Wear 260 (2006) 567572

Short communication

On the application of Dang Van criterion to rolling contact fatigue

H. Desimone, A. Bernasconi, S. BerettaPolitecnico di Milano, Dipartimento di Meccanica, via La Masa 34, I-20156, Milano, Italy

Received 24 August 2004; received in revised form 18 February 2005; accepted 8 March 2005Available online 15 April 2005

Abstract

In this note, the problem of the calibration of the Dang Van multiaxial fatigue criterion is addressed. The discussion is based on uniaxialfatigue tests performed with different stress ratios. Results show that the usual technique for calibrating the constants of the Dang Van criteriondoes not agree with experimental evidence, especially for negative stress ratios. For this reason, a different fatigue failure locus made of twostraight line segments is proposed and typical three-dimensional rolling contact stress histories are analyzed using the traditional and proposedmethods. Results show that the conventional technique does not agree with knowledge coming from shakedown approaches of rolling contactwhile the proposed method seems to constitute a more appropriate limit.

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2005 Elsevier B.V. All rights reserved.

eywords: Rolling contact fatigue; Multiaxial fatigue; Shakedown; Residual stresses; Stress ratio

. Introduction

High cycle contact fatigue failure has considerable indus-rial relevance for those applications where contact loads ap-ear, as for example, gears, cams, rolling bearings and railheel systems. In particular, rolling contact fatigue (RCF) isaybe one of the most difficult problems regarding out-of-hase multiaxial fatigue because all six components of thetress tensor may arise. In recent years, a high number ofapers have been dealing with the use of the Dang Van multi-xial criterion for RCF (see, for example, references[111]).he aim of this note is to discuss the suitability of this method

o RCF, by verifying its response to several uniaxial tests andxtrapolating data to three-dimensional contact stress histo-ies.

.1. Dang Vans fatigue criterion

Before detailing the calibration of the Dang Van locus, anutline of the practical application of this criterion will beiven. The basis of the following relationships is the appli-

cation of the elastic shakedown principles at the mesosscale, which will be shortly explained in this article. For mtheoretical details, the interested reader is referred to[12,13].The Dang Van criterion can be expressed by:

max(t) + aDVH(t) = W (1)with aDV being a constant to be determined,W the fatiguelimit in reversed torsion,H(t) the instantaneous hydrostacomponent of the stress tensor andmax(t) the instantaneouvalue of the Tresca shear stress, i.e.,

max(t) = sI (t) sIII (t)2

(2)

evaluated over a symmetrized stress deviator, which itained by subtracting from the stress deviator:

sij(t) = ij(t) ijH(t) (3)a constant tensor,sij,m, i.e.,

sij(t) = sij(t) sij,m (4)The constant tensorsij,m is defined by the relationship: Corresponding author. Tel.: +39 02 2399 8213; fax: +39 02 2399 8202.E-mail address: hernan.desimone@polimi.it (H. Desimone).

maxt

[(sij(t) sij,m)(sij(t) sij,m)]

= minsij

maxt

[(sij(t) sij)(sij(t) sij)] (5)

d.043-1648/$ see front matter 2005 Elsevier B.V. All rights reserveoi:10.1016/j.wear.2005.03.007

568 H. Desimone et al. / Wear 260 (2006) 567572

The constant tensorsij,m may be regarded as the part of thestress deviator, which has no influence on the fatigue cracknucleation, and therefore, is eliminated through the mini-mization process of Eq.(5). One of the consequences of thismethod is the correct prediction of the absence of any effectof a mean shear stress upon the torsional fatigue limit.

In Dang Vans original proposal, the existence of the con-stant stress deviatorsij,m is justified by the assumption thatthe stress deviator defined by Eq.(4) is the mesoscopic stressdeviator, i.e., the stress state found at the grain scale. Closeto the fatigue limit, some unfavorably oriented grains maystill undergo cyclic plasticity, although the macroscopic be-haviour appears elastic. For crack nucleation to be avoided,these grains necessarily have to reach an elastic shakedownstate. The presence of the residual stress deviator defined byEq.(5) allows fulfillment of this condition.

Defined at the mesoscopic scale, these residual stressesare different from those developing as a consequence of astructural elastic shakedown. In this case, the minimizationprocedure of Eq.(5) would not be enough and the evaluationof residual stresses would require the classical procedures ofthe theory of plasticity.

