creep analysis of thin rotating disc under plane stress with no edge load

8/8/2019 Creep Analysis of Thin Rotating Disc Under Plane Stress With No Edge Load

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Creep Analysis of Thin Rotating Disc under Plane Stress with no

Edge loadSANJEEV SHARMA, MANOJ SAHNI

Department of MathematicsJaypee Institute of Information Technology University

A-10, Sector 62, Noida-201307, UPINDIA

E-mail: [email protected] , [email protected]

Abstract: - Creep stresses and strain rates for a transversely isotropic and isotropic materials have been obtained fora thin rotating disc using Seths transition theory. Results obtained have been discussed numerically and depictedgraphically. It is seen that a disc made of transversely isotropic material rotating with higher angular speedincreases the possibility of fracture at the bore as compared to a disc made of isotropic material and possibility of fracture further decreases with the increase in measure N. The deformation is significant for transversely isotropicdisc for the measure N =7.

Key words : Creep, Transition, Stresses, Strain Rates, Transversely Isotropic and Rotating Disc.

Nomenclature of Symbols:a and b : Internal and external radii of discD : a constantR : Radial distance

z y x ,, : Cartesian co-ordinatesu, v, w : Displacement components

zr ,, : Polar co-ordinatesije and : Strain and stress tensorijT

iie : First strain invariant : Function of r onlyP : Function of only

ijC : Material constants

and : Lames constants) / (), / ( 0 ba Rbr R ==

66 / C T rr r = - Radial stress components

66 / C T = -Circumferential stress components

research in this field. The analytical procedurespresently available are restricted to problems withsimplest configurations. The use of rotating disk inmachinery and structural applications has generatedconsiderable interest in many problems in domain of solid mechanics. Solutions for thin isotropic disks canbe found in most of standard creep text books [1-4].Han [5] has investigated elastic and plastic stresses forisotropic materials with variable thickness. Wang [6]has investigated deformation of elastic half rings.Enescu[7] give some numerical methods fordetermining stresses in rolling bearings whileMahri[8] calculated the stresses by using finiteelement method for wind turbine rotors. Wahl [9] hasinvestigated creep deformation in rotating disks

assuming small deformation, incompressibilitycondition, Trescas yield criterion, its associated flowrule and a power strain law. Transition theory [10-11]does not require any assumptions like a yieldcriterion, incompressibility condition and thus posesand solves a more general problem from which casespertaining to above assumption can be worked out

SEAS TRANSACTIONS onAPPLIED and THEORETICAL MECHANICS SANJEEV SHARMA, MANOJ SAHNI
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


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Hencky and others, is given by [12]. The generalizedprincipal strain measure [12] is defined as

[ ] ( )

== 20

12 211121

n Aii

e Aii

n Aiiii en

deee

A

ii

(1)

where is the measure and is the principal

Almansi finite strain components [13]. This theory isapplied to a large number of elastic-plastic and creepproblems [14-20]. In this paper an attempt has beenmade to study the behaviour of transversely isotropic

thin rotating disc using transition theory [10-11].

n Aije

2. Governing EquationsWe consider a thin disc of constant density made of transversely isotropic material with internal andexternal radii a and b respectively. The disc isrotating with angular velocity about an axisperpendicular to its plane and passing through thecentre of the disc. The disc is thin and is effectively ina state of plane stress.The displacement components in cylindrical polar co-ordinates are given [13] by,

,,0),1( dzwvr u === (2)

where is a function of 22 y xr += only andis a constant.d

The finite components of strain [13] are

0

],)1(1[21

],1[21

],)(1[21

2

2

2

===

=

=

+=

A zr

A z

Ar

A zz

A

Arr

eee

d e

e

r e

(3)

where dr d / = Substituting (3) in (1), the generalized components of strain are

])(1[1 nre += ]1[

1 ne =

0331313 =++= zzrr zz eC eC eC T

eC

C eC

C e rr zz

33

13

33

13

=

0=== r z zr T T T (5)where are material constants.ijC

Using equation (4) in equation (5) we have

)1(2

)])1(1(2[ 66 nnnrr nC

Pn

AT ++=

[ ]nnnn PnC Pn AT )1(12)])1(1(2[66 +++=

0==== zz zr zr T T T T (6)where ( )3321311 / C C C A = .The equations of equilibrium are all satisfied except

