International Journal of Pure and Applied Mathematics————————————————————————–Volume 71 No. 3 2011, 403-414

THEOMOELASTIC PROBLEM OF

A THIN ANNULAR DISC DUE TO RADIATION

Hiranwar Payal1, N.W. Khobragade2 §

1,2Department of MathematicsMJP Educational CampusRTM Nagpur UniversityNagpur, 440 033, INDIA

Abstract: An attempt has been made to study thermoelastic response ofa thin annular disc occupying the space D : a ≤ r ≤ b, −h ≤ z ≤ h, withboundary conditions are of radiation type. We apply transformation techniquesto find the thermoelastic solution. Numerical calculations are carried out andresults are depicted graphically.

AMS Subject Classification: 74I25, 74H39, 74D99Key Words: thermoelastic problem, annular disc, thermal stress

1. Introduction

Nowacki in [1] has determined steady-state thermal stresses in a thick circularplate subjected to an axisymmetric temperature distribution on the upper facewith zero temperature on the lower face and circular edge. Roy Choudhary in[2] has succeeded in determining the quasi-static thermal stresses in a circularplate subjected to transient temperature along the circumference of circularupper face with lower face at zero temperature and the fixed circular edgethermally insulated.

Wankhede in [3] has determined the quasi-static thermal stresses in circularplate subjected to arbitrary initial temperature on the upper face with lowerface at zero temperature..

This paper is concerned with transient thermoelastic problem of a thinannular disc occupying the space D : a ≤ r ≤ b, −h ≤ z ≤ h, with boundaryconditions of radiation type.

Received: May 19, 2011 c© 2011 Academic Publications, Ltd.§Correspondence author

404 H. Payal, N.W. Khobragade

2. Statement of the Problem

Consider thin annular disc of thickness 2h occupying the space D : a ≤ r ≤ b,−h ≤ z ≤ h, the material is homogeneous and isotropic. The differentialequation governing to the displacement function U(r, z, t) is

∂2U

∂r2+

1

r

∂U

∂r= (1 + v) atT, (1)

Ur = 0 at r = a, b, (2)

where v and at are Poison’s ratio and the linear coefficient of thermal expansionof the material of the plate and T is the temperature of the plate satisfying thedifferential equation

∂2T

∂r2+

1

r

∂T

∂r+∂2T

∂z2=

1

k

∂T

∂t, (3)

subject to initial condition

Mt(T, 1, 0, 0) = 0, for all a ≤ r ≤ b, −h ≤ z ≤ h. (4)

The boundary conditions are

Mr(T, 1, k1, a) = 0, for all − h ≤ z ≤ h, t > 0, (5)

Mr(T, 1, k2, b) = 0, for all − h ≤ z ≤ h, t > 0, (6)

Mz(T, 1, k3, h) = e−ωtf(z) δ(r − r0), (7)

Mz(T, 1, k4, −h) = e−ωtg(z)δ(r − r0), for all a ≤ r ≤ b, t > 0. (8)

The general expression for these conditions can be written as

Mv(f, k,¯̄k, s) = (k̄f + ¯̄kf̂)v=s,

where the prime (̂) denotes differentiation with respect to v, δ(r − r0) is theDirac Delta function having a ≤ r0 ≤ b, ω > 0 is constant, e−ωtδ(r − r0) is theadditional sectional heat available on its surface at z = −h, h.

The stress function σrr and σθθ are given by

σrr = −2µ1

r

∂U

∂r, (9)

σθθ = −2µ∂2U

∂r2, (10)

where µ is the Lame’s constant, while each of the stress functions σrr, σzz andσθz are zero within the plate in the plane state of stress.

The equations (1) to (10) constitute the mathematical formulation of theproblem under consideration.

THEOMOELASTIC PROBLEM OF... 405

3. Solution of the Problem

3.1. Transient Heat Conduction Analysis

In order to solve equation (2) under the boundary condition (4), we first intro-duce the method of Marchi- Zgrablich transform and Marchi- Fasulo transformof order n over the variable r. Let n be the parameter of the transform, thenthe integral transform and its inversion theorems are written

g∗(ξn, z, t) =

b∫

a

g(r, z, t) S0(k1, k2, µnr)dr ,

g(r, z, t) =∞∑

n=1

g∗(ξn, z, t) S0(k1, k2, µnr)

µn

,

(11)

and

f∗(r, λn, t) =

b∫

a

f(r, z, t) Pn(z)dz ,

f(r, z, t) =

∞∑

n=1

f∗(r, λn, t) Pn(z)

λn,

(12)

Applying the transform defined in equation (11) to the equation (3), one obtains

k

[

−µ2mT (m, z, t) +

d2T (m, z, t)

dz2

]

=dT (m, z, t)

dt(13)

Applying the transform defined in equation (12) to the equation (13), we obtain

k[

−p2T (m, n, t)]

=dT (m,n, t)

dt, (14)

where p2 = (µ2m + λ2

n)Equation (14) is a first order differential equation whose solution is given

by

T =∞∑

m=1

∞∑

n=1

Pn(z)

