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BY 4.0 license Open Access Published by De Gruyter November 19, 2020

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

Rakesh Kumar Sahu, Lakshman Sondhi, Shubhankar Bhowmick and Amit Kumar Thawait

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 uniform mechanical 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.

1 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 viscoelastic 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.

Figure 1 Hollow spherical body

Figure 1

Hollow spherical body

2 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]

(1)εr=dudr=1Erσr2ϑσt+αTr
(2)εt=ur=1Erσt1ϑϑσr+αTr

Stress-strain relations are given by [1]

(3)σr=Er1+ϑ12ϑεr1ϑ+2ϑεt1+ϑαTr
(4)σt=Er1+ϑ12ϑϑεr+εt1+ϑαTr

Here, ϵi and σi (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,

(5)rddrσr+2σrσt+ρω2gar2=0

The material properties of sphere are described by power law function which are given by [20]

(6)Er=Earn1
(7)αr=αarn2
(8)kr=karn3
(9)ρr=ρarn4
(10)qr=qarn5

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 n1, n2, n3, n4, n5 are material index respectively.

Using above equations (1) to (10), the Navier equation, in terms of displacement, is given by

(11)rddrEλ1ϑdudr+2ϑur1+ϑαT+Eλ1ϑdudr+2ϑur1+ϑαTEλϑdudr+ur1+ϑαT+ρ(ω2ga)r2=0

In above eq. 11,

(12)E=Er,T=Tr,α=αrandρ=ρrandλ=11+ϑ12ϑ

3 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]

(13)1r2ddrr2krddrTr+q=0

Subject to boundary conditions,

(14)Tr=Taatr=aand
(15)Tr=Tbatr=b

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.

(16)A1r2T+B1rT+C1T=γ1rn5n3+2

Where,

(17)A1=ka
(18)B1=kan3+2
(19)C1=0
(20)γ1=qa
(21)P3=0
(22)P4=A1B1A1=n31

P3 and P4 are roots of general solution of eq. 16. Solving eq. 16 analytically yields

(23)Tr=Q3+Q4rP4+β1rn5n3+2
(24)dTdr=Q4P4rP41+β1(n5n3+2)rn5n3+1

Where,

(25)β1=γ1A1(n5n3+2)(n5n3+1)+B1n5n3+2+C1

Using the boundary conditions, the value of Q3 and Q4 yields [1]

(26)Q4=TaTbaP4bP4β1an5n3+2bn5n3+2aP4bP4
(27)Q3=Taβ1an5n3+2Q4aP4

4 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

(28)Ar2u+Bru+Cu=Urn2+P4+1+Vrn2n3+n5+3+Wrn2+1+Srn4n1+3

Where,

(29)A=Eaλ1ϑ
(30)B=n1Eaλ1ϑ+2Eaλ1ϑ
(31)C=2Eaλϑn1+2Eaλϑ2Eaλ
(32)U=112ϑEaαaQ4P4+EaαaQ4n1+EaαaQ4n2
(33)V=β1Eaαa12ϑn5n3+n1+n2+2
(34)W=Q3Eaαa12ϑn1+n2
(35)S=ρω2ga

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,

(36)ugr=QrP

Substituting eq. (36) in homogeneous form of eq. (28), one gets,

(37)AP2+BAP+C=0

Eq. (37) has two real roots P1 and P2 as,

(38)P1,2=AB±BA24AC2A

Thus, the general solution is given by,

(39)ugr=Q1rP1+Q2rP2

The particular solution up (r) is assuming in the form,

(40)upr=Irn2+P4+1+Jrn2n3+n5+3+Lrn2+1+Mrn4n1+3

Substituting eq. (40) in eq. (28), one obtains,

(41)[An2+P4+1n2+P4+Bn2+P4+1+C]Irn2+P4+1+An2n3+n5+3n2n3+n5+2+Bn2n3+n5+3+CJrn2n3+n5+3+[An2+1n2+Bn2+1+C]Lrn2+1+[An4n1+3n4n1+2+Bn4n1+3+C]Mrn4n1+3=Urn2+P4+1+Vrn2n3+n5+3+Wrn2+1+Srn4n1+3

Equating the coefficient of identical power,

(42)I=UA(n2+P4+1)(n2+P4)+Bn2+P4+1+C
(43)J=VDenominator

Where

Denominator=A(n2n3+n5+3)(n2n3+n5+2)+B(n2n3+n5+3)+C
(44)L=WA(n2+1)(n2)+B(n2+1)+C
(45)M=SA(n4n1+3)(n4n1+2)+B(n4n1+3)+C

