1 Introduction

The effects of the fully coupling interactions among the magnetic, thermal, and mechanical fields in the diverse industrial areas in designing heating machine, heat exchangers, space shuttles, rockets, etc. has aroused much attraction in many engineering branches^[1-2]. The mechanical components in the mentioned areas under severe temperature environment and various loadings impose different kinds of stresses in the structures. For this reason, stress assessment must be in the top priorities for engineering designers.

To control and reduce the stresses in mechanical structures and improve their capabilities, functionally graded materials (FGMs) are introduced. FGMs are characterized by the variations in the composition and structure over the volume to the specific function to conquer the delamination caused by the sharp changes in traditional composite materials. FGMs offer great promise in withstanding high thermal gradient and toughness and avoiding stress corrosion cracking, fatigue, and fracture in the structures.

The thermoelectricity analyses of FGMs are brought. Xin et al.^[3] investigated the thermos-elastic behavior of the FGM thick-walled tube under mechanical and thermal loads. The elasticity solution based on the Voigt method was presented, and the effects of the volume fraction and inhomogeneity on the parameters were shown.

Alibeigloo^[4] carried out the thermo-elastic analysis of a functionally graded composite cylindrical panel under the combination of thermal and mechanical loads, and used the Fourier series expansion and state space technique to solve the problem. Jabbari et al.^[5] performed a semi-analytical solution for the axially functionally graded rotating thick cylindrical shell subjected to thermal and internal non-uniform pressure, and used the multi-layer method (MLM) to solve the governing equations.

Kiani^[6] investigated the effects of the surface energy, temperature, power law index of FGMs on the thermo-elastic responses of a rotating FGM nanoplate. The nanoplate was divided into N annular domains. With the consideration of the interface boundary conditions, the direct method was used to solve the problem. Jamaludin et al.^[7] performed the thermo-elastic analysis of an FGM rectangular plate, and used the finite element method (FEM) to solve the problem. The effects of different values of the grading parameter determining the FGM material properties on the displacement and stresses were presented. Prakash and Ganapathi^[8] investigated the asymmetric free vibration characteristic and thermo-elastic stability of functionally graded circular plates with the FEM.

Feng et al.^[9] demonstrated the effectivity of an edge-based smoothed FEM to analyze the thermo-mechanical of FGM cylindrical vessels. Ching and Yen^[10] used the meshless local Petrov-Galerkin method (MLPG) to analyze the two-dimensional (2D) thermo-elastic problem of FGMs.

Chen and Lin^[11] carried out a stress analysis of the thick FGM cylindrical and spherical pressure vessels, solved the ordinary differential equation numerically, and showed the effects of the FGM index on the stress distribution along the radial direction. Zafarmand and Kadkhodayan^[12] studied the three-dimensional (3D) static and dynamic responses of a thick sector made of 2D-FGMs, and used the 3D graded FEM based on Hamilton's principles and the Rayleigh-Ritz energy method to solve the problem. Jabbari et al.^[13] studied a one-dimensional (1D) thermo-elastic analysis in an FGM cylindrical shell, and used the direct method to solve the Navier and heat conduction equations.

The magneto-thermo-elastic properties of FGMs have been widely studied. Loghman and Pars^[14] presented a magneto-thermo-elastic stress analysis of a thick double-walled cylinder composed of FGMs, studied the homogeneous layers, and used the direct method to solve the existing equations. Mehditabar^[15] investigated a 3D magneto-thermo-elastic analysis of an FGM truncated conical shell, and used the differential quadrature method (DQM) to solve the equations.

Zenkour and Abbas^[16] discussed the electro-magneto-thermo-elastic problem of an infinite FGM cylinder shell, and used the FEM to solve the problem. Zhang and Li^[17] presented the buckling and vibration of the FGM magneto-electro-thermo-elastic cylindrical shell, and used the function approximation corresponding to a layer wise theory to solve the problem.

Ezzat and Atef^[18] investigated a 1D transient problem of generalized magneto-thermo-elasticity in a viscoelastic FGM layer, transformed the resulting governing equations into the Laplas transform domain, and used the direct method to obtain the numerical solution. Kumar et al.^[19] investigated the interaction of the thermo-mechanical sources in a magneto-thermo-elastic rotating medium based on the Gren-Naghdi thermo-elastic theories, and used the Laplace and Fourier transforms and then the numerical inversion technique to obtain the numerical results. Rad and Shariyat^[20] modelled the FGM annular plate with variable thickness on the non-uniform elastic foundation permeated in magnetic field and undergoing the transverse temperature gradient, heat flux, and asymmetric mechanical loading, and used the DQM to achieve the numerical results.

