a green’s function approach for the thermoelastic … · a novel method named as modiﬁed...

IJAAMMInt. J. Adv. Appl. Math. and Mech. 5(2) (2017) 30 – 40 (ISSN: 2347-2529)

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International Journal of Advances in Applied Mathematics and Mechanics

A Green’s function approach for the thermoelastic analysis of anelliptical cylinder

Research Article

Tara Dhakatea, ∗, Vinod Vargheseb, Lalsingh Khalsaa

a Department of Mathematics, M.G. College, Armori, Gadchiroli (MS), Indiab Department of Mathematics, Smt Sushilabai Rajkamalji Bharti Science College, Arni, Yavatmal (MS), India

Received 31 August 2017; accepted (in revised version) 23 November 2017

Abstract: In this paper, we deal with the axisymmetric temperature distribution of an elliptical cylinder subjected to the activ-ity of an internal heat source in which sources are generated according to a linear function of the temperature. Thesolution of heat conductivity equation and the corresponding initial and boundary conditions are obtained in an ana-lytical form by employing the Green’s function method in the elliptical coordinate system. The outcomes are obtainedin a series form in terms of ordinary Mathieu and modified Mathieu functions. The solution is further verified bycomparing them to the limiting case of a circle as a special kind of ellipse. Finally, the governing equation for ther-mally induced deflection results and its associated stresses are found during bending of a simply supported ellipticalcylinder. The numerical results obtained using these computational tools are accurate enough for practical purposes.

MSC: 35B07 • 35G30 • 35K05 • 44A10

Keywords: Elliptical cylinder • Temperature distribution • Thermal stresses • Green’s function • Bending moment

© 2017 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

As far as theoretical mechanics is concerned, the solution methods for nonlinear differential equation plays a verycrucial role as many problems have been modeled using such equations. In particular, bending analysis for anystructural profile subjected to pressure load or thermal impact can be described by the same nonlinear differentialequation. The problem of thermoelastic bending and its associated stresses has attracted much attention, and fewdifferent theories like higher order shear and normal deformation theory [1], trigonometric shear deformation theory[2], four variable refined plate theory [3] have been suggested to solve it. Due to the complexity of the thermoelas-ticity problems, mostly analytic solutions were preferred for axisymmetrical problems and other simple problems,whereas for general non-axisymmetrical problems the numerical computation was adopted as the main method. Forbending problems using the Green function for thermal elastic problems has been investigated by many researchersdue to several advantages [like (i) it is a powerful and flexible method (ii) it has a systematic procedure which is eas-ily available, and so forth]. Though there are several methods outlined for finding Green’s functions associated withany partial differential equation for arbitrary domains, including the method of images, separation of variables, andLaplace transforms [4]. Even few other highly cited literature reviews with Green’s functions approaches which were

∗ Corresponding author.E-mail address(es): [email protected] (Tara Dhakate), [email protected] (Vinod Varghese),[email protected] (Lalsingh Khalsa)

http://www.ijaamm.com/

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Tara Dhakate et al. / Int. J. Adv. Appl. Math. and Mech. 5(2) (2017) 30 – 40 31

