modelling of the effect of thermal and material properties on ......repeated wheel passes on rail...

Modelling of the effect of thermal and

material properties on rail wear

Anna Maria Sri Asih

This thesis is submitted for the degree of

Doctor of Philosophy

Faculty of Engineering and Industrial Science

Swinburne University of Technology, Australia

2013

Abstract

Abstract

Repeated wheel passes on rail can cause the accumulation of plastic strain in a thin

subsurface layer of material through a process called ratcheting. If the accumulated

plastic strain reaches the critical strain to failure material fails by ratcheting failure. This

failure takes the form of either generating wear debris or causing rolling contact fatigue

crack initiation. The severity of wear and the initiation of rolling contact fatigue crack

depend on the contact conditions during service and the properties of the rail material. A

computer simulation has been developed to simulate the effect of contact and thermal

stresses and different material properties to predict the wear rate and the rolling contact

fatigue initiation.

The operating conditions, such as the maximum contact pressure, the friction

coefficient, the slip/roll ratio and the train speed will determine the severity of the

contact stress in the rail material. These contact conditions will also influence the

amount of frictional heating that occurs at the interface during the rolling/sliding. The

heat generated is transferred into both the wheel and the rail and increases their

temperature. The temperature rise in the rail material can be higher if the wheel bulk

temperature also increases due to continuous heating during rolling/sliding contact.

When the hot wheel touches the rail, there will be additional heat flows in the rail in

addition to the heat from frictional heating. The temperature rise in the rail material

causes the development of thermal stress and can also cause thermal softening if the

temperature is high enough. This thermal effect increases the accumulated plastic strain

and leads to more severe wear and the initiation of rolling contact fatigue cracks.

However, the greater wear rate may reduce the length of the crack-like flaw and prevent

a crack being initiated at the surface, which brings the subsurface flaws closer to the

surface.

In addition to the contact conditions, different rail materials with different initial

hardness, strain hardening behaviour, critical strain to failure, and thermal softening

behaviour have shown different results in wear and the initiation of rolling contact

fatigue. The material with greater initial hardness has been found to have a higher

Abstract

resistance to failure. However, during contact the initial hardness is altered by the

hardening and softening process. As the material hardens, the plastic deformation

reduces and hence reduces the failure. Likewise greater resistance to thermal softening

results in material having a greater final hardness, and this can resist further failure. The

material with the greatest final hardness was found to have better resistance to wear.

The severity of rolling contact fatigue crack initiation depends on the dominancy of the

wear rate over the growth of failures. If the wear rate is relatively low, then the crack

may be initiated sooner as the contact loading becomes severe. On the other hand, a

greater wear rate may truncate the length of a crack-like flaw and lead to later crack

initiation.

Acknowledgement

Acknowledgement

I wish to sincerely thank my supervisor, Prof. Ajay Kapoor, and Dr. Kan Ding who

were extremely helpful and offered invaluable assistance, support and guidance. Prof

Ajay Kapoor has always given me a huge amount of support, since the very beginning

of my study. I am very grateful for all the discussions regarding my research problems.

He provided me with the experiences of writing and presentation, and encouragement

throughout my study. This research would not have been possible without his help. I

also offer him many thanks for his care regarding my family, which motivated me to

finish this degree.

I would like to thank Mr. Peter Mutton (IRT, Monash University, Australia) for his

useful discussions, and for sharing his knowledge about railways in practice. I would

like to thank to Dr FJ. Franklin (Newcastle University, UK) for his helpful advice

regarding the program for the research model I used. I also would like to thank Dr. IM.

Widiyarta (Udayana University, Indonesia) for his great help in understanding the

model from his research. I would like to thank Dr. Tracy Ruan Dong and Mr. Stephen

Guillow for giving me the opportunity to learn and work in the structural mechanics

laboratory. I would also like to thank Ms. Melissa Cogdon and Ms. Shophia Haccou for

their assistance and support regarding administration during my PhD program. I would

like to thank Mrs. Christine De Boos (De Boos Editing & Educational Services) for

copy editing this thesis in line with the ‘Editing Guidelines for the Editing of Australian

Theses’. I would like to thank Dr. Sylvia Mackie as a language advisor of my thesis. I

would like to thank Iman Salehi and Sagheer Ranjha, my colleagues who were always

willing to discuss the research.

I am so grateful to have funding from Swinburne University of Technology for my

study and to be able to live in Australia. I would like to thank Prof. Pamela Green for

her support at the end of my study. I would like to thank the Department of Mechanical

and Industrial Engineering, Gadjah Mada University, for giving me the opportunity to

undertake my doctoral degree.

Acknowledgement

Finally, I would like to thank my lovely husband and son, who have always been with

me in happiness and sadness. The time I spent with them during my study gave me such

a great experience, and enabled me to grow tougher and wiser.

Declaration

Declaration

I declare that this thesis does not contain any material which has been submitted or

accepted for the award of any other degree or diploma in any University. To the best of

my knowledge the thesis does not contain any material previously published or written

by another person except where due reference is made in the text.

Anna Maria Sri Asih

May 2013

Publications

Publications

Asih, A.M.S., Ding, K., Kapoor, A, Modelling rail wear transition and mechanism due

to frictional heating, Wear, Vol. 284-285, pp. 82-90, 2012

Asih, A.M.S., Ding, K., Kapoor, A. Modelling the effect of steady state wheel

temperature on rail wear, Tribology Letters, Vol. 49, No.1, pp.239-249, 2013

Contents

Contents

Abstract

Acknowledgement

Publications

Contents

Nomenclature

List of Tables

List of Figures

Chapter 1 Introduction ..................................................................................................... 1

Chapter 2 Literature Review ............................................................................................ 7

2.1. Rail-wheel contact .................................................................................................. 7

2.1.1. Hertz-line contact theory ................................................................................. 9

2.1.2. Line loading of elastic half space .................................................................. 10

2.1.3. Rolling-sliding contact of elastic bodies ........................................................ 14

2.2. Thermo-elastic contact ......................................................................................... 16

2.3. Rail-wheel operating conditions ........................................................................... 18

2.3.1. Contact pressure ............................................................................................. 18

2.3.2. Friction / Adhesion ........................................................................................ 21

2.3.3. Creepage (Slip/roll ratio) ............................................................................... 26

2.3.4. Train speed ..................................................................................................... 28

2.4. Rail material properties ........................................................................................ 30

2.4.1. Rail steel specifications ................................................................................. 31

2.4.2. Pearlitic steel .................................................................................................. 32 2.4.2.1. The hardness of pearlitic steel ................................................................. 34 2.4.2.2. Temperature effect on the pearlitic steel’s strength ................................ 36

2.4.3. The effect of various rail steel microstructures on wear and RCF ................ 38

2.5. Material response ................................................................................................. 40

Contents

2.5.1. Material response to cyclic loading ............................................................... 40

2.5.2. Shakedown ..................................................................................................... 41

2.5.3. Plastic ratcheting ............................................................................................ 42

2.6. Rail deterioration .................................................................................................. 44

2.6.1. Wear ............................................................................................................... 45 2.6.1.1. Wear mechanism ..................................................................................... 45 2.6.1.2. Wear mechanism maps for steel ............................................................. 47 2.6.1.3. Wear transition ........................................................................................ 49

2.6.2. Rolling contact fatigue ................................................................................... 50 2.6.2.1. Crack initiation ........................................................................................ 51 2.6.2.2. Crack propagation ................................................................................... 52 2.6.2.3. Wear and fatigue interaction ................................................................... 55

Chapter 3 Model Development ...................................................................................... 59

3.1. Wear ..................................................................................................................... 60

3.2. Temperature.......................................................................................................... 63

3.2.1. Temperature rise due to frictional heating ..................................................... 63

3.2.2. Temperature rise due to steady state wheel temperature ............................... 65

3.3. Thermal stresses ................................................................................................... 67

3.3.1. Thermal stress due to frictional heating .................................................... 68

3.3.2. Thermal stress due to steady state wheel temperature ................................... 71

3.4. Thermal softening................................................................................................. 71

3.5. Material properties ............................................................................................... 74

3.6. Simulation flow diagram ...................................................................................... 76

Chapter 4 Rail wear due to frictional heating ................................................................ 79

4.1. Introduction .......................................................................................................... 79

4.2. Rail temperature rise ............................................................................................ 81

4.3. Thermo-mechanical stresses................................................................................. 81

4.4. Plastic strain accumulation ................................................................................... 83

Contents

4.5. Rail wear due to frictional heating ....................................................................... 84

4.5.1. The effect of variation on wear rate ............................................................... 86 4.5.1.1. Slip/roll ratio ........................................................................................... 87 4.5.1.2. Friction coefficient .................................................................................. 88 4.5.1.3. Peak pressure ........................................................................................... 89 4.5.1.4. Vehicle speed .......................................................................................... 90

4.5.2 Wear transition ................................................................................................ 91

4.5.3. Factorial design .............................................................................................. 93

4.6. Conclusion ............................................................................................................ 97

Chapter 5 Rail wear due to steady state wheel temperature........................................... 99

5.1. Introduction .......................................................................................................... 99

5.2. Rail temperature ................................................................................................. 102

5.3. Thermal stresses due to steady state wheel temperature .................................... 110

5.4. Plastic strain accumulation ................................................................................. 113

5.5. Rail wear ............................................................................................................. 114

5.5.1. The effect of and Sr variation on wear rate .............................................. 114

5.5.2. The effect of p0 and v0 variation on wear rate .............................................. 116

5.6. Conclusion .......................................................................................................... 119

Chapter 6 The effect of material properties on rail wear ............................................. 121

6.1. Introduction ........................................................................................................ 121

6.2. Temperature rise ................................................................................................. 122

6.3. Maximum orthogonal shear stress ...................................................................... 125

6.4. Thermal softening ............................................................................................... 127

6.5. Wear rate ............................................................................................................ 128

6.5.1. UIC 1100 rail steel ....................................................................................... 128

6.5.2. UIC 900A rail steel ...................................................................................... 130

6.6. Discussion .......................................................................................................... 133

6.7. Conclusion .......................................................................................................... 143

Contents

Chapter 7 Rail Rolling Contact Fatigue (RCF) ............................................................. 145

7.1. Introduction ........................................................................................................ 145

7.2. Effect of frictional heating on crack initiation ................................................... 148

7.3. Effect of material properties on crack initiation ................................................. 152

7.3.1. UIC 1100 rail steel ....................................................................................... 153 7.3.1.1. Surface crack initiation ......................................................................... 153 7.3.1.2. Subsurface crack initiation .................................................................... 155

7.3.2. UIC 900A rail steel ...................................................................................... 158 7.3.2.1. Surface crack initiation ......................................................................... 158 7.3.2.2. Subsurface crack initiation .................................................................... 160

7.3.3. Comparison of UIC 1100 and UIC 900A for surface crack initiation ......... 163

7.3.4. Comparison of UIC 1100 and UIC 900A for subsurface crack initiation ... 165

7.4 Discussion ........................................................................................................... 167

7.4.1. The effect of frictional heating on crack initiation ...................................... 167

7.4.2. The effect of material properties on crack initiation ................................... 170

7.5 Conclusion ........................................................................................................... 172

Chapter 8 Conclusion .................................................................................................... 175

Chapter 9 Future Work ................................................................................................. 181

References ..................................................................................................................... 183

Appendix: Source Code ................................................................................................ 195

Nomenclature

Nomenclature

a Semi contact width, mm

c Specific heat capacity, Jkg-1K-1

C Ratcheting constant

dz Wear depth, µm

E Young modulus, GPa

G Shear modulus, GPa

k0 Initial shear yield stress, MPa

keff Effective shear yield stress, MPa

keff(softening) Effective shear yield stress due to softening, MPa

L Peclet number

N Number of cycle

Nf Number of cycles to rupture by fatigue, cycle

Nr Number of cycles to failure by ratcheting, cycle

Nx Number of bricks at x direction

Nz Number of bricks at z direction

P’ Total load per unit length, N/m

Mean pressure, GPa

p0 Peak pressure

p(x) Normal pressure distribution, N/m2

q(x) Tangential traction distribution, N/m2

Heat flow rate for rail, W

Heat flow rate due to frictional heating, W

Heat flow rate due to steady state wheel temperature, W

R1, R2 Radii of curvature of the contacting body(1) and body(2), m

r Roller’s radius, mm

Sr Slip/roll ratio

t Contact width

T Tractive force

v0 Vehicle speed

Nomenclature

vs Sliding speed

Hardening rate

t Thermal expansion coefficient, K-1

Ratio of limiting hardness to the original hardness

r Thermal penetration coefficient for rail

w Thermal penetration coefficient for wheel

Thermal penetration depth, µm

Heat partitioning factor

Accumulated plastic shear strain

c Critical shear strain for failure

Plastic strain increment

Thermal diffusivity, m2/s

Thermal conductivity, W/Km

Friction coefficient

Density, mg/mm3

y Yield stress, MPa

yref Yield stress reference at room temperature, MPa

yi Yield stress at elevated temperature, MPa

Crack orientation to the surface, degree

r_FH Temperature rise in rail due to frictional heating, C

r_SSWT Temperature rise in rail due to steady state wheel temperature, C

r_actual Total temperature rise in rail, C

wo Initial wheel temperature, C

Steady state wheel temperature, C

xz Orthogonal shear stress, MPa

xz(max) Maximum orthogonal shear stress, MPa

v Poisson ratio

x Horizontal brick size, µm

z Vertical brick size, µm

List of Tables

List of Tables

Table 2.1 Friction coefficient measurement from [1] …………………… 22

Table 2.2 Adhesion coefficient for wide range conditions [1] ………….. 22

Table 2.3 Älvsjö track data [2] ………………………………………...... 23

Table 2.4 Friction coefficient of Älvsjö track data; measured during

daytime, no rain, no sunshine, and at air temperature of 19C

[2]……………………………………………………………….

23

Table 2.5 Friction coefficient of Älvsjö track data; measured during

daytime, rain during measurement of sites C and D, no rail

during measurement of sites A and B, no sunshine, and at air

temperature of 15C [2] ...……………………………………...

24

Table 2.6 Friction coefficient investigation under various set method [3] 25

Table 2.7 Properties of pearlitic steel in Figure 2.17 from [4] ................. 35

Table 2.8 Distinction between mild and severe wear [5] ........................... 48

Table 3.1 Chemical composition and mechanical properties of BS11 rail

[6] ………………………………………………………………

74

Table 3.2 Referenced data for thermal properties of rail steel [7] ………. 75

Table 3.3 Mechanical properties of UIC 900A grade rail [8] ………… 75

Table 3.4 Mechanical properties of UIC 1100 grade rail [8] ………… 75

Table 4.1 Simulation series for wear transition due to frictional heating .. 80

Table 4.2 Maximum rail temperature rise at the surface [C] due to Sr

and µ variation………………………………………………….

81

Table 4.3 Average wear rate (nm/cycle) due to frictional heating after

200,000 cycles …………………………………………………

86

Table 4.4 Two level factorial design of wear rate and flash temperature .. 96

Table 5.1 Operating conditions used in simulations .................................. 101

Table 5.2 Maximum rail surface temperature of , , and

due to variation of slip/roll ratio and friction

coefficient for p0=1.5GPa, a = 5.88 mm, and v0 = 30m/s =

104

List of Tables

144kph …………………………………………………………

Table 5.3 Maximum rail surface temperature of , , and

due to variation of peak contact pressure (GPa) and

vehicle speed (m/s) for =0.4, a = 5.88 mm, and Sr=-3% ……

105

Table 5.4 Average wear rate (nm/cycle) for conditions (1) and (2) over

200,000 cycles (w/= with and w/o = without thermal softening)

116

Table 5.5 Average wear rate (nm/cycle) for µ = 0.4 and Sr = -3% ……... 118

Table 6.1 Operating conditions used in simulation ................................... 122

Table 6.2 Maximum temperature rise at the surface due to frictional heating

(C) ………………………………………………………………….. 124

Table 6.3 Average wear rate (nm/cycle) after 200,000 cycles ………………. 130

Table 6.4 Average wear rate (nm/cycle) for UIC 900A rail steel ……………. 132

Table 6.5 Ratio of wear rate (w_UIC 900A / w_UIC 1100), normalized yield

stress (

) of UIC 900A at z = 0.5µm, and normalized yield

stress (

) of UIC 1100 at z = 0.5µm …………………………

139

Table 7.1 Effect of friction coefficient and slip/roll ratio on the number

of cycles to initiate a surface crack and the wear rate for

100,000 cycles (w/o = without; w/ = with) ……………………

149

Table 7.2 The number of cycles for surface crack initiation of UIC 1100 rail

steel ………………………………………………………………….. 155

Table 7.3 Number of cycles to initiate subsurface crack for UIC 1100 rail steel 157

Table 7.4 The number of cycles of surface crack initiation for UIC 900A rail

steel ………………………………………………………………….. 160

Table 7.5 Number of cycles of subsurface crack initiation of UIC 900A rail

steel ………………………………………………………………….. 162

List of Figures

List of Figures

Figure 1.1 Micrograph of an etched sectioned test sample from twin disc

test from [9] ……………………………………………………

2

Figure 1.2 Research flow diagram from [10] …………………………...... 4 Figure 2.1 Wheel-rail contact (a) Side view of the rail track; (b) cross-

section view of the rail track from [11] ……………………….

7

Figure 2.2 Contact patch shape due to lateral shift of the wheel-set from

[12] …………………………………………………………….

8

Figure 2.3 Line contact of two cylindrical bodies ……………………….. 11

Figure 2.4 Sliding contact from [13] ……………………………………… 12 Figure 2.5 An elastic half-space loaded by normal pressure and tangential

traction …………………………………………………………

13

Figure 2.6 Schematic depiction of traction curve with stick and slip

regions from [14] ………………………………………………

14

Figure 2.7 The distribution of tangential traction in cylindrical contact

from [13] ……………………………………………………….

15

Figure 2.8 Ultrasonic measurement for unused wheel/rail contact from

[15] …………………………………………………………….

19

Figure 2.9 The wear data points under various contact conditions from

[16] …………………………………………………………….

20

Figure 2.10 (a) Single contact; (b) double contact from [15] ……………… 20

Figure 2.11 Creep motion of railway wheel from [13, 17] ………………… 27 Figure 2.12 Characteristics of four High Speed Train (HST) models from

[18] …………………………………………………………….

29

Figure 2.13 The effect of hardness on the wear rate for several rail grades

from [19] ………………………………………………………

31

Figure 2.14 Iron-carbon phase diagram and the formation of pearlitic

structure from [20] …………………………………………….

33

Figure 2.15 Pearlitic structure from [4] .………………………………….... 33

List of Figures

Figure 2.16 The effect of lamellar spacing to the hardness from [21]…...... 34 Figure 2.17 Microstructure of pearlitic steel with detail properties as in

Table 2.1 from [4] ……………………………………………..

35

Figure 2.18 Yield strength of rail steel at elevated temperature from [22] .. 37

Figure 2.19 Material response to cyclic loading in rolling-sliding contact

from [23] ……………………………………………………….

41

Figure 2.20 Shakedown map for repeated sliding of a rigid cylinder over an

elastic-plastic half space from [24] ……………………………

42

Figure 2.21 Repeated sliding contact: (a) stress state on the surface; (b)

plastic strain cycles from [24] …………………………………

43

Figure 2.22 Competing modes of failure: low cycle fatigue (LCF) and

ratchetting failure (RF) from [25] ……………………………...

44

Figure 2.23 Wear mechanism map for steel. The shaded regions show

transitions from [26, 27] ……………………………………….

48

Figure 2.24 Wear regimes from twin disc testing of BS11 rail material vs

Class D wheel material from [16, 28] …………………………

49

Figure 2.25 Wear regimes and mechanisms from [29] …………………….. 50 Figure 2.26 Rail cross section normal to the surface crack orientation,

showing crack regularity and crack branching from [30] …….

51

Figure 2.27 Fluid assisted mechanism growth from [31]: (a) shear

mechanism, (b) hydraulic transmission of contact pressure, (c)

fluid is trapped and pressurized inside the crack, (d) squeeze

film pressure generation ……………………………………….

54

Figure 2.28 Crack growth phases from [1] ………………………………… 55 Figure 2.29 Three phases of crack life from [32] ………………………….. 55

Figure 2.30 The relationship between wear and surface crack. The surface

crack can be truncated by wear and become short or even

disappear ……………………………………………………….

56

Figure 2.31 The effect of wear rate on crack growth: Level 1 (very high

wear rate) allows no crack formation; Level 2 (regular wear

rate) shortens the crack length, enabling the crack candidate to

disappear; Level 3 (very low wear rate) has no effect on the

crack growth from [33] ………………………………………

57

List of Figures

Figure 2.32 Rail life against material removal rate from [1] ………………. 57 Figure 3.1 Brick model …………………………………………………… 60

Figure 3.2 (a) Representation of brick state at cycle (n); (b) at cycle (n+1)

- when all bricks in 1st layer became debris, this layer was

detached. All layers under the 1st layer moved up, and (c) a

new layer was added at the bottom with zero strain …………...

62

Figure 3.3 Scenarios to remove a brick as wear debris, taken from [34] … 62 Figure 3.4 The rail temperature rise was caused by (a) frictional heating

and (b) the conduction from the hot wheel into the cold rail …

66

Figure 3.5 Rolling sliding contact configuration …………………………. 68

Figure 3.6 Reduction of material yield strength due to temperature rise … 72 Figure 3.7 The process of hardening and softening at y = 406 MPa with

material hardening parameters of =25 and =1.88 (the

maximum shear yield stress of the material will be

k0=440MPa). Softening and hardening process: (1) the shear

yield stress reduces as temperature increases; and (2) the shear

yield stress recovers to its previous value as temperature

returns to normal; (3) the increment plastic strain hardens the

material ………………………………………………………...

73

Figure 3.8 Flow diagram of the simulation ……………………………….. 77 Figure 4.1 Maximum orthogonal shear stress,zx(max), against depth with

variation of: (a) peak pressure (=0.4, Sr =-3%, v0=30m/s); (b)

friction coefficient (p0=1.5GPa, Sr =-3%, v0=30m/s); (c)

slip/roll ratio (p0=1.5GPa, =0.4, v0=30m/s); (d) vehicle speed

(p0=1.5GPa, =0.4, Sr =-3%) …………………………………

82

Figure 4.2 The thermal softening is confined to a thin sub-surface layer of

depth 40 m for p0=1.5 GPa, =0.4, slip/roll ratio=-4% and

v0=60m/s after 7500 cycles; (a) variation of temperature with

depth, (b) thermal softening with depth, (c) variation of shear

strain increment with depth ……………………………………

83

Figure 4.3 The relationship between wear rate and the energy from

frictional work normalized with the contact area (T/A) in

85

List of Figures

Wear type II region for BS11 rail: , experiment from [28]; ,

simulation with thermal at p0=1.5GPa; +, simulation without

thermal at p0=1.5GPa; , simulation with thermal at

p0=1.3GPa; , simulation without thermal at p0=1.3GPa ……...

Figure 4.4 The comparison of simulation results and the experiment

conducted by Bolton and Clayton [28] for Wear type II and III

at p0=1.5GPa …………………………………………………..

85

Figure 4.5 The effect of slip/roll ratio on flash temperature and wear rate

with = 0.4, p0 = 1.5 GPa, v0 = 30 m/s ……………………….

87

Figure 4.6 The effect on wear rate and flash temperature due to variation

of with slip/roll ratio = -3%, p0 = 1.5 GPa, v0 = 30 m/s …………..

88

Figure 4.7 The effect on wear rate and flash temperature due to variation of p0

with slip/roll ratio = -3%, µ = 0.4, v0 = 30 m/s ……………………… 89

Figure 4.8 The effect on wear rate and flash temperature due to variation of v0

with slip/roll ratio = -3%, p0 = 1.5 GPa, = 0.4 …………………....

90

Figure 4.9 The effect of varying, p0, and v0 by 20% on (a) flash

temperature and (b) wear rate ………………………………….

95

Figure 5.1 (a) The friction occurs at the interface of the contacting bodies

and generates heat; (b)The heat generated from frictional work

is conducted into the rail, raising its temperatures during the

contact; (c) The rail cools down after passage of the train but

the wheel which is continuously heated attains a higher steady

state temperature. When the hot wheel contacts the rail

additional temperature rise occurs in the rail ……………….....

100

Figure 5.2 The contact patch location of rail – wheel contact can occur at

any points between rail head (A) and rail gauge (B). As the

contact location shifts from the rail head to the rail gauge the

contact pressure reduces whereas the amount of slip increases.

The wheel – rail contact was simplified into 2D brick model

with plane at the centre line of the contact……………………..

102

Figure 5.3 Wheel and rail temperature rise profile at the surface for

p0=1.5GPa, =0.3, Sr=-2%, a = 5.88 mm and v0 = 30m/s (a)

with frictional heating only (b) with frictional heating and

107

List of Figures

steady state wheel temperature effect …………………………. Figure 5.4 The temperature profile below surface for p0 = 1.5GPa, =

0.4, Sr = -3%, a = 5.88 mm, and v0 = 30m/s: (a) due to the

frictional heating only; (b) due to the steady state wheel

temperature, without the frictional heating; (c) due to

combination of frictional heating and steady state wheel

temperature …………………………………………………….

109

Figure 5.5 Orthogonal shear stress at certain depth: (a) 0.5m depth (b)

z/a = 0.1 (c) z/a= 0.5 ………………………………………….

111

Figure 5.6 Maximum orthogonal shear stress at each depth (p0=1.5GPa,

=0.4, Sr=-3%, a = 5.88 mm, and v0 = 30m/s) ……………….

112

Figure 5.7 The effect of friction coefficient variation on zx(max) at each

depth by considering the steady state wheel temperature effect

112

Figure 5.8 (a) The effect of frictional heating and steady state wheel

temperature on accumulated plastic strain after 3,000 cycles

and (b) the maximum temperature at each depth (with

frictional heating and steady state wheel temperature effect),

with p0=1.5GPa, =0.3, Sr=-2%, a = 5.88 mm, and v0 = 30m/s

113

Figure 5.9 Wear rate due to variation of for p0 = 1.5GPa, Sr=-3%, a =

5.88 mm, and v0 = 30m/s (FH = Frictional Heating; SSWT =

Steady State Wheel Temperature) ……………………………..

114

Figure 5.10 Wear rate due to variation of Sr for p0 = 1.5GPa, = 0.4, a =

5.88 mm, and v0 = 30m/s (FH = Frictional Heating; SSWT =

Steady State Wheel Temperature) ……………………………

115

Figure 5.11 Wear rate due to variation of p0 with thermal softening for =

0.4, Sr=-3%, a = 5.88 mm, and v0 = 30m/s (FH = Frictional

Heating; SSWT = Steady State Wheel Temperature) …………

117

Figure 5.12 Wear rate due to variation of v0 with thermal softening for =

0.4, Sr=-3%, a = 5.88 mm, and p0 = 1.4 Pa (FH = Frictional

Heating; SSWT = Steady State Wheel Temperature) ………....

117

Figure 6.1 Temperature rise due to Sr variation (p0 = 1.5GPa, µ = 0.5) ……….. 123 Figure 6.2 Temperature rise of the three cases (Sr = -0.5%, µ = 0.5) ………….. 123

List of Figures

Figure 6.3 Effect v0 on temperature rise (p0 = 1.5GPa, µ = 0.4). The

thermal penetration depth is shown clearly in the inset figure ...

125

Figure 6.4 Thermo-mechanical stresses due to Sr variation (p0 = 1.5GPa, µ =

0.5) …………………………………………………………………... 126

Figure 6.5 Thermo-mechanical stresses due to µ variation (p0 = 1.5GPa, Sr = -

0.5%) ………………………………………………………………… 126

Figure 6.6 Thermo-mechanical stresses for different cases (Sr = -0.5%, µ = 0.5) 127

Figure 6.7 The effect of temperature rise on shear yield stress, plotted

against depth; (a) temperature rise with depth, (b) shear yield

stress of UIC 1100 after softening, (c) shear yield stress of UIC

900A after softening …………………………………………...

128

Figure 6.8 Average wear rate of UIC1100 rail steel: (a) Case 1; (b) Case 2; (c)

Case 3 ………………………………………………………………... 129

Figure 6.9 Average wear rate of UIC 900A rail steel ………………………… 131 Figure 6.10 Wear rate comparison (a) case 1 (b) case 2 (c) case 3 ………………. 134

Figure 6.11 Hardening behaviour of UIC 1100 and UIC 900A rail steel .. 135 Figure 6.12 Maximum orthogonal shear stress (zx(max)) and effective shear yield

stress after softening (keff_soft) over different cycles (p0 = 2.1GPa, Sr =

-3%, v0 = 15m/s, and µ = 0.3) ………………………………………..

136

Figure 6.13 (a) The comparison of maximum shear yield stress (zx(max)) and

shear yield stress during softening (keff_soft) (b) the temperature rise

(p0 = 2.7GPa and µ = 0.3) …………………………………………..

137

Figure 6.14 Plastic strain increment for Sr =-1.5% and Sr = -3% (p0 = 2.7GPa

and µ = 0.3) ………………………………………………………….. 138

Figure 6.15 Comparison of shear yield stress during softening process

(case 2, Sr = -3%, µ = 0.3); (a) at z/a -0.06; (b) at z/a -0.9

141

Figure 6.16 Comparison of shear yield stress during softening (case 2, Sr =

-3%, µ=0.4) ……………………………………………………

142

Figure 7.1 At cycle (n) the cluster of failed brick may form crack-like

flaw (a). In the next cycle (n+1) the additional failed bricks

may increase and lead the fatigue growth at the flaw’s tip as

seen in (b). When the wear occurs the layers at the surface are

detached and cause all layers below to move up (c-d) ………..

146

Figure 7.2 Cluster of bricks and the angle of deformed structure at the 147

List of Figures

surface [22] ……………………………………………………. Figure 7.3 Schematic representation of brick model to identify subsurface

crack …………………………………………………………

147

Figure 7.4 The interaction between the number of cycles required to

initiate the crack and the wear rate with variation of slip/roll

ratio ( = 0.6, v0 = 30 m/s, p0 = 1.5 GPa). When the wear rate

has a rapid increase, the number of cycles required to initiate

the crack also increases ……………………………………….

150

Figure 7.5 Surface crack is identified when the maximum depth of flaw

exceeds 50µm. Maximum depth of flaw depends on the growth

of flaw tip and the removal of failed surface layers by wear (

= 0.6, slip/roll ratio = -5%, p0 = 1.5 GPa, and v0 = 30 m/s) ….

151

Figure 7.6 The number of cycles for crack initiation of UIC 1100 rail steel …. 154

Figure 7.7 The number of cycles to subsurface crack initiation for UIC

1100 rail steel ………………………………………………….

156

Figure 7.8 The number of cycles of crack initiation for UIC 900A rail steel …. 159 Figure 7.9 Number of cycles of subsurface crack for UIC 900A rail steel ……. 161

Figure 7.10 Comparison of number of cycles till crack initiation (a) case 1 (b)

case 2 (c) case 3 ……………………………………………………... 164

Figure 7.11 Comparison of the number of cycles of subsurface crack initiation

(a) case 1 (b) case 2 (c) case 3 ………………………………………. 166

Figure 7.12 The flaws may occur at the surface or at the subsurface (a).

The wear removes the layers at the surface and may cause the

flaw at the subsurface move up (b) and become surface flaw

(c) ………………………………………………………………

169

Chapter 1. Introduction

1

Chapter 1

Introduction

Railways have undergone major development since the nineteenth century. The increase

in total traffic has led to a rise in train loads and also an increase in train speeds. Hence

wheels and rails face more severe contact conditions. The dynamics of load and speed,

the variety of environmental conditions, and the rail profile can have an effect on the

different mechanical loadings transmitted through the contact. Experimental

investigations showed that a different contact patch location due to worn profiles can

influence the contact patch size, the geometry and the amount of pressure during the

contact [15]. The dynamic simulations also showed that a different contact pressure and

sliding velocity occurred in different sites of rail track [2, 35]. The amount of stress

experienced by the wheels and rails due to these dynamic conditions will govern the

material failures. If the magnitude of stress exceeds the yield strength of the rail

material there will be plastic flow in each cycle. If the load is still below the plastic

shakedown limit, the rail may experience cyclic loading resulting in repeated plastic

strains. When the plastic shakedown limit is exceeded the plastic strain accumulates in a

process called plastic ratcheting. If the accumulated plastic strain reaches the critical

strain to failure the ratcheting failures occur. Two of the inevitable failures in wheel-rail

contact due to plastic ratcheting are the material loss from the rail surface known as

wear and rolling contact fatigue cracking (see Figure 1.1). Rolling contact fatigue

cracking is associated with the fatigue phenomenon either at the surface or subsurface

and is due to repeated loading by the passing wheels [22]. Studies have found that the

growth of rolling contact fatigue crack is linked to the wear rate [36]. If the wear rate

exceeds the crack propagation rate the surface cracks will shorten or even disappear.

Therefore the prediction of wear and rolling contact fatigue crack initiation is vital in

rail maintenance strategies.


2

Figure 1.1 Micrograph of an etched sectioned test sample from twin disc test from [9]

Simulation for predicting wear and rolling contact fatigue crack initiation is important

to reduce the cost of rail maintenance. A model based on ratcheting failure, which is

caused by ductility exhaustion, has been developed by Kapoor et.al [9, 37, 38].

Widiyarta et.al [39] added the thermal effect due to frictional heating into the model and

found that the wear is sensitive to the temperature rise due to frictional heating in the

contacting bodies. The temperature rise in the rail causes the development of thermal

stresses and the reduction of material strength (thermal softening). Widiyarta [22, 39]

has shown that the thermal stresses and thermal softening from frictional heating have a

significant effect on the increasing the accumulated plastic strain resulting in higher

wear rate and increased tendency for rolling contact fatigue. However the evaluation of

the temperature rise due to frictional heating on wear transition has not been explored in

detail. Moreover the frictional heat in rail is conducted away easily and causes its


3

temperature to drop to the ambient temperature after the train has passed. On the other

hand the wheel may undergo frictional heating continuously which causes its bulk

temperature to become higher than that of the rail. When the hot wheel touches the rail,

there is an additional heat flow to the rail besides the heat generated from frictional

work. Consequently, additional thermal stresses develop in rail, especially near the

surface. The additional temperature rise also may cause enhanced thermal softening.

Greater thermal stresses and thermal softening can result in even higher wear and even

earlier crack initiation.

The severity of wear and rolling contact fatigue also depends on the material response

against the stress, which is determined by the material properties. One of the rail

material properties which is important in determining the rail life is the hardness [40].

The material that has greater hardness has been found to have a superior wear resistance

[41, 42]. However by having less wear, the crack is easier to initiate and grow. The

laboratory experiments have shown that other material properties besides hardness can

also influence wear performance and rolling contact fatigue [43-45]. Some of the

material properties such as strain hardening behaviour and critical strain to failure have

been included in the ratcheting model [45] but without the thermal effect. In order to be

able to evaluate the thermal effect on different rail materials, the thermal softening

behaviour should be included. This property is indicated by the reduction of yield stress

at elevated temperature.

In the current research the effect of operating conditions and material properties on wear

and rolling contact fatigue crack initiation were simulated using a two-dimensional

model based on ratcheting failure [34, 38, 46, 47] as shown in Figure 1.2. The effect of

frictional heating and the steady state wheel temperature were added into the model to

evaluate the thermal effect. In the model the wearing material was divided into the

smaller brick elements. Several material properties were assigned to each of them, i.e.

critical strain at failure (c), the accumulated plastic shear strain (), the initial shear

yield stress (k0), and the effective shear yield stress (keff). The initial shear yield stress

and critical strain to failure were randomly varied by 5% from their average values so

that each element could fail at a different time. The wear was calculated from the

amount of material loss from the surface over a number of cycles. The crack initiation


4

was evaluated from the number of cycles for initiation both at the surface and below

surface.

The work presented in this thesis first investigated the thermal effect on the wear. The

effect of the temperature rise from frictional heating and the steady state wheel

temperature became a specific focus. The evaluation of wear transition due to frictional

heating is also presented for BS 11 rail steel. Next two different rail materials, i.e. UIC

1100 and UIC 900A rail steel were investigated to see how they performed against

wear. The rail material properties included hardness, critical strain to failure, strain

hardening behaviour, and thermal softening behaviour. The operating conditions

consisted of the maximum contact pressure, the friction coefficient, the vehicle speed,

the slip/roll ratio, and the temperature rise. The investigation was finalized by

presenting the effect of temperature rise and material properties on rolling contact

fatigue crack initiation.

Figure 1.2 Research flow diagram from [10]

This thesis consists of 8 chapters. Chapter 2 contains a literature review of studies on

rail-wheel contact, thermo-elastic contact, the material response to the cyclic loading,

Hardness

Hardening behaviour

Critical strain to failure

Thermal softening

behaviour

Material Properties

Brick Model

Model

Failure: Wear, RCF

Overall performance

Maximum contact pressure

Friction coefficient

Vehicle speed

Slip/roll ratio

Temperature

Operating conditions


5

rail deterioration and different rail material properties. The model developed for this

thesis is presented in Chapter 3. This chapter describes the two-dimensional ratcheting

model for wear including the thermal effects. Chapter 4 presents the effect of frictional

heating on rail wear. The temperature rise due to frictional heating and its effect on the

thermal stresses and the plastic strain were also presented in this chapter. Chapter 5

presents the effect of steady state wheel temperature on rail wear. Chapter 6 presents the

effect of a rails material properties and the variation of operating conditions on wear.

