AN INVERSE THERMOELASTIC PROBLEM OF CIRCULAR PLATE

The aim of this work is to determine the unknown temperature, displacement and thermal stresses on the upper surface of the circular plate subjected to arbitrary known interior temperature under Steady-state field. The fixed circular edge is thermally insulated and temperature of a lower surface of plate is kept at zero.The governing heat conduction equation has been solved by using the Hankel transform technique. The results are obtained in series form in terms of Bessel’s functions and results have been computed numerically and illustrated graphically.


Introduction
As known, thermal behaviors of structures must be considered in many situations. Study of thermal effect on deformations and stresses of a plate, especially a circular plate is increasingly important. Firstly, the problems of circular plates are more complicated and thus more attractive to many scientists. Secondly, there are practical requirements for thick plates in various modern projects, such as high building, raceway, high-way, container wharf, and so on. Ashida et al. [1] discussed the inverse transient thermoelastic problem for a composite circular plate. Deshmukh et al. [2] are also discussed on an inverse transient problem of quasi-static thermal deflection of a thin clamped circular plate. Grysa et al [3] studied the one dimensional problem of temperature and the heat flux at the surface of a thermo elastic slab. Kulkarni et al. [5] studied an inverse transient problem of quasi-static thermal stresses in a thick circular plate. Noda [6] discussed an analytical method for an inverse problem of coupled thermal stress fields in a circular cylinder. Also Roy Choudhary [7] studied a rate of quasi-static stress in a thin circular plate due to transient temperature applied along the circumference of a circle over the upper face. Recently Gaikwad et al. [4] studied an exact solution of unsteady-state thermoelastic problem of a circular plate.
The present paper deals with the determination of unknown temperature, displacement and thermal stresses on the upper surface of circular plate subjected to arbitrary known interior temperature under steady state field. The fixed edge is thermally insulated and temperature of a lower surface of plate is kept at zero. The governing heat conduction equation has been solved by using the Hankel transform technique. The results are obtained in series form in terms of Bessel's functions and results have been computed numerically and illustrated graphically. This paper contains new and novel contribution of stresses in circular plate under steady state. The results presented here will be more useful in engineering problem particularly in the determination of the state of strain in circular plate constituting foundations of containers for hot gases or liquids, in the foundations for furnaces etc.

Statement of the problem
Consider a circular plate of radius a and thickness 2h occupying space D : 0  r  a ,h  z  h .The thermoelastic displacement function as in [6] is governed by Poisson's equation with U  0 at r  0 and r  a .
(2) where  and a t 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) Subject to the boundary conditions and where k 1 and k 2 are the radiation constants on the plane two surfaces. The stress functions  rr and   are given by, where  is the Lame's constant, while each of the stress functions  rz ,  zz and   z are zero within the plate in the plane state of stress. The equations (1) to (9) constitute the mathematical formulation of the problem under consideration.

Solution of the Problem
To obtain the expressions for temperature T (r, z) introduce the finite Hankel transform over the variable r and its inverse transform defined in [8] as where (17)

Determination of Thermoelastic Displacement
On putting the values of temperature T (r, z, t ) from Eq. (16) in Eq. (1), one obtain the thermoelastic displacement function U (r, z, t) as, The Bulletin of Society for Mathematical Services and Standards Vol. 1 3

Special Case and Numerical Results
Set  [9].

Concluding Remarks:
In this paper, we have discussed the steady-state thermoelastic problem for a circular plate on outer curved surface of the circular plate. The fixed circular edge is thermally insulated and zero temperature is maintained on lower surface.
The finite Hankel 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 (15), (18)-(20).

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BSMaSS Volume 1 Fig.1.The unknown temperature decreases with the thickness of the circular region 0  r  0.8 where as it is an increase within the annular region 0.8  r  1 .

Fig.2.
The radial displacement increases with the thickness of the circular plate. Fig.3. The radial stress function  rr is increases with the circular plate and it shows the normal curve. Also it develops the tensile stresses in radial direction. Fig.4. The stress function   is decreases with the thickness of the circular plate and it is negligible for small thickness. Also it develops the tensile stresses in radial direction.