mostly used to obtain heat conduction solutions was either in classical position [5–7] or new ones such as [8–11] intheir books for many decades. Of most recent literature, some authors have undertaken the work on bending analysis,which can be summarised as given below. Kim and Noda [13] derived Green’s functions for solving the deflection andthe transient temperature using the Galerkin method and the laminate theory. Feng and Michaelides [12] introduceda novel method named as modified Green’s functions (MGFs) to obtain the transient temperature field in a homoge-neous or a composite solid body. Lu et al. [14] presented an analytical method leading to the solution of transienttemperature field in the multidimensional composite circular cylinder using separation of variables. Kidawa-Kukla[15, 16] proposed the heat conduction solutions for circular as well as for thin annular plate subjected to movingheat source which changes its place with time along a concentric circular trajectory be obtained by using the Green’sfunction. Recently, Kukla and Siedlecka [17] derived from the Green’s function using eigenproblem method for theheat conduction problems in a finite multi-layered hollow cylinder. Similarly, very recently, Bialecki and Bulinski [18]further investigated by a generalisation of Green’s idea by obtaining heat conduction solution using Green’s functiontechnique (GFT). However, it was observed from the previous literature that almost all researcher has considered pointinstantaneous heat source either as a function of Dirac Delta or Heaviside. Things get further complicated when inter-nal heat generation persists on the object under consideration is according to the linear function of the temperature,and further becomes unpredictable when sectional heat supply is impacted on the body. Unfortunately, we cannotcomprehend each differential equation, and almost all majority phenomena are governed by nonlinear differentialequations, of which most aren’t tractable [19]. Thus, both analytical and numerical procedures have turned out tobe the best technique to take care of such problems. In any case, numerical solutions are favoured and prevalent inpractice, due to either non-accessibility or mathematical complexity of the corresponding exact solutions. Rather,limited utilization of analytical solutions should mustn’t diminish their merit over numerical ones; since exact solu-tions, if available, provide an insight into the governing physics of the problem, that is often missing in any numericalsolution [20]. One such problem is to determine the exact temperature distribution and thermoelastic behaviour inelliptical coordinates with objects subjected to the internal heat sources which are according to the linear function ofthe temperature in mediums. It was learn that there are numerous applications involving elliptical geometry requirean evaluation of temperature distribution and its thermal effect on it. One distinctive example is an elliptical nuclearfuel rod in nuclear reactors.

However, to the best of authors’ knowledge, no work has concerned with the thermoelastic bending analysis takinginto consideration of thermal moments and resultant forces in the strain energy equation are investigated. Owing tothis gap of research in this field, the authors have been motivated to conduct this study. In this present paper, therealistic problem of a thin elliptical cylinder subjected to arbitrary initial temperature on the upper lower face, withsimply supported thermally insulated are studied using a typical superposition technique. This manuscript intendsto present and illustrate a unified solution method, namely eigenvalue and the method of the integral transform forthe Green’s function derived for the differential equation with a non-homogeneous term of a point source. The ther-mal moment is derived on the basis of temperature field, whereas maximum normal stresses are derived based onresultant bending moments per unit width [21, 22]. The theoretical calculation has been considered using the dimen-sional parameter, whereas, graphical calculations are carried out using the dimensionless parameter. The successof this novel research mainly lies on the new mathematical procedures which present a much simpler approach foroptimisation of the design in terms of material usage and performance in engineering problem, particularly in thedetermination of thermoelastic behaviour in elliptical cylinder engaged as the pressure vessels, furnaces, etc. Finally,by considering a circle as a special kind of ellipse, it is seen that the temperature distribution in a circular solution canbe derived as a special case from the present mathematical solution.

2. Formulation of the problem

Consider an elliptical cylinder occupying the space D = {(ξ,η, z) ∈ R3 :0 < ξ < a, 0 < η < 2π, −1 ≤ z ≤ 1} de-fined by the transformation ξ+ iη = cosh−1[(x + i y)/c], z = z with length 2c is the distance between their commonfoci, which can be defined as 2c = 2

pa2 −b2. Explicitly, we have x = c coshξ cos η, y = c sinh ξ sinη z = z and

h2 = 2/[c2 (cosh 2ξ − cos 2η)]. The curves η =constant represent a family of confocal hyperbolas while the curvesξ=constant represent a family of confocal ellipses. Thus both sets of curves intersect each other orthogonally at everypoint in space. Now, we assume that the elliptical cylinder along the semi-major axis a, whereas semi-minor axis b.The parameter ξ defines the interfocal lines having the range ξ ∈ (0, a), and that can be given as ξ = tanh−1(b

/a) as

shown in Fig. 1.

