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Stress Distributions In Cooling Fins OfVariable Thickness With And WithoutRotationAhmet N. Eraslan a & Mehmet E. Kartal ba Department of Engineering Sciences , Middle East TechnicalUniversity , Ankara, Turkeyb Materials Engineering Department , The Open University , WaltonHall, Milton Keynes, EnglandPublished online: 01 Sep 2006.

To cite this article: Ahmet N. Eraslan & Mehmet E. Kartal (2005) Stress Distributions In Cooling FinsOf Variable Thickness With And Without Rotation, Journal of Thermal Stresses, 28:8, 861-883, DOI:10.1080/01495730590964864

To link to this article: http://dx.doi.org/10.1080/01495730590964864

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STRESS DISTRIBUTIONS IN COOLINGFINS OF VARIABLE THICKNESS WITH ANDWITHOUT ROTATION

Ahmet N. EraslanDepartment of Engineering Sciences, Middle East Technical University,Ankara, Turkey

Mehmet E. KartalMaterials Engineering Department, The Open University, Walton Hall,Milton Keynes, England

A computational model is developed to predict elastic and elastic-plastic stress distribution

in a nonlinearly hardening cooling fin of variable thickness subject to centrifugal force. The

model is based on a realistic conduction-convection mechanism, von Mises yield criterion,

Henky’s deformation theory and a Swift-type strain hardening law. Temperature

dependency of modulus of elasticity, uniaxial yield limit, coefficient of thermal expansion,

and thermal conductivity of the fin material is taken into account.

Keywords: Thermoelastoplasticity; Cooling fin; Variable thickness; Nonlinear strain hardening; von Mises

criterion

INTRODUCTION

A variable thickness annular fin mounted on a rigid shaft is considered. Thegeometry of the shaft-fin assembly and the coordinate system are depicted inFigure 1. Heat is transferred from hot shaft to annular fin and from fin tosurroundings. Moreover, the assembly may rotate about the axis of the shaft.The objective is to predict the elastic and elastic-plastic behavior of the variablethickness nonlinearly hardening annular fin under nonuniform heating or bothheating and centrifugation, taking temperature dependent physical properties intoaccount. A computational model is developed for this purpose. The model is basedon von Mises yield criterion, deformation theory of plasticity, and a Swift-typestrain hardening law. A realistic heat transfer model is proposed that accountsfor conduction, thickness variation, and convective heat loss from peripheral andedge surfaces of the fin.

Received 7 June 2004; accepted 25 March 2005.

Address correspondence to Ahmet N. Eraslan, Department of Engineering Sciences, Middle East

Technical University, Ankara 06531, Turkey. E-mail: aeraslan@metu.edu.tr

861

Journal of Thermal Stresses, 28: 861–883, 2005

Copyright # Taylor & Francis Inc.

ISSN: 0149-5739 print/1521-074X online

DOI: 10.1080/01495730590964864

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Theoretical investigation of the stresses in disks induced by centrifugation ornonuniform heating or both is an important topic due to a large number of appli-cations in structural engineering [1–4]. Accordingly, the interest of researchers onthe subject has never ceased. However, there appears only a few investigationsin the literature on the elastic or elastic-plastic deformations of disks=fins subjectedto both centrifugal force and radial temperature gradients. Yielding of a rotatinghyperbolic annular disk for a given temperature distribution was discussed byAlujevic et al. [5–6]. Later, Parmaksizoglu and Guven [7] studied the combinedeffect of nonuniform heating and centrifugal force in the fully plastic state. In theirwork, the convective heat loss from the peripheral surface of the disk wasneglected. Recently, Eraslan and Akis [8] investigated nonisothermal elastic-plasticdeformations of rotating uniform thickness disk by putting forward a realistic heattransfer model that accounts for the convective heat transfer. However, in all of theseinvestigations the physical properties of the disk=fin material were assumed to be tem-perature independent and the latter two comprehensive studies considered linearlyhardening uniform thickness disks.

This work is the extension of the theoretical work of Eraslan and Akis [8] toinclude (i) variable thickness with any functional form of thickness variability, (ii)temperature dependency of the material and thermal properties of the fin, (iii) vonMises yield criterion, and (iv) nonlinear strain hardening in the form of theSwift-type hardening law.

Figure 1 The geometry of the shaft-fin assembly and the coordinate system.

862 A. N. ERASLAN AND M. E. KARTAL

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MODEL DEVELOPMENT

Thickness Variation

In this work, we consider a thin annular fin whose thickness varies continu-ously in the form of a general parabolic function hðrÞ [9]:

hðrÞ ¼ h0 1� r

bþ n

� �k" #

; ð1Þ

where h0, n, and k are parameters (h0 > 0, n > 0, k > 0), and b is the outer radius ofthe annular fin. With this profile function, a wide range of nonlinear and continuousprofiles to describe thickness variation may be achieved. A uniform thickness fin isobtained by letting n ! 1 and a linearly decreasing fin thickness is obtained byusing k ¼ 1. If k < 1 the profile is concave and if k > 1 it is convex. Furthermore,the shape of the profile is smoothed as n increases. The convex annular fin profileshown in Figure 2 corresponds to n ¼ 0:7, k ¼ 2 .

