IJAAMMInt. J. Adv. Appl. Math. and Mech. 3(2) (2015) 100 – 112 (ISSN: 2347-2529)

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International Journal of Advances in Applied Mathematics and Mechanics

Finite difference approach for non-homogeneous problem of thermalstresses in cartesian domain

Research Article

B.B. Pandita, ∗, V.S. Kulkarnib

a Department of Mathematics, Deogiri Institute of Engineering and Management Studies, Aurangabad, Maharashtra, Indiab PG Department of Mathematics, University of Mumbai, Mumbai-400098, Maharashtra, India

Received 14 August 2015; accepted (in revised version) 04 November 2015

Abstract: The thermal conductivity of metals varies with the temperature. In most of the heat conduction problems, to avoidnonlinearity, thermal conductivity is usually assumed to be temperature independent. This assumption could not bereasonable when large variation of temperature is under consideration because structures of the materials are alsoaffected by variation in thermal conductivity. This is an attempt to study the effect of variable thermal conductivityin thermal stress analysis of rectangular plate subjected to temperature variation. As a special case, the mathematicalmodel of thermoelastic problem is constructed for Copper (pure) plate. The results for temperature distribution,displacement and thermal stresses are illustrated graphically and interpreted technically.

MSC: 35K05 • 35K61 • 65M06 • 65N06

Keywords: Rectangular plate • Thermal stresses • Nonlinear boundary value problem • Finite difference method© 2015 The Author(s). This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/3.0/).

1. Introduction

A challenging problem faced by the Engineers and Mathematicians is to find solutions of the governing equa-tions representing physical systems. There could be nothing more desirable than to find exact closed form solutionsof these equations. However due to non-linearity occurring in the most of real life engineering problems one has toadopt numerical techniques to obtain the solution.

Youssef et al (2007) proposed general finite element model to analyze transient phenomena in thermoelasticmodel in the context of the theory of generalized thermoelasticity with one relaxation time with variable thermalconductivity. Hsin-Ping Chu and Cheng- Ying Lo [1] outlined the differential transformation technique and thenprocedures for transforming and discretizing the governing equations as well as the boundary conditions are given intwo numerical examples. Mashat [2] presented finite different scheme as well as least-square method for the magneto-thermo analysis of an infinite functionally graded hollow cylinder. Bin Shen et al. [3] a heat transfer model based onthe finite difference method is developed to study the thermal aspects in grinding. Kedar G.D. et al. [4] done ThermalStresses in a Semi-Infinite Solid Circular Cylinder .Somayyeh Sadri et al. [5] done Efficiency analysis of straight fin withvariable heat transfer coefficient and thermal conductivity. Sherief and Abd El-Latief [6] observed effect of variablethermal conductivity on a half-space under the fractional order theory of thermoelasticity. Kulkarni V. S. et al. [7] havedone heat transfer and thermal stress analysis of cylinder due to internal heat generation under steady temperatureconditions. Verma Shubha et al. [8] have studied a finite element solution to transient asymmetric heat conductionin multilayer annulus. Recently Pranab Kanti Roy et al. [9] studied application of homotopy perturbation method fora conductive-radiative fin with temperature dependent thermal conductivity and surface emissivity.

In this paper, a thin clamped rectangular plate with variable thermal conductivity and internal heat generationis considered. Finite difference method is proposed to find the temperature distribution, displacement and thermalstresses. The role of variable thermal conductivity on thermal stress analysis is observed mathematically and repre-sented graphically.

∗ Corresponding author.E-mail address: panditbhagwat51@gmail.com (B.B. Pandit), vinayakskulkarni1@rediffmail.com (V.S. Kulkarni)

B.B. Pandit, V.S. Kulkarni / Int. J. Adv. Appl. Math. and Mech. 3(2) (2015) 100 – 112 101

2. Mathematical formulation

Fig. 1. Geometry of heat conduction problem.

