Content uploaded by Alejandro Cornejo

Author content

All content in this area was uploaded by Alejandro Cornejo on Nov 20, 2021

Content may be subject to copyright.

On the numerical study of fatigue process in rail heads

by means of an isotropic damage based high-cycle

fatigue constitutive law

S. Jim´eneza,b,∗

, L. G. Barbua,b, S. Ollera,c , A. Cornejoa,b

aCentre Internacional de M`etodes Num`erics en Enginyeria (CIMNE), Campus Norte UPC,

08034 Barcelona, Spain

bUniversidad Polit´ecnica de Catalu˜na (UPC), Campus Norte UPC, 08034 Barcelona, Spain

cConsejo Nacional y de Investigaciones Cient´ıﬁcas y T´ecnicas (CONICET), Facultad de

Ingenier´ıa, Universidad Nacional de Salta, Av. Bolivia 5150, 4400, Salta, Argentina

Abstract

The fatigue phenomenon has been historically related to the railway indus-

try. Nowadays, thanks to the high quality of the materials used, most of the

recorded problems have been overcome although there are still degradation pro-

cesses which are associated to fatigue and need to be considered for the deﬁnition

of the maintenance campaigns and the general progress of the sector. In this

paper, the latest improvements in the isotropic damage based high-cycle fatigue

constitutive law proposed by Oller et al. [1] are presented and the approach

is used for the study of two regions of the railway path where fatigue mecha-

nisms are experimentally detected: a straight section and a crossing element.

The analysis of the aﬀected areas is performed through a ﬁnite element simu-

lation identifying the critical regions liable to the fatigue degradation when the

structure interacts with high speed vehicles and predicting the initiation of the

degradation at the rail head while capturing the physics of the problem. The

potential of the methodology is shown through the case studies and the current

shortcomings and the future lines of research are clearly stated.

Keywords: railway, ﬁnite element method, high-cycle fatigue simulation,

isotropic damage, advance in time strategy, pearlitic steel, Hadﬁeld steel

∗Corresponding author.

Email address: sjimenez@cimne.upc.edu (S. Jim´enez)

Preprint submitted to Engineering Failure Analysis November 20, 2021

1. Introduction

Since the second half of the 19th century, the fatigue phenomenon has been

deeply studied, originally associating it to the recurrent deterioration detected

in the railway system which led to unexpected failures [2]. In this process the

mechanical properties of the material are progressively degraded when cyclic5

loads below the ultimate material strength are applied [3, 4].

Nowadays, fatigue is determining in the failure of elements and structures

in aerospace, naval and car industries as well as in some structures within the

civil engineering ﬁeld [4, 5]. The variety and complexity of the scenarios where

these failures take place [6, 7, 8, 9, 10, 11, 12, 13, 14] make it necessary to10

develop numerical tools that allow for the correct study and prediction of the

phenomenon. In this paper, a constitutive law based on the continuum damage

mechanics and dedicated to the study of High-Cycle Fatigue (HCF) processes

is presented and used for the study of one of these failure scenarios. HCF is

a speciﬁc typology of fatigue characterized by the large number of cycles that15

lead to failure (105−107cycles) and by the conditions at which it takes place.

This fatigue process occurs without the presence of plastic deformations and

failure takes place in a brittle manner, driven by the coalescence of internal

pores [1, 15]. Therefore, the problem has been historically approached from the

damage mechanics point of view in several ways [4, 16, 17].20

The main objective of the undergone research has been to confront the pre-

sented HCF model towards real railway structures where fatigue degradation is

detected, in particular it is devoted to study the degradation observed at the

wheel-rail interaction region.

The paper is organized in several sections, starting by introducing in Sections25

2 and 3 the problem that motivates the undergone research and the method-

ology applied, respectively. The degradation process, the eﬀect over the rail

components and the implications on the structure maintenance are presented in

Section 2 while Section 3 is dedicated to review the numerical approach, pointing

2

out its main capabilities and introducing the latest developments with respect30

to the original constitutive model [1]. Section 4 focuses on the pre-processing

operations that precede the simulations, i.e., the materials characterization is

described paying special attention to their fatigue response and the built nu-

merical models are presented including a description of the geometry, the ﬁnite

element mesh used and how the train-induced loads are applied. The results35

obtained in the simulations are given in Section 5 and similarities are estab-

lished between them and literature records. Finally, conclusions are drawn for

the conducted research and the capabilities and the limitations of the applied

methodology are stated in order to provide a full review and set the framework

where reliable results can be obtained.40

2. Problem statement

Fatigue problems associated to railway infrastructure have been numerous

through the years and these actually control to a certain degree the mainte-

nance operations at the rail level like grinding or ballast cleaning and replace-

ment due to the unacceptable material degradation caused during the normal45

infrastructure operation [18]. This phenomenon is not exclusive of rails and it is

convenient to distinguish between the regions where these fatigue problems are

detected because, although the origin is similar in all the cases, the correspond-

ing technical and economic consequences are diﬀerent. Fatigue degradation is

detected in vehicle components, mainly wheels but also engines, axles, body50

shells and suspensions, at subrail elements, including the fastening system, slip-

pers and foundation, at surrounding infrastructure [19] and in rails, mostly on

rail heads where contact with wheels takes place. There are many recent pub-

lications devoted to the study of these fatigue processes, mainly studying the

degradation at the train axles and rails [20, 21, 22, 23, 24, 25, 26, 27, 28], but55

also on the bogies, the chassis and the engine [29, 30, 31], indicating the rele-

vance of the topic for the railway community. The current research focuses on

the HCF degradation processes at rail level, where the proposed constitutive law

3

has been used to reproduce the fatigue developed at two points of the railway

path: a straight section and a crossing of a high-speed track.60

2.1. Straight section

Fatigue degradation on these regions can develop at diﬀerent points of the

rail depending on the underlying trigger [18, 32, 33]. Due to the improvements in

the steel manufacturing process, the degradation induced by internal inclusion

or hydrogen shrinkage defects is not an issue anymore and failure is more related65

now with fatigue evolution on railheads, i.e. the rolling area. The predominant

phenomenon is known as rolling contact fatigue (RCF) [32, 34, 35] which is pro-

duced by the normal wheel rolling along the rail. This means that the associated

degradation can potentially appear at any point of the rail track although those

areas with higher stress concentration, e.g. the inter-space between sleepers,70

sections in turns or welded sections, are the ones which concentrate the major

part of the cases. This results in a superﬁcial deterioration at the railhead level

as shown in Figure 1, which can be solved through grinding operations if the

problem is rapidly detected, otherwise the rail needs to be replaced.

(a) (b)

Figure 1: Rolling contact fatigue deterioration reported in bibliography: a) RCF microcracks

reported by Kalousek [36] and b) fatigue degradation detected in the Medell´ın subway reported

by Alarc´on et al. [34].

4

2.2. Crossing element75

Crossings are the elements of the railway system that allow switching the

trains from one rail to another when required. These are composed by two

parts: the ﬁrst one controls the transition between tracks through switch units

that redirect the train movement and the second part takes charge of accom-

modating the train on its transition from one rail to the other. Ensuring the80

smoothness along this transition is not always possible and this can lead to the

fatigue degradation of the metal piece that interacts with the train wheel, the

so called frog nose. The deterioration here is similar to the one observed on

the straight elements, i.e., a spalled region is formed along the frog nose being

more important on those points where the ﬁrst contact with the wheel takes85

place. Figure 2 shows the degradation experienced by a frog nose placed in the

Puertollano high-speed railtrack in Spain.

