Fatigue and Fracture Behaviour of Aluminothermic Rail Welds Under High Axle Load Conditions

FATIGUE AND FRACTURE BEHAVIOUR OF ALUMINOTHERMIC RAIL WELDS UNDER HIGH AXLE LOAD CONDITIONS

by Iman Salehi B.Sc., M.Sc.

A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy

Centre for Sustainable Infrastructure Faculty of Engineering and Industrial Sciences Swinburne University of Technology

March 2013 ABSTRACT

Among different rail welding methods Aluminothermic welding (ATW) is the oldest and simplest procedure widely used for re-railing and replacement of defective rails. Since ATW is a cast welding process in which several aspects are operator-dependent it suffers from the variability of the produced weld quality, presence of casting defects and inconsistencies in the microstructure and mechanical properties. Previous observations have shown that field-welded ATWs have been major sources of fatigue and overload failures in Australian heavy haul railway systems. The most common failure modes are categorized into straight breaks (transverse fissure) and horizontal split webs (HSW). Straight breaks initiate from stress concentration sites at the edge of the weld collar, in the foot, lower web and underhead regions, and propagate in vertical direction under Mode I loading. HSW failures involve the development of a horizontal fatigue crack which initiates from a surface or near-surface gross defect in the weld collar, generally in the mid- or upper-web region.

In this study analysis of fatigue crack initiation is performed at the edge of the weld collar of an aluminothermic weld, in order to examine the formation of straight break under high axle load conditions. The fatigue assessment is accomplished using a thermo-structural finite element simulation in ANSYS package followed by a shear based multi-axial fatigue critical plane criterion implemented in a MATLAB computer code. The influence of several parameters including wheel-rail contact patch eccentricity, contact tractions, residual stress distribution, seasonal temperature variation and track support condition is investigated. The analysis identifies the underhead region of a defect-free weld as the most critical location which is subject to severe fatigue damage under harsh curving and hunting behaviours.

A further study is performed on the influence of geometric features or the design of the collar edge (flank angle and toe radius). Two geometrically different aluminothermic welds, one of which is widely used in Australian heavy haul railways and the other one recently developed, are investigated in terms of fatigue crack initiation risk. The results

I confirm that the amount of fatigue damage is critically dependent on the geometric features of the collar edge, particularly at the underhead radius. A well-designed collar edge can also enhance the fatigue performance of the base region under poor track support conditions.

A specific type of defect known as cold lap or finning resulting from the leak of the molten metal out of the mould has been observed to be associated with the initiation of straight breaks at the top of the rail foot. To study the formation of straight break influenced by cold lap, the defect is considered as a crack and its propagation is studied using Linear Elastic Fracture Mechanics. The results show that the probability of crack initiation from a cold lap defect largely depends on the lap thickness, lap tip location and the wheel-rail contact conditions.

Eventually, damage tolerance analysis is performed for the two weld collar designs on specific large defects located at the surface of the web region in relation to the formation of HSWs. Two approaches, multi-axial fatigue analysis of the defect surface and fracture mechanics, are utilised each of which applicable to certain types of defects and sizes. The results suggest that the collar design in the web region can affect the weld damage tolerance for particular web defects. For crack like defects considered in this study a near flat design of the web can decrease the equivalent stress intensity factor range by up to 15%. A combination of reinforcement and the adoption of a near flat section geometry at the web region can also result in improved performance in terms of fast fracture or overload failure.

ACKNOWLEDGEMENTS

First of all, I would like to express my sincere gratitude and appreciation to my main supervisor, Professor Ajay Kapoor for his in-depth and constant supervision in pursuing this research. Throughout the years he has been providing me with excellent theoretical and technical guidance and valuable feedback on my work and publications. I would like to acknowledge his kind support, encouragement and enthusiasm that helped me perform such challenging task.

I would also like to thank my second supervisor Mr. Peter Mutton from the Institute of Railway Technology (IRT), Monash University who introduced the first idea of this study and supported me with his constructive suggestions during the course of this research. I benefited greatly from his extensive in-field knowledge and his previous studies pertaining to rail weld failures. I would also like to acknowledge the IRT for providing access to some of their experimental results and occasional financial support.

My PhD study was sponsored by Swinburne University of Technology through a SUPRA scholarship. Hereby, I greatly appreciate their financial contribution and provision of IT and laboratory equipment which facilitated this research.

My sincere thanks and appreciation goes to my beloved parents for their utmost support and encouragement while I was away from them. I am deeply indebted to them for their concern and sacrifice without which I would not have reached this stage. My thanks are also due to my brothers and sister who have always been my inspiration. Finally, I wish to express my special thanks to my wife, Zohreh Heidarirad for walking with me on this journey with her infinite love, support and encouragement.

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Dedicated to:

My beloved parents

My wonderful wife

DECLARATION

I declare that this thesis represents my own work and contains no material which has been accepted for the award of any other degree, diploma or qualification in any university. To the best of my knowledge and belief this thesis contains no material previously published or written by any other person except where due acknowledgment has been made.

Iman Salehi March 2013

TABLE OF CONTENTS

ABSTRACT ...... I

ACKNOWLEDGEMENTS ...... III

DECLARATION ...... V

TABLE OF CONTENTS ...... VI

LIST OF PUBLICATIONS ...... X

LIST OF FIGURES ...... XI

LIST OF TABLES ...... XXIV

LIST OF NOTATIONS AND ACRONYMS ...... XXV

CHAPTER 1 INTRODUCTION ...... 1 1.1 Continuous Welded Rail ...... 1 1.2 Aluminothermic Weld ...... 2 1.3 Aluminothermic Weld as a Major Source of Failure ...... 4 1.4 Aims and Objectives ...... 7 1.5 Methodology ...... 9 1.6 Thesis Structure ...... 10

CHAPTER 2 LITERATURE REVIEW ...... 12 2.1 Aluminothermic Weld Failure Mechanisms ...... 12 2.1.1 Local Plastic Deformation (Batter) ...... 14 2.1.2 Rolling Contact Fatigue and Wear ...... 17 2.1.3 Straight Breaks (Transverse Fissure) ...... 19 2.1.4 Horizontal Split Webs ...... 24 2.2 Multi-axial Fatigue ...... 29 2.2.1 Approaches to Fatigue Analysis ...... 30 2.2.2 Multi-axiality ...... 33 2.2.3 Critical Plane Approaches ...... 35 2.3 Damage Tolerance Investigation ...... 40

2.3.1 Defects as Notches ...... 41 2.3.2 Multi-axial Fatigue Criteria...... 41 2.3.3 Defects as Cracks ...... 43 2.3.4 Murakami’s Approach ...... 45

CHAPTER 3 IN-TRACK BENDING BEHAVIOUR ...... 47 3.1 Finite Element Modeling ...... 47 3.1.1 Weld Geometrical Modelling ...... 47 3.1.2 Track Modelling ...... 48 3.1.3 Loading ...... 50 3.1.4 Seasonal Thermal Load ...... 53 3.1.5 Meshing ...... 54 3.1.6 Track Length Sensitivity Analysis ...... 58 3.1.7 The Effect of Simplification in Contact Pressure ...... 59 3.1.8 Model Validation Using Experimental Results...... 61 3.2 Free Rolling Condition (Tangent Track) ...... 63 3.2.1 Longitudinal Residual Stresses ...... 68 3.3 Contact Patch Lateral Location ...... 69 3.4 Contact Tractions ...... 76 3.5 Track Support ...... 84

CHAPTER 4 MULTI-AXIAL FATIGUE ANALYSIS ...... 87 4.1 Dang Van Original Criterion ...... 87 4.2 Minimum Circumscribed Circle (MCC) ...... 89 4.3 Estimation of Fatigue Parameters ...... 95 4.4 Residual Stresses ...... 99 4.5 Free Rolling Condition (Tangent Track) ...... 102 4.6 Contact Patch Lateral Location ...... 105 4.7 Contact Tractions ...... 107 4.7.1 Contact Patch Eccentricity, Tractions and In-Service Observations ...... 109 4.8 Track Support ...... 111 4.9 Sensitivity to Residual Stresses ...... 113 4.10 Sensitivity to Seasonal Temperature ...... 117

