Exact solution for functionally graded variable-thickness rotating disc with heat source

1

Exact solution for functionally graded variable-thicknessrotating disc with heat sourceM Bayat1,2∗, A H Mohazzab3, B B Sahari1,2, and M Saleem4

1Department of Mechanical Engineering, University of Malaya, Lembah Panti, Kuala Lumpur, Malaysia2Automotive Laboratory, Institute of Advanced Technology, University Putra Malaysia, Selangor, Malaysia3South Tehran Branch, Islamic Azad University, Tehran, Iran4Department of Applied Mathematics, Z.H. College of Engineering and Technology, Aligarh Muslim University, Aligarh,India

The manuscript was received on 27 June 2009 and was accepted after revision for publication on 5 March 2010.

DOI: 10.1243/09544062JMES1812

Abstract: Exact elastic solutions for axisymmetric variable-thickness hollow rotating discs withheat source made of functionally graded (FG) materials under free–free and fixed–free boundaryconditions are presented. Material properties and disc thickness profiles are assumed to be rep-resented by specified power law distributions. The effect of the heat source and the geometry ofthe disc on stress and displacement fields are investigated. It is found that the location of maxi-mum radial stress owing to thermal load does not tend towards the outer surface like radial stressowing to mechanical load for free–free FG discs with an increase in parameter m related to thethickness profile. The temperature distribution in a disc with hyperbolic thickness profile is thesmallest compared with other thickness profiles. The FG disc with hyperbolic convergent thick-ness profile has smaller stresses because of thermal load compared with the disc with uniformthickness profile.

Keywords: functionally graded material, rotating disc, variable thickness, heat source

1 INTRODUCTION

Functionally graded materials (FGMs) are heteroge-neous materials, in which the material propertieschange gradually and continuously from one surfaceto the other as a function of position along certaindimension(s) of the structure [1, 2]. These materialsmainly constructed to operate in high-temperatureenvironments have high fracture toughness. Heatsource and related thermal stresses occur in thesematerials. One of the applications of discs is in therotor of electrical motors, and the shape of the rotoris cylindrical or the same as disc. The effect of heatsource in FGM disc of rotor can also be found inmany advanced devices such as turbine, centrifuge,flywheels, and gears, just to mention a few.

∗Corresponding author: Automotive Laboratory, Institute of

Advanced Technology, University Putra Malaysia, Serdang,

Selangor, Darul 43400UPM, Malaysia.

email: [email protected]

A number of investigations dealing with thermalstresses and deformation in circular structures suchas cylinders and rotating discs have been reportedin the scientific literature. Praveen et al. [3] usedthe finite-element formulation of axisymmetric heattransfer equation for the thermoelastic analysis of anFG ceramic–metal cylinder. Zimmerman and Lutz [4]presented solutions for thermal stresses in an FG circu-lar cylinder with uniform heating using the Frobeniusseries method and assuming temperature and stressesas linear functions of radius. Obata et al. [5] pre-sented thermal stresses in FG sphere and cylinderusing perturbation theory. They used the separationof variables’ and Laplace-transform methods to findtwo thermoelastic displacement potential functions.Tarn [6] presented exact solutions for FG cylindersunder extension, torsion, shearing, pressuring, andtemperature changes, assuming material properties tobe cylindrically anisotropic. Ruhi et al. [7] gave semi-analytical thermoelastic solutions for thick-walledfinitely long cylinders made of FGMs. Jabbari et al. [8,9] presented an analytical solution for thermome-chanical response in an FG cylinder under radially

JMES1812 Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science

2 M Bayat, A H Mohazzab, B B Sahari, and M Saleem

symmetric and non-axisymmetric steady-state loads.Based on the method reported in reference [8], Eslamiet al. [10] and Jabbari et al. [11] presented the ther-momechanical response for an FGM sphere and athick long FGM cylinder, respectively, under radiallysymmetric loads. Jabbari et al. [12] presented the the-oretical analysis of three-dimensional mechanical andthermal stresses for a short FG hollow cylinder understeady-state temperature.

Many earlier studies on rotating discs ([13] andreferences therein) have considered discs with uni-form thickness. In recent years, FG rotating discs havebeen considered. Durodola and Attia [14, 15] provideda finite-element analysis for FG rotating discs withconstant thickness. Kordkheili and Naghdabadi [16]presented a semi-analytical thermoelastic solution forFG discs with constant thickness under plane stressconditions. Bayat et al.[17–19] applied series solutionand semi-analytical methods to study the thermome-chanical response in FG rotating discs using sheardeformation theory.

Several authors ([20, 21] and references therein)have emphasized the importance of variable thicknessin rotating discs. They showed that the mechanicalresponses in rotating discs with variable thicknesswere much lower than those with uniform thicknessat the same angular velocity. Bayat et al. [22–24] stud-ied the thermomechanical response in FG rotatingdiscs with variable thickness under symmetry andsteady-state thermal loads. They used semi-analyticalmethod. Bayat et al. [25] obtained more realistic ther-momechanical results in FG rotating discs with vari-able thickness by considering temperature-dependentmaterial properties under symmetry and steady-stateconditions.

