Wheel-Rail Interaction Analysis Tanel Telliskivi

TRITA-MMK 2003:21ISSN 1400-1179

ISRN KTH/MMK/R2003/21--SE

MMK

Stockholm 2003

Doctoral Thesis Department of Machine Design

Royal Institute of Technology, KTH SE – 100 44 Stockholm, Sweden

Wheel-Rail Interaction Analysis

Tanel Telliskivi

TRITA-MMK 2003:21ISSN 1400-1179

ISRN KTH/MMK/R2003/21--SE

Stockholm 2003 Doctoral Thesis

Dept. of Machine Design Royal Institute of Technology, KTH SE – 100 44 Stockholm, Sweden

Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan i Stockholm framläggs till offentlig granskning för avläggande av teknisk doktorsexamen den 28 maj 2003 kl 10.00 i Kollegiesalen, Administrationsbyggnaden, Kungliga Tekniska Högskolan, Valhallavägen 79, Stockholm.

© Tanel Telliskivi 2003

Universitetsservice US AB, Stockholm 2003

1

Abstract A general approach to numerically simulating wear in rolling and sliding contacts is presented in this thesis. A simulation scheme is developed that calculates the wear at a detailed level. The removal of material follows Archard’s wear law, which states that the reduction of volume is linearly proportional to the sliding distance, the normal load and the wear coefficient. The target application is the wheel-rail contact.

Careful attention is paid to stress properties in the normal direction of the contact. A Winkler method is used to calculate the normal pressure. The model is calibrated either with results from Finite Element simulations (which can include a plastic material model) or a linear-elastic contact model. The tangential tractions and the sliding distances are calculated using a method that incorporates the effect of rigid body motion and tangential deformations in the contact zone. Kalker’s Fastsim code is used to validate the tangential calculation method. Results of three different sorts of experiments (full-scale, pin-on-disc and disc-on-disc) were used to establish the wear and friction coefficients under different operating conditions.

The experimental results show that the sliding velocity and contact pressure in the contact situation strongly influence the wear coefficient. For the disc-on-disc simulation, there was good agreement between experimental results and the simulation in terms of wear and rolling friction under different operating conditions. Good agreement was also obtained in regard to form change of the rollers. In the full-scale simulations, a two-point contact was analysed where the differences between the contacts on rail-head to wheel tread and rail edge to wheel flange can be attributed primarily to the relative velocity differences in regard to both magnitude and direction. Good qualitative agreement was found between the simulated wear rate and the full-scale test results at different contact conditions.

Keywords: railway rail, disc-on-disc, pin-on-disc, Archard, wear simulation, Winkler, rolling, sliding

1

Preface The work was carried out at the Department of Machine Design, Division of Machine Elements at the Royal Institute of Technology, KTH, Sweden, within the Swedish research programme SAMBA. The research programme was financially supported by the Swedish National Board for Industrial and Technical Development (NUTEK), Bombardier Transportation, the Swedish National Rail Administration, Swedish State Railways, Traintech Engineering, Green Gargo, and Stockholm Local Traffic.

I would like to thank everyone in the Tribology Workgroup in the Machine Elements Division. The opportunities you provided to listen to presentations and to talk through ideas have been invaluable. While ideas normally surface when one is alone, feedback from others is needed to establish the completeness and realisation possibilities.

I am also indebted to Professor Sören Andersson and to my supervisor, Dr. Ulf Olofsson, for admitting me to the research programme and for their support.

Finally, many thanks to Maria Wolff, my fiancée, for her love during this period and to my infant daughter Sigrid for her patience (!).

Thesis This thesis applies the modelling and simulation of wear in a rolling-sliding contact to wheel-rail analysis. It contains an introduction and the following papers:

Paper A. T. Telliskivi and U. Olofsson, ‘Contact mechanics analysis of measured wheel-rail profiles using the finite element method’ Journal of Rail and Rapid Transit, Proc. Instn. Mech. Engrs., 215 Part F, 2001.

Paper B. U. Olofsson and T. Telliskivi, ‘Wear, friction and plastic deformation of two rail steels: Full-scale test and laboratory study’ Proceedings of World Tribology Conference, Vienna, 4–7 Sep., 2001, Wear 254, 2003.

Paper C. T. Telliskivi, ‘Simulation of wear in a rolling sliding contact by a semi-Winkler model and Archard’s wear law’, Proceedings of OST-01 Symposium on Machine Design, Tallinn, 4–5 Oct., 2001. Submitted for publication 2002.

Paper D. T. Telliskivi and U. Olofsson ‘Wheel-Rail Wear Simulation’, accepted for publication at CM2003, Gothenburg, June 10-13, 2003.

Paper E. T. Telliskivi ‘Half-space solutions for frictionless elastic normal indentation originating at a point contact’, ISRN/KTH/MMK/R-03/10-SE.

2

Division of work between authors Paper A. Olofsson prepared the FE model and Telliskivi performed the FE

calculations. Olofsson assisted in structuring and editing the manuscript.

Paper B. Olofsson and Nilsson [1] conducted the full-scale tests and Telliskivi performed the dry pin-on-disc tests. The manuscript was written mainly by Olofsson, and Telliskivi assisted in structuring and data treatment.

Paper C. Written by Telliskivi.

Paper D. Telliskivi wrote the manuscript and developed the simulation. Olofsson assisted in structuring and editing the manuscript.

Paper E. Written by Telliskivi.

1

1

Contents Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4

1.1 Classification in relation to severity of wear . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Studies in wheel-rail contact analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6

1.3 The goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2. Research Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 FE analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Contact locality rolling-sliding analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Normal solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

2.4 Tangential solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.1 Need for modelling the influence of the neighbouring cell . . . . . . . . . . . . .16

2.4.2 Simulation example with the railway wheel. . . . . . . . . . . . . . . . . . . . . . .18

3. Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22

4. Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24

4.1 Randomness, time-dependence and rough surface . . . . . . . . . . . . . . . . . . 24

4.2 Wear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24

4.3 Perspectives on plastic flow and fatigue analysis . . . . . . . . . . . . . . . . . . . 25

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1

1

Background This thesis deals with the modelling and simulation of wear in a rolling and sliding contact with special emphasis on wheel-rail contact. The work is presented in the five appended papers that are summarised below.

Paper A: Contact mechanics analysis of measured wheel-rail profiles using the finite element method

All machine elements working in contact are subject to various degradation mechanisms. The main differences between contacting machine elements lie in their geometries and the motion dynamics to which they are subject. As regards materials, there are at least four main stages of complexity that have to be taken into account when attempting a realistic problem analysis.

