on the use of multiaxial fatigue criteria for fretting fatigue life assessment

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International Journal of Fatigue 30 (2008) 32–44

JournalofFatigue

On the use of multiaxial fatigue criteria for fretting fatiguelife assessment

Carlos Navarro *, Sergio Munoz, Jaime Domınguez

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

Received 3 July 2006; received in revised form 3 January 2007; accepted 26 February 2007Available online 2 March 2007

Abstract

This paper analyses the influence of different multiaxial fatigue criteria on life assessment in fretting fatigue conditions. Five groups offretting fatigue tests are used for analysis, with different materials, geometries, sizes, and contact forces. The materials are aluminium andtitanium, and the geometries are spherical and cylindrical contact. The life prediction model applied combines initiation with propaga-tion, without a prior definition of when one begins and the other ends. Results are also shown for the application of other life predictionmodels. Four aspects are analysed with these models: the influence over estimated life when using the value of stresses and strains eitherin a critical dimension or on the surface; the effect of the initiation length considered over the life obtained; the importance of one phaseversus the other; and the influence of the multiaxial criterion employed.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Fretting fatigue; Life estimation; Initiation life; Multiaxial fatigue criteria

1. Introduction

During fretting fatigue processes, multiaxial stressstates are produced close to the contact zone. Near thecontact surfaces there are areas in which there is also anon-proportional variation of the stresses [1]. This iswhy there are different proposals regarding the applica-tion of multiaxial fatigue criteria to study the fatigue limitin fretting conditions [2], the duration of the initiationprocess of the crack [3,4], or even the duration of thecomplete process [5,6].

There are various studies and proposals regarding theuse of the local strain method with multiaxial fatigue crite-ria in the analysis of the fatigue limit in fretting conditions.In general, the application of the method requires the con-sideration of the stress field not only on the surface of theelement, but also in a zone within the proximities of thepoint with the highest stress level. The fatigue strength isanalysed as a function of the value of the damage parame-

0142-1123/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijfatigue.2007.02.018

* Corresponding author. Tel.: +34 1 617 253 9825.E-mail address: [email protected] (C. Navarro).

ter of the multiaxial criterion evaluated for that zone,which can be a distance, an area, or a critical volume

[7,8]. These critical dimensions seem to be related to themicrostructural characteristics of the material [2,3]. How-ever, the main problem associated with the application ofthese criteria lies in the definition of the critical dimensionto be used in estimating the damage parameter. At presentthere is no clear criterion that allows for a definition a pri-

ori of the value of this dimension. Although in certain casessome multiaxial criteria show better behaviour than others,depending mainly on the material being used, a slight var-iation of the defined critical dimension could make theadjustment produced by another criterion the best result.In general, it can also be said that if the applied damageparameter is the value produced in the least favourablepoint, results will usually be conservative with any of thecommonly accepted multiaxial criteria.

In the analysis of life in fretting fatigue the process isusually considered as a combination of two phases: initi-ation and propagation of the crack. Depending on themethod employed, the initiation phase is considered as aprocess lasting up to a determined length of the crack.

mailto:[email protected]

QQ

P

KK

NN

Fig. 1. Scheme of the fretting setup used in testing.

C. Navarro et al. / International Journal of Fatigue 30 (2008) 32–44 33

Some authors, assuming a short propagation phase, takeonly the initiation process into account when determiningfatigue life [6]. Contrary to this, other authors considerthe propagation of the crack as practically the whole ofthe process, except for a small part of the fatigue life, ded-icated to the nucleation of the crack, which is disregardedin calculations [9,10]. In these cases, the propagationphase, which begins when the crack is a few micrometerslong, is analysed through one of the techniques associatedto fracture mechanics. At a midpoint between these twoapproaches are the authors that combine initiation withpropagation [3,4,11,12]. This article will confirm that, ingeneral, one phase cannot be disregarded against theother.

There are various approaches that consider the localstrain method and multiaxial fatigue criteria in the calcula-tion of the initiation life. A number of the multiaxial fati-gue criteria that are normally used in fatigue have beenproposed for fretting fatigue: Fatemi and Socie [13],Smith–Watson–Topper [14], McDiarmid [15], etc. Someauthors consider the initiation process to last until thecrack has sufficient length to be practically at the edge ofthe influence zone of the contact [3,4]. They calculate theduration of the initiation process from the value of thestresses and strains produced on the contact surface [4] ornear it [3]. Others, as indicated earlier, consider that prac-tically the entire process corresponds to initiation [5,6]. Yetanother approach is that of authors that do not establish aprior definition of the length of the crack at which the ini-tiation phase ends and the propagation phase begins. Theseauthors determine such a length as a function of the evolu-tion of stresses and strains, the characteristics of the mate-rial, and the growth law being considered [12]. With thismethod, the duration of the initiation phase will dependon the multiaxial criteria and crack growth law appliedand on the levels and gradients of the stresses and strainsproduced.

This paper analyses some of the characteristics of themethods that combine initiation and propagation in orderto calculate life in fretting fatigue. It takes into consider-ation the influence of some of the parameters of the modelson estimated life (both initiation life and total life). Thereare four main aspects to be analysed: how estimated lifeis influenced by the use of the value of stresses and strainsin a critical dimension or on the surface; how the initiationlength considered affects the life obtained; the importanceof one phase as opposed to the other; and the influenceof the multiaxial fatigue criteria applied. For this purpose,the paper is organised in four parts. First there is anaccount of the experimental results that will be employedfor comparison of the different approximations. Next thereis a brief introduction of the methods to be used in thecomparison and a description of the multiaxial criteriaapplied. After this, the results obtained with the differentmodels and multiaxial criteria are shown and comparedwith the experimental results. Finally, results are discussedand conclusions extracted.

