Prediction of wear and plastic flow in rails—Test rig results, model calibration and numerical prediction

Wear 271 (2011) 92–99

Contents lists available at ScienceDirect

Wear journal homepage: www.elsevier.com/locate/wear

Prediction of wear and plastic flow in rails—Test rig results, model calibration and numerical prediction Jim Brouzoulis a,∗ , Peter T. Torstensson a , Richard Stock b , Magnus Ekh a a b

CHARMEC, Department of Applied Mechanics, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden voestalpine Schienen GmbH, Kerpelystrasse 199 8700 Leoben, Austria

a r t i c l e

i n f o

Article history: Received 26 August 2010 Accepted 3 October 2010 Available online 16 October 2010 Keywords: Test rig Conformal contact Wear Plasticity Profile updating

a b s t r a c t Conventionally, laboratory measurements under idealized conditions are used to establish parameters needed in different kinds of wear models. This paper presents a procedure for determining the Archard’s wear coefficient from data collected in a full-scale wheel–rail test rig, i.e. under realistic loading conditions. Moreover, a simulation procedure capable of simulating rail profile evolution in conformal contacts incorporating both wear and plasticity is presented. In each simulation step, dynamic responses are calculated using the commercial vehicle–track interaction software GENSYS. The conformal contact is treated by applying a multi-Hertzian approach. To account for plastic deformations, a 2D elasto-plastic FE analysis is carried out in conjunction with a 3D local contact analysis in the commercial finite element (FE) software ABAQUS. It is shown that, due to the conformal contact, elastic shakedown is obtained after only a few load cycles and is therefore disregarded in the subsequent analyses. Quantitatively good results, in terms of worn-off area and shape of the worn profile, are presented for simulations between 20 k and 100 k load cycles. However, dependence between the chosen wear step length and the profiles obtained from the simulations is found. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Rail and wheel deterioration are important issues in railway maintenance engineering. Infrastructure managers have to face the challenge to deal with these effects in a technical and consequently also in a financial way. The most important mechanisms that cause the profile of wheels and rails to change with time are wear and plastic deformations [1] (cf. Fig. 1). This profile change affects the vehicle–rail system in a number of ways. The safety against derailment may be reduced as a change in profile may lead to a wheel-climbing situation. Further, the dynamic behaviour of operating vehicles is influenced by the altered contact geometry. This in turn may lead to poor steering, further aggregated wear, rolling contact fatigue, etc. These and other effects raise the need for maintenance which directly influences operational efficiency and costs. Thus, having accurate and efficient tools for simulating wheel–rail profile development is very valuable for vehicle operators and track authorities in order to optimize maintenance and (re)investments. An important issue to consider in the simulations of profile development is the contact modelling between the rail and the wheel. In general, the normal and tangential stresses developed

∗ Corresponding author. E-mail address: [email protected] (J. Brouzoulis). 0043-1648/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2010.10.021

by bodies in contact are coupled. However, if the bodies are assumed to behave elastically and are geometrically symmetric (quasi-identical) a decoupling of the normal and tangential contact problem is possible. To fulfil the geometrical requirements, the half-space assumption is commonly used [2]. To increase the computational efficiency, the exact solution to the normal problem is usually replaced with an approximate solution. Computer programs for the simulation of vehicle–track interaction often adopt the Hertzian contact model [3–5]. This solution relies on the half-space assumption and the capability to describe the geometry of the contacting surfaces by a second-order polynomial. Both conditions may be violated in the wheel–rail contact, for example in situations with contact between the rail gauge face and the wheel flange. For such cases, Wu and Wang [6] have shown that significant errors in maximum contact stress and contact area may arise when applying the Hertzian theory. Several studies have compared traditional contact models, relying on the elastic half-space assumption (Hertz and Kalker’s program CONTACT), with finite element (FE) models. Telliskivi and Olofsson [7] showed that differences are obtained in maximum contact stress and contact area for a contact between the wheel flange and the rail gauge corner. However, for contact geometries violating the half-space assumption, but which still fulfil the assumption of constant curvature over the contact patch, Wiest et al. [8] and Yan and Fischer [9] have shown good agreement between the two methods.