(a) (b)

Figure 2: Fatigue degradation detected at the frog nose element placed at the Puertollano

line. a) General and b) detail views. Source: CIMNE.

The study undergone for the straight section and the crossing element in this

paper focuses on the behaviour of the rail and the frog nose, respectively, while

trains go through. Therefore, the corresponding numerical simulations intend90

to reproduce the superﬁcial degradation patterns observed along the rail and

exempliﬁed on Figures 1 and 2. The spalling observed on the studied elements

appears on these simulations as a progressive loss of the material strength and

stiﬀness at those regions where fatigue degradation is predicted, driving the

5

material to a fully fatigued-damaged state with fully dissipated fracture energy.95

The studied scenarios have been selected for their simplicity compared with

other areas of the railtrack where fatigue could appear as a second order eﬀect.

Therefore, the study here can be focused on the fatigue phenomenon without

major interactions of other wearing processes. Finally, although at this stage

some simpliﬁcations have been taken into account, e.g., nor thermal eﬀects, nor100

plasticity, nor residual stresses, etc. have been taken into account, the present

research can be considered as a starting point in the direction of building a tool

for the advanced study of fatigue processes at rail level and it can be useful in

the planning of the maintenance campaigns.

3. HCF constitutive law105

The HCF constitutive law presented [1] is an extension of the isotropic dam-

age model proposed by Oliver et al. [37]. The underlying model predicts the

progressive degradation of the material strength and stiﬀness once a certain

stress threshold is reached.

The isotropic damage model is based on the deﬁnition of the damage scalar110

variable d, which accounts for the stiﬀness degradation in the material with

absence of plastic strains. Therefore, the constitutive response of a material is

σ=m0∂Ψ

∂ε= (1 −d)C0:ε=C0:ε

|{z}

predictive stress, σ0

−d·C0:ε

| {z }

damage correction

(1)

where σand εare the stress and strain tensors, respectively, C0is the undam-

aged constitutive tensor, m0is the material density and Ψ is the free Helmholtz

energy which can be formulated in the reference conﬁguration as a function of115

strains and the damage internal variable [38].

Ψ (ε, d) = (1 −d)1

2m0(ε:C0:ε) (2)

This is a direct constitutive model where dis explicitly deﬁned for each stress

level and no iterative process is required during the calculation at the integra-

tion point level. The damage variable takes values [0 −1] from intact to fully

6

damaged, remaining constant, ˙

d= 0, while the yielding criterion is satisﬁed,120

i.e.,

F(σ0) = f(σ0)− K ≤ 0 (3)

where f(σ0) is the equivalent predictive stress and Kthe material threshold

function which is a historical variable of the maximum stress level reached on the

material. The equivalent stress is a uniaxial measure of the tensorial stress state

which depends on the yield surface chosen. For the purpose of this work, the125

Von Mises yield surface has been adopted for the steel elements and equivalent

stresses are computed as

f(σ0) = p3J2(4)

where J2is the second invariant of the stress deviator tensor [38, 39]. Therefore,

the equivalent predictive stress is computed at each integration point of the

structure from the predictive stress σ0and governs the response of the material130

along the simulation.

When Eq. (3) is not satisﬁed, the evolution of the damage internal variable

is controlled through an exponential softening function [37, 40]

d= 1 −f0(σ0)

f(σ0)·exp A·1−f(σ0)

f0(σ0) (5)

where f0(σ0) = K0is the initial equivalent stress threshold and A,

A= gf·E

(f0(σ0))2−1

2!−1

≥0 (6)

is a parameter that guarantees that the available energy for the non-linear pro-135

cess at each Gauss point is equal to the given volumetric fracture energy of

the material, gf=Gf

lc=R∞

t=0 Ψ0˙

d dt, where Eis the Young’s modulus, Gf

is the fracture energy, lcis the characteristic length of the ﬁnite element and

Ψ0=f0(σ0)

2Eis the free energy. Figure 3 schematically reﬂects the main fea-

tures of this constitutive law in a uniaxial stress-strain chart.140

7

Figure 3: Uniaxial strain-stress scheme of the isotropic damage model.

The absence of plastic strains in HCF processes [1, 15] is the key point in

deciding to use the isotropic damage model as the underlying model in the

proposed HCF constitutive law. The main modiﬁcation introduced by this

routine aﬀects the isotropic damage yielding criterion deﬁnition (Eq. (3)), which

is now145

F(σ0) = f(σ0)

fred (Nc, R, Smax,)− K ≤ 0 (7)

where fred is the so called fatigue reduction factor and ranges from 1 to 0. This

parameter takes into account the eﬀect of the acting cyclic load and, conse-

quently, ampliﬁes the stress state depending on the number of cycles applied,

Nc, the reversion factor, R=Smin

Smax

, and the maximum stress generated by the

applied load, Smax. The number of cycles, Nc, along the simulation is updated150

based on the evolution of the equivalent stresses at the integration point level,

i.e., when maximum and minimum values of the stress are detected this indi-

cates that a new cycle has overcome. Therefore, Smax ,Smin and Rvariables

are updated at each new cycle, if necessary.

Figure 4 compares the performance of the HCF constitutive law assuming that155

a value of fred = 0.8 has been reached in the material with the isotropic damage

prediction.

8

0.E+00

2.E+08

4.E+08

6.E+08

8.E+08

1.E+09

0 0.005 0.01 0.015 0.02

Strain [‐]

Damage curve

HCF curve

Figure 4: Behaviour at a Gauss Point level with mechanical properties f0(σ0) = 924M P a,

E= 210GP a and gf= 5MJ/m3and fatigue reduction factor, fred = 0.8.

This example helps to understand the immediate eﬀects of the HCF model

over the material behaviour. On one hand, the damage threshold diminishes to

a new value K∗=K · fred ensuring that non-linear processes start below the160

original yield threshold and, on the other hand, the fracture energy is dissipated

not only through the damage process but also through the fatigue mechanism.

This is captured in Figure 4 as a variation of the enclosed areas at the stress-

strain curves, wmax, i.e.,

Damage: wmax =gf

HCF: wmax =gf·f2

red

(8)

A complete study with the mathematical derivation of the fracture energy dis-165

sipated in the HCF process is included in Appendix A.

The deﬁnition of the fred factor in the model is done in such a way that it

captures the information provided by the widely used S-N curves [41], i.e. pre-

dicts the progressive degradation of the material while a cyclic load is applied,

resulting in the material failure. In general, this failure condition is materialized170

accompanied by a rapid evolution of the damage internal variable, d. Mathe-

matically, this is achieved in this work through an exponential function of the

9

form

fred (Nc, R, Smax) = exp n−B0(R, Smax )·(log10 Nc)β2

fo(9)

being

B0(R, Smax) = ln (Smax/Su)

(log10 Nf)β2

f

(10)

a coeﬃcient of the model, Nfthe number of cycles to reach failure conditions,175

Suthe ultimate strength of the virgin material and βfa material property to be

set according to the S-N curves. Therefore, when a cyclic load induces stresses

greater than the fatigue limit and below the yield stress, the proposed approach

predicts nonlinearities at the corresponding integration points coming from two

sources: a ﬁrst one that induces gradual strength and fracture energy reduction180

due to the evolution of fred variable and a second one, triggered by the ﬁrst

one, once Nc=Nf, where damage propagates and the stiﬀness reduces. In

general, the ﬁrst non-linear process concentrates the major part of the cycles of

the material life while the second non-linear process is rapid and failure takes

place soon due to the brittleness of the material at that point. Both of them185

constitute the whole HCF life and the propagation of this process throughout

the numerical model characterizes the degradation mechanism that control the

global failure of the studied element.