2.1. Temperature distribution analysis

The governing equation of temperature distribution T (ξ,η, z, t ) of the cylinder is described by the differential equa-tion is given as follows

h2(∂2T

∂ξ2 + ∂2T

∂η2

)+ ∂2T

∂z2 + ℘(ξ,η, z, t ,T )

λ= 1

κ

∂T

∂t(1)

32 A Green’s function approach for the thermoelastic analysis of an elliptical cylinder

Fig. 1. Elliptic problem geometry

subjected to the initial and boundary conditions

T (ξ, η, z, t )∣∣

t=0 = 0, ∂∂ξ T (ξ, η, z, t )

∣∣∣ξ=a

= 0,

λ ∂∂z T (ξ,η, z, t )

∣∣∣z=1

= α0[T0 −T (a,η,1, t )],

λ ∂∂z T (ξ,η, z, t )

∣∣∣z=−1

= −α0[T0 −T (a,η,−1, t )]

(2)

in which℘(ξ,η, z, t ,T ) is the volumetric energy generation, thermal diffusivity asκ=λ/ρCv , whereλ being the thermalconductivity of the material, ρ is the density and Cv is the calorific capacity, α0 is the heat transfer coefficient, T0 isthe observed temperature of the surrounding medium, assumed to be constant.

Now we assume that ℘(ξ,η, z, t ,T ) as the superposition of the simpler function [[23], p.194]

℘ (ξ, η, z, t , T ) = Φ (ξ, η, z, t ) + ψ (t ) θ (ξ, η, z, t ),θ (ξ, η, z, t ) = T (ξ, η, z, t ) exp[−∫ t

0 ψ (τ) dτ] ,χ (ξ, η, z, t ) = Φ (ξ, η, z, t ) exp[−∫ t

0 ψ (τ) dτ]

(3)

in whichΦ (ξ, η, z, t ) is a function of coordinates and the time, but ψ (t ) is a function of the time only.Substituting Eq. (3) in the heat conduction Eq. (1) and boundary conditions (2), we assume the equivalent form as

h2(∂2θ

∂ξ2 + ∂2θ

∂η2

)+ ∂2θ

∂z2 + χ(ξ,η, z, t )

λ= 1

κ

∂θ

∂t(4)

θ(ξ,η, z, t )∣∣

t=0 = 0, ∂∂ξθ(ξ,η, z, t )

∣∣∣ξ=a

= 0,

λ ∂∂z θ(ξ,η, z, t )

∣∣∣z=1

= α0[θ0 −θ(a,η,1, t )],

λ ∂∂z θ(ξ,η, z, t )

∣∣∣z=−1

= −α0[θ0 −θ(a,η,−1, t )]

(5)

In this study, it is assumed for the sake of brevity that the thermal energy is provided on the cylinder surface has theform

χ(ξ,η, z, t ) = θ δ(ξ−ξ0,η−η0)δ(z − z0)δ(t − t0) (6)

in which 0 < ξ0 < a, 0 < η0 < 2π,−1 < z0 < 1, 0 < t , t0, θ characterises the stream of the heat and δ( ) is the Dirac deltafunction.


2.2. Thermal deflections formulation

The result of the above heat conduction gives resultant moment as

MT =αE∫ 1

−1z T d z (7)

in whichα and E denoting coefficient of linear thermal expansion and Young’s Modulus of the material of the cylinderrespectively.

The differential equation for normal deflection ω (ξ, η, t ) of the cylinder is given as

D∇4ω=−∇2MT

1−υ (8)

in which ∇2 denotes the two-dimensional Laplacian operator in (ξ, η) [24], υ denotes the Poisson’s ratio and D is theflexural rigidity of the cylinder given as D = Eh3/[12(1−υ2)].