The motivation for the present paper stems out ofthe fact that no work exists in the literature till datethat concerns the thermoelastic analysis of a variablethickness FG disc with heat source. The present studyconsiders an FG disc of variable thickness (Fig. 1) sub-jected to centrifugal force and thermal loading owing

Fig. 1 Cross-section of a hollow disc with variable thick-ness

to heat source. The loading and geometry of the discare assumed to be axisymmetric and independent ofthe out-of-plane coordinate, and the plane stress con-ditions are assumed. The symmetry with respect tothe axis and the mid-plane is also assumed. To bemore specific, this work aims to present exact solutionsfor the non-dimensional temperature distribution andthe displacement field and to investigate the effect ofsome basic factors such as property gradation and thegeometry of the hollow disc on stress and displace-ment fields under free–free and fixed–free boundaryconditions.

2 GRADATION RELATION

In this study, the property variation P of the materialin the FG disc along the radial direction is assumed tobe of the following form [23]

P(r) = Po

(rro

)�

(1a)

Here Po is the property of the material at the outerradius of the disc (i.e. at r = ro, and � is a parameterwhose value depends on the material and geometricproperties of the disc but not on its radius). In thisstudy, the Poisson’s ratio ν is assumed to be constant,and other material properties such as the thermalconductivity, the elastic modulus, and the density areassumed to vary according to the gradation relation(1a), for example, the assumed form for the thermalconductivity (k) is

k(r) = ko

(rro

)γ

(1b)

The value of the parameter γ depending on thematerial and geometric properties of the disc is speci-fied in the following sections.

The thickness profile (h) of the disc is assumed tovary radially according to the following form [23]

h(r) = ho

(rro

)−m

(2)

Here m is a geometric parameter that (also referredto in the literature as the order of the hyperbola) canbe positive or negative. In equation (2), the thicknessof the disc is characterized by m. The profile is diver-gent when m < 0 and convergent when m > 0. Theconstant thickness can be obtained by setting m = 0.

3 THERMOELASTIC EQUATION

Consider a hollow axisymmetric FG disc with variablethickness with inner radius ri and outer radius ro, asshown in Fig. 1. The disc rotates at an angular velocity

Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science JMES1812

Exact solution for FG variable-thickness rotating disc 3

ω. As, the field variables, the loading and geome-try are independent of the out-of-plane coordinate,the plane stress conditions are assumed. Owing toaxial symmetry assumptions in geometry and loading,the cylindrical coordinate system (r, θ , z) is used. Theinner and outer surfaces of the FG disc are assumedto be metal- and ceramic-rich, respectively. Betweenthese two surfaces, material properties vary accordingto equation (1a).

It may be mentioned that the assumption for ametal-rich inner surface of the disc can be well jus-tified for situations considered in this study, in whichthe FG disc is mounted on the shaft support and duc-tility plays an important role. It is worth mentioningthat the fixed–free and free–free boundary conditionsof this study correspond, respectively, to situations inwhich the disc is connected to shaft rigidly or by meansof splines where small axial movement is allowed.

Using the following strain–displacement and stress–strain relations

εr = dudr

, εθ = ur

(3)

σr = E(r)

1 − ν2[εr + νεθ − (1 + ν)α(r)T (r)] (4)

σθ = E(r)

1 − ν2[εθ + νεr − (1 + ν)α(r)T (r)] (5)

from the infinitesimal theory of elasticity and the rota-tional symmetry into the equation of motion givenin reference [26], the Navier equation for the radialdisplacement can be obtained as

rhr Erd2udr2

+(

rErdhr

dr+ rhr

dEr

dr+ Er hr

)dudr

+(

νErdhr

dr+ νhr

dEr

dr− 1

rEr hr

)u

+ (1 − ν2)hrρr r2ω2 − r(1 + ν)d(hr ErαrTr)

dr= 0

(6)

Here for brevity, symbols hr , Er , ρr , αr , and Tr

have been used for the functions h(r), E(r), ρ(r), α(r),and T (r), respectively. In equation (6), the displace-ment u is a function of r only due to axial symmetryand plane stress conditions. Details of derivation ofequation (6) can also be seen in Bayat et al. [24].

4 EXACT SOLUTION FOR DISPLACEMENT FIELDDUE TO MECHANICAL LOAD: A CASE WITH NOHEAT SOURCE

This case has been studied in Bayat et al. [22], in whichequation (6) minus the last term has been considered

in its non-dimensional form as

RHRERd2UdR2

+(

RERdHR

dR+ RHR

dER

dR+ ERHR

)dUdR

+(

νERdHR

dR+ νHR

dER

dR− 1

RERHR

)U

+ (1 − ν2)HRρRR2 = 0 (7)

where different non-dimensional variables have beendefined as

R = rro

, HR = hr

ho, U = u

uo, ER = Er

Eo, ρR = ρr

ρo

(8a)

with

uo = ρor3oω2

Eo(8b)

It has been shown in Bayat et al. [22] that for specificforms of material properties and thickness profile ofthe disc given according to equations (1) and (2) as

ER = Rη, ρR = Rβ , HR = R−m (9)

equation (7) has an exact solution as under

U (R) = D1Rs1 + D2Rs2 + Up (10)

where s1 and s2 are the two real roots of the quadratic

s2 + (η − m)s + (νη − νm − 1) = 0 (11)

and Up is the particular solution of equation (7) suchthat

Up = −(1 − ν2)R3+β−η

(3 + β − η)(2 + β − η) + (3 + β − η)

(1 + η − m) + (νη − νm − 1)

(12)

The three parameters η, β, and m in relation (9) andconsequently in the solution (10) can be determinedfrom the material properties and the geometry of thedisc as follows

η = ln ERi

ln Ri, β = ln ρRi

ln Ri, m = − ln HRi

ln Ri(13)

The arbitrary constants D1 and D2 in equation (10)can be determined by using the boundary conditions.