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CHAPTER 5 FATIGUE AND WELD COLLAR DESIGN ...... 120 5.1 Design Alternatives and Geometric Features ...... 120 5.2 Residual Stress Distribution ...... 123 5.3 Performance in Tangent Tracks...... 123 5.4 Contact Patch Lateral Location ...... 127 5.5 Contact Lateral Traction ...... 130 5.6 Track Support Condition ...... 132 5.7 Web Fatigue Behaviour ...... 134

CHAPTER 6 COLD LAP DEFECT ...... 139 6.1 Cold Lap Defect ...... 140 6.2 Analysis of Cold Lap ...... 142 6.2.1 Virtual Crack Closure Technique (VCCT) ...... 144 6.2.2 Mixed-Mode Fracture Criteria ...... 147 6.2.3 Local Stress Intensity Factors ...... 149 6.2.4 Cold Lap Finite Element Model ...... 150 6.2.5 Element Size of the Crack Tip ...... 153 6.3 Influence of Cold Lap on Fatigue Behaviour ...... 154 6.3.1 Sensitivity to Lap Thickness ...... 160 6.3.2 Sensitivity to Lap Unfused Length and Width ...... 162 6.3.3 Sensitivity to Edge Offset ...... 164 6.3.4 Contact Patch Displacement and Lateral Traction ...... 166 6.3.5 Track Support ...... 169 6.3.6 Crack Kinking ...... 170

CHAPTER 7 TOLERANCE TO WEB DEFECTS ...... 173 7.1 Defects as Notches (Multi-axial Fatigue) ...... 174 7.1.1 Spherical Defects ...... 176 7.1.2 Ellipsoidal Defects ...... 181 7.1.3 Coin-Shape Defects ...... 184 7.2 Fracture Mechanics Approach ...... 188

CHAPTER 8 CONCLUSIONS AND FUTURE WORK ...... 194 8.1 Conclusions ...... 194 8.1.1 In-Track Bending Behaviour...... 194

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8.1.2 Fatigue Behaviour ...... 195 8.1.3 Straight Break at Top of the Rail Foot ...... 196 8.1.4 Collar Design ...... 197 8.1.5 HSW and Web Defects ...... 198 8.2 Future Work ...... 200

REFERENCES ...... 202

APPENDIX: Implementation of the Multi-axial Dang Van Criterion in a MATLAB Program ...... 220

LIST OF PUBLICATIONS

1. Salehi, I., Kapoor, A., and Mutton, P., Multi-axial fatigue analysis of aluminothermic rail welds under high axle load conditions. International Journal of Fatigue, 2011. 33 (9): p. 1324-1336.

2. Salehi, I., Mutton, P., and Kapoor, A., The effect of geometric features on multi- axial fatigue behaviour of aluminothermic rail welds. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, 2012. 226 (4): p. 360-370.

3. Salehi, I., Mutton, P., and Kapoor, A., Analysis of straight break formation in aluminothermic rail welds using multi-axial fatigue criterion and fracture mechanics. Journal of Engineering Fracture Mechanics, Accepted, Under Review.

4. Salehi, I., Kapoor, A., Mutton, P.J., and Alserda, J. Improving the reliability of aluminothermic rail welds under high axle load conditions . in Proceedings of the Rail Rejuvenation and Renaissance Conference on Railway Engineering (CORE 2010) . 2010. Wellington, New Zealand, ISBN 978-0-908960-55-2.

5. Salehi, I., Mutton, P., and Kapoor, A., Analysis of damaging factors in thermite welds through multi-axial fatigue criterion , in Proceedings of the International Heavy Haul Association Conference 2011 . June 19-22, 2011, International Heavy Haul Association (IHHA): Calgary, Canada.

6. Salehi, I., Mutton, P., and Kapoor, A., Analysis of straight break formation in thermite rail welds under heavy axle load conditions , Accepted for presentation in International Heavy Haul Association Conference 2013 . Feb 4-6, 2013, International Heavy Haul Association (IHHA): New Delhi, India.

LIST OF FIGURES

Figure 1-1 Rail joining methods: (a) Fish plate and bolting; and (b) Welding 1 Figure 1-2 (a) Section view of ATW process during pouring and solidification; and (b) On-site ATW installation 3 Figure 1-3 Section view of ATW process using single-use crucible 4 Figure 1-4 Failure statistics from September 1997 to October 2000 for SKV-F welds 5 Figure 1-5 Main ATW failure modes in Australian heavy haul railways: (a) Straight break; and (b) Horizontal split web in SKV-F weld 6 Figure 2-1 Histogram for 244 weld failures in a North American railway a) Failure locations; and (b) Cause of failures 13 Figure 2-2 Failure modes (outer layer) and the cause of damage (inner layer) in Japanese railways 13 Figure 2-3 (a) Weld profile (visible regions of the weld section) and; (b) Hardness measurement on different regions of the weld and the illustrated softened region (HAZ) 14 Figure 2-4 Qualitative presentation of P1 and P2 dynamic forces induced by a rail weld surface irregularity; and are wheel-rail dynamic force and time respectively 15 Figure 2-5 Dynamic impact factor versus vehicle speed for a tangent track 16 Figure 2-6 Squats formed on the battered regions of an aluminothermic weld 18 Figure 2-7 Longitudinal section of rail head illustrating transverse defect formation from head check defect in the HAZ of an aluminothermic weld 18 Figure 2-8 Straight break failure in an Australian heavy haul railway: (a) Initiated from upper-foot (top of the foot); and (b) Initiated from the underhead radius (head-web fillet) 20 Figure 2-9 The effect of flank angle and toe radius on the stress concentration of the collar edge 21

Figure 2-10 (a) Cold lap formation at the underhead region of a weld; and (b) Fatigue crack emanated from the apex of the cold lap 22 Figure 2-11 Schematic view of cold lap formation when no leak of molten steel has occurred 23 Figure 2-12 Effect of welding parameters on melt-back depth: (a) Preheating time; and (b) Liquid temperature; the horizontal dotted line is the boundary of cold lap and no cold lap regions 23 Figure 2-13 (a) Horizontal split web fracture; and (b) Fracture face of the weld buttress showing an area of a shrinkage defect 24 Figure 2-14 Vertical residual stress distribution: (a) Flash butt; and (b) Aluminothermic weld 26 Figure 2-15 Predicted threshold crack depth and the results of testing: (a) Compressive loading of the collar web; and (b) Tensile loading; is the vertical residual stress 27 Figure 2-16 (a) Typical HSW crack path and the three points for which stress intensities are calculated; α is the angle of kinking; and (b) Finite element model of the rail and the HSW 28 Figure 2-17 Stress intensity factor range for possible kinking angles and the real crack propagation angle observed in service: (a) Step 1 showing mode II crack growth, (b) Step 2 showing mode II growth; and (c) Step 3 suggesting a Mode I growth 28 Figure 2-18 Fatigue crack nucleation and propagation: (a) Formation of material slip bands under cyclic loading; and (b) Schematic view of fatigue crack growth stages 30 Figure 2-19 Structural response to cyclic loading and the corresponding type of failure: (a) Perfectly elastic (HCF), (b) Elastic shakedown (HCF), (c) Plastic shakedown (LCF); and (d) Ratchetting mechanism (incremental collapse or ratchetting) 33 Figure 2-20 (a) Fatigue crack under pure shear and the resulting interlocking effect; and (b) Effect of normal stress in enhancing fatigue crack growth by reducing the closure effect 38 Figure 2-21 Graphic presentation of Dang Van fatigue criterion in a failure

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condition showing a sample loading or stress path crossing the criterion line during a cycle 40 Figure 2-22 Fatigue damage prediction using Findley criterion (RAHELS model) and comparison with experimental data. Shaded regions show different rail head damages predicted with respect to variable geometrical features of defects 43 Figure 2-23 Schematic of experimental data for notches with different root radius and predictions with notch method (defect modelled as notch) and crack method (defect modelled as crack) 44 Figure 2-24 A semi-elliptical surface crack under reversed pure torsion 46 Figure 3-1 PLK weld, (a) Original weld, (b) Laser scanned geometry; and (c) Model constructed in Solidworks and used in FE analysis 48 Figure 3-2 Schematic of the modelled track with aluminothermic weld at the midway between the two middle sleepers, concrete sleepers, and vertical and horizontal ballast stiffness 49 Figure 3-3 Model of ballast used in the calculation of ballast stiffness. Ballast stiffness is the equivalent stiffness of the two shaded areas 49 Figure 3-4 Track model used in FE analysis; displacement constraint in Y direction is applied to the red colour shaded areas of the sleepers and constraint in X direction is applied to the rail ends 51 Figure 3-5 Schematic of the rail-wheel Hertzian contact and the dimensions used for calculation of the contact patch semi-axis and . (Figure adopted from Iwnicki 52 Figure 3-6 Wheel passage representation in FE model 53 Figure 3-7 Finite elements used is study: (a) SOLID185 for sleepers and rail; and (b) SOLID187 for aluminothermic weld 55 Figure 3-8 Finite element mesh of the structure: (a) Track model, (b) Weld region magnified; and (c) Collar edge magnified 57 Figure 3-9 Track models used for the length sensitivity analysis 58 Figure 3-10 Finite element simulation of real contact between rail and wheel: Equivalent (von-Mises) stress distribution 60

