Bending analysis of a functionally graded rotating disk based on the first order shear deformation theory

Applied Mathematical Modelling 33 (2009) 4215–4230

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Applied Mathematical Modelling

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Bending analysis of a functionally graded rotating disk based on thefirst order shear deformation theory

M. Bayat a,b,*, B.B. Sahari a,b, M. Saleem c, Aidy Ali a, S.V. Wong a

a Mechanical and Manufacturing Engineering Department, University Putra Malaysia, 43400 UPM, Serdang Selangor, Malaysiab Institute of Advanced Technology (ITMA), Universiti Putra Malaysia, 43400 UPM, Serdang Selangor, Malaysiac Department of Applied Mathematics, Z.H. College of Engineering and Technology, AMU 202002, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 October 2007Received in revised form 26 February 2009Accepted 2 March 2009Available online 11 March 2009

Keywords:Functionally graded materialsRotating diskShear deformation

0307-904X/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.apm.2009.03.001

* Corresponding author. Address: Mechanical andMalaysia. Tel.: +60 172087765; fax: +60 3 8656606

E-mail address: [email protected] (M. Bay

The theoretical formulation for bending analysis of functionally graded (FG) rotating disksbased on first order shear deformation theory (FSDT) is presented. The material propertiesof the disk are assumed to be graded in the radial direction by a power law distribution ofvolume fractions of the constituents. New set of equilibrium equations with small deflec-tions are developed. A semi-analytical solution for displacement field is given under threetypes of boundary conditions applied for solid and annular disks. Results are verified withknown results reported in the literature. Also, mechanical responses are comparedbetween homogeneous and FG disks. It is found that the stress couple resultants in a FGsolid disk are less than the stress resultants in full-ceramic and full-metal disk. It isobserved that the vertical displacements for FG mounted disk with free condition at theouter surface do not occur between the vertical displacements of the full-metal and full-ceramic disk. More specifically, the vertical displacement in a FG mounted disk with freecondition at the outer surface can even be greater than vertical displacement in a full-metaldisk. It can be concluded from this work that the gradation of the constitutive componentsis a significant parameter that can influence the mechanical responses of FG disks.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

Functionally graded materials (FGMs) are microscopically inhomogeneous composite materials, in which the volumefraction of the two or more materials is varied smoothly and continuously as a function of position along certain dimen-sion(s) of the structure from one point to the other [1,2]. These materials are mainly constructed to operate in high temper-ature environments.

Rotating disks have extensive practical engineering applications such as in steam and gas turbines, turbo generators, fly-wheel of internal combustion engines, turbojet engines, reciprocating engines, centrifugal compressors and brake disks.Brake disk and clutch are examples of solid rotating disks where body force and bending load are involved. Gas turbines rotorcan be assumed as a clamped-free condition by ignoring thermal expansion. In gas turbine rotors, it is the pressure difference(in addition to rotational stresses) across the rotors that cause bending. In clutches the bending is caused by the force respon-sible for maintaining contact between the plates. These examples certainly emphasize on the role of bending in the design ofany such components. In all these applications, the performance of the components in terms of efficiency, service life and

. All rights reserved.

Manufacturing Engineering Department, University Putra Malaysia, 43400 UPM, Serdang Selangor,1.at).

4216 M. Bayat et al. / Applied Mathematical Modelling 33 (2009) 4215–4230

power transmission certainly depends on the material, speed of rotation and operating conditions. Normally, these compo-nents can be fabricated by using any metal. However, for some specific applications such as in aerospace where light-weightand durability becomes crucial in high temperature environment, the components need to be fabricated using special mate-rial such as a functionally graded material (FGM). As the use of FGMs increases, new methods need to be developed to char-acterize, analyze and design structural components made of these materials.

FGMs are usually made of a mixture of ceramic and metals. The ceramic constituent of the material provides the hightemperature resistance due to its low thermal conductivity. The ductile metal constituent, on the order hand, prevents frac-ture caused by stress due to high temperature gradient in a very short period of time [3].

Thermal stresses and deformation have been active research areas for long time but in recent years there has been a re-newed interest in the thermo elastic analysis of plates, beams or cylinders made of FGMs [4–11].

Tanigawa et al. [12] studied the linear thermal bending of FGM plate in steady-state condition. They also studied thestress in transient heat conduction with temperature-dependent material properties. Using the first order shear deformationtheory (FSDT), Praveen and Reddy [13] analyzed nonlinear static and dynamic responses of FG ceramic–metal plates in asteady temperature field. They used finite element method. Reddy et al. [3,13] studied axisymmetric bending and stretchingof FG solid and annular circular plates using the FSDT. The solutions for deflections, force and moment resultants were pre-sented in terms of the corresponding quantities of isotropic plates based on the classical Kirchhoff plate theory. Lanhe [14]used the FSDT and derived equilibrium and stability equations of a moderately thick rectangular plate made of FGM underthermal loads. He assumed that the material properties varied as a power law of thickness. Park and Kim [15] investigatedthermal postbuckling and vibration behaviors of the FG plate. The nonlinear finite element equations based on the FSDT wereformulated and the von Karman nonlinear strain–displacement relationship was used to account for the large deflection ofthe plate. Zenkour [16] presented the static response of a simply supported FG rectangular plate subjected to a transverseuniform load. He obtained the equilibrium equations based on a generalized shear deformation plate theory. Arciniegaand Reddy [17] presented a geometrically nonlinear analysis of FG shells. The two-constituent FG shell consisting of ceramicand metal was graded through the thickness from one surface of the shell to the other. A tensor-based finite element formu-lation with curvilinear coordinates and FSDT were used to develop the FG shell finite element. Bayat et al. [18] developed anew set of equilibrium equations with small and large deflections in a FG rotating disk with axisymmetric bending and stea-dy-state thermal loading. The material properties of the disk were varied in the thickness direction. The FSDT and von Kar-man theories were used.

