EFFECT OF CONVECTION BOUNDARY CONDITION ON HYPERBOLIC HEAT CONDUCTION IN THERMOELASTIC MEDIUM

This paper aims to study the effect of heat transfer by convection on thermoelastic rectangular solid medium in the context of hyperbolic heat conduction. The studied geometry is a two dimensional finite thin rectangular plate without internal heat generation, which is initially at a uniform temperature and subjected to convection heat transfer from the extreme edge(y=a) while the opposite side is kept insulated and remaining two sides are at a constant temperature. The material properties are assumed to be constant. The differential transform method is applied to solve the hyperbolic heat conduction equation to obtain temperature distribution in the spatial and temporal domain. Then by applying the obtained temperature in the thermoelastic equation, the displacement component, and the stress field are calculated. Also, the effect of convection boundary conditions on temperature distribution and thermoelastic field are illustrated numerically and graphically for a copper plate. It is observed that the compressive and tensile stress occurs along y-direction due to heat transfer through convection at one end.


INTRODUCTION
As a result of increased usage of nanomaterials, laser heat sources, microwaves in industries and the medical field, the interest in the hyperbolic heat conduction model have grown considerably.
Most of the practical applications are explained in the context of Classical theory of heat conduction which supports the infinite speed of heat propagation but in case of high temperature gradients, high heat fluxes or very short time duration where the speed of heat propagation is finite, the theory of Fourier heat conduction fails. To overcome these difficulties Cattaneo [1] and Vernotte [2] proposed the hyperbolic heat conduction model by introducing a new term called relaxation time or phase lag in heat flux in solids, which is given by the relation Various researches have been done on thermoelastic behaviour of solids, associated with convection heat transfer in the Fourier heat conduction domain. Thermoelastic behaviour of isotropic rectangular plate with heat transfer by convection have studied, using the finite element method under the classical Fourier heat conduction model by Shubha and Kulkarni [3]. Chen [4] has studied the thermal stress in a rectangular plate subjected to non-uniform heat transfer in terms of power series using Lanczos-Chebyshev and discrete least square methods. Sugano [5] has investigated the thermal stresses in the orthotropic rectangular plate under different third kind boundary conditions at four edges. In their study, it is observed that the thermal stresses are dependent on the anisotropic properties of a material.  [17]. Thus due to the advancement in the field of technology and medical science, it is essential to study the hyperbolic heat conduction with convective heat transfer in the field of thermoelasticity.
In the literature different analytical and numerical methods like Fourier transform, Laplace transform, Finite element method, Differential Quadrature method, and Differential transform method have been used to solve the hyperbolic heat conduction equation as well as an equation involving in the calculation of thermal stresses and displacement. In which, the differential transform method based on Taylor series is a semi-analytical method, which was first proposed by Zhou [18] in 1986 for the solution of linear and nonlinear initial value problems that appear in electrical circuits. Due to its effectiveness and simplicity in the past few years, many authors have applied this method to solve different types of partial and ordinary differential equations.
Sobhan Mosayebidorcheha et al. [19] applied DTM to find the convergent solution of nonlinear heat conduction on a fin. Lo & Chen [20] presented the solution of HHCE by applying a hybrid differential transform and volume control method by choosing appropriate shape function and concluded that this method is easy and reliable for engineering problems. Yih et.al [21] have investigated Pennes bio-heat transfer equation using DTM along with finite difference method for temperature distribution in human eyes. Mahesh Kumar et.al [22] studied the temperature distribution under convective heat transfer over a stretching sheet by applying DTM. The n-dimensional differential transform of function T of n variables [23] as follows: Where ( 1 , 2 , … … ) is the transformed function T.
The inverse differential transform of ( 1 , 2 , … … )is defined by In this article, the temperature distribution and thermal stresses under the hyperbolic heat conduction model subjected to convection type heat transfer through the boundary surface are semi-analytically investigated. The displacement component is also calculated in a spatial direction. The governing, partial differential equations are solved by applying a semi-analytical differential transform method. The copper material is used to illustrate the results numerically and graphically.

