0	−5.433386	−2.064120	−2.041591	−2.222318	−2.257551
	FSDT	Mindlin 8	εz = 0	−5.070483	−1.289996	−1.262833	−1.462341	−1.501357
1	TSNDT	Present	εz ≠ 0	−4.290670	−0.255430	−0.248900	−0.517620	−0.568530
	TSDT	Present	εz = 0	−4.563760	−1.057831	−1.103344	−1.363754	−1.410309
	HSDT	Krishna Murty 48	εz ≠ 0	−4.446990	−0.286390	−0.272490	−0.543210	−0.594890
	PSDT	Reddy46,47	εz = 0	−4.505107	−0.942408	−0.987956	−1.251893	−1.299110
	FSDT	Mindlin 8	εz = 0	−4.049298	−0.053187	−0.101390	−0.393050	−0.445342
2	TSNDT	Present	εz ≠ 0	−3.535450	0.469850	0.4398170	0.143420	0.089313
	TSDT	Present	εz = 0	−4.180908	−0.716960	−0.794474	−1.079627	−1.129111
	HSDT	Krishna Murty 48	εz ≠ 0	−3.742330	0.410067	0.386828	0.086798	0.031645
	PSDT	Reddy46,47	εz = 0	−4.117598	−0.596302	−0.674258	−0.963296	−1.013488
	FSDT	Mindlin 8	εz = 0	−3.624153	0.334914	0.251062	−0.068432	−0.124039
4	TSNDT	Present	εz ≠ 0	−2.730510	1.297220	1.246789	0.934254	0.878179
	TSDT	Present	εz = 0	−3.903652	−0.360816	−0.447693	−0.741534	−0.792153
	HSDT	Krishna Murty 48	εz ≠ 0	−3.004730	1.199144	1.157309	0.841557	0.784421
	PSDT	Reddy46,47	εz = 0	−3.836718	−0.234396	−0.321833	−0.619709	−0.671057
	FSDT	Mindlin 8	εz = 0	−3.311568	0.745331	0.650694	0.320969	0.264013
8	TSNDT	Present	εz ≠ 0	−1.982190	2.219389	2.175393	1.858454	1.801135
	TSDT	Present	εz = 0	−3.777024	0.019271	−0.050424	−0.339877	−0.390596
	HSDT	Krishna Murty 48	εz ≠ 0	−2.316960	2.087787	2.055948	1.737821	1.679633
	PSDT	Reddy46,47	εz = 0	−3.707955	0.151851	0.081849	−0.211598	−0.263053
	FSDT	Mindlin 8	εz = 0	−3.163118	1.184023	1.108290	0.782894	0.725735
10	TSNDT	Present	εz ≠ 0	−1.770660	2.512662	2.474895	2.157974	2.100298
	TSDT	Present	εz = 0	−3.760130	0.140354	0.079666	−0.207038	−0.257725
	HSDT	Krishna Murty 48	εz ≠ 0	−2.115800	2.375555	2.350969	2.033537	1.975063
	PSDT	Reddy46,47	εz = 0	−3.690739	0.274730	0.213871	−0.076786	−0.128207
	FSDT	Mindlin 8	εz = 0	−3.142802	1.322438	1.256802	0.934398	0.877259

Tables 10–13 show the numerical values of non-dimensional transverse shear stresses for an FGM shallow shell with and without considering the effects of transverse normal strain. It reveals maximum transverse shear stresses in spherical shells and minimum in cylindrical shells. It also shows a decrease in transverse shear stresses with an increase in the values of power-law indices in all four types of shells. It is also observed that the transverse shear stress increases with an increase in the value of the radius of curvature.

Table 10.

Effects of power-law indices and radii of curvature on the non-dimensional transverse shear stress $\left({\overline{\tau }}_{xz}\right)$ of the FG spherical shell (R₁ = R₂ = R) subjected to non-linear thermomechanical load (a = 10h, a = b, q₀ = 100, t₀ = 0, t₁ = 100, t₂ = −100).

