A combined wear and crack nucleation–propagation methodology for fretting fatigue prediction

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International Journal of Fatigue 30 (2008) 1509–1528

JournalofFatigue

A combined wear and crack nucleation–propagation methodologyfor fretting fatigue prediction

J.J. Madge, S.B. Leen *, P.H. Shipway

School of Mechanical, Materials and Manufacturing Engineering, University Technology Centre in Gas Turbine Transmission

Systems, University of Nottingham, University Park, Nottingham NG7 2RD, UK

Received 2 August 2007; received in revised form 5 November 2007; accepted 7 January 2008Available online 20 January 2008

Abstract

This paper presents a finite element based methodology for predicting the effects of fretting wear on crack nucleation and propagationunder fretting fatigue conditions. The method combines wear modelling with critical plane, multiaxial prediction of crack nucleation andlinear elastic fracture mechanics prediction of short and long crack propagation. A global-sub modelling finite element approach isemployed for efficient wear and crack propagation simulation. The results are compared with available fretting fatigue test data and pre-vious critical plane wear-based predictions (which neglect the propagation phase) for Ti–6Al–4V across a range of slip amplitudes. It isshown that the separate modelling of nucleation and propagation in the presence of wear can significantly affect the prediction of fatiguelife under both partial slip and gross sliding conditions. It is predicted that wear under gross sliding conditions significantly retards prop-agation rate whereas wear under partial slip conditions is predicted to increase crack propagation rates across the slip zone. These resultsare consistent with observed phenomena of fretting fatigue cracking.� 2008 Elsevier Ltd. All rights reserved.

Keywords: Fretting fatigue; Wear modelling; Crack propagation; Crack nucleation; Finite element; Life prediction

1. Introduction

1.1. General

Fretting is a phenomenon which is associated with con-tacting surfaces nominally static with respect to each other,but which in fact, under operation, experience a relativesliding motion. This can lead to fretting wear in generaland, in the presence of fatigue loads in one of the compo-nents, to fretting fatigue. Traditionally, fretting has beenloosely described as corresponding to situations in whichthere is ‘small relative motion’, where ‘small’ is oftendefined as being less than approximately 100 lm, butVarenberg has pointed out [1], via comparisons betweenmacroscopic fretting tests and nano-scale fretting testsfrom an atomic force microscope, that since fretting can

0142-1123/$ - see front matter � 2008 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijfatigue.2008.01.002

* Corresponding author. Fax: +44 0115 9513800.E-mail address: [email protected] (S.B. Leen).

occur in the latter with relative displacements of only about5 nm to about 500 nm, a more specific definition isrequired. Varenberg has suggested that it is more preciseto say that the relative motion surfaces should have ‘anon-uniform distribution of local relative displacement attheir contact’ [1]. In terms of avoiding confusion betweenreciprocating sliding (non-fretting) and gross sliding (fret-ting) cases, the latter has also proposed the concept of aslip index, and it was argued that when the slip index valueis above 10, fretting conditions do not occur, i.e. recipro-cating sliding pertains. Many engineering applications suf-fer from fretting, causing substantial reductions inoperational life. Fig. 1 shows an example of fretting fatiguecracking in a laboratory scale spline coupling whichoccurred under an overload rotating bending momentloading [2]. The phenomenon is a complex one, and despitean increasing interest in the research topic, relatively littleis known about the mechanisms that drive failure, parti-cularly the interaction of the many variables which have

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Fig. 1. Fretting fatigue cracking location for a laboratory scaled splinecoupling under combined torque, axial and bending moment overload [2].

1510 J.J. Madge et al. / International Journal of Fatigue 30 (2008) 1509–1528

been observed to affect fretting fatigue life experimen-tally. Of the fifty fretting variables that have been sug-gested, the ‘primary’ variables are generally accepted tobe: contact pressure, slip amplitude, coefficient of fric-tion and bulk stress level. However, even in terms ofthese ‘primary’ variables, it is difficult to predict keyphenomena.

A number of different approaches have therefore beenapplied to develop understanding and predictive capability.Critical plane fatigue damage parameters and fracturemechanics techniques have proved to be relativelysuccessful.

1.2. Critical plane parameter approaches

Szolwinski and Farris [3] first applied the critical planeapproach to fretting. The method maximises a fatigue dam-age parameter (FDP) over a number of different planes andpredicts life based on the most damaging plane. TheSmith–Watson–Topper (SWT) [4] and Fatemi–Socie (FS)[5] parameters have been used for life predictions in a num-ber of studies with the critical plane approach [6,7],although others such as the modified shear stress range(MSSR) [8], and Crossland [9] parameters have alsoreceived attention. The Dang Van [10] parameter is a spe-cial case as it is argued to be microstructurally-based, butit does not provide a finite life prediction, only distinguish-ing between cases of failure and non-failure.

1.3. Fracture mechanics approaches

A number of investigators have applied fracturemechanics to fretting fatigue problems [11–13]. One benefitof this approach is that stress gradient effects can bedirectly addressed, if crack growth is explicitly modelledas in [12], so that phenomena such as crack self arrest

[14] can be predicted. Crack self arrest is an attractivefeature in fretting fatigue where nucleation is commonlyunavoidable under the severe stresses and surface damageat the contact interface. FDP methods cannot predictself-arrest, since they predict damage to continuously accu-mulate. However, a component that can be designed tocreate only self arresting cracks is inherently safe. Moo-bola et al. [14] state that to create self arrest in fretting, arelatively low fatigue load is required; indeed sometimesa bulk compressive load is required to suppress the actionof contact stresses on crack growth. Of course, there willbe some problems where highly stressed components area necessity (e.g. due to space or mass limitations), so thatit is not possible to achieve self arrest anyway. Previousinvestigations have tended to concentrate on either cracknucleation or crack propagation. One notable exceptionis the work of Araujo and Nowell [6], where an analyti-cal approach was employed to solve for both the nucle-ation life (to a 1 mm crack) and the propagation life (tofailure).

1.4. Wear effects

Direct modelling of the nucleation phase is difficult,although one possible approach is the application of poly-crystal plasticity techniques, e.g. [15]. However, thisapproach is computationally very intensive, particularlyfor application to industrial components and as a designtool, and the work of Goh at al. [15] was primarily focussedon ratchetting phenomena, which may be a contributorymechanism in crack nucleation. One objective of the pres-ent work is to develop a practical design tool for designingcomplex aeroengine components which experience frettingdamage.