If the macroscopic stress state exceeded the yield limit,because of the assumption of the elastic behaviour of thecrystalline aggregate surrounding the unfavorably orientedgrains, the Dang Van criterion would not be applicable, un-l at them canb idedt n oft

oft ort thisr limitsi cingi cto so bed statics

plieds in-ii h theh -f

T ngV urebtt ingb a V-s

Fig. 1. The Dang Van diagram: a scheme of typical calibration with bendingand torsion fatigue test.

the fatigue locus is then assumed as the straight line tangent tothese two lines, with a constant slope given by the followingexpression:

aDV = 3(W

W 1

2

)(7)

Peridas and Hills[14] have recently pointed out the impor-tance of using more than these two tests. The first aim of theirproposal is to obtain a more accurate fatigue limit domain. Infact, as the limit line is tangent to the paths corresponding topure bending and torsion tests, which are close to each other,small errors may have a profound effect on the slope of thelocus line. The second and maybe more important aim of theirproposal is to confirm, on the basis of a more extended amountof data coming from experiments, the suitability of the DangVan criterion to reproduce simple experimental cases.

On this context, in this work, the Dang Van criterion istested with a new data set coming from uniaxial experimentsperformed at different stress ratios and the results are inter-preted in terms of fatigue failure predictions for typical rollingcontact stress histories.

2. Calibration of Dang Van fatigue failure locus withsmooth specimens

urel pre-v d atd peci-m l witha nga-t ltso er s am-

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SN 8ess the material shakes down to the elastic state alsoacroscopic level. For this reason, Dang Van criterione applied to some rolling contact fatigue problems, prov

hat the contact conditions allow for the elastic shakedowhe material subjected to contact stresses.

Going back to Eq.(1) because of the symmetrizationhe stress deviator, the termmax(t) alone cannot account fhe effect of normal stresses upon the fatigue limit. Foreason, the effect of a mean normal stress upon fatiguen bending and torsion is taken into account by introdunto Eq.(1) the termDVH(t). This term represents the effef the hydrostatic stress on crack nucleation and it can alemonstrated that macroscopic and mesoscopic hydrotresses are the same[12].

If residual stresses are superimposed on the aptresses, the termmax(t) is not altered because of the mmization process of Eq.(5). However, the termDVH(t)s modified by the presence of residual stresses througydrostatic part of the residual stress tensorH,res, and there

ore, Eq.(1) becomes:

max(t) + aDV[H(t) + H,res] = W (6)

he constantaDV appearing in the expression of the Daan criterion is usually calibrated with two fatigue tests, pending (or tensioncompression) and pure torsion[1]. In

he space constituted by the symmetrized shear (max) andhe hydrostatic stress (H), reversed torsion and alternatending are represented by a vertical line segment andhape curve, respectively. As shown schematically inFig. 1,In this section, the calibration of Dang Van fatigue failocus will be performed using smooth specimens. In aious work[15], bending and axial tests were performeifferent stress ratios, either on smooth and notched sens. The material was a quenched and tempered steen ultimate tensile strength of 1350 MPa, and 0.6% elo

ion at fracture. In this work, we will focus only on the resubtained with smooth specimens.Table 1summarises thesesults. It can be observed that the values of the stres

able 1ummary of normalised (toa,R=1) fatigue limita/a,R=1tress ratio (R) 2 1 0.1 0.3ormalised fatigue limit 1 1 0.76 0.6

H. Desimone et al. / Wear 260 (2006) 567572 569

Fig. 2. Traditional and proposed locus for the Dang Van criterion for smooth specimens.

plitude forR=2 and1 are the same, in other words, thecompressive mean stress seems to have no influence on thealternating fatigue stress.

Fig. 2shows the results of the fatigue tests reported ontothe Dang Van (max, H) space. The V-shaped curves rep-resent the stress paths corresponding to the stress history (interms of hydrostatic stress and symmetrized shear) experi-enced by the specimens. It is possible to observe that, as thefatigue limit is the same forR=1 and2, a horizontalsegment is expected as the locus for (small) positive val-ues of hydrostatic stress. This horizontal segment means thatin the interval of hydrostatic stress (x-axis) values betweena,R=1/4.5 anda,R=1/3, the multiaxial fatigue limit doesnot depend on hydrostatic stress, and therefore, the Dang VanparameteraDV, which measures this dependence, is close tozero.

