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Tribology International 39 (2006) 1149–1157

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Propagation in fretting fatigue from a surface defect

Carlos Navarro�, Sergio Munoz, Jaime Domınguez

Dpto. Ingenierıa Mecanica y de los Materiales, Escuela Superior de Ingenieros de Sevilla, Camino de los Descubrimientos s/n 41092, Sevilla, Spain

Available online 23 March 2006

Abstract

This paper analyses the growth of cracks in fretting fatigue from an initial flaw at the surface. Different crack growth laws are used in

order to take into consideration the particular behaviour of short cracks. This methodology is applied to estimate life in various fretting

fatigue tests with spherical contact characterized by two different geometries. The material used in the experiments is Al7075. The two

geometries present significant differences in the evolution of the stresses, crack growth, etc. which are discussed. The approaches used to

model short crack growth give different results, some of them being in good agreement with the experiments.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Fretting fatigue; Fatigue life prediction; Fracture mechanics

1. Introduction

Different models have been proposed to determinefatigue life in cases of fretting. Some determine life byconsidering that the fatigue process is merely a growthprocess [1,2]. Others consider that the process is dividedinto two phases: crack initiation and propagation. Themethods that consider life as only a growth process bearthe inconvenience of having to start from an initial cracklength that is defined more or less arbitrarily, based on theexperience and considering some parameters as surfaceroughness, internal defects, size of the process zone, etc. Onthe other hand, the methods that combine initiation andpropagation have to define where initiation stops andwhere propagation begins. In this case, the separationbetween initiation and propagation (crack initiationlength) can be a fixed quantity defined a priori [3,4] or itcan be a variable value obtained as a result of a specificmethod [5]. The analysis of these two forms of combininginitiation with propagation is dealt with elsewhere [6].

This paper analyses the capability of some fatigue crackgrowth models to predict the fretting fatigue behaviour. Itconsiders the whole fatigue process as crack propagationfrom an initial flaw at the surface, and uses differentmodels to represent the growth of short cracks. Most of the

ee front matter r 2006 Elsevier Ltd. All rights reserved.

iboint.2006.02.004

ing author. Fax: +3495 448 7295.

ess: cnp@us.es (C. Navarro).

tests analysed in this paper were also studied by the authorsin [5]. It was shown that initiation was in the order of 10%of the total life, therefore, in this group of tests theestimations will be very similar when only fracturemechanics is used. For some combinations of the loadsapplied in a fretting test it may happen that the initiationlife is important compared to the total life. In these cases itwould be more appropriate to use the method proposedin [5].In the first place, different parameters (flaw size,

initiation site, angle, etc.) are defined in order to applythe models. Then, the different crack growth models usedare explained. Later, the experimental details of the testsused to compare the models are described. Following it, themodels are applied to simulate the fretting fatigue teststrying to estimate life, and the results obtained areanalysed, comparing the ability of each model to representthe fretting fatigue crack growth behaviour. Finally, someconclusions are obtained.

2. The models

As said before, the models for fretting fatigue lifedetermination to be analysed consider the propagationfrom a surface defect, therefore, the size of the defect has tobe defined and this depends on the material. The depthfrom where to consider propagation is somewhat arbitraryand the results will depend on it. For the cases under

ARTICLE IN PRESSC. Navarro et al. / Tribology International 39 (2006) 1149–11571150

analysis this influence is small and it will be shown later on.In any case, following the observations made by Lankford[7], it appears that a reasonable value for this materialwould be the radius of a typical inclusion, 5 mm. In thiscase, it is assumed that an inclusion on the surface wouldcrack during the first applications of the loads.

The different models explained in this paper to estimatefatigue life use the evolution of the stresses. The stress fieldis assumed to be elastic and the contact condition partialslip. It can be observed that the maximum stresses in thetests analysed are below the yield stress of the materialsused and the loads are applied in each test so that a partialslip condition is forced. The stresses produced by sphericalcontact are calculated analytically using the explicitexpressions obtained by Hamilton [8].

