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**Influence des irrégularités de la voie sur la fatigue du rail**

### Alfonso Panunzio

**To cite this version:**

Alfonso Panunzio. Influence des irrégularités de la voie sur la fatigue du rail. Autre. Université Paris-Saclay, 2018. Français. �NNT : 2018SACLC016�. �tel-01754867�

## Influence des irrégularités de la

## voie sur la fatigue du rail

Thèse de doctorat de l'Université Paris-Saclay préparée à CentraleSupélec

École doctorale n°579 Sciences mécaniques et énergétiques, matériaux et géosciences (SMEMAG)

Spécialité de doctorat: Transport et génie civil

Thèse présentée et soutenue à Gif-sur-Yvette, le 16 Mars 2018, par

** Alfonso Panunzio **

Composition du Jury :

**Habibou Maitournam **

Professeur, ENSTA ParisTech Président

**Denis Duhamel **

Professeur, ENPC Rapporteur

**Geert Lombaert **

Professeur, KU Leuven Rapporteur

**Thi Mac-Lan Nguyen-Tajan **

Ingénieur de recherche, SNCF Examinatrice

**Guillaume Puel **

Professeur, CentraleSupélec Directeur de thèse

**Régis Cottereau **

Chargé de recherche CNRS, CentraleSupélec Co-encadrant

**Samuel Simon **

Ingénieur de recherche, RATP Invité

NNT : 2 0 1 8 S A CL C0 1 6

iii

## Abstract

**Influence of the track irregularities on the rail fatigue**

by Alfonso PANUNZIO

The dynamical response of a train rolling on a real track depends on several parameters. Most of them cannot be accurately identified and have to be considered as uncertain. This implies that the wheel-rail contact interactions, the train dynamics, and all the quantities related to them, cannot be completely characterized using deterministic models. The aim of this thesis is the construction of a probabilistic model of the rail fatigue life considering the track geometry and the rail wear as random fields. These latter are built in order to be statistically representative of the considered railway network.

The features of these random fields are identified using measurement data. With the Karhunen-Loève expansion (KLE), a random field can be approximated by its truncated projection on an orthogonal basis. The KLE requires the computation of the eigen-functions and the eigen-values of the covariance operator for its modal representation. This step can be very expensive if the domain is much larger than the correlation length. To deal with this issue, an adaptation of the KLE, consisting in splitting the domain in sub-domains where this modal decomposition and the sample generation can be comfortably computed, is proposed. A correlation between the KLE coefficients of each sub-domain is imposed to ensure the desired correlation structure. This technique can also be parallelized and applied for non-stationary random fields.

The multivariate distributions of the random projection coefficients of the KLE basis are characterized using a Polynomial Chaos Expansion (PCE) calibrated on measurements data of the track irregularities. The curve radius, the rail age and the train operational velocity introduce non-stationary effects that have to be taken into account to model the track. The tracks thus generated are introduced as the input of a railway dynamics software to simulate the passage of the vehicle on a track with irregularities. A validation of the random models is therefore performed using a set of measurements of the wheel-rail contact forces.

A preliminary global sensitivity analysis is performed on some dynamical quantities of interest in order to quantify the impact of the random fields on the vehicle dynamics. Using the surface contact stresses as boundary condition, a finite element calculation allows to compute the stresses inside the rail in a fixed location of the spatial domain. A fatigue criterion is applied on theses stresses. Since this step is computationally expensive, a PCE-based meta-modelling technique is employed to estimate the fatigue index in the complete spatial domain. Then, through a variance-based global sensitivity analysis, the influence of the track irregularities and the wear on the variation of the rail fatigue criterion is analysed. This study is replicated for different curve classes (by varying the design curve radius and the associated superelevation) and train operational conditions (passengers load and velocity).

iv

## Acknowledgements

This doctoral thesis has been carried out within the framework of a CIFRE contract be-tween the laboratory MSSMat at CentraleSupélec and the department of RATP operating the track maintenance.

I would like to thank my committee members, professor Habibou Maitournam, pro-fessor Denis Duhamel, propro-fessor Geert Lombaert, Dr. Thi Mac-Lan Nguyen-Tajan for serving as my committee members. Your presence at my defence was an honour for me. I also want to thank you for letting my defence be an enjoyable moment, and for your precious comments and suggestions, thanks to you.

I would like to thank everybody who contributed to this work. In particular my supervisors at RATP and CentraleSupélec. At RATP the thesis was followed by Samuel Simon and Xavier Quost and at Ecole Centrale Paris by Guillaume Puel and Régis Cottereau. Thanks to their enormous interest in this work and their availability for discussions, they brought the thesis forward and helped me in every moment. I would like to thank them for all their ideas and proposals, all the time spend for discussions and the enjoyable relationships we had. Without their involvement and support in every step throughout the project, this thesis would have never been accomplished. I would like to thank you very much for your support and understanding. I also would like to express my gratitude to all my colleagues in RATP. The complexity of subjects in the field of railway engineering treated there and all the discussions with my colleagues have been contributed to create a very stimulating environment. It has been and will be a pleasure to work with them. And finally, I would like to address my thanks to the staff and PhD students at MSSMat for their support and company. Thank you for your friendship.

v

**Contents**

**Abstract**

**iii**

**Acknowledgements**

**iv**

**Contents**

**v**

**List of Figures**

**ix**

**List of Tables** **xvi**

**List of Abbreviations** **xix**

**List of recurrent Symbols** **xix**

**General context** **1**

**1** **Overview on the rail rolling contact fatigue** **3**

Introduction . . . 3

1.1 Wheel rail-contact . . . 4

1.1.1 Analytical models . . . 5

1.1.1.a Normal problem . . . 5

1.1.1.b Tangential problem . . . 8

1.1.2 Semi-Hertzian model. . . 10

1.1.3 Consideration of rail plasticity . . . 11

1.2 Dynamic evolution of the rail-wheel surface contact stresses . . . 12

1.2.1 Vehicle model . . . 12

1.2.2 Track model . . . 14

1.2.3 Dynamic simulations . . . 15

1.3 Rail internal stress cycle due to the rolling contact . . . 17

1.3.1 Consequences of the rail-wheel contact on the rail structure . . . 17

1.3.2 Rail internal steady stress cycle computation . . . 18

1.4 Initiation of the rail contact fatigue . . . 20

1.4.1 Studies concerning the initiation of the rail fatigue . . . 21

1.4.2 Dang Van fatigue criterion. . . 22

1.4.3 Parameters influencing the rail RCF initiation . . . 26

1.5 Consequences of the rail rolling contact fatigue . . . 27

1.6 Objectives and contributions of the thesis . . . 30

**2** **Random fields generation** **33**
Introduction . . . 34

vi

2.1 Standard 1D Karhunen-Loève expansion . . . 35

2.2 Karhunen-Loève expansion for large scale 1D random processes . . . 37

2.2.1 Principles of the generation method on a large domain . . . 37

2.2.2 Correlation structure across the coupled sub-domains. . . 40

2.2.3 Continuity of the generated samples . . . 40

2.2.3.a Continuity of random processes . . . 40

2.2.3.b Definition of a continuity error at the breaking points . 41 2.2.4 Extension of the expansion on an arbitrary large domain . . . 43

