Document Type: Research Paper
Authors
^{1} Department of Mechanical Engineering, University of Sistan and Baluchestan, Zahedan,Iran
^{2} Department of Mechanical Engineering, University of Sistan and Baluchestan, Zahedan, Iran
Abstract
Keywords
Elastic analysis of Shrinkfitted Thick FGM Cylinders Based on Linear Plane Elasticity Theory
M.R. Salehi Kolahi, M. Karamooz, H. Rahmani^{*}
Department of Mechanical Engineering, University of Sistan and Baluchestan, Zahedan,Iran
Paper INFO 

ABSTRACT 
FGM Plane elasticity Shrinkfit Thickwalled cylinder 
Nowadays, functionally graded materials (FGM) are widely used in many industrial, aerospace and military fields. On the other hand, the interest in the use of shrinkfitted assemblies is increasing for designing composite tubes, highpressure vessels, rectors and tanks. Although extensive researches exist on thickwalled cylindrical shells, not many researches have been done on shrinkfitted thick FGM cylinders. In this paper, an analytical formulation for shrinkfitted of axisymmetric thickwalled FGM cylinders based on the linear plane elasticity theory is presented. The stresses and displacement fields in thick cylindrical shells are calculated using the real, Repeated and complex roots of characteristic equation. The displacements and stresses resulted are depicted for a case study. The results show that the material composition variation had evident effects on shrinkfit pressure in the intersection area of two fitted tubes. The value of this pressure affects radial and hoop stress distribution in FG circular cylinders walls. 
Functionally Graded Materials (FGMs) are advanced composite materials in which two or more different material ingredients change continuously and gradually. This results in elimination of interface problems, and consequently, uniformity of stress and temperature distribution. FGM materials are made from ceramic and metal. Usually, the composition varies from a ceramic surface to a metal surface. These changes are considered as power or exponential functions in the thickness directions [13].
To date many theorists and practitioners have intensively studied shell structures (e.g., cylindrical, spherical, etc.). Axisymmetric thickwalled cylindrical shells are common structural elements in many engineering applications, including pressure vessels, submarine hulls, rocket chambers, fuel storage tanks and many other structures. In order to optimize the weight, mechanical strength, displacement and stress distribution of a shell structure one approach is to use shells with functionally graded materials [48].
Shrink fitting is a process in which an interference fit is achieved by a relative size change after assembly. This is usually achieved by heating or cooling one component before assembly and allowing it to return to the ambient temperature after assembly [9, 10]. Heating the outer part may take a long time, and in some cases metallurgical reasons may prevent heating altogether. However, cooling the inner part will not affect the property of the material, and can be achieved much faster. Shrink fitting is a method used in making mechanical joints when axles, bushings and similar are to be fitted inside a part with a hole. Shrinkfitting is an effective way of assembling machine elements such as a gear to a shaft to transmit torque. In addition, shrinkfitted assemblies are widely used for steel rolling in the production of Ibeams and rail sections. The advantage of a shrinkfitted assembly is that it is simple to manufacture, has high integrity and is capable of being subject to high loads. Shrink fitting of layered cylinders is a way of extending the fatigue lifetimes [1115].
In 1852 for the first time, lame’ proposed the exact solution of axisymmetric thickwalled cylindrical shells made of homogeneous and Isotropic material [16]. He used plain elasticity theory to obtain the stress distribution. Lechnitsky in 1950 formulated the elasticity theory of composite structures [17]. In 1984, Niino et al. presented the concepts of FGM materials [18]. In 2006 Hongjun et al. derived the exact solution of FGM hollow cylinders in the state of plane strain with exponential function of elasticity modulus along the thickness of the cylinder [8]. In 2012, Ghannad and Zamani Nejad proposed a complete elastic solution of pressurized thick cylindrical shells made of heterogeneous FG materials [6]. They used plane elasticity theory. In 2018, Pourasghar et al. analyzed the three‐dimensional thermoelastic deformation of FGM cylindrical shells subjected to thermal load [19].
