Static Analysis of Temperature-Dependent FGM Spherical Shells Under Thermo-Mechanical Loads

Abstract

Static analysis is conducted for functionally graded material (FGM) spherical shells under thermo-mechanical loads, based on the three-dimensional thermo-elasticity theory. The material properties, which vary with both the radial coordinate and temperature, introduce nonlinearity to the problem. To address this, a layer model is proposed, wherein the shell is discretized into numerous concentric spherical layers, each possessing uniform material properties. Within this framework, the nonlinear heat conduction equations are first solved iteratively. The resulting temperature field is then applied to the thermo-elastic equations, which are subsequently solved using a combined state space and transfer matrix method to obtain displacement and stress solutions. Comparison with existing literature results demonstrates good agreement. Finally, a parametric study is presented to investigate the effects of material temperature dependence and gradient index on the thermo-mechanical behaviors of the FGM spherical shells.

1. Introduction

Owing to their geometric efficiency, spherical shells can withstand significant loads with relatively thin walls. This makes them ubiquitous in modern engineering applications, including pressure vessels, architectural domes, submersibles, and nuclear reactor containment structures [1,2,3]. Given their frequent operation under elevated temperatures and pressures, a comprehensive thermo-mechanical analysis is crucial during the design phase. Typically, large temperature variations not only induce significant thermal stresses within the structures [4] but also cause noticeable changes in material properties [5]. Consequently, the effects of material temperature dependence must be incorporated into such analyses.

Many researchers have studied the thermo-mechanical behaviors of spherical shells. Eslami et al. [6] performed the stability analysis of thin spherical shells under uniform and linear thermal loads across the wall thickness, utilizing deep and shallow shell theories. Vu et al. [7] employed the Donnell shell theory to analyze the nonlinear thermal buckling of spherical caps reinforced with graphene platelet. Zhu et al. [8], using a modified iteration method, investigated the effects of transverse shear deformation on the nonlinear buckling behaviors of symmetrically laminated cylindrically orthotropic shallow spherical shells. According to the first-order shear deformation theory (FSDT) and Donnell kinematics, Esmaeili and Kiani [9] derived the governing equations for the vibration of doubly curved shells reinforced with graphene platelet under rapid surface heating. These equations were discretized using the Ritz method with Chebyshev polynomials and solved via the Newmark time marching scheme. According to the FSDT, Javani et al. [5] investigated the dynamical snap-through of FGM spherical shells under thermal loads considering the material temperature dependence. The nonlinear governing equations were solved using the generalized differential quadrature method.

While FSDT reliably predicts global structural responses like deflections, natural frequencies, and buckling loads, it has limitations. The three-dimensional (3-D) elasticity theory indicates that shear stresses through the shell thickness exhibit at least a quadratic distribution, a feature not captured by FSDT, leading to deviations from actual structural behaviors. To address the limitations of the FSDT, Reddy and Liu [10] extended the higher-order shear deformation theory (HSDT) to the doubly curved shells. Similarly, Ghugal et al. [11] developed a trigonometric shear and normal deformation theory for the static analysis of layered spherical shells under thermo-mechanical loads.

Functionally graded materials (FGMs) are innovative composites whose composition gradually changes in space, resulting in corresponding gradient in material properties. This unique feature enables the FGM structures to mitigate the interfacial stress mismatch, which is a common limitation in traditional laminated composite structures [12,13,14,15,16,17,18]. Consequently, the FGM structures have broad application prospects in modern engineering, particularly in high-temperature applications [19,20,21]. Although the thermo-mechanical behaviors of FGM shells have been extensively studied using various two-dimensional (2-D) shell theories (e.g., FSDT and HSDT) [22,23,24,25,26,27,28,29], these theories become inadequate for more accurate analysis. In contrast, the 3-D elasticity theory provides greater accuracy in analyzing the mechanical behaviors of FGM shells, particularly in capturing stress and displacement distributions through the shell thickness [30]. Using the 3-D elasticity theory, researchers have developed semi-analytical and analytical solutions for FGM spherical shells. Meuyou et al. [31] applied the state space method, Runge–Kutta method, and transfer matrix method to obtain a semi-analytical solution for the static behaviors of FGM spherical shells under thermo-electric loads. Liu et al. [32] investigated the pyroelectric behaviors of functionally graded piezoelectric spherical shells under thermal loads. Analytical solutions were obtained by expanding the variables in the state equations into the spherical harmonic functions.

The preceding literature review on FGM shells indicates that most studies assume temperature-independent (T-I) material properties. While this simplification introduces negligible errors under small temperature variations, it leads to significant errors when variations are sufficiently large. Some studies employ symmetry assumptions for material properties, loading, and geometry to simplify the 3-D problem into a one-dimensional or 2-D problem. Furthermore, most studies are based on the 2-D shell theories, which incorporate assumptions about the shear deformation. Consequently, these theories are inherently less accurate than the 3-D elasticity theory.

To the authors’ best knowledge, the thermo-mechanical analysis of FGM spherical shells considering both the material temperature dependence and the 3-D thermo-elasticity theory has not been studied. Hence, this serves as the motivation for the current work. To account for the material temperature dependence, we propose a layer model for discretization. Within this framework, the heat conduction problem and the thermo-elasticity problem are solved sequentially. Finally, the effects of material temperature dependence and gradient index on the thermo-mechanical behaviors of the FG spherical shells are comprehensively investigated.

2. Layer Model for Spherical Shells

A spherical shell with its external radius $r_{\mathit{P}}$ and internal radius $r_0$ is displayed in Figure 1. The shell is under distributed pressures and is heated from a uniform initial temperature ${\mathit{T}}_0$. Finally, the temperature field within the shell is stable. To characterize the thermo-mechanical behaviors of the shell, a spherical coordinate system (r, θ, φ) is established with its origin at the shell’s center. The shell is made of a FGM with its properties varying with the r-coordinate and temperature, i.e.,

$$E=E(r,\mathit{T}), \mu =\mu (r,\mathit{T}), \alpha =\alpha (r,\mathit{T}), \mathit{k}=\mathit{k}(r,\mathit{T})$$

(1)

where E, μ, α, k are Young’s modulus, Poisson’s ratio, thermal expansion coefficient, and thermal conductivity of the FGM, respectively.

When the thermal boundary conditions are uniformly distributed on the external and internal surfaces, the temperature field varies solely along the r-direction [33]. Consequently, the temperature-dependent (T-D) material properties are variable along the r-direction. Under these conditions, the governing equations for the thermo-mechanical behaviors of the shell possess r-dependent coefficients, rendering exact solutions nearly intractable. To obtain asymptotic solutions of the equations, a layer model (see Figure 2) is proposed where the shell is divided into p concentric spherical layers of equal thickness. Each layer is sufficiently thin such that those r-dependent coefficients can be treated as constants, evaluated at the layer’s mean radius ${\overline{r}}_i$ (i = 1, 2, …, p).

3. Heat Conduction Analysis

Within the framework of the layer model, the nonlinear heat conduction equations induced by the material temperature dependence are solved iteratively. Prior to the iteration, we first obtain an exact solution for the T-I case.

