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Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials



G.H. Rahimi and M. Zamani Nejad
 
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ABSTRACT

In this study, thermal stresses in a hollow rotating thick-walled cylinder made of functionally graded material under internal and external pressure are obtained as a function of radial direction to an exact solution by using the theory of elasticity. Material properties are considered as a function of the radius of the cylinder and the Poisson`s ratio as constant. The distributions of the thermal stresses are obtained for different values of the powers of the module of elasticity.

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  How to cite this article:

G.H. Rahimi and M. Zamani Nejad, 2008. Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials. Journal of Applied Sciences, 8: 3267-3272.

DOI: 10.3923/jas.2008.3267.3272

URL: https://scialert.net/abstract/?doi=jas.2008.3267.3272
 

INTRODUCTION

By using the continuous change in the physical and the mechanical properties of a material, it is possible to prevent from the fracture in composite materials, which causes stress concentration and yield in such materials. These materials which possess gradual change in their material properties are known as Functionally Gradient Materials (FGM). The problems of rotating annular disks or cylinders have been investigated under various assumptions and conditions. This is a topic which can be readily found in most standard elasticity books (Boresi and Chong, 1999). Obata and Noda (1994), through the application of a perturbation approach, investigated the thermal stresses in an FGM hollow sphere and in a hollow circular cylinder. Assuming that the material has a graded modulus of elasticity, while the Poisson`s ratio is a constant, Tutuncu and Ozturk (2001) investigated the stress distribution in the axisymmetric structures. They obtained the closed-form solutions for stresses and displacements in functionally graded cylindrical and spherical vessels under internal pressure. Based on approximate solutions of temperatures and thermal stresses, the optimization of the material composition of FGM hollow circular cylinders under thermal loading was discussed (Ootao et al., 1999). Applying the Frobenius series method, Zimmerman and Lutz (1999) found a way round the problem of the uniform heating of functionally graded circular cylinder. Another general analysis of one-dimensional steady-state thermal stresses in a hollow thick cylinder made of functionally graded material was obtained (Jabbari et al., 2002). An analysis of the thermomechanical behavior of hollow circular cylinders of functionally graded materials was presented (Liew et al., 2003). They worked out a solution based on the solutions obtained by a novel limiting process that makes use of the solutions of homogeneous hollow circular cylinders, without resorting to the basic theory or the equations of non-homogeneous thermoelasticity. Tarn and Wang (2004) studied heat conduction in circular cylinders of functionally graded materials and laminated composites. They focused on the end effects and by means of matrix algebra and eigenfunction expansion, the decay length that characterizes the end effects on the thermal filed was assessed. An accurate method for conducting elastic analysis of thick-walled spherical pressure vessels subjected to internal pressure was devised (You et al., 2005). It is necessary to point out that two kinds of pressure vessel are considered: one is made up of two homogeneous layers near the inner and outer surfaces of the vessel and the other functionally graded layer in the middle; the latter consists of the functionally graded material only. Jabbari et al. (2007), making use of the generalized Bessel function and Fourier series solved the temperature and Navier equations analytically and offered a general theoretical analysis of three-dimensional mechanical and thermal stresses for a short hollow cylinder made of functionally graded material. Given the assumption that the material is isotropic with constant Poisson`s ratio and exponentially varying elastic modulus through the thickness, Tutuncu (2007), obtained power series solutions for stresses and displacements in functionally-graded cylindrical vessels subjected to internal pressure alone. Argeso and Eraslon (2008) assuming the different states of material properties including Poisson`s ratio υ, modulus of elasticity E, the yield strength σ0, the coefficient of thermal expansion α and the thermal conductivity k, assessed the thermoelastic response of cylinders and tubes.

MATERIALS AND METHODS

The thermal stresses distribution in a hollow rotating thick-walled cylinder in the plane strain condition will be calculated. Consider a thick-walled FGM cylinder with an inner radius a and an outer radius b, subjected to an internal pressure Pi and external pressure Po that are axisymmetric and rotating at a constant angular velocity ω about its axis. The cylindrical coordinates (r, θ, z) are chosen so, that r and z are the radial and axial coordinates, respectively.

