Mathematical Modeling of Quasi Static Thermoelastic Transient behavior of thick circular plate with Internal Heat Generation

C. M. Bhongade Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 9( Version 2), September 2014, pp.38-45
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Mathematical Modeling of Quasi Static Thermoelastic Transient behavior of thick circular plate with Internal Heat Generation C. M. Bhongade * and M. H. Durge** *Department of Mathematics, Shri Shivaji College, Rajura- 442905, Maharashtra, India ** Department of Mathematics, Anand Niketan College, Warora, Maharashtra, India ABSTRACT The present paper deals with the determination of displacement and thermal transient stresses in a thick circular plate with internal heat generation. External arbitrary heat supply is applied at the upper surface of a thick circular plate, whereas the lower surface of a thick circular plate is insulated and heat is dissipated due to convection in surrounding through lateral surface. Here we compute the effects of internal heat generation of a thick circular plate in terms of stresses along radial direction. The governing heat conduction equation has been solved by using integral transform method. The results are obtained in series form in terms of Bessel’s functions and the results for temperature change and stresses have been computed numerically and illustrated graphically. Keywords Thermal stresses, internal heat generation, quasi static.
I. Introduction
Non isothermal problems of the theory of elasticity became increasingly important. This is due to their wide applications in diverse fields. Nowacki [1] has determined the temperature distribution on the upper face, with zero temperature on the lower face and the circular edge thermally insulated. Sharma et al. [2] studied the behavior of thermoelastic thick plate under lateral loads. Kulkarni and Deshmukh [3] studied the thermoelastic behavior of thick circular plate with the help of arbitrary initial heat supply on the upper surface. Kedar and Deshmukh [4] has determined the quasi-static thermal stresses due to an instantaneous point heat source in a circular plate subjected to time dependent heat flux at the fixed circular boundary. Most recently Bhongade and Durge [5] considered thick circular plate and discuss, effect of internal heat generation on steady state behavior of thick circular plate.
In this paper thick circular plate is considered and discussed its thermoelasticity with the help of the Goodier’s thermoelastic displacement potential function and the Michell’s function. To obtain the temperature distribution integral transform method is applied. The results are obtained in series form in terms of Bessel’s functions and the temperature change, displacement function and stresses have been computed numerically and illustrated graphically. Here we compute the effects of internal heat generation in terms of stresses along radial direction. A mathematical model has been constructed of a thick circular plate with the help of numerical illustration by considering copper (pure) circular plate. No one previously studied such type of problem. This is new contribution to the field. The direct problem is very important in view of its relevance to various industrial mechanics subjected to heating such as the main shaft of lathe, turbines and the role of rolling mill, base of furnace of boiler of a thermal power plant, gas power plant.
II. Formulation of the Problem
Consider a thick circular plate of radius a and thickness h defined by 0 ≤푟≤푎, −푕 2≤푧≤ 푕 2. Let the plate be subjected to external arbitrary heat supply is applied on the upper surface, the lower surface is insulated and heat convection is maintained at fixed circular edge (푟=푎). Assume the circular boundary of a thick circular plate is free from traction. Under these prescribed conditions, the quasi-static thermal transient stresses, temperature and displacement in a thick circular plate with internal heat generation are required to be determined. The differential equation governing the displacement potential function 휙 푟,푧,푡 is given by, 휕2휙 휕푟2+ 1 푟 휕휙 휕푟 + 휕2휙 휕푧2= 퐾휏 (1) where K is the restraint coefficient and temperature change 휏=푇− 푇푖, 푇푖 is initial temperature. Displacement function 휙 is known as Goodier’s thermoelastic displacement potential.
