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Challenge G: An even more competitive and cost efficient railway

A new methodology for the estimation of the density of contact fatigue defects in rails

T.M.L. Nguyen-Tajan*, C. Funfschilling*

*: SNCF, Research and Innovation Department, 45, rue de Londres, 75379 Paris Cedex 08, France

[email protected] Keywords Rolling-sliding contact fatigue of rails, squat and head-checking, parametric analysis, cumulative damage law Abstract

Due to the repeated passages of the wheels, rolling contact fatigue cracks can appear in the surface of the rails. These defects, such as squats and head-checks, can propagate and lead to the rail fracture and potentially to a derailment. To assure the security, heavy monitoring and maintenance procedures for the fatigue of rails are set up by infrastructure managers. Quite important costs are generated and they may increase with the continuous traffic intensification and the global commercial running speed rise. In order to go towards a more cost efficient railway system, it is essential to optimize the maintenance of rails: frequency of monitoring, rail replacement strategy, grinding policy. One key to reach this target is to have a better understanding of the physical phenomena occurring within a fatigue crack initiation and propagation, to identify the main parameters that drive this damage and to have a methodology to estimate the density of cracks and the residual lifetime of the rail in terms of the traffic and the track characteristics.

To progress in this area, a numerical modelling tool has been developed thanks to a long-term

collaboration between railway organizations (SNCF, RFF, RATP), rail producer (Tata Steel) and research institutes and universities (INRETS, LMS, MECAMIX, INSA) within the IDR2 consortium (Initiative for Development and for Research on Rail). This modelling starts with a dynamical simulation of the vehicle rolling on a track, from which the cyclic mechanical state of the rail is calculated by means of a 3D finite element simulation and an original and time-cost efficient direct stationary algorithm. Finally, a fatigue analysis of the rail is performed with the Dang Van criterion.

To go further on the development of methods and tools which can help the rail maintenance to

be more efficient, a research project is led by SNCF in order to develop a model which gives an estimation of the density of contact defects in the rail, depending on its geometry, its material properties, the track and the traffic characteristics. The study starts with a parametric numerical analysis of the crack initiation. Different variables are studied, such as the track geometry (alignment, curves, defects), the friction coefficient, the profiles of rails and wheels, the velocity and the mass of the train. A comparison with experimental observations is made in order to validate the model and to gain knowledge on the system. From this exhaustive parametric analysis, global damage laws can be identified, as a relationship between a fatigue danger coefficient and loads parameters. Combined with a Miner cumulative damage model, a fatigue assessment of the rail submitted to variable loads can be made. An application to a rail subjected to the different wheels of an entire train is shown in this paper.


Introduction

Due to the repeated passages of the wheels, rolling contact fatigue cracks can appear in the surface of the rails. These defects, such as squats and head-checks, can propagate and lead to the rail fracture and potentially to a derailment. Hence, when a rail break is detected, the traffic can be stopped or a circulation speed limitation is imposed until a rail replacement is carried out. To avoid such a nuisance, heavy monitoring and maintenance procedures for the fatigue of rails are set up by infrastructure managers. Quite important costs are generated and they may increase with the continuous traffic intensification and the global commercial running speed rise. In order to go towards a more cost efficient railway system, it is essential to optimize the maintenance of rails: frequency of monitoring, rail replacement strategy, grinding policy... One key to reach this target is to have a better understanding of the physical phenomena occurring within a fatigue crack initiation and propagation, to identify the main parameters that drive this damage and to have a methodology to estimate the density of cracks and the residual lifetime of the rail in terms of the traffic and the track characteristics.

To progress in this area, a numerical modelling tool has been developed thanks to a long-term

collaboration between railway organizations (SNCF, RFF, RATP), rail producer (Tata Steel) and research institutes and universities (INRETS, LMS, MECAMIX, INSA) within the IDR2 consortium (Initiative for Development and for Research on Rail). This modelling starts with a dynamical simulation of the vehicle rolling on a track, from which the cyclic mechanical state of the rail is calculated by means of a 3D finite element simulation and an original and time-cost efficient direct stationary algorithm. Finally, a fatigue analysis of the rail is performed with the Dang Van criterion.