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Figure 3-11 3-point bending experimental setup for measurement of longitudinal and vertical stresses on the weld region 61 Figure 3-12 Comparison of finite element results with strain gauge measurements: (a) Vertical stress on the centerline of the weld collar, (b) Longitudinal stress on the rail surface 5 mm distant from the weld collar edge for central loading, (c) The same stress component for eccentric loading 62 Figure 3-13 Equivalent stress distribution on the weld exterior surface; High stress concentration is observed throughout the collar edge with maximum values at the base region 64 Figure 3-14 Longitudinal stress contour under central loading (tangent track) 65 Figure 3-15 Local bending behaviour of rail head on web and the resulting

longitudinal stress S X under a bending moment M Y. The local stress

SX is superimposed with the longitudinal stress developed by rail section bending behaviour to form the total longitudinal stress at the rail head 66 Figure 3-16 Variation of longitudinal stress at the rail underhead (on a path located 31 mm from rail centreline) under central loading (tangent track) inclusive and exclusive of the seasonal thermal effects: (a) Rail with no weld installed; and (b) Rail with aluminothermic weld. Shaded area shows the rail head length affected by the local bending of the rail head 67 Figure 3-17 Residual stress range and total longitudinal stress range at three locations of the weld suspected for straight break formation 68 Figure 3-18 Possible contact patch locations during vehicle steering in a medium radius left hand curve 69 Figure 3-19 Contact patch locations on the rail running surface for the study of aluminothermic weld stress distribution under eccentric loading 70 Figure 3-20 Longitudinal stress contour at the gauge side of the weld: eccentric loading, contact patch 20 mm offset from rail centerline towards the gauge side 71 Figure 3-21 Developed twisting moment and the resulting deformation 72

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Figure 3-22 Local lateral deformation of the rail head at the location of wheel

load induced by T x: (a) Top view; and (b) Front view 72 Figure 3-23 Variation of longitudinal stress at the rail underhead (on a path located 31 mm from rail centreline) under eccentric loading with different eccentricities 73 Figure 3-24 Strain gauge measurement of the longitudinal stress at the gauge side underhead of a wide gap aluminothermic weld; data for a high rail of a 68 kg/m section in a 918 m radius curve subjected to high axle load conditions. Figure shows several wheel passages 74 Figure 3-25 Contour of vertical stress on the field side of the rail under an eccentric load located 20 mm from the rail centreline towards the gauge side 74 Figure 3-26 Variation of longitudinal stress (L.S.) and vertical stress (V.S.) at different regions of the weld with respect to the load eccentricity 75 Figure 3-27 Longitudinal and lateral tractions acting on the leading and trailing wheelsets of a two-bogie passenger coach in a 1000 m curve. The position of each wheelset is schematically shown with respect to the flange way gap and the value of forces is shown by the length of the corresponding arrows 77 Figure 3-28 Longitudinal stress contour at the field side of the weld for an eccentric load located 15 mm offset from the rail centerline towards the gauge side with a lateral traction directed towards the field side (outward) with a coefficient of 0.4 78 Figure 3-29 Lateral deformation of the rail section under a tractive load applied 15 mm offset from the rail centreline and L/V ratio of 0.4: (a) Inward traction, and (b) Outward traction 79 Figure 3-30 Longitudinal stress contour at the gauge side of the weld for an eccentric load located 15 mm offset from the rail centerline towards the gauge side with a lateral traction directed towards the gauge side (inward) with a coefficient of 0.4 80 Figure 3-31 Variation of longitudinal stress (L.S.) at the underhead, base and upper foot region and vertical stress (V.S) at the mid web with

respect to different coefficients of lateral traction and the direction: (a) Field side; and (b) Gauge side of the weld. All the measurement points are on one side and the dimensions are based on Figure 3-26 81 Figure 3-32 Variation of longitudinal and vertical stresses with respect to the longitudinal traction 83 Figure 3-33 Variation of longitudinal and vertical stresses versus different ballast stiffness under the weld adjacent sleepers: (a) Tangent track (central loading); and (b) Field side in a curved track (eccentric load located 15 mm offset from the rail centreline towards the gauge side including an outward traction coefficient of 0.3) 85 Figure 4-1 Definition of macroscopic and mesoscopic scale and the associated macroscopic stress ( ) and the mesoscopic stress ( ) 88 Figure 4-2 The stress components (acting) on the plane ( ∆) passing through () the material point O subjected to cyclic loading 90 Figure 4-3 Definition of the shear stress amplitude ( ) and mean shear stress ( ): (a) The longest projection method; and (b) The longest chord method 91 Figure 4-4 Definition of the minimum circumscribed circle (MCC); is the time dependent shear stress amplitude ( ) and ( in) −the figure is the maximum amplitude () 92 Figure 4-5 Samples of circles for the combination of vertices of the shear polygon Ψ: (a) Two-point circles; and (b) Three-point circles 94 Figure 4-6 Tensile properties of weld material with respect to the hardness; red line represents the trend line for the mean values of the ultimate strength 97 Figure 4-7 Hardness distribution on the central longitudinal plane of the rail in head, web and foot regions of the weld versus longitudinal distance from the weld centreline 98 Figure 4-8 Trepanning technique for residual stress measurements and the measurement locations 100 Figure 4-9 Residual stress measurements using strain gauge method and the piece-wise linear models applied in analysis (Mode 1 horizontal

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and Mode 2 vertical residual stresses are considered for fatigue analysis and other modes are implemented for residual stress sensitivity analysis as will be described later): (a) Longitudinal residual stress at the collar edge; and (b) Vertical residual stress on the centerline of the weld collar surface 101 Figure 4-10 DV damage parameter on collar edge versus height from rail foot in a tangent track 102 Figure 4-11 Formation of cold lap at the underhead radius on both field and gauge sides of an AT weld indicated by the white arrows. The cold lap can sometimes contain pores as magnified at the bottom left figure. Pores at the tip of a cold lap could facilitate fatigue crack initiation 103 Figure 4-12 Variation of shear stress amplitude , hydrostatic stress and the DV damage parameter on the (most) damaging plane at () underhead and top of the rail foot versus longitudinal distance of the axle load from the weld centreline; maximum DV damage values at the underhead radius and upperfoot are depicted by small circles 104 Figure 4-13 DV damage parameter at the gauge side collar edge for both central and eccentric loading located 25 mm distant from rail centerline towards the gauge side with no shear tractions; Fatigue region represents the region in which fatigue crack initiation is expected 106 Figure 4-14 DV damage parameter versus the contact patch lateral displacement for the underhead radius, top of the rail foot and base fillet depicted by circles in figure 4-13; Shaded area shows the fatigue region where crack initiation is expected 107 Figure 4-15 DV damage parameter at the collar edge of the field side for an eccentric load located 15 mm offset from the rail centerline with lateral traction coefficients of 0 and 0.4 outwards; shaded area shows the fatigue crack initiation region 108 Figure 4-16 DV damage parameter versus the lateral traction coefficient for the underhead radius, top of the rail foot and base fillet depicted by

