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Volume-5, Issue-3, June-2015

International Journal of Engineering and Management Research

Page Number: 354-360

Elastic Analysis of Exponentially varying FGM Rotating Disk by Finite

Element Method

Amit Kumar Thawait1, Lakshman Sondhi2

1

Department of Mechanical Engineering, Shri Shankaracharya Technical Campus, SSGI, Bhilai (C.G.), INDIA

2

Department of Mechanical Engineering, Shri Shankaracharya Technical Campus, SSGI, Bhilai, (C.G.), INDIA

ABSTRACT

The objective of the current work is to analyze displacement, stress and strain of uniform thickness rotating annular disk by finite element method. The disk is made up of functionally graded material instead of homogeneous material. Its mechanical and physical properties vary exponentially in radial direction. 4-noded quadrilateral finite element is used to model the disk and variation of components of displacement, stress and strain along the radial direction are evaluated using ANSYS Mechanical APDL.

Key words---- Functionally graded material (FGM), Rotating Disk, Finite element method (FEM).

I.

INTRODUCTION

Many engineering components may be modeled as rotating circular plates or disks. Some examples may be found in the marine, mechanical and aerospace industry including gas turbines, gears, turbo-machinery, flywheel systems and centrifugal compressors, power transmission systems, machining devices, circular saws, microwave or baking ovens, support tables etc. The stresses due to centrifugal load can have important effects on their strength and safety. Thus, control and optimization of stresses and displacement fields can help to reduce the overall payload in industries.

On the other hand, the main advantage of using the functionally graded materials (FGMs) is providing the capability of accurately monitoring changes of the local material properties to optimize the component strength. Therefore, achieving a uniform effective stress to strength ratio in the whole component can be an objective. Depending on the function of the component, it is possible

to utilize one, two or three-directional distributions of the material properties.

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worked on Stress Analysis of Nonhomogeneous Rotating Disc with Arbitrarily Variable Thickness Using Finite Element Method. He has formulated a variable thickness rotating disk and analyzed it on ANSYS Workbench.

In this paper the rotating disk of uniform thickness made up of functionally graded material whose material properties vary in one direction that is radial direction, is analyzed by finite element method.

II.

PROBLEM FORMULATION

A. Equillibrium equation

Assuming plane stress condition (σz = 0) along with axisymmetric condition the equilibrium equation in radial direction for the rotating disk of uniform thickness is given by [8]

…(1)

Where:

B. Graded material properties:

The disk is made up of Al2O3/Al functionally

graded material and has exponentially varying material properties [9]. The young’s modulus of elasticity and density has following exponential variation:

…(2) …(3) Where

a and b are inner and outer radius, EA, EB, are young’s

modulus of elasticity for Al and Al2O3 while ρA and ρB are

densities for Al and Al2O3 respectively. The inner surface

of the disk (r = a) consists of 100% material A that is Al whereas the outer surface of the disk (r = b) is 100% material B that is Al2O3, Properties of Al and Al2O3 are

given in table 1

C. Elastic and Finite element equations

For Plane axisymmetric problem, an (r-θ) plane (analogous to x-y plane for plane elasticity) can be considered. Two independent non-zero strains exist in plane axisymmetric problems [6]:

,

Where

In standard finite element notation, strain displacement relationship can be written as [6]

…(4)

where [B] is strain displacement relation matrix which depends on element taken and contains derivatives of shape functions.

The stress – strain relationship for the problem, is given by

…(5)

Because of the symmetry τrθ and γrθ

In standard finite element notation

vanishes.

…(6)

Where D(r) is stress - strain relationship matrix and is a function of radius r.

The system level Finite element equation is given by …(7) Where

N = no. of elements.

The summation indicates assembly of individual elemental matrices following the standard procedure of assembly.

Defining element stiffness matrix [K]e and element load vector {f}e as [6]

…(8) …(9)

D. Calculation of elemental displacement vector, stiffness matrix, stress and strain

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…(10)

where

,

and

Element strain is given by:

…(11)

…(12)

…(13)

…(14)

…(15) Where

[J] = Jacobian matrix

Element stiffness matrix is given by

…(16) Or

…(17) Or

…(18) For an axisymmetric element t=2πr therefore

…(19) and Elemental stress is given by

…(20) E. Boundary Conditions

1) Constraints:

In an axisymmetric model rigid body motion in

the r and θ directions does not need to be constrained

because it is controlled by the definition of axisymmetric elements [11].However, at least one node must be constrained to prevent rigid body motion in the z- direction. Symmetry boundary condition is applied at the bottom edge of the half cross section of the disk. Inner surface is fixed along radial direction and at outer surface is free.

Other constraints are also applied based on the boundary conditions such as fixed, free or simply supported at inner or outer radius etc. [10]

2) Body forces:

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III.

