1 Introduction

Functionally graded materials (FGMs) represent a new class of non-homogeneous materials whose physical properties are functions of spatial variables and vary continuously over the whole volume. These materials offer some advantages over conventional composites, such as a reduction in delamination and residual and thermal stresses. The most common type of FGM is composed of ceramic and metal phases. Metal constituents increase their resistance against impacts, while ceramic constituents can contribute to their resistance against heat conduction, corrosion, and erosion. These novel materials have many applications in areas such as heat exchangers, turbine gas blades, military industries, cutting tools, rocket motor components, human prosthesis, dental implants, and both high voltage electrical and magnetic components^[1-7].

Uniform magnetic fields have important applications in various fields of scientific and technological knowledge. Magnetic resonance imaging, magnetic resonance spectroscopy, magnetic induction tomography, and cancer treatment by magnetic fluid hyperthermia are examples of biomedical applications that require a highly uniform magnetic field. In addition, aerospace applications for testing of satellites and sensor calibration need a uniform magnetic field^[8-11]. Researchers have conducted some works in the area of magneto-thermo-elastic for the functionally graded (FG) cylinders and spheres, some of which are referred to here. Analytical and semi-analytical solutions to hollow and solid thick-walled FG cylinders under different loadings were studied by some researchers^[12-18]. Magneto-thermo-elastic analyses of FG cylinders and disks under axisymmetric loading were investigated in order to assess the effects of the non-homogeneity constant, and thermal and magnetic fields on the distributions of stresses and strains^[19-28]. Assessment of thermal, magnetic, and mechanical non-axisymmetric loadings on the behavior of FG cylinders is available in the literature^[29-33]. Recently, an investigation of the effect of hygrothermal loading has drawn great interest from researchers. Some researchers studied the elastic response of FG piezoelectric cylinders under multi-physical loadings, including hygrothermal, mechanical, magnetic, and electric loads^[34-37].

An analytical solution of electro-magneto-elastic problems for FG piezoelectric cylinders and spheres exposed to a uniform magnetic field and subject to mechanical and electrical loadings was carried out by Dai et al.^[38]. Obata and Noda^[39] investigated the effect of the volumetric ratio of constituents and porosity on the steady thermal stresses in hollow FG cylinders and spheres in order to optimize the cylinder and sphere design. Dai et al.^[13] presented the closed-form solution of magneto-elastic behavior of FG cylinders and spheres subject to a magnetic field. In their study, an exact solution was developed for displacements and stresses. They assumed the power-law form for characterizing the material properties of the FGM.

Lutz and Zimmerman^[40] studied the effective thermal expansion and stresses of the FG spheres, and presented an analytical solution for the uniform heat transfer in the spherical bodies whose mechanical and thermal properties were a function of radius. Saadatfar and Rastgoo^[41] developed an analytical solution to the axisymmetric problem of a radially polarized, spherically isotropic piezoelectric hollow sphere subject to uniform internal and external pressures and thermal gradient. A two-dimensional analytical piezothermoelastic solution for an FG hollow sphere with integrated piezoelectric layers as a sensor and actuator subject to non-axisymmetric loads was investigated by Barati and Jabbari^[42]. They employed a feedback gain control algorithm for active control of stress and displacement. Eslami et al.^[43] analyzed the thermo-elastic problem of thick-walled FG spheres. In their work, the temperature varied through the radius. They determined the temperature distribution and the radial and circumferential stresses using heat transfer and Navier's equations. Iaccarino and Batra^[44] analyzed the radial expansion of hollow spheres made of second-order isotropic, elastic, inhomogeneous, and incompressible materials. The static behavior of the electro-magneto-elastic hollow FG spheres subject to hygrothermal loading was studied by Saadatfar and Aghaie-Khafri^[45].

In this paper, an exact solution is presented for the displacement, strain, and stress distributions of rotating FG thick-walled spheres under both mechanical and thermal loadings based on the magneto-thermo-elasticity theory. All material properties are considered to vary along the radial direction according to the power-law function. A second-order ordinary differential equation is obtained by combining equilibrium and constitutive equations. The temperature distribution results from the steady-state heat transfer equation in the spherical coordinates. The effects of parameters such as the non-homogeneity constant, the rotation, the magnetic field, the internal pressure, and the thermal loading on the distributions of stress, strain, and displacement are studied.