2.2.5 Positive definiteness condition . . . 44

2.3 Generation of multi-dimensional random fields . . . 46

2.3.1 General correlation function. . . 46

2.3.2 Tensorizable correlation function . . . 49

2.4 Parallel computing of the random field generation . . . 51

2.4.1 1D processes generation parallel computing . . . 52

2.4.2 Multi-dimensional fields generation parallel computing . . . 52

2.5 Extension to non-Gaussian random fields . . . 53

2.5.1 Gaussian translation . . . 53

2.5.2 Polynomial Chaos expansion . . . 55

2.5.3 Conditioned KLE for non-Gaussian random fields . . . 55

2.6 Extension to non-stationary random fields . . . 56

2.7 Numerical applications . . . 57

2.7.1 Example of generation on a large domain . . . 57

2.7.2 Computational complexity . . . 58

2.7.2.a Kernel modal decomposition . . . 60

2.7.2.b Conditioning matrices computation . . . 61

2.7.2.c Random field sampling . . . 61

2.7.3 Influence of the correlation kernel . . . 61

2.7.4 Influence of the sub-domain size . . . 65

Conclusion . . . 66

**3** **Random model of the track geometry and the rail wear** **69**
Introduction . . . 70

3.1 Stochastic model of the track geometry irregularities . . . 71

3.1.1 Track geometry measurement data description . . . 71

3.1.1.a Definition of the track geometry . . . 71

3.1.1.b Measurement system . . . 74

3.1.1.c Detection of the circulation direction . . . 75

3.1.1.d Signal deconvolution . . . 77

3.1.1.e Measurement campaigns . . . 78

3.1.2 Statistical description of the track geometry measurements . . . . 79

3.1.3 Parametric model of the random irregularities . . . 84

3.1.3.a Karhunen-Loève expansion . . . 84

3.1.3.b Polynomial Chaos expansion. . . 85

3.1.4 Random generation of the track geometry. . . 87

3.1.5 Comparison with measurements . . . 88

3.2 Stochastic model of the rail wear . . . 88

vii

3.2.2 Modal decomposition of the wear random field . . . 91

3.2.3 Correlation with the track features . . . 93

3.2.4 Random generation of the rail wear . . . 96

3.3 Random models validation through dynamical simulations . . . 97

3.3.1 Wheel-rail contact forces experimental data . . . 97

3.3.1.a Measurement system . . . 97

3.3.1.b Measurement campaigns . . . 98

3.3.2 Comparison of the power spectral densities of the contact forces: simulations with random geometry irregularities . . . 99

3.3.2.a Straight track . . . 100

3.3.2.b Curved track . . . 101

3.3.2.c Acceleration and braking . . . 102

3.3.2.d Influence of the number of Karhunen-Loève terms . . . 102

3.3.3 Simulations including the random rail wear . . . 103

3.3.4 Simulation with wheel irregularities . . . 104

Conclusion . . . 105

**4** **Sensitivity analysis on railway dynamics and rail fatigue** **107**
Introduction . . . 108

4.1 Sensitivity analysis on the vehicle dynamics . . . 110

4.1.1 Dynamical computation of the rail-wheel contact forces and kine-matic parameters . . . 110

4.1.2 Vehicle dynamics quantities of interest . . . 111

4.1.3 Global sensitivity indices . . . 112

4.1.4 Results . . . 113

4.1.4.a Wheel-rail lateral position - y . . . 113

4.1.4.b Wheel-rail vertical position - z . . . 115

4.1.4.c Wheelset yaw angle - ψW S . . . 116

4.1.4.d Leading outer rail lateral force - Y . . . 117

4.1.4.e Leading outer rail vertical force - Q . . . 118

4.1.4.f Leading Bogie lateral force -P Y . . . 119

4.1.4.g Wear number - WN . . . 119

4.1.4.h Simulations with constant profile . . . 120

4.2 Sensitivity analysis on the rail fatigue . . . 123

4.2.1 Input random space of the metamodel. . . 124

4.2.2 Polynomial Chaos Expansion basis. . . 124

4.2.3 Polynomial Chaos Expansion coefficients . . . 125

4.2.4 Results . . . 126

4.2.4.a Rail fatigue index . . . 127

4.2.4.b Life cycles. . . 131

4.2.4.c Simulations with constant rail profile . . . 132

4.2.4.d Fatigue on the inner rail. . . 134

4.2.4.e Simulations including the trailing wheel . . . 136

Conclusion . . . 137

**Conclusion and perspectives** **139**

viii

A.1 Friction coefficient random variability . . . 143

A.2 Suspension stiffnesses random variability . . . 143

A.3 Sensitivity analysis on the railway dynamics . . . 144

A.4 Sensitivity analysis on the fatigue index . . . 146

A.5 Conclusion . . . 148

**B Wheel-rail contact positions** **151**

**Bibliography** **157**

**Résumé** **175**

ix

**List of Figures**

1 Histogram of the rail (red) and the welding (green) failures in the recent years concerning the RATP RER line A. Ordinate axis scale not shown

for reasons of confidentiality. . . 1

1.1 Curvatures definition in a two-solid contact problem, from [Quo05]. . . 5

1.2 Two-solid interpenetration and Hertzian ellipse, from [Quo05]. . . 7

1.3 Rail (blue) and wheel (red) profiles represented when a contact on the surface tread occurs. Rail cant angle α = 1 20 rad. . . 7

1.4 Contact curvature A (black solid line) and B (green dashed line) relative to the profiles in Figure 1.3. . . 8

1.5 Slipping and adhesion in rolling contact, from [Quo05]. . . 8

1.6 Slipping and adhesion zones in a rolling contact patch. . . 9

1.7 Limitation of the normal stress (red solid line) with respect to the elastic stress distribution (blue dashed line) along a stripe of the rail. . . 12

1.8 Sketch of a common railway vehicle, from [Sim14]. . . 13

1.9 Sketch of a common railway track, from [Dah06].. . . 14

1.10 Wheel-rail relative position. . . 17

1.11 Responses of a material subjected to a cyclic load, from [Sim14]. . . 18

1.12 Rail subjected to surface contact stresses, inspired from [DGM89]. . . 19

1.13 Bilinear stress-strain law. . . 20

1.14 Sketch of a dislocation (left) and movement of a dislocation in a crystal structure (right), from [Lou10]. . . 21

1.15 Centre of the smallest hypersphere circumscribed to the macroscopic stress deviator: 2D representation. . . 23

1.16 Dang Van fatigue criterion representation. . . 24

1.17 Wöhler curve of the rail material. . . 25

1.18 Shakedown map, from [DM02]. . . 25

1.19 Rail crack propagating, from [Mag11]. . . 27

1.20 Rail spalling on the wheel tread, from [Sim14]. . . 28

1.21 View of an oval flaw with weak (left) and large (right) crack propagation, from [Sim14]. . . 29

1.22 View of a head check on the rail, initial (left) and advanced (right) phase of development, from [Sim14]. . . 29

1.23 View of a squat, from [Sim14]. . . 30

2.1 Example of generation of a random process with Gaussian correlation. Before (black solid line) and after (red dashed line) conditioning. Corre-lation length lc = 0.15L. KLE error 2KL = 0.001. Number of KLE terms N = 12. . . 39

x

2.2 Example of a 2D domain subdivision. Sub-domain numbering indicated in the grid. . . 47 2.3 Optimal ˜T (red cross markers) and uniform T (blue circle markers)

trun-cation sets. KLE error for the optimal set 2

KL = 0.001. Tensorizable

Gaussian correlation function with correlation lengths lc1 = 0.25Land

lc2= 0.15L. . . 50

2.4 Example of generation of a random process with Gaussian correlation. Before (black solid line) and after (red dashed line) conditioning. Correla-tion length lc = 0.15L. KLE error 2KL = 0.001. . . 53

2.5 Example of parallel 2D random field generation steps. Gaussian 2D correlation function, with correlation lengths equal to lc1 = 0.2L and

lc2= 0.1. KLE truncation error 2KL = 0.001. Number of terms N = 139. . 54

2.6 Wiener (left) and Brownian (right) bridge covariance function. . . 57 2.7 Example of generation of a non-stationary random process with Wiener

(left) and Brownian bridge (right) covariance. Before (black solid line) and after (red dashed line) conditioning. KLE error 2

KL= 0.001. . . 57

2.8 Correlation function of Eqn. (2.76). . . 58 2.9 Sample of the random fields having the correlation function of Eqn. (2.76),

generated using the conditioned KLE, on the domain [0, M L]2_{, with}

M = 21. The right figure presents a zoom on the left lower corner. The black dotted lines delimit the sub-domains. . . 59 2.10 Correlation function of the generated random field for s1 = s2 = Land

t ∈ [0.5L, 1.5L]2, on the left, and absolute difference with the theoretical correlation on the right. The blue dotted lines delimit the sub-domains. . 60 2.11 Correlation function of Eqn. (2.77). . . 62 2.12 Computational time of the standard KLE (blue circle markers) and the

generation by KLE prolongation (red cross markers). Tensorizable cor-relation function in Eqn. (2.77) on the left. Non-tensorizable corcor-relation function of Eqn. (2.76) on the right. KLE truncation error 2

KL = 0.001.