τxy.m z’’ + k z∂∆ = N L/E/A τxy, τyz, τxz σ = σ(ε)

Figure 1. Evolution of complexity of structural problems

As can be seen in figure 1, there is a progression from great simplification to increasing complexity of analysis. If a body is considered as rigid, a one-dimensional point mass analysis is used. Two-dimensional solutions are more elegant, but are still only halfway to elastic body analysis. In three-dimensional analysis, numerical methods are often needed.

The only general tool currently available for material plasticity analysis is the finite element method (FEM), which has been used for complex material analysis and is presented in paper A. The geometric flexibility of FEM and the availability of material models make it a good basis for understanding, even though faster and simpler methods are sometimes preferred. The limits and possibilities of FEM are studied, and attention is drawn to the significant increase in contact area and the lowering of pressure when geometric effects are taken into account when modelling an elastic-plastic material.

2

Paper B: Wear, friction and plastic deformation of two rail steels: Full-scale test and laboratory study

Extensive material testing is necessary to improve our knowledge of the behaviour of materials in various contact situations. Paper B presents the results obtained from the three types of experiments that are most commonly used to analyse wheel-rail interaction. Pin-on-disc tests were undertaken to establish a sliding wear coefficient and disc-on-disc tests were used to analyse rolling-sliding wear with relatively constant creep over the contact area. The disc-on-disc tests and the field tests were then simulated using the same wear model in order to be able to draw conclusions about what happens in wheel-rail contact. These two simulations are presented in papers C and D.

Paper C: Simulation of wear in a rolling sliding contact by a semi-Winkler model and Archard’s wear law

A fast model of a new method of contact analysis was developed and applied for a disc-disc simulation. The goal was to find a realistic relation between the tangential stresses and the sliding distances for a simply modelled realistic normal contact and for a linear-elastic material with generalised tangential displacement for a non-elliptical shape of the contact area. The half-space assumption that is essential for potential function solutions was taken as valid because of the almost flat contact and by not permitting the breadth of contact to reach the disc corners during simulation. The biggest obstacle to accepting this restriction is the non-smooth surfaces caused by wear. The friction limit that causes a contact to slide was modelled, thus enabling the simulation of Archard’s wear law. Contact locality simulation was extended and a qualitative match of important parameters in the simulation and in the experimental data was obtained. A comparison of elliptical contacts with a well-known numerical method (Fastsim) showed differences in only one constant.

Paper D: Wheel-Rail Wear Simulation

The model presented in paper C was modified with a more general geometry treatment so as to be able to simulate a wheel-rail contact. The stress in normal direction (normal solution by a Winkler method) was calibrated using the results from FEM modelling of the wheel-rail contact with the elastic-plastic material model. This approach produced more valid results in regard to tangential stress, with the half-space assumption being compensated for by the transformation of the contact surface into a half-space due to the profile curvature of the rail. As in paper C, the largest numerical error is due to uneven wear on the rail surface. Wear in a typical wheel-rail two-point contact was simulated and the results were compared with those from a full-scale field test.

3

Paper E: Half-space solutions for frictionless elastic normal indentation originating at a point contact

The general use of the solutions of potential theory is presented for normal indentation analysis of arbitrary body shapes. This application is analogous to the tangential solutions in previous papers.

4

1. Introduction One of the basic tasks in the study of machine elements has traditionally been the characterisation of wear. Wear is defined as the material loss or change in surface texture occurring when two or three surfaces of mechanical components contact each other. There are many different types of wear and a widely varying range of working conditions, making wear a very complex problem.

Recent studies have shown the importance of the association between the wear analyses of different machine elements such as roller-bearings [2], cam followers [3] and gears [4]. While the dynamics and geometry are different, the material is more or less the same. What is known as Archard’s linear wear law has traditionally been used in the study of both sliding and rolling-sliding contacts. This law assumes that wear is proportional to normal load, the sliding distance and a wear coefficient, divided by the surface hardness.

Contact of a friction pair- material parameters- surface parameters (lubrication)- loading- relative local velocity

Wear simulation- FEM simulations- fast numerical methods(Winkler, BEM,combinations)

Experiments- in field- pin-on-disc- disc-on-disc

Figure 2. Wear analysis scheme

The basic approach adopted in this thesis (outlined in figure 2) does not differ much from the standard methodology for investigating component wear. The qualitative wear models generated by direct physical interpretation thirty years ago made a creative contribution to the body of wear modelling. However, they had not stood yet the tests of time and experiment. Since the 1980s, wear modellers have begun to use relevant theories from other fields of engineering to explain such wear phenomena as plastic deformation, fatigue, heat generation, oxidation, and crack formation and propagation. Many of these phenomena have been studied in detail in other fields and validated theories have been developed. The adopted theories have also been used to describe variations in working conditions and some single phenomena during the wear process.

5

Fatiguemechanisms

PlasticityShakedownRatchetting

”Archard”process

Adhesion, immediatemechanical mass loss

Flow and otherchanges in

the material

Crack initiation after some cycles, fatigue

Figure 3. Categorisation of surface degradation processes

Surface degradation, especially that caused by a curving train, involves various phenomena that are difficult to separate. The first of these phenomena is the immediate adhesive mass loss. The mechanism of wear is clarified by the Archard’s linear wear law [5] (see figure 3). Due to the complexity of wear systems, it is important to begin with relatively simple methods and then to enhance these methods where possible. The wear system itself involves a number of interacting mechanisms.

1.1 Classification in relation to severity of wear Wearing systems have been classified in terms of the severity of wear on the wearing surfaces. Archard and Hirst [6] proposed two broad types of wear phenomena: severe wear and mild wear. Severe wear is characterised by high wear rates, extensive plastic deformation, transfer of material to the harder counterface, and flake-like metallic wear debris. Mild wear, by contrast, is characterised by low wear rates, minimal plastic deformation, formation of a surface film protecting against metal-to-metal contact, and oxide wear debris.

A wearing system consists of a number of mechanisms that need to be precisely defined in order to avoid overlaps in wear analysis. For example, the severity of wear needs to be defined in terms of precise, well-accepted definitions of such features as the amount of mass loss, the coefficient of friction and the surface roughness in order to accurately distinguish mild and severe wear. Without such precise definitions, models of wear may produce different results on the basis of different classifications. This is clear from the confusion already introduced by the considerable overlap in the classification of wearing systems. Erosion, for example, has been classified as wear based on relative motion [7] and as a wear mechanism. However, the literature suggests that several mechanisms contribute to erosion, implying that erosion itself is not a mechanism. The mechanisms proposed include cutting [8], thermal melting [9], brittle fracture [10] and low cycle fatigue [11]. The mechanism category is the lowest category in a hierarchy. A wear mechanism involves basic atomic and molecular interactions such as atomic diffusion, monolayer film formation, adhesion due to surface roughness, dislocation interaction, surface chemical reactions and the like.