Undoubtedly, wear damage plays a key role in cracknucleation under fretting fatigue conditions. Even if themechanism of wear damage cannot yet be modelleddirectly, simulation of material removal due to wear canhelp to capture key phenomena related to both nucleationand propagation. One example is the observed dependenceof fretting fatigue life on slip amplitude, particularly therecovery of life in the gross sliding cases. The classicalmechanistic explanation of this phenomenon has been thatthe ablation of the surface serves to remove micro-cracksbefore they can nucleate and/or propagate. Recent finiteelement (FE) studies at the University of Nottingham[16,17] have combined a wear modelling technique similarto that developed by McColl et al. [18], which simulatesthe cycle-by-cycle evolution of the contact surfaces due tomaterial removal, with a multiaxial SWT-based fatigue lif-ing approach, using a damage accumulation approach toaccount for the cycle-by-cycle evolution of stresses anddamage. The method successfully predicted the measuredeffect of slip amplitude on fretting fatigue life for a cylinderon flat Ti–6Al–4V contact, as reported by Jin and Mall[19], for a fretting test set-up as shown schematically inFig. 2. It was shown that two phenomena are responsible

Fig. 2. Schematic of cylinder on flat fretting fatigue test configuration.

J.J. Madge et al. / International Journal of Fatigue 30 (2008) 1509–1528 1511

for the effect of slip on life. One is the removal of damagedmaterial due to wear and the second is the redistribution ofcontact stresses, favourably in the case of gross sliding,unfavourably in the case of partial slip, due to the geome-try evolution. Fig. 3 shows the contrasting cases of (a)gross sliding where the peak contact pressure is graduallyreduced and the contact patch increased with increasingnumbers of fretting cycles and (b) partial slip where a local-ised peak develops at the stick-slip interface. However, thiswork did not distinguish between crack nucleation andpropagation. Therefore, the purpose of the present workis to present an FE-based fretting fatigue prediction meth-odology which combines the effects of cyclic wear simula-tion with crack nucleation and propagation prediction,including short crack growth prediction. The advantageof the FE basis is that it facilitates application to morecomplex geometries for more direct application to indus-trial design problems, e.g. [20].

The new methodology will permit prediction of the fol-lowing key phenomena, which cannot be captured usingeach technique in isolation:

(1) Crack self arrest.(2) Dependence of fretting fatigue life on slip amplitude.(3) Effect of wear on crack growth.(4) Relative importance of nucleation and propagation

behaviour.

2. Methodology

2.1. Overview

Fig. 4 shows a schematic representation of the proposednew methodology, which combines wear modelling withcrack nucleation and crack propagation prediction. Thetotal fatigue life is defined as

N f ¼ N i þ Np ð1Þwhere Ni is the predicted number of cycles for nucleation ofa short crack, typically about 10 lm depth, and Np is thepredicted number of cycles for this short crack to growto failure, which is defined here as the number of cyclesat which the crack length becomes equal to half the fatiguespecimen depth. The wear analysis is used to determine theevolution of both contact and subsurface stress and straindata, which are used to determine the nucleation life Ni andlocation. A fracture mechanics sub-model is employed tosimulate crack growth from the predicted nucleation site.As described below, both short and long crack growthare included. Surface tractions are applied to the sub-mod-el which mimic the contact stresses that are predicted bythe wear model. Hence, the effect of wear on both nucle-ation and propagation can be investigated.

2.2. Nucleation life prediction

Nucleation is modelled using a critical-plane implemen-tation of the SWT fatigue damage model, described indetail in [17]. According to the SWT model, for any givenloading cycle, defined by a maximum strain range over onecycle, De, and a peak normal stress rmax, on any givenplane orientation, the predicted damage per cycle is 1/Ni,where Ni is defined by

SWT ¼ rmaxDe2¼ r02f

Eð2N iÞ2b þ r0fe

0fð2N iÞbþc ð2Þ

where r0f is the fatigue strength coefficient, E is Young’smodulus, b is the fatigue strength exponent, e0f is the fatigueductility coefficient and c is the fatigue ductility exponent.An inconsistency in the use of Eq. (2) for predicting nucle-ation of a 10 lm crack relates to the fact that the fatigueconstants on the right hand side are normally obtainedfrom fatigue tests where the associated number of cyclesis taken as either corresponding to specimen failure ordetection of a 1 mm crack. This is discussed further below,where a method is presented to redress this inconsistency.

Due to the evolving nature of the stress–strain historyfrom cycle to cycle, as a consequence of material removal(see Section 2.4) and the associated stress redistributions,it is necessary to employ a damage accumulation frame-work. As in [17] Miner’s rule is employed, whereby nucle-ation is defined to have occurred at a material pointwhen the total accumulated damage x reaches a value of1, where x is defined as

x ¼XNi

j¼1

1

Ni;jð3Þ

where Ni,j is the critical-plane SWT predicted number ofcycles to failure for fretting cycle j (found using Eq. (2))and Ni is the total number of fretting cycles to nucleation,which is the key output from this calculation. This methodcan also provide information on the orientation of thenucleated crack.

Fig. 3. Contact pressure evolutions due to wear for: (a) gross sliding for calculated contact slip (d) of 8.7 lm, and (b) partial slip for d of 2.8 lm [17].


2.3. Crack growth

The stress intensity factor K is used to characterise thecrack tip stress field. As mentioned, the contour integralmethod is used to assess the stress intensity factors for boththe opening mode (K1) and the sliding mode (KII) aroundeach fretting cycle, furnishing a maximum and minimumvalue for any given cycle analysed. The effective stressintensity factor range DKeff is then found [21]

DKI ¼ KI;max � KI;min ðwhere KI P 0Þ ð4ÞDKII ¼ KII;max � KII;min ð5Þ

DKeff ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðDK2

I þ DK2IIÞ

qð6Þ

The well-known Paris law can be used to correlate crackgrowth rate with stress intensity factor range. A range ofdevelopments have been suggested for the Paris crackgrowth law since its original conception to allow more gen-

eral application. The version used here makes use of mod-ifications which allow better description of near threshold-and short crack growth behaviour. Cracks are found toself-arrest if the stress intensity factor range is below somecritical threshold DKth. Short cracks are often found topropagate at rates significantly higher than a long crackwith an equivalent SIF. In addition short cracks can prop-agate at stress intensity factor ranges below DKth. El-Had-dad et al. [22] proposed that the growth of a short crack oflength l can be described by analysing the crack as if it wereactually of length (l + l0) so that