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circles in figure 4-13; contact patch located 15 mm offset from the rail centerline towards the gauge side 109 Figure 4-17 Straight break in the considered AT weld initiated at the underhead radius 110 Figure 4-18 DV damage parameter for two longitudinal traction coefficients in a tangent track 110 Figure 4-19 DV damage parameter at the collar edge versus ballast horizontal stiffness (HS) and vertical stiffness (VS) in a tangent track 112 Figure 4-20 DV damage parameter at the collar edge versus different longitudinal (L) and vertical (V) residual stress modes depicted in figure 4-9 for an eccentric load located 25 mm from rail centerline towards the gauge side 114 Figure 4-21 (a) DV damage parameter on the critical plane at the underhead and upper foot regions versus the relative location of axle load with respect to the weld centerline for different longitudinal residual stress (LRS) values; vertical solid lines define the location of axle load when the damage parameter at the mentioned regions achieves its highest value; and (b) Value of shear stress amplitude, hydrostatic stress and damage parameter versus LRS 115 Figure 4-22 Application of ultrasonic impact treatment on collar edge of an aluminothermic weld using the specialized hand tool 116 Figure 4-23 Damage parameter at the collar edge in a tangent track for three rail temperatures: neutral temp=35 oC, 17 oC above neutral temp=52 oC, and 16 oC below neutral temp=19 oC 117 Figure 4-24 Variation of shear stress amplitude, hydrostatic stress and DV damage parameter on the most damaging shear plane at the underhead and top side of the foot (depicted by circles in figure 4- 23) for different rail temperatures; vertical solid line illustrates the neutral temperature 118 Figure 5-1 Two weld collar designs under investigation: (a) Type A sample weld; (b) Type B sample weld; (c) Type A computer model; (d) Type B computer model; and (e) Section view from top of the rail

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foot indicating the toe radius and flank angle in the two collar designs 121 Figure 5-2 Finite element mesh of the Type B weld with the collar edge of the upperfoot magnified at the bottom right figure 122 Figure 5-3 Residual stress measurements using trepanning and strain gauge method and the piece-wise linear model applied in fatigue analysis: (a) Longitudinal residual stress at 3 mm offset from the collar edge; and (b) Vertical residual stress measured on the surface of the collar mid-web 124 Figure 5-4 Longitudinal stress contour for a tangent track when the load is located at the centerline of the weld: (a) Type A weld; and (b) Type B weld 125 Figure 5-5 DV damage parameter versus height above the rail foot at the collar edge of the two weld types in a tangent track 126 Figure 5-6 Collar shape with the flank angle at the underhead region of the two weld types 127 Figure 5-7 Longitudinal stress contour on the gauge side for a curved track with contact patch eccentricity of 25 mm with no tractions: (a) Type A weld; and (b) Type B weld 128 Figure 5-8 Variation of longitudinal stress at the underhead radius (at 31mm from rail centerline) of the gauge side under central loading (tangent track) and eccentric load located 25mm from rail centerline towards the gauge side 128 Figure 5-9 DV damage parameter on the collar edge versus height above rail foot for a contact patch located 25mm offset from rail centerline towards the gauge side 129 Figure 5-10 Longitudinal stress contour at the field side of the weld for an eccentric load located 15 mm from the rail centerline with lateral traction coefficient of 0.4: (a) Type A; and (b) Type B 130 Figure 5-11 Variation of longitudinal stress at the under head and base fillet of the two weld types under different lateral traction coefficients 131 Figure 5-12 DV damage parameter at the collar edge of the two weld types for

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an eccentric load located 15mm from the rail centerline with lateral traction coefficient of 0.4 towards the field side 132 Figure 5-13 Damage parameter at the collar edge versus ballast vertical and horizontal stiffness for a tangent track (a) Type A; and (b) Type B 133 Figure 5-14 Vertical stress contour on the field side of the weld for an eccentric load located 25mm from the rail centerline with no lateral traction: (a) Type A weld, (b) Type B weld; and (c) Section view of the collar at the location of maximum bending stress and the effective bending moment 135 Figure 5-15 Vertical stress versus height from rail base and the contribution of residual stresses for an eccentric load located 25 mm from the rail centerline 136 Figure 5-16 DV damage parameter on the centerline of the web surface for an eccentric load located 25 mm from the rail centerline 137 Figure 6-1 Fatigue crack propagation at top of the rail foot underneath the associated cold lap defects in three weld samples; arrows show the fatigue crack propagation regions 140 Figure 6-2 Section view from the top of the rail foot illustrating two types of cold lap defect: (a) Leaking of weld material and formation of an unfused appendix (finning); and (b) Lack of fusion on the rail surface inside collar boundaries (cold lap) 141 Figure 6-3 The unfused surfaces of a cold lap and its apex may resemble the faces and tip of an existing crack which can propagate under fatigue loading and form a straight break fracture 143 Figure 6-4 Crack closure technique (CCT) using two step finite element simulations: (a) Step one, original crack (separation forces are calculated at ); and (b) Step two, crack is grown and the displacements at are obtained 145 Figure 6-5 Virtual crack closure technique (VCCT or MCCT) for 8-node brick elements 146 Figure 6-6 Crack deflection angles and for mixed-mode crack problem 148

Figure 6-7 Kinking at a crack tip and the local stress intensity factors (a) Original crack; and (b) Crack with a kink at the tip 150 Figure 6-8 Cold lap model as a simple rectangular parallelepiped and its geometric features 151 Figure 6-9 Finite element mesh and the refined block at the crack tip 152 Figure 6-10 Equivalent (von Mises) stress in the cold lap region under a central load (tangent track) and the lap cross section showing crack mouth opening 155 Figure 6-11 Variation of stress intensity factors and the resulting stress intensity factor ranges at the middle point of the cold lap during one wheel passage (loading cycle) 156 Figure 6-12 Variation of energy release rate (G) with respect to the kink angle. Maximum energy release rate corresponds to the kink angle of 55 o 159 Figure 6-13 Variation of stress intensity factor range at the tip of a kink with respect to the kink angle. Note that the kink angle predicted using different criteria is consistent to the maximum 160 Figure 6-14 Variation of stress intensity factors and the resulting∆ stress intensity factor ranges at the middle point of the cold lap for two lap thickness values 161 Figure 6-15 Equivalent stress intensity factor range at the middle point of the lap versus the cold lap thickness. The two threshold values are included in the figure 162 Figure 6-16 Equivalent stress intensity factor range at the middle point of the lap with respect to lap unfused length 163 Figure 6-17 Equivalent stress intensity factor range at the middle point of the lap with respect to lap width 164 Figure 6-18 Variation of stress intensity factor ranges and the equivalent parameter with respect to the distance of the lap tip from the collar edge. Negative values relate to lap tips inside the collar while positive values imply lap tips located outside 165 Figure 6-19 Variation of separated stress intensity factor ranges as well as the equivalent value with respect to the contact patch eccentricity from

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the centerline towards the gauge side for the lap located on the field and gauge side of the weld 166 Figure 6-20 Variation of , , and with respect to lateral traction coefficient∆ ∆ of both∆ inward∆ and outward directions for a lap located on the field side of the weld 168 Figure 6-21 Variation of , , and with respect to lateral traction coefficient∆ ∆ of both∆ inward∆ and outward directions for a lap located on the gauge side of the weld 168 Figure 6-22 Variation of , , and with respect to ballast vertical stiffness∆ in∆ a tangent∆ track∆ (central loading) 170 Figure 6-23 Variation of , , and at the tip of an existing kink with respect to∆ the ∆ kink ∆length ∆ 171 Figure 6-24 Straight break at the top of the rail foot associated with a cold lap defect with high content of porosity at its apex 172 Figure 7-1 Variation of the DV damage parameter on the web centerline of the Type A and Type B welds and the corresponding locations of the maximum damage where the defects are modelled 175 Figure 7-2 Defect models used in the damage tolerance analysis: (a) Spherical defect; (b) Ellipsoidal defect; and (c) Coin-shape defect 175 Figure 7-3 Finite element mesh of the 3 mm radius spherical defect 176 Figure 7-4 Vertical stress contour on a spherical defect when the contact load is located exactly at the top of the weld: (a) R=0.5 mm Type A weld, (b) R=0.5 mm Type B, (c) R=4 mm Type A; and (d) R=4 mm Type B 178

Figure 7-5 Variation of vertical stress ( σz) at two points on the surface of the defect with respect to the defect radius: point D at the deepest point of the defect and Point S at the intersection of the defect and collar web surfaces 179 Figure 7-6 Lines of force on the surface of the collar around a spherical defect and the location of stress concentration: (a) Type A weld; and (b) Type B weld 179