In order to apply the different models, other parameters,besides the initial crack length, need to be defined. Forexample, the point of initiation is chosen where themaximum of a multiaxial fatigue parameter is reached. Inthe case of spherical or cylindrical contact, this point layson the limit of the contact zone if the stresses are evaluatedon the surface, but it shifts inside the contact when thestresses are evaluated underneath, close to the surface. Thisresult can be obtained analytically when the combinationof the axial stress and the contact load does not producereversed slip in the slip zone. In this situation, it can beshown that it does not depend on the values of the loadsapplied. For the case of spherical contact, Fig. 1 shows thedistribution of the McDiarmid multiaxial fatigue para-meter (equivalent stress) [9], along the symmetry plane ofthe contact surface for one of the tests analysed in thispaper [10]. The equivalent stresses along other lines of thesymmetry plane at different distances, z, below the surfaceare also shown. Distances are normalized to the radius, b,of the contact zone, and equivalent stresses to the shearfatigue limit, t1. Selecting different points of the initiationband (Fig. 1) as starting points will produce differentfatigue lives. However, the fatigue life obtained varies very

-1.0 -0.5 0.0 0.5 1.00.0

0.5

1.0

1.5

2.0

2.5

initiationband

z =0 z/b = 0.05 z/b = 0.1 z/b = 0.2 z/b = 0.5

McD

iarm

id e

quiv

alen

t str

ess

σ eq/t 1

x/b

Fig. 1. McDiarmid equivalent stresses produced at different depths (z)

on the plane of symmetry of the contact zone for one of the tests

considered [10].

lightly from one case to another, and the worst case is whenthe crack initiates at the limit of the contact zone. Thus, forsimplicity, all the analysis will be carried out consideringthe limit of the contact zone as starting point.There is yet another parameter to consider: the crack

path. Fig. 2 shows a typical fretting fatigue crack. At thepoint of initiation these cracks grow with a high inclina-tion, but after reaching a depth of about 20 mm they rotateand grow almost perpendicular to the surface.Different crack paths can be used. A possibility is to

assume two phases in the crack growth, the first one beingan inclined crack with an angle obtained by applying the‘‘crack analogue’’ and the second phase being a crackperpendicular to the surface. The transition point betweenthe two of them can be defined using different criteria, e.g.the crack length where the SIF equals the threshold [11].Another option is to determine the crack growth directionat each point by calculating the orientation where Dsy ismaximum and growing the crack by a certain amount [12].Nevertheless, the simplest assumption is to consider thatthe crack grows in a straight line, and for most of the lifethat is true. In this case, different angles can be used but theone producing the shortest and most conservative lives is901 from the surface.Once all these parameters have been set, the SIF has to

be calculated. It can be used a weight function proposed byBueckner [13] for a through crack:

wðtÞ ¼1ffiffi

tp 1þm1

t

aþm2

t

a

� �2� �, (1)

where a is the crack length and t is an integrating variable thatruns from the crack tip to the surface. The parameters m1 andm2 are functions depending on the ratio a/W where W is thespecimen thickness. With this weight function the stressintensity factor (SIF) can be obtained from the expression

KI ¼

ffiffiffi2

p

r Z a

0

wðtÞsxðtÞdt, (2)

Fig. 2. Fretting fatigue crack [9].