λ2n

ψ

p2[1 − e−p2kt]

S0(k1, k2, µmr)

µ2m

, (15)

where ψ = Pn(−h)k2

e−ωt δ(r − r0) −Pn(h)

k1e−ωt δ(r − r0),

Pn(z) = Qn cos(anz) −Wn sin(anz),

406 H. Payal, N.W. Khobragade

Qn = an(α1 + α2) cos(anh) + (β1 − β2) sin(anh),

Wn = (β1 + β2) cos(anh) + (α2 − α1)an sin(anh),

λn =

h∫

−h

P 2n(z) dz = h

[

Q2n +W 2

n

]

+sin(2anh)

2an

[

Q2n −W 2

n

]

The eigen values an are the solutions of the equation

[α1a cos(ah) + β1 sin(ah)] × [β2 cos(ah) + α2a sin(ah)]

= [α2a cos(ah) − β2 sin(ah)] × [β1 cos(ah) − α1a sin(ah)] ,

where α1, α2, β1 and β2 are constants.

The kernel function Sp(α, β, µm r) can be defined as

Sp(α, β, µm r) = Jp(µmr) [Yp(α, µma) + Yp(β, µmb)]

− Yp(µmr) [Jp(α, µma) + Jp(β, µmb)],

and Jp(µ r) and Yp(µ r) are Bessel function of first and second kind respectively.

The eigen values µm are the positive roots of the characteristic equation

J0(k1, µa) Y0(k2, µb) − J0(k2, µb) Y0(k1, µa) = 0.

Equation (15) is the desired solution of the given problem.

4. Thermoelastic Displacement Function

Substituting value of temperature distribution T (r, z, t) from (15) in equation(1), one obtains the thermoelastic displacement function U(r, z, t) as

U = −(1 + v) at

∞∑

m=1

∞∑

n=1

Pn(z)

λ2n

ψ

p2[1 − e−p2kt]

S0(k1, k2, µmr)

µ2m

. (16)

THEOMOELASTIC PROBLEM OF... 407

5. Determination of Stress Functions

Substituting the value of thermoelastic displacement function U(r, z, t) fromequation (16) in equations (9) and (10), one obtain the stress functions

σrr =

(

2µ (1 + v) at

r

) ∞∑

m=1

∞∑

n=1

Pn(z)

λ2n

ψ

p2[1 − e−p2kt]

S′0(k1, k2, µmr)

µ2m

, (17)

σθθ = 2µ (1 + v) at

∞∑

m=1

∞∑

n=1

Pn(z)

λ2n

ψ

p2[1 − e−p2kt]

S?0(k1, k2, µmr)

µm. (18)

6. Special Case and Numerical Results

Set T = δ(r − r0)(r − a)2(r − b)2ez .Take k=0.86, h = 2cm, b = 4 cm, t = 1 sec, r0 = 1 cm ω = 1. Substitute

this values in (15), one obtains

T =

∞∑

m=1

∞∑

n=1

Pn(z)

λ2n

ψ

p2[1 − e−0.86p2

]S0(0.25, 0.25, µmr)

µ2m

.

7. Alternative Solution of the Problem

Applying the transform defined in equation (11) to the equation (3), one obtains

k

[

−µ2mT (m, z, t) +

d2T (m, z, t)

dz2

]

=dT (m, z, t)

dt. (19)

Applying the Laplace transform to the equation (19), we get

d2T∗(m,n, t)

dz2−

(

µ2m +

s

k

)

T∗

= 0. (20)

Equation (20) is a second order differential equation whose solution is given by,

T∗

= Aeqz +Be−qz, (21)

where A and B are constants to be determined and

q2 =(

µ2m +

s

k

)

.

408 H. Payal, N.W. Khobragade

Using boundary conditions we obtain the values of A and B. Substituting thesevalues in equation (21) and then applying inversion of Laplace transform andMarchi- Zgrablich transform, one obtains the expression for temperature dis-tribution T(r,z,t) as

T =∞∑

m=1

∞∑

m=1

(−1)m+1m sin(mπ

2h

)

(z − h)S0(k1, k2, µnr)

µn

×

t∫

0

g(r)e−k

“

µ2m+ m

2π2

4h2

”

(t−t′)dt′

−

∞∑

m=1

∞∑

n=1

(−1)m+1m sin(

mπ2h

)

(z + h)S0(k1, k2, µnr)

µn

×

t∫

0

f(r)e−k

“

µ2m+ m

2π2

4h2

”

(t−t′)dt′. (22)

8. Thermoelastic Displacement Function

Substituting the value of temperature distribution T (r, z, t) from (22) in equa-tion (1), one obtains the thermoelastic displacement function U(r, z, t) as

U = −(1 + v) at

∞∑

m=1

∞∑

n=1

(−1)m+1m sin(

mπ2h

)

(z − h)S0(k1, k2, µnr)

µn

×

t∫

0

g(r)e−k

“

µ2m+ m

2π2

4h2

”

