investigation of the e ect of pad geometry on flat and rounded fretting...

Scientia Iranica, Vol. 15, No. 3, pp 332{339c Sharif University of Technology, June 2008

Investigation of the E�ect of Pad Geometryon Flat and Rounded Fretting Fatigue

A. Mohajerani1 and G.H. Farrahi�

In this paper, the e�ect of geometrical parameters on the fretting fatigue of a half-plane in contactwith a at and rounded pad is studied. This is accomplished by calculating and comparing thestress states and stress intensity factors of fretting cracks for a number of pad geometries. Thepad geometry is represented by the radius of its rounded corners and the width of its central atpart. The distribution of dislocation method is employed to calculate the stress intensity factorsof fretting induced cracks of di�erent lengths for di�erent values of geometrical parameters. Theresults of this study can be insightful for improving the geometrical design of an aero-enginecompressor disk, as similar contact and damage are prone to occur in its dovetail region.

INTRODUCTION

Fretting fatigue is damage caused by a reduction of fa-tigue strength, due to low-amplitude oscillatory slidingmotions occurring between contacting surfaces. Thisdamage is responsible for much premature failure ofdisks and blades in turbine engines and, therefore, hasbeen the focus of many studies [1-10]. The assemblageof these components is normally undertaken by usingdovetail or �r-tree shaped attachments. The dovetailregion of aeroengine compressor disks experiences fret-ting in a at-on-at contact with a relatively largeradius at the edge of the contact [11]. Some studieshave been conducted to directly investigate cracks andstress states in the dovetail region [1,12-14]. Sinceexperimental setups are costly and stress states aremostly unknown, simpli�ed models of the dovetailregion are widely used for analysis purposes. One suchmodel is that of a at and rounded pad in contactwith a half-plane placed under the fretting loading.This model is favorable because of its similarity to themechanism of contact in the dovetail region and alsoits analytical tractability [11,15-20].

In the present work, the e�ect of pad geometryon the fretting fatigue of a half-plane, in the afore-mentioned model, is studied. This is accomplishedby calculating and comparing stress states and Stress

1. Department of Mechanical Engineering, Sharif Univer-sity of Technology, Tehran, I.R. Iran.

*. Corresponding Author, Department of Mechanical Engi-neering, Sharif University of Technology, Tehran, I.R.Iran. E-mail: [email protected]

Intensity Factors (SIF) of fretting cracks for a numberof pad geometries. The geometry of the pad isrepresented by the radius of its rounded corners and thewidth of its central at part. Knowledge of the e�ect ofthese geometrical parameters on the stress states andthe SIFs of the fretting cracks will help improve thedovetail attachment geometry designs.

To calculate the SIFs of fretting fatigue cracks,the distribution of dislocations method is used, whichis based on the numerical solution of integral equa-tions [21]. First, the stress states in the half-plane arecalculated for each of the pad geometries. Next, crackswith a variety of lengths at the contact boundary areconsidered and their SIFs are calculated in respect toeach of the pad geometries. A sample model is analyzedusing the Finite Element Method (FEM) to verify theresults.

STRESS STATE IN THE HALF-PLANE

Normal and Shear Traction Calculation

In this section, the method used to calculate stressstates are described, in respect to a typical geometryof the pad. Figure 1 shows the con�guration of thecontact model. The contact area has a width of 2a andthe stick zone width is 2c. The pad has a at centralpart of width 2b and two rounded corners of radius R.These corners are approximated by parabolic curves.As shown, the half-plane is subjected to, not only thenormal force, P , and the tangential force, Q, but also abulk stress, �b. Due to the exertion of bulk stress to thehalf-plane, the stick zone shifts by the value of e and

E�ect of Pad Geometry on Fretting Fatigue 333

Figure 1. A schematics illustration of a at and roundedcontact con�guration.

the shear traction distribution becomes asymmetric.Although the presence of bulk stress results in morecomplications in analytical solutions, it should be takeninto account, as the fretting fatigue normally takesplace in the presence of bulk stresses within one orboth of the contacting bodies, due to loadings otherthan the contact itself.

