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**Study of an Exact Solution of
Steady-State Thermoelastic problem of an Annular Disc**
^{ }
**Gaikwad P. B.**^{1*} **and Ghadle K. P.**^{2}
^{1}Department
of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad
(MS) India.
^{2}Department
of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad
(MS) India.
^{*}Corresponding
Addresses
[email protected]
*
Abstract:* In this paper, an attempt is made to determine the unknown
temperature, displacement and stress functions on outer curved surface,
where as third kind boundary condition maintained on upper surface and
zero temperature is maintained on lower boundary surface. The governing
heat conduction has been solved by using finite Hankel transform
technique. The results are obtained in series form in term of Bessel’s
functions. The results for unknown displacements and stresses have been
computed numerically.
*
Key words: **
Steady- state, Thermoelastic problem, Hankel
transforms, Annular Disc***.**
**
Introduction:**
During the second half of the twentieth
century, non –isothermal problems of the theory of elasticity became
increasingly important. This is due mainly to their many applications in
diverse fields. First, the high velocity of modern aircrafts give rise
to an aerodynamic heating, which produce intense thermal stresses
reducing the strength of aircrafts structure.
Two dimensional transient problems for a
thick annular disc in thermoelasticity studied by (Dange et al.,
2009). An inverse temperature field of theory
of thermal stresses investigated by (Grysa et al; 1981)
while A note of quasi –static thermal stresses in steady state thick
annular disc and an inverse quasti-static thermal stresses in thick
annular disc are studied by (Gaikwad et al; 2010).
An inverse problem of coupled thermal stress fields in a circular
cylinder considered by (Noda; 1989).
In this paper, in the first problem, an
attempt is made to determine the unknown temperature, displacement and
stress functions on curved surfaces, where an arbitrary heat is applied
on the upper surface (z = h) and maintained zero on lower surface (z = -
h) while in second problem third kind boundary condition is maintained
on lower and upper surface of an annular disc. The governing heat
conduction equation has been solved by using Hankel transform technique.
The results are obtained in series form in terms of Bessel’s functions
and illustrated numerically.
---------------------------------------------
*
*2000 Mathematics Subject Classifications:
primary 35A25, secondary* *74M99,
74K20*.
This paper contains new and novel
contribution of thermal stresses in an annular disc under steady state.
The above results were obtained under steady state field. The result
presented here are useful in engineering problem particularly in the
determination of the state of strain in an annular disc constituting
foundations of containers for hard gases or liquids, in the foundations
for furnaces etc.
**2.
Results Required:**
** **
The
finite Hankel transform over the variable r and it inverse transform
defined in [7].
(2.1)
(2.2)
where (2.3)
(2.4)
The normality constant
(2.5)
and
,,………..are the roots transcendental equation,
(2.6)
is Bessel function of the first kind of the
order and
Bessel function of the second kind of the order
.
** **
**3. Formulation of the
Problem:**
Consider an annular disc of
thickness occupying space, .
The differential equation governing the displacement potential function
is given in [7] as,
(3.1)
with at and
(3.2)
here
and
are the poisson’s ratio and the linear
coefficient of thermal expansion of the material of the plate and T is
the temperature of the plate satisfying the differential equation
(3.3)
Subject to the boundary conditions
at
(3.4)
at
(3.5)
at
(3.6)
The
stress functions and are given by,
(3.7)
(3.8)
where
is the Lame’s constant, while each of the
stress functions and
are zero within the plate in the plane state
of stress. The equations (3.1) to (3.8) constitute the mathematical
formulation of the problem under consideration.
**4. Solution of the Problem:**
** **
Applying
finite Hankel transform defined in [6] to the equations (3.1) to Eq.
(3.6), one obtain (4.1)
where
is the Hankel transform of T.
On
solving Eq. (4.1) under the conditions given in Eq. (3.5) and Eq.
(3.6).one obtains
(4.2)
On
applying the inverse Hankel transform defied in Eq. (2.2) to Eq. (4.2),
one obtain the expression for the temperature as
(4.3)
(4.4)
where
(4.5)
Equation
(4.3) and (4.4) are the desired solution of the given problem.
** **
**
Determination of Thermoelastic Displacements:**
Substituting the values of T(r, z) from Eq. (4.3) in Eq. (3.1) one
obtains the thermoelastic displacement function U(r, z) as,
(4.6)
** **
**
Determination of Stresses Functions:**
Using Eq.
(4.6) in Eq. (3.7) and (3.8), one obtains the stress functions
and
as,
(4.7)
(4.8)
** **
**
Special Case and Numerical Calculations:**
** **
Set in (4.4) one obtains
(4.9)
The numerical calculation have
been carried out for steel (SN 50 C) plate with parameters
,, ,thermal diffusivity and
poisons ratio ,while, ,,,, being
the positive roots of transcendental equation as in [6].
**Concluding Remarks:**
** **
In this paper, we have
discussed the steady –state thermoelastic problem for an annular disc on
outer curved surface of the annular disc, whereas arbitrary heat is
applied on the upper surface and zero temperature is maintained on lower
surface.
The finite Hankel transform
transform technique is used to obtain the numerical results. The
thermoelastic behavior is examined such as unknown temperature,
displacement and stresses that are obtained can be applied to the design
of useful structures or in engineering applications. Also any particular
case of special interest can be derived by assigning suitable values to
the parameters and functions in the expressions [4.3], [4.4],
[4.6]-[4.9].
**References:**
** **
[1]
Dange, W.K., Khobraade, N.W., and Varghese, V., Two dimensional
transient problems for a thick annular disc in thermoelasticity. Far
East Journal of Applied Mathematics, Vol.43, 205-219, 2009.
[2]
Grysa, K., Cialkowski, M. J and Kaminski H., An inverse
temperature field of theory of thermal stresses. NUCL. Eng. Des. Vol.64.
161-184, 1981.
[3]
Gaikwad K.R., Ghadle K.P., A note of Quasi-Static Thermal
Stresses in Steady State Thick Annular Disc.Acta Ciencia
Indica,Vol.XXXVI M,No.3,385, 2010.
[4]
Gaikwad K.R.,Ghadle K.P., An Inverse Quasi-Static Thermal
Stresses in Thick Annular Disc., International Journal of Applied
Mathematics & Statistics. Vol.19. D10, 2010.
[5]
Nowacki, W., Thermoelasticity Mass. Addition-Wesley Publication
Co. Chapter-1, 1962.
[6]
Ozisik N.M., Boundary value problem of Heat Conduction,
International text book, Comp. Scranten, Pennsylvania, pp.481-492, 1968.
[7]
Sneddon, I.N., The use of integral transform, McGraw Hill, New
York, 235-238, 1972.
[8]
Noda, N., An inverse problem of coupled thermal stress fields in
a circular cylinder, JSME,Intermational Journal,Ser.A.32, pp.791-797,
1989. |