6

1.2 Studies in wheel-rail contact analysis Wheel-rail analysis has focused mainly on what is known as rolling contact fatigue. This phenomenon arises primarily where there are contacts with low relative sliding, such as on the rail head on straight tracks. A simulation of a so-called ratchetting model using finite elements carried out by Ringsberg et al. [12] identified asymptotic values of the friction coefficient at which crack initiation would occur. In northern Sweden, 60% of rail replacements were found to be due to problems caused by rolling contact fatigue and surface defects, while only 5% were due to flange wear [13]. In southern Sweden the statistics are slightly different. An additional problem that has arisen with high-speed trains has been vertical transient motions, including the response from the foundation, that has resulted in periodic degradation on the rail head and led to failures [14, 15]. Sometimes the contribution of sliding wear is totally ignored, yet it is still evidently accepted as a factor in wheel wear due to braking [16] and in the general understanding of wear [17]. Preventive grinding, in which rails are ground at regular intervals, is one method adopted by rail operators to extend the service life of rails. A model of the effects of adhesive wear makes it clear why preventive grinding is necessary.

Continuum rolling contact theory started with a publication by Carter [18], in which he approximated the wheel by a cylinder and the rail by an infinite half-space. The analysis was two-dimensional and an exact solution was found. Carter showed that the difference between the circumferential velocity of a driven wheel and the translational velocity of the wheel has a non-zero value as soon as an accelerating or a braking couple is applied to the wheel. This difference increases as the couple increases until the maximum value according to Coulomb’s law is reached. Carter formulated a creep-force law relating the driving–braking couple and the velocity difference. Carter’s theory is adequate for describing the action of driven wheels (for example, it is capable of predicting the frictional losses in a locomotive driving wheel). However, it is not sufficient for vehicle motion simulations that involve lateral forces as well as the motion in rolling direction [19].

Johnson [20] generalised Carter’s results to circular contacts and longitudinal and lateral creep. Vermeulen and Johnson [21] generalised this theory to elliptical contact areas. Shen et al. [22] improved the results by replacing the approximate values for the creep coefficients given by Vermeulen and Johnson with more accurate values. All of this work is Hertzian-based, giving contact solutions for a class of geometrical objects satisfying the half-space restriction [23].

In the development of wear modelling in the railway context, an Archard-like wear model developed by Li and Kalker [24] has also been used in which the normal load is replaced by the frictional load or, in other words, the normal load is multiplied by the friction coefficient. An analogous approach by Li et al. [25] was practically applied in the analysis of a city train railway, but no direct measures such as profile changes are presented. Linder and Brauschli [26] did analyse the profile change of a

7

train wheel, noting the qualitative difference in the wear rate between the wheel flange and the wheel tread.

New wear models do not tend to produce any simplifications in terms of wear coefficients and the like. For example, Fries and Davila’s [27] wear coefficient based on energy does not solve the problem of the wear coefficient and is still dependent on the same influences as other wear models, such as Archard’s. Furthermore, energy itself is not the mechanism and there is a problem with dimensions in physical relations.

There is a lack of work connecting different phenomena, and too many oversimplifications that attempt to deal with the whole issue in terms of stresses only or, at the opposite extreme, attempt to apply various wear coefficients in applications without having much of a theoretical structure and an understanding of the sources and circumstances of wear. It is important to remember that shearing and stress-related failures happen around the sticking region of the contact because of the static hooking of asperities that can move back and forth. Adhesive wear occurs primarily under sliding conditions, where asperities are beating each other under a transient load and stress-related effects may also be present.

1.3 The goal Accurate wear modelling requires detailed descriptions of the many different wear phenomena that occur simultaneously on wearing surfaces if the analytical model is to explain wear phenomena in a wear system.

The goal of this thesis is to standardise the mathematical expression of different wear phenomena. The long-term goal must be to devise a wear classification scheme based primarily on the mechanisms by which the asperities deform and particles are detached. In such a scheme, crack initiation would be explained in terms of surface imperfections and the irregular ductility of the continuous bulk structure that may act to increase stress and lead to the initiation of contact fatigue. The first steps will be taken towards constructing a flexible simulation model in which the nature of the wear mechanism can change depending on various geometric, kinematic and structural parameters rather than produce a new wear equation.

8

2. Research Approach The simulation tool for wear analysis represents an attempt to achieve an integrated understanding of wear and other degradation mechanisms. It is not intended only for work related to railways. Analysis of metal-on-metal contact is a common element in machine design, as is clear from the large number of wear tests conducted every year in different institutions. The data obtained from such tests needs a tool that attempts to systematise it and responds flexibly to the working conditions that produce wear.

Since the early 1970s, there have been numerical simulations of the behaviour of rail vehicles and of the interactions between vehicles and track. Specialised software has been developed, including Vampire, developed by British Rail, Medyna developed by Deutsche Luft und Raumfahrt, and Nucars in the USA. All of these are highly specialised and optimised for a reasonable turn-around time for a simulation (Andersson et al. [28]). An example of recently developed commercially available software is Gensys [29]. General-purpose software for dynamic simulations of multibody systems (MBS), such as Adams, Simpack, and Dads, has recently included features that enable efficient dynamic simulation of railway vehicles and vehicle-track interaction. One-dimensional beam models are usually sufficient for the frequency range up to three kHz (Knothe et al. [30]). Software for vehicle motion simulations is normally concerned with the orientation of each wheel relative to the track, and thus with the point-contact between the wheel tread and the rail head and the contact forces that are caused by the dynamic interaction, with time-consumption as the prime restriction. There are also potential function-based fast numerical methods such as simplified theory (Fastsim – [31]). Fastsim, developed by J. J. Kalker, treats the material as linear-elastic and the contact as elliptical, with constant creep (velocity) over a whole contact. In the simplified theory, the surface displacement at one unique point depends only on the surface traction at that point (Winkler model). The simplified theory was implemented in early, special purpose computer codes.

What is often referred to as the complete theory was implemented in a computer program called Contact [32], which is based on the boundary element (BE) method. Kalker extended his theory of rolling contact between arbitrary bodies to the case where the shape of the contact area is non-elliptical, and thus non-Hertzian. In order to get an approximate solution, the contact area is divided into rectangular elements. The Contact program is roughly 400 times slower than routines based on the simplified theory, but it has been used to validate the linear and simplified theories as well as to validate the theory of Shen et al. [22], which has been implemented in a program that runs significantly faster than Contact.

The form change of curves can be large over time (see paper B). Figure 4 shows the form change in two UIC 60 high rails over 2 years and 3 years respectively in a narrow curve on a commuter train track. As part of a wider study, this track was studied over a period of 2 years and the form and hardness of the track were

9

characterised in terms of its two-dimensional profile and surface hardness measurements. New rails of 20m apiece were inserted in two narrow curves. A length of the old rail was left in place as a test rail, enabling the study of both new and 3-year-old rail.