DK ¼ QDr1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipðlþ l0Þ

pð7Þ

where Q is the geometric form factor, Dr1 is the far fieldstress and l0 is a ‘fictitious’ crack length which is foundto correspond with the Kitagawa and Takahashi [23] ‘crit-ical crack size’ from threshold fatigue behaviour and is de-fined as follows:

Fig. 4. Schematic showing three stages in the proposed wear–nucleation–propagation fretting fatigue methodology.


l0 ¼1

pDK th

re

� �2

ð8Þ

where re is the fatigue limit.As a finite element formulation is used to determine the

stress intensity factor, it is necessary to approximate the El-Haddad stress intensity factor DKeff,EH as a perturbation ofthe FE determined value DKeff, as follows:

DKeff ;EH ¼ DKeff

ffiffiffiffiffiffiffiffiffiffiffiffilþ l0

l

rð9Þ

This approach means that the effective threshold SIF belowlo increases linearly with increasing (short) crack length.The crack growth law used in this study is thus formulated

dldN¼ C DKeff

ffiffiffiffiffiffiffiffiffiffiffiffilþ l0

l

r !m

� Kmth

!ð10Þ

where dldN is the cyclic growth rate, C and m are the Paris

constants. In a study specifically aimed at comparativelyassessing a number of different approaches for modellingshort and long crack growth in fretting fatigue, Navarroet al. [24] concluded that Eq. (10) gave the best correlationwith measured life, compared with a number of alternativecrack growth laws for fretting fatigue of Al-7075.

Under the multiaxial stress state created by frettingloads it is possible for cracks to change direction. A num-ber of different criteria have been suggested in the litera-ture. The approach used here is the Sih [25], minimumstrain energy density function. For two-dimensional prob-lems the strain energy density function S is given by

S ¼ c11K2I þ c12KIKII þ c22K2

II ð11Þ

where

c11 ¼1

16pGð3� 4m� cos hÞð1þ cos hÞ ð12Þ

c12 ¼1

8pGsin h � ðcos h� 1þ 2mÞ ð13Þ

c22 ¼1

16pGð4ð1� mÞðcos h� 1þ 2mÞð1� cos hÞ

þ ð3 cos h� 1Þð1þ cos hÞÞ ð14Þ

with h as the angle between the existing crack propagationdirection and the new propagation direction, m is Poisson’sratio, taken here as 0.32, and G is the shear modulus. h isselected such that S forms a local minimum. If more thanone minimum occurs within the domain �p < h < p, theminima chosen is that which gives the maximum absolutevalue of S [26].

2.4. Wear modelling

The geometry evolution of the contact due to wear ismodelled using a modified version of the Archard equation[17]

Dhðx; tÞ ¼ k � pðx; tÞdðx; tÞ ð15Þwhere Dh(x, t), p(x, t) and d(x, t) are the incremental weardepth, contact pressure, and relative slip at a node at hor-izontal position x, where the x-axis origin is at the centre ofthe contact, and time t, respectively, and k is the wear coef-ficient.

To model each individual cycle would be too computa-tionally expensive, so a cycle jumping technique isemployed. Wear is assumed to be linear over a given cycle


jump DN. In this way one FE cycle simulates the cumula-tive wear effect of DN cycles

Dhðx; sÞ ¼ DNkpðx; sÞdðx; sÞ ð16Þwhere s is the time within one ‘cycle’ corresponding to DNwear cycles. The spatial adjustment of the contact nodes isachieved via a user subroutine called UMESHMOTIONand within an adaptive meshing framework in the generalpurpose, non-linear FE code ABAQUS [27]. When this cy-cle jumping technique is combined with Eq. (3), the nucle-ation damage rule becomes

x ¼XN t

DN

j¼1

DNNi;k

ð17Þ

where Nt is the total number of fretting cycles simulatedand Ni,k is the critical-plane SWT predicted number of cy-cles to failure for fretting cycle-jump k. Given the adoptionof a critical-plane approach to calculate fatigue damage, itis necessary to make an assumption about how damage ona given plane interacts with damage on other planes fromcycle to cycle. One approach is to calculate the criticalplane SWT for each wear increment and simply accumulatethis value from wear increment to increment, irrespective ofthe fact that the associated orientations of the critical planemay change with wear, i.e. assume the critical-plane SWTdamage to have an isotropic damaging effect. An alterna-tive approach is to calculate SWT values for all plane ori-entations for each wear increment and to accumulate theincremental damage on each plane. The latter approachis significantly more computationally intensive, requiring

Fig. 5. Schematic showing transfer of traction dat

storage of damage on 36 different planes at each elementcentroid (note that a rectangular grid of 1700 elements ismonitored for damage here) for each wear increment (typ-ically about 100 wear increments). A more rigorous alter-native again would be an anisotropic damage mechanicsapproach, which could include interaction effects betweendamage on different planes, but this is beyond the scopeof the present paper.

Due to the use of adaptive meshing to update the meshon removal of material, specific elements and nodes are nolonger linked uniquely to actual material points through-out the analysis. Consequently, a ‘material point mesh’(MPM) is created as the global reference for damage accu-mulation; the nodes of the MPM have fixed coordinatesthroughout the analysis. Cyclic damage is calculated atthe centroid of each element, over an area correspondingto 1.1a0 into the contact depth and 4a0 across the contactwidth (a0 is the initial contact semi-width for an unworngeometry) and linearly interpolated back to the MPM foraccumulation. In this way, nodes on the MPM correspond-ing to removed material (due to wear) do not accumulateany further fatigue nucleation damage.

3. Finite element modelling

3.1. Overview

A global-sub model approach is employed to solve thecrack propagation problem in the presence of wear-induced geometry modification. Fig. 5 shows the global

a from global wear model to crack submodel.


model and crack growth sub-model of the fretting fatiguetest arrangement, as discussed in more detail by Madgeet al. [17], and based on the experimental work of Jinand Mall [8], the specimen geometry of which is shown inFig. 6. The time histories of contact pressure p(x, t) andshear traction q(x, t) are calculated using the global modelover Nt fretting cycles, along with crack nucleation data forinput to the SWT critical plane calculations. p(x, t) andq(x, t) are then passed to the sub-model and crack growthis simulated over Nt fretting cycles. Fig. 7 shows a moredetailed flowchart of the methodology, which has beenautomated via a MATLAB program, created to drive thecrack growth analysis, in conjunction with ABAQUS.