0)( 22 =+ r T rT dr d

rr (7)

where is the density of the material.Using equation (6) in equation (7), we get a non-lineardifferential equation in as,

dP

d

PP

P

nPnA

C

Ar

PPn

nnnn

+++

+

+=+ +

))1(1(

))1(

1(2

)1(

66

2211

(8)where .Pr ='

The transition points of in equation (8)are 1 Pand P .The boundary conditions are

0=rr T at ar = and (9)br =

3. Solution through Principal StressDifference

It has been shown that the asymptotic solution throughh i i l diff [13 14 16 18 19] h



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( ) ( )2

1exp

exp2

1 2213

11

3

1

20

22

0

2

RdR f R

dR f R

R

R

g

R

gr

+=

(21)( )( )

=1

31

320

2

0

2

2

exp2

exp1

R

g

g

r

dR f R

f R R (22)

where( )

( )n

nn

Dn

Rbn f

ng

24

,

2

13 2232

+

=+=+

These equations are same as obtained by Gupta et.al[16].

4. Strain RatesThe stress-strain relation can be written as,

( )ijijij T H

C A

H

C A

e6666 22

= (23)

where

( )

33

213

11

6666332211 ,4,

C

C C A

AC C H T T T

=

=++=

When the creep sets in, strain should be replaced bystrain rates. The stress-strain relation (23) becomes

( )ijijij T H

C A

H

C Ae 6666

22

= & (24)

where is the strain rate tensor with respect to flow

parameter t.ije&

Differentiating second equation of (4) with respect tot, we get

&& 1= ne (25)

For Swainger measure (n=1), we have from equation(25)

&& = (26)

From equation (10), the transitional value of is,

n

rr T T Cn

1

)(2

=

(27)

=

r rr & (28)

=

r & (29)

+=

r zz&(30)

where1

1

)(2

= nr

n ,

3311

2131C C

C =

=

11

66

3311

213 21

C

C

C C

C ,

+=

3311

213

11

66 14C C

C

C

C

For isotropic material, equations (28), (29) and (30)becomes

=

C C

r C C

rr 21

232

2& (31)

= r C

C C C 2

123

22

& (32)

(

+

=

r C

C C C

zz 21

232

2& ) (33)

These equations are same as obtained by Odquist [2]provided we put N n / 1= ,For incompressible materials, i.e. , equations

(31), (32), (33) becomes

0C

=23

r rr & (34)

=23r

& (35)

[ ]r zz += 6& (36)

Numerical Illustration and Discussion



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Elastic constants for transversely isotropic

material (Magnesium) and isotropic material (Brass)have been given in table 1.

ijC

Table 1: Elastic Constants (in terms

of )

ijC 210 / 10 m N

44C 11C 12C 13C 33C TIM(Mg)

1.64 5.97 2.62 2.17 6.17

IM(Brass)

1.0 3.0 1.0 1.0 3.0

Curves have been drawn in figures 1 4 betweenstresses and radii ratio R= r/b for a disc rotating withdifferent angular speed and for measure N = 3, 5and 7.

2

For a disc made of transversely isotropic

material rotating with angular speed = 0.1,circumferential stress is maximum at the internalsurface for measure n = 1/3 (N = 3) as compared todisc made of isotropic material and this value of circumferential stress decreases at the internal surfacewith increase in measure (i.e., N = 5, 7). Thecircumferential stress increases at internal surfacewith increase in angular velocity ( = 1, 5, 10). It

means that a disc made of transversely isotropicmaterial rotating with higher angular speed increasesthe possibility of fracture at the bore as compared to adisc made of isotropic material. Possibility of fracturedecreases with the increase in measure N. It can beconcluded that a rotating disc made of isotropicmaterial for measure N = 7 is on the safer side of thedesign in comparison to a disc made of transverselyisotropic material.

2

2

In figures 58, curves have been drawnbetween strain rates and radii ratio (R = r/b) at angularspeed and measure N = 3, 5, 7. It has beenobserved that a transversely isotropic disc experiencesa significant deformation for the measure N = 7 ascompared to a rotating disc made of isotropic

2

References

[1]. H. Kraus, Creep Analysis , Wiley, New York,1980.