When the at rounded pad and half-plane havethe same modulus of elasticity, E, and Poisson ratio,v, the equation used to calculate the normal traction,p(x), can be written as [16]:

�ARp(x) = x ln�bpa2 � x2 + xpa2 � b2�2

a2 jx2 � b2j

� b ln�pa2 � b2 +pa2 � x2�2

jx2 � b2j

+ 2pa2 � x2 arccos b

a; (1)

where A is given by:

A =�+ 12G

;

� =

(3� 4v for plane strain3�v1+v for plane stress

(2)

As G is the modulus of rigidity. In this paper, onlycases of moderate bulk stresses are considered, forwhich tangential traction has the same sign over theentire contact area. Using the relation between sheartraction, q(x), and the tangential relative slip of surfacepoints, g(x), the unknown q(x) is calculated:

1A@g@x

=1�

Z a�a

q(�)� � xd� �

�b4: (3)

The shear traction can be de�ned as:

q(x) = q�(x) + f jp(x)j; jx� ej � c;

q(x)=f jp(x)j; a�x��c+e [ c+e�x�a; (4)where f is the coe�cient of friction. In the stick zone,there should be no relative movement (@g=@x = 0).Thus, by substituting Equation 4 into Equation 3 inthe stick zone, one obtains:

�b4� 1�

Z a�a

f jp(x)j� � x d� =

Z e+ce�c

q�(�)� � xd�: (5)

The Gauss-Chebyshev quadrature method can be usedto solve Equation 5 [22]. The same procedure that isused in [23] is applied here to �nd the shear tractionat certain points. The stick zone shift and �b arerelated variables and only one of them can be chosenarbitrarily. The relation between these variables canbe found by invoking the consistency condition [24].Through the calculation of normal and shear tractionson the contact surface, the values of normal andtangential forces can be found using the equilibriumconditions of the pad [16].

Interior Stress State

The stress state in the half-plane is given by thesuperposition of the contact-induced stress state andthe bulk stress. In order to evaluate the contact-induced stress state, the pressure and the shear trac-tions are considered as arrays of overlapping triangulartraction elements. The use of such elements results in apiecewise linear approximation to the surface tractionsand is, thus, free from discontinuities associated withthe piecewise method [25]. The stress state due toeach traction element is calculated and used to obtainthe total contact-induced stress state. For convenience,only the �rst component of the stress state, �x, is usedfor demonstration purposes in the following sections.However, it should be noted that this method can beapplied e�ciently to calculate all components of thestress.

FINITE ELEMENT MODELING

The con�guration of the �nite element model is shownis Figure 2. This model has been extensively usedin [11,23-27] because of its similarity to most frettingfatigue experimental setups. In the �nite elementmodeling of this con�guration, the lower plate wastaken as large enough to appropriately approximate ahalf-plane.

The model was loaded in two steps. In the�rst load step, normal force was applied to the padand, in the second step, the bulk stress, �b, wasapplied to the lower plate. Note that the bulk stressautomatically brings about the tangential force, Q. Asshown in Figure 3, in order to create the moment, M ,

334 A. Mohajerani and G.H. Farrahi

Figure 2. The con�guration of the �nite element modelunder fretting loading. Note that in order to introduce thetangential force, Q, the pad is constrained in x direction.

Figure 3. A schematic view of the pad loading. Staticequilibrium requires the pressure distribution to beasymmetrical to make the moment (M), opposite to themoment created by the tangential force around thecontact center (o).

necessary for the equilibrium of the pad, the pressuredistribution over the contact surface should be asym-metric. However, pressure distribution in homogeneouscontacts should be independent of tangential loads.The symmetry of pressure distribution is a signi�cantconcern in the �nite element modeling, and can besigni�cantly overcome by coupling the nodes of thepad's upper side in the y direction. The size andnumber of the crack tip singular elements were chosenaccording to [26].

The model was meshed using 8-node plane ele-ments. Mesh re�nement was undertaken in multiplesteps. At each step, the model was run and the stressvalues at some points of interest were recorded. Themesh re�nement was quitted when these stress valueswere stabilized.