It should be noted that the computational model developed herein is not onlylimited to this thickness variation. It can be used for variable thickness with anyfunctional form of thickness variability. The main reason for the choice of thepreceding parabolic function is that, plane stress analytical solution of a rotatingisothermal disk exhibiting thickness variation of this type is available in theliterature [9].

Physical Properties

Formulations in the following sections use the notation and basic equations ofthermoelasticity given in Boley and Weiner [1]. To obtain more realistic predictionsin thermoelastoplastic calculations, the temperature dependency of the mechanicaland thermal parameters must be taken into account. To this aim, we consider ahigh-strength low-alloy steel for which the modulus of elasticity E and uniaxial yieldlimit r0 vary with temperature T according to the empirical relations [10]

EðTÞ ¼ E0 1þ T

2000 lnðT=1100Þ

� �½N=m2�; ð2Þ

Figure 2 Convex fin profile for a=b ¼ 0:2, n ¼ 0:7, k ¼ 2.

STRESS DISTRIBUTIONS IN COOLING FINS 863

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r0ðTÞ ¼ r0C 1þ T

600 lnðT=1630Þ

� �½N=m2�; ð3Þ

where E0 ¼ 200 GPa and r0C ¼ 410 MPa. In addition, the thermal conductivity kand the coefficient of thermal expansion a for the steel alloy considered can be fitin quadratic forms as

kðTÞ ¼ k0 þ k1T þ k2T2

¼ 45� 0:018T � 1:0� 10�5T2 ½W=m2 �C�; ð4Þ

aðTÞ ¼ a0 þ a1T þ a2T2

¼ 11:7� 10�6 þ 3:0� 10�9T þ 2:5� 10�12T2 ½1=�C�: ð5Þ

Variation of the nondimensional forms of the physical properties with temperaturein the range 0�400�C is depicted in Figure 3. Although all properties vary with tem-perature to some extent, the highest impact is on the yield limit r0ðTÞ, which isexpected to affect the thermoplastic response of the fin considerably.

Figure 3 Variation of physical properties with temperature.

864 A. N. ERASLAN AND M. E. KARTAL

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Temperature Distribution in the Fin

An energy balance over an infinitesimally small ring in the thin fin yields [8, 11]

d2Hdr2

þ 1

rþ h0ðrÞ

hðrÞ

� �dHdr

� 2HCðr;xÞkðTÞhðrÞ H ¼ 0; ð6Þ

where HðrÞ¼TðrÞ�T0 is the temperature difference between the surface of the finand the ambient temperature, HC the heat transfer coefficient, and a prime denotesdifferentiation with respect to the radial coordinate r. The heat transfer coefficientHC is a function of both the radial position r and angular speed x given byHCðr;xÞ ¼ Aþ Bxr, with A and B being parameters [8]. Energy equation (6) is tobe solved subjected to the boundary conditions:

HðaÞ ¼ Tb � T0 and �kðTÞ dHdr

����r¼b

¼ HCðb;xÞHðbÞ; ð7Þ

where Tb represents the temperature of the hot shaft (see Figure 1). Equation (6) isnonlinear because of kðTÞ in the denominator of the third term and hence does notassume analytical solution. The solution should be found by numerical means. Forthis purpose Eq. (6) is put into the form

d2Hdr2

¼ fT r;H;dHdr

� �: ð8Þ

A nonlinear shooting method using Newton iterations, as described in the Appendix,is of great advantage to solve nonlinear two point boundary value problems of thistype [12].

Analytical Temperature Distribution

Analytical solution of Eq. (6) for the thickness variation given by Eq. (1) maybe found by reduction of order if the thermal conductivity k of the material isassumed constant and if the geometric parameter k is assigned a numerical value.Such a solution can then be used to verify the numerical solution procedure. Usingk ¼ k0, the general solution of the energy equation, Eq. (6), for TðrÞ is obtained as

TðrÞ ¼ C1PðrÞ þ C2QðrÞ þ T0; ð9Þ

where Ci is an arbitrary constant and the functions P(r), Q(r) are determined in theform of infinite series:

PðrÞ ¼X1s¼0

psrs; ð10Þ

and

QðrÞ ¼ PðrÞ ln r

ðbþ nÞkþX1s¼0

qsrs

" #: ð11Þ

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For k ¼ 1, first few coefficients are

p0 ¼ 1; p1 ¼ 0; p2 ¼A

2k0h0; p3 ¼

3Aþ 2Bxðbþ nÞ9ðbþ nÞk0h0

; ð12Þ

p4 ¼3A2ðbþ nÞ2 þ 4k0h0½3Aþ 2Bxðbþ nÞ�

48ðbþ nÞ2k20h20; ð13Þ

p5 ¼Aðbþ nÞ2½69Aþ 52Bxðbþ nÞ� þ 60k0h0½3Aþ 2Bxðbþ nÞ�

900ðbþ nÞ3k20h20; ð14Þ

and

q0 ¼ 0; q1 ¼1

ðbþ nÞ2; q2 ¼

1� 2p2ðbþ nÞ2

2ðbþ nÞ4; q3 ¼

1� 2p2ðbþ nÞ2 � 2p3ðbþ nÞ3

3ðbþ nÞ4;

