co

F. M400g, Sysiaelang

Article history:

Keywords:Functionally graded materialBrake diskFinite element methodThermoelastic

of disk are assumed to be represented by power-law distributions in the radial direction. The inner and

conditions. Normally, brake disks are fabricated by using any metalor ber-reinforced composites. However, the gradual change ofmechanical properties can be tailored to different applicationsand working environments such as in aerospace where lightweight and durability becomes crucial in high temperature. Mate-rials in which the volume fraction of two or more materials is var-

plastic deformation in semi-innite medium by using FEM. Keand Wang [5] divided FGM into several sub-layers to develop amulti-layered model for sliding frictional contact analysis witharbitrarily varying shear modulus under plane strain-state defor-mation. Yevtushenko and Kovalenko [6,7] studied a solution ofthe axisymmetric contact problem for a half-space and slidingbody, taking into account transient heat generation due to frictionand wear. They found that heat generation from the action of fric-tion forces in frictional sliding contact leads to a signicant redis-tribution of contact pressure.

* Corresponding author. Address: Mechanical Engineering Department, Univer-sity of Malaya (UM), 50603 Kuala Lumpur, Malaysia. Tel.: +60 1 72087765; fax: +603 79675317.

Composite Structures 92 (2010) 15911602

Contents lists availab

S

sevE-mail address: Bayat@eng.upm.edu.my (M. Bayat).Brakes disks are mostly designed by using two sliding contactsurfaces. The values of pressure on both sides of the disk can be dif-ferent and this causes deection in brake disk. The contact surfacesalso generate heat. These disks when subjected to bending andthermal expansion may become misaligned. Maximum stressesand deections in brake disk can be controlled by designing thedisk with material properties that vary with position. The diskmay have uniform thickness. Brake disk is an example of solidand hollow rotating disks subjected to body force, thermal, frictionand bending loads. In this application, the performance of the com-ponents in terms of efciency, service life and heat dissipationcapacity depend on the material, speed of rotation and operating

certain dimension(s) of the structure from one point to the other[1,2] are called functionally graded materials (FGMs). These mate-rials which are mainly constructed to operate in high temperaturewere introduced as ultra light temperature-resistant materials forspace vehicles [3]. In the present study, the brake disk is made ofmetal-ceramic FGMs to take advantage of metal strength and heatresistant ceramic. The coefcient of friction of the brake varieswith radius. Thus, the present study aims at two objectives; deter-mination of displacement and stress elds, and evaluating the con-tact status. Finite element method (FEM) is used.

Some works on contact friction and frictional heating in homog-enous brake disk have been published. Hasan and Alaettin [4] ana-lyzed the effects of thermomechanical surface load on elastic1. Introduction0263-8223/$ - see front matter 2009 Elsevier Ltd. Adoi:10.1016/j.compstruct.2009.11.022outer surfaces considered are metal and ceramic, respectively. Pure material is considered for the brakepad. Coulomb contact friction is assumed as the heat source. It is divided into two equal parts betweenpad and brake disk which leads to thermal stresses. Mechanical response of FG disks are compared andveried with the known results from the literatures. The results show that the maximum value of radialdisplacement in mounted FG brake disk is not at outer surface. It is found that the all areas between padand brake disk is in full-contact status when the ratio of pad thickness to brake disk thickness is 0.66. It isobserved that the total strain due to thermomechanical load is negative for some parts of the disks,whereas, the thermal strains are always positive. It can be concluded that gradation index of themetal-ceramic has signicant effect in the thermomechanical response of FG disks.

2009 Elsevier Ltd. All rights reserved.

ied smoothly and continuously as a function of position alongAvailable online 3 December 2009An analysis of thermoelastic contact problem of functionally graded (FG) rotating brake disk with heatsource due to contact friction is presented. Finite element method (FEM) is used. The material propertiesFinite element analysis of thermoelasticgraded axisymmetric brake disks

M.M. Shahzamanian a,b, B.B. Sahari a,b, M. Bayat b,c,*,aMechanical and Manufacturing Engineering Department, Universiti Putra Malaysia, 43b Institute of Advance Technology (ITMA), Universiti Putra Malaysia, 43400 UPM, SerdancMechanical Engineering Department, University of Malaya, 50603 Kuala Lumpur, MaladAerospace Engineering Department, Universiti Putra Malaysia, 43400 UPM, Serdang, S

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

Composite

journal homepage: www.elll rights reserved.ntact problem in functionally

ustapha d, Z.N. Ismarrubie a

UPM, Serdang, Selangor, Malaysiaelangor, Malaysia

or, Malaysia

le at ScienceDirect

tructures

ier .com/locate /compstruct

s equivalent frictional stress (N/m2)x angular velocity (rad/s)Subscripts

Unless stated otherwise, the subscripts connote the fol-lowing meaning when applied with these variable P, K,E, r, R, a, l, q, h, q, U, u, and r.

