Chalmers University of Technology Fatigue initiation in ...anek/research/CM2000.pdf · Chalmers...

CHARMEC

Department of Solid MechanicsChalmers University of Technology

1

* Corresponding author

Fatigue initiation in railway wheels – on the influence of defects

Elena Kabo* Anders Ekberg

[email protected] [email protected]

Dept of Solid Mechanics / CHARMECChalmers University of TechnologySWEDEN

J

ULY

18, 2000

Abstract

Stresses around a material defect in the rim of a railway wheel are studied during an over-rolling contact using elasto-plastic finite element analysis. Results in the form of animations show a complex evolution of the stress field. Based on multi-parametric studies, conclusions of practical interest to railway engineers are drawn. Insights into the physical process have been gained.

1 Introduction

Wheel failures caused by propagating fatigue cracks may manifest themselves in a violent manner where a part of the wheel (or the entire wheel) breaks off. Consequences of such failures are damages to rails and sleepers and to train suspensions and bearings. In rare cases derailments may occur. The failures may be very costly in economical terms and in terms of delays and human injuries. Fatigue of railway wheels is a subject of increasing interest.

Prevention of fatigue failures of railway wheels requires an understanding of the underlying physical mechanisms. A desirable approach in gaining such knowledge would be to perform physical experiments. Full-scale experiments would, however, be extremely complicated and expensive. Scaled experiments have been performed, see

e g

[

1

], and the results achieved provide valuable information.


2

2 B

ACKGROUND

Another way of approaching the fatigue issue is to study statistics of the causes behind reprofiling of operational railway wheels. Such studies have been carried out,

e g

[

2

]. The task is complicated by the fact that fatigue only occurs on a very small minority of wheels in a large fleet that operates under fairly similar conditions. To find wheels with fatigue damages and, in particular, to identify differences in operational conditions as compared to non-damaged wheels has proved to be very difficult.

In this paper, fatigue initiation of railway wheels is studied from a theoretical point of view using a computer analysis of the state of stress and strain. The drawback of computational simulations is that the validity of the results is completely relying on the adequacy of the theoretical models. However, the approach permits a fairly fast analysis of the influence of different parameters such as load level, contact geometry, defect position

etc

.

2 Background

The work in the present paper is part of a project within

charmec

“Prediction of lifetime of railway wheels” that was started in 1994. The ultimate aim of the project is to develop predictive models for the deterioration of railway wheels based on a knowledge of load spectra, contact geometries and material strengths. Previous results from the project have been presented in references [

2

] to [

7

]. In the first three papers, a predictive model for fatigue initiation in a ‘defect-free material’ was developed. Naturally, all railway wheels contain material defects of some shape. However, the assumption made was that the influence of existing defects was limited to a local decrease in fatigue strength and that they did not influence the global stress distribution. Indeed it turned out, based on numerical simulations in [

4

], on evaluations of reprofiling statistics in [

2

] and on statistical simulations of fatigue impact due to an applied load spectrum in [

5

], that such a decrease in fatigue resistance is more or less a requirement for a fatigue crack to initiate. Estimated levels of fatigue impact for ‘ideal’ contact geometry between wheel and rail and a defect-free material were, in themselves, not sufficient to lead to fatigue failures. Consequently, either an increase in induced stress magnitudes,

e g

due to a bad contact geometry, or a decrease in material strength,

e g

due to material defects, is required for fatigue cracks to initiate. This was further emphasized by subsequent studies of wheel failures occurring in practice, as presented in [

6

] and [

7

].

The question of what impact on fatigue resistance a material defect imposes, should be clarified. A first estimation based on a semi-empirical model developed by Murakami and coworkers, [

8

], was tried in [

6

] and [

7

]. This approximate model suggests that the reduced fatigue resistance, in terms of a fatigue limit, can be expressed as a function of the defect size as:


3 M

ATERIAL

DEFECTS

3

(1)

Here, is the fatigue limit of the material containing a defect of size . Further, is the fatigue limit in the absence of large defects and the defect size corresponding to fatigue initiation at a stress level .