3.2. Material, loading history and model details

As shown in Figs. 2 and 6 [8], a pair of cylindrical(radius 50.8 mm) fretting pads are held in contact with aflat, uniaxially-loaded fatigue specimen (depth 3.8 mm).The material used in the experiments is a dual phaseTi–6Al–4V alloy which consists of a hexagonal closepacked and b body centred cubic phases. The materialwas solution heat treated at 935 �C for 1.75 h, cooled inair and annealed at 700 �C for 2 h in vacuum and cooledin argon. A symmetry plane parallel to the specimen axisis employed in the global FE model so that only one padneeds to be modelled (Fig. 5). A material model of Ti–6Al–4V with a Young’s modulus of 126 MPa, a yield stressof 930 MPa and a Poisson’s ratio of 0.32 was used.

Fig. 8 shows the load cycle history for the first few loadcycles. In the first analysis step a normal load P of 208N/mm is applied to the cylinder, resulting in a peak Hertz-ian contact pressure of 302 MPa. In the next step the spec-imen is loaded by a cyclic fatigue load r(t) with a maximumvalue rb of 550 MPa and a stress ratio R of 0.03. The pad isalso loaded with a prescribed maximum displacementdapp,m, which is varied for each simulation to cover a rangeof slip amplitudes including partial slip through to gross

Fig. 6. Schematic of: (a) the fretting fatigue specimen, and (b) frettingpad. After Jin and Mall [8].

sliding conditions. Note that dapp(t) is the distance bywhich the fretting pad is displaced relative to the globalcoordinate axis. The ‘slip range’ d is the maximum slipobserved on the contact surface over one cycle. To modelthe effect of wear, dapp(t) and r(t) are applied cyclically inthe FE model. Using the von Mises yield criterion, themacroscopic material behaviour is predicted to remainelastic throughout the loading history.

The Lagrange multiplier contact algorithm was used tostrictly enforce the stick condition when the shear stressis less than the critical value according to the Coulomb fric-tion law. In the fretting experiments, the coefficient of fric-tion (COF) typically starts low, and rises to a higher valuewithin the first few thousand cycles (Jin and Mall [8]).Sabelkin and Mall [28] have suggested that a constantCOF value of 0.8 is representative for the tests studied here.

A value for wear coefficient k of 2.75 � 10�8 MPa�1 hasbeen estimated as representative of the Ti–6Al–4V frettingcontact pair of the present work (see Madge et al. [17] fordetails), using measured wear scar data from Magazineret al. [29]. The values of the Paris constants C and m

employed here are: C = 1.25 � 10 �11 MPa�2.59 mm�0.295

and m = 2.59 [6]. DKth is taken as 133 MPap

mm [12]. Arange of different l0 values between 10 and 60 lm have beensuggested for Ti–6Al–4V at different stress ratios [30]. Inthis study the value of l0 is taken as 20 lm, based on valuesof 133 MPa mm1/2 and 569 MPa for DKth and re, from[31], following the approach used in [32].

The accuracy of the global FE model has been validatedin [17] against available theoretical solutions for both par-tial slip and gross sliding.

The FE-implementation of the critical-plane SWTparameter follows the method described by Sum et al. [7].In practice this is achieved by transforming the time histo-ries of element centroidal stresses and strain ranges ontoplanes at 5� intervals over a 180� range using the two-dimensional transformation (Mohr’s circle) equations forstress and strain. The maximum normal stress rmax withrespect to time, and the corresponding strain range Deare determined for each of the 36 planes in each element.De is the difference between the maximum and minimumvalues of strain normal to the candidate plane over thecomplete loading cycle. Thus, SWT values are obtainedfor each candidate plane in each element. These valuesare then employed to establish the maximum critical planeSWT value with respect to plane orientation in each ele-ment, which in turn is used with Eq. (2) to furnish a num-ber of cycles to nucleation, Ni. The SWT implementationhas been successfully validated against corresponding FE-based predictions by Sum et al. [7] and volume-averagedresults of Araujo and Nowell [6] for partial slip frettingconditions of an Al4%Cu alloy.

3.3. Combining critical plane and LEFM approaches

Table 1 shows the Ti–6Al–4V SWT constants availablefrom Dowling [33], which have been derived from fatigue

Fig. 7. Flow chart showing computational sequence for submodel analysis.

σ(t)

δapp(t)

PP

b

app,m

min

Time

Fig. 8. Normal load, tangential displacement and substrate fatigue stresshistories implemented.

Table 1SWT constants for Ti–6Al–4V corresponding to a 1 mm crack (fromDowling [33])

r0f (MPa) b e0f c

2030 �0.104 0.841 �0.688


tests where failure is defined as the occurrence of a 1 mmcrack. However, the length scales of damage relevant tocrack nucleation in the present work are significantly

shorter than this; specifically, a crack is assumed to havenucleated here (i.e. x = 1) when it has reached a depth of10 lm. The element depth at the surface is 16 lm, so thatthe centroid, where damage is monitored is at 8 lm. Thisassumed to be sufficiently close to 10 lm to assume a10 lm crack, with negligible effect on the resulting predic-tions. Hence a modified set of fatigue constants is derivedcorresponding to 10 lm nucleation. The SWT equation is


based on the combination of the Basquin high cycle fatigue(HCF) equation, for stress to life relation, and the Coffin–Manson low cycle fatigue (LCF) equation, for plastic strainto life relation, as follows, respectively

Dr2¼ r0fð2N fÞb ð18Þ

Dep

2¼ e0fð2N fÞc ð19Þ

with consideration of the peak stress to account for themean stress effect. The approach adopted here is asfollows:

Table 2Estimated Basquin constants corresponding to 10 lm crack nucleation inTi–6Al–4 V

r0f (MPa) b

1817.2 �0.0978

200

250

300

350

400

450

500

550

4 6 8log 2N i

S (

MP

a)

1 mm crack0.01 mm crack

5 7 9

Fig. 9. Comparison of Basquin stress-life curves corresponding tonucleation of a 10 lm crack and development of 1 mm crack.

Fig. 10. An example mesh showing initial 10 lm surface defect and

1. Establish Nf across the 300–500 MPa range of stressamplitudes, corresponding to a 1 mm crack length,using the Basquin constants of Table 1.