3.1. Heat Conduction Analysis for the T-I Case

In the T-I case, the heat conduction along the r-direction for the ith layer is [34]

$${\displaystyle \frac{\mathit{d}}{\mathit{d}r}}\left(r^2{\displaystyle \frac{\mathit{d}{\mathit{T}}_i}{\mathit{d}r}}\right)=0, i=1, 2, \dots , \mathit{p}$$

(2)

where ${\mathit{T}}_i$ is the temperature of the ith layer. The general solution to Equation (2) is

$${\mathit{T}}_i={\displaystyle \frac{A_i}{r}}+B_i$$

(3)

where $A_i$ and $B_i$ are unknows determined by the boundary conditions.

Since the layer model is fictitious, the temperature and flux are continuous at the layer interfaces, i.e.,

$${\mathit{T}}_i(r_i)={\mathit{T}}_{i+1}(r_i), {\overline{\mathit{k}}}_i{\left.{\displaystyle \frac{\mathit{d}{\mathit{T}}_i}{\mathit{d}r}}\right|}_{r=r_i}={\overline{\mathit{k}}}_{i+1}{\left.{\displaystyle \frac{\mathit{d}{\mathit{T}}_{i+1}}{\mathit{d}r}}\right|}_{r=r_i}, i=1, 2, \dots , \mathit{p}-1$$

(4)

where $r_i$ and ${\overline{\mathit{k}}}_i$ are the external radius and the uniform thermal conductivity of the ith layer, respectively.

When the temperature field within the shell is stable, the surface temperatures are constants, i.e.,

$${\mathit{T}}_{\mathit{P}}(r_{\mathit{P}})={\mathit{T}}_{ex}, {\mathit{T}}_1(r_0)={\mathit{T}}_{in}$$

(5)

where ${\mathit{T}}_{ex}$ and ${\mathit{T}}_{in}$ are the external and internal surface temperatures, respectively.

Substituting Equation (3) into Equations (4) and (5), the unknows $A_i$ and $B_i$ in Equation (3) can be obtained as

$$A_i={\displaystyle \frac{{\mathit{T}}_{ex}-{\mathit{T}}_{in}}{\frac{{\overline{\mathit{k}}}_i}{r_{\mathit{P}}{\mathit{k}}_{\mathit{P}}}+{\overline{\mathit{k}}}_i{\displaystyle \sum _{j=1}^{\mathit{p}-1}\frac{{\overline{\mathit{k}}}_{j+1}-{\overline{\mathit{k}}}_j}{r_j{\overline{\mathit{k}}}_{j+1}{\overline{\mathit{k}}}_j}-\frac{{\overline{\mathit{k}}}_i}{r_0{\overline{\mathit{k}}}_1}}}}, B_i={\mathit{T}}_{in}-{\displaystyle \frac{A_1}{r_0}}+{\displaystyle \sum _{j=1}^{i-1}\frac{A_j-A_{j+1}}{r_j}}$$

(6)

Substituting Equation (6) back into Equation (3), the temperature field across the shell thickness can be obtained for the T-I case.

3.2. Heat Conduction Analysis for the T-D Case

In this subsection, we use an iteration method to solve the heat conduction problem for the T-D case. Before initiating the iteration, we assume a linear temperature distribution across the shell thickness. During each iteration step, the exact solution Equation (3) is used to generates an updated temperature distribution. The iteration is detailed in the flowchart displayed in Figure 3.

In should be mentioned that the temperature varies only radially; however, the angular thermal gradients, which can cause circumferential displacements, are not considered. Furthermore, only Dirichlet conditions are considered in the present analysis. It is convenient to include the other thermal boundary conditions such as convection/radiation boundary conditions, which can be explored in future research.

4. Static Analysis

4.1. Basic Equations and Superposition Principle

Based on the 3-D thermo-elasticity theory and the layer model, the strain–displacement relation for the ith layer is

$$S_{rr}^i=r{\epsilon }_r^i={\nabla }_2{u}_r^i, S_{\mathsf{\theta }\mathsf{\theta }}^i=r{\epsilon }_{\mathsf{\theta }}^i={\displaystyle \frac{\partial u_{\mathsf{\theta }}^i}{\partial \mathsf{\theta }}}+u_r^i, S_{\phi \phi }^i=r{\epsilon }_{\phi }^i={\displaystyle \frac{1}{\sin \mathsf{\theta }}}{\displaystyle \frac{\partial u_{\phi }^i}{\partial \phi }}+u_r^i+u_{\mathsf{\theta }}^i\cot \mathsf{\theta },$$

$$2S_{r\mathsf{\theta }}^i=r{\gamma }_{r\mathsf{\theta }}^i={\displaystyle \frac{\partial u_r^i}{\partial \mathsf{\theta }}}+{\nabla }_2{u}_{\mathsf{\theta }}^i-u_{\mathsf{\theta }}^i, 2S_{r\phi }^i=r{\gamma }_{r\phi }^i={\displaystyle \frac{1}{\sin \mathsf{\theta }}}{\displaystyle \frac{\partial u_r^i}{\partial \phi }}+{\nabla }_2{u}_{\phi }^i-u_{\phi }^i,$$

$$2S_{\mathsf{\theta }\phi }^i=r{\gamma }_{\mathsf{\theta }\phi }^i={\displaystyle \frac{\partial u_{\phi }^i}{\partial \mathsf{\theta }}}-u_{\phi }^i\cot \mathsf{\theta }+{\displaystyle \frac{1}{\sin \mathsf{\theta }}}{\displaystyle \frac{\partial u_{\mathsf{\theta }}^i}{\partial \phi }}, i=1, 2, \dots , \mathit{p}$$

(7)

where ${\nabla }_2=r\partial /\partial r$; $u_r^i$, $u_{\mathsf{\theta }}^i$, and $u_{\phi }^i$ are displacement components of the ith layer; ${\epsilon }_r^i$, ${\epsilon }_{\mathsf{\theta }}^i$, ${\epsilon }_{\phi }^i$, ${\gamma }_{r\mathsf{\theta }}^i$, ${\gamma }_{r\phi }^i$, and ${\gamma }_{\mathsf{\theta }\phi }^i$ are strain components; and $S_{j\mathit{k}}^i$ (j, k = r, θ, φ) are the generalized strain components.

The stress-strain relation is

$${\Sigma }_{\mathsf{\theta }\mathsf{\theta }}^i=r{\sigma }_{\mathsf{\theta }}^i=c_{11}^i{S}_{\mathsf{\theta }\mathsf{\theta }}^i+c_{12}^i{S}_{\phi \phi }^i+c_{13}^i{S}_{rr}^i-r{\mathit{t}}_i,$$

$${\Sigma }_{\phi \phi }^i=r{\sigma }_{\phi }^i=c_{12}^i{S}_{\mathsf{\theta }\mathsf{\theta }}^i+c_{11}^i{S}_{\phi \phi }^i+c_{13}^i{S}_{rr}^i-r{\mathit{t}}_i,$$

$${\Sigma }_{rr}^i=r{\sigma }_r^i=c_{13}^i{S}_{\mathsf{\theta }\mathsf{\theta }}^i+c_{13}^i{S}_{\phi \phi }^i+c_{33}^i{S}_{rr}^i-r{\mathit{t}}_i,$$

$${\Sigma }_{r\mathsf{\theta }}^i=r{\tau }_{r\mathsf{\theta }}^i=2c_{44}^i{S}_{r\mathsf{\theta }}^i, {\Sigma }_{r\phi }^i=r{\tau }_{r\phi }^i=2c_{44}^i{S}_{r\phi }^i, {\Sigma }_{\mathsf{\theta }\phi }^i=r{\tau }_{\mathsf{\theta }\phi }^i=2c_{66}^i{S}_{\mathsf{\theta }\phi }^i,$$