Remark 1

Equation (4.53) of reference [27] is a special case ofequation (7) of this paper for homogeneous materials.Consequently, results of equation (7) can be comparedwith those reported in reference [27] for pure materi-als. Such a comparison will be a part of the case-studyresults reported in section 8.

JMES1812 Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science

4 M Bayat, A H Mohazzab, B B Sahari, and M Saleem

5 EXACT SOLUTION FOR THE THERMAL LOAD

The heat conduction equation

ddr

[rhr kr

dT (r)

dr

]= −q(r) (14)

with heat source represented by q(r) and symbols hr,kr, and Tr used for the functions h(r), k(r), and T (r),respectively, can be written in the non-dimensionalform as

ddR

(RHRKR

dTR

dR

)= −QR (15)

where different non-dimensional variables have beendefined as

KR = kr

ko, TR = Tr

T ∗o

, QR = q(r)

qo(16a)

with

T ∗o = qoro

hoko(16b)

It has been shown [24] that for the particular formsfor the thermal conductivity and the thickness profileof the disc according to relations (1) and (2) given as

KR = Rγ , HR = R−m (17a)

with parameter γ defined as

γ = ln KRi

ln Ri(17b)

equation (15) can be solved to give exact solution as

TR = −∫

F (R)

Rγ+1−mdR + A1

1(m − γ )

Rm−γ + A2 (18)

where

F (R) =∫

QR dR

and A1 and A2 are arbitrary constants that can bedetermined by using boundary conditions.

6 BOUNDARY CONDITIONS

6.1 Thermal boundary conditions and heat sourcefunction

The mixed boundary conditions are [12, 28, 29]

C11TRi + C12d(TRi)

dr= f1, at R = Ri = ri

ro

C21TRo + C22d(TRo)

dr= f2, at R = Ro = ro

ro= 1

(19)

By choosing suitable values for the parametersCij (i, j = 1, 2), different types of the thermal boundary

conditions including prescribed temperature differ-ence, heat flux, and convection may be considered forthe disc. Functions f1 and f2 are given at the inside andoutside surfaces of the disc, respectively.

The disc is heated by the rate of energy generationper unit time per unit length with q(r) given as

q(r) = 100(ri − ro)2

[−r2 + (ro + ri)r − riro]

×Amean︷ ︸︸ ︷

2π(ro + ri)

2× hr

∣∣r=(ro+ri)/2

(20)

The assumed energy generation provides zero ratesat the inside and outside surfaces and 25(w/m) atmid-radius. Using equation (20), the non-dimensionalform of the heat source function to be considered inthis paper becomes

Q(R) = q(r)

qo= q(r)

qo = 100π (ro + ri) × hr

∣∣r=(ro+ri)/2

= − 1(1 − Ri)2

[R2 − (1 + Ri)R + Ri

](21)

6.2 Mechanical boundary conditions

Hollow disc free–free. The following traction conditionson the inner and outer surfaces of the rotating hollowdisc must be satisfied

σR = ER

1 − ν2

(dUdR

+ νUR

)= 0, at R = Ri = ri

ro

σR = 0, at R = Ro = ro

ro= 1

(22)

Hollow disc fixed–free. The following conditions mustbe satisfied

U = 0, at R = Ri

σR = 0, at R = Ro(23)

7 EXACT SOLUTION FOR DISPLACEMENT FIELDDUE TO THERMAL LOAD: A CASE WITH NOMECHANICAL LOADING

Navier equation (6) for thermal load can be writ-ten in the non-dimensionalized form by using thenon-dimensional variables of equation (8) along withadditional variables as given below

UT = uu∗

o

, α = αr

αo(24)

where

u∗o = roαoT ∗

o

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Exact solution for FG variable-thickness rotating disc 5

The non-dimensional form of equation (6) is thengiven by

RHRERd2UT

dR2+

(RER

dHR

dR+ RHR

dER

dR+ ERHR

)dUT

dR

+(

νERdHR

dR+ · · · νHR

dER

dR− 1

RERHR

)UT

= R(1 + ν)d(ERHRαTR)

dR(25)

For the dimensionless thermal expansion property,α is assumed to vary according to equation (1a) as

α = Rξ , where ξ = ln αRi

ln Ri(26)

It can be seen that the exact solution of the elasticequation (25) is possible.

Using relations (9) and (26) in equation (25), itreduces to an Euler equation as follows

R2 d2UT

dR2+ R(1 + η − m)

dUT

dR+ (ην − mν − 1)UT

= (1 + ν)

[(η − m + ξ)Rξ+1TR + Rξ+2 d(TR)

dR

]

(27)

Using equation (21) for the heat source functionQ(R), equation (18) yields

TR = R3+m−γ

3(1 − Ri)2(3 + m − γ )

− (1 + Ri)R2+m−γ

2(1 − Ri)2(2 + m − γ )

+ Ri

(1 − Ri)2(1 + m − γ )R1+m−γ

+ A11

m − γRm−γ + A2 (28)

Supposing that there is no convection at thetop and bottom surfaces of the disc, the tempera-ture difference, TR, between the side surfaces andthe environment of the disc are assumed to be[27, 30, 31]

TR = TRi , at R = Ri

TR = TRo , at R = Ro = 1(29a)

Remark 2

A special case of equation (15) has been consideredin reference [27]. More specifically, it can be shownthat under boundary conditions (29a), the tempera-ture rise at any distance R from the centre in a constant

thickness disc (m = 0) without heat source (QR = 0)

made of homogeneous materials and having constantthermal expansion is the same as that reported in ref-erence [27]. Interested readers may refer to Bayat et al.[22, 23] for further details.