[2]. F.K.G. Odqvist, Mathematical theory of creepand creep structure , Claredon Press, Oxford , 1974.

[3]. F.R.N. Nabarro, H.L.de.Villers, Physics of creep , Taylor & Francis, PA, 1995.

[4]. H. Altenbach, J.J. Skrzypek, Creep and Damage in Materials and Structures , SpringerVerlag, Berlin, 1999.

[5]. R.P.S. Han, Kai-Yuan Yeh, Analysis of High-Speed Rotating Disks with VariableThickness and Inhomogenity, Transactions of the ASME, 61, 1994, pp. 186-191.

[6] Xiue Wang, Xianjun Yin, On LargeDeformations of Elastic Half Rings , WSEASTransactions on Applied and Theoretical

Mechanics , Volume 2, Issue 1, January 2007,pp. 24.

[7] Ioan Enescu, Dan Lepadatescu, HoratiuTeodorescu, Numerical Methods forDetermination the Elastic Stresses in RollingBearings, WSEAS Transactions on Applied and Theoretical Mechanics , Volume 1, Issue1, November 2006, pp. 121-124.

[8] Z.I. Mahri,, M.S. Rouabah, Calculation of Dynamic Stresses using Finite ElementMethod and Prediction of Fatigue Failure forWind Turbine Rotor, WSEAS Transactions on

Applied and Theoretical Mechanics, Volume3, Issue 1, 2008, pp. 28-41.

[9]. A.M. Wahl, Analysis of creep in rotatingdiscs based on tresea criterion and associatedflow rule, Jr. Appl. Mech ., 23, 1956, pp. 103.

[10]. B.R. Seth, Creep Transition, Jr. Math. Phys .Sci., 6, 1972, pp. 73-81.

[11]. B.R. Seth, Transition theory of Elastic-PlasticDeformation, Creep and Relaxation, Nature ,195, 1962, pp. 896-897.

[12]. B.R. Seth, Measure concept in mechanics,Inter Jr Non linear Mechanics 1 1966 pp



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under internal Pressure, Def. Sc. Journal , Vol.54(2), 2004, pp. 135-141.

[16]. S.K. Gupta, Pankaj, Creep Transition in aThin Rotating Disc with rigid inclusion, Def.Sc. Journal , 57(2), 2007, pp. 185-195.

[17]. S.K. Gupta, Pankaj, Thermo Elastic-plastictransition in a Thin Rotating Disc with rigidinclusion, Thermal Science Scientific Journal ,2007, pp. 103-118.

[18]. S.K.Gupta, R.L.Dharmani, V.D. Rana, CreepTransition in Torsion, Int. Jr. Non-linear

Mechanics , Vol. 13, 1979, pp. 303-309.[19]. S.K. Gupta, R.L. Dharmani, Creep Transition

in thick-walled cylinder under internalpressure, ZAMM , Vol. 59, 1979, pp. 517-521.

[20] Sanjeev Sharma, Creep Transition in Non-homogeneous thick-walled rotating cylinder ,

Accepted for Publication in Def. Sci. Journal..



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Figure 1: Creep Stresses in a Thin Rotating Disc along the Radius R for Angular Speed 1.02

=


ISSN: 1991-8747 731 Issue 7, Volume 3, July 2008


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Figure 3: Creep Stresses in a Thin Rotating Disc along the Radius R for Angular Speed 52

=



102 = Figure 4: Creep Stresses in a Thin Rotating Disc along the Radius R for Angular Speed


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10=Figure 4: Creep Stresses in a Thin Rotating Disc along the Radius R for Angular SpeedSEAS TRANSACTIONS on

APPLIED and THEORETICAL MECHANICS SANJEEV SHARMA, MANOJ SAHNI



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Figure 5: Strain Rates in a Thin Rotating Disc along the Radius R for Angular Speed 1.02 =




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Figure 6: Strain Rates in a Thin Rotating Disc along the Radius R for Angular Speed 12

=




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Figure 7: Strain Rates in a Thin Rotating Disc along the Radius R for Angular Speed 52 =



gure : ra n a es n a n o a ng sc a ong e a us or ngu ar pee =


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creep analysis of thin rotating disc under plane stress with no edge load

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