The present work aims to investigate the 3D functionally graded cylindrical shell immersed in primary magnetic field and under non-uniform internal pressure with a rapid temperature change at the inner surface. The boundaries and the governing heat conduction equations in terms of the displacement, magnetic, and temperature parameters are derived. An efficient discretization technique, i.e., the DQM, is proposed to convert a set of partial differential equations to a set of algebraic equations. By numerically solving the resulting transformed algebraic equations, the effects of the inhomogeneity and thermal gradient on the distributions of the stresses, displacement, and induced magnetic field are graphically displayed.

2 Governing equations

The schematic configuration of a cylindrical FGM shell subjected to the combined magnetic, thermal, and mechanical loading is demonstrated in Fig. 1. The cylindrical coordinate system (r, θ, and z) for the geometrical description is considered. The axial length, inner radius, and thickness of the shell are determined by L, R₁, and h_sh, respectively.

The inner surface of the perfectly conducting cylinder shell is assumed to be permeated by an initial magnetic vector H=(H₀, 0, 0) in the cylindrical coordinate uniformly distributed along the circumferential direction and the z-direction (see Fig. 1).

Moreover, the shell subjected to the temperature gradient field caused by the temperature difference between the inner surface and the outer surface of the cylinder and the inside of the shell is exposed to the asymmetric internal pressure as a function of the circumferential coordinate, i.e., P₀ cos θ.

The effective Young's modulus, heat conductivity, thermal expansion, and magnetic permeability of an FGM shell are supposed to vary continuously in the radial direction according to the power law rule as follows:

(1)

where E₀, K₀, α₀, and μ₀ are the corresponding material properties at the inner surface, and n is the power law index, whereas Poisson's ratio υ is assumed to be constant within the thickness.

Considering u, v, and w as the corresponding mechanical displacements in the r-, θ-and z-directions, respectively, we can express the six components of the strain in the cylindrical coordinate as follows:

(2)

The constitutive thermo-elastic relations between the stress and the strain for isotropic materials in the matrix form are

(3)

where σ_ij, τ_ij, and ε_ij are the dimensional normal and shear Kirchhoff stress components and the strain components, respectively, and T and α are the temperature distribution along the thickness and the thermal expansion coefficient, respectively.

Considering the isotropic material properties and introducing λ and G in terms of the elastic module E and Poisson's ratio υ, we can express the elastic constants c_ij as follows:

(4)

Using the strain-displacement equation (2), we can define the stress components of an isotropic shell in terms of the displacement components and the temperature parameter as follows:

(5)

where

To obtain the force inserted by the magnetic field on a perfectly conducting elastic body (the Lorentz force), we need to use the constitutive linearized electromagnetic Maxwell equations as follows (the electric displacement is neglected)^[21]:

(6)

where h and J represent the induced magnetic field and current electric density vectors, respectively.

To compute the induced magnetic field within the shell according to the above formula, the following identify must be invoked^[22]:

(7)

Considering an initial magnetic vector H=(H₀, 0, 0) and the corresponding mechanical displacements u (x, y, z), v (x, y, z), and w (x, y, z) in the r-, θ-, and z-directions, respectively, the components of the induced magnetic field h and the current electric density J are obtained as follows:

(8)

Consequently, the Lorentz body force can be represented as follows:

(9)

where μ is the magnetic permeability.

The 3D stress equilibrium equations in the cylindrical coordinate with the body force are expressed as follows:

(10a)

(10b)

(10c)

Substituting Eq. (5) into Eqs. (10a), (10b), and (10c), we can derive the governing equilibrium equations in terms of the displacement components, magnetic parameters, and temperature of the shell (the Navier equations) as Eqs. (11a), (11b), and (11c), respectively.