The fatigue limit can then be introduced in a standard fatigue design process,

e g

as proposed in [

2

] to [

5

].

Although extremely appealing due to its simplicity, it should be observed that this model was originally developed for uniaxial tension/compression loading. Thus the physical basis for the use of the model in rolling contact applications is fairly weak even though some arguments as to why it might be appropriate can be found, see [

6

].

Below, an attempt is made to further investigate the influence of defects in rolling contact with focus on wheel/rail applications. This is made by elasto-plastic finite element (FE) simulations of the material response in the vicinity of defects owing to passing contact loads. In particular, the influence of parameters such as load magnitude, contact geometry and depth of the defect below the wheel tread is studied.

3 Material defects

3.1 Types of material defects and their impact

Occurring material defects in railway wheels may be of various types depending on the manufacturing process. Types of material defects reported in the literature include non-metallic inclusions, such as

MnS

, and also dross from brick lining and oxides, such as

SiO

, see [

6

,

7

] and figure 1. Further, metallic inclusions, such as

AlO

, may appear and there may exist pores in the wheel material.

The impact of the different defects will vary depending on the type of inclusion and its interaction with the surrounding material. Of the types shown in figure 1, dross will manifest itself in fairly big and brittle defects that are easily crushed during the rolling contact. If cracks occur, they will here propagate from the vertical sides of the defect.

Silicon oxide is a type of hard inclusions that will not normally be crushed. Also in this case, possible cracks will start to propagate from the vertical sides of the defect.

Manganese sulphide is soft and will deform into ‘strings’ during the forging process and under rolling contact. Due to its small size, it could be argued that cracks are less likely to initiate from these

σw

σe-------

dd0-----

1 6⁄–=

σwd σed0σe

σw


4

3 M

ATERIAL

DEFECTS

inclusions. However, a crack already initiated would easily grow through a band of manganese sulphide inclusions. Unfortunately, these hypotheses are difficult to verify by metallographic examinations since

MnS

-inclusions are soft and thus likely to wear off during the process of crack propagation.

figure 1

Typical defects in wheel materials: (

a

) dross from brick lining of casting pipes (size

≈

1 mm)(

b

) silicon oxide (size

≈

50 µm)(

c

) manganese sulfide (size

≈

10 µm)Picture courtesy: Peter Sotkovszki

In the computer simulations presented in this paper, material defects are modelled as pores. The reason is that a pore seems to be one of the worst types of inclusions, see [

9

,

10

]. It should also be noted that if an inclusion becomes crushed, its further resistance to compressive loading can be neglected. Such an inclusion may then, from a mechanical point of view, be regarded as a pore.

In the two-dimensional numerical simulations, the defect is modelled as circular with a diameter of 1 mm. This is a simplification of the shape of real defects (with the exception of

SiO

-inclusions). However, sharp-edged defects will soon be embedded in a zone of plastically deformed material whereby the influence of the original defect shape diminishes. In future studies, also the influence of shape and size of defects will be investigated.

a b

c


4 C

OMPUTATIONAL

MODEL

5

4 Computational model

4.1 Overview

The computer model of the wheel is two-dimensional and plane strain conditions are assumed. A ‘physical realization’ of the model is shown in figure 2. During a load cycle, the rail ‘moves’ in the direction indicated by the arrow. Note that an exact representation would depict the wheel and rail as infinitely wide.

figure 2

‘Physical representation’ of the computer model.

4.2 Loading

The loading is applied as a Hertzian contact that passes over the defect, see figure 3. The finite element mesh was designed to provide a high mesh density in the contact zone as well as around the defect.