2. Estimate Np, the numbers of cycles for a 10 lm crack togrow to 1 mm, for a series of stress levels across therange 300–500 MPa, using LEFM with the El-Haddadcorrection for short crack growth, using the solutionfor a semi-elliptical surface crack in a round bar undertension from [34]

KI ¼ F Irffiffilp

ð20Þwhere FI is taken to correspond to the position of max-imum SIF along the crack-front.

3. Calculate Ni = Nf � Np across the same range of stressamplitude levels and hence obtain the new Basquin con-stants corresponding to a 10 lm crack.

Table 2 shows the resulting modified Basquin constantsand Fig. 9 shows the 10 lm nucleation and the 1 mm crackstress-life curves.

3.4. Stress intensity factor determination

The model primarily uses modified pressure quadratictriangular elements, as these provide a more versatile re-meshing capability than quadrilaterals for potentially irreg-ular crack paths. The meshes are generated using the Dist-Mesh Matlab suite [35]. The contour integral facility withinABAQUS is used to evaluate the stress intensity factors; tothis end, a rosette of quadratic elements is implanted at thecrack tip to enable crack tip parameter evaluation (Fig. 10).For the rosette elements a quadratic quadrilateral formula-tion is used, the elements sharing the crack tip node use aquarter-point node spacing to provide the correct 1/

pr

strain singularity for LEFM, where r is radial distancefrom the crack tip. Figs. 11 and 12 show that this approachgives very good agreement with benchmark solutions from

detail of rosette of quadrilateral elements around the crack tip.

Fig. 11. Comparison of theoretical [36] and FE-based predictions of crack propagation in SENT specimen for transverse crack (plain fatigue).


the literature, for a range of slanted cracks in SENT spec-imens under plain fatigue [36,37].

3.5. Contact stress evolution treatment

The impact of wear on fretting fatigue crack growth isstudied here through the effect of evolving stresses on prop-agation. In the cases studied here the crack propagationrate is significantly higher than the wear rate, so that thereduction in length of a crack due to wear is negligible.The stress evolution aspect of wear is investigated byupdating the sub-model boundary tractions p(x, t) andq(x, t) derived from the global model wear analysis. Froma computational viewpoint, it is desirable to use largecycle-jump sizes for the wear simulation, in this case DN

of 1500 cycles, whilst ensuring that accuracy and stabilityare not compromised. However, the forward predictiontechnique used in the crack propagation simulationrequires a much smaller cycle-jump size, e.g. 100 cycles.Consequently, a linear interpolation scheme was imple-mented to determine the pressure and shear traction distri-butions for each crack growth cycle-jump, which areintermediate to the wear cycle-jump p(x, t) and q(x, t) dataavailable from the global wear model.

4. Results

4.1. Comparison with test data

Fig. 13 shows the effect of slip dapp,m on fretting fatiguelife, as obtained from: (i) the present methodology (referredto as ‘wear–nucleation–propagation’), which separately

predicts nucleation of a 10 lm crack and propagation ofthat crack to specimen half-depth, including the effects ofgeometry evolution due to wear, (ii) previous life predic-tions from [17] (referred to as ‘wear-SWT total life’), whichdid not distinguish between nucleation and propagation,but did include the effects of geometry removal due to wearand predicted life to a 1 mm crack, (iii) the predicted life tonucleation of a 10 lm crack (referred to as ‘wear–nucle-ation’) including the effects of geometry evolutions due towear and (iv) the measured lives from the work of Jinand Mall [8]. Note that the results of [8] have been cor-rected in order to relate the remote slips recorded experi-mentally to the more local slip dapp,m, as described in [17]and based on the relationship presented by Sabelkin andMall in [28]. Further, due to two slightly different appara-tus being used in [8], only those results obtained using aconstant normal load are considered here as this matchesthe boundary condition used in the model more closely.Fig. 14 shows an example of the wear depth evolutionfor the fatigue specimen under gross sliding conditions withan applied displacement dapp,m of 8.7 lm.

4.2. Effect of nucleation period

The combined nucleation–propagation predictions ofFig. 13 are predicated on the assumption of an initially per-fect material, with no inherent flaws, so that the nucleationlife is defined as the number of fatigue cycles from zero (ini-tial) damage to a damage level equivalent to a 10 lm crack.However, real materials will not be perfect but will haveinitial flaws and weaknesses which can be modelled as ini-tial damage. One of the benefits of the present approach is

Fig. 12. Comparison of benchmark solution [37] and FE-based solutions for SIF for slanted (angle shown is angle of crack to specimen surface) cracks inSENT specimens for: (a) Mode I, and (b) Mode II. (Note: curves show benchmark solution, crosses show authors’ FE data).


the ability to separate the nucleation and propagation lives,so that the nucleation life can be varied to simulate differ-ent numbers of cycles to nucleation. As the contact stressescontinuously evolve due to material removal effects, thecrack propagation life will be affected by the assumed (orcalculated) nucleation life. Therefore the effect on subse-quent fretting fatigue crack propagation of varying Ni isinvestigated here; this can be thought of as considering dif-ferent degrees of inherent damage, so that for Ni = 0, forexample, the material is assumed to already have a 10 lmcrack present, whereas for Ni = 10,000, the assumption isthat the 10 lm crack has occurred after 10,000 cycles, viz.the nucleation response of the material has been variedaccordingly, or alternatively, inherent damage in the mate-rial is assumed so that the additional number of cyclesrequired for a 10 lm crack is Ni = 10,000. Thus, for

Ni = 0, the crack is assumed to have nucleated immediatelyand initially grows under the influence of the unworn sur-face tractions, whereas for Ni = 10,000, for example, theinitial flaw grows in a worn contact stress field relating tothe 10,000th cycle.

In the present work, two dapp,m values have been stud-ied, one corresponding to an applied displacement dapp,m

of 8.7 lm and another corresponding to dapp,m = 2.9 lm.The FE-predicted contact slip distributions correspondingto both of these cases have been given in [17], where it isclear that the dapp,m = 2.9 lm case gives rise to a centralstick region from about x = �0.027 mm to aboutx = 0.1 mm, with non-zero slip elsewhere reaching a maxi-mum value of about 2.9 lm at one contact edge (x is dis-tance from the centre of contact). In contrast, thedapp,m = 8.7 lm gave no stick region, but still gave a

Fig. 14. Predicted evolution of wear depth distribution on fatigue specimen for different numbers of fretting cycles for the dapp,m = 8.7 lm case.