(8)

where ${\sigma }_r^i$, ${\sigma }_{\mathsf{\theta }}^i$, ${\sigma }_{\phi }^i$, ${\tau }_{r\mathsf{\theta }}^i$, ${\tau }_{r\phi }^i$, and ${\tau }_{\mathsf{\theta }\phi }^i$ are stress components of the ith layer; ${\Sigma }_{j\mathit{k}}^i$ (j, k = r, θ, φ) are the generalized stress components; $c_{j\mathit{k}}^i$ (j, k = 1, 2, …, 6) are the stiffness coefficients and ${\mathit{t}}_i$ is the thermal stress term which are given by

$$c_{11}^i=c_{33}^i={\displaystyle \frac{(1-{\overline{\mu }}_i){\overline{E}}_i}{(1+{\overline{\mu }}_i)(1-2{\overline{\mu }}_i)}}, c_{12}^i=c_{13}^i={\displaystyle \frac{{\overline{\mu }}_i{\overline{E}}_i}{(1+{\overline{\mu }}_i)(1-2{\overline{\mu }}_i)}},$$

$$c_{44}^i=c_{66}^i={\displaystyle \frac{{\overline{E}}_i}{2(1+{\overline{\mu }}_i)}}, {\mathit{t}}_i={\displaystyle \frac{{\overline{E}}_i{\overline{\alpha }}_i\left({\overline{\mathit{T}}}_i-{\mathit{T}}_0\right)}{1-2{\overline{\mu }}_i}}$$

(9)

where ${\overline{\mathit{P}}}_i$ (P = E, μ, α) are the uniform material properties of the ith layer.

The equilibrium equation is

$${\nabla }_2{\Sigma }_{rr}^i+{\displaystyle \frac{\partial {\Sigma }_{r\mathsf{\theta }}^i}{\partial \mathsf{\theta }}}+\csc \mathsf{\theta }{\displaystyle \frac{\partial {\Sigma }_{r\phi }^i}{\partial \phi }}+{\Sigma }_{rr}^i-{\Sigma }_{\mathsf{\theta }\mathsf{\theta }}^i-{\Sigma }_{\phi \phi }^i+{\Sigma }_{r\mathsf{\theta }}^i\cot \mathsf{\theta }=0,$$

$${\nabla }_2{\Sigma }_{r\mathsf{\theta }}^i+{\displaystyle \frac{\partial {\Sigma }_{\mathsf{\theta }\mathsf{\theta }}^i}{\partial \mathsf{\theta }}}+\csc \mathsf{\theta }{\displaystyle \frac{\partial {\Sigma }_{\mathsf{\theta }\phi }^i}{\partial \phi }}+\left({\Sigma }_{\mathsf{\theta }\mathsf{\theta }}^i-{\Sigma }_{\phi \phi }^i\right)\cot \mathsf{\theta }+2{\Sigma }_{r\mathsf{\theta }}^i=0,$$

$${\nabla }_2{\Sigma }_{r\phi }^i+{\displaystyle \frac{\partial {\Sigma }_{\mathsf{\theta }\phi }^i}{\partial \mathsf{\theta }}}+\csc \mathsf{\theta }{\displaystyle \frac{\partial {\Sigma }_{\phi \phi }^i}{\partial \phi }}+2{\Sigma }_{\mathsf{\theta }\phi }^i\cot \mathsf{\theta }+2{\Sigma }_{r\phi }^i=0$$

(10)

Since the present analysis is based on linear elasticity theory, the superposition principle applies. Consequently, the static behaviors induced by pressure and thermal load can be analyzed separately. Furthermore, each problem allows for simplification: (i) for the pressure problem, the thermal load term in Equation (8) can be omitted; (ii) for the thermal load problem, due to its spherical symmetry, the induced displacements and stresses depend solely on the coordinate r, and, in particular, the displacements $u_{\mathsf{\theta }}$, $u_{\phi }$ and shear stresses ${\tau }_{r\mathsf{\theta }}$, ${\tau }_{r\phi }$, ${\tau }_{\mathsf{\theta }\phi }$ vanish. Solutions to both problems will be presented subsequently.

4.2. Solution to the Pressure Problem

4.2.1. State Space Method

To solve the problem, we assume that the displacements and stresses are expressed by [18]

$$u_{\mathsf{\theta }}^i=-{\displaystyle \frac{1}{\sin \mathsf{\theta }}}{\displaystyle \frac{\mathit{\partial }{\psi }_i}{\mathit{\partial }\phi }}-{\displaystyle \frac{\mathit{\partial }{\mathit{G}}_i}{\mathit{\partial }\mathsf{\theta }}}, u_{\phi }^i={\displaystyle \frac{\mathit{\partial }{\psi }_i}{\mathit{\partial }\mathsf{\theta }}}-{\displaystyle \frac{1}{\sin \mathsf{\theta }}}{\displaystyle \frac{\mathit{\partial }{\mathit{G}}_i}{\mathit{\partial }\phi }},$$

$${\Sigma }_{r\mathsf{\theta }}^i=-{\displaystyle \frac{1}{\sin \mathsf{\theta }}}{\displaystyle \frac{\mathit{\partial }{\Sigma }_1^i}{\mathit{\partial }\phi }}-{\displaystyle \frac{\mathit{\partial }{\Sigma }_2^i}{\mathit{\partial }\mathsf{\theta }}}, {\Sigma }_{r\phi }^i={\displaystyle \frac{\mathit{\partial }{\Sigma }_1^i}{\mathit{\partial }\mathsf{\theta }}}-{\displaystyle \frac{1}{\sin \mathsf{\theta }}}{\displaystyle \frac{\mathit{\partial }{\Sigma }_2^i}{\mathit{\partial }\phi }}$$

(11)

where ${\psi }_i$ and ${\mathit{G}}_i$ are displacement functions; ${\Sigma }_1^i$ and ${\Sigma }_2^i$ are stress functions.

Using Equation (11), two state equations can be obtained from Equations (7), (8), and (10):

$${\nabla }_2\left\{\begin{array}{c}{\Sigma }_1^i\\ {\psi }_i\end{array}\right\}=\left[\begin{array}{cc}-2& -c_{66}^i\left({\nabla }_1^2+2\right)\\ 1/c_{44}^i& 1\end{array}\right]\left\{\begin{array}{c}{\Sigma }_1^i\\ {\psi }_i\end{array}\right\}$$

(12)

$${\nabla }_2\left\{\begin{array}{c}{\Sigma }_{rr}^i\\ {\Sigma }_2^i\\ {\mathit{G}}_i\\ u_r^i\end{array}\right\}=\left[\begin{array}{cccc}2{\beta }_i-1& {\nabla }_1^2& {\mathit{k}}_1^i{\nabla }_1^2& -2{\mathit{k}}_1^i\\ {\beta }_i& -2& {\mathit{k}}_2^i{\nabla }_1^2-2c_{66}^i& -{\mathit{k}}_1^i\\ 0& 1/c_{44}^i& 1& 1\\ 1/c_{33}^i& 0& \beta {\nabla }_1^2& -2{\beta }_i\end{array}\right]\left\{\begin{array}{c}{\Sigma }_{rr}^i\\ {\Sigma }_2^i\\ {\mathit{G}}_i\\ u_r^i\end{array}\right\}$$