Lee et al. [19] obtained an elastic solution for pure bending problem of simply supported transversely isotropic circularplates with elastic compliance coefficients being arbitrary functions of the thickness coordinate. Chen et al. [20] obtained athree-dimensional analytical solution for transversely isotropic FG disk rotating at a constant angular velocity. Using the ba-sic equations for axisymmetric problems of transversely isotropic elastic materials, five equations governing the displace-ment functions were derived. Chen and Chen [21] derived three-dimensional analytical solution of the elastic equationsfor transversely isotropic FG rotating plate by means of direct displacement method. The displacement components wereassumed as a linear combination of certain explicit functions of the radial coordinate. Li et al. [22] used stress function meth-od and presented a set of elasticity solutions for the axisymmetric problem of transversely isotropic simply supported andclamped edge FG circular plates subjected to a transverse load. They showed the effect of material inhomogeneity on theelastic field in FG plates.

Durodola and Attia [23,24] presented a finite element analysis for FG rotating disks using commercial software package.The disks were modeled as non-homogeneous orthotropic materials such as those obtained through non-uniform reinforce-ment of metal matrix by long fibers. They considered three types of gradation distributions for the Young’s modulus in thehoop direction relative to matrix material modulus. Kordkheili and Naghdabadi [25] presented a semi-analytical thermoelastic solution for hollow and solid rotating axisymmetric disks made of functionally graded materials under plane stresscondition. The results were compared with those of Durodola and Attia [23,24] under the centrifugal loading.

Jahed and Sherkatti [26] applied the variable material properties (VMP) method and obtained stresses for an inhomoge-neous rotating disk with variable thickness under steady temperature field assuming the material properties as field vari-ables. Jahed and Shirazi [27] evaluated the temperature in a rotating disk during heating and cooling using VMP method.Farshi et al. [28] also used VMP method and obtained optimal profile for an inhomogeneous non-uniform rotating disk.

It may be noted that the existing literature on circular disks as cited above concerns with combination of thermal loadingwith either body force or bending. To the best of our knowledge, no work has been reported till date that has analyzed FGdisk with combination of body force and bending. This study attempts to incorporate both body and bending forces in theanalysis of FG rotating disks. In this paper, an FG annular rotating disk as shown in Fig. 1 with inner radius, ri, outer radius, r0,and thickness, h, axisymmetric with respect to z axis and subjected to mechanical loading is considered. The mechanicalloading is obtained by having pressure load in the z direction and centrifugal load in the radial direction. The material prop-erties of the constituent components of the disk are assumed to be represented by a power law distribution along the radialdirection in the disk. The first order shear deformation theory is used. To be more specific, this work aims to investigate theeffect of some basic factors such as material property gradation and the mechanical loading on displacement and stress fieldsin the solid disk with roller supported boundary conditions and in an annular disk with boundary conditions with clampedsupport at the inner edge r ¼ ri and free or roller supported at outer edge r ¼ ro. A semi-analytical solution for displacementfield is given for small deflection. Numerical results for normalized displacement and stress resultant components along theradius and thickness of the disk are presented.

Fig. 1. Two dimensional and three-dimensional views of the hollow disk of inner radius ri , outer radius ro and thickness h.

M. Bayat et al. / Applied Mathematical Modelling 33 (2009) 4215–4230 4217

In the semi-analytical method, the radial domain is divided into some virtual sub-domains where in each sub-domain, themechanical property is assumed to be constant. This assumption yields the governing equations in each sub-domain as ODEwith constant coefficients. Imposing the continuity at the interface of the adjacent sub-domains together with global con-ditions, a set of linear algebraic equations is derived. Increasing the number of sub-domains (divisions) in the radial directionincreases the accuracy in the solution.

2. Gradation relation

In this study, the property variation PðrÞ, of the material in the FG disk along the radial direction is assumed to be of thefollowing form (Ruhi et al., [29]):

PðrÞ ¼ ðPo � PiÞr � ri

ro � ri

� �n

þ Pi; ri < r < ro: ð1aÞ

Here Po and Pi are the corresponding properties of the outer and inner surfaces of the disk; ro and ri are the outer and innerradii of the disk, respectively; n P 0 is the volume fraction exponent (n is also called grading index in this paper). The powerlaw as given in Eq. (1a) is widely accepted form of property variation, and reflects a simple rule of mixtures in terms of thevolume fraction of the materials. In this study, the Poisson’s ratio, m, is assumed to be constant. The elastic modulus, E, andthe mass density, q; are assumed to vary according to the Eq. (1a). Accordingly, the assumed form for the modulus of elas-ticity, E, in the following sections will be:

EðrÞ ¼ ðEo � EiÞr � ri

ro � ri

� �n

þ Ei; ri < r < ro; ð1bÞ

3. Theoretical formulation and equilibrium equations

Consider a thin axisymmetric FG disk with constant thickness, h, inner radius, ri, and outer radius, ro, as shown in Fig. 1.This disk rotates at an angular velocity, x, and is subjected to an axisymmetric transverse loading, qzðrÞ. Due to axial

4218 M. Bayat et al. / Applied Mathematical Modelling 33 (2009) 4215–4230

symmetry in geometry and loading, cylindrical coordinates ðr; h; zÞ are considered. The inner and outer surfaces of the FG diskare assumed to be metal-rich and ceramic-rich, respectively. Between these two surfaces, the material properties varyaccording to Eq. (1a).

It may be mentioned that although a metal-rich at the inner surface and ceramic-rich at the outer surface gradient hasbeen considered for all the disks in this paper, the method of solution is however, independent of such gradient and maybe applied to other gradients as well. However, several applications considered in this paper such as an FG disk mountedon a shaft support justify consideration of metal-rich inner surface of the disks. Not to mention, with FG disk mounted ona shaft support ductility plays an important role and the inner surface needs to be metal dominated.