The boundary condition for simply supported cylinder is taken as

ω |ξ=a = ∂ω

∂ξ

∣∣∣∣ξ=a

= 0 (9)

2.3. Thermal bending stresses

The maximum normal stresses acting on those sections are parallel to ξz or ηzplanes. Furthermore, the thermalstress components can be determined using small deflection and resultant moment as

σξξ = 6`2

{D h2

[(∂2ω∂ξ2 +υ ∂2ω

∂η2

)− (1−υ)sinh2ξ

(cosh2ξ−cos2η)∂ω∂ξ +

(1−υ)sin2η(cosh2ξ−cos2η)

∂ω∂η

]+ MT

1−υ}

σηη = 6`2

{D h2

[(υ ∂

2ω∂ξ2 + ∂2ω

∂η2

)+ (1−υ)sinh2ξ

(cosh2ξ−cos2η)∂ω∂ξ −

(1−υ)sin2η(cosh2ξ−cos2η)

∂ω∂η

]+ MT

1−υ}

σξη = 6`2

{D h2

[∂ω∂ξ sin2η+ ∂ω

∂η sinh2ξ− ∂2ω∂ξ∂η (cosh2ξ−cos2η)

]} (10)

The Eqs. (1) to (10) constitute the mathematical formulation of the problem under consideration.

3. Solution for the Problem

3.1. Temperature distribution

The solution to the governing equation which is in an analytical form is derived by using the properties of theGreen’s function approach. The Green’s function for the heat conduction problem describes the temperature distri-bution induced by the temporary, local energy impulse. The Green’s function is a solution to the differential Eq. (4) as

h2(∂2G

∂ξ2 + ∂2G

∂η2

)+ ∂2G

∂z2 − 1

κ

∂G

∂t= δ(ξ−ξ′,η−η′)δ(z −ζ)δ(t −τ)

λ(11)

in which 0 ≤ ξ′ ≤ a, 0 < η′ < 2π and −1 ≤ z ′ ≤ 1.Moreover, the Green’s function also satisfies the initial and homogeneous boundary conditions analogous to con-

ditions (2) as

G(ξ, η, z, t )∣∣

t=0 = 0, ∂∂ξG(ξ, η, z, t )

∣∣∣ξ=a

= 0,

∂∂z G(ξ, η, z, t )

∣∣∣z=1

= µoG(ξ,η,1, t ),∂∂z G(ξ, η, z, t )

∣∣∣z=−1

= −µoG(ξ,η,−1, t )

(12)

where µo = αo/a.In order to solve Eq. (11) under the boundary condition (12), we firstly assume the solutions in the form of a series

as

G(ξ,η, z, t ) =∞∑

m=0

∞∑n=1

∞∑p=1

Θ2m,n(ξ, η, t )ψp (z) (13)

Substituting Eq. (13) in the differential Eq. (11) and the boundary conditions (12), yields the eigenvalue

∂2ψp

∂z2 +β2ψp = 0 (14)


and the boundary conditions along z-directional as

dψp

∂z+µ0ψp

∣∣∣∣z=1

= 0,dψp

∂z−µ0ψp

∣∣∣∣z=−1

= 0 (15)

in whichΨp (z) are the eigenfunction corresponding to the eigenvalue.The expressionΨp (z) can be expressed as [25]

ψp (z) =βp cos(βp z)+µ0 sin(βp z), p = 1,2, ... (16)

in which eigenvaluesβp are the roots of the positive roots of the characteristic equation 2µ0βp cosβp − (β2p −

µ20)sinβp = 0.

These functions are pairwise orthogonal so that the following conditions are satisfied∫ 1

−1ψp (z)ψq (z)d z =

{0 for p 6= qQp for p = q

(17)

in which

Qp =∫ 1

−1ψ2

p (z)d z = (β2p +µ2

0)+(β2

p −µ20)cosβp sinβp

βp(18)

Now, the Dirac function δ(z −ζ) in Eq. (11) can be re-expressed in the form

δ(z −ζ) =∞∑

p=1

ψp (z)ψp (ζ)

Qp(19)

Substituting Eq. (13) and (19) into Eq. (11) gives

h2[(∂2

∂ξ2 + ∂2

∂η2

)−β2

p

]Θ2m,n − 1

κ

∂Θ2m,n

∂t= δ(ξ−ξ′, η−η′)δ(t −τ)ψp (ζ)