0 20 40 60

0

10

20

30

rail new at test start

rail three years old at test start

wear

plasticdeformationhe

ight

(mm

)

length (mm)

0 20 40 60

0

10

20

30

plasticdeformation

wear

heig

ht (m

m)

length (mm)

Figure 4. Form change of a UIC 60 high rail in a 303m curve over a period of 2 years. Top figure: Solid line = 3-year-old rail at test start; dotted line = after 1 year of use; dashed line = after 2 years of use. Bottom figure: Solid line = new rail at test start; dotted line = after 1 year of use; dashed line = after 2 years of use.

The experimental form measurements showed that there was a significant change in the rail profile due to both wear and plastic deformation and that both processes influence the form of a rail that has been in use for more than 5 years. The surface hardness measurement showed that the hardness of the new rail increased, but that after 2 years’ use it had not yet reached the hardness of the old rail. These experimental results show that plastic deformation is a necessary element in wheel-rail contact analysis.

The study in paper B showed that there are three important elements that are not addressed in present methods of wheel-rail wear analysis:

- shakedown and plasticity effects, which operate continuously in real contacts;

- the non-elliptical shape of the contact zone, especially for worn profiles;

- the velocity difference between the rail head and rail edge, which can be more than 1m/s, and which changes direction rotationally, causing spin.

In order to model these observations correctly, it is necessary to investigate them fully.

10

2.1 FE analysis First, to overcome the limitations inherent in traditional approaches and their lack of ability to analyse plasticity, a tool for FE-based quasi-static wheel-rail contact modelling and simulations was developed (see paper A and [33]. 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 [34] 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 and the material models are treated as elastic-plastic with kinematic hardening. The quasi-static loads were obtained from train dynamic calculations with special purpose MBS software.

In the finite element method, plasticity is modelled according to established plasticity theories, but the time taken to do this is impractical for wear analysis. Further drawbacks of this approach are that the surface discretisation is a lengthy procedure and that the contact element requires extra stiffness that is not physically correct. The contact nodes interfered in the final output. The rolling problem (even in quasi-static form) could be solved only after several discrete steps in which the sizes in rolling steps and the contact elements had a significant influence on the final result in the stick-slip region. Slip detection is defined in the pre-processing and had several options such as full-sliding, small motions, sticking etc. By choosing different modes resulted in different output.

2.2 Contact locality rolling-sliding analysis The FE method has undergone significant improvement, but the parallel progress of the FE method and faster numerical methods are obviously of interest.

The contact locality is the rectangular area that is large enough to envelope the true contact region. The benefits of solutions focusing only on the contact locality are as follows:

- It is not necessary to model the bodies as only the surface is discretised;

- The tangential solution can be achieved by one computation, analogous to that in the program Fastsim.

In paper C some supplementary steps have been added to enhance the modelling capabilities of the standard Winkler brush (see paper C for more details of what is called the semi-Winkler model and paper D for Winkler coefficient in normal direction). The transformations of the contact locality are performed in six consecutive stages:

1) To create three-dimensional geometries of wheel and rail. The geometrical data for analysis is expected to be 2-dimensional discrete samples of wheel and rail contours in the y-z plane (see paper B). The two-dimensional profile samples are extruded rotationally for the wheel and linearly for the rail.

2) To rotate wheel and rail geometries for the normal solution. Both profiles are rotated to the resultant normal load direction and the contact localities of the wheel

11

and rail are set together. The curving train has sideways centrifugal forces that cause the normal force to incline. The profiles are rotated according to this known inclination. A two-point contact at the rail head and rail corner involves large deviations in the normal direction compared to the normal load direction of the contacting cells, which are taken into account by the cosine rule.

3) To move the wheel and the rail back to their original position in space after the normal solution. The penetrations calculated in the normal solution are now included and the velocities are calculated.

4) To transform the three-dimensional geometries to a flat surface (half-space). The velocity and creep system is rotated so that the Euler angles become zero. This simplifies calculation of the tangential tractions since only the components in the x- and y-directions are used.

5) To rotate the rail and wheel profiles again to the resultant normal load direction where the original 2-dimensional data is updated with the form change due to wear.

6) To transform back to the original position in order to create the new contact localities of the worn rail and the wheel.

The application of rigid body velocities is particularly important when updating the geometries and developing a sort of ‘rough surface’ during the wear. The time-step is associated with length and is important when using time-dependent equations:

)1(arctan

Ω

−= Rx

t

where x is the length in the rolling direction [m], R is the radius of the wheel [m], and Ω is the constant rotation velocity [rad/s]. Dimensional compatibility shows that both Ωt and arctan(x/R) are angles [rad]. The piecewise approach and stick-slip analysis solve the problem of rolling friction using only the pure sliding friction coefficient obtained in pin-on-disc tests and provide an approximate upper limit for linear motion. The results are assumed to be valid for the so-called quasi-linear motion (Normal and tangential solution procedures are described in more detail in later sections). The velocity difference between the rail head and the rail edge leads to several wear mechanisms being applied concurrently. At different velocities and load conditions, different wear coefficients apply, providing a practical application of the ‘wear maps’ for steels initiated by Lim and Ashby [35]. The removal of material follows Archard's law, that is, the removal volume, W=ΣW [m3], is linearly proportional to the sliding distance, ur SLIDING [m], and the normal load, P [N], and is updated for every time-step in the solution procedure:

uuP

W2

ji,y2

ji,xji,ji, H

k SLIDINGSLIDING +⋅⋅= (2)

12

where H [Pa] is the hardness of material. The wear coefficient, k, was chosen on the basis of a pin-on-disc test in earlier work (see paper B). Using the formula W=ΣW [m3], the following parameters can be calculated:

- wear volume per metre: determined from the volume, W knowing that the steady-state solution gives the wear volume per discretisation unit ∆x as W/∆x [m3/m].

- mass loss per metre: determined by the combination of wear volume and material density, ρW/∆x [kg/m].

- the wear volume per wheel revolution: found by 2π⋅reW/∆x [m3/rev.].

- wear depth: derived by dividing the wear volume by the discretisation area (∆x·∆y). This is a vertical length for updating the profiles per breadth, ∆y (the wear rate is constant for each sample within the length, ∆y). The formula for wear depth is Wj/(∆x·∆y) [m], where Wj is the index of an array after the summation of matrix W only in x-direction (i).