Chapter 7 presents the thermal effect and material properties on rolling contact fatigue

crack initiation. Chapter 8 presents a summary of all the work, and the direction of

future research relevant to the thesis is presented in Chapter 9.


6

Chapter 2. Literature review

7

Chapter 2

Literature Review

2.1. Rail-wheel contact

The wheel-rail rolling contact problem has become a major concern in the effort to

improve rail life. The characterization of wheel-rail contact depends on the wheel and

rail profile, as shown in Figure 2.1. This figure shows a typical schematic diagram of

wheel-rail contact from the side view of a track (a) and from a cross section of the track

(b). In the cross section view, the wheel is made conical in order to accommodate the

movement in the curved track without slipping [11].

Figure 2.1 Wheel-rail contact (a) Side view of the rail track; (b) cross-section view of

the rail track from [11]


8

Marshall et. al [15] investigated the size of contact area and found it to be very small,

about 80-120 mm2, compared to the wheel and rail dimension. The length of the semi-

contact patch in the longitudinal direction is about 8-12 mm [48]. This dimension is

greater when the contact patch location moves to the rail gauge. The geometry of both

bodies also influences the size of the contact patch. The worn rail was shown to have

greater contact area compared to the new rail because the contact was more conformal.

Marshal et. al [15] used ultrasound to characterize the wheelrail contact patch

evolution. The results showed a good agreement with the numerical calculation using

Hertzian contact theory.

Figure 2.2 Contact patch shape due to lateral shift of the wheel-set from [12]


9

Contact can occur on the rail head. However, due to the wheel’s conical shape and its

lateral shift, the contact can move from the rail head to the rail gauge, which causes

flange contact, see Figure 2.2. On a straight track, the contact mostly occurs on the rail

head, and flange contact rarely occurs. On a curving track, the flange contact is difficult

to avoid. When it does occur, most of the contacts are sliding, which leads to higher

dissipation of energy due to friction. This frictional energy transforms into heat,

increases the temperature, causing thermal softening and thermal stress to develop and

results in excessive wear. Due to the wear and plastic flow, the wheel and rail profiles

are altered and can cause multiple contacts. These are common contacts between the rail

head-wheel tread and rail gauge-wheel flange. In multiple contacts the total contact area

increases and hence causes the total load to be distributed over a greater area [15].

2.1.1. Hertz-line contact theory

Hertz first introduced contact theory in 1882 with the hypothesis that the contact area

between the two bodies is elliptical. In order to be able to calculate the deformation, he

did simplifications in which each body in contact is considered to be an elastic half

space loaded over a small elliptical region on its plane surface [13]. When two bodies

are in contact and subjected to normal load, they deform elastically and give rise to a

contact area. His theory is valid under some assumptions [13, 49-51]:

1. The contact radius is much smaller than the radii of curvature of the contacting

bodies. Each of the contacting bodies is approximated by an elastic half space

loaded over the plane.

2. The dimensions of the contact area must be small compared to the dimensions of

each body so that the stress vanishes as the distance increases from the contact

surface.

3. The material of both bodies is assumed to be linearly elastic and hence the

material behaviour is based on Hooke’s law. The strains are sufficiently small

for linear elasticity to be valid

4. The contacting surfaces are smooth. There is no friction and hence only normal

pressure is transmitted.


10

5. The strain in the contact region is sufficiently small so that all deformations

occur in the elastic range.

2.1.2. Line loading of elastic half space

The line contact occurs when two cylindrical bodies are lying parallel at the y-axis, as

shown in Fig 2.3. The two bodies are pressed in contact with the normal load P per unit

length, and have a contact with a size of 2a in x-direction. The size b is much larger

compared to the size a, such a contact is referred to as line contact or two-dimensional

contact. The pressure distribution between the surface is given as,

( ) ( (

)

)

(2.1)

Where p0 is the peak pressure at the centre of contact and a is the semi-contact patch.

The total load per unit length is equal to

(2.2)

where E* is the contact modulus and R* is the equivalent radius, defined as:

(2.3)

(2.4)

Here, E is the Young modulus, is the Poisson’s ratio and R is the radius of the

cylinder. The subscripts refer to body 1 and body 2 respectively.

The semi-contact patch therefore is equal to:

(2.5)


11

When pm is equal to the mean pressure, the maximum contact pressure is equal to:

(

)

(2.6)

Figure 2.3 Line contact of two cylindrical bodies

By substituting Equation (2.6) into Equation (2.1), the pressure distribution becomes:

( )

( (

)

)

(2.7)

The relationship between the tangential traction and the normal pressure is given by

Amonoton’s Law of sliding friction, as follows:

( )

( )

(2.8)

is the constant friction coefficient. Combining Equation (2.7) and (2.8) results in the

tangential traction of:

( )

( ) (2.9)

The negative sign represents the positive velocity V, as shown in Figure 2.4.

y

z p(x)

x

P

Body 1

Body 2 L

Body 1

Body 2

R1

R2

x

z

2a

p(x)


12

Figure 2.4. Sliding contact from [13]

The stress field in the body can be calculated by substituting the normal pressure

distribution (Equation (2.7)) and the tangential traction (Equation (2.9)). Figure 2.5

shows that the amount of traction at point B on an elemental area ds is equal to the

amount of force p ds normally acting on the surface and the tangential force of q ds.

These forces determine the amount of stress in the body, see Figure 2.5 point A, which

can be evaluated by substituting the distribution of p(x) and q(x) into:

∫

( )( )

{( ) }

∫

( )( )

{( ) }

(2.10)

∫

( )

{( ) }

∫

( )( )

{( ) }

(2.11)

∫

( )( )

{( ) }

∫

( )( )

{( ) }

(2.12)

Because normal traction p(x) and tangential traction q(x) are proportional in Equations

(2.10) – (2.12), the analogy between the two can be expressed by

( )

( )

(2.13)

( )

( )

(2.14)

with .

a a

Q

Q V

x

z

P


13

Figure 2.5 An elastic half-space loaded by normal pressure and tangential traction

The stress components due to normal pressure are evaluated using the plane strain

condition, and they are equal to

( )

{ (

) } (2.15)

( )

(

) (2.16)

( )

(

) (2.17)

The stress components due to tangential tractions are given by

( )

(

) (2.18)

( )

{ (

) } (2.19)

The stress component ( ) is evaluated independently and is equal to [13]:

( )

{ (

) } (2.20)

s

B

A(x,z)

q(s)

p(s)

x

z

ds

a a

O


14

2.1.3. Rolling-sliding contact of elastic bodies

With a rolling/sliding contact between wheel and rail, when the tangential traction is

transmitted in the interface, some points in the contact patch may slip and others may

stick. The difference in tangential strain in the stick and slip area is known as the

slip/roll ratio or creepage. Hence the contact patch can have three situations, i.e. pure

stick, pure slip, or combinations between stick and slip. Figure 2.6 shows that when the

creepage is equal to 0, all areas in the contact patch stick, which is typically known as

pure rolling. As the value of creepage increases, the slip region expands and the

frictional force also increases. As the creepage reaches 100%, the traction force reaches

the maximum value and is equal to the frictional force. With this condition the entire

contact patch is in pure sliding.

Figure 2.6 Schematic depiction of traction curve with stick and slip regions from [14]

Figure 2.7 shows the distribution of traction of the three states mentioned previously.

With pure rolling, all regions of the contact patch are sticking; hence the value of c, half

the width of the stick region, is equal to a, half width of the contact region. The


15

distribution of the tangential traction is merely influenced by the traction distribution

acting over the strip –c x c, which is equal to:

( )

( ) (2.21)

for contact of cylinders. As the torque increases, the area of slip expands, which reduces

the value of d, and increases the slip region. The total traction will be equal to: q’(x) +

q”(x), with:

( ) ( ) (2.22)

When the stick region vanishes, the value of d is equal to 0 and the contact is in full slip.

Thus the tangential traction distribution will be equal to q’(x). This condition is

achieved when the traction force is equal to the limiting friction force. The creep curve

of tractive rolling contact for cylinders shows that at a certain stage the traction

coefficient reaches the maximum and remains constant.

Figure 2.7 The distribution of tangential traction in cylindrical contact from [13]


16

2.2. Thermo-elastic contact

The variation temperature has an effect on the contact between two bodies by altering

the stress field inside the two bodies and may change the contact conditions due to the

changes to the surface profiles [13]. When two non-conforming bodies with different

temperatures are in contact, heat will flow from the hot body to the cold body through

the contact area. It hence increases the temperature of the colder body and the thermal

expansion can cause the surface profile to become more convex. On the other hand, the

body that has a higher temperature will have thermal contraction, hence causing the

surface profile to be more concave. The expansion and contraction will have exactly the

same amount if both bodies have the same material properties, elastically and thermally.

Therefore, with the thermo-elastic problem, it is important to determine the contact

temperature distribution because it will affect the thermal stresses developed in the body

and may cause more deformation. The temperature rise also may alter the material

properties of both bodies, such as the material hardness. The reduction of material

strength, known as thermal softening, may increase the amount of deformation in each

loading cycle [35, 39].

In the wheel-rail contact, the temperature rise in the contact patch is mainly caused by

the sliding which increases the frictional energy between the two surfaces. The friction

that occurs between the two surfaces generates heat. This heat flows into both bodies in

contact and increases the temperature, especially around the contact surface. The

maximum value of the temperature due to frictional heating is known as the flash

temperature and occurs near the trailing edge of the contact. However, the heat

generated by frictional heating at the contact patch only temporarily increases the rail

temperature at the contact spot. When the wheel has passed, the rail temperature returns

to ambient. On the other hand, the wheels are continuously heated up by friction over

many cycles and hence accumulate heat in their body. The accumulated heat may

increase the wheel bulk temperature. After continuous running the wheel bulk

temperature reaches a steady state value dependent on many factors, including the

ambient temperature and the natural or forced cooling of the wheel. When a hot wheel

makes contact with the rail, there is an additional temperature rise within the rail at the

contact surface [7].


17

The flash temperature concept was first investigated by Blok [52]. Ertz & Knothe [7]

introduced a semi-analytical and numerical method, based on Blok’s formula, to

investigate the temperature rise at the wheel-rail contact for a Hertzian case. The

temperature rise occurs in a very thin layer as the sliding occurs and causes friction

between the two surfaces. They also investigated the increase in the wheel bulk

temperature as a result of the continuous frictional heating.

The changes of temperature in the body can also give rise to severe thermal stress. This

thermal stress was investigated by Goshima [53], Fischer et.al. [54], and Ertz and

Knothe [55]. Goshima conducted an analytical approximation of thermo-mechanical

stress for moving contact sources at constant velocity. The heat flow rate was assumed

to be equal to the work done against friction at the contact surface. The thermo-

mechanical solutions were provided with the contact pressure distribution as an input to

the model. Fischer et. al [54] conducted an analytical solution for temperature

distribution in a dimensionless form and the corresponding thermal stress was evaluated

using finite element modelling. Their results showed that very high thermal stress is

confined in a very thin layer. The magnitude of the thermal stress can be of the same

order as the mechanical stress. Its value depends on the contact conditions, such the

contact stress, the vehicle velocity, the amount of friction and slip, the size of the

contact patch, and the thermal properties of the material, i.e. thermal conductivity,

thermal expansion, and thermal diffusivity. Ertz and Knothe [55] did an approximate

analytical solution for the thermal stresses provided using the line contact model. The

effect of wheel bulk temperature was also taken into account. They found that thermal

stress reduced the elastic limit of the material and hence caused the yielding to occur at

lower mechanical loads.

The increased temperature also may modify the mechanical and metallurgical properties

of the contacting surface [27]. When the temperature reaches 200C the surface is

oxidized resulting in the formation of a magnetite layer [56]. This prevents the metal to

metal contact between the two bodies which can influence the rate of wear [27, 56].

This layer also limits the amount of heat conducted from the hot body to the colder one.

Thus it can influence the effective thermal conductivity of the steel and the oxide layer

[57]. The thickness of the oxide layer can reach 20µm for a friction coefficient of 0.6

and contact pressure of 900MPa. However, this layer may become too weak to resist


18

higher friction and contact pressure. The oxide layers can be penetrated at higher

contact pressure and friction [27] resulting in direct metal to metal contact. Lim and

Ashby [27] referred to this process as low delamination wear in their wear map.

2.3. Rail-wheel operating conditions

The efficiency of railway transportation depends on the many factors that influence the

contact between wheel and rail. Although the contact patch is only the size of a

thumbnail, undesired phenomena often occur in this contact area. High contact stresses

due to a high load may lead to the yielding of the material or rolling contact fatigue. The

presence of frictional force also may result in surface mass loss, which will alter the

profile of contacting bodies. The vehicle speed, and traction and braking can alter the

value of friction and sliding, which can have an effect on the development of heat and

hence increase the temperature. Therefore thermal stress and the possible thermal

softening may advance the wear and RCF failure. The worn profile and the irregularities

of both bodies can cause poor dynamic conditions for wheel rail contact in following

cycles. As the surface becomes conformal, the contact stresses drop because they are

distributed over a larger area. This has an influence on passenger comfort and, in the

worst-case scenario, may threaten safety as failure may cause a derailment and the like

if the rail is poorly maintained. Therefore improvement in bogie performance, rail

grinding, lubrication, and material selection should be considered in order to maintain

the wheel-rail operating conditions in a safe range [1].

2.3.1. Contact pressure

In railway operation, the amount of load from the bogies passing on the rail drives the

contact stresses, in particular the maximum contact pressure. Besides the load, the

contact pressure also depends on the wheel-rail profile, which determines the location

and size of the contact patch that occurs in each contact. Marshal et.al [15] conducted

ultrasonic measurements on contacts between unused wheels and rails. The


19

measurements showed that the maximum contact pressure increases for smaller contact

patches, see Figure 2.8. On a straight track, the wheel is normally in contact with the

rail head. This commonly applies to new profiles. The contact is elliptic and about 18

mm longitudinally and 11 mm laterally [1]. Yan & Fischer [48] used the Hertzian

contact to predict the contact geometry, which results in a longitudinal semi-contact

patch of around 8 mm for the rail head contact where the maximum contact pressure is

around 1,250 MPa. Lewis & Olofsson [16] mapped the wear of several rail materials

under various contact conditions, as shown in Figure 2.9 From this figure, it can be seen

that the maximum contact pressure for a wheel tread/rail head contact lies in the range

of 500-1,500 MPa. With a worn wheel/rail, contact between the wheel’s tread and the

rail gauge may occur, known as multi-point contacts or double contacts [15], see Figure

2.10. In multi-point contacts, the contact pressure reduces because it is distributed over

the larger area in which all the contacts happened.

Figure 2.8 Ultrasonic measurement for unused wheel/rail contact from [15]


20

Figure 2.9 The wear data points under various contact conditions from [16]

Figure 2.10 (a) Single contact; (b) double contact from [15]

With a curved track, contact between the wheel flange and rail gauge corner frequently

occurs. It can be seen in Figure 2.9 that the maximum contact pressure in the rail

gauge/wheel flange contact can reach 2.7 GPa. Yan & Fischer [48] also showed similar

values at 2.5-2.6 GPa with a semi-contact patch size of about 10 mm and contact radius

of 13 mm. Telliskivi et.al. [35] also demonstrated that with a contact radius of 20 mm,

the maximum Hertzian contact pressure is predicted to be 3.6 GPa.


21

Studies on contact pressure have shown its effect on wear and rolling contact fatigue.

Franklin et.al. [38] found that the level of steady state wear rate increases as the peak

pressure increases. Bolton & Clayton [28] also did an experimental investigation of

rolling sliding damage in several rail materials and tyre steel. The results showed that

the mass loss during testing increased with higher pressure and sliding. The wear map

based on [28] and [29] shows that the wear coefficient increases with contact pressure,

with greater sensitivity as the sliding increases. The effect of contact pressure on rolling

contact fatigue crack initiation was investigated experimentally by Tyfour et.al [6] who

carried out a twin disc test of rolling sliding contact on pearlitic rail steel. They found

that the RCF life is strongly correlated with the degree of ratcheting. Their experimental

results showed that the RCF life drops as a higher peak pressure is applied.

2.3.2. Friction / Adhesion

Friction is the resistance that hinders the tangential motion of two interacting surfaces

[1]. A friction coefficient is defined simply by dividing the tangential force by the

normal load:

. The value of the friction coefficient depends strongly on the

weather and contamination conditions. Low values of friction coefficient may lead to

vehicle slip on rails, whereas a high friction coefficient leads to excessive wear.

However, when braking or accelerating, high friction is necessary. Low friction is also

needed to prevent noise and corrugation in sharp curves.

The friction coefficient covers a wide range of values. Table 2.1 shows field

measurements using a salient system tribometer. It shows that the values can vary

between 0.05-0.7, depending on the weather and contact conditions. In a wheel rail

contact, the term adhesion or traction coefficient, tc, is used to define the ratio of the

traction force to the applied normal force. If the traction force is less than the limiting

friction force, the contact is in a state of partial slip [58], with some areas of the contact

region in stick and others in slip. Table 2.2 lists the adhesion coefficient for broader

conditions.


22

Table 2.1 Friction coefficient measurement from [1]

Condition Coefficient of

friction

Sunshine dry rail, 19C 0.6-0.7

Recent rain, 5C 0.2-0.3

With a lot of grease on rail, 8C 0.05-0.1

Damp leaf film on rail, 8C 0.05-0.1

Table 2.2 Adhesion coefficient for wide range conditions from [1]

Condition of rail surface Adhesion

coefficient

Dry rail (clean) 0.25-0.30

Dry rail (with sand) 0.25-0.33

Wet rail (clean) 0.18-0.20

Wet rail (with sand) 0.22-0.25

Greasy rail 0.15-0.18

Moisture on rail 0.09-0.15

Light snow on rail 0.10

Light snow on rail (sand) 0.15

Wet leaves on rail 0.07


23

Olofsson & Telliskivi [2] also compared field measurements of friction coefficients in a

laboratory study. The field measurements were carried out on Älvsjö track at the rail

head and rail gauge for curved track. The details are in Table 2.3. This track carries

almost entirely unidirectional commuter trains with an average speed of 75 km/h. It

contains two different pearlitic steels in the same curve: UIC 900A grade rail (0.7% C,

1% Mn with ultimate strength of 880 N/mm2) and UIC 1100 grade rail (0.7% C, 1%

Mn, 1% Cr with ultimate strength of 1080 N/mm2). The amount of mega tonne traffic

on each curve is: 2 years (12.3 Mt), 3 years (12.8 Mt), and 5 years (24.7 Mt). Table 2.4

& 2.5 show the measurements of the friction coefficient on different occasions. There is

a significant difference between lubricated and unlubricated values. Similar results were

shown between field measurements and laboratory tests for unlubricated conditions but

were a little lower in the test on a full-scale lubricated rail.

Table 2.3 Älvsjö track data from [2]

Table 2.4 Friction coefficient of Älvsjö track data; measured during daytime, no rain, no

sunshine, and at air temperature of 19C from [2]


24

Table 2.5 Friction coefficient of Älvsjö track data; measured during daytime, rain

during measurement of sites C and D, no rail during measurement of sites A and B, no

sunshine, and at air temperature of 15C from [2]

The value of friction and traction coefficient depends on the contact conditions of a

wheel and rail and also on the wheel and rail material. Baek et.al. [59] conducted an

experimental study of the traction coefficient at the wheel/rail interface under dry

conditions with varying humidity and temperature. They found that rolling speed,

contact pressure, slip ratio, temperature and humidity did influence traction

characteristics. The maximum traction coefficient reduces with increasing rolling speed,

temperature and humidity, but increases with an enhancement of the contact pressure.

On the other hand, the steady traction coefficient increases when the rolling speed and

contact pressure rise, but reduces as the temperature and humidity decrease. The effect

of the slip ratio on the traction coefficient can be seen in that the maximum and the

steady traction coefficient decrease gradually with an increase in the slip ratio. The

results of investigations into the friction coefficient under various test methods are

given in Table 2.6.

The value of the friction coefficient can be controlled by sanding the rail, by lubricating

the wheel flange and rail gauge corner in curves, and by using friction modifiers. The

addition of fluid lubricant or friction modifier in the contact area can maintain the level

of friction in the following categories [1]:

a. Low coefficient friction modifiers: reduce the friction coefficients to 0.1 at the

wheel flange-rail gauge corner contact surface

b. High coefficient friction modifiers: give a value of friction coefficient of 0.2-0.4

at the wheel tread-rail head interface


25

c. Very high coefficient friction modifiers, i.e. friction enhancers: typically used to

rise the adhesion for both traction and braking

Table 2.6 Friction coefficient investigation under various set method from [3]

Author Test apparatus Load/Contact

pressure

Rolling

speed

(km/h)

Test

condition Peak Slip at

peak

(%)

Stable

(5% slip)

Zhang

et. al.

[60]

Full-scale roller

rig (using an

actual bogie)

44 kN

10-70 Dry 0.57-0.5 2 0.57-0.5

67 kN

10-70 Dry 0.55-0.44 1-2 0.52-0.44

44 kN

120-240 Wet 0.13-0.07 0.5-1 0.12-0.065

67 kN

80-240 Wet 0.11-0.05 0.5-1 0.105-0.05

Jin et.al.

[61]

Full-scale roller

rig (using an

actual bogie)

67 kN 140-300 Oil 0.055-0.045 0.052-0.044

135 kN 140-300 Oil 0.05-0.04 0.048-0.037

Harrison

et.al.

[14]

Triborailer

(used on actual

track)

Push tribometer

Dry

0.52 0.5

Dry

0.7 0.7

Nagase

[62]

Instrumented

bogie on test

vehicle (run on

test track and

actual routes)

Variable Variable

Dry

Range of 0.2-0.4

Wet

Range of 0.05-0.2

Oil

Range of 0.05-0.07

Leave

Range of 0.05-0.1

Gallardo

et.al. [3] Twin-discs

1500 MPa

(7,7 kN) 3.54

Dry

0.6 2 0.54

Wet

0.2 1 0.17

Oil 0.07 1 0.06

The introduction of friction modifiers may reduce the frictional force at the contact

surface. They can reduce the amount of traction and generate less heat. Hence the

degree of wear reduces [39]. However, the presence of fluid at the interface can enhance

the crack propagation by pressuring mechanism [31, 63]. Fletcher et.al. [31]

investigated the effect of fluid pressurisation on rolling contact fatigue cracks. They


26

found an indication of very high growth rates for cracks containing pressurised fluid.

With larger crack sizes, when the contact could not seal the fluid entrapped inside the

crack, the crack growth rate was much lower, but sufficiently fast, at around ten times

the wear rate.

2.3.3. Creepage (Slip/roll ratio)

When two elastic bodies make contact, the conditions of the contact region may

combine stick and slip [58]. If there is no tangential force, all points in the contact

region are stick, known as pure rolling. When the tangential force emerges, the

condition of the contact patch can be either partial slip or pure slip, both known as

tractive rolling. The presence of slip in the contact patch is known as creep or creepage

[17]. The creep is important in maintaining the stability of a running vehicle. There are

three kinds of creep: longitudinal creep, lateral creep, and spin creep. Longitudinal

creep, x, is caused by traction or braking forces, see Figure 2.11. It occurs when there is

a difference in the relative velocity of both bodies, as in Equation (2.23) where Vw is the

relative velocity of the wheel and Vr is the velocity of the rail relative to the train.

Lateral creep, x, arises from the yaw of the wheel, and spin creep, , emerges from the

conicity of the wheel. All creeps are derived by the following relationships [13]:

Longitudinal creep ratio,

(2.23)

Lateral creep ratio,

(2.24)

Spin parameter, ( )

{

( )

} (2.25)


27

Figure 2.11 Creep motion of railway wheel from [13, 17]

The creepage is saturated when the stick region disappears and the traction force

reaches its maximum at the limiting friction force (Q = P), and remains constant,

theoretically. The relationship between creepage and friction coefficient is described by

Carter’s equation [58], assuming that the friction coefficient is constant:

[ {

}

] (2.26)

Matsumoto et.al. [64] investigated rail corrugation on a curved track. They found that

corrugation is caused mainly by large creepage and vertical force fluctuation on the

wheel/rail contact surface. The magnitude for longitudinal creep is 1.8% and 0.6% for a

trailing wheel-set and leading wheel-set, respectively; whereas the lateral creep is 0.3%

and 1.3% for a trailing wheel-set and leading wheel-set, respectively. Together with the

vehicle speed, the amount of creepage is responsible for the magnitude of the sliding

velocity on the wheel-rail contact. Olofsson & Telliskivi [2] found that the sliding

velocity was never above 0.1 m/s and 0.9 m/s for the rail head-wheel tread and rail

gauge-wheel flange, respectively.

Vw

Vr

Vw

Vr


28

2.3.4. Train speed

In rail transportation, various trains are designed for specific purposes. Heavy axle loads

involve an application of railway transportation commonly used for freight. Increasing

axle loads can be an effective way to reduce freight costs but this may induce greater

wear and gauge corner cracking in rails. Increasing axle loads also results in a reduction

in vehicle speed. One example of this is the ore lines in Sweden, where for an empty

train, the maximum speed is around 60-70 km/h, whereas when the train is loaded up to

30 tonnes, the maximum speed is reduced to 50-60 km/h [1]. In addition to freight

trains, there are commuter trains with a common speed of between 3.6 km/h – 75.6

km/h [65]. Commuter trains are typically used for short distance travel. For long

distance travel, High Speed Trains (HST) are a convenient way to travel instead of road

or air transport. HST typically have a speed of at least of 200 km/h [18]. In the last

decade, the maximum operating speed on the Tokaido line in Japan has commercially

increased to 270 km/h while the TGV Atlantique line in France reaches a maximum

speed of 300 km/h and a short MAGLEV in China runs at maximum speed of 430 km/h.

The standard of maximum speeds for new lines is even higher for new HST lines such

as the Madrid-Barcelona line, which is 350 km/h. Higher speeds seem to be

commercially unfeasible due to noise problems, operating costs, and other technical

problems.

The base model for HST is the Japanese Shinkansen, followed by three other models:

TGV (Train à Grande Vitesse); tilting HST; and MAGLEV. The difference between the

Shinkansen and the TGV is that the latter has a locomotive while the other has not [66].

TGV also can operate on conventional tracks as well. The tilting HST allows for higher

speeds on conventional lines with tight curves. The bogies remain firmly attached to the

rail while the carriage body tilts, and hence compensates for the centrifugal force while

the train runs at high speed during curving. This model costs less than the Shinkansen or

TGV. However it only can reach a maximum speed of 250 km/h. Today, the tilting

mechanism has also been adopted in TGV trains and Shinkansen. In terms of speed, the

MAGLEV can operate at speeds of 500 km/h. It relies on electromagnetic forces in

order to make the vehicle hover above the track and move forward at high speeds. A


29

comparison of these models, in terms of operating speed, compatibility with the existing

track, and construction costs, is presented in Figure 2.12.

Figure 2.12 Characteristics of four High Speed Train (HST) models from [18]

With a sharp curved track, the magnitude of vehicle speed will depend on the actual

elevation of the outside rail and the degree of curvature as in Equation 2.27 [67]:

√

(2.27)

Where Vmax is the maximum allowable operating speed (miles per hour); Ea is the actual

elevation of the outside rail (inches), and D is the degree of curvature (degrees). A

standard table of vehicle speed for various Ea and D is provided in [67]. The onsite

measurements made by Saulot et.al [68] showed that for a curve radius of 300, the

vehicle speed is about 60 km/h. Similar measurements were also shown by Zhai &

Wang [69], for a curved track of 287 m with a speed of 65km/h.


30

2.3.5. Rail roughness

Surface roughness is important in the mechanism of rolling/sliding contact between

surfaces. It was shown by Kapoor and Johnson [70] that even under a small load,

surface roughness can cause severe contact stresses. They also showed that the contact

pressure at the asperities is much higher than the nominal (average) value [71]. Hence

the shakedown limit can be easily exceeded leading to severe plastic flow.

The flash temperature generated by the friction also can increase significantly at the

asperities contact in a rough surface. As the wheel continuously accumulates the

frictional heat, its bulk temperature may also significantly increase. This heat will be

transferred to the rail as asperities contacts occur. Nevertheless, the consideration of the

asperities contact increases the complexity of the problem. Besides higher contact

pressure at the asperities, Chen et.al [48] also found that the amount of traction

coefficient reduces as the amplitude of surface roughness decreases. Tomberger et.al

[12] showed that the traction coefficient can increase for rougher surface. After repeated

contact, the asperities are plastically deformed and hence their height variation is

reduced [24]. As the asperities are flattened, this may lead to changes in the friction

coefficient. The variation of the friction coefficient will certainly modify the amount of

frictional heat generated during sliding. Wang and Liu [72] also found that the thermal

distortion due to frictional heating may influence the contact pressure.

2.4. Rail material properties

In a wheel-rail operation, selection of rail materials with regard to the operating

conditions, the environment, and the geometry of the track is vital in order to optimize

the cost and to prolong the rail service life. The selection of rail material cannot be

separated from the material properties of rail, i.e. composition and microstructure. The

improvement of rail material has focused on wear resistance, as this is inevitable in any

rail operation. Many investigations have shown that material with a higher strength and

better strain hardening provides good wear resistance [4, 42, 45]. Fig 2.13 shows the

laboratory results of several rail grades with different levels of hardness. It shows that


31

the material with greater hardness has greater wear resistance. For many years pearlitic

steel has been used in railway track because it has good resistance to wear and plastic

flow. The development of this steel continues through the modification of its

composition, such as in the carbon and manganese content.

Figure 2.13 The effect of hardness on the wear rate for several rail grades from [19]

2.4.1. Rail steel specifications

There are three types of rail steels commonly used in railway operations, based on the

tensile strength [19]:

1. Normal grades

These grades have a minimum tensile strength of 700 N/mm2. They are used for

normal service conditions, including high and medium speed passenger trains

going at 200 km/h and 100 km/h respectively. They are also used with heavy

axle load freight. Examples of these grades are BS11 Normal grade and UIC


32

860-0 Grade 70 with a detailed composition of 04-0.6 % C, 0.05-0.35 % Si, and

0.8-1.25 % Mn.

2. Wear-resisting grades

These grades have a minimum tensile strength of 880NM/mm2. These grades

have greater hardness and hence greater wear resistance compared to normal

grades. The content of carbon and manganese has been increased in various

combinations to achieve finer pearlite lamellae. These grades are normally used

for track with high wear, such as heavy axle load, high-density traffic, and for a

sharp curved track. Examples of these grades include BS11 wear-resisting grade

A/B and UIC 860-0 Grade 90A/B with detail composition of 0.55-0.8 %C, 0.1-

0.5 %Si, and 0.8-1.7 %Mn.

3. High-strength grades

These grades have a minimum tensile strength of 1080-1200 Nm/mm2. These

grades have further refined the pearlite structure to gain increased strength. This

has been achieved through high-alloy additions or the use of accelerated cooling

from the austenite range. These types of rail are used for the track under very

heavy axle loads and in a tightly curved track. For track that has exceptionally

high wear rates, the austenitic 14% Mn rail is considered with a composition of

0.75-0.9 %C, 0.2-0.4 %Si, and 1-14 %Mn.

2.4.2. Pearlitic steel

Pearlitic steel is a kind of steel with a carbon content of 0.77%. The pearlite structure

consists of a mixture of two phases, ferrite (Fe) and cementite (Fe3C). This structure is

shown in a dark shadow in Figure 2.14, where ferrite is represented as , and cementite

as Cm. The pearlite structure is derived from the cooling process of austenite grains to a

temperature below A1, known as the eutectoid point or pearlite point. Below this point,

the steel microstructure varies widely and pearlite is just one such structure [20]. The

structure of pearlite grain is shown in Figure 2.15.


33

Figure 2.14 Iron-carbon phase diagram and the formation of pearlitic structure from

[20]

Figure 2.15 Pearlitic structure from [4]


34

2.4.2.1. The hardness of pearlitic steel

Many studies have shown that pearlitic steel has a higher hardness that is found in the

finer pearlite structure characterized by the reduction of lamellar spacing. To obtain this

finer structure of pearlite, a higher degree of cooling is needed. The addition of

manganese can also lower the pearlitic transition temperature, which results in the

reduction of lamellar spacing and the size of the pearlite colonies [21]. When the

thickness of the cementite lamellas is reduced, the hardness increases, and hence the

wear resistance improves [4, 21], see Figure 2.16. The microstructure of pearlitic steel

used by Yokoyama et.al [4], see Figure 2.17 and Table 2.7, also revealed that hardness

can be increased by controlling the colony size, lamellar spacing, and the volume

fraction of cementite. The level of hardness tends to increase when the colony size and

lamellar spacing are decreased and the volume fraction of cementite increased.

Figure 2.16 The effect of lamellar spacing to the hardness from [21]


35

Figure 2.17 Microstructure of pearlitic steel with detail properties as in Table 2.7 from

[4]

Table 2.7 Properties of pearlitic steel in Figure 2.17 from [4]

Decreasing the lamellar spacing in pearlitic structure is typically conducted by raising

the carbon content. Ueda, et. al [73] found that an increase in the carbon content of

pearlitic steel promotes the grain refinement of the matrix ferrite. This results in the

hardness and work hardening rate of the contact surface increasing and hence the wear

resistance is improved. Tomota, et.al [74] and Herian et.al [21] also found that the

lamellar spacing depends on the carbon content and the pearlite temperature point.


36

According to the Hall-Petch relation, the 0.2% proof stress was observed to be

proportional to the inverse square root of the interlamellar spacing [75]:

√ (2.28)

with d is the inter-lamellar spacing in mm.

Gomes, et.al [76] also investigated thoroughly the effect of microstructure on the

mechanical properties of pearlitic steels. They found that interlamellar spacing, d,

increases as the austenising temperature increases, and with the addition of Nobium.

The tensile strength increases for larger prior austensite grain size and for smaller

interlamellar spacing. The ductility increases for smaller prior austenite grain size. They

also found that the interlamellar spacing has an important effect on crack initiation as

the material with the largest spacing has the lowest resistance to crack initiation,

although its yield stress is not the lowest. Crack propagation is insensitive to variation in

the interlamellar spacing.

Beynon et.al. [77] investigated the rolling contact fatigue of three pearlitic steels in the

laboratory under water lubricated conditions. They found that hard oxide inclusion did

contribute to crack initiation. Head-hardened grade eutectoid steels had the highest

resistance to rolling contact fatigue, followed by naturally hard eutectoid steel. The

lowest strength steel had the worst performance on the rolling contact fatigue.

2.4.2.2. Temperature effect on the pearlitic steel’s

strength

Temperature has a great influence on altering the steel material properties. At certain

levels, it may change the microstructure and hence change the value of properties such

as yield stress. Nakkalil, et.al [78] established that the yield stress drops as the testing

temperature increases. They evaluated the effect of temperature on several plain carbon

and low alloy eutectoid rail steels by performing compression testing at high strain rates

in the temperature range 25-680C. Figure 2.18 also shows the plot of yield stress


37

against elevated temperature for three different rail grades, i.e. grade 900A, 1100, and

HSH) [22]. All steel grades show a reduction of yield stress due to temperature rise.

Figure 2.18 Yield strength of rail steel at elevated temperature from [22]

The level of temperature rises during rail-wheel slide may reach the austenite

transformation close to the contact surface. While the wheel is continuously rolling, the

rail cools rapidly after contact occurs. Very rapid cooling prevents the austenite from

transforming to the ferrite + cementite structure and hence a new structure, martensite,

may form in the surface layer. The formation of martensite is responsible for the

formation of white etching layers [75] which have a high strength level [20]. However,

in this transformed layer, a crack is often formed and may propagate deeper into the

material [79]. The cracks are even worse if a stretched MnS inclusion takes place in the

highly deformed layers as they can propagate rapidly through the brittle martensite

around them.


38

2.4.3. The effect of various rail steel microstructures on

wear and RCF

The material properties of rail steel determine the ability to resist damage. Many studies

have shown that different properties may affect the wear resistance and rolling contact

fatigue differently [4, 30, 43, 45, 76, 80, 81]. The ability of various microstructures to

resist wear was investigated thoroughly by Wang et.al [82]. They found that wear

resistance increased in the following order: martensite + carbide + retained austenite,

spheroidized structure, martensite, bainite, lamellar pearlite. They concluded that wear

resistance for various microstructures is strongly related to micro-structural thermal

stability, resistance to plastic deformation, and resistance to nucleation and crack

propagation. They also found that wear resistance does not depend on the initial

hardness of the material. In their findings, although the hardness of lamellar pearlite was

less than martensite, the lamellar pearlite had better wear resistance than the martensite.

Pearlitic steel, the most commonly used rail steel, has better wear resistance and rolling

contact fatigue especially for a fine microstructure. However, there are limitations in

producing fine grains of pearlite in the manufacturing and heat treatment processes. It is

believed that the hardness of pearlitic has reached its limit and hence it would be

difficult to improve wear performance beyond the current state [45, 69]. Bainitic steel is

a rail steel that offers comparable strength, derived from an ultrafine structure with

many dislocations that do no damage but offer high strength [80]. It is an alternative

material that has greater hardness and ductility yet a comparable performance to

premium pearlitic steels [62]. It is used in special track applications, such as high axle

crossing diamonds. Investigations of these two rail steels have been carried out by many

researchers [41, 42, 45, 62, 80, 81, 83].