(t−t′)dt′

+ (1 + v) at

∞∑

m=1

∞∑

n=1

(−1)m+1m sin(

mπ2h

)

(z + h)S0(k1, k2, µnr)

µn

×

t∫

0

f(r)e−k

“

µ2m+ m

2π2

4h2

”

(t−t′)dt′. (23)

THEOMOELASTIC PROBLEM OF... 409

9. Determination of Stress Functions

Substituting the value of thermoelastic displacement function U(r, z, t) fromequation (23) in equations (9) and (10), one obtain the stress functions as

σrr =

(

2µ (1 + v) at

r

) ∞∑

m=1

∞∑

n=1

(−1)m+1m sin(mπ

2h

)

(z − h)S′0(k1, k2, µnr)

×

t∫

0

g(r)e−k

“

µ2m+ m

2π2

4h2

”

(t−t′)dt′

−

(

2µ (1 + v) at

r

) ∞∑

m=1

∞∑

n=1

(−1)m+1m sin(mπ

2h

)

(z + h)S′0(k1, k2, µnr)

×

t∫

0

f(r)e−k

“

µ2m+ m

2π2

4h2

”

(t−t′)dt′, (24)

σθθ = 2µ (1 + v) at

∞∑

m=1

∞∑

n=1

(−1)m+1m sin(mπ

2h

)

(z − h)µnS′′0 (k1, k2, µnr)

×

t∫

0

g(r)e−k

“

µ2m+ m

2π2

4h2

”

(t−t′)dt′

− 2µ (1 + v) at

∞∑

m=1

∞∑

n=1

(−1)m+1m sin(mπ

2h

)

(z + h)µnS′′0 (k1, k2, µnr)

×

t∫

0

f(r)e−k

“

µ2m+ m

2π2

4h2

”

(t−t′)dt′. (25)

410 H. Payal, N.W. Khobragade

10. Special Case and Numerical Results

Setf(z) = e−he−wt(1 − e−t)(r − b)δ(r − r0) and g(z) = ehe−wt(1 − e−t)(r −b)δ(r − r0).

Take k = 86, h = 2cm, b = 4 cm, t = 1 sec, r0 = 1 cm ω = 1.Substitute these values in (22), one obtains

T =∞

∑

m=1

∞∑

n=1

[

(−1)mm sin (0.79m) (z + 2)S0(0.25, 0.25, µnr)

µn

]

S0(0.25, 0.25, µn)

×

1∫

0

(0.06)e−0.86(µ2m+0.62m2)(1−t′)dt′

+

∞∑

m=1

∞∑

n=1

[

(−1)m+1m sin (0.79m) (z − 2)S0(0.25, 0.25, µnr)

µn

]

S0(0.25, 0.25, µn)

×

1∫

0

(1.2)e−0.86(µ2m+0.62m2)(1−t′)dt′. (26)

11. Conclusion

In this paper, the temperature distribution, displacement function and thermalstresses have been determined for thin annular disc. The finite Marchi-Fasulotransform, March-Zgrablich transform and Laplace transform techniques havebeen used to obtain numerical results. The temperature, displacement and ther-mal stresses that are obtained can be applied to the design of useful structuresor machines in engineering applications

Acknowledgments

The authors are thankful to University Grant Commission, New Delhi for pro-viding the partial financial assistance under major research project scheme.

THEOMOELASTIC PROBLEM OF... 411

References

[1] E. Marchi, A. Fasulo, Heat conduction in sector of hollow cylinder withradiation, Atti. della Acc. Sci. di. Torino, 1 (1967), 373-382.

[2] M. El-Maghraby Nasser, Two dimensional problems with heat sources ingeneralized thermoelasticity, Journal of Thermal Stresses, 27 (2004), 227-239.

[3] W. Nowacki, The state of stress in thick circular plate due to temperaturefield, Ball. Sci. Acad. Polon Sci. Tech., 5 (1957), 227.

[4] N. Noda, R.B. Hetnarski, Y. Tanigawa, Thermal Stresses, Second Edition,Taylor and Francis, New York (2003), 260pp.

[5] S.K. Roy Choudhary, A note on quasi-static thermal deflection of a thinclamped circular plate due to ramp-type heating on a concentric circularregion of the upper face, J. of the Franklin. Institute, 206 (1973), 213-219.

[6] P.C. Wankhede, On the quasi-static thermal stresses in a circular plate,Indian J. Pure and Appl. Math., 13, No. 11 (1982), 1273-1277.

[7] E. Marchi, G. Zgrablich, Vibration in hollow circular membranes withelastic supports, Bulletin of the Calcutta Mathematical Society, 22, No. 1(1964), 73-76.

412 H. Payal, N.W. Khobragade

Appendix A: Figures

Figure 1: Graph of equation (16)

THEOMOELASTIC PROBLEM OF... 413

Figure 2: Graph of equation (17)

Figure 3: Graph of equation (22)

414 H. Payal, N.W. Khobragade

Figure 4: Graph of equation (23)

Figure 5: Graph of equation (24)