A single parameter to evaluate stress state in rail head for rollingcontact fatigue analysis

C. L. PUN1, Q. KAN2, P. J. MUTTON3, G. KANG2 and W. YAN1

1Department of Mechanical and Aerospace Engineering, Monash University, Clayton, VIC 3800, Australia, 2Department of Applied Mechanics andEngineering, Southwest Jiaotong University, Chengdu, 610031, China, 3Institute of Railway Technology, Monash University, Clayton, Vic 3800, Australia

Received Date: 25 September 2013; Accepted Date: 27 December 2013; Published Online: 7 February 2014

ABSTRACT Based on the Smith-Watson-Topper (SWT) method, a phenomenological approach formultiaxial fatigue analysis, the maximum SWT parameter is proposed as a single param-eter to evaluate the stress state in the rail head for assessing the fatigue integrity of thestructure. A numerical procedure to calculate the maximum SWT parameter from afinite element analysis is presented and applied in a case study, where the stress and strainfields due to wheel/rail rolling contact are obtained from a three-dimensional finiteelement simulation with the steady-state transport analysis technique. The capability ofthe SWT method to predict fatigue crack initiation in the rail head is confirmed in thecase study. Analogous to von Mises stress for strength analysis, the maximum SWTparameter can be applied to evaluate the fatigue loading state not only in rail head dueto rolling contact fatigue but also in a generic structure subjected to a cyclic loading.

Keywords stress state in rail head; rolling contact fatigue; SWT method; maximum SWTparameter; finite element modelling.

NOMENCLATURE Δεe¼ elastic strain rangeΔεp¼ plastic strain rangeΔε¼ normal strain rangeNf¼ fatigue lifeb¼ fatigue strength exponentc¼ fatigue ductility exponent

σ ’f ¼ fatigue strength coefficientE¼ elastic modulusε’f ¼ fatigue ductility coefficient

σmax¼ maximum normal stressσ11, σ22, σ33¼ normal stressesτxy, τyz, τxz¼ shear stressesεxx, εyy, εzz¼ normal strainsγxy, γyz, γxz¼ shear strains

n¼ unit normal to the candidate critical planenx, ny, nz¼ direction cosines of unit normal n

σ’¼ critical plane normal stressε’¼ critical plane normal strain

SWT¼ Smith-Watson-Topper parameterSWTmax ¼ maximum Smith-Watson-Topper parameterθi¼ candidate critical plane anglesx¼ running direction along the rail heady¼ transverse direction across the rail headh¼ the depth of the rail head

C1, C2¼ the two crack surfaces

Correspondence: Wenyi Yan. E-mail: wenyi.yan@monash.edu

© 2014 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct, 2014, 37, 909–919 909

doi: 10.1111/ffe.12157

I NTRODUCT ION

The demanding conditions imposed by rail transportwith higher axle loads and increasing annual haulagerates lead to increased rates of rail degradation, suchas wear and rolling contact fatigue. Although thesesituations can be mitigated by the development andapplication of higher strength rail steels, rolling contactfatigue, for example, in the form of dense surface cracksas shown in Fig. 1, is commonly found on the rail, espe-cially in heavy haul railways, because of the combinationof high axle loads and frequent train operations.1

Therefore, the study of the degradation behaviour ofrail steel has become an attractive research field inrecent years. Some experimental studies and fieldobservations2–6 reported that the rolling contact fatiguestrength/behaviour is dependent on rail grades, micro-structure, composition and hardness of the rail steels.For instance, the improvement of the wear resistance byincreasing the carbon content of the rail steels can influ-ence the relative rates at which wear and rolling contactfatigue damage occur in rail head, and in some cases alterthe morphology of the rolling contact fatigue damage.Additionally, Cookson and Mutton7 found that theenvironmental factors play an important role in rollingcontact fatigue cracking processes.

