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Abstract

The fatigue phenomenon has been historically related to the railway industry. Nowadays, thanks to the high quality of the materials used, most of the recorded problems have been overcome although there are still degradation processes 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 mechanisms are experimentally detected: a straight section and a crossing element. The analysis of the affected areas is performed through a finite element simulation 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.
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 (105107cycles) 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=m0Ψ
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·1f(σ0)
f0(σ0) (5)
where f0(σ0) = K0is the initial equivalent stress threshold and A,
A= gf·E
(f0(σ0))21
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 nB0(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)+(SuSth (R)) ·exp αt(R)·(log10 Nc)βf(11)
where Sth (R) is the fatigue limit function
Sth (R) :
|R| ≤ 1Sth (R) = Sth (R=1) + (SuSe)·1 + R
2Sth,R1
|R|>1Sth (R) = Sth (R=1) + (SuSe)·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| ≤ 1αt(R) = αf+1 + R
2·AUX R1
|R|>1αt(R) = αf1 + 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
+1Sth (R)
Su·exp αt(R)·(log10 Nf)βf
Nf(R, Smax) = 10
1
βf·αt(R)·ln
Smax Sth (R)
SuSth (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< Nffred (Nc, R, Smax)> S (R, Nc)/SuFatigue regime
NcNffred (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 (105107cycles). 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)// dis 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
Ballast
Gravel
Subsoil
Ties
Rail
2.3m
3.1m
1.8m
1.0m
1.3m
0.3m
Elastomeric pads
Figure 6: Numerical model of the straight railway section. Main elements and dimensions.
(a)
16.5
150
72
172
(b)
Figure 7: UIC 60 profile view, a) EN standard scheme [51] and b) model detail. Units in
[mm].
16
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 55.9559MPa. 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
Gσ)
¯σd¯σ=gf(A.1)
where ¯σf(σ0) is the predictive equivalent stress, ¯σ0f0(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
Gdis 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
1AA=¯σ02
2gfE
Exponential case: d= 1 ¯σ0
¯σ·exp hA·1¯σ
¯σ0i A=1
gfE
σ0)21
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
Gσ)
¯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
Gσ)
¯σd¯σ=
=Z+
¯σ0·fred
σ]2
2E
A
¯σ+fred ¯σ0
σ]2·exp A·1¯σ
fred ¯σ0d¯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= ∆dis 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=1fyield
f(σ)·1
1Afred = [1 d·(1 + A)] f(σ0)
f0(σ0)(B.1)
Exponential case
d= 1 f0(σ0)·fred
f(σ0)·exp A·1f(σ0)
f0(σ0)·fred 
(1 d)·f(σ0)
f0(σ0)·fred
= exp A·1f(σ0)
f0(σ0)·fred 
ln (1 d)·f(σ0)
f0(σ0)+ ln 1
fred =A·1f(σ0)
f0(σ0)·fred
A·f(σ0)
f0(σ0)·fred
+ ln 1
fred =Aln (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)
1d
(B.2)
Where ω(·) = P
n=1
(n)n1
n!(·)nis the Lambert function. And the fatigue
reduction factor:510
fred = exp nB0·(log10 Nc)β2
foNc= 10
ln (fred)
B0
12
f
(B.3)
Therefore,
Linear case: Nc= 10
ln [1 d·(1 + A)] ·f(σ0)
f0(σ0)
B0
12
f
Exponential case: Nc= 10
ln
A·f(σ0)
f0(σ0)·1
ωA·exp (A)
1d
B0
12
f
(B.4)
33
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