Fig. 13. Comparison of predicted fretting fatigue life as a function of local slip amplitude, dapp,m, (see Fig. 5) against test data of [8], including results of‘wear–nucleation–propagation’ method, along with ‘wear–nucleation’ and ‘wear-SWT total life‘ method of [17].


significantly non-uniform distribution on contact slip,again reaching a maximum value of about 8.7 lm at onecontact edge. It should be noted that in the present work(see Fig. 5) the point of application of the applied displace-ment is very close to the contact surface. Varenberg [1] haspointed out the importance of understanding the relativedifferences between applied displacement and contact slip,in assessing which fretting regime pertains in a given situa-tion. To this end the concept of a slip index was introduced,

with a view to uniquely defining the fretting regime and thepartial slip and gross sliding fretting regimes, with identifi-cation of each regime (and hence the associated values ofslip index) being based on the observed friction loops. Asdiscussed in more detail in [17], the two cases of gross slid-ing and partial slip studied here, correspond directly to twocases in [8], which have been identified as gross sliding andpartial slip also via the associated characteristic frictionloops.

Fig. 15. Predicted effect of nucleation time on the gross sliding propagation behaviour (xi/a0 = 1.02).


Fig. 15 shows the predicted effect of varying Ni on thesubsequent crack propagation under gross sliding condi-tions. If nucleation is assumed to occur instantly the prop-agation period is 4300 cycles, whereas if nucleation isjudged to have occurred at 1 � 105 cycles, the total propa-gation time is 7600 cycles. This is longer than that found inthe plain fatigue case of the same bulk stress load. Fig. 16shows that towards the final stage of propagation, the pres-ence of the contact slows growth appreciably compared tothe plain fatigue case. This is attributed to the widening ofthe contact width with wear, such that the crack is eventu-ally under the contact and experiencing compressive stres-ses over an increasing portion of the loading cycle. It isimportant to note that the propagation period is small incomparison to the total life (�1%) for these gross sliding

Fig. 16. Comparison of predicted gross sliding crack propagation rates withprediction.

cases, so that for these cases, the nucleation period is themore important consideration in terms of fretting fatiguelife prediction.

Fig. 17 shows the predicted effect of varying Ni for thepartial slip case, for two different assumed nucleation loca-tions, namely at the stick-slip interface and at the edge ofcontact, while Fig. 18 shows the effect of varying Ni acrossthe complete (evolving) slip zone for the partial slip case. Ifthe nucleation period is assumed to be negligible, i.e.Ni = 0, the propagation life is observed to be a relativelystrong function of the nucleation location, with the edgeof contact found to provide the most rapid propagation.However, as wear is allowed to advance by increasing theassumed nucleation period, the propagation life is foundto become a much weaker function of the nucleation loca-

different nucleation periods (xi/a0 = 1.02) against plain fatigue growth

Fig. 17. The predicted effect of varying the nucleation period upon partial slip propagation behaviour at different cracking locations: (a) xi/a0 = 0.57, (b)xi/a0 = 1.02.


tion, i.e. propagation in the slip region has been acceleratedsignificantly by wear, specifically the development of thestick-slip interface pressure peak (see Fig. 3b). Conversely,the effect of wear at the edge of contact is to retard propa-gation slightly, which is also consistent with the reducingpressure at the initial contact edge (Fig. 3b). Under partialslip conditions wear creates a wider region over whichcracking could be expected, which helps to explain whycracking is often observed at multiple points along the slipregion in experimental studies.

4.3. Effect of bulk stress: crack arrest

In the experimental validation, the cases reported by Jinand Mall [8] use a relatively high bulk fatigue load of550 MPa. The results presented here suggest that under

these high stress conditions the majority of the fatigue lifeis spent in the initiation phase. The phenomenon of crackarrest arises when the stress intensity at the crack tip dropsbelow the threshold value for growth (Eq. (10)). This canoccur either due to the crack having grown into a lowerstress region or due to the stresses on the crack tip reducingdue to stress redistribution. The sub-modelling approachadopted for the crack propagation analysis obtains theevolving contact tractions from the global model, whichexplicitly models the geometry removal due to wear. It istherefore possible to decouple the evolving contact trac-tions p(x,t) and q(x, t) from the bulk fatigue stress r(t), toexplore the effect of varying the bulk stress under a con-stant surface traction evolution.

In order to investigate this effect and thereby illustratesome key fretting fatigue phenomena, a wear-induced con-

Fig. 18. Partial slip predicted crack propagation life as a function of crack nucleation position.

Table 3Effect of bulk stress variation on gross sliding case results

Bulk stress(MPa)

Final length(mm)

Comments

0 With wear 0.01 No growthNo wear 0.01 No growth

100 With wear 0.026 Arrested 13,000 cyclesNo wear 1.8 (Failed) Failed 115,000 cycles

200 With wear 1.8 (Failed) Failed 40,000 cyclesNo wear 1.8 (Failed) Failed 35,000 cycles


tact stress evolution associated with a 550 MPa peak bulkfatigue load, is employed with two slip amplitudes;dapp,m = 8.7 lm (gross sliding) and dapp,m = 4.2 lm (partialslip). Each case is studied with a varying bulk fatigue stressapplied to the sub-model crack propagation analysis. Thepropagation behaviour is then investigated using ‘withwear’ and ‘without wear’ analyses. The ‘without wear’analyses use the unworn pressure distribution, whereasthe ‘with wear’ analyses use the evolving surface tractions,as predicted by the global model. For each case the crackis placed at the point of maximum damage as predictedby the nucleation model. For the gross sliding case this isat the edge of contact and for the partial slip case it is atthe edge of contact for the ‘without wear’ analyses and atthe stick-slip interface (xi/a0 = 0.121) for the ‘with wear’analyses.

Table 3 summarises the results of the gross sliding cases.Generally, wear has the predicted effect of slowing crackgrowth compared to when wear is neglected; this is demon-strated at the 200 MPa fatigue load. However, if the fatigueload (rb) is dropped to 100 MPa, the role of wear becomescritical in failure prediction; Fig. 19 compares the crackgrowth curves for the ‘with wear’ and ‘without wear’assumptions. The ‘without wear’ assumption predicts thatthe crack propagates to failure, whereas the ‘with wear’assumption causes sufficient attenuation of stresses thatthe crack arrests at a length of 26 lm after 1.3 � 104 cycles,having grown out of the most severe high stress regionlocal to the contact surface. If rb is reduced to zero it isfound that both assumptions lead to predictions of nocrack growth.

