IJAAMMInt. J. Adv. Appl. Math. and Mech. 5(2) (2017) 16 – 24 (ISSN: 2347-2529)

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International Journal of Advances in Applied Mathematics and Mechanics

A mathematical approach to solving an inverse thermoelastic problemin a thin elliptic plate

Research Article

Ishaque Khana, ∗, Lalsingh Khalsaa, Preetam Nandeshwarb

a Department of Mathematics, M.G. College, Armori, Gadchiroli (MS), Indiab Department of Mathematics, RTM Nagpur University, Nagpur (MS), India

Received 13 August 2017; accepted (in revised version) 03 November 2017

Abstract: This article investigates the inverse thermoelasticity of an elliptical plate for determining the temperature distributionand its associated thermal stresses by mean of integral transform techniques. Furthermore, by considering a circle asa special kind of ellipse, it is seen that the temperature distribution and history in a circular solution can be drawn asa special case of the present mathematical solution. The numerical results obtained using these computational toolsare accurate enough for practical purposes.

MSC: 35B07 • 35G30 • 35K05 • 44A10

Keywords: Elliptical plate • Temperature distribution • Thermal stresses • Elliptical co-ordinate • Integral transform

© 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

Recently, there has been an emergent tendency to utilise elliptical structures as they have proven to be of consid-erable practical importance in a wide range of sectors such as mechanical, aerospace and food engineering fields forthe past few decades to attain desired functions. Unfortunately, there are only a few studies concerned with steadyand transient state heat conduction problems in elliptical objects owing to its mathematical complications. For ex-ample, McLachlan [3] obtained a mathematical solution of the heat conduction problem for an elliptical cylinder inthe form of an infinite Mathieu function series considering special case by ignoring surface resistance. Kirkpatic et al.[4] extended the McLachlanâAZs solution with the help of numerical calculation. Gupta [5] introduced a finite trans-form involving Mathieu functions and used it for obtaining the solutions of boundary value problem involving ellipticcylinders. Choubey [6] also introduced a finite Mathieu transform whose kernel is given by Mathieu function to solveheat conduction in a hollow elliptic cylinder with radiation. Sugano et al. [7] dealt with transient thermal stress ona confocal hollow elliptical structure with both face surfaces insulated perfectly and obtained the analytical solutionwith couple-stresses. Erdogdu et al.[8, 9] investigated the heat conduction within an elliptical cylinder using a finitedifferential method. El Dhaba et al. [10] used boundary integral method to solve the problem of the plane, uncoupledlinear thermoelasticity with heat sources for an infinite cylinder with elliptical cross section, subjected to a uniformpressure and a thermal radiation condition on its boundary. Sato [11] subsequently obtained heat conduction prob-lem of an infinite elliptical cylinder on heating and cooling considering the effect of the surface resistance. Recently

∗ Corresponding author.E-mail address(es): iakhan_get@rediffmail.com (Ishaque Khan), lalsinghkhalsa@yahoo.com (Lalsingh Khalsa),preetam.b.nandeshwar@gmail.com (Preetam Nandeshwar)

Ishaque Khan et al. / Int. J. Adv. Appl. Math. and Mech. 5(2) (2017) 16 – 24 17

Nomenclature

ξ,η Elliptical Coordinates

q Parameter of Mathieu equation

cen (η, q) Ordinary Mathieu function of first kind of order n

Cen (ξ, q) Modified Mathieu function of second kind of order n

h Interfocal distance [and 1/h2 = c2(coshξ−cos2η)/2]

k Thermal conductivity

Cen (k j ,ξi , q) Mathieu function defined in equation

Fe yn (k j ,ξo , q) Mathieu function defined in equation

Sn,m (k1,k2,ξ,η, qn,m ) Mathieu function defined in equation

qn,m Parametric roots of equation

f (qn,m ) Mathieu transform of f (ξ,η)

θ, (ξ,η, t ) The temperature distribution at any time t

φ The Airy’s stress functions

f (η, t ) The heat supply available on curved surface

2c Focal length = 2√

a2i −b2

i = 2√

a2o −b2

o

ξi , ξo = tanh−1(bi /ai ), = tanh−1(bo /ao )