J. Brouzoulis et al. / Wear 271 (2011) 92–99

93

simulate plastic deformations of the rail, FE analyses of the rail with the constitutive model mentioned above are performed. The applied contact loading in these FE analyses are based on local contact analyses in the commercial FE software ABAQUS, taking into account the plastic deformation of the material. Furthermore, as the profile develops due to wear and plasticity, the vehicle–track interaction will be influenced. Therefore, the important issue of how often the dynamic simulations need to be updated is also investigated in terms of stability and accuracy of the simulated profile development. 2. Test rig measurements 2.1. Test rig conditions

Fig. 1. Example of profile deterioration of a rail cross-section caused by plastic deformation and wear.

The contact modelling is further complicated by the fact that multiple contact points often occur in railway vehicle–track interaction. Since the assumptions in the Hertzian theory limit its use to single-point contact situations, multiple contacts belong to the category of non-Hertzian contacts [10]. Here, a multi-Hertzian method was proposed, which treats multi-point and non-elliptical contacts by replacing them with a set of Hertzian contact ellipses [10]. This approach will be pursued in the current study. Early pin-on-disc tests performed by Archard and Hirst identified two wear regimes, in terms of the wear rate and type of wear debris; mild and severe [11]. Later the identification of a third regime denoted catastrophic wear was made by Bolton and Clayton [12]. Different parts of a rail profile may be subjected to different wear regimes. The work by Olofsson and Telliskivi showed that a rail profile may be exposed to mild wear on the rail crown and severe wear on the gauge corner. The wear rate at the gauge corner was found to be up to ten times that on the rail crown [1]. Lim and Ashby suggested a two parameter mapping method that relates the wear regimes and their mutual transitions to the contact pressure and the sliding velocity [13]. The same method has later been adopted for visualizing the non-dimensional wear coefficient used in the commonly known Archard’s wear law [11]. The large contact forces transmitted at the wheel–rail interface, may lead to plastic flow and work hardening of the rail. This is the case for situations when the stress in the rail exceeds the yield stress of the rail material, for example at the gauge corner. The stress is influenced by factors such as the magnitude of the contact force, the friction coefficient, the size of the contact patch and also the rail material behaviour. Finite element simulations with a constitutive model of the rail material behaviour can be used to predict the plastic flow and the work hardening of the rail. In the present study R260 material in the rail is considered. A constitutive model is proposed in [14] that take into account large deformations and ratcheting behaviour for R260. This model will be used in the present work. The main purpose of this paper is to propose a method on how to calibrate the parameters in Archard’s wear law against experimental data in terms of profile development from a full-scale wheel–rail test rig. In order to simulate profile development due to wear and plasticity, several cross-disciplinary models are combined. The wear simulation is based on the common Archard’s law for sliding wear. Relative motions between rail and wheel, contact forces and positions, are obtained from simulations of the dynamic vehicle–track interaction using the commercial code GENSYS. To

Experimental results from a full-scale test rig (cf. Fig. 2), are used as data for the calibration of wear parameters [15]. The test rig was designed by voestalpine Schienen GmbH and offers the possibility to simulate many operational wheel–rail contact conditions. Important parameters like the inclination of the rail, angle of attack between rail and wheel, friction in the wheel–rail contact, etc. can be adjusted. Underneath a full wagon wheel a rail segment is moved by hydraulic actuation, pressing against the wheel which generates a stationary vertical and lateral force. Simulation data in the current study is based on a test with no angle of attack, no inclination of the rail and dry contact conditions. Consistent vertical and lateral load conditions were chosen to ensure flange contact between wheel and rail (curve simulation) and to provide sufficient wear levels. In the tests a R260 rail with a 60E1 profile in combination with a standard freight wagon wheel (diameter 920 mm) with a S1002 profile and the grade R7 were used. Rail and wheel profiles were measured with a Miniprof system after the following number of wheel passages: 2 k, 5 k, 10 k, 20 k, 50 k, 80 k, and 100 k. Calibration of the model for simulation of profile development is performed between 20 k and 100 k. In this range conformal contact is fully developed. 3. Profile updating The simulation procedure for determining the profile development when subjected to a large number of (load) cycles is presented below. The suggested procedure combines several numerical tools in order to simulate plastic deformation and surface wear. As described in [4], the large difference in time scales between wear development and the dynamic problem has led to the concept of wear steps. In such a wear step all the phenomena are assumed to

Fig. 2. voestalpine Schienen test rig.