During the past years, many scholars have studied the stress and strain distributions in hollow cylindrical shells made of FG materials but to the best of our knowledge, there is no complete analytical formulation for shrinkfitted thick FGM cylinders with a power variation of material properties. In 2006, Jehed et al. [13] studied the fatigue life time of multilayer shrinkfitted cylinders made of homogenous materials. In 2012, Sharifi et al. investigated optimum design of shrinkfit multilayer compound cylinders made of homogenous materials employing an analytical method [15]. In 2015, Arslan and Mack [20] presented an analytical formulation for rotating shrink fitted cylinders with considering thermal effects. In their study the outer cylinder is made of FG material and the inner part is a homogenous shaft. They only considered the real distinct roots of the characteristic function. In 2017, Apatay et al. [21] investigated elastic–plastic design of a rotating shrink fit with functionally graded hub using semianalytical formulation. Motivated by the aforementioned remarks the focus of this research is to develop a complete analytical formulation for the behavior of shrinkfitted thick FGM cylinders based on linear plain elasticity theory. The final formulation is a closedform solution of the differential equations therefore a purely analytical discussion of the problem is possible. The presented formulation is general as it considers all the possible roots of the characteristic function. So it can be used for any grading index and any end boundary condition. The remaining of this article is organized as follows: In section ‘‘2,’’ analytical model is derived. In section ‘‘3,’’ a case study is presented. Finally, in section ‘‘2,’’ concluding remarks are made.
According to the plane elasticity theory, it is assumed that all the plane sections perpendicular to the central axis of cylinder, after applying pressure, remain flat and perpendicular to the central axis. The resultant deformations are symmetric with respect to the central axis and their values do not change along the length of the cylinder. Thus, the radial displacement u_{r} is only a function of radius r and both longitudinal stress σ_{x} and strain ε_{x} have constant values and normal stresses are principal stresses [22].
(1) 
The stress equation in the absence of body forces is defined by Eq. (2).
(2) 
Moreover, straindisplacement relation for axisymmetric case is defined as
(3) 
And for isotropic and heterogeneous material the stress is related to the strain as
(4) 
where A and B are functions of Poisson’s ratio and they are defined based on boundary conditions. In this paper, we assume that the Young’s modulus is a power function of radius [23].
(5) 
where E_{i} and r_{i} are Young’s modulus and radius at inner side of the cylinder. n is heterogeneity constant of material.
Substituting Eqs. (3) and (4) in Eq. (2), one can obtain
(6) 
Eq. (6) is a homogeneous, linear second order ODE (CauchyEuler equation), in which the roots of the characteristic polynomial
(7) 
could be real distinct, repeated and complex. The roots of the characteristic polynomial are
(8) 
In this case, Δ > 0, the general solution of Eq. (6) is:
(9) 
Using Eqs. (3) and (4), we get
(10) 
The general equations of FGM cylindrical shells subjected to external and internal uniform pressure can be found in [6]. In this study, the derived equations for σ_{r }and σ_{θ} are simplified version of those equations as the outer cylinder is only subjected to internal pressure and the inner cylinder is subjected to external pressure. According to Fig. 1, we can consider the following loading conditions. It is assumed that both members have the same length.
Outer cylinder:
(11) 
Inner cylinder:
(12) 
Therefore, based on Eqs. (10), (11) and (12), the constants C_{1} and C_{2} can be obtained. Eqs. (13) and (14) show the constants C_{1} and C_{2} for outer and inner cylinders respectively.
(13) 
(14) 
where
(15) 
Substituting C_{1} and C_{2} in Eqs. (4) and (9), we have the following equations.
Outer cylinder:
(16) 

(17) 

(18) 
Inner cylinder:
(19) 

(20) 