Zapata et.al. [44] investigated the behaviour of pearlitic and bainitic steel using a disc-

on-disc tribometer with a low load regime. The results showed that mass losses were

similar for both steels, regardless of their initial hardness. However, with high sliding

the pearlitic steel showed higher sliding resistance compared to bainitic steel [42],

which occurs because the pearlitic steel has better strain hardening than bainitic steel.

Also, the wear mechanism that occurs in the pearlitic was observed to be oxidative


39

wear, whereas adhesive wear was the main removal mechanism, which causes more

mass loss on the surface. Alwahdi et.al [45] studied the effect of the roughness of

pearlitic and bainitic rail steel on wear. The results showed agreement that the level of

wear of pearlitic steel is lower than that of bainitic steel.

Aglan [80] revealed that bainitic rail steel shows more ductile fracture behaviour, such

as tearing, and has more resistance to crack initiation and propagation. Its performance

is superior over standard head hardened pearlitic steel as less cracking occurs compared

to the pearlitic steel. This result is supported by the findings by Yokoyama et.al [41].

They revealed that bainitic steel has better resistance to RCF damage for all angles of

attack tested. Nevertheless, the manufacturing and welding difficulties are drawbacks

of bainitic steels.

Wang et. al [43, 82] also investigated surface hardness in sliding wear, finding that it

depends on the frictional temperature and the thermal conductivity of the material. The

material with lower thermal conductivity has a higher surface temperature because the

heat at the surface is transmitted less into the body. The thermal conductivity of the

pearlite is greater than that of the martensite and much higher than that of cementite,

with values in the following order: 51.9, 29.3 and 4.2 W/mK. Material with a higher

thermal conductivity was found to have better wear resistance. However, although

pearlite has higher wear resistance, its resistance to fatigue is less than martensite. It

becomes worse when the material contains large incoherent particles, such as carbides,

because this can initiate a crack and promote propagation [43]. Cvetkovski et. al [84]

evaluated the effect of temperature on the thermal softening of different alloying levels

of silicon and manganese in pearlitic steels. Thermal softening causes micro-structural

changes, which alters the mechanical behaviour and decreases the fatigue life. Material

with a higher content of Si and Mn has better resistance to thermal softening for both the

original and plastically deformed material. Hence they present a greater fatigue life than

that of the lower alloyed steel.


40

2.5. Material response

2.5.1. Material response to cyclic loading

How a ductile material responds to cyclic loading depends on the magnitude of applied

load on the material as shown in Figure 2.19 [6]. The material responds in four different

ways:

a. If the stress is under the elastic limit of the material, its response will be

perfectly elastic.

b. Elastic shakedown occurs if the stress exceeds the elastic limit and causes plastic

deformation but, due to the development of residual stress and strain hardening,

the steady state behaviour is perfectly elastic. The stress does not exceed the

elastic shakedown limit and hence forms a closed loop. In this case the material

will fail through high cycle fatigue.

c. Plastic shakedown occurs if the stress exceeds the elastic shakedown limit but is

less than the plastic shakedown limit. There is no accumulation of plastic strain

and hence it is referred to as cyclic plasticity. The material will most likely fail

through low cycle fatigue.

d. Ratcheting occurs if the stress exceeds the plastic shakedown limit, known as the

ratcheting threshold. There is an open elastic-plastic loop, and the material

accumulates in the uni-directional plastic strain in each cycle, by a process

known as ratcheting. In this case the material will fail through either ratcheting

failure or low cycles fatigue, depending upon which mechanism results in the

shorter life. It is also possible that the two mechanisms can be additive providing

an even shorter life [34].


41

Figure 2.19 Material response to cyclic loading in rolling-sliding contact from [23]

2.5.2. Shakedown

The shakedown theorems stated that if the residual stresses found in any system, in

combination with the stresses due to cyclic load, do not exceed yield at any time, then

elastic shakedown will take place [24, 46]. When the material is subjected to a cyclic

loading, its response depends on the magnitude of the maximum stress to the yield

stress, as shown in Figure 2.20. As long as the contact pressure is under the elastic limit,

then the material responds elastically. When the contact pressure increases above the

yield limit, the material flows plastically. However, after the load passes, the residual

stresses developed in the material protect it from the tendency of plastic flow. The

plastic flow which occurs in each loading also causes the material to harden up. Strain

hardening also prevents the material from further plastic flow. Figure 2.20 reveals that

below 0.25, the shakedown is controlled by subsurface stresses, and above this

value plastic flow occurs at the surface, forming cyclic plasticity or ratcheting. The

shakedown limit decreases for higher . However, for perfectly plastic material and

material that hardens, the shakedown limit is raised due to the existence of residual

stresses and strain hardening. With the development of residual stress and the strain

hardening of the material shakedown could be assisted by a change in geometry which

can increase the contact area and reduce the maximum contact stress. Kapoor et.al [85]

found that plastic flow in the lateral direction alters the contact geometry. The


42

conforming contact causes reduction of the stress level because it is distributed over a

larger area and may lead the shakedown to a steady state.

Figure 2.20 Shakedown map for repeated sliding of a rigid cylinder over an elastic-

plastic half space from [24]

2.5.3. Plastic ratcheting

From the shakedown map, plastic ratcheting is enhanced when exceeds 0.25 where

plastic flow occurs at the surface. Figure 2.21(a) shows the elastic stress due to

frictional traction at the surface, q, using Hertz contact pressure. When the maximum

contact pressure, p0, exceeds the shakedown limit, p0s, the plastic strain cycles occur in

the material, as shown in Figure 2.21(b). Hence the ratcheting strain per cycle is a

function of the factor p0/p0s, by which the maximum Hertz contact pressure exceeds the

shakedown limit [24].

Under cyclic loading, failure of the material can occur due to low cycle fatigue or

ratcheting. The criterion of rupture used in both processes is determined by the number


43

of cycles to failure. If the reversing strain is acting alone, then the number of cycles to

failure is given by the Coffin Manson equation:

(

)

(2.29)

where is the number of cycles to failure, is the range of reversing plastic strain,

C is constant with a magnitude comparable to the fracture strain in monotonic loading,

and n is equal to 0.5.

Figure 2.21 Repeated sliding contact: (a) stress state on the surface; (b) plastic strain

cycles from [24]

The failure due to plastic ratcheting is determined by the number of cycles of

accumulation of unidirectional plastic strain to reach the critical value of failure ( )

[25]. Therefore, this number of cycles to failure is given by

(2.30)

where is the ratcheting strain per cycle.


44

Kapoor [25] further hypothesized that the modes of failure, i.e. low cycle fatigue (LCF)

and ratcheting failure (RF) are competitive. The one that has the lower number of cycles

to failure is activated. This hypothesis is supported by the data in Figure 2.22, which

reveals that the ratcheting failure will dominate the failure.

Figure 2.22 Competing modes of failure: low cycle fatigue (LCF) and ratcheting failure

(RF) from [25]

2.6. Rail deterioration

The wheel-rail contact has a dynamic interaction that determines the contact forces and

other contact conditions. This dynamic contact may induce damage in both wheel and

rail, especially at the contact surface. The combination of local contact stresses, thermal

stresses, and residual stress contribute to the rail failure. High cycle loading by the

wheels on the rail can cause the material near the surface to accumulate plastic


45

deformation by the ratcheting process. When the plastic strain accumulation reaches the

critical shear strain, the material fails in the form of wear or rolling contact fatigue

cracks. Surface-initiated RCF may form head checks, gauge corner cracks or squats [1].

2.6.1. Wear

According to Ashby [26], wear is defined as the loss of material when the contacting

surface slides, whereas the wear rate is defined by Williams [5] as the volume lost from

the wearing surface per unit sliding distance. The magnitude of wear depends on the

operating conditions on the contact patch, which include the environmental conditions,

the presence of contaminants, and the material properties. The magnitude of wear is

commonly described using the Archard wear equation, as in Equation (2.31). The wear

rate, w, is proportional with the load P on the contact and the wear coefficient, K, but

inversely proportional to the surface hardness, H.

(2.31)

2.6.1.1. Wear mechanism

Wear in metal can be classified based on its mechanisms, i.e. oxidative wear, adhesive

wear, abrasive wear, delamination wear, fatigue wear, fretting wear and by erosive wear

[5]. Each of the mechanisms is explained in more detail in the following section.

Oxidative wear

Oxidative wear is the wear of dry unlubricated metals in the presence of

air or oxygen. Atmospheric oxygen may change the wear debris from

metallic iron to iron oxides. Thicker film oxide leads to mild wear, and

when it is broken down adhesive wear is inevitable.


46

Adhesive wear

Adhesive wear occurs when two bodies adhere on the contact surfaces

and cause a loss from either surface. The load in the contacting asperities

is so high as to make the two bodies adhere and create micro-joints. The

friction may rupture these joints and lead to scuffing or galling.

Abrasive wear

Abrasive wear occurs when the solid objects scrub each other or rub

against particles that have equal or greater hardness [86]. Abrasive wear

with only two bodies involved in the friction process is known as two-

body wear, whereas when hard particles are trapped between rubbing

surfaces, it is known as three-body wear. Wear occurs when the softer

material faces the asperities of the harder material.

Delamination wear

Delamination wear occurs when the sub-surface crack grows and joins

the surface causing detachment of a piece of material [47]. The surface

crack is driven by plastic deformation from ratcheting process which

occurs and accumulates in each cycle of repeated load [24].

Fatigue wear

Fatigue wear occurs from the process of cyclic loading during friction. It

happens when the applied load exceeds the fatigue strength of the

material. As the fatigue cracks start at the surface, they may grow and

connect with each other, leading to the delamination of material pieces if

the crack goes towards the surface or propagates into the material and

results in fracture. In this type of wear, the worn surfaces have

extensiveplastic deformation compared to the unworn materials.

Fretting and corrosion wear

Fretting wear occurs when the two surfaces have a relative oscillatory

motion of small amplitude, usually only few tens of microns. It can be

recognized by the appearance of reddish-brown debris. The fretting wear

process follows this pattern: the mechanical action disrupts the protective

film, and hence exposes the reactive metal. This surface rapidly oxidizes

within the presence of oxygen from the atmosphere and is disrupted

again in the next cycles of the mechanical action. The particles of the


47

disrupted film are trapped in the contact and cause an additional

contribution to the abrasive wear. If the areas of the surfaces that are not

completely oxidized come into direct contact, adhesive wear may occur

as well.

Erosive wear

Erosive wear is the progressive material loss caused by the impact of

particles of a solid or liquid against the object’s surface [86]. The

particles that impact the surface gradually remove the material from the

surface through repeated deformations and cutting actions. Kapoor &

Johnsons [23] investigated plastic ratcheting due to erosive wear. They

found that if the contact stresses during impact exceed the elastic limit,

then localized plastic flow will occur. Despite the hardening of the

material, plastic ratcheting is expected to emerge if the impact stresses

are high enough and exceed the shakedown limit. The small increments

of plastic strain are accumulative and cause the progressive extrusion of

a thin layer of plastically deformed material.

2.6.1.2. Wear mechanism maps for steel

The wear mechanism map for steel, introduced by Lim and Ashby [26, 27] and shown

in Figure 2.23, shows a summary of wear behaviour over a wide range of loads and

sliding velocity. It also identifies the dominant mechanisms and the overall wear rate.

This figure reveals that the lightest wear is the ultra-mild wear with very low pressure

and sliding speed. When the load and sliding speed increases plastic strain accumulation

is likely to occur and lead to delamination wear. When the load increases more, the

mechanism of oxidation starts to occur and causes oxidational wear. With an increase in

the sliding speed, the oxidational wear becomes severe. With a higher sliding speed, the

temperature increases and can lead to the material melting and hence causing the melt

wear to occur. The most severe wear occurs with a very high pressure and sliding speed,

and is known as seizure.


48

Generally, the wear regimes are classified as mild or severe wear, as the wear mostly

happens in these regions. Olofsson et.al. [2] found that mild wear usually occurs at the

rail head whereas severe wear dominates in the rail gauge as higher sliding occurs there.

The difference between the two wear regimes are shown in Table 2.8.

Figure 2.23 Wear mechanism map for steel. The shaded regions show transitions from

[26, 27]

Table 2.8 Distinction between mild and severe wear from [5]

Mild wear Severe wear

Results in an extremely smooth surface –

often smoother than original

Results in rough, deeply torn surface –

much rougher than the original

Debris extremely small, typically only

100nm diameter

Large metallic wear debris, typically up to

0.01nm diameter

High electrical contact resistance, little

true metallic contact

Low contact resistance, true metallic

junction formed


49

2.6.1.3. Wear transition

The severity of rail/wheel wear has been studied and classified into several types of

wear based on its mechanisms [26, 87, 88]. One type of wear map was introduced by

Bolton and Clayton [28]. Their experimental study summarized the wear regimes into

three types: I, II, and III, as shown in Figure 2.24. Wear type I occurs when the limiting

friction is reached and thus the wear magnitude is solely determined by the tangential

force. Wear type II shows the requirement for some activation energy to generate wear

particles in each passage of the wheel through the contact zone. The wear in this regime

depends on the contact pressure and the slip/roll ratio (creepage). In wear type III, the

increased slip/roll ratio causes ploughing of one surface by the other, resulting in a rapid

increase of wear rate and rough surface.

Figure 2.24 Wear regimes from twin disc testing of BS11 rail material vs Class D wheel

material from [16, 28]

The wear mechanism associated with the rail/wheel was studied carefully by Lewis &

Dwyer [29]. Their experimental results, as seen in Figure 2.25, show that the first wear

regime, mild wear, is caused by the oxidation process, whereas the second regime is

caused by plastic ratcheting and the third regime is due to a severe delamination

process. The results also show that temperature has a significant effect on the transition


50

between mild and severe wear [16, 89]. This temperature rise corresponds to the

activation energy required in wear type II and it is proportional to the tractive force

times the slip (T). With a high slip/roll ratio, severe thermo-mechanical stress and

material softening may cause a rapid increase in wear rate and lead to severe or

catastrophic wear (Type III) [90]. This type of wear transition is what has been

modelled in the current work.

Figure 2.25 Wear regimes and mechanisms from [29]

2.6.2. Rolling contact fatigue

Rolling contact fatigue (RCF) is the kind of material damage that occurs due to

repeating high loads on the contact surface that causes cracks to emerge. This failure

occurs at the surface as well as at the sub-surface [22]. The RCF crack grows from the

micro-cracks generated in the deformed surface layer and may propagate into the rail

due to the cyclic deformation of the rail [91]. The RCF is governed by many factors, i.e.

environmental conditions, rail and wheel profiles, track curvatures, grades, lubrication

practices, rail metallurgy, vehicle characteristics, track geometry defects and rail

grinding practices.


51

2.6.2.1. Crack initiation

Based on the contact loads, working conditions and material properties, cyclic plastic

deformation can produce cracks due to the ratcheting process [6] or due to low cycle

fatigue. Kapoor [34, 47] stated that if the load is greater than the elastic shakedown limit

but less than the plastic shakedown limit, there will be a closed cycle of plastic flow,

and no accumulation of plastic strain. Hence the failure will be caused by low cycle

fatigue (LCF). When the load exceeds the plastic shakedown limit, plastic strain will be

accumulated by the ratcheting process and hence the material will fail when the plastic

strain accumulation reaches the ductility of the material. The crack can still be initiated

after a higher number of cycles due to the LCF without ratcheting occuring.

As the ductility of the material is exhausted, surface rolling contact fatigue cracks

initiate and typically propagate at shallow angles, around 10 to 30 into the rail head

until 4-5 mm depth from the surface, where cracks may branch up towards the rail

surface or down towards the rail foot, as shown in Figure 2.26 [30].

Figure 2.26 Rail cross section normal to the surface crack orientation, showing crack

regularity and crack branching from [30]

Two kinds of surface initiated cracks due to RCF failure in the rail head are head checks

and squats. Head checks are angle cracks that form near the rail gauge corner, usually in

the high rail of curves and crossing rails. This kind of failure is caused by the repeated

plastic deformation and consecutive accumulated damage at the surface of the rail head.

It is also possible for the crack to propagate in a transverse direction and lead to rail


52

fracture. Squats are visible as a widening of the wheel/rail contact band and a small

depression of the rail surface. In this area a small crack may emerge with a circular or v-

shape. If it propagates at a shallow angle to the surface, it can cause detachment of the

surface of the material, whereas for cracks with a length of around 3-5 mm below the

surface, they can propagate downwards to form a progressive transverse crack. [92]

The criteria used to determine a fatigue crack initiation are: the Dang Van criterion, the

Coffin Manson relation, the Smith-Watson-Topper (SWT) relation, and ratcheting

failure [92].

1. Dang Van criterion: this criterion uses a shear-stress-based multi-axial fatigue

limit criterion for high-cycle-fatigue (HCF) conditions, which is applicable for a

steady state of elastic shakedown where no further damage occurs.

2. Coffin-Manson and SWT criterion: these criteria are used for low-cycle-fatigue

(LCF). The initial damage process in the Coffin-Manson criterion was based on

assumptions of shear strain dominated damage. The SWT criterion was

interpreted as an energy-based approach and applicable to materials that fail

with cracks perpendicular to the principal strain amplitude (tensile mode).

3. Ratcheting failure: this criterion uses the critical strain to failure by ratcheting as

a value of the accumulation of unidirectional plastic strain to be reached in order

to be considered as the initiation of a crack. The number of cycles to failure due

to ratcheting is equal to the critical strain for failure by ratcheting of the material

divided by the equivalent ratcheting plastic strain per cycle, which are the

average ratcheting axial and shear strains per cycle.

2.6.2.2. Crack propagation

Due to cyclic deformation of the rail, the crack initiated at the surface may grow with an

angle of less than 25 to the surface and reach a depth of around 100m. It has been

revealed by many studies that liquid plays an important role in the mechanism of RCF

crack growth. Some characteristics of the crack growth show that the cracks always

propagate in conditions of pure rolling or in the presence of sliding, with the direction


53

of load motion at a shallow angle of around 15-25 to the surface. They propagate

faster in the presence of a liquid [63], descibed by Bower [93] as the “fluid entrapment

effect”. The process of a fluid-assisted mechanism of crack growth is illustrated in

Figure 2.27. At the beginning, the shear mode crack growth is influenced by friction

between the crack faces. Due to the hydraulic transmission of contact pressure, the

crack is filled with fluid, and hence this fluid could be trapped in the next wheel pass.

The fluid trapped in the crack is driven to the crack tip and becomes pressurised until

the contact has moved away and allowing the crack to open and let the fluid out. The

fluid entrapment, as seen in Figure 2.27(c), causes the crack to grow [31].

Fischer et.al [91] also studied some factors that influence crack behaviour, namely

loading and material strength. The development of surface cracks and their propagation

behaviour due to rail operation are unavoidable. However, the variation of material

strength can increase the rail resistance against crack propagation. Therefore it is

certainly important to investigate the microstructure of the rail material in order to

increase the fatigue life of the rail. Garnham et.al [94] found that many surface micro-

cracks are initiated along the border between the strained pro-eutectoid ferrite phase and

strained pearlite. RCF crack propagation was facilitated running along these pearlites,

and along highly strained, pro-eutectoid ferrite border zones within the constraint of the

overall strain field. The results from a twin disc test, micro and nano-hardness tests and

metallurgical examinations revealed that the higher the amount of pro-eutectoid ferrite

in the rail microstructure, the lower the mean RCF crack initiation life, due to the more

rapid straining of the pro-eutectoid ferrite when relatively constrained by neighbouring

pearlite nodules. However, low pro-eutectoid ferrite samples gave both the highest

mean life and the widest spread of RCF lives [30, 94].


54

Figure 2.27 Fluid assisted mechanism growth from [31]: (a) shear mechanism, (b)

hydraulic transmission of contact pressure, (c) fluid is trapped and pressurized inside the

crack, (d) squeeze film pressure generation

Figures 2.28 and 2.29 show the three phases of crack life [1, 32]. The crack initiation is

mainly driven by shear stress (phase (i)). Crack initiation occurred in this phase is

completed with a crack length of about 0.1-0.5 mm. As the crack gets longer and

deeper, the cracks are driven by the stresses near the crack tip (phase (ii)). Beyond a

certain critical crack length, the stress intensity drops and leads to a reduction in the

crack growth rate. In phase (iii) the crack growth increases rapidly. The crack growth in

this phase is driven by bending stresses and causes the crack to turn downwards. At

critical length the crack at this stage can cause quick fracture resulting in rail break.


55

Figure 2.28 Crack growth phases from [1]

Figure 2.29 Three phases of crack life from [32]

2.6.2.3. Wear and fatigue interaction

Under normal rolling contact conditions, rail operates mostly under low creepage, and

hence it is possible for wear and RCF to occur concurrently. The effect of the wear and

fatigue crack on rail life needs to be avoided. Wear, as it causes the material to be

removed from the surface, can partially or entirely remove the surface cracks, thereby

counteracting crack propagation [36, 95]. Figure 2.30 shows that the wear of vertical

depth d will shorten the crack by length t, where t = d/sin(). If at the same time the

crack grows by increment da, the net change of crack length is da-t.

phase III

phase II

phase I


56

Figure 2.30 The relationship between wear and surface crack. The surface crack can be

truncated by wear and become short or even disappear

Figure 2.31 also shows the effect of wear and crack on the total damage of the material.

In this figure, if the level of wear rate is higher than the crack growth rate (wear rate

level 1), it will not allow any crack formation. At wear rate level 2, the crack candidate

is truncated by the wear or even disappears. At the very low wear rate (wear rate level

3), the crack growth is not influenced at all by wear. This can lead the crack to grow

deeper into the base and may cause damage downwards leading to a rail break [34].

Figure 2.32 shows the effect of the material removal rate on the rail life. With a low

level of material removal, the rail life due to wear is high, but its effect in reducing

crack growth is low. Hence life due to fatigue failure is low. This is an unsafe condition,

because if the crack propagates too deep into the base, a rail break may occur. In

contrast, when the material removal rate is high, rail life due to wear is low and the life

due to fatigue is high. If the material loss at the surface is too great, the effect is still

visible on the changed profile. Hence, further failure can still be prevented and failure

by wear at this stage is a safe failure. It is necessary, because of this behaviour, to

predict the wear rate and fatigue crack initiation and its combination to optimize the rail

performance and to reduce the cost.

crack length truncation (t)

= d / sin ()

crack tip advance (da)

wear of vertical depth (d)


57

Figure 2.31 The effect of wear rate on crack growth: Level 1 (very high wear rate)

allows no crack formation; Level 2 (regular wear rate) shortens the crack length,

enabling the crack candidate to disappear; Level 3 (very low wear rate) has no effect on

the crack growth from [33]

Figure 2.32 Rail life against material removal rate from [1]


58

Chapter 3. Model Development

59

Chapter 3

Model Development

The model developed in this work is based on the brick model that was used to predict

the wear rate by other researchers [34, 38, 39, 58]. The brick model is based on

ratcheting failure and considers the wearing material as a two-dimensional mesh of

brick elements, as seen in Figure 3.1 [34, 38, 39, 96]. Basically the wearing material is

divided into two regions: the first is close to the contact surface and is made finer in

order to accurately capture thermal and contact stresses; the second region below the

finer mesh is made coarser to reduce the simulation time. Figure 3.1 also shows that it is

possible to add more regions with a larger brick size in the region at greater depth. The

depth of the model was set to 4a, with ‘a’ as the semi contact width in the longitudinal

direction. This depth was needed in order to include all possible plastic strain occurring

in all the material elements.

Each brick element in the model was associated with several parameters, such as the

critical strain at failure (c), the accumulated shear strain (), the initial shear yield stress

(k0), and the effective shear yield stress (keff). The rail material was assumed to be BS11

with the material properties listed in [6]. For the work presented in this thesis, the shear

yield stress and critical strain at failure were varied in each brick using a normal

distribution with a standard deviation equal to 5% of the mean value of each parameter,

as in reference [38]. With this variation, different elements will accumulate strain at

different rates and fail at different times.


60

Figure 3.1 Brick model

3.1. Wear

The rail material was subjected to the wheel load typically modelled as Hertzian contact

stress, refer to Equation (2.12). The incremental plastic strain () was calculated as a

function of the maximum orthogonal shear stress at each depth (zx(max)), the effective

shear yield stress (keff), the material strain hardening constants ( and ) and the

material constant (C). The value of C was equal to 0.00237, as in [6], whereas and

depended on the material properties of each steel. The plastic strain accumulation was

defined as in the following equations, which have been used in several previous studies

[34, 38, 39, 96]:

[( ( )

) ] (3.1)

Rail Frictional heat

Wheel v0

z 1

z 2

z n

z =

4a

……

……

……

……

……

……

……

……

Rail Heat flux

Hertzian contact

pop(x)

Contact zone 2a

p0p(x)

z n

> …

>

z 2 >

z 1

z 1

z 2

z n

Nx

Nz

x


61

(3.2)

{ √ } (3.3)

The plastic strain occurred when the maximum orthogonal shear stress at any material

element exceeded the shear yield strength of the material ( ( ) ) at that

point. The amount of plastic strain in each cycle was calculated using Equation (3.1).

When the plastic strain occurred the material hardened up (Equation (3.3)), and thus the

yield stress increased and was taken into account in the next wheel pass. The material

hardening followed Voce equation [97] with two parameters involved: speed of

hardening, , and the ratio of limiting hardness and original hardness, . The material

failed when the accumulated plastic strain () reached the critical strain at failure (c).

The failed material was detached from the surface based on some scenarios [34] to

remove an element as wear debris (Figure 3.2(a)). These scenarios are shown in Figure

3.3. The wear rate in each cycle was calculated by summing up the volume of the debris

elements. When all the bricks in the top layer became debris, this layer was detached

and the layers below moved up (Figure 3.2(b)). A new layer was added at the bottom

with zero strain (Figure 3.2(c)).

The wear rate was calculated as in the following equations:

∑

(3.4)

Where Nx was the number of bricks in one layer and N was the number of cycles.


62

Figure 3.2 (a) Representation of brick state at cycle (n); (b) at cycle (n+1) - when all

bricks in 1st layer became debris, this layer was detached. All layers under the 1st layer

moved up, and (c) a new layer was added at the bottom with zero strain

Figure 3.3 Scenarios to remove a brick as wear debris, taken from [34]

(c)

(b)

(a)

Cycle (n+1)

1st layer failed

New layer is added at the bottom with zero strain

All layers below 1st layer move up

Cycle (n)

= wear debris (n) = wear debris (n+1) = failed element = weak/healthy element

(a) (b) (c)

(d) (e) (f)

(g) (h)


63

3.2. Temperature

3.2.1. Temperature rise due to Frictional Heating (FH)

The heat generated at the contact patch due to sliding friction was transferred to the rail,

as seen in Figure 3.1. The temperature rise in the rail material due to frictional heating

was investigated, based on moving the heat source in a rolling/sliding contact, as

described by Knothe et.al. [7]. In the contact region, the tangential force was equal to

T=N, while was kept constant. With very little contact as the wheel passes on the

rail, the heat penetrates into a very thin layer with a thermal penetration depth calculated

using the following equations [7]:

√ (3.5)

(3.6)

(3.7)

where a is the semi contact patch, v is the speed of the moving heat source, is the

thermal diffusivity, is the density, and c is the specific heat capacity. The Peclet

number, L, is a dimensionless number, defined as the ratio of speed on the surface to the

rate of diffusion of heat into the solid [13]. For L > 5, the heat only diffuses a short

distance into the body and hence the heat flow can be approximated perpendicular to the

surface [13], i.e. in z-direction.

The maximum temperature at each depth was calculated by Equations (3.8 – 3.14)

which have been taken from [7]. By using a constant heat flow rate, with constant

friction and sliding velocity, the frictional energy is equal to:

() √ (3.8)

It was assumed that all the frictional energy was transformed into heat and that this heat

would flow into the wheel and rail with the heat partitioning factor, :


64

√

√ √ (3.9)

with as the thermal penetration depth

√

√ (3.10)

and its value for rail was equal to that for the wheel, r = w.

The average heat flow rate at the wheel surface was:

∫ ()

(3.11)

and the heat flow rate at the surface of the rail was equal to:

( )

( ) (3.12)

In order to find the temperature inside the rail, the thermal penetration depth, r, and the

heat flow rate at the surface of the rail, , are substituted in the heat general solution

for rail temperature based on Carslaw and Jaeger [98],

( )

√ ∫ ( )

(

)

√ (3.13)

By substituting t with the current position x in a co-ordinate system fixed to the contact

patch (Fig 2 from [7]) and using the dimensionless variables of

, the

temperature in the rail body was determined to be [7]:

( )

√

{√

( )

(

( )) (

√ ( ))} ; for -1 1 (3.14)


65

( )

√

{[√

( )

(

( )) (

√ ( ))] [√

( )

(

( ))

(

√ ( ))]} ; for > 1 (3.15)

3.2.2. Temperature rise due to Steady State Wheel

Temperature (SSWT)

The temperature within the contact area is a result of the frictional heating and the

conduction of heat from the hot wheel (see Figure 3.4). The model of current research

did not include the effect of surface roughness as described in Section 2.3.5 for several

reasons:

1. The surface roughness keeps on changing due to repeated contact under varied

conditions [99, 100]. The surface can be flattened or roughened during contact.

The measurement of surface roughness is difficult due to issues of sampling

length and its fractal nature.

2. High contact pressure on a small concentrated contact implies that most of the

area is in contact. Therefore the estimates of temperatures in the current model

are also quite reasonable.

3. The variation of surface roughness causes the operating conditions, such as

contact pressure and friction, to change. Therefore the heat generated due to

frictional heating is also varied. If there is a temperature difference between

wheel and rail, the degree of heat conduction also will be influenced by the area

in contact. When the asperities are flattened, the area in contact increases and

hence there will be more heat conducted from the hotter body to the colder one.

The model is also for contamination free contact. If the contamination or oxide layer

exists, metal to metal contact is reduced and the temperature effect also decreases (see

Section 2.2). Thus the smaller heat partitioning factor and effective thermal conductivity

can be considered in future work.


66

Figures 3.4(a) and 3.4(b) show the temperature rise due to frictional heating and a

steady state wheel temperature, respectively. For this thesis, the temperature in the rail

due to frictional heating, , was calculated by using the analytical solution for a

constant heat flow rate (Equations (3.16) and (3.17) from [7]) where is the heat

partitioning factor, is the friction coefficient, p0 is the peak pressure, vs is the sliding

speed, r is the thermal penetration coefficient for rail, and vr is equal to vehicle speed.

The variables of and are the dimensionless coordinates, equal to

and

respectively, with being the thermal penetration depth.

( )

(3.16)

( )

√

{√

( )

(

( )) (

√ ( ))}

(3.17)

Figure 3.4 The rail temperature rise was caused by (a) frictional heating and (b) the

conduction from the hot wheel into the cold rail

The effect of steady state wheel temperature on rail temperature was calculated using

Equations (3.18) (3.20) below, which were taken from Ertz and Knothe [7], sections 6

and 8. In these sections, they calculated the case for a two-dimensional model solution.

Equation (3.18) was used to calculate the steady state wheel temperature ( ). When

the wheel made contact with the rail, the heat flow rate was equal at every point in the

(a) (b)

w0= p0p(x)

r_FH

(fh)

rail rail

+

r_SSWT

wheel wheel

v0


67

contact patch for both bodies. The surface temperature of the wheel and rail had the

same value, (Equation (3.19)). For the calculation, the initial wheel temperature

when making the contact, , was equal to the steady state wheel temperature, . As

and were known, then any variation of temperature within the rail was due to the

steady state wheel temperature, , and can be calculated as is given by Equation

(3.20).

√

(3.18)

√

√ √ (3.19)

( ) { (

√ ( ))}

{ (

√ ( ))}; (3.20)

The actual temperature of the rail, , was the summation of and

(Equation (3.21)).

( ) ( ) ( ) (3.21)

3.3. Thermal stresses

The temperature rise, as described in the previous section, was used to evaluate the

thermal stress developed in the rail body. The thermal stress developed in the rail

material was caused by the temperature rise due to heat from frictional heating and due

to conduction of heat from the hot wheel in a steady state condition. Both were added to

the mechanical stress to make the total stress developed in the contacting body.


68

3.3.1. Thermal stress due to frictional heating

Thermal stresses in the model were calculated in terms of the applied normal and shear

traction and the contact temperature. The thermal stresses due to frictional heating were

evaluated by using the thermo-mechanical stress solutions derived by Goshima [53,

101]. He used a fully slipping contact to determine contact stresses due to normal

pressure, shear traction and contact temperature. Figure 3.5 shows the configuration of a

rolling sliding contact with contact pressure p0p(x), tangential load p0p(x), and when

the wheel is moving with contact velocity V over the surface of half space. The

following are the dimensionless parameters used to calculate the thermo-mechanical

stresses [53]. The coordinates of ( ) were fixed to the moving wheel as shown in

Figure 3.5.

Figure 3.5 Rolling sliding contact configuration

(3.22)

(3.23)

(3.24)

Hertzian contact p0p(x)

Contact zone

p0p(x)

Rail Frictional heat

Wheel v0

x’

z’(t2) z’(t1)

a a v0(t2-t1)


69

(3.25)

( )

( ) (3.26)

The amount of heat flow rate was calculated from the work done against friction in the

contact patch as follows:

() ( ) (3.27)

If ( ) was assumed as Hertz contact pressure distribution, i.e. ( ) √( ( )

),

the heat flow rate became:

() √ (3.28)

With the thermal boundary conditions given by [53]:

( )

{

( )

(3.29)

( ) (3.30)

And the mechanical boundary conditions as the following [53]:

{ ( )

(3.31)

{ ( )

(3.32)

( ) (3.33)


70

By substituting the contact pressure distributions, P(t), such as Hertzian or parabolic,

the thermo-mechanical stresses in the contacting bodies were calculated as follows [53]:

{

∫ ( )

∫ ( )

∫ ( )

∫ ( )

∫ ( )

(3.34)

where

√ {( ) } { ( )( ) }

{( ) } (3.34a)

( ) ( )

( ) (3.34b)

( ) (3.34c)

( ) (3.34d)

( ) (3.34e)

( )

(3.34f)

(3.34g)

(3.34h)

( ) (3.34i)

(3.34j)

( ) (3.34k)

( ) ( ) (3.34l)

( ) (3.34m)

( ) (3.34n)


71

3.3.2. Thermal stress due to steady state wheel

temperature

Using the same thermal and mechanical boundary conditions from Goshima [53, 101],

the thermal stress due to the steady state wheel temperature was evaluated by using

Equations (3.35) (3.37). The heat flow rate for the rail due to heat conducted from the

wheel is given in Equation (3.35) from reference [7]. By substituting Equations (3.18)

and (3.19) into Equation (3.35), the heat flow rate due to conduction of heat from the

hot wheel into the rail can be simplified into Equation (3.36). The total stresses were the

sum of both the mechanical stresses, and the thermal stresses due to frictional heating

and the steady state wheel temperature, as in Equation (3.37).

() √

( ) (3.35)

() √

( ) (3.36)

(3.37)

3.4. Thermal softening

The effect of the temperature rise on the yield stress reduction was approximated using

Equations (3.38) (3.40). Equation (3.38) was the reduction of yield stress used by

Widiyarta et.al [22] to investigate BS11 rail steel. It was derived from the yield stress

reduction of three pearlitic steels, UIC 900A, UIC 1100, and HSH. For the current

presented here, the evaluation of frictional heating effect and steady state wheel

temperature used this model with the material properties of BS11 rail. For Chapter 6,

which focuses on the effect of material properties on rail wear and rolling contact

fatigue crack initiation, the reduction of yield stress against the temperature rise was

evaluated separately for UIC 900A and UIC 1100. The relationship refers to Figure 2.18

and its curve fitting was presented in Equations (3.39) and (3.40) for UIC 900A and


72

UIC 1100 rail material, respectively. The plot of yield stress against the temperature rise

for these materials is shown in Figure 3.6.

{

(

) (

)

(3.38)

where: Tref1 = 250; Tref2 = 750C

{

(3.39)


{

(3.40)


Figure 3.6 Reduction of material yield strength due to temperature rise

0 100 200 300 400 500 600 700 8000

0.2

0.4

0.6

0.8

1

1.2

Temperature (C)

Nor

mal

ized

yie

ld s

tress

from [20]UIC 900AUIC 1100


73

The process of thermal softening in each cycle is represented in Figure 3.7. When the

wheel was in contact with the rail, frictional heating occurred and increased the flash

temperature. When the temperature exceeded 250C, the material strength was reduced

and shown as keff(softening) (Figure 3.7, number 1). When the wheel passed over, the rail

surface cooled down. The temperature returned to room temperature and the material

strength recovered to its previous value (Figure 3.7, number 2). Due to the plastic strain

in the load cycle, the rail material hardened, as shown by Figure 3.7 (number 3). This

cycle was repeated with each wheel pass. Hence the thermal softening temporarily

occurred in each cycle due to temperature rise, while the hardening process due to

plastic strain occurred permanently in each cycle. The material hardening and thermal

softening was repeated over many cycles until the material failed. The thermal softening

at a high temperature can cause larger plastic strain accumulation and lead to a greater

wear rate.

Figure 3.7 The process of hardening and softening at y = 406 MPa with material

hardening parameters of =25 and =1.88 (the maximum shear yield stress of the

material will be k0=440MPa). Softening and hardening process: (1) the shear yield

stress reduces as temperature increases; and (2) the shear yield stress recovers to its

previous value as temperature returns to normal; (3) the increment plastic strain hardens

the material


74

3.5. Material properties

There were several rail materials used in this work. One was the same material used in

the twin disc test by Tyfour, et.al. [6, 102], i.e. BS11 rail grade. This material was used

for the simulation of the frictional heating effect on rail wear and RCF (Chapters 4 and

7) and also the steady state wheel temperature effect on rail wear (Chapters 5). The

properties of BS11 rail grade are summarized in Table 3.1. The thermal properties used

for rail steel in regard to the simulation of thermal effect on wear and RCF are listed in

Table 3.2.