3.1. Displacement field and strains

The first order shear deformation plate theory (FSDT) is the simplest theory that accounts for nonzero transverse shearstrain. It is based on the displacement field

ur ¼ urðr; zÞ ¼ u0ðrÞ þ z wðrÞ; ð2aÞuz ¼ uzðr; zÞ ¼ wðrÞ; uh ¼ 0; ð2bÞ

where u0 ¼ u0 is the in-plane displacement of the mid-plane, ur , uh and uz are the radial, circumferential and vertical dis-placements, respectively, w ¼ wðrÞ denotes rotation of a transverse normal in the plane h = constant, w ¼ wðrÞ is the displace-ment along the thickness. The first order theory includes a constant state of transverse shear strain to the thicknesscoordinate and hence requires a shear correction factor which depends not only on the material and geometric parametersbut also on the loading and boundary conditions.

The strain–displacement equations of the small displacement theory are given by [26]:

er ¼@ur

@r¼ duo

drþ z

dwdr; ð3aÞ

eh ¼ur

r¼ uo

rþ z

wr; ð3bÞ

crz ¼ 2erz ¼@ur

@zþ @uz

@r¼ wþ dw

dr; ð3cÞ

crh ¼ 2erh ¼ 0; chz ¼ 2ehz ¼ 0; ez ¼@uz

@z¼ 0: ð3dÞ

Also, the linear constitutive elastic equations in the cylindrical coordinates are used in the form of:

rr

rh

� �¼ EðrÞ

1� m2

1 mm 1

� � er

eh

� �; ð4aÞ

rrz ¼EðrÞ

2ð1þ mÞ crz: ð4bÞ

3.2. Equilibrium equations

For a rotating disk, if U1 is the total strain energy of the body and V1 is the total external work done on the body by thetotal specified external forces as shown in Fig. 2, then the total energy P can be represented as P � U1 � V1.

The principle of minimum total energy states that dP ¼ 0 and this yields,

ri

Nh �ddrðrNrÞ � q1r2x2

� �duo þ Mh þ rQ r �

ddrðrMrÞ

� �dw� d

drðrQrÞ þ rqzðrÞ

� �dw

� �dr ¼ 0; ð5Þ

where

Nr ¼ NrðrÞ;Nh ¼ NhðrÞ are the stress resultants per unit length,Mr ¼ MrðrÞ;Mh ¼ MhðrÞ are the stress couples per unit length,Qr ¼ Q rðrÞ is the transverse shear resultant per unit length,

and q1 ¼ q1ðrÞ is proportional to the mean of the density along the thickness.Various entities in (5) are defined as:

ðNr ;Nh;Q rÞ ¼Z h=2

�h=2ðrr ;rh;rrzÞdz; ð6aÞ

ðMr;MhÞ ¼Z h=2

�h=2ðrr;rhÞzdz; ð6bÞ

q1 ¼ q1ðrÞ ¼Z h=2

�h=2qðrÞdz ¼ hqðrÞ: ð6cÞ

Fig. 2. Forces and moments associated with stresses. Symbols � and � indicate arrows in the þz (downward) and �z (upward) directions, respectively inaxisymmetric plate.

M. Bayat et al. / Applied Mathematical Modelling 33 (2009) 4215–4230 4219

In Eqs. (6), rr ;rh are the radial and circumferential stresses, respectively.Minimum energy condition (5) yields the following equilibrium equations:

duoðrÞ : � ddrðrNrÞ þ Nh � q1r2x2 ¼ 0; ð7aÞ

dwðrÞ : � ddrðrMrÞ þMh þ rQ r ¼ 0; ð7bÞ

dwðrÞ :ddrðrQ rÞ þ rqzðrÞ ¼ 0: ð7cÞ

It may be noted that Eqs. (7a)–(7c) are different from the one considered by Reddy and Huang [30]. Here Eq. (7a) has oneextra term representing the effect of body force due to rotation.

Substituting for rr ;rh and rrz from Eq. (4) into (6) gives:

Nr ¼ hEðrÞ

1� m2

du0ðrÞdr

þ mu0ðrÞ

r

� �; ð8aÞ

4220 M. Bayat et al. / Applied Mathematical Modelling 33 (2009) 4215–4230

Nh ¼ hEðrÞ

1� m2 mdu0ðrÞ

drþ u0ðrÞ

r

� �; ð8bÞ

Mr ¼h3

12EðrÞ

1� m2

dwðrÞdrþ m

wðrÞr

� �; ð8cÞ

Mh ¼h3

12EðrÞ

1� m2 mdwðrÞ

drþ wðrÞ

r

� �; ð8dÞ

Qr ¼ hEðrÞ

2ð1þ mÞ ks wðrÞ þ dwdr

� �; ð8eÞ

where ks is shear correction factor. It can be noted that the assumption that transverse shear strain is constant with respectto the thickness coordinate in FSDT implies that the transverse shear stresses will also be constant. The shear deformationcan be modified by considering ks ¼ 5=6 as introduced by Reddy [31]. For the present case, the material properties of FG var-ied in radial direction. Therefore, in any section through the thickness the material is homogenous. The shear correction fac-tor for FG plate whose material properties vary along the thickness also has value equal to 5=6 ðffi 0:8333Þ in [3,22]. However,according to Table 4 in [32] the values of the shear correction factor for FG plate whose material properties vary along thethickness with ECeramic=EMetal ¼ 5700=2700 ffi 2:11 ranges from 0.784 to 0.842.

Substituting for various terms from Eqs. (8) into Eqs. (7) yields a set of three ordinary differential equations for displace-ment field.

rEðrÞd2uo

dr2 þ EðrÞ þ rdEðrÞ

dr

� �duo

drþ m

dEðrÞdr� EðrÞ

r

� �uo þ ð1� m2Þr2x2qðrÞ ¼ 0; ð9aÞ

rEðrÞd2w

dr2 þ EðrÞ þ rdEðrÞ

dr

� �dwdrþ m

dEðrÞdr� EðrÞ

rþ 6ksðm� 1Þ rEðrÞ

h2

� �wþ 6ksðm� 1Þ rEðrÞ

h2

dwðrÞdr

¼ 0; ð9bÞ

rEðrÞ d2w

dr2 þ EðrÞ þ rdEðrÞ

dr

� �dwdrþ rEðrÞdw

drþ EðrÞ þ r

dEðrÞdr

� �wþ 2ð1þ mÞ

hksrqzðrÞ ¼ 0: ð9cÞ

4. Boundary conditions

The following boundary conditions will be used in this paper (Reddy et al. [3]).