If one tries to extrapolate this null slope to the origin(H = 0) a kind of contradiction arises because it would meanthat W/W is equal to 0.5. This case is far from reality, asappointed by Ekberg et al.[16] because steels generally ex-hibit ratios ofW/W larger than 0.5. This also implies thatthe fatigue limit corresponding to alternating torsion (H = 0)is larger than the one resulting from the projection of the re-ferred horizontal segment onto themax axis. Nevertheless,it is always possible to take this horizontal projection as aconservative fatigue limit, asFig. 2shows.

ratioR n bed orre-s sumea ori-zaa -t athc ut-

side this locus, the limit presented herein seems to be a con-servative multiaxial fatigue limit.

The difference with traditional Dang Van literature onRCF [111] is that the limit is composed by two straightsegments with different slopes instead of a single line whoseslope is defined by the unique parameteraDV. Logically, ifstress histories coming from the problem under considerationfall in the right part of the curve (for the material here con-sidered, for values ofH > 0.3W,1), the locus could still beconsidered as conformed by an unique negative-slope straightline, but this is not the case of RCF, as it will be clarified innext section.

It is worth noting that the fatigue experiments on smoothspecimens of a high strength steel reported herein are inagreement with some previous literature about the Haigh di-agram. In fact, classical literature about the Haigh diagramusually shows a flat response for the stress amplitude in fa-tigue when negative stress ratios are considered (see, for ex-ample, Heywood[17] or Dalan[18]). The Dang Van criteriondoes not predict this behavior. In other words, the Dang Vancriterion predicts an increase of the stress amplitude to fail-ure, which disagrees with a flat response for negative stressratios. In this way, the experiments considered herein seemto be consistent with previous knowledge.

Indeed, it would clearly be better to calibrate multiaxialcriteria using multiaxial instead of uniaxial tests (although,o uni-a entlyto hases erredp sucha epre-s iguer . Thisi

Regarding the values of the fatigue limits at stress=1, 0.1 and 0.3, a negative-slope straight line carawn between the vertices of the V-shaped curves cponding to these test results. Then, it is possible to asfatigue limit locus made of two different segments, a h

ontal one up to the limit given by experiment atR=1, andnegative-slope one, passing from the results forR=1, 0.1nd 0.3. This locus is shown inFig. 2. Although the con

radiction previously mentioned is still present, i.e., the porresponding to pure torsion fatigue test is likely to fall of course, a multiaxial criterion should be able to predictxial failures). In this sense, one of the authors has rec

ested multiaxial criteria with out of phase loading[19], inrder to reproduce some of the features of the out-of-ptresses generated by rolling contact loading. In the refaper, even if the steel includes some micro-defects (s inclusions), and therefore, the material cannot be rented as smooth, some flattening behavior in the fatesponse for negative hydrostatic pressure is observeds in agreement with the ideas here proposed.

570 H. Desimone et al. / Wear 260 (2006) 567572

3. Application of the new and original formulationsto RCF

In this section, the application of the proposed formulationand the original Dang Van criterion are analysed and com-pared with traditional information coming from the shake-down theories of rolling contact. First of all,Fig. 3a showsan application of the proposed locus to a three-dimensionalrolling/sliding point contact (i.e., a sphere rolling on a halfspace). The radius of the spherical contact is 1 mm. The fric-tion factor was set equal to 0.1. The stresses were com-puted using the equations proposed by Sackfield and Hills[20]. The residual stresses were approximated following thesuggestions of Hills and Sackfield[21]. It is worth notingthat for the Dang Van method, it is not necessary to take intoaccount the residual shear stresses as explained previously

Fap(sw

in the introduction. The contact conditions are summarizedby the ratiopo/k= 3.5,po being the maximum pressure andk the cyclic yield shear stress. It can be observed that, if theproposed conservative locus is used, failure is predicted un-der these conditions. On the other hand, if the conventionalDang Van formulation is considered, failure would not bepredicted.