Basically, development of rolling contact fatiguedamage can be divided into three stages, (i) initiation ofcracks; (ii) crack propagation; and (iii) final failure.Among these three stages, prediction of fatigue crackinitiation due to wheel/rail rolling contact is of foremostinterest in railway management. First of all, such atheoretical analysis is applicable to prolonging the lifefor fatigue crack initiation. Second, the results of the pre-diction incorporating the analysis of crack propagationcan be applied to estimate the appropriate rail grindinginterval, which relies on the quantitative information of

wear and rolling contact fatigue development. Asrolling contact fatigue is a stress-related damage phe-nomenon, it is essential to investigate and quantifythe stress state in the rail head by conducting stressanalysis. Additionally, the results of the stress analysiscan be used to design experimental tests, such as cyclicbi-axial compression-torsion test, in order to obtainthe material parameters for the study of the ratchetingbehaviour of rail steels using a three-dimensionalcyclic plastic model.

Tournay et al.8 have identified controlling the stressstate in the rail head as of importance. Appropriatecontrol of stress state of the railway can not only preventunexpected failure, such as rolling contact fatiguedamage, but also improve the life of both wheel and railand the service reliability.9 Fröhling10 also emphasisesthat the stress state at the wheel/rail interface must becontrolled within an acceptable limit for safer and morecost effective railway operations. Some previous studieswere carried out to analyse the contact stresses or thestress field in the rail head under different rolling contactconditions. Influence of the tangential surface traction onthe contact stresses was investigated by Jiang et al..11 Xuand Jiang12 and Wen et al.13 studied the influence of par-tial slip conditions in the contact area on the stress field inthe rail using two-dimensional and three-dimensionalfinite element model, respectively. Study of the stressdistribution on the white etching layer on rail head wascarried out using two-dimensional finite element modelby Seo et al..14 The influence of non-steady rollingcontact on the contact stresses was investigated by Wenet al..15 Mutton et al.1 also investigated the influence ofloading conditions and increased head loss on longitudi-nal stresses in the rail head and the tendency for rollingcontact fatigue cracks to cause rail failure.

Although the importance of controlling the stressstate of the railway has been highlighted, most of thestress analyses conducted so far aimed to study the influ-ence on either the contact stresses or the stress field inthe rail head, which could be extracted from the finiteelement simulation directly. The basic question of howto evaluate the stress state in the rail head due to cyclicrolling contact for the assessment of structural integrityof the rail has not been fully investigated yet. Clearly,von Mises stress, which is used to evaluate the stress statefor strength analysis of a structure, is not suitable forpurpose for fatigue analysis of a structure under cyclicloading. The objective of this research is to identify a sin-gle parameter to evaluate the stress state in the rail headduring actual rolling contact situations without carryingout a complicated fatigue life analysis. Such a parametershould have the capacity to correctly characterise thepossible rail degradation level, such as the initiation ofrolling contact fatigue.

Fig. 1Dense surface cracks on the surface of the rail head in a heavyhaul rail.1

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The Smith-Watson-Topper (SWT) method,16 whichconsiders stress, elastic strain and plastic strain compo-nents, has demonstrated the capacity to predict thelocation, the orientation of crack initiation and thefatigue life of a structure under multiaxial cyclic loading,for example,17 and 18 Additionally, Klemenc et al.19

combined the SWT method as an analytical model witha hybrid multi-layer perceptron to model the dependencyof the SWT damage parameter on the number of cyclesto fatigue failure. To improve the accuracy for theestimation of multiaxial fatigue life, two fatigue damagemodels, were successfully developed by modifying theSWT method.20,21 Therefore, the maximum SWT pa-rameter from the SWT method for fatigue analysis isproposed to effectively evaluate the stress state in the railhead in the paper. A case study indicates that this param-eter and the SWTmethod have the capability to not onlyevaluate the stress state but also predict the initiation offatigue crack in the rail head.