λQp(20)

and the boundary condition (12) is rewritten as

Θ2m,n(ξ, η, t )∣∣

t=0 = 0,∂

∂ξΘ2m,n(ξ, η, t )

∣∣∣∣ξ=a

= 0 (21)

Now in order to derive the eigen functions Θ2m,n(ξ, η, t ) from the differential Eq. (20), we first introduce the integraltransform [26] of order m and n over the variable ξ and η as

f (q2m,n) = ∫ a0

∫ 2π0 (cosh2ξ−cos2η) Ce2m(ξ, q2m,n )

×ce2m(η, q2m,n ) f (ξ,η)dξdη(22)

in which q2m,n is the root of the transcendental equation Ce2m(ξ, q2m,n) = 0, ce2m(η, q) [[24], p.21] is a Mathieu func-tion of the first kind of order m, Ce2m(ξ, q) [[24], p.27] is a modified Mathieu function of the first kind of order m. Theinversion formula was given [26] as

f (ξ,η) =∞∑

n=0

∞∑m=1

f (q2m,n)Ce2m(ξ, q2m,n)ce2m(η, q2m,n)/C2m,n (23)

where C2m,n = π∫ a

0 (cosh2ξ−ℵ2m,n)Ce22m(ξ, q2m,n)dξ [[24], p.21] and ℵ2m,n = 1

π

∫ 2π0 cos2ηce2

2m(η, q2m,n)dη [[24],p.21].If, therefore, we multiply both sides of equation (20) by Ce2m(ξ, q2m,n) ×ce2m(η, q2m,n) and integrate with respect to ηfrom 0 to 2π and ξ with respect to ξ from 0 to a, we find that Θ2m,n which satisfies the differential equation

∂

∂tΘ2m,n +κ (α2

2m,n +β2p )Θ2m,n =−κ

(ψp (ζ)

λQp

)exp(−α2

2m,n ξ′) exp(−α2

2m,n η′)δ(t −τ) (24)

with the initial condition

Θ2m,n(q2m,n , t )∣∣

t=0 = 0 (25)

in which Θ2m,n is the transformed function ofΘ2m,n andα22m,n = 4q2m,n/c2 and q2m,n are the root of the transcenden-

tal equation Ce2m(a, q2m,n) = 0.


Taking the Laplace transform of Eq. (24) and bearing in account the Eq. (25), we get mathematical simplification as

_

Θ2m,n=−κ(ψp (ζ)

λQp

) [exp(−α2


2m,n η′)

s +κ (α22m,n +β2

p )

]exp(−τ s) (26)

in which_

Θ2m,n is the transformed function of Θ2m,n ,moreover, the Laplace inversion formula gives

Θ2m,n =−κ(ψp (ζ)λQp

)exp(−α2


2m,n η′)

×∫ t0 δ(u −τ) exp

[−κ (α2

2m,n +β2p )

](t −u)du

(27)

then accomplishing inversion theorems of the transform rules defined by Eqs. (23) on Eq. (27), yields

Θ2m,n =∑∞n=0

∑∞m=1

{−κ

(ψp (ζ)λQp

)exp(−α2


2m,n η′)


[−κ (α2

2m,n +β2p )

](t −u)du

}×Ce2m(ξ, q2m,n)ce2m(η, q2m,n)/C2m,n

(28)

Finally substituting Eqs. (16) and (28) into Eq. (13), the Green function has the form

G(ξ,η, z, t |ξ′,η′,ζ,τ) =∑∞m=0

∑∞n=1

∑∞p=1[βp cos(βp z)+µ0 sin(βp z)]

×{−κ

(ψp (ζ)λQp

)exp(−α2


2m,n η′)


[−κ (α2

2m,n +β2p )

](t −u)du

}×Ce2m(ξ, q2m,n)ce2m(η, q2m,n)/C2m,n

(29)