Slope of the dashed lines indicated on the figures. . . 62 2.13 Correlations functions used in Section 2.7.3, from the left: exponential,

triangular, damped sine and Gaussian correlation. Their analytical ex-pression is indicated in Table 2.4. . . 63 2.14 Power spectral densities related to the correlations functions used in

Section 2.7.3, from the left: exponential, triangular, damped sine and Gaussian correlation. Different scales for the ordinate axis. . . 63 2.15 Example of the coupling matrix K, Eqn. (2.11), related to the correlation

functions in Table 2.4, from the left: exponential, triangular, damped sine and Gaussian correlation. . . 64 2.16 Example of generation of a random process using the conditioned KLE.

Correlation functions in Table 2.4, from the top left to the bottom right: exponential, triangular, damped sine and Gaussian correlation. Before (black solid line) and after (red dashed line) conditioning.. . . 64

xi 2.17 Evolution of the KLE truncation error (solid lines) and the continuity

error (dashed lines) for a random process with correlation functions in Table 2.4, from the top left to the bottom right: exponential, triangular, damped sine and Gaussian correlation. Correlation length lc = 0.05L

(black), lc= 0.15L(dark grey), lc = 0.25L(light grey). . . 65

2.18 Minimal eigen-value of the matrix I − KKT_{− K}T_{K} _{according to the}

number of KLE terms and the correlation lengths for the correlation functions listed in Table 2.4. The step used to discretize the fields is chosen to be equal to 0.01L. White areas indicate a negative minimal eigen-value. The blue line indicates the limit. From the left to the right right: exponential, triangular, damped sine, and Gaussian correlation functions. . . 66 3.1 Reference frame (zoom on the right). . . 72 3.2 Sketch of track geometry irregularities (design position in black,

irreg-ular position in green). From top to bottom: gauge, horizontal level, superelevation, vertical level. . . 73 3.3 Measurement train V355 and its laser measurement system. Bottom part

of the picture frommermecgroup.com. . . 74 3.4 Versine measurement . . . 75 3.5 Curvature to versine transfer function (H(f ) in Eqn. 3.8): (left) module

and (right) phase. . . 76 3.6 Example of reconstruction of track gauge signal (Eqn. 3.9): (left) measured

and reconstructed gauge filtered on [0.19 − 0.21]m−1and (right) Fourier transform of the difference of the measured and reconstructed signals (filtered on some bands) calculated in a measurement campaign.. . . 77 3.7 Direction error indicator, Eqn. (3.10), probability distribution. . . 77 3.8 signal-to-noise ratio estimation: PDFs of the signal-to-noise ratios (left)

and PDF of the correlation coefficients between reconstructed and mea-sured signals (right). . . 78 3.9 Example of track geometry measurement with corresponding theoretical

values (dashed red line), from the top left to the right bottom: gauge, superelevation, horizontal curvature and vertical curvature. . . 79 3.10 Correlations (at spatial lag equal to zero) between the design horizontal

curvature (|CHd(s)|), the gauge measure (G) and the irregularities fields

(gauge Girr, superelevation Eirr, horizontal curvature CHirrand vertical

curvature CV irr). . . 80

3.11 Experimental PDFs of geometry irregularities, from the top left to the right bottom: gauge, superelevation, horizontal curvature and vertical curvature. . . 81 3.12 Joint PDFs of the track irregularities (at zero lag). Logarithmic color scale. 81 3.13 Correlation function of geometry irregularities, from left: gauge,

superel-evation, horizontal curvature and vertical curvature. Different scales of the lag axis (x). . . 82 3.14 Experimental PSD of geometry irregularities, from left: gauge,

superele-vation, horizontal curvature and vertical curvature. . . 82 3.15 Correlation function of design horizontal curvature. . . 83

xii

3.16 Histogram (normalized) of the length between two welds superposed with the correlation of the horizontal curvature irregularity. . . 83 3.17 PSDs of horizontal (left) and vertical level (right). The dashed black

lines represents two German PSDs standard [Ber13,ZLZ+_{10}_{,}_{LCS04}_{].}

Experimental PSDs in red. . . 83 3.18 Karhunen-Loève truncation errors. . . 85 3.19 First (most energetical) three Karhunen-Loève modes (blue, red and

yellow lines in the energetical order), from left: gauge, superelevation, horizontal curvature and vertical curvature. . . 85 3.20 Percentage error in the PCE identification (εP CE in Eqn. (3.20)) of the

vertical curvature irregularity (100 KLE terms). The dashed red line delimits the area in which the identification is not possible since the number of PCE terms is lower than the number of KLE terms to model. The dotted white line delimits the area in which the PCE error is lower than 1%. . . 87 3.21 Evaluation of the continuity error, expressed in Equation 2.29, for

differ-ent numbers of Karhunen-Loève terms and for each generated geometry irregularity field. . . 88 3.22 PSDs of geometry irregularities, from left: gauge, superelevation,

hori-zontal curvature and vertical curvature. Measurements (red line), gener-ations with 10% (dotted green line with circle markers), 1% (dashed blue line) and 0.1% (dotted violet line with square markers) KLE truncation error, 1% of the mean spectral density (horizontal dashed black line). . . 89 3.23 PDFs of geometry irregularities, from left: gauge, superelevation,

hori-zontal curvature and vertical curvature. Measurements (red line), gener-ations (dashed blue line). Truncation error equal to 1%. . . 89 3.24 Example of geometry irregularities, from left: gauge, superelevation,

horizontal curvature and vertical curvature. Measurements (red line), generations (blue line). . . 90 3.25 Examples of measured rail profile (left) and examples of wear curve

for three different track layouts (right). Rail profile curvilinear abscissa growing from the interior to the exterior of the track. . . 91 3.26 Average (left) and covariance (right) of the wear, defined in Eqn. (3.24) . 92 3.27 1st (blue solid), 2nd (red dashed) and 3rd (green dotted) wear modes (left),

resulting from KLE decomposition in Eqn. (3.25), and their respective rail profile forms (right). Theoretical profile (UIC60) plotted with the black dash-dotted line. The wear modes are adimensional. . . 92 3.28 Wear KLE truncation error, resulting from the decomposition in Eqn. (3.25). 93 3.29 PDFs (left) and covariance functions (right) of the 1st (blue solid), 2nd

(red dashed) and 3rd (green dotted) wear modes coefficients ˜ηi(s, θ),

resulting from the regression in Eqn. (3.28). . . 95 3.30 Examples of wear generation with varying curve radius (left) and tonnage

between (right). Colormap indicating the wear in mm. . . 96 3.31 Contact forces decomposition (red) and track reference frame (black). . . 97 3.32 Measurement vehicle schema. . . 98 3.33 Position of the strain gauges used for the contact forces measurement on

xiii 3.34 Example of the measured contact forces (front wheelset, front bogies,

leading car) with the corresponding horizontal curvature, superelevation and train velocity. . . 99 3.35 Example of contact positions in a curved track . . . 100 3.36 PSD of contact forces, front wheelset (1st row straight track, 2nd row

outer rail, 3rd row inner rail), constant velocity. Experimental data (red line), numerical simulations with geometry irregularities (dashed blue line), with geometry irregularities and rail wear irregularities (dotted dashed green line) and without any irregularities (dotted black line). . . 101 3.37 PSD of contact forces, front wheelset in tanget track, variable velocity

(1st row acceleration, 2nd row breaking). Experimental data (red line), numerical simulations with (dashed blue line) and without (dotted black line) geometrical irregularities. . . 102 3.38 PSD of contact forces, front wheelset in tangent track, constant velocity.