i related to the inner surface of FG brake disko related to the outer surface of FG brake diski, pad related to the inner surface of pad brake disko, pad related to the outer surface of pad brake diskr related to the radial directionh related to the hoop directionz related to the vertical or transverse directionrh, hr, rz related to the shear (polar coordinate)m related to the metalmax related to the maximum value

site Structures 92 (2010) 15911602The effect of deection on the stresses in brake disks should beconsidered for accurate design and long life. Deection problemshave been solved by using shear deformation theory [812]. The

Nomenclature

D stressstrain matrix (N/m2)Dc bending stiffness related to ceramicE modulus of elasticity (N/m2)G Shear modulus (N/m2)h thickness (mm)K thermal conductivity (W/(m C))n volume fractionq pressure (kPa)P material propertyQT heat generation due to frictionr radius (mm)R non-dimensional radiusT temperature (C)Hf frictional dissipated energy converted into heatV sliding rate between the brake disk and pad brake (m/s)u displacement vector (m)U displacement (m)w vertical displacement (m)a thermal expansion coefciente straink factor of friction coefcientl friction coefcientleffect effective friction coefcientt Poissons ratioq mass density (kg/m3)r stress (N/m2)

1592 M.M. Shahzamanian et al. / Compolarge bending deection of annular plates with variable thicknesshas been analyzed in detail by Reddy and Huang [8]. They usedthe general Reisner plate equations as well as Von Karman plateequations and solved them by FEM. They studied the effects of ra-dius-to-thickness ratio (i.e. shear deformation), and nonlinearity inmaterial properties on stresses and deformations. Reddy et al. [9]applied the rst order shear deformation theory (FSDT) to indicatethe axisymmetric bending and stretching for FG circular plates. Thesolutions for deections, force and moment resultants of the FSDTwere presented in terms of the corresponding quantities of isotro-pic plates based on the classical Kirchhoff plate theory. The exactrelationship between the bending solutions of the classical platetheory (CPT) and the FSDT was developed for FG circular plates.Bayat et al. [10] used exact and semi-analytical solution to obtainthe small and large deection in FG rotating disk when the mate-rial properties change in thickness direction. Moreover, Bayatet al. [11] employed FSDT to study the FG rotating disk with axi-symmetric bending load. Bayat et al. [12] extended the previouswork to thermoelastic analysis of FG rotating disks with variablethickness. They compared their results for bending case with thatobtained by Reddy et al. [9].

Durodola and Attia [13,14] presented a nite element analysisfor FG rotating disks. A non-homogeneous orthotropic materialwas used to model the disks. Kordkheili and Naghdabadi [15] ap-plied semi-analytical method to obtain the thermoelastic solutionsfor axisymmetric FGMs rotating disks under plane stress condition.They compared their results with those of Durodola and Attia[13,14] under the centrifugal loading. Bayat et al. [16,17] used ex-act solution and semi-analytical method to obtain the elastic solu-tion for FG rotating disk with variable thickness subjected tomechanical load.

Many studies on FGMs were related to the analysis of thermalstresses and deformations. In recent years, Jabbari et al. [18] stud-ied the axisymmetric mechanical and thermal stresses in thick andshort FGM cylinders. They developed the exact solution of steady-state 2-D axisymmetric mechanical and thermal stresses for a

cr related to the ceramicref related to the stress-freeSuperscripts

Unless stated otherwise, the superscripts will have thefollowing meaning when used with the variable e, Eand k

el elasticth thermal non-dimensionalshort hollow cylinder made of FGM. Matsunaga [19] presented a2-D higher-order deformation theory for the evaluation for the dis-placements and stresses in FG plates subjected to thermal andmechanical loads. Bayat et al. [20] applied variable thickness FGrotating disk due to radial symmetry loads to obtain the mechan-ical and thermal stresses. Bayat et al. [21] also analyzed the ther-moelastic response of a uniform and variable thickness FGrotating disk with temperature-dependent material properties.

To the best of the authors knowledge, nowork has been reportedtill datewhich concernswith the analysis of FG brake disk subjectedto frictional heating. In the present study, a FG hollow rotating brakedisk, as shown in Fig. 1,with inner radius ri, outer radius ro, thickness

Fig. 1. Conguration of brake and pad disks.

h and axisymmetric with respect to z-axis subjected to contact withone homogenous material hollow disk is analyzed. The materialproperties of the constituent components of the brake disk are as-sumed tobe representedby apower lawdistributionwith the radiusof disk. Friction is considered to be the heat source that causes thethermal stresses. Thermal, bending, friction andbody forces are con-sidered on a FG brake. The effect of some basic factors such asmate-rial property gradation and ratio of sizes on deection, stress anddisplacement elds are investigated. However, the determinationof brake capacity will be the subject in our future paper.