Table 3.1 Chemical composition and mechanical properties of BS11 rail from [6]

Chemical composition (wt.%)1 value

C 0.52

Si 0.2

Mn 1.07

Ni 0.03

Cr < 0.01

Mo < 0.01

S 0.018

P 0.013

Mechanical properties mean Assumed standard deviation

2Tensile yield strength, y, MPa 406 20.3 (5%)

2Young modulus, E, GPa 209 -

2Poisson ratio 0.3 -

2Hardening parameter -

25 -

1.88 -

2Critical shear strain (c) 11.5 0.575 (5%)

2Constant (C) 0.00237 -

3Density, , kg/m3 7850

1 from Ref [6], 2 from Ref [9], 3 from Ref [7]


75

Table 3.2 Referenced data for thermal properties of rail steel from [7]

Thermal properties value

Thermal diffusivity, , m2/s 14.2 x 10-6

Thermal conductivity, Kt, W/Km 50

Coefficient of thermal expansion, t, K-1 1.2 x 10-5

Thermal penetration coefficient, r, Ws0.5/Km2 13290

To evaluate the effect of different rail material properties, UIC 900A rail grade and UIC

1100 rail grade were used in simulating its material property effects on wear (Chapter

6). The tensile yield strength, hardening parameters and the critical shear strain of the

two rails are listed in Tables 3.3 and 3.4, for UIC 900A and UIC 1100 respectively. The

other parameters, such as Young modulus, Poisson ratio, constant (C), and steel density

follow the values in Table 3.1. The thermal properties also used the same values as in

Table 3.2.

Table 3.3 Mechanical properties of UIC 900A grade rail from [8]


Tensile yield strength, y, MPa 507 25.35 (5%)

Hardening parameter -

63.94 -

2.31 -

Critical shear strain (c) 8.8 0.44 (5%)

Table 3.4 Mechanical properties of UIC 1100 grade rail from [8]


Tensile yield strength, y, MPa 710 35.5 (5%)

Hardening parameter -

95.18 -

1.86 -

Critical shear strain (c) 12 0.6 (5%)


76

3.6. Assumptions

In order to simplify the model, there are some assumptions made:

1. The contact surface between wheel and rail is assumed to be smooth. In fact it is

rough and this may lead to higher contact pressure at the peak roughness

(asperity) [71].

2. The contact condition between wheel and rail is assumed to be fully slipping

although the contact surface commonly has stick and slip areas [58].

3. The operating conditions, i.e. friction coefficient, slip/roll ratio, peak pressure,

and vehicle speed are assumed to be constant. In reality they fluctuate depending

on the characteristics of the contact, such as the wheel/rail geometry, material

properties, the surface roughness, contaminants, and the temperature rise at the

interface.

4. The thermal properties are considered constant. Actually specific heat capacity

(c), and thermal conductivity () may change due to temperature changes [103].

5. The ratcheting rate does not depend on the elevated temperature. In fact it will

vary by temperature changes [104].

6. The wheel temperature at steady state does not change with operating condition

variation.

7. The model is for contamination free contact and did not include the effect of

oxide layers formed during contact.

3.7. Simulation flow diagram

The flow diagram of the simulation is presented in Figure 3.8. The simulation was built

using the Matlab code. The listing code is provided in Appendix. The variables

considered in the simulations are defined in each chapter, which are sufficient to present

the effects of frictional heating, steady state temperature and crack initiation.


77

START

Input:Operating condition,

Mechanical propertiesThermal properties

Evaluate temperature rise due to frictional heating

number of cycles reached ?

Evaluate plastic strain increment and plastic strain

accumulation(,)

Accumulate debris

≥ c

Evaluate maximum shear yield stress at each depth

zx(max)

Evaluate contact stress Evaluate thermal stress

Evaluate thermal softening(keff(softening))

Evaluate strain hardening(keff)

Evaluate temperature rise due to steady state wheel

temperature

Yes

Evaluate surface crack initiation

Evaluate wear rate

Yes

No

END

Depth of surface crack candidate > 50

Sub-surface crack:Major axis length ≥ 5 x minor

axis length

Evaluate sub-surface crack initiation

Yes

Yes

No

Evaluate wear debris(Fig. 3.3)

NoNo

Figure 3.8 Flow diagram of the simulation


78

Chapter 4. Rail wear due to frictional heating

79

Chapter 4

Rail wear due to frictional heating

4.1. Introduction

In the rolling/sliding contact between a wheel and rail, heat is generated from the

friction between the two surfaces. This heat is conducted to the wheel and rail and hence

increases their temperature during contact. The degree of the rise in temperature

governs the magnitude of both the thermal stress that develops and the thermal

softening. Both affect the increase in the plastic strain in the material and hence the rate

of wear. This chapter explores the effect of temperature rise on wear rate due to

frictional heating, including its effect on wear transition.

The values of the variables used in the simulation included wear transition due to the

thermal effect. The four variables considered were peak pressure, friction coefficient,

vehicle speed and the slip/roll ratio. Their normal operating values were used for the

simulations described in this thesis. The peak pressure was varied in the range 1 – 2

GPa. The friction coefficient was varied between 0.3 and 0.6, representing dry

conditions [3]. The slip/roll ratio was varied between -1% to -5% and the vehicle speed

considered was in the range of 108-216 km/h or 30-60 m/s. The simulation used the

data in several series, as shown in Table 4.1. This table compares three conditions:

(i) without thermal stress and without thermal softening

(ii) with thermal stress and without thermal softening; and

(iii) with both thermal stress and thermal softening

The thermal parameters of thermal expansion, specific heat, and thermal conductivity

referred to [7] and listed in Table 3.2. These parameters were independent of the

temperature. The tensile yield strength (y), critical shear strain (c), hardening


80

parameter ( and ) and material constant (C) used the data from BS11 rail studied in

[96] and [6] respectively. These values are listed in Table 3.1.

Table 4.1 Simulation series for wear transition due to frictional heating

SERIES 1. The thermal effect on temperature and wear rate

Each series (1.a – 1.d) consists of 3 simulations:

without thermal stresses and without thermal softening

with thermal stresses and without thermal softening

with thermal stresses and with thermal softening

1.a. Sr * = -1%, -2%, -3%, -4%, -5%, -6%; with =0.4, p0 = 1.5 GPa, v0 =30 m/s

1.b. = 0.3, 0.4, 0.5, 0.6; with p0 = 1.5 GPa, Sr=-0.03, v0 =30 m/s

1.c. p0 (GPa ) = 1, 1.2, 1.4, 1.6, 1.8; with =0.4, Sr=-0.03, v0 =30 m/s

1.d. v0 (m/s) = 30, 40 50 60; with =0.4, Sr=-0.03, p0 = 1.5 GPa

SERIES 2. Sensitivity analysis by changing the variables value (20%) from the reference value

(with thermal stresses and with thermal softening)

reference Sr = -1%, -2%, -3%, -4%, -5%; with =0.4, p0 = 1.5 GPa, v0=30 m/s

2.a. p0 (GPa ) = 1.2, 1.8; others: reference

2.b. = 0.32, 0.48; others: reference

2.c. v0 (m/s) = 24, 36; others: reference

SERIES 3. Factorial design two-level to check the main and interaction effects of variables

The simulation used the combination of the following parameters:

(with thermal stresses and with thermal softening)

Sr = -1%, -5%;

p0 (GPa ) = 1, 2;

= 0.3, 0.6;

v0 (m/s) = 30, 60;


81

4.2. Rail temperature rise

Table 4.2 summarizes the temperature rise due to the variation of Sr from -1% to -5%

and µ from 0.3 to 0.6. The results show that the temperature rises as Sr or µ increases.

Higher friction at the interface leads to an increase in frictional energy, which is

transformed into heat. The amount of sliding also plays an important role in increasing

heat in the contact region. When the temperature rise at µ = 0.3 and Sr = -3% reaches

250, it is high enough to cause the thermal softening of rail material.

Table 4.2 Maximum rail temperature rise at the surface [C] due to Sr and µ variation

Sr

-1% -2% -3% -4% -5%

0.3 88.9 177.4 265.4 352.9 440.2

0.4 118.5 236.5 353.8 470.6 586.9

0.5 148.2 295.6 442.3 588.3 733.6

0.6 177.8 354.7 530.8 705.9 880.3

4.3. Thermo-mechanical stresses

The subsurface material experiences contact stress as well as thermal stress generated

by frictional heating in the contact area. The maximum orthogonal shear stress, zx(max),

increases and its maximum at each depth is shown in Figure 4.1(a-d). Based on the

elastic superposition, the thermal effect is superposed to the contact stress and hence the

zx(max) increases with the value of the variable. For all operating variables, i.e. peak

pressure, friction coefficient, slip/roll ratio, and vehicle speed, the zx(max) increased by

10 MPa / 100C at 0.5 m depth, but at vehicle speed of 60 m/s it slightly reduced to

7.3 MPa / 100C. These figures also show that the zx(max) without thermal effects


82

increased as the peak pressure or friction coefficient increased. In the case of the

slip/roll ratio and the vehicle speed, the zx(max) without thermal effects remained the

same for all values.

(a) (b)

(c) (d)

Figure 4.1 Maximum orthogonal shear stress,zx(max), against depth with variation of: (a)

peak pressure (=0.4, Sr =-3%, v0=30m/s); (b) friction coefficient (p0=1.5GPa, Sr =-

3%, v0=30m/s); (c) slip/roll ratio (p0=1.5GPa, =0.4, v0=30m/s); (d) vehicle speed

(p0=1.5GPa, =0.4, Sr =-3%)

400 450 500 550 600 650 700 750 800-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

Dep

th (m

m)

zx(max) (MPa)

= 0.4; Sr = -0.03; v0 = 30 m/s

C DBA

A = contact stress; p0=1.4GpaB = contact stress + thermal stress; p0=1.4GPaC = contact stress; p0=1.8GPaD = contact stress + thermal stress; p0=1.8GPa

400 450 500 550 600

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Dep

th (m

m)

zx(max) (MPa)

A B C DA = contact stress; =0.3B = contact stress + thermal stress ; =0.3C = contact stress ; =0.4D = contact stress + thermal stress ; =0.4

480 500 520 540 560 580 600 620 640

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

zx(max) (MPa)

Dep

th (m

m)

= 0.4; Sr = -0.03; P0 = 1.5GPa

contact stress; v0=30m/s

contact stress; v0=60m/s

contact stress + thermal stress; v0=30m/s

contact stress + thermal stress; v0=60m/s

480 500 520 540 560 580 600 620 640 660 680-6

-5

-4

-3

-2

-1

0

zx(max) (MPa)

Dep

th (m

m)

P0 = 1.5GPa; = 0.4; v0 = 30 m/s

Maximum of zx

A = contact stress; Sr = -0.03 = contact stress; Sr = -0.05B = contact stress + thermal stress; Sr = -0.03C = contact stress + thermal stress; Sr = -0.05

B CA

A = contact stress; Sr=-0.03 = contact stress; Sr = -0.05B = contact stress + thermal stress; Sr=-0.03C = contact stress + thermal stress; Sr=-0.05


83

4.4. Plastic strain accumulation

When the thermal effect was considered, the temperature became an input with which to

alter the thermo-mechanical stresses and thermal softening. The maximum temperature

at each depth was calculated based on [7]. The variation of temperature with depth is

shown in Figure 4.2(a). The calculated temperature was used to evaluate the thermo-

mechanical stresses using a thermo-elastic model [53, 101]. The temperature rise was

also used to calculate the reduction of the yield stress which is approximated by using

Equation (3.38) [22] and is shown in Figure 3.6. Based on this reduction, Figures 4.2(b)

and 4.2(c) show the variation of effective shear yield stress and ratchet strain per cycle

after 7,500 cycles for peak pressure (p0) =1.5 GPa, friction coefficient ()=0.4, slip/roll

ratio (Sr)=-4%, and vehicle speed (v0)=60m/s. These figures show that the effect of

thermal softening in increasing the ratchet rate is confined to a thin surface layer of

depth 40 m.

Figure 4.2 The thermal softening is confined to a thin sub-surface layer of depth 40 m

for p0=1.5 GPa, =0.4, slip/roll ratio=-4% and v0=60m/s after 7500 cycles; (a) variation

of temperature with depth, (b) thermal softening with depth, (c) variation of shear strain

increment with depth

0 100 200 300 400 500-100

-80

-60

-40

-20

0

Dep

th (

m)

Temperature (C)

p0=1.5GPa,=0.4, Sr = -4%, v0 =60m/s

250 300 350 400 450-300

-250

-200

-150

-100

-50

0

Dep

th (

m)

keff (MPa)

without thermal softeningw/ thermal softening

0.5 1 1.5 2 2.5 3

x 10-3

-300

-250

-200

-150

-100

-50

0

Dep

th (

m)

/cycle

without thermal softeningwith thermal softening

(a) (b) (c)


84

4.5. Rail wear due to frictional heating

The wear rate is proportional to the amount of frictional work done at the contact

surface [16, 28, 105]. Bolton and Clayton [28] use TA to represent the energy from

frictional work, where T is the tractive force, is the amount of slip, which corresponds

to the value of the slip/roll ratio used in the present study, and A is the contact area.

Their results, shown as solid circles in Figure 4.3, demonstrate that the wear rate is

proportional to T. The simulations considers a few sample of operating conditions with

p0=1.3GPa and p0=1.5GPa. The simulations were for cases with frictional heating only

to mimic the experiments where the rollers were cooled down during the test. The

conditions without thermal were also included. Figure 4.3 shows that the wear

simulated at p0=1.5GPa has a better match with the experimental results. For both cases

the temperature rise does not exceed 250C resulting in no thermal softening. Thus the

thermal effect on wear rate is solely driven by the magnitude of thermal stress. As the

thermal stress increases as the contact stress increases, the thermal effect on wear rate at

p0=1.5GPa is higher than that at p0=1.3GPa. Nevertheless, the results from simulations

are far from the experimental results for higher TA in Wear type III or in catastrophic

region (see Figure 4.4). It is because the effect of surface roughness was not included in

the current model. A rougher surface topography was already found in wear type II

(Bolton and Clayton, [28]) and ploughing tracks on worn surfaces in wear type III.

Roughness produces high asperity stresses which can be several times the smooth

Hertzian contact pressure [71, 106]. The thermal effects would be higher and the wear

rate would be expected to be much higher compared to prediction using smooth Hertz

assumption. Therefore it is important to obtain the correct operating conditions which

can represent the experiments.


85

10 15 20 25 30 35 40 450

100

200

300

400

T/A (N/mm2)

wea

r rat

e (

g/m

/mm

2 )

0 20 40 60 80 100 1201

10

100

1000

10000

100000

T/A (N/mm2)

wea

r rat

e (

g/m

/mm

2 )

BS11 rail vs Class D tyresimulation with thermalsimulation without thermal

Type I, Mild

Type II, Severe

Type III, Catastrophic

Figure 4.3 The relationship between wear rate and the energy from frictional work

normalized with the contact area (T/A) in Wear type II region for BS11 rail: ,

experiment from [28]; , simulation with thermal at p0=1.5GPa; +, simulation without

thermal at p0=1.5GPa; , simulation with thermal at p0=1.3GPa; , simulation without

thermal at p0=1.3GPa

Figure 4.4 The comparison of simulation results and the experiment conducted by

Bolton and Clayton [28] for Wear type II and III at p0=1.5GPa


86

The wear rate results due to frictional heating are summarized in Table 4.3. The effect

of thermal softening is also included. This table shows that the wear rate increases as µ

or Sr increases. The thermal softening effect is not pronounced at µ < 0.4 but shows an

obvious increase with a greater µ especially at Sr = -5%, by up to 50%. The contribution

of variation of Sr and µ on the wear rate is discussed in more detail in the next section.

Table 4.3 Average wear rate (nm/cycle) due to frictional heating after 200,000 cycles

without thermal softening With thermal softening

Sr

-1% -2% -3% -4% -5% -1% -2% -3% -4% -5%

FH effect

0.3 3.3x10-3

7.2x10-3

1.9x10-2

5.3x10-2

9.8x10-2

4.9x10-3

1.3x10-2

1.6x10-2

0.35 1.34

0.4 39.00 43.67 60.20 75.10 81.72 41.36 47.1 61.57 79.06 103

0.5 166.17 176.17 189.41 200.47 214.05 166.57 176.13 197.16 211.4 256

0.6 284.64 298.47 315.06 330.04 342.37 284.95 300.68 324.26 367.58 507

4.5.1. The effect of variation on wear rate

The effects of the thermal stress and thermal softening on wear rate for all variables are

shown in this section. The variables varied are slip/roll ratio, friction coefficient, peak

pressure and vehicle speed. The results compare three conditions as described in Section

4.1. The effect of increased temperature on thermal stress is captured by condition (ii),

and the effect of temperature rise on thermal stress and material softening is captured by

condition (iii). Condition (i) is for comparison and considers no thermal effect i.e. no

rise in temperature.


87

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

20

40

60

80

100

120

Wea

r ra

te (n

m/c

ycle

)

-Sr (%)

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

150

300

450

600

750

900

Flas

h te

mpe

ratu

re (

C)

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

150

300

450

600

750

900

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

with thermal stress & with thermal softeningwith thermal stress & without thermal softeningwithout thermal stress & without thermal softening

___ wear rate- - - temperature

4.5.1.1. Slip/roll ratio

Figure 4.5 shows that without a thermal effect the wear rate remains flat regardless of

changes in the slip/roll ratio. When the temperature rises, the thermal stress increases

and produces a greater wear rate. When both effects are combined, the wear rate is

significantly altered after slip/roll ratio of -3%. At a slip/roll ratio of -3% the

temperature rises up to 350C and causes a reduction of the yield stress by 12%. The

wear rate increases by 70% without thermal softening and by 120% with thermal

softening. For a slip/roll ratio of -4%, the temperature reaches a value of around 500,

and it increases the wear rate by 175% with thermal stress and 600% with a thermal

softening effect.

Figure 4.5 The effect of slip/roll ratio on flash temperature and wear rate with = 0.4,

p0 = 1.5 GPa, v0 = 30 m/s


88

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.60

50

100

150

200

250

300

350

Wea

r ra

te (n

m/c

ycle

)

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.60

110

220

330

440

550

660

770

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.60

110

220

330

440

550

660

770

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.60

110

220

330

440

550

660

770

Flas

h te

mpe

ratu

re (C

)



4.5.1.2. Friction coefficient

The results in Figure 4.6 show that without a thermal effect the wear rate increases as

the friction coefficient increases. This is because increased friction causes the zx(max) to

increase at each depth (Figure 4.1(b)). The results show that at = 0.3 the wear rate is

very small, lower than 1 nm/cycle, with and without a thermal effect. Above this value,

the effects of thermal stress and thermal softening increase only slightly as the friction

coefficient increases from 0.3 to 0.4. However, for a friction coefficient above 0.4 the

wear rate increases rapidly (1380% from =0.4 to 0.5, without thermal effect). This

behaviour is similar under conditions B and C, with a temperature of 352C. Figure 4.6

shows that at = 0.55 the thermal stress increases the wear rate by 21% and the thermal

softening increases it even more, by 6.6%. At = 0.6 the effect of thermal softening

increases to 8.8% and thermal stress causes an 18% increase in wear rate, which is 3%

lower than = 0.55. The thermal softening’s role increases as the temperature increases,

which causes a reduction in shear yield stress from 32% to 42% at = 0.55 to = 0.6,

respectively.

Figure 4.6 The effect on wear rate and flash temperature due to variation of with

slip/roll ratio = -3%, p0 = 1.5 GPa, v0 = 30 m/s

6.6% increase due to thermal softening

21% increase due to thermal stress




89

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.80

50

100

150

200

250

300

350

Wea

r ra

te (n

m/c

ycle

)

p0 (GPa)

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.80

110

220

330

440

550

660

770

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.80

110

220

330

440

550

660

770

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.80

110

220

330

440

550

660

770

Flas

h te

mpe

ratu

re (C

)with thermal stress & with thermal softeningwith thermal stress & without thermal softeningwithout thermal stress & without thermal softening


4.5.1.3. Peak pressure

Figure 4.7 shows that the wear rate without thermal stress increases as the contact

pressure increases. This is because the increased peak pressure causes the zx(max) to

increase, as shown in Figure 4.1(a). The wear rate increases slightly from 1 GPa to 1.4

GPa and the gradient changes after 1.4 GPa. The wear rate without a thermal effect

rapidly increases after 1.6 GPa, but with a thermal effect, the wear rate significantly

increases earlier, at 1.4 GPa with a temperature of 328C. The wear rate is very low, up

to 1.2 GPa, which is lower than 1 nm/cycle. At a peak pressure of 1.4 GPa, the wear rate

is 2.6 nm/cycle without thermal stress, and increases to 5.7 nm/cycle without softening,

and by 7.1 nm/cycle with a softening effect. At 1.7 GPa the wear rate increases by 67%

due to thermal stress and by 4.9% due to thermal softening. With the higher peak

pressure of 1.8 GPa, the effect of thermal stress is reduced to 38% whereas thermal

softening is only slightly reduced to 4.4%.

Figure 4.7 The effect on wear rate and flash temperature due to variation of p0 with slip/roll

ratio = -3%, µ = 0.4, v0 = 30 m/s






90

30 35 40 45 50 55 600

5

10

15

20

25

30

Wea

r ra

te (n

m/c

ycle

)

v0 (m/s)

30 35 40 45 50 55 600

100

200

300

400

500

600

Flas

h te

mpe

ratu

re (

C)

30 35 40 45 50 55 600

100

200

300

400

500

600

30 35 40 45 50 55 60



4.5.1.4. Vehicle speed

Figure 4.8 shows that the wear rate is independent of the vehicle speed in the condition

without a thermal effect. At vo = 30 m/s, the wear rate slightly increases by 6.5 nm/cycle

without softening and by 11.4 nm/cycle with thermal softening. When the vehicle speed

increases to 60 m/s the wear rate increases only around 14.2 nm/cycle without thermal

softening and 18.8 nm/cycle with thermal softening. Although the temperature is

increased from 351C at 30 m/s to 496C at 60 m/s and causes the material strength to

reduce from 12% to 35%, the thermal stress does not change, as shown in Figure 4.1(d).

This leads to slow plastic strain accumulation and hence a low wear rate. It corresponds

to the experiment results, which show that an increase in vehicle speed leads to a lower

wear volume and results in higher wear resistance [107].

Figure 4.8 The effect on wear rate and flash temperature due to variation of v0 with slip/roll

ratio = -3%, p0 = 1.5 GPa, = 0.4


91

4.5.2 Wear transition

The investigation of the slip/roll ratio, peak pressure, friction coefficient and vehicle

speed on temperature and wear rate, as shown in Section 4.4.1, reveals that the

minimum temperature needed for a transition to occur is around 350C. Below this

temperature there is hardly any increase in the wear rate. At low temperatures (below

250C), the thermal stress is insufficient to generate major plastic deformation and

hence the wear rate is low. Thermal softening begins to influence material strength at

250C and causes a 12% reduction of the shear yield stress when the temperature

reaches 350C. However, the wear rate still remains low. Above this temperature,

thermal stress and thermal softening lead to changes in the gradient of the wear rate

curve. In all cases (Figures 4.5 – 4.8), the simulation was conducted for 200,000 cycles.

These results show that thermal stress had a greater effect than thermal softening. It is

possible to have different results if the number of cycles is greater [39]. This is because

high shear stress also occurs at the subsurface, as well as at the surface. When the top

layers fail, the accumulated plastic strain in the layers beneath reach the critical strain.

Thus, once the top layers wear, the layers beneath also wear quickly, resulting in a sharp

peak in the wear rate. Hence, with a greater number of cycles, the average wear rate

may increase. Another factor that causes the softening effect to not be pronounced is

that the softening effect is confined to a very thin sub-surface layer. Fig 4.2(c) shows

that the average plastic strain increment in the thermally softened surface layer is about

1.6x10-3. It drops linearly in a thick layer of 7.1 mm with an average value of 6.3x10-4.

As wear takes place, the material elements move from the bulk of the material to the

surface. The non-thermally softened layers that have a lower plastic strain increment

move up and lead to a lower failure rate. As the depth of thermally softened layers is

only 0.5% of the 7.1 mm depth where plastic strain occurs, the number of cycles needed

for the material to fail and form the wear is still large, although the plastic strain

increment in the thermally softened layers is 2.5 times higher than the non-thermally

softened layers. On the other hand, the effect of thermal stress goes much deeper into

the material. Figure 4.1 shows that, for this condition, it is around 5 mm, which is nearly

the full depth of the material undergoing plastic ratcheting.


92

By considering both thermal stress and thermal softening, the wear rate range after a

transition was obtained from the simulations shows the value to be between 25-300

nm/cycle. As the volume of wear of a disc is equal to 2rtdz, when r is the disc radius, t

is the contact width, and dz is the wear depth, the weight of wear per distance rolled

becomes tdz/N ( is the density and N is the number of cycles). The result can be

normalized by dividing it with the contact area of the wearing material (2ta), which

results in dz/2aN. When the value of the steel density is 7,850 kg/m3, the wear rate

range can be converted to 0.02 – 0.2 mg/mm2 contact area/m rolled. These results

correspond to wear type II from Bolton & Clayton’s experiment [28]. Thus, the present

model indicates thermal stress and material softening cause a wear rate transition from

wear type I to wear type II or from mild wear to severe wear.

The possibility of thermal effects resulting in catastrophic wear needs another

mechanism to be considered. The characteristic of catastrophic wear found in the

experiment indicates the role of surface roughness in this regime. A rougher surface

topography and an increase in the slip/roll ratio cause surface ploughing or machining.

Sundh [90, 108] also found that catastrophic wear is characterized by heavy scoring

marks, which penetrate the top oxide layer and result in wear of the bulk material. A

high contact temperature with a high slip/roll ratio makes the oxide thicker and more

brittle and it is easy to break down the layer [88]. The rough surface is also responsible

for the highly localized temperature and peak pressure. Kapoor et.al [71] found that

roughness of the surfaces can produce localized peaks of very high contact pressure, i.e.

about 10 times higher than the nominal Hertz pressure. On the other hand, when the

surface becomes rough, the friction increases and consequently leads to an even higher

temperature rise. When the temperature rises above the austenitization temperature,

phase transformation occurs and causes microstructural changes. This process was

investigated by Sundh [90], who found it led to catastrophic wear. Gaard [109] also

found that a temperature rise could lead to an increase in the adhesive force on the

rough surface that can cause severe surface damage. The effect of surface roughness and

microstructural changes need to be considered in any future modelling studies.

Figures 4.5-4.8 show that wear transition is specified for each variable (slip/roll ratio,

peak pressure, friction coefficient and vehicle speed). In all cases in this thesis, the

results indicate that the transition occurs above a temperature of 350C, which can be


93

reached with a slip/roll ratio of at least -3%, a peak pressure of at least 1.4 GPa, a

friction coefficient of at least 0.4 and a vehicle speed of at least 30 m/s. The transition

can still be reached for any combination of the four variables if the temperature

generated is above 350C. The transition is unlikely to occur if the temperature is below

this value.

Figures 4.6 and 4.7 also show that the wear rate depends on the friction coefficient and

peak pressure, with or without the thermal effect. The wear rate still increases even

though the thermal effect is neglected. This is because these two variables contribute to

the increase of the mechanical stress field. When the peak pressure and friction

coefficient increase, the tangential force (p0) increases, and increases the maximum

shear stress at each depth. Equation (3.1) shows that when the maximum shear stress at

each depth increases, the plastic strain increment per cycle will be higher, producing

quicker failure of the material and a higher wear rate. Hence the wear transition can be

reached faster.

The other two variables (slip/roll ratio and vehicle speed) both relate to sliding between

the surfaces. Most of the friction during sliding is transformed into heat [26] and causes

the temperature to increase. Through thermal stress and thermal softening, these two

variables consequently influence the wear rate. Many studies show that sliding is

responsible for wear [16, 26, 110]. The different effects of the slip/roll ratio and vehicle

speed create their individual contributions to the temperature rise and wear rate, as

shown in Figures 4.5 and 4.8. When the operating condition is lower than -6% of the

slip/roll ratio and in the range of 30 m/s to 60 m/s of the vehicle speed, the slip/roll ratio

has more effect on the wear rate than the vehicle speed. A comparison of the way the

four variables contribute to temperature rise and wear rate around the transition is

described more fully in the next section.

4.5.3. Factorial design

Figure 4.9(a) shows that as the peak pressure and friction coefficient are changed by the

same percentage, the effect on the temperature is the same, whereas the vehicle speed


94

has less effect. When the friction coefficient and the peak pressure are reduced by 20%,

the wear rate drops to nearly zero for both. However, when the variable’s value is

increased by 20%, the wear rate due to an increase in peak pressure has a higher value

compared to the friction coefficient, which is 6 times and 3.5 times greater than the

reference value, respectively. By using the factorial design level two, shown in Table 2,

the main effects of the peak pressure and friction coefficient on the wear rate are greater

with the other two parameters because of their effect on the longitudinal friction force,

which directly contributes to the mechanical stress.

However the slip/roll ratio makes the greatest contribution to any increase in

temperature inside the contact patch. This main effect is followed by the same pattern in

the interaction effects. A combination of any of the parameters with peak pressure and

friction coefficient tends to increase the wear rate, whereas any combination with the

slip/roll ratio will result in an increase in temperature. A significant effect can be seen

with the combination of the peak pressure-friction coefficient and the peak pressure-

slip/roll ratio. The results from the twin disc testing [105] and the two-roller model

[107] showed that although an increase in peak pressure causes a slight reduction in the

friction coefficient the wear rate still increased. Figure 4.9(b) shows that when the peak

pressure is increased by 20% it causes the wear rate to increase by 600%, and when the

friction coefficient is reduced by 20% it causes the wear rate to be reduced by 100%,

but the net result is still an increase in the wear rate by five times. The vehicle speed,

and any combinations with this parameter, have a low effect on temperature and wear

rate.


95

Figure 4.9 The effect of varying, p0, and v0 by 20% on (a) flash temperature and (b)

wear rate

1 1.5 2 2.5 3 3.5 4 4.5 50

100

200

300

400

500

600

700

800

-Slip/roll ratio (%)

Flas

h T

empe

ratu

re

(C)

reference120% 120% p0120% v0

80% 80% p0

80% v0

(a)

(b)

1 1.5 2 2.5 3 3.5 4 4.5 50

50

100

150

200

250

300

350

Wea

r ra

te

(nm

/cyc

le)

-Slip/roll ratio (%)

reference120% 120% p0

120% v0

80% 80% p080% v0


96

By using the factorial design level two, the main and interactive effects of these

variables (peak pressure, slip/roll ratio, friction coefficient, and vehicle speed) can be

analysed within a certain range, as shown in Table 4.4. This table shows that the main

effects of peak pressure and friction coefficient on wear rate are greater among the other

two parameters because the effect on the traction force occurred at the contact patch,

which directly contributes to the mechanical stress. However the slip/roll ratio makes

the greatest contribution to the increase in temperature inside the contact patch. This

main effect is followed by the same pattern in the interaction effects. A combination of

any parameters with the peak pressure and friction coefficient will increase the wear

rate, whereas any combination with the slip/roll ratio will result in an increase in the

temperature. A significant effect can be seen when the peak pressure-friction coefficient

and peak pressure-slip/roll ratio are combined. The vehicle speed, and any combination

with this parameter, has a low effect on both the temperature and wear rate.

Table 4.4 Two level factorial design of wear rate and flash temperature

Effect on wear rate

[nm/cycle]

Effect on flash temperature

[C]

Main effects

553.0 316.9

p0 607.0 317.0

Sr 220.0 631.4

v0 47.7 162.0

Two-factor interactions

x p0 441.0 105.7

x Sr 111.0 210.5

x v0 -12.3 54.0

p0 x Sr 198.0 210.5

p0 x v0 36.0 54.0

v0 x Sr 22.0 107.6


97

These results show that the peak pressure, friction coefficient and slip/roll ratio make a

significant contribution to the increase in wear rate and, hence, cause wear transition to

occur. However, in reality, the value of the friction coefficient could be changed by a

variation of the peak pressure and vehicle speed. From the two roller model [107], it

was found that an increase in the peak pressure or vehicle speed reduced the friction

coefficient. When the peak pressure was increased, the friction coefficient was reduced

slightly but the wear rate increased intensely. This shows a decrease in wear-resistance

with a growth in peak pressure. Different behaviour was observed when the vehicle

speed increased. Although the friction coefficient was reduced with an increase in the

vehicle speed, it still resulted in a low wear volume and greater wear-resistance. Further

investigation is needed in regard to the effect of other parameters on the friction

coefficient.

4.6. Conclusion

The wear rate has been shown to be influenced by the thermal effect that occurs due to

frictional heating. The heat generated influences the thermo-mechanical stresses and

causes material softening if the temperature is high enough. By varying some of the

variables that contribute to the thermal effect, namely the peak pressure, slip/roll ratio,

friction coefficient, and vehicle speed, then the wear transition occurs when the flash

temperature is at least 350C. It is possible for the transition to be reached with any

combination of the slip/roll ratio, peak pressure, friction coefficient or vehicle speed

when the temperature is reached. The transition is unlikely to occur when the all the

parameters lead to a temperature below this value. The present model indicates that

thermal stress and thermal softening cause the wear rate transition from wear type I to

wear type II, that is mild wear into severe wear. The possibility of the thermal effect

causing catastrophic wear needs to take into consideration other mechanisms, such as

surface roughness. The peak pressure and friction coefficient are direct contributors to

the thermo-mechanical stress field, and have significant effects on the temperature rise

and wear rate, whereas the slip/roll ratio and vehicle speed contribute to thermal stress

only. However, the slip/roll ratio has a significant effect on the temperature rise despite


98

having a slightly lower effect on wear rate compared to the peak pressure. Therefore

these variables are important in causing wear transition. Vehicle speed has less effect on

the temperature and wear rate compared to other variables and hence contributes less to

the transition.

Chapter 5. Rail wear due to steady state wheel temperature

99

Chapter 5

Rail wear due to steady state wheel

temperature

5.1. Introduction

During wheel – rail contact, the sliding friction causes the frictional heating at the

contact patch [52, 111]. The heat conducts to the wheel and rail and hence increases

their temperature. The frictional heat in the rail is conducted away easily and hence its

temperature drops down to ambient after the train has passed. The rail temperature

increases and drops again when the next train passes. On the other hand the wheel

undergoes frictional heating continuously which causes its bulk temperature to become

higher than that of the rail. After continuous running the wheel bulk temperature reaches

a steady state value dependent on many factors including the ambient temperature and

natural or forced cooling of the wheel. When a hot wheel makes contact with the rail,

there is additional temperature rise within the rail at the contact surface [7]. These two

effects are illustrated in Figure 5.1.

The high temperature is confined in a very thin layer near the contact surface [7]. The

thermal stresses in the contacting bodies are significantly increased near the contact

surface. They are superimposed on the contact stresses resulting in higher stresses in the

rail [55]. The temperature rise also produces thermal softening and reduces the yield

stress of rail material [39]. The higher contact stresses and reduced yield stress may

result in early failure of the material, which can make the rail more prone to damage in

the form of wear and rolling contact fatigue.

The effect of temperature on wear was studied by Lewis et. al [29, 105] by conducting

test on a twin disc rig. The bulk temperature rose due to frictional heating and was

measured. They found that the temperature rise can significantly affect the wear


100

transition from mild to severe wear. The influence of the frictional heating on rail wear

was modelled by Widiyarta et. al [39] by varying the slip/roll ratio and friction

coefficient. It was found that the accumulated plastic shear strain increased as both the

slip/roll ratio and friction coefficient increased. For = 0.4 and Sr from -1% to -5%, the

wear rate increased around 11 times compared to the condition without thermal stress

and thermal softening. However, the effect of bulk temperature of the wheel on rail

wear, as explained above, was not included in the model.

Figure 5.1 (a) The friction occurs at the interface of the contacting bodies and generates

heat; (b)The heat generated from frictional work is conducted into the rail, raising its

temperatures during the contact; (c) The rail cools down after passage of the train but

the wheel which is continuously heated attains a higher steady state temperature. When

the hot wheel contacts the rail additional temperature rise occurs in the rail

In this chapter the effect of both frictional heating and wheel bulk temperature on rail

wear were evaluated. The first effect has been termed frictional heating (FH) and the

second effect has been termed steady state wheel temperature (SSWT). The SSWT is

the maximum temperature that the wheel can reach after continuous running at constant

operating conditions. Detail calculation of SSWT can be seen in Section 3.2.2 and

reference [7]. When the hot wheel touches the rail surface the rail temperature rises.

This effect is additional to FH effect. The effect of both flash temperature and steady

state wheel temperature on thermal stresses, thermal softening, and plastic strain and

hence on the wear rate were investigated.

