Transient Thermoelastic Problem for a Thick Annular Disc With Radiation

Int. J. of Appl. Math and Mech. 7 (7): 57-73, 2011.

TRANSIENT THERMOELASTIC PROBLEM FOR A THICK

ANNULAR DISC WITH RADIATION

V. Varghese1 and L. Khalsa

2

1Reliance Industries Limited, Mauda, Nagpur (MS) 440 104, India

2Department of Mathematics, M.G. College, Armori, Gadchiroli (MS), 441208, India

Email: [email protected]

Received 3 February 2010; accepted 31 May 2010

ABSTRACT

We apply integral transformation techniques to study thermoelastic response of a thick

annular disc, in general in which sources are generated according to the linear function of the

temperature, with boundary conditions of the radiation type. The results are obtained as series

of Bessel functions. Numerical calculations are carried out for a particular case of a thick disc

made of Aluminum metal and the results are depicted in figures.

Keywords: Transient response, thick disc, temperature distribution, thermal stress, integral

transform

1 INTRODUCTION

As a result of the increased usage of industrial and construction materials the interest in

isotropic thermal stress problems has grown considerably. However there are only a few

studies concerned with the two-dimensional steady state thermal stress. Nowacki [9] has

determined steady-state thermal stresses in a thick circular plate subjected to an axisymmetric

temperature distribution on the upper face with zero temperature on the lower face and

circular edge. Wankhede [6] has determined the quasi-static thermal stresses in circular plate

subjected to arbitrary initial temperature on the upper face with lower face at zero

temperature. However, there aren’t many investigations on transient state. Roy Choudhuri [7]

has succeeded in determining the quasi-static thermal stresses in a circular plate subjected to

transient temperature along the circumference of circular upper face with lower face at zero

temperature and the fixed circular edge thermally insulated. Noda et al. [5] have considered a

circular plate and discussed the transient thermoelastic-plastic bending problem, making use

of the strain increment theorem. In a recent work, some problems have been solved by Noda

et al. [5] and Kulkarni et al. [8]. In all aforementioned investigations an axisymmetrically

heated plate has been considered. Recently, Nasser [3, 4] proposed the concept of heat sources

in generalized thermoelasticity and applied to a thick plate problem. They have not however

considered any thermoelastic problem with boundary conditions of radiation type, in which

sources are generated according to the linear function of the temperatures, which satisfies the

time-dependent heat conduction equation. From the previous literatures regarding thick

annular disc as considered, it was observed by the author that no analytical procedure has

been established, considering internal heat sources generation within the body. The success of

V. Varghese and L. Khalsa


58

this novel research mainly lies with the new mathematical procedures with a much simpler

approach for optimization for the design in terms of material usage and performance in

engineering problem, particularly in the determination of thermoelastic behaviour in thick

disc engaged as the foundation of pressure vessels, furnaces, etc.

This paper is concerned with the transient thermoelastic problem in a thick annular disc in

which sources are generated according to the linear function of temperature, occupying the

space :),,{( 3RzyxD ,)( 2/122 byxa },hzh where 2122 )( yxr with radiation type

boundary conditions.

2 STATEMENT OF THE PROBLEM

In the first instance, we consider a thick annular disc in which sources are generated

according to the linear function of temperature. The material of the disc is isotropic,

homogenous and all properties are assumed to be constant. Heat conduction with internal heat

source and the prescribed boundary conditions of the radiation type, the quasi-static thermal

stresses are required to be determined. The equation for heat conduction is ),,,( tzr the

temperature, in cylindrical coordinates, is:

ttzr

zrr

rr

),,,(

1

2

2

(1)

where ),,,( tzr is the internal source function, and ,/ C λ being the thermal

conductivity of the material, ρ is the density and C is the calorific capacity, assumed to be

constant. For convenience, we consider the undergiven functions as the superposition of the

simpler function [1]:

),,()(),,(),,,( tzrttzrtzr (2)

