stress analysis in a functionally graded disc n

8/11/2019 Stress Analysis in a Functionally Graded Disc n

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Stress Analysis in aFunctionally Graded DiscUnder Mechanical Loads

and a Steady StateTemperature Distribution

ByShrikant M Dhole(13ME63R24)


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Content

1. Introduction

2.

Stress analysis3.

Conclusion

4. References


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IntroductionFunctionally graded material (FGM) may becharacterized by the variation in composition and structuregradually over volume , resulting in corresponding changesin the properties of the material.

Fig: FGM made up of metal and ceramic


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Introductioncont..

Features -:1. Gradual transition in

microstructure.2. Minimize stressconcentration.

Application-:

Turbine components,rocket nozzle, chemicalreactor tubes, criticalfurnace parts etc.


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Introductioncont..

When shuttle enters in earth atmosphere, it is subjected tovery high temperature and load.

Fig: Entry of shuttle in earth atmosphere

Outer body of space shuttle is made up of FGM.

Outer surface of outer body is pure ceramic for sustaining hightemperatureInner surface of outer body is pure metal for high strength


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Introductioncont.. Applications of FGMs

1) Structural walls that combine two or morefunctions including thermal and soundinsulation

2) Enhanced sports equipment such as golfclubs, tennis rackets, and skis with addedgraded combinations of flexibility,elasticity, or rigidity


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Stress Analysis

Fig: A functionally graded rotating disc under the mechanicalloads and steady state temperature distribution

Po = External PressurePi = Internal PressureA = Inner Radius

b = External Radius = Angular VelocityT0 = Temperature ofinside surface


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Stress AnalysisCont..

The solution can be efficiently handled by using aspecial stress function that automatically satisfies theequilibrium equation (1) .

The particular stress stress function relation withthis property is given by

(2)where, F = F(r) is the stress function


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The governing equation for the stress function F isdetermined from the compatibility condition .

For this axi-symmetric case, the displacement field is of the form


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The strain field is given by

(3)

where,= Strain in radial direction

= Strain in tangential directionu = Displacement in radial direction

On eliminating u from equation (3) develops the

simple compatibil i ty conditi on .


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Simple compatibility condition which is derived fromequation (3)

(4)

( )


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where,E(r) = Elasticity modulus(r) = Thermal expansion coefficientT(r) = Temperature change

= Poissons ratio

Using Hookes Law for plane stress case, thestrains are given by

(5)


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It is assumed that the disc is of material properties (E, and ) and thermal change (T) varying throughout theradial section .

Poissons ratio is assumed as a constant because itsvariation has much less practical significance thanvariation in the other material properties.


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On substituting equation (2) and equation (5) incompatibility equation (4), we get

(6)

{ }

equation (2), stress-stress function relation equation (5), Hookes law relation

equation (4), compatibility equation


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It is assumed that

(7)where,

n1 , n 2 & n3 are grading indexes andT

0 is the temperature in the inner surface of the disc


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On substituting equation (7) in to the differentialequation (6), we get

(8)

On solving equation (8), we get the relation for stressfunction .


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where,

(9)

10 (a)

10 (b)

10 (c)


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Stress AnalysisCont.. The stress components can be obtained from the stress

function in equation (9) as ,

11(a)

11(b)equation (2) , stress-stress function relation

[ ]


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Conclusion Functionally graded materials are very useful where

situation arises such that the material should have more thanone mechanical properties.

Stress analysis is carried out on functionally gradedrotating hollow discs subjected to internal, external pressures,centrifugal force due to a constant angular velocity and steadystate temperature by using an analytical solution including

small deformation theory of elasticity.


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ReferencesHasan Callioglu; Stress analysis in a functionally graded discunder mechanical loads and a steady state temperaturedistribution ; Sadhana, Vol. 36, Part 1, 2011, pp. 53 64.

Indian Academy of SciencesMehdi Bayat, M. Saleem, B.B. Sahari, A.M.S. Hamouda, E.Mahdi; Mechanical and thermal stresses in a functionallygraded rotating disk with variable thickness due to radially

symmetry loads ; International Journal of Pressure Vessels andPiping 86; (2009): 357 372http://www.docstoc.com/docs/48428992/Functionally-graded-materials-%28FGM%29-aim-at-optimizing-the-performance


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Thank You!

stress analysis in a functionally graded disc n

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