ð15Þ

q4 ¼1� 2p2ðbþ nÞ2 þ 3p22ðbþ nÞ4 � 2p3ðbþ nÞ3 � 2p4ðbþ nÞ4

4ðbþ nÞ5; ð16Þ

q5 ¼1� 2p2ðbþ nÞ2 � 2p3ðbþ nÞ3 � 2p4ðbþ nÞ4 � 2p5ðbþ nÞ5

5ðbþ nÞ6

þ 3p22 þ 6p2p3ðbþ nÞ5ðbþ nÞ2

;

ð17Þ

and for k ¼ 2 they are

p0 ¼ 1; p1 ¼ 0; p2 ¼A

2k0h0; p3 ¼

2Bx9k0h0

; p4 ¼A½Aðbþ nÞ2 þ 4k0h0�

16ðbþ nÞ2k20h20; ð18Þ

p5 ¼Bx½13Aðbþ nÞ2 þ 30k0h0�

225ðbþ nÞ2k20h20; ð19Þ

q0 ¼ 0; q1 ¼ 0; q2 ¼1� 2p2ðbþ nÞ2

2ðbþ nÞ2; q3 ¼ � 2p3

3ðbþ nÞ2; ð20Þ

q4 ¼1� 2p2ðbþ nÞ2 þ 3p22ðbþ nÞ4 � 2ðbþ nÞ4p4

4ðbþ nÞ6; ð21Þ

q5 ¼ � 2p3 � 6p2p3ðbþ nÞ2 þ 2p5ðbþ nÞ2

5ðbþ nÞ4: ð22Þ

866 A. N. ERASLAN AND M. E. KARTAL

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Integration constants in Eq. (9) are determined by the application of boundary con-ditions (7). The result is

C1 ¼ � ðTb � T0Þ½HCðb;xÞQðbÞ þ k0Q0ðbÞ�HCðb;xÞPðbÞQðaÞ þ k0QðaÞP0ðbÞ � PðaÞ½HCðb;xÞQðbÞ þ k0Q0ðbÞ� ; ð23Þ

C2 ¼ðTb � T0Þ½HCðb;xÞPðbÞ þ k0P0ðbÞ�

HCðb;xÞPðbÞQðaÞ þ k0QðaÞP0ðbÞ � PðaÞ½HCðb;xÞQðbÞ þ k0Q0ðbÞ� : ð24Þ

To compare numerical and analytical temperature profiles it is convenient to intro-duce the following dimensionless variables. Radial coordinate: �rr ¼ r=b, temperature:h ¼ T=Tb, heat load: �qq ¼ E0a0Tb=½r0C logðb=aÞ�, angular speed: X ¼ bx

ffiffiffiffiffiffiffiffiffiffiffiffiffiq=r0C

p,

heat transfer coefficient: HC ¼ 1þ BX�rr with B ¼ ðB=AÞffiffiffiffiffiffiffiffiffiffiffiffiffir0C=q

pand finally tem-

perature gradient GT ¼ ðdT=drÞ=ðdT=drjfr¼a;x¼0gÞ. Using the parameters n ¼ 2,

k ¼ 1, �aa ¼ a=b ¼ 0:2, �hh0 ¼ h0=b ¼ 0:2 (thin fin), B ¼ 1:5, and assigning �qq ¼ 2,temperature profiles and gradients are calculated in stationary as well as rotating finsat different angular speeds. The results of these calculations are shown inFigures 4(a) and (b). In these figures, dots show analytical solution obtainedretaining 25 terms in series, dashed lines numerical results for constant thermalconductivity (k ¼ k0) and solid lines numerical results for variable thermal conduc-tivity. For constant k, numerical and analytical results are in perfect agreement.The largest difference between constant and variable k solutions occurs inthe stationary fin (X ¼ 0). As seen in Figure 4(a), due to increasing rate of convectiveheat transfer, the lowest temperature at the edge of the fin is obtained for thelargest angular speed. Moreover, larger gradients are obtained as the angularspeed is increased, as depicted in Figure 4(b). Another comparison of numericaland analytical results is made in Figure 5. This figure corresponds to n ¼ 0:7,k ¼ 2, �qq ¼ 2, other parameters being the same. Again, perfect agreement isobtained between numerical and analytical solutions which assumed constantthermal conductivity.

Elastic Deformations

A state of plane stress (rz ¼ 0) and small deformations are assumed. The basethickness h0 of the fin must be taken sufficiently small compared to its diameter sothat the plane stress assumption is justified [2]. The strain displacement relations:Er ¼ u0, Eh ¼ u=r, the equation of motion

d

drðrhrrÞ � hrh ¼ �hqx2r2; ð25Þ

the compatibility relation

d

drðrEhÞ � Er ¼ 0; ð26Þ

STRESS DISTRIBUTIONS IN COOLING FINS 867

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and the equations of generalized Hooke’s law

Er ¼1

Eðrr � nrhÞ þ

Z T

T0

adT þ Epr þ Eperr ; ð27Þ

Eh ¼1

Eðrh � nrrÞ þ

Z T

T0

adT þ Eph þ Eperh ; ð28Þ

form the basis for the analysis. In the preceding equations, Ej denotes the totalstrain, rj the stress, q the mass density, and n is Poisson’s ratio. The super-scripts p and per have been used to designate plastic and permanent strains,respectively. In a plastically predeformed region Eperj is nonzero and it is alwayszero otherwise.