2. Gradation relation

In the present study, the property variation, P, of the material inthe FG disk along the radial direction is assumed to be of the fol-lowing form [11,12]:

Pr Po Pi r riro ri

n Pi ri < r < ro 1

Here Po and Pi are the corresponding properties of the outer and in-ner surfaces of hollow disk respectively. Whereas, ro and ri are theouter and inner radius of the hollow brake disk respectively, andnP 0 is a grading index of the material. In this paper, the Poissonsratio, m, is assumed to be constant. The modulus of elasticity, E, themass density, q, thermal conductivity, K, thermal expansion coef-cient, a, and friction coefcient, l, are assumed to vary according togradation equation (1). As an example, the assumed form for themodulus of elasticity, E, is given by,

Er Eo Ei r riro ri

n Ei ri < r < ro 2

In the present paper, the disk has dimension ro = 100 (mm), ri = 20(mm) and h = 10 (mm). By using the non-dimensional radius Rand non-dimensional modulus of elasticity E, dened by,

R rro

; E EE0

3

.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1.0

0.9

0.8

0.7

0.6

0.5

0.46

Non-dimensional Radius (R=r/ro)

No

n-d

ime

nsi

on

al M

odu

lus

of E

last

icity

n = 0.5

n = 0.8

n = 1.0

n = 3.0 n = 5.0

Fig. 2. Gradation diagram of modulus of elasticity.

5

M.M. Shahzamanian et al. / Composite Structures 92 (2010) 15911602 15930.2 0.3 0.4 0.50.54

0.6

0.7

0.8

0.9

1.0

Non

-dim

ensio

nal F

rictio

n Co

effic

ient

n = 0.

n = 0.8

n = 1.0Non-dimensional

Fig. 3. Variation of friction coefcient with radi0.6 0.7 0.8 0.9 1.0

n = 3.0 n = 5.0Radius (R=r/ro)us for different values of grading index (n).

where 0.2 6 R 6 1. The variation of E versus R is shown in Fig. 2.Following the same method as [17], the assumed form for the

friction coefcient, l, becomes:

lr lo lir riro ri

n li ri < r < ro 4

The non-dimensional friction coefcient, dened as l, can be repre-sented as:

l 1 li R Ri1 Ri

n li 5

where li lilo and Ri riro. The variation of l versus R is shown in

Fig. 3.

3. Finite element analysis (FEA)

3.1. Finite element modeling

A nite element package ANSYS 10 is used in the present work.The FG brake disk is divided into two hundred elements as shownin Fig. 4. For thermomechanical loads plane 13 element is used forthe brake disk and brake pad.

By using ANSYS 10, FG brake is considered as target and brakepad as penetrator segments in order to dene the contact condition

between pad and FG brake disks. Therefore, Targe 169 and Conta171 are applied for brake and pad disks respectively. The materialproperties of FG brake disk are evaluated at the mean radius of theeach element. Increasing number of the elements improves theaccuracy of the results. MATLAB software is used for thepreprocessing.

3.2. Elastic relationships

Consider a hollow axisymmetric FG disk with uniform thicknesswith inner radius ri and outer radius ro, as shown in Fig. 1. The diskrotates at an angular velocity x. Due to the axial symmetry ingeometry and loading, cylindrical coordinate system (r, h, z) isused. The inner and outer surfaces of the FG disk are assumed tobe metal-rich and ceramic-rich, respectively. The elastic stressand strain relation can be written as:

frg Dfeelg 6

where

frg stress vector rr rh rz rrh rhz rrz T 7aD Stress strain matrix 7bfeelg Elastic strain vector feg fethg 7c

me

0

n=3

- -

1594 M.M. Shahzamanian et al. / Composite Structures 92 (2010) 15911602Fig. 4. Finite element

0.1 0.2 0.3 0.40

1

2

3

4

5

6

7

8

9

10

Non

-dim

ensi

onal

Dis

plac

emen

t in

Verti

cal D

irect

ion(

z)

Ceramic

Metal

n=10.0

n=1.0 n=0.5Non-dimens

Fig. 5. Non-dimensional displacements in vertical direction by present study, the contsh for the brake disk.

.5 0.6 0.7 0.8 0.9 1

.0

:Present study. - - - - : Presented by Bayat et al (2009)ional (R=r/ro)

inued lines are present study and dashed lines are presented by Bayat et al. [20].

current temperature and Tref is the reference temperature.

0 0 0 0 1=G 064 75

frictional dissipation QT is given by:

The friction coefcient depends on factors such as temperature,

siteQT Hf lr q V 12where Hf is the frictional dissipated energy converted into heat, q isthe contact pressure and V (V = r x) is the rate of sliding velocity.In the present case, it is assumed that the frictional energy is totallyconverted into heat and therefore Hf = 1.