94

J. Brouzoulis et al. / Wear 271 (2011) 92–99

ing the maximum appropriate wear step length is presented in Section 5.3.

4. Dynamic wheel–rail interaction model

Fig. 3. Schematic figure showing the simulation procedure adopted in the current study.

be decoupled, i.e. the profile shape is updated due to wear without considering the influence it has on the dynamic behaviour of the system. However, in the current study the profile development due to plastic deformations is also considered.

In order to resemble the conditions in the test rig, a mathematical model has been created in the commercial dynamic vehicle–track interaction software GENSYS [16] (cf. Fig. 4). The model consists of a wheel and a rail, both modelled as rigid bodies. The kinematics of the bodies are formulated using a track-following coordinate system (xyz). This is a common approach in railway multi-body dynamic simulations to achieve a linearization of the kinematics of the vehicle bodies for track geometries generating large body rotations with respect to the inertial coordinate system (XYZ). The use of a track-following coordinate system in the current study is in accordance with the formulation used in GENSYS. The rail is modelled as rigidly attached to the ground and is therefore constrained in all degrees-of-freedom. The wheel can translate in the lateral (y) and vertical (z) directions as well as rotate around the y-axis (pitch). To resemble the conditions in a curve, the test rig enables run conditions with the wheel in a fixed angle of yaw  with respect to the rail. However, only the case with zero yaw is considered in this study. Furthermore, the external load vector, F, is prescribed with components in the vertical and lateral directions. Each simulation was performed for a time period sufficiently long to reach a quasi-static state. The model parameters and run conditions were chosen in accordance with the test rig. The mass of the wheel was 366 kg, and the mass moment of inertia in pitch was estimated to be 39 kg m2 . The travelling speed  of the wheel was taken as 0.5 m/s, and a friction coefficient of 0.6 (dry conditions) was used. To incorporate the correct contact geometry, wheel and rail profiles measured with Miniprof [17] were adopted. The applied load F, taken from the measurements in the test rig, was in the order of magnitude 235 kN and 38 kN in the vertical and lateral directions, respectively.

3.1. Simulation procedure Based on a set of input data, such as the wheel and rail geometry and the load conditions, dynamic simulations are performed (cf. Section 4). Outputs from the simulations are contact point locations, lengths of the Hertzian semiaxes, normal contact force and lateral, longitudinal and spin creepages. From these quantities, the generated wear and plastic deformation (cf. Sections 5 and 6), are computed for a simulation step. The rail profile change from both these mechanisms is then combined and the profile is updated. This marks the end of a simulation step and the procedure is repeated until the predefined running distance is achieved (cf. Fig. 3). During the simulation no evolution of the wheel profile is taken into account. However, to incorporate the small differences in profile shape around the wheel circumference, profiles measured at different locations of the wheel are used interchangeably in the dynamic simulations. Conventionally, in wear evolution simulations a smoothing of the computed worn profile is applied [3,4]. This is usually supported by the fact that it is not possible, in a mathematical model, to capture the wide spread of load cases occurring in a realistic traffic situation. Another reason may be numerical problems in the dynamic simulations originating from discontinuous curvatures of the profile associated with sharp edges on the non-smoothed profile. The smoothing is usually performed after each wear step as a post-processing procedure [3,4]. In the current study, the ambition is to establish a simulation model that does not require profile smoothing. This has been achieved by computing responses from the dynamic simulations using frequently updated rail profiles. An investigation concern-

Fig. 4. Illustration of the dynamic model. The track-following coordinate system (xyz) moves with the speed of the wheel () in the longitudinal direction with respect to the inertial system (XYZ). The vectors rw and rr are position vectors of the centre of gravity of the wheel and rail, respectively,  is the yaw angle of the wheel with respect to the track-following coordinate system and F represents the applied load with components in the vertical and lateral directions.