The structure of this paper is as follows. The SWTmethod for rolling contact fatigue analysis, which corre-sponds to the maximum SWT parameter, is brieflydescribed in the section on SWT Method for RollingContact Fatigue Analysis and the Maximum SWTParameter. The finite element model and the numericalresults from the SWT method in a case study ispresented and discussed in the section on NumericalExample. Conclusions are given in the last section.

SWT METHOD FOR ROLL ING CONTACT FAT IGUEANALYS IS AND THE MAX IMUM SWTPARAMETER

SWT method

Rolling contact fatigue is one of the two typical raildegradation modes. The other degradation mode, wear,is also the result of ratcheting mechanisms,22 with theoverall degradation of the rail reflecting the relative ratesat which these two processes occur. Research has beenextensively carried out to address all the three stages ofrolling contact fatigue. Generally, two approaches, aphenomenological approach and a fracture mechanicsapproach, can be applied to study fatigue.23 The aim ofcurrent research is to evaluate the stress state in rail headfor assessing and controlling the rail integrity withoutcarrying out a complicated fatigue life analysis. For thispurpose, the phenomenological approach for fatigueanalysis is adopted.

Socie24 has reported that the SWT method can give agood correlation with in-phase and out-of-phase fatiguelives under multiaxial cyclic loadings. Some previousstudies have already shown the applicability of the critical

plane approach in the prediction of both crack plane andthe fatigue life.25,26 Ringsberg27 proposed a fatigueparameter from a combined energy-density based andcritical plane models to predict the fatigue life of wheel/rail cyclic loading. The SWT method in conjunctionwith the critical plane approach has also been successfullyapplied in a number of studies to predict the crack initia-tion and crack location of fretting specimens.17,28–31 TheSWT method in conjunction with the critical planeapproach is applied to predict the fatigue resistance ofthe rail steel due to rolling contact in the rail head incurrent study. This method is briefly presented in thesucceeding text.

The SWT method belongs to the strain-life phenom-enological approach, which includes both elastic strainrange and plastic strain range.17 The contribution ofthe elastic strain range, Δεe, on the fatigue life, Nf isoriginally described by the Basquin’s equation for highcycle fatigue,

Δεe2

� �¼

σ ’f

E2Nf� �b (1)

where σ ’f is the fatigue strength coefficient and b is the fa-

tigue strength exponent. Under uniaxial compression-tension cyclic loading with the loading ratio as �1, Eq.(1) can be converted into the stress formula,

σmax ¼ Δεe2

� �E ¼ σ ’

f 2Nf� �b (2)

where σmax is the stress amplitude under the uniaxialcompression-tension test. The contribution of the plasticstrain range, Δεp, is originally described by the Coffin-Manson equation for low cycle fatigue,17,23

Δεp2

� �¼ ε’f 2Nf

� �c (3)

where ε’f is the fatigue ductility coefficient and c is thefatigue ductility exponent. The summation of Eqs. (2)and (3) gives the strain-life equation as follows,23

Δε2

¼σ ’f

E2Nf� �b þ ε’f 2Nf

� �c (4)

The philosophy of the SWT method is to multiplythe Basquin’s formulation for maximum normal stress,Eq. (2), by the strain-life equation, Eq. (3), whichleads to,

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σmaxΔε2

¼σ ’f

� �2

E2Nf� �2b þ σ ’

f ε’f 2Nf� �bþc (5)

For a given material, the material coefficients, b, c andYoung’s modulus, E in Eq. (5) can be obtained from ex-perimental tests. To apply the SWT method Eq. (5) topredict the fatigue life of a structure under three-dimen-sional cyclic loading, the following SWT parameter,SWT, is introduced.