4. Boundary conditions

4.1. Mechanical boundary conditions

The following traction conditions on the inner and outer sur-faces of the rotating hollow brake disk must be satised;

ur 0 at r ri 14and

rr 0 at r ro 15

4.2. Thermal boundary conditions

The heat generated QT is assumed to be divided equally betweenbrake and pad disks. The contact area is between 0.8 6 R 6 1 atupper surface z h. Hence, for the brake pad, the heat generatedis

Qbrake or pad QT2

lr V q2

16

5. Validation

For the validation of the results of this paper, the semi-analyti-cal method presented by Bayat et al. [20] and the exact solution re-ported by Reddy et al. [9] are used. Bayat et al. [20] considered FG0 0 0 0 0 1=G

and G is shear modulus.

3.3. Thermal relationships

In order to consider heat generation due to friction, a thermalstructural analysis is performed.

3.3.1. Heat generationIn the coupled thermalstructural contact modeling, the rate ofUsing Eq. (7c), total strain vector can be found as:

feg fethg D1frg 9where the exibility matrix [D]1 is:

D1

1=E t=E t=E 0 0 0t=E 1=E t=E 0 0 0t=E t=E 1=E 0 0 00 0 0 1=G 0 0

26666666

37777777 10with

feg Total strain vector 8afethg Thermal strain vector DT a a a 0 0 0 8bwhere a is the coefcient of thermal expansion, DT = TTref, T is the

M.M. Shahzamanian et al. / Comporoller-supported rotating solid disks with uniform pressure loadper unit area of qo = 0.14 GPa, x = 1000.0 (rad/s), EmEC 0:464;

qmqC

0:474 and t = 0.3. Bayat et al. [20] demonstrated that the non-moisture and lubricants. These parameters can reduce the effectivefriction coefcient. The percentage of friction coefcient (k) isintroduced and is given in Tables 38 to show the effect of frictionwith contact status.

Tables 38 show the contact status between pad and disk fordifferent values for pad thickness, grading index and percentageof friction coefcient. The effective friction coefcient is leffect = kland in this study leffect is used.

It can be seen that contact status changes from sticking to con-tact and then to near contact with the increase of the pad thicknessfor all values of k and n. In all Tables 38 the shape of contact area(C) is like a cone that means the contact between pad and disk in-creases with the decrease in the value of percentage of frictioncoefcient. As expected, the contact area between full-ceramic diskand pad (Table 3) is greater than those for other disks due to smal-dimensional vertical displacement wmaxw 64Dcq r4

is as given inFig. 5, when Dc Ech

3

121t2 and Ec 151:0 MPa.Plots of vertical displacement using the present FEM for FG

brake disks and semi-analytical method [20] are given in Fig. 5.It can be seen from Fig. 5 that the results are comparable withthose produced by mathematical method.

Reddy et al. [9] presented an expression for the non-dimension-alized maximum deection as wmaxw 64Dcq r4 with x = 0 (rad/s),qo = 0.14 GPa, EmEC 0:396; Ec 151:0 MPa and t = 0.288 of homog-enous roller-supported FG solid disk. The results of Reddy et al. [9]is given in Table 1 together with the present results.

It can be seen that the results of the present study as shown inTable 1 are very well comparable with those of Reddy et al. [9].Hence, the present nite element method is suitable.

6. Numerical results

For the numerical illustration of the thermoelastic solution ofthis study, a hollow brake disk with xed-free boundary conditionwith Ro = 5Ri and hbrake = 10 mm is considered. Aluminum is usedas inner-surface metal and zirconia as outer-surface ceramic, thesame as that considered in Ref. [17]. The corresponding materialproperties are listed in Table 2.

The brake disk is assumed to be subjected to a uniform centrif-ugal force due to x = 1000.0 rad/s and uniform vertical pressureq = 1000.0 KPa. Aluminium is considered for pad disk.

The friction coefcient between pad and brake disk is in therange of 0.75 6 l 6 1.4. A pad disk with (Ri,pad)/(Ro,pad) = 0.8 is con-sidered when Ro,pad = Ro. The suitable pad thickness will be deter-mined in the following section to achieve proper conditionbetween pad and brake disks.

6.1. Contact status for different values of pad thickness and gradingindex

This section is devoted to investigate the different types of con-tact status by considering different pad thickness and grading indi-ces to nd a suitable value of pad thickness. There are three typesof contact status namely contact (C), near contact (N) and sticking(S). Contact means that all elements of the disk and pad are in con-tact. Near contact means that some elements are not in contact andthey are close to contact. Sticking means that some elements of thedisk and pad stick together, even if only at one element [22]. Thefollowing results about contact status have been presented basedon element [22]. The contact condition used is that establishedby ANSYS 10. The contact status between pad and brake disks se-verely depends on the effective friction coefcient.