As stated before, the deﬁnition of the fred variable has been done according

to the S-N curves which can be ﬁtted in a surface of the type [1]190

S(R, Nc) = Sth (R)+(Su−Sth (R)) ·exp −αt(R)·(log10 Nc)βf(11)

where Sth (R) is the fatigue limit function

Sth (R) :

|R| ≤ 1⇒Sth (R) = Sth (R=−1) + (Su−Se)·1 + R

2Sth,R1

|R|>1⇒Sth (R) = Sth (R=−1) + (Su−Se)·1 + R

2RSth,R2

(12)

that establishes the minimum stress value below which the cyclic load being

applied does not induce any fatigue eﬀect in the material, i.e., ˙

fred = 0, αt(R)

10

is a parameter that depends on the cyclic load being applied

αt(R) :

|R| ≤ 1⇒αt(R) = αf+1 + R

2·AUX R1

|R|>1⇒αt(R) = αf−1 + R

2R·AUX R2

(13)

and Sth,R1,Sth,R2,αf, AUX R1 and AUX R2 are material properties to be195

calibrated according to the S-N curves.

Through this description of the S-N space, an expression for the Nffunction

can be obtained, imposing that for any load the fred function and the normalized

expression of the S-N curves, S(R, Nc)/Suare equal, i.e.,

fred (Nf, R, Smax) = exp (−ln (1/R)·(log10 Nf)β2

f

(log10 Nf)β2

f)

S(R, Nf)

Su

=Sth (R)

Su

+1−Sth (R)

Su·exp −αt(R)·(log10 Nf)βf

⇒

⇒Nf(R, Smax) = 10

−

1

βf·αt(R)·ln

Smax −Sth (R)

Su−Sth (R)

(14)

Therefore, the derived fatigue formulation predicts the material behaviour while200

the cyclic load is applied and its use requires the calibration of 4 to 6 parameters,

i.e., αf,βf,Sth,R1and/or Sth,R2and AUX R1 and/or AUX R2. Mathematical

meaning can be given to these parameters through Eqs. 9 to 13; αfand βffully

characterize the S-N curve for R=−1.0 while Sth,Ri and AUX Ri with i= 1,2

deﬁne the S-N surface in the remaining Rdomain. In particular, Sth,Ri deﬁnes205

the fatigue limit function, i.e., the slice of the S-N surface for high Ncvalues.

An approach based on the native ﬁtting functions available in Matlab [42]

is used for the calibration of these parameters. These functions are fed with

experimental S-N data to ﬁt the mathematical description of the W¨ohler surface

stated in Eq. 11. The accuracy of this ﬁtting process depends on the amount210

of available experimental results, being important to have data for diﬀerent

Rand Smax values, covering the whole surface domain. In addition to that,

as the underlying constitutive law is dedicated to HCF analysis, experimental

data obtained for that regime should be weighted to ensure a better ﬁtting

11

there. Finally, in those scenarios where the amount of experimental data is215

not suﬃcient for a good calibration in all the domain, e.g., only values for

one particular Rare available, the calibrated parameters can still be used to

simulate fatigue processes on the surrounding regions of these well-ﬁtted areas.

The results of the calibration process done for the materials of interest are shown

in Section 4.220

Figure 5: Normalized W¨ohler curve and fred evolution for a generic material and a generic

load being applied.

Figure 5 helps to understand the relation between the fred and the normal-

ized W¨ohler curve functions showing their evolution with respect to the number

of cycles, Nc. The normalized S-N curve is generic for the reversion factor, R,

of the cyclic load being applied but the fred curve plotted is only valid for the

Smax level considered. From this sketch, the following relations are deduced225

Nc< Nf⇒fred (Nc, R, Smax)> S (R, Nc)/SuFatigue regime

Nc≥Nf⇒fred (Nc, R, Smax)≤S(R, Nc)/SuFatigue+damage regime

(15)

which can be extrapolated to the whole reversion factor space.

Finally, this constitutive model takes advantage of the information used to

12

characterize the fatigue behaviour of metals, i.e., the S-N curves, and predicts

the material behaviour during its life before and after reaching Nf, accommo-

dating if necessary any change in the load that is being applied.230

The use of this approach is destined to HCF cases where no plastic defor-

mations are expected but where a high amount of cycles are required before

reaching failure conditions (105−107cycles). This induces high computational

costs and prevents the use of this model even for small cases. In order to over-

come this issue, the constitutive law works together with an advance in time235

(AIT) strategy that allows to quickly skip load cycles before and after damage

initiation (see Algorithm 1). By using this strategy, the computational costs

signiﬁcantly reduce and the calculation attention is redirected to the non-linear

process. The strategy has been updated since its presentation in Barbu et al. [15]

by including new stabilization and jump criteria that control the AIT calcula-240

tion and by adapting it to allow multi-loaded cases where various cyclic loads are

applied, deﬁning independent cycle and period counters per integration point.

In addition to this, the eﬃciency has been improved by parallelizing not only

the AIT strategy but the whole HCF constitutive law using the OpenMP ap-

plication.245

13

Algorithm 1: Advance In Time (AIT) strategy - basic layout

// Calculation at Gauss Point level, i.

η1,i =

Ri+1 −Ri

Ri+1 // Reversion factor stabilization norm

η2,i =

Si+1

max −Si

max

Si+1

max // Maximum stress stabilization norm

if (di>0) then

damage = true // Non-linear process has started at any GP

end

NFi=Nf,i −Nc,i // Cycles to failure

// Loop for the elements.

if (Piη1,i <tol. ∧Piη2,i <tol.)then

// Stable conditions reached (AIT strategy ON ).

if damage == false then

AIT = min {N Fi} · period // Skipping the linear phase

else

AIT ∼f(∆d∗)// ∆d∗is a user defined variable that

controls the advance in time once damage has

initiated at any GP (see Appendix B).

end

else

// Unstable conditions (AIT strategy OF F )

AIT = 0

end

time =time +AIT // Updating simulation time

This constitutive law is used in this paper to characterize the behaviour

of two types of steels: the pearlitic Grade 900Asteel and the Hadﬁeld steel.

Pearlitic steels are used in the railway industry due to their high strength,

high wear resistance and low cost [43, 44, 45]. In particular, 900A steel is250

characterized by a high fatigue resistance with a high fatigue limit but a poor

fatigue crack growth resistance [43]. This material is used in the construction

14

of UIC 60 proﬁles which are used in those tracks with speed limits >160km/h,

including high-speed rail systems [46]. On the other hand, Hadﬁeld steels are

alloys with a high content of manganese. This component provides them with255

high impact strength [47, 48] and thus these steels have been traditionally used

in railway crossings.

4. Computational model and materials characterization

The analysis of the two regions described in Section 2 has been done through

the Finite Element Method (FEM). Two models have been created in order260

to study the HCF eﬀects that the continuous traﬃc circulation has over the

rails. The geometries and the ﬁnite element meshes have been created using the

pre and post-processor tool GiD [49] and the calculations have been performed

using the open-source code Kratos Multi-physics [50] where the HCF algorithm

presented in Section 3 has been implemented.265

4.1. Straight section

The straight section study has been undergone building the 3.1mlong rail-

way straight section shown in Figure 6. The model includes the main elements

of a standard high-speed railway track, i.e., the rail, the elastomeric pads, the

ties, the ballast and the main sub-structure ground layers. The rail proﬁle cor-270

responds to a UIC 60 deﬁned in the EN 13674-1:2002 [51] and a detailed view

of the modelled rail is shown in Figure 7.