4.1. Roller supported solid disk

For a solid disk with a roller support that restrained in z direction at r ¼ ro,

At r ¼ 0 : u ¼ 0; w ¼ 0; Q r ¼ 0; ð10aÞAt r ¼ ro : w ¼ 0; Nr ¼ 0; Mr ¼ 0: ð10bÞ

4.2. Clamped-free annular disk

For an annular disk with a clamped support at the inner edge r ¼ ri and free at outer edge r ¼ ro,

At r ¼ ri; u ¼ 0; w ¼ 0; w ¼ 0; ð11aÞAt r ¼ ro : Q r ¼ 0; Nr ¼ 0; Mr ¼ 0: ð11bÞ

4.3. Clamped-roller supported annular disk

For an annular disk with a clamped support at the inner edge r ¼ ri and roller supported at outer edge r ¼ ro,

At r ¼ ri; u ¼ 0; w ¼ 0; w ¼ 0; ð12aÞAt r ¼ ro : w ¼ 0; Nr ¼ 0; Mr ¼ 0: ð12bÞ

5. Elastic solution

A closed-form solution of Eqs. (9) with variable coefficients seems to be difficult to obtain. Hence, in this work a semi-analytical solution is attempted. In this method, a disk is divided into some virtual sub-domains (say m), with tðkÞ denotingthe radial-width of the kth sub-domain as shown in Fig. 3. Evaluating the coefficients of Eqs. (9) at r ¼ rðkÞ, the mean radius ofthe kth division, these equations reduce to the following set of three ordinary differential equations valid for kth sub-domain.

tk

tk+1

tk-1

r(k)

(k+1)th division

ro kth division

(k-1)th division

ri

Fig. 3. Dividing radial domain into some finite sub-domains.

M. Bayat et al. / Applied Mathematical Modelling 33 (2009) 4215–4230 4221

cðkÞ1d2

dr2 þ cðkÞ2ddrþ cðkÞ3

!uðkÞ0 þ cðkÞ4 ¼ 0; ð13aÞ

cðkÞ1d2

dr2 þ cðkÞ2ddrþ cðkÞ3 þ cðkÞ5

!wðkÞ þ cðkÞ5

ddr

wðkÞ ¼ 0; ð13bÞ

cðkÞ1ddrþ cðkÞ2

� �wðkÞ þ cðkÞ1

d2

dr2 þ cðkÞ2ddr

!wðkÞ þ cðkÞ6 ¼ 0; ð13cÞ

where

cðkÞ1 ¼ rðkÞEðrðkÞÞ; ð14aÞ

cðkÞ2 ¼ rðkÞdEðrÞdr

����r¼rðkÞ

þ EðrðkÞÞ; ð14bÞ

cðkÞ3 ¼ mdEðrÞdr

����r¼rðkÞ

�EðrðkÞÞrðkÞ

; ð14cÞ

cðkÞ4 ¼ ð1� m2ÞqðrðkÞÞðxrðkÞÞ2; ð14dÞ

cðkÞ5 ¼ 6ksðm� 1ÞrðkÞEðrðkÞÞ

h2 ; ð14eÞ

cðkÞ6 ¼ 2ðmþ 1ÞrðkÞqzðrðkÞÞ

hks: ð14fÞ

Using the same technique for each sub-domain, a system of m ordinary differential equations with constant coefficients isobtained with m being the number of virtual sub-domains.

The solution for Eq. (13a) can be written as:

uðkÞo ¼ XðkÞ1 expðkðkÞ1 rÞ þ XðkÞ2 expðkðkÞ2 rÞ � cðkÞ4

cðkÞ3

; ð15Þ

where,

kðkÞ1 ; kðkÞ2 ¼�cðkÞ2 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðcðkÞ2 Þ

2 � 4cðkÞ3 cðkÞ1

q2cðkÞ1

; ð16Þ

and XðkÞ1 and XðkÞ2 are unknown constants for kth sub-domain.Operating on Eq. (13b) by the operator cðkÞ1

d2

dr2 þ cðkÞ2ddr

� and on Eq. (13c) by the operator cðkÞ5

ddr and then eliminating the

term with wðkÞ from Eqs. (13b) and (13c) yields a fourth-order linear homogeneous differential equation in wðkÞ as follows:

Fig. 4.

Compa

Table 1Non-dim

Materia

PartiallAlumin

4222 M. Bayat et al. / Applied Mathematical Modelling 33 (2009) 4215–4230

ddr

cðkÞ1ddrþ cðkÞ2

� �cðkÞ1

d2

dr2 þ cðkÞ2ddrþ cðkÞ3

!wðkÞ ¼ 0 ð17Þ

The general solution of Eq. (17) can be written as:

wðkÞ ¼ XðkÞ3 expðkðkÞ1 rÞ þ XðkÞ4 expðkðkÞ2 rÞ þ XðkÞ5 expðkðkÞ3 rÞ þ cðkÞ6 cðkÞ5

cðkÞ2 cðkÞ3

; ð18Þ

where kðkÞ1 , kðkÞ2 are the same as given in Eq. (16) and kðkÞ3 ¼ �cðkÞ2

cðkÞ1

.