The temperature distribution θ(ξ, η, z, t ) is expressed by Green’s function as

θ(ξ, η, z, t ) =∫ t

0dτ

∫ a

0dξ′

∫ 2π

0dη′

∫ 1

−1χ(ξ′,η′,ζ,τ)G(ξ,η, z, t |ξ′,η′,ζ,τ)dζ (30)

Taking into account the second equation of Eq. (3), the temperature distribution is finally represented by

T (ξ, η, z, t ) = exp[∫ t

0 ψ (τ) dτ]∫ t

0 dτ∫ a

0 dξ′∫ 2π

0 dη′∫ 1−1χ(ξ′,η′,ζ,τ)

×G(ξ,η, z, t |ξ′,η′,ζ,τ)dζ(31)

Substituting the value of Eq. (29) into Eq. (30), one yield

T (ξ, η, z, t ) =∑∞m=0

∑∞n=1

∑∞p=1

{[βp cos(βp z)+µ0 sin(βp z)] exp[

∫ t0 ψ (τ) dτ]

× ∫ t0 dτ

∫ a0 dξ′

∫ 2π0 dη′

∫ 1−1χ(ξ′,η′,ζ,τ)

(−κψp (ζ)λQp

)× exp(−α2


2m,n η′)Ce2m(ξ, q2m,n)ce2m(η, q2m,n)

}dζ


[−κ (α2

2m,n +β2p )

](t −u)du/C2m,n

(32)

The above function is given in Eq. (32) represents the temperature at every instance and at all points of an ellipticalcylinder of finite height under the influence of boundary conditions.

3.2. Thermal deflections analysis

Substituting the value of Eq. (32) into Eq. (7), one yields resultant moment

MT =αE∑∞

m=0∑∞

n=1∑∞

p=1[2µ0(−βp cosβp )+ sinβp ]⟨

exp[∫ t

0 ψ (τ) dτ]

× ∫ t0 dτ

∫ a0 dξ′

∫ 2π0 dη′

∫ 1−1

{−κ

(ψp (ζ)λQp

)χ(ξ′,η′,ζ,τ)exp(−α2

2m,n ξ′)

×exp(−α22m,n η

′)Ce2m(ξ, q2m,n)ce2m(η, q2m,n)}

dζ


[−κ (α2

2m,n +β2p )

](t −u)du

⟩/β2

pC2m,n

(33)

Now substituting the value of Eq. (33) into Eq. (8), one gets

ω=∑∞m=0

∑∞n=1

∑∞p=1

⟨[A2m,n cos2nξsin2mη+B2m,nCe2m(ξ, q2m,n)

×ce2m(η, q2m,n)]+ αED(1−υ)

{exp[

∫ t0 ψ (τ) dτ]

∫ t0 dτ

∫ a0 dξ′

∫ 2π0 dη′

×∫ 1−1

{−κ

(ψp (ζ)λQp


2m,n ξ′)

× exp(−α22m,n η

′)Ce2m(ξ, q2m,n)ce2m(η, q2m,n)}

dζ


[−κ (α2

2m,n +β2p )

](t −u)du

⟩/β2

pC2m,n

(34)


in which A2m,n and B2m,n are the constants to be determined from Eq. (34) and taking into account the boundarycondition (9), one obtains

A2m,n = 0,

B2m,n =⟨

[− cosh2na Ce ′2m(a, q2m,n)+ sinh2na Ce2m(a, q2m,n)]

× αED(1−υ)

{exp[

∫ t0 ψ (τ) dτ]

∫ t0 dτ

∫ a0 dξ′

∫ 2π0 dη′

×∫ 1−1

{−κ

(ψp (ζ)λQp


2m,n ξ′)

× exp(−α22m,n η

′)ce2m(η, q2m,n)}

dζ


[−κ (α2

2m,n +β2p )

](t −u)du

⟩/ce2m(η, q2m,n)Z2m,n

(35)

in which

Z2m,n =β2pC2m,n[cosh2naCe ′2m(a, q2m,n)+ sinh2na Ce2m(a, q2m,n)]

Finally, by substituting the values of A2m,n and B2m,n from Eq. (35) into Eq. (34) results in the required expression forthe thermal deflection.