Load magnitudes [MN/m] were used in the study. This corresponds to a highest contact pressure of [MN/m

2

]. In a shakedown diagram for a

defect-free material

, see [

11

], these three magnitudes correspond to elastic response, elastic shakedown and subsurface flow, respectively.

rail

wheel

defectmoving

direction

F 7 12 18, ,=p0 925 1211 1484, ,=


6

4 C

OMPUTATIONAL

MODEL

figure 3

FE-model of a railway wheel (upside down as in figure 2) containing a defect subjected to a moving Hertzian contact load. The inset shows the region around the 1 mm diameter defect. In the statement of load magnitudes in MN/m, the length [m] refers to the axial direction, which is perpendicular to the figure plane

4.3 Material model

The employed material model is elasto-plastic with non-linear hardening. The model was originally proposed by Chaboche and Lemaitre in 1990 and is included in the commercial FE-package ABAQUS [

12

] used for the calculations.

Kinematic and isotropic hardening are governed by the equations

(2)

and

(3)

Here is the backstress, the plastic multiplier and the dragstress. Further, where ‘ ’ denotes double scalar product and where with denoting the deviatoric part of the stress tensor. The material parameters are, in the present study, taken as GPa, , MPa and MPa. The yield limit is MPa, Young’s modulus GPa and Poisson’s ratio . Details about the constitutive equations and their accuracy in modelling material response due to compressive cyclic loading are presented in [

13

].

Moving Hertzian load

120 elements in the area around the defect

200 mm

100 m

m

≈10700 8-node elements

in the FE mesh

X λ c23---ndev γX–

=

K λb 1 KK∞-------–

=

X λ Kndev τdev τdev:τdev⁄≡ :

τdev σdev X–= σdev

c 6.49= γ 0.81= b 0.47=K∞ 22.8= σy 543=

E 210= ν 0.29=


5 STRESSES AND STRAINS IN THE VICINITY OF A DEFECT 7

5 Stresses and strains in the vicinity of a defect

The computational model presumes a state of plane strain. In addition, the defect will introduce a free surface. Consequently, the only non-vanishing stress components at the position indicated by the black square in figure 4 are the normal stresses and , leading to a local biaxial state of stress. In addition, the only non-vanishing strain components are , and .

figure 4 Stress and strain components at the surface of a pore defect

5.1 Stabilization of the state of stress and strain

It is found that when stress magnitudes in the vicinity of the defect exceed the plastic limit, the material response of a virgin wheel stabilizes after a few load cycles. For the chosen combination of contact geometry, load levels and material parameters, this stabilization occurs fairly fast, see figure 5. Based on this observation, the subsequent results were chosen to display the response during the fifth load cycle.

It is further found, see figure 6, that when plastic loading occurs, the local strain ratio will approach . This was one of the assumptions made in [6] to justify the use of an approximate model for fatigue strength reduction as discussed above.

σy σz

εx εy γxy

σz ≠ 0

σy ≠ 0

Materialdefectx

y

z

εx ≠ 0

εy ≠ 0γxy ≠ 0

R εmin εmax⁄ 1–= =


8 5 STRESSES AND STRAINS IN THE VICINITY OF A DEFECT

figure 5 Maximum and maximum at the edge of the

defect, respectively

figure 6 Maximum and minimum plastic strains during the fifth load cycle as functions of the depth of the defect below the wheel tread

6.5

7.5

8.5

9.5

2 3 4 5 6 7 8 9 10

F = 7 MN/mF = 12 MN/mF = 18 MN/m

1.0

2.0

3.0

4.0

F = 12 MN/mF = 18 MN/mF = 12 MN/m, residualF = 18 MN/m, residual

Number of passage

1 2 3 4 5 6 7 8 9 10

1

Number of passage

ma

x T

res

ca

str

es

s [

N/m

2]

ma

x p

las

tic

pri

nc

ipa

l s

tra

in [

–]

[x108]

5.5

[x10-2]

0

τT σ1 σ3–= ε1p

-8

-6

-4

-2

10 15 20 25

0

2

4

6

8F = 7 MN/mF = 12 MN/mF = 18 MN/mF = 12 MN/m, residualF = 18 MN/m, residual

Depth to defect [mm]

min

an

d m

ax p

last

ic p

rin

cip

al s

trai

n [

–]

[x10-2]



5.2 Stresses and strains during a load passage

The following examples are derived for a loading of MN/m and a 1 mm defect located at a depth of 15 mm.