In contrast to the contact mechanics approach ofanalyzing the stress state, in which the Q value is aninput to the problem, in this �nite element model, Qcould not be prescribed directly into the model. Thevalue of Q is a�ected by many factors, such as the

length and elasticity modulus of the pad arm (i.e. theextended side of the pad in Figure 2) and the bulkstress. In the current model, the pad arm was de�nedby a new set of material properties, so that its E valuescould be adjusted for obtaining the desired Q value.The adopted trial and error approach required severalruns of the model for the number of E values of thepad arm and for recording the corresponding Q values.Finally, an E value for the pad arm was found thatcorresponds to an acceptable value of Q.

DISTRIBUTION OF DISLOCATIONSMETHOD

The distribution of dislocations method is a convenientapproach for calculating the SIFs of cracks in half-planes under complicated stress states [21,27]. Thismethod is based on the numerical solution of integralequations that relates the relative displacement of thecrack faces to the stress state of an uncracked half-plane. Once the relative displacements of the crackfaces are calculated, the SIFs can be obtained usingthe well-known fracture mechanics relations of the SIFsand the relative displacement of the crack faces. Inthis method, �rst the stress values along the line of thecrack are obtained in its absence. Then, a distributionof dislocations is introduced along the line of the crack.Let N(x) be introduced as the total normal tractionacross the crack, and �T (x) as the stress along the lineof the crack in its absence. Based on the coordinatesystem depicted in Figure 4, the mentioned integralequation for cracks of length b can be written as [21]:

N(x) = �T (x) +G

�(�+ 1)

Z b0By(c)K(x; c)dc: (6)

Figure 4. A normal crack in a half-plane with the lengthb.


It should be noted that for traction free crack faces,N(x) is zero. In the above equation, By(c) is thedensity of the distribution of dislocations given by:

By(c) =dbydx

(c); (7)

where by is the length of the dislocation burger vector,which in the case of the present work is directed inthe y direction. K(x; c) is a generalized Cauchy kernelgiven by:

K(x; c)=2�

1x�c�

1x+c

� 2c(x+c)2

+4c2

(x+c)3

�: (8)

The �rst step involved in the numerical solution ofEquation 6 is to normalize the variables by introducingnew variables given by:

s =2xb� 1; r = 2c

b� 1: (9)

And by switching to a series representation, such thata system of simultaneous linear algebraic equationsremains, the following is obtained:

b2� G�(k+1)

nXi=1

2�(1+ri)2n+ 1

K(sk; ri)�(ri)=��T (sk):

By calculating the values of �(ri), �(1) can be obtainedas follows:

�(1) =2

2n+ 1

nXi=1

cot��2

�sin (n�)�(ri); (10)

where � = (2i�1)�2n+1 . Finally, the �rst mode stressintensity factor, denoted by KI , can be obtained:

KI = 2p

2p�b

G(�+ 1)

�(1): (11)

A similar approach can be applied to obtain the modeII stress intensity factor, KII .

DISCUSSION AND RESULTS

The role of pad geometry on fretting fatigue is studiedby observing the e�ect of varying each geometricalparameter on the half-plane stress states and SIFs offretting.

Three values are considered for the radius of thepad corners, R, as 50, 100 and 140 mm. For each Rvalue, the width of the central at part of the pad, 2b,is taken as 3, 6, 10 or 16 mm.

Consideration of the mentioned values for b and Rresults to twelve pad geometries. For each of these 12pad geometries, the SIFs were calculated for a numberof cracks with lengths from 0.1 mm to 1 mm. The plane

Table 1. The speci�cations of the contact model used inthe study.

E(MPa)

v f P(N)

Q(N)

�b(MPa)

126000 0.3 0.5 896 376 23.7

strain state was assumed. The other speci�cations ofthe models are shown in Table 1.

Since the values of a, c and e vary in accordancewith R and b, their values should �rst be calculated, us-ing the contact equilibrium equations and consistencyconditions for each of the contact geometries [16,23].

The e�ect of pad geometry on stress states in thehalf-plane are studied for each of the four cases shownin Table 2, where the highest and lowest values for Rand b have been chosen in their mentioned intervals toaccentuate the e�ect of parameter variations. Usingthe method described previously, the value of the x-component of the normal stress, �x, at the depth of0.1 mm (y = 0:1 mm), was calculated for each case.Figure 5 depicts the value of �x for the four cases.A peak value of �x can be observed near the contactboundary. It can be seen that increasing b and Rreduces the peak value of �x. Also, note that, as the

at part width of the pad increases, the e�ect of cornerradius on the peak value of �x is weakened.