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Figure 7-7 Variation of DV damage parameter at two points on the surface of the spherical defect with respect to the defect radius 180 Figure 7-8 Equation of the considered ellipsoidal defect with conditions: b=c and a=cte 181 Figure 7-9 Vertical stress contour on an ellipsoidal defect (a) b=0.25 mm Type A weld, (b) b=0.25 mm Type B, (c) b=1.25 mm Type A; and (d) b=1.25 mm Type B 182

Figure 7-10 Variation of vertical stress ( σz) at two points on the surface of the defect with respect to the semi-minor axis b 183 Figure 7-11 Variation of DV damage parameter at two points on the surface of the defect with respect to the semi-minor axis b 183 Figure 7-12 Geometric parameters of the coin-shape defect considered in this study (w is variable) 184 Figure 7-13 Vertical stress contour on a coin-shape defect: (a) w=1 mm Type A weld, (b) w=1 mm Type B, (c) w=6 mm Type A; and (d) w=6 mm Type B weld 185 Figure 7-14 Variation of the equivalent (von Mises) stress at two points on the surface of the coin-shape defect with respect to the defect width (w) 186 Figure 7-15 Determination of relationship between hardness and yield strength of rail welds 187 Figure 7-16 Variation of the DV damage parameter at two points on the surface of the coin-shape defect with respect to the defect width (w) 188 Figure 7-17 Finite element mesh of the crack block and the application of radial meshing suitable for virtual crack closure technique 189 Figure 7-18 Vertical stress contour on the crack block and the lower crack face when contact load is located exactly at the top of the weld: (a) R=5 mm Type A weld, (b) R=5 mm Type B weld 190 Figure 7-19 Variation of equivalent stress intensity factor range with respect to the crack radius: point D at the deepest point of the crack and Point S at the intersection of the crack front and the collar surface 191

Figure 7-20 Variation of mode I stress intensity factor ( KI) with respect to the crack radius 192

XXIII

LIST OF TABLES

Table 3-1 Summary of the finite element model parameters 54 Table 3-2 Stress values in two locations of the weld collar with respect to element size for the case of a vertical load located at the centreline of the weld 56 Table 3-3 Stress values in three regions of the weld versus the track length modelled 58 Table 3-4 Stress values in three regions of the weld versus the contact pressure distribution 60 Table 6-1 Summary of the finite element model parameters 153 Table 6-2 Crack tip stress intensity factors versus the element size at the crack tip block 154

XXIV

LIST OF NOTATIONS AND ACRONYMS

Chapter 1

CWR Continuous Welded Rail ATW Aluminothermic Welding FBW Flash Butt Welding GPW Gas Pressure Welding EAW Enclosed Arc Welding HSW Horizontal Split Web

Chapter 2

HAZ Heat Affected Zone Dynamic wheel load Static wheel load Depth of weld dip Unsprung mass per wheel Vehicle speed Wavelength of the track irregularity RCF Rolling Contact Fatigue Fracture toughness Fatigue notch factor Web or base thickness Weld flank angle Weld toe radius Depth of the surface roughness at the weld toe root Tensile strength of the toe root material Threshold crack depth Threshold stress intensity factor range ∆ Crack shape factor

XXV

Stress range ∆ Ratio of minimum to maximum total vertical stress Vertical residual stress Stress intensity factor range under mode I loading ∆ Stress intensity factor range under mode II loading HCF∆ High Cycle Fatigue LCF Low Cycle Fatigue Fatigue strength of material Number of cycles to failure under a reversed loading True stress amplitude Slope of S-N line Alternating applied stress Mean applied stress Fully reversed fatigue strength Plastic strain amplitude Total strain amplitude Elastic strain amplitude Modulus of elasticity Critical strain in ratchetting Cyclic plastic strain accumulated in each loading cycle ∆ Equivalent stress amplitude Principal stress amplitudes , , Equivalent mean stress Principal mean stresses , ,, Mean stresses in x,y,z directions Poisson’s ratio , , Principal alternating strains Equivalent alternating strain Maximum range of shear stress on the shear plane ∆ Maximum normal stress acting on the shear plane , Fatigue limit in alternate bending

XXVI

Fatigue limit in reversed pure torsion Maximum shear strain amplitude ∆ ⁄ 2 Normal strain range on the shear plane ∆ Yield strength of material Normal stress amplitude on the critical plane , Shear stress amplitude on the critical plane , Hydrostatic stress amplitude , Mean normal stress on the critical plane , Dang Van damage parameter Time dependent shear stress amplitude on the shear plane () Time dependent hydrostatic stress () Constant in Dang Van criterion Constant in Dang Van criterion Stress concentration factor Amplitude of the second invariant of the deviatoric stress tensor , Maximum value of the hydrostatic stress including gradient effect ∗ Gradient of hydrostatic stress Maximum hydrostatic stress in a load cycle Threshold stress intensity factor Square root of the crack area √ Vickers hardness Fatigue limit of a defective material ∆ and Dimensions of semi-elliptical surface crack Chapter 3

Sleeper sitting width Effective support length of the sleeper per rail Sleeper span Ballast thickness ℎ Ballast stress distribution angle Ballast modulus of elasticity

XXVII

Vertical applied load Rolling radius of the wheel Transverse radius of the wheel Transverse radius of the rail and Contact patch semi axes in x and y directions Mean contact pressure Contact Pressure ( , ) Maximum contact pressure Vertical wheel load MBS Multi Body System L/V Ratio of the lateral to vertical wheel load

Chapter 4

Deviatoric part of the mesoscopic stress () Deviatoric part of the macroscopic stress ( ) Deviatoric part of the residual stress ∗ Second invariant of the deviatoric stress tensor Instantaneous value of the maximum mesoscopic Tresca shear stress () Instantaneous mesoscopic hydrostatic stress MCC() Minimum Circumscribed Circle Total stress vector Normal stress vector on the shear plane and Shear stress vector on the shear plane and () Shear stress amplitude vector () Mean shear stress vector Number of vertices of the shear path polygon Number of circles formed by two vertices of the shear path polygon Number of circles formed by three vertices of the shear path polygon Brinell hardness Ultimate tensile strength UIT Ultrasonic Impact Treatment

XXVIII

Chapter 6

LEFM Linear Elastic Fracture Mechanics COD Crack Opening Displacement VCCT Virtual Crack Closure Technique MCCT Modified Crack Closure Technique Crack length Crack extension ∆, , Separated energy release rates Crack front element width , , Crack tip Stress intensity factors Shear modulus Equivalent stress intensity factor and Crack kinking angles Equivalent stress intensity factor range ∆MERR Maximum Energy Release Rate

Chapter 7

Dynamic fracture toughness

XXIX

CHAPTER 1 INTRODUCTION

1.1 Continuous Welded Rail

It is a long time since continuous welded rail (CWR) has replaced the traditional bolting method to join rail sections. Previously, fixed lengths of rails were joined together by bolting the rail ends using metal perforated fish plates or joint bars (Figure 1-1a). This method incorporated small gaps to accommodate expansion of the rails in hot seasons and because of this gap the passage of train induced impact forces on the joint causing different failures on the rail ends, joint components and railway track. Nowadays, continuous rails are produced by welding the rail sections using different welding techniques to form a continuous welded rail which may extend several kilometres in length (Figure 1-1b). The history of CWR goes back to the early 20 th century with the construction of 7,000 metres of continuous rail by Krefeld railway in Germany in 1924 [1]. Since 1950’s this method has become common especially in main line applications.

(a) (b)

Figure 1-1: Rail joining methods: (a) Fish plate and bolting; and (b) Welding

CWR has resulted in several improvements in the performance of railways thanks to the elimination of the expansion gap and the associated rail surface discontinuity. The benefits attributed to the CWR include reduced maintenance requirements as a result of lower bolt hole and fish plate failures, lower impact forces on railway due to better

1 integrity, reduced track deterioration, better wear performance and generally increasing rail lives. CWR has also largely contributed to ride quality enhancement and passenger comfort.

Flash butt welding (FBW) and Aluminothermic welding (ATW) are the most common welding methods used all over the world. Other alternatives such as gas pressure welding (GPW) and enclosed arc welding (EAW) are also used, however to a lesser extent [2-4]. FBW incorporates an automatic welding machine in which the rail ends are heated to fusion temperature by running a strong electric current through the high resistance contact of the rail ends as they periodically touch each other. The rails are then forged together with high pressure forming a strong bond. As the portability of the equipment is restricted the FBW is the preferred method to weld rail sections in welding plants prior to transportation to track construction site. However with the development and more widespread availability of mobile flash butt welding equipment, flash butt welding is increasingly used in place of aluminothermic welding for in-track welding.