Structures 92 (2010) 15911602 1595ler friction coefcient between pad and full-ceramic disks. Fromthe numerical results for different pad thicknesses given in Tables

Table 1Comparison of non-dimensional vertical displacement wmaxw 64Dcq r4 of full-metal and full-ceramic roller supported solid disk in this study and Reddy et al. [9].

Dimensionless thickness (h/a) (wmax/w) for n = 0, (full-metal) (wmax/w) for n = 105 (full-ceramic)

Present study Reddy et al. [9] Present study Reddy et al. [9]

0.05 10.356 10.396 0.05 10.3560.1 10.430 10.481 0.1 10.4300.15 10.517 10.623 0.15 10.5170.2 10.708 10.822 0.2 10.708

Table 2Material property.

Material property E (GPa) t qkg=m3 aC1 K W=mC l between pad and pure material disk

Partially stabilized zirconia(PSZ), ceramic 151.0 0.3 5700 10 106 2.0 0.75Aluminum, metal 70.0 0.3 2700 23 106 209 1.4

Table 3Contact status of full-ceramic (n = 0).

Pad thickness (10 mm) 10 15 20 25 40 45 50 55 60 65 66 70 75 80 100 105 110

k = 1 S S S C C C C C C C C C C C C N Nk = 0.9 S S S C C C C C C C C C C C C C Nk = 0.8 S S C C C C C C C C C C C C C C Nk = 0.7 S S C C C C C C C C C C C C C C Nk = 0.6 S C C C C C C C C C C C C C C C N

Table 4Contact status of FG (n = 0.5).

Pad thickness (10 mm) 35 40 45 50 55 60 65 66 70 75 80 85 90 95 100 105 110

k = 1 S S S S S S C C C C N N N N N N Nk = 0.9 S S S S S C C C C C C C C N N N Nk = 0.8 S S S C C C C C C C C C C C C N Nk = 0.7 S S C C C C C C C C C C C C C C Nk = 0.6 S C C C C C C C C C C C C C C C N

Table 5Contact status of FG (n = 0.8).

Pad thickness (10 mm) 35 40 45 50 55 60 65 66 70 75 80 85 90 95 100 105 110

k = 1 S S S S S S C C C N N N N N N N Nk = 0.9 S S S S C C C C C C C C N N N N Nk = 0.8 S S S C C C C C C C C C C C C N Nk = 0.7 S S C C C C C C C C C C C C C C Nk = 0.6 S C C C C C C C C C C C C C C C N

Table 6Contact status of FG (n = 1.0).

Pad thickness (10 mm) 35 40 45 50 55 60 65 66 70 75 80 85 90 95 100 105 110

k = 1 S S S S S S C C N N N N N N N N Nk = 0.9 S S S S C C C C C C C C N N N N Nk = 0.8 S S S C C C C C C C C C C C N N Nk = 0.7 S S C C C C C C C C C C C C C C Nk = 0.6 S C C C C C C C C C C C C C C C N

Table 7Contact status of FG (n = 1.5).

Pad thickness (10 mm) 30 35 40 45 50 55 60 65 66 70 75 80 85 90 95 100 110

k = 1 S S S S S S C C N N N N N N N N Nk = 0.9 S S S S C C C C C C C C N N N N Nk = 0.8 S S S C C C C C C C C C C C N N Nk = 0.7 S S C C C C C C C C C C C C C N Nk = 0.6 S C C C C C C C C C C C C C C C N

1596 M.M. Shahzamanian et al. / Composite Structures 92 (2010) 15911602

60

SCCC

siteTable 8Contact status of full-metal (n =1).

Pad thickness (10 mm) 25 30 35 40 45 50 55

k = 1 S S S S S S Sk = 0.9 S S S S C C Ck = 0.8 S S S C C C Ck = 0.7 S S C C C C C

M.M. Shahzamanian et al. / Compo38, it can be suggested that by considering hpad = 6.6 mm, the con-tact status is C for all cases.

6.2. Results and discussions

In the following sections, the results are presented in non-dimensional form by normalizing the temperature, displacement,stress and strain by factors Tmax Tmin, qcr x

2r3oEcr

; qcrx2r2o andqcr

x2r2oEcr

respectively.

k = 0.6 S C C C C C C C

0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Non-dimensiona

Non

-dim

ensio

nal T

empe

ratu

re

n = 1.5

n = 1.0

n = 0.8

n = 0.5

Fig. 6. Non-dimensional temperature versus th

0.2 0.3 0.4 0.5-0.1

-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

Non-dimension

Non

-dim

ansio

nal V

ertic

al D

ispla

cem

ent

n = 0.5n = 0.8

n =

Fig. 7. Non-dimensional vertical displaceme65 66 70 75 80 85 90 95 100 105

S C N N N N N N N NC C C C C N N N N NC C C C C C N N N NC C C C C C C C N N

Structures 92 (2010) 15911602 15976.2.1. Temperature variationThe non-dimensional temperature distribution, T/(Tmax Tmin),

along the radial direction for FG brake disks mounted on a rigidshaft for different values of the grading index n is presented inFig. 6.