				R/a
p	Theory	Reference	ε z	1	5	10	50	100
0	TSNDT	Present	εz ≠ 0	−0.694920	0.207306	0.322913	0.364640	0.365986
	TSDT	Present	εz = 0	−0.679802	0.165435	0.265687	0.301455	0.302605
	HSDT	Krishna Murty 48	εz ≠ 0	−0.562590	0.195163	0.292624	0.327821	0.328956
	PSDT	Reddy46,47	εz = 0	−0.578533	0.163859	0.251918	0.283336	0.284346
	FSDT	Mindlin 8	εz = 0	0.040987	0.142576	0.154546	0.158813	0.158950
1	TSNDT	Present	εz ≠ 0	−0.658420	0.205479	0.315571	0.353743	0.354770
	TSDT	Present	εz = 0	−0.607084	0.178815	0.268626	0.297420	0.297826
	HSDT	Krishna Murty 48	εz ≠ 0	−0.536060	0.192935	0.286653	0.319486	0.320421
	PSDT	Reddy46,47	εz = 0	−0.515833	0.175577	0.254707	0.280154	0.280526
	FSDT	Mindlin 8	εz = 0	0.040238	0.142208	0.154438	0.158808	0.158949
2	TSNDT	Present	εz ≠ 0	−0.622660	0.201149	0.302522	0.336866	0.337683
	TSDT	Present	εz = 0	−0.569761	0.177927	0.260403	0.286048	0.286285
	HSDT	Krishna Murty 48	εz ≠ 0	−0.507600	0.188856	0.275308	0.304910	0.305665
	PSDT	Reddy46,47	εz = 0	−0.483074	0.174340	0.247002	0.269696	0.269924
	FSDT	Mindlin 8	εz = 0	0.039164	0.135443	0.146714	0.150728	0.150857
4	TSNDT	Present	εz ≠ 0	−0.590300	0.199102	0.293175	0.324571	0.325252
	TSDT	Present	εz = 0	−0.546287	0.175689	0.253323	0.277440	0.277674
	HSDT	Krishna Murty 48	εz ≠ 0	−0.481560	0.186826	0.267125	0.294194	0.294824
	PSDT	Reddy46,47	εz = 0	−0.462240	0.172080	0.240412	0.261726	0.261948
	FSDT	Mindlin 8	εz = 0	0.038598	0.130172	0.140596	0.144294	0.144412
8	TSNDT	Present	εz ≠ 0	−0.580780	0.202775	0.295385	0.326266	0.326924
	TSDT	Present	εz = 0	−0.548339	0.176682	0.254783	0.279653	0.280009
	HSDT	Krishna Murty 48	εz ≠ 0	−0.474420	0.190562	0.269740	0.296370	0.296975
	PSDT	Reddy46,47	εz = 0	−0.464305	0.173555	0.242350	0.264312	0.264636
	FSDT	Mindlin 8	εz = 0	0.039671	0.132601	0.143075	0.146785	0.146904
10	TSNDT	Present	εz ≠ 0	−0.580000	0.204131	0.297098	0.328219	0.328898
	TSDT	Present	εz = 0	−0.550831	0.177036	0.255919	0.281323	0.281736
	HSDT	Krishna Murty 48	εz ≠ 0	−0.473860	0.191986	0.271491	0.298316	0.298936
	PSDT	Reddy46,47	εz = 0	−0.466702	0.174099	0.243615	0.266048	0.266421
	FSDT	Mindlin 8	εz = 0	0.040110	0.134201	0.144817	0.148578	0.148699

Table 11.

Effects of power-law indices and radii of curvature on the non-dimensional transverse shear stress $\left({\overline{\tau }}_{xz}\right)$ of the FG cylindrical shell (R₁ = R and R₂ = ∞) subjected to non-linear thermomechanical load (a = 10h, a = b, q₀ = 100, t₀ = 0, t₁ = 100, t₂ = −100).