0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20.2

10.4

10.45

10.5

10.55

10.6

10.65

10.7

10.75

10.8

10.85

Nondimensional thickness (h/ro)

Max

imum

non

dim

ensi

onal

ver

tical

di

spla

cem

ent i

n th

e m

iddl

e pl

ane Reddy et al. [3]

Present Study

10.396

10.481

10.623

10.822Results for Full - Metal Disk

0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20.24.05

4.1

4.15

4.2

4.25

4.3

Nondimensional thickness (h/ro)

Max

imum

non

dim

ensi

onal

ver

tical

di

spla

cem

ent i

n th

e m

iddl

e pl

ane Reddy et al. [3]

Present Study

4.117

4.151

4.207

4.285Results for Full - Ceramic Disk

(a) Comparison of non-dimensional vertical displacement 64Dcqo r4

ow of full-metal roller supported circular plate in this study and Reddy et al. [3]. (b)

rison of non-dimensional vertical displacement 64Dcqo r4

ow of full-ceramic roller supported circular plate in this study and Reddy et al. [3].

ensional maximum vertical displacement 64Dcqo r4

ow of homogeneous roller-supported circular plates.

l property EðGPaÞ m qðKgm3Þ

y stabilized zirconia (PSZ), ceramic 151.0 0.3 5700um, metal 70.0 0.3 2700

M. Bayat et al. / Applied Mathematical Modelling 33 (2009) 4215–4230 4223

Now either Eq. (13b), (13c) can be used to find wðkÞ as:

wðkÞ ¼ �XðkÞ3expðkðkÞ1 rÞ

kðkÞ1

� XðkÞ4expðkðkÞ2 rÞ

kðkÞ2

� XðkÞ5cðkÞ3 þ cðkÞ5

kðkÞ3 cðkÞ5

!expðkðkÞ3 rÞ � cðkÞ6 ðc

ðkÞ3 þ cðkÞ5 Þ

cðkÞ2 cðkÞ3

r þ XðkÞ6 ð19Þ

It may be noted that the solution of Eqs. (15), (18), and (19) is valid for:

rðkÞ � tðkÞ

26 r 6 rðkÞ þ tðkÞ

2; ð20Þ

where rðkÞ and tðkÞ are the mean radius and the radial-width of the kth sub-domain, respectively. The unknowns XðkÞ1 ..., XðkÞ5 andXðkÞ6 can be determined by applying the necessary conditions between each two adjacent sub-domains. For this purpose, thecontinuity of the radial displacement ur , vertical displacement uz ¼ wðrÞ, slope of vertical displacement duz

dr ¼dwðrÞ

dr , the stress

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

10

r/ro

Non

dim

ensi

onal

dis

plac

emen

t in

verti

cal d

irect

ion(

z)

Ceramic

Metal

n=10.0

n=3.0

n=1.0n=0.5

Fig. 5. Non-dimensional vertical displacement 64Dcqo r4

ow along the radial direction of FG solid disk.

0.1 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

0.018

r/ro

Non

dim

ensi

onal

radi

al s

tress

resu

ltant

Metal

Ceramic

n=0.5n=1.0

n=3.0

n=10.0

Fig. 6. Non-dimensional radial stress resultant Nrqc r3

o x2 along radial direction of FG solid disk.

4224 M. Bayat et al. / Applied Mathematical Modelling 33 (2009) 4215–4230

couple Mr ; the transverse shear resultant Q r and radial stress rr are imposed at the interfaces of the adjacent sub-domains.These continuity conditions at interfaces are:

uðkÞr

��r¼rðkÞþtðkÞ

2¼ uðkþ1Þ

r

��r¼rðkþ1Þ�tðkþ1Þ

2; ð21aÞ

uðkÞz

��r¼rðkÞþtðkÞ

2¼ uðkþ1Þ

z

��r¼rðkþ1Þ�tðkþ1Þ

2; ð21bÞ

duz

dr

����r¼rðkÞþtðkÞ

2

¼ duz

dr

����r¼rðkþ1Þ�tðkþ1Þ

2

; ð21cÞ

MðkÞr

���r¼rðkÞþtðkÞ

2

¼ Mðkþ1Þr

���r¼rðkþ1Þ�tðkþ1Þ

2

; ð21dÞ

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

r/ro

Non

dim

ensi

onal

stre

ss c

oupl

e in

r di

rect

ion Metal or Ceramic

n=0.5

n=1.0

n=3.0

n=10.0

Fig. 7. Non-dimensional radial stress couple Mrpr2

o qoalong radial direction of FG solid disk.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

r/ro

Non

dim

ensi

onal

tran

sver

se s

hear

res

ulta

nt

Metal or Ceramic

n=0.5n=1.0

n=3.0

n=10.0

Fig. 8. Non-dimensional transverse shear resultant 2pro hDc

Qr along radial direction of FG solid disk.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

10

r/ro

Non

dim

ensi

onal

dis

plac

emen

t in

verti

cal d

irect

ion(

z)

Ceramic

Metal

n=1.0

n=3.0n=10.0

n=0.5

Fig. 9. Non-dimensional vertical displacement 64Dcqo r4

ow along the radial direction of FG clamped-free disk.

M. Bayat et al. / Applied Mathematical Modelling 33 (2009) 4215–4230 4225

Q ðkÞr

���r¼rðkÞþtðkÞ

2

¼ Q ðkþ1Þr

���r¼rðkþ1Þ�tðkþ1Þ

2

; ð21eÞ

rðkÞr

��r¼rðkÞþtðkÞ

2¼ rðkþ1Þ

r

��r¼rðkþ1Þ�tðkþ1Þ

2: ð21fÞ

The continuity conditions (21) together with the global boundary conditions (10) or (11) or (12) yield a set of linear alge-braic equations in XðkÞ1 . . ., XðkÞ5 and XðkÞ6 . Solving these equations for XðkÞ1 . . ., XðkÞ5 and XðkÞ6 and substituting them in Eqs. (15), (18),and (19), the displacement components ur and uz are determined in each sub-domain. Increasing the number of divisionsimproves the accuracy of the results.