And

t

t

dtzrtzr

dtzrtzrT

0

0

)(exp),,(),,(

,)(exp),,(),,(

or for the sake of brevity, we consider (3)

),exp(2

)()(),,(

0

00 tr

zzrrtzr

,0 bra ,0 hzh 0

Substituting equations (2) and (3) to the heat conduction equation (1), one obtains

t

Ttzr

z

T

r

Tr

rr

),,(

1

2

2

(4)

Transient Thermoelastic Problem For A Thick Annular Disc With Radiation


59

where is the thermal diffusivity of the material of the disc (which is assumed to be

constant), subject to the initial and boundary conditions

0)0,0,1,( TTM t for all bra , hzh (5)

,0),,1,( 1 akTM r 0),,1,( 2 bkTM r for all hzh , 0t (6)

),()exp(),,1,( 03 rrthkTM z 0),,1,( 4 hkTM z for all bra , 0t (7)

The most general expression for these conditions can be given by

sfkfkskkfM )ˆ(),,,(

where the prime ( ^ ) denotes differentiation with respect to ; )( 0rr is the Dirac Delta

function having bra 0 ; 0 is a constants; )()exp( 0rrδtω is the additional sectional heat

available on its surface at z = h; 0T is the reference temperature; k and k are radiation

coefficients of the disc, respectively.

The Navier’s equations without the body forces for axisymmetric two-dimensional

thermoelastic problem can be expressed as [5]

021

)1(2

21

1

021

)1(2

21

1

2

2

zz

eu

rr

e

r

uu

tz

tr

r

(8)

where ru and zu are the displacement components in the radial and axial directions,

respectively and the dilatation e as

z

u

r

u

r

ue zrr

The displacement function in the cylindrical coordinate system are represented by the

Goodier’s thermoelastic displacement potential and Michell’s function M as [5]

,2

zr

M

rur

(9)

z

MM

zu z 2

22)1(2

(10)

in which Goodier’s thermoelastic potential must satisfy the equation

t

1

12 (11)

and the Michell’s function M must satisfy the equation



60

0)( 22 M (12)

where

2

22 1

zrr

rr

The component of the stresses are represented by the use of the potential and Michell’s

function M as

2

222

2

2

2r

MM

zrGrr

, (13)

r

M

rM

zrrG

112 22

, (14)

2

222

2

2

)2(2z

MM

zrGzz

, (15)

And

2

22

2

)1(2z

MM

rzrGrz

(16)

where G and υ are the shear modulus and Poisson’s ratio respectively.

The equations (1) to (16) constitute the mathematical formulation of the problem under

consideration.

3 SOLUTION OF THE PROBLEM

3.1 Transient heat conduction analysis

In order to solve fundamental differential equation (4) under the boundary condition (6), we

firstly introduce the integral transform [2] of order n over the variable r. Let n be the

parameter of the transform, then the integral transform and its inversion theorem are written

as

,),,()()( 21b

a np drrkkSrgrng

1

21 ),,()/)(()(n

npnp rkkSCngrg (17)

where )(ng p is the transform of )(rg with respect to nucleus ),,( 21 rkkS np .

Applying the transform defined in equation (17) to the equations (3), (4), (5) and (7), and

taking into account equation (6), one obtains



61

t

tznTtzn

z

tznTtznTn

),,(),,(

),,(),,(

2

22 (18)

0)0,0,1,( TTMt (19)

),,,()exp(),,1,( 021003 rkkSrthkTM nz 0),,1,( 4 hkTM z (20)

)exp()(),,(),,( 002100 tzzrkkSrtzn n (21)

where T is the transformed function of T and n is the transform parameter. The eigenvalues

n are the positive roots of the characteristic equation

0),(),(),(),( 10202010 akYbkJbkYakJ

The kernel function ),,( 210 rkkS n can be defined as

)],(),([)()],(),([)(),,( 2010020100210 bkJakJrYbkYakYrJrkkS nnnnnnn

With

)()(),(

)()(),(

000

000

rYkrYrkY

rJkrJrkJ

ii

ii

for 2,1i

And

b

a nn drbkkSrC 2210 )],,([

in which )(0 rJ and )(0 rY are Bessel functions of first and second kind of order 0p

respectively.