Experimental data (red line), numerical simulations with track geometry irregularities with KLE error equal to 10% (dotted green line with circle markers), 1% (dashed blue line) and 0.1% (dotted violet line with square markers). . . 103 3.39 Experimental PSD of the leading wheelsets of the passenger car (red line)

and motor coach (dotted grey line). . . 104 3.40 Wheel untrue used for the numerical simulations. . . 105 3.41 PSD of contact forces, front wheelset in tangent track, constant velocity.

Experimental data (red line), numerical simulations with track geometry irregularities without (dashed blue line) and with (dotted brown line) wheel untrueness. . . 105 4.1 Schema: from the track irregularities to the fatigue criterion. . . 110 4.2 Level curves of the joint PDF of the wheel-rail relative position y and the

lateral contact force Y ; without wear (new rail profile 60E1), tight curve, slow empty train. . . 115 4.3 Average worn profile for a 1000 m curve radius (left) and anti head-check

profile (right). 60E1 represented by the black dashed line. . . 121 4.4 First-order marginal and the joint PDFs of the components y and ψ of the

random field u for the test cases with empty (thick lines) and full (thin lines) vehicle: slow wide curve (solid very light gray), fast wide curve (dashed light gray), slow tight curve (dashed-dotted dark gray), fast tight curve (dotted black). . . 126 4.5 Mean value of the fatigue index in the rail section: empty slow

vehi-cle; wide (top) and tight curve (bottom); without (left) and with (right) random wear included. . . 127 4.6 Standard deviation value of the fatigue index in the rail section: empty

slow vehicle; wide (top) and tight curve (bottom); without (left) and with (right) random wear included.. . . 128 4.7 Mean value of the fatigue index in the rail section: empty slow vehicle;

tangent track; without (left) and with (right) wear included. . . 129 4.8 Standard deviation value of the fatigue index in the rail section: empty

xiv

4.9 PDFs of the maximal value of the fatigue index: slow wide curve (solid very light gray), fast wide curve (dashed light gray), slow tight curve (dashed-dotted dark gray), fast tight curve (dotted black); empty vehicle, without (left) and with (right) wear included. . . 130 4.10 PDFs of the number of life cycles: slow wide curve (solid very light gray),

fast wide curve (dashed light gray), slow tight curve (dashed-dotted dark gray), fast tight curve (dotted black); empty (left) and full (right) vehicle, without (left) and with (right) wear included. . . 131 4.11 PDFs of the maximal value of the fatigue index: slow wide curve (solid

very light gray), fast wide curve (dashed light gray), slow tight curve (dashed-dotted dark gray), fast tight curve (dotted black); empty vehicle, 60E1 (left) and worn or anti head-check (right) profiles. . . 133 4.12 Mean value of the fatigue index in the rail section: empty slow vehicle;

tight curve; 60E1 profile (left), anti head-check profile (right). . . 134 4.13 PDFs of the maximal value of the fatigue index: slow wide curve (solid

very light gray), slow tight curve (dashed-dotted dark gray); empty vehicle, without (left) and with (right) wear included. . . 134 4.14 Mean value of the fatigue index in the rail section: empty slow vehicle;

wide curve (top), tight curve (bottom); without wear (left), with wear (right). . . 135 4.15 PDFs of the maximal value of the fatigue index: slow wide curve (solid

very light gray), slow tight curve (dashed-dotted dark gray); empty vehicle, without (left) and with (right) trailing wheel load cycle. . . 136 4.16 Mean value of the fatigue index in the rail section: empty slow vehicle;

wide (top) and tight curve (bottom); without (left) and with (right) trailing wheel load cycle. . . 137 A.1 PDFs of the friction coefficient on the surface tread (left) and the gauge

corner (right) areas. On the right plot, the blue solid line and the red dashed line refer respectively to the tangent and the curved track portions.144 A.2 PDFs of the stiffnesses of the primary suspension components. From the

top left: first vertical, second vertical, lateral, longitudinal, pitch, roll-yaw stiffness. Red dashed line indicating the nominal design value. . . 144 A.3 PDFs of the fatigue Dang Van index without (left) and with (right)

ran-dom variable friction coefficient: slow wide curve (solid very light gray), fast wide curve (dashed light gray), slow tight curve (dashed-dotted dark gray), fast tight curve (dotted black); empty vehicle. . . 149 B.1 Average wheel-rail contact positions. Wide curve with small cant

defi-ciency (slow velocity). Distribution of the surface normal forces indicated on the rail profile. . . 152 B.2 Average wheel-rail contact positions. Wide curve with high cant

defi-ciency (fast velocity). Distribution of the surface normal forces indicated on the rail profile. . . 153 B.3 Average wheel-rail contact positions. Tight curve with small cant

defi-ciency (slow velocity). Distribution of the surface normal forces indicated on the rail profile. . . 154

xv B.4 Average wheel-rail contact positions. Tight curve with high cant

defi-ciency (fast velocity). Distribution of the surface normal forces indicated on the rail profile. . . 155 B.5 Average wheel-rail contact positions. Tangent track. Distribution of the

xvi

**List of Tables**

2.1 Computational time for generating a 2D random field with correlation
function as in Eqn. (2.76) on an extended domain [0, M L]2_{, with M = 21}

using the conditioned KLE. Computed on Intel® Xeon® CPU E5-2695 v3 @2.30GHz. 1 processor used. . . 59 2.2 Numerical complexity of the standard KLE and the conditioned KLE.

Non-tensorizable covariance kernel. ns=number of discretization steps

of a segment of length L. M = number of prolongations in one direction. d =dimension. N = total number of retained KLE terms for M = 1. . . . 60 2.3 Numerical complexity of the standard KLE and the conditioned KLE.

Tensorizable covariance kernel. ns =number of discretization steps of a

segment of length L. M = number of prolongations in one direction. d = dimension. ¯N =maximal number of retained KLE terms among all the directions (dimensions) for M = 1. . . 60 2.4 Correlation functions used in Section 2.7.3 and relative truncation errors,

number of retained KLE terms and continuity errors, with lc= 0.15Lin

all the cases and τ = |s − t|. . . 63 3.1 Number of terms used in the random fields model. KLE truncation error

εKL = 0.01. . . 86

3.2 Mean and standard deviation of the wear modes coefficients, Eqn.(3.27), according to the track horizontal layout. . . 94 3.3 Correlation between the wear modes coefficients, Eqn.(3.27), and the

track features. . . 94 3.4 Number of terms used in the random geometry irregularities model

with two different KLE truncation error, where d and ng indicate the

polynomial chaos degree and the number of germs. . . 103 4.1 Operating scenarios test cases.. . . 113 4.2 Mean (µ), standard deviation (σ), coefficient of variation (cv) and 1st

order Sobol indices of the wheel-rail lateral position y. "Inter" indicates the indices whose order is higher than one (interactions). . . 114 4.3 Mean (µ), standard deviation (σ), coefficient of variation (cv) and 1st

order Sobol indices of the wheel-rail vertical position z. "Inter" indicates the indices whose order is higher than one (interactions). . . 115 4.4 Mean (µ), standard deviation (σ), coefficient of variation (cv) and 1st

order Sobol indices of the wheelset yaw angle ψW S. "Inter" indicates the

xvii 4.5 Mean (µ), standard deviation (σ), coefficient of variation (cv) and 1st order

Sobol indices of the leading outer rail lateral force Y . "Inter" indicates the indices whose order is higher than one (interactions). . . 117 4.6 Mean (µ), standard deviation (σ), coefficient of variation (cv) and 1st order

Sobol indices of the leading outer rail vertical force Q. "Inter" indicates the indices whose order is higher than one (interactions). . . 118 4.7 Mean (µ), standard deviation (σ), coefficient of variation (cv) and 1st order

Sobol indices of the leading bogie lateral forceP Y . "Inter" indicates the indices whose order is higher than one (interactions). . . 119 4.8 Mean (µ), standard deviation (σ), coefficient of variation (cv) and 1st order

Sobol indices of the wear number related to the leading outer wheel WN.

