On the Analysis of Two-Dimensional Functionally Graded Rotating Thick Hollow Cylinder

Subject Areas : Mechanical Engineering

Mostafa Omidi Bidgoli ^{1 *} , Mohammad Hosseini ² , Ali Fata ³

1 - Department of Mechanical Engineering, Islamic Azad University, Badroud branch, Badroud, Iran
2 - Department of Mechanical Engineering, University of Hormozgan, Bandar Abbas, Iran
3 - Department of Mechanical Engineering, University of Hormozgan, Bandar Abbas, Iran

Received: 2024-09-10 Accepted : 2025-03-18 Published : 2025-06-22

Keywords: 2D Functionally Graded Material, FEM, Rotating Cylinder, Thick Hollow Cylinder ,

Abstract :

Rotary components are widely used in industries. There is both mechanical and thermal loading in most rotating cylinder applications. On the other hand, functionally graded material has better performance under different loads. Therefore, a finite length two dimensional functionally graded material (2D-FGM) thick hollow cylinder under angular velocity is investigated, in this research. Volume fraction distribution of functionally graded material and geometry of the cylinder are axisymmetric but not uniform along the radial and axial directions. The finite element method based on Rayliegh-Ritz energy formulation is applied to obtain the governing Equation and associated boundary conditions of a 2D-FG thick hollow cylinder. Using this method, the effects of the power law exponents and angular velocity on the displacements and distribution of stresses are investigated for simply supported 2D-FG thick hollow cylinder. The results indicate that 2D-FGM facilitates improved design, allowing for better control of both maximum stresses and stress distribution through material distribution.

References:

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Full-Text:



Int.  J.   Advanced Design and Manufacturing Technology, 2025, Vol. 18, No. 2, pp. 71-80

DOI:  10.71644/admt.2025.1106718                                     ISSN: 2252-0406 https://admt.isfahan.iau.ir

 

On the Analysis of Two-Dimensional Functionally Graded Rotating Thick Hollow Cylinder

Mostafa Omidi Bidgoli1, *

Department of Mechanical Engineering, Islamic Azad University, Badroud branch, Badroud, Iran

 E-mail: [e-mail]

*Corresponding author

Mohammad Hosseini2, Ali Fata3

Department of Mechanical Engineering, University of Hormozgan, Bandar Abbas, Iran

E-mail: [e-mail], [e-mail]

Received: 10 September 2024, Revised: 25 February 2025, Accepted: 18 March 2025

Abstract: Rotary components are widely used in industries. There is both mechanical and thermal loading in most rotating cylinder applications. On the other hand, functionally graded material has better performance under different loads. Therefore, a finite length two dimensional functionally graded material (2D-FGM) thick hollow cylinder under angular velocity is investigated, in this research. Volume fraction distribution of functionally graded material and geometry of the cylinder are axisymmetric but not uniform along the radial and axial directions. The finite element method based on Rayliegh-Ritz energy formulation is applied to obtain the governing Equation and associated boundary conditions of a 2D-FG thick hollow cylinder. Using this method, the effects of the power law exponents and angular velocity on the displacements and distribution of stresses are investigated for simply supported 2D-FG thick hollow cylinder. The results indicate that 2D-FGM facilitates improved design, allowing for better control of both maximum stresses and stress distribution through material distribution.

Keywords: 2D Functionally Graded Material, FEM, Rotating Cylinder, Thick Hollow Cylinder 

Biographical notes: Mostafa Omidi Bidgoli is an assistant professor and accomplished mechanical engineer specializing in solid mechanics, with expertise in FGM, stress and strain analysis, and creep analysis. He received his Ph.D. in Mechanical Engineering from the University of Kashan in 2019. He is an assistant professor at IAU, Badroud branch. Mohammad Hosseini is an assistant professor and accomplished mechanical engineer specializing in solid mechanics, with expertise in nano-structures, MEMS, and NEMS, stress analysis, and finite element methods. He received his Ph.D. in Mechanical Engineering from Shahid Chamran University of Ahvaz in 2017. Ali Fata is an associate professor of Mechanical Engineering at Hormozgan University, specializing in Manufacturing Engineering and advanced materials processing. He received his Ph.D. in Mechanical Engineering from Amir Kabir University in 2012.