For specific thermal boundary conditions, it isassumed that the temperature differences at inner andouter surfaces are [30, 31]

TRi = 0, at R = Ri

TRo = 0, at R = Ro

(29b)

In order to show the effect of heat source only onthe behaviour of the FG structure (like [30, 31]), it isassumed that the heat source placed at the mid-radiusof the disc does not affect the thermal conditions atthe inner and outer radii of the disc. It is worth men-tioning that the method of this study is general innature and can be used for other boundary conditionsalso.

It may be noted that A1 and A2 in equation (28) canbe obtained by using boundary conditions (29b) as

A1 = (xi − xo)(m − γ )

1 − Rm−γ

i

, A2 = xoRm−γ

i − xi

1 − Rm−γ

i

(30)

where

xi = 1(1 − Ri)2

[− (5 + m − γ )

6(3 + m − γ )(2 + m − γ )R3+m−γ

i

+ (3 + m − γ )

2(2 + m − γ )(1 + m − γ )R2+m−γ

i

](31a)

and

xo = 1(1 − Ri)2

[− (5 + m − γ )

6(3 + m − γ )(2 + m − γ )

+ (3 + m − γ )

2(2 + m − γ )(1 + m − γ )Ri

](31b)

Substituting for TR from equation (28) intoequation (27), it becomes

R2 d2UT

dR2+ R(1 + η − m)

dUT

dR+ (ην − mν − 1)UT

= (1 + ν)(3 + η + ξ − γ )

3(1 − Ri)2(3 + m − γ )R4+m+ξ−γ

− (1 + ν)(1 + Ri)(2 + η + ξ − γ )

2(1 − Ri)2(2 + m − γ )R3+m+ξ−γ

+ (1 + ν)Ri(1 + η + ξ − γ )

(1 − Ri)2(1 + m − γ )R2+m+ξ−γ

JMES1812 Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science

6 M Bayat, A H Mohazzab, B B Sahari, and M Saleem

+ (1 + ν)(η + ξ − γ )

m − γA1R1+m+ξ−γ

+ (1 + ν)(η − m + ξ)A2R1+ξ (32)

where A1 and A2 are the same as given in equation (30).The general solution of equation (32) can be writ-

ten as

UT(R) = B1Rs1 + B2Rs2 + UT|P (33a)

where

UT|P = UT|p

)1+ UT|p

)2+ UT|p

)3

+ UT|p

)4+ UT|p

)5

(33b)

In equation (33a), B1 and B2 are the two arbitraryconstants; s1 and s2 are the two real roots of thequadratic equation (11); and UT|P is the particularsolution of equation (32) such that

UT|p

)1

=[(1 + ν)(3 + η + ξ − γ )/3(1 − Ri)

2(3 + m − γ )]

R4+m+ξ−γ[(4 + m + ξ − γ )(3 + m + ξ − γ ) + (4 + m + ξ − γ )(1 + η − m) + (ην − mν − 1)

] (34a)

UT|p

)2

= −[(1 + ν)(1 + Ri)(2 + η + ξ − γ )/2(1 − Ri)2(2 + m − γ )]R3+m+ξ−γ[

(3 + m + ξ − γ )(2 + m + ξ − γ ) + (3 + m + ξ − γ )(1 + η − m) + (ην − mν − 1)] (34b)

UT|p

)3

= [(1 + ν)Ri(1 + η + ξ − γ )/(1 − Ri)2(1 + m − γ )]R2+m+ξ−γ[

(2 + m + ξ − γ )(1 + m + ξ − γ ) + (2 + m + ξ − γ )(1 + η − m) + (ην − mν − 1)] (34c)

UT|p

)4

= [(1 + ν)(η + ξ − γ )/m − γ ]A1R1+m+ξ−γ[(1 + m + ξ − γ )(m + ξ − γ ) + (1 + m + ξ − γ )(1 + η − m) + (ην − mν − 1)

] (34d)

UT|p

)5

= (1 + ν)(η − m + ξ)A2R1+ξ

[(1 + ξ)ξ + (1 + ξ)(1 + η − m) + (ην − mν − 1)](34e)

8 NUMERICAL RESULTS

For numerical illustration of the elastic solutions ofthis study, it is assumed that all the discs consideredhave the same volume. This can be achieved by suit-ably choosing the value of ho. It can be noted that theresults obtained in this study are based on the non-dimensional formulation and thus are independentof the absolute value of ho. Two cases considered are,namely, hollow disc free–free and hollow disc fixed–free. The analysis is conducted using aluminium asthe inner surface metal and ceramic as the outer sur-face material, same as considered in reference [24].The material properties are

Ei = EAl = 70.0 GPa, Eo = ECer = 151.0 GPa

ρi = ρAl = 2700.0 kg/m3, ρo = ρCer = 5700.0 kg/m3

αi = αAl = 23.0 × 10−6/◦C,

αo = αCer = 10.0 × 10−6/◦C

Ki = KAl = 209.0W/m ◦C, Ko = KCer = 2.0W/m ◦C

ν = 0.3(35)

A hollow disc with Ro = 5Ri or a solid disc sub-jected to uniform centrifugal force is considered. Allother parameters such as η, m, β, γ and ζ can foundby using equations (9), (17), (26), and (35). Arbitraryconstants A1, A2, B1, B2, D1, and D2 can be obtained byapplying boundary conditions. Different cases for the

thickness profiles used in these illustrations obtainedfrom equation (2) are listed in Table 1.

