Effect of density variation parameter in a solid...

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Vol. 16, br. 3 (2016), str. 143–148

STRUCTURAL INTEGRITY AND LIFE

Vol. 16, No 3 (2016), pp. 143–148

143

Pankaj Thakur1, Suresh Kumar2, Joginder Singh3, Satya Bir Singh4

EFFECT OF DENSITY VARIATION PARAMETER IN A SOLID DISK

UTICAJ PARAMETRA PROMENE GUSTINE KOD ČVRSTOG DISKA

Originalni naučni rad / Original scientific paper

UDK /UDC: 539.37

Rad primljen / Paper received: 24.10.2016.

Adresa autora / Author's address: 1) ICFAI University Baddi, Faculty of Science and Technol-

ogy, Department of Mathematics, Solan, Himachal Pradesh,

India, email: [email protected] 2) I.K. Gujral Punjab Technical University Jalandhar, Depart-

ment of Applied Sciences (Mathematics), Punjab, India 3) Chandigarh Engineering College, Department of Math-

ematics, Landran, Mohali, Punjab 4) Punjabi University Patiala, Department of Mathematics,

Punjab, India Keywords

• solid disk

• stresses

• displacement

• angular speed

• yielding

Abstract

The purpose of this paper is to present the study of

density variation in a solid disk by using Seth’s transition

theory. Seth’s transition theory is applied to the problem of

creep stresses and strain rates in non-homogeneous spheri-

cal shell under steady-state temperature. Neither the yield

criterion nor the associated flow rule is assumed here. The

results obtained here are applicable to compressible mate-

rials as well as incompressible material. It has been seen

that radial stress has a maximum value at the inner surface

but infinite at the origin of a solid disk. With the introduc-

tion of a density parameter, the values of radial stresses are

increased at the inner surface and the circumferential

stress at the outer surface of the solid disk. Compressible

materials increase the values of plastic stresses at the

centre of the disk.

Ključne reči

• čvrsti disk

• naponi

• pomeranje

• ugaona brzina

• tečenje

Izvod

U radu je predstavljena studija promene gustine kod

čvrstog diska primenom teorije prelaznog stanja Seta.

Teorija prelaznog stanja Seta je primenjena na probleme

napona puzanja i brzine deformacija kod nehomogene

sferne ljuske u uslovima ravnomerne temperature. Ovde se

ne pretpostavlja ni kriterijum puzanja, a ni odgovarajući

zakon protoka. Dobijeni rezultati se mogu primeniti na

stišljive i na nestišljive materijale. Uočava se da napon u

radijalnom pravcu ima najveću vrednost na unutrašnjoj

površini, a postaje beskonačan u osi čvrstog diska. Uvođe-

njem parametra gustine, vrednosti radijalnih napona se

povećavaju na unutrašnjoj površini i tangencijalnog napo-

na na spoljnjoj površini čvrstog diska. Stišljivi materijali

povećavaju vrednosti napona u plastičnoj oblasti u središtu

diska.

INTRODUCTION

The Solid-State Disk Drive technology has been around

for a couple of decades already in the form of memory sticks,

thumb drives, and so on. Only recently, they have developed

up to an enterprise level with characteristics required by

modern large-scale commercial applications on IBM Power

Systems with IBMi. The problems of rotating solid disks

have been performed under various interesting assumptions

and the topic can be easily found in most of the standard

elasticity and plasticity books /1, 2, 3/. Rotating disks have

received a great deal of attention because of their wide use in

many mechanical and electronic devices. They have

extensive practical engineering applications such as in steam

and gas turbines, turbo generators, flywheel of internal

combustion engines, turbojet engines, reciprocating engines,

centrifugal compressors and brake disks. The theoretical and

experimental investigations on the rotating solid disks have

been widespread attention due to the great practical

importance in mechanical engineering. For a better utilization

of the material, it is necessary to allow variation of the

effective material or thickness properties in one direction of

the solid disk. Most of the research works are concentrated

on the analytical solutions of rotating isotropic disks with

simple cross-section geometries of uniform thickness and

specifically variable thickness. Gamer /4/ found the ‘elastic-

plastic deformation of the rotating solid disk under the as-

sumptions of Tresca’s yield condition, its associated flow

rule and linear strain hardening. To obtain the stress distribu-

mailto:[email protected]

Effect of density variation parameter in a solid disk Uticaj parametra promene gustine kod čvrstog diska