2. Experimental results

Some of the previously cited life estimation models wereapplied to five groups of fretting tests: three with sphericalcontact and two with cylindrical contact. The tests werecarried out on a setup such as the one shown in Fig. 1.The contact elements are pressed against the specimen witha force N. An axial force P is applied to the specimen,which produces the global stress experimented by it, r. Inaddition, this same force, P, generates a tangential forcein the contact, Q, which is higher or lower, depending onthe stiffness of the supports, K: the higher the stiffness,the higher the force. Regardless of the value of stiffnessK, for each constant value of it, the force Q will vary prac-tically linearly with P [16]. All the tests were performedunder conditions of partial slip, i.e. Q < lN, where l isthe friction coefficient.

The properties of the materials and geometric character-istics of the elements employed in the five test groups areshown in Table 1. For each material tested the table showsthe following parameters: Young modulus, E; Poissonratio, m; tensile strength, ru; and yield strength, ry; con-stants C and n of the crack growth law, with units ofMPa and m/cycle; fatigue crack growth threshold, DKth;fatigue limit, DrFL; El Haddad parameter, a0; parametersr0f and b of the Basquin equation, r ¼ r0fð2NÞb; frictioncoefficient, l, between the contact elements and the speci-men; average grain size, D, of the material of the specimenin the direction perpendicular to the surface (approximatedirection of the growth of the crack); radius, R, of the con-tact elements; thickness of the specimens, and geometry ofthe contact employed in testing with those materials.

In the first group of tests with spherical contact (G1) thematerial used is an aluminium alloy 7075 T6 [16,17]. Thefriction coefficient, l, considered for these tests, is the oneappearing in the slip zone of the contact. This value is lar-ger than the one existing between the two surfaces whenthey come into contact for the first time, and is measuredusing a technique described in [18]. This technique consistsin carrying out a fretting test with a determined force, N,and a cyclical axial force with amplitude, P, which slowlyincreases from a small value until the increase of Q gener-ated by the increase of P produces global sliding, with no

Table 1Properties and characteristics of the materials employed in the five analysed test groups

G1 7075 T6 G2 7075 T651 G3 Ti6Al4V G4 Al4%Cu G5 2024 T351

E (GPa) 71 71 115.7 74.1 74.1m 0.33 0.33 0.321 0.33 0.33ru (MPa) 572 572 965 500 470ry (MPa) 503 503 925 465 310C 8.831 · 10�11 8.831 · 10�11 1.8 · 10�13 1.74 · 10�10 6.529 · 10�11

n 3.322 3.322 5 4 3.387DKth (MPa m0.5) 2.2 2.2 5 2.1 2.1DrFL (MPa) 216 169 525 124 235a0 (lm) 33 54 29 91 25r0f (MPa) 1090 1610 1933 1015 714b �0.1122 �0.1553 �0.1 �0.11 �0.078l 1.2 1.27 0.5 0.75 0.65Grain size D (lm) 35 50 17 50 50R (mm) 25.4 100 12.7 12.5–150 127–229Specimen thickness (mm) 5 10 5 12.5 12.7Geometry Spherical Spherical Spherical Cylindrical Cylindrical

34 C. Navarro et al. / International Journal of Fatigue 30 (2008) 32–44

recuperation of partial slipping. The parameters of thecurve r–N ; r0f and b, shown in the table, correspond withR = �1 [19] and those of crack growth, C and n, forR = 0 [20]. The parameters are employed for R = �1because the multiaxial fatigue criterion uses the initial datafor that ratio, later to modify the equation r � N as a func-tion of the mean stresses in each case. On the other hand,the curve da/dN for R = 0 is used because it is supposedthat compressive stresses do not contribute to crackgrowth. At this point in the development of the model itis supposed that crack closure is produced when the stressintensity factor (SIF) is null.

In the second group of tests (G2) the material used is Al7075 T651, very similar to the preceding material. Thesetests were carried out in the Mechanical Engineering Lab-oratory of the University of Seville and the loads appliedcan be found in [21]. The main difference with the previ-ously described tests is found in the radius of the contactsphere and some of the mechanical properties of the mate-rial used. In the case of parameters r0f and b, the tableshows the values obtained in tests carried out with the samelot of material used in the fretting tests [22].

The third group of tests (G3) is that carried out by Kirk-patrick [17] with spherical contact and the titanium alloy Ti–6Al–4V. In the tests presented by Kirkpatrick, tests number1 and 8 show a tangential force Q > lN, which is impossibleby the very definition of the friction coefficient. In these twocases it will be assumed that the loads shown are the actualones applied and measured during the test and that testing iscarried out on the limit of global sliding. This implies assum-ing, in these two tests, a friction coefficient, l = 0.6, some-what different from the value shown in Table 1, used forthe rest of the simulations with this material.

Regarding the fourth group of tests (G4), the material isAl4%Cu and the contact elements are cylindrical [3]. Inthese tests different values, varying between 12.5 mm and150 mm, were used for the radius of the contact pad, seeTable 1.

In the fifth group of tests to be analysed (G5), the mate-rial, Al2024 T351, is very similar to the preceding one andcontact is also cylindrical [4]. Different radii were also usedfor the cylinders, varying between 127 mm and 229 mm.

3. Prediction models for initiation and total life

A number of the previously cited methods are used topredict initiation and total life. These models can bedivided into two groups; (i) those that take only initiationcriteria into account when assessing total life, and evaluatestress on the surface or near it, (ii) those that combine ini-tiation and propagation. This second group can in turn bedivided into two subgroups. On one hand there are themodels that consider propagation to begin at a previouslyfixed length, defined for each of the test series [3,4,11]. Onthe other, the model proposed by the authors [12] where itis considered that this length is not previously fixed, butdepends on conditions in each of the tests and is obtainedthrough the application of the model. This model, desig-nated as the variable initiation length model, is explainedin more detail below.

