Dag 2 Pass 1 Johan Sandstrom Del 1

8/10/2019 Dag 2 Pass 1 Johan Sandstrom Del 1

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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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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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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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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



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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



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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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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



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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



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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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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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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


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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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



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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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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



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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

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