Consider a rectangular plate with its dimensions 0 ≤ x ≤ a , 0 ≤ y ≤ b as shown in Fig. 1 and initial temperatureTi .The nonlinear boundary value problem for heat conduction is given as,

ρcp∂T

∂t=O(k(T )OT )+ g (x, y, t ) (1)

Subjected to the boundary conditions,

∂T

∂x= 0 at x = 0, t > 0 (2)

∂T

∂x−hT = 0 at x = a, t > 0 (3)

∂T

∂y= 0 at y = 0, t > 0 (4)

∂T

∂y=φ(t ) at y = b, t > 0 (5)

and initial condition,

T = Ti at t = 0 (6)

where T =T (x, y, t ), φ(t )=T0e−mt , k(T )=k0(1+βT ) is the thermal conductivity varies linearly with temperature, k0 isthe thermal conductivity at the reference temperature, β is the temperature coefficient of thermal conductivity, cp isspecific heat, ρ is the density of the material, T0 is the temperature strength, h is heat transfer coefficient g (x, y) =gsδ(x −x ′)δ(y − y ′) is an instantaneous heat source located at (x ′, y ′) with temperature strength gs .

Since the plate is in plane stress state without bending, Airy’s stress function method is applicable to the analyt-ical development of the thermoelastic field. The fundamental equation is given as( ∂2

∂2x+ ∂2

∂2 y

)2U =−µE

( ∂2

∂2x+ ∂2

∂2 y

)T (7)

The plate is clamped at all sides gives the boundary conditions,

U = 0,∂U

∂x= 0 at x = 0, x = a (8)

U = 0,∂U

∂y= 0 at y = 0, y = b (9)

where µ is linear coefficient of the thermal expansion, E is Young’s modulus, U is Airy’s Stress function.

102 Finite difference approach for non-homogeneous problem of thermal stresses in cartesian domain

The displacement components ux and uy in x and y directions are represented in the integral form and thestress components in terms of U are given by,

ux =∫ [ 1

E

(∂2U

∂2 y−ν∂

2U

∂2x

)+µT

]d x (10)

uy =∫ [ 1

E

(∂2U

∂2x−ν∂

2U

∂2 y

)+µT

]d y (11)

σxx = ∂2U

∂2 y(12)

σy y = ∂2U

∂2x(13)

σx y =− ∂2U

∂x∂y(14)

and

σx y =σy y = 0 at y = b

σxx =σx y = 0 at x = a

where ν is the Poisson’s ratio of the material of the plate.Equations (1) to (14) constitute the mathematical formulation of the problem.

3. Mathematical solution

3.1. Heat transfer analysis

Following Ozisik [10] the Crank Nicolson finite difference method is applied to solve the nonlinear boundaryvalue problem defined by (1) to (6). Divide the x,y ,t domain into small intervals 4x,4y ,4t such that

x = i4x, i = 1,2,3...., N (N4x = a)

y = i4y, j = 1,2,3...., M(M4y = b),

t = n4t ,n = 1,2,3....

The dependent variable at the nodal points (i4x, j4y) at time n4t is denoted by T (i4x, j4y)=T ni , j

The Crank Nicolson finite difference representation by Ozisik [11] for two dimensional nonlinear heat Eq. (1) isgiven by,

ρcp

T n+1i , j −T n

i , j

4t= 1

2

[ki− 1

2 , j

T n+1i−1, j −T n+1

i , j

4x2 +ki+ 12 , j

T n+1i+1, j −T n+1

i , j

4x2

]+ 1

2

[ki− 1

2 , j

T ni−1, j −T n

i , j

4x2 +ki+ 12 , j

T ni+1, j −T n

i , j

4x2

]+ 1

2

[ki , j− 1

2

T n+1i , j−1 −T n+1

i , j

4y2 +ki , j+ 12

T n+1i , j+1 −T n+1

i , j

4y2

]+ 1

2

[ki , j− 1

2

T ni , j−1 −T n

i , j

4y2 +ki , j+ 12

T ni , j+1 −T n

i , j

4y2

]+ g

n+ 12

i , j

(15)

The subscript i ± 1

2for the thermal conductivity denotes that a mean value of thermal conductivity between the nodal

points i ±1 and i .Solving Eq. (15) for T n+1

i , j and setting square grids 4x=4y one gets the recursive relation as,

T n+1i , j = Ai j T n+1

i−1, j +Bi j T n+1i+1, j +Ci j T n+1

i , j−1 +Di j T n+1i , j+1 −Ei j T n+1

i , j +bi j +g

n+ 12

i , j 4t

ρcp(16)

B.B. Pandit, V.S. Kulkarni / Int. J. Adv. Appl. Math. and Mech. 3(2) (2015) 100 – 112 103

where

Ai j =ki− 1

2 , j4t

24x2ρcp

Bi j =ki+ 1

2 , j4t

24x2ρcp

Ci j =ki , j− 1

24t

24y2ρcp

Di j =ki , j+ 1

24t

24y2ρcp

Ei j = Ai j +Bi j +Ci j +Di j

bi , j = Ai j T ni−1, j +Bi j T n

i+1, j +Ci j T ni , j−1 +Di j T n

i , j+1(1−Ei j )T ni , j (17)

