Exact solution for the thermo-elastic deformation and stress states of FG rotating spherical body

Abstract: In this paper, a generalized solution for 1-D steady-state mechanical and thermal deformation and stresses in rotating hollow functionally graded spherical body is presented. Spherical shells are treated under mechanical and thermal loads in the form of rotational body force with heat generation. Temperature distribution is assumed to vary along the radial direction due to variable heat generation. General uniformmechanical boundary condition at inner and outer surfaces along with prescribed temperatures at both the ends are assumed as boundary conditions. In the present study, material properties are taken as power function of radius with grading parameter ranging between −2 to 3. Governing differential equation with variable coefficient is developed and solved to find deformation and stresses. The obtained results are verified with benchmark results and are found to be in good agreement. Results show that deformation and stresses decrease with an increase in the value of grading parameter and are less as compared to the homogeneous body.


Introduction
FGMs are advanced class of composite materials wherein mechanical properties vary continuously at macroscopic level from surface to surface. Thermo-mechanical stresses in FG thick sphere is reported by M.R. Eslami, M.H. Babaei, R. Poultangari [1] wherein they considered a thick hollow spherical body of FG material under one dimensional steady state distributed temperature with general type of boundary conditions (mechanical and thermal). Deformation and stresses in rotating FG material pressurized thick hollow cylindrical body under thermal load is given by, M. Zamani Nejad and G.H. Rahimi [2]. Effect of material gradient on stresses of thick FG spherical pressure vessels using exponentially varying grading properties are given by M. Zamani Nejad, M. Gharibi [3]. A novel approach to stress analysis of pressurized FGM cylinder, disc and spheres is given by Naki Tutuncu, Beytullah Temel [4]. FG hollow cylindrical under thermal and pressure loading effects due to material parameters on stresses and temperature distributions are reported by Celal Evci, Mufit Gulgec [5]. Rotating disk, cylinder and sphere with variable thickness are analysed and reported in [6]. Thermo-elastic, thermomechanical stress analysis has been conducted in few literatures [7][8][9][10]. Semi exact solution of non uniform disk was analysed and presented in [11] and [12]. Some exact solution of cylinders are analysed [13] and [14]. The dynamic analysis of rotating doubly-curved shell structures made of FGMs on critical speed is analyzed in [15]. Thermo-elastic analysis of rotating multilayer FG-GPLRC truncated conic based on a coupled TDQM-NURBS scheme is reported by [16]. A multi objective optimization of a FG sandwich panel with mechanical loading was carried out in [17]. Indentation of materials with a linear yield strength gradient by spherical indenters are analyzed in [18]. Temperature dependent vibration analysis of functionally graded vis-coelastic cylindrical micro-shell is analyzed and reported in [19].
In the present work report, to determine the stress and deformation state of hollow functionally graded spherical body, the problem is moulded using Navier equation including rotational body force and variable heat generation. The validation of the present exact solution is carried out with existing literatures. Corresponding to rotational speed, body force and variable heat generation in spherical body, the stress and deformation in the FG spherical body is estimated. The existing results are reported in dimensionless form.

Mathematical formulation
A rotating hollow spherical body of 'a' inner radius and 'b' outer radius and made of FGM material is considered. The variation of material properties of hollow sphere are function of radius 'r'. Let 'u' be the displacement in the radial direction. The relation between strain and displacement are given by [1] Stress-strain relations are given by [1] σr = E (r) Here, ϵ i and σ i (i = r, t) are the strain and stress tensor. Heat conduction equation is used to determine distribution of the temperature T(r), α is the thermal expansion coefficient. The equilibrium equation in radial direction, including the inertia term and body force, is given by, The material properties of sphere are described by power law function which are given by [20] E (r) = Ea(r) n1 (6) α (r) = αa(r) n2 (7) k (r) = ka(r) n3 (8) ρ (r) = ρa(r) n4 (9) q (r) = qa(r) n5 (10) Where, E (r), α (r), k (r), ρ (r), q(r) are elastic modulus, thermal expansion coefficient, thermal conduction coefficient density and heat generation at any radius respectively. Ea, αa, ka, ρa, qa are material properties as described above at inner radius and n 1 , n 2 , n 3 , n 4 , n 5 are material index respectively. Using above equations (1) to (10), the Navier equation, in terms of displacement, is given by In above eq. 11, E = E (r) , T = T (r) , α = α (r) and ρ = ρ (r) (12) and λ = 1