Cn,m =∫ ξo

ξi

∫ 2π

0(cosh2ξ−cos2η)S2

n,m (k1,k2,ξ,η,±qn,m )dξdη

Bhad et al. [13–16] and Dhakate et al. [17] found an exact solution for the thermoelastic responses in an elliptical discwith an internal heat source, under thermal boundary conditions which are subjected to arbitrary initial temperatureon the upper and lower face at zero temperature, with radiation boundary conditions on both surfaces using integraltransformation technique. From the previous literature regarding elliptical plate, it was observed that no analyticalprocedure had been established for the elliptical plate, considering inverse thermoelastic analysis. Actually, by con-sidering a circle as a special kind of ellipse, it is seen that the temperature distribution and history in a circular solutioncan be derived as a special case of the present mathematical solution. The success of this research mainly lies with theanalytical procedures which present a much simpler approach for optimisation of the design regarding material usageand performance in engineering problem, particularly in the determination of thermoelastic behaviour in ellipticalplate engaged as the foundation of pressure vessels, furnaces, etc. In this paper, we have determined the tempera-ture distribution of a finite elliptical plate using integral transform occupying the space D = {(ξ,η, z) ∈ R3 : 0 ≤ ξ≤ ξo ,0 ≤ η≤ 2π , 0 ≤ η≤ 2π}. For illustrating the practical usage of this research, a particular case with the realistic exampleis explained for further clarification.

2. Formulation of the problem

The thermoelastic issue of an elliptical membrane subjected to radiation type boundary conditions on the outsideand inside surfaces can be rigorously analysed by introducing the elliptical coordinates(ξ, η), which are related to therectangular coordinates (x, y) of the relation

x = c coshξcosη, y = c sinhξsinη (1)

in which c is the semi-focal length as shown in Fig. 1. From the above equations, one obtains a group of confocalellipses and hyperbolas with the common foci for various values of ξ and η, respectively.

2.1. Transient heat conduction analysis

The governing equation of heat conduction with the initial condition and boundary conditions in elliptical cylin-drical coordinates are given, respectively as

h2 (θ,ξξ+θ,ηη ) (ξ, η, t ) = (1/κ)θ,t (ξ, η, t ) (2)

θ(ξ, η, t )∣∣

t=0 = θ0 (3)

18 A mathematical approach to solving an inverse thermoelastic problem in a thin elliptic plate

Fig. 1. Shows the geometry of the problem

θ(ξ, η, t )+k1θ(ξ, η, t ),ξ= 0, (4)

θ(ξ, η, t )+k2θ(ξ, η, t ),ξ= 0, (5)

in which θ (ξ, η, t ) is the temperature function, ki (i = 1, 2) are radiation coefficients, κ = λ/ρC represents thermaldiffusivity in which λ being the thermal conductivity of the material, ρ is the density and C is the calorific capacity,assumed to be constant. The Eqs. (2) to (5) constitute the mathematical formulation for temperature change withinthe elliptical membrane.

2.2. Displacement and thermal stress analysis

Following Gosh [2] and Jeffery [1], the displacements are given by

(2µ)u/h = −φ (ξ, η, t ),ξ+P (ξ, η, t ),η /h2,

(2µ) v/h = −φ (ξ, η, t ),η+P (ξ, η, t ),ξ /h2

}(6)

in which (u, v) are displacements in the directions normal to the curves (ξ, η) constant, P satisfies the equations

∇2P = 0,

(λ+µ) [(h−2P,η),ξ+ (h−2P,ξ),η] = (λ+2µ)[(φ,ξξ+φ,ηη ) ]

}(7)

and stress function in Eq. (6) satisfies the following equation of the fourth order

h2∇2h2∇2φ=−h2∇2θ (8)

The components of the stresses are represented as

σξξ = h2φ,ηη+ (e2h4/2) sinh2ξ φ,ξ − (e2h4/2) sin2η φ,η ,

σηη = h2φ,ξξ− (e2h4/2) sinh2ξ φ,ξ + (e2h4/2) sin2η φ,η ,

σξη = −h2φ,ξη+ (e2h4/2) sin2η φ,ξ + (e2h4/2) sinh2ξ φ,η

(9)

For traction free surface the stress functions

σξξ = σξη = 0 at ξ= ξi , ξo (10)

The set of Eqs. (8) to (10) constitute mathematical formulation for displacement and thermal stresses developedwithin solid due to temperature change.