SWT ¼ σ maxΔε2

(6)

For a given material point, σmax is the maximumnormal stress component in a specified direction duringa fatigue loading cycle and Δε is the difference of themaximum and minimum normal strain components inthe same specified direction during the loading cycle.Therefore, the parameter, SWT is a function of the spaceorientation at a given material point. Among all theorientations, a maximum SWT, denoted as SWT

max , can befound for this material point. Calculation should becarried out for the whole structure to search for thehighest SWT

max during the loading cycle. It is the highestSWTmax that is then applied in Eq. (5) to predict the fatigue

life to crack initiation in the structure,Nf. The location ofthe highest SWT

max corresponds to the site of the fatiguecrack initiation. The orientation corresponding to thehighest SWT

max at the critical site represents the normal di-rection of the predicted crack surface because the crack isnormally perpendicular to the maximum normal stresscomponent. The predicted crack surface was designatedas the critical plane in the literature.17

Numerical procedure to determine the maximumSWT parameter

The parameter, SWT can be calculated on the basis ofstress and strain fields from a three-dimensional finiteelement analysis. The determination of the maximum

SWT parameter SWTmax in the SWT method from a finite

element simulation for fatigue analysis requires somebasic material parameters, such as elastic modulus E,yield strength Y, elastic Poisson’s ratio ν, which can beobtained from a relatively simple testing method, thatis, the monotonic tensile test. The procedure to numeri-cally determine the highest SWT

max for wheel/rail rollingcontact is shown in Fig. 2. The calculation starts fromthe identification of a critical zone, which is large enoughto cover both large vonMises stress values and large prin-cipal strain values, for example, the top 30% to 40% ofthe von Mises stress values and of the maximum principalstrain values. This is because the highest SWT

max should bewithin a zone with large von Mises stress and largeprincipal strain. After that, the values of the stress andstrain components at the integrations points of the ele-ments within the selected critical zone are collected.Three-dimensional critical plane implementation is thenapplied to define the direction cosines of a unit normal, n,to the candidate critical plane as described by Eqs. (7–9).17

nx ¼ � sinθv sinθh (7)

ny ¼ cosθh (8)

nz ¼ � sinθh cosθv (9)

where θv and θh are the two angles to define the orientationof a specific candidate plane as shown in Fig. 3. Differentcandidate critical planes are considered by varying the an-gles of the plane from 0° to 360° with the increment of5°. Three-dimensional stress and strain transformation asdescribed by Eqs. (10) and (11) are then applied to deter-mine the normal stress σ’ and normal strain ε’ to differentcandidate critical plane.

σ ’ ¼ σ xxn2x þ σ yyn2y þ σ zzn2z þ 2τxynxny þ 2τyznynzþ2τxznxnz (10)

Fig. 2 Procedure to calculate the highest value of the maximum Smith-Watson-Topper parameter, SWTmax.

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ε’ ¼ εxxn2x þ εyyn2y þ εzzn2z þ 2γxynxny þ 2γyznynzþ2γxznxnz

(11)

After that, the maximum normal stress component,σmax, along the specified normal direction n, and thestrain range, Δε, which is the difference between the max-imum and minimum values of the normal strain along thespecified normal direction n, that is, the direction normalto the candidate critical plane of an integration point canbe determined. Finally, multiplying the maximum stressamplitude by the strain range gives the SWT value foreach candidate critical plane. The plane which gives themaximum SWT value, SWT

max is the critical plane of thatintegration point. The same calculation should be re-peated for all the integration points within the identifiedcritical zone. The integration point that gives the highestSWTmax value is the predicted location where the first fatigue

crack will initiate in the structure. The critical plane atthe critical integration point for the highest SWT

max is thecrack initiating surface of the predicted fatigue crack. Acase study to determine the highest SWT

max is presented inthe next section.

