A tool and a method for FE analysis of wheel and rail interaction

A TOOL AND A METHOD FOR FE ANALYSIS OF WHEEL AND RAIL INTERACTION Tanel Telliskivi, Ulf Olofsson, Ulf Sellgren and Patrik Kruse Machine Elements, Department of Machine Design Royal Institute of Technology (KTH), Stockholm, Sweden ABSTRACT 1. INTRODUCTION Damage mechanisms such as surface cracks, plastic deformation and wear can significantly reduce the service life of railway track and rolling stock. They also have a negative impact on the rolling noise as well as on the riding comfort. A proper understanding of these mechanisms require a detailed knowledge of physical interaction between wheel and rail. Furthermore, demands for higher train speeds and increased axle loads implies that the consequences of larger contact forces between wheel and rail must be thoroughly investigated. Two methods have traditionally been used to investigate the rail-wheel contact, that is the Hertz analytical method and simplified numerical methods based on the boundary element (BE) method. These methods rely on a half-space assumption and a linear material model. To overcome these limitations, a tool for FE-based quasi-static wheel-rail contact simulations has been developed. The tool is a library of Ansys macro routines for configuring, meshing and loading of a parametric wheel-rail model. The meshing is based on measured wheel and rail profiles. In order to reduce the size of the computational model, the superelement technique is utilized. The wheel and rail materials in the contact region are treated as elastic-plastic with kinematic hardening. The kinematic constraints are enforced with the Ansys contact element Contac49. By controlling the values of the configuration parameters, representations of various driving cases can be generated. The quasi-static loads are obtained from train motion simulations with special purpose software. Interaction phenomena such as rolling, spinning and sliding can be included. The modeling tool and a methodology are described in the presented paper. Simulation results are compared with Hertzian and BE solutions. Significant differences in the calculated state between the FE solution and the traditional approaches can be observed. These differences are most significant in situations with flange contact. Most railway wheels are rigidly mounted on a steel shaft. A typical wheelset on a straight track is shown in figure 1. The axle load may be as high as 220 kN and the contact area between a wheel and the rail is roughly 1 cm2. The contact region is thus very highly stressed. The interaction in the contact zone between wheel and rail is determined by the global dynamic behavior of the vehicle and by various physical phenomena that occur in the contact zone. The profiles of wheel treads and railheads are transformed by the wheel and rail interaction. This transformation, which may be severe in curves, has a significant effect on the contact state. FIGURE 1. A WHEELSET ON A STRAIGHT TRACK. Since the early 1970s, numerical simulations of the dynamic behavior of rail vehicles and the interaction between vehicle and track have been performed. Software, such as Vampire developed by British Rail, Medyna by Deutsche Luft und Raumfahrt, and Nucars in the USA, were developed for that purpose. They were all highly specialized and optimized for a reasonable a turnaround time for a simulation (Andersson et al., 2000). An example of recently developed commercially available software is Gensys (1999). General-purpose software for dynamic simulations of multi-body-systems (MBS), such as Adams, Simpack, and Dads, have recently included features that enable efficient dynamic simulations of railway vehicles and vehicletrack interaction. Adams, Gensys, Nucars, Simpack and Vampire have recently been benchmarked (Iwnicki, 1998)(Iwnicki, 1999). One-dimensional beam models are usually sufficient for the frequency range up to three kHz (Knothe et al., 1994). Software for vehicle motion simulations, are normally concerned with the orientation of each wheel relative to the track, and thus the pointof-contact between wheel tread an rail head and the contact forces that are caused by the dynamic interaction (see figure 2). z y x FX MZ FY FN reached. Carter formulated a creepage-force law connecting the driving–braking couple and the velocity difference: − kξ + 0.25k 2ξ ξ FT = µFN  − sign(ξ ) kξ ≤ 2 kξ ≥ 2 (1) where FT and FN is the total tangential and normal contact force, respectively, per unit lateral length, µ is the coefficient of friction, ξ=2(VT - VC)/(VT + VC) is the creepage, and k is the creepage coefficient. ξ is a function of µ, the radius of the wheel, and the semilength of the contact area measured in the rolling direction. Carter’s theory is sufficient when we consider the action of driven wheels, e.g. it is capable of predicting the frictional losses in a locomotive driving wheel, but it is not sufficient for vehicle motion simulations. For that we must treat forces in the lateral direction together with the motion in the rolling direction (Kalker, 1991). Johnson (1958) generalized Carter’s result to circular contacts and longitudinal and lateral creepage. Vermeulen and Johnson (1964) generalized this theory to elliptical contact areas. Shen et al. (1984) improved the results by replacing the approximate values for the creepage coefficients given by Vermeulen and Johnson with more accurate values. In these theories, which are all Hertzian-based, it turned out that the contact area is elliptic in form, with semiaxes a and b in the rolling and lateral directions respectively. The ratio of the axes, a/b depends only on the curvatures of the wheel and the rail. Furthermore, the size of the contact area depends on the normal force FN but it is independent of the tangential force FT. A FIGURE 2. CONTACT FORCES FROM AN MBSSIMULATION. Damage mechanisms such as surface cracks, plastic deformation and wear, see for example Kalousek et al. (1999) for a survey, can significantly reduce the service life of railway track and rolling stock. Furthermore, they can have a negative impact on the rolling noise as well as on the riding comfort. A proper understanding of these mechanisms require a detailed knowledge of the wheel and rail interaction. Five theories of rolling are presently in use: the two-dimensional theory of Carter (1926), the linear theory (Kalker, 1967), the complete contact theory (Kalker, 1983), the theory of Shen et al. (1984), and the simplified theory (Kalker, 1982b). All of these theories have limitations and they can be viewed as complementary. Continuum rolling contact theory started with a publication by Carter (1926), where he approximated the wheel by a cylinder and the rail by an infinite half-space. The analysis was twodimensional and the exact solution was found. Carter showed that the difference between the circumferential velocity VC of a driven wheel and the translational velocity VT of the wheel has a nonzero value as soon as an accelerating or a braking couple is applied to the wheel. This difference increases with increasing couple until the maximum value according to Coulomb is B FIGURE 3. CONTACT ON STRAIGHT RAIL (A) AND FLANGE CONTACT ON HIGH RAIL IN A CURVE (B). In vehicle motion simulations, usually only the global contact forces, as shown in figure 2, are required. A linearization of the relation between the tangential contact forces and the creepage, in railway mechanics usually referred to as the linear theory (Kalker, up to and above 3 GPa when the wheel is in contact with the gauge corner of the rail. The former result was based on Hertzian theory and the latter was calculated with Contact. Such results highlight the need for a nonlinear elastic-plastic material model and a thorough three-dimensional approach. A rail three years old at test start height (mm) 30 initial profile (3 years) one year of traffic wear two years of traffic 20 10 plastic deformation 0 0 20 40 60 length (mm) B rail new at test start 30 height (mm) 1967), is thus extensively used in this class of simulations. The linear theory is valid for small motions, without flange contact (see figure 3A). Flanging may occur on high rail, i.e. the outer rail in a curve, see the rightmost wheel in figure 3B. Due to the conicity of the wheel profile, flanging results in a large spin. In the simplified theory, wheel and rail are modeled as rigid bodies with a set of three-orthogonal springs located at discrete points on the interacting surfaces (Kalker, 1973). In the simplified theory, the surface displacement at one unique point depends only on the surface traction at the same point, i.e. it is a so-called Winkler model. The simplified theory was early