Temperature of the plate at time t satisfying heat conduction equation as follows,
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휕2푇 휕푟2+ 1 푟 휕푇 휕푟 + 휕2푇 휕푧2+ 푞 푟,푧,푡 푘 = 1 훼 휕푇 휕푡 (2) with the boundary conditions 푇 = 0 푎푡 푡=0,0≤푟≤푎 (3) 푞 푟,푧,푡 =0 푎푡 푡=0, (4) 푇=푓 푟,푡 푎푡 푧 = 푕 2, 0≤푟≤푎 (5) 휕푇 휕푍 =0 푎푡 푧=− 푕 2,0≤푟≤푎 (6) 푕1푇+ 휕푇 휕푟 =0 푎푡 푟=푎 (7) where 훼 is the thermal diffusivity of the material of the plate, k is the thermal conductivity of the material of the plate, q is the internal heat generation and 푕1 is heat transfer coefficient. The Michell’s function M must satisfy 훻2훻2푀=0 (8) where, 훻2= 휕2 휕푟2+ 1 푟 휕 휕푟 + 휕2 휕푧2 (9) The components of the stresses are represented by the thermoelastic displacement potential 휙 and Michell’s function M as
휍푟푟=2퐺 휕2휙 휕푟2− 퐾휏+ 휕 휕푧 푣훻2푀− 휕2푀 휕푟2 (10)
휍휃휃=2퐺 1 푟 휕휙 휕푟 − 퐾휏+ 휕 휕푧 푣훻2푀− 1 푟 휕푀 휕푟 (11)
휍푧푧=2퐺 휕2휙 휕푧2− 퐾휏+ 휕 휕푧 (2−푣)훻2푀− 휕2푀 휕푧2 (12)
푎푛푑 휍푟푧=2퐺 휕2휙 휕푟휕푧 + 휕 휕푟 (1−푣)훻2푀− 휕2푀 휕푧2 (13) where G and v are the shear modulus and Poisson’s ratio respectively. The boundary conditions on the traction free surfaces of circular plate are 휍푟푟=휍푟푧=0 at 푟=푎 (14) Equations (1) to (14) constitute mathematical formulation of the problem.
III. Solution of the Heat Conduction Equation
To obtain the expression for temperature T( r, z, t ), we introduce the finite Hankel transform over the variable r and its inverse transform are 푇 훽푚,푧,푡 = 푟′ 퐾0 훽푚,푟′ 푎 푟′=0 푇(푟′ ,푧,푡) 푑푟′ (15) 푇(푟,푧,푡) = 퐾0 훽푚,푟 ∞푚 =1 푇 훽푚,푧,푡 (16) where, 퐾0 훽푚,푟 = 2 푎 훽푚 (푕12+ 훽푚 2) 12 퐽0(훽푚푟) 퐽0(훽푚푎) (17) and 훽1,훽2,….. are the positive roots of the transcendental equation 훽푚 퐽0’ (훽푚푎)+ 푕1 퐽0 훽푚푎 =0 (18) where 퐽푛 푥 is the Bessel function of the first kind of order n. The Hankel transform-H, defined in Eq. (15), satisfies the relation 퐻 휕2푇 휕푟2+ 1 푟 휕푇 휕푟 = −훽푚 2푇 훽푚,푧,푡 (19)