To go further on the development of methods and tools which can help the rail maintenance to

be more efficient, a research project is led by SNCF in order to develop a model which gives an estimation of the density of contact defects in the rail, depending on its geometry, its material properties, the track and the traffic characteristics. The study starts with a parametric numerical analysis of the crack initiation. Different variables are studied, such as the track geometry (alignment, curves, defects), the friction coefficient, the profiles of rails and wheels, the velocity and the mass of the train. A comparison with experimental observations is made in order to validate the model and to gain knowledge on the system. From this exhaustive parametric analysis, global damage laws can be identified, as a relationship between a fatigue danger coefficient and loads parameters. Combined with a Miner cumulative damage model, a fatigue assessment of the rail submitted to variable loads can be made. An application to a rail subjected to the different wheels of an entire train is shown in this paper.

In the two first parts of the paper, a brief description of the defects and the numerical strategy and models is given. Then, the parametric analysis of the rolling contact fatigue of rails and the identified global damage laws are described. In the last part, a new method for estimating the fatigue behaviour of a rail submitted to variable loads is proposed. The main ideas of the methodology are explained and the first results obtained with a high-speed train are shown.

The rolling fatigue contact defects on rails

It is well known that due to the repeated passes of the wheels on the rails, defects such as squats and head-checks can appear on the surface of the rail. Squats (Fig.1a), which consist in fatigue cracks with a V-shape propagating in a direction slightly parallel to the free surface before turning off down and leading to a fracture, mainly appear in alignments on the rolling surface of the rail head. Head-checks (Fig.1b) appear as a network of small parallel fatigue cracks, with a direction that forms a 45° angle with the circulation direction. They can lead to peeling and exclusively appear in the external rail in a curve.


1.a : squat leading to a rail fracture 1.b : head-check leading to peeling

Figure 1: two major rolling contact defects on rails

The numerical modelling strategy for the rolling contact fatigue of rails

Track observations and previous research works have shown that crack initiation and propagation depends on many factors, such as the material properties, the vehicle characteristics (mass, suspension architecture, etc.), the running speed, the track geometry and quality (curve diameter, surface level, track gauge, etc.), the friction coefficient, environmental conditions, the maintenance policy and so on. To have a better understanding of the mechanisms and to assess the rail fatigue behaviour in terms of the running conditions, a numerical strategy has been defined and numerical tools have been developed within a coordinate research program on fatigue of rails, called IDR2 (Initiative for the Development and the research on Rail). This long-term collaboration aims at bringing together railway organizations (SNCF, RFF, RATP), a rail producer (Tata Steel) and research institutes and universities (INRETS, LMS from Ecole Polytechnique, MECAMIX, INSA de Lyon).

The numerical approach devoted to the modelling of the fatigue defects of rails (squats and head-checks) consists in a sequential modelling process, as drawn in the diagram in Figure 2. It includes: a dynamic train-track interaction simulation to evaluate the contact loads on the rail, a three-dimensional finite element calculation to determine the stabilized cyclic mechanical

stresses and strains in the rail, a fatigue analysis to assess the risk of a crack initiation and to determine its location in the rail.


Figure 2: the numerical modelling process for the rail rolling contact fatigue assessment

The dynamic train-track interaction simulation is performed using a multi-body model coupled with a 3D finite element code which takes into account the track and the rail dynamics in both longitudinal and vertical directions. The contact modelling is solved with specific models based on the theories of Hertz and Kalker to calculate the normal and tangential stresses in a semi-analytic way, in order to ensure fast and reliable simulations of the contact stresses in the presence of dry friction. The software used is called VOCO; it is developed by INRETS and is described in [1][3][4]. It predicts the dynamic loads encountered on the rail in terms of vehicle characteristics, velocity, rail and wheel geometries, friction coefficient and track geometry. Simulations done for a train rolling on a real track show that the contact loads vary along the rail: the contact forces distribution, the localization and even the number of contact zones (Fig. 3).