Ishaque Khan et al. / Int. J. Adv. Appl. Math. and Mech. 5(2) (2017) 16 – 24 19

3. Solution for the problem

3.1. Transient heat conduction analysis

To solve the fundamental differential equation, we firstly introduce the extended integral transformation of ordern and m over the variable ξ and η as

f (±qn,m) =∫ ξo

ξi

∫ 2π

0f (ξ,η)(cosh2ξ−cos2η)Sn,m(k1,k2,ξ,η,±qn,m)dξdη (11)

in which the kernel can be given as

Sn,m(k1,k2,ξ,η,±qn,m) = Cen(ξ,±qn,m) cen(η,±qn,m) [Fe yn(k1,ξi ,±qn,m)

+Fe yn(k2,ξo ,±qn,m)]+ Fe yn(ξ,±qn,m)cen(η,±qn,m)

×[Cen(k1,ξi ,±qn,m)+Cen(k2,ξo ,±qn,m)]

Inversion theorem of (11) is

f (ξ,η) =∞∑

n=0

∞∑m=1

f (±qn,m)Sn,m(k1,k2,ξ,η,±qn,m)/Cn,m (12)

in which ±q are the roots of the transcendental equation

Cen(k1,ξi ,η,±q)Fe yn(k2,ξo ,η,±q)−Cen(k2,ξ0,η,±q)Fe yn(k1,ξi ,η,±q) = 0 (13)

and

Cn,m =∫ ξo

ξi

∫ 2π

0(cosh2ξ−cos2η) S2

n,m(k1,k2,ξ,η,±qn,m)dξdη (14)

Performing above integral transformation under the conditions (4) and (5), we obtain

θ,t (n,m, t ) +α2n,m θ (n,m, t ) = 0 (15)

in which θ(n,m, t ) is the transformed function of θ(ξ,η, t ) andα2n,m = 2κqn,m h2. On solving (15) under initial bound-

ary condition given in Eq. (3), one obtain

θ(n,m, t ) = θ0 exp(−αn,m t ) (16)

On applying inversion theorems defined in (12), one obtain the expression for temperature as

θ(ξ,η, t ) = θ0

∞∑n=0

∞∑m=1

Sn,m(k1,k2,ξ,η, qn,m) exp(−αn,m t )/Cn,m (17)

The function given in Eq. (17) represents the temperature at every instant and at all points of elliptical membraneunder the influence of radiation.

3.2. Thermoelastic solution

Assuming Airy’s stress function φ(ξ, η, t ) which satisfies condition (8) as,

φ(ξ, η, t ) = θ0

∞∑n=0

∞∑m=1

(ξ+Xn,m ξ2 +Yn,m η2

Cn,m (ω−αn,m)

)Sn,m(k1,k2,ξ,η, qn,m) exp(−αn,m t ) (18)

Arbitrary functions Xn,m and Yn,m are determined using Eqs. (18) and (9) in Eq. (10). Thus Eq. (18) becomes

φ(ξ, η, t ) = ∑∞n=0

∑∞m=1

{(−ω+αn,m )exp[t (−ω+αn,m )] C1 Sn,m (ξ,η)+C2 Sn,m (ξ0,η)

2Cn,m (−ω+αn,m )2ξ0 (2A1+C3 Sn,m [ξ0,η]+C4)+4ηC5 Sn,m [ξ0,η]

}×exp(−αn,m t ) (19)

Now using Eq. (19) in (9), one obtains the resulting equations of stresses (i.e. σξξ, σηη and σξη) which are ratherlengthy, and consequently are omitted here for the sake of brevity, but considered during graphical discussion de-scribed in below section.

20 A mathematical approach to solving an inverse thermoelastic problem in a thin elliptic plate

4. Numerical results, discussion and remarks

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

bo = bo/ao , ao = ao/ao , e = c/ao , h2 = h2a2o , τ= κ t/a2

o ,

θ(ξ,η, t ) = θ(ξ,η, t )/θk , (θi ,θo) = (θi ,θo)/θk (k = i ,o),

φ(ξ, η, t ) =φ(ξ, η, t )/Eαtθk a2o , σi j =σi j /Eαtθk (i , j = ξ,η)

(20)