NUMER ICAL EXAMPLE

Finite element model

In this numerical example, the commercial finite elementpackage Abaqus with the capability of steady-state trans-port analysis is used to simulate the wheel/rail contact

under free rolling situation. The steady-state transportanalysis technique applies the arbitrary LagrangianEulerian (ALE) formulation, which was developed byNackenhorst,32 to model rolling and sliding contactproblems. As the ALE formulation is applied, this tech-nique is a time-independent method. A steady movingcontact is converted into a pure spatial problem bydecomposing the total deformation of the rolling wheelinto a rigid body motion and material deformation.33,34

The rigid bodymotion is described by the Eulerianmethodwhile the material deformation is described by theLagrangian method. Therefore, the dynamics of the wheelduring rolling contact can be captured, although the rotat-ing body does not undergo the large rigid body spinningmotion. This method has been practically validated andsome recent numerical studies of wheel/rail contact prob-lems based on ALE formulation can be found in.33,35

A geometrically nonlinear three-dimensional finiteelement model is generated as shown in Fig. 4. Thegeometrical parameters of the wheel include the flangeradius of 16mm, the width of 145mm, and the diameterof 965mm; see Fig. 5. The technical parameters of therail include the profile of a flat bottom rail, which hasthe mass of 68 kg/m with crown radius of 254mm, gaugeradius of 31.75mm, and rail cant of 1/40; see Fig. 6.

The three-dimensional wheel model was generated byrevolving an axisymmetric model of the wheel profileabout its axis of revolution. The inner radius of the wheelis clamped to a reference point, which is located at thecentre of the wheel. The normal load L, which is the axleload, is applied on that reference point; see Fig. 4. The

Fig. 3 Definition of angles θv and θh for calculating SWTmax.

17

Fig. 4 Finite element model for simulating the steady-state wheel/rail rolling contact.

S INGLE PARAMETER TO EVALUATE STRESS STATE FOR ROLL ING CONTACT FAT IGUE 913

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spinning motion of the wheel about its axis and the transla-tional motion of the wheel with respect to the rail aredenoted as ω and v, respectively, as illustrated in Fig. 4.The initial contact position between the wheel and the rail,which is the origin of the global coordinate system of themodel, is calibrated by using a commercial mathematicalsoftware package MATLAB. The origin point (0,0,0) is usedas the reference for presenting the results in following sec-tions. The three-dimensional rail model was generated byextruding a two-dimensional rail profile. The mesh at thecontact zone is refined in order to obtain accurate stressand strain results while coarse mesh is applied to the otherpart of the rail model in order to reduce the computationaltime. The surface-based mesh tie constraint, which makesthe active degrees of freedom equal for a pair of surfaceswith uneven mesh densities,34 was applied to connect lowand high density mesh regions. The wheel-rail finite ele-ment model consists of 88 631 C3D8 elements and it has290 643 degrees of freedom in total.

Both wheel and rail are treated as deformable body,and they are described by the classical von Miseselastic-plastic constitutive material model with the con-sideration of isotropic hardening throughout the analysis.The Young’s modulus and yield strength for the rail are207GPa and 751MPa while those for the wheel are210GPa and 910MPa. The Poisson’s ratio and the workhardening exponent for both wheel and rail are 0.3 and

0.05, respectively. Coulomb friction law, which relatesthe frictional stresses to the slip velocity, is used as thefriction model.34

Numerical results and discussion

The SWT method, which considers the effect of elasticstrain, plastic strain and stress, incorporating criticalplane approach is proposed to fully evaluate the stressstate in the rail head. Following Fig. 2, the results andthe details to determine the highest maximum SWTparameter, SWT

max in the rail head of this wheel/rail rollingcontact example under heavy haul loading situation arepresented in the succeeding text.

Based on the finite element analysis, the von Misesstress and the maximum principal strain fields in the railhead for a free rolling case of friction coefficient of 0.3,and axle load of 30 t are shown in Figs. 7 and 8, respec-tively. The maximum von Mises stress zone is consistentwith the maximum principal strain zone. Importantly, thecritical zone for the SWT calculation can be determinedas the zone with the top 35% of the von Mises stressvalues, which is outlined in Fig. 9. It is worth mentioningthat the critical zone must be reasonably large to coverboth large von Mises stress values and the large principalstrain values if the large von Mises stress zone is not con-sistent with the large principal strain zone. After that, all

Fig. 5 Cross-section of the wheel profile.

