Document Type : Research Article
Authors
^{1} Mechanical Engineering Department, Shri Shankaracharya Technical Campus SSGI Bhilai, India
^{2} Mechanical Engineering Department, Shri Shankaracharya Technical Campus, SSGI Bhilai, India
^{3} Department of Mechanical Engineering, National Institute of Technology Raipur, CG, India
^{4} Mechanical Engineering Department, Indian Institute of Technology Bombay, Mumbai India
Abstract
Keywords
Main Subjects
Thermo–Elastic Stresses and Deformation Analysis of FG Rotating Hollow Spherical Body
a Department of Mechanical engineering, SSTC, SSGI, Bhilai, 490020, India
b Department of Mechanical engineering, NIT Raipur, Raipur, 492010, India
c Department of Mechanical engineering, IIT Bombay, Mumbai, 400076, India
KEYWORDS 

ABSTRACT 
Functionally graded material; Hollow spherical body; Axisymmetric body; Analytical method; Grading index. 
In this paper, a generalized solution for 1D steadystate thermomechanical analysis of the FG rotating hollow spherical body is presented. Deformation and stresses are calculated for a spherical body subjected to rotation, gravitation force, and uniform heat generation. Temperature distribution with uniform heat generation to the spherical body is assumed to vary along the radius. General thermal and mechanical boundary conditions at the inner surface and outer surfaces of the hollow spherical body are applied. Material properties are assumed as a power function of the radius with grading indices ranging from 2 to 3. Governing differential equation for the FG spherical body is developed and solved analytically. The obtained results are verified with benchmark results and are found to be in very good agreement. The result shows that deformation and stresses in the FG body are less compared to the homogeneous material body and the same is reported to decrease with increasing value of the grading parameter. 
Functionally graded materials (FGM) are a special group of heterogeneous composite materials with mechanical properties varying continuously from surface to surface at a macroscopic level. Thermomechanical stresses in the functionally graded thick spherical body are investigated by, R. Poultangari, M. R. Eslami, M.H. Babaei [1] wherein the performance of a thick hollow spherical body of functionally graded material under 1dimensional steady state distributed temperature with a general thermomechanical type of boundary conditions is reported. Stresses and deformations in rotating functionally graded material pressurized thick hollow cylinder under thermal loading are investigated by, G. H. Rahimi and M. Zamani Nejad [2]. Effect of material gradient on stresses of the thick functionally graded spherical pressurised body using exponentially distributed material grading is reported by M Gharibi and M. Zamani Nejad [3]. A novel approach to stress analysis of pressurized functionally graded disc, cylinder, and spheres is established by N. Tutuncu, B. Temel[4]. A functionally graded hollow cylindrical body under pressure and thermal loading conditions under the effect of material parameters on stresses and temperature distributions are presented by M. Gulgec and C. Evci [5]. Elastic analysis of rotating spherical body, cylindrical body, and disc of variable thickness are reported by A. M. Zenkour [6]. Thermomechanical and thermoelastic stresses were analysed in [7  10]. 2D thermal and elastic behaviour of functionally graded cylindrical body is studied and reported by Ghannad, Yaghoobi [11], wherein axisymmetric functionally graded cylindrical body subjected to external heat and pressure in the inner surface is reported. A spherical pressure vessel is designed and analysed using FEM by Afkar et al. [12]. Multiscale hybrid disc resting on nonlinear elastic foundation under nonlinear frequency and extremely large oscillation is investigated by Ali Shariati et al. [13]. A shear deformation theory (refined quasi3D theory) for the thermal and mechanical study of FG sandwich plates resting on two parameters elastic foundation is presented by Abdelkader Mahmoudi et al. [14].
Fig. 1. Hollow spherical body
Nonlinear thermalinstability of electrically functionally graded GPLRC disc using GDQ method is investigated by M.S.H.AlFurjan et al. [15]. Four variable quasi 3D HSDT is used for analytical modelling of vibration and bending of thick composite plates by Mokhtar Khiloun et al. [16]. Using a solidstate process, an application review of a functionally graded fabricated disk is reported by R. Madan and S. Bhowmick [17]. Limit elastic analysis of EFG materialmodeled rotating disc subjected to thermomechanical properties is studied by R. Madan et al. [18]. Thermal stresses induced due to internal heat generation (nonuniform heat) in FG hollow spherical bodies are reported by S. P. Pawar et al. [19]. Investigation of numerical, analytical, and experimental stress of spherical body having large volume is presented by Radovan Petrovic et al. [20].
In the present study, the deformation and changes in stresses of the functionally graded hollow spherical body are investigated. The problem is analytically solved using an inhouse source code implementing the Navier equation for body force, rotation, and constant heat generation. The validation of the present study is carried out with existing literature. Corresponding to rotational speed, body force, and uniform heat generation in the spherical body, stress, and deformation are estimated. The existing results are reported in dimensionless form. There is a vast application of functionally graded spherical bodies such as submarine, pneumatic and hydraulic reservoirs, storage vessels, oil refineries, petrochemical plants, domestic hot water tanks, pressure reactors, autoclaves, etc.
Considering a rotating hollow spherical body made of functionally graded material, wherein the material properties of the body are assumed to be a power function of radius ‘r’. Stress, strain, and displacement relations are given by[1]