J. Brouzoulis et al. / Wear 271 (2011) 92–99

Quasi-identity [2] (i.e. bodies in contact are elastic and geometrically symmetric) was assumed for the wheel–rail contact, and hence the normal and tangential contact problems were treated separately. In the present study, a linear contact stiffness of 600 MN/m was used for the normal problem. In the tangential direction, Kalker’s simplified theory of rolling contact, realized through the algorithm FASTSIM, was used [18]. A maximum number of three simultaneous contact patches could be treated by the contact force algorithm. In the test rig, conformal contact was developed since the wheel flange and the rail gauge face were in contact for all cycles. Hence, the half-space assumption as well as the assumption regarding a non-varying curvature over the contact patch was violated. At a certain time step in the time integration, this may cause significant differences with regard to for example contact forces and the size of the contact patch between the current approach and a FE solution [7]. However, how this affects the evolution of wear during a large number of load cycles has not, to the authors’ knowledge, been studied in literature. This study makes an attempt to predict rail profile development in a conformal contact situation, by using a multiHertzian contact approach in combination with a frequent updating of the dynamic simulations in the main simulation scheme (cf. Fig. 3).

4.1. Contact detection problem The location and orientation of the contact patch is determined by the position of the bodies in contact, their geometry and their deformations. In [10] several simplifications were proposed in order to accelerate the solution of the multi-Hertzian contact problem in the dynamic simulations. For example, the pre-calculation of data files which stores the necessary inputs for the Hertzian contact theory [10]. To achieve a computational efficient solution, GENSYS adopts this procedure and solves the contact detection problem in its pre-processor KPF. In KPF, the computed contact point functions are only dependent on the relative lateral displacement, y, between the wheel and rail. The algorithm takes into account the roll angle of the wheelset created while it is displaced laterally (not used in this work), but the algorithm neglects the influence of the angle of yaw  . The contact point functions contain the parameters used in the computation of the contact problem. For example, to calculate the tangential forces using FASTSIM in the current application, the lateral difference in curvature, Cy , is used to determine the semiaxes of the elliptical contact patch. The location of the contact point on the wheel, ycw , and the rail, ycr , and the difference in rolling radius, r, are used to calculate the creepages. The resulting tangential forces act in the plane defined by the contact angle,  (also contained in the contact point functions) and the longitudinal direction.

95

5.1. Archard’s wear law According to Archard’s wear law, the worn-off material volume Vwear [m3 ] can be calculated as Vwear = kw

Nd H

(1)

where kw is the wear coefficient [−], N the normal force [N], d the sliding distance [m] and H the hardness of the softer material in the contact [N/m2 ]. The non-dimensional wear coefficient, kw , depends on the prevailing wear mechanism. Since the wear mechanism can be related to the contact pressure and sliding velocity in the contact, kw may be plotted in a wear map (cf. Fig. 6). In the FASTSIM solution of the tangential contact problem, the elliptical contact patch is discretized into a grid of elements. Depending on the normal force and creepages acting on the contact patch, a distribution of stick and slip is obtained. The wear depth in the contact patch local coordinate system, z, of one such sliding element is given by the following expression z = kw

pz d H

(2)

where the sliding distance, d, can be thought of as the distance that a particle on the wheel slides during its passage through the concerned grid element. The wear depth is computed for each grid element individually. The amount of wear, a specific lateral section of the rail experiences during a wear step, is determined by summing the wear contributions from all grid elements along a strip in the longitudinal direction (x-direction) (see Fig. 5). The normal contact pressure, pz , is taken from the solution of the normal contact problem according to Hertz theory. The calculated wear depths act in the contact surface between the deformed rail and wheel, at an angle  with respect to the lateral axis of the profile. However, in the current study it is assumed that the wear acts in the direction normal to the undeformed rail profile. 5.2. Determination of the wear coefficient For a certain material combination and contact condition (i.e. friction and moist), a wear map associating the wear coefficient to the contact stress and sliding velocity can be established. Jendel [3] presented a wear map with different wear regimes for the R260 material. The wear map was constructed based on laboratory measurements performed by Olofsson and Telliskivi [1] using discon-disc and pin-on-disc machines. The transition between the mild and severe wear regimes occurred at a sliding velocity of 0.2 m/s.

5. Wear modelling In the wheel–rail contact, a number of wear mechanisms may be present; abrasive wear, adhesive wear, corrosive wear and surface fatigue or delamination wear [19,20]. Mild wear (low wear rate in the wheel–rail contact) has been found to be caused by oxidation and abrasion, whereas severe and catastrophic wear (high or very high wear rate) are due to delamination wear and surface fatigue [21]. In this study, the commonly applied Archard’s law for sliding wear is adopted [11] and calibrated towards measurements from the test rig. FASTSIM is used to determine the distribution of slip velocities and sliding distances in the contact patch used as input in Archard’s law.