"Inter" indicates the indices whose order is higher than one (interactions).120 4.9 Mean (µ), standard deviation (σ), coefficient of variation (cv) and 1st

order Sobol indices of the wheel-rail relative lateral position y (left) and the vertical position z (right). "Inter" indicates the indices whose order is higher than one (interactions). . . 122 4.10 Mean (µ), standard deviation (σ), coefficient of variation (cv) and 1st

order Sobol indices of the wheel-rail lateral force Y (left) and the vertical force Q (right). "Inter" indicates the indices whose order is higher than one (interactions). . . 122 4.11 Mean (µ), standard deviation (σ), coefficient of variation (cv) and 1st

order Sobol indices of the sum of the bogie lateral forcesP Y (left) and the wear number WN (right). "Inter" indicates the indices whose order is

higher than one (interactions). . . 123 4.12 Mean (µ), standard deviation (σ), coefficient of variation (cv) and 1st

order Sobol indices of Dang Van fatigue index on the rail section. "Inter" indicates the indices whose order is higher than one (interactions). . . . 130 4.13 Mean (µ), standard deviation (σ), coefficient of variation (cv) and 1st

order Sobol indices of the number of life cycles of the rail. Mean value
expressed in 105_{cycles. "Inter" indicates the indices whose order is higher}

than one (interactions). . . 132 4.14 Mean (µ), standard deviation (σ), coefficient of variation (cv) and 1st order

Sobol indices of the rail fatigue index (maximal value in the rail section). "Inter" indicates the indices whose order is higher than one (interactions).133 A.1 Mean (µ), standard deviation (σ), coefficient of variation (cv) and 1st

order Sobol indices of the wheel-rail lateral position y. . . 145 A.2 Mean (µ), standard deviation (σ), coefficient of variation (cv) and 1st

order Sobol indices of the wheel-rail vertical position z. . . 145 A.3 Mean (µ), standard deviation (σ), coefficient of variation (cv) and 1st

order Sobol indices of the wheelset yaw angle ψW S. . . 146

A.4 Mean (µ), standard deviation (σ), coefficient of variation (cv) and 1st

order Sobol indices of the leading outer rail lateral force Y . . . 147 A.5 Mean (µ), standard deviation (σ), coefficient of variation (cv) and 1st

order Sobol indices of the leading outer rail vertical force Q. . . 147 A.6 Mean (µ), standard deviation (σ), coefficient of variation (cv) and 1st

xviii

A.7 Mean (µ), standard deviation (σ), coefficient of variation (cv) and 1st order

xix

**List of Abbreviations**

**CDF** **Cumulative Density Function**

**KLE** **Karhunen-Loève Expansion**

**FFT** **Fast Fourier Transform**

**MBS** **Multi-Bodies System**

**MCS** **Monte Carlo Simulations**

**PCE** **Polynomial Chaos Expansion**

**PDF** **Probability bfDensity Function**

**PSD** **Power Spectral Density**

**QoI** **Quantity of Interest**

**RCF** **Rolling Contact Fatigue**

**List of recurrent Symbols**

CH Horizontal curvature [1/m]

CV Vertical curvature [1/m]

E Superelevation [m]

G Track gauge [m]

r Rail profile curvilinear abscissa [m]

s Track curvilinear abscissa [m]

w Rail wear [m]

δij Kronecker delta [any]

G Standard normal CDF [any]

E Expectation [any]

P Probability [-]

1

**General context**

Rail rolling contact fatigue (RCF) is a common and dangerous issue on all types of rail-way systems. The RCF represents a dominant cause of maintenance and rail replace-ments for the heavy-haul railway networks and is a significant economic and safety challenge for suburban and metro railway networks.

With the RER (A and B) subway lines, RATP (Régie Autonome des Transports Parisiens) operates some of the busiest urban railway networks in Europe. Because of the train passage frequency (high overall tonnage), the large acceleration and breaking actions, the rails and track are extremely solicited. This thesis is carried out in the RATP department that operates the maintenance of the railway tracks. The increase of the service demand has encouraged the company to modify its track maintenance policy, gradually passing from a corrective to a preventive approach.

In recent years, some optimisation techniques, such as the choice of the rail steel and the modification of the rail profile, have increased the life of the rails in the critical zones of the network. Thanks to these improvements, combined with the increase of the preventive rail replacements cycles, the rail breaking has become a rare event during the standard circulation of the trains (Figure1, where the ordinate axis scale is censored for reasons of confidentiality.). Although rare, it remains a dangerous and expensive event.

FIGURE 1: Histogram of the rail (red) and the welding (green) failures in the recent years concerning the RATP RER line A. Ordinate axis scale not shown for reasons of confidentiality.

Some works concerning the RATP railway network [Sim14] tend to show that some un-expected phenomena seem to be related to small variations from the nominal dynami-cal behaviour of the train. Small fluctuations of the track features (such as geometry,

2 General context rail profiles, soil stiffness) and the vehicle set up can lead to important variations of the wheel-rail contact stresses and consecutive premature degradation of the track. Be-cause of the scarcity of rail breaking events, it is sometimes difficult to establish a rela-tionship between the rail fatigue life and the characteristics of the track and the rolling stock. Some numerical models, whose inputs can be uncertain, have been proposed in the literature to deal with this issue.

In this thesis, the rail RCF is studied by constructing a random model (representing the track geometry and rail wear irregularities) and by propagating uncertainties into numerical models able to estimate a fatigue criterion index. The aim is to obtain the probability density functions of the fatigue index (for different operating scenarios) and quantity the influence of the irregularities on it.

This thesis has been carried out in the Department of RATP operating the track mainte-nance (80 % of time) and the laboratory MSSMat in CentraleSupélec (20 % of time). This work has been founded under the form of a CIFRE agreement.

3

**1 Overview on the rail rolling contact**

**fatigue**

**Contents**

**Introduction** **. . . .** **3**

**1.1** **Wheel rail-contact** **. . . .** **4**

1.1.1 Analytical models . . . 5

1.1.1.a Normal problem . . . 5

1.1.1.b Tangential problem . . . 8

1.1.2 Semi-Hertzian model. . . 10

1.1.3 Consideration of rail plasticity . . . 11

**1.2** **Dynamic evolution of the rail-wheel surface contact stresses** **. . . . .** **12**

1.2.1 Vehicle model . . . 12

1.2.2 Track model . . . 14

1.2.3 Dynamic simulations. . . 15

**1.3** **Rail internal stress cycle due to the rolling contact** **. . . .** **17**

1.3.1 Consequences of the rail-wheel contact on the rail structure . . 17

1.3.2 Rail internal steady stress cycle computation . . . 18

**1.4** **Initiation of the rail contact fatigue** **. . . .** **20**

1.4.1 Studies concerning the initiation of the rail fatigue. . . 21

1.4.2 Dang Van fatigue criterion. . . 22

1.4.3 Parameters influencing the rail RCF initiation . . . 26

**1.5** **Consequences of the rail rolling contact fatigue** **. . . .** **27**

**1.6** **Objectives and contributions of the thesis** **. . . .** **30**

**Introduction**

In this chapter the mechanisms leading to the rolling contact fatigue and their numerical modelling are described starting from the wheel-rail contact to the estimation of the rail fatigue life.

The rail RCF is originated by the repeated passages of loaded wheels on the rails. This rolling contact implicates a complex multi-axial state of stress dominated by a

4 Chapter 1. Overview on the rail rolling contact fatigue combination of shear and compressive stresses. The generalities concerning the wheel-rail contact description and surface stresses computation are given in Section1.1. These surface stresses evolve along the track and their numerical dynamical computation is described in Section1.2.

When the rail is subjected to high surface contact stresses overtaking the steel plastic limit, corresponding to a wheel passage, its micro-structure and geometry evolve at each load cycle [Joh85,KJ92]. The rail response consists in reacting to these surface stresses with its elastic and plastic deformations (interrelated through internal stress cycle). The steps leading to estimate the rail internal stresses corresponding to this stage are reported in Section1.3.