On applying the finite Hankel transform defined in the Eq. (15) and its inverse transform defined in Eq. (16) to the Eq. (2), one obtains the expression for temperature as
푇 푟,푧,푡 = 2 푎 훽푚 (푕12+ 훽푚 2) 12 퐽0 훽푚푟 퐽0 훽푚푎 ∞ 푚=1
× 2푛−1 휋훼 −1 푛푕2푐표푠 2푛−1 휋 2푕 푧+ 푕 2 × −퐹 훽푚, 푡 푒 −훼 훽푚 2+ 2푛−1 2휋24푕2 푡 −1 훼 훽푚 2+ 2푛−1 2휋24푕2 2 –푓1 푡 + 2훼 (−1)푛 푕 푠푖푛 2푛−1 휋 2푕 푧+ 푕 2 푓2(푡)+퐶1(훽푚,푧 ,푡) ∞푛 =1 (20) where, 퐶1 훽푚,푧 ,푡 =퐿−1[퐴1 훽푚,푧 ,푠 ], 퐴1 훽푚,푧 ,푡 is particular integral of differential equation, 푓1 푡 = 퐴1( 푡 0훽푚, 푕 2,푡−푢)푒 −훼 훽푚 2+ 2푛−1 2휋24푕2 푢 푑푢 and 푓2 푡 = − 퐴1( 푡 0훽푚,− 푕 2,푡−푢)푒 −훼 훽푚 2+ 2푛−1 2휋24푕2 푢 푑푢 Michell’s Function M We select M which satisfy Eq. (8) is given by 푀= 퐵푚푛 퐽0 훽푚푟 + 퐶푚푛 훽푚 푟 퐽1(훽푚푟) ∞푛 =1∞푚 =1푐표푠푕[훽푚(푧+ 푕 2)] (21) where 퐵푚푛 and 퐶푚푛 are arbitrary functions, which can be determined by using condition (14). Goodiers Thermoelastic Displacement Potential and Thermal Stresses Displacement function 휙(푟,푧,푡) which satisfies Eq. (1) as 휙(푟,푧,푡)= 2 푎 퐾 훽푚 (푕12+ 훽푚 2) 12 퐽0 훽푚푟 퐽0 훽푚푎 ∞ 푚=1 × − 2푛−1 휋훼 −1 푛푕2푐표푠 2푛−1 휋 2푕 푧+ 푕 2 × −퐹 훽푚, 푡 [푒 −훼 훽푚 2+ 2푛−1 2휋24푕2 푡 −1] 훼 훽푚 2+ 2푛−1 2휋24푕2 2 –푓1(푡) − 2훼 −1 푛 푕 푠푖푛 2푛−1 휋 2푕 푧+ 푕 2 푓2 푡 훽푚 2+ 2푛−1 2휋24푕2 − 2푛−1 휋훼 푓2 푡 푒 [ 퐾+훽푚 2 푧+ 푕 2 ] –1 푛 푕2 퐾+훽푚 2 훽푚 2+ 2푛−1 2휋24푕2 ∞푛 =1 (22) Now using Eqs. (20), (21) and (22) in Eq. (10), (11), (12) and (13), one obtains the expressions for stresses respectively as For convenience setting 푔1 푟 = − 퐽1’ 훽푚푟 훽푚 훽푚 2+ 2푛−1 2휋24푕2 +퐽0 훽푚푟 , 푔2 푟 = − 퐽1 훽푚푟 훽푚 푟 훽푚 2+ 2푛−1 2휋24푕2 +퐽0 훽푚푟 , 퐴= − 퐽1’ 훽푚푎 훽푚 2 훽푚 2+ 2푛−1 2휋24푕2 +퐽0 훽푚푎 and 푅= 2푣−1 훽푚 3퐽0 훽푚푎 + 푎 퐽0 훽푚푎 훽푚 4 퐽1 훽푚푎 훽푚 3− 푎 퐽0 훽푚푎 훽푚 4− 2 1−푣 훽푚 3퐽1 훽푚푎 퐽1’ 훽푚푎 훽푚 3