ARTICLE IN PRESSC. Navarro et al. / Tribology International 39 (2006) 1149–1157 1151

where sx is the normal stress in the direction perpendicular tothe plane of the crack and t is the distance to the crack tip.This method of calculating the SIF incorporates the effect ofthe finite thickness of the specimen and is valid only for acrack smaller than the half-thickness ða=Wo0:5Þ. Moreover,the weight function of Eq. (1) was developed for an edgethrough crack, therefore the SIF has to be modified in orderto take into account the real shape of the crack which is semi-elliptical. The SIF obtained with Eq. (2) is divided by thefactor F, which was given by Irwin [14] and it depends on theaspect ratio of the crack

F ¼Z p

2

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 1�

a

c

� �2� �sin2j

sdj. (3)

The aspect ratio of the crack, a/c, seems to vary along thecrack path, starting with a value of around 0.5 and becomingmore and more circular reaching a value close to 1. This facthas been observed in different tests stopped before failureoccurred (cracks in the range of 40–150mm). But it has beenobserved as well in tests that failed, as seen on Fig. 3. In thetype of set-up analysed in this paper the contact loads areapplied symmetrically on opposite sides of the specimen,therefore there are two possible sites for the initiation ofcracks. Usually cracks develop and grow on both sides butonly one of them causes the failure of the specimen. Fig. 3shows the failure surface of one of the tests analysed,test 5A (see Section 3). This figure shows the initiationpoint of the crack that produced the failure, but it can alsobe seen the semi-elliptical crack on the opposite side. This‘‘secondary crack’’ can be used to calculate the aspect ratio ofthe crack, a/c.

Independent of the latter parameters considered, differ-ent crack growth laws can be used to simulate the growthprocess. For example, experimental data of crack growthrate obtained for the whole range of the SIF can be used.Another approximation is to use simple equations that

Fig. 3. Failure surface in a fretting fatigue test.

match the data obtained in the crack growth rate tests.Among the laws found in the literature, some take intoaccount the crack growth threshold and others do not.Moreover, when analysing the growth of short cracks,some laws modify the crack growth threshold by reducingit and others introduce an effective crack length (which ishigher than the real crack length) in the calculation ofthe SIF.The consideration of the crack growth threshold effect

can be done in several ways, only two of them areconsidered in this paper. The simplest is substituting DK byDK�DKth in the crack growth rate equation. This type ofmodification will be referred to as model a. Therefore, theParis law, given by

da

dN¼ CDKn (4)

changes to

da

dN¼ C DK � DK thð Þ

n. (5)

Another form of including the threshold is subtractingDKn

th from DKn. Thus, in model b, Eq. (4) is transformed to

da

dN¼ C DKn � DKn

th

� �. (6)

The three laws shown in Eqs. (4)–(6) are presented in Fig. 4for the material used in the experiments. The fatigue crackgrowth data for long cracks used for the simulation weretaken from the NASA/FLAGRO program for Al 7075-T6[15]. The material constants are C ¼ 4:82537� 10�11, n ¼

3:517 and DK th ¼ 2:5MPam0:5 for R ¼ 0 and for units in mand MPa. This figure shows the comparison between Parislaw and both models, a and b. It can be seen that model b

produces a faster transition from the threshold to the Parisregime than model a. This difference will produce differentestimates of life as it will be seen on the results.Apart from the use of the threshold, another modifica-

tion to the growth law should be made, as mentioned

1 1010-11

10-10

10-9

10-8

10-7

10-6

10-5

Paris lawlong crack model along crack model b

da/d

N (

m/c

ycle

)

∆K (MPa m1/2)

Fig. 4. Crack growth laws for the Paris law, models a and b.

ARTICLE IN PRESS

σ

N

N

Q

Fig. 5. Scheme of loads in the fretting tests.

C. Navarro et al. / Tribology International 39 (2006) 1149–11571152

before, introducing the behaviour of short cracks. Thismodification can also be made in different ways. Thedifferent laws used in the estimations have in common thatthey only need: Paris law, threshold for long cracks andfatigue limit. Following are detailed all these different lawsused in the calculations.

2.1. Model 1

The first option is to use Paris law. Assuming that theSIF depends on the crack length of each test analysed, thelaw takes the form

da

dN¼ CDKðaÞn, (7)

where a is the crack length. This crack growth law is verysimple but it has the disadvantage that it does not take intoaccount the crack growth threshold, nor the behaviour ofshort cracks, making it unrealistic.