(13)

and the stress components ${\Sigma }_{\mathsf{\theta }\mathsf{\theta }}^i$, ${\Sigma }_{\phi \phi }^i$, and ${\Sigma }_{\mathsf{\theta }\phi }^i$ can be derived as

$${\Sigma }_{\mathsf{\theta }\mathsf{\theta }}^i-{\Sigma }_{\phi \phi }^i=2c_{66}^i\left({\nabla }_1^2{\mathit{G}}_i-2{\displaystyle \frac{{\mathit{\partial }}^2{\mathit{G}}_i}{\mathit{\partial }{\mathsf{\theta }}^2}}+2\cot \mathsf{\theta }\csc \mathsf{\theta }{\displaystyle \frac{\mathit{\partial }{\psi }_i}{\mathit{\partial }\phi }}-2\csc \mathsf{\theta }{\displaystyle \frac{{\mathit{\partial }}^2{\psi }_i}{\mathit{\partial }\mathsf{\theta }\mathit{\partial }\phi }}\right)$$

$${\Sigma }_{\mathsf{\theta }\mathsf{\theta }}^i+{\Sigma }_{\phi \phi }^i=2{\beta }_i{\sum }_{rr}^i+{\mathit{k}}_1^i{\nabla }_1^2{\mathit{G}}_i-2{\mathit{k}}_1^i{u}_r^i$$

$${\Sigma }_{\mathsf{\theta }\phi }^i=-c_{66}^i\left({\nabla }_1^2{\psi }_i-2{\displaystyle \frac{{\mathit{\partial }}^2{\psi }_i}{\mathit{\partial }{\mathsf{\theta }}^2}}-2\cot \mathsf{\theta }\csc \mathsf{\theta }{\displaystyle \frac{\mathit{\partial }{\mathit{G}}_i}{\mathit{\partial }\phi }}+2\csc \mathsf{\theta }{\displaystyle \frac{{\mathit{\partial }}^2{\mathit{G}}_i}{\mathit{\partial }\mathsf{\theta }\mathit{\partial }\phi }}\right)$$

(14)

where ${\nabla }_1^2={\displaystyle \frac{{\mathit{\partial }}^2}{\mathit{\partial }{\mathsf{\theta }}^2}}+\cot \mathsf{\theta }{\displaystyle \frac{\partial }{\partial \mathsf{\theta }}}+{\csc }^2\mathsf{\theta }{\displaystyle \frac{{\partial }^2}{\partial {\phi }^2}}$ is the spherical Laplacian, ${\beta }_i=c_{13}^i/c_{33}^i$, ${\mathit{k}}_1^i=2c_{13}^i{\beta }_i-(c_{11}^i+c_{12}^i)$, ${\mathit{k}}_2^i={\mathit{k}}_1^i/2-c_{66}^i$. It should be mentioned that the generalized thermal stress term $-r{\mathit{t}}_i$ in Equation (8) is omitted in the development of Equations (12)–(14), since we only focus on the pressure problem in this subsection.

To solve Equations (12) and (13), we assume that the state variables are expressed by

$$\left\{\begin{array}{c}{\Sigma }_1^i\\ {\psi }_i\\ {\Sigma }_{rr}^i\\ {\Sigma }_2^i\\ {\mathit{G}}_i\\ u_r^i\end{array}\right\}={\displaystyle \sum _{\mathit{m}=0}^n{\displaystyle \sum _{n=0}^{\infty }\left\{\begin{array}{c}{\Sigma }_{1n}^i(\xi )S_n^{\mathit{m}}(\mathsf{\theta },\phi )\\ {\psi }_n^i(\xi )S_n^{\mathit{m}}(\mathsf{\theta },\phi )\\ {\Sigma }_{rn}^i(\xi )S_n^{\mathit{m}}(\mathsf{\theta },\phi )\\ {\Sigma }_{2n}^i(\xi )S_n^{\mathit{m}}(\mathsf{\theta },\phi )\\ {\mathit{G}}_n^i(\xi )S_n^{\mathit{m}}(\mathsf{\theta },\phi )\\ u_{rn}^i(\xi )S_n^{\mathit{m}}(\mathsf{\theta },\phi )\end{array}\right\}}}$$

(15)

where $S_n^{\mathit{m}}(\mathsf{\theta },\phi )={\mathit{P}}_n^{\mathit{m}}(\cos \mathsf{\theta })\exp (i\mathit{m}\phi )$ are the spherical harmonics; ${\mathit{P}}_n^{\mathit{m}}(x)$ are the associated Legendre polynomials; and ξ is given by

$$\xi =\ln (r/r_{i-1}), 0\le \xi \le {\xi }_i$$

(16)

in which ${\xi }_i=\ln (r_i/r_{i-1})$.

Applying Equation (15) to Equations (12) and (13) yields

$${\displaystyle \frac{\mathit{d}}{\mathit{d}\xi }}{\mathit{T}}_{1n}^i(\xi )={\mathit{M}}_{1n}^i{\mathit{T}}_{1n}^i(\xi ), 0\le \xi \le {\xi }_i, n=1, 2, 3,$$

(17)

$${\displaystyle \frac{\mathit{d}}{\mathit{d}\xi }}{\mathit{T}}_{2n}^i(\xi )={\mathit{M}}_{2n}^i{\mathit{T}}_{2n}^i(\xi ), 0\le \xi \le {\xi }_i, n=0, 1, 2,$$

(18)

where

$${\mathit{M}}_{1n}^i=\left[\begin{array}{cc}-2& c_{66}^i\left(l-2\right)\\ 1/c_{44}^i& 1\end{array}\right], {\mathit{M}}_{2n}^i=\left[\begin{array}{cccc}2{\beta }_i-1& -l& -{\mathit{k}}_1^{i}l& -2{\mathit{k}}_1^i\\ {\beta }_i& -2& -{\mathit{k}}_2^{i}l-2c_{66}^i& -{\mathit{k}}_1^i\\ 0& 1/c_{44}^i& 1& 1\\ 1/c_{33}^i& 0& -{\beta }_{i}l& -2{\beta }_i\end{array}\right]$$

$${\mathit{T}}_{1n}^i(\xi )={\left\{\begin{array}{cc}{\Sigma }_{1n}^i(\xi )& {\psi }_n^i(\xi )\end{array}\right\}}^{\mathrm{T}}, {\mathit{T}}_{2n}^i={\left\{\begin{array}{cccc}{\Sigma }_{rn}^i& {\Sigma }_{2n}^i& {\mathit{G}}_n^i& u_{rn}^i\end{array}\right\}}^{\mathrm{T}}, l=n(n+1)$$

(19)

4.2.2. Solution Strategy

According to the matrix theory, we can obtain the general solutions to Equations (17) and (18) as follows:

$${\mathit{T}}_{1n}^i(\xi )={\mathit{G}}_{1n}^i(\xi ){\mathit{T}}_{1n}^i(0), 0\le \xi \le {\xi }_i, n=1, 2, 3, \dots$$

(20)

$${\mathit{T}}_{2n}^i(\xi )={\mathit{G}}_{2n}^i(\xi ){\mathit{T}}_{2n}^i(0), 0\le \xi \le {\xi }_i, n=0, 1, 2, \dots$$

(21)

where ${\mathit{G}}_{1n}^i(\xi )=\exp ({\mathit{M}}_{1n}^i\xi )$ and ${\mathit{G}}_{2n}^i(\xi )=\exp ({\mathit{M}}_{2n}^i\xi )$.