We introduce the another integral transform [1] that responds to the boundary conditions of

type (7)

,)(),(),( dzzPtzftmfh

hm

1

)(),(

),(m

mm

zPtmf

tzf

(22)

Further applying the transform defined in equation (22) to the equations (18), (19) and (21),

and using equation (20) one obtains

dt

tmnTdtmntmnTatrkkSr

k

hPtmnT mn

mn

),,(),,(),,()exp(),,(

)(),,(

***2

021003

*2

(23)

*

0* )0,0,1,( TTM t (24)

)exp()(),,(),,( 002100* tzPrkkSrtmn mn (25)



62

where *T is the transformed function of T and m is the transform parameter. The symbol ( ¯

) means a function in the transformed domain, and the nucleus is given by the orthogonal

functions in the interval hzh as

)sin()cos()( zaWzaQzP mmmmm

Where

),cos()( 43 hakkaQ mmm

),sin()()cos(2 43 haakkhaW mmmm

][2

)2sin(][)( 22222

mmm

mmm

h

hmm WQ

a

haWQhdzzP

The eigenvalues am are the positive roots of the characteristic equation

)]sin()[cos()]sin()cos([

)]sin()[cos()]sin()cos([

34

43

ahakahahahak

ahakahahahak

After performing some calculations on equation (23) and using equation (25), the reduction is

made to linear first order differential equation as

),(*,

*

mnmn aHTdt

Td (26)

Where

22

, mnmn a

And

)exp(),,()()(

),( 0210003

trkkSrzPk

hPaH nm

mmn

The general solution of equation (26) is a function

CtH

ttmnT mnmn

mnmn

)exp(

),()exp(),,( ,

,,

*

(27)

Using equations (24) in equation (27), we obtain the values of arbitrary constants C.

Substituting these values in (27) one obtains the transformed temperature solution as

)exp(),(

)exp(),(

),,( ,,

*0

,

* taH

TtaH

tmnT mnmn

mn

mn

mn

(28)



63

Applying inversion theorems of transformation rules defined in equations (17) to the equation

(28), there results

1

,,*

0, )()]exp()()exp([1

),,(m

mmnmnmnm

zPtTttznT

(29)

and then accomplishing inversion theorems of transformation rules defined in equations (22)

on equation (29), the temperature solution is shown as follows:

),,()()]exp()()exp([11

),,( 2101 1

,,*

0, rkkSzPtTtC

tzrT nn m

mmnmnmnmn

(30)

Where

mn

mnmn

aH

,,

),(

Taking into account the first equation of equation (3), the temperature distribution is finally

represented by

1 1

,,*

0, )()]exp()()exp([11

),,(n m

mmnmnmnmn

zPtTtC

tzr

tn drkkS

0210 )(exp),,( (31)

The function given in equation (31) represents the temperature at every instant and at all

points of thick annular disc of finite height when there are conditions of radiation type.

3.2 Thermoelastic solution

Referring to the fundamental equation (1) and its solution (31) for the heat conduction

problem, the solution for the displacement function are represented by the Goodier’s

thermoelastic displacement potential governed by equation (11) are represented by

1 1,,

*0,

,

)()]exp()()exp([11

1

1),,(

n mmmnmnmn

mnmnt zPtTt

Ctzr

tn drkkS

0210 )(exp),,( (32)

Similarly, the solution for Michell’s function M are assumed so as to satisfy the governed

condition of equation (12) as

1 1,,

*0,

,

)]exp()()exp([11

1

1),,(

n mmnmnmn

mnmnt tΛκTtω

ΛλCα

υ

υtzrM



64

tnnn drkkSzzz

0210 )(exp),,()]cosh()[sinh( (33)