It can be seen that the maximum temperature decreases withthe decrease of the grading index n and the maxima of radial tem-perature for FG brake disks occurs after the rst contact pointRi;padRo

0:8

and that point is close to R = 0.804.

C C C C C C C C C N

0.9 1.0

l Radius (R=r/ro)e non-dimensional radius at contact area.

0.6 0.7 0.8 0.9 1

al Radius (R=r/ro)

Full ceramic

Full metal

1.0n = 1.5

nt versus the non-dimensional radius.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

Non

-dim

ensio

nal M

id-P

lane

Rad

ial D

ispla

cem

ent

Non-dimensional Radius (R=r/ro)

Full metal

Full ceramic n = 0.5 n = 0.8 n = 1.0 n = 1.5

Fig. 8. Non-dimensional radial displacement versus the non-dimensional radius.

0.8 0.9 1.00.70.60.50.40.30.2

0.04

0.035

0.03

0.025

0.02

0.015

0.01

0.005

0

-0.005

-0.01

Non-

dim

ansio

nal R

adia

l Stre

ss

Non-dimensional Radius (R=r/ro)

Full metal

Full ceramic

n = 0.5 n = 0.8

n = 1.0 n = 1.5

Selected area is shown in Fig. 10

Fig. 9. Non-dimensional radial stress versus the non-dimensional radius.

0.8 0.9 1.00.76-6

-5

-4

-3

-2

-1

0

1

Non

-dim

ansi

onal

Rad

ial S

tress

Non-dimensional Radius (R=r/ro)

Full metal

Full ceramicn = 0.5

n = 0.8

n = 1.0

n = 1.5

R = 0.804 R=0.854

Fig. 10. Non-dimensional radial stress versus the non-dimensional radius at contact area.

1598 M.M. Shahzamanian et al. / Composite Structures 92 (2010) 15911602

site-3

-2

-1

0

hear

Stre

ss of

out-p

lane

n = 0.5n = 0.8n = 1.0n =1.5

R=0.229

M.M. Shahzamanian et al. / CompoIt is observed that there are two intersection points(R = 0.856, R = 0.9958) at which the temperature is close to zeroand also for R 6 0.795 temperature is zero. This phenomenon canbe explained by the presence of interactive effect between heatgeneration, vertical pressure, centrifugal force and boundary con-ditions. It can be shown that the temperature is not zero beforethe rst contact point (0.795 6 R 6 0.8) due to heat conduction.

6.2.2. Thermomechanical resultsThe variations of non-dimensional vertical displacement due to

thermomechanical load for different values of the grading index nin FG brake disk are shown in Fig 7.

As expected, the absolute vertical displacement values for full-metal (aluminum) disk are greater than those for full-ceramic (zir-conia) disk due to higher modulus of elasticity of the latter. For FG

0.2 0.3 0.4 0.5-6

-5

-4

Non-dimansion

Non-

dimen

siona

l S R = 0.225

S

Fig. 11. Non-dimensional out-plane shear st

0.80.75-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

Non-dimansional

Non

-dim

ansio

nal S

hear

Stre

ss o

f out

-pla

ne

Full ceramic

Full metal

R = 0.76 R = 0.83 R = 0.837

597.0=R

R = 0.79

Fig. 12. Non-dimensional out-plane shear stress ver Full ceramic

Full metal

Structures 92 (2010) 15911602 1599disks, the displacements occur in between the values of full-cera-mic and full-metal. It can be noticed that the slope of the displace-ment curves is zero at inner surface due mounted boundarycondition.

Fig. 8 illustrates the non-dimensional radial displacement in theFG brake disk along its radius for different values of the grading in-dex n.

It can be seen that the radial displacement increases with theincrease of the grading index n from zero (homogenized zirconiabrake disk) up to its maximum value for n?1 (homogenized alu-minum brake disk). Also it is observed that the behavior of the ra-dial displacements in mounted FG brake disks is similar to purematerial disks. It is seen that for brake disk maximum radial dis-placement occurs in between the inner and outer surfaces due toexistence of thermomechanical loads.

0.6 0.7 0.8 0.9 1

al Radius (R=r/ro)

21.giFninwohssiaeraelected

ress versus the non-dimensional radius.

0.9 1 Radius (R=r/ro)

n = 0.5n = 0.8n = 1.0n =1.5

R = 0.903 R = 0.942

R = 1.0

sus the non-dimensional radius at contact area.

