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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
1Department
of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad
(MS) India.
2Department
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. |