=

(a) (b) (c)

p0p(x)

rail rail + Heat flux due to

conduction from hot wheel Heat flux due to frictional

heating

Hot wheel

v0

rail

Friction at the interface


101

The simulation was run with operating conditions as listed in Table 5.1. The effect of

friction coefficient and slip/roll ratio was investigated with constant p0 of 1.5 GPa and

v0 of 30m/s. The effect of thermal is investigated and compared to the case with no

thermal. The effect of maximum contact pressure and vehicle speed was also

investigated with constant µ of 0.4 and Sr = -3% to see the influence of steady state

wheel temperature on wear compared to that due to frictional heating only. When the

thermal effect is considered, the effect of thermal stresses and thermal softening are

evaluated for both case, which are the effect of frictional heating only and the effect of

steady state wheel temperature.

Table 5.1 Operating conditions used in simulations

Variables

Duration, cycles 200,000

Peak pressure, p0, GPa 1 – 2

Vehicle speed, v, m/s 30 – 110

Semi contact patch, a, mm 5.88

Initial rail temperature, T0, C 15

Friction coefficient, 0.3, 0.4, 0.5, 0.6

Slip/roll ratio, Sr -1%, -2%, -3%, -4%, -5%

The wheel – rail contact may occur at different locations as illustrated in Figure 5.2. In

straight track the contact normally occurs between wheel tread and rail head. However,

due to hunting mechanism the contact location may be shifted and causes wheel flange

– rail gauge contact. The conditions at the rail head – wheel tread contact normally has

high contact pressure and low slip whereas at the rail gauge – wheel flange contact the

contact pressure is relatively lower and the amount of slip increases. To simplify the

real situation the contact location was chosen to be around rail gauge corner – wheel

flange contact. The simulation was run in a 2D brick model at the maximum contact

pressure of 1.5GPa which is typical for wheel – rail contact [2]. The semi contact patch

size is 5.88mm and slip/roll ratio is varied from -1% to -5% to cover the severe wear

due to ratcheting failures. With the vehicle speed of 30 m/s, which is typical for

passenger train, the contact conditions is shown to be in the area of rail gauge – wheel


102

flange contact [16]. The rolling/sliding contact was simulated using a Hertz line contact

in a fully slipping condition with constant friction coefficient and constant velocity over

the surface of the half space. The friction coefficient is taken in the range of 0.3 – 0.6 to

represent the operating conditions that might be possible to occur during the service

especially in dry conditions. From the field measurement the friction coefficient can be

detected up to 0.6 for condition without lubrication [2].

Figure 5.2 The contact patch location of rail – wheel contact can occur at any points

between rail head (A) and rail gauge (B). As the contact location shifts from the rail

head to the rail gauge the contact pressure reduces whereas the amount of slip increases.

The wheel – rail contact was simplified into 2D brick model with plane at the centre line

of the contact

5.2. Rail temperature

Tables 5.2 and 5.3 show the results of variation of friction coefficient, slip/roll ratio,

peak contact pressure, and the vehicle speed effect on the temperature rise. These tables

present the temperature rise due to frictional heating and the temperature rise due to the

high p0 low Sr

low p0 high Sr

A

B

p0

Hertz contact pressure


103

steady state wheel temperature. The sum of these two temperatures is also shown as the

actual temperature. These values are calculated using Equation (3.17), (3.20), and

(3.21).

In Table 5.2 the actual rail temperature rises as the friction coefficient or slip/roll ratio

increases. The additional temperature due to steady state wheel temperature is only

slightly lower than the flash temperature. Hence the maximum of the total rail

temperature (actual temperature) is nearly double of the flash temperature. Without the

additional temperature from steady state wheel temperature, the temperature rise does

not exceed 200C for Sr = -1% and all range of considered. Hence it will not affect

the material softening (see Figure 3.6). If the effect of steady state wheel temperature is

considered, the softening may occur at Sr = -1% and of at least 0.5.

The effect of variation of peak contact pressure and vehicle speed on temperature rise is

shown in Table 5.3. The temperature rise as both peak contact pressure and vehicle

speed increase. Without additional heat from steady state wheel temperature, at p0 = 1.2

GPa and v0 = 30 m/s the temperature rise has exceed 250C which cause the thermal

softening. By considering the steady state wheel temperature the thermal softening has

occurred for the all ranges considered.

The rail temperature variation is dependent on the temperature condition of wheel prior

to the contact. There are two conditions: (1) the steady state wheel temperature is equal

to the steady state rail temperature, and (2) the steady state wheel temperature is higher

than the steady state rail temperature. For condition (1) the rail temperature rises due to

frictional heating only, see Figure 5.3(a). During the wheel-rail contact, when the wheel

is at ambient temperature, the rail temperature starts to increase at x/a = -1 due to

frictional heating and reaches a maximum at x/a = 1. This value can be obtained by

simplifying Equation (3.17) by taking as zero and as 1 below,

( ) ( ) √

(5.1)

Outside the contact patch (x/a > 1), the rail temperature reduces to the ambient

temperature. It is to be remembered that the wheel temperature rise is also due to the

frictional heating.


104

Table 5.2 Maximum rail surface temperature of , , and due to variation of slip/roll ratio and friction coefficient for

p0=1.5GPa, a = 5.88 mm, and v0 = 30m/s = 144kph

Frictional heating temperature ( )

(C)

Additional temperature rise due to steady

state wheel temperature ( )

(C)

Actual temperature ( )

(C)

Sr

-1% -2% -3% -4% -5% -1% -2% -3% -4% -5% -1% -2% -3% -4% -5%

0.3 88.4 176.3 263.8 350.9 437.5 69.9 140.2 210.7 281.7 352.9 158.3 316.5 474.5 632.5 790.4

0.4 117.8 235.1 351.7 467.8 583.4 93.2 186.9 280.9 375.5 470.5 211.0 421.9 632.7 843.4 1053.9

0.5 147.3 293.8 439.7 584.8 729.2 116.5 233.6 351.2 469.2 588.2 263.8 527.4 790.9 1054.2 1317.4

0.6 176.7 352.6 527.6 701.8 875.1 139.8 280.3 421.5 563.3 705.8 316.5 632.9 949.1 1265.1 1580.9


105

Table 5.3 Maximum rail surface temperature of , , and due to variation of peak contact pressure (GPa) and vehicle speed

(m/s) for =0.4, a = 5.88 mm, and Sr=-3%

Frictional heating temperature ( )

(C)

Additional temperature rise due to steady

state wheel temperature ( )

(C)

Actual temperature ( )

(C)

v0

p0 30 50 70 90 110 30 50 70 90 110 30 50 70 90 110

1 234.49 302.20 357.07 404.38 446.57 187.32 241.56 285.56 323.54 357.43 421.81 543.76 642.63 727.92 804.00

1.2 281.39 362.64 428.48 485.26 535.89 224.78 289.87 342.67 388.24 428.92 506.17 652.51 771.15 873.50 964.81

1.4 328.29 423.08 499.89 566.13 625.20 262.24 338.18 399.78 452.95 500.41 590.53 761.26 839.67 1019.08 1125.61

1.6 375.19 483.52 571.31 647.01 714.52 299.70 386.49 456.89 517.66 517.89 674.89 870.01 1028.20 1164.67 1232.41

1.8 422.08 543.96 642.72 727.89 803.83 337.17 434.80 513.99 582.37 643.38 759.25 978.76 1156.71 1310.26 1447.21

2 468.98 604.40 714.13 808.76 893.14 374.63 483.11 571.11 647.08 714.87 843.61 1087.51 1285.24 1455.84 1608.01


106

As the maximum temperature limit is approximately twice as high as the temperature

for the first contact of initially cold wheel [7, 112] the resulting contact temperature at

normal operating condition could be higher than 200C with the steady state wheel

temperature of around 250C [7, 112]. For condition (2) the rail temperature increases

due to both the frictional heating and due to conduction of heat from the hot wheel, see

Figure 5.3(b). In the case considered the steady state wheel temperature is 280C above

the ambient and the rail is at ambient temperature. At the leading edge the wheel

temperature drops and that of the rail increases sharply because the heat of the wheel is

assumed to conduct t the rail instantly. The heat partitioning factor in the current

simulation is in the range of 0.4936 – 0.4987 which has been approximated to 0.5. Since

the heat partitioning factor is about 0.5 the temperature rise for the wheel and the rail

become similar. The temperature rise for both is now around 140C above ambient.

Within the contact patch (-1 < x/a < 1) the temperature profile of the wheel is similar as

that of the rail. The temperature keeps rising because frictional heat input into the wheel

and rail. At the end of contact, the rail temperature drops because of no further

conduction of heat from the wheel into the rail and no further frictional heating.

Likewise the wheel temperature begins to return to its steady state temperature. The

highest rail and wheel temperature rise occurs at x/a = 1 and its value can be obtained

by simplifying Equation (3.21),

( ) ( ) √

√

(5.2)

It can be seen that this is 111% of the steady state wheel temperature, as obtained by

dividing Equation (5.2) by Equation (3.20).


107

(a)

Figure 5.3 Wheel and rail temperature rise profile at the surface for p0=1.5GPa, =0.3,

Sr=-2%, a = 5.88 mm and v0 = 30m/s (a) with frictional heating only (b) with frictional

heating and steady state wheel temperature effect

The frictional heating contributes about 56% of the maximum temperature rise and the

conduction from the hot wheel about 44%, Equations (3.17) and (3.21). The ratio of

(a)

(b)

-2 -1 0 1 20

50

100

150

200

Tem

pera

ture

rise

( C

)

x/a

end of contact

beginning of contact

-2 -1 0 1 20

50

100

150

200

250

300

350

Tem

pera

ture

rise

( C

)

x/a

wheel (FH+SSWT)rail (FH+SSWT)


108

maximum possible value for rail temperature due to steady state wheel temperature and

flash temperature can be obtained by dividing Equation (3.20) by Equation (3.17) and

simplifying it as seen in Equation (5.3). The maximum value for both conditions was

achieved at the surface, =0, =1, with refers to Equation (3.16). The ratio comes out

to be 0.79. If the wheel is cooled by natural or other methods to remain at the ambient

temperature, then the rail temperature would be equal to the frictional heating

temperature. Equation (3.17) (Condition 1) and Equation (3.21) (Condition 2) define the

maximum temperature range that the rail surface may experience.

( ) ( )

√

√

√

( )

(5.3)

From Table 5.2, at p0=1.5GPa, v0=30m/s, =0.3 and Sr=-2%, the maximum of frictional

heating temperature, , reaches 177C, which is not high enough to reduce the yield

stress of the material, see Figure 3.6. However, the yield stress value may drop as the

temperature rises. With the additional temperature due to steady state wheel temperature

effect (condition 2) of 140C, the actual temperature rise of the rail can reach at the

maximum value of 316.5C which can soften the material by 7%. The distribution of

actual temperature rise below the surface, contributions from frictional heating and

conduction from the hot wheel are shown in Figures 5.4(a-c), respectively. The

maximum temperature was determined at each depth and used with Figure 3.6 to obtain

the reduction in material yield stress at that depth.


109

Figure 5.4 The temperature profile below surface for p0 = 1.5GPa, = 0.4, Sr = -3%, a

= 5.88 mm, and v0 = 30m/s: (a) due to the frictional heating only; (b) due to the steady

state wheel temperature, without the frictional heating; (c) due to combination of

frictional heating and steady state wheel temperature

-1.5 -1 -0.5 0 0.5 1 1.5 20

50

100

150

200

250

300

350

Tem

pera

ture

( C

)

x/a

100 m

50 m

30 m

10 m

-2 -1 0 1 20

100

200

300

400

500

600

700

x/a

Tem

pera

ture

( C

)

10 m

0.5 m

30 m

50 m

100 m

(a)

(b)

-1.5 -1 -0.5 0 0.5 1 1.5 20

50

100

150

200

250

300

350

Tem

pera

ture

( C

)

x/a

10 m

30 m

50 m

100 m

(c)


110

5.3. Thermal stresses due to steady state wheel

temperature

The high temperature is confined in a very thin layer near the contact surface [7]. The

thermal stresses in the contacting bodies are significantly increased near the contact

surface. They are superimposed to the contact stresses results in a higher stresses in rail

[55]. The temperature rise also leads to thermal softening and reduces the yield stress of

rail material [39]. The higher contact stresses and reduced yield stress may result in

early failure of the material, which can make the rail more prone to damage in the form

of wear and rolling contact fatigue.

The combination of contact stress and thermal stress near the surface and the maximum

orthogonal shear stress at each depth are shown in Figures 5.5 and 5.6. There are three

stresses to compare: (1) without any thermal effects (zx nt), (2) with frictional heating

only (zx nt +zx FH), and (3) with frictional heating and steady state wheel temperature (zx

nt +zx FH +zx SSWT).

Figures 5.5(a)-(c) show the increase in total orthogonal shear stresses at certain depth. It

is shown in Figure 5.5(a) that at 0.5 m depth the increase of orthogonal shear stress

due to frictional heating and steady state wheel temperature effect is roughly doubled

from the increase in orthogonal shear stress due to frictional heating only, which is in

line with the temperature behaviour discussed in the previous section. The results of the

calculation lead to the infinite value at the leading edge of contact, = -1. At this point,

the model recognizes a sudden drop of temperature, which causes the discontinuity in

calculating the stresses. However, this part is ignored in calculating the total orthogonal

shear stress. Figures 5.5(b)-(c) show that the effect of thermal diminishes with depth. At

z/a=0.5 there is almost no difference between the case without and with thermal effect

including condition (1) and (2).

The maximum orthogonal stress at each depth also shows the similar behaviour as

shown in Figure 5.6. Near the contact surface the increase of maximum orthogonal

shear stress for condition (2) is at least doubled from the increase of maximum

orthogonal shear stress for condition (1). The effect of friction coefficient on zx(max) is


111

shown in Figure 5.7. The value of zx(max) mostly reached maximum at the surface except

for = 0.3 which has the maximum value of zx(max) for = 0.3 at z/a=0.37 under the

surface.

Figure 5.5 Orthogonal shear stress at certain depth: (a) 0.5m depth (b) z/a = 0.1 (c)

z/a= 0.5 (p0=1.5GPa, =0.4, Sr=-3%, a = 5.88 mm, and v0 = 30m/s)

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

zx /

p 0

x/a

zx nt

zx nt + zx FH

zx nt + zx FH + zx SSWT

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

x/a

zx /

p 0

zx at z/a=0.1

zx nt

zx nt + zx FH


-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-0.1

0

0.1

0.2

0.3

0.4

0.5

x/a

zx /

p 0

zx at z/a=0.5

zx nt

zx nt + zx FH


(b)

(a)

(c)


112

Figure 5.6 Maximum orthogonal shear stress at each depth (p0=1.5GPa, =0.4, Sr=-3%,

a = 5.88 mm, and v0 = 30m/s)

Figure 5.7 The effect of friction coefficient variation on zx(max) at each depth by

considering the steady state wheel temperature effect (p0=1.5GPa, Sr=-3%, a = 5.88

mm, and v0 = 30m/s)

0.1 0.2 0.3 0.4 0.5 0.6 0.7-3

-2.5

-2

-1.5

-1

-0.5

0zxmax SSWT

z/a

zx(max) / p0

=0.5

=0.3

=0.4 =0.6

0.2 0.25 0.3 0.35 0.4 0.45-1.5

-1

-0.5

0

zx(max) / p0

z/a

zx nt (max)

zx nt+FH (max)

zx nt+FH+SSWT (max)


113

5.4. Plastic strain accumulation

Figure 5.8 shows the accumulated plastic strain after 3,000 cycles averaged at each

depth by considering the effect of frictional heating only, and the effect of both

frictional heating and steady state wheel temperature. For comparison a case where

there is no heating is also shown. Figure 5.8(a) shows that the thermal effect increases

the accumulated plastic strain. Both thermal stresses and thermal softening of the

material as result of increased temperature are included in these calculations. The effect

of steady state wheel temperature is to further increase the rail temperature, and hence

increases the plastic strain.

Figure 5.8 (a) The effect of frictional heating and steady state wheel temperature on

accumulated plastic strain after 3,000 cycles and (b) the maximum temperature at each

depth (with frictional heating and steady state wheel temperature effect), with

p0=1.5GPa, =0.3, Sr=-2%, a = 5.88 mm, and v0 = 30m/s

Figure 5.8(b) shows the temperature variation with depth. The frictional heating

produces a maximum temperature for 177C for the case considered. This temperature

rise is less than 250C and hence no thermal softening occurs. Hence the plastic strain

(a) (b)

50 100 150 200 250 300 350-0.01

-0.008

-0.006

-0.004

-0.002

0

Temperature (C)

z/a

r FH

r FH+r SSWT


114

increment due to frictional heating shown in Figure 5.8(a) is solely caused by the

development of thermal stress. If the steady state wheel temperature effect is included,

the temperature at the rail surface is about 316C. At this temperature the rail material

loses 7% of its strength which may result in higher plastic strain, see Figure 5.8(a).

Below the surface the plastic strain is increased by thermal softening until depth of z/a =

0.04 as the temperature rise in this region is higher than 250C (see Figure 5.8(b)).

5.5. Rail wear

5.5.1. The effect of and Sr variation on wear rate

The effect of friction coefficient and slip/roll ratio variation on wear rate is shown in

Figures 5.9 and 5.10 for p0 = 1.5 GPa and v0 = 30 m/s corresponding to 108 kph. The

effect of thermal softening is also shown. Without thermal softening, the wear rate due

to frictional heating increases as Sr or increases. The wear rate becomes higher if

thermal softening effect is considered. If the steady state wheel temperature effect is

considered, the thermal softening increases the wear rate significantly. In this case the

gradient of the wear rate is steeper than the gradients of the other wear rates which

appear to be linear.

Figure 5.9 Wear rate due to variation of for p0 = 1.5GPa, Sr=-3%, a = 5.88 mm, and v0

= 30m/s (FH = Frictional Heating; SSWT = Steady State Wheel Temperature)

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.650

100

200

300

400

500

600

700

wea

r rat

e (n

m/c

ycle

)

FH only ; without thermal softeningFH only ; with thermal softeningFH + SSWT ; without thermal softeningFH + SSWT ; with thermal softening


115

Figure 5.10 Wear rate due to variation of Sr for p0 = 1.5GPa, = 0.4, a = 5.88 mm, and

v0 = 30m/s (FH = Frictional Heating; SSWT = Steady State Wheel Temperature)

The results of the wear rate due to variation of friction coefficient and slip/roll ratio are

summarized in Table 5.4. This table shows the results for both conditions (1) and (2).

The effect of thermal softening on wear rate is also included. At low friction coefficient

and slip/roll ratio the temperature rise is not high enough to cause the thermal softening.

It applies for both conditions (1) and (2). Although the thermal stress has developed, its

magnitude is relatively small. Hence it causes very small additional accumulated plastic

strain. Consequently, the wear rate is only slightly increased. As the friction or sliding

increases, the temperature increases causing the thermal softening. Both thermal

softening and thermal stresses developed in the material cause higher plastic strain and

hence enhance the wear rate. At higher temperature, the thermal softening effect is large

enough to reduce the material strength significantly. It causes the material near the

surface to easily reach the critical shear strain for failure which results in a higher wear

rate. Therefore the effect of thermal softening is more severe for condition (2). When

both the frictional heating and the steady state wheel temperature effects are considered

the wear rate increases by up to an order of magnitude. As the thermal softening has a

significant effect on the wear rate it can become an important factor regarding the

development of new rail materials. If the material can resist thermal softening it is

expected that the material will not fail quickly and both wear and rolling contact fatigue

will reduce.

1 1.5 2 2.5 3 3.5 4 4.5 50

100

200

300

400

500

-Sr (%)

wea

r rat

e (n

m/c

ycle

)

FH only ; without thermal softening

FH only ; with thermal softening

FH + SSWT effect ; without thermal softening

FH + SSWT effect ; with thermal softening


116

Table 5.4 Average wear rate (nm/cycle) for conditions (1) and (2) over 200,000 cycles

(w/= with and w/o = without thermal softening)

w/o softening w/ softening

Sr

-1% -2% -3% -4% -5% -1% -2% -3% -4% -5%

FH effect

0.3 3.25e-3 7.20e-3 1.89e-2 5.32e-2 9.81e-2 4.85e-3 1.28e-2 1.56e-2 0.35 1.34

0.4 39.00 43.67 60.20 75.10 81.72 41.36 47.1 61.57 79.06 103

0.5 166.17 176.17 189.41 200.47 214.05 166.57 176.13 197.16 211.4 256

0.6 284.64 298.47 315.06 330.04 342.37 284.95 300.68 324.26 367.58 507

FH + SSWT effect

0.3 4.25e-3 2.00e-2 7.68e-2 0.26 0.28 4.25e-3 0.11 1.95 24.07 113.32

0.4 43.28 76.72 103.17 135.18 166.18 43.57 81.77 135.46 263.1 405.57

0.5 178.01 214.91 246.33 284.55 324.79 178.01 224.12 350.29 561.59 734.86

0.6 306.45 347.21 392.27 439.93 488.16 306.78 372.05 634.92 884.35 1092

5.5.2. The effect of p0 and v0 variation on wear rate

The effect of steady state wheel temperature with variation of peak pressure and vehicle

speed was also investigated and their results are shown in Figures 5.11 and 5.12. Both

figures show the comparison of wear rate with and without steady state wheel

temperature effect. As shown in Figure 5.11, the variation of p0 causes the changes in

wear rate gradient at higher values. The wear rate increases by 50% by steady state

wheel temperature effect at p0 = 2 GPa. The effect of steady state wheel temperature on

the variation of vehicle speed is shown in Figure 5.12 to have a significant increase of

wear rate compared to the case with frictional heating only. Although the temperature

rise at v0 = 90 - 110 m/s and p0 = 1.4 GPa is higher compared to that at v0=30m/s and

p0=2 GPa, the wear rate for the latter is greater than the first. It is because v0 only has an

effect on the thermal stresses but it does not have a significant effect on changing the

contact stresses. The gradient of wear rate decreases as v0 increases because when the

vehicle speed increases, the thermal penetration depth reduces. Hence the temperature

rise has less effect in the material at greater depth. The effect of p0 and v0 variation on

wear rate is summarized in Table 5.5. These results show that the additional temperature


117

rise due to steady state wheel temperature cause the wear rate to increase for at least 1.5

times compared to the wear rate due to frictional heating only.

Figure 5.11 Wear rate due to variation of p0 with thermal softening for = 0.4, Sr=-3%,

a = 5.88 mm, and v0 = 30m/s (FH = Frictional Heating; SSWT = Steady State Wheel

Temperature)

Figure 5.12 Wear rate due to variation of v0 with thermal softening for = 0.4, Sr=-3%,

a = 5.88 mm, and p0 = 1.4 Pa (FH = Frictional Heating; SSWT = Steady State Wheel

Temperature)

1 1.2 1.4 1.6 1.8 20

200

400

600

800

p0 (GPa)

wea

r rat

e (n

m/c

ycle

)=0.4; Sr=-3%; v0=30m/s with softening

FHFH + SSWT

30 40 50 60 70 80 90 100 1100

50

100

150

200

v0 (m/s)

wea

r rat

e (n

m/c

ycle

)

=0.4; Sr=-3%; p0=1.4GPa with softening

FHFH + SSWT


118

Table 5.5 Average wear rate (nm/cycle) for µ = 0.4 and Sr = -3%

v0

p0 (m/s)

(GPa)

30 50 70 90 110

FH effect

1 - 1.2e-3 1.39e-2 6.27e-2 0.14

1.2 0.36 0.82 1.23 1.71 2.16

1.4 8.28 10.36 11.36 13.19 21.12

1.6 120.25 122.27 125.36 132.78 150.37

1.8 270.83 279.05 288.61 304.96 343.10

2 443.20 451.50 474.20 523.51 546.34

FH + SSWT effect

1 0.25 1.49 3.30 6.98 19.98

1.2 4.13 8.58 24.51 50.23 65.21

1.4 48.80 99.15 146.61 170.79 185.64

1.6 206.63 297.64 339.98 358.20 370.06

1.8 420.25 522.98 560.29 576.59 584.67

2 677.09 773.19 803.46 817.80 824.09

It is shown in the effect of operating conditions variation on the wear rate that the

thermal softening has a significant effect on the wear rate. Hence it can become an

important factor regarding with the development of new rail materials. If the material

can resist thermal softening, it is expected that the material will not fail quickly and

hence both wear and rolling contact fatigue will reduce.

The effect of thermal softening on the wear failure has been evaluated through some

experimental investigations. Sim, et al. [104] considered the wear characteristics of

steels against abrasive materials. Their experimental results showed that thermal

softening from frictional heating can enhance the wear failure of steels. Through

experiments Wang et.al [82] also showed that the more thermally stable steels generally

exhibit higher wear resistance.

Table 5.4 also shows that the steady state wheel temperature effect increases the wear

rate gradually as friction coefficient or slip/roll ratio increases. For low value of friction

and slip/roll ratio, the wear rate does not alter much due to steady state wheel

temperature. However, at high values of friction coefficient or slip/roll ratio the wear

rate has a significant increase if thermal softening is ignored and the increment is even


119

larger if the thermal softening is considered. The effect of steady state wheel

temperature can be reduced by avoiding temperature rise of wheel body as it passes over

the rail. When the wheel bulk temperature is kept low, the temperature rise in the rail

material is only mainly due to frictional heating. If the temperature does not exceed

600C, the thermal softening reduces the material strength by less than 50% and the

phase transformation does not occur [113]. An alternative to reduce steady state wheel

temperature could be the lubrication of wheel rail interface which can reduce the value

of friction between the two surfaces. As the friction reduces, the amount of heat due to

frictional heating reduces with overall lower temperature for rail and the wheel

temperature. Beside lubrication, the design of the train wheel which has self-cooling

effect can be another alternative to reduce its bulk temperature. Further investigation is

needed in this area.

5.6. Conclusion

During rolling/sliding contact, the wheel and rail temperature rise due to frictional

heating. The rail cools down after the train has passed, but the wheel is continuously

heated up and hence its steady state temperature increases. When the steady state wheel

temperature is higher than that of the rail, the rail surface temperature can be up to 79%

higher which develops higher stresses. At higher temperature the thermal softening is

also possible. The increasing thermal stresses and thermal softening may result in higher

plastic strain which leads to a greater wear rate. For higher temperature the effect of

thermal softening on the wear rate was shown to be more pronounced than the effect of

thermal stresses. This can increase the wear rate by up to an order of magnitude for the

case considered. It was shown in [105] that the new material shifted the wear transition

to catastrophic wear to higher values of T/A. While this transition was caused by the

thermal effect the new material has shown a better ability to resist thermal softening.

Beside of improving material resistance to thermal softening the level of wear rate due

to thermal effect can also be reduced by preventing the increase of the temperature of

the wheel. Alternatively lubrication could be applied which may reduce the frictional


120

heat, or a self-cooling effect could be designed for the wheel which may reduce its

temperature rise during service.

Chapter 6. The effect of material properties on rail wear

121

Chapter 6

The effect of material properties on rail

wear

6.1. Introduction

The ongoing development of rail material is now an important consideration due to

today’s increase in axle load and speed combined with dynamic environmental

conditions. Rail in service experiences more severe loading than previously, which may

result in higher wear and rolling contact fatigue failure. The variation in operating

conditions at different locations of wheel–rail contact also makes the situation more

complex. The lower axle load and high speed of a passenger train and heavy haul

operations with a higher axle load and lower speed also lead to various combinations of

operating conditions. This may result in different levels of damage to the rail. During

rail operations, the rail head–wheel tread and the rail gauge–wheel flange contacts may

occurs as the contact locations shift from the rail head to the rail gauge. The rail head–

wheel tread contact normally has relatively lower contact pressure and slip compared to

those occurring at the rail gauge–wheel flange contact. The magnitude of friction is

likely to be higher at the rail head due to the accelerating and breaking conditions,

whereas at the rail gauge, the friction is managed to be kept low using lubrication to

reduce wear due to the high slip situation [60].

The level of wear rate in rail–wheel contact is influenced by the contact loading and the

properties of the material. In order to see the effect of material properties on the wear

rate, two pearlitic rail steels, i.e. UIC 1100 and UIC 900A were used in the current

work. The material properties considered are the hardness, the critical strain to failure,

the hardening behaviour and the thermal softening behaviour. Both UIC 1100 and UIC

900A have different properties and different behaviour as obtained from previous

studies [8, 22, 45]. These properties are described in Section 3.6.


122

Wear simulations using these material properties were carried out for several contact

conditions, listed in Table 6.1. The three cases represent the contact conditions from the

rail head to the rail gauge. These three cases are based on previous studies that

investigated wheel–rail contact at different contact locations [2, 15, 65, 114]. The values

used are the operating conditions of a commuter train on the Alsvjö track, Sweden. The

friction coefficient varied from 0.2 to 0.7 for all cases because of the environmental

conditions on the contacting rail and wheel surfaces, such as the weather, season, or the

dirt and leaves attached to it.

Table 6.1 Operating conditions used in simulation

CASE 1 CASE 2 CASE 3

Peak pressure

(p0, GPa) [16] 1.5 2.1 2.7

Semi contact

patch

(a, mm) [114]

8.2 9.37 10.48

Slip/roll ratio

(Sr) [1] -0.5% -1.5% -3% -0.5% -1.5% -3% -0.5% -1.5% -3%

Vehicle speed

(v0, m/s) [65] 20 17.5 15 20 17.5 15 20 17.5 15

Friction

coefficient

(µ) [2]

0.2 – 0.7 0.2 – 0.7 0.2 – 0.7

6.2. Temperature rise

Chapters 4 and 5 describe how the temperature of the rail depends on the frictional

heating and wheel bulk temperature. Chapters 4 and 5 have shown that the temperature

rise increases the amount of wear regardless of the source of the heat. Therefore it is

sufficient to use frictional heating only to see the effect of material properties on wear.


123

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 30

50

100

150

200

250

300

350

400p0 = 1.5GPa, = 0.5

x/a

Tem

pera

ture

rise

at d

epth

= 0

.5

m (

C)

Sr = -0.5%

Sr = -1.5%

Sr = -3%

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 30

50

100

150Sr = -0.5%, = 0.5

x/a

Tem

pera

ture

rise

at d

epth

= 0

.5

m (

C)

CASE 1p0 = 1.5GPa

a = 8.2mm

CASE 2p0 = 2.1GPa

a = 9.37mm

CASE 3p0 = 2.7GPa

a = 10.48mm

The temperature rise near the surface, due to variation of Sr, is shown in Figure 6.1.

This figure shows the temperature rise near the surface at a depth of 0.5µm for p0 =

1.5GPa and µ = 0.5. The temperature rise is shown to be proportional to the increase in

Sr. Figure 6.2 also shows the comparison of the temperature rises for different contact

pressures corresponding to the three cases when Sr = -0.5% and µ = 0.5. The higher the

peak pressure the greater the amount of frictional energy generated at the interface. This

frictional energy is also influenced by the size of the contact patch, and will be higher as

more area experiences the friction. The value of a increases from 8.2mm to 10.48mm.

Thus the temperature rise increases as the contact patch location shifts from the rail

head to the rail gauge.

Figure 6.1 Temperature rise due to Sr variation (p0 = 1.5GPa, µ = 0.5)

Figure 6.2 Temperature rise of the three cases (Sr = -0.5%, µ = 0.5)


124

Table 6.2 summarizes the temperature rises for the three cases shown in Table 6.1. The

results show that the temperature rise increases as the friction coefficient or slip/roll

ratio increases. Greater contact patch and higher maximum contact pressure also

enhance the temperature rise. In addition, Figure 6.3 shows that the reduction of v0

causes the temperature rise to infiltrate at a greater depth. This is indicated by the value

of thermal penetration depth. Using Equations (3.5 – 3.7), the thermal penetration depth

for v0 of 20m/s, 17.5m/s and 15m/s are 9.14µm, 9.7µm and 10.6µm, respectively.

Table 6.2 Maximum temperature rise at the surface due to frictional heating (C)

p0 = 1.5GPa ; a = 8.2 mm

Sr v0

(m/s)

µ =

0.2

µ =

0.3

µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 28.6 42.9 57.2 71.5 85.5 99.7

-1.5% 17.5 80.1 119.7 159.5 199.4 239.3 279.2

-3.0% 15 147.7 220.8 294.42 368.0 441.6 515.2

p0 = 2.1GPa ; a = 9.37 mm

Sr v0

(m/s)

µ =

0.2

µ =

0.3

µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 42.8 63.9 85.3 106.6 127.9 149.3

-1.5% 17.5 119.8 179.1 238.8 298.5 358.2 417.9

-3.0% 15 221.0 330.5 440.7 550.9 661.1 771.2

p0 = 2.7GPa ; a = 10.48 mm

Sr v0

(m/s)

µ =

0.2

µ =

0.3

µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 58.2 87.0 116.0 145.0 174.0 203.0

-1.5% 17.5 162.9 243.6 324.8 406.0 487.2 568.4

-3.0% 15 300.5 449.5 599.4 749.2 899.0 1048.9


125

0 50 100 150 200 250 300-0.1

-0.08

-0.06

-0.04

-0.02

0

Temperature rise (C)

z/a

v0 = 20m/s

v0 = 17.5m/s

v0 = 15m/s

0 5 10

-0.1

-0.08

-0.06

-0.04

-0.02

0

Temperature rise (C)

z/a

Figure 6.3 Effect v0 on temperature rise (p0 = 1.5GPa, µ = 0.4). The thermal penetration

depth is shown clearly in the inset figure

6.3. Maximum orthogonal shear stress

The increasing temperature in the rail will enhance the development of the thermal

stress inside the material. Figures 6.4 and 6.5 show the maximum orthogonal shear

stress (zx(max)) at each depth due to variation of Sr and µ respectively. These figures

represent zx(max) in case 1. Figure 6.4 shows that the maximum orthogonal shear stress

increases as Sr increases. The effect of µ variation shown in Figure 6.5 indicates that the

value of zx(max) also increases as µ increases. In the specified operating conditions the

increment is about 150MPa as µ increases by 0.1.


126

Figure 6.4 Thermo-mechanical stresses due to Sr variation (p0 = 1.5GPa, µ = 0.5)

Figure 6.5 Thermo-mechanical stresses due to µ variation (p0 = 1.5GPa, Sr = -0.5%)

A comparison of the thermal stress for the three cases is shown in Figure 6.6. The

graphs in this figure represent the condition in all cases with µ = 0.5. It can be seen that

as p0 increases by 0.6GPa, the value of zx(max) at depth of 0.5µm increases by 200MPa.

However, this increment is actually a combination effect of p0 and semi contact patch

480 500 520 540 560 580 600 620 640

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0p0 = 1.5GPa, = 0.4 --> Sr variation

z/a

zx (max) (MPa)

Sr = -0.5%

Sr = -1.5%

Sr = -3%

200 300 400 500 600 700 800 900 1000 1100-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0p0 = 1.5GPa, Sr = -0.5%, v0 = 30m/s --> variation

z/a

zx (max) (MPa)

= 0.2

= 0.3

= 0.4 = 0.5

= 0.7 = 0.6


127

size, a. Because both variables affect the amount of contact stress and thermal stress, the

value of zx(max) increases as the contact patch location shifts from rail head to rail gauge.

Figure 6.6 Thermo-mechanical stresses for different cases (Sr = -0.5%, µ = 0.5)

6.4. Thermal softening

The effect of temperature rise on the shear yield stress for UIC 1100 and UIC 900A rails

is based on the thermal softening model presented in Figure 3.6. In this figure, the

reduction of yield stress for the UIC 1100 rail shows a decreasing curve as the

temperature rises. On the other hand, the reduction of yield stress of UIC 900A has a

slight fluctuation under 300C before the curve goes down. The curve of UIC 900A

shows a reduction of yield stress up to 100C and an increase with a further rise in

temperature up to 300C. Using this thermal softening (with thermal hardening for UIC

900A) behaviour, Figure 6.7 shows the comparison of shear yield stress against depth

for both UIC 1100 and UIC 900A. The temperature rise in Figure 6.7(a) is the

maximum temperature calculated at each depth. Figure 6.7(b) shows that the shear yield

stress of UIC 1100 decreases as the depth decreases. This is because the temperature

rise is greater closer to the surface. Therefore the layers near the surface experience

greater thermal softening. For UIC 900A, the shear yield stress shows a fluctuation for

z/a < 0.06. At the depth of z/a > 0.03 the temperature rise is less than 100C overall,

300 400 500 600 700 800 900 1000 1100-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0 = 0.4, Sr = -0.5%, v0 = 20m/s --> p0 variation

z/a

zx (max) (MPa)

CASE 1p0 = 1.5GPa

a = 8.2mm

CASE 2p0 = 2.1GPa

a = 9.37mm

CASE 3p0 = 2.7GPa

a = 10.48mm


128

resulting in a softening of material strength (see Figure 3.6). The yield stress increases

at 0.008 < z/a < 0.03 when the temperature rise is in the range of 100-300C. At z/a <

0.008, the temperature rise is higher than 300C and results in the reduction of yield

stress (see Figure 3.6). The different softening behaviour of these two rail steels affects

the final hardness of the material and may result in a different level of wear rate.