6. Numerical results and discussion

6.1. Verification of the solution

For the verification of the results of this paper, the exact solution reported by Reddy et al. [3] for a solid disk with x ¼ 0when m ¼ 0:288 is considered. Reddy et al. [3] presented an expression for non-dimensionalized maximum deflection asWmax ¼ 64wDc

qor4o

with Dc ¼ Ec h3

12ð1�m2Þ and Ec ¼ 151:0 GPa of FG roller supported circular plate.Graphs of the non-dimensionalized vertical displacement against thickness as obtained from semi-analytical method are

given in Fig. 4a and b for full-metal disk and full-ceramic disk, respectively with EmEc¼ 0:396 and qo ¼ 0:14 GPa. It can be seen

that the results of this study as shown in Fig. 4a and b are very well comparable with those of Reddy et al. [3].

6.2. Case study for FG disk

For numerical illustration of the elastic solutions of this study, three cases are considered, namely, hollow disk clamped-free, hollow disk clamped-roller and solid disk roller supported. The analysis is conducted using aluminum as inner surfacemetal and zirconia as outer surface ceramic. The details of material properties are shown in Table 1.

The disk with ro ¼ 5ri and ro ¼ 5h is assumed to be subjected to a uniform centrifugal force (x ¼ 1000:0 rads ), uniform

transverse load per unit area (qzðrÞ ¼ 0:14GPa).

6.2.1. Results and discussionIn this section, results are presented in non-dimensional form normalizing vertical displacement, stress resultant, trans-

verse shear resultant and stress couple by factors qor4o

64Dc;qcx2r3

o ;Dc

2proh and pr2oqo, respectively.

6.2.2. Roller supported solid diskFig. 5 illustrates the non-dimensional vertical displacement 64Dc

qor4o

w for different values of the grading index n.

It can be seen that the vertical displacement is maximum for full-metal disk, minimum for full-ceramic disk, and for dif-ferent solid FG disks, these vertical displacements occur in between. It is observed that the vertical displacement increaseswith the increase of the grading index n from zero (homogenized zirconia disk) up to the maximum value for n!1(homogenized aluminum disk).

4226 M. Bayat et al. / Applied Mathematical Modelling 33 (2009) 4215–4230

The variation of non-dimensional radial stress resultant with radius is shown in Fig. 6. It is evident that the full-metalsolid disk has smaller stress resultants in comparison with FG and full-ceramic disks. It can be noticed that close to the innersurface, the radial stress resultant for some specific value of the grading n (n ¼ 0:5) in FG disk is even larger than that for full-ceramic disk. This phenomenon can be explained by the presence of interactive effect between material properties such asstiffness, density and mechanical load of the disk.

The variation of non-dimensional radial stress couples with radius in the FG solid disk for different values of the gradingindex n is shown in Fig. 7. It is worth noting that the stress couple resultant in full-metal is the same as in full-ceramic disks.It is observed that the stress couple resultants in FG solid disks are lower than the stress resultants in full-ceramic and full-metal disks.

The transverse shear resultant distributions for solid FG disk mounted on a roller at outer surface, for different values ofthe grading index n are shown in Fig. 8.

It is seen that the transverse shear resultant in full-metal is the same as in full-ceramic disks and its values (in absolutesense) are bigger than those of FG disk throughout. It can also be seen that for some specific values of the grading index n

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

r/ro

Non

dim

ensi

onal

radi

al s

tress

resu

ltant

Metal

Ceramic

n=0.5

n=1.0

n=3.0

n=10.0

Fig. 10. Non-dimensional radial stress resultant Nrqc r3

o x2 along radial direction of FG clamped-free disk.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

r/ro

Non

dim

ensi

onal

str

ess

coup

le in

r d

irect

ion

Metal or Ceramicn=0.5

n=1.0

n=3.0

n=10.0

Fig. 11. Non-dimensional radial stress couple Mrpr2

o qoalong radial direction of FG clamped-free disk.

M. Bayat et al. / Applied Mathematical Modelling 33 (2009) 4215–4230 4227

(n ¼ 10 in Fig. 8) the transverse shear resultant may have a local minimum close to the outer surface of FG solid disk unlikefull-metal and full-ceramic disk. In pure material (full-metal or full-ceramic) disk, transverse shear resultant decreases lin-early along the radial direction.

6.2.3. Clamped-free annular diskThe non-dimensional transverse displacements for FG disks mounted on a rigid shaft, for different values of the grading

index n are shown in Fig. 9.As expected, the transverse displacement values for full-metal (aluminum) disk are greater than those for full-ceramic

(zirconia) disk due to higher modulus of elasticity of the latter. It is seen that the transverse displacement for FG mounteddisk may not lie in between the values for full-metal and full-ceramic disk as shown in Fig. 5 for solid disk with a roller sup-port at outer. It is interesting to note that for the grading index value n ¼ 0:5 the transverse displacement remains in be-tween the displacements for pure material disks. Thus, it may be suggested that for mechanical design where lowertransverse displacement is needed, n ¼ 0:5 can be used.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

r/ro

Non

dim

ensi

onal

tran

sver

se s

hear

resu

ltant

Metal or Ceramic

n=0.5

n=1.0

n=3.0

n=10.0

Fig. 12. Non-dimensional transverse shear resultant 2pro hDc

Qr along radial direction of FG clamped-free disk.

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

r/ro

Non

dim

ensi

onal

dis

plac

emen

t in

verti

cal d

irect

ion(

z)

Ceramic

Metal

n=10.0

n=3.0

n=1.0n=0.5

Fig. 13. Non-dimensional vertical displacement 64Dcqo r4

ow along the radial direction of FG clamped-roller disk.

4228 M. Bayat et al. / Applied Mathematical Modelling 33 (2009) 4215–4230

The non-dimensional radial stress resultant in mounted FG disk is shown in Fig. 10. For FG mounted disk the behavior ofthe radial stress resultants are the same as those for full-metal and full-ceramic disk. It is noticed that, close to the innersurface, the radial stress resultants of FG disks for some specific values of grading index n (n ¼ 0:5 or n ¼ 1:0 in Fig. 10)are larger than those for full-ceramic disk. The curves of Fig. 10 show that the radial stress resultants for FG mounted diskare greater compared with those for FG solid disk with roller at the outer (see Fig. 6).