2.2. Model 2

This model uses a crack growth law for long cracks, i.e.introducing a threshold for long cracks, DKthN. Asmentioned before, it has been introduced in two differentforms:

model 2ada

dN¼ C DKðaÞ � K th1ð Þ

n, (8a)

model 2bda

dN¼ C DKðaÞn � Kn

th1

� �. (8b)

This law is more realistic but it does not show thebehaviour of short cracks.

2.3. Model 3

This model employs the same law as model 2 butmodifying DKthN in Eq. (8) to take into account theempirical fact that the threshold for short cracks is lower.This modification has been used before by Faanes andFernando [1] in the form of Eq. (9a). The crack growthequations are

model 3ada

dN¼ C DKðaÞ � DK th1

ffiffiffiffiffiffiffiffiffiffiffiffiffia

aþ a0

r� �n

,

(9a)

model 3bda

dN¼ C DKðaÞn � DK th1

ffiffiffiffiffiffiffiffiffiffiffiffiffia

aþ a0

r� �n� �,

(9b)

where a0 is the parameter given by El Haddad et al. [16]:

a0 ¼1

pDK th1

DsFL

� �2

, (10)

with DsFL being the fatigue limit.

2.4. Model 4

In this case, instead of modifying the long crackthreshold, the SIF modified as shown in Eq. (11) to takeinto account that short cracks grow faster [2].

model 4a da=dN ¼ C DKðaÞ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiaþ a0

a

r� DK th1

!n

,

(11a)

model 4b da=dN ¼ C DKðaÞ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiaþ a0

a

r !n

� DKnth1

!.

(11b)

2.5. Model 5

Finally, this model is the same as model 2 butintroducing an effective crack length as proposed by ElHaddad et al. [16].

model 5ada

dN¼ C DKðaeff Þ � DK th1ð Þ

n, (12a)

model 5bda

dN¼ C DKðaeff Þ

n� DKn

th1

� �. (12b)

The effective crack length is

aeff ¼ aþ a0. (13)

3. Experimental details

The different models have been applied to frettingfatigue tests with spherical contact. Fig. 5 shows aschematic of the loads applied in this type of tests, wherethe normal load, N, is constant and the axial load, s, isvariable. Due to the stiffness of the supports the tangentialload, Q, is induced through the contact pads and varies inphase with s. The tangential load, Q, applied in every testproduces a contact condition of partial slip.The tests are divided into two groups. The first group of

tests is addressed as group A. These tests used the materialAl7075-T651. The specimens had a square section of10� 10mm, the contact pads were spheres with a radius of100mm and the loads applied are detailed in Table 1. This

ARTICLE IN PRESSC. Navarro et al. / Tribology International 39 (2006) 1149–1157 1153

table also shows the maximum contact pressure, p0, due tothe normal load, N, the maximum stress on the surface dueto all the loads applied and the number of cycles to failure.The radii of the contact zone range from 500 to 900 mm.The friction coefficient measured experimentally and usedfor simulation was 1.27. This value of the coefficient offriction is the one experienced in the slip zone in a frettingtest, which is higher than the mean value in the wholecontact zone. The procedure to obtain this value is detailedin [17] by method number one.

The other group of tests, group B, used the materialAl7075-T6 and they are described elsewhere [5,18]. Thespecimens had a square section of 5� 5mm, the contactpads were spheres with a radius of 25.4mm and thecoefficient of friction was 1.2 [17]. The radii of the contactzone ranged from 150 to 240 mm. The size of the grain

Table 1

Loads and cycles applied in the tests of group A

Test Normal

load N

(N)

Tangential

load Q (N)

Axial load

s (MPa)

p0 (MPa) sx max

(MPa)

No. of

cycles

1A 230 117 82.5 192 394 413 374

2A 230 127 111.6 192 440 210 934

3A 120 67 111 154 381 221 537

4A 70 29 110 129 325 326 391

5A 420 207 68.3 235 434 587 516

Table 2

Loads and cycles applied in the tests of group B

Test Normal

load N

(N)