The material properties are assumed to be radically dependent. The module of elasticity, thermal conductivity, linear expansion coefficient and density through the wall thickness are assumed to vary as follows:

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(1)

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(2)

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(3)

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(4)

where, E, k, α and ρ are module of elasticity, thermal conductivity, linear expansion coefficient and density. E0, k0, α0 and ρ0 are the material constants and β, ξ, η and γ are the power law indices of the material.

It could be assumed that:

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(5)

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(6)

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(7)

where, m1, m2 and m3 are constant. In the current study, a range of –2≤β≤2 is employed which consists of all the values which has widely been used in the research mentioned above.

To show the effect of inhomogeneity on the stress distributions, different values were considered for β. Since, the variation of Poisson`s ratio, υ, for engineering materials is small, it is assumed constant.

It is assumed that plane strain εz = 0. The radial strain εr and circumferential strain εθ are related to the radial displacement u by:

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(8)

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(9)

The equilibrium equations in the absence of body forces reduce to:

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(10)

where, σr and σθ are the radial and circumferential stress components, respectively.

In the steady state case, the heat conduction equation for the one-dimensional problem in polar coordinates simplifies to:

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(11)

Boundary condition of temperature is as follows:

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(12)

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(13)

where, Ta and Tb are temperatures of the surrounding media, ha and hb are the heat transfer coefficients and subscripts a and b correspond to surfaces r = a and r = b, respectively.

The general solution of Eq. 11 with considering relation of thermal transfer coefficient Eq. 2 and boundary conditions into Eq. 12 and 13 is:

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(14)

with considering special case in which there is no heat transfer taking place between the inner surface and outer surface with the surrounding medium and that the surface temperature at the inner and outer surfaces prescribed as Ta and Tb, respectively. Thus:

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(15)

Applying Hooke,s law, the equations for radial and circumferential stresses would be:

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(16)

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(17)

where, A1, A2, A3, A4, A5 and A6 are

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(18)

Using Eq. 1-10 and Eq. 16-18, the Navier equation in term of the radial displacement is:

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(19)

where, Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials and B1, B2 and B3 are

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(20)

Equation 19 is the non-homogeneous Euler-Caushy equation whose complete solution is:

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(21)

where, f1, f2, C1, C2 and C3 are:

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(22)

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(23)

By substituting Eq. 21 into Eq. 8 and 9 and results in Eq. 16 and 17, the stresses are obtained as:

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(24)

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(25)

To determine the constants C4 and C5 consider the boundary conditions for stresses given by:

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(26)

By substituting the boundary conditions Eq. 26 into Eq. 24, the constants becomes:

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(27)

where, the parameters of g1, g2, g3, g4, g5 and g6 are as follows:

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
(28)

RESULTS AND DISCUSSION

Here, the exact solution obtained from the previous part of the study will be followed up by an example. A hollow thick-walled cylinder with the internal radius of a = 100 cm, the outer radius of b = 115 cm is considered, which is rotating around the z-axis at the constant angular velocity of ω = 10 rad sec-1. The modulus of elasticity, the thermal coefficient of expansion and density at internal radius, respectively, have the values of Ei = 200 GPa, αi = 11.7(10-6)/°C, ρi = 7810 kg m-3.

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
Fig. 1: Distribution of temperature versus radius

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
Fig. 2: Distribution of radial displacement versus radius

It is also assumed that the Poisson`s ratio, υ, has a constant value of 0.33. For boundary conditions, the internal and external surfaces of the cylinder are taken to be under the pressure of Pi = 40 MPa, Po = 0 MPa, respectively. In addition, the temperature of the internal and the external surfaces are considered constant as Ta = 15°C, Tb = 5°C. Furthermore, it is assumed that m1 = m2 = m3 = 1. In Fig. 1-7, the changes are shown according to the different values for β in the radial direction within the range of –2≤β≤2.

The variations of the temperature in the radial direction for different values of the β are shown in Fig. 1.

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
Fig. 3: Distribution of radial stress versus radius

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
Fig. 4: Distribution of circumferential stress versus radius

The Fig. 1 shows that with increasing β, the temperature decreases. The radial displacement along the radius is shown in Fig. 2. There is a decrease in the value of the radial displacement as β increases. Figure 3 and 4 show the distribution radial and circumferential stresses in the radial direction. As β is increases, so does the magnitude of the radial stress. For β>0, the circumferential stress increases as the radius increases whereas for β<0 the circumferential stress along the radius decreases. Given that β = 0, the circumferential stress remains nearly constant along the radius.