The equilibrium equation in the r-direction can be expressed as follows:

(11a)

The equilibrium equation in the θ-direction is defined by

(11b)

The equilibrium equation in the z-direction is written as follows:

(11c)

3 Boundary equations

For the sake of simplicity, the mechanical, thermal, and magnetic boundary conditions are expressed as follows:

(12)

The corresponding boundary conditions to the top and bottom of the FGM cylindrical shell are assumed to be as follows:

(13)

The heat conduction steady state equations without considering the internal heat source in the cylindrical coordinate^[23] are defined by

(14)

4 Method of solution

The DQM is used as the distinct numerical technique to convert a set of resultant partial differential equations to a set of algebraic equations. The partial derivatives of the functions with respect to the coordinate system are approximated by a weighted linear sum of the functional values at all grid points within the domain under consideration. The nth-order derivatives of the function f (x) in the cylindrical coordinate are defined by^[24]

(15)

where Aⁿ, Bⁿ, and Cⁿ denote the weighting coefficients for the nth-order derivatives of the function f (x) in the radial direction at the mesh point r_i, the circumferential direction at the mesh point θ_i, and the axial direction at the mesh point z_i, respectively.

The important step in this method is to determine the weighting coefficient. In this work, Quan and Chang's polynomial expansion based differential quadrature (PDQ) is used to compute the weighting coefficients in the r-and z-directions, and the Fourier expansion based differential quadrature (FDQ) is used to obtain the weighting coefficients in the θ-direction. The explicit computation details of the differential quadrature coefficients for the first-and second-order derivatives are brought in the following equations^[23]:

(16a)

(16b)

All the spatial derivatives in the governing, boundary, and heat transfer partial differential equations are discretized in the series form by use of the above defined weighting coefficients, which are derived in Eqs. (17a), (17b), (17c), (17d), and (17e), respectively.

The differential quadrature form of Eq. (11a) is

(17a)

Using the DQM to discretize the derivatives in Eq. (11b), we have

(17b)

Using the DQM method to discretize the spatial derivatives in Eq. (11c), we have

(17c)

The discretized form of the boundary conditions is

(17d)

The differential quadrature form of the stead-state heat conduction equation is

(17e)

5 Numerical results and discussion

The distribution of the grid point selected as the roots of the Chebyshev-Gauss-Lobatto polynomial in the domain can be determined by the following equations^[23].

In the r-direction,

(18a)

In the θ-direction,

(18b)

In the z-direction,

(18c)

In this examined case, the properties of the geometrical shape for the shell are considered as follows:

The shell is subjected to the magnetic field H₀=5 × 10⁸ A/m, and the magnetic constant or permeability of free space has the value u₀=4π × 10^-7 H/m. In this study, the material chosen for the numerical computation is aluminum. The thermal and mechanical properties of the material are

(19)

For simplicity, let us define the normalized parameters as follows:

(20)

First, the convergence and accuracy of the present method for the 1D thick cylindrical shell with the geometrical and mechanical properties R₁=1, h_sh=0.2, E₀=200 MPa, α₀=1.2 × 10^-6, υ=0.3 and the thermal and mechanical boundary conditions σ_rr (R₁)=-50 MPa, σ_rr (R₁+h_sh)=0, T_i=10 ℃, T_o=0 ℃ are examined and demonstrated in Fig. 2. It is found that the magnitudes of the dimensionless radial displacement by increasing the number of the sampling points along the radial direction are rapidly converged. The proper number of the grid point 20 is selected. The converged result is in excellent agreement with the available data in Ref. ^[13]. Figure 3 shows more validation of the present method by comparing the circumferential variations of the non-dimensional stress along the thickness carried out by the DQM with the results reported in Ref. ^[13]. We can see that reasonably good agreement between the results exists. To check the obtained numerical results with the DQM, another analysis is carried out with the commercial finite element method code ANSYS. The cylindrical shell is discretized by use of the Solid5 element which is defined by eight nodes and has the combined capability of 3D magnetic, thermal, and mechanical couplings. In the case of n=0 and in the positions of z=L/2 and r=R₁+h_sh/2, the circumferential distribution of the dimensionless displacement u comparison is made among the results of the DQM and those of the FEM in Fig. 4. From the figure, it is found that excellent solution agreement between the two methods exists.

The numerical results are evaluated for different values of the power law index n in Figs. 5-10 to demonstrate the effects of the material inhomogeneity on the distribution of the above-mentioned parameters through the circumferential direction. It should be noted that all the numerical results are computed and given for the middle of the axial length z=L/2 and the thickness r=R₁+h_sh/2. The thermal boundary conditions, inside the shell temperature T_i=100 ℃ and outside the shell temperature T_o=0, are considered, and the internal pressure is P₀=150 cos θ MPa.