R

site0.30.2 0.4 0.5-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Non

-dim

ansio

nal T

otal

Rad

ial S

train

Full metal

Full ceramic

n = 1.5 n = 1.0

n = 0.8n = 0.5 R = 0.393

1600 M.M. Shahzamanian et al. / CompoThe non-dimensional radial stresses for FG brake disks mountedon a rigid shaft, for different values of the grading index n due tothermomechanical load are described in Figs. 9 and 10.

It is seen that the radial stress for FG mounted brake disk maynot lie in between the values for full-metal and full-ceramic diskas shown in Figs. 9 and 10. It is interesting to note that the radialstress has a local minimum close to the rst contact point (atR = 0.8). It is also noticed that close to R 0.7, the radial stressesfor disks are negative. Taking into account the boundary condi-tions, this phenomenon can be explained by the presence of theinteractive effects between the vertical pad pressure, centrifugalforce, heat generation and friction load. It is observed fromFig. 10 that for 0.804 6 R 6 0.854, the radial stress values for full-metal (aluminum) disk are greater than those for full ceramic (zir-conia) disk.

Figs. 11 and 12 depict the variation of non-dimensional out-of-plane shear stress due to thermomechanical load in the middleplane of brake disk.

It is noticed that the absolute maximum out-of-planes shearstresses (srz) for the brake disks occur at the inner surface. It can

Non-dimension

Fig. 13. Non-dimensional total radial strai

0.8-0.015

-0.01

-0.005

0

Non-dimensio

Non

-dim

ansio

nal T

otal

Rad

ial S

train

n = 1.5 n = 1.0n = 0.8n = 0.5

Fig. 14. Non-dimensional total radial strain versu0.6 0.7 0.8 0.9 1.0

41.giFninwohssiaera

= 0.68

Selected

Structures 92 (2010) 15911602be seen that the absolute shear stress (srz) in FG disks are smallerthan those in pure material brake disks when 0.209 6 R 6 0.79. It isalso shown that the shear stress values for pure material disks areclose to zero when 0.229 6 R 6 0.76 and 0.837 6 R 6 1. Fig. 12shows that, for 0.795 6 R 6 0.83 and 0.942 6 R 6 1, the shear stress(srz) in FG brake disks are greater than those in pure material brakedisks. It can be observed from Fig. 12, that for 0.837 6 R 6 0.903the shear stress in full-metal disk is greater than full-ceramic disk.

Figs. 13 and 14 demonstrate the non-dimensional radial straindue to thermomechanical load.

It is seen that the radial strains in FG brake disks are smallerthan those of full-metal disk when 0.2 < R < 0.68. The total radialstrain for FG mounted brake disks lie in between the values forfull-ceramic and full-metal disk for 0.2 6 R 6 0.393 as shown inFig. 13. It can be noted that the total radial strain values for full-metal (aluminum) brake disk are greater than those for other brakedisks when 0.2 6 R 6 0.62. In addition, as observed in Figs. 13 and14, that the extreme values for the radial strains occur at aboutR = 0.8. It is also observed that the total strain takes negative valueswhen 0.680 6 R 6 1.0.

al Radius (R=r/ro)n versus the non-dimensional radius.

0.9 1.0

nal Radius (R=r/ro)

Full metal Full ceramic

s the non-dimensional radius at contact area.

0ion

vers

siteThe variations of non-dimensional radial thermal strains due tothermomechanical load of the mounted FG brake disks for differentvalues of the grading index n are shown in Fig. 15. It is seen that thethermal radial strain is zero for 0.2 6 R 6 0.795. This phenomenoncan be explained by the presence of interactive effects betweenthermomechanical load and contact area. The temperature distri-bution is shown in Fig. 6. From Figs. 14 and 15 it can be observedthat the behavior of thermal radial strain is completely different ascompared with the behavior of total radial strain.

7. Conclusions

Functionally graded (FG) axisymmetric brake disks subjectedto thermal, bending, friction and body forces has been analyzedusing nite element method. The thermoelastic analysis of hollowFG brake disk (with outer radius = 5 times inner radius) with heatsource due to dry contact friction has been presented. Materialproperties are assumed to be represented by power-law distribu-

0.80

1

2

3

4

5

6

7

Non

-dim

ansio

nal T

herm

al R

adia

l Stra

in

Non-dimans

n = 1.5

n = 1.0

n = 0.8

n = 0.5

Fig. 15. Non-dimensional thermal radial strain

M.M. Shahzamanian et al. / Compotions in the radial direction while the inner surface is full-metaland the outer surface is full-ceramic and pure material is consid-ered for the hollow pad disk. Non-dimensional stress and dis-placement elds for the brake disk clamped at inner edge areobtained by applying coulomb contact friction as the heat source.These results are compared with those for rotating disk withbending load.