The truncation error is of order 0[4x2,4y2,4t 2] and the scheme is unconditionally stable.The finite difference ap-proximation for initial and boundary conditions,Initially at t = 0

T 0i , j = Ti (18)

At initial edge (x = 0)

T ni−1, j = T n

i+1, j (19)

At the extreme parallel edge (x = a)

T nN+1, j = T n

N+1, j +24xhT nN , j (20)

At extreme edge along horizontal axis (y = b)

T ni ,M+1 = T n

i ,M−1 +24yT0e−mn4t (21)

The initial edge (y = 0)along horizontal axis

T ni , j−1 = T n

i , j+1 (22)

Equation (18) gives the initial value of temperature T at each grid point of the plate (att = 0).Assuming that thecoefficients Ai j ,Bi j ,Ci j ,Di j ,Ei j at each iteration, Eq. (16) with the boundary conditions (19) to (22) gives a set oflinear equations,one can apply the successive over relaxation method to solve these equations. The recursive relationis given by,

T n+1i , j = (1−ω)T n

i , j +ω(Ai j T ni−1, j +Bi j T n

i+1, j +Ci j T ni , j−1 +Di j T n

i , j+1 −Ei j T ni , j +bi j +

gn+ 1

2i , j 4t

ρcp) (23)

where relaxation factor ω lies between 1 and 2.

104 Finite difference approach for non-homogeneous problem of thermal stresses in cartesian domain

3.2. Thermal Stresses and Displacement Component

The 13- point difference scheme for equation (7) is given by(Ui , j+2 +Ui , j−2 +Ui+2, j +Ui−2, j

)+2(Ui+1, j+1 +Ui+1, j−1 +Ui−1, j+1 +Ui−1, j−1

)−8

(Ui , j+1 +Ui , j−1 +Ui+1, j +Ui−1, j

)+20Ui , j

=−4x2µE(Ti , j+1 +Ti , j−1 +Ti+1, j +Ti−1, j −4Ti , j

) (24)

The difference scheme for boundary conditions (8) and (9) are given by

U0, j = 0,UN , j = 0 at x = 0, x = a (25)

Ui ,0 = 0,Ui ,M = 0 at y = 0, y = b (26)

Ui−1, j =Ui+1, j at x = 0, x = a (27)

Ui , j+1 =Ui , j−1 at y = 0, y = b (28)

Substituting the boundary conditions (25) to (28) in Eq. (24) one gets the set of simultaneous linear equations.Thesystem of simultaneous linear Eq. (24) has the unique solution by using the following theorem due to Thomas [12] ,

Theorem 3.1.

If A is an irreducible matrix for which |ai j | >M∑

k=1,k 6= j|a j ,k | for at least one j then A is invertible.

The finite difference approximation for displacement components for sufficiently small value of 4x are givenby,

ux =[ 1

E

(Ui , j+1 +Ui , j−1 −2Ui , j

4y2 −νUi+1, j +Ui−1, j −2Ui , j

4x2

)+µT n

i , j

]4x (29)

uy =[ 1

E

(Ui+1, j +Ui−1, j −2Ui , j

4x2 −νUi , j+1 +Ui , j−1 −2Ui , j

4y2

)+µT n

i , j

]4y (30)

Equations (29) and (30) give the displacement components at each nodal point. The difference scheme for Eqs. (12)to (14) is given by

σxx = Ui , j+1 +Ui , j−1 −2Ui , j

4y2 (31)

σy y =Ui+1, j +Ui−1, j −2Ui , j

4x2 (32)

σx y =−(Ui+1, j+1 −Ui+1, j−1 −Ui−1, j+1 +Ui−1, j−1

4x4y

)(33)

Equations (31) to (33) give the thermal stresses at each nodal point.

4. Numerical calculations

The plate is divided in equal grids of 4x = tr i ang le y = 0.1 meters. Fifty iterations has been performed for eachtime step of 4t=0.1 seconds.The simultaneous equations formed by using (25) to (28) in Eq. (24) are solved by using MATLAB programming.