Figure 4 (a) Temperature distributions in stationary and rotating variable thickness fins for n ¼ 2, k ¼ 1,�aa ¼ 0:2, �hh0 ¼ 0:2, �BB ¼ 1:5, and �qq ¼ 2, (b) corresponding normalized temperature gradients. Dots represent

analytical solution, dashed lines numerical solution for constant thermal conductivity, and solid lines

numerical solution for variable thermal conductivity.

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Now let the stress function Y be defined in terms of radial stress as

Y ðrÞ ¼ rhrr; ð29Þ

so that

rr ¼Y

hrand rh ¼ qx2r2 þ 1

h

dY

dr: ð30Þ

In purely elastic deformations of the fin Epj ¼ Eperj ¼ 0. Hence, in terms of the stressfunction Y elastic strains become:

Er ¼1

E

Y

hr� n

h

dY

dr� nqx2r2

� �þZ T

T0

a dT ; ð31Þ

Eh ¼1

E� nY

hrþ 1

h

dY

drþ qx2r2

� �þZ T

T0

a dT : ð32Þ

Figure 4 Continued

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Governing equation for elastic deformations is obtained by the substitution ofelastic strains in the compatibility relation (26) to give

d2Ye

dr2þ 1

r� E0

E� h0

h

� �dYe

dr� 1

r2� nE0

rE� nh0

rh

� �Ye

¼ � 3þ n � rE0

E

� �hqx2r� Eh

d

dr

Z T

T0

a dT

� �; ð33Þ

in which Ye is the elastic stress function and E0 represents the derivative with respectto r and by chain rule

dE

dr¼ dE

dT

dT

dr: ð34Þ

Although elastic Eq. (33) is linear, its analytical solution does not exist because ofhighly complicated coefficients of Ye and Y 0

e. However, its numerical solution posses-sing high-order accuracy may be found by the shooting method (see Appendix).

Figure 5 Temperature distributions in stationary and rotating variable thickness fins for n ¼ 0:7, k ¼ 2,

�aa ¼ 0:2, �hh0 ¼ 0:2 , �BB ¼ 1:5, and �qq ¼ 2. Dots represent analytical solution, dashed lines numerical solution

for constant thermal conductivity, and solid lines numerical solution for variable thermal conductivity.

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Analytical Stresses in the Elastic State

For constant properties, aðTÞ ¼ a0, EðTÞ ¼ E0, the general solution of elasticequation, Eq. (33), may be obtained with the aid of homogeneous solution given in[9]. The result is

uðrÞ ¼ C3PðrÞ þ C4QðrÞ þ RðrÞ; ð35Þ

rr ¼E0

1� n2C3

nPrþ P0

� �þ C4

nQr

þQ0� �

þ nRr

þ R0� �

� E0a0T1� n

; ð36Þ

rh ¼E0

1� n2C3

P

rþ nP0

� �þ C4

Q

rþ nQ0

� �þ R

rþ nR0

� �� E0a0T

1� n; ð37Þ

where R is the particular integral solution and

PðrÞ ¼ rF aH ; bH ; dH ;r

bþ n

� �k !; ð38Þ

QðrÞ ¼ 1

rF aH � dH þ 1; bH � dH þ 1; 2� dH ;

r

bþ n

� �k !: ð39Þ

The function FðaH ; bH ; dH ; zÞ in Eqs. (38) and (39) is the hypergeometric function[13], the arguments of which are determined as

aH ¼ 1

2þ 1

k� 1

2k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ 4ð1� knÞ

q; ð40Þ

bH ¼ 1

2þ 1

kþ 1

2k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ 4ð1� knÞ

q; ð41Þ

dH ¼ 1þ 2

k: ð42Þ

The particular solution RðrÞ is obtained by the method of variation of parametersand expressed in the form

RðrÞ ¼ UU1ðrÞPðrÞ þUU2ðrÞQðrÞ; ð43Þ

in which

UU1ðrÞ ¼ �Z r

a

QðnÞ f ðnÞWroðnÞ

dn; ð44Þ

UU2ðrÞ ¼Z r

a

PðnÞ f ðnÞWroðnÞ

dn; ð45Þ

WroðrÞ ¼ PðrÞQ0ðrÞ �QðrÞP0ðrÞ; ð46Þ

STRESS DISTRIBUTIONS IN COOLING FINS 871

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and

f ðrÞ ¼ � ð1� n2Þqx2r

E0þ a0kð1þ nÞT

1� rbþn

� ��k� �

r

þ a0ð1þ nÞ dTdr

: ð47Þ

Since the integrands in UUi are polynomials, they may be evaluated analyticallyby expanding them in Taylor series. Thermoelastic solution is completed by theapplication of boundary conditions. For an annular fin mounted on a rigid shaft,boundary conditions are uðaÞ ¼ 0 and rrðbÞ ¼ 0. By noting that RðaÞ ¼ 0, the inte-gration constants are evaluated as