휍푟푟 퐾 =2퐺 ∞푚 =1 − 2 푎 훽푚 푕21+훽푚 21 퐽0 훽푚푎
× 2푛−1 훼휋 −1 푛푕2 푐표푠 2푛−1 휋 2푕 푧+ 푕 2 × −퐹 훽푚 ,푡 푒 −훼 훽푚 2+ 2푛−1 2휋24푕2 푡 −1 − 푓1 푡 훽푚 2+ 2푛−1 2휋24푕2 훼 훽푚 2+ 2푛−1 2휋24푕2 2 + 2훼 −1 푛 푕 푠푖푛 2푛−1 휋 2푕 푧+ 푕 2 푓2 푡 ×푔1 푟 + 2푛−1 휋 훼 –1 푛 푕2 훽푚 2 퐽1′ 훽푚푟 푓2(푡) 훽푚 2+ 2푛−1 2휋24푕2 푒 [ 퐾+훽푚 2 푧+ 푕 2 ] 퐾+훽푚 2+퐽0 훽푚푟 퐶1 훽푚,푧 ,푡 ∞푛 =1 + 퐵푚푛 퐽1’ 훽푚푟 + 퐶푚푛 훽푚푟 퐽1 훽푚푟 +(2푣−1)퐽0 훽푚푟 ∞ 푚=1훽푚 3푠푖푛푕 훽푚 푧+ 푕 2 (23) 휍휃휃 퐾 =2퐺 ∞푚 =1 − 2 푎 훽푚 푕21+훽푚 21 퐽0 훽푚푎 × 2푛−1 훼휋 −1 푛푕2 푐표푠 2푛−1 휋 2푕 푧+ 푕 2 × −퐹 훽푚 , 푡 푒 −훼 훽푚 2+ 2푛−1 2휋24푕2 푡 −1 − 푓1 푡 훽푚 2+ 2푛−1 2휋24푕2 훼 훽푚 2+ 2푛−1 2휋24푕2 2 + 2훼 −1 푛 푕 푠푖푛 2푛−1 휋 2푕 푧+ 푕 2 푓2 푡 ×푔2 푟 + 2푛−1 휋 훼 –1 푛 푕2 훽푚 2 퐽1′ 훽푚푟 푓2(푡) 훽푚 2+ 2푛−1 2휋24푕2 푒 [ 퐾+훽푚 2 푧+ 푕 2 ] 퐾+훽푚 2+퐽0 훽푚푟 퐶1 훽푚,푧 ,푡 ∞푛 =1 + [ 퐶푚푛 훽푚 3 퐽0 훽푚푟 ∞푚 =1 + 퐵푚푛 훽푚 2 푟 퐽1 훽푚푟 ] 푠푖푛푕 훽푚 푧+ 푕 2 (24) 휍푧푧 퐾 =2퐺 2 푎 훽푚 푕21+훽푚 2 퐽0 훽푚푟 퐽0 훽푚푎 ∞푚 =1 × 2푛−1 휋훼 −1 푛푕2 2푛−1 2 휋2 4푕2[훽푚 2+ 2푛−1 2 휋2 4푕2] −1 푐표푠 2푛−1 휋 2푕 푧+ 푕 2 × −퐹 훽푚 , 푡 푒 −훼 훽푚 2+ 2푛−1 2휋24푕2 푡 −1 − 푓1 푡 훽푚 2+ 2푛−1 2휋24푕2 훼 훽푚 2+ 2푛−1 2휋24푕2 2 + 2훼 −1 푛 푕 푠푖푛 2푛−1 휋 2푕 푧+ 푕 2 푓2 푡 + 2푛−1 휋 훼 –1 푛 푕2 퐾+훽푚 2 푓2(푡) 훽푚 2+ 2푛−1 2휋24푕2 푒 [ 퐾+훽푚 2 푧+ 푕 2 ]+ 퐶1 훽푚,푧 ,푡 ∞푛 =1 + 퐶푚푛 2 1−푣 +훽푚 푟 퐽1 훽푚푟 − 퐵푚푛 퐽0 훽푚푟 훽푚 3 푠푖푛푕 훽푚 푧+ 푕 2 (25)