3.3. Thermal bending stresses

The resulting equations of stresses can be obtained by substituting the resultant moment (33) and deflection equa-tions (34) in Eq. (10). The equations of stresses are rather lengthy, and consequently the same have been omitted herefor the sake of brevity, but have been considered during the graphical discussion using MATHEMATICA software.

4. Transition to circular cylinder

When the elliptical cylinder tends to a circular structure of the radius a, the semi-focal c → 0 and then αn is theroots of the transcendental equation J0(αn) = 0.Also e → 0 [as ξ→∞], cosh2ξdξ→2cosh2ξsinh2ξdξ→2r dr /c2, sinhξ→ coshξ, h coshξ→ r [as h → 0], coshξdξ→r dr,h sinhξdξ→ dr,Using results from [24]

Ce0(ξ, q0,n) → p ′0 J0 (λnr ) ,ce0(η, q0,n ) → 1/

p2,ℵ2m → 0, λ2

0,n =α20,n/a2 =α2

n/a2 =λ2n , p ′

0 =Ce0(0, q0,n )ce0(2π, q0,n )/A(0)0 .

Taking into account the aforesaid parameters, the temperature distribution in cylindrical coordinate is finally repre-sented by

T (r, z, t ) =∑∞n=0

∑∞p=1

{[βp cos(βp z)+µ0 sin(βp z)] exp[

∫ t0 ψ (τ) dτ]

× ∫ t0 dτ

∫ a0 dr ′ ∫ 1

−1χ(r,ζ,τ)(−κψp (ζ)

λQp

)exp(−λ2

n r ′) p ′0 J0 (λnr )

}dζ

×∫ t0 δ(u −τ) exp[−κ (λ2

m +β2p )] (t −u)du/Cn

(36)

The aforementioned degenerated result agrees with the previous result [15, 16].

5. Special case and Numerical calculations

For the sake of simplicity of calculation, we introduce the following dimensionless values

ξ= ξ/a, z = [z − (−`/2)]/a, e = c/a, τ= κ t/a2,

θ = θ/θ0, ω=ω/αθ0ξ0 , σi j =σi j /Eαθ0 (i , j = ξ,η)

}(37)

Substituting the value of Eq. (37) in temperature Eq. (32), deflection Eq. (34) and components of stresses, we obtainedthe expressions for the temperature, deflection and thermal stresses respectively for our numerical discussion. Thenumerical calculation have been carried out for Aluminum metal with physical parameter a = 1 m, ` = 2 m, Modulusof Elasticity E = 70 GPa, Poisson’s ratio υ = 0.35, Thermal expansion coefficient, α = 23× 10−6 /0C, Thermal diffusivityκ = 84.18 ×10−6m2/s−1, Thermal conductivity λ = 204.2 Wm−1 K−1 with q2m,n =0.0986, 0.3947, 0.8882, 1.5791, 2.4674,3.5530, 4.8361, 6.3165, 7.9943, 9.8696, 11.9422, 14.2122, 16.6796, 19.3444, 22.2066, 25.2661, 28.5231, 31.9775, 35.6292are the positive & real roots of the transcendental equation Ce2m(ξ, q2m,n) = 0. In order to examine the influenceof heating on the cylinder, the numerical calculation for all variables was performed. Numerical calculations aredepicted in the following figures using MATHEMATICA software.


Fig. 2. Temperature distribution along ξ - direction fordifferent values of τ.

Fig. 3. Temperature distribution along time τ for variousvalues of z.

Fig. 4. Temperature distribution along z - direction fordifferent value of τ.

Fig. 5. Temperature distribution along η - direction forvarious value of τ.