Fig. 5. Example of wear distribution for a contact patch discretized into a number of grid elements.

96

J. Brouzoulis et al. / Wear 271 (2011) 92–99

Fig. 6. Resulting contact pressures and sliding velocity magnitudes for the three contact points obtained from dynamic simulations corresponding to 80 k load cycles.

In the present study, normal contact forces and slip velocities associated with the three contact patches (1, 2 and 3) resulting from the dynamic simulations of the test rig during 80 k load cycles are plotted in Fig. 6. Low magnitudes of sliding velocity were obtained for all contact points. The contact patch number increases with the distance from the centre of the rail crown. Accordingly, “contact point 3” is the contact furthest out on the gauge face. Sliding velocities on the gauge corner and gauge face (contact points 2 and 3) are observed to exceed those on the rail crown (contact point 1). All samples have a sliding velocity significantly below 0.2 m/s. Therefore, according to the wear map in [3] the generation of wear occurs in the mild regime and the wear coefficient is suggested to be in the range of kw = 1 − 10 × 10−4 . In the current study, kw is assumed to vary linearly between two wear coefficients k1 and k2 corresponding to sliding velocities 0 m/s and 0.03 m/s, respectively. The reason is that it was found necessary in order to achieve good agreement between test rig data and simulations (cf. Section 7). 5.3. Discretization study To obtain efficient (with respect to computation time), reliable and robust simulations, the issue of discretization error must be investigated. Here, the discretization error is defined as the error in profile development with respect to the size of the wear step. In the simulations of wheel wear on a straight track, using a nonHertzian contact algorithm, a wear step of 0.01 mm was found to be sufficient for both the contact and dynamic simulations [19]. For the application of wheel wear simulation (using Hertz and FASTSIM), Jendel [3] concluded that an appropriate wear step is 0.1 mm. In order to assure discretization-independent results when simulating the test-rig conditions between 20 k and 80 k load cycles, a study to determine the maximum allowed length of a wear step (cf. Section 3) has been conducted. Convergence of the simulated results are based on an error function (measuring the difference in profile between simulations and experiments), defined as f (kw ) = w1

N  i=1

2

(dyi · zi ) + w2

N 

¯ i )2 (1 − ni · n

(3)

i=1

The rail profiles are divided into N strips of width dyi and height zi . The vertical difference between simulated and measured profiles for strip i is defined as, zi = zi − z¯ i where over-bar denotes simulated values. Further, kw = [k1 , k2 ]T is a vector containing the wear

Fig. 7. Worn-off area with respect to the measured 20 k profile as function of the wear step length, total of 80 k load cycles.

coefficients, and w1 , w2 are weight factors. Eq. (3) is a weighted function between the error in worn-off area and the error in colinearity between the surface normals (ni is the measured normal of ¯ i is the corresponding simulated normal). Since converstrip i and n gence with respect to the co-linearity between the surface normals was found already for long wear steps, only the error in worn-off area was used in the present discretization study (i.e. w1 = 1 and w2 = 0). In Fig. 7 the simulated worn-off area after 80 k load cycles is shown for different wear step lengths together with the measured worn-off area from the three different measurement locations (taken at 10 cm intervals along the rail). The length of the wear steps denotes the number of extrapolated load cycles between two dynamic simulations in the simulation procedure (cf. Fig. 3). For wear step lengths below 500, a stabilization in worn-off area is observed. However, a stationary magnitude and hence convergence is not reached. In this range the maximum peak-to-peak difference in worn-off area for different wear step lengths is 0.80 mm2 . To evaluate the practical implications of this unstable behaviour in the wear simulations, the total simulated worn-off area (depending on wear step length) is compared with experimental results. From Fig. 7, it is observed that the spread in experimental wornoff area exceeds that from simulations. Hence, in the calibration of the model with respect to the wear coefficient, the influence of the wear step length is moderate compared to the choice of location from where the measurements are taken. The relation between the simulated results and the wear step length was investigated further by reducing the total number of load cycles to 1000. Then another convergence study was performed with the possibility to include a wear step length as small as one (no extrapolation). This means that a new dynamic simulation is performed for each load cycle. Neither for this case a stable convergence was achieved. Accordingly, it is not possible to choose a wear step length without at the same time affecting the result in terms of worn-off area. Based on Fig. 7, a wear step length of 160 (corresponding to a maximum perpendicular wear depth of about 0.002 mm) was chosen for the continued study. To investigate how the wear is distributed in the lateral direction of the rail, the difference in simulated worn-off area between two chosen wear step lengths is evaluated. The chosen wear step lengths are 1.00 and 1.11, corresponding to 1000 and 900 updates in the dynamic simulations, respectively. Fig. 8 shows the difference in worn-off area along the lateral direction of the rail after