It is very interesting to compare these results with theshakedown map. The theory of shakedown maps for rollingcontact is thoroughly described by Johnson[22]. On thesemaps, available for two- and three-dimensional contacts,po/kvalues are at the ordinates and the friction coefficient (or theratio between traction and normal loads) are at the abscissa.For example, inFig. 4, the shakedown map for elliptical con-tacts is presented[23]. Lines A and C represent the limit forelastic shakedown. In other words, if for a given friction fac-tor the loadpo/k is smaller than the one found on the line(A or C), a fully elastic behaviour will be the final materialresponse after a few cycles. During these first cycles, residualstresses and strains could arise. If thepo/k is larger than thelimit provided by line A (or C), alternating plasticity (or in-cremental growth) will take place. It is worth remarking herethat a working point in the elastic region does not guaranteethat fatigue failure will not take place. In fact, the shakedownlimit only guarantees that, up to this level, the final responseof the material will be fully elastic. However, a fully elasticr ailure.I ail-u way,o forv s Aa sseds wnig. 3. Application of Dang Vans criterion to rolling/sliding contact, forspherical contact of radius 1 mm and friction coefficient= 0.1; (a)

o/k= 3.5: the original Dang Vans failure locus does not predict failure;b) po/k= 7: although this condition corresponds to occurrence of RCF, thetress paths fall below the Dang Vans limit locus, and therefore, failureould not be predicted.

F lidingc atingp entalg entalg yclef angV resentt ected.L

esponse does not guarantee the absence of fatigue fn fact, for a normal pushpull axial fatigue test, fatigue fre occurs below the yield stress value. In the samene should expect the limit of high cycle fatigue failurealues ofpo/k smaller than the ones represented by linend C in the shakedown map. This limit is also exprechematically inFig. 4, where the shadow region was dra

ig. 4. The shakedown map for a general three-dimensional rolling-sontact. (A) Upper bound to elastic shakedown limit against alternlasticity. (B) Upper bound to plastic shakedown limit against incremrowth. (C) Upper bound to elastic shakedown limit against incremrowth of surface strain. HCF: just for illustration scope, a line of high c

atigue, below curves A and C is shown. The prediction with original Dan locus is also shown (identified as DV), whereas shadow region rep

he part of the shakedown map where high cycle fatigue may be expines A, B and C after Ponter et al.[23].

H. Desimone et al. / Wear 260 (2006) 567572 571

just for illustration purposes and does not represent any realmultiaxial high cycle fatigue limit for this material.

Now, it is very important to point out that with the two-slope locus, failure is predicted forpo/k= 3.5, while with theoriginal approach failure is not predicted. Therefore, the con-tact load should be increased in order to reach the fatiguelocus, but difficulties arise when trying to establish the exactlimit of po/k for the conventional locus. This is because it isdifficult to obtain the residual stress distribution for highervalues ofpo/k, mainly because cyclic plastic deformation islikely to occur. Nevertheless, it is possible to estimate thelimit above a value ofpo/k= 7. In fact, the stress paths on theDang Van map corresponding to this load are represented inFig. 3b. It can be observed that although some small vari-ation could be expected if a more sophisticated method isused in order to determinate the residual stresses, the stresspaths are still well below the failure locus. For this reason, itis possible to affirm that the limit tends to fall well over thevaluepo/k= 4.7, which represents the onset of cyclic plastic-ity for frictionless contacts, as shown in the shakedown mapof Fig. 4, where a value of abovepo/k= 7 is indicated.

In this way, if the traditional application of the Dang Vancriterion were used, the high cycle fatigue limit would befar over the elastic shakedown limit; a situation that is incontrast with the initial assumptions of both mesoscopic andmacroscopic shakedown. Moreover, one might even wonderw be-c re isl ionc it.

4

et locusb -m eed,a linep whenc e limitv es-t ei-t esent.I f theD stics yclef cksp vi-o d tor

ights d: (a)t ereina ree-m

Fig. 5. Change of stress histories in the (max, H) plane when residualstresses are considered.

It is also worth noting one additional issue interesting forcomputational reasons. If a horizontal segment (withaDV = 0)is assumed as the limit forH

572 H. Desimone et al. / Wear 260 (2006) 567572

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[ truc-li, G.lop-,

[ etric

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[23] A. Ponter, A. Hearle, K. Johnson, Application of the kinematicalshakedown theorem to rolling and sliding point contacts, J. Mech.Phys. Solids 33 (1984) 339364.

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On the application of Dang Van criterion to rolling contact fatigueIntroductionDang Vans fatigue criterion

Calibration of Dang Van fatigue failure locus with smooth specimensApplication of the new and original formulations to RCFConcluding remarksAcknowledgementsReferences