implemented in special purpose computer codes (Kalker, 1982b). It has shown to be able to efficiently interpret a large number of contact phenomena as long as the contact is Hertzian (Kalker, 1991). Kalker (1979) generalized the three-dimensional rolling contact problem of two elastic bodies for combinations of longitudinal, lateral, and spin creepage. Spin creepage, or spin for short, is a significant phenomenon in curves. It is caused by a rotational velocity around the vertical and longitudinal axes of the wheelset due to different rolling radius of the two wheels in a wheelset and the conicity of the wheel profile. Spin due to conicity is comparable to camber in the automotive industry. Kalker (1982) extended his theory of rolling contact between arbitrary bodies for the case where the shape of the contact area is nonelliptical, and thus non-Hertzian. This theory is often referred to as the complete theory, although it is limited to contact problems between linear elastic bodies that can be described by half-spaces. In order to get an approximate solution, the contact area is divided into rectangular elements. This theory was implemented in a computer program called Contact (Kalker, 1983), which is based on the boundary element (BE) method. Contact which is roughly 400 times slower than routines based on the simplified theory (Kalker, 1991) has been used to validate the linear and simplified theories as well as to validate the theory by Shen, Hendrick and Elkins (1984) which also is significantly faster than Contact. In strength and fatigue calculations, a common practice is to assume an elliptic contact area and to use the Hertzian traction for the normal pressure, while the tangential distribution is found by multiplying the normal pressure distribution by the coefficient of friction (Kalker, 1991). This approach is also used in more recent studies, e.g. by Ekberg (2000). Higher train speeds and increased axle loads have led to larger contact forces between wheel and rail. The half-space assumption in the traditional approaches puts geometrical limitations on the contact i.e. the significant dimensions of the contact area must be small compared to the relative radii of curvature of each body. Especially in the gauge corner of the rail profile the half-space assumption is not valid since the contact radius here is of the same order of size to the contact zone i.e. approximately around 1cm. The form change of rail can be large over time, see Olofsson and Nilsson (1998) and Olofsson (1999). Figure 4 shows the transformation of the profile of a UIC 60 high rail over a period of two years in a narrow curve with a radius of 303m and trafficked by commuter trains. These experimental results show that plastic deformation must be treated thoroughly in wheel-rail contact analysis. Previously presented results by Cassidy (1996) and Knothe et al. (1999) reveal contact stresses initial profile (new) one year of traffic two years of traffic 20 10 wear plastic deformation 0 0 20 40 60 length (mm) FIGURE 4. FORM CHANGE OF AN UIC HIGH RAIL IN A NARROW CURVE. THREE YEAR OLD RAIL AT START (A). NEW RAIL AT START (B). To overcome the limitations inherent in the traditional approaches, a tool for FE-based quasi-static wheel-rail contact modeling and simulations has been developed. The tool is a library of macro routines for configuring, meshing and loading a parametric wheel-rail model. The routines are written in the Ansys programming language. The meshing can be based on measured wheel and rail profiles, i.e. worn profiles. The kinematic constraints are enforced with the Ansys contact element Contac49 (Ansys, 1997). The material models are treated as elastic-plastic with kinematic hardening. By controlling the values of the configuration parameters, representations of various driving cases can be generated. The quasi-static loads are obtained from train dynamic calculations with special purpose MBS software. Interaction phenomena such as rolling, spinning and sliding can be included. In order to reduce the size of the computational model, the superelement technique is utilized for the linear model features. The modeling and simulation tool and a methodology are described below, and simulation results are compared with solutions obtained with traditional methods. 