(1) 

(2) 
Stress strain relations [1] being:

(3) 

(4) 
T(r) is determined from the heat conduction equation in section 3. The equilibrium equation in the radial direction, including the inertia term and body force, is given by,

(5) 
The power law is used to describe the material properties of a hollow spherical body which are given by[19],

(6) 

(7) 

(8) 

(9) 

(10) 
Here, , , q(r) are modulus of elasticity, thermal expansion coefficient, thermal conduction coefficient, density, and heat generation at radius ‘r’ respectively. , , are material properties as described above at radius ‘a’
Using eq.(1) to (10), the Navier equation, in terms of displacement, is given by

(11) 





in above eq. (11),

(12) 
The ‘onedimensional heat conduction equation’ with ‘heat generation’ in ‘steady–state’ condition in spherical coordinate is as follows [19].

(13) 
Thermal boundary conditions[19] are:

(14) 

(15) 
where, is the temperature at radius ‘r’, is the temperature at ‘a’ and is the temperature at ‘b’.
Differentiating above eq. (13) of heat conduction, we get the Navier equation for temperature
^{ } 
(16) 
where,

(17) 

(18) 

(19) 

(20) 

(21) 

(22) 
P_{3} and P_{4} are roots of the general solution of Eq. (16). After solving Eq. (16) analytically it gives,

(23) 
^{ } 
(24) 
where,


Using the boundary conditions, the value of and yields

(25) 

(26) 
After solving the function T(r) in above section 3, the value of T(r) is put in eq. (11)

(27) 

where,

(28) 

(29) 

(30) 

(31) 

(32) 

(33) 

(34) 
Equation (27) is the Navier equation, which is a nonhomogeneous Euler differential equation. Assuming general solution, as

(35) 
Substituting the eq. (35) in homogeneous form of eq. (27)

(36) 
The above eq. (36) has two real roots And As,

(37) 
Now, the general solution is

(38) 
Assuming the particular solution in the form,

(39) 
Substituting the above Eq. (39) in eq. (27), we get,





(40) 



Equating the coefficients of identical power in above eq. (40), we have

(41) 

(42) 

(43) 

(44) 
Complete solution for displacement function is given by,

(45) 
Thus

(46) 

Substituting eq. (46) in eq. (1) to (2), the strains and stresses are obtained as,

(47) 

(48) 

(49) 
To determine the constants and , the boundary conditions for stress profile are used. Considering the mechanical boundary conditions in the radius ‘a’ and outer radius ‘b’ [1]

(50) 
Substituting the above eq. (50) in eq. (49), the integration constants become:

(51) 

(52) 
where,

(53) 

(54) 

(55) 

(56) 
, 
(57) 

(58) 

(59) 
The numerical values of different parameters considered in the work are as follows: the inner and outer radius of the hollow spherical body are
a = 1 m, b = 1.2 m, Poisson's ratio 𝜗 = 0.3 since it is considered that material properties are in accordance with eq. (6) to (10). The internal properties of the hollow spherical body are as follows: modulus of elasticity GPa, thermal coefficient of expansion 1.2*10^{6} per ^{0}C, thermal conduction coefficient k_{a} = 15 W/mK, Density kg/m^{3}, Heat generation q = 50*10^{3 }kJ/m^{3} and Gravity g = 9.81 m/s^{2 }Rotation ω = 50 rad/s. Thermal boundary conditions are taken as T (a) = 10 ^{0}C and T (b) = 0 ^{0}C. Mechanical Boundary conditions are taken as internal pressure of 50 MPa and External pressure = 0; material grading indices ‘n’ are choosen as 2 to 3 and are identical but [1].