Some salient conclusions of this study can be summarized as:

The absolute vertical displacements in FG brake disks remainbetween the minimum displacement value for the full-ceramicdisk and maximum displacement value for the full-metal disk.

The maximum values of the radial displacements for mountedFG brake disks are not at outer surface, this maximum valuesare between inner and outer surfaces.

The radial displacement increases with the increase of the grad-ing index n from zero (homogenized zirconia brake disk) up toits maximum value for n?1 (homogenized aluminum brakedisk).

The radial stress in FG mounted brake disk does not lie inbetween the full-metal and full-ceramic disks.

The absolute out-of-plane shear stress values for FG disks aresmaller than those for pure material brake disks when0.209 6 R 6 0.79.

Thermal radial strain is positive in FG brake disk throughout.From the numerical results for FG brake disks presented in thisstudy, it can be concluded that the gradation of the constitutivecomponents is a signicant parameter in the thermomechanicalresponses of FG brake disks.

References

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[2] Reddy JN. Analysis of functionally graded plates. Int J Numer Methods Eng2000;47:66384.

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[5] Ke LL, Wang YS. Two-dimensional sliding frictional contact of functionallygraded materials. Euro J Mech A Solids 2007;26:17188.

[6] Yevtushenko AA, Kovalenko YV. The interaction of frictional heating and wear

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al Radius (R=r/ro)us the non-dimensional radius in contact area.

Structures 92 (2010) 15911602 1601at a transient sliding contact. Int Appl Math Mech 1995;59(3):45966.[7] Yevtushenko AA, Kulchytsky-Zhyhailo RD. Axi-symmetrical transient contact

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[9] Reddy JN, Wang CM, Kitipornchai S. Axisymmetric bending of functionallygraded circular and annular plates. Euro J Mech A Solids 1999;18(2):18599.

[10] Bayat M, Saleem M, Sahari BB, Hamouda AMS, Mahdi E. Thermo elasticanalysis of a functionally graded rotating disk with small and large deections.Thin-Walled Struct 2007;45:67791.

[11] Bayat M, Sahari BB, Saleem M, Ali A, Wong SV. Bending analysis of afunctionally graded rotating disk based on the rst order shear deformationtheory. Appl Math Model 2009;33(11):421530. doi:10.1016/j.apm.2009.03.00.

[12] Bayat M, Sahari BB, Saleem M, Ali A, Wong SV. A thermo-elastic solution of afunctionally graded variable thickness rotating disk with bending based on therst order shear deformation theory. Thin-Walled Struct 2009;47(5):56882.

[13] Durodola JF, Attia O. Deformation and stresses in FG rotating disks. Int JCompos Sci Technol 2000;60:98795.

[14] Durodola JF, Attia O. Property gradation for modication of response ofrotating MMC disks. Int J Mater Sci Technol 2000;16:91924.

[15] Hosseini Kordkheili SA, Naghdabadi R. Thermoelastic analysis of a functionallygraded rotating disk. Compos Struct 2007;79:50816.

[16] Bayat M, Saleem M, Sahari BB, Hamouda AMS, Mahdi M. On the stress analysisof functionally graded gear wheels with variable thickness. Int J Comput EngSci Mech 2008;9(2):12137.

[17] Bayat M, Sahari BB, Hamouda AMS, Saleem M, Mahdi E. Analysis offunctionally graded rotating disks with variable thickness. Mech ResCommun 2008;35(5):283309.

[18] Jabbari M, Bahtui A, EslamiMR. Axisymmetric mechanical and thermal stressesin thick short length FGM cylinders. Int J Pres Ves Pip 2009;86(5):296306.

[19] Matsunaga H. Stress analysis of functionally graded plates subjected tothermal and mechanical loadings. Compos Struct 2009;87:34457.

[20] Bayat M, Saleem M, Sahari BB, Hamouda AMS, Mahdi M. Mechanical andthermal stresses in a functionally graded rotating disk with variable thicknessdue to radial symmetry loads. Int J Press Vess Piping 2009;86(6):35772.

[21] Bayat M, Sahari BB, Saleem M, Hamouda AMS, Reddy JN. Thermo-elasticanalysis of functionally graded rotating disks with temperature-dependentmaterial properties: uniform and variable thickness. Int J Mech Mater Des2009:117. doi: 10.1007/s10999-009-9100-z.

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1602 M.M. Shahzamanian et al. / Composite Structures 92 (2010) 15911602

Finite element analysis of thermoelastic contact problem in functionally graded axisymmetric brake disksIntroductionGradation relationFinite element analysis (FEA)Finite element modelingElastic relationshipsThermal relationshipsHeat generation

Boundary conditionsMechanical boundary conditionsThermal boundary conditions

ValidationNumerical resultsContact status for different values of pad thickness and grading indexResults and discussionsTemperature variationThermomechanical results

ConclusionsReferences