Overall solution for u (r) is given by,

(46)ur=ugr+upr

Thus

(47)ur=Q1rP1+Q2rP2+Irn2+P4+1+Jrn2n3+n5+3+Lrn2+1+Mrn4n1+3

Substituting eq. (47) in eq. (1)-(4), the strains and stresses are obtained as,

(48)εr=Q1p1rP11+Q2p2rP21+In2+P4+1rn2+P4+Jn2n3+n5+3rn2n3+n5+2+Ln2+1rn2+Mn4n1+3rn4n1+2
(49)εt=Q1rP11+Q2rP21+Irn2+P4+Jrn2n3+n5+2+Lrn2+Mrn4n1+2
(50)σr=EλQ11ϑP1+2ϑrn1+P11+Q21ϑP2+2ϑrn1+P21+Irn1+n2+P41ϑ(n2+P4+1+2ϑ]+Jrn1+n2n3+n5+21ϑ(n2n3+n5+3+2ϑ]+Lrn1+n21ϑ(n2+1+2ϑ]+Mrn4+21ϑ(n4n1+3+2ϑ]1ϑαQ3rn1+n2+Q4rn1+n2+P4+β1rn1+n2+n5n3+2

To determine the constant Q1 and Q2, 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]:

(51)σra=paandσrb=pb

Upon substituting eq. (51) in eq. (50), the integration constants becomes

(52)Q1=ϕ22Xϕ12Yϕ11ϕ22ϕ12ϕ21
(53)Q2=ϕ11Yϕ21Xϕ11ϕ22ϕ12ϕ21

Where,

(54)ϕ11=EaλP11ν+2νan1+P11
(55)ϕ12=EaλP21ν+2νan1+P21
(56)ϕ21=EaλP11ν+2νbn1+P11
(57)ϕ22=EaλP21ν+2νbn1+P21
(58)X=paZa,Y=pbZb
(59)Za=EaλIan1+n2+P41ϑn2+P4+1+2ϑ+Jan1+n2n3+n5+21ϑn2n3+n5+3+2ϑ+Lan1+n21ϑn2+1+2ϑ+Man4+21ϑn4n1+3+2ϑ1ϑαQ3an1+n2+Q4an1+n2+P4+β1an1+n2+n5n3+2
(60)Zb=EaλIbn1+n2+P41ϑn2+P4+1+2ϑ+Jbn1+n2n3+n5+21ϑn2n3+n5+3+2ϑ+Lbn1+n21ϑn2+1+2ϑ+Mbn4+21ϑn4n1+3+2ϑ1ϑαQ3bn1+n2+Q4bn1+n2+P4+β1bn1+n2+n5n3+2

5 Results and discussion

5.1 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 = 1.2 m, Poisson’s ratio, ϑ = 0.3, since material properties are taken in accordance with equation 6 to 10. The internal properties are assumed as, modulus of elasticity Ea = 200 GPa, thermal coefficient of expansion αa = 1.2 × 10−6 per C, thermal conduction coefficient ka = 15 W/mk, density ρa = 7800 kg/m3, heat generation q = 50 × 103 kJ/m3 and rotation ω = 50 rad/s and Gravity g = 9.81 m/s2. The boundary condition for temperature are taken as, T(a) = 10C and T(b)=0 C Boundary conditions for stress calculations under mechanical loading are taken as, internal pressure = 50 MPa and external pressure = 0, Grading parameter n is chosen as −2 to 3 and identical for all (n1 = n2 = n3 = n4 = n5 = n) [1].

Figure 2 to 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.

Figure 2 Radially distributed elastic modulus

Figure 2

Radially distributed elastic modulus

Figure 3 Radially distributed thermal expansion coefficient

Figure 3

Radially distributed thermal expansion coefficient

Figure 4 Radially distributed density

Figure 4

Radially distributed density

Figure 5 Radially distributed thermal conduction coefficient

Figure 5

Radially distributed thermal conduction coefficient

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 Radially distributed temperature

Figure 6

Radially distributed temperature

Figure 7 Radially distributed displacement

Figure 7

Radially distributed displacement

Figure 8 Radially distributed radial stress

Figure 8

Radially distributed radial stress

Figure 9 Radially distributed tangential stress

Figure 9

Radially distributed tangential stress

Figure 10 Radially distributed von-Mises stress for b/a = 1.2

Figure 10

Radially distributed von-Mises stress for b/a = 1.2

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 11 Radially distributed temperature

Figure 11

Radially distributed temperature

Figure 12 Radially distributed displacement

Figure 12

Radially distributed displacement

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 = 1 shows that tangential stress is uniform along the radial direction. Stress distribution along the radial direction is investigate in von-Mises stress distribution which shows in Figure 15 for aspect ratio b/a = 1.2. It is clear from figure that for r/a < 1.084 (approx.), as the grading parameter increases, the von-Mises stress decreases but for r/a > 1.084 (approx.), the situation is reversed and the von-Mises stress increases as the grading parameter increases.