Taking $\xi ={\xi }_i$ into Equations (20) and (21) gives

$${\mathit{T}}_{1n}^i({\xi }_i)={\mathit{G}}_{1n}^i({\xi }_i){\mathit{T}}_{1n}^i(0), n=1, 2, 3, \dots$$

(22)

$${\mathit{T}}_{2n}^i({\xi }_i)={\mathit{G}}_{2n}^i({\xi }_i){\mathit{T}}_{2n}^i(0), n=0, 1, 2, \dots$$

(23)

Therefore, the state variables located at the internal and external surfaces of each layer are connected by Equations (22) and (23). Because the state variables are continuous at each interface, we have

$${\mathit{T}}_{1n}^{\mathit{p}}({\xi }_{\mathit{p}})={\displaystyle \prod _{\mathit{k}=\mathit{p}}^1\left[{\mathit{G}}_{1n}^{\mathit{k}}({\xi }_{\mathit{k}})\right]}{\mathit{T}}_{1n}^1(0), n=1, 2, 3, \dots$$

(24)

$${\mathit{T}}_{2n}^{\mathit{p}}({\xi }_{\mathit{p}})={\displaystyle \prod _{\mathit{k}=\mathit{p}}^1\left[{\mathit{G}}_{2n}^{\mathit{k}}({\xi }_{\mathit{k}})\right]}{\mathit{T}}_{2n}^1(0), n=0, 1, 2, \dots$$

(25)

Therefore, the state variables located at the internal and external surfaces of the shell are connected by Equations (24) and (25). When the boundary loads are specified, i.e., ${\Sigma }_{1n}^1(0)$, ${\Sigma }_{1n}^{\mathit{P}}({\xi }_{\mathit{P}})$, ${\Sigma }_{rn}^1(0)$, ${\Sigma }_{rn}^{\mathit{P}}({\xi }_{\mathit{P}})$, ${\Sigma }_{2n}^1(0)$, and ${\Sigma }_{2n}^{\mathit{P}}({\xi }_{\mathit{P}})$ are known, we can finally obtain from Equations (24) and (25) that

$${\psi }_n^1(0)={\displaystyle \frac{{\Sigma }_{1n}^{\mathit{p}}({\xi }_{\mathit{p}})-S_{1n11}{\Sigma }_{1n}^1(0)}{S_{1n12}}}, n=1, 2, 3, \dots$$

$$\left\{\begin{array}{c}{\mathit{G}}_n^1(0)\\ u_{rn}^1(0)\end{array}\right\}={\left[\begin{array}{cc}{S}_{2n13}& S_{2n14}\\ S_{2n23}& S_{2n24}\end{array}\right]}^{-1}\left(\left\{\begin{array}{c}{\Sigma }_{rn}^{\mathit{p}}({\xi }_{\mathit{p}})\\ {\Sigma }_{2n}^{\mathit{p}}({\xi }_{\mathit{p}})\end{array}\right\}-\left[\begin{array}{cc}{S}_{2n11}& S_{2n12}\\ S_{2n21}& S_{2n22}\end{array}\right]\left\{\begin{array}{c}{\Sigma }_{rn}^1(0)\\ {\Sigma }_{2n}^1(0)\end{array}\right\}\right), n=0, 1, 2, \dots$$

(26)

where

$$\left[\begin{array}{cc}{S}_{1n11}& S_{1n12}\\ S_{1n21}& S_{1n22}\end{array}\right]={\displaystyle \prod _{\mathit{k}=\mathit{P}}^1\left[{\mathit{G}}_{1n}^{\mathit{k}}({\xi }_{\mathit{k}})\right]}, n=1, 2, 3, \dots$$

$$\left[\begin{array}{cccc}{S}_{2n11}& S_{2n12}& S_{2n13}& S_{2n14}\\ S_{2n21}& S_{2n22}& S_{2n23}& S_{2n24}\\ S_{2n31}& S_{2n32}& S_{2n33}& S_{2n34}\\ S_{2n41}& S_{2n42}& S_{2n43}& S_{2n44}\end{array}\right]={\displaystyle \prod _{\mathit{k}=\mathit{P}}^1\left[{\mathit{G}}_{2n}^{\mathit{k}}({\xi }_{\mathit{k}})\right]}, n=0, 1, 2, \dots$$

(27)

By using Equation (26) the state variables located at the internal surface of the shell are determined, which are then used to obtain the state variables at any point as follows:

$${\mathit{T}}_{1n}^i(\xi )={\mathit{G}}_{1n}^i(\xi ){\displaystyle \prod _{\mathit{k}=i-1}^1\left[{\mathit{G}}_{1n}^{\mathit{k}}({\xi }_{\mathit{k}})\right]}{\mathit{T}}_{1n}^1(0), 0\le \xi \le {\xi }_i, n=1, 2, 3, \dots$$

$${\mathit{T}}_{2n}^i(\xi )={\mathit{G}}_{2n}^i(\xi ){\displaystyle \prod _{\mathit{k}=i-1}^1\left[{\mathit{G}}_{2n}^{\mathit{k}}({\xi }_{\mathit{k}})\right]}{\mathit{T}}_{2n}^1(0), 0\le \xi \le {\xi }_i, n=0, 1, 2, \dots$$

(28)

Substituting ${\mathit{T}}_{1n}^i(\xi )$ and ${\mathit{T}}_{2n}^i(\xi )$ back to Equation (15) and using Equations (11) and (14), the displacement components $u_r^i$, $u_{\mathsf{\theta }}^i$, and $u_{\phi }^i$ and the generalized stress components ${\Sigma }_{j\mathit{k}}^i$ (j, k = r, θ, φ) induced by the pressure can be obtained. Finally, substituting ${\Sigma }_{j\mathit{k}}^i$ back to Equation (8) and omitting the thermal stress term, the stress components ${\sigma }_r^i$, ${\sigma }_{\mathsf{\theta }}^i$, ${\sigma }_{\phi }^i$, ${\tau }_{r\mathsf{\theta }}^i$, ${\tau }_{r\phi }^i$, and ${\tau }_{\mathsf{\theta }\phi }^i$ can be obtained.