MATHEMATICAL MODELING FOR TEMPERATURE FIELD
The present investigation concerns a finite thin rectangular plate, initially kept at constant temperature 0 and subjected to convection type heat transfer at the edge y=b, the other side opposite to it is kept insulated and the other two parallel sides are at a constant temperature ∞ .
Here two-dimensional heat conduction and constant thermal properties have considered. A plate of rectangular shape occupying the space D: {( , ): ∈ [0, ], ∈ [0, ]} has taken into consideration. Where the unsteady state temperature distribution under the hyperbolic heat conduction in the plate with no internal heat generation given in Eq. (2) is subjected to initial and boundary conditions, which are given as: For convenience we shall employing the following dimensionless variables: Using the above dimensionless variables, the governing heat conduction equation becomes: And the initial and boundary conditions are transformed as:

SOLUTION FOR TEMPERATURE FIELD
Applying differential transform method defined in Eq.(3) on non-dimensional heat transfer Eq.
To find the differential transform of initial and boundary condition, here first initial condition and  (19) Now assuming ( , 0, Θ) = and applying above transformed initial and boundary conditions from Eqs.(16)- (19) in Eq. (15) , one can obtain the transformed parameter for temperature using a recursive method and then applying inverse differential transform defined in Eq.(4) and neglecting the higher-order terms,we get the expression for temperature distribution

MATHEMATICAL MODELING FOR THERMOELASTIC FIELD
Considering the thermal stress function defined as [24], in the rectangular coordinate system for the plane thermoelastic problem. The fundamental equation is given as: With the following mechanical boundary conditions: Also, the fundamental equation for displacement function defined as [24] for plane problem in a rectangular coordinate is given as:  ( )

SOLUTION FOR THERMOELASTIC FIELD
The complementary function , the general solution of Eq.(33) may be taken in the following form: Now the solution of particular integral ̅ is obtained by applying differential transform and its inverse on Eq.(34).The transformed form of Eq.(34) is given by Where is a Dirac delta function. Now using recursive method and applying inverse differential transform, one can obtain ̅ as : Similarly, using Eqs.(36)~(38),we will get the solution for displacement components as:    Temperature distribution in a finite plate for specific values of dimensionless time along spatial direction X and Y has shown in Fig.1 and Fig2. From Fig.1 the small variation in temperature is observed for a small part of the plate, but as move along y-direction and as reaches near to the boundary, large variation in temperature has observed due to heat transfer by convection at that end.
In Fig.2 Fig. 3 that the plate experienced a small amount of stress initially but as the value of dimensionless time increase, the amount of stress in X-direction also increases, and the left part of the plate experienced maximum stress. In Fig.4 variation of dimensionless stress component ̅ along Y-direction is plotted. The stress fluctuates between compressive and tensile stress throughout the plate. It is clear from the graph that, the plate experienced peak stress at the mid of the plate initially and the compressive stress occurs in mid of the plate as the time increases, due to the phase lag in heat flux. is observed that the tensile stress occurs in a plate, suddenly after mid of the plate at = 0. For the increasing values of time, the behaviour of stress changes due to the hyperbolic nature of temperature distribution. In Fig.6 variation of ̅ in Y-direction has been depicted. It can be easily seen that the amount of stress increase as increasing the value of time. Also, it is observed that the boundaries exert compressive stress in this case.
As shown in Fig.7, the thermal stress component ̅ along x-direction does not occur up to mid part of the plate and then suddenly plate experienced the compressive stress at the boundary. In  and y direction has been plotted. From both graphs, it is observed that the maximum displacement occurs at the boundary x=a and y=b. It is clear that the displacement along y-direction varies linearly.

CONCLUSION
In the present work, the effects of convection type boundary condition on thermoelastic behaviour of rectangular plate under the framework of the hyperbolic heat conduction model have been investigated. The distribution of temperature, thermal stresses, and the displacements in a plate is obtained in the differential transform domain. The solution is obtained in the form of an infinite series. From the study, it is observed that the peak temperature is experienced at the boundary where heat is transfer by convection. The study also shows that tensile and compressive stresses occur in the plate along Y-direction. Also, it is observed that the behaviour of stress changes as time increases. This study may be useful for designing and engineering applications where the heat transfer by convection along with temperature gradient is high or the time duration is very short.