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
Fig. 5: Distribution of equivalent stress versus radius for b/a = 1.15

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
Fig. 6: Distribution of equivalent stress versus radius for b/a = 1.2

For the purpose of studying the stress distribution along the cylinder radius, the Von mises effective stress Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials is plotted in the radial direction for various and b/a and β values. Figure 5-7 are plotted for b/a = 1.15, 1.2 and 1.25, respectively. It must be noted from Fig. 5 that for β = 1 the effective stress remains almost uniform along the radius of the cylinder.

Image for - Exact Solutions for Thermal Stresses in a Rotating Thick-Walled Cylinder of Functionally Graded Materials
Fig. 7: Distribution of equivalent stress versus radius for b/a = 1.25

As Fig. 6 and 7 suggest, with increasing b/a ratios, the curves related to β = 1 produce an almost uniform distribution for the effective stress.

CONCLUSION

In the present study, by the application of the elasticity theory, thermal stresses are obtained for an FG rotating cylinder. It is assumed that the material properties change as graded in radial direction to a power law function. The values of ξ, η and γ are taken as coefficients of the value of β. Change in value of β may or may not bring about changes in these coefficients, depending on the value of m1, m2 and m3. Numerical results show that value of β has a great effect on the thermoelastic stresses. This solution could be the most general solution for such problems. The findings of this study can be used to find optimum values of β, ξ, η and γ which can, in turn, be used to minimize and uniform stresses and prevent from yielding.

REFERENCES
1:  Argeso, H. and A.N. Eraslan, 2008. On the use of temperature-dependent physical properties in thermomechanical calculations for solid and hollow cylinders. Int. J. Thermal Sci., 47: 136-146.
CrossRef  |  Direct Link  |  

2:  Boresi, A.P. and K.P. Chong, 1999. Elasticity in Engineering Mechanics. 2nd Edn., Wiley-Interscience, New York, ISBN-10: 0471316148.

3:  Jabbari, M., S. Sohrabpour and M.R. Eslami, 2002. Mechanical and thermal stresses in a functionally graded hollow cylinder due to radially symmetric loads. Int. J. Pressure Vessels Piping, 79: 493-497.
CrossRef  |  Direct Link  |  

4:  Jabbari, M., A.H. Mohazzab, A. Bahtui and M.R. Eslami, 2007. Analytical solution for three-dimensional stresses in a short length FGM hollow cylinder. ZAMM. Z. Angew. Math. Mech., 87: 413-429.
Direct Link  |  

5:  Liew, K.M., S. Kitipornchai, X.Z. Zhang and C.W. Lim, 2003. Analysis of the thermal stress behaviour of functionally graded hollow circular cylinders. Int. J. Solids Struct., 40: 2355-2380.
CrossRef  |  Direct Link  |  

6:  Obata, Y. and N. Noda, 1994. Steady thermal stresses in a hollow circular cylinder and a hollow sphere of a functionally gradient material. J. Thermal Stresses, 17: 471-487.
CrossRef  |  Direct Link  |  

7:  Ootao, Y., Y. Tanigawa and T. Nakamura, 1999. Optimization of material composition of FGM hollow circular cylinder under thermal loading: A neural network approach. Composites Part B., 30: 415-422.
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8:  Tarn, J.Q. and Y.M. Wang, 2004. End effects of heat conduction in circular cylinders of functionally graded materials and laminated composites. Int. J. Heat Mass Transfer, 47: 5741-5747.
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9:  Tutuncu, N. and M. Ozturk, 2001. Exact solutions for stresses in functionally graded pressure vessels. Compos. Part B., 32: 683-686.
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10:  Tutuncu, N., 2007. Stresses in thick-walled FGM cylinders with exponentially-varying properties. Eng. Struct., 29: 2032-2035.
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11:  You, L.H., J.J. Zhang and X.Y. You, 2005. Elastic analysis of internally pressurized thick-walled spherical pressure vessels of functionally graded materials. Int. J. Pressure Vessels Piping, 82: 347-354.
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12:  Zimmerman, R.W. and M.P. Lutz, 1999. Thermal stresses and thermal expansion in a uniformly heated functionally graded cylinder. J. Thermal Stresses, 22: 177-188.
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