For all analysis carried out in the paper, the stresseshave been calculated analytically. The explicit expressionsproposed by Hamilton [23] have been used for sphericalcontact. For the stresses produced in cylindrical contact,the analytical expressions have been taken from the litera-ture [24]. All calculations have been done assuming that thesurfaces are perfectly smooth, and, in case of cylindricalcontact, that the load is uniformly distributed along thecontact line. Isotropic material has been considered.

3.1. Life prediction with variable initiation length model

The model presented here for the estimation of life infretting was proposed by the authors [12]. It is assumedthat two different mechanisms act upon a material sub-jected to fatigue: initiation and propagation.


It can be supposed that the initiation mechanism acts asfollows. The initiation phenomenon begins by nucleating acrack at the point of maximum stresses, normally on thesurface, Fig. 2. The number of necessary cycles, N0, isobtained from the Basquin equation. Once the crack is ini-tiated at that point, as the number of cycles increases, otherfarther points, with lower stress, reach the number of nec-essary cycles for initiation to begin at them. For any depth,l, in the crack path, Fig. 2, the number of initiation cyclesNi(l), is calculated using the process represented in Fig. 3.Once the stress variation with the depth is known,Fig. 3a, by means of the Basquin curve, Fig. 3b, it is easyto determine the value of Ni(l), Fig. 3c. Therefore, oncethe crack is initiated at points progressively farther, thiswill make it grow from the first initiation point to thenew initiation points. That increase in the length of thecurve can be considered equivalent to a propagation pro-cess. If the slope in each point is calculated for the curvel–Ni in Fig. 3c, an equivalent growth rate is obtained by

Crack path

Crack initiationpoint

N N

Q Q

Fig. 2. Location of the initiation of the crack.

Fig. 3. Calculation process for what is

consecutive initiations, as a function of the depth,dl/dNi � l, Fig. 3d.

The second phase, propagation, uses Linear ElasticFracture Mechanics (LEFM), combining the curve da/dN � DK with the evolution of the SIF with crack length,DK � a, in order to obtain the growth rate as a functionof crack length, da/dN = f(a). Comparing the two crackgrowth rates obtained, Fig. 4, it can be observed that nearthe surface the growth rate by initiation is larger than therate in propagation, dl/dNi > da/dN, whereas the contraryoccurs when the crack is long. This means that when thecrack is short it will advance faster by the initiation mech-anism than by propagation. The crossing between the twocurves is produced for a length value l = a, called initiationlength, li. At that length is where the change from onemechanism to the other is produced. In other words, thispoint determines where initiation ends and propagationbegins. Before the initiation length is reached, the crackis initiated at each point before it has time to propagatein accordance with LEFM, while beyond this initiationlength, growth is dominated by fracture mechanics. There-fore, total life will be that of initiation, which corresponds

termed as growth rate by initiation.

da/dN i

Depth

da/dN |

|

p

Initiationlength

Fig. 4. Initiation and propagation rates.

Depth

NT

Ni

Np

Initiationlength

Fig. 5. Application of the variable initiation length method.


to the number of cycles necessary to start the crack at adepth equal to the initiation length, plus the propagationof the crack from the initiation length until final failureusing LEFM.

In practice this model is applied in a different way,although it can be confirmed as mathematically equivalent,Fig. 5. On one hand, the mentioned curve of the number ofinitiation cycles at each depth, Ni � l, is obtained. On theother, if the crack growth law da/dN = f(a) is known, itis possible to obtain the number of cycles, Np, that a crackof length a requires in order to reach the final length of fail-ure af:

NpðaÞ ¼Z af

a

daf ðaÞ ð1Þ

Note that Np(a) does not represent the number of cycles toreach a crack length a, but the number of cycles for thecrack of length a to grow until final failure. This means thatthe curve Np(a) monotonously decreases. Combining initi-ation and propagation, by adding the two curves, Ni–l andNp–a, total life is obtained, NT–a. The minimum of thiscurve corresponds to the crack initiation length, li, previ-ously defined.

The number of cycles to failure, Nf, in a specific test willnot depend on a, according to the previous definitions, ifgeometry, loads and final crack length are defined. There-fore it can be represented by

N f ¼ NðaÞ þ NpðaÞ ð2Þwhere N(a) is the number of cycles to reach a crack lengtha. Derivating the latter expression, the following relation-ship is found:

dN f

da¼ 0 ¼ dNðaÞ

daþ dNpðaÞ

da! dNðaÞ

da¼ � dNpðaÞ

dað3Þ

The minimum of the curve NT in Fig. 5 can be obtainedderivating the expression of that curve:

NTðaÞ ¼ N i þ Np ð4ÞdN TðaÞ

da¼ 0! dN i

dl¼ � dNp

dað5Þ

It means that the minimum is produced when

dN i

dl¼ dN

dað6Þ

In other words, it is produced exactly at the previously de-fined initiation length and the value of this minimum coin-cides with the life that is estimated using the methoddescribed in the preceding paragraph.

In both the propagation and initiation phases it isassumed that the crack initiates at the limit of the contactzone and grows perpendicularly to the surface. Experimen-tally, it can be checked that these suppositions are not farfrom reality in the cases analysed in this article [4,12].

Given the complexity of the stress field, the initiationphase analysis should employ a multiaxial fatigue criterion,combined with an equation of the type e–N. Still, since thecalculation of stresses in this analysis is elastic, the Basquinequation will be used

Dr2¼ r0f ð2N fÞb ð7Þ

The following section describes the different criteria used inthe comparative analysis carried out in this article.