Temperature formulation
The '1-D spherical heat conduction equation with heat generation' under 'steady-state' condition and thermal boundary conditions for FG hollow spherical bodies, are given by [20] 1 Subject to boundary conditions, T (r) = T (a) at r = a and (14) Where, T(r) is the temperature at any radius, T (a) and T (b) are the temperature at inner and outer radius respectively. Differentiating eq. (13) gives the Navier equation for temperature as follows. Where, P 3 and P 4 are roots of general solution of eq. 16. Solving eq. 16 analytically yields Where, Using the boundary conditions, the value of Q 3 and Q 4 yields [1]

Solutions of displacement equation
Navier equation given in eq. (11) needs a separate calculation for function T (r). Once the function T (r) is known, the equation is solved analytically. Substituting T (r) in eq. 11 gives Where, The Navier equation in terms of radial displacement u, in eq. (28) is Euler differential equation of non-homogeneous form which possesses general as well as particular solutions.
The general solution, ug is assuming, Substituting eq. (36) in homogeneous form of eq. (28), one gets, Eq. (37) has two real roots P 1 and P 2 as, Thus, the general solution is given by, The particular solution up (r) is assuming in the form, Substituting eq. (40) in eq. (28), one obtains, Equating the coefficient of identical power, Overall solution for u (r) is given by, Thus Substituting eq. (47) in eq. (1)-(4), the strains and stresses are obtained as, To determine the constant Q 1 and Q 2 , the boundary conditions arising out of mechanical loading may be used. The mechanical boundary conditions in the inner surface and outer surface are as follows [1]: Upon substituting eq. (51) in eq. (50), the integration constants becomes Where,

Internal pressure and temperature
The numerical values of different system parameters considered in the work are as follows: Inner and outer radii of hollow sphere are assumed to be a = 1 m, b = 1.     Figure 5 shows the variation of material properties such as modulus of elasticity, thermal expansion coefficient, density, thermal conduction coefficient respectively for different values of n. It is clear from the figure, that for grading parameter n = 0, all mechanical properties are constant whereas for positive grading parameter, the material properties increase from inner to outer radius while for negative grading parameter, the material properties decrease from inner to outer radius.

Comparison of current results with benchmark reports [1]
Validation of present work is carried out with M.R. Eslami et al. [1] and comparisons are presented in Figure 6 to Figure 10. The results obtained are found to be good agreement with [1]. This establishes validity of the mathematical formulation and the MATLAB source code and is further used for the investigation of FG hollow spherical having different non-linear material behaviour under the effect of rotation, gravitational force and internal heat generation.    Figure 6 shows distribution of temperature for different grading parameter. It is clear from figure that the temperature reduces from inner to outer radius and it is clear from figure that as the grading parameter increases, the temperature further reduces. Figure 7 shows that as the grading parameter increases, radial displacement also decreases from inner to outer radius. Figure 8 shows the as  the grading parameter increases, radial stress at any point in the sphere decreases but the distribution of radial stress for any given value of grading parameter increases along radius from inner to outer of hollow sphere body. Figure 9 shows the plot of tangential stress. In this figure, for n < 1 the tangential stress decreases along radial direction from inner to outer but when n > 1, the tangential stress is increases along radial direction from inner to outer and for n = 1, tangential stress is uniform along the radial direction. Stress distribution along the radial direction is investigated in terms of von-Mises stress distribution which is reported in Figure 10 for aspect ratio b/a = 1.2. It is clear from figure that for r/a < 1.09 (approx.), as the grading parameter increases, the von-Mises stress decreases but for r/a > 1.09 (approx.), the situation is reversed and the von-Mises stress increases as the grading parameter increases. The von-Mises stress is almost uniform for grading parameter n = 3 along the radial direction.

Case 1: Rotating spherical body
In this case study a hollow spherical body with a rotational motion is investigated. The effect in hollow spherical body due to the rotation motion shows in graphs. The outcomes values of all are higher as compared to without rotation case as in benchmark. Figure 11 shows the distribution of temperature for different grading parameter. It is clear from figure that the temperature reduces along radial direction from inner to outer and it is also observed that as the grading parameter increases, the temperature reduces. Figure 12 shows that as the grading parameter increases, radial displacement also decreases from inner to outer radii. Figure 13 shows that as the grading parameter increases, radial stress decreases but radial stress increases from inner to outer radii of hollow sphere body. Figure 14 shows the plot of tangential stress. In this figure for n < 1 the tangential stress decreases along radial direction from inner to outer but when n > 1, the tangential stress is increases along radial direction from inner to outer and for n