1.2 Aluminothermic Weld

The invention of thermite process and aluminothermic weld is considered a milestone in the development of continuously welded rail. The thermite reaction and its applications were first patented by German chemist Hans Goldschmidt in 1895 when he was investigating the reaction of metal oxides with aluminium powder [5-7]. The thermite reaction is an exothermic process in which large amount of thermal energy is released; the quantity sufficient to provide molten metal for casting and welding processes. The thermite process associated with ATW incorporates reaction between fine aluminium and iron oxide powders in a crucible. The following reactions are widely used in the ATW processes [1]:

Fe 2O3 + 2Al 2Fe + Al 2O3 + 181.5 Kcal (1-1)

3Fe 2O3 + 8Al 9Fe + 4Al 2O3 + 719.3 Kcal (1-2)

The practical temperature achieved by the second reaction is about 1930 oC, which is well above the melting point of iron. After the abovementioned reaction is complete and the slag (mainly aluminium oxide) has floated on the top of the crucible, the molten steel is poured (through an automatic tap at the underneath of the crucible) in a two or a three piece hardened sand mould which has already been clamped and sealed around the two rail ends. The rail ends which have been heated through a preheating stage (a pre- reaction stage in which the rail ends are heated by gas torches) are partially melted in presence of the hot molten steel and fuse with it forming the complete weld (Figure 1- 2). Afterwards, the setup is allowed to cool down, the moulds are then removed, the risers are trimmed and the weld head is finally ground to the consistent rail head profile. All the above-mentioned procedures are performed in a timely manner and according to certain guidelines to ensure consistent quality of the produced welds [8].

Variations of the ATW process include different preheating duration, post-weld cooling condition, portion (thermite mixture) hardness, weld gap width and weld collar design (reinforcement shape design), and more recently, single-use crucibles (Figure 1-3). In any case, the applicability of a new AT weld (process and consumables) must be approved by strict guidelines and performance tests according to adopted railway and welding standards such as AS1085-Part 20 [9] or BS-EN 14730 [10].

(a) (b)

Figure 1-2: (a) Section view of ATW process during pouring and solidification [11]; and (b) On-site ATW installation

Figure 1-3: Section view of ATW process using single-use crucible [11]

Although, the ATW process is quite old and largely operator dependent it is the preferred method for on-site applications due to the low cost of equipments and consumables, fast installation and lower delay in normal railway operation, portability of equipment and the possibility to weld different rail sections together. ATW is mainly used for the following applications:

• Joining new rails on site as part of the railway construction • Replacing defective or broken rails as a maintenance procedure • Installation of insulated rail joints • Renewal of railway crossings

1.3 Aluminothermic Weld as a Major Source of Failure

Despite the main role as a welding procedure and its operational preference, ATW is considered a manual cast welding process and as a result suffers from the variations of the produced weld quality. ATW processes involve several stages such as rail end alignment, gap adjustment, mould installation, sealing and preheating in which the operator plays an important role and any departure from the guidelines could result in a defective weld. In the meantime, since ATW is a casting process the fused material features coarser microstructure (columnar grains with predominantly pearlitic microstructure) and sometimes includes traces of non-metallic inclusions and porosity which are contributing factors to lower ductility and fracture toughness of the ATWs

4 compared to the parent rail. In general ATW is considered a discontinuity in rail line due to the following factors [12]:

• Inconsistency of the weld microstructure and material properties (static and fatigue strength, hardness, ductility, etc.) to the rail material. • Presence of high residual stresses as a result of welding process and their difference with those of the parent rail. • Difference in geometrical shape and dimensions of the ATW collar from the parent rail section. • Propensity of the ATW process to produce cast related defects such as inclusions, porosity, shrinkage cracks and hot tears and their possible contribution to fatigue and premature failure of ATWs.

Previous studies have shown that field welded ATWs have been frequent sources of previous failures in Australian heavy haul railway systems [13-15]. One investigation shows that for a period of 18 months prior to June 2001 failures in one particular type of aluminothermic weld resulted in approximately 75% of all broken rail reports for the Newman mainline of the BHP Billiton Iron Ore railway system, and the majority of these failures occurred in rail welds which were less than 6 months old [15].

Figure 1-4: Failure statistics from September 1997 to October 2000 for SKV-F welds [15]

Figure 1-4 illustrates the number of failed welds and the failure rate with respect to the service life of the installed welds. Around 70% of the installed welds failed in the early stage of their service life (less than 3 months) while not many welds survived more than 3 years in service. Most of the failures were due to straight breaks (vertical or near vertical fracture of the weld) and horizontal split webs (horizontal fracture at the mid- web region) associated with shrinkage and other large defects and inclusions at the web region (Figure 1-5).

Figure 1-5: Main ATW failure modes in Australian heavy haul railways: (a) Straight break; and (b) Horizontal split web in SKV-F weld [16]

Another study performed on North American railroad infrastructure implies that around 40% of all failures in a class I railroad has been due to field welded ATWs with around 90% of the failures initiated at the base or web regions [17]. It is worth mentioning that railhead defects may be more frequent than base or web-initiated failures but rail head fatigue cracks are less likely to cause any broken rails as the defect can be readily detected through standard NDT techniques such as ultrasonic and X-ray inspections and removed before they result in total fracture. Similar statistical study in Japanese railways shows that out of 121 damaged welds (comprising four different welding techniques) in 17 years prior to 2002, 43% has been associated to ATWs [18].

The information provided on the failure rates proves the importance of ATW as a key maintenance and safety issue. ATW has been under constant improvements since its invention in 1895 and because of that the current failure rates have dropped

6 significantly compared to early installations. However, by development of the new high strength hypereutectoid and bainitic rail materials there is a big challenge to keep ATW lives compatible to those of the parent rails in which they are installed. Thorough engineering investigations and improvements are still needed especially if ATW aims to remain the preferred on-site welding method for higher speed and axle load demands of the railway industry.

1.4 Aims and Objectives

As previously mentioned, straight breaks and horizontal split webs are the most common ATW failure modes in Australian heavy haul railway system. Straight breaks initiate from stress concentration sites at the edge of the weld collar, at the foot, lower web and underhead regions, and propagate in vertical direction under Mode I loading. On the other hand, horizontal split web failures involve the development of a horizontal crack which initiates from a surface or near-surface defect at the weld collar, generally in the mid or upper-web region. Understanding the mechanism and the contributing factors in formation of these failures is vital to prevent their occurrence and before any modifications of the process design could be made.

‹ Bending and Fatigue Behaviour

The main objective of this study is to investigate the fatigue behaviour of the welded rail section with respect to the abovementioned failure modes under heavy axle load conditions. The risk of fatigue crack initiation and the associated critical locations of the weld collar (particularly with respect to straight break formation) are analyzed and the effect of the following operational and process parameters is quantified:

• Track curving or hunting through variable wheel–rail contact patch location and introduction of wheel lateral tractions • Longitudinal and vertical residual stress distribution as a result of welding process, preheating and post-weld cooling conditions • Seasonal-dependent stresses arising from the difference between the current rail temperature and the temperature at the time of weld installation

• Unsupported or weakly supported sleepers in the vicinity of the weld as a result of track deterioration processes arising from rail surface irregularities

‹ Collar Design

Amongst different metallurgical and mechanical parameters important in initiation and propagation of these failures, geometrical design of the collar is a determining factor in the fatigue performance and service life of ATW. Part of the study is devoted to the analysis of geometrical features (flank angle and toe radius) particularly at the edge of the weld collar (which is responsible for or assistive in straight break formation). Two geometrically different aluminothermic welds, one of which is widely used in Australian heavy haul railways and the other one recently developed, are investigated in terms of fatigue crack initiation risk. The role of the collar shape at the mid web region in mitigating or increasing the risk of horizontal split web formation is also studied.