Table 4 shows the results of the partial slip cases. Inthe 0 MPa case, neither the ‘with wear’ nor ‘withoutwear’ analyses predict failure. However, Fig. 20 showsthat the ‘with wear’ analysis predicts that the crack is ini-

tially dormant, until about 4 � 104 cycles, at which pointthe contact stresses have evolved to a condition, which isfavourable for crack growth, after which the crack growsto a length of about 0.8 mm, at which point it arrests,due to it having grown out of the fretting fatigue contactstress field. When the bulk stress is increased to 50 MPa,the ‘no wear’ analysis predicts no growth, i.e. infinite life.However, as shown in Fig. 21, the ‘with wear’ analysispredicts that although crack growth is relatively slow,the pressure peak which develops at the stick-slip inter-face is enough to drive the crack through to failure. Atthe higher stress level of 100 MPa, both analyses predictfailure, after 1.51 � 105 cycles for the ‘without wear’model, and after 1.06 � 105 cycles for the ‘with wear’model.

5. Discussion

In the experimental cases modelled, the predicted prop-agation lives are short when compared to the totalexpected life, and they are similar for both gross slidingand partial slip, even though the stress gradients are sig-nificantly steeper and the peak stresses are higher forthe worn partial slip cases. In the gross sliding cases the

Fig. 19. Comparing ‘with wear’ and ‘without wear’ predictions of crack length vs. number of cycles for gross sliding case with dapp,m = 8.7 lm andrb = 100 MPa.

Table 4Effect of bulk stress variation on partial slip case results

Bulk stress level (MPa) Analysis type Final length (mm) Comments

0 With wear 0.8 Reactivated 30,000 cycles, arrested 400,000 cyclesNo wear 0.01 No growth

50 With wear 1.8 (Failed) Reactivated 30,000 cycles, failed 364 000 cyclesNo wear 0.01 No growth

100 With wear 1.8 (Failed) Failed 151 000 cyclesNo wear 1.8 (Failed) Failed 106 000 cycles

Fig. 20. With wear’ predicted crack growth curve for dapp,m = 4.2 lm partial slip case with rb = 0 MPa.


Fig. 21. ‘With wear’ predicted crack growth curve for dapp,m = 4.2 lm partial slip case with rb = 50 MPa.


propagation life is typically only around 1–2% of the fati-gue life. The rapid propagation of fatigue cracks underthis loading regime is consistent with the plain fatigueresults: simple 2D theoretical predictions of crack propa-gation life show that reducing the fatigue load from 550to 250 MPa would increase life from 7500 to 80,000 cyclesfor a similar single-edge notched tension (SENT) geome-try. In contrast, for the partial slip cases and the low life,low slip gross sliding cases, the predicted propagation lifeforms a greater proportion of the total life, e.g. up toabout 20% for a slip of about 3.5 lm. Recent work oncrack growth prediction in Ti–6Al–4V for fretting fatigueby Fadag et al. [31], without considering the effects ofwear, has analysed the proportion of crack propagationas a function of load, assumed initial crack length andfor both flat and cylindrical pad fretting pad geometries.It was shown that, for the intermediate and high cycleregimes (with total life greater than about 105 cycles), pre-dicted propagation life formed only a small part of totallife (less than 10%). This was reported to be in agreementwith experimental observations from [38] where the initia-tion and propagation proportions were estimated frompost-mortem microscopic observations of striations.Clearly, these latter results are consistent with the pre-dicted propagation proportions of Fig. 13.

Fig. 13 shows that the wear–nucleation–propagationmethodology predicts the same general trend with respectto slip amplitude as the results of Madge et al. [17] (whichare wear-SWT total life results) and the experimentaldata; specifically, a critical range of slip amplitudes is pre-dicted corresponding to the partial slip regime, whichresults in short fatigue lives. However, some differences

are apparent between the results of the present methodol-ogy and those of the wear-SWT total life approach. In thepartial slip region, the wear–nucleation–propagationmethod is more accurate than the wear-SWT total lifemethod, whereas in the gross sliding region, the converseis true.

Figs. 15 and 17 indicate that the propagation period isless sensitive to slip amplitude than the initiation period.Fig. 13 (‘wear–nucleation’ curve) shows that initiation lifecan vary by an order of magnitude with slip amplitude,whereas the maximum variation in propagation life is onlyaround a factor of 2.0. In the dapp,m > 1 lm partial slipregime the reduction in initiation life through the adoptionof short-scale constants is more than offset by the associ-ated propagation life, so that the wear–nucleation–propa-gation methodology predicts longer life than the wear-SWT total life approach. In the gross sliding regime, thepropagation life is small with respect to the reduction innucleation life caused by the use of short-scale fatigueparameters. Hence, under gross sliding conditions, thewear–nucleation–propagation methodology results in asmaller predicted life than the wear-SWT total lifeapproach of Madge et al. [17]. Due to the complexity ofthe interactions between wear, nucleation and propagation,it is difficult to identify precisely why this is the case. Oneplausible explanation is that the 10 lm nucleation life isunder-predicted, e.g. due to the simplifying assumptionsmade in Section 3.3, and this, compounded with the sensi-tivity of predicted nucleation life to slip amplitude, resultsin the differences observed in Fig. 13.

It is also worth pointing out a number of other relatedissues, as follows:


� The Coffin–Manson terms have not been modified in theprocess of identifying the ‘short-scale’ SWT constants.The 10 lm nucleation life may thus be underestimated,since the high cycle fatigue crack propagation behaviouris thus assumed to apply across the full range of stresslevels.� Crack closure and mean stress effects are not included in

the crack propagation model; this will also contribute todifferences between the wear-SWT total life and thewear–nucleation–propagation life.� The sub-model (crack propagation) analyses of this

paper, do not explicitly model the change of geometryassociated with wear, but only the associated (globalmodel) predicted surface traction evolution. Conse-quently, the predicted (sub-model) surface stress evolu-tion with fretting cycles will be slightly in error.� The use of a symmetry assumption means that it is

assumed that there are two cracks growing at the sametime from both sides of the specimen. This is not unrea-sonable when the cracks are short, since multiple crackinitiation sites are typical of fretting experiments, butas the cracks become longer, e.g. within linear elasticfracture mechanics regime, this assumption may becomeless realistic, since typically only one dominant crackgrows to failure. However, comparison of SENT anddouble-edge notched tension (DENT) SIF expressionssuggests that for short cracks, l/W < 0.05, the symmetryassumption will lead to only a slightly over-estimated(about 2%) SIF, which will in turn lead to only a slightlyunderestimated propagation life (by about 3%).