Case 2: Spherical body with gravity
In this case, a hollow spherical body with gravitational force is investigated. The effect in hollow spherical body due to the gravitational force is shown in the following graphs. Figure 16 shows distribution of temperature for different grading parameter. Similar to previous results, Figure 17 shows that as the grading parameter increases, radial displacement decreases from inner to outer radii. Figure 18 shows that as the grading parameter increases, radial stress decreases but radial stress increases along radius from inner to outer radii of hollow sphere body. In Figure 19, tangential stress is plotted. In this figure, for n < 1 the tangential stress decreases along radial direction from inner to outer radii but for n > 1, the tangential stress increases along radial direction from inner to outer radii and at n = 1, tangential stress is uniform along the radial di- Figure 17: Radially distributed displacement rection. Stress distribution along the radial direction is investigated in terms of von-Mises stress distribution which is shown in Figure 20 for aspect ratio b/a = 1.2. It is clear from figure that for r/a < 1.095 (approx.), as the grading parameter increases, the von-Mises stress decreases but for r/a > 1.095 (approx.), the situation is reversed and the von-Mises stress increases as the grading parameter increases.  The von-Mises stress is almost uniform for grading parameter n = 3 along the radial direction.

Case 3: Spherical body with variable heat generation
In this case, a hollow spherical body with variable heat generation is investigated and reported in Figure 21 to 24 wherein displacement and stresses are plotted. It is observed from figure that Figure 24 and 25, that in case of variable heat generation in the hollow sphere, the tangential stress and von-Mises stress are larger than as reported in [1].

Case 4: Spherical body with rotation and gravitational force
In this case, a hollow spherical body subjected to rotation and gravitational force is investigated. The results obtained for displacement and stresses shown in figure

Case 5: Spherical body with rotation and internal heat generation
Under this case study, a hollow spherical body with a rotational and variable heat generation is investigated. The results obtained for displacement and stresses is shown in figure Figure 31 to 35. It is clear from Figure 34 the tangential stress increases when the body subjected to internal heat generation.  In this case study a hollow spherical body subjected to gravitational force and variable heat generation is investigated. Similar observations are obtained for displacement and stresses under this case as in case 3.  In this case, a hollow spherical body with a rotational, gravitation force and variable heat generation is investigated. The radial distribution of temperature is shown in    In the graphs presented in all the cases reported above, it has been observed that nature of von-Mises stresses for r/a in the range of 1.08 to 1.1 (approx.) is reversed i.e. for r/a < 1.08 (approx.) the von-Mises stress is inversely proportional to the grading parameters and for r/a > 1.1 (ap-    Table 1 for b/a = 1.08 for corresponding to grading parameters, and different loading condition.

Conclusion
The present paper reports the exact solution for thermoelastic deformation and stress state of functionally graded rotating spherical body. Power law function is used for material grading along the radial direction. Stresses are ob- tained from the solution of Navier equation through direct method and the effect due to grading parameter, rotational speed, gravitational force and variable heat generation are studied on the stresses and displacement field. This paper shows the general mathematical formulation and solution technique based on material grading law for the FG spherical body.
• The effect of grading parameter was studied on internally pressurised hollow functionally graded spherical body and found that with the increments of grading parameter, the strength of hollow spherical body improves. • Effect of grading parameter on radial displacement of hollow functionally graded spherical body was investigated and noticed that as grading parameter is increased, the displacement of hollow spherical body reduces continually. • Displacement reduces continuously along radial direction from inner to outer side of hollow spherical body under all cases reported for the present study. • Effect of grading parameter on radial stress of hollow functionally graded spherical body was investigated and noticed that as grading parameter increases, the radial stress recedes. • Effect of grading parameter on tangential stress of hollow functionally graded spherical body was investigated and noticed that for n < 1 the tangential stress decreases along radial direction from inner to outer radii but for n > 1, the tangential stress increases along radial direction from inner to outer radii and at n = 1, tangential stress is almost uniform along the radial direction. • To study the overall stress distribution along the radial direction, the von-Mises stress is plotted along the radial direction for b/a = 1.2. The von-Mises stress is almost uniform along radial direction for n = 3. It is reported that von-Mises stress is higher at inner radius and continuously reduces along til the outer radius is reached. • It is clear from all graphs of von-Mises stress that the von-Mises stress decreases as grading parameter increases till a critical value of b/a is reached. Beyond this, von-Mises stress is increases as the grading parameter increases.