2. FE MODELLING OF WHEEL-RAIL INTERACTION A complete FE model of one wheel, a piece of rail, a set of contact elements and the quasi static loads on the wheel are interactively created with a library of Ansys macro routines that are accessed with the Ansys Graphical User Interface (GUI), see figure 5. Sets of keypoints that define the rail head and wheel tread profiles are assumed to exist as separate files. Keypoints that describe the shape of a profile are preferably generated with a standard Miniprof instrument. The two sets of keypoints are converted to 2-D spline curves by an Ansys macro routine. FIGURE 6. SPLINE CURVES GENERATED FROM MEASURED POINT SETS. Center node Superelement Wheel Contact region Superelement Rail 600 mm FIGURE 5. THE MODELING TOOL IS ACCESSED THROUGH THE ANSYS GUI. The basic steps in the creation of a wheel-rail interaction model for an X1 motor coach in a curve with a radius of 303m and a standard UIC60 rail is presented below. The three-dimensional geometric model of the wheel is generated by revolving the two-dimensional spline curves that describe the profile of the wheel tread. The rail model is created by extruding railhead profile curves a distance of 600 mm, which is the distance between the sleepers. Two sets of curves for a new wheel and a new rail are shown in figure 6. The two solid bodies are shown in figure 7. To get a reasonable configure model, the wheel is spatially oriented relative to the rail according to the quasi-static state calculated by an MBS software, e.g. Gensys or Medyna. To aid an efficient discretization of the contact region, a measure of the expected contact length and the number of expected contact patches have to be supplied by the user. FIGURE 7. SOLID MODEL GENERATED FROM THE PROFILE CURVES. The contact region, i.e. the small portion of the two bodies that are close to the anticipated contact patch, is meshed with the Ansys linear isoparametric element Solid45 (Ansys, 1997). For these elements, which almost exclusively are hexahedrons (see figure 8), an elastic-plastic material model with kinematic hardening is defined (see figure 9). The size of the contact zone is approximately 30mm in the lateral direction of the rail, 50mm in the longitudinal direction, and 10mm in the normal direction. The size in the longitudinal direction is based on the size of the contact zone and the distance of rolling required for the simulation. The main parts of the wheel and rail bodies are meshed with degenerated linear isoparametric elements. The hub surface is covered with shell elements. These two submodels are condensed to superelements. The size of a typical wheel-rail model is given in table 1. The nodes on the hub surface are connected to a center node with constraint equations (see figure 10). 800 B Rolling direction 640 A ≈10 480 320 160 ≈50 mm Contact region Wheel 0 0.0 Contact region Rail FIGURE 9. THE WHEEL-RAIL CONTACT REGION. Superelements Form Element Feature type Rail SOLID45 Wheel SOLID45 Hub SHELL63 Number of elements 35131 284 52785 Number of nodes 14796 212741 171 Master DOFs 1701 2709 0 Contact region Rail SOLID45 Wheel SOLID45 Center MASS21 Contact CONTAC49 4560 5920 1 2978 5439 7056 1 1391 16317 21168 6 0 0.2 0.4 0.6 0.8 1.0 ε (%) FIGURE 9. NON-LINEAR MATERIAL MODELS FOR THE RAIL (A) AND WHEEL (B) PORTIONS OF THE CONTACT REGION. 3. A METHODOLOGY TABLE 1. TYPICAL SIZE OF A WHEEL-RAIL MODEL. Here, the term methodology is used for a collection of methods and tools, the use of which is governed by a process superimposed on the whole (Coleman, 1994). Generally, a method, which is an organized, single purpose discipline or practice, evolve as a distillation of the best-practices experience in a particular domain of cognitive or physical activity (IDEF4, 1995). A tool refers to a software system, such as a FE system, designed to support the method. Contact region Rail Angel 1/30 Wheel Constraint equations connect center node to hub FIGURE 10. THE WHEEL HUB IS CONNECTED TO THE CENTER NODE WITH CONSTRAINT EQUATIONS. Contact forces Motion simulation Wheel location Gensys/ Medyna Simulation condition definition Boundary conditions Macros Rail profile Rail measurement Contact state history FE modeling of wheel-rail interaction Miniprof Wheel measurement Simulation control Wheel profile Ansys, Macros FE model FE simulation Ansys, Macros Miniprof µ measurement Salient tribometer µ Stress-strain curves