Vol. 16, br. 3 (2016), str. 143–148


Vol. 16, No 3 (2016), pp. 143–148

144

tion, they matched the plastic stresses at the same radius r = z

of the disk. The solution of a rotating solid disk with constant

thickness is obtained by Gamer /4, 5/ taking into account the

linear strain hardening material behaviour. L.H. You et al. /6/

discuss the problem of elastic-plastic stresses in a rotating

solid disk. Ahmet N. Eraslan /7/ et al. analysed the problem

of rotating elastic-plastic solid disks of variable thickness

having concave profiles. Thakur et al. /16/ investigated the

problem of infinitesimal deformation in a solid disk using

Seth’s transition theory. Seth’s transition theory /8, 9/ does

not acquire any assumptions like a yield condition, incom-

pressibility condition, and thus poses and solves a more

general problem from which cases pertaining to the above

assumptions can be worked out. It utilizes the concept of

generalized strain measure and asymptotic solution at critical

points or turning points of the differential equations defining

the deforming field and has been successfully applied to a

large number of problems /12-16/. Seth /10/ has defined the

generalized principal strain measure as:

12

0

2

1 2

11 1 2 , ( 1, 2,3)

A

ii

ne A A

ii ii ii

nA

ii

e e d e

e in

(1)

where n is the measure. The density of the disk varies along

the radius in the form

= (r/b)m, (2)

where is the constant density at r = b and m is the density

parameter. In this paper, we investigate the problem of a

solid disk having variable density by using Seth’s transition

theory. Results have been discussed numerically and are

depicted graphically.

MATHEMATICAL MODEL OF SOLID DISK

We consider a state of plane stress and assume infinitesi-

mal deformation. Suppose a solid disk of variable density

with radius b. The disk is rotating with angular velocity

about an axis perpendicular to its plane and passing through

the centre as shown in Fig. 1. The thickness of the disk is

assumed to be constant and is taken to be sufficiently small

so that the disk is effectively in a state of plane stress that

is, the axial stress Tzz is zero.

Origin r = b

Figure 1. Geometry of a solid disk.

Governing equations

The cylindrical polar coordinates are given by Seth’s

/10/:

u = r(1 – ); v = 0; w = dz (3)

where is function of r = 2 2x y only, and d is a

constant.

The strain components for infinitesimal deformation are:

1 ,

1 ,

,

0

A

rr

A

A

zz

A A A

r z zr

ue r

r

ue

r

we d

z

e e e

(4)

Using Eq.(4) in Eq.(1), the generalized components of

strain are:

2

2

2

11 2 1 ,

11 2 1 ,

11 (1 2 ) ,

0

n

rr

n

nzz

r r zr

e rn

en

e dn

e e e

(5)

where = d/dr.

Stress-strain relation: the stress-strain relations for iso-

tropic media are given by /11/:

1 2 , ( , 1,2,3)ij ij ijI e i j (6)

where: ij are stress components; eij – strain components; ,

are Lame’s constants; I1 = ekk is the first strain invariant;

and ij is the Kronecker delta. Equation (6) for this problem

becomes:

22 ,

2

2 22 ,

2 2

0

rr rr rr

rr

zz zr r z

e e e

e e e

(7)

Substituting Eq.(4) in Eq.(6), the strain components in

terms of stresses are obtained as /11/:

1 1 1,

2

1 1 1,

2

1 1

2

rr rr rr

rr rr

zz rr rr

u ce

r E E c

u ce

r E E c

ce

E c E

(8)

where:

3 2E

,

2

, and

2



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Vol. 16, No 3 (2016), pp. 143–148

145

Substituting Eq.(5) in Eq.(7), stresses are obtained as:

2 2

2 2

23 2 2 ( 1) 1 (2 ) (2 1) (1 ) ,

23 2 2 ( 1) 1 (1 ) (2 1) (2 ) ,

0 (9)

n nrr

n n

zz zr r z

c P c cn

c P c cn

where:

r = P, and

2

2c

.

Equation of equilibrium: the equations of equilibrium are

all satisfied except:

2 2 0rr

dr r

dr (10)

Non-linear differential equation: using Eq.(9) in Eq.(10),

we get a non-linear differential equation in as:

122

2 22

2

(2 ) 2 ( 1) 1

2 ( 1) 12

( 1)(2 )1

2 ( 1) 1

(1 )2 1 1

2 1

n

n

n

dPc n P P

d

n rP

n P P c

P

n P c

(11)

Transition points: transition points of in Eq.(11) are

P 0 and P . P 0 gives nothing of importance.