In figure 7, stress components at a node corresponding to the position of the the black square in figure 4 are depicted for the first load cycles. It can be seen that a residual tensile is introduced after the first load cycle. Further, even though the stresses are derived in a Gaußian point very close to the side of the defect (<0.1 mm), steep stress gradients (and numerical approximations) make the stress components and non-zero.

figure 7 Magnitudes of stress components during the first five load passages

The evolution of the hydrostatic pressure (positive in compression and defined as ) and the Tresca effective stress (defined as , where and are the algebraically largest and smallest principal stress, respectively) are shown in figure 8. The somewhat unexpected evolution of the Tresca effective stress is due to the transition compression/tension of .

figure 8 Evolution of hydrostatic pressure and Tresca effective stress during the first five load passages

F 12=

σy

σx τ xy

sx

txy

sy

sz

[x108]

2

0

2

-4

-6

Stre

sses

at t

he s

ide

of th

e de

fect

[N/m

2 ]

Number of passage1 2 3 4 5

-8

4

σh σx σy σz+ +( ) 3⁄=τT σ1 σ3–= σ1 σ3

σy

[x108]


0

4

2

Hyd

rost

atic

pre

ssu

re a

t th

e si

de

of

the

def

ect

[N/m

2 ]

-1

1

3

[x108]

2

6

4

0

Tres

ca e

ffec

tive

str

ess

at t

he

sid

e o

f th

e d

efec

t [N

/m2 ]



10 5 STRESSES AND STRAINS IN THE VICINITY OF A DEFECT

The evolution of the strain magnitudes are shown in figure 9. As can be seen, a tensile residual and a compressive residual are induced during the first load passage.

figure 9 Magnitudes of strain components during the first five load cycles

The sequence in figure 10 shows the development of the Tresca effective stress during the fifth load passage. The deformation scale factor is 50. The inset shows a magnified view of stresses around the defect with deformation scale factor 20.

In a, the contact load is located fairly far away from the defect and the magnitude of the Tresca stress is lower at the defect than at a position closer to the the surface.

In b, the contact load is right above the defect, resulting in an almost symmetric stress pattern.

In c, the contact load has passed the defect. Note how the location of maximum stress at the side of the defect rotates along the defect edge during the entire load passage. Details are more easily seen in an animation of the time-dependent stress field [14].

In d, finally, the residual Tresca stress is shown. Note the symmetry of this residual stress field.

From the animation of the Tresca effective stress and other stress/strain measures, it is seen that the load has to be applied close (in the horizontal direction in figure 3) to the defect in order to result in a large stress at the defect. One conclusion is that defects outside the running track on the tread should be of no importance to the rolling contact fatigue performance of the wheel.

εx εy

ex

exy

ey

5.0

2.5

0

-2.5

-5.0

-7.5

Str

ain

s at

th

e si

de

of

the

def

ect

[–]


[x10-3]



figure 10 Tresca effective stress during the fifth load passage. Loading 12 MN/m, defect size 1 mm, defect located 15 mm below the surface

+6.34e+04+7.37e+04+6.71e+07+1.34e+08+2.01e+08+2.68e+08+3.35e+08

(Ave. Crit.: 75%)S, Tresca

+4.02e+08+4.69e+08+5.36e+08+6.03e+08+6.70e+08+7.37e+08

+8.31e+03+7.37e+04+6.71e+07+1.34e+08+2.01e+08+2.68e+08+3.35e+08


+4.02e+08+4.69e+08+5.36e+08+6.03e+08+6.70e+08+7.37e+08

+7.37e+04+6.15e+07+1.23e+08+1.84e+08+2.46e+08+3.07e+08+3.69e+08


+4.30e+08+4.91e+08+5.53e+08+6.14e+08+6.76e+08+7.37e+08

+7.37e+04+6.15e+07+1.23e+08+1.84e+08+2.46e+08+3.07e+08+3.69e+08


+4.30e+08+4.91e+08+5.53e+08+6.14e+08+6.76e+08+7.37e+08

a

b

c

d

[Pa]