In this manner two displacement functions in the cylindrical coordinate system and M are

fully formulated. Now, in order to obtain the displacement components, we substitute the

values of thermoelastic displacement potential and Michell’s function M in equations (9)

and (10), one obtains

1 1,,

*0,

,

)]exp()()exp([11

1

1

n mmnmnmn

mnmntr tTt

Cu

tnnnnnm drkkSzzzzP

0210 )(exp),,()sinh()cosh()1()( (34)

1 1,,

*0,

,

)]exp()()exp([11

1

1

n mmnmnmn

mnmntz tTt

Cu

))]cosh()(sinh()sin(4))cos()sin(([ 2 zzzzzaWzaQa nnnnnmmmmm

tn drkkS

0210 )(exp),,( (35)

Thus, making use of the two displacement components, the dilation is established as

1 1,,

*0,

,

)]exp()()exp([11

1

1

n mmnmnmn

mnmnt tTt

Ce

)cosh()14()()sinh()cosh()1()([ 22 zzPazzzzP nnmmnnnnm

tnnnn drkkSzzz

02103 )(exp),,())]sinh()(cosh( (36)

Then, the stress components can be evaluated by substituting the values of thermoelastic

displacement potential from equation (32) and Michell’s function M from equation (33) in

equations (13), (14), (15) and (16), one obtains

1 1,,

*0, )]exp()()exp([

11

1

12

n mmnmnmn

mntrr tTt

CG

),,()cosh(2[)],,(),,([)({ 21021

,2102101, rkkSzrkkSrkkSzP nnnmnnnmnm

tnnnnn drkkSzzz

0210 )(exp)]},,()]sinh()cosh()1[( (37)

1 1,,

*0, )]exp()()exp([

11

1

12

n mmnmnmn

mnt tTt

CG



65

),,()cosh(2[)],,(),,([)({ 21021

,2102101, rkkSzrkkSrkkSrzP nnnmnnnmnm

tnnnnn drkkSrzzz

02101 )(exp)]},,()]sinh()cosh()1[( (38)

1 1,,

*0, )]exp()()exp([

11

1

12

n mmnmnmn

mntzz tTt

CG

)cosh()2[()],,(),,([)({ 2210210

1, zrkkSrkkSzP nnnnnmnm

tnnn drkkSzz

02103 )(exp)},,()]sinh( (39)

1 1,,

*0,

,

)]exp()()exp([11

1

12

n mmnmnmn

mnmntrz tTt

CG

)sinh(2[),,()]cos()sin([{ 210 zrkkSzaWzaQa nnnmmmmm

tnnnn drkkSzzz

02102 )(exp)},,()]]cosh()[sinh( (40)

4 SPECIAL CASE

Set

,)( 00 T (41)

,2/)( 20

tdt

0*0 T (42)

Substituting the value of equation (42) into equation (31) to (40), one obtains the expressions

for the temperature and stresses respectively as follows:

]2/exp[),,()()]exp()exp([ 2210

1 1,

,trkkSzPtt

Cn

n mmmn

mn

mn

(43)

1 1

2102101,,

,0 )],,(),,([)({)]exp()exp([

n mnnmnmmn

mn

mnrr rkkSrkkSzPtt

C

)]sinh()cosh()1[(),,()cosh(2[ 21021

, zzzrkkSz nnnnnnnmn

]2/exp[)]},,( 2210 trkkS n (44)

)],,(),,([)({)]exp()exp([ 2102101,

1

1 1,

,0 rkkSrkkSrzPtt

Cnnmnm

n mmn

mn

mn



66

1

21021

, )]sinh()cosh()1[(),,()cosh(2[ rzzzrkkSz nnnnnnnmn

]2/exp[)]},,( 2210 trkkS n (45)