휍푟푧 퐾 =2퐺 − 2 푎 훽푚 2 푕21+훽푚 2 퐽1 훽푚푟 퐽0 훽푚푎 ∞푚 =1 × 2푛−1 2휋2훼 2 −1 푛푕3 푠푖푛 2푛−1 휋 2푕 푧+ 푕 2 × −퐹 훽푚 , 푡 푒 −훼 훽푚 2+ 2푛−1 2휋24푕2 푡 −1 − 푓1 푡 훽푚 2+ 2푛−1 2휋24푕2 훼 훽푚 2+ 2푛−1 2휋24푕2 2 − 2푛−1 훼휋 −1 푛푕2푐표푠 2푛−1 휋 2푕 푧+ 푕 2 푓2 푡 훽푚 2+ 2푛−1 2휋24푕2 + 2푛−1 휋 훼 –1 푛 푕2 푓2 푡 훽푚 2+ 2푛−1 2휋24푕2 푒 퐾+훽푚 2 푧+ 푕 2 ∞푛 =1 + 퐶푚푛 2푣−2 퐽1 훽푚푟 +푟 훽푚 퐽0 훽푚푟 + 퐵푚푛 퐽1 훽푚푟 훽푚 3푐표푠푕 훽푚 푧+ 푕 2 ∞푚 =1 (26) In order to satisfy condition (14), solving Eqs. (23) and (26) for 퐵푚푛 and 퐶푚푛 , one obtains
퐵푚푛 = ∞푚 =1 2 푅 푎 퐾 훽푚 푕21+훽푚 2 1 퐽0 훽푚푎 2푣−2 퐽1 훽푚푎 + 푎 훽푚 퐽0 훽푚푎 푠푖푛푕[훽푚(푧+ 푕 2)] × 2푛−1 휋훼 −1 푛푕2 푐표푠 2푛−1 휋 2푕 푧+ 푕 2 × −퐹 훽푚 , 푡 푒 −훼 훽푚 2+ 2푛−1 2휋24푕2 푡 −1 − 푓1 푡 훽푚 2+ 2푛−1 2휋24푕2 훼 훽푚 2+ 2푛−1 2휋24푕2 2 + 2훼 −1 푛 푕 푠푖푛 2푛−1 휋 2푕 푧+ 푕 2 푓2 푡 ×퐴 + 2푛−1 휋 훼 –1 푛 푕2 훽푚 2 퐽1′ 훽푚푎 푓2(푡) 훽푚 2+ 2푛−1 2휋24푕2 푒 [ 퐾+훽푚 2 푧+ 푕 2 ] 퐾+훽푚 2+퐽0 훽푚푎 퐶1 훽푚,푧 ,푡 ∞푛 =1 − 훽푚 퐽1 훽푚푎 퐽0 훽푚푎 2푣−1 훽푚 3퐽0 훽푚푎 + 푎 퐽1 훽푚푎 훽푚 4 푐표푠푕[훽푚(푧+ 푕 2)]
× 2푛−1 2휋2훼 2 −1 푛푕3푠푖푛 2푛−1 휋 2푕 푧+ 푕 2 × −퐹 훽푚,푡 [푒 −훼 훽푚 2+ 2푛−1 2휋24푕2 푡 −1] 훼 훽푚 2+ 2푛−1 2휋24푕2 2–푓1(푡) − 2푛−1 휋훼 −1 푛푕2푐표푠 2푛−1 휋 2푕 푧+ 푕 2 푓2 푡 훽푚 2+ 2푛−1 2휋24푕2 + 2푛−1 휋훼 푓2 푡 푒 [ 퐾+훽푚 2 푧+ 푕 2 ] –1 푛 푕2 훽푚 2+ 2푛−1 2휋24푕2 ∞푛 =1 (27)
퐶푚푛 = 2 푎 푅 퐾 훽푚 푕21+훽푚 2 1 퐽0 훽푚푎 ∞푚 =1 휷풎 ퟑ 푱ퟏ 휷풎풂 풔풊풏풉[휷풎(풛+ 풉 ퟐ )]

× 2푛−1 휋훼 −1 푛푕2 푐표푠 2푛−1 휋 2푕 푧+ 푕 2 × −퐹 훽푚 , 푡 푒 −훼 훽푚 2+ 2푛−1 2휋24푕2 푡 −1 − 푓1 푡 훽푚 2+ 2푛−1 2휋24푕2 훼 훽푚 2+ 2푛−1 2휋24푕2 2 + 2훼 −1 푛 푕 푠푖푛 2푛−1 휋 2푕 푧+ 푕 2 푓2 푡 ×퐴 + 2푛−1 휋 훼 –1 푛 푕2 훽푚 2 퐽1′ 훽푚푎 푓2(푡) 훽푚 2+ 2푛−1 2휋24푕2 푒 [ 퐾+훽푚 2 푧+ 푕 2 ] 퐾+훽푚 2+퐽0 훽푚푎 퐶1 훽푚,푧 ,푡 ∞푛 =1 − 퐽1’ 훽푚푎 퐽1 훽푚푎 훽푚 4 푐표푠푕[훽푚(푧+ 푕 2)]
× 2푛−1 2휋2훼 2 −1 푛푕3푠푖푛 2푛−1 휋 2푕 푧+ 푕 2 × −퐹 훽푚,푡 [푒 −훼 훽푚 2+ 2푛−1 2휋24푕2 푡 −1] 훼 훽푚 2+ 2푛−1 2휋24푕2 2–푓1(푡) − 2푛−1 휋훼 −1 푛푕2푐표푠 2푛−1 휋 2푕 푧+ 푕 2 푓2 푡 훽푚 2+ 2푛−1 2휋24푕2 + 2푛−1 휋훼 푓2 푡 푒 [ 퐾+훽푚 2 푧+ 푕 2 ] –1 푛 2푕2 훽푚 2+ 2푛−1 2휋24푕2 ∞푛 =1 (28) 4. Special case and Numerical Calculations Setting
(1) 푓 푟 =훿 푟−푟1 푒−푡, 푟1 =0.5푚,푎=1푚.