Boundary condition: the boundary conditions are:

u = 0 at r = 0, and

Trr = 0 at r = b (12)

Solution of the problem

It has been shown /12-20/ that the asymptotic solution

through the principal stress leads from elastic to plastic

state at the transition point. We define the transition func-

tion R as:

2

2

2

(3 2 ) 2 ( 1) 1

(1 ) (2 1) (2 )

n

n

nR

C P

c c

(13)

Taking the logarithmic differentiation of Eq.(13) with

respect to r and using Eq.(11), we get Eq.(14):

Taking the asymptotic value of Eq.(14) as P and

after integration we get:

1 (2 )1

cR A r (15)

where: A1 is a constant of integration which can be deter-

mined by boundary condition. From Eqs.(13) and (15), we

have:

1 (2 )1

2 cA rn

(16)

Substituting Eq.(16) in Eq.(10) and using Eq.(2), then

after integration, we get:

2 2

1 (2 ) 0 11

2 (2 )

(1 ) (3 )

mc

rr m

r BcA r

n c rm b

(17)

where: B1 is a constant of integration which can be deter-

mined by boundary condition. Substituting Eqs.(16) and

(17) in Eq.(8), we get:

1 (2 )1

2 20 1

1 2

3 2

(1 )(2 ) (3 )

c

m

m

uA r

r E n

r Bc

c c rm b

(18)

2 2

0 1(1 )

(2 ) (3 )

m

m

r Bu c

r E c rm b

, (19)

where 2 (3 2 )

(2 )

cE

c

is the Young’s modulus.

Integrating Eq.(18) with respect to r, we get:

1

21 2

2 30

12

3 21 2

1

log3

c

c

m

m

cu A r

E n c

rB r D

m b

(20)

where: D is a constant of integration which can be deter-

mined by boundary condition. Comparing Eqs.(19) and

(20), one gets:

1

21 2

2 30

1

2 3 2

(1 )

1 (1 )

(3 ) (2 )(3 )

(1 ) (2 ) log

(2 )

c

c

m

m

cA r

n c

r c

m cm b

c c rB DE

c

(21)

and 2 3

01

(1 )

(2 ) (3 )

m

m

rcu B

E c m b

. (22)

2 22 2 2 2

2 2

12 ( 1) (2 1) (1 )(2 1) (2 )(2 1)

2 2(log )

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

n n n n

n

n n

c n rP n P c n P c

cd R

dr r c P c c

(14)



Vol. 16, br. 3 (2016), str. 143–148


Vol. 16, No 3 (2016), pp. 143–148

146

Using boundary conditions Eq.(12) in Eqs.(22), we get:

B1 = 0. Putting Eqs.(21) in Eq.(17) and using boundary

condition (12), we get:

2 2

0 1 1 (3 2 )

(3 ) 3 2 (1 )(2 )

b c cD

m E m c c c

.

Substituting values of B1, D and using Eqs.(21) in Eqs.(16)

and (17) respectively, we get the plastic stresses and dis-

placement as, Eqs.(23), (24), and (25):

22 20

2

(1 )(2 )

(3 )(3 2 )

1 1 1 1 3 2

3 2 3 2 (1 )(2 )

3 2, (23)

(1 )(2 )

m

rr

m

C C b r

m C b

C b C C

m C r m C C C

C r

C C b

22 2 20 (1 ) 1 1

(3 )(3 2 ) 3 2

1 1 3 2

3 2 (1 )(2 )

mC b r C

m C b m C

b C C

r m C C C

(24)

and 2 3

0(1 )

(2 ) (3 )

m

m

rCu

E C m b

. (25)

Initial yielding: for a solid disk the stress at the centre is

given when r = 0. With r equal to zero the above Eq.(23)

will yield infinite stresses whatever the speed of rotation,

these stresses are not meaningful. Yielding begins at the

centre, r = 0 and the compressive radial stress rr is greater

than compressive circumferential stress . It has been seen

that rr has a maximum value at the inner surface r < r1 and

r > 0. For yielding at r < r1 (= 0.05; numerically say),

Eq.(23) becomes:

1

22 20 1

1

2

1

(1 )(2 ) 1 1

(3 )(3 2 ) 3 2

1 1 3 2

3 2 1 2

3 2( )

1 2

m

rr r r

m

C C b r C

m C b m C

b C C

r m C C C

rCY say

C C b

where Y denotes the initial yield stress. The angular speed

necessary for inner-plastic-zone is given by Eq.(26):

2

2 21 1

1

(3 )(3 2 )

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

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

m m

m C

C C C CC C x x

m C x m C C C C C

(26)

where

1 1(0 ) / ( 0.05)x r r a , 2 2

2 0 b

Y

.

The stresses and displacement from Eqs.(23)-(25) are

obtained in non-dimensional form as, Eqs.(27)-(29):

2

2 21 1

1

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

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

m mr

C C C C C Cx x

m C m C x m C C C C C

, (27)

2 2

21

1

(1 ) 1 1 1 1 1 3 2

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

mC C C Cx

m C m C x m C C C

, (28)

and 2 3

1(1 )

(2 ) (3 )

mxCu

C m

. (29)

where: ; ; rrr

T T uEu

Y Y Yb

.