)],,(),,([)({)]exp()exp([ 2102101,

1 1,

,0 rkkSrkkSzPtt

Cnnmnm

n mmn

mn

mnzz

]2/exp[)},,()]sinh()cosh()2[( 2210

32 trkkSzzz nnnnnn (46)

),,()]cos()sin([{)]exp()exp([ 2101 1

,,

0 rkkSzaWzaQattC

nmmmmmn m

mnmn

mnrz

]2/exp[)},,()]]cosh()[sinh()sinh(2[ 2210

2 trkkSzzzz nnnnnn (47)

Where

tG

1

120

5 NUMERICAL RESULTS, DISCUSSION AND REMARKS

To interpret the numerical computations, we consider material properties of Aluminum metal,

which can be commonly used in both, wrought and cast forms. The low density of aluminum

results in its extensive use in the aerospace industry, and in other transportation fields. Its

resistance to corrosion leads to its use in food and chemical handling (cookware, pressure

vessels, etc.) and to architectural uses.

Table 1: Material properties and parameters used in this study.

Property values are nominal.

Modulus of Elasticity, E (dynes/cm2) 6.9 10

11

Shear modulus, G (dynes/cm2) 2.7 10

11

Poisson ratio, 0.281

Thermal expansion coefficient, t (cm/cm-0C) 25.5 10

-6

Thermal diffusivity, (cm2/sec) 0.86

Thermal conductivity, (cal-cm/0C/sec/ cm

2) 0.48

Inner radius, a (cm) 1

Outer radius, b (cm) 4

Thickness, h (cm) 2

In the foregoing analysis are performed by setting the radiation coefficients constants,

)3,1(86.0 iki and )4,2(1 iki , so as to obtain considerable mathematical simplicities. The

other parameters considered are r0 = 2.5, z0 = 1 and .



67

The derived numerical results from equation (43) to (47) has been illustrated graphically

(refer figs. 2 to 5) for both (a) with internal heat source, as well as (b) without internal heat

source, with available additional sectional heat on its flat surface at z = 1.

Figure 1: Temperature distribution along r- and z-direction for t =0.25

(a) with internal heat source (b) without internal heat source

Figure 1 shows the temperature distribution along the radial and thickness direction of the

thick annular disc at t = 0.25. It is observed that due to the thickness of the disc, a steep

increase in temperature was found at the beginning of the transient period. As expected,

temperature drop becomes more and more gradually along thickness direction. It is also

observed that, without internal heat source, magnitude of temperature gradient decreases.

-1

-0.5

0

0.5

1

z

1

2

3

4

r

-0.5

0

0.5

-1

-0.5

0

0.5

1

z

-1

-0.5

0

0.5

1

z

1

2

3

4

r

-0.2

0

0.2

-1

-0.5

0

0.5

1

z



68

Figure 2: Radial stress distribution for varying along r-axis and z-axis for t =0.25


Figure 2 shows the radial stress distribution rr along the radial and thickness direction of the

thick annular disc at t =0.25. From the figure, the location of points of minimum stress occurs

at the end points through-the-thickness direction, while the thermal stress response are

maximum at the interior and so that outer edges tends to expand more than the inner surface

leading inner part being under tensile stress.

1

2

3

4

r

-1

-0.5

0

0.5

1

z

-2 ́ 10 8

-1.5 ´ 10 8

-1 ́ 10 8

-5 ́ 10 7 0

rr

1

2

3

4

r

1

2

3

4

r

-1

-0.5

0

0.5

1

z

-1.5 ´ ́10 8

-1 ́ 10 8

-5 ́ 10 7

0

rr

1

2

3

4

r



69

Figure 3: Tangential stress distribution for varying along r-axis and z-axis for t =0.25


Figure 3 shows the tangential stress distribution along the radial and thickness direction of

the thick annular disc at t =0.25. The tangential stress follows a sinusoidal nature with high

crest and troughs at both end i.e. r = 1 and r = 4.