Fig. 2. Variation of elasticity modulus along the radius 

Fig. 3. Radial variation of thermal expansion coefficient 

Fig. 4. Variation of density along the radius 

Fig. 5. Radial variation of thermal conduction coefficient 
The above graphs show the material property variation wherein Fig. 2 is for elastic modulus, Fig. 3 is for thermal expansion coefficient, Fig. 4 is for material density and Fig. 5 for thermal conduction coefficient for n = 2 to 3.
For n = 1 to 3 the value of material properties are in increasing order from radius a to radius b while for n = 1 to 2 the value of material properties are in decreasing order from radius a to radius b.

Fig. 6. Variation of temperature along the radius 

Fig. 7. Variation of radial displacement 

Fig. 8. Variation of radial stress 

Fig. 9. Variation of tangential stress 

Fig. 10. Variation of vonMises stress for b/a = 1.2 
Graphs plotted in Fig. 6 to Fig. 10 show the validation of the present study, which are found to be in good agreement with reference [1] for grading indices ranging from n = 2 to 3. It is shown in Fig. 6 that the temperature varies inversely with the radius and grading indices, too. In Fig. 7, radial displacement is plotted and is observed to be inversely proportional to the grading index ‘n’. In Fig. 8, radial stresses are plotted and are reported to vary directly with radius. Fig. 9 reports tangential stress that is decreasing radially outward for n < 1, but increasing outward for n > 1 and constant throughout the thickness for n = 1. The vonMises stress is investigated and plotted in Fig. 10. It is clearly observed from the graph that VonMises stress is inversely proportional to the grading index up to r/a = 1.10 beyond which there is a reversal in variation.
Case 1: Rotating FG Spherical Body
Herein, a functionally graded rotating hollow spherical body is investigated under the effect of rotation in a hollow spherical body. Fig. 11 reports the temperature distribution for grading indices ranging from n = 2 to 3 and it is observed that temperature varies inversely to the grading index for FG rotating hollow sphere.
In Fig. 12, it is shown that radial displacement is inversely proportional to the grading index ‘n’. Fig. 13 shows the distribution of radial stress that is directly proportional to the radius i.e. temperature at outer radius is higher than inner radius but radial stress is inversely proportional to the grading index ‘n’. Fig. 14 reports that tangential stress is decreasing radially outward for n < 1, but increasing outward for n > 1 and constant throughout the thickness for n = 1. The vonMises stress is investigated in Fig. 15. It is evident from the figure that VonMises stress is inversely proportional to the grading index up to r/a = 1.08 (approx.) beyond which the vonMises stress becomes directly proportional to the grading index.

Fig. 11. Variation of temperature 

Fig. 12. Variation of radial displacement 

Fig. 13. Variation of radial stress 

Fig. 14. Variation of tangential stress 

Fig. 15. Variation of vonMises stress 
Case 2: FG Hollow Spherical Body Subjected to Gravitational Force
The effect of gravitational force on the functionally graded hollow spherical body is investigated and reported in this subsection. The distribution of temperature for different grading indexes ‘n’ is shown in Fig. 16. Similar to case 1, the temperature distribution is inversely proportional to the radius of the hollow spherical body and grading index too. The distribution of radial displacement shown in Fig. 17 is inversely proportional to the radius of the hollow spherical body and grading index too. The radial stress distribution is shown in Fig. 18 wherein it is observed to be directly proportional to the radius but inversely proportional to the grading index of the hollow spherical body as in the previous case. The tangential stress distribution is plotted in Fig. 19 wherein it is observed to decrease along the radius for n < 1, but increases outward for n > 1 and remains constant throughout the thickness for n = 1 similar to case 1.
The variation in VonMises stress is plotted in Fig. 20. It is clearly observed from the graph that VonMises stress is inversely proportional to the grading index up to r/a = 1.09 (approx.) beyond which there is a reversal in variation.

Fig. 16. Variation of temperature 

Fig. 17. Variation of displacement 

Fig. 18. Variation of radial stress 

Fig. 19. Variation of tangential stress 

Fig. 20. Variation of vonMises stress 
Case 3: FG Spherical Body Subjected to Constant Heat Generation
In this section, the effect of constant heat generation on the displacement and stresses is investigated for the hollow spherical body. The results are reported in a similar sequence of graphs as mentioned for the remaining cases. (Fig. 21 – Fig. 25). The reversal in the variation of VonMises stress is observed to occur at r/a = 1.09 (approx.) as shown in Fig. 25.

