Available online 6 January 2009

Keywords:Stationary methodFatigueStructural Paris lawCrack propagation

pattern. This paper presents the main steps of such an approach devoted to the modeling of

erent

Step 1. It is rst necessary to estimate the initial state of the new rail. Due to the fabrication process, an initial residualstress pattern exists which may greatly inuence the rail fatigue phenomena. However, these stresses are difcult to mea-sure. A computational approach can help to evaluate their level, in order to take into account their inuence on the railperformance. Different modeling attempts exist. For instance, in order to predict the residual stress distribution induced

0013-7944/$ - see front matter 2009 Elsevier Ltd. All rights reserved.

* Corresponding author.E-mail address: dangvan@lms.polytechnique.fr (K. Dang Van).

Engineering Fracture Mechanics 76 (2009) 26262636

Contents lists available at ScienceDirect

Engineering Fracture Mechanicsdoi:10.1016/j.engfracmech.2008.12.020risks which implies to have high rail quality and good maintenance policy; second, continuous demands increase in trainfrequency, running speed and axle load. Both of these objectives must be satised with ever limited resources. In order toreach this goal, the development of predictive methods on conditions of crack formation and propagation in the rail, andparticularly in the rail head, in relation with the type and quality of the rail, the trafc (type of material, speed, characteristicof the line, etc.) the resulting internal mechanical states (plastic deformation, residual stresses, etc.) is of primary interest(see for instance [1]). The mechanical response and the resistance of the rails depend of course also on the type of materialand its behavior (constitutive equations, fatigue limit, crack propagation threshold, etc.). In order to help railway engineers, anew efcient methodology based on an original comprehensive approach for studying rail damages is proposed. The approachcan be decomposed in different steps corresponding to different computation tools. These steps are:1. Introduction

Railway companies are facing diffdefects induced in the rails by the trafc. Special attention is paid to some of the principaldifculties met as well as to the proposed solutions. Examples of applications for the pre-diction of initiation as well as propagation of some defects are presented. It is shown thatnumerical simulations predict very well the locus of crack initiation as well as its propaga-tion in the rail. Our approach presents at least three main originalities: rst, it is a globalapproach starting from the evaluation of the initial state of the rail to the simulation ofthe crack propagation under complex loading including multiaxial residual stresses. Sec-ond, special and original numerical methods for the evaluation of the initial states, theoverloads and the elastoplastic state under service loading have been developed. Third, anew concept based on a structural Paris law has been developed and used in the crackpropagation simulations.

2009 Elsevier Ltd. All rights reserved.

objectives: rst, improving railway safety by reducing rail failures and associatedA comprehensive approach for modeling fatigue and fracture of rails

K. Dang Van a,*, M.H. Maitournam a, Z. Moumni b, F. Roger b

a Laboratoire de Mcanique des Solides, UMR 7649 CNRS, Ecole polytechnique, F-91128 Palaiseau, FrancebUnit de Mcanique, Groupe Matriaux et Structures, Ecole Nationale Suprieure de Techniques Avances, 91761 Palaiseau Cedex, France

a r t i c l e i n f o

Article history:Received 20 February 2008Received in revised form 13 November 2008Accepted 23 December 2008

a b s t r a c t

A comprehensive approach is developed for studying the fatigue phenomena (crack initia-tion and propagation) induced by repeated rolling or rollingsliding contacts betweenwheel and rail. Cracks initiate and propagate in the rail head in a complex varying multi-axial stress regime due to Hertzian or non-Hertzian contacts generating 3D residual stress

journal homepage: www.elsevier .com/locate /engfracmech

is achieved by using micro-indentation techniques associated with inverse calculations.