				R/a
p	Theory	Reference	ε z	1	5	10	50	100
0	TSNDT	Present	εz ≠ 0	−0.254650	0.322913	0.355293	0.365986	0.366323
	TSDT	Present	εz = 0	−0.252857	0.265687	0.293462	0.302605	0.302893
	HSDT	Krishna Murty 48	εz ≠ 0	−0.193460	0.292624	0.319935	0.328956	0.329240
	PSDT	Reddy46,47	εz = 0	−0.203547	0.251918	0.276315	0.284346	0.284599
	FSDT	Mindlin 8	εz = 0	0.092451	0.154546	0.157860	0.158950	0.158985
1	TSNDT	Present	εz ≠ 0	−0.240280	0.315667	0.345597	0.354779	0.354965
	TSDT	Present	εz = 0	−0.208225	0.268626	0.291844	0.297826	0.297754
	HSDT	Krishna Murty 48	εz ≠ 0	−0.184270	0.286739	0.312394	0.320430	0.320620
	PSDT	Reddy46,47	εz = 0	−0.165048	0.254707	0.275204	0.280526	0.280470
	FSDT	Mindlin 8	εz = 0	0.091491	0.154438	0.157831	0.158949	0.158984
2	TSNDT	Present	εz ≠ 0	−0.218570	0.302615	0.329721	0.337692	0.337798
	TSDT	Present	εz = 0	−0.185464	0.260403	0.281290	0.286285	0.286148
	HSDT	Krishna Murty 48	εz ≠ 0	−0.166970	0.275391	0.298671	0.305673	0.305793
	PSDT	Reddy46,47	εz = 0	−0.145323	0.247002	0.265456	0.269924	0.269814
	FSDT	Mindlin 8	εz = 0	0.088078	0.146714	0.149831	0.150857	0.150889
4	TSNDT	Present	εz ≠ 0	−0.197460	0.293261	0.318151	0.325260	0.325318
	TSDT	Present	εz = 0	−0.171030	0.253323	0.272949	0.277674	0.277553
	HSDT	Krishna Murty 48	εz ≠ 0	−0.149840	0.267201	0.288587	0.294832	0.294908
	PSDT	Reddy46,47	εz = 0	−0.132668	0.240412	0.257731	0.261948	0.261850
	FSDT	Mindlin 8	εz = 0	0.085697	0.140596	0.143468	0.144412	0.144442
8	TSNDT	Present	εz ≠ 0	−0.188670	0.295464	0.319963	0.326932	0.326982
	TSDT	Present	εz = 0	−0.170824	0.254783	0.274834	0.280009	0.279950
	HSDT	Krishna Murty 48	εz ≠ 0	−0.142530	0.269810	0.290873	0.296982	0.297047
	PSDT	Reddy46,47	εz = 0	−0.132263	0.242350	0.260041	0.264636	0.264591
	FSDT	Mindlin 8	εz = 0	0.087674	0.143075	0.145957	0.146904	0.146934
10	TSNDT	Present	εz ≠ 0	−0.187780	0.297172	0.321837	0.328905	0.328965
	TSDT	Present	εz = 0	−0.172386	0.255919	0.276319	0.281736	0.281705
	HSDT	Krishna Murty 48	εz ≠ 0	−0.141670	0.271557	0.292759	0.298942	0.299014
	PSDT	Reddy46,47	εz = 0	−0.133601	0.243615	0.261617	0.266421	0.266398
	FSDT	Mindlin 8	εz = 0	0.088689	0.144817	0.147739	0.148699	0.148729

Table 12.

Effects of power-law indices and radii of curvature on the non-dimensional transverse shear stress $\left({\overline{\tau }}_{xz}\right)$ of the FG hyperbolic shell (R₁ = R and R₂ = -R) subjected to non-linear thermomechanical load (a = 10h, a = b, q₀ = 100, t₀ = 0, t₁ = 100, t₂ = −100).