The model has been constrained according to the conﬁning conditions of the

surrounding soil, i.e., limiting the displacements on the bottom of the subsoil

material and on the four sides of the ballast and ground layers, and the same275

hypothesis has been considered to the rail, where the longitudinal displacement

has been blocked on both ends. These boundary conditions are suﬃcient for

the undergone analysis considering the localized eﬀect that the applied loads

introduce in the model and because of the assumed simpliﬁcations (see Section

2).280

15

The mechanical properties used in the simulation to characterize the be-

haviour of the straight section model are included in Table 1. These basic

properties characterize the elastic response of the structure. This has been con-

sidered suﬃcient for all the elements of the structure except for the rail, where

considerably higher stresses are concentrated.

Table 1: Mechanical properties of the materials included in the straight section model.

Element Density [kg/m3]Young modulus [MPa] Poisson

Rail UIC60 7850 210000 0.32

Elastomeric pad 1200 100 0.45

Concrete tie 2200 25000 0.32

Ballast 1390 300 0.27

Gravel 1390 280 0.27

Subsoil 1390 2750 0.27

285

Therefore, these properties have been complemented with the fatigue char-

acterization done for the pearlitic Grade 900A steel used in the rail. This

calibration of the non-linear response has been done by adjusting the control

variables of the model to experimental results, as explained in Section 3. For

this material, the R= 0.1 tensile fatigue tests undergone by Christodoulou290

et al. [43] have been used. The samples tested there were obtained from rail

heads in unused conditions and thus the properties exhibited by the material

are suitable for the study of the rail fatigue behaviour. Despite this, the eﬀect

that the maintenance operations introduce in the rail along its life [52, 53] are

not reﬂected in these original properties and have not been considered in the295

undergone study.

Results of the calibration process are summarized in Table 2 and Figure

8 where the values of the HCF constitutive law control variables and the S-

N curves for the reversion factor spectra R∈[0 −1] are shown. Considering

that the ﬁtting has been done using only values corresponding to R= 0.1,300

the use of this calibrated material should be limited to fatigue cases where the

reversion factor of the applied cyclic loads is close to the experimental one. This

condition is satisﬁed in the undergone analysis as the oscillating load applied in

17

the straight section model has R= 0.0.

Table 2: Pearlitic Grade 900A steel fatigue characterization. HCF model variables for the

spectra R∈[−1; 1].

Variable Value

Su 924 MPa

Sth(R=−1) 369.1 MPa

αf0.0068

βf3.20

Sth,R12.00

AUX R1 0.01

400

500

600

700

800

900

1000

1E+04 1E+05 1E+06 1E+07

Maximum stress [MPa]

Number of cycles, Nc

R 1 . 0

R 0 . 0

Experimental values, R=0.1

Numerical adjustment

Figure 8: Fatigue calibration of the pearlitic Grade 900A steel according to the experimental

results obtained by Christodoulou et al. [43] for R= 0.1. Run-outs points are highlighted

with arrows.

The mechanical rolling over eﬀect of the train is considered by imposing305

a cyclic vertical displacement at the mid-span of the space between ties where

maximum stresses are generated. This displacement has been introduced through

the model of a train wheel as shown in Figure 9 in order to guarantee the correct

force distribution over the rail during the simulation. Other actions like thermal

or chemical eﬀects are out of the scope of this research and only a displacement310

which generates a stress state compatible with fatigue is being considered.

18

(a) (b)

Figure 9: Load application at the straight section model: a) general view and b) section view.

Finally, the ﬁnite element mesh of the model has been built using 8-nodded

hexahedral elements. Higher mesh density has been provided to the rail ele-

ment, particularly to the region between ties where the imposed displacement is

applied as this is the region where fatigue deterioration is expected. Figure 10315

shows this reﬁnement, where 71,400 elements of the total of 142,309 hexahedra

are concentrated.

(a) (b)

Figure 10: Finite element mesh detail of the rail element, a) elevation view of the reﬁned

region between ties where the displacement is imposed and b) section view.

4.2. Crossing element

The crossing element study has been undergone building a 2.1mx 2.0m

model which focuses on the frog nose where fatigue degradation is studied. The320

model includes the main elements of the crossing system, i.e., elastomeric pads,

concrete ties and a 0.3mballast layer as shown in Figure 11. No sub-ballast

19

soil layers have been considered for this simulation due to the high localization

of the phenomenon as will be seen on the results section. The dimensions of all

the elements have been adjusted to the Puertollano line crossing element shown325

in Figure 2.

The corresponding constrictions have been set according to the conﬁning

conditions as in the straight section model however, only the downstream end

of the frog nose has its longitudinal displacement blocked, while the tip is free.

These boundary conditions are again suﬃcient for the crossing element model330

due to the localization of the applied loads and because of the assumed simpli-

ﬁcations (see Section 2).

1.3m

2.1m

2.0m

Ballast

Ties

Frog nose

Elastomeric pads

0.3m

Figure 11: Crossing element numerical model. Main parts and dimensions

The same basic mechanical properties used for the straight section model and

shown in Table 1 are used to characterize the materials of the crossing element

model and a new calibration of the HCF constitutive law has been done for the335

manganese steel present in the frog nose. The R= 0.1 bending fatigue results

obtained by Kang et al. [48] have been used to do this fatigue characterization,

where the diﬀerences between tension and bending fatigue [54, 55] have been

taken into account by deﬁning a R= 0.1 S-N curve which predicts earlier

failure than the ones obtained through the bending experiments. Results of340

this calibration process are summarized in Table 3 and Figure 12 where the

20

values of the HCF constitutive law control variables and the S-N curves for the

reversion factor spectra R∈[0 −1] are shown. As for the straight section case,

the use of this calibrated material should be limited to fatigue cases where the

applied cyclic loads have reversion factors close to the experimental one, i.e.,345

R= 0.1. This condition is satisﬁed in the undergone analysis as the oscillating

load considered in the frog nose model has R= 0.0.

Table 3: Hadﬁeld steel fatigue characterization. HCF model variables for the spectra R∈

[−1; 1].

Variable Value

Density 7850 kg/m3

Young modulus 210000 MPa

Poisson 0.30

Su 1100 MPa

Sth(R=−1) 300 MPa

αf0.0008

βf4.46

Sth,R11.58

AUX R1 -0.00032

500

600

700

800

900

1000

1100

1200

1E+04 1E+05 1E+06 1E+07

Maximum stress [MPa]

Number of cycles, Nc

R 1 . 0

R 0 . 0

Experimental values, R=0.1

Numerical adjustment

Figure 12: Fatigue calibration of the Hadﬁeld steel used for the frog nose according to the

experimental results obtained by Kang et al. [48] for R= 0.1. Run-outs points are highlighted

with arrows.

The interaction between the wheel and the frog nose has proven to take

21

place at a distance of 0.47mfrom the tip and it is at this point where the

cyclic action has been applied. The application of this external displacement350

has been performed analogously to the straight section, as shown in Figure 13,

and neither thermal nor chemical eﬀects have been considered in this case.

(a) (b)

Figure 13: Load application at the crossing element model: a) general view and b) section

view.