4.3. Solution to the Thermal Load Problem

Recalling that the thermal load problem has spherical symmetry, the basic equations—Equations (7), (8), and (10)—can be simplified. Consequently, the corresponding state equation for the thermal load problem is

$${\displaystyle \frac{\mathit{d}}{\mathit{d}\xi }}{\mathit{T}}_3^i(\xi )={\mathit{M}}_3^i{\mathit{T}}_3^i(\xi )+{\mathit{N}}_i(\xi ), 0\le \xi \le {\xi }_i$$

(29)

where

$${\mathit{T}}_3^i(\xi )=\left\{\begin{array}{c}{u}_r^i(\xi )\\ {\Sigma }_{rr}^i(\xi )\end{array}\right\}, {\mathit{M}}_3^i=\left[\begin{array}{cc}-2{\beta }_i& 1/c_{33}^i\\ -2{\mathit{k}}_1^i& 2{\beta }_i-1\end{array}\right], {\mathit{N}}_i(\xi )=\left\{\begin{array}{c}{\mathit{t}}_i{r}_{i-1}\exp (\xi )/c_{33}^i\\ 2{\mathit{t}}_i{r}_{i-1}\left({\beta }_i-1\right)\exp (\xi )\end{array}\right\}$$

(30)

The stresses ${\Sigma }_{\mathsf{\theta }\mathsf{\theta }}^i$ and ${\Sigma }_{\phi \phi }^i$ can be determined by

$${\Sigma }_{\mathsf{\theta }\mathsf{\theta }}^i(\xi )={\Sigma }_{\phi \phi }^i(\xi )=-{\mathit{k}}_1^i{u}_r^i+{\beta }_i{\Sigma }_{rr}^i+{\mathit{t}}_i{a}_i({\beta }_i-1)\exp (\xi )$$

(31)

By applying matrix theory again, we can obtain the general solution to Equation (29) as follows:

$${\mathit{T}}_3^i(\xi )={\mathit{G}}_3^i(\xi ){\mathit{T}}_3^i(0)+{\mathit{K}}_i(\xi ), 0\le \xi \le {\xi }_i$$

(32)

where

$${\mathit{G}}_3^i(\xi )=\exp ({\mathit{M}}_3^i\xi ), {\mathit{K}}_i(\xi )={\displaystyle {\int }_0^{\xi }\exp [{\mathit{M}}_3^i(\xi -\tau )]{\mathit{N}}_i\mathit{d}\tau }$$

(33)

Similarly to the recurrence relation Equations (24) and (25) for the pressure problem, the recurrence relation for the thermal load problem can be obtained as

$${\mathit{T}}_3^{\mathit{p}}({\xi }_{\mathit{p}})={\displaystyle \prod _{\mathit{k}=\mathit{P}}^1\left[{\mathit{G}}_3^{\mathit{k}}({\xi }_{\mathit{k}})\right]}{\mathit{T}}_3^1(0)+{\displaystyle \sum _{\mathit{k}=1}^{\mathit{p}-1}\left\{{\displaystyle \prod _{j=\mathit{p}}^{\mathit{k}+1}\left[{\mathit{G}}_3^j({\xi }_j)\right]}{\mathit{K}}_{\mathit{k}}({\xi }_{\mathit{k}})\right\}}+{\mathit{K}}_{\mathit{p}}({\xi }_{\mathit{p}})$$

(34)

It is clear that for the thermal load problem, all the boundary loads vanish, i.e., ${\Sigma }_{rr}^1(0)={\Sigma }_{rr}^{\mathit{p}}({\xi }_{\mathit{p}})=0$. Consequently, we can finally obtain from Equation (34) that

$$u_r^1(0)=-{\overline{S}}_2/S_{21}$$

(35)

where

$$\left[\begin{array}{cc}{S}_{11}& S_{12}\\ S_{21}& S_{22}\end{array}\right]={\displaystyle \prod _{i=\mathit{P}}^1{\mathit{G}}_3^i({\xi }_i)}, {\left\{\begin{array}{cc}{\overline{S}}_1& {\overline{S}}_2\end{array}\right\}}^{\mathit{T}}={\displaystyle \sum _{i=1}^{\mathit{p}-1}\left\{{\displaystyle \prod _{j=\mathit{p}}^{i+1}\left[{\mathit{G}}_3^j({\xi }_j)\right]}{\mathit{K}}_i({\xi }_i)\right\}}+{\mathit{K}}_{\mathit{p}}({\xi }_{\mathit{p}})$$

(36)

By using Equation (35), the state variables of the internal surface of the shell are determined, which are then used to obtain the state variables at any point as follows:

$${\mathit{T}}_3^i(\xi )={\mathit{G}}_3^i(\xi )\left\{{\displaystyle \prod _{\mathit{k}=i-1}^1\left[{\mathit{G}}_3^{\mathit{k}}({\xi }_{\mathit{k}})\right]}{\mathit{T}}_3^1(0)+{\displaystyle \sum _{\mathit{k}=1}^{i-2}\left\{{\displaystyle \prod _{j=i-1}^{\mathit{k}+1}\left[{\mathit{G}}_3^j({\xi }_j)\right]}{\mathit{K}}_{\mathit{k}}({\xi }_{\mathit{k}})\right\}}+{\mathit{K}}_{i-1}({\xi }_{i-1})\right\}+{\mathit{K}}_i(\xi )$$

(37)

Substituting ${\mathit{T}}_3^i(\xi )$ back to the first term of Equation (30) and using Equation (31), the displacement components $u_r^i$ and the generalized stress components ${\Sigma }_{jj}^i$ (j = r, θ, φ) induced by the thermal load can be obtained. Substituting ${\Sigma }_{jj}^i$ back to Equation (8), the stress components ${\sigma }_r^i$, ${\sigma }_{\mathsf{\theta }}^i$, and ${\sigma }_{\phi }^i$ can be obtained. Finally, superimposing the results from Section 4.2 and Section 4.3, the complete displacement and stress fields are determined.

It should been mentioned that the current model assumes one-way coupling, i.e., solving heat conduction is independent of mechanical responses and thermo-elastic coupling effects (e.g., stress-dependent conductivity) are neglected.

5. Results and Discussion

In the following examples, numerical results are presented for the thermo-mechanical behaviors of an FGM spherical shell. Unless otherwise stated, the geometry and material properties are given by [34]:

$$r_0=0.5 \mathrm{m}, r_{\mathit{p}}=1 \mathrm{m} ;$$

$$\mathit{E}=(2.1 \times {10}^5-27.5\mathit{T}-0.141{\mathit{T}}^2) {\left({\displaystyle \frac{r}{r_{\mathit{P}}}}\right)}^{\mathit{m}} \mathrm{MPa}$$

$$\alpha =(1.2 \times {10}^{-5}+1 \times {10}^{-8}\mathit{T}) {\left({\displaystyle \frac{r}{r_{\mathit{P}}}}\right)}^{\mathit{m}} °{\mathrm{C}}^{-1}, \mathit{k}=(50.16-0.0293\mathit{T}) {\left({\displaystyle \frac{r}{r_{\mathit{P}}}}\right)}^{\mathit{m}}$$

(38)

where m is the gradient index. The Poisson’s ratio of the FGM is fixed at $\mu =0.3$. As displayed in Figure 4, two loading configurations are analyzed:

(i) load A: Uniform pressure ${\mathit{P}}_{in}$ distributed over the entire internal surface combined with thermal load;

(ii) load B: Uniform pressure ${\mathit{P}}_{ex}$ distributed within the ranges $\mathsf{\theta }\in \left[0, {\mathsf{\theta }}_0\right]$ and $\mathsf{\theta }\in \left[\pi -{\mathsf{\theta }}_0, \pi \right]$ on the external surface.