In summary, the wear–nucleation–propagationapproach gives better correlation with the test data forthe lower slip amplitudes (partial slip), whereas the wear-total life approach gives better correlation for the higher

Fig. 22. Comparison of ‘with wear’ predicted DKeff history with thresh

slip amplitudes (gross sliding). There are several works inthe literature, where reasonable life predictions have beenachieved considering only the propagation period, i.e.neglecting the nucleation period completely (e.g. [24]).However, the present work shows that the adoption of apropagation-only prediction model can lead to significantinaccuracies. The balance between nucleation and propaga-tion periods is dependent not only on the prevailing load-ing conditions, but also on the way in which the localstresses evolve with time.

The bulk stress study shows how critical it is to considerthe wear and propagation aspects of fretting modelling.Under gross sliding, it is found that the attenuation ofstresses due to wear can result in crack arrest whereas the‘without wear’ analysis would fail to predict the potentialbeneficial self arrest condition: the attenuation of stressesdue to wear is sufficient to stop the crack at a short length.Fig. 22 shows that initially the crack tip has sufficient stressintensity to drive propagation, but the reduction in contactpressure due to the gross sliding condition causes the crackto arrest.

The threshold envelope DKth,eff depicted is bounded bythe line

K th;eff ¼ K th

ffiffiffiffiffiffiffiffiffiffiffiffil

lþ l0

sð21Þ

This shows the value for DKeff at which crack growth willoccur at the current crack length, demonstrating a reduc-tion in the threshold value at short crack lengths.

The partial slip cases of Table 4 highlight that it wouldbe incorrect to assume that neglecting wear just results inmore conservative estimates for life when compared to anequivalent ‘no-wear’ analysis.

old envelope, for dapp,m = 8.7 lm gross sliding with rb = 100 MPa.

Fig. 23. Fretting map indicating the various wear induced phenomenaand their relation to slip and bulk stress level.


The predicted effects of wear are complex. Depending onthe slip distribution across the surface, and the substratestress, not only is accelerated or retarded failure predicted,but different predictions with respect to finite or infinite lifeare obtained. Fig. 23 shows the qualitative occurrence ofthese wear-induced phenomena in terms of the slip andrb variables on a fretting map.

It is important to note that the present methodologyemploys several parameters that are purely empirical andhave been obtained from fretting or other tests, such asstress- or strain-controlled fatigue tests, crack growth testsetc. Examples of these include coefficient of friction, wearcoefficient, SWT constants, Paris constants and the shortcrack growth threshold. This is advantageous in the sensethat this type of data is typically readily available, butnonetheless, it means that the present methodology is nota purely theoretical tool. It should be pointed out thatthe empirical parameters used in the model may be affectedby normal load, material properties, surface roughness,surface contamination etc. Hence, due attention wouldneed to be given to these considerations for different situa-tions, since the values could be expected to be different tothose employed here.

The present methodology shows that material removaldue to wear can have a significant effect of crack initiationand propagation, so that in gross sliding, for example, thecontact area increases due to material removal and hencethe contact pressure reduces. However, as discussed in[39], for example, contact area can also increase due tojunction growth, resulting from the combination of normaland tangential loading. This phenomenon is not includedin the present methodology, but its absence may contributeto the discrepancies between the predicted and measureddata and this aspect would need to be included in a morepurely theoretical methodology.

6. Conclusions

The wear–nucleation–propagation model presented inthis paper is the first attempt, to the authors’ knowledge,to address the interaction between crack nucleation andgrowth and wear effects in a finite element context. Thepaper has presented a study of the effects of wear on fret-ting fatigue crack propagation (and nucleation). Theresults show that the propagation of fretting fatigue crackscan be strongly affected by changes in the contact stress dis-tributions which are caused by wear. The conclusions areas follows:

� It is shown that the model captures the key effect of aminimum predicted fretting fatigue life with respect toa certain range of slip (or displacement) amplitude. Thisphenomenon has not been captured by other methodol-ogies, excluding the previously-published wear-total lifemethodology of the authors. Furthermore, the inclusionof crack propagation effects with the wear–nucleationapproach leads to improved life prediction accuracy inthe critical range of slip (minimum life), where the prop-agation life is predicted to form up to 20% of total life.� The evolution of contact stress under partial slip condi-

tions increases the rate of propagation of cracks in theslip zone to levels similar to those at the edge of contact,thus capturing the experimentally-observed phenome-non of cracking within the slip zone, as well as at thecontact edge. This is in contrast to models that neglectwear, which predict the highest propagation rates atthe edge of contact.� For the loading conditions considered, the study of

propagation in isolation does not give an accurate indi-cation of total fatigue life. Nucleation can be a signifi-cant component of total life.� A method has been suggested for deriving a set of short-

scale (nucleation) SWT fatigue constants, by combiningfracture mechanics with traditional long-scale SWT con-stants, to allow a more meaningful application of multi-axial fatigue damage models to problems involving highstress gradients, as found in contact fatigue.� It is shown that different loading combinations of the

same initial geometry can produce markedly differentrelative contributions of propagation and nucleation tototal life. However, it has been shown that the notionthat the distinction between nucleation and propagationaspects of life can be neglected has merit under certainconditions.� The complex interaction between wear and fatigue has

been highlighted by analyses using lower bulk fatigueloads:– The wear resulting from gross sliding can cause self

arrest in a case where a ‘without wear’ analysis wouldpredict crack propagation and component failure.

– Partial slip can result in a dormant crack propagatingfrom the stick-slip boundary to cause failure. Criti-cally, similar loading conditions used in a ‘without


wear’ analysis predicted that the crack tip loadingremains below the threshold SIF, indicating infinitelife.

Acknowledgements

The authors wish to thank Rolls-Royce plc, AerospaceGroup, for their financial support of the research, whichwas carried out at the University Technology Centre inGas Turbine Transmission Systems at the University ofNottingham. The views expressed in this paper are thoseof the authors and not necessarily those of Rolls-Royceplc, Aerospace Group.

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A combined wear and crack nucleation–propagation methodology for fretting fatigue prediction