K. Dang Van et al. / Engineering Fracture Mechanics 76 (2009) 26262636 2627 Step 3. The third step is devoted to the evaluation of the envelope of the loads encountered on the track. For this purpose,it is important to model the dynamical overloads induced by potential geometric defects of the rail. This evaluation musttake into account the characteristics of the track, the rolling stock and the rolling speed. Measuring these parameters is adifcult task and it is therefore necessary to have a modeling tool in order to simulate the inuence of a great number ofparameters varying in certain possible domains. A coupled vehicle-track computational model of the overload is estab-lished which makes it possible to take into account the geometric defects, the rigidity of the rail, the sleepers and theballast.

Step 4. The evaluation of the mechanical state in the rail due to the contact stress induced by the railway trafc is crucialfor the modeling of the rail resistance: plastic deformations occur in the region near the contact zone due to repeated roll-ingsliding contacts between the wheels and the rail. As a consequence, starting from the initial residual stress distribu-tion (step 1), a residual stress pattern built up progressively modies the present stress distribution. To be realistic, it isnecessary to take into account this phenomenon which may be very signicant for crack initiation and propagation in therail head. However, using classical softwares, the evaluation of the distributions of plastic deformation and residual stres-ses is difcult. This is due, rstly to the rolling contactwhich necessitates special integration schemes in order to make thecalculations tractable and reliable, and secondly, to that the stabilized state may necessitate a great number of passages(often hundreds or even more). It is well known that reaching this state cannot be achieved with the classical numericaltools.

Step 5. In order to predict crack initiation, from the stress history which includes contact stress, stabilized residual stress,bending stress and thermal stress, a multiaxial fatigue criterion is used which has proven its efciency in many industrialapplications and particularly in automotive industry. It is based on a multi-scale approach and on a shakedown limithypothesis. Prediction of crack nucleation and propagation induced in rails (and wheels) by rolling contact problem differsfrom classic fatigue initiation and propagation problems in several aspects because the rolling contact loading causes amultiaxial state of stress (and strain) with out-of-phase stress components and varying principal stress directions. Thus amultiaxial fatigue criterion is necessary for predicting the fatigue limit and the crack initiation. The Dang Van multiaxialfatigue criterion [4] is chosen and programmed in specic fatigue software. It is used to predict the risk of crack initiationand its locus in the rail head with respect to the material behavior and the type of trafc.

Step 6. Simulation of crack propagation requires taking into account thermal stresses, residual stresses and elastic stressesinduced by the trafc (bending and contact stresses). The introduction of residual stresses presents some difculties, sincethe measured residual stresses are not self equilibrated. Moreover, this residual stress eld is modied by the presence ofa crack. Instead of stresses, we propose to introduce the incompatible plastic strain distribution which is evaluated pre-viously in the step 4 and which is not modied by the presence of the crack. The proposed rail crack propagation modelingdiffers completely from existing approaches. In particular, only mode I cracking is involved and there is no need of very highbending stress to make the crack propagate in the rail head due to trafc loading. This result matches the observationsmade on the track. Finally, in order to evaluate the crack propagation, we propose a structural Paris law obtained by inter-preting experiments on articially cracked rail.

For clarity seeking, this paper will be mainly focused on some aspects of the problemmainly those presented in steps 4, 5and 6. However the tools we have developed cover all the steps mentioned previously.

2. Evaluation of the asymptotic mechanical state due to the trafc: step 4

The objective of this section is to determine the evolution of the mechanical state of the rail under repeated rolling andsliding contact induced by the trafc. In some cases, especially in curves, there is evidence of severe surface plastic defor-mation in the rail. This plastic deformation can lead to wear and generate residual stresses which must be taken into accountfor an accurate prediction of the rail damage.

It is well known that under repeated rolling contacts, different asymptotic mechanical states could occur in the structure:elasticity, elastic shakedown, plastic shakedown or ratcheting [5]. In the elastic shakedown regime, although plastic ow oc-curs during the early cycles (the number of which could be quite high, depending on the load and on the chosen constitutiveequations for the elastoplastic behaviour of the rail), the load will be supported elastically and failure, if any, would occur inby the straightening process using rollers, the classical approaches simulate only the bending effects induced by these roll-ers. In these modeling, the moving contacts imposed by the rollers of the straightening machine are not taken intoaccount. The obtained results (compression in the rail head and tension in the foot) are quite different from the measuredstresses (tension in the rail head as well as in the foot). This difculty is avoided by the use of a stationary algorithm [3,8].The numerical results obtained by this model are consistent with the measurements. It will be shown that the same ori-ginal method can be used in order to predict the resulting distribution of plastic strains which are necessary for the furthercalculations presented in the step 4.

