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 closedform 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 onedimensional steadystate 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 nonhomogeneous 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 thickwalled 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 threedimensional 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 functionallygraded 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 σ_{0}, 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 thickwalled cylinder
in the plane strain condition will be calculated. Consider a thickwalled
FGM cylinder with an inner radius a and an outer radius b, subjected to
an internal pressure P_{i} and external pressure P_{o}
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:
where, E, k, α and ρ are module of elasticity, thermal conductivity,
linear expansion coefficient and density. E_{0}, k_{0},
α_{0} and ρ_{0} are the material constants and
β, ξ, η and γ are the power law indices of the material.
It could be assumed that:
where, m_{1}, m_{2} and m_{3} 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 ε_{z} = 0. The radial strain
ε_{r} and circumferential strain ε_{θ}
are related to the radial displacement u by:
The equilibrium equations in the absence of body forces reduce to:
where, σ_{r} and σ_{θ} are the radial
and circumferential stress components, respectively.
In the steady state case, the heat conduction equation for the onedimensional
problem in polar coordinates simplifies to:
Boundary condition of temperature is as follows:
where, T_{a} and T_{b} are temperatures of the surrounding
media, h_{a} and h_{b} 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:
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 T_{a} and T_{b}, respectively. Thus:
Applying Hooke^{,}s law, the equations for radial and circumferential
stresses would be:
where, A_{1}, A_{2}, A_{3}, A_{4}, A_{5}
and A_{6} are
Using Eq. 110 and Eq. 1618, the
Navier equation in term of the radial displacement is:
where,
and B_{1}, B_{2} and B_{3} are
Equation 19 is the nonhomogeneous EulerCaushy equation
whose complete solution is:
where, f_{1}, f_{2}, C_{1}, C_{2} and
C_{3} are:
By substituting Eq. 21 into Eq. 8
and 9 and results in Eq. 16 and 17,
the stresses are obtained as:
To determine the constants C_{4} and C_{5} consider the
boundary conditions for stresses given by:
By substituting the boundary conditions Eq. 26 into
Eq. 24, the constants becomes:
where, the parameters of g_{1}, g_{2}, g_{3},
g_{4}, g_{5} and g_{6} are as follows:
RESULTS AND DISCUSSION
Here, the exact solution obtained from the previous part of the study
will be followed up by an example. A hollow thickwalled cylinder with
the internal radius of a = 100 cm, the outer radius of b = 115 cm is considered,
which is rotating around the zaxis 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 E_{i} = 200 GPa, α_{i} = 11.7(10^{6})/°C,
ρ_{i} = 7810 kg m^{3}.

Fig. 1: 
Distribution of temperature versus radius 

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 P_{i} = 40 MPa, P_{o} = 0 MPa, respectively.
In addition, the temperature of the internal and the external surfaces
are considered constant as T_{a} = 15°C, T_{b} = 5°C.
Furthermore, it is assumed that m_{1} = m_{2} = m_{3} = 1. In Fig. 17, 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.

Fig. 3: 
Distribution of radial stress versus radius 

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.

Fig. 5: 
Distribution of equivalent stress versus radius for
b/a = 1.15 

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 is plotted in the radial direction for various and b/a and β values. Figure
57 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.

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 m_{1},
m_{2} and m_{3}. 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.