This internal load cycle is used to estimate the initiation of the RCF. In Section1.4, after a review on the RCF formation literature, a fatigue multi-axial criterion is applied on the rail internal stresses to determine the RCF criticity, the location of the RCF initiation and the rail fatigue life is presented. Some of the consequences of the RCF and the maintenance operations usually adopted to prevent this phenomenon are described in Section1.5.

Since most of the parameters involved in the RCF determination are uncertain, a stochastic approach is developed in this thesis to build a probabilistic model of the rail fatigue life and determine and quantify the influence of these parameters on the RCF. The main objectives and scientific contributions of this thesis are reported in the final section of this chapter.

**1.1**

**Wheel rail-contact**

The aim of this section is to provide the reader with a basis for understanding the wheel–rail interactions phenomena that are hereinafter described in this thesis. The railway wheelset is essentially composed of two conical wheels, linked together with a rigid axle. The wheels are equipped with a flange on the internal side, in order to prevent derailment. The contact is generally localised on the tread surface in a straight line track, while, in a curved track, the flange zone can be subjected to contact.

The wheel-rail contact interface is a small horizontal contact patch, where high contact pressures are concentrated. With respect to the centre of the contact patch, the contact forces are usually locally decomposed into three components corresponding to three directions [CA06]:

• ~n normal to the contact surface → Normal force FN

• ~tX normal to the rail transversal section → Traction or braking force FX

• ~tY orthogonal to ~n and ~tX → Lateral creep force FY

The vector ~tY completes a right-handed frame. The forces FX and FY constitute the

tangential force: FT =pFX2 + FY2. These forces act on the wheelset, which is connected

1.1. Wheel rail-contact 5

**1.1.1**

**Analytical models**

The computation of the contact forces and stresses and the determination of the con-tact patch can be achieved by studying the concon-tact with finite element methods. How-ever, analytical methods are more advisable in computational dynamics when the con-tact problem has to be solved at each iteration time. The goal is to determine some contact parameters: the contact surface, the pressures and the tangential forces. This determination is generally separated into two steps corresponding to the two directions:

• The normal problem (Hertz’s theory) • The tangential problem (Kalker’s theory)

**1.1.1.a** **Normal problem**

Let us consider two elastic bodies being in contact (pressed together) without fric-tion as in Figure 1.1. Let suppose that their surfaces are characterized by constant curvatures and that their dimension is infinite compared to the size of the contact patch. Hertz solved the contact problem for this configuration with the following consequences [Her82,Joh85]:

• The contact surface is an ellipse • The contact surface is flat

• The contact pressure distribution field is semi-ellipsoid

6 Chapter 1. Overview on the rail rolling contact fatigue Let us define the contact relative curvatures A and B, with respect to Figure1.1:

A = 1 2 1 R1,X + 1 R2,X , B = 1 2 1 R1,Y + 1 R2,Y (1.1) The two elastic bodies enter into contact with each other in a single point M where the normal distance between them is minimal. Near this contact point the surfaces of the bodies can be represented by two second order polynomials. When in contact, the vertical separation of the two bodies is:

z = Ax2+ By2 − δ0 (1.2)

where δ0, called depth of indentation, represents the indentation between the two bodies

due to a normal force FN that is applied to them.

The Hertzian theory allows to link the depth of indentation to the normal force to determine the size of the contact ellipsoid patch, through its two semi-axis parameters a and b, and the distribution of the normal pressure stresses acting on the contact surface:

a = m
3
2FN
1 − ν2
E(A + B)
1_{3}
b = n
3
2FN
1 − ν2
E(A + B)
1_{3}
δ0 = ρ
"
3
2FN
1 − ν2
E
2
(A + B)
#1_{3}
σzz =
3FN
4ab
r
1 −x
a
2
−y
b
2
(1.3)

with E being the Young’s elastic modulus and ν the Poisson’s ratio, assuming the same material for the two bodies. The quantities m, n, and ρ are the non-dimensional Hertzian parameters [Her82], resulting from elliptic integral and depending on the A

B ratio. The parameters of the ellipse and the depth of indentation are sketched in Figure1.2. The

b

a ratio varies in the same way as the A

B ratio. Note that the sign of the curvatures is important, for determining the shape of the contact patch. The curvature radii are assumed to be positive when the curvature centre is inside the body, i.e. the shapes are convex. The pressure distribution is elliptical, with maximal value equal to 1.5FN

πab. In the railway case, the curvatures are deduced from the wheel and the rail profiles, represented in Figure1.3. Let (yw, zw)and (yr, zr)be the coordinates representing the

profiles of the wheel and the rail. By assuming that the curvature of the rail along its longitudinal direction is null (the rail is straight) and that the wheel is a perfect solid of revolution (with nominal radius equal to R0), the contact curvatures parameter A and

1.1. Wheel rail-contact 7

FIGURE1.2: Two-solid interpenetration and Hertzian ellipse, from [Quo05].

B are equal to [CA06]:

A = 1
2zw+ R0
1
s
1 + ∂zw
∂yw
2
, B = ∂
2_{z}
w
∂r2 +
∂2_{z}
r
∂r2 (1.4)

with r corresponding to the curvilinear abscissa of the considered profile. These curva-tures are shown in Figure1.4

Y [m] -0.05 0 0.05 0.1 Z [m ] -0.03 -0.02 -0.01 0 0.01

FIGURE 1.3: Rail (blue) and wheel (red) profiles represented when a contact on the surface tread occurs. Rail cant angle α = 1

20 rad.

Note that the rail transverse curvature is positive (the rail has a convex shape), while the wheel curvature at the contact point can be positive or negative (concave shape). When the contact is localized on the surface tread, the rail and the wheel can have the same curvature value with opposite sign, i.e. B = 0. In this configuration the contact is known as conformal, meaning that the wheel and the rail perfectly fit together.

8 Chapter 1. Overview on the rail rolling contact fatigue Curvilinear abscissa r [m] 0.02 0.04 0.06 0.08 0.1 0.12 0.14 C u rv at u re A [1/m ] 0 0.5 1 1.5 C u rv at u re B [1/m ] 0 20 40 60

FIGURE1.4: Contact curvature A (black solid line) and B (green dashed line) relative to the profiles in Figure1.3.

**1.1.1.b** **Tangential problem**

The classical Coulomb dry friction laws cannot correctly describe a railway rolling con-tact problem, in which the relative velocity between the two bodies is not small enough for neglecting the elastic deformation of the solids [Quo05]. With respect to Figure1.5, the two points M1 and M2, getting in contact at the same time, remain at the same

rel-ative position because of the deformability of the solids, i.e. the adhesion condition occurs. However, when the shear stresses become large enough, the two points sepa-rate initiating the slipping phenomenon. The tangential stresses corresponding to this situation are saturated.

FIGURE1.5: Slipping and adhesion in rolling contact, from [Quo05].

The tangent components of the contact force, FX and FY, depend on the creepages

(relative velocities between the two bodies) at the contact point: • νx, longitudinal creepage (direction ~tX)

• νy, lateral creepage (direction ~tY)

• φs, spin creepage (relative to the rotation around ~n

Analytical models exist for describing the rolling contact in adhesion and slipping conditions. When the adhesion condition occurs, the tangential forces can be expressed

1.1. Wheel rail-contact 9
~
n
~_{t}
X
~t_{Y}
F_{Y}
FX
FT
Adhesion
Slip

FIGURE1.6: Slipping and adhesion zones in a rolling contact patch.

through the linear theory of Kalker [Kal79]: Fx = −abc11J νx Fy = −abc22J νy− 3 √ abJ c23φs Mz = −abc32J νy− (ab)2J c33φs (1.5)

where cij are the Kakler’s non-dimensional coefficients and J =

E

2(1 + ν) is the Coulomb shear modulus.

When the saturation is considered, the tangential forces are limited. The saturation condition proposed in [Car26] limits the tangential stresses to a value equal to the normal stress multiplied by the friction coefficient µ. This method has been extended in [Kal72] to elliptic contact patches. The saturated tangential force ¯FT is calculated as:

¯
FT
µFN
=
3
2ζ arccos(ζ)(1 − (1 + 0.5ζ
2_{)p1 − ζ}2_{)} _{if 0 ≤ ζ ≤ 1}
1 if ζ > 1
(1.6)
where ζ = 4FT
3πµFN

, with FT being the unsaturated tangential force, Eqn. (1.5).