C3 ¼ � QðaÞ½nRðbÞ þ bR0ðbÞ � ba0ð1þ nÞTðbÞ�QðaÞ½nPðbÞ þ bP0ðbÞ� � PðaÞ½nQðbÞ þ bQ0ðbÞ� ; ð48Þ

C4 ¼PðaÞ½nRðbÞ þ bR0ðbÞ � ba0ð1þ nÞTðbÞ�

QðaÞ½nPðbÞ þ bP0ðbÞ� � PðaÞ½nQðbÞ þ bQ0ðbÞ� : ð49Þ

On the other hand, for plane stress, the von Mises yield criterion takes the form [14]

rY ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2r � rrrh þ r2h

q; ð50Þ

in which rY represents the yield stress. The fin will yield, that is, the plastic defor-mation in the fin will begin as soon as rY > r0 and elastic limit heat load q orangular speed x is to be calculated from rY ¼ r0ðTÞ.

For the presentation of results, we use formal nondimensional variables; stress:�rrj ¼ rj=r0C and radial displacement: �uu ¼ uE0=ðbr0CÞ. Constant thermal and materialproperties are assumed to allow comparison of numerical and analytical solutions. Thevalues of the parameters used are �aa ¼ 0:2, �hh0 ¼ 0:2, n ¼ 0:3, and B ¼ 1:5. The elasticlimit heat load qe of the stationary fin (X ¼ 0) possessing the geometric parametersn ¼ 0:7 and k ¼ 2 is computed as �qqe ¼ 1:22036. The stresses and displacement at thiscritical heat load are plotted in Figure 6. Dots represent analytical solution. Numericaland analytical solutions agree perfectly. Since the expansion at the rigid shaft-annularfin interface is not allowed, the circumferential stress turns out compressive in a largeportion of the fin. The nondimensional stress variable / in the figure is computed from

/ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�rr2r � �rrr�rrh þ �rr2h

q; ð51Þ

which corresponds to the yield stress �rrY in the plastic core. Note that / ¼ 1 at the plas-tic-elastic border and / < 1 in the elastic region. As seen in Figure 6, /ðaÞ ¼ 1, whichindicates an elastic-plastic border at this surface. Hence, for the stationary fin, the innersurface is critical and yielding commences at this location. The effect of geometric para-meters n and k on elastic limit heat load for stationary fins is shown in Figure 7. Dashedlines belong to constant property solutions and n ! 1 represents uniform thicknessfin. Both solutions give similar trends for corresponding values of parameters, but finswith temperature dependent physical properties yield at much lower heat loads. This is

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as expected because the yield limit r0ðTÞ sharply decreases with increasing tempera-tures (see Figure 3). It is observed in Figure 7 that, elastic limit heat load decreases withincreasing values of both n and k. Lowest limits belong to uniform thickness fins.More-over, elastic limits for stationary concave fins are higher than those of convex ones.

Elastic-Plastic Deformations

Substituting total strains from Eqs. (27) and (28) in the compatibility relation(26) we obtain the governing equation for the plastic region in the form

d2Yp

dr2þ 1

r� E0

E� h0

h

� �dYp

dr� 1

r2� nE0

rE� nh0

rh

� �Yp ¼ � 3þ n � r

E0

E

� �hqx2r

� Ehd

dr

Z T

T0

a dT

� �þ Eh

rEpr � Eph þ Eperr � Eperh � r

dEphdr

þ dEperh

dr

� �� �; ð52Þ

in which Yp denotes the stress function in the plastic region. Note that in the elasticregion, the plastic strains Epj , E

perj and hence their derivatives vanish and this equation

Figure 6 Stresses and displacement in stationary variable thickness fin at the elastic limit heat load

�qqe ¼ 1:22036 for n ¼ 0:7, k ¼ 2, �aa ¼ 0:2, �hh0 ¼ 0:2, n ¼ 0:3, and B ¼ 1:5. Dots belong to analytical solution

and solid lines numerical solution with constant physical properties.

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reduces to the elastic equation given by Eq. (33). Furthermore, at the elastic-plasticborder rep, for the stresses rr and rh to be continuous, from Eq. (30) it is requiredthat the stress function and its first derivative must be continuous, that isYeðrepÞ ¼ YpðrepÞ and Y 0

eðrepÞ ¼ Y 0pðrepÞ. Since the plastic strains and their first deri-

vatives on the right hand side of Eq. (52) are not known a priori, further elaborationis necessary, which is done next.

According to Henky’s deformation theory (total deformation theory), the plas-tic strains Epr , E

ph, and Epz are given as [14, 15]

Epr ¼EEQrY

rr �1

2rh

� �; ð53Þ

Eph ¼EEQrY

rh �1

2rr

� �; ð54Þ

Epz ¼ �ðEpr þ EphÞ; ð55Þ

Figure 7 Variation of elastic limit heat load with geometric parameters for �aa ¼ 0:2, �hh0 ¼ 0:2, n ¼ 0:3,�BB ¼ 1:5, and X ¼ 0. Dashed lines show constant property solution.