Figure 13 Radially distributed radial stress

Figure 13

Radially distributed radial stress

Figure 14 Radially distributed tangential stress

Figure 14

Radially distributed tangential stress

Figure 15 Radially distributed von-Mises stress

Figure 15

Radially distributed von-Mises stress

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 direction. 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.

Figure 16 Radially distributed temperature

Figure 16

Radially distributed temperature

Figure 17 Radially distributed displacement

Figure 17

Radially distributed displacement

Figure 18 Radially distributed radial stress

Figure 18

Radially distributed radial stress

Figure 19 Radially distributed tangential stress

Figure 19

Radially distributed tangential stress

Figure 20 Radially distributed von-Mises stress

Figure 20

Radially distributed von-Mises stress

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].

Figure 21 Radially distributed temperature

Figure 21

Radially distributed temperature

Figure 22 Radially distributed displacement

Figure 22

Radially distributed displacement

Figure 23 Radially distributed radial stress

Figure 23

Radially distributed radial stress

Figure 24 Radially distributed tangential stress

Figure 24

Radially distributed tangential stress

Figure 25 Radially distributed von-Mises stress

Figure 25

Radially distributed von-Mises stress

Figure 26 Radially distributed temperature

Figure 26

Radially distributed temperature

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 Figure 27 to 29. Similar observations are obtained for displacement and stress as in case 1.

Figure 27 Radially distributed displacement

Figure 27

Radially distributed displacement

Figure 28 Radially distributed radial stress

Figure 28

Radially distributed radial stress

Figure 29 Radially distributed tangential stress

Figure 29

Radially distributed tangential stress

Figure 30 Radially distributed von-Mises stress

Figure 30

Radially distributed von-Mises stress

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.

Figure 31 Radially distributed temperature

Figure 31

Radially distributed temperature

Figure 32 Radially distributed displacement

Figure 32

Radially distributed displacement

Figure 33 Radially distributed radial stress

Figure 33

Radially distributed radial stress

Figure 34 Radially distributed tangential stress

Figure 34

Radially distributed tangential stress

Figure 35 Radially distributed von-Mises stress

Figure 35

Radially distributed von-Mises stress

Case 6: Spherical body with variable heat generation and gravity force

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.

Figure 36 Radially distributed temperature

Figure 36

Radially distributed temperature

Case 7: Spherical body with rotation, gravitational force and variable heat generation

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 Figure 41 and displacement and stresses are shown in figures Figure 42 to 45.

Figure 37 Radially distributed displacement

Figure 37

Radially distributed displacement

Figure 38 Radially distributed radial stress

Figure 38

Radially distributed radial stress

Figure 39 Radially distributed tangential stress

Figure 39

Radially distributed tangential stress

Figure 40 Radially distributed von-Mises stress

Figure 40

Radially distributed von-Mises stress

Figure 41 Radially distributed temperature

Figure 41

Radially distributed temperature

Figure 42 Radially distributed displacement

Figure 42

Radially distributed displacement

Figure 43 Radially distributed radial stress

Figure 43

Radially distributed radial stress

Figure 44 Radially distributed tangential stress

Figure 44

Radially distributed tangential stress

Figure 45 Radially distributed von-Mises stress

Figure 45

Radially distributed von-Mises stress

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 (approx.) the von-Mises stress is directly proportional to the grading parameter. The values are given in Table 1 for b/a = 1.08 for corresponding to grading parameters, and different loading condition.

Table 1

For different cases 1-7 the von-Mises stresses (σ*) at b/a =1.08

Grading parameter
CASES−2−10123
Case 10.210.210.210.210.210.21
Case 20.200.200.190.190.190.19
Case 30.190.190.190.190.190.18
Case 40.210.210.210.210.210.21
Case 50.210.210.210.210.210.21
Case 60.190.190.190.190.190.18
Case 70.210.210.210.210.210.21

6 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 obtained 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.

Nomenclature

ϵr

Radial strain

ϵt

Tangential strain

ω

Rotation

ρ

Density of material

σr

Radial stress

σt

Tangential stress

G

Gravity

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Received: 2020-07-27
Accepted: 2020-09-25
Published Online: 2020-11-19

© 2020 R. Kumar Sahu et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.