In order to calculate the tangential stresses in each location of the contact patch, the FASTSIM method [Kal82], which is an evolution of the Kalker’s linear theory, is em-ployed. The idea behind this technique is to calculate the tangential stresses incremen-tally starting from the leading edge of the contact patch (where the stresses are null):

σzx(x + δx, y) = σzx(x, y) + δσzx= σzx(x, y) + 3 8Gc11νx− 4 π r a bGc23φsy δx a σzx(x + δx, y) = σzy(x, y) + δσzy = σzy(x, y) + 3 8Gc22νy+ 4 π r a bGc23φsx δx a (1.7)

10 Chapter 1. Overview on the rail rolling contact fatigue To take into account the saturation, the norm of the tangential σT stresses is compared

with the traction limit equal to: σzz(x, y) = µ 2FN πab 1 −x a 2 −y b 2 (1.8) If the norm of the tangential stresses is larger than this limit, the slip occurs and the stresses are expressed as:

σzx(x + δx, y) = σzx(x + δx, y) σt(x + δx, y) σzz(x + δx, y) σzy(x + δx, y) = σzy(x + δx, y) σt(x + δx, y) σzz(x + δx, y) (1.9)

**1.1.2**

**Semi-Hertzian model**

The use of the Hertzian theory allows us to consider one contact elliptic patch. However, in railway contact applications, in some situation the contact can occur in more locations along the rail profile and the multiple contact patches are not elliptic. Moreover, the curvatures A and B, defined in Eqn. (1.1), can be considered as constant in some regions, especially when the contact is localized on the surface tread. In other contact configurations, for instance when the wheel flange is involved in the contact, the curvature variations are very steep. When the curvature cannot be considered as constant along the whole contact zone, the Hertzian theory loses its validity. In these cases, a semi-Hertzian model of the contact can be used to overcome this issue.

The method proposed in [PK08] allows us to consider multiple contact patches solving the problem by considering the virtual penetration of the wheel and the rail profile. The STRIPES method [AC05], which is used in this thesis to model the wheel-rail contact, has the particularity of considering variable contact curvatures along the rail profile. The basic idea of this method is the subdivision of the rail profile in stripes along the lateral direction. Then, in each stripe, the contact problem is solved as Hertzian. In this section the main steps of this method are reported.

The first step is to find a link between the contact ellipse defined in the Hertzian case and the geometrical ellipse resulting from the intersection of two solids. In fact, when two solids get in contact with a depth of indentation defined in Eqn. (1.2), the ellipse resulting from the geometrical intersection is not equal to the Hertzian ellipse whose parameters are defined in Eqn. (1.3). The ratio between the geometrical and the Hertzian aspect ratios is equal to n

m r B

A. To impose an equivalence between the two ellipses,
the curvatures and the depth of indentation of the geometrical problem are slightly
modified:
˜
A =n
m
2
B
˜
δ0 = δ0
n2_{B}
ρ(A + B)
(1.10)

1.1. Wheel rail-contact 11
With these modifications the two ellipses are equal. The curvature correction is
com-bined by a filtering smoothing step, calibrated by comparison with finite element models
of non-Hertzian cases [QSE+_{06}_{].}

At this stage, the contact problem is treated as an equivalent geometrical problem. The rail profile is subdivided in stripes having a small width (∼ 0.5 mm). Each stripe is char-acterized by a possible (if the contact occurs in the considered stripe) interpenetration. The contact angle and curvature also vary in each stripe. Then the problem is solved in each stripe using the formulation described in Section1.1.1.b.

Being based on the geometrical interpenetration of the rail and the wheel profiles, this semi-Hertzian contact model allows us to consider more separated contact patches. For instance, the contact can occur, at the same time, on the gauge corner and the surface tread of the rail.

**1.1.3**

**Consideration of rail plasticity**

In a wheel-rail contact, the maximum contact pressure can sometimes be larger than
1 GPa [QSE+_{06}_{]. Hertzian and semi-Hertzian methods are based on an elastic }
formu-lation of the problem. In order to take into account the rail plasticity, for considering
a realistic plastic law including a stress hardening law, the assessment of the strains,
which would increase the computational time, would be required. The method
pro-posed in [SCAC12], deals with this issue by limiting the surface contact stresses (the rail
internal stresses are not considered) and by modifying their distribution shape. This
method is combined with the STRIPES semi-Hertzian contact model (Section1.1.2).
The von Mises yield criterion is evaluated at the centre of each stripe (identified by the
subscript i) of the rail and a perfect plastic condition is formulated to limit the maximal
value of the stresses along the considered stripe (ˆσi):

ˆ σi ≤ σE r ((1 − 2ν)αi)2+ FT FN 2 (1.11)

where σE is the elastic limit of the material and αi are coefficients, depending on the

local profile geometry and interpenetration, that are calibrated with comparison with finite element simulations [SCAC12]. The stresses acting on the considered stripe are therefore limited to this value. When the limit is reached, a correction of the length of the contact patch is needed in order to obtain the same contact force, as for the elastic case, when the stresses are integrated along the stripe (Figure1.7). The width of the stripe is slightly increased for this purpose and the resulting contact area is larger than in the elastic case.

Note that this value is over the elastic limit of most steels, because it also depends on the friction coefficient and the shape of the contact patch.

12 Chapter 1. Overview on the rail rolling contact fatigue
xof the stripe [m] _{×10}-3
-6 -4 -2 0 2 4 6
Nor
m
al
st
re
ss
[P
a]
×108
0
2
4
6
8
10
12
14

FIGURE1.7: Limitation of the normal stress (red solid line) with respect to the elastic stress distribution (blue dashed line) along a stripe of the rail.

**1.2**

**Dynamic evolution of the rail-wheel surface contact**

**stresses**

In the previous section, the basis for understanding the wheel-rail contact description and the estimation of the surface contact stresses have been given. A computational model of the surface contact stresses is generally used for dynamic simulations, in order to estimate the evolution along the track of the contact patches and stresses when a vehicle is running. The wheel-rail contact model represents an interface between the vehicle and the track on which it is running. Therefore the contact patches depend on the vehicle and the track features. In this section the main elements concerning the modelling of the vehicle and the track are firstly introduced in Sections1.2.1and1.2.2. Then, some discussions about the numerical simulations are reported in Section1.2.3.

**1.2.1**

**Vehicle model**

Railway vehicles consist of many linked components whose mechanical properties need to be identified and modelled for vehicle-track dynamic simulations. A general main sub-division of the vehicle components can be made into body components and suspensions components which connect all the bodies. The most important body components, in terms of inertial properties, are the car body (motor or passenger coaches), bogie frames and wheelsets (Figure 1.8). The masses and the moments of inertia for all bodies have to be specified. The weight of the car body can strongly vary because of the passenger load, especially in case of urban railway networks, where it can represent about one third of the total weight. The bodies are usually considered as rigid, although, in many railway applications the body structural flexibility also needs to be considered, e.g. the flexibility of the car body should be considered when performing ride comfort studies while the flexibility of the bogies should be taken into account when simulating highly twisted track [SS01]. The flexibility of the bodies,

1.2. Dynamic evolution of the rail-wheel surface contact stresses 13 when introduced, can be represented with a limited number of eigen-frequencies and eigen-modes. Modal properties can be measured or simulated using finite element tools.

FIGURE1.8: Sketch of a common railway vehicle, from [Sim14].