Fig. 11 shows the variation of non-dimensional radial stress couple resultants for different values of grading index n in themounted FG disk.

It can be seen that the radial stress couple resultant along the radial direction in a full-ceramic or full-metal disk is thesame, whereas, for a FG disk it can have greater or smaller values than the resultant for homogeneous disk. It is observed thatthe homogeneous disk has smaller absolute radial stress couple resultants in comparison with FG disk at the inner surface. Itis also observed that the minimum radial stress couple resultant increases with the increase of the grading index n.

The variation of non-dimensional transverse shear resultants of the mounted FG disk for different values of the gradingindex n are shown in Fig. 12. It is seen that the transverse shear resultant in full-metal is the same as in full-ceramic disks.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

r/ro

Non

dim

ensi

onal

radi

al s

tress

resu

ltant

Metal

Ceramic

n=0.5

n=1.0

n=3.0

n=10.0

Fig. 14. Non-dimensional radial stress resultant Nrqc r3

o x2 along radial direction of FG clamped-roller disk.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

r/ro

Non

dim

ensi

onal

stre

ss c

oupl

e in

r di

rect

ion

Metal or Ceramicn=0.5

n=1.0

n=3.0n=10.0

Fig. 15. Non-dimensional radial stress couple Mrpr2

o qoalong radial direction of FG clamped-roller disk.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.5

0

0.5

1

1.5

2

2.5

3

r/ro

No

nd

ime

nsi

on

al t

ran

sve

rse

sh

ea

r re

sulta

nt

Metal or Ceramic

n=0.5

n=1.0

n=3.0

n=10.0

Fig. 16. Non-dimensional transverse shear resultant 2pro hDc

Qr along radial direction of FG clamped-roller disk.

M. Bayat et al. / Applied Mathematical Modelling 33 (2009) 4215–4230 4229

However, it is smaller than that in FG disk. It is also seen that the maximum transverse shear resultant decreases with theincrease of the grading index n.

6.2.4. Clamped-roller supported annular diskFig. 13 shows the non-dimensional vertical displacement values of a clamped-roller FG disk for different values of the

grading index n. It is seen that the vertical displacement is maximum for full-metal disk and minimum for full-ceramic disk.It is noticed that the maxima of vertical displacements for FG disks occur in between maxima of full-metal and full-ceramicdisks with clamped-roller supported condition.

Fig. 14 shows the non-dimensional radial stress resultant values for different values of the grading index n in a clamped-roller disk. The graphs of Fig. 14 show that the radial stress resultants for FG clamped-roller disk are the same as for mountedFG disk (Fig. 10) but are greater than those for solid FG disk with roller at the outer (Fig. 6). It is observed that the stressresultant in a FG disk may not lie in between the stress couple resultants in full-metal and full-ceramic disk.

Fig. 15 shows the variation of non-dimensional radial stress couple resultants for different values of grading index n. Itis seen that the radial stress couple resultants go from negative to positive and then to zero. This phenomenon can beexplained by the presence of interactive effect between centrifugal force and bending load of disk by considering boundaryconditions.

Fig. 16 shows the variation of non-dimensional transverse shear resultants with radius for different values of the gradingindex n in a clamped-free disk. It is seen that the transverse shear resultant in full-metal disk or full-ceramic disks is less thanthose in FG disks. The transverse shear resultants are positive close to inner surface and negative near to outer surface. It canbe noticed that it is zero around r ¼ 0:7ro and in FG disk this ratio depends on the value of grading index n.

7. Conclusions

New set of equilibrium equations for a functionally graded (FG) axisymmetric rotating disk with bending are developed.The material properties of FG disks are assumed to vary continuously along the radial direction of the disk graded accordingto a power law distribution of the volume fraction of the constituents. First order shear deformation theory (FSDT) is used. Asemi-analytical solution for displacement field is given. Dimensionless deflection, stress and moment resultants are com-puted for different combinations of functionally graded disks. Elastic stresses and displacement for the solid disk with a roll-er support at outer radius and the hollow disk (when ro ¼ 5ri) with clamped inner edge and roller at the outer edge or freeboundary conditions are obtained. Results are presented for solid and hollow disks with thickness and outer radius ratio ta-ken as h ¼ 0:2ro and for various values of the grading index n of material properties. Numerical results are presented for theFG disk using aluminum as the inner surface metal and zirconia as the outer surface ceramic.

Some general observations of this study can be summarized as follows:

� For a given pair of materials there is a particular volume fraction that maximizes a specific mechanical response underbody force and bending load. As an example n ¼ 10 in Fig. 8 and n ¼ 0:5 in Fig. 9 are found to be crucial values of gradingindex.

4230 M. Bayat et al. / Applied Mathematical Modelling 33 (2009) 4215–4230

� The vertical displacements in FG solid disks with roller support at outer surface have values in between the maximumvalue for full-metal disk and the minimum value for full-ceramic disk.

� The stress couple resultants in FG solid disks have values smaller than those of stress couple resultants in full-ceramic andfull-metal disks.

� The transverse shear resultants may have a local minimum close to the outer surface of FG solid disks unlike pure materialdisks.

� The vertical displacement values for FG mounted disk with free condition at outer surface may not lie in between the val-ues for full-metal and full-ceramic disks.

� The maxima of vertical displacement for FG disks occur between maxima of full-metal and full-ceramic disks withclamped-roller supported condition.

� The stress resultants in a FG disk may not lie in between the stress couple resultants in full-metal and full-ceramic disk.� The transverse shear resultants in homogeneous disks are smaller than those in FG disks.

From the semi-analytical results for FG disks given in this study, it can be suggested that the gradation of the constitutivecomponents is a significant parameter in the mechanical responses of rotating FG disks.