Figure 6.7 The effect of temperature rise on shear yield stress, plotted against depth; (a)

temperature rise with depth, (b) shear yield stress of UIC 1100 after softening, (c) shear

yield stress of UIC 900A after softening

6.5. Wear rate

6.5.1. UIC 1100 rail steel

A comparison of wear rate for the three cases of UIC 1100 is shown in Figure 6.8. This

figure shows that there is no wear for µ = 0.2. At µ 0.3, the wear rate in case 3 is

significantly higher than that in the other two cases and the wear rate in case 2 is higher

than that in case 1. In case 1 the surface curve of wear rate looks flat, while in case 2 the

surface curve increases up to 200 - 300 nm/cycle. The surface curve for case 3 increases

0 200 400-0.08

-0.06

-0.04

-0.02

0

z/a

temperature rise (C)

max temp rise at each depth p0=1.5Gpa; Sr = -3%; - 0.5

2.6 2.8 3 3.2

x 108

-0.08

-0.06

-0.04

-0.02

0

k0(softening) (Pa)

z/a

k0 UIC 900A p0=1.5Gpa; Sr = -3%; - 0.5

3 3.5 4 4.5

x 108

-0.08

-0.06

-0.04

-0.02

0

z/a

k0(softening) (Pa)

k0 UIC 1100 p0=1.5Gpa; Sr = -3%; - 0.5

(a) (b) (c)


129

significantly, especially at µ = 0.7 and Sr = -3%, which is around 3 times higher than

that at µ = 0.7 and Sr = -3% in case 2. This indicates that an increase in p0 and a can

lead to a significant increase in the wear rate, particularly with a higher friction

coefficient. The slip/roll ratio also contributes to an increase in the wear rate especially

in case 3 with a significant increment at µ = 0.7 and Sr = -3%.

Figure 6.8 Average wear rate of UIC1100 rail steel: (a) Case 1; (b) Case 2; (c) Case 3

The average wear rate for UIC 1100 rail is summarized in Table 6.3. This table shows

the detailed results for the three cases with µ and Sr variation. It shows that the wear

rate increases as Sr or µ or p0 increases. There is no wear detected in case 1 at µ < 0.5

and at µ < 0.4 for case 2 except at µ = 0.3 and Sr = -3% with very low wear rate at a

value of less than 1 nm/cycle. The wear rate in case 3 occurs earlier at µ = 0.3 compared

to the other two cases, although only with values of less than 6 nm/cycle. The higher


130

values of p0 and a in case 3 cause higher stresses and plastic strain accumulation

resulting in wear at lower friction coefficient.

Table 6.3 Average wear rate (nm/cycle) after 200,000 cycles

Case 1: p0 = 1.5GPa ; a = 8.2 mm

Sr v0 (m/s) µ = 0.2 µ = 0.3 µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 0 0 0 8.25 x 10-3 2.97 16.3

-1.5% 17.5 0 0 0 2.31 x 10-1 6.12 23.83

-3.0% 15 0 0 0 1.28 11.77 35.86

Case 2: p0 = 2.1GPa ; a = 9.37 mm

Sr v0 (m/s) µ = 0.2 µ = 0.3 i = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 0 0 1.10 20.09 110.32 235.46

-1.5% 17.5 0 0 3.33 30.46 131.29 259.64

-3.0% 15 0 2.2 x 10-3 7.45 51.17 177.95 354.61

Case 3: p0 = 2.7GPa ; a = 10.48 mm

Sr v0 (m/s) µ = 0.2 µ = 0.3 µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 0 4.62 x 10-1 136.8 360.41 572.79 800.72

-1.5% 17.5 0 2.03 172.66 389.21 608.42 852.48

-3.0% 15 0 5.87 224.81 477.12 887.68 1349.9

6.5.2. UIC 900A rail steel

The average wear rate for the three cases of UIC 900A rail steel is plotted in Figure 6.9.

This figure clearly shows that in all cases the average wear rate increases as Sr or µ

increases. This normally occurs with sliding and friction as this means that more heat is

generated at the contact patch. This causes the temperature to rise and leads to greater

stress and thermal softening. Thus the wear rate increases. A comparison of the three

cases shows that the wear rate in case 1 is the lowest, followed by case 2 and then case

3. As the value of a increases, the wear rate increases. The increase of p0 also has a


131

significant role because it produces greater stress and higher temperature, which results

in more plastic strain and hence higher failure. Therefore the wear commences at lower

µ with at a higher value as p0 and a increase. Figure 6.9 shows that the increases in wear

in case 1 is very small compared to case 2 and 3. In case 3, the magnitude of wear rate

at µ = 0.7 is roughly 3 times higher than in case 2, and 30 times higher compared to

case 1.

Figure 6.9 Average wear rate of UIC 900A rail steel

The average wear rate for UIC 900A is summarized in Table 6.4. In all 3 cases, the wear

rate increased as µ or Sr increased. No wear rate was detected in case 1 for µ < 0.5. The

value of µ and Sr in these conditions is not high enough to cause a significant plastic

strain increment. Hence the brick elements cannot accumulate plastic strains high

enough to exceed the critical strain to failure. At µ = 0.5 the wear rate is still zero for Sr

= -0.5% and has a very low value at Sr = -1.5% and -3% which is less than 1 nm/cycle.


132

At µ = 0.6 the wear rate increases more than 15 nm/cycle and it is at least tripled at µ =

0.7.

Table 6.4 Average wear rate (nm/cycle) for UIC 900A rail steel

p0 = 1.5GPa ; a = 8.2 mm

Sr v0 (m/s) µ = 0.3 µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 0 0 0 17.30 58.42

-1.5% 17.5 0 0 2171 x 10-1 16.64 67.72

-3.0% 15 0 0 1.30 x 10-1 28.04 87.61

p0 = 2.1GPa ; a = 9.37 mm

Sr v0 (m/s) µ = 0.3 µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 1.25 x 10-3 8.43 161.0 364.67 582.35

-1.5% 17.5 1.5 x 10-4 6.91 181.63 383.31 606.94

-3.0% 15 0 18.85 216.72 464.97 797.15

p0 = 2.7GPa ; a = 10.48 mm

Sr v0 (m/s) µ = 0.3 µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 5.68 511.29 875.12 1221.5 1608.6

-1.5% 17.5 2.48 x 10-1 557.13 908.83 1279.4 1668.1

-3.0% 15 49.98 596.39 1051.4 1748.0 2498.5

In case 2, the wear rate generally increased as µ or Sr increased except at µ = 0.3. It can

be seen that when µ = 0.3, the wear rate reduced as Sr increased. It even reached zero

when Sr = -3%. This interesting pattern can be linked to the thermal softening behaviour

of this material. Figure 3.6 shows that there is an increment of yield stress when the

temperature rise is in the range of 100 – 300C. When the yield stress increases, the

material can prevent greater plastic strain. From Table 6.2 the temperature rises are

64C, 179C and 330C for Sr = -0.5%, -1.5%, and -3% (case 2, µ = 0.3), respectively.

Therefore the increasing of yield stress occurred at Sr = -1.5% and -3%. Moreover, if

the shear yield stress is higher than the maximum shear stress then there will be no

plastic deformation occurring. Hence the wear rate could be zero.

In case 3, the wear rate was similar to case 2. The wear rate increased as µ or Sr

increased except at µ = 0.3, which showed a reduction from Sr = -0.5% to Sr = -1.5%


133

and an increase from Sr = -1.5% to Sr = -3%. The wear rate reduction at Sr = -1.5% was

for the same reason as for case 2, which is thought to be caused by the increase in yield

stress in the temperature range of 100-300C. As shown in Table 6.2 the temperature

rose by 87C, 243C and 449C for Sr = -0.5%, -1.5% and -3% (case 3, µ=0.3),

respectively. Hence there was more reduction of yield stress for Sr = -0.5% and -3% at

Sr = -1.5% the yield stress near the surface was dominated by hardening. When Sr = -

3%, the wear rate increased because the temperature rise exceeded 300C and the

material had softened again.

6.6. Discussion

The performance of UIC 1100 and UIC 900A on wear shows that both rail materials

present an increase in wear rate as µ, Sr, p0 or a increases, with better wear resistance in

UIC 1100 compared to UIC 900A. The comparison of this performance is shown in Fig.

6.10. In all three cases UIC 900A has higher wear rate than UIC 1100. As the operating

conditions were the same, the difference in wear rate results was driven by the

properties of the rail material. The mechanical properties listed in Table 3.3 and 3.4

show that UIC 1100 has higher initial hardness compared to UIC 900A, with a value of

tensile yield stress at 710MPa for UIC 1100 and 507MPa for UIC 900A. Besides the

initial hardness, the rail will undergo a hardening process as more plastic strain emerges

with each cycle. The hardening process alters the original hardness to a greater value.

Figure 6.11 shows the yield stress for UIC 900A and UIC 1100 due to strain

hardening over different cycles. The hardening behaviour of material is determined by

the rate of hardening and the limiting hardness to the original hardness. The hardening

behaviour of UIC 1100 and 900A shown in Table 3.3 and 3.4 reveals that UIC 1100 rail

has a faster hardening rate at = 95.18 than UIC 900A where = 63.94. However the

ratio of limiting hardness and original hardness, , of UIC 1100 is lower than UIC

900A, with the value of 1.86 and 2.31, respectively. It can be seen in Figure 6.11 that

the yield stress of UIC 1100 reaches k0 at around 200 cycles whereas UIC 900A shows

a higher value of around 400 cycles to reach maximum. This figure also shows that

although the ratio of limiting hardness to original hardness for UIC 900A is higher than


134

UIC 1100, the final hardness (k0) is 762 MPa for UIC 1100 and 676 MPa for UIC

900A. These characteristics clearly show that UIC 1100 has higher initial and final

hardness.

Figure 6.10 Wear rate comparison (a) case 1 (b) case 2 (c) case 3

(c)

(b)

(a)


135

0 100 200 300 400 500 600 700620

640

660

680

700

720

740

760

780p0 = 1.5GPa, Sr = -1.5%, = 0.5

shea

r yie

ld s

tress

(M

Pa)

Cycles

UIC 1100 railUIC 900A rail

k0

k0

Figure 6.11 Hardening behaviour of UIC 1100 and UIC 900A rail steel

The hardness of the rail material is also affected by the temperature rise during

rolling/sliding contact. The temperature rise causes softening of the material. For UIC

900A there is some hardening between 100C-300C. The elements that are softened

become weaker and more prone to failure. Unlike the hardening process which occurs

continuously due to plastic strain increment in each cycle, thermal softening process

only occurs when the temperature rise is high enough to reach the level to soften the

material. If thermal softening occurs, it will follow the hardening process in each cycle

until the material fails. Figure 6.12 shows an example of the final shear yield stress after

strain hardening and thermal softening up to a certain cycles. As the number of cycles

increases the shear yield stress after hardening and softening process (keff soft) also

increases. However, after a certain point, this value does not increase any further as it

has reached the limiting hardness. With operating conditions of p0 = 2.1 GPa, Sr = -3%

and µ = 0.3, the effective shear yield stress reaches a steady value at 2,000 cycles.


136

Figure 6.12 Maximum orthogonal shear stress (zx(max)) and effective shear yield stress after

softening (keff_soft) over different cycles (p0 = 2.1GPa, Sr = -3%, v0 = 15m/s, and µ = 0.3)

Different behaviour as regards the thermal softening of the two rails, UIC 1100 and UIC

900A, shown in Figure 3.6, leads to different responses to the loading. This figure

shows that the yield stress of UIC 1100 shows a decreasing curve along the elevated

temperature. Therefore it will cause higher plastic strain when the temperature is

elevated. However, the yield stress reduction for UIC 900A in Figure 3.6 shows a

fluctuation curve. It starts with a decrease in yield stress as the temperature rises up to

100C, and is then followed by an increase in yield stress when the temperature range is

100 - 300C. The yield stress decreases again when the temperature rises above 300C.

The effect of this fluctuation on shear yield stress is shown in Figure 6.13(a) which

plots keff(soft) and zx(max) for Sr = -1.5% and -3% at an elevated temperature (Figure

6.13(b)). Figure 6.13(a) shows that the maximum shear stress for Sr = -1.5% is lower

than that for Sr =-3% because of the lower thermal stress. However keff(soft) for Sr = -

1.5% has an increasing curve to the surface which means that the yield stress hardens

4.5 5 5.5 6 6.5 7 7.5 8 8.5 9

x 108

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0p021 sr=03% 0.3 UIC900A

Stress (Pa)

Nor

mal

ized

dep

th (z

/a)

keff softat 10 cycles



keff soft

at 200,000 cycles

zx(max)


137

up. The temperature rise plotted in Figure 6.13(b) shows that at Sr = 1.5% temperature

rise is in the range of 100C - 250C for -0.01 z/a 0. Thus the material will

experience increased yield stress at this depth. When Sr = -3%, the curve of keff(soft) also

shows an increase in yield stress for z/a between -0.025 and -0.0075. However it then

reduces towards the surface once the elevated temperature at -0.0075 < z/a is more than

300C.

Figure 6.13 (a) The comparison of maximum shear yield stress (zx(max)) and shear yield stress

during softening (keff_soft) (b) the temperature rise (p0 = 2.7GPa and µ = 0.3)

As the magnitude of plastic strain is driven by the ratio of (zx(max)/keff soft) the curve of

also follows this ratio. It can be seen in Figure 6.14 that the value of at Sr = -1.5%

decreases for z/a 0.01 with the value of = 4x10-4 at a depth just below the surface.

The plastic strain increment for Sr = -3% shows an increase below z/a = -0.025 and a

decrease between z/a = -0.025 to -0.0075. It increases again above z/a = -0.0075 with

= 11.5x10-4near the surface, which is almost three times higher than that when Sr = -

0 200 400 600-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

temperature rise (C)

Nor

mal

ized

dep

th (z

/a)

p021 sr=-1.5% vs -3% 0.3 UIC900A

Sr = -1.5%

Sr = -3%

(a) (b)


138

1.5%. Because the plastic increment near the surface is quite low for Sr = -1.5%, the

plastic strain accumulates very slowly, which results in a low wear rate, i.e.

0.25nm/cycle (see Table 6.4). At Sr = -3%, the plastic strain accumulates faster and

leads to more failed bricks that have the potential to be detached as wear debris or to

initiate a crack. The amount of wear rate in this case is much higher at 49nm/cycle (see

Table 6.4).

Figure 6.14 Plastic strain increment for Sr =-1.5% and Sr = -3% (p0 = 2.7GPa and µ = 0.3)

From the characteristics described above, UIC 1100 shows greater strength in terms of

hardness and hardening behaviour. However, in terms of softening behaviour, UIC

900A gives a better performance. Despite the fluctuation in yield stress reduction at

elevated temperatures, the normalized yield stress is higher in UIC 900A than in UIC

1100. Table 6.5 shows the ratio of the wear rate of UIC 900A to UIC 1100 rail (w_UIC

900A / w_UIC 1100), and also the normalized yield stress (

) for both materials for all

three cases considered. The ratio of wear rate includes the effect of both hardness and

thermal softening. The overall performance shows that UIC 1100 has higher wear

resistance to UIC 900A. This is shown by the wear rate ratio between the two rail types,

which is more than 1. The reason for this is that UIC 1100 has greater original and final

2 4 6 8 10 12

x 10-4

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

Nor

mal

ized

dep

th (z

/a)

p021 sr=-1.5% vs -3% 0.3 UIC900A

Sr = -3%

Sr = -1.5%


139

hardness than UIC 900A. However the ratio of wear rate decreases as the load increases,

shown in this table, with the least value being seen in case 3. It is thought that thermal

softening plays a role by lowering the yield stress in UIC 1100 more than in UIC 900A

(see Figure 3.6). It can also be seen in Table 6.5 that the UIC 1100 yield stress is mostly

reduced by a greater amount than in UIC 900A.

Table 6.5 Ratio of wear rate (w_UIC 900A / w_UIC 1100), normalized yield stress (

) of

UIC 900A at z = 0.5µm, and normalized yield stress (

) of UIC 1100 at z = 0.5µm

Sr P0 = 1.5Gpa ; a = 8.2 mm

v0 (m/s) µ = 0.2 µ = 0.3 µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 - - 1.08 1.02

- 1.08 1.02

23.9 0.95 0.93

5.8 0.96 0.93

3.6 0.97 0.92

-1.5% 17.5 - - 1.08 1.02

- 1.04 0.91

0.94 1.09 0.91

2.7 1.14 0.91

2.8 1.17 0.90

-3.0% 15 - - 1.08 1.02

- 1.17 0.90

0.10 1.13 0.86

2.4 1.01 0.78

2.4 0.86 0.66

Sr p0 = 2.1GPa ; a = 9.37 mm

v0 (m/s) µ = 0.2 µ = 0.3 µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 - Inf* 0.95 0.94

7.6 0.96 0.93

8.0 0.97 0.92

3.3 0.99 0.91

2.4 1.03 0.91

-1.5% 17.5 - Inf* 1.07 0.91

2.0 1.14 0.91

5.9 1.17 0.89

2.9 1.14 0.86

2.3 1.06 0.80

-3.0% 15 - 0** 1.16 0.88

2.5 1.02 0.78

4.2 0.77 0.59

2.6 0.49 0.38

2.2 0.16 0.18

Sr p0 = 2.7GPa ; a = 10.48 mm

v0 (m/s) µ = 0.2 µ = 0.3 µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 - 12.3 0.96 0.93

3.7 0.98 0.92

2.4 1.02 0.91

2.1 1.06 0.91

2.0 1.1

0.91

-1.5% 17.5 - 0.12 1.15 0.91

3.2 1.16 0.89

2.3 1.08 0.82

2.1 0.92 0.71

1.9 0.73 0.56

-3.0% 15 - 8.5 1.00 0.77

2.6 0.65 0.50

2.2 0.23 0.22

2.0 0.05 0.07

1.8 0.05 0.13

* no wear in UIC 1100 ** no wear in UIC 900A


140

There are also certain conditions where the ratio of wear rate is less than 1, as shown in

the shaded areas. This means that there is more wear in UIC 1100 than in UIC 900A.

The value of normalized yield stress shows that, under these conditions, the yield stress

reduction in UIC 1100 is around 25% lower than in UIC 900A. This may cause a higher

plastic strain increment in UIC 1100. In order to demonstrate the details of the yield

stress reduction under these conditions, Figure 6.15(a) shows the plot of shear yield

stress after softening in case 2 with µ = 0.3 and Sr = -3%. This figure reveals that the

k0(softening) of UIC 1100 near the surface is lower than that of UIC 900A but higher at z/a

< -0.01. Because of the lower shear yield stress near the surface, the plastic strain

accumulates faster, resulting in greater wear in UIC 1100. Although UIC 900A has

much lower k0(softening) at -0.05 < z/a < -0.01 and -0.8 < z/a < -0.2, greater accumulated

plastic strain in this region stays at the same depth because the plastic strain at the

surface accumulates very slowly (Figure 6.15(b)). In the current simulation, it does not

reach the critical strain to failure unless reaching more than 200,000 cycles. Therefore

no wear can be detected in the current result.

The wear rate ratio around the shaded area in Table 6.5 shows that, although the amount

of yield stress reduction in UIC 1100 is lower than that in UIC 900A, the wear is much

higher in UIC 900A. For example, in case 2, when µ = 0.4 and Sr = -3% the wear rate of

UIC 900A is 2.5 times higher than the wear rate of UIC 1100. Figure 6.16 reveals that

although the value of k0(softening) of UIC 1100 is lower than that of UIC 900A, this

condition only occurs at z/a > -0.02. With greater depth, the value of k0(softening) of UIC

900A is much lower than that of UIC 1100 up to z/a = -1. This is because the

temperature rise is confined to a thin layer near the surface. Thus the effect of softening

is also localized to this region. By having greater hardness, UIC 1100 can resist failure

more at greater depth.


141

(a)

(b)

Figure 6.15 Comparison of shear yield stress during softening process (case 2, Sr = -3%,

µ = 0.3); (a) at z/a -0.06; (b) at z/a -0.9

6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8

x 108

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

Stress (Pa)

z/a

k0 softening 900A 1100 sr = -3% = 0.3 case 2

zx(max)

k0(softening) UIC 1100

k0(softening) UIC 900A

5 5.5 6 6.5 7 7.5 8

x 108

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Stress (Pa)

z/a



zx(max)



142

Figure 6.16 Comparison of shear yield stress during softening (case 2, Sr = -3%, µ=0.4)

A comparison of wear rates between full-scale field tests and laboratory tests was

conducted by Lewis et.al [16] and showed that rail steel with higher initial hardness has

a lower magnitude of wear. The investigation of the wear of different rail steels at

different sites by Alwahdi [8, 45] also shows the wear superiority of material with a

higher initial hardness. Compared to the results from [8], the current simulation results

show a good agreement with the magnitude of wear rate of UIC 900A of at least double

than that of UIC 1100.

Although some studies have shown that the level of original hardness plays a significant

role in increasing the wear [21, 73, 115] resistance, there are other investigations that

show that the ability to resist wear also depends on the work hardening ability of the

material [73, 82]. This is because the important material property for resistance to wear

occurring is cyclic plastic resistance. Ueda et.al [73] showed that the work hardening

rate at the surface is the dominating factor in determining wear resistance ability in

pearlitic steel. The investigation by Wang et al [82] on different rail types also showed

5.5 6 6.5 7 7.5 8 8.5 9

x 108

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Stress (Pa)

z/a




zx(max)


143

that pearlitic steel has a superior performance in wear resistance, due to its great work

hardening wear in rolling sliding contact, although with equal hardness.

Besides the hardening process, surface hardness can be decreased by frictional heating

in the sliding wear [82]. The heat generated is transferred to both contacting bodies and

this increases their temperature. The hardness of the softened elements decreases and

hence they are more prone to failure. Nevertheless, the effect of temperature elevation

on material softening has not been widely tested experimentally. To explain the

microstructural thermal stability on wear behaviour, Wang et al [82] used the

relationship of hardness vs. tempering temperature rises up to 700C. The results

revealed that the wear resistance for various microstructures depends on its thermal

stability.

6.7. Conclusion

The effect of the material properties of two different pearlitic rail steels on wear has

been presented. The material properties included in the simulation included material

hardness, critical strain to failure, hardening behaviour and thermal softening behaviour.

These four material properties are basically related to one another as the hardening

behaviour will increase the yield stress but the thermal softening will reduce it.

Therefore the effect of hardness becomes more complex because it does not depend on

the initial hardness. In this current work, a comparison of UIC 1100 and UIC 900A

performance showed that UIC 1100, which has greater hardness and hardening

behaviour but a lower thermal softening ability, mostly presents superior resistance to

wear. The superiority of UIC 1100 over 900A is caused by a greater difference in

hardness, 40% higher in UIC 1100 than UIC 900A. The work hardening ability in UIC

1100 is actually greater on the hardening speed but lower in the ratio of limiting

hardness to original hardness. As the original hardness of UIC 1100 is relatively greater

than UIC 900A, the final hardness of UIC 1100 is still greater than that of UIC 900A.

Moreover, although UIC 900A has better resistance to thermal softening, the reduction

of yield stress is confined to a very thin layer near to the surface. Therefore, although

the hardness at the surface in UIC 900A is greater under certain conditions, the hardness


144

of UIC 1100 at a greater depth is much higher than UIC 900A. Thus the accumulated

plastic strain in UIC 900A is actually higher than UIC 1100. The results from the

current work are specific for certain conditions where the hardness and hardening

behaviour in one material is better in one material but the thermal softening behaviour is

less than the other. There are other combinations of these three properties that may lead

to different effects on wear. Further investigation is required in this area.

Chapter 7. The effect of material properties on rail wear and RCF crack initiation

145

Chapter 7

Rail Rolling Contact Fatigue (RCF)

7.1. Introduction

The accumulation of the unidirectional plastic deformation driven by sliding and high

tangential forces may cause material fatigue at the surface [32]. Continuous cyclic

loading can also produce material ductility exhaustion at the subsurface, resulting in the

formation of clusters of failed elements. These clusters have the potential to initiate a

subsurface crack which can grow to cause a failure. As the sliding and tangential forces

are responsible for the increase in friction at the interface, a rise in temperature actually

plays a part in causing the elements to fail and hence in crack initiation. It can be

expected that the thermal effect will lead to additional levels of thermal stress and

thermal softening in the material, resulting in a higher plastic strain increment in each

cycle. Thus the critical strain for failure can be reached sooner, leading to the quicker

failure of element clusters, which appear as flaws (see Figure 1(a)). Over higher number

of cycles the additional failed brick increases and may lead to the fatigue growth at the

flaw’s tip (Figure 7.1(b)). The increasing of the severity of the contact loading can cause

quicker failure and additional failed bricks, which may cause early crack initiation.

However, it also increases the wear rate, as shown in Chapters 4 and 5. The wear

removes the failed layers at the surface as seen in Figure 7.1(c) and causing the layers

below to move up (Figure 7.1(c)-(d)).

The model used to investigate wear, i.e. brick model [39] also has been used to

investigate a surface breaking crack and subsurface fatigue damage without a thermal

effect [9]. In the brick model the failed material at the surface could be detached as wear

debris or, if still surrounded by healthy material, formed the weak path recognized as

rolling contact fatigue crack initiation [9, 92].


146

Figure 7.1 At cycle (n) the cluster of failed brick may form crack-like flaw (a). In the

next cycle (n+1) the additional failed bricks may increase and lead the fatigue growth at

the flaw’s tip as seen in (b). When the wear occurs the layers at the surface are detached

and cause all layers below to move up (c-d)

To identify the surface crack, the clusters of failed bricks were traced down from the

failed bricks recognized at the surface. These clusters were identified as surface cracks

if their depth reaches or exceeds 50µm. From the twin disc test experiment [6], it was

found that the material at this depth showed large shear deformation of the pearlitic

microstructure. Figure 7.2 shows the representation of failed brick cluster at the surface.

In this figure the white brick represent the healthy brick ( < c) whereas the black one

for the failed brick ( c). Some failed bricks that connected each other may form a

cluster. When the plastic strain occurred, the bricks were deformed at a certain angle, as

shown in this figure. The tangential value of the deformed structure’s angle,, was

equal to the shear strain value, , which was equal to x/y. The crack angle,, hence

could be calculated from ( ) With critical shear strain of 11.5, the resulting

angle is at a value of about 5. As the plastic deformation continuous the crack angle

keeps reducing.

Cluster of failed bricks called flaw

wear depth

All layers below move up

(a) (b) (c) (d) flaw tip


147

Figure 7.2 Cluster of bricks and the angle of deformed structure at the surface [22]

The identification of subsurface cracks was carried out by converting the bricks

property into binary code, which is 1 for failed brick and 0 for healthy brick. From this

conversion the ellipse of a cluster of failed bricks can be identified (see Figure 7.3). The

properties of these regions, i.e. bounding box, centroid, major axis length, and minor

axis length, were used to define the subsurface crack. The clusters of failed bricks

identified were defined as subsurface crack if the major axis length of the ellipse after

shear exceeded five times the minor axis length [9] and no failed bricks touch the

surface as seen in Figure 7.3.

Figure 7.3 Schematic representation of brick model to identify subsurface crack

An investigation of the thermal effect on these surface and subsurface cracks was

conducted under various operating conditions, i.e. friction coefficient, slip/roll ratio,

Healthy/weak element failed element centroid of the ellipse - - - bounding box

major axis length

surface

damage depth

minor axis length


148

maximum contact pressure, and vehicle speed. The effect of frictional heating on crack

initiation was evaluated in Section 7.2 using the same operating conditions as those

described in Chapters 4 and 5. The effect of a variation of the maximum contact

pressure and vehicle speed was also investigated within the range of 1 to 2 GPa for p0

and 30m/s to 110m/s for v0. The effect of material properties, i.e. the initial hardness,

the critical strain to failure, and the hardening and softening behaviour, on crack

initiation was evaluated in Section 7.3. The operating conditions followed those

described in Chapter 6 by including the effect of frictional heating temperature.

7.2. Effect of frictional heating on crack initiation

The effect of temperature rise on surface crack initiation due to frictional heating is

summarized in Table 7.1. The effect of thermal softening has also been included. The

results in this table show that as and Sr increase the number of cycles needed to

initiate a crack reduces. This applies both with and without a thermal effect. The

temperature rise causes additional thermal stresses that can generate higher plastic strain

and cause the bricks to fail sooner. As more bricks fail, the flaw can form earlier. When

depth of a flaw reaches or exceeds 50µm, surface crack initiation can be identified. This

formation can be detected at lower cycles in conditions with a frictional heating effect

compared to those without it.

Table 7.1 also shows that the effect of thermal softening on the initiation of a crack

generally presents a slightly reduced number of cycles to initiation compared to when

there is no thermal softening. In some cases, however, where the friction coefficient and

slip/roll ratio are high there is a tendency for a crack to commence later. It can be seen

that at = 0.5 and Sr = -5% the number of cycles for crack initiation increased from

5,700 to 6,000 and for = 0.6 and Sr = -5% the number of cycles increased from 4,000

to 6,300, with and without thermal softening, respectively. The increase in the friction

coefficient and slip/roll ratio in these cases has led to a higher wear rate and has

impacted upon crack initiation.


149

Table 7.1 Effect of friction coefficient and slip/roll ratio on the number of cycles to

initiate a surface crack and the wear rate for 100,000 cycles (w/o = without; w/ = with)

The relationship between crack initiation and a wearing surface was explained

previously and is described in Figure 7.4. This figure plots the number of cycles

required to initiate a crack alongside the wear rate for the variation of the slip/roll ratio.

The wear rate results were taken from simulation in Chapter 5. At slip/roll ratios of -2%

and -1% the wear rate and the number of cycles between the two conditions, with and

without thermal softening, are similar. With this condition, the wear rate only slightly

increases and this may have no significant effect on the growing flaw. Thus the crack is

detected earlier and shown in this figure by the decreasing number of cycles. For a

slip/roll ratio between -2% to -5%, the wear rate’s gradient increases when there is

thermal softening. Without thermal softening, the wear rate increases linearly.

Behaviour associated with the number of cycles to crack initiation also shows a

difference. The number of cycles for crack initiation without thermal softening linearly

decreases but when thermal softening occurs, the number of cycles shows a slight

increment from Sr = -3% to -4% followed by a rapid jump from Sr = -4% to -5%. The

Number of cycle to initiate a surface crack [x103 cycles], wear rate [nm/cycle] (bottom of each cell)

w/o thermal Sr = -0.01 Sr = -0.02 Sr = -0.03 Sr = -0.04 Sr = -0.05

w/o thermal softening

0.3 -

1.4 x 10-3

193.0

1.4 x 10-3

102.0

6.8 x 10-3

69.0

1.7 x 10-2

53.6

4.8 x 10-2

42.0

8.3 x 10-2

0.4 15.2

10.2

12.8

36.4

11.9

41.3

11.2

46.4

10.9

50.3

9.9

76.3

0.5 7.4

147

6.8

161

6.5

170

6.2

184

5.9

196

5.7

207

0.6 4.9

267

4.6

280

4.5

295

4.3

310

4.2

322

4.0

340

w/o thermal w/ thermal softening

0.3 -

1.4 x 10-3

193.0

2.7 x 10-3

102.0

6.8 x 10-3

69.0

2.1 x 10-2

53.6

3.1 x 10-1

42.0

1.29

0.4 15.2

10.2

12.8

36.4

11.9

41.3

11.2

49.7

10.8

76.5

9.7

91.0

0.5 7.4

147

6.8

162

6.4

171

6.2

185

5.9

212

6.0

251

0.6 4.9

267

4.6

280

4.4

295

4.3

321

4.2

360

6.3

502


150

wear rate slope also increases rapidly from Sr = -4% to-5%. These results indicate that

the wear rate may reduce the flaw depth or even make the surface flaw disappear, thus

causing the crack to commence later. This behaviour can be analysed in detail by

looking at the crack mouth truncation and the crack growth in each cycle, as shown in

Figure 7.5.

Figure 7.4 The interaction between the number of cycles required to initiate the

crack and the wear rate with variation of slip/roll ratio ( = 0.6, v0 = 30 m/s, p0 = 1.5

GPa). When the wear rate has a rapid increase, the number of cycles required to initiate

the crack also increases

Figure 7.5 shows the depth of flaw truncation (which is calculated from the depth of the

detached layers) and vertical flaw growth in each cycle plotted for a specific range of

cycles. The vertical flaw growth is calculated from the incremental of flaw depth in

each cycle. This figure shows that the depth of flaw truncation may cause the crack

candidate to be shortened or even disappear, but the crack can still propagate at a depth

when the vertical flaw growth exceeds the rate of flaw truncation. The maximum depth

of a flaw increases when the vertical flaw growth is positive and higher than the depth

of the wearing surface. In contrast, the maximum depth of a flaw decreases when the

wear depth is higher than the vertical flaw growth. When both values are the same, the

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.054000

5000

6000

7000

8000

Slip/roll ratio (-%)

Num

ber o

f cyc

les

for c

rack

initi

atio

n

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05250

350

450

550

650

Wea

r rat

e (n

m/c

ycle

)

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05250

350

450

550

650

Crack initiation; without softeningWear; without softeningCrack; with softeningWear; with softening


151

6200 6400 6600 6800 7000 7200 7400-50

0

50

100

150

(m

)

Cycles

mu=0.6; v0=30 m/s; P0= 1.5 GPa; Sr = -0.05

growth of flaw tipdepth of surface material removed by wearmaximum depth of surface flaw50m

maximum depth of a flaw is constant. For the condition specified, no surface crack was

identified below 10,000 cycles. This is because, although the vertical growth of the flaw

is positive, the maximum depth of the flaw is still lower than 50m. Hence it cannot be

recognized as a surface crack yet. Moreover, some wear occurs in this range and causes

flaw growth to penetrate slowly into the material. At 10,186 cycles, the maximum flaw

depth exceeds 50m for the first time, and therefore this value becomes the point that

the first crack can be identified at the surface. At higher cycles, as shown, the crack

may disappear between 6340 – 6480 and between 6760 – 6900, but after that the flaw

growth is higher than the rate of surface truncation and leads to an increase in the

maximum depth of the surface crack.

Figure 7.5 Surface crack is identified when the maximum depth of flaw exceeds

50µm. Maximum depth of flaw depends on the growth of flaw tip and the removal of

failed surface layers by wear ( = 0.6, slip/roll ratio = -5%, p0 = 1.5 GPa, and v0 = 30

m/s)


152

The effect of maximum contact pressure and vehicle speed on wear rate and the number

of cycles for crack initiation is shown in Figures 7.6(a) and 7.6(b), respectively. Figure

7.6(a) shows that that as p0 increases the wear rate increases. In Figure 7.6(b) the crack

starts to be detected at p0 = 1.2GPa and shows a reduction in the number of cycles as p0

increases. Higher contact pressure increases the thermo-mechanical stresses in the

contacting material and may also cause more severe thermal softening. The plastic

strain accumulates faster and hence the cluster of failed elements expands. Although the

wear rate increases significantly as p0 increases, the flaw growth in the vertical direction

is faster than the removal of the surface layer. Thus the depth of the flaw at the surface

reaches the limit for a surface crack, i.e. 50µm sooner. Therefore the number of cycles

for crack initiation decreases.

Unlike the peak pressure effect, the vehicle speed did not contribute significantly either

in the wear rate or the number of cycles required to initiate a crack. As the vehicle speed

increased the wear rate only slightly increased (Figure 7.6(a)), while the number of

cycles was similar for all v0 variations (Figure 7.6(b)). When the vehicle speed

increased the contact stress also increased, however, the effect of frictional heating

inside the contacting bodies reduced. It was because the value of Peclet number (L)

increased at higher v0 and led to a lower thermal penetration depth, as shown in

Equations (3.5)(3.7). The thermal penetration depth at v0 = 30m/s was 75µm compared

to that at v0 = 110m/s which was 39µm. The thermal penetration depth indicates how

deep the temperature has penetrated into the material. Thus the temperature rise is only

localized near to the surface when the thermal penetration is shallower. This condition

causes the thermal stress and thermal softening to occur only in this region. When

failure occurs mainly at a lower depth, it prevents flaw growth in the vertical direction.

Hence, flaw growth was affected by localized temperature and by the removal of the

surface layer through wear.

7.3. Effect of material properties on crack initiation

Different material properties has been shown to have a different effect on wear rate (see

Chapter 6). Two pearlitic steel, i.e. UIC 1100 and UIC 900A with different initial


153

hardness, critical strain to failure, hardening behaviour, and thermal softening behaviour

have been compared. UIC 1100 as the material that has higher critical strain to failure

and also higher initial and final hardness showed a higher wear resistance compared to

UIC 900A. However, the higher values of wear rate in UIC 900A may have the

advantage to truncate the flaw, resulting in late crack initiation as shown in Section 7.1.

In order to see the effect of these material properties on rolling contact fatigue crack

initiation the same properties in Chapter 5 and contact conditions in Chapter 6 were

used in the current simulation. The crack initiation at the surface and subsurface was

indicated by the number of cycles leading to crack initiation.

7.3.1. UIC 1100 rail steel

7.3.1.1. Surface crack initiation

Figure 7.6 shows the number of cycles for crack initiation plotted against Sr and µ. The

number of cycles to crack initiation decreases as µ or Sr increases. The value of µ

plotted started at 0.3 because no surface crack was detected at µ = 0.2 in all cases.

Under this condition, the maximum contact pressure and the friction coefficient were

not high enough to cause a significant ratcheting failure. In case 1, the crack was

initiated at µ = 0.5 and Sr = -0.5% with the number of cycles of around 115,000.

However the conditions are mild and only some ratcheting is observed. Thus resulted in

a low wear rate (less than 1nm/cycle, Table 6.3) and the cluster of failed elements grew

very slowly towards greater depth. Hence it needed a higher number of cycles to reach

50µm and be recognized as a surface crack.

In case 2, the crack was detected from µ = 0.4 with the number of cycles at around

27,000 at µ = 0.4 and then reduced for µ > 0.4. Case 3 also shows a similar behaviour

with the number of cycles for crack initiation of 34,000 at µ = 0.3 and then decreasing

at µ > 0.3. It can also be seen in this figure that the number of cycles till crack initiation

in case 3 decreased more than that in case 2 and the number of cycles in case 2

decreased more than that in case 1. Higher contact loading, i.e. p0 and a, led to higher

stresses and thermal softening, and produced more failures and early crack initiation.