In the propagation phase, the SIF in mode I is calcu-lated through a weight function [25], which is modifiedfor the tests with spherical contact to take into account thatin this case fretting cracks are semi-elliptical [26]. For thegrowth law in the propagation phase, various proposalsby the authors to consider the growth of small cracks couldbe applied, modifying either the growth threshold for longcracks or the SIF [27]. However, with the aim of exclusivelycomparing the multiaxial fatigue criteria in this paper, onlythe Paris law

dadN¼ CDKn; ð8Þ

will be applied, for its simplicity and because it does notintroduce new variables, such as the threshold DKth.

In any case, it has already been confirmed that althoughthe Paris law is not able to model the growth of smallcracks, it does offer good results in fretting fatigue underspherical contact life assessment for lives shorter than 106

cycles [27].

4. Multiaxial fatigue criteria

As commented earlier, the analysis of the initiationphase in any of the proposed calculation methods shouldbe carried out using a multiaxial fatigue criterion. Fourof the most relevant criteria were chosen, based on calcula-tion of stresses, strains, or a combination of both.

4.1. McDiarmid

This criterion [15] was developed for multiaxial fatiguein cases where crack initiation is governed by shear stresses.It can be included within the group of the so-called ‘‘critical


plane’’ criteria. In this criterion the critical plane is that inwhich the variation range of the shear stresses throughoutone load cycle becomes maximum. Thus, the equivalentstress is defined at the point of initiation as

req ¼Dsmax

2þ t

2ru

rmax ð9Þ

where Dsmax is the maximum increment of shear stresses,rmax is the maximum normal stress in the direction perpen-dicular to the plane where Ds is maximum, t is the shear fa-tigue limit, and ru is the tensile strength. Actually,McDiarmid mentions two shear fatigue limits, tA and tB,depending on whether the crack grows along the surfaceor towards the inside of the surface, respectively. Normallythere is only one value for t, which will be the one to use.The way to combine this criterion with the r–N curve isto apply it in the case of fatigue tests over unnotched spec-imens and a symmetrical cycle, ±r. In this case, the equiv-alent stress becomes

req ¼r2þ t

2ru

r2¼ Dr

2� 12

1þ t2ru

� �¼ Dr

2� f ð10Þ

The combination of Eqs. (7) and (10) renders the equationthat provides the number of initiation cycles as a functionof McDiarmid equivalent stress

req ¼ f � r0f ð2N fÞb ð11Þ

4.2. Fatemi–Socie

The Fatemi and Socie criterion [13] is also focused onmaterials whose initiation and initial crack growth are pro-duced by a growth process in mode II (shear stresses). Asopposed to McDiarmid, it uses a range of shear strainsinstead of stresses. It also incorporates a term that reflectsthe opening of the crack, which is the normal maximumstress perpendicular to the plane of the maximum shearstrain increments. The parameter is therefore

FS ¼ Dcmax

21þ k

rmax

ry

� �ð12Þ

where Dcmax is the shear strain increment in the planewhere it is maximum, k is a constant that is fitted fromthe uniaxial and torsion fatigue tests data, rmax is the nor-mal stress perpendicular to the plane where the maximumof Dc is produced, and ry is the yield strength. Just as inthe previous criterion, once the k constant is known, thisparameter can be applied to the case of simple fatigue withsymmetric cycle, ±r, and it can be combined with Eq. (7)to obtain

FS ¼ ð1þ mÞ r0f

Eð2N fÞb þ

k2ð1þ mÞ r02f

Ery

ð2N fÞ2b ð13Þ

The two parameters, FS and McDiarmid, produce verysimilar results when the local dominating stresses andstrains are elastic. In fact, in the life estimations carriedout in the present study, where stress analysis is elastic,

the same results were obtained with both criteria. For thisreason, and for the purpose of brevity, the results of thiscriterion will not be included in some cases.

4.3. Smith–Watson–Topper

The criterion defined by Smith et al. [14] is applied tomaterials in which cracks grow practically from the begin-ning in mode I. In this case the fatigue parameter, usuallycalled Smith–Watson–Topper (SWT), is expressed as

SWT ¼ rmax1

De1

2ð14Þ

where De1 is the maximum range of principal strain andrmax

1 is the maximum normal stress in the plane wherethe maximum range of principal strain is produced.The parameter is applied according to the followingequation:

SWT ¼ r02fEð2N fÞ2b ð15Þ

When the load cycle is not proportional, as occurs in fret-ting, it is more complicated to apply this parameter as a re-sult of the rotation of the principal directions. In this case,the SWT parameter is defined as the maximum, among allpossible directions, of the product of the strain amplitudetimes the normal maximum stress [28]:

SWT ¼ rmaxDe2

� �max

ð16Þ

In this way there is greater simplicity and the results turnout to be the same as those obtained with the first defini-tion of this parameter, Eq. (14).

4.4. Crossland

The Crossland criterion [29] is different from the previ-ous ones in the sense that it is no longer a critical plane cri-terion, but a global one, based on the values of an invariantof the stresses. While previous criteria require a maximumvalue among all directions projecting the stresses(or strains), in the Crossland criterion the calculation ofstresses can be done in any direction, and it only requiresthe calculation of an invariant. This criterion has the obvi-ous advantage of a much shorter computation time, withthe inconvenience of the loss of a physical sense of theproblem. The stress, or equivalent stress parameter, is

Cross ¼ffiffiffiffiffiffiffiJ 2;a

pþ I1;max

3

3tb�

ffiffiffi3p� �

ð17Þ

where J2,a is the amplitude of the second invariant of thedeviatoric stress tensor and I1,max is the maximum of thefirst invariant of the stress tensor, t is the shear fatigue limitand b is the bending fatigue limit. The application of thisparameter to the case of a uniaxial fatigue test providesthe necessary expression for its application to any case inthe calculation of the number of cycles:


Cross � bt¼ r0fð2N fÞb ð18Þ

Fig. 7. Life estimations in the two test groups with cylindrical contactusing the McDiarmid criterion.