where 훿 푟 is well known dirac delta function of argument r. 퐹 훽푚 = 2 푎 훽푚 훽푚 2+ 푕12 푟1 퐽0 훽푚푟1 퐽0 훽푚푎 푒−푡 (2) 푞 푟,푧,푡 = 훿 푟−푟0 푡 푐표푠⁡( 2휋푧 푕 ) , 푟0 =0.5푚,푕=0.25푚 푞 훽푚,푧,푡 = 2 푎 훽푚 훽푚 2+ 푕12 푐표푠 2휋푧 푕 푟0 퐽0 훽푚푟0 푡 퐽0 훽푚푎 푞 훽푚,푧,푠 = 2 푎 훽푚 훽푚 2+ 푕12 푐표푠 2휋푧 푕 푟0 퐽0 훽푚푟0 퐽0 훽푚푎 1 푠2 Material Properties The numerical calculation has been carried out for a copper (pure) circular plate with the material properties defined as, Thermal diffusivity 훼 = 112.34× 10−6 푚2푠−1, Specific heat 푐휌=383퐽/퐾푔, Thermal conductivity k = 386 W/m K, Shear modulus 퐺=48 퐺 푝푎, Poisson ratio 휗=0.3. Roots of Transcendental Equation The 훽1=1.7887, 훽2=4.4634,훽3=7.4103, 훽4=10.4566, 훽5=13.543,훽6=16.64994 are the roots of transcendental equation 훽푚 퐽0’ 훽푚푎 + 푕1 퐽0 훽푚,푎 =0. The numerical calculation and the graph has been carried out with the help of mathematical software Matlab.
IV. Discussion
In this paper a thick circular plate is considered and determined the expressions for temperature, displacement and stresses due to internal heat generation within it and we compute the effects of internal heat generation in terms of stresses along radial direction. As a special case mathematical model is constructed for considering copper (pure) circular plate with the material properties specified above.

Fig. 1 Temperature distribution 푞=0 Fig. 2 Temperature distribution ( 푞≠0)
Fig.4Radial stress function σrrK(푞≠0) Fig. 3 Radial stress function σrrK(푞=0)
Fig. 5 Angular stress function σθθK 푞=0 Fig. 6 Angular stress function σθθK 푞≠0
Fig.7 Axial stress function σzzK 푞=0 Fig. 8 Axial stress function σzzK 푞≠0

Fig.9 Stress function σrzK(푞=0) Fig. 10 Stress function σrzK(푞≠0) From Fig. 1 and 2, it is observed that the temperature vary with internal heat generation. From Fig. 3 and 4, it is observed that the radial stresses σrrK vary along radial direction due to internal heat generation, it develops compressive stress in the radial direction. From Fig. 5 and 6, it is observed that the angular stresses σθθK vary along radial direction due to internal heat generation, it develops tensile stress in the radial direction. From Fig. 7 and 8, it is observed that the axial stresses σzzK vary along radial direction due to internal heat generation, it develops compressive stress in the radial direction. From Fig. 9 and 10, it is observed that the stresses σrzK vary along radial direction due to internal heat generation.
V. Conclusion
We can summarize that due to internal heat generation in thick circular plate the radial stress develops compressive stress, whereas the angular stress is tensile. Also, it can be observed from the figures that the value of temperature and stresses due to internal heat generation are higher as compared to plate with no heat generation. The results obtained here are useful in engineering problems particularly in the determination of state of stress in thick circular plate and base of furnace of boiler of a thermal power plant and gas power plant. ACKNOWLEDGEMENTS The authors are thankful to UGC, New Delhi India to provide the partial finance assistance under minor research project scheme File No.47 747/13(WRO). REFERENCES [1] W. Nowacki, The state of stresses in a thick circular plate due to temperature field, Bull. Acad. Polon. Sci., Scr. Scl. Tech.,5, 1957, 227. [2] J. N. Sharma, P. K.Sharma and R. L.Sharma, Behavior of thermoelastic thick plate under lateral loads, Journal of Thermal Stresses, 27, 2004, 171-191. [3] V. S. Kulkarni and K.C. Deshmukh, Quasi-static transient thermal stresses in a thick circular plate, Journal of Brazilian Society of Mechanical Sciences and Engineering, 30(2), 2008, 172-177. [4] G.D. Kedar and K. C. Deshmukh, Estimation of temperature distribution and thermal stresses in a thick circular plate, African Journal of Mathematics and Computer Science, Research, 4(13), 2011, 389-395. [5] C. M.Bhongade and M.H.Durge, Mathematical modeling of quasi-static thermoelastic steady state behavior of thick circular plate with internal heat generation, International Journal of Physics and Mathematical Science, 3(4), 2013, 8-14. [6] M. N. Ozisik, Boundary value problems of heat conduction (International Text Book Company, Scranton, Pennsylvania, 1968).

Mathematical Modeling of Quasi Static Thermoelastic Transient behavior of thick circular plate with Internal Heat Generation

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