It can be mentioned that the method of solutionof this study is general in nature, but the numericalresults reported in the following are dependent on thematerial properties and the geometry of the disc.

8.1 Results for thermomechanical loading

Figure 2 presents the non-dimensional radial stressesowing to heat source and centrifugal load for differentvalues of the geometric parameter m.

It can be seen from Fig. 2 that when R < 0.5, the non-dimensional radial stress takes the maximum valuefor hyperbolic divergent thickness profile and theminimum for hyperbolic convergent thickness pro-file, and for FG disc with uniform thickness, radialstress occurs in-between. It is seen that the FG discwith uniform thickness takes the maximum thermo-mechanical stress when 0.65 < R < 0.75. It is observedthat the FG disc with hyperbolic convergent thicknessprofile has smaller stress in most parts of the disc. Itis also observed that close to the inner surface radial

Table 1 Different cases of thickness profiles

Equation (2) m < 0 m = 0 m > 0 m = −1

Case (a):hyperbolicdivergent

Case (b):constantthickness

Case (c):hyperbolicconvergent

Case (d):linear

Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science JMES1812

Exact solution for FG variable-thickness rotating disc 7

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

R

Non

-dim

ensi

onal

The

rmom

echa

nica

l Rad

ial S

tres

s

m = - 2.0

m = - 1.0

m = 0.0

m = 1.0

m = 2.0

Fig. 2 Non-dimensional radius stress σR in the hollow disc (free–free) with variable thickness dueto thermomechanical load for different values of the geometric parameter m

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

R

No

n-d

ime

nsi

ona

l Th

erm

om

ech

an

ica

l Ho

op

Str

ess

Pure Material, m = - 2.0

FG, m = - 2.0

FG, m = - 1.0

Pure Material, m = 0.0

FG, m = 0.0

FG, m = 1.0

Pure Material, m = 2.0

FG, m = 2.0

Fig. 3 Non-dimensional circumferential stress σθ in the hollow disc (free–free) with variablethickness due to thermomechanical load for different values of the geometric parameter m

stresses decrease with an increase in the geometricparameter m.

Figure 3 demonstrates the non-dimensional hoopstresses due to heat source and centrifugal load fordifferent values of the geometric parameter m.

It is seen that thermomechanical hoop stressdecreases by increasing the value of m. The effect ofmaterial property on the hoop stress is shown by fixingthe value of the geometric parameter m = −2.0 (say)and by considering results for pure material and FGdisc, as given in Fig. 3. For discs with the same thick-ness profiles, it is seen that close to the inner surface,the FG discs have smaller hoop stresses comparedwith those with homogeneous material, but towardsthe outer surface, the stresses for FG discs can belarger than the circumferential stress for pure materialdiscs.

It can be seen that the graphs obtained by con-sidering only mechanical loading (Figs 4 and 5) andthe those with thermomechanical loading (Figs 2

and 3) are very similar. It is indeed not surprisingbecause when one considers the combined loadingof centrifugal force and thermal effects, it is the cen-trifugal force that dominates and suppresses the effectof thermal loading by superposition law [25]. Similarconclusion can be drawn by comparing the scales ofmechanical radial and hoop stresses in Figs 4 and 5and thermal radial and hoop stresses in Figs 11 and 12. Q1Thus, for a purpose of showing the application of thepresent exact solution for analysing an FG rotatingdisc, two cases of loading may be considered: one withuniform centrifugal force and the other with thermalloading due to heat source.

8.2 Results for body force

8.2.1 Hollow disc (free–free)

Figures 4 and 5 illustrate the non-dimensional radialand circumferential stresses, respectively, for different

JMES1812 Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science

8 M Bayat, A H Mohazzab, B B Sahari, and M Saleem

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

R

Non

-dim

ensi

onal

Mec

hani

cal R

adia

l Str

ess

X: 0.45Y: 0.264

Pure Material, m = - 2.0

Pure Material, m = 0.0

FG, m = - 2.0

FG, m = - 1.0

FG, m = 0.0

FG, m = 1.0

Pure Material, m = 2.0

FG, m = 2.0

Fig. 4 Non-dimensional radius stress σR in the hollow disc (free–free) with variable thickness dueto centrifugal load for different values of the geometric parameter m

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

R

Non

-dim

ensi

on

al M

ech

an

ica

l Circ

um

fere

ntia

l Str

ess

Pure Material, m = - 2.0

FG, m = - 2.0

FG, m = - 1.0

Pure Material, m = 0.0

FG, m = 0.0

FG, m = 1.0

Pure Material, m = 2.0

FG, m = 2.0

Fig. 5 Non-dimensional circumferential stress σθ in the hollow disc (free–free) with variablethickness due to centrifugal load for different values of the geometric parameter m

values of the geometric parameter m in the FG rotatingdisc due to body force.

As expected, FG discs with hyperbolic conver-gent thickness profile have smaller maximum radialstresses compared with those with hyperbolic diver-gent thickness profile owing to body force. It can beobserved that the maximum of the radial stress movesfrom the inner surface to the outer surface when mis increased. It is seen that the radial stress for theFG disc with hyperbolic convergent thickness profileis smaller than that of the corresponding case with.uniform thickness. It may be mentioned here thatQ2for the same thickness profile, the mechanical radialstresses for the FG disc are smaller than those for purematerial disc, except the region close to the outer sur-face Calculating the maximum for radial stress givenby equation (4.42a) of reference [27], it turns out thatit occurs at

√RiRo. The same value

√RiRo = 0.447 is

shown in Fig. 4 for maximum radial stresses in purematerial disc with uniform thickness (m = 0).