J. Brouzoulis et al. / Wear 271 (2011) 92–99

97

Fig. 9. Magnification of a section of the wheel and rail profiles measured with Miniprof.

Fig. 8. Difference in worn-off area between simulations performed with a wear step equal to 1.00 and 1.11 and plotted at different number of total load cycles. The worn-off area is summed up over eight sections of the rail profile.

500, 750 and 1000 load cycles. Small differences in worn-off area are observed up to 500 load cycles. Then, for an increasing number of load cycles, significant differences emerge. These are mainly orientated towards the gauge side of the rail and clearly illustrate the sensitivity of the simulation model. To fully explain the reason for the non-converging behaviour of the simulation model, a more thorough study would be needed. However, based on the result in Fig. 8, one hypothesis is that it is related to the multi-Hertzian approach used in the modelling of the conformal contact at the gauge side of the rail. Rather than a wear depth smoothly distributed over an extended conformal contact, it is localized to (in this case) three discrete contact patches. Surface irregularities of varying magnitude can be observed in the wheel and rail profiles obtained from Miniprof measurements (cf. Fig. 9). To investigate the sensitivity of the contact point detection problem, a study of the influence of surface irregularity on the contact point position has been conducted using wheel and rail profiles measured after 20 k load cycles. By performing 2D finite element wheel–rail contact analyses in ABAQUS [22], applying elastic and hyper-elastoplastic [14] material models, the contact point positions can be determined. In the FE model, plane strain is assumed with a model thickness of 10 mm and the load of the wheel is applied as surface tractions. By introducing different amounts of smoothing of the profiles (in terms of reduction of peaks) and performing the same wheel–rail contact analysis, the lateral shift of a given contact point may give an indication of how sensitive the contact position is with regard to surface irregularities. In Fig. 10, this lateral shift is presented for a varying degree of profile smoothing. It is observed that for an elastic material model and with no smoothing applied, a large lateral shift of the contact point is obtained. However, for an elasto-plastic material model, no such movement of the contact point is seen, regardless of surface irregularity. This is caused by the irreversible compression (i.e. plastic deformation) of the observed irregularities in the Miniprof measurements. From this study it can be concluded that the position of the con-

tact points are highly sensitive to small variations of the surface shape. 6. Modelling of plastic deformation Plastic deformations are simulated by using a 2D FE model, with plane strain assumption. This is clearly an idealization of the actual 3D case. To obtain a realistic response from the 2D model, the width of the contact patch and the thickness of the 2D model are based on results from local 3D contact simulations in ABAQUS. To be specific, the width of the contact patch in the 2D model is assumed to be the same as the width obtained from the local 3D simulation. Further, the thickness of the 2D model is chosen such that the same maximum von Mises stress is obtained as in the local 3D simulation (cf. Figs. 11 and 12). The 3D contact simulations are local in the sense that small blocks (side lengths around 20 mm) of the rail and the wheel (with constant curvature in the contact interface)

Fig. 10. Lateral movement of the contact point for added smoothing of profiles. Zero percent corresponds to no smoothing with a maximum surface roughness of around 0.01 mm. One hundred percent corresponds to totally smooth profiles (plane geometry).