Step 2. A characterization of the mechanical properties of the rail steel, particularly the local elastoplastic behavior isneeded. The rail being a massive structure, these properties may differ at points situated at different depths resultingin a spatial gradient of the mechanical parameters. It is then important to characterize the material behavior locally. This

tigueevalua

Dir

Froing fro

Inwhenstant,vioushomoated bond paccou

2628 K. Dang Van et al. / Engineering Fracture Mechanics 76 (2009) 26262636in order to obtain the asymptotic state, if it exists (the regime corresponds to elastic shakedown or plastic shakedown).It is clear that using this classical integration scheme requires considerable computing resources: the description of the

contact load and its translation along the rail until the stabilization requires a ne mesh or remeshing to be accurate enough;when the number of passages required to reach the asymptotic state is large, this calculation cannot be performed using thismethod within an acceptable computation time. In conclusion, we propose alternative algorithms which are much easier touse and which provide better accuracy.

2.2. The stationary method

The main idea is to take advantage of the time independence of mechanical quantities if the phenomena are observed in areference (x,y,z) moving with the load. More precisely, if (X,Y,Z) is a xed reference, with x = X Vt, y = Y, z = Z, and if B is amaterial tensorial quantity related to the rail, one has:

_B dBdt

V @B@x

7

In the reference (x,y,z), the wheel has no translation movement, so that the load is xed. Thus, the time derivative equationsgoverning the inelastic behavior of the rail in the rolling sliding contact problem are replaced by spatial derivative in the oppo-site direction of the movement of the wheel. Thanks to this idea, calculations become easier since they need only comparisonm a practical point of view the problem is solved by replacing the rates _B dB=dt by their increments DB _BDt, start-m the initial state of the structure and following the load evolution.the case of the rolling contact problem, the different steps are the following: rst, compute the elastoplastic responsethe load is xed, which necessitates to increase progressively the load; then move the load of Dx, the load being con-compute the new elastoplastic response taking account of the state (stresses, plastic strains, . . .) obtained in the pre-step; repeat the translation of the load until a steady state is reached, i.e. the solution far behind the load isgeneous in the moving direction. The steady state plastic strain eld corresponds to the inelastic deformation gener-y the rst passage and the residual stress eld corresponds to the stress eld down-stream, far from the load. The sec-assage will create new plastic deformation which means that it is necessary to repeat the same calculations takingnt of the plastic strain pattern generated previously. These calculations must be repeated a certain number of timesnot permit to calculate the stress and strain tensor eld which are necessary for the fatigue analysis. The stress distributionsat the contact site are very complex and require the use of nite element method (FEM) in an adequate manner. Two numer-ical strategies are possible for the evaluation of the asymptotic mechanical state. For instance shakedown limits can beestablished by running FEM simulations for a great number of cycles until a steady state has been achieved. Using a classicalintegration scheme is very lengthy and not accurate enough. We have proposed alternative methods, the pass by pass sta-tionary method and the direct cyclic method [2] which make it possible to avoid these drawbacks. These methods are re-called in the following. Different applications to rail modeling are presented in [79].