Tangential

load Q (N)

Axial

load s(MPa)

p0

(MPa)

sx max

(MPa)

No. of

cyclesa

1B 13 7 83 183 382 10 00 0000

(40)

2B 7.3 6.6 83 151 377 3 450 000

(162)

3B 20 16 62.5 211 448 2 190 000

(65)

4B 20 15 56 211 431 1 540 000

(150)

5B 20 15 63 211 440 2 940 000

(110)

6B 20 15 59 211 435 1 780 000

(150)

7B 20 15 84.2 211 463 549 000

8B 10.3 7.5 83.6 171 388 2 940 000

9B 30 15 85 241 463 480 000

10B 20.8 15 83 214 463 449 000

11B 15.6 15 85 194 469 395 000

12B 12.5 15 83 180 472 361 000

13B 18.5 13.6 77 206 444 551 000

14B 16 11.7 83 196 434 530 000

15B 13.9 10 83 187 417 803 000

16B 8.3 8 83 158 396 616 000

17B 20 15 70 211 448 516 000

aFor tests stopped before failure, the crack depth in microns measured

after test is shown in parenthesis.

perpendicular to the surface, measured according tostandard ASTM E 112-88, [19], was 35 mm. Table 2 showsdetails of the tests in group B. From this set of results therewere 11 tests that ended with the failure of the specimenand six other tests that were stopped before failureoccurred. The latter tests were then sectioned, mounted,polished and etched to reveal the cracks present in thespecimen.

4. Results and discussion

The objective of this work is to find the model thatproduces the best approximation to the fretting fatiguebehaviour obtained experimentally. The parameter used tocompare the behaviour of the models if the specimen fails isthe fatigue life predicted. If it does not fail, the comparisonparameter is the size of the largest crack present in thespecimen.

4.1. Failure

The following results are obtained applying all thesemodels to the tests that failed. Models 2a, 3a, 4a and 5agive infinite lives for the tests analysed, therefore they arenot valid for simulation. Fig. 6 shows the estimated vs.experimental lives using the models 1, 2b, 3b, 4b and 5b.Groups A and B of tests are shown with open and solidsymbols, respectively.It can be observed that using model 1 (Paris law)

reasonable results are obtained, except for one test ofgroup B, even though this law does not model thebehaviour of short cracks. As a matter of fact, in a sense,Paris law could be closer to the real crack growth law forshort lives. This law does not have a threshold and in thecase of short cracks with high stresses the crack growth rateis much higher than expected for a long crack with thesame SIF. For longer lives the short crack growth rate maydecrease as the crack grows, and this effect cannot be

105 106

105

106

2x

1 A3b A4b A5b A1 B3b B4b B5b B

Nf (

estim

ated

)

Nf (experimental)

Fig. 6. Estimated lives for the tests that failed.

ARTICLE IN PRESSC. Navarro et al. / Tribology International 39 (2006) 1149–11571154

predicted by the Paris law. Probably this is the reason whythe worst result with this model corresponds to the testwith a considerable longer life. The results correspondingto model 2b (long crack threshold) do not appear in thefigure because the estimated lives obtained are infinite.Therefore, the behaviour of short cracks has to be takeninto account in some way in order to predict life in frettingfatigue using a fracture-mechanics-based methodology.The quality of the estimates obtained with model 3b isdifferent for groups A and B. In the first case, it givessimilar lives as model 1, meanwhile for group B it givesbetter results than model 2 but worse than model 1. Model4b gives similar results to model 1, except for two tests, thetest with the longest life is now better estimated but anothertest falls now outside of the 2x band shown in Fig. 6. Thesetwo tests belong to group B. Model 5b gives, for group A,similar results to models 1b, 3b and 4b, but, for group B,only two of the eleven tests have an acceptable estimatedlife. The other nine results are not represented in Fig. 6because they are outside the range of lives of the plot.