Here E stands for Young’s modulus, αt for Thermal expansion coefficient, respectively. Substituting the values inEqs. (17), (19) and (20), we obtained the expressions for the temperature, displacement and stresses respectively forour numerical discussion. The numerical computations have been carried out for Aluminum metal with parametera=2.65 cm, b=3.22 cm, h=2.00 cm, Modulus of Elasticity E= 6.9 × 106N/cm2, Shear modulus G= 2.7 × 106N/cm2, Pois-son ratio υ = 0.281, Thermal expansion coefficient αt = 25.5 × 106cm/cm-0C, Thermal diffusivity κ= 0.86 cm2/sec,Thermal conductivity λ = 0.48 calsec −1/cm 0C with qn,m =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, 35.6292, 39.4784 arethe positive & real roots of the transcendental equation (A). The foregoing analysis is performed by setting the ra-diation coefficients constants, ki = 0.86 (i = 1, 2) so as to obtain considerable mathematical simplicities. In orderto examine the influence of uniform heating on the membrane, we performed the numerical calculation for timeτ = 0.001, 0.05, 0.12, 0.30, 0.70..∞ and numerical calculations are depicted in the following figures with the help ofMATHEMATICA software. The theoretical analysis on the heat conduction & its thermal stress in a confocal hollowelliptical plate without internal heat source subjected to non-axisymmetric heating on internal and outer ellipticalboundaries was investigated by integral transform by Sugano et al. [7], where kernel was expressed in the form ofMathieu and modified Mathieu functions. The thermoelastic effects on the temperature, displacements, and thermalstresses without internal heat source are fully discussed in research paper [7]. For the sake of brevity, discussion

Fig. 2. Temperature distribution versus ξ at η=900 fordifferent values of time

Fig. 3. σηη versus ξ at η= 900 for different values of time

of these effects is omitted here and graphical illustration is investigated for thermoelastic responses for an ellipticalmembrane considering interior heat generation. Figs. 2-6 illustrate the numerical results of dimensionless tempera-ture, displacement and stresses of elliptical membrane under thermal boundary condition that are subjected to arbi-trary initial temperature on the upper and lower face at zero temperature and boundary conditions of radiation type

Fig. 4. σξξ versus ξ at η= 900 for different values of time Fig. 5. σξη versus ξ at η= 900 for different values of time

Ishaque Khan et al. / Int. J. Adv. Appl. Math. and Mech. 5(2) (2017) 16 – 24 21

Fig. 6. Temperature distribution versus η at ξ= 0.45 fordifferent values of time

Fig. 7. σξξ versus η at ξ= 0.45 for different values of time

Fig. 8. σηη versus η at ξ= 0.45 for different values of time Fig. 9. σξη versus η at ξ= 0.7 for different values of time

on the outside and inside surfaces, with independent radiation constants in radial direction at η=900for different val-ues of time. As shown in Fig. 2, the temperature falls as the time proceeds along radial direction and is greatest in asteady & initial state. From Fig. 2, it can be seen that the temperature change on the heated surface decreases whenthe radius of plate increases. Fig. 3 shows the variation of displacement in the radial direction. It can be seen fromFig. 3 that the displacement increases when the radius increases. The variation of normal stresses σξξ, σηη, and σξηis shown in Figs. 4, 5 and 6 respectively. From Fig. 4, the large compressive stress occurs on the inner heated surfaceand the tensile stress occurs on the inner surface which drops along the radial direction. From Fig. 5, the compres-sive stress occurs on the outer edge of the ellipse and the absolute value rises as the time proceeds. From Fig. 6, themaximum tensile stress occurs during uniform heating inside the core of the membrane which follows assumed trac-tion free property. Figs. 6 to 9 shows dimensionless temperature, displacement and thermal stresses along angulardirection. Fig. 7 shows the time variation of temperature distribution along angular direction of the membrane. Thetemperature increases with time, and the maximum value of temperature magnitude occurs at higher steady statewith available internal heat energy.

Fig. 10. σξξ versus η at ξ= 0.45 for different values of time Fig. 11. σηη versus η at ξ= 0.45 for different values of time

22 A mathematical approach to solving an inverse thermoelastic problem in a thin elliptic plate

Fig. 12. σξη versus η at ξ= 0.7 for different values of time Fig. 13. σξξ versus η at ξ= 0.45 for different values of time

Fig. 14. σηη versus η at ξ= 0.45 for different values of time

The aforementioned results agree with the results [7]. The distribution of the dimensionless temperature gradientat each time decreases in the unheated area of the outer ellipse boundary tending below zero in one direction. Fig. 8shows the thermal displacement in η direction of the membrane. It is noted from this graph that the values of thermaldisplacement decreases over time and their maximum values are located beyond the neighborhood from central part.The stress distributions are shown from Figs. 8 and 9. It is observed that the stress patterns from elliptical inner holeto mid core part which follows the similar pattern of the applied mechanical boundary conditions. The radial stressσξξ, circumferential stress σηη and shear stress σξη at inner surface are nearly zero due to the assumed traction freeboundary conditions. It is noted that maximum tensile stress occurs near the outer surface and the compressive stressoccurs inside the membrane and its absolute value increases with time.