2.1. Recall of classical elastoplastic scheme and application to the rolling contact problem

It is well known that the response of an elastoplastic structure S depends strongly on the loading path. It is thus necessaryto integrate the rate equations governing the boundary value problem. These equations are summarized hereafter:

_eij 12 _ui;j _uj;i 1_e _ee _ep 2_r ktrace _ee 2l _ee 3div _r 0 in S 4_r n _F on SF and _u _ud on Su 5_ep K @f

@r; _a K @f

@AK 0; f r;A 0; Kf r;A 0 6

u is the displacement vector, e the strain tensor, ee the elastic strain tensor, ep the plastic strain tensor, r the stress tensor, kand l the elastic coefcients, F the surface tractions on SF which is a part of the surface of S, ud the prescribed displacementson Su which is a part of the surface of S, f the yield function, and A is the thermodynamic force associated to the hardeningparameter a.or even to severe wear. In order to predict the damages (initiation and propagation of the cracks), it is necessary tote the stress and strain cycles in the rail head induced by the repeated contacts between rail and wheels.ect shakedown approaches as proposed by Johnson et al. [6] to derive upper and lower bounds of shakedown limit dothe high cycle fatigue regime. Above elastic shakedown limit, plastic shakedown occurs and corresponds to closed cycles ofplastic deformation; the structure likely experience low cycle fatigue. Finally, if the contact loads are above the plastic shake-down limit, the plastic strain increases cycle after cycle continuously (ratcheting); this regime corresponds to low cycle fa-

The PPSMmethod is very exible to explore the rate of convergence for each pass in order to derive the mechanical steadystate.

K. Dang Van et al. / Engineering Fracture Mechanics 76 (2009) 26262636 2629the following:

It is necessary to take account of the motion; the stabilized solution of the rolling load differs from the stabilized solu-tion with repeated static load (i.e. repeated indentation).

The rapidity of convergence toward the stabilized asymptotic mechanical state, if it exists (shakedown), depends onthe level of the loading and the behavior of the material. In those cases, the use of classical elastoplastic computationalalgorithms do not provide good results if a great number of passes is required.

An adaptation of this method was used by Maitournam [3] to calculate the initial residual stress and the plasticstrains induced in the rail by the manufacturing process due to the moving contacts imposed by the rollers of thestraightening machine which corresponds to step 1 of our global approach. The results obtained correspond quite wellto the measured distribution of residual stress, i.e. tension near the upper and lower surface of the rail and compressionin the foot.

If it is not necessary to simulate the generation of the residual stress distribution (and the associated plastic strains), thedirect stationary method (DSM) is an easier and faster way to obtain directly the stabilized mechanical state. It consists intaking into account the two following additional conditions:

ep1; y;z ep1; y; z and a1; y; z a1; y; z 8

where a represents the internal strain hardening parameters in the previous stationary method.The convergence of such a scheme models shakedown. This shakedown is elastic if the plastic strain and the hardening

parameters are constant along each streamline; otherwise, it is plastic. Non convergence of the algorithm indicates ratchet-ing. This procedure is very convenient for investigating the nature of the steady state cycle, since this method directly pro-vides the steady state solution.

For illustration, results of a 2D example are presented in Fig. 1 where a residual longitudinal stress distribution inducedby repeated contact between a rigid cylinder rolling on the surface of an elastoplastic media is shown.

The stabilized state obtained either by using the PPSM or the DSM methods are equal but they differ from the stabilizedstate of repeated static loading with the same maximum load value.

2.3. Application to the rail problem

The previously described method has been applied to determine the stabilized state in the rail by using sequentially VOC-OLIN software [11] and the stationary algorithm. First, the contact between wheel and rail is evaluated by means of VOCO-LIN. Its characteristics, which are number and dimensions of contact areas, normal and tangential pressure, can be Hertzianor non-Hertzian depending on the position on the rail, as shown on Fig. 2.

Second, using the stationary algorithm, the stabilized mechanical state (residual stress and plastic strain distribution) iscomputed and shown in Fig. 3. This simulation corresponds to the repeated passes of the contact distribution area indicatedby a circle in Fig. 2. An elastic shakedown is obtained; all components of the plastic deformation tensor are constant along allthe streamlines of the gauge corner. As a consequence, high cycle fatigue is likely to occur.Numerous calculations have been done using this method. The main conclusions derived from the obtained results areof the values of B induced by the xed load in two adjacent elements situated at the same depth. It is possible to consider com-plex rolling-sliding conditions, because the load is now xed. It is also possible to take account of local constitutive equationvarying with each point of the rail section, as determined in the step 2.