References

[1] S. Suresh, A. Mortensen, Fundamentals of Functionally Graded Materials, IOM Communications Limited, London, 1998.[2] J.N. Reddy, Analysis of functionally graded plates, Int. J. Numer. Methods Eng. 47 (2000) 663–684.[3] J.N. Reddy, C.M. Wang, S. Kitipornchai, Axisymmetric bending of functionally graded circular and annular plate, Eur. J. Mech. A/Solids 18 (1999) 185–

199.[4] Y. Obata, N. Noda, T. Tsuji, Steady thermal stresses in a functionally gradient material plate, JSME 58 (1992) 689–1695.[5] K. Tanaka, Y. Tanaka, H. Watanabe, V.F. Poterasu, Y. Sugano, An improved solution to thermo elastic material design in functionally gradient materials:

Scheme to reduce thermal stresses, Comput. Methods Appl. Mech. Eng. 106 (1993) 377–389.[6] Y. Tanigawa, Some basic thermoplastic problems for nonhomogeneous structural material, J. Appl. Mech. 48 (1995) 377–389.[7] R.W. Zimmerman, M.P. Lutz, Thermal stresses and thermal expansion in a uniformly heated functionally graded cylinder, J. Thermal Stresses 22 (1999)

177–188.[8] Z.Q. Cheng, R.C. Batra, Three-dimensional thermoelastic deformations of a functionally graded elliptic plate, Composites Part B 31 (2000) 97–106.[9] B.V. Sankar, J.T. Tzeng, Thermal stresses in functionally graded beams, AIAA J. 40 (2002) 1228–1232.

[10] S.S. Vel, R.C. Batra, Exact solution for thermoelastic deformations of functionally graded thick rectangular plates, AIAA J. 40 (2002) 1421–1433.[11] L.F. Qian, R.C. Batra, L.M. Chen, Analysis of cylindrical bending thermoelastic deformations of functionally graded plates by a meshless local Petrov–

Galerkin method, Comput. Mech. 33 (2004) 263–273.[12] Y. Tanigawa, T. Akai, R. Kawamura, N. Oka, Transient heat conduction and thermal stress problems of a nonhomogeneous plate with temperature-

dependent material properties, J. Thermal Stresses 19 (1996) 77–102.[13] G.N. Praveen, J.N. Reddy, Nonlinear transient thermoelastic analysis of functionally graded ceramic–metal plates, Int. J. Solids Struct. 35 (1998) 4457–

4476.[14] W. Lanhe, Thermal buckling of a simply supported moderately thick rectangular FGM plate, Compos. Struct. 64 (2004) 211–218.[15] J.S. Park, J.H. Kim, Thermal postbuckling and vibration analyses of functionally graded plates, J. Sound Vib. 289 (2006) 77–93.[16] A.M. Zenkour, Generalized shear deformation theory for bending analysis of functionally graded plates, J. Appl. Math. Model. 30 (2006) 67–84.[17] R.A. Arciniega, J.N. Reddy, Large deformation analysis of functionally graded shells, Int. J. Solids Struct. 44 (2007) 2036–2052.[18] M. Bayat, M. Saleem, B.B. Sahari, A.M.S. Hamouda, E. Mahdi, Thermo elastic analysis of a functionally graded rotating disk with small and large

deflections, Thin-Walled Struct. 45 (2007) 67–691.[19] X.Y. Lee, H.J. Ding, W.Q. Chen, Pure bending of simply supported circular plate of transversely isotropic functionally graded material, J. Zhejiang

University (Sci. A) 7 (8) (2006) 1324–1328.[20] J.Y. Chen, H.J. Ding, W.Q. Chen, Three-dimensional analytical solution for a rotating disc of functionally graded materials with transverse isotropy, Arch.

Appl. Mech. 77 (4) (2007) 241–251.[21] J.Y. Chen, W.Q. Chen, 3D analytical solution for a rotating transversely isotropic annular plate of functionally graded materials, J. Zhejiang University

(Sci. A) 8 (7) (2007) 1038–1043.[22] X.Y. Li, H.J. Ding, W.Q. Chen, Elasticity solutions for a transversely isotropic functionally graded circular plate subject to an axisymmetric transverse

load qrk, Int. J. Solids Struct. 45 (2008) 191–210.[23] J.F. Durodola, O. Attia, Deformation and stresses in FG rotating disks, Compos. Sci. Technol. 60 (2000) 987–995.[24] J.F. Durodola, O. Attia, Property gradation for modification of response of rotating MMC disks, J. Mater. Sci. Technol. 16 (2000) 919–924.[25] S.A.H. Kordkheili, R. Naghdabadi, Thermoelastic analysis of a functionally graded rotating disk, Compos. Struct. 79 (2007) 508–516.[26] H. Jahed, S. Sherkatti, Thermoelastic analysis of inhomogeneous rotating disk with variable thickness, in: Proc. of the EMAS Conference of Fatigue,

Cambridge, England, 2000.[27] H. Jahed, R. Shirazi, Loading and unloading behavior of thermoplastic disk, Int. J. Press. Vess. Piping 78 (2001) 637–645.[28] B. Farshi, H. Jahed, A. Mehrabian, Optimum design of inhomogeneous non-uniform rotating disks, Int. J. Comput. Struct. 82 (2004) 773–779.[29] M. Ruhi, A. Angoshtari, R. Naghdabadi, Thermo elastic analysis of thick-walled finite-length cylinders of functionally graded materials, J. Thermal

Stresses 28 (2005) 391–408.[30] J.N. Reddy, C.L. Huang, Nonlinear axisymmetric bending of annular plates with varying thickness, Int. J. Solids Struct. 17 (1981) 811–825.[31] J.N. Reddy, Mechanics of Laminated Composite Plates and Shells Theory and Analysis, CRC Press, USA, 2004.[32] T.K. Nguyen, K. Sab, G. Bonnet, First-order shear deformation plate models for functionally graded materials, Compos. Struct. 83 (2007) 25–36.