The main suspension components are usually sub-divided into primary (connecting the wheelsets to the bogies) and secondary (connecting the bogies to the car body) suspen-sions (Figure1.8). They have low masses (compared to the main bodies) and serve to connect the vehicle components. The suspensions consist of various physical springs and dampers whose forces essentially are related to the displacements and velocities of the components. In a railway vehicle, most of the suspension components have a non-linear response, e.g. stiffness varying according to the loads or the displacements. Trac-tion rods, bump stops, anti-roll bars, trailing arms, linkages, also belong to this group of components. Some points on the main bodies are defined as connection locations. The suspensions play an important role for the reduction of the bogies and car body acceler-ations as well as dynamic wheel-rail forces. They also allow a proper curve negotiation and ride comfort [OB06]. Bump stops and anti-roll bars are therefore usually employed to mitigate this kind of problems. The bars also reduce quasi-static lateral accelerations on the car body floor. The traction rods can also be denoted as suspension components although their main task is to transfer longitudinal forces between bogie and car body during acceleration or braking phases. Other linkages act between the car bodies for transmitting the traction generated by the motor coach.

The wheelsets are the components concerned by rolling contact. They provide the guidance that determines the motion within the rail gauge and ensure the necessary distance between the vehicle and the track. They are also employed for transmitting traction and braking forces to the rails to accelerate and decelerate the vehicle. A plain wheelset normally consists of one solid unit with two wheels linked by a wheel axle. A typical mass for a plain wheelset is 1000 to 1500 kg. Additional mass can be introduced through brake discs, mounted on the axle or wheels, and traction gear. The axle boxes, also add some mass. Indeed, they are sometimes treated as separate bodies in the modelling.

In summary, the vehicle model is represented by a set of bodies (usually considered as rigid) connected to the suspensions elements. This leads to a multi-body system (MBS), where most of the vehicle degrees of freedom are assigned to the motion of the vehicle

14 Chapter 1. Overview on the rail rolling contact fatigue bodies. By the knowledge of the inertial properties of the bodies and the features of the suspensions, it is possible to derive the matrices of mass, damping and stiffness that allow to write a system of equations (depending on the time variable) where the unknowns are the degrees of freedom of each body.

**1.2.2**

**Track model**

By interacting with the vehicle, the main functions of the track are ensuring the guiding of the wheelsets, keeping the train into the prescribed gauging, absorbing the static and dynamics forces due to the passages of the vehicles and ensuring the electric continuity used for the circulation and the signalisation.

A sketch of a common railway track is shown in Figure1.9. The track is composed of two rails connected by equally spaced transversal sleepers (made of wood or concrete). The railpads protect the sleepers from wear and impact damage, and they provide electrical insulation of the rails. Wooden sleepered tracks may not have rail pads. All these components lie on a ballast layer or directly on a concrete platform. The ballast is a granular material typically made of crushed stone whose main functions are absorbing and distributing the loads (for avoiding the soil compaction), damping the vibrations originated by the circulation and ensuring the draining of the rain. Some sub-layers can also be present: sub-ballast and sub-grade. Track flexibility is usually integrated in railway simulations models. A simple moving track model for lateral, vertical, and roll flexibilities of the track may be sufficient for analysis of the vehicles interactions with the track [Dah06]. Stiffness and damping can have linear or non-linear characteristics.

FIGURE1.9: Sketch of a common railway track, from [Dah06].

Concerning the rails, modern track typically uses hot-rolled steel with a profile of an asymmetrical rounded I-beam. Unlike some other uses of iron and steel, railway rails are subject to very high stresses and have to be made of very high-quality steel alloy. In Europe, one commonly used rail profile is the 60E1 rail (Vignoles type), where 60 refers to the mass of the rail in kg per meter. The rails should provide smooth running surfaces for the train wheels and they should guide the wheelsets in the direction of the track. The rails also carry the vertical load of the train and distribute the load over the sleepers. Lateral forces from the wheelsets, and longitudinal forces due to traction and braking of the train should also be transmitted to the sleepers and further down into

1.2. Dynamic evolution of the rail-wheel surface contact stresses 15 the ballast layers. The rails also act as electrical conductors for the signalling system. The rail profile is subjected to wear during its life and, therefore, the form of its profile is not constant along the track. The wear evolves along both the track position and the track profile coordinate. This evolution is modelled as a random field in Chapter3. The position of the rails with respect to a global (or moving local) reference frame defines the track geometry. The nominal track geometry (layout) is defined by circular curve radii and lengths, types of transition curves, and track cant angles. Track irregularities are normally given as lateral and vertical deviations of the track median line (from nominal geometry), and by deviations in track cant angle and track gauge (see Figure3.2 on page73). These irregularities can be included into the simulation model by varying the position of the rail along the track curvilinear abscissa. A statistical analysis and modelling of track irregularities associated to the rail wear will be treated in Chapter3.

**1.2.3**

**Dynamic simulations**

The vehicle-track system is finally represented by a network of bodies connected to each other by flexible, massless elements representing the linkages. The complexity of the MBS can be varied according to the simulation and the results required. Indeed, depending on the purpose of the simulation a wide range of outputs, for example, displacements, accelerations, forces, at any point of the track can be extracted. The first stage in setting up a computational model is to prepare the set of mathematical equations that represent the vehicle-track system. These are the equations of motion and are usually second order differential equations that can be combined into a set of matrices:

M¨x + C ˙x + Kx = F (1.12)

where M, C, K are the mass, damping, stiffness matrices, F is the forces vector and x gathers all the displacements of the degrees of freedom.

All the rigid bodies can be considered to have a maximum of six degrees of freedom, three translational and three rotational. Physical constraints may mean that not all of these movements are possible and the system can be accordingly simplified. Application of the constraint equations results in a set of equations of motion which are ordinary differential equations or linear algebraic equations. All of the interconnections may include non-linearities.

Non-linearities also occur at the wheel–rail contact interface1_{, due to creep and flange}
contact, in the suspensions responses and into the model of the track flexibility. Because
of the presence of these non-linearities the full equations of motion cannot normally be
solved analytically. It is sometimes possible to linearise the equations of motion but
otherwise a numerical method must be used to integrate the equations at small time
intervals over the simulation time interval [PBI06]. The numerical solutions at each
point are used to predict the behaviour of the system at the next time step.

A large range of numerical methods are available for this type of simulation. Some solvers use a varying time step which is automatically adjusted according to the current

16 Chapter 1. Overview on the rail rolling contact fatigue
state of the simulation. At each time step every term of the system of the equations of
motion is evaluated. At each time step the equations of motion are set up and all the
system non-linearities, the creepages and creep forces between the wheels and rails are
evaluated. Finally, the resulting accelerations of each body for each degree of freedom
are computed2_{.}

The displacements and velocities are calculated through the integration routine and stored, the elapsed time is increased, and the complete calculation step repeated. The whole integration process is stopped until the chosen maximum time period or distance on the track is reached.

More specifically, the surface rolling contact stress tensors acting on the rail are extracted at each time integration step for of each wheel of the vehicle. Those tensors will determine the contact fatigue initiation at a given time step (at which a track curvilinear position s corresponds).

Employing the semi-Hertzian contact model, described in Section 1.1.2, the surface contact stresses will be entirely determined by the knowledge of: the shape (profiles) of the wheels and the rails, the wheel-rail relative position and velocity, and the wheel-rail friction coefficient. These quantities evolve along the position on the track, identified by the track curvilinear abscissa s, because of the variability of the track condition and the vehicle-track dynamic interactions. The rail and wheel profiles shapes essentially vary because of the wear. The wear phenomenon is due to the rolling contact and influences the contact patch location and the intensity of the stresses. In other words there is a two-way relationship between wear and surface contact stresses.

Concerning the wheel-rail relative position, expressed in the rail reference frame, on which the computation of the contact stresses depends, it is determined by the following parameters (shown in Figure1.10):

• y(s) Wheelset-rail lateral relative position. • z(s) Wheelset-rail vertical relative position. • ψ(s) Wheelset-rail yaw angle.

• φ(s) Wheelset-rail roll angle. • ω(s) Wheelset-rail rotational angle

The wheel-rail relative velocity is determined by the six following components: • ˙x(s) Wheelset longitudinal velocity.

• ˙y(s) Wheelset lateral velocity. • ˙z(s) Wheelset vertical velocity. • ˙ψ(s)Wheelset-rail yaw rate. • ˙φ(s)Wheelset-rail roll rate.

• ˙ω(s) Wheelset-rail rotational rate.