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where rY is the yield stress defined earlier by Eq. (50) and EEQ is the equivalentplastic strain. Furthermore, using Swift’s expression for nonlinear isotropic strainhardening, the relation between the yield stress rY and the equivalent plastic strainEEQ can be expressed as [16]

rY ¼ r0ð1þ gEEQÞ1=m; ð56Þ

where g is the hardening parameter and m the material parameter. The inverserelation is

EEQ ¼ rYr0

� �m

�1

� �1

g: ð57Þ

A polynomial relationship in expressing the yield stress-equivalent plasticstrain relation instead of Eq. (56) is also possible [17] and can easily be incorporatedinto the present model. However, Swift’s hardening law will be retained here for itsconvenience.

Using Eqs. (50), (57), and (54) one arrives at

dEphdr

¼ rY ðr0r � 2r0hÞ � r0Y ðrr � 2rhÞ2gr2Y

� 1

2gr2Y

rYr0

� �mðm� 1Þr0Y ðrr � 2rhÞ

þrY ðr0r � 2r0hÞ � rYmðrr � 2rhÞr00r0

�; ð58Þ

in which the derivative r0Y may conveniently be expressed as

r0Y ¼ rrrY

r0r �1

2r0h

� �þ rhrY

r0h �1

2r0r

� �; ð59Þ

and by chain rule

dr0dr

¼ dr0dT

dT

dr: ð60Þ

The governing equation for the plastic region, Eq. (52), can now be rewritten interms of Yp and its derivatives by virtue of Eqs. (50), (57), (53), (54), and (58) andof the definitions

rr ¼Yp

hrand r0r ¼ �Yp

hr

1

rþ h0

h

� �þY 0

p

hr; ð61Þ

rh ¼ qx2r2 þ 1

h

dYp

drand r0h ¼ 2qx2r� 1

h2dh

dr

dYp

drþ 1

h

d2Yp

dr2: ð62Þ

Hence, Eq. (52) as well as Eq. (33) can be put into the following general form

d2Yi

dr2¼ fi r;Yi;

dYi

dr

� �: ð63Þ

The numerical solution is obtained by the use of the shooting method.

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SAMPLE CALCULATIONS

The values of the parameters: �aa ¼ 0:2, �hh0 ¼ 0:2, n ¼ 0:3, n ¼ 1, k ¼ 1:5, andB ¼ 1:5 are used throughout the computations and all critical loads reported hererefer to the calculations that consider the temperature dependency of physicalproperties. Constant property solutions accompany for comparison. The nonrotat-ing fin (X ¼ 0) yields at the shaft-fin interface when the heat load reaches�qq ¼ �qqe ¼ 1:12950. The plastic region formed at this surface propagates toward theedge of the fin for the values of the heat load q > qe. Taking hardening parametersm ¼ 0:5 and H ¼ gr0C=E0 ¼ 0:5 and assigning �qq ¼ 2 > �qqe, the distribution of stres-ses, displacement, and plastic strains are computed in the partially plastic fin andplotted in Figure 8. In this figure, dashed lines belong to constant property solution.Plastic strains are normalized using �EEpj ¼ Epj E0=r0C before they have been plotted.The plastic-elastic border radius is determined as �rrep ¼ rep=b ¼ 0:263013. For con-stant property solution it is �rrep ¼ 0:247226. If the heat load is kept constant at�qq ¼ 2 and the angular speed X is increased slowly, the plastic core propagates withincreasing rate toward the edge of the fin until the fin becomes fully plastic atX ¼ Xfp.

Figure 8 Stresses, displacement and plastic strains in a partially plastic fin at �qq ¼ 2 for n ¼ 1, k ¼ 1:5,

X ¼ 0, m ¼ 0:5, and H ¼ 0:5. Dashed lines show constant property solution.

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The propagation of plastic core with increasing values of X for �qq ¼ 2 is shownin Figure 9. As seen in this figure, rep remains almost constant in the range0 < X < 1:6. For X > 1:6, the plastic core advances very rapidly and eventuallyreaches the fully plastic limit. It is also seen in this figure that, for the fin whoseproperties vary with temperature, the plastic core propagates more rapidly incomparison to constant properties. The fully plastic stress state at Xfp ¼ 2:17507for �qq ¼ 2 is plotted in Figure 10. Constant property solution still predicts a par-tially-plastic fin. All stresses are tensile in contrast to nonrotating fin (Figures 6and 8). As the angular speed is increased slowly, a transition for rh from compressivestate to tensile state takes place and as a consequence the plastic core propagatesvery slowly during this period (see Figure 9).

Variation of fully plastic limit angular speed Xfp with geometric parameters nand k are depicted in Figure 11 for �qq ¼ 2. Fully plastic limit angular speeds ofvariable thickness fins are much higher than those of constant thickness fins. Finalparametric analysis is carried out to investigate the effect of hardening parametersH and m. Figure 12 shows variation of Xfp with material parameter m using H asa parameter. The heat load is still �qq ¼ 2. It is noted that the shape of the nonlinear

Figure 9 Propagation of elastic-plastic border radius with increasing angular speeds X for �qq ¼ 2, n ¼ 1,

k ¼ 1:5, m ¼ 0:5, and H ¼ 0:5. Dashed lines show constant property solution.