Table 2. The geometrical parameters of the pads for thecases shown in Figure 5. In each case, the highest orlowest value of each parameter is chosen for a bettercomparison of the resulting stress states.

Case No. 1 2 3 4

b 8 8 1.5 1.5

R 50 140 140 50

Figure 5. The values of �x in respect to four padgeometries, in the depth of 0.1 mm.


A sample case was studied by FEM to verify theresults. For this sample model, parameters are chosento be b = 5 and R = 100. As shown in Figure 6,the values of �x, at the depth of 0.1 mm, are in goodagreement with FEM results. The values of KI forcracks with di�erent lengths in this model, are shownin Figure 7.

In order to investigate the e�ect of pad geometryon the SIF of fretting cracks, the �rst mode stressintensity factor, KI , versus crack length, is illustratedin Figures 8, 9 and 10 for R = 50, 100 and 140 mm,respectively.

As shown in Figures 8, 9 and 10, the e�ect ofpad at part width, b, on the value of KI , is highlya�ected by crack length. For shorter cracks, increasingb results in the decrease of KI . However, for crackslonger than a critical value, the e�ect of b on KI is

Figure 6. �x in the depth of 0.1 mm in the sample model.

Figure 7. The KI values vs. crack length in the samplemodel.

Figure 8. KI values vs. crack length for R = 50 mm.




reversed. This critical crack length varies, according tothe values of R.

Although KI and its variations have been themain concern of most fatigue studies, knowledge aboutKII values is essential too, for fretting fatigue studies.Therefore, the e�ect of pad geometry on KII valueshas also been investigated using the same methods andassumptions employed for the study on the �rst modeof stress intensity factors.

The KII values of the cracks in the sample model,calculated by FEM, and the distribution of dislocationsmethod are shown in Figure 11. The KII values ofcracks with di�erent lengths for R = 50, 100 and 140mm are shown in Figures 12, 13 and 14, respectively.As shown, increasing R and b can signi�cantly reducethe KII values.

Figure 11. The KII values vs. crack length in the samplemodel.

Figure 12. KII values vs. crack length for R = 50 mm.



CONCLUSIONS

In this paper, the e�ect of geometrical parameters onthe fretting fatigue of a half-plane in contact with a

at and rounded pad is studied. Some of the keyobservations made in this work can be summarized asfollows:

1. For the pads with small at parts, the �rst modestress intensity factor of the fretting crack, KI , isalmost independent of the crack length. It was alsoobserved that increasing the pad width decreasesthe KI values of short cracks and increases the KIvalues of longer ones. The crack length at thisturning point is highly a�ected by the pad cornerradius. Indeed, this is the only noticeable e�ect ofthe pad radius on the values of KI .

2. In contrast to the observations made on the vari-


ations of KI values, a relatively consistent patternwas observed on the e�ect of the pad geometry onthe second mode stress intensity factors, KII . Asthe crack length increases, a consistent increase inthe KII values takes place. Increasing each of thegeometrical parameters decreases the values of KII .Although this e�ect is less noticeable for very shortcracks, it can be clearly acknowledged for the longerones.

Results of the study show that no general rule can beestablished for the assignment of a pad at part widthto minimize fretting fatigue. However, increase of thepad corner radius can consistently be a good choice forweakening the fretting fatigue.

NOMENCLATURE

a half of the contact widthb half the width of the pad at partb crack lengthby length of burger vector of a normal

dislocationB(c) density of distribution of dislocationsc half of the stick zone widthe stick zone shift from the center of

contactE modulus of elasticityf coe�cient of frictionG modulus of rigidityg(x) tangential relative slip of contact

pointsK generalized Cauchy kernelKI �rst mode stress intensity factorKII second mode stress intensity factorM moment created by asymmetrical

pressure distributionN total normal traction on crack pathp(x) normal traction on contact areaP normal contact forceq(x) shear traction on contact surfaceQ tangential contact forceR radius of the pad cornersv Poisson's ratio�b bulk stress�T normal traction on crack path due to

loading

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investigation of the e ect of pad geometry on flat and rounded fretting...

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