4.2. No failure

All these models can also be applied to estimate fatiguelife on the unbroken specimens. Table 3 shows theexperimental and estimated results for these tests. The firstcolumn represents the number of cycles where the test wasstopped and the depth of the longest crack found in thespecimen after analysing it. The rest of the columns showthe lives estimated by each model as well as the predictednonpropagating crack depths. This estimation of thenonpropagating crack can be done through the graph ofthe crack growth rate for each test and model as will beseen in the following section.

The data in Table 3 show that model 1, as expected,cannot predict the stopping of the crack. Models 2 and 5give infinite lives, which is logical since they gave infinitelives also with the tests that failed, and do not predict anygrowth of the initial cracks before become nonpropagatingcracks. Finally, models 3 and 4 give similar results. Theypredict infinite lives for all the tests except for one, and theyalso predict values of the nonpropagating cracks on thesame order of the cracks found in the experiments.

Table 3

Results for the tests with no failure from group B: Life/nonpropagating

crack

Test Experiment Model 1 Model 2b Model 3b Model 4b Model 5b

1B 41e7/40 mm 1.13e6/

Fa

N/5 mm N/81 mm N/81 mm N/5 mm

2B 43.5e6/162 mm 0.88e6/F N/5 mm 2.87e6/F 1.39e6/F N/5 mm3B 42.19e6/65 mm 1.05e6/F N/5 mm N/192 mm N/192 mm N/5 mm4B 41.5e6/150 mm 1.55e6/F N/5 mm N/111 mm N/111 mm N/5 mm5B 42.9e6/110 mm 1.15e6/F N/5 mm N/150 mm N/150 mm N/5 mm6B 41.8e6/150 mm 1.36e6/F N/5 mm N/124 mm N/124 mm N/5 mm

aF ¼ Failure

It is worth adding two comments about the models. Oneis that the prediction of length for nonpropagating cracksused in this paper does not take into account the size of thegrains, or microstructural barriers, but it is known thatthese ‘‘barriers’’ make the crack slow down and even stop.Therefore, it could be argued that for the same loadsapplied, the length of nonpropagating cracks has a scatterin the order of the grain size. Another aspect to consider isthe value of the parameter a0 used in models 3–5. Theconstant a0 is a material constant since it relates twomaterial parameters: the fatigue limit and the threshold ofthe SIF. This constant establishes a separation between theuse of the S–N curve and fracture mechanics in a crackwith no stress gradient. An aluminium alloy presents theproblem that it is a material, which does not exhibit a realfatigue limit, however, a value of it is needed to define theconstant a0. The value of the fatigue limit used in Eq. (10)is the one for 106 cycles since the number of cycles at whichthe unbroken tests were stopped was between 106 and 107

cycles and also 106 cycles is a typical value for the fatiguelimit.

4.3. Evolution of crack growth rate

In order to show the different evolutions of the crackgrowth rates, and estimated lives, five tests from group Aand B representing different situations will be analysed.These tests are 2A and 5A, shown in Table 1 and 1B, 7Band 8B, shown in Table 2.Test 7B has the highest contact and axial loads among

the three tests of group B, and the crack grows faster andthe specimen breaks earlier. Test 8B has lower contactloads, this implies that the initiation of the crack will beslower, but, after reaching a certain depth, it will grow atthe same rate as test 7B. Test 1B is very similar to 8Bconsidering the loads but it has a slightly higher normalload, making the load ratio more negative and theinitiation even slower. The test was stopped with 107

cycles. After analysing the specimen, the longest crackfound in it was 40 mm long. Tests 2A and 5A cannot becompared directly to the other three because the size of thecontact pads is very different.Figs. 7–10 show the estimated crack growth rate for

these tests using four of the different models. Model 2 hasbeen excluded because of the poor results obtained withthis method and because it does not show any differentcharacteristics. The evolution for groups A and B is verydifferent, probably due to the scale effect. The large contactzone for tests in group A makes the influence of the contactstresses go deeper into the specimen, and this in turn makeit easier for a specimen to fail. There is another differencebetween these two groups, the specimens in group A are10mm thick, meanwhile in group B they are 5mm thick.Therefore, the finite thickness of the specimen influencesearlier the crack growth rate in the tests of group B thanthose in group A. This makes the crack grow faster ingroup B than A for the same axial stress and the same