(21) 
Now we should deal with the contact pressure P. Prior to assembly, the outer radius of the inner cylinder was larger than the inner radius of the outer cylinder by the radial interference δ. For Shrinkfitting the input of the problem is the radial interference, therefore the contact pressure is the result of δ.
(22) 
Substituting Eqs. (18) and (21) into (22), the contact pressure can be obtained.
Fig 1. Schematic model of inner and outer cylinders and their notations
In this case, Δ = 0, the general solution of Eq. (6) is
(23) 
Following the same steps as section 2.1, we achieve to the Eqs. (24) to (31).
Outer cylinder:
(24) 
(25) 

(26) 

(27) 
Inner cylinder:
(28) 

(29) 

(30) 

(31) 
In this case, Δ < 0, so we have
(32) 
Therefore, the general solution of Eq. (6) is
(33) 
Again following the same steps as section 2.1, we achieve to the Eqs. (34) to (42).
Outer cylinder:
(34) 
where
(35) 

(36) 

(37) 

(38) 

Inner cylinder:
(39) 
where
(40) 
(41) 

(42) 

(43) 
In this section as a case study, we consider two thick cylinders, which their characteristics are tabulated in table 1. Here we consider that for both of the cylinders the end boundary conditions are freefree. Eq. (44) shows the constants A and B for these boundary conditions. The analytical solution is carried out by writing an appropriated program in MATLAB.
(44) 
Figure 2 shows the distribution of the elasticity modulus along the cylinder's thickness. In this case study the inner surface of both cylinders is softer. Fig. 3 shows the distribution of radial stress along the thickness of both inner and outer cylinders. It can be seen from Fig. 3 that by increasing the heterogeneity constant n from 2 to 2 the radial stress increases, which means that the contact pressure value, is increasing too.
Figure 4 shows the distribution of the hoop stress along the thickness of both cylinders. It is obvious that the distribution of the hoop stress in both cylinders strongly depends on the value and sign of the heterogeneity constant n. According to Fig. 4 for negative values of heterogeneity constant n, the behavior of FG material is similar to homogeneous material. Next, we investigate the radial displacement in both cylinders. Fig.5 shows the radial displacement for the inner and outer cylinders. It can be concluded from Fig. 5 that as the heterogeneity constant n increases the radial displacement of the outer cylinder increases too. According to this Fig the behavior of the inner cylinder is complicated. For n≤1 the behavior of FG material is similar to the homogeneous material but for n>1 the behavior is completely different.
It is worth noting that in the case of homogeneous material (n=0) we can achieve to the conventional relations [22, 24] for hollow thickwalled cylinders under uniform internal and external pressure. Our present results for homogenous and isotropic material are in coincidence with reference [25].
Table 1. Geometrical and mechanical characteristics 

Parameter 
Value 
E_{i} 
200 GPa 
r_{1} 
60 mm 
r_{2} 
40 mm 
r_{3} 
20 mm 
ν 
0.3 
δ 
0.1 mm 
n 
Fig 2. Distribution of elasticity modulus in the outer cylinder 
Fig 3. Continues distribution of radial stress 
Fig 4. Continues distribution of hoop stress 
In this paper, based on plane elasticity theory, an analytical solution has been obtained for shrinkfitted axisymmetric hollow thickwalled cylindrical shells made of FG materials and the effects of compositional gradient exponent n of FG circular cylinder were investigated. The variation of FGM properties is supposed to be a power function along the thickness of cylinders. The results show that with increasing the softness (elastic module) of intersection area of fitted cylinders, the shrinkfit pressure reduced.
Fig 5. Continues distribution of radial displacement
Radial stress in inner surface of outer cylinder and outer surface of inner cylinder are equal to the shrinkfit pressure and in free internal and external surfaces of each cylinder are equal to zero. Hoop stress distribution is also dependent to the shrinkfit pressure. This stress has a variable direction and magnitude in cylinders. For n=2,1, 0 and 1 the hoop stress in inner and outer cylinders have values close together while for n=2 the hoop stress in inner cylinder is much larger than outer one. Future work will involve the effects of the temperature on the elastic behavior of shrinkfitted thick FGM cylinders.
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