Fig. 8. Life estimations in the two test groups with cylindrical contactusing the Crossland criterion.

5. Results of the models

The first part of this section will present and discuss theresults obtained from the model proposed by the authors,which combines initiation plus propagation with variableinitiation length and the different multiaxial fatigue criteria.Based on these results, the analysis will focus on the resultsobtained from other prediction models that analyse the lifeconsidering the initiation and propagation phase.

5.1. Total estimated life with the variable initiation length

model

Following are the total life estimations obtained withthe model that combines initiation plus propagation withvariable initiation length. Fig. 6 shows the results of thesimulations with the three groups of tests with sphericalcontact (G1–G3). This figure compares the simulationsemploying the four multiaxial fatigue criteria with theexperimental results. Figs. 7 and 8 show the resultsobtained for cylindrical contact (G4 and G5) with two ofthe multiaxial fatigue criteria that were employed, McDiar-mid and Crossland respectively. Fatemi–Socie criteriumgives the same result as McDiarmid and with SWT crite-rium the estimated lives lie between the ones obtained withMcDiarmid and Crossland.

For a quantitative comparison of the results obtained, aseries of statistical parameters are defined, which determinewhether the results are centred around real life values andtheir dispersion. For this purpose, in each test, i, a ratio iscalculated between each life estimation, Nei, and the exper-imental value, Nti, and the logarithm of each ratio isobtained

ai ¼ logN ei

N tið19Þ

Fig. 6. Life estimations in the three test groups with spherical contact.

The average value, �a, is then calculated, as well as the stan-dard deviation, ra:

�a ¼ 1

N

XN

i¼1

ai ð20Þ

ra ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N � 1

XN

i¼1

ðai � �aÞ2vuut ð21Þ

Finally, the results employed for comparison are the anti-logarithms of these values:

�x ¼ 10�a ð22Þrx ¼ 10ra ð23Þ

The use of logarithmic scale for the calculation of the meanvalue and standard deviation of a is justified by the factthat the variable being analysed is a ratio. This is the samereason why Figs. 6–8 employ logarithmic scales. With thesecomparison parameters, the perfect adjustment of estima-


tions would produce values such as �x ¼ 1 and rx = 1. If thedistribution of a could be represented by a Gaussian distri-bution, a defined rx value indicates that the h% scatterband around the average value lies between rp

x and 1=rpx ,

where h represents the percentage of data on the scatterband and p is a constant which can be obtained from theGaussian distribution tables.

Table 2 shows the values of �x and rx obtained with theproposed model for the five test groups. In the analysedcases it was confirmed that approximately 70% of theresults obtained are situated within the range rx–1/rx,around the mean value.

Based on Figs. 6–8, as well as Table 2, it can be deducedthat life estimations are acceptable in the case of sphericalcontact but not quite satisfactory in the case of cylindricalcontact. In some cases dispersion is larger and values aresomewhat distanced from reality, as occurs with the testsin group G4. For the purpose of clarity in the graphics,Figs. 7 and 8, for tests with cylindrical contact, only showthe most differentiated values, corresponding with two ofthe multiaxial fatigue criteria: McDiarmid and Crossland.In view of the expressions of the parameters of the McDi-armid and Fatemi–Socie criteria, it is confirmed that forthe case of elastic calculation of stresses, the two producevery similar results. Table 2 illustrates how the results arepractically the same in the analysed cases. It can also beobserved that the multiaxial fatigue criterion chosen hasless influence on the dispersion of results, rx in Table 2,than on the average value, �x. Certain criteria show betterbehaviour in some of the test groups and other criteriawork better in others. This implies that there is not an opti-mum criterion that can be used for all of them. In fact, thevariability found in the literature regarding the propertiesof the materials employed could be of larger influence overthe life obtained in fretting tests than using one criterion oranother. In any case, in general, it can be said that Cross-land and SWT produce better average results than McDi-armid and Fatemi–Socie.

Table 2 shows that the choice of the multiaxial fatiguecriteria has practically no influence on life estimation in testgroups G1 and G2, with Al 7075 and spherical contact.The influence on the estimated life of the fatigue criteriain test group G3, with the titanium alloy, is larger. Thehighest influence is found in test groups with cylindricalcontact, G4 and G5, where it is significant. This behaviouris related to the percentage of life dedicated to the initiation

Table 2Average values and dispersion in the five test groups with the variable initiati

Spherical

G1 Al 7075 T6 G2 Al 7075 T651 G3

�x rx �x rx �x

McD 0.60 1.82 0.62 1.46 1.1FS 0.60 1.82 0.62 1.46 1.1SWT 0.61 1.82 0.63 1.46 1.2Cross 0.63 1.82 0.66 1.46 1.2

phase: the longer the initiation phase the higher the influ-ence of the fatigue criteria on the estimated life. As willbe seen later on, the initiation phase in cylindrical contactis longer than in spherical, and in test group G3 longerthan in groups G1 and G2.

In some of the analysed tests, failure was produced attwo or three million cycles. This is an area in which thelife of the test is invariably underestimated. In this areawith a high number of cycles, the crack is strongly heldback in the initial part of its growth, when it is short.The Paris law used in the propagation phase does not cor-rectly reflect this behaviour, so that the lives estimated arenotably shorter than in reality. To solve this problem itwould be necessary to use a propagation law that consid-ered the growth of short cracks. Some examples can befound elsewhere [27]. However, as was mentioned earlier,only Paris law is applied in this analysis to limit the num-ber of variables. In any case, the majority of the analysedtests are below one million cycles, and for this range thementioned growth laws produce values that are similarto Paris.