[Pa]

[Pa]

[Pa]


12 6 INFLUENCE OF SOME PARAMETERS

5.3 Residual stresses

In figure 11, the residual stress field, as represented by the hydrostatic pressure, is shown. It is seen that the passing contact load will introduce compressive residual stresses at a depth corresponding to that of the maximum shear stress (in the absence of defects) and at the outside of the defect sides. Further, a tensile residual stress (detrimental) is introduced at the very side of the defect. Note the steep gradient of this residual stress as well as of the Tresca effective stress (figure 10) near the vertical sides of the defect. This issue will be further discussed below.

figure 11 Residual hydrostatic pressure, for input data as in figure 10

6 Influence of some parameters

6.1 Defect position and load magnitude

In figure 12, the influence of the vertical position of the defect (defined as the vertical distance between wheel surface and centre of defect) on the maximum Tresca effective stress at the defect is shown for three load levels. The response can be divided into three ‘stages’ (cf figure 6). In the first stage, the response at the defect is elastic and the magnitude of the shear stress increases for decreased depth to the defect and/or increased load magnitude. Such a response is obtained for deep defects and a load of MN/m.

In the second stage, plastic deformations will start to occur. This will lead to a redistribution of stresses at the defect so that the stress magnitude remains almost constant when load magnitudes are increased and/or the depth to the defect is decreased. Such a response is obtained for shallow defects when MN/m and for deep defects when MN/m.

In the third stage, there will be large plastic deformations at the edge of the defect and the stress magnitude will increase with increased load and/or decreased depth to the defect. Such a response is found for shallow defects when MN/m and for MN/m.

-1.95e+08-1.64e+08-1.33e+08-1.02e+08-7.17e+07-4.08e+07-1.00e+07

(Ave. Crit.: 75%)S, Pressure

+2.08e+07+5.17e+07+8.25e+07+1.13e+08+1.44e+08+1.75e+08

[Pa]

F 7=

F 7=F 12=

F 12= F 18=


6 INFLUENCE OF SOME PARAMETERS 13

A planned study of defects more shallow than 10 mm had to be cancelled due to numerical problems.

figure 12 Maximum Tresca effective stress vs vertical position of the material defect in the wheel

6.2 Contact geometry

Figure 13 shows the maximum Tresca effective stress and the maximum plastic principal strain versus the radius of the contact cylinder corresponding to the contact stress distribution in figure 3. It is seen that a variation of the contact geometry will have little effect on the response. For a radius of 0.1 m, gross deformations and pertinent numerical problems occurred. Consequently, the results for 0.1 m radius in figure 13 should be used with precaution.

figure 13 Influence of radius of contact cylinder for a 1 mm defect located at a depth of 15 mm

0.35

0.6

0.85

1.1

1.4

Depth to defect [mm]

F = 7 MN/mF = 12 MN/mF = 18 MN/m

10 15 20 25

[x109]

max

Tre

sca

stre

ss [

N/m

2 ]

3.5E+08

6.0E+08

8.5E+08

1.1E+09

1.4E+09

0.1 0.2 0.3 0.4 0.5Cylinder radius [m]

F = 12 MN/m

F = 12 MN/m

F = 12 MN/m, residual

0.1 0.2 0.3 0.4 0.5

2.0E-02

4.0E-02

6.0E-02

8.0E-02

Cylinder radius [m]

max T

resca s

tress [

N/m

2]

max p

lasti

c p

rin

cip

al

str

ain

[–]


14 7 CONCLUDING REMARKS

7 Concluding remarks

7.1 Practical implications

Some preliminary conclusions of practical interest may be drawn from the above results:

· The state of stress at the defect is locally biaxial and exhibits a complex evolution during a load cycle (i e, a load passage).