 

 

1        Introduction 

Rotating cylinders with functionally graded materials (FGMs) have attracted much attention recently as an innovation in materials, engineering, and structural design. FGMs are heterogeneous composite materials whose mechanical properties vary continuously between several different materials. Also, the rotary cylinder is widely used in various industries. Functionally graded cylinders are used in internal combustion engines, gas turbines, medicine, etc. The presence of variable stresses and thermal fields in rotating cylinders requires careful design and the use of materials with specific properties. FGMs have the ability to adapt to these challenges by gradually changing the composition and properties of the materials along the design. In most rotating cylinder applications, there is both mechanical and thermal loading. The cylinder can be made of functionally graduated to perform optimally under different loads. It is necessary to theoretically examine the behavior of the functionally graded structure before the construction process, considering the high costs of laboratory tests. In the last few years, much research has been carried out to investigate the behavior of the rotating functional graduated cylinder, which will be discussed in the following.

Shi and Xie [1] presented an exact solution for a hollow FG cylinder under the simultaneous action of an applied magnetic field and internal pressure. They analyzed residual stresses caused by unloading internal pressure and the effect of some parameters on the plastic zone size. Li et al. [2] derived the governing Equation of a functionally graded cylinder or circular disk. The structure was under axisymmetric mechanical and thermal loads. They investigate the effect of some parameters, such as the inhomogeneity parameter, thermal and magnetic fields, internal pressure, and rotating velocity. Using the Pseudospectral Chebyshev Method, Yarimpabuç and Çalişkan [3] investigated the elastic behavior of FG rotating hollow cylindrical pressure vessels. Daghigh et al. [4] discussed the time-dependent behavior of FGM rotating disks of variable thickness subject to mechanical load and uniform temperature. In the frameworks of first-order shear deformation theory, Sedighi et al. [5] studied the time-dependent creep behavior of thick-walled cylinders under internal pressure, and heat flux at the inner and outer surfaces has been investigated. Omidi Bidgole et al. [6] used first-order shear deformation theory to analyze transient stress and deformations of short-length FG rotating cylinders subjected to thermal and mechanical loads on a friction bed. Based on the finite element method, Maitra et al. [7] presented an analysis for rotating truncated radially FG conical shells under constant and variable internal pressure. Creep analysis of an FG rotating disc of variable thickness was discussed by Saadatfar et al. [8]. They explained the effect of design parameters on the creep behavior of the disc. Using the Biot poroelastic law, Babaei et al. [9] derived the governing Equation and discussed the dynamic behavior of the FG-saturated porous rotating thick truncated cone. Xu et al. [10] presented a model to explain elastic wave propagation in a 2D-FG thick hollow cylinder. Yarımpabuç [11] used a closed-form solution to study the transient thermal stress analysis of a radially FG hollow cylinder subjected to high-temperature difference and periodic rotation effect. Jabbari and Zamani Nejad [12] investigated the effect of thermal and mechanical loads on the strain and stress distribution of rotating radially FG shells of variable thickness. Mehditabar et al. [13] studied the mechanical behavior of radially FG rotating hollow cylindrical shells under dynamic loading. Gharibi et al. [14] derived and solved the governing Equation of radially FG rotating thick cylindrical pressure vessels. Jabbari et al. [15] analyzed axially FG rotating thick cylindrical of variable thickness subject to thermo-elastic loads. They used higher-order shear deformation theory and the multilayer method to derive the governing Equation.

Jafari Fesharaki et al. [16] presented a 2D solution for the electro-mechanical behavior of functionally graded piezoelectric hollow cylinders under 2D electro-mechanical loads.

Zafarmand and Hasani [17] studied the elastic behavior of a 2D-FGM rotating disk of variable thickness. They investigate the effect of the thickness profile and material inhomogeneity index on the displacement distribution and stresses. Dai et al. [18] analyzed displacements and stresses of radially FG piezoelectric rotating hollow cylinders. Also, much research has been done on rotating structures in addition to the ones reviewed in this article [19-44].

As can be seen, much research has been done on rotating functionally graduated cylinders. Most of the articles have assumed material changes either in the radial direction or in the axial direction. In this paper, the stress analysis of 2D-FGM rotating thick hollow cylinders has been investigated for the first time. The problem is modeled using 2D axisymmetric elasticity theory and the finite element method based on the Rayleigh-Ritz energy formulation. The influence of power law exponents and rotational velocity on the distribution of displacements and stresses is considered.