The Winkler brush model is used for the normal problem in order to compensate for the differences between the linear-elastic and elastic plastic material model. The normal problem is solved separately from the tangential problem. The displacement matrix with size m×n is expressed as

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

⋅⋅⋅

=

nmm

ji

n

,1,

,

,11,1

uuu

uu

u

and is calculated with the help of the (2m+1)×(2n+1) coefficient matrix C:

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

⋅⋅⋅

=

+++

++

+

12,121,12

1n1,m

12,11,1

CC

C

CC

nmm

n

C

where the matrix C is real-valued and symmetric. The known surface tractions, qr (p = |qz|) with size m×n can be represented by two different equation systems, 3a and 3b. Equation 3a (see paper C) consists of three sets of programming ‘for loops’ that fill the ur ELASTC (index r = x, y and z directions) values at indices, ur i-1-mi+nj,j-1-mi+nj in the specified locations summarised in the block matrix shown here:

13

+

=

+−+−

+−+−+−+−

+−+−+−+−

+−−+−−

+−−+−−

+−−+−−

ji

jiy

jix

njminnjmim

njminnjmimnjminnjmimyx

njminnjmimxynjminnjmim

njmijnjmiiz

njmijnjmiiy

njmijnjmiix

ELASTICz

ELASTICy

x ELASTIC

,

,

,

,z

,y,

,,x

1,1

1,1

1,1

pqq

CCCCC

uuu

uu

u

(3a)

where ur ELASTC starts with the m×n zero values and every index picked in right hand side is updated by local summation indexwise, i=1…m, j=1…n, mi is the number of neighbour levels (see figure 5) used in computation and nj=1…2⋅mi+1. A control must be made that the neighbouring indices are inside the boundary m×n, otherwise a matrix of size (m+2⋅mi)×(n+2⋅mi) is updated in equation 3a.

Figure 5 Neighbouring levels mi for the contact locality and for the influence matrix.

Equation 3b is a complete superposition of matrices with equal size m×n Displacements at the contact locality are updated at every point in every summation step i,j by the coefficient matrix C (see paper D and E) that is cut to the size of m×n from the size (2⋅m+1)×(2⋅n+1) by:

∑∑= =

++++

++++++++

++++++++

=

m

1i

n

1j,

,

,

j-1j...2n-2ni,-1i...2m-2m

j-1j...2n-2ni,-1i...2m-2m j-1j...2n-2ni,-1i...2m-2m

j-1j...2n-2ni,-1i...2m-2m j-1j...2n-2ni,-1i...2m-2m

pqq

ji

jiy

jix

z

yyx

xyx

ELASTICz

ELASTICy

x ELASTIC

CCCCC

uu

u

(3b)

The coefficient matrices are built from influence functions by:

14

( )

( )G

yxyxyxyx

yxyxG

yxG

yx

yxxy

yx

⋅++−++++−−+−−−++⋅

==⋅

+⋅−=

⋅⋅−+

=

πηξηξηξηξν

πν

πν

2)()()()()()()()(

),(C),(C

,2

gBxxgAxx1),(C,2

gBxx)1(gAxx),(C

22222222

(4)

where gAxx and gBxx represent the components of analytic influence coefficients for linear-elastic material for constant tractions and rectangular elements:

)()(

)()(log )(

)()(

)()(log )(

gBxx

)()(

)()(log )(

)()(

)()(log )(

gAxx

22

22

22

22

22

22

22

22

−+−++−

++−++−+

−++++−

++++++

=

−+−++−

−++++−+

++−++−

++++++

=

ηξη

ηξηξ

ηξη

ηξηξ

ηξξ

ηξξη

ηξξ

ηξξη

yxy

yxyx

yxy

yxyx

yxx

yxxy

yxx

yxxy

(5)

where ξ and η are the half-lengths of the discretised cell and the Cxy and Cyx are not included in this work.

The function of Cz is left open, pending analysis in the next section.

2.3 Normal solution In the earlier studies presented in paper A, a finite element method was employed to predict the changes in contact properties when subjected to high loads. A particular elastic-plastic material model was simulated. There are a number of different material models, and therefore the results are qualitative, predicting a rise in the contact area and a decrease in the maximum contact pressure. A comparable method as regards computation time versus accuracy of the structural properties is the Winkler mattress method used in papers C and D. The normal displacement uz is related to the normal contact pressure by

KNpu =z (6)

where KN is the linear modulus of the foundation. According to equation 3, the influence function becomes a constant in equation 6, i.e., Cz = 1/KN. KN can be determined by experimental work, by FEM analysis or by comparison with another calculation theory, such as Hertzian theory [36].

The method outlined in paper E is valuable when focusing on linear-elastic contacts. The Winkler brush model (used in papers C and D) leaves some parameters as unknowns. The method adopted in paper E establishes the link between a pure linear-elastic solution and the Winkler brush method that has to be adjusted for different solutions. The method is based on several assumptions that challenge the assumptions in previous implementations of influence functions [32, 37]. These assumptions are:

15

- that the pressure distribution on each rectangular cell is approximately constant. Love’s [38] solutions cover the area using rectangular elements. This solution meets the boundary conditions and the superposition principle is valid. Although there are still restrictions on the curvature, Hertzian geometries are certainly valid;

- that the solution is to treat the initial overlapping of two bodies as a purely geometric problem. The overlap is gradually eliminated by proceeding in discrete steps, each a predetermined fraction (1/1000 or the like) of the maximum overlap;

- that every discrete step may consist of several equal lengths in different places (the corresponding pressure is automatically found) and that the order of succession within this step is not important;

- that every discrete subtraction length also successively subtracts the influence lengths at neighbouring cells so that the total subtraction is made for the entire overlap (bodies);

- that those discrete lengths (without the neighbour effects) are accumulated because they are directly proportional to pressure (force). The addition of those discrete lengths can be stopped if any total load restriction is met, enabling computation to be either load-based or approach-based.

The normal component in equation 3 is expressed in this case as

( ) ( )G

yxz ⋅+⋅−

=π

ν2

gBxxgAxx1),(C (7)

where gAxx and gBxx are given in equation 5 and are the functions that scale the displacements for the neighbouring cell. The problem is solved using only geometric parameters. Exact Hertzian solutions are obtained. Moreover, the bodies may be described as general polynomials (with variable curvatures) in any order and several concurrent contacts can be solved solely on the basis of geometric overlapping.

2.4 Tangential solution This section contains a review of selected aspects of wear analysis focusing on the formulation of tangential contact problems for deformable discrete surfaces. In contact problems, frictional effects are generally accounted for by the introduction of a friction law that relates the sliding velocity to the contact forces. The tangential component of the contact tractions, or frictional traction, can be exerted without sliding, i.e., under stick conditions, until a certain threshold is overcome to allow sliding. According to Coulomb’s law, the threshold is proportional to the magnitude of the normal pressure. When sliding occurs, the frictional tractions always oppose the sliding velocity and are, therefore, dissipative.