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stress-strain curve is described by the exponent m in Eq. (56) and its slope by normal-ized hardening parameter H. The curve is concave for m < 1, linear for m ¼ 1 andconvex for m > 1. As seen in Figure 12, the parameters m and H both affect Xfp con-siderably as the temperature dependent physical properties do.

CONCLUDING REMARKS

A realistic computational model to predict the thermoplastic response of a cool-ing fin of variable thickness with or without rotation is outlined. The verification ofthe computational model is done in comparison with the analytical solution employ-ing temperature independent physical properties in the elastic range. Calculationsindicate that limiting loads leading to transition from elastic to plastic and from elas-tic-plastic to fully plastic stress states are much higher for variable thickness fins thanthat of uniform thickness ones (Figures 7 and 11). Higher limits mean lower stresses incomparison to those of constant thickness fin under the same loading conditions.

Convective heat transfer from the peripheral and edge surfaces of the fin istaken into consideration. The heat transfer coefficient is assumed to be a linear

Figure 10 Stresses, displacement and plastic strains in fully plastic fin at X ¼ 2:17507, for �qq ¼ 2, n ¼ 1,

k ¼ 1:5, m ¼ 0:5, and H ¼ 0:5. Dashed lines show constant property solution.

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function of angular speed to account for the effect of rotation. This linear relation-ship enables us to obtain analytical solution of the energy equation. However, thecomputational model can easily handle nonlinear variations of the heat transfercoefficient. The effect of rotation on the thermal response of the fin is assessed(Figures 4(a) and (b), and 5). As the rotation speed increases, temperature-dependentthermal conductivity solutions get closer to those of constant thermal conductivitysolutions indicating the increasing effect of convective heat transfer. Under purelythermal loading of the fin, the circumferential stress component is compressive inthe majority of the fin. A transition from compressive to tensile stress states occursas the thermally loaded fin is started to rotate.

It is well known that physical properties of engineering materials vary consider-ably with temperature (Figure 3). Accordingly, the temperature dependency of themechanical and thermal properties must be taken into account for dependablepredictions. The difference between constant property and variable property calcula-tions can clearly be evaluated in the results of this work (Figures 7–11). Moreover,computations for nonlinearly hardening fins depicted that the stresses, displacements

Figure 11 Variation of fully plastic limit angular speed Xfp with geometric parameters for �qq ¼ 2, m ¼ 0:5,

and H ¼ 0:5. Dashed lines show constant property solution.

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and strains in them are affected considerably by the material and hardening para-meters (Figure 12).

ACKNOWLEDGMENTS

The authors take this opportunity to thank Professor Ismail Tosun in theDepartment of Chemical Engineering at METU for his valuable help and many use-ful discussions on the subject. Ms. Defne Akıncı in the Academic Writing Center atMETU edited the manuscript with care and patience for its language and style. Weare indebted to her.

REFERENCES

1. B. A. Boley and J. H. Weiner, Theory of Thermal Stresses, Wiley, New York, 1960.2. S. Timoshenko and J. N. Goodier, Theory of Elasticity, 3rd ed., McGraw–Hill, New York,

1970.3. D. W. A. Rees, The Mechanics of Solids and Structures, McGraw-Hill, New York, 1990.

Figure 12 Variation of fully plastic limit angular speed Xfp with material parameter m using hardening

parameter H as a parameter for �qq ¼ 2, n ¼ 1, k ¼ 1:5. Dashed lines show constant property solution.

880 A. N. ERASLAN AND M. E. KARTAL

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4. A. C. Ugural and S. K. Fenster, Advanced Strength and Applied Elasticity, 3rd ed.,Prentice-Hall International, London, 1995.

5. A. Alujevic, J. Legat, and J. Zupec, Thermal Yield of a Rotating Hyperbolic Disk, J. Appl.Math. Mech. (ZAMM), vol. 73, pp. T283–T287, 1993.

6. A. Alujevic, P. Les, and J. Zupec, Plasticity of a Thermally Loaded Rotating HyperbolicDisk, J. Appl. Math. Mech. (ZAMM), vol. 73, pp. T287–T290, 1993.

7. C. Parmaksızoglu and U. Guven, Plastic Stress Distribution in a Rotating Disk withRigid Inclusion Under a Radial Temperature Gradient, Mech. Structures Machines, vol.26, pp. 9–20, 1998.

8. A. N. Eraslan and T. Akis, On the Elastic-plastic Deformation of a Rotating Disk Sub-jected to a Radial Temperature Gradient, Mechanics Based Design of Structures andMachines, vol. 31, pp. 529–561, 2003.

9. A. N. Eraslan, Elastoplastic Deformations of Rotating Parabolic Solid Disks UsingTresca’s Yield Criterion, Eur. J. Mech. A=Solids, vol. 22, pp. 861–874, 2003.