On the contrary of the classical methods which require rened meshing of the whole surface where the moving contactoccurs, in this case rened meshing is only necessary in the vicinity of the xed contact zone.

Details of this algorithm are given in [2] which can be used for any kind of elastoplastic behavior. Uniqueness of the stea-dy state solution is proven in [2]. Convergence of the algorithm is guaranteed thanks to its implicit character and to the use ofthe orthogonal projection as proven in [10]. Computations can be done either by using the pass by pass stationary method(PPSM) or the direct stationary method (DSM).

In the PPSM, one numerical steady state computation gives an instantaneous picture of the rail passing under the wheel.The plastic strain state when x tends toward +1 (which in the discretized problem corresponds to the farthest points locatedahead of the load) is the initial state of the structure; the plastic strain state at 1 (which is represented by the points lo-cated far behind the loads) corresponds to the permanent strain after the wheel pass. Thus, in order to obtain the residualstresses and strains in the rail head after the wheel pass, one performs an elastoplastic analysis with the plastic strains atx = 1 as initial strains. To obtain the stabilized state after a nite number of cycles, one simulates the successive passes.Each pass is computed by taking the residual state obtained at the previous pass as initial state. The stabilized regime isreached when at x = 1, the same plastic strain at x = +1 is found. An elastic shakedown occurs if all components of theplastic strain tensor are constant on any (horizontal) streamline. A plastic shakedown occurs if at least one of those compo-nents varies between 1 and +1. Ratcheting occurs if from one pass to the next, it keeps on increasing in the absolute valueat 1.

2630 K. Dang Van et al. / Engineering Fracture Mechanics 76 (2009) 262626363. Pre

Foris baseefciethe rwherestresspoints

s(t) anbe detnate b

Fig. 1.static cdiction of crack initiation: step 5

the prediction of crack initiation, theDangVanmultiaxial fatigue criterion is used. This criterion, described in detail in [4],d on a multi-scale approach which assumes that shakedown occurs before crack initiation. This criterion has proven itsncy in many industrial applications, particularly in automotive industry [12]. In this approach two scales are introduced:st one corresponds to engineering scale; the second corresponds to the scale of metallic grains (called mesoscopic scale),the rst cracks initiate. Thanks to the shakedown assumption at that local scale, it is possible to estimate themesoscopicfrom the engineering stress cycle. The criterion is then expressed as an inequality related to this mesoscopic stress at allin time t of the cycle so that the damaging load can be precisely characterized. The criterion used is expressed as:

maxtfst aptg b 9

d p(t) are the instantaneous mesoscopic shear stress and hydrostatic stress, a and b are material constants which canermined by two classical fatigue tests. For instance, they are related to classical experimental fatigue limits f1 (alter-ending) and t1 (alternate twisting) by [4]:

Longitudinal residual stress distributions obtained by different methods. Note the difference between the stabilized stress distribution for repeatedontact and rolling contact [9].

Fig. 2. Example of calculated rail/wheel contact area obtained by simulation (VOCOLIN, INRETS).

Fig. 3. Stabilized longitudinal plastic strain distribution.

Fig. 4. Crack initiation observed on the rail and prediction by modeling: case of trafc on tangent track.

Fig. 5. Crack initiation observed (the picture of the cracked rail has been supplied by R.A.T.P. (Rgie Autonome des Transports Parisiens)) on the rail andprediction by modeling: case of trafc on a curve.