				R/a
p	Theory	Reference	ε z	1	5	10	50	100
0	TSNDT	Present	εz ≠ 0	0.366433	0.366435	0.366435	0.366435	0.366435
	TSDT	Present	εz = 0	0.302989	0.302989	0.302989	0.302989	0.302989
	HSDT	Krishna Murty 48	εz ≠ 0	0.329335	0.329335	0.329335	0.329335	0.329335
	PSDT	Reddy46,47	εz = 0	0.284683	0.284683	0.284683	0.284683	0.284683
	FSDT	Mindlin 8	εz = 0	0.158996	0.158996	0.158996	0.158996	0.158996
1	TSNDT	Present	εz ≠ 0	0.355887	0.355127	0.355032	0.354956	0.354946
	TSDT	Present	εz = 0	0.297498	0.297498	0.297498	0.297498	0.297498
	HSDT	Krishna Murty 48	εz ≠ 0	0.321484	0.320800	0.320715	0.320646	0.320638
	PSDT	Reddy46,47	εz = 0	0.280253	0.280253	0.280253	0.280253	0.280253
	FSDT	Mindlin 8	εz = 0	0.158996	0.158996	0.158996	0.158996	0.158996
2	TSNDT	Present	εz ≠ 0	0.338628	0.337891	0.337798	0.337724	0.337715
	TSDT	Present	εz = 0	0.285841	0.285841	0.285841	0.285841	0.285841
	HSDT	Krishna Murty 48	εz ≠ 0	0.306563	0.305908	0.305826	0.305761	0.305753
	PSDT	Reddy46,47	εz = 0	0.269554	0.269554	0.269554	0.269554	0.269554
	FSDT	Mindlin 8	εz = 0	0.150900	0.150900	0.150900	0.150900	0.150900
4	TSNDT	Present	εz ≠ 0	0.326044	0.325362	0.325277	0.325208	0.325200
	TSDT	Present	εz = 0	0.277272	0.277272	0.277272	0.277272	0.277272
	HSDT	Krishna Murty 48	εz ≠ 0	0.295580	0.294977	0.294901	0.294841	0.294833
	PSDT	Reddy46,47	εz = 0	0.261612	0.261612	0.261612	0.261612	0.261612
	FSDT	Mindlin 8	εz = 0	0.144452	0.144452	0.144452	0.144452	0.144452
8	TSNDT	Present	εz ≠ 0	0.327624	0.327004	0.326927	0.326865	0.326857
	TSDT	Present	εz = 0	0.279734	0.279734	0.279734	0.279734	0.279734
	HSDT	Krishna Murty 48	εz ≠ 0	0.297649	0.297095	0.297026	0.29697	0.296963
	PSDT	Reddy46,47	εz = 0	0.264406	0.264406	0.264406	0.264406	0.264406
	FSDT	Mindlin 8	εz = 0	0.146944	0.146944	0.146944	0.146944	0.146944
10	TSNDT	Present	εz ≠ 0	0.329568	0.328986	0.328913	0.328855	0.328848
	TSDT	Present	εz = 0	0.281516	0.281516	0.281516	0.281516	0.281516
	HSDT	Krishna Murty 48	εz ≠ 0	0.299583	0.299059	0.298994	0.298942	0.298935
	PSDT	Reddy46,47	εz = 0	0.266236	0.266236	0.266236	0.266236	0.266236
	FSDT	Mindlin 8	εz = 0	0.148739	0.148739	0.148739	0.148739	0.148739

Table 13.

Effects of power-law indices and radii of curvature on the non-dimensional transverse shear stress $\left({\overline{\tau }}_{xz}\right)$ of the FG elliptical shell (R₁ = R and R₂ = 1.5 R) subjected to non-linear thermomechanical load (a = 10h, a = b, q₀ = 100, t₀ = 0, t₁ = 100, t₂ = −100).

				R/a
p	Theory	Reference	ε z	1	5	10	50	100
0	TSNDT	Present	εz ≠ 0	−0.594910	0.251920	0.335920	0.365188	0.366123
	TSDT	Present	εz = 0	−0.580296	0.204327	0.276860	0.301923	0.302722
	HSDT	Krishna Murty 48	εz ≠ 0	−0.478850	0.232764	0.303593	0.328283	0.329072
	PSDT	Reddy46,47	εz = 0	−0.491139	0.198020	0.261732	0.283747	0.284449
	FSDT	Mindlin 8	εz = 0	0.053009	0.147222	0.155879	0.158869	0.158964
1	TSNDT	Present	εz ≠ 0	−0.564990	0.248261	0.327743	0.354184	0.354859
	TSDT	Present	εz = 0	−0.513848	0.214091	0.278215	0.297637	0.297822
	HSDT	Krishna Murty 48	εz ≠ 0	−0.457640	0.229301	0.297062	0.319881	0.320508
	PSDT	Reddy46,47	εz = 0	−0.433855	0.206646	0.263165	0.280350	0.280526
	FSDT	Mindlin 8	εz = 0	0.052137	0.146949	0.155803	0.158866	0.158963
2	TSNDT	Present	εz ≠ 0	−0.531560	0.240713	0.313604	0.337229	0.337744
	TSDT	Present	εz = 0	−0.479054	0.210471	0.269090	0.286202	0.286258
	HSDT	Krishna Murty 48	εz ≠ 0	−0.430970	0.222547	0.284803	0.305239	0.305726
	PSDT	Reddy46,47	εz = 0	−0.403378	0.202997	0.254669	0.269839	0.269904
	FSDT	Mindlin 8	εz = 0	0.050616	0.139819	0.147968	0.150780	0.150870
4	TSNDT	Present	εz ≠ 0	−0.500620	0.235920	0.303385	0.324883	0.325295
	TSDT	Present	εz = 0	−0.456918	0.206364	0.261484	0.277589	0.277651
	HSDT	Krishna Murty 48	εz ≠ 0	−0.405990	0.218212	0.275877	0.294476	0.294870
	PSDT	Reddy46,47	εz = 0	−0.383772	0.199067	0.247606	0.261862	0.261931
	FSDT	Mindlin 8	εz = 0	0.049734	0.134227	0.141752	0.144342	0.144424
8	TSNDT	Present	εz ≠ 0	−0.490640	0.239019	0.305435	0.326569	0.326964
	TSDT	Present	εz = 0	−0.458280	0.207468	0.263069	0.279841	0.280007
	HSDT	Krishna Murty 48	εz ≠ 0	−0.398300	0.221514	0.278363	0.296643	0.297016
	PSDT	Reddy46,47	εz = 0	−0.385109	0.200665	0.249656	0.264482	0.264637
	FSDT	Mindlin 8	εz = 0	0.051061	0.136677	0.144235	0.146833	0.146916
10	TSNDT	Present	εz ≠ 0	−0.489790	0.240488	0.307205	0.328529	0.328941
	TSDT	Present	εz = 0	−0.460647	0.208088	0.264326	0.281531	0.281743
	HSDT	Krishna Murty 48	εz ≠ 0	−0.397630	0.223046	0.280163	0.298594	0.298979
	PSDT	Reddy46,47	εz = 0	−0.387336	0.201457	0.251030	0.266234	0.266429
	FSDT	Mindlin 8	εz = 0	0.051632	0.138332	0.145993	0.148627	0.148711