2 Problem statement

To derive the governing equations, a hollow sphere with the inner and outer radii r_i and r_o and corresponding temperatures T_i and T_o, respectively, is considered, as shown in Fig. 1, and placed in the uniform magnetic field H=(0, 0, H_ϕ). The sphere is subject to the internal pressure P_i and rotates at a constant angular velocity ω about the z-axis. Material properties such as the elastic modulus E(r)=E₀r^β, the magnetic permeability coefficient μ(r)=μ₀ r^β, the density ρ(r)=ρ₀ r^β, the thermal conductivity k(r)=k₀r^β, and the thermal expansion coefficient α (r)=α₀r^β are assumed to vary along the radial direction. E₀, μ₀, ρ₀, k₀, and α₀ are the material properties at the outer surface, and β is a non-homogeneity constant of the FGM.

denotes the dimensionless radius, and r is a radius of sphere at an arbitrary point.

By applying the magnetic field, the magnetic field h, an electric field e, and the Lorentz force f_r result. It is assumed that the magnetic permeability on the outer surface is equal to the magnetic permeability of its surrounding environment. Therefore, the electrodynamic Maxwell's equations for a conductive elastic body are^{[13, 18]}

(1a)

(1b)

(1c)

(1d)

(1e)

where J denotes the vector of electric current density, and

Applying the magnetic field H=(0, 0, H_ϕ) in the spherical coordinates (r, θ, ϕ) to Eq. (1) yields

(2a)

(2b)

(2c)

(2d)

(2e)

where u indicates the displacement along the radial direction, and H_ϕ is the magnetic field intensity. Given the assumption of axisymmetric loadings (i.e., variables are functions of r only), the Lorentz force resulted from the magnetic field is computed as follows:

(3)

Considering the Lorentz force, the dynamic equation of electromagnetic field for the rotating FG spheres can be written as

(4)

where σ_r and σ_θ are the radial and circumferential stresses, respectively, and ρ is the density at the arbitrary point.

3 Heat transfer problem

It is necessary to determine the temperature distribution in order to compute the thermal stresses for the FG sphere. The steady state condition is considered without internal heat generation in the sphere. Therefore, the axisymmetric heat transfer equation in the spherical coordinate system is defined as

(5)

in which T(r) is the temperature gradient in terms of radius. Thermal boundary conditions on the inner and outer surfaces of the sphere are, respectively, expressed as

(6a)

(6b)

Simplifying Eq. (5), the Cauchy-Euler differential equation is obtained as follows:

(7)

The characteristic equation of Eq. (7) and its corresponding roots are

(8a)

(8b)

(8c)

Therefore, the temperature distribution is determined using the following equation:

(9)

where A and B are thermal constants, computed as follows:

(10a)

(10b)

4 Stress and strain analysis

The radial and circumferential stress relations can be written as^[43]

(11a)

(11b)

where T^* and ϑ denote the reference temperature and Poisson's ratio, respectively. By substituting Eqs. (3), (9), and (11) into Eq. (4), the Cauchy-Euler equation with nonhomogeneous part is achieved. Using

the dimensionless Cauchy-Euler equation is expressed as follows:

(12a)

(12b)

(12c)

(12d)

(12e)

The characteristic equation of Eq. (12a) with corresponding roots is determined as

(13a)

(13b)

(13c)

Therefore, the homogenous solution to Eq. (12a) is expressed as

(14)

Using changes of variable and a little simplification, a particular solution can be computed as

(15)

Regarding Eqs. (14) and (15), the solution to the Cauchy-Euler equation can be determined as follows:

(16)

Using Eq. (16), the radial and circumferential strains are obtained as

(17a)

(17b)

By employing Eqs. (11) and (16), the radial and circumferential stresses are computed as

(18a)

(18b)

The constants X₁ and X₂ are determined using the boundary conditions as follows:

(19a)

(19b)

5 Numerical results and discussion 5.1 Verification

In Fig. 2, the results of current research in the absence of the magnetic field and rotation effects are compared with those obtained from Ref. [43] with very good agreement. As shown in this figure, the variation of non-homogeneity constant β on the results has a significant effect especially on the circumferential stress σ_θ. In this paper, to achieve the results, a sphere with inner and outer radii r_i =1 m and r_o =1.2 m and the corresponding temperatures T_i =10 ℃ and T_o =0 ℃ is considered. Also, the internal and external pressures are P_i =50 MPa and P_o =0 MPa, respectively. Other properties of the outer surface, such as the elastic modulus and the thermal expansion, are considered as E₀=200 GPa and α₀ =1.2×10^-6 1/℃^[43], respectively.