‹ Straight Break at Top of the Rail Foot

Beside the role of irregularities of the collar shape, analysis of the fractured welds have shown that a specific type of defect known as cold lap is responsible for the formation of straight breaks at the upper foot region close to the collar edge. Cold lap is a condition where the weld metal seeps into the gap between the mould and the parent rail and solidifies forming an unfused metallic appendix. One of the objectives of this study is to investigate whether and under which conditions fatigue crack could grow from the tip of a cold lap and lead to a straight break failure.

‹ HSW and Web Defects

A further chapter is devoted to the role of specific defects in formation of horizontal split webs at the critical locations of the two above mentioned ATWs. The study focuses on large pores (as the most detrimental defect) with predefined geometries and variable dimensions for example semi-spherical, ellipsoidal and coin-shape surface defects. This study is considered in the context of damage tolerance analysis through which the tolerance of alternative collar shapes to the existing welding defects at the web region could be evaluated.

1.5 Methodology

Prior to any numerical investigation of fatigue crack initiation, the cyclic stress history of the weld section under service loading must be determined. Accordingly, a thermo- structural finite element analysis is performed in ANSYS 12.0 package as a preliminary stage to provide stress history for fatigue analysis and to facilitate the interaction of service and seasonal-dependent thermal stresses. A specific length of railway track comprising concrete sleepers and elastic foundation in both vertical and longitudinal directions is modelled and the passage of wheel is considered through longitudinal movement of the associated contact patch and pressure.

Analysis of fatigue behaviour is performed using the calculated stress histories combining the effect of contact, bending, seasonal-induced thermal and the weld residual stresses. Due to the complexity of the stress state, out of phase nature of the stress components and rotating principal stresses it is necessary to use a multi-axial fatigue criterion. The shear based Dang Van multi-axial criterion based on the concept of critical plane is exploited through a fatigue code developed in this study using MATLAB software. The effect of the previously mentioned service and process conditions are also examined using this methodology.

Damage tolerance analysis is performed using two techniques each of which is applicable to certain type of defects. For large predefined defects (spherical, ellipsoidal and coin-shape) considered on the mid web surface, fatigue analysis is performed through finite element analysis of the stress distribution history on the surface of the defect. The risk of crack initiation is subsequently determined using the Dang Van multi-axial fatigue criterion. For some types of defects like coin-shape and cold lap, the method ceases to be valid since the stresses at the critical locations exceed elastic limits. In these conditions, damage tolerance analysis is performed using linear elastic fracture mechanics considering the presumed defects as equivalent cracks. Virtual crack closure technique in conjunction with finite element method is used to extract stress intensity ranges and determine the risk of crack propagation.

1.6 Thesis Structure

This thesis is organized in eight chapters the content of which are summarized as follows:

Chapter 2, Literature review: the main ATW failure mechanisms, contributing factors and related background and studies are reported in this chapter. Subsequently, the methods meant to be used in fatigue analysis and damage tolerance investigation are described and the related literature review is provided.

Chapter 3, In-track bending behaviour: in this chapter the bending behaviour of ATW and related stress distributions are determined using finite element simulation. Model construction, meshing, loading and model validation are described and the effect of different service conditions such as tangent, curved tracks and track support at the vicinity of ATW are presented.

Chapter 4, Multi-axial fatigue analysis: fatigue behaviour of ATW is investigated using the critical plane criterion. The approach to estimate material fatigue parameters and the implementation of residual stresses in the fatigue code are described. The effect of operational conditions on the fatigue behaviour is quantified and sensitivity analyses on the seasonal temperature, residual stresses and weld support condition are performed.

Chapter 5, Fatigue and weld collar design: the effect of collar geometrical features on multi-axial fatigue behaviour of ATW is investigated. Two geometrically different welds are considered and their performance under variable service conditions is quantified. Eventually, the role of reinforcement design at the web region in crack initiation locus and fatigue damage value is reported.

Chapter 6, Cold lap defect: damage tolerance analysis is performed on the crack initiation risk at the tip of the cold lap defect through crack FE modeling and applications of linear elastic fracture mechanics. Sensitivity analysis is performed on cold lap geometrical dimensions and some service conditions.

Chapter 7, Tolerance to web defects: the influence of predefined surface defects on fatigue behaviour of web region is investigated. The results of two approaches using the multi-axial fatigue analysis of defect surface and linear elastic fracture mechanics are presented.

Chapter 8, Conclusions and future work: the core objectives and methodology of the study are summarized and a conclusion on the main results is presented. The limitations of the current study are pointed out and recommendations are made for future works.

CHAPTER 2 LITERATURE REVIEW

2.1 Aluminothermic Weld Failure Mechanisms

The variability of the produced weld quality due to excessive operator dependability and the cast like nature of the process bring about increased deterioration rate compared to that of the parent rail. The study of ATW failure modes is critical both to the characterization of weld performance and to the rail lives since some of the failure modes indirectly affect the adjacent parent rails.

A number of studies have been performed on the frequency of ATW failure incidents, their locations and main initiators. Lawrence [17] indicates that most of the failures (on a North American railway) have resulted from fatigue cracking of the base, web-base fillet and web locations and attributes them to the weld defects such as cold laps, slag, porosity and hot tears (Figure 2-1). Terashita [18] classifies the type and cause of the failures (on Japanese railways) in a pie chart graph indicating the transverse fissure (straight break) as the most prominent type of failure and the lack of fusion and shrinkage defect as the main cause of this failure mode (Figure 2-2). However, the proportion of failures and the main reasons largely depend on the type of ATW process, its design and also the type and location of railway in which the weld is installed. In fact, one might find a specific type of ATW failure mode as a maintenance issue in a railway system while no such failure is observed elsewhere.

In this chapter, the main ATW failure modes in Australian heavy haul railways are categorized in four sections according to the location and type of failure. A description is provided for each failure mode, the cause and contribution factors and the pertinent previous studies. Some of the failure modes such as local plastic deformation of the rail running surface, rolling contact fatigue, wear and horizontal split webs are common between ATW and FBW (flash butt weld) and so the studies related to either of these welds will be mentioned briefly.

Figure 2-1: Histogram for 244 weld failures in a North American railway a) Failure locations; and (b) Cause of failures [17]

Figure 2-2: Failure modes (outer layer) and the cause of damage (inner layer) in Japanese railways [18]

2.1.1 Local Plastic Deformation (Batter)

In aluminothermic welding process the rail ends are subjected to tremendous heat input from the preheating stage and the molten metal. As a consequence of this thermal energy, the parent rail material softens and undergoes microstructural changes which lead to lower hardness levels compared to the unaffected regions of the rail. This thermally affected region is called heat affected zone (HAZ) and is an important characteristic in terms of the performance of ATWs. Figures 2-3a and 2-3b illustrate the section view (weld profile) of an ATW and the hardness measurement at different regions of the weld. The width of the HAZ region and its hardness distribution depend on the process parameters such as the weld gap, heat input during preheat stage and more importantly the weld portion hardness which is selected to match the hardness of the parent rail.

(a) (b)

Figure 2-3: (a) Weld profile (visible regions of the weld section) and; (b) Hardness measurement on different regions of the weld and the illustrated softened region (HAZ) [19]

As the new weld is exposed to repetitive service loading the softened region depicted in Figure 2-3b is deformed plastically and as a result, local dipping (batter) forms at the weld running surface. The geometrical irregularities (due to batter and sometimes rail head misalignments) on the surface of the weld cause impact loading on the weld and the track structure underneath and contribute to their failure. The surface irregularities of the weld could also affect the adjacent parent rail in the form of corrugations which

14 develop as a result of dynamic impact loads through a combination of cyclic plastic deformations (ratchetting) and wear mechanisms of the rail running surface [20-21].

A number of studies have been performed to quantify the impact forces and accordingly characterize the weld surface irregularities and their influence on the weld and railway supporting components [22-27]. According to Jenkins et al. [23] the impact forces arising from the surface irregularities has two peaks defined as P1 and P2 which represent the high frequency and low frequency responses of the track (Figure 2-4). The first peak (P1) occurs in about half a milliseconds after the wheel passage and is attributed to the instantaneous response of the rail and sleeper inertia. This force mainly affects the rail wheel contact region (responsible for weld batter) and is damped by the rail and sleepers before it can reach the track bedding. However, a second peak is also observed after several milliseconds and is related to the delayed response of the wheel set unsprung mass. The P2 peak has a high energy compared to P1 (as it has a larger wavelength) and is the main cause of ballast deterioration since it magnifies the sleeper loads and penetrates into the track bedding.