FIGURE 11. A FE MODELING PROCESS AND THE SUPPORTING TOOLS. FE modeling of the wheel-rail interaction requires the shape of the geometric, domains, a material model, a value for the coefficient of friction, and knowledge about the contact forces. The material model is based on stress-strain curves supplied by the manufacturers of wheel and rail. The coefficient of friction is achieved from field instruments such as the Salient system tribometer. The profiles of railheads and wheel treads are measured with the Miniprof instrument. MBS-simulations provide contact point locations and quasi-static contact forces. These contact forces are transformed to global forces at the center node. With these data as input, the macro routines described above is capable of defining a complete FE model and the proper boundary conditions for a quasi-static simulation that capture the physical behavior that is caused by the combined rolling and sliding interaction. In a final step, the contact state history is extracted from the Ansys result database an exported in ASCIIformat for further analysis and manipulation. 4. A COMPARISON WITH TRADITIONAL METHODS A sharp curve in a track trafficked by commuter trains serving the Stockholm area is chosen for a comparison between the FE simulation results and results obtained with traditional methods. The chosen track carries almost exclusively unidirectional commuter trains with an average speed of 75 km/h. Two types of vehicles are used: the X1 and the X10 both operating in pairs with one powered unit and one trailing unit. This track and the rolling stock have been studied in a national Swedish program. Both rail and wheel profiles have been measured over a couple of years, see Nilsson (2000). Furthermore, has the X1 and the X10 vehicles been modeled in the train dynamic simulation software Gensys, see Jendel (2000), giving access to the necessary input data in form of wheel attitude with optional contact locations and forces resulted from simulations. Two cases were used to study the model. Both cases were from simulation of the X1 powered unit and represent the first and second wheel set in the leading boogie. The coordinate system is chosen similar to the Deutche Industrial Norm (DIN) with positive vertical (z) co-ordinate upwards, y to left and x is positive to the train motion direction. Figure 12 presents the contact points location on the wheel and the rail and table 1 shows the forces in the center point of the wheel. From a geometrical perspective the two load cases represents the contact points with a large difference in the curvature of contacting bodies. In case 1, the minimum contact radius was about 300 mm and in case 2 it was about 20 mm. New rail profiles and a wheel profile from a X1 train that has been in traffic for two years were used in the two cases. In both cases, the normal force was 80377N. The tool for the FE analysis is made and the preliminary results along with the comparison to the main concurrently available methods is outlined. In the first stage the differences between the methods are presented (see table 2 and figure 13). Case 1 Case 2 FIGURE 12. CONTACT POINT LOCATION FOR THE TWO TEST CASES. Method Case 1 Case 2 FEM with plasticity 606 MPa 577 MPa Contact 3057 MPa 715 MPa Hertzian max stress 1080 MPa TABLE 2. MAXIMUM CONTACT PRESSURE WITH THE FE AND TRADITIONAL METHODS. The main scope for this work was to enhance the knowledge of the contact pressure and the maximum stresses in bulk material. This should give an appropriate basis to study the degradation mechanism along with the wear simulation. The results in case 1 could not be compared with the Hertz method assuming one-point contact. The Hertzian solution showed approximately twice the contact length in y-direction compared with the other methods. This remained the same for the regions of the surface radii of curvature in the range of 0.01 .. 