16

We shall be concerned with the motions of a deformable body, but first the rigid body motions are determined. Rigid body kinematic expressions provide many commonly used functions for dealing with rigid body attitude coordinates. The rotation matrix includes the time dependent Euler angles ϕ, θ, ψ are roll, pitch and yaw, respectively. In the present case ϕ → γ will be the inclination matrix of the perpendicular of roller curvature and θ = Ωt is the angle of curvature of the roller radius.

The basic steps for tangential solution as described in Paper C are as follows:

- Calculate relative velocities and creep for the rigid body;

- Use an artificial displacement field created by creep ratios and enlarged by the influence of neighbouring cells. The influence of a neighbouring element is determined logarithmically by the solutions of potential theory for constant traction on a rectangular area;

- Creep times the discretisation unit, ∆x in square is linearly (in every index separately) divided by the artificial displacements;

- Cumulatively sum the result from the beginning of the contact (in the rolling direction). The results are directly proportional to the tangential surface tractions. Check surface tractions by the frictional bounds and, if applicable, reduce them to level µ·P;

- Modify the elastic displacements in line based on the previous restriction. The part that was cut is the sliding component used in Archard’s wear equation;

- Calculate the wear volume using Archard’s wear law.

2.4.1 Need for modelling the influence of the neighbouring cell

Because the numerical algorithm is rather straightforward, the method is verified by using special cases. The unsymmetrical case is taken from an industrial application. It involves a spherical roller thrust bearing, the lateral profile of which is unsymmetrical (see figure 6). Mathematically defined bodies can also be unsymmetrical, but using an industrial example shows the correlation with experimental and empirical data in the literature [39].

Figure 6. Outline and contact model of spherical roller thrust bearing.

17

Figure 7. Contact pressure, total rigid speed and local creep distribution on roller surface.

Figure 7 shows the special conditions in order to demonstrate the benefits of the proposed methodology. The contact pressure is unsymmetrical and is solved linear-elastically using the method described in paper E. The rigid body creep and velocities are solved for conforming contact where the direction of the initially zero creep changes (see figure 7, right. lower image). Theoretically, creep should be zero in the case of free rolling of rollers.

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 10-4

-4

-3

-2

-1

0

1

2

3

4

5x 10-3 Elastic displacement directions on contact surface

x [m]

y [m

]

Figure 8. Results of surface traction distribution on roller surface.

The change of direction also results in a surface traction solution by the method described in paper C, where the influence of neighbours results in there being regions of zero displacement with no sliding at all independent of the friction coefficient. These regions appear as hollows due to low distances in the wear results (see figure 8, right, upper).

This brief demonstration of the concept underlying the method presented in paper C can be illustrated by examining the effects when displacements are in opposite directions (see figure 9).

18

Figure 9. Four unit displacements (linearly related to tractions) with neighbouring effects at t=[–6 –3 0 6] resulting in the displacement distribution.

The left-hand image in figure 9 shows the superposition of four unit (so called ‘direct’, see paper E) displacements where the influence of the solid means that these do not originate at zero (i.e., if these unit displacements are linearly related to the magnitude of traction, unit tractions are applied to get the final penetration field). The starting position is unimportant, only the magnitude and direction count. In the right-hand image in figure 9, one displacement applied at t = 0 is in the opposite direction. The superposition principle is simply applied and the neighbours (discrete case) or influence (continuous in figure 9) affect the final solution. The model is no longer a simple brush model. This increase in complexity is an irreplaceable benefit in modelling tangential contact in a spherical thrust bearing.

2.4.2 Simulation example with the railway wheel at the same attitude against the rail

Two-point contact is the characteristic contact mode in curving wheel-rail contact. In a two-point contact, there is contact at the rail head due to gravity and also contact between the rail edge and the wheel flange due to the centrifugal side forces of the curving train. A wear simulation for a two-point contact on a high rail was performed with a wheel at the same attitude for each wheel passage. A worn rail profile from a curve with low radius (303 m) and a worn wheel profile from the first wheelset in the leading bogie of an X1 train were used to generate the geometry. The wear of wheel is not included. Table 1 presents the parameters used in the wear simulation of the two-point contact.

Table 1. Parameters used in the wear simulation of the

two-point contact.

The main input data

Fn [N] 80377

α1 [rad] -0.58

α2 [rad] -0.85

ψ [rad] 0.035

VTRAIN [km/h] 75 (-20.8m/s)

RCURVE [m] 303

µ 0.6

19

Two different levels of creep were used in the analysis, see table 2. The wear simulation results are presented in table 2 in parameter format after one wheel passage and after 3000 wheel passages.

Table 2. Two different levels of creep and parameters

that are derived from creep.

creep [%] 0.5 1 Ω = VTRAIN /re

d/-45.06 -44.84

re 0.4623 0.4646 The results from the wear simulation are presented in figure 10 as wear rate or mass loss for 1m of rail length and cumulative mass loss showing the tendency of wear after some time. Two other parameters important for wear simulation are the change in maximum contact pressure and area. Figure 11 shows the increase in the contact area during the simulation. At the same time the contact pressure was reduced in corresponding cells. In figure 10 the loss of mass is reduced as the contact pressure drops (see figure 11). This phenomenon is explainable in terms of Archard’s wear law, of which contact pressure over a rectangular area subdivision, is a component. The locations that cause high-pressure concentrations disappear as the shape of the contact conforms. In figure 12, the left hand side of the figure represents the initial contact where the wheel attitude for normal solution is determined according to table 1 and 2, and the solution for the two-point contact is found. The right hand side of figure 12 shows how, after simulation of 3000 wheel passages (assuming contact with only the leading wheel of the bogie), the two-point contact has been spread to a larger contact area. The maximum level of contact pressure was significantly reduced after 3000 wheel passages. The rail profile change is present although not clearly visible at figure 12. Figure 13 shows changes in the wear and the tangential surface displacement as the wear causes a contact to conform to a non-wearing wheel kept at the same attitude. The dotted-line contour represents the sum of elastic displacements in the x-direction over the whole contact locality (including outside the contact). The x-line contour shows the total sliding distance included in Archard’s wear law. The solid line represents elastic displacement and sliding that is the total rigid motion in the contact. The lower part of both cases in figure 13 shows that the wear volume at the rail gauge corner (the most positive y-axis) is highest for the first iteration.

20

Figure 10. Mass loss and cumulative mass loss versus number of wheel passages.

Figure 11. Contact area and maximum contact pressure versus number of wheel passages.

Figure 12. Visualisation of the normal pressure after the first contact solution (left) and after 3000 wheel passages (right).

21

Figure 13. Elastic displacements at a whole contact locality and wear volume from every discretisation cell after one wheel passage (upper) and after 3000 wheel passages (lower).