ARTICLE IN PRESS

0 500 1000 1500 2000 2500

2A5A7B8B1B

Crack length (µm)

10-11

10-10

10-9

10-8

10-7

10-6

da/d

N (

m/c

ycle

)

Fig. 8. Estimated crack growth rate with the model 3b.

0 500 1000 1500 2000 2500

2A5A7B8B1B

Crack length (µm)

10-11

10-10

10-9

10-8

10-7

10-6

da/d

N (

m/c

ycle

)

Fig. 9. Estimated crack growth rate with the model 4b.

0 500 1000 1500 2000 2500

2A5A7B8B1B

Crack length (µm)

10-11

10-10

10-9

10-8

10-7

10-6

da/d

N (

m/c

ycle

)

Fig. 7. Estimated crack growth rate with the model 1.

0 500 1000 1500 2000 2500

2A5A7B8B1B

Crack length (µm)

10-11

10-10

10-9

10-8

10-7

10-6

da/d

N (

m/c

ycle

)

Fig. 10. Estimated crack growth rate with the model 5b.

C. Navarro et al. / Tribology International 39 (2006) 1149–1157 1155

crack length. These figures also show that the crack growthrate for cracks far enough from the surface depend on theaxial load applied and all the models give approximatelythe same value.

Models 3b and 4b present a minimum in the curve for allthe tests. The position of this minimum is different in each

test but it seems related to the influence of the contactstresses, i.e., a bigger contact pad and higher contact loadproduce a minimum at a deeper point. This minimum isvery clear in Figs. 8 and 9. When the tangential contactloads are high, this decrease in the velocity is not veryimportant as in tests 2A, 5A and 7B. But for lower loads itmakes the crack grow much slower, as in tests 8B, or evenstop, as in test 1B. As was mentioned earlier, the differencebetween test 8B and 1B is that the normal load in the latteris a bit higher. At first sight, this is contradictory but ahigher normal load, maintaining the rest constant, makesthe load ratio more negative and the initial crack growthrate lower [17].It can be understood now, looking at Figs. 8 and 9, how

models 3b and 4b can predict a nonpropagating crack. Atthe beginning of the process they have a nonzero crackgrowth rate, but as the crack grows this velocity maydecrease more and more until it falls down to zero and thecrack stops. In cases of tests producing failure, models 3band 4b are able to predict the crack growth rate reductionproduced by the high stress gradient close to the contactzone. The minimum of the crack growth rate curvesrepresented by these models is produced at crack lengthsclose to the radii of the contact zones, where the contactstresses go down to very small values and only the globalaxial stresses are significant.

4.4. Evolution of the crack length

Figs. 11 and 12 show the evolution of the crack length vs.number of cycles for the same tests of Section 4.3, exceptfor the one that did not fail, 1B. Fig. 11 shows theevolution produced by model 1 and Fig. 12 the same formodel 4b. The final crack considered for all the tests wasthe midpoint of the specimen, i.e., 2.5mm for group B and5mm for group A. At that point KI is still lower than KIC

but the crack is growing so fast that the error is negligible.Besides, as mentioned before, the weight function used inthe calculation of KI is only valid for cracks smaller thanthe half the thickness of the specimen. These figures show

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0 1x105 2x105 3x105 4x105 5x105 6x105 7x105 8x105 9x105 1x105

0

1000

2000

3000

4000

5000

2A

5A

7B

8B

Cra

ck le

ngth

(µm

)

No. of cycles

Fig. 11. Estimated evolution of crack length using model 1.