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the elements within the critical zone are considered infinding the highest SWT

max. The results of stress and straincomponents of each integration points at each incrementwithin the loading cycle are then collected. With the useof MATLAB, SWT

max is determined by following Eqs. (6–11).

Figure 10 shows the surface contour plot of SWTmax on

the x-y plane at different locations along the depth ofthe rail head, h. For all calculations, the initial contactposition between the wheel and the rail is chosen to bethe reference point (0,0,0). Figures 10(a) and 10(b) showthat the distributions of the SWT

max on the top surface and0.5mm below the top surface of the rail head are not assmooth as those shown in Figs. 10(c) to 10(f). This factis due to the influence of the friction coefficient on bothstress and the strain fields at the contact between thewheel and the rail. Additionally, the transverse locationof the highest SWT

max on each surface varies along the depthof the rail head. ComparingSWT

max distribution in Fig. 10(a)to 10(f), the highest SWT

max of the model is 11.589MPa andit is located at 1.5mm below the top surface of the railhead; see Fig. 10d.

Figure 11 shows SWTmax versus the depth of the rail head,

h on the plane with the highest SWTmax. It can be seen that

Fig. 6 Cross-section of the rail profile.

Fig. 7 von Mises stress field in the rail head of the case study.

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SWTmax increases with h until the highest SWT

max is reached.After that, it decreases with h and approaches to zero.Figure 12 shows the distribution of SWT

max on the planeof the highest SWT

max along the transverse direction, y.The results indicate that the transverse location of thehighest SWT

max is 1.5mm from the initial contact positiontowards the gauge corner. Figure 13 demonstrates thevariation of SWT

max along the running direction x on theplane with the highest SWT

max . The results clearly demon-strate that the highest SWT

max is around the initial contactposition between the wheel and the rail. Combining theresults in Figs. 11, 12 and 13, the exact location of thehighest SWT

max of the specimen is at (�0.2, 1.5, �1.5) withrespect to the reference point. It is worth nothing thatthe orientation of the candidate critical plane is definedby the two angles θv and θh, which define the directioncosines of a normal to the candidature plane; see Eqs.(7–9). As mentioned earlier, angles of candidate criticalplanes from 0° to 360° with the increment of 5° are consid-ered in the calculation for each single integration point.According to Eqs. (7–9), four sets of angles θv and θh givethe sameSWT

max value at each single integration point. In an-other word, two crack surfaces give the sameSWT

max value foreach single integration point. In current case study, thecalculated angles θv and θh for the orientations of the

normal to the crack plane with the highest SWTmax are

summarised in Table 1. Additionally, the predicted direc-tions of two crack surfaces, C1 and C2, at the location ofthe highest SWT

max are illustrated in Fig. 14.Typically, a rolling contact fatigue crack can be origi-

nated from either a small surface crack or sub-surfacecrack.36 In the current case study, the results indicate thatthe crack initiates in the sub-surface of the rail head andpropagates on two possible crack surfaces. Both cracksurfaces are inclined with a very acute angle of 15° to therunning surface of the rail head. The predicted rollingcontact fatigue cracks with such orientations have beencommonly observed in practice, for example, Fig. 1.1

Applying a single parameter to evaluate the stress statein the rail head, the result of the critical position for crackinitiation from the von Mises stress method, is differentfrom the SWTmethod. This is because the SWTmethodincludes the elastic and plastic deformation of the materialfor fatigue analysis while the von Mises stress does not. Itis worth nothing that a reasonable critical zone, which cancover both large von Mises stress and large maximumprincipal strain values, should be identified for determin-ing the highest maximum SWT parameter and, therefore,for locating the position of crack initiation and crackdirection(s). The value of the highest maximum SWTparameter, combined with experimental data via Eq. (5),

Fig. 8 Maximum principal strain field in the rail head of the case study.