5. Transition to annular-circular plate

When the elliptical membrane degenerates into an annular circular membrane with the thickness h → 0, internalradius ξi , and external radius ξo →∞, occupying the space D ′ = {(x, y, z) ∈ R3 :a ≤ (x2 + y2)1/2 ≤ b,z = `}, wherer =(x2+ y2)1/2 in such a way that h exp(ξ)/2 → r,h exp(ξi )/2 → a, and h exp(ξ0)/2 → b [3] and taking θ independent of η.For that we take,

n = 0, q → 0,e → 0,cosh2ξdξ→ 2r h2dr, A(0)2 → 0, A(0)

0 → 1/p

2, (21)

λ20,m →α2

m ,ce0(η, q0,m) → 1/p

2,ce0(ξ, q0,m) → J0(αmr ),Fe y0(ξ0, q0,m) → Y0(αmr ), (22)

αm(=α0,m)are the roots ofJ0(k1,αa)Y0(k2,αb)− J0(k2,αb)Y0(k2,αa) = 0Where,

J0(k j ,αi r ) = J0(αi r )+k j J ′0(αi r )

Y0(k j ,αi r ) = Y0(αi r )+k j Y ′0(αi r )

}i = 1, 2 (23)

Ce0(k1,ξ,η, q0,m) →Ce0(k1,rαm)

Fe y0(k1,ξ,η, q0,m) → Fe y0(k1,rαm)

S0,m(k1,k2,ξ,η, q0,m) → S0,m(k1,k2,rαm) (= Sm(k1,k2,rαm))

(24)

Ishaque Khan et al. / Int. J. Adv. Appl. Math. and Mech. 5(2) (2017) 16 – 24 23

Eq. (17) degenerates into temperature distribution in hollow circular membrane

θ (r, z, t ) =∞∑

m=1

1

Cm

(1+ exp(αm −ω) t

αm −ω)

Sm(k1,k2,rαm) exp(−αm t ) (25)

Where,

Cm =∫ b

ar S2

m(k1,k2,rαm)dr,

and kernel as

Sm(k1,k2,rαm) = J0(rαm) [Y0(k1, aαm)+ Y0(k2,bαm) ]

−Y0(rαm) [ J0(k1, aαm)+ J0(k2,bαm) ]

The aforementioned results agree with the results [12].

6. Conclusion

The proposed analytical solution of transient plane thermal stress problem of the confocal elliptical region washandled in elliptical coordinate system. To author’s knowledge there have been no reports of solution so far inwhich sources are generated according to the linear function of the temperature in mediums in the form of ellipti-cal membrane of finite height with boundaries conditions of the radiation type. The analysis of non-stationary two-dimensional equation of heat conduction is investigated with the integral transformation method as when there areconditions of radiation type contour acting on the object under consideration. With proposed integral transformationmethod, it is possible to apply widely to analysis stationary as well as non-stationary temperatures. Also by using theAiry’s stress function induced by Sugano [7], we have proposed an exact solution theoretically and illustrated graphi-cally for better understanding. The following results were obtained to carry away during our research are:

1. The advantage of this method is its generality and its mathematical power to handle different types of mechan-ical and thermal boundary conditions.

2. The maximum tensile stress is shifting from central core to outer region may be due to heat, stress, concentra-tion under considered temperature field.

3. Finally, the maximum tensile stress occurs in the circular hole on the major axis compared to elliptical hole in-dicates that the distribution of weak heating. It may be due to insufficient penetration of heat through ellipticalinner surface. The aforementioned integral transform will also be extended to other elliptical objects havingfinite height with conditions of radiation type contour during further research work.

Acknowledgements

We are greatly indebted to Dr. Vinod Varghese (Department of Mathematics, Sm. Sushilabai Rajkamalji BhartiScience College, Arni, Yavatmal, Maharashtra, India) for concrete suggestions on improving upon the presentation ofthe work.

References

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