5.2. Influence of the initiation phase in the variable initiation

length model

This section centres on the importance, in each of thetest groups, of the initiation phase with each of the multi-axial fatigue criteria. Table 3 offers a summary of theresults. Two pieces of information are presented for eachgroup: the proportion of the number of initiation cycleswith respect to total life (Ni/NT) and the initiation length,li. The data shown for each group are mean values of allthe tests in the group.

Although the table only shows average values, the vari-ability of the initiation values is high within each group.For example, using the McDiarmid criterion, in groupG1 the initiation life, varies between 3% and 7% of the totallife, and the initiation length between 3 and 10 lm; in G2,initiation varies between 3% and 21%, and the initiationlength between 6 and 18 lm; in G3 initiation variesbetween 8% and 25%, and the initiation length between11 and 22 lm; in G4, initiation varies between 53% and77%, and the initiation length between 0.8 and 2.7 lm;lastly, in G5 initiation varies between 20% and 77%, andinitiation length between 1 and 14 lm. Something similaroccurs with the other multiaxial fatigue criteria.

on length model

Cylindrical

Ti–6Al–4V G4 Al4%Cu G5 Al 2024 T351

rx �x rx �x rx

2 1.98 0.29 2.04 0.76 1.843 1.98 0.29 2.04 0.76 1.848 1.99 0.39 2.08 1.00 1.941 1.98 0.51 2.16 1.32 2.12

Table 3Mean values of the percentage of initiation time and value of the initiation length in the five test groups with the variable initiation length model

Spherical Cylindrical

G1 Al 7075 T6 G2 Al 7075 T651 G3 Ti–6Al–4V G4 Al4%Cu G5 Al 2024 T351

% Initiation li (lm) % Initiation li (lm) % Initiation li (lm) % Initiation li (lm) % Initiation li (lm)

McD 5.0 5.3 9.9 10.5 18.5 15.0 64.0 1.7 40.1 4.5FS 5.0 5.3 9.9 10.5 18.5 14.9 63.7 1.7 40.1 4.5SWT 5.5 4.4 10.3 8.8 18.6 10.8 67.4 1.2 46.1 2.5Cross 7.1 4.0 12.8 7.6 21.8 14.0 73.7 1.1 54.9 2.2


Once again, this data underlines the difference betweenspherical and cylindrical contact. The proportion of lifededicated to initiation is much smaller in spherical contactthan in cylindrical contact, with even the initiation lengthsbeing larger in spherical contact. In cylindrical contact thestress gradients are usually lower, which is related to thefraction of life dedicated to initiation: the higher the gradi-ent, the shorter the initiation. Fig. 9 shows a typical evolu-tion of the amplitude of the normal stress in the directionof the axial load as a function of depth with the differentgroup of tests. In order to better compare the gradients,the amplitude of stresses in each test is divided by theamplitude of these stresses on the surface. The higheststress gradient appears in group G1 and the lowest ingroups G4 and G5, with cylindrical contact.

A high gradient makes the crack initiate rapidly withhigh stress levels, while, as it grows, it is soon found withlower stress levels that make it advance slowly. This isthe reason why the multiaxial fatigue criterion has lessinfluence on life assessment in testing with sphericalcontact.

In spherical contact, it can be established that, for testswith the same Q/lN ratio, there is a very close relationbetween the stress gradient and the radius of the contactsphere: the smaller the radius, the higher the stress gradi-ent. Thus, in group G2, with larger sphere radius and smal-

Fig. 9. Evolution of the amplitude of normal stresses in the direction ofaxial load in a representative test of each of the five groups. In each test thestress is adimensionalised with the amplitude of stresses at the surface.

ler stress gradient, initiation has a larger influence overtotal life than in G1. However, in the tests with titanium,G3, also with spherical contact, the proportion of initiationis larger than in previous groups, while the radius of thecontact sphere is smaller and a gradient similar to that ofgroup G2. The longer initiation phase in G3 is due to thegreat difference existing between the materials used in thetests (titanium versus aluminium). A more detailed analysiswill make it clear that the greater importance of initiationin the G3 group is attributable to some factors. On oneside, the number of initiation cycles is similar comparedto aluminium because, although the stresses are higher inthe tests with titanium, Fig. 9, the stress for a specific num-ber of cycles to initiation in the curve r–N increases in thesame amount, Fig. 10.

Also, the crack growth rate corresponding to titanium isconsiderably lower than that for aluminium at low DK. Itmakes the initiation mechanism of crack length increaseto predominate over the crack growth mechanisms for alonger period of life. In addition, the Paris law correspond-ing to titanium has a considerably higher slope than thealuminium in G1 and G2, Fig. 11. This fast increase ofthe crack growth rate reduces the time spent in propaga-tion, and makes the initiation phase in tests with titaniumpossess more relative importance than in aluminium, inspite of the higher stress gradient.

The relation between the stress gradient, radius of thecontact element and relative importance of the initiation

Fig. 10. Basquin equation r–N for each material used in the fretting tests.

Fig. 11. Crack growth rate for each material used in the fretting tests.


phase, is also reflected within each of the test groups withthe same material and geometry. Considering the relativeduration of the initiation and the propagation phases, itcan be observed that as the stress gradient increases the rel-ative duration of the crack initiation relative to the total lifedecreases. This is shown in Fig. 12 for groups G1 and G2when McDiarmid criterium is used. Knowing that theinfluence of the contact fades away at a depth similar tothe radius of the contact zone, if the size of contact is sim-ilar throughout the tests of the same group, a measure ofthe stress gradient for the same contact zone radius couldbe the ratio rmax/raxial. Thus, it can be seen that a reduc-tion of rmax/raxial produces an increase of the relative dura-tion of the initiation phase. It can be also seen in the figurethat, for the same rmax/raxial ratio, initiation in group G2 islonger than in G1. That is mainly because the radius of thecontact sphere in G2 is larger than in G1, which means alower gradient.