0.015m for the rail and -0.02.. -0.1m for the wheel. So, the normal force was split similar way as in program Contact to three parts. After that the results were very similar for these two classical methods (see figure 13). Compared to Ansys, the difference was approximately 300% for the contact area and more than 200% for the maximum pressure. For case 2 the result show that the difference in the maximum contact pressure and the size of the contact area was small when the minimum contact radius is large compared with the significant dimensions of the contact area, i.e. the half space assumption is valid. Using the linear-elastic model the differences in results are significant in regards especially to the relatively new wheel and rail shapes. Significant flattening of the contact pressure profile was found with increasing plastic deformation. The maximum pressure and plastic work moved outward in the direction of the contact edge along with the increase of friction coefficient. The distribution of the equivalent vonMises stress shows that, even with the three contact patches, significant yielding will occur (see figure 14). FIGURE 13. RESULTS FROM THE COMPARISON BETWEEN THREE DIFFERENT CONTACT MECHANICS METHODS, MAXIMUM CONTACT PRESSURE AND CONTACT AREA. Experiments have shown that the losses in the rail crosssectional area remains approximately constant in time, see Nilsson (2000). Assuming a constant rate of degradation the relationship between the plastic flow and wear changes. In the initial phase, i.e. case 1, the plastic work is very large. The maximum equivalent von Mises stress exceeds even the ultimate stress limit which for the actual plasticity model was 606MPa (see table 2). Thus, the material in this phase of the degradation process behaves perfectly plastic. The plastic flow hardens the material and makes the contact more conform. In the continuing process, other wear mechanisms will thus be significant. give a lot of information necessary for the wear-plasticity analysis. 7. ACKNOWLEDGEMENT This work was performed within the Swedish research programme SAMBA. The work was financially supported by the Swedish National Board for Industrial an d Technical Development (NUTEK), Adtranz Sweden AB, Stockholm Local Traffic, the Swedish National Rail Administration and the Swedish State Railways. 8. REFERENCES FIGURE 14. DISTRIBUTION OF VON MISES EQUIVALENT STRESS IN THREE YEARS OLD RAIL AND WHEEL. 5. CONCLUSIONS A tool for contact mechanics modeling and simulation of the wheel rail contact has been developed. The geometry of the contact can easily be changed. The model can be generated from measured wheel and rail profiles. Traditional methods and computational tools are limited by an half space assumption and a linear material model. The results from two test cases show that the difference in maximum equivalent stress between traditional methods and the FE model is small when the minimum contact radius is large compared to the significant dimensions of the contact area, i.e. when the half space assumptions is valid. However, in the test case where the minimum contact radius was of the same order as the significant dimensions of the contact area the difference between the FE results and results obtained with traditional Hertzian and BE methods was as large as 3GPa. This large difference was probably due to limitations in both the half space assumption and the linear elastic material model in the traditional methods. 6. ADDITIONAL WORK TO BE PERFORMED The regime for a rail and wheel contact is frequently characterized as either stress related or wear related (Tournay, 1996). Crack initiation and propagation are examples of damage mechanisms that are driven by stresses. In many situations, wear is a significant damage mechanism. The wear can be integrated if the the contact state history is known. A wear model includes a wear coefficient, which has a scientific base, see for example Archard (1953) and Lim and Ashby (1987). If the contact pressure distribution and the accumulated sliding distances are known, the right wear regime can be predicted for each point that has passed the contact zone. For a given wear regime, the wear coefficient can be obtained from lab-tests. Already the pure static tests along with the degraded profiles Andersson, E., Berg, M. and Stichel, S., 2000, ”Dynamics of rail vehicles (in Swedish)”, Railway Technology, Dept. of Vehicle Engineering, Royal Institute of Technology, KTH, Stockholm, Sweden. Ansys, 1997, ”ANSYS Theory Reference, release 5.4”, ANSYS, Inc.. Archard J.F., 1953, ”Contact and rubbing of flat surfaces”, Journal of Applied Physics, Vol. 24, pp. 981-988. Ekberg, A., 2000, Rolling contact fatigue of railway wheels – Towards tread life prediction through numerical modelling considering material