98

J. Brouzoulis et al. / Wear 271 (2011) 92–99

load in the 2D model has the following assumed spatial distribution



2N f = at

1−

 s 2 a

(4)

where N is the normal contact force, t is the chosen thickness of the 2D model, a is the Hertzian semiaxis in the s-direction and s is a curvlinear coordinate ranging between (−a, a) with origin at the centre of the contact patch. 6.3. Results From the simulations with the 2D model elastic shakedown was obtained after only a few wheel passes. This behaviour was expected since the contact is highly conformal for the current rail and wheel profiles. As a result it was deemed to be sufficient, in this case, to consider only wear as the driving force for profile change. 7. Calibration and numerical prediction Fig. 11. Effective von Mises stress distribution in rail block for a specific contact position. Results generated in ABAQUS. Largest effective stress can be observed a few millimetres below the contact surface (391 MPa).

are extracted at the contact points. The same constitutive model is used in the 3D FE model of the rail as in the 2D FE model. 6.1. Plasticity model In the present study a hyper-elastoplastic material model is used [14], which is calibrated for steel grade R260 (900A). The material model includes several hardening variables for the development of nonlinear kinematic and isotropic hardening. This material model permits small plastic deformations to accumulate over time (ratcheting), and is therefore expected to be able to simulate for example the material relocation often observed at the lower gauge corner (lipping).

From the calibration procedure, the values for the wear coefficients were found to be k1 = 6.0 × 10−4 and k2 = 1.5 × 10−4 . This is within the presumed range suggested in [3]. The simulated profile after 100 k load cycles, together with the measured profiles after 20 k and 100 k load cycles, are shown in Fig. 13. Based on Fig. 13, a good agreement is observed along the gauge face of the rail. Near the rail crown the amount of simulated worn-off area is underestimated and the largest model errors are observed here (cf. Fig. 13). A general observation is that the predicted amount of wear is underestimated at the rail crown and overestimated at the gauge face. The influence on the simulated profile shape by introducing smoothing has been investigated. A smoothing algorithm which conserves the calculated worn-off area in each wear step was used. This was done by distributing the computed wear, associated with a contact patch, over a wider distance while maintaining the same amount of worn-off area. It was concluded that the applied smooth-

6.2. Loading Based on the results from the dynamic simulations, three representative contact points and the pertaining contact forces have been selected. For each of these contact points local contact simulations in ABAQUS have resulted in data used for the applied load as well as for the thickness of the 2D model of the rail. The applied

Fig. 12. Effective von Mises stress distribution for maximum applied load in 2D FE simulation, corresponding to when the wheel is positioned straight above the loaded section. Largest effective stress can be observed a few millimetres below the contact surface (391 MPa).

Fig. 13. Rail profiles measured after 20 k and 100 k load cycles together with the simulated rail profile after 100 k load cycles. Magnified views near (a) the rail crown and (b) the rail gauge corner.

J. Brouzoulis et al. / Wear 271 (2011) 92–99

ing did not lead to an improvement of the simulated profile shape compared to measured profiles.

99

tional Mechanics, Chalmers University of Technology, is gratefully acknowledged for fruitful discussions. For valuable inputs, thanks are directed to Mr Vestor.

8. Concluding remarks References In this paper a wear simulation model, taking into account plasticity, has been calibrated against measurement data collected between 20 k and 100 k load cycles of realistic wheel–rail contact conditions. A multi-Hertzian approach for the contact problem, using three contact points, was adopted. Quantitatively, in terms of wornoff area and shape of worn rail profile, good agreement between predicted and measured rail profile was found. However, some differences in lateral wear distribution were observed. The highest magnitude of contact pressure was calculated for a contact located close to the gauge corner. By extending the function for the wear coefficient, kw , to also include a dependence on the contact pressure, it may be possible to achieve improved modelling accuracy. The simulation model showed a somewhat weak convergence with respect to the chosen wear step length. This, is likely to be partly caused by the significant influence that surface irregularities have on the contact detection problem for elastic material behaviour. Plastic deformations of the rail were simulated by a 2D FE analysis. Furthermore, local 3D contact analyses in ABAQUS resulted in highly conformal pressure distributions which in turn resulted in elastic shakedown in the 2D model after only a few load cycles (in the magnitude of a hundred). Therefore, because of the highly conformal wheel–rail contact, plastic deformations may be disregarded in this case. The development of an accurate and computationally efficient contact model for conformal contact will be the subject of future work. The purpose is clearly to remove the inherent sensitivity observed with the current contact model. Acknowledgements This work was performed as part of the activities within the Centre of Excellence CHARMEC (CHAlmers Railway MEChanics, www.charmec.chalmers.se). Professor Jens Nielsen of Chalmers Applied Mechanics has given great support during the project and also contributed with valuable comments on the manuscript. Mr Ingemar Persson (DEsolver) is gratefully acknowledged for his assistance when establishing the dynamic simulation model. Also Dr Fredrik Larsson at the Division of Material and Computa-