Finally, the ﬁnite element mesh of the model has been build using 8-nodded

hexahedral elements. Higher mesh density has been provided to the frog nose

element where fatigue study is intended, particularly to the section located at355

0.47mfrom the tip where the wheel-rail interaction takes place. Figure 10 shows

this reﬁned region, where 47,648 elements of the total of 82,686 hexahedra are

concentrated.

22

(a)

(b) (c)

Figure 14: Finite element mesh detail of the frog nose, a) general of the frog nose reﬁned

region where the displacement is imposed, b) section view and c) frontal view of the frog nose

ﬁnite element mesh.

5. Results

In this section the results obtained for the previous scenarios are presented.360

A preliminary control case has been included where one of the tensile experi-

ments used on the fatigue characterization of the pearlitic steel is reproduced.

The main features of the HCF constitutive law are reviewed through this ini-

23

tial case where result are already known. Figure 15 shows the model built and

an overall view of the ﬁnite element mesh used. Displacements in one end of365

the sample have been blocked while on the opposite end a cyclic displacement

has been imposed to reproduce the experiment conditions of a normalized axial

fatigue test according to ASTM E466 [56].

(a)

(b)

Figure 15: Model of the samples used in the tensile fatigue experiments undergone by

Christodoulou et al. [43]: a) geometry with main dimensions and b) ﬁnite element mesh

made of 29,051 8-nodded hexahedra.

On this simulation stresses oscillate between 55.9−559MPa. For this stress

rate, damage initiates after 318,663 cycles and rapidly evolves and propagates.370

Damage is triggered by the decrease of the fred variable on those areas where

stress exceeds the fatigue limit. This susceptible to fatigue region extends

throughout the entire narrowed section as shown in Figure 16, however, two

rings showing higher degradation level are recognizable and set the starting

point for damage propagation.375

24

Figure 16: Fatigue reduction factor distribution on the sample when damage is about to start,

NC= 318,663.

The fracture energy exhaustion caused by the fatigue process generates a

fragile fracture, which induces a rapid evolution and propagation of the damage

internal variable, as shown in Figure 17. Damage propagates symmetrically on

the sample, starting superﬁcially but evolving to the centre and ﬁnally creat-

ing a full damaged plane across the sample after 1,026 cycles. Note that these380

318 −319kcycles to failure are on the same range that the ones obtained ex-

perimentally and shown in Figure 8 when a R= 0.1 cyclic load with maximum

stress ≃560MPa is applied.

Simulation course

Figure 17: Damage internal variable, d, evolution along the simulation, from NC= 318,663

to 319,689.

Now the results for the straight section and the crossing element are pre-

sented. The introduced action is an oscillating vertical displacement (R= 0.0)385

at the rail and the frog nose heads, as stated previously. The way that this

external action has been introduced reproduces the physics of the problem but

25

the magnitude imposed has been deﬁned pursuing the main objective of the

paper, i.e., ensuring the fatigue degradation of the structure and studying it

through the HCF constitutive law presented.390

Firstly, the resultant tensile stress distribution obtained by the applied load

is presented for both scenarios. These stresses are the ones that induce fatigue

on the structure if they surpass the corresponding fatigue limit, Sth , and are

pointed out in Figure 18. Some of these areas match with the aﬀected regions

gathered in literature [33, 34], e.g., the surroundings of the region in contact395

with the train wheel, the bottom of the rail head or the union between the web

and the foot. This fact points out the suitability of the built cases to study the

phenomenon of interest.

(a)

(b)

Figure 18: Rail regions working under tension in a) the straight section model and b) the

crossing element model.

Maximum stresses are observed on the rail head surface of both models,

which will lead to superﬁcial damage as the one shown in Figures 1 and 2.400

The equivalent stress distribution in this area is complex; it is characterized by

26

a noticeable compressed bulb as consequence of the external load that rapidly

dissipates and in the vicinity of this area, small tensed regions appear induced by

the shear components of the stress. This behaviour can be observed in Figure

19 where equivalent stresses are plotted for the instant when displacement is405

maximum in absolute value. The maximum tensile stress obtained in each case

surpasses the corresponding fatigue limit, i.e., Sth(R= 0) = 508.20MPa in the

pearlitic steel and Sth(R= 0) = 568.51MPa in the manganese steel, but this is

exclusive of these superior regions where fatigue degradation will concentrate.

Despite this, this can change once damage starts and redistribution on the stress410

ﬁeld takes place, potentially creating new fatigued areas.

(a) (b)

Figure 19: Equivalent stress distribution on for the instant when maximum load is being

applied a) on the rail head of the straight section model and b) on the frog nose head.

Figure 20 shows the damage distribution on the straight section and the

crossing element models after 264,408 and 2,562,626 cycles, respectively. Be-

yond this point the degradation stabilizes for the current cases. Despite the

simpliﬁcations considered for these simulations, the type of degradation ob-415

served can explain some of the deterioration patterns registered in literature,

reﬂecting the potential of the applied formulation for the study of the HCF

phenomenon.

27

(a) (b)

Figure 20: Damage internal variable distribution by the end of the simulation on a) the

straight section model and b) the crossing element model.

Finally, the behaviour of the model can be checked by plotting the evolution

of the problem variables along the simulations as shown in Figure 21. These420

charts are built using the information from the most stressed integration point

in each model. The stress curve intersects the S-N curve at a point quite close to

the corresponding fatigue limits, which explains the small degradation observed

in Figure 20. Obtaining a generalized fatigued area through the simulation

would be possible when higher loads take place or by considering the passing-425

through eﬀect of the train along the rail. Despite this, damage initiation and its

rapid evolution due to the fracture energy dissipation motivated by the cyclic

load, can be observed in both scenarios once the S-N curve is reached, i.e., after

the corresponding Nfcycles computed as a function of the maximum stress,

Smax and the reversion factor, R, as stated in Section 3. The swap between the430

normal advancing mode and the AIT strategy is also noticeable in both cases,

allowing to take the problem right to the beginning of the damage non-linearities

after only 3 cycles. This reﬂects the amount of computational-time saved by

the use of this technology.

28

100102104106

Number of cycles, Nc(log. scale)

0

0.2

0.4

0.6

0.8

1

Normalized stress, f( ) / Su

Normalized Wöhler curve, S(R = 0,Nc) / S u

Reduction factor, fred

Damage internal variable, d

(a)

100102104106

Number of cycles, Nc(log. scale)

0

0.2

0.4

0.6

0.8

1

Normalized stress, f( ) / Su

Normalized Wöhler curve, S(R = 0,Nc) / S u

Reduction factor, fred

Damage internal variable, d

(b)

Figure 21: Evolution of the internal variables of the problem on a) the straight section model

and b) the crossing element model.

6. Conclusions435

In this research, an isotropic damage based HCF constitutive law has been

presented and its performance has been shown through several examples. Al-

though the approach was introduced on previous publications, latest improve-

ments as well as newly relevant demonstrations for the constitutive law are pre-

sented. Among others, the code parallelization, the enhancement of the AIT440

strategy and the quantiﬁcation of the dissipated fracture energy along with the

29

fatigue process are introduced in the document. In addition to that, the appli-

cation of the methodology into the railway ﬁeld has been studied for the ﬁrst

time and the arisen strengths and weakness can be pointed out.