B.B. Pandit, V.S. Kulkarni / Int. J. Adv. Appl. Math. and Mech. 3(2) (2015) 100 – 112 105

Dimensions

• Length of rectangular plate a = 2m

• Breadth of rectangular plate b = 1m

Material properties

The numerical calculation has been carried out for a Copper (Pure) rectangular plate with the material proper-ties as,

• Thermal conductivity k0 = 386W /mK ,

• Specific heat k0 = 383J/K g K ,

• Thermal diffusivity α= 112.34x10−6m2/s,

• Density ρ = 8954K g /m3,

• Poisson ratio ν= 0.35

• Linear coefficient of the thermal expansion µ= 16.5x10−6K −1

• Young’s modulus E = 130GPa

• Initial Temperature Ti = 300K

• Temperature Strength T0 = 400K

• Strength of heat generation gs = 400K at x = 1m and y = 0.5m

Temperature, displacement and stresses at grid points equally spaced with 4x=4y=0.1 meters and at time t =50x0.1=5 seconds is calculated.The surface plots are given by Figs. 2 - 7 ,

5. Concluding Remarks

In this manuscript, the attempt has been made to discuss role of temperature dependent thermal conductivityin heat transfer and thermal stress analysis. Due to consideration of temperature dependent thermal conductivity,the mathematical formulation of physical application in the form of boundary value problem is highly non-linear. Tofind mathematical solution of non-linear boundary value problem, the finite difference scheme has been developedfor governing non-linear partial differential equations. The convergence and stability analysis of finite differencesolution has been done by fundamental theorems of numerical analysis. The results obtained for heat transfer andthermal stress analysis has been validated by equilibrium and compatibility equations in classical thermoelasticity.The comparison is made for temperature, displacement and thermal stresses at each nodal point when the thermalconductivity is independent (β= 0 ) and dependent (β 6= 0 ) of temperature.

• Fig. 2 shows temperature change within rectangular plate. The convection due to dissipation can be observedthrough x = a of the transient heat flux applied at y = b and this happens since initial edges x = 0 and y = 0are thermally insulated. Also the difference in rate of heat transfer can be seen by considering with and withouttemperature dependent thermal conductivity.

• Figs. 3 and 4 shows displacement component in x and y direction respectively. Due to application of heat fluxand convection due to dissipation, the displacement takes place at extreme edges x = a and y = b . The dif-ference in displacement can be observed when the thermal conductivity are temperature independent and de-pendent.

• Figs. 5, 6 and 7, shows thermal stresses developed within rectangular plate. The development of tensile stressescan be seen around extreme edgesx = a and y = b . The difference in thermal stresses is observed with constantthermal conductivity and temperature dependent thermal conductivity.

Thus one can summaries that, the temperature dependent thermal conductivity plays important role in heat transferand thermal stress analysis, particularly when solid is subjected to large temperature variation. This work gives betteroutline for the solution of non-linear boundary value problem. Any special case of particular interest can be derivedby this approach.

106 Finite difference approach for non-homogeneous problem of thermal stresses in cartesian domain

(a) β=−0.5

(b) β= 0

(c) β= 0.5

Fig. 2. Temperature distribution T (K )

B.B. Pandit, V.S. Kulkarni / Int. J. Adv. Appl. Math. and Mech. 3(2) (2015) 100 – 112 107

(a) β=−0.5

(b) β= 0

(c) β= 0.5

Fig. 3. Displacement Component ux (m)

108 Finite difference approach for non-homogeneous problem of thermal stresses in cartesian domain

(a) β=−0.5

(b) β= 0

(c) β= 0.5

Fig. 4. Displacement Component uy (m)

B.B. Pandit, V.S. Kulkarni / Int. J. Adv. Appl. Math. and Mech. 3(2) (2015) 100 – 112 109

(a) β=−0.5

(b) β= 0

(c) β= 0.5

Fig. 5. Thermal Stresses σxx (N /m2)

110 Finite difference approach for non-homogeneous problem of thermal stresses in cartesian domain

(a) β=−0.5

(b) β= 0

(c) β= 0.5

Fig. 6. Thermal Stresses σx y (N /m2)

B.B. Pandit, V.S. Kulkarni / Int. J. Adv. Appl. Math. and Mech. 3(2) (2015) 100 – 112 111

(a) β=−0.5

(b) β= 0

(c) β= 0.5

Fig. 7. Thermal Stresses σy y (N /m2)

112 Finite difference approach for non-homogeneous problem of thermal stresses in cartesian domain

Acknowledgements

The authors are thankful to University Grants Commission, New Delhi to provide the partial financial assistanceunder major research project scheme.

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