K. Dang Van et al. / Engineering Fracture Mechanics 76 (2009) 26262636 2631

a 3t1 0:5f1f1

; b t1 10

The fatigue of the rail is checked point by point by evaluating the quantity

DV maxtfst apt b=bg 11

A positive value of DV means occurrence of fatigue. It is noteworthy that, at this stage, no information on fatigue life beforecrack initiation is provided. Applications have been made for different trafc conditions. For trafc on tangent track, crackinitiation occurs below the surface of the rail head. In the case of curves, the predicted locus of the crack is at the gauge cor-ner near the rolling surface. This matches the observations on the track as presented in Figs. 4 and 5.

4. Modelling of crack propagation: step 6

The knowledge on crack propagation rate is necessary for establishing the frequency of rail inspection. In engineering ap-proaches, the growth of rail defects is supposed to be driven by the cyclic variation of stress due to rail bending. Thermalstress and residual stress also contribute to the propagation. However, the way to take into account these residual stressesis very crude and quite arbitrary. Finally, in such case of complex structure and crack geometries, the evaluation of stressintensity factors is based on approximate formulas containing different empirical factors [13]. Due to all these approxima-tions, the engineering approaches are not sufciently predictive to be really applicable.

In the following a new crack propagation model is proposed. It starts with a presentation and an interpretation of a bend-ing fatigue test with a machined defect performed by the German Institute BAM. As a rst conclusion, it can be shown thatthe cyclic bending stress only applied to the rail is insufcient to cause the crack propagating after its initiation. The initialcrack is dened as the detectable crack by ordinary maintenance operations in railways. In this paper, its characteristiclength is taken as 3 mm.

2632 K. Dang Van et al. / Engineering Fracture Mechanics 76 (2009) 26262636Fig. 7. Meshing a rail section with different crack sizes.4.1. Determination of a structural Paris law

In order to determine a fatigue propagation law applicable to the rail, an approach based on the interpretation of exper-imental tests done by the German Research Institute BAM coupled with numerical calculations is proposed. In the test, ashort section of used rail made of 900A steel is reproled and machined in order to present a sharp defect on the rail headsurface, oriented perpendicularly to the rolling direction. The cracked rail is then subjected to three point bending, denedwith a bending nominal stress rb = 16 MPa + /141 MPa (to be compared to the real bending on the track 16+/33 MPa).The variation of the crack size is empirically correlated with the COD (crack opening displacement). The latter being obtainedby measuring the displacement of two points situated on the rail surface at each side of the defect. The test arrangement isrepresented in Fig. 6.

Fig. 6. Scheme of rail bending test of BAM.

Our approach consists of coupling numerical simulations and experimental results in order to obtain a generalized (struc-tural) fatigue crack propagation law (structural Paris law). This law is qualied as structural because it takes into accountstructural effects in contrast to the classical Paris laws obtained using CT specimen. The different steps of the analysis aregiven below:

1. For the rail studied, meshes were generated for a certain number of cracks of different sizes in the section of the rail. Fig. 7shows two examples of such meshes. For the problem at hand 14 meshes has been realized.

2. For each crack size a FEM simulation, using the experimental loading, is performed in order to calculate the correspondingstress intensity factor range DK as well as the corresponding COD. Fig. 8 gives the evolution of these two parameters withrespect to the crack size. It is shown that both DK and the COD increase with the crack size. The evolution of DKPa mp with respect to the crack size a(m) can empirically be approximated by the following formula (cf. tting in Fig. 8a):

DK 1:73E08 ap 12

1. The experimental results obtained from the BAM test are the evolution of the measured COD versus the number of loadcycles N (Fig. 9a), i.e. CODmeasured = f(N). With the numerical simulations performed for the different crack sizes a, we haveobtained the corresponding numerical COD, i.e. a = g(CODnum) (Fig. 8b). Assuming that the experimental and numericalCOD are the same, the previous relations give the crack size as a function of the number of loading cycle N, i.e.a = g(f(N)) (Fig. 9b).

2. Finally, inserting Eq. (12) in the Paris law one obtains:

dN daCDKm

da

C1:73E08 ap 13The integration of this differential equation with respect to the crack size gives:

N Z aa0

da

c1:73E08 ap m 1

c 1 m2 1:73E08m a1m2 a1m20

14

C and m can then be determined by curve tting as shown in Fig. 10.