5.2 Current results

In this section, numerical examples are presented in order to illustrate the results. A rotating FG sphere with inner and outer radii r_i =0.5 m and r_o =1 m and the corresponding temperatures T_i =25 ℃ and T_o =100 ℃, respectively, is considered. Also, the internal and external pressures are assumed to be P_i =200 MPa and P_o =0 MPa, respectively. The FGM contains the metal and ceramic phases whose properties are listed in Table 1^[46]. The dimensionless parameters shown in Figs. 4-8 are

The non-dimensional temperature distributions are shown in Fig. 3 for the rotating thick-walled sphere. It is observed that with an increase in the non-homogeneity constant, the heat transfer increases.

The effects of the non-homogeneity constant, the magnetic field, the internal pressure, the angular velocity, and the temperature gradient on the stresses, strains, and displacement are studied. Regarding Figs. 4(a) and 4(b), by raising the non-homogeneity constant, the absolute values of the radial displacement and strain increase. The minimum value of the radial strain occurs at the outer radius of the sphere. It is clear from Fig. 4(c) that the absolute values of circumferential strain are raised by increasing the non-homogeneity constant. Figure 4(d) reveals that the absolute values of radial stress decrease with an increase in the non-homogeneity constant in the range of β < 0, whereas the absolute values of the radial stress tend to be increased by raising the non-homogeneity constant in the range of β >0. It is also obvious from Fig. 4(e) that the increase in the absolute values of circumferential stress is proportional to an increase in the non-homogeneity constant.

It is shown in Fig. 5(b) that the absolute values of the radial strain decrease by increasing the magnetic field intensity. Figures 5(a), 5(c), 5(d), and 5(e) show that the absolute values of the radial displacement, circumferential strain, and radial and circumferential stresses also decrease as the magnetic field intensity increases.

Figure 6(a) displays that the increase in the absolute value of the radial displacement is associated with an increase in the internal pressure. The maximum and minimum values of the displacement occur at the outer and inner radii of the sphere, respectively. According to Fig. 6(b), the absolute values of radial and circumferential strains decrease and increase by raising the internal pressure, respectively. It can be also seen from Figs. 6(d) and 6(e) that the absolute values of radial and circumferential stresses are diminished as the internal pressure increases.

It is illustrated in Fig. 7(a) that the absolute values of the radial displacement increase with an increase in the angular velocity. Figures 7(b) and 7(c) show that with the increase in the angular velocity, the absolute values of the radial and circumferential strains decrease and increase respectively. Regarding Fig. 7(d), the absolute values of the radial stress decrease by raising the angular velocity. It is evident from Fig. 7(e) that the absolute values of the circumferential stress also decrease by increasing the angular velocity in the range of ω≤15 000 r/min, whereas it tends to reverse the behavior out of this range.

Considering Fig. 8, the effects of the temperature gradient on the displacement, strains, and stresses are investigated. It is evident from Fig. 8(a) that the absolute values of the radial displacement increase with an increase in the temperature. As can be observed from Fig. 8(b), with an increase in the temperature, the absolute values of the radial strain increase. Figures 8(c) and 8(d) show that the absolute values of the circumferential strain and the radial stress also increase by raising the temperature. It is obvious from Fig. 8(e) that by increasing the temperature, the absolute values of the circumferential stress increase.

6 Conclusions

In this paper, an exact solution to the magneto-thermo-elastic problem of rotating thick-walled FG spherical pressure vessels is presented. The effects of some parameters such as the non-homogeneity constant, the angular velocity, the internal pressure, the temperature gradient, and the magnetic field are studied on the distributions of the stresses, strains, and displacement of the FG spherical pressure vessel. The results represent that all the studied parameters have a significant effect on the distributions of the stresses, strains, and displacement. The increase in the parameters such as the non-homogeneity constant, the internal pressure, and the angular velocity increases the absolute values of stresses, strains, and displacement. The values of the displacement, strains, and stresses decrease significantly due to the increase in the magnetic field intensity which makes their distributions smooth. It is noteworthy that the effect of the magnetic field can be ignored for the values less than 10⁷ A/m. Also, the radial displacement and stress increase and decrease with the increase in the temperature, respectively. Therefore, with the suitable selection of mentioned parameters, one can achieve the proper distributions of the displacement, strains, and stresses, and design the optimum spherical pressure vessel.