Even though several simpliﬁcations have been considered for the studied445

problems, interesting results have been obtained which highlight the main fea-

tures of the applied methodology. A complete study of the fatigue phenomenon

historically observed on railway industry should include among others: plas-

tic behaviour of materials, residual stresses at rails, thermal actions, stochastic

imperfections distribution on materials that are responsible of some speciﬁc fail-450

ures, time eﬀects like delayed settlements induced by ballast degradation, eﬀect

of the aluminothermic welds on rails, etc. which were out of the scope of the

undergone research. On one hand, some of these can be indirectly addressed

through the used approach but a wider material characterization would be then

required. This is the case of these phenomena that induce a preexisting quan-455

tiﬁable degradation in the material and reduce their fatigue performance, like

the presence of residual stresses. On the other hand, the inclusion of other phe-

nomena necessary depends on the extension of the presented approach which

constitutes future lines of work. These include: switching from a determinis-

tic to a stochastic approach in the fatigue analysis by aﬀecting the calibration460

process followed in the HCF characterization, or extending the methodology

to address coupled thermo-mechanical problems which will increase the added

value of the presented methodology on the study of fatigue phenomenon for

railway elements.

Despite this, at this moment the model is appropriate for the study of a large465

number of scenarios governed by the HCF phenomenon, allowing to eﬃciently

predict the initiation and propagation of the damage on the model while loads

are applied, independently of their typology, i.e., uniform or cyclic and the

uniformity, i.e., R̸=const. Using the presented approach, the rail regions

reported in literature as susceptible to fatigue degradation have been identiﬁed,470

the degradation observed at the rail head and the frog nose has been simulated

and the diﬀered material degradation and the brittle fracture that characterize

30

the physics of the problem have been captured.

7. Acknowledgments

This work has been done within the framework of the RESILTRACK (IDI-475

20171003) project: resilencia de infraestructuras ferroviarias frente a cambio

clim´atico. This project has received funding from the Spanish Government. The

work has been also supported by the Spanish Government program FPU17/04196.

The authors gratefully acknowledge all the received support.

Appendix A. Fracture energy dissipation in the HCF process480

Here it is derived the expression for the variation of the available fracture

energy when the HCF constitutive law is used. The expression for the total

dissipated energy in the isotropic damage constitutive law [38] is

wmax

t=Z∞

t=0

Ξdt =Z∞

t=0

Ψ0˙

d dt =Z¯σmax

¯σ0

[¯σ]2

2·E

|{z}

free energy

∂ˆ

G(¯σ)

∂¯σd¯σ=gf(A.1)

where ¯σ≡f(σ0) is the predictive equivalent stress, ¯σ0≡f0(σ0) is the yield

stress, ¯σmax ≡fmax (σ0) is the predictive stress level when the fracture energy485

has been fully dissipated, Eis the Young modulus and ˆ

G≡dis the damage

internal variable function. The result, wmax

t=gfis independent of the deﬁni-

tion chosen for ˆ

G. Two cases are studied here: the linear and the exponential

descriptions of the damage.

Linear case: d=1−¯σ0

¯σ·1

1−A∧A=¯σ02

2gfE

Exponential case: d= 1 −¯σ0

¯σ·exp hA·1−¯σ

¯σ0i ∧A=1

gfE

(¯σ0)2−1

2

(A.2)

The ¯σmax value reached in each scenario is ¯σmax

lin. =¯σ0

Afor the linear description490

and ¯σmax

exp. = +∞for the exponential case. These stress values are obtained from

Eq. A.2 when d= 1.0.

31

The extrapolation to the HCF case is done by aﬀecting Eqs. A.1 and A.2

with fred. This function results in a reduction of the normal yield surface of the

underlying damage model as a consequence of the fatigue process. Finally,495

Linear:

wmax, HC F

t=Z∞

t=0

Ξdt =Z∞

t=0

Ψ0˙

d dt =Z¯σ0·fred /A

¯σ0·fred

[¯σ]2

2E

∂ˆ

G(¯σ)

∂¯σd¯σ=

=Z¯σ0·fred /A

¯σ0·fred

[¯σ]2

2E

¯σ0·fred

[¯σ]2(1 −A)d¯σ=¯σ·¯σ0·fred

2E·(1 −A)

¯σ0·fred /A

¯σ0·fred

=

=¯σ02·f2

red

2EA =gf·f2

red

(A.3)

Exponential:

wmax, HC F

t=Z∞

t=0

Ξdt =Z∞

t=0

Ψ0˙

d dt =Z+∞

¯σ0·fred

[¯σ]2

2E

∂ˆ

G(¯σ)

∂¯σd¯σ=

=Z+∞

¯σ0·fred

[¯σ]2

2E

A

¯σ+fred ¯σ0

[¯σ]2·exp A·1−¯σ

fred ¯σ0d¯σ=

=−

fred ¯σ02·fred ¯σ0+A¯σ·exp A·(1 −¯σ)

fred ¯σ0

2AE

+∞

¯σ0·fred

=

=f2

red ¯σ02(A+ 2)

2AE =gf·f2

red

(A.4)

Appendix B. Computing the advance in time (AIT) once damage

has initiated

In order to smartly advance once the internal damage variable has initiated,

i.e., Nfhas been reached at some GP, the user deﬁnes the ∆d∗-parameter which500

controls the AIT. The time increment that analytically induces a variation of

the damage internal variable ∆d= ∆d∗is computed through this input. The

relation between the time increment and the variation of the damage internal

variable is obtained using dand fred deﬁnitions (Eq. (5), (9)) and that the

stress state in the HCF regime is f(σ0)

fred

.505

32

The damage:

Linear case

d=1−fyield

f(σ)·1

1−A⇒fred = [1 −d·(1 + A)] f(σ0)

f0(σ0)(B.1)

Exponential case

d= 1 −f0(σ0)·fred

f(σ0)·exp A·1−f(σ0)

f0(σ0)·fred

⇒(1 −d)·f(σ0)

f0(σ0)·fred

= exp A·1−f(σ0)

f0(σ0)·fred

⇒ln (1 −d)·f(σ0)

f0(σ0)+ ln 1

fred =A·1−f(σ0)

f0(σ0)·fred

⇒A·f(σ0)

f0(σ0)·fred

+ ln 1

fred =A−ln (1 −d)·f(σ0)

f0(σ0)

⇒exp A·f(σ0)

f0(σ0)·fred ·1

fred

=exp (A)·f0(σ0)

(1 −d)·f(σ0)

⇒fred =A·f(σ0)

f0(σ0)·1

ωA·exp (A)

1−d

(B.2)

Where ω(·) = P∞

n=1

(−n)n−1

n!(·)nis the Lambert function. And the fatigue

reduction factor:510

fred = exp n−B0·(log10 Nc)β2

fo⇒Nc= 10

−

ln (fred)

B0

1/β2

f

(B.3)

Therefore,

Linear case: ∆Nc= 10

−

ln [1 −∆d∗·(1 + A)] ·f(σ0)

f0(σ0)

B0

1/β2

f

Exponential case: ∆Nc= 10

−

ln

A·f(σ0)

f0(σ0)·1

ωA·exp (A)

1−∆d∗

B0

1/β2

f

(B.4)

33

References

[1] S. Oller, O. Salom´on, and E. O˜nate. A continuum mechanics model for

mechanical fatigue analysis. Computational Materials Science, 32(2):175–

195, 2005. ISSN 0927-0256. doi: https://DOI.org/10.1016/j.commatsci.515

2004.08.001.

[2] H. Zenner and K. Hinkelmann. August W¨ohler – founder of fatigue strength

research. Steel Construction, 12(2):156–162, 2019. doi: https://DOI.org/

10.1002/stco.201900011.