Figure 4b displays that load B exhibits pressure distribution symmetry about the equatorial plane. By introducing the pressure distribution coefficient λ (relation between λ and ${\mathsf{\theta }}_0$ is $h=r_{\mathit{P}}/\lambda =(1-\cos {\mathsf{\theta }}_0)r_{\mathit{p}}$) for load B, the distribution ranges of the external pressure can be controlled. For example, when $\lambda \to \infty$, the pressure converges to a couple of point loads at the two poles; when $\lambda =1$, the pressure is uniformly distributed over the entire surface. Based on the Legendre polynomials, the external pressure ${\mathit{P}}_{ex}$ can be expanded as

$${\mathit{P}}_{ex}={\displaystyle \sum _{n=0}^{\infty }{a}_n{\mathit{P}}_n(\cos \mathsf{\theta })}$$

(39)

where

$$a_n=\left\{\begin{array}{l}{\mathit{P}}_{ex}/\lambda ,n=0\\ \left[{(-1)}^{n+1}-1\right]\left[{\mathit{P}}_{n+1}\left({\displaystyle \frac{\lambda -1}{\lambda }}\right)-{\mathit{P}}_{n-1}\left({\displaystyle \frac{\lambda -1}{\lambda }}\right)\right]{\displaystyle \frac{{\mathit{P}}_{ex}}{2}},n>0\end{array}\right.$$

(40)

5.1. Validation of the Temperature Solution

First, the convergence of the temperature solution should be examined, since it is obtained using the layer model and iteration method. Consider that the shell with the gradient index $\mathit{m}=2$ is under load A (${\mathit{T}}_{ex}=20 °\mathrm{C}$, ${\mathit{T}}_{in}=200 °\mathrm{C}$).

Table 1. Convergence of the temperature results (unit: °C).

Position	Layer Number	Iteration Step
		j = 1	j = 2	j = 3	j = 4	j = 5
r = 0.625 m	p = 10	98.212	97.788	97.800	97.800	97.800
	p = 50	97.597	97.148	97.161	97.162	97.162
	p = 100	97.572	97.122	97.136	97.136	97.136
	p = 200	97.571	97.121	97.135	97.135	97.135
	p = 400	97.571	97.121	97.135	97.135	97.135
r = 0.75 m	p = 10	53.670	53.764	53.773	53.773	53.773
	p = 50	53.608	53.693	53.702	53.702	53.702
	p = 100	53.606	53.690	53.700	53.700	53.700
	p = 200	53.605	53.690	53.700	53.699	53.699
	p = 400	53.605	53.690	53.699	53.699	53.699

displays the temperature results for different layer numbers p and iteration steps j. The results converge quickly and are accurate to three decimal places when p ≥ 200 and j ≥ 4. When p = 200 and j = 4, the central processing unit (CPU) time is about 0.016 s.

The correctness of the present temperature solution can be examined by comparing with the reported results [34], which are obtained by the finite difference method and Kirchhoff’s transform method. Figure 5 compares the temperature distributions for the shell with ${\mathit{m}}_{\mathit{k}}=2$, ${\mathit{T}}_{ex}=1000 °\mathrm{C}$, and ${\mathit{T}}_{in}=0 °\mathrm{C}$. Excellent agreement between our results and the reported ones can be found.

5.2. Validation of the Displacement and Stress Solutions

The convergence of the displacement and stress solutions is examined. Consider the shell with the gradient index $\mathit{m}=2$.

Table 2. Convergence of the displacement and stress results induced by load A.

Displacement or Stress	Layer Number	Position
		r = 0.6 m	r = 0.7 m	r = 0.8 m	r = 0.9 m
ur (mm)	p = 10	0.266	0.266	0.262	0.253
	p = 50	0.265	0.264	0.260	0.251
	p = 100	0.265	0.264	0.260	0.250
	p = 200	0.265	0.264	0.260	0.250
	p = 400	0.265	0.264	0.260	0.250
	p = 600	0.265	0.264	0.260	0.250
σr (MPa)	p = 10	−41.663	−33.131	−23.736	−12.877
	p = 50	−41.576	−33.029	−23.651	−12.828
	p = 100	−41.574	−33.026	−23.649	−12.827
	p = 200	−41.573	−33.025	−23.648	−12.826
	p = 400	−41.573	−33.025	−23.648	−12.826
	p = 600	−41.573	−33.025	−23.648	−12.826
σθ (MPa)	p = 10	−12.223	3.337	23.244	48.748
	p = 50	−15.638	−1.000	17.670	41.622
	p = 100	−16.064	−1.522	17.014	40.792
	p = 200	−16.278	−1.781	16.688	40.381
	p = 400	−16.384	−1.910	16.526	40.177
	p = 600	−16.420	−1.953	16.472	40.109

displays the results of $u_r$, ${\sigma }_r$, and ${\sigma }_{\mathsf{\theta }}$ induced by load A (${\mathit{P}}_{in}=50 \mathrm{MPa}$, ${\mathit{T}}_{ex}={\mathit{T}}_0=20 °\mathrm{C}$, ${\mathit{T}}_{in}=200 °\mathrm{C}$) for different layer numbers p. The accuracy of the results becomes satisfactory when p ≥ 400. When p = 400, the CPU time is about 1.394 s.

Table 3. Convergence of the displacement and stress results at r = 0.75 m, θ = π/6 rad induced by load B.

Displacement or Stress	Layer Number	Series Term
		n = 10	n = 20	n = 30	n = 40
ur (μm)	p = 10	−85.927	−86.024	−86.015	−86.015
	p = 50	−85.938	−86.034	−86.025	−86.025
	p = 100	−85.938	−86.034	−86.026	−86.025
	p = 200	−85.938	−86.034	−86.026	−86.026
	p = 400	−85.938	−86.034	−86.026	−86.026
	p = 600	−85.938	−86.034	−86.026	−86.026
σr (MPa)	p = 10	−3.817	−3.918	−3.900	−3.900
	p = 50	−3.806	−3.907	−3.888	−3.888
	p = 100	−3.806	−3.906	−3.888	−3.888
	p = 200	−3.806	−3.906	−3.888	−3.888
	p = 400	−3.806	−3.906	−3.888	−3.888
	p = 600	−3.806	−3.906	−3.888	−3.888
σθ (MPa)	p = 10	−22.220	−22.287	−22.277	−22.277
	p = 50	−21.169	−21.234	−21.224	−21.224
	p = 100	−21.040	−21.105	−21.095	−21.095
	p = 200	−20.976	−21.040	−21.031	−21.030
	p = 400	−20.943	−21.008	−20.998	−20.998
	p = 600	−20.933	−20.997	−20.988	−20.988

displays the results of $u_r$, ${\sigma }_r$, and ${\sigma }_{\mathsf{\theta }}$ at r = 0.75 m, $\mathsf{\theta }=\pi /6 \mathrm{rad}$ induced by load B (${\mathit{P}}_{ex}=10 \mathrm{MPa}$, $\lambda =4$) for different layer numbers p and series terms n. The accuracy of the results becomes satisfactory when p ≥ 400 and n ≥ 30. When p = 400 and n = 30, the CPU time is about 1.461 s.