As regards initiation length, the results show very smallvalues, in some cases several times smaller than the grain

Fig. 12. Proportion of initiation life as a function of a measure of stressgradient in two groups of spherical tests, G1 and G2.

size of each material. The exception is titanium, where initi-ation length is very similar to the grain size. Bearing in mindthe high gradients that appear in fretting, even in these smalldepths the stress level has already decreased considerably:between 10% and 50% in the tests analysed. But, accordingto the r–N curve in Fig. 10, a reduction in stress between10% and 50% will produce an initiation life between 3 and1000 times higher. This proves the great sensibility of initi-ation life regarding the depth at which the value is calcu-lated. It is important to note that, in spite of this fact,with the applied model of life prediction, the total life esti-mated is not quite so sensitive with regard to initiationlength. This is an advantage. With this model, the estimatedlife is the minimum of the curve NT(a) = Ni(a) + Np(a),Fig. 5. While the curve Ni(a) grows with the increase ofdepth in the proximities of the minimum of NT, the curveNp(a) decreases monotonically. Thus the estimated total lifevaries little in this area, in any case less than with the appli-cation of a life estimation model based exclusively on theinitiation process, or solely on LEFM.

Figs. 13 and 14 show an example of the application ofthis model, one for group G2 and another for G4, withsimilar lives. These figures are like Fig. 5, when the estima-tion model was explained, but for two specific tests. In thetest of group G2, the applied forces are N = 60 N,Q = 30 N, the axial stress produced by the bulk load,raxial = 94.7 MPa, and the radius of the spherical pad isR = 0.1 m. In the test of group G4 the applied forces areN = 2000 N, Q = 900 N, raxial = 77.2 MPa, and the radiusof the cylindrical pad is R = 0.1 m. It is observed that incylindrical contact the sensibility of estimated life (trian-gles) as a function of initiation length is larger than inspherical contact, due to the higher slope of the initiationcurve (squares). This slope is higher, in spite of the fact thatthe stress gradient is lower in cylindrical contact, becausethe stress level in cylindrical contact is also lower nearthe surface (<20 lm) in these tests, Fig. 9. Therefore, by

Fig. 13. Number of initiation and propagation cycles and total number ofcycles as a function of the initiation length in a test of group G2, withspherical contact.

Fig. 14. Number of initiation and propagation cycles and total number ofcycles as a function of the initiation length in a test of group G4, withcylindrical contact.


looking at the curve r–N, Fig. 10, and having into accountthat the scale is logarithmic, it is observed that the sameincrement in stresses within the zone of high number ofcycles implies a much larger variation in number of cyclesthan in the zone with a low number of cycles.

5.3. Life assessment model with prefixed separation of

initiation and propagation phase

This section describes the other life prediction model,used in this paper. It also combines initiation with propa-gation but previously establishes the critical distance andthe length at which propagation begins. In other words,it establishes the depth, di, at which the stresses and strainsshould be calculated for the estimation of initiation life andthe initial crack length, dp, for the estimation of propaga-tion life. Unlike the variable initiation model, this othermodel uses different values for the critical distance, di,and the initial crack length, dp, used in the propagationphase (di 6¼ dp).

For the analysis of fretting fatigue with cylindrical con-tact, some authors [3,4] take as the initial propagationlength, dp, a value approximately equal to the semiwidthof the contact zone. Beyond this length, the influence ofthe contact over the stresses produced is small. In orderto simplify calculations, since there are different test groupswith different geometries, the value, dp, is taken as a depthequal to the radius (spherical contact) or semiwidth (cylin-drical contact) of the contact zone in each particular test.

On the other hand, in order to calculate the number ofinitiation cycles, Ni, two possibilities are considered: evalu-ating the stresses on the surface and evaluating them at adetermined depth, di. There are different ways to choosethe depth, although always related to the properties ofthe material. It can be related to the size of the grain,di = D/2 [2], or, for example, to the parameter defined byEl Haddad, a0:

a0 ¼1

pDK th

DrFL

� �2

ð24Þ

where DKth is the growth threshold and DrFL the fatiguelimit. Taylor [30] uses a depth di = a0/2 to estimate total fa-tigue life in notched components. In the cases of sphericalcontact with aluminium here analysed, these two ways ofconsidering di (di = D/2 and di = a0/2) produce very similarresults in the estimation of fatigue life, since D � a0. Incontrast, in testing with titanium alloy, and mainly withcylindrical contact and aluminium alloy the results pro-duced are very different using one parameter or the other.This paper presents the results obtained for di = D/2 be-cause the life estimations obtained are better consideringthe five group of tests. In any case, the conclusions to bedrawn are the same.

5.4. Comparison of results between the two models applied

Table 4 shows the summarized values of these results,bearing in mind that for the average ratio of estimated toexperimental life and its dispersion the expressions usedare those appearing in Eqs. (22) and (23). Although theyare not shown, the variations obtained in results with thedifferent multiaxial fatigue criteria are similar to the onesshown in Table 2 for the variable initiation length model.It is necessary to take into account that the larger the ini-tiation percentage, the larger the influence of the chosencriterion. For this reason and for the purpose of clarity,the results of the McDiarmid criterion will be the only onesshown.