22

3 Summary of results The analysis of wheel-rail interactions up to this point has been preparatory in the sense that it has focused on developing the ability to simulate particular locations on a track and the optimum rail or wheel design. However, this work has given rise to thoughts about future possible applications of the new methods.

The main results of the FE modelling presented in paper A were as follows:

- A FE tool for wheel-rail contact analysis has been developed. This tool allows easy changing of the geometry of a contact. Measured wheel and rail profiles were used in generating the model. Unlike the Hertzian analytical method and the Contact program, which uses the well-known boundary element method, the FE model does not have to assume a half-space or a linear-elastic material model.

- The results of the two test cases presented show that the difference in maximum contact pressure between the Contact and the Hertzian method and the FE method was negligible where the radii of curvature of the two contacting bodies at the contact point were large compared with the significant dimensions of the contact area (in other words, where the half-space assumption was valid). However, in test case 1, where the radii of curvature of the rail edge were small compared to the dimensions of the contact area, the difference between the model used here and the Contact and Hertzian methods was as large as 3GPa, probably due to both the half-space assumption and the material model.

The experimental studies presented in paper B supported the following conclusions:

- The form of the unlubricated curves showed significant changes due to wear and plastic deformation. This was a continuing process even for rail that had been in service for five years.

- The contact situation in terms of lubrication or sliding velocity and contact pressure had more influence on form change than whether the material was UIC 900A or UIC 1100.

- Different wear mechanisms affected different parts of the rail. Mild wear was the dominant wear mechanism at the rail head, while severe wear was the dominant mechanism at the edge. The difference in wear rate between rail head and rail edge could be as great as a factor of ten.

- The plastic deformation mechanism at the rail edge was plastic ratchetting.

- Laboratory tests showed that the wear coefficient depended on the sliding velocity. The increase in the wear coefficient with increasing sliding velocity was due to a change in the wear mechanism from mild wear to severe wear.

23

The main conclusions from the disc-on-disc simulation presented in paper C were as follows:

- The tangential stress-displacement field for rolling-sliding contact with a linear-elastic material can be calculated without special coefficients that are dependent on Poisson’s ratio and the shape of the contact area. Realistic results can be obtained for any pressure distribution.

- The friction coefficient in rolling is affected by the stick-slip region as a result of creep and normal load (and probably also because of plasticity effects) and is therefore generally lower than coefficients obtained from pin-on-disc tests. The rolling friction model obtained from the simulation involves the division of the longitudinal resistance force by the total normal load and can accommodate plasticity effects, which dissipate energy.

- Good agreement was found between experiments and the simulation in terms of wear and rolling friction at different levels of normal load and creep.

- Good qualitative agreement in regard to the form change of the rollers was obtained.

The main results of the full-scale wheel-rail simulation presented in paper D were as follows:

- The normal load was validated for the two cases that were under investigation.

- Two-point contact was analysed at different attitudes giving information about attitude restrictions imposed by the bogie and the curve.

- The simple two-point contact with Archard’s wear law was simulated.

The results of the contact mechanics method presented in paper E were as follows:

- The potential function solutions for rectangular contact divisions carrying uniform pressure are employed using a simple superposition method for a non-elliptical shape of the contact area at indentation. The penetration or load was tested within only one programming loop.

- Deformed bodies resulting from penetration are obtained.

- The proportions of speed and accuracy were analysed.

24

4 Future work The focus so far has been on wheel-rail analysis. However, the work done in this field suggests future possibilities in a number of engineering fields using the methods introduced here.

4.1 Randomness, time-dependence and rough surface The use of variation in discretisation introduces the possibility of making greater use of what is known as the Monte Carlo technique if the variables change randomly based on their probabilistic distribution. In the present case, only the lengths of ∆x and ∆y are variable. MBS data such as attitude angles between the wheel and the rail and the global creep ratio νr GLOBAL, can be varied, as can the pin-on-disc data coefficient of friction µ, and the wear coefficient k. Many other parameters may be varied within their probabilistic bounds between the different cells in contact or between the computation steps, because the computation steps progress with reasonable frequency. For instance, Beckmann and Dierich [40] proposed that wear prognoses must take account of the statistical nature of hardness. The method’s robustness can be analysed by comparing input and output variation and more general relationships can also be found.

The introduced transformation by the time-dependent Euler angles and the corresponding velocities (accelerations) permits the study of transient motions.

It will also be possible to study the effects of a rough surface on a rectangular area. Such a study could be statistical, in the form of what is known as the Abbott curve implementation, or could involve precisely measured asperities mechanically attached (although this approach would be rather time-consuming).

4.2 Wear Future work in regard to wear can be divided into long-term and short-term plans. The short-term plans involve wheel-rail analysis to study how lubricated and wet conditions affect the degradation mechanisms in wheel-rail wear. With lubrication, the elastic tension in the tangential direction is shortened due to the decrease in the friction coefficient and the plastic flow effect is reduced.

In the long-term perspective, the aim will be to study range of materials to determine how the plastic limit indicated as equivalent stress on the surface affects the wear coefficient. A related problem is the ‘softening’ of the so far optionally linear-elastic tangential solution in the proposed method in paper C.

The effect of roughness on friction and the wear coefficient is not only interesting in general but also affects the validity of the method through discretisation and the half-space assumption. An important part of curved geometries is determining what metric length constituting a valid discretisation length.

25

4.3 Perspectives on plastic flow and fatigue analysis

In the approach adopted in this thesis, sliding displacements and elastic displacements are separated. The loading of a train wheel on rail is usually such that plastic flow occurs in the rail with every wheel passage, imparting a small increment of plastic strain in the opposite direction to traction. This strain accumulates until it reaches the ductility of the material, at which point rupture occurs. The incremental rise in surface sliding is obvious at the wheel flange. The relative transversal velocities for the trailing wheel in a bogie have motions in the flange contact in the opposite direction. However, the leading wheel has a greater effect on the flange contact because it is steering the bogie.

Zone A

Zone Dδz

BC

D

A

Figure 14. Visual confirmation of severe plastic flow on a gauge corner at zone D. Zone A has not been in contact with the wheel.

As can be seen in figure 14, material has been moved and the structure of the steel has been stretched. On the one hand, the von Mises equivalent stress has been very high in this location. The higher the relative motion in the contact, the more plastic flow has occurred. Figure 14 illustrates the surface traction directions that will be the directions for potential plastic flow.

The mechanism of rupture when a metal is subjected to open strain cycles (ratchetting) can be analysed qualitatively using the linear-elastic stress-strain potential function solutions for the constant tractions of a rectangular sub-area. Additional constitutive relations may be worked out in the future. The computed displacements can easily be converted into strains if needed, but as a first step the fraction of tangential displacements may be used as a simulation criterion and as a starting step an approximated tangential material flow on the surface may be simulated.