154

The number of cycles until crack initiation is summarized in Table 7.2. These results

show in detail that there is no crack detected in case 1 for µ < 0.5, whereas in cases 2

and 3 there is no crack for µ < 0.4 and µ < 0.3, respectively. The crack is detected at

µ=0.5 (case 1), µ=0.4 (case 2), and µ=0.3 (case 3). However, although there is already

surface removal caused by wear (see Table 6.3), the magnitude of the wear rate is

relatively low and thus does not have a significant effect on crack prevention.

This table also shows that in case 1 the number of cycles decreases as Sr or µ increases.

Higher values of Sr and µ cause a higher temperature to be generated, especially in the

contact area, which enhances the additional thermal stresses and increases the

occurrence of thermal softening. The material in contact becomes weaker and

accumulates plastic strain faster. Hence the elements in this area fail easily and form a

crack after fewer cycles.

Figure 7.6 The number of cycles for crack initiation of UIC 1100 rail steel

It is interesting that, in case 2, the crack at the surface is initiated earlier as µ or Sr

increases, except from µ = 0.7 and Sr = -1.5% to µ = 0.7 and Sr = -3% where the

number of cycles increases from 3,821 to 4,637. This behaviour also occurs in case 3


155

where the number of cycles increases from µ = 0.5 - 0.7 with Sr = -1.5% to µ = 0.6 - 0.7

with Sr = -3%. The increase in the number of cycles under these conditions is

influenced by the level of wear. Table 6.3 shows that at these conditions the wear rate

increases by 6-9% when Sr increases from -0.5% to -1.5% and by 22-58% when Sr

increases from -1.5% to -3%. As the wear rate increment is much higher, it causes the

removal of failed layers and hence increases the number of cycles to crack initiation.

Table 7.2 The number of cycles for surface crack initiation of UIC 1100 rail steel

p0 = 1.5GPa ; a = 8.2 mm

Sr v0 (m/s) µ = 0.2 µ = 0.3 µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 - - - 115,352 18,782 10,237

-1.5% 17.5 - - - 40,990 14,244 8,626

-3.0% 15 - - - 27,602 11,442 7,831

p0 = 2.1GPa ; a = 9.37 mm

Sr v0 (m/s) µ = 0.2 µ = 0.3 µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 - - 27,012 10,053 6,559 4,437

-1.5% 17.5 - - 18,968 8,527 5,362 3,821

-3.0% 15 - - 14,347 7,231 5,038 4,637

p0 = 2.7GPa ; a = 10.48 mm

Sr v0 (m/s) µ = 0.2 µ = 0.3 µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 - 34,114 9,099 5,237 3,678 2,837

-1.5% 17.5 - 22,6440 7,784 4,461 3,352 2,570

-3.0% 15 - 16,049 6,508 5,209 5,325 4,173

- no crack observed during simulation

7.3.1.2. Subsurface crack initiation

Figure 7.7 shows the number of cycles to subsurface crack initiation. In case 1, there is

no subsurface crack detected when µ < 0.3. The loading experienced by the material

under these operating conditions is not high enough to cause a significant plastic strain


156

increment. The subsurface crack is initiated when the number of cycles is around 1,150

cycles at µ = 0.4 and Sr = -3%. At µ = 0.5 the number of cycles decreases as Sr

increases because of the higher temperature rise with the higher Sr. When µ = 0.6 and

0.7 the effect of Sr does not show any significant change and the number of cycles for

crack initiation looks similar.

In case 2, no subsurface crack is detected when µ = 0.2. It is first detected when µ = 0.3

and Sr = -0.5% with the peak value at around 60,000 cycles. It then decreases as µ

increases. The variation of Sr causes a declining slope as Sr increases but then shows a

similar value when µ 0.5. The subsurface crack in case 3 is recognized at a lower

value of µ, compared to case 2, namely 0.2 with Sr = -0.5%. It needs around 20,000

cycles for the first subsurface crack to be detected. The number of cycles shows a

reduction as µ increases and no significant fluctuation in the number of cycles due to the

variation of Sr. As Sr has a significant contribution on the temperature rise (see Chapter

4), the thermal effect is saturated and being negligible for these cases.

Figure 7.7 The number of cycles to subsurface crack initiation for UIC 1100 rail steel


157

A comparison of these three cases shows that subsurface cracks are detected at earlier

cycles as p0 and a increase, even with a lower value of µ and Sr. As p0 and a increase,

the contact stresses and thermal softening are greater, and this causes the critical strain

for failure, c, to be reached sooner. It produces more failed bricks, which form the

clusters that can be recognized as a subsurface crack.

The number of cycles required for subsurface crack initiation is summarized in Table

7.3. These results show that the number of cycles required for subsurface crack

initiation decreases as µ or Sr increases, which occurred in all three cases. It also

decreases as the value of p0 and a increases. The number of cycles dropped to very low

values of under 1000 cycles for µ = 0.6 - 0.7 and Sr = -3% in case 2, and even under

100 cycles for µ = 0.6 - 0.7 and Sr = -3% in case 3, because the plastic strain

accumulated very quickly. Although the removal of failed layers at the surface by wear

also increases, the growth of failed materials in depth direction is faster. Also there is no

effect of wear on truncating the mouth of the flaw as these flaws are subsurface. Thus it

causes the detection of subsurface crack at earlier cycles.

Table 7.3 Number of cycles to initiate subsurface crack for UIC 1100 rail steel

p0 = 1.5GPa ; a = 8.2 mm

Sr v0 (m/s) µ = 0.2 µ = 0.3 µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 - - - 23,962 10,102 6,543

-1.5% 17.5 - - - 15,809 8,459 5,743

-3.0% 15 - - 121,462 12,010 5,533 2,918

p0 = 2.1GPa ; a = 9.37 mm

Sr v0 (m/s) µ = 0.2 µ = 0.3 µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 - 76,763 12,643 6,466 4,337 3,263

-1.5% 17.5 - 55,904 10,314 5,675 3,804 2,576

-3.0% 15 - 39,188 6,480 2,425 932 229

p0 = 2.7GPa ; a = 10.48 mm

Sr v0 (m/s) µ = 0.2 µ = 0.3 µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 19,996 12,920 5,990 3,779 2,763 2,180

-1.5% 17.5 19,570 11,554 5,240 3,053 1,798 1,051

-3.0% 15 18,959 6,769 1,747 349 98 81


158

7.3.2. UIC 900A rail steel

7.3.2.1. Surface crack initiation

The RCF crack initiation at the surface of UIC 900A rail steel is shown in Figure 7.8.

This figure shows the plot of the number of cycles to surface crack initiation for the

three cases considered. It can be seen in this figure that the number of cycles decreases

as µ increases. This is because, as µ increases, the material has greater thermal and

contact stresses, so that the bricks can fail at lower number of cycles. Thermal

softening, if any, will do cause the crack initiation to occur earlier.

This figure also shows that as p0 and a increase, the crack is detectable at a lower µ. In

case 1, it was detected at µ = 0.5, case 2 at µ = 0.4 and case 3 at µ = 0.3. The effect of

the increase in load and the size of the semi-contact patch increased the contact stress

and temperature, resulting in failures of more bricks. However, in case 3, when µ = 0.3

the cracks started later than in case 2 at µ = 0.4. When µ = 0.3, the wear rate may

exceed the growth rate of failed bricks in depth direction. Thus the depth of cluster of

failed bricks does not reach the critical 50µm.

Figure 7.8 also shows that the number of cycles mostly increases as Sr increases. The

value of Sr determines the amount of sliding, which contributes to the heat generated in

the contact patch. As the temperature rises the thermal stresses and thermal softening

may increase. Thus there will be more failed bricks as Sr increases and the crack can

commence earlier. However, the surface curve shown in Figure 3.6 reveals that for

some cases the number of cycles first increases and then decreases. The thermal

hardening is seen for this material between 100C-300C. The level of wear rate under

these conditions may reach a level that reduces the flaw depth so that the surface crack

initiates later. However, when µ = 0.7 the number of cycles appears to be slightly less in

case 3 when Sr = -3%. With this operating condition, the number of failed bricks in

depth direction may grow faster than the wearing away of the surface layer and results

in earlier surface crack detection.


159

Figure 7.8 The number of cycles of crack initiation for UIC 900A rail steel

The number of cycles to surface crack initiation of UIC 900A is summarized in Table

7.4. This table shows that the number of cycles required to initiate a surface crack

decreases as µ increases. In case 1 there is no crack detectable with low friction

coefficient. The crack appears at µ = 0.5, at µ = 0.4 and at µ = 0.3 for cases 1, 2, and 3,

respectively. In case 1, the number of cycles decreases as Sr increases. However there

are slight fluctuations in cases 2 and 3. A small increase at Sr = -1.5% can be seen for

these two cases. This behaviour is caused by thermal hardening as the temperature rise

in these operating conditions are in between 100C-300C (Table 6.2). The wear rate

also plays an important role to delay the crack initiation particularly when the

temperature rise has exceed the thermal hardening range, i.e. at µ = 0.6 and Sr = -1.5%

(case 2), and µ = 0.5 and Sr = -1.5% (case 3). At µ = 0.7 and Sr = -3% the number of

cycles drops significantly to under 100 cycles, which indicates that the failed elements

cluster grows very quickly in the vertical direction. With this condition, the surface

wear may be unable to exceed this growth. Fletcher and Beynon [60] conducted twin


160

disc test using pearlitic rail steel and showed the surface crack could initiate at about a

hundred cycles.

Table 7.4 The number of cycles of surface crack initiation for UIC 900A rail steel

p0 = 1.5GPa ; a = 8.2 mm

Sr v0 (m/s) µ = 0.3 µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 - - - 8,524 5,562

-1.5% 17.5 - - 25,295 11,243 7,271

-3.0% 15 - - 38,499 11,676 6,693

p0 = 2.1GPa ; a = 9.37 mm

Sr v0 (m/s) µ = 0.3 µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 - 11,209 5,522 3,724 2,888

-1.5% 17.5 - 15,319 7,439 4,668 3,396

-3.0% 15 - 15,904 6,227 3,674 2,835

p0 = 2.7GPa ; a = 10.48 mm

Sr v0 (m/s) µ = 0.3 µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 13,010 5,154 3,359 2,538 2,063

-1.5% 17.5 20,170 6,701 4,039 2,757 2,051

-3.0% 15 19,936 5,792 3,082 3,278 65

7.3.2.2. Subsurface crack initiation

The subsurface crack initiation for UIC 900A rail steel is also indicated by the number

of cycles before crack initiation as shown in Figure 7.9. The subsurface crack is

detectable with the high number of cycles in case 1 at µ = 0.4 and Sr = -1.5%, around

55,000 cycles. The number of cycles can be seen to decrease as µ or Sr increases. The

subsurface crack is also detectable with a high number of cycles, about 35,000 - 50,000

cycles in case 2 at µ = 0.2. It also decreases as the value of µ increases. In case 3, the

subsurface crack is detectable at less than 10,000 cycles for all the operating conditions


161

considered. It shows a reduction as µ increases and has a minor effect due to Sr

variation. The comparison of the three cases shows that the number of cycles decreases

as p0 or a increases.

Figure 7.9 Number of cycles of subsurface crack for UIC 900A rail steel

The number of cycles required for subsurface crack initiation is summarized in Table

7.5. The results in this table show that there is no subsurface crack detected at µ < 0.4 in

case 1. The subsurface crack is detectable at µ = 0.4 for case 1 and at µ = 0.2 for cases 2

and 3. This table shows in detail that the number of cycle for crack initiation reduces as

µ increases. However, the effect of Sr variation shows fluctuations which is not clearly

seen in Figure 7.9. In case 1 the reduction of the number of cycles at Sr = -1.5% and µ =

0.4-0.5 is caused by the softening of material near the surface. However, the number of

cycles shows an increment at µ = 0.6-0.7. This is because the temperature rise of the

surface layer reaches the thermal hardening range and thermal softening occurs at the

subsurface layers. However, the amount of stress at the subsurface is still relatively low

which leads to lower plastic strain. Hence the failure at the subsurface occurs at higher


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number of cycles. In case 2 the effect of Sr shows an increment at Sr = -1.5% and µ =

0.4-0.6. Similar with the behaviour in case 1, although the temperature rise results in

thermal softening at the subsurface, the amount of stresses could be relatively low at the

subsurface. Thus the critical shear strain to failure is reached at higher number of cycles

for the subsurface layers. Besides, the amount of wear also takes place especially at µ =

0.6 which reaches more than 300nm/cycle (Table 6.4). The wear detaches the surface

layers and brings subsurface layers up closer to the surface. Thus the subsurface flaw

that reaches the surface becomes surface flaw. In case 3 the delayed of subsurface crack

detection at Sr = -1.5% and µ = 04-0.5 is caused mainly by the amount of wear rate

which is more than 500nm/cycle (Table 6.4), which is expected to exceeds the growth

of subsurface flaw in depth direction. It leads the subsurface flaw to move up and

become surface flaws. However, the amount of stress at µ = 0.6-0.7 is relatively high to

cause the quick failure at the subsurface leading to the formation of subsurface crack at

earlier cycle.

Table 7.5 Number of cycles of subsurface crack initiation of UIC 900A rail steel

p0 = 1.5GPa ; a = 8.2 mm

Sr v0 (m/s) µ = 0.2 µ = 0.3 µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 - - 63,894 10,059 5,601 3,803

-1.5% 17.5 - - 55,138 9,965 5,724 4,166

-3.0% 15 - - 107,094 10,336 4,847 2,377

p0 = 2.1GPa ; a = 9.37 mm

Sr v0 (m/s) µ = 0.2 µ = 0.3 µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 50,582 19,249 6,644 3,810 2,716 2,141

-1.5% 17.5 37,245 17,818 7,035 4,424 3,077 2,167

-3.0% 15 35,077 15,560 5,769 1,924 703 76

p0 = 2.7GPa ; a = 10.48 mm

Sr v0 (m/s) µ = 0.2 µ = 0.3 µ = 0.4 µ = 0.5 µ = 0.6 µ = 0.7

-0.5% 20 9,368 6,888 3,615 2,459 1,895 1,529

-1.5% 17.5 9,235 6,658 4,164 2,625 1,417 764

-3.0% 15 8,752 5,975 1,360 179 60 51


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7.3.3. Comparison of UIC 1100 and UIC 900A for surface

crack initiation

A comparison of the surface cracks of the two materials is shown in Figures 7.10(a-c).

The surface crack of UIC 1100 mostly occurs after a greater number of cycles than UIC

900A. It happened in all cases and was due to the variation of µ and Sr except at certain

points, namely µ = 0.5, Sr = -3% (case 1), µ = 0.4, Sr = -3% (case 2), and µ = 0.3, Sr = -

3% (case 3). In these operating conditions, the crack in UIC 1100 commenced earlier

than in UIC 900A. The lower wear rate of UIC 1100 may cause the surface flaw to grow

deeper as it is less truncated. The surface crack of UIC 900A commenced later because

the wear rate under these conditions was possibly high enough to prevent the growth of

failed bricks to a greater depth. The greater number of cycles of UIC 1100 under other

operating conditions could be caused by the hardness of this material, which is greater

than UIC 900A. Thus the plastic strain increment in each cycle is lower and hence the

failure inside the material needs more time to develop.


164

Figure 7.10 Comparison of number of cycles till crack initiation (a) case 1 (b) case 2 (c) case 3

(a)

(c)

(b)


165

7.3.4. Comparison of UIC 1100 and UIC 900A for

subsurface crack initiation

Figure 7.11 shows a comparison of the number of cycles needed for subsurface crack

initiation. This figure reveals that, overall, the subsurface crack is initiated after fewer

cycles in UIC 900A than in UIC 1100. In all three cases, the difference in the number of

cycles is greater with a lower value of µ but converges to a similar value at µ = 0.7. UIC

1100 is obviously much better at resisting RCF crack initiation than UIC 900A in case

1. However, this performance becomes similar to that performed by UIC 900A in case

3. When the contact loading becomes severe, UIC 900A accumulates plastic strain

faster than UIC 1100. Thus the cluster of failed bricks is easily formed in UIC 900A.

On the other hand UIC 1100 is more resistance to failure due to its hardness. Therefore

the formation of failed brick cluster in this material is slower. As the amount of wear in

UIC 1100 is less than in UIC 900A, the failures under the surface accumulates and

forms flaws at early cycles because of less effect of wear. In UIC 900A the amount of

wear is higher and hence bring the subsurface failures closer to the surface to become

surface flaws or even truncate it and make it disappear. However, as the subsurface

failures easily occur due to its lower hardness, the subsurface crack is still detected at

early cycles too.


166

Figure 7.11 Comparison of the number of cycles of subsurface crack initiation (a) case 1 (b)

case 2 (c) case 3

(a)

(c)

(b)


167

7.4 Discussion

7.4.1. The effect of frictional heating on crack initiation

The severity of crack initiation depends on the magnitude of the maximum contact

pressure, the friction coefficient and the amount of slip. The maximum contact pressure

and the friction coefficient drive the amount of traction force at the contact patch. As the

traction force increases, the maximum orthogonal shear stress (zx(max)) also increases,

which leads to higher plastic strain. The presence of sliding accompanied by higher

traction force also contributes to the amount of frictional heating between the two

surfaces. The increasing temperature at the contact surface is responsible for developing

additional thermal stress in the material and may cause the reduction of material yield

stress, known as thermal softening or the increase of material yield stress, known as

thermal hardening. Severe thermal stresses and thermal softening/thermal hardening are

found to occur near the surface because of the higher temperature in these layers [39].

Thermal stresses and thermal softening cause a significant increase in the plastic strain

and this leads to the material failure. When the thermal hardening occurs, the plastic

strain increment decreases and reduces the material failure which consequently

influences the RCF crack initiation.

The thermal effect due to frictional heating on crack initiation exhibited some variation

due to changes in loading conditions. It has been shown that the frictional heating effect

causes a crack to commence earlier. However, the RCF crack initiation is also driven by

wear at the surface by removing the failed surface layers. The results presented in the

previous sections showed that at low contact pressure, friction coefficient and slip/roll

ratio the crack is initiated after a greater number of cycles. The small amount of loading

experienced by the material under contact, when accompanied by a lower temperature

rise near the surface, will result in a lower plastic strain increment. Thus the

accumulated plastic strain reaches the critical strain to failure following a greater

number of cycles. Under some operating conditions, the level of wear rate is small or

even zero. As the amount of wear is small, its effect on delaying the crack initiation is

insignificant.


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When the operating conditions become more severe, the crack and wear generally

become worse. This is because, as the accumulated plastic strain increases, the critical

strain to failure can be reached sooner. This means that there will be more failed bricks

inside the material. It can be anticipated that when the number of failed bricks increases

then more failed brick clusters will form as a crack. Thus the crack can be detected

following fewer cycles. However, the results also showed that when the friction

coefficient or the slip/roll ratio is moderate then the thermal effect causes no changes in

the number of cycles for crack initiation. Even with a higher value in the friction

coefficient and slip/roll ratio, the thermal stresses and the thermal softening effect has a

tendency to cause a crack to commence after a greater number of cycles. As crack

failure is proportional to the amount of stress experienced by the rail material, the

greater number of cycles before crack initiation must be caused by other mechanisms,

which are expected to be the thermal hardening as shown in UIC 900A or the wear.

When the number of failed bricks increases, especially in the surface layer, the

likelihood of a failed brick being removed as wear debris also increases. It has been

shown in the crack initiation and wear plot (see Figures 7.4–7.5) that when the

conditions of loading become severe then the wear rate increases, which may decrease

the depth at which a brick cluster is failing. This phenomenon is illustrated in Figure

7.1. The original flaw grows deeper into the material (Figure 7.1(b)). However, wear at

the surface (Figure 7.1(c)) causes crack mouth to be truncated, resulting in a crack

(Figure 7.1(d)) which has grown by the difference between crack tip growth by fatigue

(ductile fracture) and crack mouth truncation by wear. Crack initiation can be taken as

development of a crack flaw of a certain length and as a result the crack is initiated after

a greater number of cycles, if wear is taking place.

Crack initiation is also sensitive to the maximum contact pressure but less sensitive to

any variation in vehicle speed. The variation in vehicle speed also indicates an opposite

result to other variables. When the vehicle speed increases the crack tends to commence

later. This is because, as the speed increases, the depth of thermal penetration decreases.

Thus, the effect of temperature is localized in the layers closer to the surface and causes

less failed elements in the vertical direction. When the maximum contact pressure,

friction coefficient, slip/roll ratio and also vehicle speed increase, then it is possible for


169

any vertical flaws growth to decrease, due to the increase in the removal of failed

surface layers by wear and because of the lower depth of thermal penetration.

Crack initiation at the subsurface is also influenced by the contact conditions. As the

severity of the contact loading increases the cluster of failed bricks at the subsurface

forms at lower number of cycles. However, when the wear increases more layers at the

surface layer are detached, the layers below move up. Thus the depth of the subsurface

flaw also decreases and it may reach the surface and become surface flaw as shown in

Figure 7.12.

Figure 7.12 The flaws may occur at the surface or at the subsurface (a). The wear

removes the layers at the surface and may cause the flaw at the subsurface move up (b)

and become surface flaw (c)

The dominancy of wear has been studied, as this tends to prevent a crack to occur or

slows down the crack growth at a greater depth [32, 36]. In contrast, the absence of

wear often causes a crack to appear on the surface. The thermal effect due to sliding has

also been found to affect the RCF performance on steel material [84, 116]. The material

that has greater thermal stability has been proven to have a greater ability to resist

fracture. As reported by Cvetkovski [84], steel material that has a better resistance to

softening has a greater fatigue lifetime.

Surface flaw

Subsurface flaw

the subsurface flaw becomes a surface flaw

wear depth

All layers below move up

(a) (b) (c)


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The initial surface crack was studied by Ringberg and Bergkvist [32] and found to be

driven by shear stress, and known as a short crack with a length of about 0.1-0.5mm.

The growth of this short crack is affected by many factors, such as crack length, crack

orientation, normal and tangential loads, lubrication, plastic deformation, crack

coalescence, wear, etc. As it is driven by the amount of shear stress close to the surface,

the inclination of the crack is relatively low at around 5, as shown in the current

simulation when compared to the FE simulation, which is between 83< <89, while

the experimental figure was = 85 [92]. The low value of the crack angle was caused

by the amount of shear deformation, which was large when very close to the surface. It

is possible for a crack orientation to change direction, branching upward to the surface

or downward into the rail base. Ringsberg [33] and Raje, et.al [103] showed that some

cracks close to the surface may propagate parallel to the surface or change direction

upward and cause spalling failure. At a significant distance from the surface, a crack

may grow towards the base, and this can cause a rail fracture [56, 98, 103, 115].

7.4.2. The effect of material properties on crack initiation

The performance of two different materials has demonstrated that strength, work

hardening ability, and thermal softening behaviour are important factors in determining

the material response to loading. The performance of each rail type shows variations in

the RCF crack initiation behaviour at the surface and the subsurface. Because wear and

crack initiation depends on the amount of accumulated plastic strain inside the material,

the resistance to these types of damage will be determined not only by the conditions of

the contact loading but also by the material properties in response to the loading. The

results described in the previous sections reveal that, with the same operating

conditions, the amount of wear in UIC 1100 and UIC 900A is different and, therefore,

so is the crack initiation at both the surface and the subsurface. The properties

considered in the present work, such as hardness, hardening behaviour and thermal

softening behaviour have shown interesting results when both rail types were compared.

Simulations show that the surface and subsurface crack in both materials are initiated

earlier in dry conditions. The amount of sliding also reduces the number of cycles


171

required for crack initiation because of the additional frictional heating effect. Because

UIC 1100 can resist failure better than UIC 900A, due to its higher hardness, a crack in

UIC 1100 needs more time to emerge. Therefore the performance of crack initiation in

UIC 1100 is better than 900A in most cases. UIC 900A shows a better performance in

resisting surface crack over UIC 1100 when the contact conditions are moderate. The

surface crack in UIC 900A is detectable at a higher number of cycles. In this operating

condition the higher wear in UIC 900A has an advantage to reduce the flaw length or

make it disappear. On the other hand the surface flaw keeps growing in depth direction

due to less wear.

In the subsurface crack initiation, both materials show a reduction in the number of

cycles for subsurface crack initiation as µ or Sr increases. The performance of UIC 1100

in resisting subsurface crack initiation show a slightly better performance than UIC

900A despite of a different behaviour between the two. The early detection of

subsurface crack in UIC 900A is caused by its lower strength to resist failure. Although

the wear also increases as the contact loading increases, the growth of flaw in depth

direction is faster resulting in early formation of subsurface crack. On the other hand as

UIC 1100 can resist failure more than UIC 900A, the flaws in UIC 1100 can have more

space and time to accumulate and grow. As the contact loading becomes severe the

speed of wear cannot exceeds the flaws growth. It leads to the detection of subsurface

crack at early cycles.

The material hardness is one important material property that has been shown to have a

significant influence not only on wear but also on rolling contact fatigue [117, 118]. A

comparison of 900A rail steel and head hardened rails, i.e. grade 340 and 370 HB,

showed that because the head hardened rails have a higher yield point, they prove to

have greater resistance to fatigue. Because of this the rolling contact fatigue life of head

hardened sample was about an order of magnitude longer than that of grade 900A

material [118]. Franklin et.al [119] investigated the effect of an additional surface layer

with different properties to increase the railhead performance. The results showed that

the material coatings with higher initial hardness than the base material (UIC 900A)

survived 200,000 cycles of water-lubricated twin disc testing without any crack

formation. Both showed an excellent RCF performance, although one of the coating


172

materials developed cracks quickly during water-lubricated testing after 15,000 dry

cycles, and UIC 900A immediately showed cracking after only 4,000 cycles.

Cvetkovski [84, 116] also discovered that the material with better resistance to softening

has a longer fatigue life. The investigation used alloyed steel with Mn and Si content.

The higher alloyed steel had a higher strain-hardening rate, both initially and after

annealing, and a greater annealed hardness but slightly lower initial hardness to the

lower alloyed steel. The results showed that a 15% yield stress reduction on higher

alloyed steel and a 22% reduction of yield stress in lower alloyed steel resulted in a

reduction of fatigue life by a factor of 2.5 and 7, respectively.

Rolling contact fatigue initiation and crack propagation is also linked to wear [32].

When wear is dominant, the length of the surface flaws decreases or is even removed

[36, 60]. In contrast, when there is little wear, the flaw length increases and propagates

deeper into the rail. From the discussion above, it would seem that the performance of

different rail materials for both wear and a rolling contact fatigue crack is more

complex. The findings from an experimental investigation by Cannon and Pradier [118]

demonstrated that head hardened rail which had a greater hardness than 900A rail steel

showed a longer rolling contact fatigue life and also better wear resistance. However,

Zhong et.al [120, 121] found that the superior ability of materials to resist crack

depends on its strength. The lower strength rail is subjected to a greater degree of work

hardening, so that the crack may be able to propagate in the plastic strain direction

while, in material with greater strength, the crack may propagate in the grain boundary

direction. It can be seen from this discussion that the effect of hardness in sliding wear

is varied and complex [122].

7.5 Conclusion

The thermal effect due to frictional heating has been successfully simulated and shows

encouraging results. The severity of crack initiation strongly depends on the magnitude

of the elevated temperature. The temperature rise in the rail material has shown a higher

thermal stress and a greater reduction in material strength. Thus the material will be


173

more prone to failure. When one element fails it will coalesce with the adjacent bricks

to form a cluster, called as flaw, which can potentially be detected as a crack. However,

failed bricks at the surface can also be detached as wear debris. When wear exists, it can

reduce the length of the flaw or even move it away from the surface. In contrast, when

there is no wear, a crack can often appear at the surface. Thus, crack initiation is

influenced not only by the severity of loading but also by the wearing surface. When the

thermal effect is included, the wear rate also becomes greater. Thus, crack initiation

depends on whether the wear rate or the speed of failure growth is dominant. The wear

rate also affects a crack at the subsurface. When the surface layers are worn away, lower

layers move up and hence any subsurface crack is brought closer to the surface.

The comparison of UIC 1100 and UIC 900A performance showed that UIC 900A

presented a lower surface and subsurface crack resistance compared to UIC 1100. It was

found that the wear in UIC 1100 was less than in UIC 900A, due to its greater strength.

For certain contact conditions at the rail head, UIC 900A has the advantage of the wear

delaying the crack initiation by removing failed surface layers. When the contact shifts

to the rail gauge, the wear cannot overcome the growing failure due to fatigue at both

the surface and subsurface. The results have shown that the UIC 1100 can resist the

fatigue failures more than UIC 900A. The higher resistance to crack initiation in UIC

1100 over UIC 900A was mainly caused by a greater initial and final hardness of UIC

1100 compared to UIC 900A. Higher value of critical strain to failure in UIC 1100 also

played an important role to prevent failure in early cycles. Although UIC 900A has a

better thermal softening resistance and thermal hardening compared to UIC 1100, this

effect is confined in a thin layer near the surface where the temperature rises. With

lower initial and final hardness after hardening and softening process, UIC 900A

accumulated plastic strain faster than UIC 1100 resulting in greater failure in UIC

900A. Although the current results shows that the hardness is vital in determining the

failure, these results were limited to the material properties specified. Other

combinations of material properties are still open for further investigation.


174

Chapter 8.Conclusion

175

Chapter 8

Conclusion

The effects of temperature and material properties on wear and rolling contact fatigue

have been simulated using a ratcheting failure model. The results have been presented

and discussed in detail in each chapter (Chapters 4 to 7). This chapter highlights the

conclusions that have been drawn in each chapter.

The simulations with thermal effects due to frictional heating are successful in

predicting the wear transition. The results show good agreement with the experiments

conducted by Bolton and Clayton [28]. In order to compare the simulation results with

the experimental results of Bolton and Clayton [28], the data imitates the condition from

the experiments where a limiting friction of 0.5 – 0.6 was achieved for type II wear

rates. The wear rate from simulation was plotted against T, which represents the energy

from the friction. These results correspond to wear type II from the Bolton and Clayton

experiment, and showed an encouraging match. It indicates that the thermal stress and

thermal softening that was seen in the work undertaken for this thesis caused a wear

transition from wear type I to wear type II or from mild wear to severe wear. For higher

operating conditions the experimental values reached catastrophic wear and it were

higher than the predicted results. The surface roughness issues were not modelled and

are thought to be responsible for this discrepancy. The characteristic of catastrophic

wear found in the experimental investigation indicated surface ploughing by a rougher

surface topography. Sundh [108] also found heavy scoring marks in this wear type.

According to Kapoor et.al [71], the roughness of the surfaces can produce localized

peaks of very high contact pressure, i.e. about 10 times higher than the nominal Hertz

pressure leading to higher wear rate. Besides, when the surface becomes rougher the

friction increases and consequently leads to a greater temperature rise [109] as well.

Wear transition due to variation in the slip/roll ratio, maximum contact pressure, friction

coefficient and vehicle speed indicated that the minimum temperature required for the


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transition was around 350C. below this temperature the thermo-mechanical stress is

not high enough to generate large plastic deformation. Moreover the thermal softening

for the case considered started reducing the yield strength at 250C. At 350 the yield

stress decreased by 12% but the wear rate still remained low. Above this temperature,

both thermal stress and thermal softening led to changes in the gradient of the wear rate

curve. In all cases run over 200,000 cycles, the thermal stress showed a greater effect

than thermal softening. It can be presumed that the thermal softening effect only occurs

in layers very near to the surface, whereas high shear stress occurs at greater depth. It

was shown that the average plastic strain increment in the thermally softened surface

layer was about 1.6x10-3 and dropped to 6.3x10-4 in a thick layer of 7.1mm. As the

thickness of the thermally softened layer was only 0.5% of 7.1mm depth when plastic

strain occurred then the number of cycles required for the material to reach failure and

become detached as wear debris was still high, despite the higher plastic strain

increment in the thermally softened layers.

The wear transition which occurs at 350C and condition to produce it are a value of

slip/roll ratio of at least -3%, maximum contact pressure of at least 1.4GPa, friction

coefficient of at least 0.4, and vehicle speed of at least 30 m/s. The transition can still

occur for any combination of those variables as long as the temperature rise is above

350C. The transition is unlikely to occur for contact temperature below this value.

With the four variables used in various combinations in the simulation, the wear rate

was shown to be sensitive to friction coefficient and maximum contact pressure with or

without thermal effect. This was because the maximum contact pressure and friction

coefficient contributed to the magnitude of tangential force (p0) and hence the

maximum shear stress at each depth. If the maximum shear stress at each depth

increased, the plastic strain increment per cycle would be greater, causing early material

failure and a greater wear rate. Moreover, these two variables also contributed to the

amount of frictional heating at the contact surface. Hence the wear transition could be

reached faster due to the increasing of contact stresses, thermal stresses and thermal

softening. The other two variables, namely the slip/roll ratio and vehicle speed,

influenced the amount of sliding at the interface. They contributed only to the additional

thermal stresses and thermal softening but not to the contact stresses, as the friction


177

coefficient and maximum contact pressure did. Hence, their role in causing wear

transition was less than the first two variables. From the factorial design level two, it

was shown that the slip/roll ratio had a significant effect on the temperature rise despite

being a little less influential on the wear rate compared to the maximum contact

pressure. The vehicle speed had less effect on the temperature and wear rate compared

to the other variables and hence it contributed less to the transition.

During wheel-rail contact, the heat generated from frictional work is conducted to the

wheel and rail. The frictional heat in the rail is conducted easily and hence the

temperature drops to the ambient value once the train has passed. The temperature rises

again for the next contact and drops again as the next train passes. However, the wheel

is continuously heated by frictional heating over the cycles and hence its bulk

temperature may increase to a higher temperature than that in the rail. With continuous

running, the wheel bulk temperature reaches a steady state temperature, subject to

ambient temperature and natural or forced cooling of the wheel. When the hot wheel

makes contact with the rail the heat from the hot wheel is conducted to the rail and gives

an additional temperature rise within the rail but a reduction in the wheel temperature

[7].

When the hot wheel makes contact with the rail, the wheel temperature drops at the

leading edge, while that of the rail increases rapidly as the heat from the wheel conducts

to the rail. The temperature keeps rising to the trailing edge because of frictional heating

and reaches the maximum at the trailing edge at about 114% of the steady state wheel

temperature. Within the contact patch (-1<x/a<1) the temperature rise profile at the

wheel and rail surface was found to be similar because the heat-partitioning factor was

taken to be 0.5. At the end of the contact patch the rail temperature drops because no

further heat conducts from the hot wheel and, therefore, no frictional heating. The wheel

temperature goes back to its steady state temperature. It could be seen from the

temperature rise profile in the simulations that frictional heating contributed about 56%

of the maximum temperature rise, whereas conduction from the hot wheel contributed

about 44%.

The thermal stresses developed are higher when the steady state wheel temperature was

included. The increase in the total orthogonal shear stresses at 0.5µm depth, due to both


178

frictional heating and the steady state wheel temperature, was roughly double that due to

frictional heating only, which was in line with the temperature profile. Because the

thermal stress and the temperature rise doubled due to the involvement of the steady

state wheel temperature, the plastic strain increment in the layers just below the surface

increased very rapidly. The effect of thermal softening was more severe compared to the

case with frictional heating only. The magnitude of wear rate due to both frictional

heating and the steady state wheel temperature was estimated to increase by up to an

order of magnitude. Hence, if the material could resist thermal softening, then it could

be expected that failure could be reduced, which would lead to less wear. This finding is

important for developing new materials. These results are in line with the experimental

investigation by Sim et.al [123], which revealed that thermal softening from frictional

heating could increase wear failure. Wang et.al [82] also found that the material with

higher thermal stability can exhibit higher wear resistance. The effect of the steady state

wheel temperature can be reduced by avoiding a temperature rise in the wheel body as it

passes over the rail, so that the temperature rise is mainly due to frictional heating.

Simulations show that for the temperature rise below 600C the material strength

remains above 50% of its original value and there is no occurrence of phase

transformation [124]. One alternative for reducing the steady state wheel temperature

could be lubrication at the wheel-rail interface, so reducing the amount of heat due to

friction. Another alternative is to design the train wheel with a self-cooling effect to

keep the wheel bulk temperature low. Further investigation is required in this area and

beyond the scope of the current work.

The thermal effect on a rolling contact fatigue crack has been shown to be closely

related not only to the severity of loading but also to the amount of wear taking place at

the surface. The greater the thermal effect, the more likely an increase in the cluster of

failed bricks that forms a crack-like flaw. This flaw may occur at the surface or at the

subsurface. The increasing failed brick at the surface layer also increases the wear rate.

Therefore, surface crack initiation is strongly dependent upon the competitive role

between the growing failure in the vertical direction and the wear. When wear exists,

the surface flaw is shortened or even prevented. In contrast, an absence of wear may

result in the emergence of a flaw at the surface. Likewise, a crack initiation in the

subsurface is also influenced by the wear. When the wear is relatively low the


179

subsurface failures had more space to grow and hence subsurface crack could be

detected at early cycles. If the wear is high, the subsurface flaws moved up and may

become surface flaws. It can cause the subsurface crack to initiate later. The crack

orientation for a surface generally reaches an average value of 5. This indicates that the

surface crack has followed the path of the shear deformation close to the surface. This

result shows a good agreement with that from the FE simulation, which is between 83<

<89, while from experiments it was detected to be = 85 [92].