Different characteristics of the models applied are ana-lysed in the following paragraphs:

(a) Influence of the point chosen as the initiation depthon the estimated life. Table 4 shows how life estima-tion can be affected by calculating initiation eitheron the surface or at a small distance from it. Alsoshown are the results obtained from the variable ini-tiation length model for comparison. When di = D/2is used, instead of di = 0, in the tests with sphericalcontact the estimated lives are practically doubledand the importance of the initiation phase goes frombeing practically unappreciable to almost a third orhalf of the life. In the tests with cylindrical contact,the estimated life is up to ten times longer whenevaluating initiation in D/2 versus evaluating it onthe surface. In addition, initiation comes to occupypractically the totality of the process. This highlightsthe significance of the choice of the point where thestresses are evaluated when estimating initiation inthese models. In the particular case of the analysedtests, the estimated lives obtained are closer to real-ity when stresses are evaluated at a depth of half thesize of the grain instead of on the surface. Further-more, although it is not shown in the analysis, it canbe confirmed that the influence of the chosen

Table 4Mean values, �x (Eq. 22), dispersion of estimated to actual total live, ratios rx (Eq. 23), and ratios Ni/NT (%), obtained with different models

Variable initiation length Initiation at surface (di = 0) Initiation at di = D/2

�x rx % Initiation �x rx % Initiation �x rx % Initiation

G1 7075T6 0.60 1.82 5 0.33 1.85 0.9 0.61 1.88 45G2 7075T651 0.62 1.46 9.9 0.24 1.48 7.1 0.38 1.57 38G3 Ti6Al4V 1.12 1.98 18.5 0.20 2.00 4.5 0.30 1.82 35G4 Al4%Cu 0.29 2.04 64.0 0.11 2.35 91 1.10 2.30 99G5 2024T351 0.76 1.84 40.1 0.20 2.40 68 1.20 3.09 94

The multiaxial criterion applied is McDiarmid.


multiaxial fatigue criteria when stresses are evalu-ated at a certain depth is larger than when it is doneon the surface. This is due to the increase of the ini-tiation phase.

(b) Comparison between the influence of di and that ofthe multiaxial fatigue criterion. It can be establishedthat it is more important to determine the value ofthe initiation length, di, than the multiaxial fatiguecriterion to be chosen. As an example, following isconsidered the influence of choosing the McDiarmidor Crossland criterion versus using a distance di orother. In the tests with aluminium Al4%Cu, groupG4, if initiation is evaluated using the McDiarmidcriterion at a depth of di = a0/2 = 45.5 lm insteadof di = D/2 = 25 lm, the parameter �x goes from 1.1to 2.6. On the other hand, if the Crossland criterionis used instead of McDiarmid, evaluating stresses atdi = D/2 = 25 lm, the parameter �x goes from 1.1 to2.2.

(c) Mean value of estimated lives. Regarding the meanvalues, �x, in the majority of the test groups analysed,better results are obtained using the variable initia-tion length model. For spherical contact, the variableinitiation length model produces better results withany multiaxial criterion. For cylindrical contact,group G4 and G5, the model with prefixed separationof initiation and propagation with di = D/2 andMcDiarmid criterion produces better results. How-ever, Crossland and SWT with variable initiationlength produce similar results for G5. As said before,using Crossland instead of McDiarmid with the pre-fixed separation initiation–propagation makes theresults for cylindrical contact worse than thoseobtained with variable initiation length. Moreover,in the methods based on a critical distance, di, andinitial length of propagation, dp, both prefixed, thedefinition of these values must be somewhat arbi-trary. The main problem with these methods lies inthe definition of values di and dp, for which there isno established criterion. The convenience in the vari-able initiation length model, proposed here, is thequality of its results and the fact that there is no needfor a previous definition of any parameters. All that isneeded are the properties of the material and charac-teristics of the test.

(d) Dispersion of estimated lives. When analysing thedispersion of estimated lives around the experimentalones, the variable initiation length model is better inalmost all test groups. Although the dispersionobtained is of the same order with each model, exceptfor group G5, where the model with di = D/2 pro-duces a very high dispersion. This indicates that thereare factors that none of the models are taking intoaccount, or simply the characteristic variability offatigue that, with the present state of knowledge,has yet to be modelled.

6. Conclusions

The following conclusions can be extracted from theresults obtained with the different life assessment proce-dures and their comparison with the tests:

1. The analysis of the tests conducted confirms the differ-ence existing between spherical and cylindrical contact,concerning stress evolution and behaviour in frettingfatigue. It is observed that, throughout the tests thatwere analysed, the stress gradient and the stress levelare higher in spherical contact than in cylindrical con-tact. As a consequence, compared to spherical contact,in cylindrical contact the initiation phase is more impor-tant. Therefore, the value of the total estimated life ismore sensitive to the depth at which stresses are evalu-ated in order to calculate that phase, li (variable initia-tion length model) or di (fixed initiation length model).

2. The importance of the initiation phase may be differentfrom one case of fretting to another, depending not onlyon the criteria and models used, but also on the mate-rial, the applied forces, and the geometry of the contact.Therefore, this importance cannot be established previ-ously. The initiation phase cannot be disregarded a pri-

ori versus the propagation phase, or vice versa, as thiscould be the cause of significant errors in life prediction.

3. The different multiaxial fatigue criteria predict differentinitiation lives, though McDiarmid and Fatemi–Socieare quite similar. These differences decrease when the lifeprediction models used combine initiation and propaga-tion. In some cases the initiation phase is short and thefatigue criterion used has little influence on the final


result, as can be observed in the cases of spherical contactanalysed (influence on life is less than 10%). In cylindricalcontact it is the contrary (it can be almost 100%).

4. In the models that previously fix the initiation length, di,choosing the length correctly is more important thanwhat the multiaxial fatigue criterion may be. This is truein cylindrical contact, with a high proportion of life ininitiation, and also in spherical contact. This and otherreasons, such as the dispersion of the estimated livesbeing the same in all of the criteria, make it difficult inthis analysis to determine which of the multiaxial fatiguecriteria is better to estimate initiation.

5. The way to avoid having to depend on the choice of di

and dp is to use the variable initiation length model,which, in general, offers better results without dependingon arbitrarily established parameters.

Acknowledgement

The authors thank the Ministerio de Educacion y Cul-tura for their financial support through the investigationproject DPI2004-07782-C02-01.

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on the use of multiaxial fatigue criteria for fretting fatigue life assessment

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