[3] ASTM E1823-13. Standard terminology relating to fatigue and fracture520

testing. Technical report, ASTM International, West Conshohocken, PA,

2013. URL www.astm.org.

[4] L.G Barbu. Numerical simulation of fatigue processes. Application to steel

and composite structures. PhD thesis, Universitat Polit`ecnica de Catalunya

(UPC), Spain, 2015. URL http://hdl.handle.net/10803/386479.525

[5] R.I. Stephens, A. Fatemi, R. Stephens, and H.O. Fuchs. Metal Fatigue in

Engineering. 01 2000.

[6] Y. Bai and W.L. Jin. Part 3: Fatigue and fracture. In Marine Structural

Design (Second Edition), pages 477–578. Butterworth-Heinemann, Oxford,

second edition edition, 2016. ISBN 978-0-08-099997-5. doi: https://DOI.530

org/10.1016/B978-0-08-099997-5.00025-3.

[7] L.G. Barbu, X. Martinez, S. Oller, and A.H. Barbat. Validation on large

scale tests of a new hardening–softening law for the barcelona plastic dam-

age model. International Journal of Fatigue, 81:213–226, 2015. ISSN 0142-

1123. doi: https://DOI.org/10.1016/j.ijfatigue.2015.07.031.535

[8] L.G. Barbu, S. Oller, X. Martinez, and A.H. Barbat. High-cycle fa-

tigue constitutive model and a load-advance strategy for the analysis of

unidirectional ﬁber reinforced composites subjected to longitudinal loads.

34

Composite Structures, 220:622–641, 2019. ISSN 0263-8223. doi: https:

//DOI.org/10.1016/j.compstruct.2019.04.015.540

[9] H. E. Boyer. Atlas of Fatigue Curves. ASM International, 1986. ISBN

978-0-87170-214-2.

[10] J.G. Kaufman. Properties of aluminum alloys: Tensile, creep, and fatigue

data at high and low temperatures. 1 1999. URL https://www.osti.gov/

biblio/6305846.545

[11] X. Martinez, S. Oller, L.G. Barbu, A.H. Barbat, and A.M.P. de Jesus.

Analysis of ultra low cycle fatigue problems with the barcelona plastic

damage model and a new isotropic hardening law. International Journal

of Fatigue, 73:132–142, 2015. ISSN 0142-1123. doi: https://DOI.org/10.

1016/j.ijfatigue.2014.11.013.550

[12] C. Nagel, A. Sondag, and M. Brede. 4 - Designing adhesively bonded

joints for wind turbines. In Adhesives in Marine Engineering, Woodhead

Publishing Series in Welding and Other Joining Technologies, pages 46–71.

Woodhead Publishing, 2012. ISBN 978-1-84569-452-4. doi: https://DOI.

org/10.1533/9780857096159.1.46.555

[13] L. Susmel. High-cycle fatigue of notched plain concrete. Procedia Structural

Integrity, 1:2–9, 2016. ISSN 2452-3216. doi: https://DOI.org/10.1016/j.

prostr.2016.02.002.

[14] R. Talreja and C.V. Singh. Damage and Failure of Composite Materials.

Cambridge University Press, 2012. ISBN 9781139016063. doi: https://560

DOI.org/10.1017/CBO9781139016063.

[15] L.G. Barbu, S. Oller, X. Martinez, and A. Barbat. High cycle fatigue simu-

lation: A new stepwise load-advancing strategy. Engineering Structures, 97:

118–129, 2015. ISSN 0141-0296. doi: https://DOI.org/10.1016/j.engstruct.

2015.04.012.565

35

[16] R. Alessi, S. Vidoli, and L. De Lorenzis. A phenomenological approach

to fatigue with a variational phase-ﬁeld model: The one-dimensional case.

Engineering Fracture Mechanics, 190:53–73, 2018. ISSN 0013-7944. doi:

https://DOI.org/10.1016/j.engfracmech.2017.11.036.

[17] P. Carrara, M. Ambati, R. Alessi, and L. De Lorenzis. A framework to570

model the fatigue behavior of brittle materials based on a variational phase-

ﬁeld approach. Computer Methods in Applied Mechanics and Engineering,

361:112731, 2020. ISSN 0045-7825. doi: https://DOI.org/10.1016/j.cma.

2019.112731.

[18] L. Lesley. 2 - Fatigue in railway and tramway track. In Fatigue in Rail-575

way Infrastructure, pages 20–57. Woodhead Publishing, 2009. ISBN 978-

1-85573-740-2. doi: https://DOI.org/10.1533/9781845697020.20.

[19] P.M.R. Lewis and K. Reynolds. Forensic engineering: a reappraisal of

the Tay Bridge disaster. Interdisciplinary Science Reviews, 27(4):287–298,

2002. doi: 10.1179/030801802225005725.580

[20] Y. Hu, S. Wu, P.J. Withers, H. Cao, P. Chen, Y. Zhang, Z. Shen, T. Vo-

jtek, and P. Hutaˇr. Corrosion fatigue lifetime assessment of high-speed

railway axle ea4t steel with artiﬁcial scratch. Engineering Fracture Me-

chanics, 245:107588, 2021. ISSN 0013-7944. doi: https://doi.org/10.1016/

j.engfracmech.2021.107588.585

[21] T. Vojtek, P. Pokorn´y, I. Kubˇena, L. N´ahl´ık, R. Fajkoˇs, and P. Hutaˇr.

Quantitative dependence of oxide-induced crack closure on air humidity

for railway axle steel. International Journal of Fatigue, 123:213–224, 2019.

ISSN 0142-1123. doi: https://doi.org/10.1016/j.ijfatigue.2019.02.019.

[22] S.C. Wu, Z.W. Xu, G.Z. Kang, and W.F. He. Probabilistic fatigue as-590

sessment for high-speed railway axles due to foreign object damages. In-

ternational Journal of Fatigue, 117:90–100, 2018. ISSN 0142-1123. doi:

https://doi.org/10.1016/j.ijfatigue.2018.08.011.

36

[23] D. Regazzi, S. Beretta, and M. Carboni. An investigation about the in-

ﬂuence of deep rolling on fatigue crack growth in railway axles made of595

a medium strength steel. Engineering Fracture Mechanics, 131:587–601,

2014. ISSN 0013-7944. doi: https://doi.org/10.1016/j.engfracmech.2014.

09.016.

[24] H.M. El-sayed, M. Lotfy, H.N. El-din Zohny, and H.S. Riad. Prediction of

fatigue crack initiation life in railheads using ﬁnite element analysis. Ain600

Shams Engineering Journal, 9(4):2329–2342, 2018. ISSN 2090-4479. doi:

https://doi.org/10.1016/j.asej.2017.06.003.

[25] R.M. Nejad, K. Farhangdoost, and M. Shariati. Three-dimensional sim-

ulation of rolling contact fatigue crack growth in UIC60 rails. Tribology

Transactions, 59(6):1059–1069, 2016. doi: 10.1080/10402004.2015.1134738.605

[26] P. Gurubaran, M. Afendi, M.A. Nur Fareisha, M.S. Abdul Majid, I. Haftir-

man, and M.T.A. Rahman. Fatigue life investigation of UIC 54 rail proﬁle

for high speed rail. 908:012026, oct 2017. doi: 10.1088/1742-6596/908/1/

012026.

[27] Z. Popovi´c and V. Radovi´c. Rolling contact fatigue of rails. Belgrade, 04610

2016. Conference: The III International Scientiﬁc and Professional Confer-

ence. ”CORRIDOR 10 – A sustainable way of integrations“.