The circumferential stress is shown in Fig. 5. Thecircumferential stress for the FG disc with hyper-bolic convergent profile turns out to be the smallestin comparison with the other thickness profiles (i.e.linear or hyperbolic divergent). It is noticed that forsome specific values of the geometric parameter m > 0(m = 1.0), the circumferential stresses may have alocal maximum close to the outer surface. Also forthe specific value of the geometric parameter m = 2.0,the maximum of the circumferential stresses may notbe at the inner surface. The hoop stresses in the purematerial disc with constant thickness have the samevalues as those obtained from non-dimensional formof equation (4.42b) of reference [27]

σθ

(ρω2R2o)/E

= 3 + ν

8

⎛⎜⎜⎜⎜⎝

r2o

r2o

+

R2i︷︸︸︷

r2i

r2o

+R2i

R2− 1 + 3ν

3 + νR2

⎞⎟⎟⎟⎟⎠

Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science JMES1812

Exact solution for FG variable-thickness rotating disc 9

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

R

Non

-dim

ensi

ona

l Ra

dia

l Dis

pla

cem

en

t

m = - 2.0m = - 1.0m = 0.0m = 1.0m = 2.0

Fig. 6 Non-dimensional radial displacement U in the hollow disc (free–free) with variablethickness due to body force for different values of the geometric parameter m

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

R

No

n-d

imen

sio

na

l Ra

dia

l Str

ess

m = - 2.0

m = - 1.0

m = 0.0

m = 1.0

m = 2.0

Fig. 7 Non-dimensional radius stress σR in the hollow disc (fixed–free) with variable thickness dueto body force for different values of the geometric parameter m

It can be noted that for the same thickness profile,the hoop stresses for the FG disc are smaller than thosefor the pure material disc, except the area close to theouter surface.

Figure 6 shows the non-dimensional radial displace-ment in the FG disc along its radius for differentvalues of the geometric parameter m. It can be seenthat for some specific values of geometric parameter(m = −2.0), the highest radial displacement can be atthe inner surface, whereas for discs with constant orconvergent thicknesses, these values may occur at theouter surface.

8.2.2 Hollow disc (fixed–free)

The stress distributions owing to centrifugal force forFG disc with variable thicknesses mounted on a rigidshaft for different values of the geometric parameterm are presented in Figs 7 and 8.

It is seen that for the cases with hyperbolicdivergent and constant thickness profiles considered

in Fig. 7, the maximum radial stresses with fixed–freeconditions occur at the inner surface and they aregreater than their corresponding values with free–freeconditions given in Fig. 4. Comparing different thick-ness profiles, the disc with the hyperbolic convergent(i.e. m = 2.0) profile is found to have its maximumstress smallest. It can also be seen that mountedFG discs with hyperbolic convergent thickness havemaximum radial stresses close to R = 0.6. It is notedthat the shape of the radial stress changes from con-cave to convex by increasing the value of geometricparameter m.

The circumferential stress is shown in Fig. 8. It is seenthat for mounted FG discs with hyperbolic convergentthickness profile, the stress is smaller than those withother thickness profiles (i.e. constant or hyperbolicdivergent). For FG discs, the circumferential stressesdo not have maximum value at the inner surface, andthe position of this point shifts towards the outer sur-face by increasing the value of the geometric parame-ter. Figure 8 suggests that the circumferential stresses

JMES1812 Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science

10 M Bayat, A H Mohazzab, B B Sahari, and M Saleem

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

R

Non

-dim

ensi

on

al C

ircu

mfe

ren

tial S

tre

ss

m = - 2.0m = - 1.0m = 0.0m = 1.0m = 2.0

Fig. 8 Non-dimensional circumferential stress σθ in the hollow disc (fixed–free) with variablethickness due to body force for different values of the geometric parameter m

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

R

Non

-dim

ensi

onal

Rad

ial D

ispl

acem

ent

m = - 2.0

m = - 1.0

m = 0.0

m = 1.0

m = 2.0

Fig. 9 Non-dimensional radial displacement U in the hollow disc (fixed–free) with variablethickness due to body force for different values of the geometric parameter m

for mounted FG discs with hyperbolic convergentthickness profile are smaller compared with those withuniform thickness.

The radial displacement is shown in Fig. 9. Itis evident from Fig. 9 that the maximum radialdisplacement is at the outer surface. Here again,the FG disc with the hyperbolic convergent thick-ness profile has smaller radial displacement com-pared with linear, constant, or hyperbolic divergentprofiles. It can be mentioned that the differencebetween maximum radial displacements increaseswith a decrease in the geometric parameter m. It can benoted that the value of maximum radial displacementincreases with a decrease in m.

8.3 Results for thermal loading

8.3.1 Temperature variation

It is assumed that the temperature varies only alongthe radius owing to the heat source at mean radius

Rm = 0.6[(Ri + Ro)/2 = 0.6]. If the inner surface of thedisc made of metal and the outer surface of the discmade of ceramic are assumed to remain at 0 ◦C, thenon-dimensional temperature change TR along theradius for different values of geometric parameter mcan be obtained as shown in Fig. 10.

It can be seen from this figure that the maximumtemperature may not be at the mean radius (Rm = 0.6)

(i.e. the position of the heat source). The maximumvalues of the temperature tend towards the outer sur-face, with an increase in the geometric parameter m.At a given point along the radius, the temperature inthe FG hyperbolic convergent disc is lower than thatin a constant or hyperbolic divergent disc.