K. Dang Van et al. / Engineering Fracture Mechanics 76 (2009) 26262636 26330102030405060708090

100110120

0 0.05 0.1 0.15 0.2Number of loading cycles N

(million)

CO

D (

m)

05

101520

0 0.05 0.1 0.15 0.2

Number of loading cycles N (million)

Cra

ck s

ize

a

Fig. 9. Evolution of the COD (a) and the crack size (b) with respect of the number of loading cycles.25(m130140150 30m

)00 5 10 15 20 25 30 35

Crack size (mm)

0 10 20 30 40

Crack size (mm)

Fig. 8. (a) Evolution of computed and tted DK with respect to the crack size; and (b) Evolution of calculated COD with respect to the crack size.

measured COD simulated COD 355010

15

20

KI (M

P

20

40CO25a) 60D ( 30

3580

100

m)Calculated K K obtained using eq (12) 120

This laTh

2634 K. Dang Van et al. / Engineering Fracture Mechanics 76 (2009) 26262636steel determined by use of classical CT specimens.

4.2. Modelling of crack propagation in the railThthe fostressstressresidu

Thso tharesidurelatiocultistresstic strperhadue toof thebendi

The d

A nsidetrafcurw will be used for the prediction of crack growth in the rail.e obtained nal DKI before very fast propagation is 33 MPa

pmwhich is very close to KIc (35 MPa

pm) of this quality ofBy that way, crack size and then DKI are linked to the number of cycles. In these calculations the crack shape is supposedelliptic and the value of DKI is evaluated at the small axis of the ellipse (denoted by a hereafter).

The generalized Paris law for the rail structure is then determined:

dadN

1013 DK I1:245

3:0115

Fig. 10. Identication of the crack propagation law (using Eq. (16)).e presented results are applied to predict the propagation of the initiated cracks studied previously. In this modelling,llowing trafc conditions are considered: track in curve, axle load of 22t. They lead to a maximum contact normalof 1520 MPa, a maximum contact longitudinal shear stress of 130 MPa and a maximum contact transversal shearof 310 MPa. The crack propagation is controlled by elastic stresses induced by the trafc (bending, contact stresses . . .),al stresses (evaluated in step 4) and thermal stress (variation of 20 C).e scenario of fatigue crack propagation is the following: below the wheel, the stress eld is mainly severe compressiont the crack is closed; when the wheel moves off, the crack opens progressively due to the positive bending stress, theal stress and the thermal stress induced by temperature variations. The residual stresses are evaluated previously inn with the type of trafc and the material; the introduction of residual stresses in the calculations presents some dif-es because they are modied by the presence of the crack. Thus it is not correct, to consider these stresses as far eldes as it is usually assumed. In the proposed modelling, instead of stress, we introduce the stabilized incompatible plas-ain distribution [14,15] which is at the origin of the residual stresses and which does not depend on the crack, exceptps very near from the crack tip. The temperature variation is estimated as about 1520 C so that the maximum stresstemperature variation is approximately 3040 MPa. The maximum of the crack opening corresponds to the maximumpositive bending stress. The amplitude of the stress intensity factor is the sum of stress intensity factor induced byng, temperature and stabilized plastic deformation.

DK I KMaxI KMaxbending Ktemperature Kep 16ifferent steps of the calculation are the following:

umerical relationship between DKI and crack size a is rst established. For this purpose, different crack sizes are con-red and the corresponding DKI (bending, temperature and plastic deformation) are calculated (cf. Table 1). Differentc conditions were studied. The rst one corresponds to a tangent track situation whereas the second corresponds toved track. (1) Trafc on tangent track: in that case, the crack initiates in the rail at a distance of around 57 mm in

K. Dang Van et al. / Engineering Fracture Mechanics 76 (2009) 26262636 2635Table 1Numerical values of different contributions to the total stress intensity factor (Pa m1) corresponding to different sizes of the crack (mm).