[1] U. Olofsson, T. Telliskivi, Wear, plastic deformation and friction of two rail steels – a full scale test and a laboratory study, Wear 254 (2003) 80–93. [2] J Kalker, Three-Dimensional Elastic Bodies in Rolling Contact, Kluwer Academic Publishers, London, 1990. [3] T. Jendel, Prediction of wheel profile wear – methodology and verification, Licentiate thesis, Royal Institute of Technology, Department of Vehicle Engineering, Stockholm, Sweden, 2000. [4] R. Enblom, On Simulation of Uniform Wear and Profile Evolution in the Wheel – Rail Contact, Ph.d. thesis, Royal Institute of Technology, Department of Vehicle Engineering, Stockholm, Sweden, 2006. [5] J. Nielsen, A. Igeland, Vertical dynamic interaction between train and track – influence of wheel and track imperfections, Journal of Sound & Vibration 187 (5) (1995) 825–839. [6] H. Wu, J. Wang, Non-hertzian conformal contact at wheel/rail interface, in: Proceedings of the 1995 IEEE/ASME Joint Railroad Conference, 1995, pp. 137–144. [7] T. Telliskivi, U. Olofsson, Contact mechanics analysis of measured wheel–rail profiles using the finite element method, Journal of Rail & Rapid Transit 215 (Part F2) (2001) 67–72. [8] M. Wiest, E. Kassa, W. Daves, J. Nielsen, H. Ossberger, Assessment of methods for calculating contact pressure in wheel–rail/switch contact, in: Proceedings 7th International Conference on Contact Mechanics and Wear of Rail/Wheel Systems (CM2006), 2006, pp. 501–508. [9] W. Yan, F. Fischer, Application of the hertz contact theory to rail–wheel contact problems, Archive of Applied Mechanics 70 (2000) 255–268. [10] J. Piotrowski, H. Chollet, Wheel–rail contact models for vehicle system dynamics including multi-point contact, Vehicle System Dynamics 43 (2005) 455–483. [11] J. Archard, W. Hirst, The wear of metals under unlubricated conditions, Proceedings of the Royal Society of London 236 (1206) (1956) 397–410. [12] P. Bolton, P. Clayton, Rolling-sliding wear damage in rail and tyre steels, Wear 93 (1984) 145–165. [13] S. Lim, M. Ashby, Wear-mechanism maps, Acta Metal 35 (1) (1987) 1–24. [14] G. Johansson, J. Ahlström, M. Ekh, Parameter identification and modelling of large ratcheting strains in carbon steel, Computers & Structures 84 (15–16) (2006) 1002–1011. [15] D. Eadie, D. Elvidge, K. Oldknow, R. Stock, P. Pointner, J. Kalousek, P. Klauser, The effects of top of rail friction modifier on wear and rolling contact fatigue: fullscale rail–wheel test rig evaluation, analysis and modelling, Wear 265 (2008) 1222–1230. [16] DEsolver: GENSYS user’s manual, 2009. [17] C. Esveld, Miniprof wheel and rail profile measurement, in: Proceedings of the Second Mini Conference on Contact Mechanics and Wear of Rail/Wheel Systems, 1996, p. 34. [18] J. Kalker, A fast algorithm for the simplified theory of rolling contact, Vehicle System Dynamics 11 (1982) 1–13. [19] Z. Li, Wheel–Rail Rolling Contact and Its Application to Wear Simulations, Ph.d. thesis, Delft University of Technology, Delft University Press, 2002. [20] F. Alwahdii, Wear and Rolling Contact Fatigue of Ductile Materials, Ph.d. thesis, The University of Sheffield, 2004. [21] R. Lewis, R. Dwyer-Joyce, U. Olofsson, R. Hallman, Wheel material wear mechanisms and transitions, in: 14th International Wheelset Congress, 2004. [22] Abaqus, inc., abaqus/explicit (version 6.7), 2007.