8.3.2 Hollow disc (free–free)

Figures 11 and 12 illustrate the non-dimensionalradial and circumferential stresses in the FG rotatingdisc (free–free), respectively, owing to heat source fordifferent values of the geometric parameter m.

Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science JMES1812

Exact solution for FG variable-thickness rotating disc 11

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

R

No

n-d

ime

nsi

on

al T

em

pe

ratu

re D

istr

ibu

tion

m = - 2.0m = - 1.0m = 0.0m = 1.0m = 2.0

Fig. 10 Non-dimensional temperature change along the radial direction of the disc for given heatsource and the zero value of temperatures in the inner and outer surfaces

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1x 10

-3

R

Non

-dim

ensi

onal

The

rmal

Rad

ial S

tres

s

m = - 2.0

m = - 1.0

m = 0.0

m = 1.0

m = 2.0

Fig. 11 Non-dimensional thermal radial stress σR in the hollow disc (free–free) with variablethickness due to heat source for different values of the geometric parameter m

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.01

-0.005

0

0.005

0.01

R

Non

-dim

ensi

ona

l The

rma

l Circ

umfe

rent

ial S

tre

ss

m = - 2.0m = - 1.0m = 0.0m = 1.0m = 2.0

Fig. 12 Non-dimensional circumferential stress σθ due to thermal load in the hollow disc(free–free) with variable thickness for different values of the geometric parameter m

JMES1812 Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science

12 M Bayat, A H Mohazzab, B B Sahari, and M Saleem

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.002

0.004

0.006

0.008

0.01

0.012

R

Non

-dim

ensi

onal

The

rmal

Rad

ial D

ispl

acem

ent

m = - 2.0

m = - 1.0

m = 0.0

m = 1.0

m = 2.0

Fig. 13 Non-dimensional radial displacement U due to thermal load in the hollow disc (free–free)with variable thickness for different values of the geometric parameter m

It is seen that the radial stress in the FG disc owingto thermal load may have local maximum close to theinner surface as increases m up to a certain value. Theabsolute value of the maximum thermal radial stressdecreases with an increase in m. It can also be seenthat the radial stress changes from compressive to ten-sile stress by increasing the value of the geometricparameter m, and this change starts from the innersurface.

The results of Fig. 11 can be summarized to concludethat the hollow FG disc with hyperbolic convergentthickness profile and having heat source is better thanthose with other thickness profiles in terms of lowerthermal stress. This result is similar to the one reportedin reference [24].

Figure 12 shows the non-dimensional thermal cir-cumferential stresses in an FG disc with heat sourcefor different values of m. It is seen that the maximumstresses occur at the outer surface and they are tensilesimilar to the stresses at the inner surface, whereasin-between inner and outer surfaces at a certain inter-val, the stresses may be compressive. It is noticed thatthe compressive stresses decrease with an increasein m.

The variation of non-dimensional radial displace-ment with radius for different values of m is shownin Fig. 13. It is seen that the radial displacement forthe FG disc with hyperbolic convergent thickness pro-file is the smallest in comparison with other thicknessprofiles (i.e. constant, linear, or hyperbolic divergent).Comparing with the results of Fig. 6 for the same valueof m, it can be mentioned that the thermal radialdisplacement is smaller than the mechanical radialdisplacement. As expected, the radial displacementvalues close to the outer surface are greater than thoseclose to the inner surface disc because of higher ther-mal conductivity and thermal expansion at the innersurface. It is noticed that the radial displacementsfor some specific values of m = −2.0 in hyperbolic

divergent profile discs close to the inner surface canbe even smaller than radial displacements in discs withuniform thickness.

8.3.3 Hollow disc (fixed–free)

The stress distributions for the FG disc with variablethickness mounted on a rigid shaft for different valuesof the geometric parameter m are presented in Figs 14and 15.

It is seen that the absolute value of maximum radialstress for a specific value of m = −2.0 may not occur atthe inner surface, like in the disc with uniform thick-ness. Comparing uniform, hyperbolic divergent, andconvergent thickness profiles, a hyperbolic conver-gent disc is found to have smaller absolute radial stress.It can also be seen that the shape of the radial stresschanges from concave to convex with an increase inm. It is noted that it is only tensile stress for hyper-bolic convergent thickness profile discs, whereas forother thickness profile discs, it can be either tensile orcompressive.

Figure 15 shows the non-dimensional thermal cir-cumferential stresses in a mounted FG disc with heatsource. It is seen that the maximum stresses occur atthe outer surface and they are tensile in nature like thestresses at the inner surface, whereas in-between innerand outer surfaces, there can be compressive stress. Itis noted that the behaviour of thermal circumferentialstresses in FG discs under fixed–free condition is thesame as that under free–free condition, but in the latercase, stresses are greater.

Figure 16 shows the non-dimensional radial dis-placement values along the radius of a mounted discwith variable thickness for different values of thegeometric parameter m.