Discussion on the Graphical Results

Figure 4 shows the effects of radii of curvature on non-dimensional transverse displacements ( $\overline{w})$ with respect to the thickness coordinate (z/h) for FGM shells under the action of thermomechanical load. Figure 4 reveals an increase in transverse displacement with an increase in the radii of curvature. With higher radii of curvature, the transverse displacements start converging towards single values, which is attributed to the shell converting into a plate. It is interested to note that, when the R/a < 5 the shell becomes deep shells and hence the deflection, moments, and rotations are highly dependent on R/a while for shells with R/a >10, the shells behavior is more like a flat square plate. Therefore, for R/a >10 the values of transverse displacements almost become same. The same behavior can be observed in Figure 4.

Figure 4.

Effect of radii of curvature on the non-dimensional transverse displacements ( $\overline{w})$ with respect to the thickness coordinate for FGM shells under non-linear thermomechanical load ( ${\epsilon }_z\ne 0,$ a = 10h).

Figure 5 shows the effect of the power-law index on non-dimensional transverse displacement ( $\overline{w})$ with respect to the thickness coordinates (z/h) for FGM shells. Figure 5 shows that the transverse displacement increases with the power-law index ‘p.’ Figure 5 also reveals that the transverse displacement of the titanium (metal-rich) shell is larger than the corresponding one of the zirconia (ceramic-rich) shell.

Figure 5.

Effect of power-law index on the non-dimensional transverse displacements ( $\overline{w})$ with respect to the thickness coordinate for FGM shells under non-linear thermomechanical load ( ${\epsilon }_z\ne 0$ , a = 10h).

The non-linear distributions of in-plane normal stresses are observed in Figure 6. The in-plane normal stresses are tensile at the top face and compressive at the bottom. Figure 7 shows the effect of the power-law index on in-plane normal stresses ( ${\overline{\sigma }}_x)$ with respect to the thickness coordinate (z/h) for FGM shells. The homogeneous ceramic-rich shell (p = 0) shows the linear behavior of the shell. In contrast, with an increase in the value of ‘p’, the behavior becomes non-linear, attributing to the warping of the metallic shell due to the mechanical loading effect. Figure 8 shows the effect of radii of curvature on the non-dimensional in-plane normal stresses ( ${\overline{\sigma }}_x)$ with respect to the thickness coordinate (z/h). It shows the non-linear behavior of the non-dimensional in-plane normal stresses attributed to the warping effect due to the mechanical loading. It also shows that, with an increase in the values of radii of curvature, the in-plane normal stresses almost become equal in all cases of the shells.

Figure 6.