2        Material and geometry of structure

Figure 1 shows a thick hollow cylinder, where r and z are the radial, and length axis of cylindrical coordinate system.

 

Fig. 1        Geometry of the cylinder.

 

The inner surface of the cylinder is made of ceramics, and the outer surface is made of metal alloys. Material properties vary through both the radial and axial directions. The volume fraction of constituent materials can be expressed as:

 

 

Therefore, the material properties at each point can be obtained by the linear rule of mixtures as:

 

	(1)
	(2)
	(3)
	(4)

(5)

 

For instance, the volume fraction distribution of the second ceramic and the variation of modulus of elasticity through the cylinder for typical values of  are shown in “Figs. 2 and 3”, respectively. In this case, a=1 m, b=1.5 m, and L=1 m. The essential constituents of the 2D-FGM cylinder are presented in “Table 1”.

 

 

Fig. 2        Volume fraction distribution of second ceramic through the cylinder for .

 

Fig. 3        Distribution of modulus of elasticity through the cylinder for

 

Table1 Basic constituent of 2D-FGM cylinder

3        Governing Equations of Rotating Thick Hollow Cylinder

Figure 1 shows a rotating thick hollow cylinder. Based on the axial symmetry assumption, the formulation is reduced to two dimensions. Due to axisymmetric geometry, axisymmetric material properties, and axisymmetric load distribution, the circumferential displacement, shear stresses, and strains in  and  planes are zero. Governing Equations in axisymmetric cylindrical coordinates were obtained as:

 

        (6)

 

        (7)

 

Linear strain-displacement relations are:

 

        (8)

Where  and  are the radial and axial displacements, respectively. “Eq. (8)” could be rewritten as:

 

Constituent	Material	E (GPa)	Density (kg/m3)
	Ti6Al4V	115	4515
	Al 1,100	69	2715
	SiC	440	3210
	Al2O3	300	3470

(9)

 

Where:

 

 

 

Based on the Hooke’s law, the stress-strain relationship is:

 

        (10)

 

Where:

 

(11)

 

        

In “Eq. (11)”,  is the Young’s modulus, which is a function of r and z components. Also,  denotes Poisson’s ratio, and it is constant.

The cylinder has simply supported boundary conditions, so displacement boundary conditions are as follows:

 

        (12)

4        FINITE ELEMENT MODELLING

The finite element method is used to solve the governing Equations for 2D-FGM thick hollow cylinders. The displacement matrix of each element  in terms of the nodal displacement matrix  and shape function  is:

 

        (13)

 

By substituting “Eq. (13)” into “Eq. (9)”, the strain matrix of the element  is determined as:

 

        (14)

 

Where  is the strain-displacement matrix. matrix is as follows for the quadratic six nodded triangular element:

 

                                                                          (15)

 

Where:

 

 

 

 

 

        
                                                                           (16)

 

 

 

 

 

 

 

Where  and  are the radial and axial coordinates of the ith node, respectively.

 

 

 

 

And  matrix is:

 

 

                                          (17)

 

 

Which the components of  matrix are presented in Appendix A.

The material inhomogeneity of the FG hollow cylinder can be defined using nodal values. Consequently, a graded finite element approach can be utilized to accurately capture smooth variations in material properties at the element level. FGM modeling using graded elements yields more precise results than partitioning the solution domain into homogeneous elements. So, shape functions similar to those of the displacement components can be used:

 

        (18)

 

        (19)

 

where and are the modulus of elasticity and mass density corresponding to ith node. ,  and   are vectors of shape functions, modulus of elasticity, and mass densities of each element, respectively, and are defined as follows:

 

 

        (20)

 

Therefore, “Eq. (11)” may be rewritten as:

 

        (21)

 

Based on the principle of minimum potential energy and the Rayleigh-Ritz method, governing Equations can be derived. The total potential energy of the rotating 2D-FGM thick hollow cylinder can be written as:

 

        (22)

Utilizing the principle of minimum total potential energy leads to:

 

               (23)

 

In compact form:

 

        (24)

 

Where:

 

        (25)

 

        (26)

 

 is defined as follows:

 

        (27)

 

Therefore, the finite element form of the governing Equations of FG thick hollow cylinder will written as:

 

        (28)

5        RESULTS

5.1. Verification of Solving Method

To validate the results of this study, a comparison was made between the radial displacement findings of the current research and those obtained from ANSYS WORKBENCH (“Fig. 4”). Accordingly, in relationships, it is assumed that (, ) are zero. Therefore, the cylinder is homogeneous and is made of . The following parameters were used for validation.