As expected, the maximum radial displacement forthe mounted FG disc with variable thickness is at theouter surface. Comparing Figs 13 and 16, it is clear

Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science JMES1812

Exact solution for FG variable-thickness rotating disc 13

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-3

-2

-1

0

1

2

3

4

5x 10

-3

R

Non

-dim

ensi

onal

The

rmal

Rad

ial S

tres

s m = - 2.0

m = - 1.0

m = 0.0

m = 1.0

m = 2.0

Fig. 14 Non-dimensional thermal radial stress σR in the hollow disc (fixed–free) with variablethickness for different values of the geometric parameter m

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.01

-0.005

0

0.005

0.01

R

Non

-dim

ensi

onal

The

rmal

Circ

umfe

rent

ial S

tres

s m = - 2.0

m = - 1.0

m = 0.0

m = 1.0

m = 2.0

Fig. 15 Non-dimensional circumferential stress σθ due to thermal load in the hollow disc(fixed–free) with variable thickness for different values of the geometric parameter m

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.002

0.004

0.006

0.008

0.01

0.012

R

Non

-dim

ensi

onal

The

rmal

Rad

ial D

ispl

acem

ent

m = - 2.0

m = - 1.0

m = 0.0

m = 1.0

m = 2.0

Fig. 16 Non-dimensional radial displacement U due to thermal load in the hollow disc (fixed–free)with variable thickness for different values of the geometric parameter m

JMES1812 Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science

14 M Bayat, A H Mohazzab, B B Sahari, and M Saleem

that the general observations for the thermal radialdisplacements under free–free conditions (Fig. 13) arealso valid for mounted discs (Fig. 16), except close tothe inner surface where the behaviour is different andit is related to boundary conditions.

9 CONCLUSION

Exact elastic solutions for axisymmetric variable-thickness rotating discs with heat source made ofFGM are presented. Three forms of thickness profiles,namely, uniform, hyperbolic convergent, and hyper-bolic divergent are considered. Material propertiesand disc thickness profiles are assumed to be rep-resented by power law distributions. Elastic stressesand radial displacements for the hollow disc withboth free–free and fixed–free boundary conditions areobtained. Analytical solutions are given under free–free and fixed–free boundary conditions in the caseof hollow discs, based on the form of the power lawdistributions for the mechanical properties of the con-stituent components and the thickness profiles. Theeffects of the geometric parameter m and the heatsource on the stresses and the radial displacement areinvestigated. Numerical results are presented for theFG disc using aluminium as the inner surface mate-rial and ceramic as the outer surface material. Theseresults are compared with those for rotating discs withuniform thickness.

Some specific observations of this study can besummarized as follows.

1. The thermomechanical stresses for FG discs withthe same thickness profile are smaller than thosefor pure material discs close to the inner surface incontrast to the region close to the outer surface.

2. The radial stresses due to mechanical loading inFG hollow discs with variable thickness under free–free conditions are positive. Contrary to this, radialstresses for FG discs due to thermal load can benegative or positive depending on the geometricparameter. The thermal radial stress is tensile forthe FG hyperbolic convergent disc.

3. The effect of thickness profile is to shift the loca-tion and to change the value of the maximumstress. The location of maximum radial stress dueto mechanical load for the free–free FG discs tendsto shift towards the outer surface with an increasein m unlike the thermal radial stress under free–freeconditions.

4. The temperature in FG discs with hyperbolic con-vergent thickness profile is smaller in comparisonwith other thickness profiles. The maximum valuesof the temperature tend to shift towards the outersurface with an increase in m while the heat sourceis at the centre.

5. The FG disc with hyperbolic convergent thicknessprofile has smaller stresses due to centrifugal orthermal load than that with hyperbolic divergentthickness profile.

6. Mounted FG discs have tensile radial and circum-ferential stresses due to centrifugal loading for alltypes of thickness profiles. Unlike this, mounted FGdiscs may have both compressive and tensile radialand circumferential stresses due to thermal load forsome specific values of m.

7. Unlike FG discs with uniform thickness, the radialstress due to thermal load in FG discs with hyper-bolic convergent thickness profiles is tensile.

8. The slope of radial displacement in FG discs withfixed–free boundary condition due to mechanicalload is greater than that of thermal radial displace-ment in FG discs with the same condition (Figs 7and 14).

From the exact solution for FG discs given in thisstudy, it can be suggested that an efficient and opti-mal design of the FG disc calls for a variable sectionbeing thicker at the hub and tapering down to a smallerthickness towards the periphery.

© Authors 2010

REFERENCES

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APPENDIX

Notation

A1, A2, B1, B2, arbitrary constants to be obtainedD1, and D2 by applying boundary conditions

E modulus of elasticityE dimensionless modulus of

elasticityh thicknessH dimensionless thicknessk thermal conductivityK dimensionless thermal

conductivitym geometric parameterP material propertyq radial heat sourceQ dimensionless heat sourcer radius of the discR dimensionless radiuss1, s2, xi, parameters related to material

and xo properties and geometry of the FGdisc

u radial displacementU dimensionless radial displacement

α coefficient of the thermal expansionα dimensionless coefficient of the

thermal expansionβ a parameter whose value depends

on the density and radius at innerand outer surfaces of the disc

γ a parameter whose value dependson the thermal conductivity andradius at inner and outer surfacesof the disc

T radial temperature changeT dimensionless radial temperature

change

JMES1812 Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science

16 M Bayat, A H Mohazzab, B B Sahari, and M Saleem

ε strainη a parameter whose value depends

on the modulus of elasticity andradius at inner and outer surfacesof the disc

ν Poisson’s ratioξ a parameter whose value depends

on the thermal expansion andradius at inner and outer surfacesof the disc

ρ density

ρ dimensionless densityσ , σ stress and dimensionless stressω angular velocity

Subscripts

i inner surface of the disco outer surface of the discr, R radial directionsT thermal loadθ hoop direction

Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science JMES1812