Variation of the non-dimensional in-plane normal stress ( ${\overline{\sigma }}_x)$ with respect to the thickness coordinate for FGM shells under non-linear thermomechanical load (R/a = 5, p = 2).

Figure 7.

Effect of power-law index on in-plane normal stress ( ${\overline{\sigma }}_x)$ with respect to the thickness coordinate for FGM shells under non-linear thermomechanical load ( ${\epsilon }_z\ne 0$ , a = 10h).

Figure 8.

Effect of radii of curvature on the non-dimensional in-plane normal stress ( ${\overline{\sigma }}_x)$ with respect to the thickness coordinate for FGM shells under non-linear thermomechanical load ( ${\epsilon }_z\ne 0$ , a = 10h).

The transverse shear stresses are plotted in Figure 9, confirming traction-free boundary conditions at the top and bottom surfaces of the shells. The curves closely follow each other, indicating that the various refined higher-order theories provide similar predictions for the behavior of all the shells within the range considered, whereas in the case of FSDT, the transverse shear stresses developed are minimal. Figure 10 shows the effect of the power-law index on the non-dimensional transverse shear stresses ( ${\overline{\tau }}_{xz})$ with respect to the thickness coordinate (z/h) for FGM shells. Figure 10 shows a decrease in the transverse shear stresses with an increase in the power-law index. Figure 11 shows the effect of radii of curvature on the non-dimensional transverse shear stresses ( ${\overline{\tau }}_{xz})$ with respect to the thickness coordinate (z/h). It shows an increase in the transverse shear stresses with an increase in the values of radii of curvature. Figure 11 shows that the values for the transverse shear stresses are calculated at various values of R/a (=1, 5, 10, 50, 100). When the R/a < 5 the shell becomes deep shells and hence the deflection, shear stresses, moments, and rotations are highly dependent on R/a and same can be observed in figure also. For shells with R/a ≥ 5, the shells behavior is more like a flat square plate. Therefore, for R/a ≥ 5 the values of transverse displacements almost become same and this also been observed in figure.

Figure 9.

Variation of the non-dimensional transverse shear stress ( ${\overline{\tau }}_{xz})$ with respect to the thickness coordinate for FGM shells under non-linear thermomechanical load (R/a = 5, p = 2).

Figure 10.

Effect of power-law index on the non-dimensional transverse shear stress ( ${\overline{\tau }}_{xz})$ with respect to the thickness coordinate for FGM shells under non-linear thermomechanical load ( ${\epsilon }_z\ne 0$ , a = 10h).

Figure 11.

Effect of radii of curvature on the non-dimensional transverse shear stress ( ${\overline{\tau }}_{xz})$ with respect to the thickness coordinate for FGM shells under non-linear thermomechanical load ( ${\epsilon }_z\ne 0$ , a = 10h).

As current study focus on the effect of transverse normal stress, the behavior of it plotted in Figure 12 which shows the effect of radii of curvature on the non-dimensional transverse normal stress ( ${\overline{\sigma }}_z)$ with respect to the thickness coordinate (z/h) under non-linear thermomechanical load. It shows decrease in the transverse normal stress with an increase in the values of radii of curvature. Spherical shell yields maximum value of transverse normal stress whereas hyperbolic shell yields minimum value.

Figure 12.

Effect of radii of curvature on the non-dimensional transverse normal stress ( ${\overline{\sigma }}_z)$ with respect to the thickness coordinate for FGM shells under non-linear thermomechanical load ( ${\epsilon }_z\ne 0$ , a = 10h).

Conclusions

The current study, studies the static response of doubly-curved FGM shells using refined higher-order trigonometric, parabolic shear, and normal deformation theory, along with the first-order shear deformation theory. The currently used theories yield realistic boundary conditions at the top and bottom surfaces of the shells. The Navier technique has been used to calculate transverse displacements and stresses for FGM spherical, cylindrical, hyperbolic, and elliptical shells under the action of thermomechanical loads. The obtained results are compared with the various theories available. It is concluded that the present results of static analysis of FGM shells show a good agreement with all theories. Also, the obtained numerical results for hyperbolic and elliptical shells can be a benchmark for future researchers to validate their research, as very limited research is available on it. The formulation and techniques used herein should be useful in further studies and analysis of shells subjected to hygrothermomechanical loads.

For thermal loading, both E(z) and α(z) variations with temperature may be large. Therefore, a study with another material gradation rules such as sigmoidal or exponential can be done in future.