ANALYSIS

IN

ANSYS

14.0

MECHANICAL

APDL

Modeling as well as meshing of functionally graded material and geometry is done by the ANSYS Mechanical APDL coding. The rotating disk of uniform thickness is modeled here as a rectangular axisymmetric cross section. Plane182 type element is taken here which is used for 2-D modeling of solid structures. The element can be used as either a plane element (plane stress, plane strain or generalized plane strain) or an axisymmetric element. It is defined by four nodes having two degrees of freedom at each node: translations in the nodal x and y directions. Element behavior is axisymmetric. Total 135 elements of 1 mm size is taken in radial direction.

For finite element modeling of functionally graded material 135 different materials are modeled having graded properties according to exponentially grading law and are assigned to each element. Therefore material properties inside the element is constant but vary exponentially from element at inner radius to element at outer radius.

After modeling and meshing constraint boundary condition

(i

n current problem fixed boundary condition at inner surface, free boundary condition at outer surface and symmetry boundary condition in z- direction at the bottom edge of the half cross section) and inertial boundary condition (angular velocity) is applied on the disk. Finally solution is done and results are obtained.

IV.

RESULTS AND DISCUSSION

A. Validation: The data of a functionally graded thin

rotating disk [7] is used and the problem is modeled and analyzed in ANSYS. The inner and outer radii of the disk are a = 40mm and b = 100mm, and the thickness of the disk is 2.5 mm. The elasticity modulus and density vary in the r direction as below:

where E0= 72 GPa, ρ0= 2,800 kg/m3

Figure 1: Comparison of current work with reference [7]

B. Numerical Results: In this section a rotating annular disk of Al/Al

and the angular velocity is ω = 1,570.8 Rad/s. Poisson’s ratio is taken 0.3. The boundary condition is free on both the inner and outer surfaces.

Figure 1 shows the comparison of current work with reference [7], results of current work are in good agreement with pre analyzed results of literature.

2O3 functionally graded material [9] is

analyzed. Inner radius of the disk is 15 mm while outer radius is 150 mm. Disk is of 10 mm thickness. Young’s modulus and density varies exponentially according to equation (2) and (3) respectively. Poisson’s ratio is taken as 0.3. Angular velocity of the disk is 1 rad/s.

Figure 2 to Figure 13 shows the variation of radial displacement, radial stress, circumferential stress, von mises stress, radial strain and circumferential strain along the radius respectively in contour plot and graph plot. In contour plot cross sectional view of the disk is shown. In abscissa (r-a) is taken in meter, while in ordinate displacement, stress and strain components are plotted.

The value of minimum radial displacement is 0 at the inner radius due to the fixed boundary condition and maximum value is 4.23×10-3

The maximum value of radial stress is 0.0141 MPa at inner radius which is due to the maximum density and fixed boundary condition at inner radius which causes a maximum centrifugal stress. The minimum value is 1.0×10

mm at a radius of 112 mm. from inner radius its value increases continuously till the maximum in a decreasing slope manner and after maximum it decreases (negative slope of the curve) till the outer radius.

-7

MPa at outer radius which satisfy the free

boundary condition at outer radius (at r = b, σr = 0). From

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radius. The maximum value of von mises stress is 0.01243

MPa at inner radius while minimum value is 8.3×10-3 MPa at a radius of 75 mm. From inner radius to 75 mm radius it decreases continuously while from 75 mm to outer radius it starts increasing and reaches 0.01 MPa approximately at outer radius.

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V.

CONCLUSION

In this paper two dimensional analysis of FGM rotating disk is done by finite element method. A rotating disk made up of Al/Al2O3

[5] Abdur Rosyid, Mahir Es-Saheb and Faycal Ben Yahia. Stress Analysis of Nonhomogeneous Rotating Disc with

Arbitrarily Variable Thickness Using Finite Element Method. Research Journal of Applied Sciences, Engineering and Technology 2014;7(15):3114-3125.

FGM whose properties vary exponentially in radial direction is analyzed and variation of radial displacement, radial stress, circumferential stress, von mises stresses, radial strain and circumferential strain are evaluated. The objective of using FGM is to minimize the maximum stress on the disk and allowing it to rotate at a higher speed than a homogeneous disk. Figure 1 shows that the maximum radial stress for a homogeneous disk is 10 MPa approximate while for FGM disk it is 7 MPa. Therefore an FGM disk having same geometry and mass as homogeneous disk can rotate at higher speed.

REFERENCES

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[3] Hassan Zafarmand, Behrooz Hassani. Analysis of two-dimensional functionally graded rotating thick disks with variable thickness. Acta Mech 2014; 225:453–464.

[4] Manish Bhandari, Dr. Kamlesh Purohit. Analysis of Functionally Graded Material Plate under Transverse Load for Various Boundary Conditions. IOSR Journal of Mechanical and Civil Engineering (IOSR-JMCE) 2014; 10(5):46-55.

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[9] A.M. Afsar, J. Go. Finite element analysis of thermoelastic field in a rotating FGM circular disk. Applied Mathematical Modelling 2010; 34: 3309–3320. [10] M. Shariyat and R.Mohammadjani. Three-dimensional compatible finite element stress analysis of spinning two-directional FGM annular plates and disks with load and elastic foundation nonuniformities. LAJSS 2013; 10: 859 – 890.