Equations (27)-(29) give stresses and displacement in the

solid disk, having a variable density.

NUMERICAL ILLUSTRATION AND DISCUSSION

To see the effect of stress distribution in a rotating solid

disk, the following values have been taken: c = 0 (incom-

pressible material); 0.25 (compressible material); 0.5 (com-

pressible material). In Fig. 2, the curves have been drawn

between stresses required for yielding at the outer surface

of the rotating solid disk along the radii ratio x1 = (0 < r <

r1)/a (= 0.05). It has been seen that the radial stress has a

maximum value at the inner surface, but at the origin - it

goes to infinite. With the introduction of density parameter,

the radial stresses have increased at the inner surface and

the circumferential stress increases at the outer surface of

the solid disk. For a perfectly plastic material, the usual

statically determinate stress distribution is recovered but,

since the plastic stress at the axis becomes infinite, these

stresses are not meaningful. Compressible materials have

increased the values of stresses at the centre of the disk.



Vol. 16, br. 3 (2016), str. 143–148


Vol. 16, No 3 (2016), pp. 143–148

147

sigma r = r , sigma theta =

Figure 2. Graph between stresses required for yielding at the outer surface of the solid disk along the radii ratio x1.



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Vol. 16, No 3 (2016), pp. 143–148

148

REFERENCES

1. Timoshenko, S.P., Goodier, J.N., Theory of Elasticity,

McGraw-Hill, New York, 1970.

2. Blazynski, T.N., Applied Elasto-Plasticity of Solids, McMillan

Press Ltd., London, 1983.

3. Chakrabarty, J., Theory of Plasticity, McGraw-Hill, New

York, 1987.

4. Gamer, U. (1984), Elastic-plastic deformation of the rotating

solid disk, Ingenieur-Archiv, 54: 345-354.

5. Gamer, U. (1985), Stress distribution in the rotating elastic-

plastic disk. ZAMM, 65, T136-137.

6. You, L.H., Zhang, J.J. (1999), Elastic-plastic stresses in a

rotating solid disk, Int. J Mechanical Sciences, 41(3):269-282.

7. Eraslan, A.N., Orcan, Y. (2002). On the rotating elastic-plas-

tic solid disks of variable thickness having concave profiles,

Int. J Mech. Sciences, 44(7): 1445-1466.

8. Seth, B.R. (1962), Transition theory of elastic-plastic defor-

mation, creep and relaxation, Nature, 195: 896-897.

9. Johnson, W., Mellor, P.B., Plasticity for Mechanical Engi-

neers, Van-Nostrand Reinhold Company, London, 1962.

10. Seth, B.R. (1966), Measure concept in mechanics, Int. J Non-

Linear Mech., 1: 35- 40.

11. Sokolnikoff, I.S., Mathematical Theory of Elasticity, 2nd Ed.,

New York, 1952, pp. 65-79.

12. Gupta, S.K., Thakur Pankaj (2007), Creep transition in a thin

rotating disc with rigid inclusion, Defence Science J, India, 57

(2): 185-195.

13. Gupta, S.K., Thakur Pankaj (2007), Thermo elastic-plastic

transition in a thin rotating disc with inclusion, Thermal

Science, 11(1): 103-118.

14. Thakur Pankaj (2010), Elastic-plastic transition stresses in a

thin rotating disc with rigid inclusion by infinitesimal defor-

mation under steady-state temperature, Thermal Science 14(1)

: 209-219.

15. Thakur Pankaj, Singh, S.B., Jatinder Kaur (2013), Thickness

variation parameter in thin rotating disc, FME Transactions

41(2): 96-102.

16. Thakur Pankaj, Singh, S.B. (2015), Infinitesimal deformation

in a solid disk using Seth’s transition theory, Int. J Electro

Mechanics and Mech. Behaviour 1(1): 1-6.

17. Singh, S.B., Ray, S. (2001). Steady state creep behaviour of

an isotropic functionality graded rotating disc of Al-SiC com-

posites, Metallurg. and Mater. Trans. 32A(7): 1679-1685.

18. Singh, S.B., Ray, S. (2003). Newly proposed yield criterion

for residual stress and steady state creep in an anisotropic

composite rotating disc, J Mater. Proc. Tech., Elsevier Publ.,

143 (144C): 623-628.

19. Thakur Pankaj, Singh, S.B., Kaur, J. (2016). Thermal creep

stresses and strain rates in a circular disc with shaft having

variable density, Eng. Comp. 33(3): 698-712.

20. Pankaj Thakur, Gupta, N., Singh, S.B. (2016). Creep strain

rates analysis in cylinder under temperature gradient for differ-

ent material, accepted for publ. Eng. Computation, 2016.

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