 

a=1 (m), b=1.5 (m), L=1, E=69GPa, ρ=2715,  =and ν =0.3.

 

A good agreement was observed for the radial displacement of the cylinder at z=.

 

 

Fig. 4        Comparison between the radial displacement of the present study and Ansys Workbench.

 

5.2. Numerical Results

Numerical results have been studied for 2D-FGM rotating thick hollow cylinders. The effects of angular velocity and various power law exponents on the displacements and stress distributions have been investigated. The values of the parameters are:

 

a=1 (m), b=1.5 (m), L=1 (m),

 =  and ν =1/3.

 

The effect of the different power law exponents on the radial displacement at z= of the simply supported 2D-FGM thick hollow cylinder is depicted in “Fig. 5”.

 

 

Fig. 5        Radial displacement at z=L/2 for different power law exponents.

 

It can be seen that the maximum radial displacement belongs to  =0. In this case, the cylinder is made of m2, and it has minimum stiffness. Also, the minimum displacement occurs for  =5 with maximum stiffness. Figures 6-8 explain the effect of power law exponents on the radial, tangential, and axial stresses at z=, respectively. Results show that the axial stresses are less affected by power law exponents, but tangential and radial stresses are mainly affected. Also, results show that maximum and minimum stresses belong to nr=5, nz=1, and nr=nz=0, respectively.

 

 

Fig. 6        Radial stress at z=L/2 for different power law exponents.

 

 

Fig. 7        Tangential stress at z=L/2 for different power law exponents.

 

 

Fig. 8        Distribution of the axial stress at z=L/2 for different power law exponents.

 

Distribution of radial displacement, radial, tangential, and axial stresses for power law exponents nr=1, nz=5 are shown in “Fig. 9”. The effect of the circular velocity on the radial stress at z= for the simply supported 2D-FGM thick hollow cylinder is shown in “Fig. 10”. Radial stress is increased as angular velocity increases.

 

 

 

 

Fig. 9        Radial displacement, radial, tangential, and axial stresses for nr=1, nz=5.

 

 

 

Fig. 10        Radial stress at z=L/2 for different rotational velocities.

6        CONCLUSIONS

This рionееring аnаlysis introduсеs 2D ахisymmеtriс еlаstiсity modeling to еvаluаtе strеss distributions in rotаting two-dimensional funсtionаlly grаdеd thiсk hollow сylindеrs for the first time. The graded finite еlemеnt method and Rаylеigh-Ritz еnеrgy formulation were strаtеgiсаlly employed to solvе governing Equаtions.

The effects of power law exponents and angular velocity on displacement and stress distributions were examined. Based on the results, 2D-FGM utilization leads to a better design to control both the maximum stresses and stress distribution by changing the material distribution.

The main results of the present study are:

o       Maximum radial displacement belongs to nr=nz=0. Also, the minimum displacement belongs to nr=nz=5 with maximum stiffness.

o       The axial stresses are less affected by power law exponents, but tangential and radial stresses are mainly affected. Also, maximum and minimum stresses belong to nr=5, nz=1, and nr=nz=0, respectively.

o       Radial stress increased by increasing the angular velocity.

APPENDIX A

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameters

References

[1]       Shi, P., Xie, J., Exact Solution of Magneto-Elastoplastic Problem of Functionally Graded Cylinder Subjected to Internal Pressure, Applied Mathematical Modelling, Vol. 123, 2023, pp. 835-855,

[2]       Li, X., Xie, J., and Shi, P., Magneto-Thermal–Mechanical Analysis of Functionally Graded Rotating Cylinder and Circular Disk, Archive of Applied Mechanics, Vol. 93, 2023, pp. 1449-1457. 

[3]       Yarımpabuç, D., Çalıskan, A. J. C. M., Computations, Applications: An International Journal, "Elastic Analysis of Functionally Graded Rotating Pressurized Cylinders, Vol. 15, 2023.

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