Figure 3: contact forces variations along the rail [12]

The next stage is to calculate the cyclic stresses and strain in the rail when submitted to a great number of these contact loads. The difficulty lies on the fact that plastic deformations and high stress gradients can occur in the area near the rail-wheel contact zone due to the high level of the rolling-sliding contact loads. Therefore, very refined finite element meshes are used to capture correctly the stresses gradients under the contact load and simulate the wheel passages. To make the fatigue analysis of the rail, it is necessary to determine the stabilized cyclic stresses in the rail, which would be too time-consuming with a classical time incremental method. To circumvent the numerical


difficulty using this approach, an original method is used and has been implemented in the software called STARAIL. This approach, called the direct stationary algorithm developed by Nguyen QS, Dang Van and Maitournam and precisely described in [6][7][9][11], is based on the following assumptions:

the simulation is made using the wheel as reference instead of the rail and; the rail material flow is modelled using Eulerian formulations;

in this reference frame, the regime is supposed to be stationary; the assumption of an elastic or plastic shakedown is made so that the cyclic mechanical fields

(stresses and plastic strains) are assumed to be periodic. These assumptions allow an algorithm which is fast. Instead of simulating many cycles until stabilization, a unique time step calculation is needed to directly evaluate the stabilized cyclic stresses and plastic strains in the structure, as shown in Figure 4. For the rail modelling, an elastic-plastic law with a linear kinematic hardening is used to model the behaviour of the steel.

Figure 4: stabilized cyclic stresses in the rail after a great number of wheel passages calculated

with the direct stationary algorithm [11]

Once the stabilized cyclic stresses are obtained, a fatigue analysis can be performed. Under the rolling-sliding contact load, multiaxial stresses with out-of phase components and varying principle directions are generated. Hence, a multiaxial fatigue criterion which has proven its efficiency in many industrial applications [2][8][13][14] is chosen. The Dang Van fatigue criterion is described in details in [5]. This fatigue criterion is based on a multi-scale approach and a shakedown limit hypothesis: the fatigue phenomenon occurs at the scale of the grain structure of the steel, so that the

quantities driving the fatigue behaviour are the stresses (and the strains) at the mesoscopic scale; with the assumption of an elastic shakedown of the stresses in high-cycle fatigue, the use of the

elastic shakedown theorem of Mandel [10] and mesoscopic-macroscopic passage model (for example the Lin Taylor model) enables to estimate the mesoscopic stresses (which drives the fatigue) in terms of the macroscopic stresses (which are calculated by the Finite Element simulations).

The criterion is then expressed in terms of mesoscopic quantities as an inequality as written as follow:

If btPat ht )(max there is no crack initiation

where

tPh is the instantaneous hydrostatic pressure at the macroscopic and mesoscopic scale

)(t is the instantaneous shear stress amplitude at the mesoscopic scale a and b are two material parameters which can be determined from the

endurance limits identified from two classical fatigue Wöhler curves.

This criterion is generally represented in the ))(),(( tPt h diagram, called the Dang Van diagram. A Dang Van danger coefficient cd can be calculated, being proportional to the distance between the


stresses path in this diagram and the material straight line whose equation is )( baxy and which defines the fatigue domain, as shown in Figure 5. This Dang Van danger coefficient can be used to estimate the risk of a crack initiation.

b

btPatc

htd

)(max

initiationcrackaistherecifinitiationcracknoistherecif

d

d

00

Figure 5: Dang Van Diagram and danger coefficient values in the rail

The parametric numerical analysis of the fatigue of rails: identification of global fatigue laws The numerical tools have previously been validated thanks to simulations performed by Mecamix [7], RATP and SNCF [12] which have been compared to experimental observations.

A parametric analysis of the contact fatigue crack initiation in rails has been performed, using the comprehensive numerical tool going from the dynamical vehicle modelling to a multi-axial fatigue assessment described below, in order to: have a better understanding of the rolling-sliding contact fatigue phenomena occurring in the rail analyze some specific parameters that helps the design and the maintenance of the rails identify a global fatigue law which is the first step towards a methodology to estimate the density

of squats and head-checks in terms of the running conditions (traffic and vehicles characteristics), rail and track characteristics.