imperfections, probabilistic loading and operational data”, Thesis, department of Solid Mechanics, Chalmers University of Technology, Göteborg, Sweden. Carter, F.W., 1926, ”On the action of a locomotive driving wheel”, Proc R. Soc. London. A, 112, pp.151-157. Cassidy, P.D., 1996, ”Variation of normal contact stresss for different wheel/rail profile combinations”, European Rail Research Institute, ERRI D 173/DT 336. Coleman, D., Arnold, P., Bodoff, S., Dollin, C. and Gilchrist, H., 1994, ”Object-Oriented Development: The Fusion Method”, Englewood Cliffs, NJ, Prentice Hall. Gensys, 1999, ”Gensys User’s Manual”, Release 9910, Desolver. IDEF4, 1995, ”Information integration for concurrent engineering (IICE) IDEF4 object-oriented design method report”, Knowledge Based Systems, Inc., Draft January 1995. Iwnicki, S.D. (ed.), 1998, ”The Manchester Benchmarks for Rail Vehicle Simulation”, Special Issue of Vehicle System Dynamics (Computer Simulation of Rail Vehicle Dynam ics), Vol. 30, NO. 3-4, pp. 295-313. Iwnicki, S.D. (ed.), 1999, ”The Manchester Benchmarks for Rail Vehicle Simulation”, Supplement to Vehicle System Dynamics, Vol. 31. Jendel T., 2000, “Prediction of wheel profile wear – comparison with field measurements”, submitted to Contact Mechanics and Wear of Rail/Wheel Systems, Tokyo, 25-27 July,2000. Johnson, K.L., 1958, ” The effect of a tangential contact force upon the rolling motion of an elastic sphere on a plane”, J. Appl. Mech., 25, pp. 339-346. Johnson, K.L., 1985, ”Contact University Press, Cambridge. Mechanics”, Cambridge Kalousek, J., Masel, E. and Grassie, S., 1999, ”Perspective on metallurgy and contact mechanics”, International Heavy Haul Association STS Conference Wheel/Rail Interface, Moscow, June 14-17. Kalker, J.J., 1967, ”On the rolling contact of two elastic bodies in the presence of dry friction”, Thesis, Delft. Kalker, J.J., 1973, ”Simplified theory of rolling contact”, Delft Progress Report 1, pp. 1-10. Kalker, J.J., 1979, ”The computation of three dimensional contact with dry friction”, Int. J. Num. Meth. Eng. Vol. 14, pp. 12931307. pp. 243-261. Knothe, K.L., Strzyakowski, Z., and Willner, K., 1994, "Rail vibrations in the high frequency range", Journal of Sound and Vibration, 169(1), pp. 111-123. Knothe K., Theiler A. and Güney S., 1999, ”Investigation of contact stresses on the wheel/rail system at steady-state curving”, 16th IAVSD Conference, Pretoria RSA, 30 August- 3 Sept. Lim, S.C., Ashby, M.F.,1987, “Wear mechanism maps”. Acta metall. Vol. 35, No. 1 , pp. 1-24. Nilsson R., 2000, “Wheel and rail wear – measured profiles and hardness changes during 2.5 years for Stockholm commuter traffic”, Hannover EUROMECH 409 Colloquium 6-9 mars, 2000. Olofsson, U. and Nilsson, R., 1998, ”Initial wear of a commuter train track”, Nordtrib’98, June 8-10, Ebeltoft, Denmark. Kalker, J.J., 1982, ”The contact between wheel and rail”. Report of the Department of Mathematics and Informatics No. 82/27, TH Delft. Olofsson, U., 1999, ”on the form change due to wear and plastic deformation in narrow curves on a commuter train track”, submitted for publication. Kalker, J.J., 1982b, ”A fast algorithm for the simplified theory of rolling contact”, Vehicle Syst. Dyn., 11, pp.1-13. Shen, Z.Y., Hedrick, J.K., and Elkins, J.A., 1984, ”A comparison of alternative creep-force models for rail vehicle dynamic analysis”, The Dynamics of Vehicles, Proc. 8th IAVSD Symp., Hedrick,J:K: (ed.), Cambridge, MA, Swets and Zeitlinger, Lisse, pp. 591-605. Kalker, J.J., 1983, ”Two algorithms for the contact problem in elaststatics”, Proc. Int. Symp. On Contact Mechanics and Wear of Rail-Wheel Systems I Vancouver BC, July 1982, University of Waterloo Press, Waterloo, Ontario, pp. 101-120. Kalker, J.J., 1987, ”Wheel-rail wear calculations with the program CONTACT”, in Gladwell, G:M:L:, Ghonen, H. and Kalonsek, J. (eds.), Proc. Int. Symp. On Contact Mechanics and Wear of Rail-Wheel Systems II, Kingston, RI, July 1986, University of Waterloo Press, Waterloo, Ontario, 1987, pp. 3-26. Kalker, J.J., 1991, ”Wheel-rail rolling contact theory”, Wear, 144, Tournay, H.M., 1996, ”Managing rail/wheel interaction on a railroad for optimum operating performance”, Rail International, Februaray-March 1996, pp. 42-45. Vermeulen, P.J. and Johnson, K.L., 1964, ”Contact of nonspherical bodies transmitting tangential forces”, J. Appl. Mech., 31, pp. 338-340.