[28] L. Xin, V.L. Markine, and I.Y. Shevtsov. Numerical procedure for fatigue

life prediction for railway turnout crossings using explicit ﬁnite element

approach. Wear, 366-367:167–179, 2016. ISSN 0043-1648. doi: https:615

//doi.org/10.1016/j.wear.2016.04.016. Contact Mechanics and Wear of Rail

/ Wheel Systems, CM2015, August 2015.

[29] J.W. Seo, H.M. Hur, H.K. Jun, S.J. Kwon, and D.H. Lee. Fatigue design

evaluation of railway bogie with full-scale fatigue test. Advances in Materi-

als Science and Engineering, page 11, 2017. doi: https://doi.org/10.1155/620

2017/5656497.

37

[30] A. Ekberg and E. Kabo. Risk of fatigue of train car chassis due to pressure

waves between meeting trains. Technical report, Chalmers University of

Technology, 2020.

[31] A. Ktari, N. Haddar, and H.F. Ayedi. Fatigue fracture expertise of train625

engine crankshafts. Engineering Failure Analysis, 18(3):1085–1093, 2011.

ISSN 1350-6307. doi: https://doi.org/10.1016/j.engfailanal.2011.02.007.

[32] R.A. Smith. 1 - Fatigue and the railways: an overview. In Fatigue in

Railway Infrastructure, pages 1–19. Woodhead Publishing, 2009. ISBN

978-1-85573-740-2. doi: https://DOI.org/10.1533/9781845697020.1.630

[33] S. Tamargo, J.M. Duart, and J.I. Verdeja. El carril. Soldabilidad y fatiga.

In III Congreso de Historia Ferroviaria. Fundaci´on de los Ferrocarriles

Espa˜noles, 2003.

[34] G.I. Alarc´on, J. Sanchez, J. Santa, and A. Toro. Identiﬁcaci´on de mecan-

ismos de desgaste en rieles de v´ıa comercial del metro de Medell´ın. Revista635

Colombiana de Materiales, pages 72–77, may 2014. ISSN 2256-1013.

[35] E. Magel. Rolling contact fatigue: a comprehensive review. NRC Publi-

cations Archive. U.S. Department of Transportation. Federal Railroad Ad-

ministration, 2011. doi: 10.4224/23000318.

[36] J Kalousek. Keynote address: light to heavy, snail to rocket. Wear, 253(1):640

1–8, 2002. ISSN 0043-1648. doi: https://DOI.org/10.1016/S0043-1648(02)

00076-5.

[37] J. Oliver, M. Cervera, S. Oller, and J. Lubliner. Isotropic damage models

and smeared crack analysis of concrete. In Second International Conference

on Computer Aided Analysis and Design of Concrete Structures, volume 2,645

pages 945–958, 1990.

[38] S. Oller. Nonlinear dynamics of structures. 01 2014. doi: 10.1007/

978-3-319-05194-9.

38

[39] E.S. Neto, D. Peric, and D.R.J. Owen. Computational methods for plastic-

ity. ISBN: 978-0-470-69452-7. Wiley, 2008.650

[40] S. Oller, A.H. Barbat, E. O˜nate, and A. Hanganu. A damage model for the

seismic analysis of building structures. In Congreso Mundial de Ingenier´ıa

Antis´ısmica, Madrid, 1992.

[41] C. Li, S. Wu, J. Zhang, L. Xie, and Y. Zhang. Determination of the

fatigue P-S-N curves – A critical review and improved backward statistical655

inference method. International Journal of Fatigue, 139, 2020. ISSN 0142-

1123. doi: https://DOI.org/10.1016/j.ijfatigue.2020.105789.

[42] MATLAB. version 9.7.0 (R2019b). The MathWorks Inc., Natick, Mas-

sachusetts, 2019.

[43] P.I. Christodoulou, A.T. Kermanidis, and G.N. Haidemenopoulos. Fatigue660

and fracture behavior of pearlitic Grade 900A steel used in railway appli-

cations. Theoretical and Applied Fracture Mechanics, 83:51–59, 2016. ISSN

0167-8442. doi: https://DOI.org/10.1016/j.tafmec.2015.12.017. ICEAF-IV

Engineering Against Failure.

[44] J. M. Duart, J. A. Pero-Sanz, and J. I. Verdeja. Carriles para alta velocidad.665

comportamiento en fatiga. Revista de Metalurgia, 41(1):66–72, 02 2005. doi:

10.3989/revmetalm.2005.v41.i1.188.

[45] S. Maya Johnson, A. Ramirez, and A. Toro. Fatigue crack growth rate of

two pearlitic rail steels. Engineering Fracture Mechanics, 138, 03 2015. doi:

10.1016/j.engfracmech.2015.03.023.670

[46] A. L´opez. Infraestructuras ferroviarias. Barcelona : Edicions UPC, 2010,

2010. ISBN 978-84-9880-415-7.

[47] S. Bhattacharyya. A friction and wear study of hadﬁeld manganese steel.

Wear, 9(6):451–461, 1966. ISSN 0043-1648. doi: https://DOI.org/10.1016/

0043-1648(66)90136-0.675

39

[48] J. Kang, F.C. Zhang, X.Y. Long, and B. Lv. Cyclic deformation and fatigue

behaviors of Hadﬁeld manganese steel. Materials Science and Engineering:

A, 591:59–68, 2014. ISSN 0921-5093. doi: https://DOI.org/10.1016/j.msea.

2013.10.072.

[49] R. Rib´o, M. Pasenau, E. Escolano, J.S. Ronda, and A. Coll. GiD user680

manual. CIMNE, 2007.

[50] P. Dadvand, R. Rossi, and E. O˜nate. An object-oriented environment for

developing ﬁnite element codes for multi-disciplinary applications. Archives

of Computational Methods in Engineering, 17:253–297, 2010. ISSN 1886-

1784. doi: https://DOI.org/10.1007/s11831-010-9045-2.685

[51] EN 13674-1:2002. Railway applications - Track - Rail - Part 1: Vignole

railway rails 46kg/m and above. Standard, European Committee for Stan-

dardization, Brussels, November 2002.

[52] C. Zhao, J. Li, W. Fan, Y. Liu, and W. Wang. Experimental and simulation

research on residual stress for abrasive belt rail grinding. The International690

Journal of Advanced Manufacturing Technology, 109:129–142, 2020. ISSN

1433-3015. doi: 10.1007/s00170-020-05664-5.

[53] L.M. Huang, H.H. Ding, S.Y. Zhang, K. Zhou, J. Guo, Q.Y. Liu, and W.J.

Wang. Simulation research on temperature ﬁeld and stress ﬁeld during

rail grinding. Proceedings of the Institution of Mechanical Engineers, Part695

F: Journal of Rail and Rapid Transit, pages 1–13, 2021. doi: 10.1177/

0954409720984568.

[54] T. Hassan and Z. Liu. On the diﬀerence of fatigue strengths from rotating

bending, four-point bending, and cantilever bending tests. International

Journal of Pressure Vessels and Piping, 78(1):19–30, 2001. ISSN 0308-700

0161. doi: https://DOI.org/10.1016/S0308-0161(00)00080-6.

[55] S. S. Manson. Fatigue: A complex subject—Some simple approximations.

40

Experimental Mechanics, 5:193–226, 1965. ISSN 1741-2765. doi: https:

//DOI.org/10.1007/BF02321056.

[56] ASTM E466-15. Standard practice for conducting force controlled constant705

amplitude axial fatigue tests of metallic materials. Technical report, ASTM

International, West Conshohocken, PA, 2015. URL www.astm.org.

41