26

References [1] Nilsson, R., ‘Wheel and rail wear – measured profile and hardness changes during 2.5 years for Stockholm commuter traffic, Railway Engineering – 2000, London, 5-6 July, 2000

[2] Olofsson, U., Andersson S. and Björklund S. ‘Simulation of mild wear in boundary lubricated spherical roller thrust bearings’ Wear 241 pp. 180–185, 2000

[3] Hugnell, A. B.-J. ‘Simulation of the Dynamics and Wear in a Cam-Follower Contact’ Ph.D. thesis, Stockholm, 1995

[4] Flodin, A. ‘Wear of spur and helical gears’ Ph.D. thesis, Stockholm, 2000

[5] Archard, J. F. ‘Contact and rubbing of flat surfaces’ Journal of Applied Physics, Vol. 24 pp. 981–988, 1953

[6] Archard, J. F. and Hirst, W. ‘Wear of metals under unlubricated conditions’ Proc. R. Soc., London. A, 1956, 236, 3–55

[7] Peterson, M. B. and Winer, W. O., Wear Control Handbook ASME, New York, 1980

[8] Finnie, I. ‘Some observations on the erosion of ductile metals’ Wear 19 pp. 81, 1972

[9] Jennings, W. H., Head, W. J. and Manning, C. R., Jr. ‘A mechanistic model for the prediction of ductile erosion’ Wear 40 pp. 93, 1976

[10] Evans, A. G. ‘Impact damage Mechanics: Solid Projectiles’ Treatise on material science and technology’, Vol. 16, Erosion, C. M. Preece (ed.) , Academic, New York, p. 1, 1979

[11] Hutchings, I. M. ‘A model for the erosion of metals by spherical particles at normal incidence’ Wear 70 pp. 269–281, 1981

[12] Ringsberg, J. W., Loo-Morrey, M., Josefson, B. L. and Beynon, J. H. ‘Prediction of fatigue crack initiation for rolling contact fatigue’, Int. J. of Fatigue, 22, pp. 205–215, 1999

[13] Waara, P. ‘Wear prediction performance of rail flange lubrication’ Licentiate thesis, Lulea, 2001

[14] Knothe, K. and Grassie, S. L. ‘Modelling of railway track and vehicle/track interaction at high frequencies’ Vehicle System Dynamics, 22(3–4): pp. 209–262, 1993

[15] Igeland, A. and Ilias, H. ‘Rail head corrugation growth predictions based on non-linear high frequency vehicle/track interaction’ Wear 213 pp. 90–97, 1997

[16] Jendel, T. ‘Prediction of wheel profile wear: Methodology and verification’ Licentiate thesis, Royal Institute of Technology, Sweden, TRITA-FKT 2000:49

[17] Jendel, T. ‘Prediction of wheel profile wear: Comparison with field measurements’, Proceedings of Contact Mechanics and Wear of Rail/Wheel Systems, Tokyo, 25–27 July, 2000

[18] Carter, F. W. On the action of a locomotive driving wheel, Proc. Roy. Soc., London, Ser. A, pp. 151–157, 1926

[19] Kalker, J. J. ‘Wheel-rail rolling contact theory’, Wear 144 pp. 243–261, 1991

[20] Johnson, K. L. ‘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, 1958

[21] Vermeulen, P. J. and Johnson, K. L. ‘Contact of non-spherical bodies transmitting tangential forces’, J. Appl. Mech., 31, pp. 338–340, 1964

27

[22] Shen, Z. Y., Hedrick, J. K., and Elkins, J. A. ‘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, 1984

[23] Johnson, K. L. Contact Mechanics, Cambridge University Press, 1985

[24] Li, Z. L. and Kalker, J. J. ‘Simulation of Severe Wheel-Rail Wear’, Proc. of the 6th International Conference on Computer Aided Design, Manufacture and Operation in the Railway and Other Advanced Mass Transit Systems: Computational Mechanics Publications, Southampton, pp. 393–402, 1998

[25] Li, Z. L., Kalker, J. J., Wiersma, P. K. and Snijders, E. R. ‘Non-Hertzian wheel-rail wear simulation in vehicle dynamical systems’ Proc. of the 4th Int. Conf. on Railway Bogies and Running Gears, Budapest, 1998

[26] Linder, C. and Brauschli, H. ‘Prediction of wheel wear’ Proc. of 2nd Mini Conf. on Contact Mechanics and Wear of Rail/Wheel Systems, Budapest, July 29–30, 1996

[27] Fries, R. H. and Davila, C. G. ’Analytical methods for wheel and rail wear prediction’ Proceedings 9th IAVSD Symposium, Linköping, 1985

[28] Andersson, E., Berg, M. and Stichel, S. Dynamics of Rail Vehicles (in Swedish), Railway Technology, Dept. of Vehicle Engineering, Royal Institute of Technology, KTH, Stockholm, Sweden, 2000

[29] Gensys, Gensys User’s Manual, Release 9910, Desolver, 1999

[30] Knothe, K. L., Strzyakowski, Z., and Willner, K. ‘Rail vibrations in the high frequency range’, Journal of Sound and Vibration, 169(1), pp. 111–123, 1994

[31] Kalker, J. J. ‘A fast algorithm for the simplified theory of rolling contact’, Vehicle System Dynamics, 11, pp. 1–13, 1982

[32] Kalker, J. J. ‘Wheel-rail wear calculations with the program CONTACT’, in Gladwell, G. M. L., Ghonen, H. and Kalousek, 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, pp. 3–26, 1987

[33] Telliskivi, T., Olofsson, U., Sellgren, U. and Kruse, P. ‘A tool and a method for FE analysis of wheel and rail interaction’ Proceedings ANSYS Conference in Pittsburgh, Pennsylvania, 2000

[34] Ansys, ANSYS Theory Reference, release 5.4, ANSYS, 1997

[35] Lim, S. C. and Ashby, M. F. ‘Wear-mechanism maps’ Acta Metal. vol. 35, no. 1, pp. 1-24, 1987

[36] Põdra, P. and Andersson, S. ‘Wear simulation with the Winkler surface model’, Wear 207 pp. 79–85, 1997

[37] Johnson, K. L. 'The application of shakedown principles in rolling and sliding contact' Eur. J. Mech., A/Solids, 11, pp. 155–172, 1992

[38] Love, A. E. H. ‘The stress produced in a semi-infinite solid by pressure on part of the boundary’, Phil. Trans. Royal Society, A228, 377, pp. 54–55, 1929

[39] Olofsson, U. ‘Characterisation of wear in boundary lubricated spherical roller thrust bearings’, Wear 208 pp. 194–203, 1997

[40] Beckmann, G. and Dierich, P. ‘A representation of the microhardness distribution and its consequences for wear prognosis’, Wear 107 pp. 195–212, 1986