The performance of rail steel in service has been shown to depend on the conditions of

service, such as the maximum contact pressure, friction coefficient, slip/roll ratio and

vehicle speed, especially when a thermal effect is involved. In addition to these

operating conditions, the response of the rail material to this loading also plays an

important role in determining the level of damage that may occur during service. This

response is closely related with the properties of the rail steel. Hardness is an important

material property that needs to be considered when developing rail material. Previous

studies have also shown that the material with the greatest hardness presents greater

resistance to wear [73, 82, 121]. A comparison of wear rates between full-scale field

tests and laboratory tests was conducted by Lewis et.al [16] and showed that rail steel

with higher initial hardness has a lower magnitude of wear. The investigation of the

wear of different rail steels by Alwahdi [8, 45] also shows the wear superiority of

material with a higher initial hardness. By using the same rail material, i.e. UIC 900A

and UIC 1100, there is a good agreement for the wear results from the current

simulation and the result from [8] with less wear on UIC 1100 rail steel.

In addition to hardness, the hardening behaviour and thermal softening behaviour

investigated during the research for this thesis were also found to contribute to wear and

RCF initiation. These material properties are closely related to one another, because the

hardening process increases the yield stress as the plastic deformation occurs whereas

the material softening causes a yield reduction if the temperature is high enough.

Therefore the effect of hardness as a cause of failure becomes more complex as it is not

determined by the original hardness. In the research for this thesis, the properties of UIC

1100 showed greater hardness and hardening behaviour but a lower thermal softening

ability compared to UIC 900A. However the results showed that UIC 1100 generally

had a greater resistance to wear and RCF crack initiation. In addition to the greater


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original hardness, the work hardening ability of UIC 1100 is actually greater with

regard to hardening speed but with a lower ratio between limiting hardness and original

hardness compared to UIC 900A. However the final hardness of UIC 1100 is still

greater because its original hardness is much greater than UIC 900A. Moreover, while

UIC 900A has a better resistance to thermal softening, the reduction in yield stress

mainly occurs in thin layers very close to the surface. Therefore, although the surface

hardness of UIC 900A under certain conditions can be greater than that found in UIC

1100, its hardness at a greater depth is much less than that found in UIC 1100. Thus

UIC 900A actually has greater accumulated plastic strain than UIC 1100. An interesting

result was that although the wear rate of UIC 900A was greater than that of UIC 1100,

the ability to resist a crack initiation was also lower than that of UIC 1100. The growth

of a failed brick cluster in UIC 900A was greater than the wear rate for the conditions

specified in the simulation. The flaws are easier to grow in UIC 900A than that in UIC

1100, so the RCF crack initiation occurs faster in UIC 900A. From these results,

hardness was found to be the most important factor to cause this effect. These

simulations cover only some combinations of operating conditions and material

properties; hence the inferences drawn are true only for the limited conditions. Further

investigations are required to see how the combinations of material properties affect

wear and a rolling contact fatigue crack.

Chapter 9.Future Work

181

Chapter 9

Future Work

Based on the results from this thesis, there are many factors that can be considered in

order to improve the model. From the analysis of wear transition, it was found that the

thermal effect could cause transition from mild to severe wear, whereas the

experimental investigation found it possible to reach catastrophic wear. The surface

roughness proved to be an important factor causing this wear type. The current model

showed that brick removal at the surface actually causes a non-smooth profile.

However, the effect of this roughness on operating conditions has not been investigated

yet. The investigation by Kapoor et.al [71] showed that the roughness of the surface can

produce localized peaks of very high contact pressure, about ten times higher than the

nominal Hertz pressure. As the surface becomes rougher, the friction increases and may

cause greater tangential stress and greater temperature rise. Thus, it is possible that the

austenitization temperature could be reached sooner and cause microstructural changes.

Some of these effects have been studied experimentally by Sundh [90] and Gaard [109],

who found that these can lead to catastrophic wear. Therefore the effect of surface

roughness will be an important consideration in any future modelling studies.

The line contact model used in the current model can also be advanced by using a three-

point contact model. In reality, the rail and wheel profiles have a complex geometry and

may cause various shapes of contact patch. By using this three-dimensional model, it is

possible to evaluate the wear that occurs across the width of the contact patch [34].

Moreover a more accurate model of crack initiation and propagation could be obtained

if the grain structure is involved. Therefore another important development for

consideration is the involvement of the microstructure. This is because a different

microstructure will lead to different behaviours regarding wear and rolling contact

fatigue failure. The steel microstructure may include a variation in lamellar spacing and

its orientation, and a simple representation of the pearlitic grain structure with

proeutectoid ferrite grain boundary. These features could give a better representation of

Chapter 9.Future Work

182

the wearing material rather than randomly assigning all elements with the same

properties. Moreover, as the thermal effect may reach the phase transformation

temperature, this may cause microstructural changes. Therefore, a more comprehensive

analysis of the process of failure could be obtained. It is recommended that the current

model be further developed with these features. This was also proposed by Franklin

et.al [125] and Eden et.al [30], who investigated the metallurgical aspect using twin disc

testing.

The thermal effect, which focuses on the steady state wheel temperature in the current

model, can also be developed to analyse the effect of wheel bulk temperature on wear

and RCF failure in a wheel. Further research on the thermal model will make it

applicable to wheel damage.

Another possible development is in the combining of material properties considered in

the model. The results from this thesis have only considered one combination, and show

the advantages of higher hardness, ductility and hardening behaviour over thermal

softening behaviour. There are many other combinations of material properties, which

may lead to different effects on wear and rolling contact fatigue behaviour. Various

kinds of rail steel could also be involved in an investigation of the advantages and

disadvantages of specific properties on the service conditions experienced by the rail.

Further investigation in this area looks promising finding the search for rail material

with a longer lifetime.

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[117] H. Muster, H. Schmedders, K. Wick, and H. Pradier, "Rail rolling contact

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fatigue and wear behaviour of the infrastar two-material rail," Wear, vol. 258 pp.

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[120] W. Zhong, J. J. Hu, Z. B. Li, Q. Y. Liu, and Z. R. Zhou, "A study of rolling

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[121] W. Zhong, J. J. Hu, P. Shen, C. Y. Wang, and Q. Y. Lius, "Experimental

investigation between rolling contact fatigue and wear of high-speed and heavy-

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[122] D. A. Rigney, "Comments on the sliding wear of metals," Tribology

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characteristics of two abrasion-resistant steels: Q/T C1095 and 15B37H,"

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[124] K. Knothe and S. Liebelt, "Determination of temperatures for sliding contact

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265 (9-10), pp. 1332-1341, 2008.

References

194

Appendix: Source Code

195


%% Wear Simulation.m

clear all; clc;

%% VARIABLES INITIALIZATION Sr = -0.03; % slip/roll ratio v0 = 30; % vehicle speed (m/s) p0 = 1.5e9; % Hertzian peak pressure (Pa) mu = 0.4; % friction coefficient alpha = 25; % hardening parameter beta = 1.88; % hardening parameter constant = 0.00237; % material constant cycles = 50000; % number of wheel pass a = 5.88e-3; % contact patch size (mm) % the thermal parameters lambda = 50; % thermal conductivity (W/mK) alphat = 1.2e-5; % thermal expansion coefficient (/K) density = 7850; % steel density (kg/m^3) cp = 450; % specific heat capacity (J/kgK) kt = lambda/(density*cp); betar = sqrt(lambda*density*cp); %thermal penetration coefficient v = 0.3; % Poisson ratio E = 210e9; % Young modulus (Pa) % rail type: 1=BS11 2=900A 3=1100 rail_type = 3;

%% brick in -direction stepx = 0.0001*4*a; x = -(1*a) : stepx : (1*a); xnorm = x./a;

%% brick in z-direction stepz = [1e-6 (a-100e-6)/250 a/50]; limitz = [100e-6 a 2.9*a]; z=[]; for i=1:length(stepz) if i==1 depth = 0 : stepz(i) : limitz(i); num_brick = round(limitz(i) / stepz(i)); else depth = limitz(i-1)+stepz(i) : stepz(i) : limitz(i); num_brick(i) = int16((limitz(i)-limitz(i-1)) / stepz(i)); end z = cat(2,z,depth); end temp=0; for i = 1:length(num_brick) temp = temp + num_brick(i); lim_brick(i) = temp; end for i = 1:length(z)-1 zavg(i) = (z(i)+z(i+1))/2; end


196

znorm = zavg./(a); clear ('depth'); clear ('temp'); z = single(z); znorm = single (znorm); zavg = single (zavg); x = single (x); xnorm = single (xnorm);

%% support parameters num_of_area = 64; % number of colour gradation between

% white(0) and black(1)

map(num_of_area+1,:)=[1 1 1]; wear_count = 0; count_wear1 = 0; count_wear2 = 0; counter = 1; total_debris = zeros(1,cycles,'single'); h_debris = zeros(1,cycles,'single'); counter_debris = 1; counter_dd = 1; dam_depth(counter_dd) = 0; number_failed (counter_dd) = 0; theta (counter_dd) = 0; max_failed_depth (counter_dd) = 0;

%% softening effect [tflashmaxeq,tetar,tflashmax,tflashavg,tetaw_cond_avg,tetar_cond_avg,t

bulk,ttot,tetaw_conduction,tetar_conduction] =

contact_temp(p0,mu,a,v0,Sr,zavg,kt,betar,stepx);

%% yield stress sigma_y = single (sigma_y); mean_sy = 406e6; std_sy = 0.05*mean_sy; sigma_y = random('Normal',mean_sy,std_sy,length(znorm),length(x)-1); sigma_y_ori = sigma_y; k0 = sigma_y/sqrt(3); k_eff = k0;

%% gamma_c (critical strain) mean_g = 11.5; std_g = 0.05*mean_g; gamma_c = random('Normal',mean_g,std_g,length(znorm),length(x)-1); gamma_c_ori = gamma_c;

% list_area: define the colour area based on gamma_c if std_g == 0 list_area = 0:(gamma_c/(num_of_area-1)):gamma_c; else list_area = zeros(1,num_of_area); end list_area = single(list_area);

%% Maximum orthogonal shear stress at each depth [Q,Qmaxp0,Qmax] =

stress_thermal(Sr,mu,v0,a,kt,lambda,alphat,v,E,p0,zavg,stepx); tau_zx_max = Qmaxp0;


197

%% MAIN PROGRAM

for k=1:(cycles) k [k0_softening,k_eff,gamma,delta_gamma ] = ...

non_f_brick( beta,alpha,gamma,tau_zx_max,k_eff,constant,k0,...

tflashmax,m_area,rail_type); [m_area] = color_fill(map,num_of_area,gamma_c,gamma,r,c); [m_area,count_debris,sigma_y,k0,k_eff,gamma,counter_sigmay,...

gamma_c,count_wear1,count_wear2,num_layer_off] = ...

percent_brick_check_remlayer(map,sigma_y,k0,k_eff,gamma,m_area,...

counter_sigmay,lim_brick,gamma_c,count_wear1,count_wear2,stepz,...

sigma_y_ori,gamma_c_ori); num_layer_off1(k) = num_layer_off;

%% calculating the wear if ~isempty(count_debris) total_debris(counter_debris) = count_debris; h_debris(counter_debris) = counter_debris = counter_debris + 1; end

[r,c] = find(m_area==65);

%% surface crack [failed,failed_65] = crack_dd4(m_area); avg_failed = sum(failed)/length(find(failed>0)); max_failed = max(failed); failed_depth_ori(:,k) = failed; if max_failed >= 50 crack_failed = (failed .* stepz(1) ./ 50e-6); crack_failed1 = (crack_failed>=1); number_crack1(counter_dd) = length(crack_failed1); crack_gamma1 = mean(mean(gamma(1:max_failed,2))); theta1(counter_dd) = (pi/2 - atan (crack_gamma1)) * (180/pi); max_failed_depth(counter_dd) = max_failed; avg_failed_depth (counter_dd) = mean(failed(find(failed>=50))); failed(failed<50) = 0; failed_depth(:,counter_dd) = failed; crack_cycle1(counter_dd) = k; else if counter_dd ==1 number_crack1 (counter_dd) = 0; theta1 (counter_dd) = 0; max_failed_depth (counter_dd) = 0; else number_crack1 (counter_dd) = 0; theta1 (counter_dd) = 0; max_failed_depth (counter_dd) = 0; end crack_cycle1(counter_dd) = k; end

%% subsurface crack [crack_depth, max_centroids_crack,num_crack,max_majoraxislength,...

max_boundingboxes,orientations] = crack_dd3(m_area,gamma); if crack_depth ~= 0 dam_depth(counter_dd) = crack_depth; centroidsx(counter_dd) = max_centroids_crack(1); centroidsy(counter_dd) = max_centroids_crack(2);


198

number_crack2(counter_dd) = num_crack; max_length (counter_dd) = max_majoraxislength;

limit1 = ceil(max_boundingboxes(2)); limit2 = ceil(max_boundingboxes(2)); crack_gamma2 = mean(mean(gamma(limit1:limit2+max_boundingboxes(4)),..

2)); theta2(counter_dd) = (pi/2 - atan (crack_gamma2)) * (180/pi); crack_cycle2(counter_dd) = k; elseif counter_dd == 1 dam_depth(counter_dd) = 0; centroidsx(counter_dd) = 0; centroidsy(counter_dd) = 0; number_crack2(counter_dd) = 0; max_length (counter_dd) = 0; theta2(counter_dd) = 0;

else dam_depth(counter_dd) = 0; centroidsx(counter_dd) = 0; centroidsy(counter_dd) = 0; number_crack2(counter_dd) = 0; max_length (counter_dd) = 0; theta2(counter_dd) = 0; centroids(counter_dd) = 0; end crack_cycle2 (counter_dd) = k; end counter_dd = counter_dd + 1; end

%% stress_thermal.m

function [Q_hertz,Q_bulk,Q_bulk0,Q_total,Qmax_total,Qmaxp0_total] =

stress_thermal(Sr,mu,V,a,kt,lambda,alphat,v,E,P0,z1,stepx)

G0=E/(2*(1+v)); H0=2*alphat*G0*kt*(1+v)/(lambda*(1-v)); Pe=V*a/kt;

x1=-(3*a):stepx:(3*a); z=z1/a; x=x1/a;

rows = size (z,2); cols = size (x,2);

for j=1:rows for i=1:cols Q_hertz(j,i) = quadgk(@(t)myfun(t,x(i),z(j),Sr,mu,H0,Pe),-1,1); Q_bulk(j,i) = quadgk(@(t)myfun_ertz(t,x(i),z(j),Sr,mu,H0,Pe),-1,1); end end

Sr = 0; for j=1:rows for i=1:cols Q_bulk0(j,i) = quadgk(@(t)myfun_ertz(t,x(i),z(j),Sr,mu,H0,Pe),-1,1);


199

end end

Q_total = Q_hertz + (Q_bulk - Q_bulk0); for j=1:rows for i=1:cols Qmax_total(j) = max(Q_total(j,:)); end end

Qmaxp0_total = Qmax_total*P0; end

%% myfun.m

function Sigmapq = myfun(t,x,z,Sr,mu,H0,Pe)

theta0 = atan((x-t)./z); theta2 = 2.5.*theta0; [Apq,Bpq,Dpq] = tauxz (theta0,theta2,x,t,Pe,z); e = 2.718281828459045;

Pt = sqrt(1-t.^2); Fpq = H0.*Sr.*mu.*Apq ./ (sqrt(2.*pi.*Pe).*((x-t).^2+z.^2).^(3/4)) ... + (2 .* (z + (2.*H0.*Sr-1) .* (x-t).*mu).*Bpq) ./ (pi.*((x-...

t).^2+z.^2).^2); Hpq = 0.5.*mu.*Sr.*Dpq .* ((pi.*Pe).^(-0.5)) .* ((x-t).^(-2.5)) .* ...

e.^(-Pe.*z.^2./(4.*(x-t)));

if x<=-1 Sigmapq = Pt.*Fpq; elseif ((x>-1) & (x<=1)) if t<=x Sigmapq = (Pt.*Hpq) + (Pt.*Fpq); else Sigmapq = (Pt.*Fpq); end elseif x>1 Sigmapq = (Pt.*Hpq) + (Pt.*Fpq); end

%% myfun_ertz.m

function Sigmapq = myfun_ertz(t,x,z,Sr,mu,H0,Pe)

theta0 = atan((x-t)./z); theta2 = 2.5.*theta0; [Apq,Bpq,Dpq] = tauxz (theta0,theta2,x,t,Pe,z); e = 2.718281828459045; % natural logarithm ???

Pt = pi.*sqrt(1./(2.*(x+1))) ./4; Fpq = H0.*Sr.*mu.*Apq ./ (sqrt(2.*pi.*Pe).*((x-t).^2+z.^2).^(3/4)) ... + (2 .* (z + (2.*H0.*Sr-1) .* (x-t).*mu).*Bpq) ./ (pi.*((x-...

t).^2+z.^2).^2); Hpq = 0.5.*mu.*Sr.*Dpq .* ((pi.*Pe).^(-0.5)) .* ((x-t).^(-2.5)) .* ...


200

e.^(-Pe.*z.^2./(4.*(x-t)));

if x<=-1 Sigmapq = Pt.*Fpq; elseif ((x>-1) & (x<=1)) if t<=x Sigmapq = (Pt.*Hpq) + (Pt.*Fpq); else Sigmapq = (Pt.*Fpq); end elseif x>1 Sigmapq = (Pt.*Hpq) + (Pt.*Fpq); end

%% contact_temp.m

function

[tflashmaxeq,tetar,tflashmax,tflashavg,tetaw_cond_avg,tetar_cond_avg,t

bulk,ttot,tetaw_conduction,tetar_conduction,tetaw0] =

contact_temp(p0,mu,a,v0,Sr,z,kt,betar,stepx)

x = (-2*a):stepx:(2*a); Sr = -Sr; vs = v0*(Sr); betaw = betar; vw = v0+vs; vr = v0;

part_factor = betaw * sqrt(vw) / (betaw*sqrt(vw) + betar*sqrt(vr)); qr = (1-part_factor)*pi*mu*vs*p0/4;

Pe = a*v0/(2*kt); delta = a/sqrt(Pe);

xnorm = x./a; znorm = z./delta;

[tetar,tflashmaxeq,tflashmax,tflashavg] =

tflash_max_avg(mu,p0,a,qr,vs,vw,vr,betar,betaw,part_factor,xnorm,znorm

);

[tetaw_cond_avg,tetar_cond_avg,tetaw_conduction,tetar_conduction,tetaw

0] = conduction (mu,p0,a,v0,vs,betar,part_factor,xnorm,znorm);

tbulk = tflashavg+tetar_cond_avg'; ttot = tflashmax+tbulk;

%% tflashmaxavg.m

function

[tetar,tflashmaxeq,tflashmax,tflashavg]=tflash_max_avg(mu,p0,a,qr,vs,v

w,vr,betar,betaw,part_factor,xnorm,znorm) znormori = znorm';


201

xnormori = xnorm; znorm = repmat(znormori,1,length(xnorm)); xnorm = repmat(xnormori,length(znorm),1); [rows,~] = size (znorm);

[~,c1] = find(real(xnorm(1,:)) == -1); [~,c2] = find(real(xnorm(1,:)) == 1);

znorm1 = znorm(:,1:c1-1); znorm2 = znorm(:,c1:c2); znorm3 = znorm(:,c2+1:end);

xnorm1 = xnorm(:,1:c1-1); xnorm2 = xnorm(:,c1:c2); xnorm3 = xnorm(:,c2+1:end);

f1(1:rows,1:c1-1)=0; f2 = sqrt((2.*(xnorm2+1))./pi) .* exp(-znorm2.^2./(2.*(xnorm2+1))) -

znorm2.*erfc(znorm2./sqrt(2.*(xnorm2+1))); f3 = (sqrt((2.*(xnorm3+1))./pi) .* exp(-znorm3.^2./(2.*(xnorm3+1))) -

znorm3.*erfc(znorm3./sqrt(2.*(xnorm3+1)))) ... - (sqrt((2.*(xnorm3-1))./pi) .* exp(-

znorm3.^2./(2.*(xnorm3-1))) - znorm3.*erfc(znorm3./sqrt(2.*(xnorm3-

1)))); f=cat(2,f1,f2,f3); tetar = qr./betar.*sqrt((2.*a)./vr).*f;

tflashmaxeq=1.253*part_factor*mu*vs*p0*sqrt(a/vw)/betaw; max(tetar(1,:)); tflashmax= max(tetar,[],2);

[~,c]=find(xnorm>-1.001,1,'first'); xpos=c; tflashavg=mean(tetar(:,xpos:end),2);

%% conduction.m

function

[tetaw_cond_avg,tetar_cond_avg,tetaw_conduction,tetar_conduction,tetaw

0] = conduction (mu,p0,a,v0,vs,betar,part_factor,xnorm,znorm)

tetaw0 = mu*vs*p0*(sqrt(pi^3*a/(32*v0)))/betar; tetam = part_factor*tetaw0;

rows = size(znorm,2); cols = size(xnorm,2);

for i = 1 : rows for j = 1 : cols if ((xnorm(j) >= -1) & (xnorm(j) <= 1)) tetaw_conduction(i,j)= ...

tetam+(tetaw0-tetam)*erf(znorm(i)/sqrt(2*(xnorm(j)+1))); tetar_conduction(i,j) = ...

tetam*(1-erf(znorm(i)/sqrt(2*(xnorm(j)+1))));


202

else tetar_conduction(i,j) = 0; tetaw_conduction(i,j) = 0; end end end for i = 1 : rows tetaw_cond_avg(i) = mean(nonzeros(tetaw_conduction(i,:))); tetar_cond_avg(i) = mean(nonzeros(tetar_conduction(i,:))); end

tf = isnan(tetar_cond_avg); [r,c] = find(tf==1); for i = 1 : length(c) tetar_cond_avg(r,c(i)) = 0; end

tetaw_surface = ...

tetaw0.*(1-(2.*(1-part_factor).*asin(sqrt(2./(xnorm+1)))./pi) );

%% non_f_brick.m

function [ k_eff_softening,k_eff,gamma,delta_gamma ] = non_f_brick(...

beta,alfa,gamma,tau_zx_max,k_eff,constant,k0,temp,m_area,rail_type)

k_eff_softening = softening_cycle(k_eff,temp,rail_type); [~,cols] = size (k_eff); tau_max = repmat(tau_zx_max',1,cols); strain = tau_max > k_eff_softening; notfailed = (m_area > 1) & (m_area < 65); delta_gamma = notfailed .* strain .* constant .* ((tau_max ./

k_eff_softening)-1); gamma = gamma + delta_gamma; k_eff = k0 .* (max(1,beta .* sqrt(1 - exp(-alfa .* gamma))));

%% softening_cycle.m

function [k_eff_softening]=softening_cycle(k_eff,temp,rail_type)

sigma_y = k_eff * sqrt(3); [Yi] = softening1(sigma_y,temp,rail_type); k_eff_softening = Yi / sqrt(3);

%% softening1.m

function [Yi] = softening1(Y0,temp,rail_type) if rail_type == 1 Tref1 = 250; Tref2 = 750;

f = temp;

f(f <= Tref1) = 1; f((f > Tref1) & (f < Tref2)) = 1 - (0.741 .* ((f((f > Tref1) & ...


203

(f < Tref2)) - Tref1) ./ Tref2)) - (1.026 .* (((f((f > Tref1) & ...

(f < Tref2)) - Tref1) ./ Tref2 ).^2)); f(f >= Tref2) = 0.05;

elseif rail_type == 2 Tref1 = 21; Tref2 = 763; f = temp;

f(f <= Tref1) = 1; f((f > Tref1) & (f < Tref2)) = -7.111e-14.*f((f > Tref1) & ...

(f < Tref2)).^5 + 1.593e-10.*f((f > Tref1) & (f < Tref2)).^4 - ...

1.279e-7.*f((f > Tref1) & (f < Tref2)).^3 + 4.044e-5.*...

f((f > Tref1) & (f < Tref2)).^2 - 0.0044.*f((f > Tref1) & (f < ...

Tref2)) + 1.075; f(f >= Tref2) = 0.05;

elseif rail_type == 3 Tref1 = 13; Tref2 = 807; f = temp;

f(f <= Tref1) = 1; f((f > Tref1) & (f < Tref2)) = -9.342e-15.*f((f > Tref1) & (f < ...

Tref2)).^5 + 3.091e-11.*f((f > Tref1) & (f < Tref2)).^4 - ...

3.319e-8.*f((f > Tref1) & (f < Tref2)).^3 + 1.269e-5.*...

f((f > Tref1) & (f < Tref2)).^2 - 0.002117.*f((f > Tref1) & ...

(f < Tref2)) + 1.024; f(f >= Tref2) = 0.05; end

[~,cols] = size(Y0); f_repmat = repmat(f,1,cols); Yi = Y0 .* f_repmat;

%% color_fill.m

function [m_area]=color_fill(map,num_of_area,gamma_c,gamma,r,c)

d_area = gamma_c/(num_of_area-1); m_area = num_of_area - floor(gamma ./ d_area); m_area (m_area < 1) =1;

if ~isempty(r) for i = 1 : length(r) m_area(r(i),c(i)) = length(map); end end

%% percent_brick_check_remlayer.m

function[m_area,count_debris,sigma_y,k0,k_eff,gamma,counter_sigmay,...

gamma_c,count_wear1,count_wear2,num_layer_off] =...

percent_brick_check_remlayer(map,sigma_y,k0,k_eff,gamma,m_area,...


204

counter_sigmay,lim_brick,gamma_c,count_wear1,count_wear2,stepz,...

sigma_y_ori,gamma_c_ori)

[row,col] = size(m_area); count_debris = 0; x_brick = size(gamma,2); num_layer_off = 0; nomatch = 0; j = 1;

while nomatch < col nomatch = 0; for i = 1 : col if j == 1 if i == 1 object = [65 65 65 ; 1 m_area(j,1:2) ; 1 m_area(j+1,1:2)]; elseif i == col object = [65 65 65;m_area(j,col-1:col) 1;m_area(j+1,col-1:col) 1]; else object = [65 65 65 ; m_area(1:2 , i-1 : i+1)]; end

elseif i == 1 object = [1 m_area(j-1,1:2) ; 1 m_area(j,1:2) ; 1 m_area(j+1,1:2)]; elseif i == col object = [m_area(j-1,col-1:col) 1 ; m_area(j,col-1:col) 1 ; ...

m_area(j+1,col-1:col) 1];

else

object = m_area(j-1:j+1 , i-1 : i+1); end end end

value = heuristic_check(object); if value == 1 if (m_area(j,i) == length(map)) continue; else m_area(j,i) = length(map); count_debris = count_debris + 1;

%% REGION 2 % checking whether 1 layer from reg 2 must up count_wear1 = count_wear1 + 1; wear_thickness1 = (count_wear1/x_brick) * stepz(1); if wear_thickness1 > stepz(2) gamma (lim_brick(1)+1 : lim_brick(2)-1 , :) = ...

gamma (lim_brick(1)+2 : lim_brick(2) , :); gamma_c (lim_brick(1)+1 : lim_brick(2)-1 , :) = ...

gamma_c (lim_brick(1)+2 : lim_brick(2) , :); m_area (lim_brick(1)+1 : lim_brick(2)-1 , :) = ...

m_area (lim_brick(1)+2 : lim_brick(2) , :); k0(lim_brick(1)+1 : lim_brick(2)-1 , :) = ...

k0(lim_brick(1)+2 : lim_brick(2) , :); k_eff (lim_brick(1)+1 : lim_brick(2)-1 , :) = ...

k_eff (lim_brick(1)+2 : lim_brick(2) , :); sigma_y (lim_brick(1)+1 : lim_brick(2)-1 , :) = ...

sigma_y (lim_brick(1)+2 : lim_brick(2) , :);


205

% the last layer of region 2 for m = 1:floor(x_brick/10) rand_val = randperm(10); gamma (lim_brick(2),(10*(m-1))+1 : (m*10)) = ...

gamma (lim_brick(2)+1 , (10*(m-1)) + rand_val); gamma_c (lim_brick(2),(10*(m-1))+1 : (m*10)) = ...

gamma_c (lim_brick(2)+1 , (10*(m-1)) + rand_val); m_area (lim_brick(2),(10*(m-1))+1 : (m*10)) = ...

m_area (lim_brick(2)+1 , (10*(m-1)) + rand_val ); k0(lim_brick(2),(10*(m-1))+1 : (m*10)) = ...

k0(lim_brick(2)+1 , (10*(m-1)) + rand_val); k_eff(lim_brick(2),(10*(m-1))+1 : (m*10)) = ...

k_eff (lim_brick(2)+1 , (10*(m-1)) + rand_val); sigma_y (lim_brick(2),(10*(m-1))+1 : (m*10)) = ...

sigma_y (lim_brick(2)+1 , (10*(m-1)) + rand_val); end

wear_thickness1 = wear_thickness1-stepz(2); end count_wear1 = ceil(wear_thickness1 * x_brick/ stepz(1));

%% REGION 3 % checking whether 1 layer from reg 2 must up

count_wear2 = count_wear2 + 1; wear_thickness2 = (count_wear2/x_brick) * stepz(1);

if wear_thickness2 > stepz(3) gamma (lim_brick(2)+1 : lim_brick(3)-1 , :) = ...

gamma (lim_brick(2)+2 : lim_brick(3) , :); gamma_c (lim_brick(2)+1 : lim_brick(3)-1 , :) = ...

gamma_c (lim_brick(2)+2 : lim_brick(3) , :); m_area (lim_brick(2)+1 : lim_brick(3)-1 , :) = ...

m_area (lim_brick(2)+2 : lim_brick(3) , :); k0(lim_brick(2)+1 : lim_brick(3)-1 , :) = k0 ...

(lim_brick(2)+2 : lim_brick(3) , :); k_eff (lim_brick(2)+1 : lim_brick(3)-1 , :) = ...

k_eff (lim_brick(2)+2 : lim_brick(3) , :); sigma_y (lim_brick(2)+1 : lim_brick(3)-1 , :) = ...

sigma_y (lim_brick(2)+2 : lim_brick(3) , :);

% the last layer of region 1 gamma (lim_brick(3) , :) = 0; if counter_sigmay > row counter_sigmay=1; end sigma_y (lim_brick(3) , :) = sigma_y_ori(counter_sigmay,:); gamma_c (lim_brick(3) , :) = gamma_c_ori(counter_sigmay,:); counter_sigmay = counter_sigmay+1;

m_area (lim_brick(3) , :) = length(map)-1; k0(lim_brick(3) , :) = sigma_y(lim_brick(3),:)./sqrt(3); k_eff (lim_brick(3) , :) = k0(lim_brick(3),:); wear_thickness2 = wear_thickness2-stepz(3); end count_wear2 = ceil(wear_thickness2 * x_brick / stepz(1)); else nomatch = nomatch + 1;


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end end

%% REGION 1 fail_m_area1 = m_area < 65; percent_fail = mean(fail_m_area1,2); ref_percent_fail = 0.0; add_debris = 0;

loop1 = 0; for n = 1 : length(percent_fail) if percent_fail(n) <= ref_percent_fail add_debris = add_debris + (percent_fail(n)* x_brick); loop1 = loop1 + 1; num_layer_off = num_layer_off + loop1; else break; end end count_debris = count_debris + add_debris; count_wear1 = count_wear1 + add_debris; count_wear2 = count_wear2 + add_debris;

if loop1 == 0; j = j + 1; num_layer_off=num_layer_off + 0; end

for n = 1 : loop1 gamma (1:lim_brick(1)-1 , :) = gamma (2:lim_brick(1) , :); gamma_c (1:lim_brick(1)-1 , :) = gamma_c (2:lim_brick(1) , :); m_area (1:lim_brick(1)-1 , :) = m_area (2:lim_brick(1) , :); k0 (1:lim_brick(1)-1 , :) = k0 (2:lim_brick(1) , :); k_eff (1:lim_brick(1)-1 , :) = k_eff (2:lim_brick(1) , :); sigma_y (1:lim_brick(1)-1 , :) = sigma_y (2:lim_brick(1) , :);

% the last layer of region 1 for m = 1:floor(x_brick/10) rand_val = randperm(10); %ceil(rand(1,10)*10); gamma (lim_brick(1),(10*(m-1))+1 : (m*10)) = ...

gamma (lim_brick(1)+1 , (10*(m-1)) + rand_val); gamma_c (lim_brick(1),(10*(m-1))+1 : (m*10)) = ...

gamma_c (lim_brick(1)+1 , (10*(m-1)) + rand_val); m_area (lim_brick(1),(10*(m-1))+1 : (m*10)) = ...

m_area (lim_brick(1)+1 , (10*(m-1)) + rand_val ); k0(lim_brick(1),(10*(m-1))+1 : (m*10)) = ...

k0(lim_brick(1)+1 , (10*(m-1)) + rand_val); k_eff(lim_brick(1),(10*(m-1))+1 : (m*10)) = ...

k_eff(lim_brick(1)+1 , (10*(m-1)) + rand_val); sigma_y (lim_brick(1),(10*(m-1))+1 : (m*10)) = ...

sigma_y (lim_brick(1)+1 , (10*(m-1)) + rand_val); end end end


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%% crack_dd4.m

function [failed,failed_65] = crack_dd4(m_area)

[~,col] = size(m_area); failed = zeros(1,col); failed_65 = zeros(1,col);

for i = 1 : col test = 0; row = 1; while m_area(row,i) == 65 row = row + 1; failed_65(i) = failed_65(i) + 1; end if m_area(row,i) == 1 while (m_area(row,i) == 1) failed(i) = failed(i) + 1; row = row + 1; end

if (i > 2) & (i < col-1) active_while = 0; while (m_area(row,i-1) == 1) | (m_area(row,i+1) == 1) failed(i) = failed(i) + 1; row = row + 1; active_while = 1; end if active_while == 1 while (m_area(row,i-2) == 1) | (m_area(row,i) == 1) | ...

(m_area(row,i+2) == 1) failed(i) = failed(i) + 1; row = row + 1; end end test = 1; elseif (i==1) | (i==2) active_while = 0; while (m_area(row,i+1) == 1) failed(i) = failed(i) + 1; row = row + 1; active_while = 1; end if active_while == 1 while (m_area(row,i) == 1) | (m_area(row,i+2) == 1) failed(i) = failed(i) + 1; row = row + 1; end end test = 1;

elseif (i == 99) | (i==98) active_while = 0; while (m_area(row,i-1) == 1) failed(i) = failed(i) + 1; row = row + 1; active_while = 1; end if active_while == 1


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while (m_area(row,i-2) == 1) | (m_area(row,i) == 1) failed(i) = failed(i) + 1; row = row + 1; end end test = 1; end else failed(i) = 0; end end end

%% crack_dd3.m

function [crack_depth, max_centroids_crack,num_crack,...

max_majoraxislength,max_boundingboxes,orientations] = ...

crack_dd3(m_area,gamma)

m_area_failed = (m_area==1); gauss = fspecial('Gaussian',[3 3],0.5); gauss_temp = filter2(gauss,m_area_failed,'full'); m_area_failed = (gauss_temp>0.5);

stat = ... regionprops(m_area_failed,'BoundingBox','Centroid',...

'MajorAxisLength','MinorAxisLength','Orientation'); boundingboxes = cat (1, stat.BoundingBox); centroids = cat (1, stat.Centroid); majoraxes = cat (1, stat.MajorAxisLength); minoraxes = cat (1, stat.MinorAxisLength); orientations = cat (1, stat. Orientation); orientations = orientations (orientations > 0);

centroids_crack1 = centroids (orientations>0,:); major_crack1 = majoraxes (orientations>0,:); minor_crack1 = minoraxes (orientations>0,:); bb_crack1 = boundingboxes (orientations>0,:); orient_crack1 = orientations;

for i = 1: size(bb_crack1,1) mean_gamma (i) = mean(gamma(ceil(bb_crack1(i,2)),2)); major_crack2(i) = (sin(orient_crack1(i)*pi/180) / sin((pi/2)-...

atan(mean_gamma(i)))) * major_crack1(i); minor_crack2(i) = minor_crack1(i) /(sin(orient_crack1(i)*pi/180)/...

sin((pi/2)-atan(mean_gamma(i)))) ; ratio(i) = major_crack2(i) / minor_crack2(i); end

if ~isempty(bb_crack1) centroids_crack3 = centroids_crack1 (major_crack2 >= ...

5*minor_crack2,:); major_crack3 = major_crack1(major_crack2 >= 5*minor_crack2,:); minor_crack3 = minor_crack1 (major_crack2 >= 5*minor_crack2,:); bb_crack3 = bb_crack1 (major_crack2 >= 5*minor_crack2,:); orient_crack3= orient_crack1 (major_crack2 >= 5*minor_crack2,:); num_crack = size(centroids_crack3,1); else


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bb_crack3=[]; end

if ~isempty(bb_crack3) [r,~] = find(bb_crack3 == max(bb_crack3(:,2))); crack_depth = ceil(bb_crack3(r(1),2)); max_centroids_crack = ...

centroids_crack3(major_crack3==max(major_crack3),:); max_majoraxislength = max (major_crack3); max_boundingboxes = ...

bb_crack3(major_crack3==max(major_crack3),:); else crack_depth = 0; max_centroids_crack = 0; max_majoraxislength = 0; max_boundingboxes = 0; num_crack = 0; end

modelling of the effect of thermal and material properties on ......repeated wheel passes on rail...

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