Figs. 2–14 illustrates the numerical results of temperature distribution, deflection and thermal stresses on ellipticalcylinder due to internal heat generation within the solid under thermal boundary condition that are subjected toknown temperature at any instant time τ. Figs. 2–5 illustrates the numerical results for temperature distribution ofthe cylinder, under thermal boundary condition. Figs. 2–3 shows that the temperature distribution along the radialdirection for different values of time τ and along time τ for various values of z at any instant which maximise itsmagnitude towards outer edge due to internal heat supply. Fig. 4 depicts that the temperature distribution along z -direction for a different time τ, in which it is observed that the maximum temperature distribution occurs at outer coreof the cylinder. Fig. 5 illustrates temperature distribution along η - direction. The temperature distribution approachto zero at both extreme ends due to the more compressive force acting along the edges, whereas temperature attainsthe maximum value at the centre due to tensile force along the mid part for different values of τ following a bell-shaped temperature distribution curves. Fig. 6 depicts the thermal deflection along the radial direction, and initially

Fig. 6. Thermal deflection along ξ - direction for variousvalue of τ.

Fig. 7. Thermal deflection along time τ - for different valueof ξ.


Fig. 8. Thermal stress σξξ along ξ- direction for variousvalue of τ.

Fig. 9. Thermal stress σηη along ξ- direction for differentvalue of τ.

Fig. 10. Thermal stress σξη along ξ- direction for variousvalue of τ.

Fig. 11. Thermal stress σξξ along τ for different value of ξ.

Fig. 12. Thermal stress σηη along τ for various value of ξ. Fig. 13. Thermal stress σξη along τ for different value of ξ.

Fig. 14. Radial stress σξξ along η- direction for different τ.


the deflection attains maximum expansion due to the accumulation of thermal energy dissipated by internal heatgeneration and attains zero at the outer boundary which automatically satisfies the first equation of Eq. (9). In Fig. 7as expected gradual increase in thermal deflection along time parameter for various value of ξ was observed.

Figs. 8–10 indicated that the thermal stresses σξξ, σηη and σξη along the radial direction attain zero at the inner edgedue to the compressive stress, whereas, it has maximum tensile stress at the outer surface due to maximum expansionat the outer part and its absolute value increases with a time parameter. Fig. 9 explains that the tangential stress σηηis initially on the negative side due to the compressive stress occurring at the inner region which goes on increasingas we move towardsξ- direction. It was observed from Fig. 10 that the thermal stress σξη gradually increases along theradial direction, attains maximum magnitude at the mid-core then starts decreasing towards the outer edge and againattending higher side of the outer edge, thus following a Gaussian curvature shape. Figs. 11–13 illustrates the thermalstresses σξξ, σηη and σξη for different value ofξ, initially the stresses attain zero, and maximum stresses occurs as wemove towards the time parameter τ. Fig. 14 illustrates the radial stress distribution σξξ along the angular directionfor a different time; it is evident from the figure that at the early stage, radial stress increases gradually and maximumradial stress occurs at the mid-core and suddenly attains minimum value, thus curves follow a dome shape. Similarcurve nature was observed for σηη and σξη along the angular direction for different timeτ and has been omitted herefor the sake of brevity.

6. Conclusion

The proposed analytical solution of transient thermal stress problem of an elliptical cylinder was handled in ellip-tical coordinate system with the presence of a source of internal heat. To the author’s knowledge, there have been noreports of solution so far in which sources are generated according to the linear function of the temperature in medi-ums in the form of elliptical cylinder of finite height. The analysis of non-stationary three-dimensional equation ofheat conduction is investigated with the unified solution method, namely eigenvalue and the method of the integraltransform for the Green’s function. The following results were obtained to carry out during our research are:

1. The advantage of this technique is its generality and its mathematical power to handle differing types of me-chanical and thermal boundary conditions.

2. The maximum tensile stress shifting from central core to outer region may be due to heat, stress, concentrationor available internal heat sources under considered temperature field.

3. Finally, the maximum tensile stress occurs in the circular core on the major axis compared to elliptical centralpart indicates the distribution of weak heating. It might be due to insufficient penetration of heat through theelliptical inner surface.

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