1

2

3

4

r

-1

-0.5

0

0.5

1

z

-1 ́ 10 8

0

1 ́ 10 8

1

2

3

4

r

1

2

3

4

r

-1

-0.5

0

0.5

1

z

-5 ́ 10 7

0

5 ́ 10 7

1

2

3

4

r



70

Figure 4: Axial stress distribution for varying along r-axis and z-axis for t =0.25


Figure 4 shows the axial stress distribution zz, which is similar in nature, but small in

magnitude as compared to radial stress component.

1

2

3

4

r

-1

-0.5

0

0.5

1

z

-4 ́ 10 10

-2 ́ 10 10

0 zz

1

2

3

4

r

1

2

3

4

r

-1

-0.5

0

0.5

1

z

-3 ́ 10 10

-2 ́ 10 10

-1 ́ 10 10

0

zz

1

2

3

4

r



71

Figure 5: Shear stress distribution for varying along r-axis and z-axis for t =0.25


Figure 5 shows the shear stress distribution rz along the radial and thickness direction of the

thick annular disc at t =0.25. Shear stress also follows more sine waveform with high pecks

and troughs along the radial direction at r = 1 and r = 4, but minimum at the center part along

thickness direction.

1

2

3

4

r

-1

-0.5

0

0.5

1

z

-1 ́ 10 10

-5 ́ 10 9 0

5 ́ 10 9

1 ́ 10 10

rz

1

2

3

4

r

1

2

3

4

r

-1

-0.5

0

0.5

1

z

-5 ́ 10 9

0

5 ́ 10 9

rz

1

2

3

4

r



72

In order for the solution to be meaningful the series expressed in equation (43) should

converge for all bra and hzh . The temperature equation (43) can be expressed as

]2/exp[),,()()]exp()exp([ 2210

1 1,

,trkkSzPtt

Cn

M

n

M

mmmn

mn

mn

(48)

We impose conditions so that ),,( tzr converge in some generalized sense to ),( srg as 0t

in the transform domain. Taking into account of the asymptotic behaviors of

),(zPm ,n ),,( 210 rkkS n and nC [1, 2], it is observed that the series expansion for ),,( tzr will

be theoretically convergent due to the bounded functions.

As convergence of the series for br implies convergence for all br at any value of z. An

exact solution requires use of infinite number of terms in the equations. The effects of

truncating of terms are brought out by the comparison table for solutions of different

functions for 5 and 10 terms. For the convergence test of the present method, we calculate the

temperature values at the point r = 4, z =1 with different time for 5 terms and 10 terms, as

shown in the table 1.

Table 1: The temperature under different time and different number of terms at r = 4, z = 1

Functions 5 terms 10 terms

Time = 0.1 0.24912 0.46965

Time = 1.0 0.04513 0.06836

Time = 2.0 0.00140 0.00215

From Table 1, we can see that series solutions converge rapidly provided we take sufficient

number of terms in the series.

CONCLUSION

In this study, we treated the two-dimensional thermoelastic problem of a thick annular disc in

which sources are generated according to the linear function of the temperature. We

successfully established and obtained the temperature distribution, displacements and stress

functions with additional sectional heat, )()exp( 0rrt available at the edge hz of the

disc. Then, in order to examine the validity of two-dimensional thermoelastic boundary value

problem, we analyze, as a particular case with mathematical model for )( and

numerical calculations were carried out. Moreover, assigning suitable values to the

parameters and functions in the equations of temperature, displacements and stresses

respectively, expressions of special interest can be derived for any particular case. We may

conclude that the system of equations proposed in this study can be adapted to design of

useful structures or machines in engineering applications in the determination of

thermoelastic behaviour with radiation.



73

ACKNOWLEDGMENT

Authors are thankful to University Grant Commission, New Delhi to provide the partial

financial assistance under major research project scheme.

REFERENCES

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Naotake Noda, Richard B. Hetnarski, Yashinobu Tanigawa, Thermal stresses, Second ed.

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Documents

Transient Thermoelastic Problem for a Thick Annular Disc With Radiation