2A5A7B8B

0.0 5.0x105 1.0x106 1.5x106 2.0x106 2.5x106

0

1000

2000

3000

4000

5000

Cra

ck le

ngth

(µm

)

No. of cycles

Fig. 12. Estimated evolution of crack length using model 4b.

0 50 100 150 200

0.1

0.2

0.3

0.4

0.50.60.70.80.9

1

2

3

4

1A2A3A4A5A7B8B9B10B11B12B13B14B15B16B17B

initial crack length (µm)

Nes

timat

ed/N

real

Fig. 13. Influence of the depth from where to start the propagation for

model 1.

0 50 100 150 200

0.1

0.2

0.3

0.4

0.50.60.70.80.9

1

2

3

4

1A2A3A4A5A7B8B9B10B11B12B13B14B15B16B17B

initial crack length (µm)

Nes

timat

ed/N

real

Fig. 14. Influence of the depth from where to start the propagation for

model 4b.

C. Navarro et al. / Tribology International 39 (2006) 1149–11571156

that the initiation of the cracks takes place very early in life.After a region of stable growth, which is approximatelyuntil 20% of the specimen thickness, the crack rapidlygrows and in a few cycles it produces the failure of thespecimen.

4.5. Influence of the initial crack length

The initial length from where to consider the propaga-tion has been chosen from [7] as 5 mm for this material, butthis value is somewhat arbitrary. It is interesting to checkthe influence of this choice on the results. New calculationshave been done using different values of this parameterranging from 2 to 200 mm. This evolution will be shownonly for the two models that give best results and for everytest that failed. The results of model 1 are shown on Fig. 13and the results for model 4b are shown on Fig 14. In model1, the initial crack length has some influence in the rangeconsidered, but not too much. Anyway, there is acontinuous decrease in the estimated value of life whenthe initial crack length is increased.On the other hand, in model 4b the influence of the

initial crack length is negligible up to a certain value, whichis different for each test. The minimum of these valuesseems to be around 130 mm related to the test 8B. As couldbe seen on Fig. 9 this test has a high drop of crack growthrate. This model is more robust and any value of the initialcrack length up to 130 mm gives an acceptable result. It canbe noted that El Haddad characteristic length for thismaterial, a0 ¼ 8mm could be used as the initial crack lengthproducing almost the same results with this model.

5. Conclusions

From the previous analysis, some conclusions can beobtained:

1.

Regarding the estimated life for the tests that failed itcan be said that: assuming a long crack behaviour(model 2) infinite lives are obtained; modifying the cracklength by a certain parameter (model 5) produce areasonable result for only two tests, the rest beinginfinite; modifying the crack growth threshold toconsider the effect of short cracks (model 3) give finiteresults but not good enough; models 1 (Paris law) and 4(modified SIF) give the best results.

ARTICLE IN PRESSC. Navarro et al. / Tribology International 39 (2006) 1149–1157 1157

2.

Regarding the tests that were stopped before failureoccurred: model 1 gives always finite lives; models 2 and5 give infinite lives but no nonpropagating cracks;finally, models 3 and 4 produce infinite lives for all thetests except for one and also nonpropagating crackswith values close to the ones measured on the specimens.

3.

Model 4b produces the best results for the entire set oftests analysed and it is adequate to estimate the totalfretting fatigue life.

4.

Model 1 deserves a special comment. It is surprising thata crack growth law that is not valid in the short crackregime give such good results. Other tests with a widerrange in lives should be analysed.

5.

Regarding the initial crack length considered, it can besaid that its effect in the model 4b results is very smallwhen the selected initial length is smaller than a0 and theradius of the contact zone.

Acknowledgements

The authors wish to thank the Ministerio de Educacion yCultura for their financial support through investigationproject DP1 2001-2400-C02-01.

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