· Stress gradients at a defect are large, a fact which raises questions regarding fatigue design. If stresses/strains at nodes (or integration points) in a FE-model are used in the fatigue analysis, the results may be overly conservative. There are proposals that stresses and strains should be averaged over a characteristic area, such as the grain size. However, due to the large gradients, the results will depend heavily on the area chosen.

· The type of response at the defect depends on the combination of load magnitude and vertical position of the defect. This has to be taken into account when identifying admissible load magnitudes (or defect sizes).

· In contrast to the case of a defect-free material, see [4], the contact geometry seems to be of minor importance for the fatigue impact of a defect.

7.2 Future work Subsequent studies will broaden the scope to include also parameters such as size and shape of defects.

The results obtained from the present and the future studies will be introduced in a predictive fatigue design model for railway wheels.

AcknowledgmentsThe current work has been carried out within the Centre of Excellence CHARMEC (CHAlmers Railway MEChanics). The authors would like to thank Profs Hans Andersson, Roger Lundén and Bengt Åkesson, and also Dr Peter Sotkovszki, Mr Jonas Ringsberg and Mr Johan Marais as well as many other friends and colleagues for their valuable help.

References1. TYFOUR W R, BEYNON J H & KAPOOR A, Deterioration of rolling

contact fatigue life of pearlitic rail steel due to dry-wet rolling-sliding line contact, Wear, vol 197, pp 255–265, 1996

2. EKBERG A, Rolling contact fatigue of railway wheels - Computer modelling and in-field data, Proc 2nd MiniConference on Contact Mechanics and Wear of Rail/Wheel Systems, Budapest, pp 154-163, 1996

3. EKBERG A, BJARNEHED H & LUNDÉN R, A fatigue life model for general rolling contact with application to wheel/rail damage, Fatigue & Fracture of Engineering Materials & Structures, vol 18, no 10, pp 1189–1199, 1995


7 CONCLUDING REMARKS 15

4. EKBERG A, Rolling contact fatigue of railway wheels – a parametric study, Wear, vol 211, pp 280–288, 1997

5. EKBERG A, LINDQVIST R & OLOFSSON M, Multiaxial fatigue – a probabilistic analysis of initiation in cases of defined stress cycles, Fatigue '99 – Proc 7th International Fatigue Congress, Beijing, China, pp 923–928, 1999

6. EKBERG A & MARAIS J, Effects of imperfections on fatigue initiation in railway wheels, IMechE, Journal of Rail and Rapid Transit, vol 214, pp 45–54, 2000

7. EKBERG A & SOTKOVSZKI P, Anisotropy and rolling contact fatigue of railway wheels, accepted for publication in International Journal of Fatigue, 2000

8. MURAKAMI Y & ENDO M, Effects of defects, inclusions and inhomogeneities on fatigue strength, International Journal of Fatigue, vol 16, pp 163–182, 1994

9. FRY G T, LAWRENCE F V & ROBINSON A R, A model for fatigue defect nucleation in thermite rail welds, Fatigue & Fracture of Engineering Materials & Structures, vol 19, pp 655–668, 1996

10. MELANDER A, Simulation of the behaviour of short cracks at inclusions under rolling contact fatigue loading – specially the effect of plasticity, in Bearing Steels: into the 21st century, ASTM STP 1327, 1998

11. JOHNSON K L, The strength of surfaces in rolling contact, Proc Institution of Mechanical Engineers (IMechE), vol 203, pp 151–163, 1989

12. HIBBITT, KARLSSON & SORENSEN, ABAQUS v 5.8, 199813. EKH, M, JOHANSSON A, THORBERNTSSON H & JOSEFSON B L,

Models for cyclic ratchetting plasticity – integration and calibration, ASME Journal of Engineering Materials and Technology, vol 122, pp 49–55, 2000

14. KABO E, QuickTime-movies of stresses and strains at a defect during rolling contact, www.solid.chalmers.se/anek/research/cm2000/

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