Fig. 9 Critical zone in the rail head of the case study.

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(a) (b)

(c) (d)

(e) (f)

Fig. 10 Surface contour plot of the maximum Smith-Watson-Topper parameter, SWTmax , on the x-y plane at (a) h = 0mm; (b) h = 0.5mm; (c)

h = 1mm; (d) h = 1.5mm; (e) h = 2mm; (f) h = 2.5mm.

0 3 6 9 120

3

6

9

12

max

(MP

a)

h (mm)

h

Fig. 11 Distribution of the maximum Smith-Watson-Topper, SWTmax,

along the depth of the rail head, h, on the plane of the highest SWTmax.

-6 -3 0 3 60

3

6

9

12

max

(M

Pa)

y (mm)

Fig. 12 Distribution of the maximum Smith-Watson-Topper, SWTmax,

on the plane of the highest SWTmax along the transverse direction, y.

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can be used to predict the fatigue life. Additionally, thecase study demonstrates that to find the highest maximumSWT parameter, the location of fatigue crack initiationand the crack surfaces for crack propagation can also bepredicted. Therefore, the maximum SWT parameter isproposed as a single parameter to evaluate the stress statein the rail head for rolling contact fatigue, which is analo-gous to use of von Mises stress as a single parameter toevaluate the strength of a structure. Practically, knowingthe value of the highest maximum SWT parameter in arail head of a specified railway can provide a quantitativemeasure about the structural integrity of the rail headwithout carrying a complicated failure analysis. For exam-ple, if the value of the highest maximum SWT parameter

in Case A is higher than that in Case B for the same rail,then Case A is more vulnerable to fatigue failure. Activelymonitoring the highest maximum SWT parameter in arail head can provide the information for rail maintenance.This single parameter can also be applied at the designstage of a railway rail. Furthermore, analogous to vonMises stress for strength analysis under static loading, themaximum SWT parameter can be applied to evaluatethe fatigue loading state in a generic structure subjectedto a cyclic loading.

CONCLUS IONS

To search for a single parameter to evaluate the stressstate in the rail head for rolling contact fatigue analysis,the maximum SWT parameter, which is originated fromthe SWT method for multiaxial fatigue analysis, wasanalysed in current study. The SWT method includesthe contribution of stress, elastic strain and plastic straincomponents in a multiaxial fatigue analysis. It can predictnot only the fatigue life but also the location of thepossible rolling contact fatigue and the direction of theinitiated fatigue crack, which has been demonstratedsuccessfully in a case study. Consequently, the maximumSWT parameter is proposed as a single parameter toevaluate the stress state in the rail head for rolling contactfatigue analysis. Practically, the value of this singleparameter can be applied as an indicator to control thestress state and to maintain the rail in a railway system.

Acknowledgements

The present work has been partly funded by an Austra-lian Research Council Linkage Project (LP110100655)with support from Rio Tinto Iron Ore. The computa-tional simulations were carried out at the NCI NationalFacility in Canberra, Australia, which is supported bythe Australian Commonwealth Government.

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-8 -4 0 4 80

3

6

9

12

max

(MP

a)

x (mm)

Fig. 13 Distribution of the maximum Smith-Watson-Topper, SWTmax,

on the plane of the highest SWTmax along the running direction, x.

Table 1 Angles, θv and θh, of the crackplane at the highest SWT

max of the model.

θv θh

15° 85°15° 265°195° 95°195° 275°

Fig. 14 Illustration of the predicted directions of the crack surfaces,C1 and C2, at the location of the highest SWT

max.

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S INGLE PARAMETER TO EVALUATE STRESS STATE FOR ROLL ING CONTACT FAT IGUE 919

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