Figure 5 illustrates the variations of the non-dimensional radial displacement field u along the circumferential direction for different values of the power law index. From the figure, it can be figured out that by changing n from 2 to -2, the absolute value of the displacement increases, and creates more uniform radial displacement along the circumferential direction.

The distribution of the dimensionless radial stress through the circumferential direction is depicted in Fig. 6. It can be observed that by changing the inhomogeneity constant n from -2 to 2, the absolute value of the radial stress will increase along the circumferential direction. In general, the individual effects of the power law index on the radial stress are relatively small. Figure 7 reveals the variations of the dimensionless circumferential stress along the circumferential direction. It is evident that changing n from -2 to 2 leads to an increase in the absolute amplitude of the dimensionless hoop stress in most regions across the circumferential direction. For all the cases under consideration, the achieved results reveal that the hoop stress has larger maximum values than other stress components inferring plays pronounced in the structural failures.

Figure 8 plots the circumferential variations of the shear stress τ_rz with different values of the power law index n. From the plotted results, it can be inferred that, the effect of inhomogeneity on the shear stress τ_rz is minor, and the power law index changes, causing slightly changes in the magnitude of the shear stress. The circumferential distribution of the induced magnetic field for various values of the inhomogeneity constants is plotted in Fig. 9. It is not difficult to see that the absolute values of the induction caused by the magnetic field in the shell increases by changing n from 2 to -2.

The temperature distribution versus the radial direction for different values of the inhomogeneity constant is shown in Fig. 10. The non-dimensional temperature distribution reduces with changing the power law index from -2 to 2. It is worthwhile mentioning that the prescribed boundary conditions for the temperature at the inner tube and the outer tube are fully satisfied. The effects of the thermal load or temperature difference at the two surfaces of the cylinder on the radial displacement, stress components, temperature, and induced magnetic field for the material property identified by n=1 are presented in Figs. 11-15. In all the cases, it is assumed that the outer surface temperature is assumed to be fixed, i.e., T_o=0 ℃, and the inner temperature value changes as a parameter.

Figure 11 reveals the effect of the temperature gradient between the inner surface and the outer surface on the distribution of the radial non-dimensional displacement u through the θ-direction. As we expected, increasing the magnitude of the temperature gradient accompanies with a sharp increase in the radial displacement absolute values. The normalized radial stress for various thermal loadings across the circumferential direction is demonstrated in Fig. 12. A close scrutiny of the graphs indicates that increasing the temperature difference between the two sides of the shell adds compressive stresses in the shell.

The variations of σ_θθ along the circumferential direction due to the temperature gradient are indicated in Fig. 13. It is clearly shown that the normalized stress σ_θθ in the functionally graded cylindrical shell decreases when the temperature difference increases, which means that a compressive stress is added. Besides, Figs. 12 and 13 show that the effect of the temperature difference is more significant on the circumferential stress than the radial stress.

To examine the effects of the thermal loading magnitude on the circumferential distribution of the shear stress τ_rz, Fig. 14 is plotted. It is obvious that the graph shifts towards the positive direction of the vertical coordinate. It means that the amounts of the shear stress τ_rz and the absolute value of the shears stress in most regions increase. Figure 15 presents the effects of the thermal loading on the evolution of the induced magnetic field throughout the circumferential direction in the functionally graded cylindrical shell. From the figure, we can see that increasing the temperature difference leads to an increase in the magnetic field and an decrease in the absolute magnitude of the cylindrical shell.

6 Concluding remarks

The 3D magneto-thermo-elastic interaction of the cylindrical shell made FGM is studied. The properties of the FGM are regarded as heterogeneous through the thickness coordinate and according to the power law distribution. The formulation begins with the extraction of the governing equations in terms of the displacement components and the magnetic and temperature parameters, and then the DQM is used to discretize the existing partial differential equations and transform them to a set of algebraic equations. With the convergence study, we obtain the rapid convergence and relatively small grid point toward the analytical results, and prove the effectiveness of the present method. Subsequently, the individual effects of the material inhomogeneity and rapid temperature change on the magneto-thermo-elastic response of the cylindrical shell are displayed and discussed. From the obtained results, it can be concluded that, the amplitude of the stress components and the perturbation of the magnetic field vector can be controlled by selecting an appropriate inhomogeneity constant n. This new composite class can be a perfect choice for engineering designers to tailor the FGM cylindrical shell for special requirements.