Inverse transient thermoelastic problem with heat source in an annular disc

This article deals with formulating an inverse problem of heat conduction to analyse thermoelastic stress in a thick annular disc due to internal heat generation within the solid. The integral transformation techniques, Marchi-Zgrablich transform and finite MarchiFasulo integral transform, are employed in order to solve governing transient heat conduction equation, under stated thermal boundary conditions. The outcomes obtained from analytical solution are represented in the form of infinite series consisting Bessel’s functions. Numerical verification is done for a particular case of a thick annular disc made of aluminummetal and results are depicted graphically.


Introduction
Thermoelasticity is the domain of science that deals with the study of temperature distribution, thermal stresess and strain developed in a material body as an effect of application of thermo-mechanical load. Thermoelasticity describes a broad range of phenomena and hence it is of paramount importance in the stress analysis. In extensive engineering applications, mainly in the field of design of a structural elements,thermoelastic behaviour of a material plays a primary and decisive role. Due to change in temperature , thermal stresses occur in the material body, which in turn results in failure of a structural element. Thermal stresses , being one of the most important factor that affect the life of the material body, its analysis is very important in plenty of engineering applications . As a consequence, the demand for the study of thermal behaviour of the solids increases.
With the progress of science and technology, thermoelasticity becomes more and more attractive subject for researchers. A lot of investigators are interested to deal with the thermoelastic behaviour of the annular discs constituting the foundation of containers for hot gases, furnaces and many more applications.In addition, the inverse problems of thermoelasticity are of keen interest in view of its relevance in aerospace engineering.
A thorough literature review shows that, though many studies have reported the direct thermoelastic problems, very little research has been done in the area of inverse thermoelastic problems of thick solid bodies. Marchi  hollow cylinder with radiation is discussed by Marchi E. and Fasulo A. [2].An inverse unsteady thermoelasic problem is analysed by Noda et. al. [3] for a transversely isotropic body. Thermal stress analysis of a direct thermoelastic problem of a thick annular disc with radiation conditions is elaborated by Khobragade N. [4].A thin annular disc has assessed by Khobragade K et. al [5] to study an inverse transient thermoelastic problem. Analysis of thermal stresses in a thick annular disc is done by Kulkarni and Deshmukh [ 6]. An axisymmetric inverse steady state problem of thermoelastic deformation of a finite length hollow cylinder is discussed by Deshmukh and Wankhede [7]. Recently, influence of internal heat generation on thermal stresses is determined by Shinde A. et. al. [8] by considering transient thermoelastic problem of a thick circular plate.
The purpose of this paper is to analyze the impact of internal heat source on thermoelastic behaviour of a thick annular disc. Marchi-Zgrablich and finite MarchiFasulo integraltransforms are employed to tackle the expression of heat conduction. The expressions are derived for different field parameters such as temperature distribution, displacementand thermalstresses.To demonstrate the analytical solution,Numerical computations and graphic plots are done for thick annular disc ofaluminum material. A special case of interest have also been presented.

Mathematical formulation
Consider a thick annular disc having thickness 2ℎ, defined by ܽ ≤ ‫ݎ‬ ≤ ܾ, −ℎ ≤ ‫ݖ‬ ≤ ℎ, which is initially at zero temperature. For times ‫ݐ‬ > 0, heat is generated within the disc at the rate ‫,ݎ(݃‬ ‫,ݖ‬ ‫)ݐ‬and dissipated by convection from its fixed circular edges ‫ݎ‬ = ܽ and ‫ݎ‬ = ܾ into a surrounding at zero temperature. On the lower surface ‫ݖ‬ = −ℎ , homogeneous boundary condition of the third kind ismaintained.With given interior condition and stated boundary conditions, the unknown temperature ‫,ݎ(ܩ‬ ‫)ݐ‬ on upper surface ‫ݖ‬ = ℎ and the thermal stresses need to be determined.
The displacements, ‫ݑ‬ and ‫ݑ‬ ௭ , in radial and axial directionsrespectively, are expressed in terms of the Goodier's thermoelastic displacement potential ߶and Love's function ‫ܮ‬as ߥis Poisson's ratio and ߙ is linear coefficient of thermal expansion of the material of the disc.
Also, ‫ܮ‬must satisfy the equation∇ ଶ ∇ ଶ ‫ܮ‬ = 0 (2.11) The components of the stress are On the traction free surfaces of the thick annular disc , the boundary conditions are ߪ ௭௭ = ߪ ௭ = 0 ‫ݐܽ‬ ‫ݖ‬ = ±h( 2.16) This system of equations (2.1) to (2.16) constitutes the mathematical formulation of the thermoelastic problem for displacement and associated thermal stresses developed within the thick annulardisc as a effect of temperature distribution.

The integral transforms required to find analytical solution 3.1FiniteMarchi-Zgrablich integral transform
The finite Marchi-Zgrablich integral transform of ݂(‫)ݎ‬of order ‫‬is defined as The inversion formula is and the property of this transform is where ℎ ଵ , ℎ ଶ are constants in the boundary conditions  where ‫ܬ‬ (ߤ ‫)ݎ‬is Bessel function of first kind and ܻ ( ߤ ‫)ݎ‬ is Bessel function of second kind.

Determination of displacement components
Assume the Love function L , that satisfy equation (2.11) as Then from (2.10) and (4.1.10), the displacement potential ߶is and ,

Determination of stress components
Substituting eq.(4.2.1) and ( 4.2.2) in equations . (2.12)to (2.15),we have   It is observed that the temperature decreases uniformly from inner boundary (r = 1) to the outer boundary (r = 4) . It attains peak at the inner circular boundary , become zero at‫ݎ‬ = 2.5 and then gradually decreases towards the outer circular boundary . As time increases, temperature also increases.      From figure7, the axial stresses ߪ ௭௭ shows varying nature. It develops tensile stresses near inner circular boundary , but in most of the region, in middle and outer circular boundary, it develops compressive stresses.

Figure8
. Distribution of resultant stress ߪ ௭ /Jversus radius ‫.ݎ‬ From figure8, the resultant stress function ߪ ௭ develops tensile stresses in annular region of disc. At center, it attains peak value, and attains equilibrium condition at both circular boundaries , i.e. at r = 1 and r = 4. It develops tensile stresses in the annular region of disc and as time increases, resultant stresses also increases.

Conclusion
The present paper focuses on the thermal stressanalysis . The unknown temperature, displacement components and associated thermal stresses in thick annular disc under unsteady state conditions with radiation type boundary conditions are determined. The internal heat source has an essential and vital role in distribution of temperature and thermal stresses. The displacement and stress fields have both increasing and decreasing effects due to internal heat generation .
Any special Scenarios of thermoelastic behaviour of annular discs can be achieved by allocating particular values to the field parameters and functions in the derived expressions. The results presented here will be useful and applicable to many practical engineering applications where thermal environment plays a primary and decisive role.