10. Y. Orcan and A. N. Eraslan, Thermal Stresses in Elastic–plastic Tubes with TemperatureDependent Mechanical and Thermal Properties, J. Thermal Stresses, vol. 24, pp.1097–1113, 2001.

11. V. S. Arpaci, Conduction Heat Transfer, Addison-Wesley, Reading, Pennsylvania, 1966.12. A. N. Eraslan and M. E. Kartal, A Nonlinear Shooting Method Applied to Solid

Mechanics: Part 1. Numerical Solution of a Plane Stress Model, Inter. J. Nonlinear Anal.Phenomena, vol. 1, pp. 27–40, 2004.

13. M. Abramowitz and A. I. Stegun (eds.), Handbook of Mathematical Functions, U.S.Government Printing Office, Fifth Printing, Washington, DC, 1966.

14. A. Mendelson, Plasticity: Theory and Application, Macmillan, New York, 1968.15. W. F. Chen and D. J. Han, Plasticity for Structural Engineers, Springer, New York, 1988.16. W. Johnson and P. B. Mellor, Engineering Plasticity, Van Nostrand Reinhold, London,

1973.17. L. H. You, S. Y. Long, and J. J. Zhang, Perturbation Solution of Rotating Solid Disks

with Nonlinear Strain-Hardening, Mechanics Res. Comm., vol. 24, pp. 649–658, 1997.18. J. R. Rice, Numerical Methods, Software, and Analysis, 3rd ed., McGraw-Hill, Singapore,

1987.

APPENDIX

The Nonlinear Shooting Method

Details of the method may be found in ref. [12]. A summary is provided here.Consider a two-point boundary value problem in the form

d2Wdr2

¼ G r;W;dWdr

� �; ð64Þ

subject to the boundary conditions of type I:

WðaÞ ¼ c1 and a2dWdx

����x¼b

þ b2WðbÞ ¼ c2; ð65Þ

or of type II:

a1dWdx

����x¼a

þ b1WðaÞ ¼ c1 and WðbÞ ¼ c2; ð66Þ

where ai, bi and ci are constants.

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Letting /1 ¼ W and /2 ¼ W0, Eq. (64) is converted into a system of initial valueproblems (IVP)

d/1

dr¼ /2; ð67Þ

d/2

dr¼ Gðr;/1;/2Þ; ð68Þ

subject to the initial conditions

/01 ¼ Wðr ¼ aÞ and /0

2 ¼ W0ðr ¼ aÞ: ð69Þ

Depending on the boundary conditions (65) or (66), either one of /01 or /0

2 or bothmay be unknowns. These unknowns are computed iteratively by the application ofNewton method.

In the case of BC type I, /1ðaÞ ¼ c1, that is, /01 is known but /0

2 is not, Newtoniterations begin with an initial estimate /0

2 and at the i-th iteration cycle, the IVP sys-tem is solved three times with

I: /02 ¼ /i

2ðaÞ to give f1 ¼ a2/2ðbÞ þ b2/1ðbÞ � c2;

II: /02 ¼ /i

2ðaÞ þ D/ to give f2 ¼ a2/2ðbÞ þ b2/1ðbÞ � c2;

III: /02 ¼ /i

2ðaÞ � D/ to give f3 ¼ a2/2ðbÞ þ b2/1ðbÞ � c2;

in which D/ is a small increment. Two of these solutions involving /i2ðaÞ � D/ are

performed for the purpose of generating tangents numerically. Via central differ-ences this tangent is ðf2 � f3Þ=2D/ and hence, a better approximation for/02 ¼ /2ðaÞ can now be obtained from

/02 ¼ /iþ1

2 ðaÞ ¼ /i2ðaÞ �

ð2D/Þf1f2 � f3

: ð70Þ

Iterations are repeated until j/iþ12 ðaÞ � /i

2ðaÞj<eT , where eT represents the specifiederror tolerance.

On the other hand, in the case of boundary conditions defined by Eq. (66),both initial conditions /0

1 and /02 are unknowns. In this case, /0

1 is estimated, /02

is calculated from /02 ¼ ðc1 � b1/

01Þ=a1 and again at the i–th iteration cycle, the

IVP system is solved three times with

I: /01 ¼ /i

1ðaÞ and /02 ¼ ðc1 � b1/

01Þ=a1 to give f1 ¼ /1ðbÞ � c2;

II: /01 ¼ /i

1ðaÞ þ D/ and /02 ¼ ðc1 � b1/

01Þ=a1 to give f2 ¼ /1ðbÞ � c2;

III: /01 ¼ /i

1ðaÞ � D/ and /02 ¼ ðc1 � b1/

01Þ=a1 to give f3 ¼ /1ðbÞ � c2:

Newton iteration equation similar to Eq. (70) is then used to successively correct /01.

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The initial value system defined by Eqs. (67) and (68) are solved numerically bythe use of Runge-Kutta Fehlberg predictor corrector method [18]. The main advan-tages of this procedure are accuracy provided by higher-order Runge-Kutta methodsand rate of convergence. Only a few iterations are performed to arrive at conver-gence and this rate depends weakly on the initial estimates to start the computations.

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