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Division of Material and Computational Mechanics
Department of Applied Mechanics
Johan SandstrmCHARMEC
Probability of subsurface fatigue
initiation in rolling contact
Johan Sandstrm
Anders EkbergJacques de Mar
Department of Applied Mechanics, Department of Mathematical Sciences
Chalmers University of Technology
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Division of Material and Computational Mechanics
Department of Applied Mechanics
Johan SandstrmCHARMEC
Outline of method
Probabilities of
fatigue initiation ina rolling wheel
Stress field under a
Hertzian contact
Probability of
fatigue initiation
for
inhomogeneous
stress fields
Occurrence and
size distribution of
material defects,
(truncation of
large defects)
Fatigue criterion
f C
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Division of Material and Computational Mechanics
Department of Applied Mechanics
Johan SandstrmCHARMEC
Occurrence and size distribution of material
defects
The sizes Dof defects are assumed to be exponentially
distributed with mean m.The probability distribution of
defects being larger than d is
The occurrences of defects in material volumes are
assumed to be Poisson distributed with the intensity 0
The intensity 0can be estimated from results ofultrasonic scans of wheels, where the ratio of wheels with
defects is designated asA
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Di i i f M t i l d C t ti l M h i
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Division of Material and Computational Mechanics
Department of Applied Mechanics
Johan SandstrmCHARMEC
Fatigue criterion
When a material defect is present, the fatigue limit
decreases A simplified Murakami relation gives the relation
between the reduced fatigue limit and defect size
The equivalent stress measure to quantify the fatigue
is here taken as the Dang Van stress DV
Di i i f M t i l d C t ti l M h i
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Division of Material and Computational Mechanics
Department of Applied Mechanics
Johan SandstrmCHARMEC
Probability of fatigue initiation for
inhomogeneous stress fields
For a material point with a certain stress state and an existing
defect, the probability of fatigue initiation is established with
the fatigue relation between stress and defect size
The total probability of fatigue initiation in a volume is
achieved with the Poisson process and integration of thevarying probability density over the volume
Di i i f M t i l d C t ti l M h i
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Division of Material and Computational Mechanics
Department of Applied Mechanics
Johan SandstrmCHARMEC
Stress field under a Hertzian contact
With Hertzian contact theory
the subsurface stress
components can be
determined With these the Dang Van
stress field can be formulated
for a typical railway contact
patch and varying contactforce as
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Division of Material and Computational Mechanics
Department of Applied Mechanics
Johan SandstrmCHARMEC
Dang Van stress evaluation under a rolling
contact
The stress field evolution due toa rolling contact is complex andinvolves all six components andare non-proportional
The computerized methodsneeded to find the Dang Vanstress that predicts fatigueinitiation, are examined
The resulting Dang Van stresscan be given as a surface overthe plane normal to the rollingdirection
12
34
0
1
2
0.05
0.1
0.15
z/b
Dang Van stress
y/b
-2 -1 0 1 2-0.8
-0.6
-0.4
-0.2
0
0.2
t
/p0
stress components
x
y
z
xy
yz
xz
Division of Material and Computational Mechanics
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Division of Material and Computational Mechanics
Department of Applied Mechanics
Johan SandstrmCHARMEC
Probabilities of fatigue initiation
in a rolling wheel
The total probability of fatigue initiation in a rolling wheelis achieved by integration of the probability density overthe wheel volume
With cylindrical coordinates, the integrand for theinitiation probability becomes
This integrand can be computed numerically at arelatively low cost
Division of Material and Computational Mechanics
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Division of Material and Computational Mechanics
Department of Applied Mechanics
Johan SandstrmCHARMEC
Results
The results are presented in the form of the probability
Pvthat fatigue initiates in the wheel during rolling Apart from examining the effect of varying contact force
or maximum contact pressure, the effect of other
parameters are studied
Contact geometry relevant for railway applications is
usedwheel radius 440 mm and rail head radius 286
mm
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Division of Material and Computational Mechanics
Department of Applied Mechanics
Johan SandstrmCHARMEC
Truncation and
normal contact force
When the normal contact
force reaches 50 kN, an
increase of probability is
noticed
The difference between no
truncating and truncating
for defects greater than 1
mm is small
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Division of Material and Computational Mechanics
Department of Applied Mechanics
Johan SandstrmCHARMEC
Continuation on truncation
The effect of truncation can
increase, if truncation can
be made for smaller defects
The probability is seen to
decrease when truncating is
made for 0.5 mm or smaller
To test for such small
defects may not be practical
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Division of Material and Computational MechanicsDepartment of Applied Mechanics
Johan SandstrmCHARMEC
Wheel steel quality theAmeasure
Ais the fraction of wheels
that during ultrasonic scans,
is found to have a defect
greater than 1 mm
The decrease of probabilityis rapid whenAdecreases
below 0.01
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Division of Material and Computational MechanicsDepartment of Applied Mechanics
Johan SandstrmCHARMEC
Probability of fatigue failure
The method can be
taken further to also
estimate probability offatigue failure
The fatigue life is
estimated with a
Whler curve and
Palmgren-Miner linear
damage accumulation
Division of Material and Computational Mechanics
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Division of Material and Computational MechanicsDepartment of Applied Mechanics
Johan SandstrmCHARMEC
Load distributions
In fatigue life evaluations,
the variation of the load
must be considered
From dynamic train
tracksimulations, realistic load
spectrums can be
extracted
With use of probabilitytheory, probability of
fatigue failure can be
estimated
0 500 1000 1500 20000
1
2
3
4
5
6x 10
-3
p0[MPa]
P
robdensity[MPa-1]
speed = 200 kmhcorrugation = -6 dB
0 500 1000 1500 20000
1
2
3
4
5
6x 10
-3
p0[MPa]
P
robdensity[MPa-1]
speed = 200 kmhcorrugation = 3 dB
0 500 1000 1500 20000
1
2
3
4
5
6x 10
-3
p0[MPa]
Probdensity[M
Pa-1]
speed = 300 kmh
corrugation = -6 dB
0 500 1000 1500 20000
1
2
3
4
5
6x 10
-3
p0[MPa]
Probdensity[M
Pa-1]
speed = 300 kmh
corrugation = 3 dB
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Division of Material and Computational MechanicsDepartment of Applied Mechanics
Johan SandstrmCHARMEC
Resulting fatigue life
The results can be
presented as fatigue
failure probability after
a rolled distance
By varying different
parameters, their
impact can be evaluated
0 100 200 3000
0.05
0.1
0.15
0.2
0.25
0.3
1000 km
probability%
200 km/h
250 km/h
300 km/h
Division of Material and Computational Mechanics
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Division of Material and Computational MechanicsDepartment of Applied Mechanics
Johan SandstrmCHARMEC
Conclusions
The probability of subsurface
fatigue initiation due to
material defects in a rolling
wheel can be established with
relatively easy computations
Truncation of large material
defects yields a small effect
It is also possible to assessprobability of fatigue failure
Future and ongoing
work
Finding material data for the
involved material parameters
Wheel re-profiling and avarying contact position will
affect the probability of
fatigue initiation, which can be
accounted for by modifying
the probability densityfunction