The correctness of the displacement and stress solutions can be examined by comparing with the analytical results reported by Eslami et al. [33]. In the authors’ work, the geometry and material properties of the FGM shell are taken as

$$r_0=1 \mathrm{m}, r_{\mathit{p}}=1.2 \mathrm{m};$$

$$E=200 {\left({\displaystyle \frac{r}{r_0}}\right)}^{\mathit{m}} \mathrm{GPa}, \alpha =1.2\times {10}^{-6} {\left({\displaystyle \frac{r}{r_0}}\right)}^{\mathit{m}} °{\mathrm{C}}^{-1}, \mathit{k}={\mathit{k}}_0{\left({\displaystyle \frac{r}{r_0}}\right)}^{\mathit{m}}, \mu =0.3$$

(41)

where ${\mathit{k}}_0$ is the thermal conductivity at the internal surface. Figure 6 displays the stress distribution obtained by the present method and the reported results for the shell with different values of gradient index m, where the shell is assumed to be under load A (${\mathit{P}}_{in}=50 \mathrm{MPa}$, ${\mathit{T}}_{ex}={\mathit{T}}_0=0 °\mathrm{C}$, ${\mathit{T}}_{in}=10 °\mathrm{C}$). Excellent agreement between our results and the reported ones can be found.

The correctness of the displacement and stress solutions can also be examined by comparing with the finite element method (FEM) solutions obtained using ABAQUS. Here, a FGM shell under load B (${\mathit{P}}_{ex}=10 \mathrm{MPa}$, $\lambda =1$) is considered. The geometry and material properties of the shell are taken as

$$r_0=0.5 \mathrm{m}, r_{\mathit{p}}=1 \mathrm{m}, E=200{\displaystyle \frac{r_{\mathit{p}}}{r}} \mathrm{GPa}, \mu =0.3$$

(42)

In the FEM simulation, the continuous variation in material properties can be achieved by using an auxiliary (non-physical) temperature-dependence [35], and the elastic analysis is performed using the 8-node element C3D8R.

Table 4. Displacement and stress solutions obtained by the present method and the FEM.

Variable	Method	r = 0.6 m	r = 0.7 m	r = 0.8 m	r = 0.9 m
ur (μm)	Present	36.060	34.958	35.931	38.312
	FEM	36.056	34.954	35.927	38.307
	Error (%)	0.011	0.011	0.011	0.011
σr (MPa)	Present	5.962	8.393	9.434	9.862
	FEM	5.975	8.402	9.438	9.863
	Error (%)	0.214	0.104	0.041	0.009
σθ (MPa)	Present	16.835	13.771	12.051	10.974
	FEM	16.861	13.783	12.059	10.981
	Error (%)	0.155	0.086	0.064	0.063

displays the displacement and stress solutions obtained by the present method and the FEM. Excellent agreement can be found between the two methods and the maximum error is only about 0.214%.

5.3. Effects of Material Temperature Dependence

Since the material properties of FGM are functions of temperature (see Equation (38)), the effects of material temperature dependence on the thermo-mechanical behaviors of the shell should be studied. Consider the FGM shell with the gradient index $\mathit{m}=2$ under load A (${\mathit{T}}_{ex}={\mathit{T}}_0=20 °\mathrm{C}$, ${\mathit{P}}_{in}=50 \mathrm{MPa}$). Figure 7 displays the distributions of T, $u_r$, ${\sigma }_r$, and ${\sigma }_{\mathsf{\theta }}$ across the shell thickness for different values of the internal temperature ${\mathit{T}}_{in}$, where the results of T-D and T-I cases are both performed. Note: the T-I material properties are evaluated by the initial temperature ${\mathit{T}}_0$. We can find that the deformations and stresses of the shell are generally increased with ${\mathit{T}}_{in}$. Furthermore, raising ${\mathit{T}}_{in}$ amplifies the discrepancy between the T-D and T-I results, since raising the temperature amplifies the discrepancy of material properties between the two cases.

To further clearly demonstrate the effects of material temperature dependence on the thermo-mechanical behaviors, we plot in Figure 8 the variation in T, $u_r$, and ${\sigma }_r$ at r = 0.75 m and ${\sigma }_{\mathsf{\theta }}$ at r = 0.5 m with the internal temperature ${\mathit{T}}_{in}$ for both the T-D and T-I cases. As expected, T, $u_r$, ${\sigma }_r$, and ${\sigma }_{\mathsf{\theta }}$ linearly change with ${\mathit{T}}_{in}$ in the T-I case yet nonlinearly change in the T-D case. When ${\mathit{T}}_{in}$ reaches 200 °C, the discrepancies in T, $u_r$, ${\sigma }_r$, and ${\sigma }_{\mathsf{\theta }}$ between the two cases reach 8.03%, 1.08%, 5.03%, and 23.06%, respectively. Therefore, the material temperature dependence is a crucial factor in the high-temperature analysis.

5.4. Effects of Gradient Index

Due to the through-thickness gradation of material properties, the effects of gradient index m on the thermo-mechanical behaviors should also be studied. Firstly, the shell under load A (${\mathit{T}}_{ex}={\mathit{T}}_0=20 °\mathrm{C}$, ${\mathit{T}}_{in}=200 °\mathrm{C}$, ${\mathit{P}}_{in}=50 \mathrm{MPa}$) is considered. Figure 9 displays the radial distributions of T, $u_r$, ${\sigma }_r$, and ${\sigma }_{\mathsf{\theta }}$ for different values of m. We can find from Figure 9a that the temperature within the shell is decreased with the increase in m. Therefore, the displacements and stresses are generally decreased with the increase in m. Furthermore, the displacements and stresses vary more sharply across the shell thickness with the decrease in m.

Then the shell under load B (${\mathit{P}}_{ex}=10 \mathrm{MPa}$, $\lambda =4$) is considered. Figure 10 displays the meridian distributions of $u_r$ and ${\sigma }_r$ at r = 0.75 m for different values of m. Due to the symmetry, only a half of the distributions, i.e., $\mathsf{\theta }\in \left[0, \pi \right]$, is plotted. It can be seen that $u_r$ is negative near the two poles (θ = 0, π rad) yet is positive near the equator (θ = 0.5π rad), and ${\sigma }_r$ near the two poles is larger than that near the equator. This is due to the fact that the pressure is acted near the two poles. It is also noticed that $u_r$ is generally increased with the increase in m, whereas ${\sigma }_r$ is generally decreased with the increase in m.

6. Conclusions

Based on the 3-D thermo-elasticity theory, the static analysis of T-D FGM spherical shells under thermo-mechanical loads is conducted. By proposing a layer model, asymptotic temperature, displacement, and stress solutions are derived. The convergence and correctness of the asymptotic solutions are examined. The effects of material temperature dependence and gradient index on the thermo-mechanical behaviors are studied. The results show:

(1)

When the material temperature dependence is neglected, the displacements and stresses exhibit linear variation with the surface temperature. Conversely, consideration of the material temperature dependence induces nonlinear thermo-mechanical behaviors. The discrepancy between the two cases grows progressively with raising surface temperature.

(2)

The gradient index significantly influences the temperature, displacement, and stress distributions. Specifically, the stresses are decreased with the increase in the gradient index.

(3)

When the external pressure is symmetrically distributed near the two poles of the spherical shell, the deformations and stresses exhibit greater magnitude in polar regions than near the equatorial plane.

It should be mentioned that although the thermo-mechanical behaviors of FGM spherical shells are studied, the present model has some limitations. Firstly, the model only focuses on the static behaviors of the shells in the linear elastic regime; however, the creep and dynamic behaviors are not considered. Secondly, the layer model enables asymptotic results of the temperature, displacements, and stresses, but introduces inherent discrete approximation. Higher precision could be achieved by using ‘graded elements’ which enable a continuous variation in properties [35]. These aspects can be explored in future research.