a b DKbending DKep DKtemp DKI

4 5 1.31E+06 8.79E+06 1.23E+06 1.13E+078 10 2.23E+06 8.58E+06 1.98E+06 1.28E+07

12 15 2.77E+06 1.64E+06 2.58E+06 6.98E+0616 20 3.25E+06 1.28E+06 3.00E+06 7.53E+0620 25 3.80E+06 1.03E+06 3.78E+06 8.61E+0624 30 4.46E+06 1.31E+06 4.36E+06 1.01E+07depth and propagates downwards. The crack shape is approximated by an ellipse (with a the minor semi-axis, and b themajor semi-axis) and it is supposed that during its propagation this shape does not change. (2) Trafc in curves: the initialcrack initiates at the surface of the gauge corner of the rail. The crack is now supposed to initiate at the surface and itsshape does not change during propagation. The corresponding mesh is given in Fig. 11. An analytical expression of thisrelation is obtained by curve tting.

Introducing this expression in the global structural Paris law, one can derive the number of cycles before fracture by inte-grating the obtained differential equation.

As an example, Fig. 12 represents the number of cycles versus the crack size characterized by a in the case of a gaugecorner surface crack. Meshing used for the simulation of crack propagation is also presented in Fig. 11. It is thus possibleto calculate the number of cycles up to fracture corresponding to an acceleration of propagation rate.

Fig. 11. Meshing used for crack propagation simulation in the case of gauge corner surface crack.

0

5

10

15

20

25

30

35

0.5 0.7 0.9 1.1 1.3 1.5 1.7

Number of loading cycles (million)

Cra

ck s

ize a

(mm

)

Fig. 12. Evolution of crack size versus number of cycles for a gauge corner surface crack.

5. Conclusion

Improving railway safety by reducing failures is an objective of all railway companies. The primary methods are based ontesting. However they are time consuming, expensive and cannot cover all the parameters and situations possible. Virtualmodeling by numerical approaches may be an interesting option [16,17]. However many theoretical, modelling and compu-tational difculties must be overcome. This paper focus on some main obstacles and presents the solutions we propose.

1. The evaluation of global forces acting on the rail, particularly dynamic overloads, is performed using a computationalapproach coupling the track, the rail and the vehicle characteristics. One notes that this aspect has not presented inthis paper.

2. Efcient numerical tools, taking account of the movements of complex contact loads, the initial state and the inelastic

2636 K. Dang Van et al. / Engineering Fracture Mechanics 76 (2009) 26262636behavior of the materials, permits to evaluate the asymptotic mechanical state of the rail due to trafc (stabilized plas-tic strain, residual stresses . . .). In particular, non-Hertzian contact pressures (normal and tangential) distribution aretaken into account.

3. The use of the Dang Van multiaxial fatigue criterion permits to determine the locus of the rst crack initiation in therail for different trafc conditions.

4. By simulating and interpreting tests performed by the German Research Institute BAM, a structural Paris law, basedonly on mode I, is derived and successfully used to predict the crack propagation under 3D complex loading conditionsinvolving contact, bending, thermal and residual stresses.

Applications to the analysis of some rail defects encountered on the French railway show that these proposals are rele-vant. A maintenance approach is thus available for railways administrations for the estimation of intervals of rail inspection.

Acknowledgement

This work has been partially done within the framework of the project NOVUM IDR2 and supported nancially by SNCF,RATP, and CORUS.

References

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A comprehensive approach for modeling fatigue and fracture of railsIntroductionEvaluation of the asymptotic mechanical state due to the traffic: step 4Recall of classical elastoplastic scheme and application to the rolling contact problemThe stationary methodApplication to the rail problem

Prediction of crack initiation: step 5Modelling of crack propagation: step 6Determination of a structural Paris lawModelling of crack propagation in the rail

ConclusionAcknowledgementReferences