Many parameters have been studied in this project, such as the vehicle mass, speed and

accelerations, the ratio between the sprung masses and the unsprung masses, the friction coefficient, the rails and wheels profiles (theoretical and measured ones), the curvature radius of the track, etc. All the results can not be described in this paper. First, an illustration on a comparison of different rails profiles is described. Then the general results obtained with the exhaustive parametric analysis are shown.

The experimental feedback shows that the fatigue behaviour of the rail is very sensitive to the rail profile, which is also confirmed by simulations. In order to extend the lifetime of rails, two different anti-head-check profiles have been proposed by the engineers of SNCF and have been simulated.


They will be called the AHC1 and AHC2 solutions (Fig.6a). The global modelling process described in the previous part has been performed on these two anti-head-check rails and the original rail (new rail without any modification of the profile). The results (Fig.6b) show that, for two curves with different radii, the AHC1 solution improves the fatigue resistance of the rail, while the AHC2 solution tends to be worst than the original profile. The analysis of the dynamical modelling shows that the AHC1 induces little swaying compared to the non-modified profile, which may explain the reduced values of the shear forces in the contact zone in the flank, while the AHC2 increases the bounce and the swaying. Globally, with the AHC1, the flank is less stressed, at the expense of the rolling surface. For the AHC2, the conclusions are inversed. As head-checks occur in curves and generate cracks on the flank of the external rail, the AHC1 seems to be a better solution. These conclusions from simulations have been confirmed by experiments, even if precautions have to be taken since wear has not been taken into account in this approach and may have an important influence on the global lifetime of the rail.

Figure 6a : anti-head-check profiles AHC1 and AHC2

Figure 6b: anti-head-check profiles AHC1 and AHC2 fatigue danger coefficients

As explained below, intensive calculations have been performed in order to identify the main parameters that influence the fatigue behaviour of the rail and to identify global fatigue laws, which will be the first step of the method for estimating the density of squats and head-checks.

The results show that, for fatigue analysis, the situations can be split into two categories: the cases when only one contact zone can be found, on the rolling surface of the rail, and the cases when the wheel flange touches the rail and induces many contact zones. The case of two contact zones is shown in Fig. 7.

Parametric simulations have been performed, and the analysis has been led by separating the results into these two situations (depending on the radius of the curves, the vehicle speed, the friction coefficient, the superelevation.). In Figure 8, results obtained with all the wheels of a whole TGV, for different geographic points of the track, corresponding to different curve radii, vehicle speeds, are given. Two global fatigue laws have been identified, giving a relationship between the Dang Van fatigue coefficient cd and global values of load: Pmax which is the maximum value of the pressure in the contact zone and Tmax which is the maximum value of the shear forces in the contact zone. These two


values can be reached at different points of the contact area. This result means that, for each type of rail (material and profile), the local fatigue danger coefficient cd can be directly estimated from the contact load knowledge. From now, the dynamical simulation of a train can be sufficient to assess the risk of a fatigue crack initiation on the rail, as long as the global fatigue law has been previously identified. It simplifies the process a lot and will be a key point for the development of a methodology for the squats and head-check densities estimation in terms of the running conditions.

Figure 7: mono-contact situation and bi-contact situation

Figure 8: global fatigue law giving the fatigue danger coefficient in terms of load values


The fatigue of rail under loads with variable amplitudes: application to a TGV

In this last part, a complete TGV train is studied in order to evaluate the variations of the loads and their influence on the fatigue of the rail. The test TGV IRIS of SNCF has been modelled. It contains 2 engine cars and 8 passenger cars. The values of Pmax and Tmax obtained for the different wheels of the train, at a specific point of the track are given in Figure 9. One can observe that the loads induced by the engine wheels are quite more severe than the ones generated by the coaches’ wheels. The leading axles are always more stressing than the led axles. The weakest values of Pmax and Tmax are logically obtained for the axle which is not divided into two cars. Thanks to the global fatigue laws for mono-contact and multi-contact situations identified and given in the previous part, the fatigue danger coefficients (one for the rolling contact surface of the rail and one for the flank) induced by each wheel can then be directly estimated. It can be easily seen that the variations in terms of the wheels of the whole train are quite important. It shows that to make assessments of the rail fatigue, it is necessary to take into account this variability.