Fig. 21. Variation of temperature 

Fig. 22. Variation of displacement 

Fig. 23. Variation of radial stress 

Fig. 24. Variation of tangential stress 

Fig. 25. Variation of vonMises stress 
Case 4: FG Spherical Body Subjected to Gravitational Force and Rotation
Herein, the effect of gravitation and rotational force on displacement and stresses of the functionally graded hollow spherical body is investigated and the results are reported in Fig. 26  Fig. 30 in a similar sequence of graphs.

Fig. 26. Variation of temperature 

Fig. 27. Variation of displacement 

Fig. 28. Variation of radial stress 

Fig. 29. Variation of tangential stress 

Fig. 30. Variation of vonMises stress 
Case 5: Rotating Spherical Body with Constant Heat Generation
The displacement and stresses for a functionally graded hollow spherical body under the effect of centrifugal loading and constant heat generation are reported herein (Fig. 31Fig.35).

Fig. 31. Variation of temperature 

Fig. 32. Variation of displacement 

Fig. 33. Variation of radial stress 

Fig. 34. Variation of tangential stress 

Fig. 35. Variation of vonMises stress 
Case 6: FG Spherical Body Subjected to Constant Heat Generation and Gravitation
Herein, the effect of constant heat generation and gravitation on the stresses and displacement field of the hollow spherical body is investigated and reported (Fig. 36Fig. 40).

Fig. 36. Variation of temperature 

Fig. 37. Variation of displacement 

Fig. 38. Variation of radial stress 

Fig. 39. Variation of tangential stress 

Fig. 40. Variation of vonMises stress 
Case 7: Rotating FG Hollow Sphere Under Gravitation and Constant Heat Generation
The stresses and displacements of a rotating hollow spherical body with gravitation and constant heat generation are investigated and reported in Fig. 41Fig. 45.

Fig. 41. Variation of temperature 

Fig. 42. Variation of displacement 

Fig. 43. Variation of radial stress 

Fig. 44. Variation of tangential stress 
The variation of VonMises stresses for various loading conditions as reported in cases 17 in FG hollow spherical body for b/a = 1.2 is reported in Fig. 15, Fig. 20, Fig. 25, Fig. 30, Fig. 35, Fig. 40, Fig. 45 with respect to grading indices ranging from n=2 to 3.

Fig. 45. Variation of vonMises stress 
It is evident that for a hollow sphere subjected to rotation in presence of other gravitation and/or constant heat generation, the distribution of VonMises stresses along the radial direction attains maximum value as shown in Fig. 35 and Fig. 45. The vonMises stress distribution is lowest for case 2 as shown in Fig. 20.
It has been also observed that, for r/a in the range of 1.08 to 1.1(approx.), the reversal of the gradient of VonMises stresses is obtained in relation to grading indices, i.e. for r/a < 1.08 (approx.) the vonMises stress is inversely proportional to the grading index and for r/a > 1.1 (approx.) the vonMises stress is directly proportional to the grading index. The VonMises stress at b/a =1.08, corresponding to grading indices and different loading conditions, is reported in Table 1.
Table 1: For cases 17 the VonMises stresses ( )
at b/a =1.08
Cases 
Grading index 

2 
1 
0 
1 
2 
3 

Case 1 
0.21 
0.21 
0.21 
0.21 
0.21 
0.21 
Case 2 
0.20 
0.20 
0.19 
0.19 
0.19 
0.19 
Case 3 
0.19 
0.19 
0.19 
0.19 
0.19 
0.19 
Case 4 
0.21 
0.21 
0.21 
0.21 
0.21 
0.21 
Case 5 
0.21 
0.21 
0.21 
0.21 
0.21 
0.21 
Case 6 
0.19 
0.19 
0.19 
0.19 
0.19 
0.19 
Case 7 
0.21 
0.21 
0.21 
0.21 
0.21 
0.21 
The present study reports the exact solution for elastic and thermoelastic deformation and stresses of the FG rotating spherical body. Power law grading of material properties along the radial direction has been considered in the formulation. Stresses and displacement are obtained through the direct solution of the Navier equation and the effect due to grading index, rotational, gravitation force, and constant heat generation are investigated for the hollow spherical body.
Nomenclature
a 
inner radius of the hollow sphere 
b 
outer radius of the hollow sphere 
u 
displacement in the radial direction 
(i = r, t) 
strain tensor 
(i = r, t) 
stress tensor 
α 
thermal expansion coefficient 

material density 

rotation 
g 
gravitational force 
(j = 1 to 5) 
grading index 
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