Figure 9: variations of loads and fatigue coefficients along a TGV train for the rail flank


From these global fatigue laws and the Wöhler curve of the steel grade of the rail, giving the

lifetime of the material in terms of the stress amplitude, a relationship between the function f(Pmax, Tmax) and the number of cycles at crack initiation can be deduced (Fig. 10). In this diagram, the histogram of the loads imposed to the rail can be plotted. A classical Miner rule is then used to calculate the cumulative damage D undergone by the rail:

i

i

i

NnD

where ni is the number of occurrences of the load f(Pmax, Tmax)i and Ni the number of cycles before crack initiation for a cyclic load f(Pmax, Tmax)i. This approach has been applied to the IRIS TGV for a specific point of the track. The next step of the research work will be to apply the comprehensive approach to a whole track, by considering the traffic data (various types of trains and frequencies).

Figure 10: Fatigue law giving the number of cycles before crack initiation in terms of the loads parameters

Conclusions and perspectives

In this paper, a description of the numerical modelling of the rolling-sliding contact fatigue of the rail developed by SNCF and its partners of IDR2 has been given. Simulations of different situations have been performed. In particular, different anti-head-check profiles of rails have been compared. Finally, an exhaustive parametric analysis has led to the identification of two global fatigue laws – one for the rail rolling surface and one for the flank - giving the fatigue danger coefficients in terms of contact load values determined by the dynamical simulations. A proposal of a methodology to take into account the variability of the loads in the fatigue assessment of rails is given and a first application to a TGV train is shown.

This work is the first step to a comprehensive approach for the estimation of the densities of

squats and head-checks along real tracks. Some short term perspectives of this work are the following: how taking into account the variable localization of the critical point which depends on the contact

load ? to enrich the parametric numerical simulations in order to analyze more deeply the sensitivity of

the rail fatigue to some variables, such as the track defaults, the material, the wheels and the rails profiles

to analyze the aggressiveness of a distributed motorization train compared to a classical one : the values of the contact loads induced by a distributed motorization train should be less but as all axles are powered, the cumulative damage generated can be greater than the damage caused by a classical train.

to make a statistical treatment of real service loads to assess the density of contact cracks in rails.


More long-term perspectives to this work are to develop a model of the fatigue crack propagation in order to evaluate, for a measured squat or head-check, the residual lifetime of the rail and to develop wear laws and models in order to understand the physical phenomenon and have tools to evaluate the effects of combined wear and fatigue phenomena. Acknowledgment The authors are grateful to the IDR2 partners (RATP, Tata Steel, RFF, LMS, Mecamix, INRETS, INSA) with whom the numerical tools have been developed. The authors also thank Y. Le Guennec, G. Van Kalsbeek, F. Elhoudaigui, S. Dieudonné and C. Rivron for their contributions. References [1] Ayasse JB, Chollet H, Determination of the wheel rail contact patch in semi-Hertzian conditions,

Vehicle Ssystems Dynamics, Vol 43 Number 3 March 2005 [2] Bignonnet, A., Fatigue Design in Automotive Industry, High Cycle Metal Fatigue, FromTheory to

Applications, C.I.S.M. Courses and Lectures N° 392, Ed. Ky Dang Van & Ioannis V. Papadopoulos, Springer 1999 pp. 145-168

[3] Chollet H, Contact roue-rail : vérification expérimentale de la théorie de Kalker, rapport de

recherche INRETS n° 135, 87 pages, juin 1990 [4] Chollet H, Ayasse JB, Wheel Rail Contact, in Handbook of railway vehicle dynamics", sept 2005 [5] Dang Van K and Papadoupoulos I, Eds., High-Cycle Metal Fatigue in the Context of Mechanical

Design, CISM Courses and Lectures, No.392, Springer-Verglas, 1999

[6] Dang Van K and Maitournam MH, Steady-state flow in classical elastoplasticity: Application to repeated rolling and sliding Contact. J. Mech. Phy. Solids, 41 (11), 1691-1710, 1993

[7] Dang Van K, Maitournam H, Moumni Z, Roger F, A comprehensive approach for modeling fatigue

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