Figure 2-4: Qualitative presentation of P1 and P2 dynamic forces induced by a rail weld surface irregularity; and are wheel-rail dynamic force and time respectively [26] Perhaps the most applicable methods for design engineers to calculate the impact force components (as mentioned previously) are the ones proposed by Jenkins et al. [23] and Frederick [24] which are based on simplified theoretical relations. For instance,

Frederick estimates the influence of irregularities through the definition of a total dynamic force:

(2-1) 60 ( )/ Where is the dynamic wheel load (N), is the static wheel load at weld (N), is the depth of weld dip (m), is the unsprung mass per wheel (kg), is the vehicle speed (m/s) and L is the wavelength of the track irregularity (m). However, for a better understanding of the impact load and the contributing factors the more rigorous vehicle- track dynamic modellings such as the ones performed by Steenbergen [25] and Ishida [27] seems necessary. Recently, a sophisticated model has been developed by Wen [28] in which the vehicle-track dynamic interaction model is combined with finite element analysis of the plastic deformation (batter formation) at the weld running surface.

The results achieved by strain gauging of the rail and measurement of the dynamic force however indicate that under heavy axle load conditions an amplification factor (coefficient by which the static wheel load is magnified) of 1.2 to 1.5 occurs at welds with about 0.5 mm dip depth supported by concrete sleepers [29]. According to Mutton and Alvarez [15] there is a near linear relation between the dynamic impact factor (amplification factor) and the vehicle speed for both the dipped and peaked welds and it generally varies between 1.1 to 1.6 (Figure 2-5).

Figure 2-5: Dynamic impact factor with respect to vehicle speed for a tangent track [15]

2.1.2 Rolling Contact Fatigue and Wear

Since the rail running surface is under repetitive service loading with high contact stresses, incremental plastic deformation (under ratchetting mechanisms due to exceedance of shakedown limits [30]) develop in the rail material. Plastic (shear) strain is accumulated until the ductility of material is exhausted and the material fails to support further deformation. The failed material either is detached from the surface in the form of wear debris or develops micro-cracks which subsequently propagate and branch into material forming rolling contact fatigue cracks [30-32].

Rolling contact fatigue (RCF) and wear are among failure modes which are common between different welding methods and rail sections. However, aluminothermic welds are more susceptible to these failures mostly due to lack of sufficient ductility in the fusion zone as a result of the casting nature and the lower strength of the softened HAZ [33-34]. Additionally, the existence of HAZ and the associated batter (or generally surface irregularities) lead to impact forces which contribute to the initiation and higher propagation rate of RCF cracks. The relatively lower fracture toughness ( ) of aluminothermic rail welds [35] also promotes lower weld life as a result of faster transition of weld RCF failures to brittle fractures.

RCF defects usually appear on the running surface in the form of head checks (gauge corner cracking), spalling (flaking), shelling and as localised defects such as squats. Figure 2-6 shows squat defects in the two HAZ bands of an aluminothermic weld [22]. The RCF defects at the rail running surface may propagate and form transverse defects (detail fracture), vertical and horizontal split heads and eventually lead to total rail section break. According to Mutton et al. [36] increased tensile stress areas in the rail head specifically under severe head wear will lead to higher crack growth rates and RCF cracks to turn down and form transverse defects (see Figure 2-7).

To investigate the fatigue behaviour of wheel rail contact region and the effect of material properties several studies have been performed using finite element modelling, fracture mechanics and application of multi-axial fatigue criteria [37-42]. These

17 procedures can be extended to aluminothermic welds through incorporation of weld HAZ and fusion zone material characteristics and the effect of amplified service loadings. Nevertheless, no such analysis has been performed so far.

Figure 2-6: Squats formed on the battered regions of an aluminothermic weld [22]

Figure 2-7: Longitudinal section of rail head illustrating transverse defect formation from head check defect in the HAZ of an aluminothermic weld [36]

2.1.3 Straight Breaks (Transverse Fissure)

Straight break or transverse fissure is a fatigue failure (less likely an overload or brittle fracture) which initiates from a defect, geometrical discontinuity or stress concentration in the weld section and develops in a vertical or near vertical direction (under mode I crack propagation) eventually leading to total section break. The main locations of the weld in which straight breaks initiate are the collar edge or its vicinity (HAZ) at the base, web-base fillet or web regions of the weld [17]. However, in the weld type currently used in Australian heavy haul, the underhead (head-web fillet) and upper-foot (top of rail foot) regions are very prone to straight break formation (Figures 2-8a and 2- 8b). It is reported that in some defective welds, the straight break may also initiate from a shrinkage defect at the weld centreline or a lack of fusion defect on the fusion boundary of the weld [13]. However, the straight breaks initiating form such gross defects propagate very quickly in the form of a brittle or overload fracture with no sign of fatigue crack growth (polished fracture surface, striations and beach marks). The initiation and propagation of straight break is largely under the influence of the following longitudinal stress components:

• Tensile cyclic stress (particularly at the rail base region) as a result of service loading and the bending behaviour of rail section. Poor rail support condition due to ballast deterioration in the vicinity of aluminothermic welds can increase this cyclic stress component. • Seasonal thermal stresses in the form of static tensile stress in cold months of the year when the ambient temperature is much lower than the neutral (stress free) temperature. According to strain gauge measurements, longitudinal stress values up to ±80 to ±100 MPa can be observed for temperature deviations of ±35 to ±45 oC around the neutral temperature. • Tensile residual stress as a result of the welding process, pre-heat and post cooling conditions. According to Mutton et al. [43] the longitudinal residual stress at the top of the foot and underhead region of some aluminothermic welds can reach up to 300 MPa.

(a) (b)

Figure 2-8: Straight break failure in an Australian heavy haul railway: (a) Initiated from upperfoot (top of the foot) [44]; and (b) Initiated from the underhead radius (head-web fillet) [16]

The presence of stress concentration is vital for the occurrence of straight breaks and as mentioned the collar edge which is the boundary of parent rail surface and the weld collar is very prone to this type of failure. The two parameters used to characterize the geometry of collar edge are flank angle and toe radius which represent the angle between the free surface of rail with the collar and the notch radius respectively. Ross [45] has performed some finite element simulations on the effect of flank angle and toe radius. The results show the significant dependence of stress concentration factor on the flank angle and toe radius (Figure 2-9). The effect of flank angle however reduces as the toe radius increases and ceases to be influential for high angles above 70 degrees.

According to Lawrence et al. [46] even in absence of defects, most collar edges are critical and can potentially be straight break initiators. They have developed a model for fusion welded butt joints which can be applied to quantify the severity of fatigue (fatigue notch factor, ) at the collar edge of aluminothermic welds based on the geometrical and surface conditions:

(2-2) . 0.27 tan 1 + 0.1054 √ − 1 1

Where is the web or base thickness, is the weld flank angle, is the weld toe radius, is the depth of the surface roughness at the weld toe root and is the tensile strength of the toe root material. Based on the equation, the fatigue performance is improved through reduction of the flank angle, enlarging the toe radius and enhancing the surface smoothness of the collar edge i.e. reducing the value of surface roughness ( ).

Figure 2-9: The effect of flank angle and toe radius on the stress concentration of the collar edge [45]

Beside the stress concentration associated with the collar edge features, a type of welding defect referred to as cold lap has been reported to be responsible for a majority of straight breaks. Cold lap is not exclusive to aluminothermic welds and is considered a common defect in any welding method which incorporates a filler metal e.g. arc welding. According to Dimitrakis [47] cold laps greatly reduce the fatigue life of fusion weldments by accelerating and sometimes eliminating the nucleation stage of fatigue cracks as a result of a very high stress concentration at the cold lap notch root.

The cold lap defect generally occurs in locations where the welding mold does not perfectly fit the exterior surface of the rail ends. The resulting gap allows the molten metal to leak out of the mold and solidify forming an unfused region on the rail surface. The reason for lack of fusion between the leaked material and the rail surface is mainly due to insufficient heat input to melt the rail surface and fuse the two surfaces.

However, presence of contamination on the surface of parent rail may also be responsible for the lack of fusion [17]. Figure 2-10 shows a typical cold lap formed at the underhead radius of an aluminothermic weld and the resultant fatigue crack developed from its apex (notch root). Figure 2-8a also illustrates a cold lap formed at the top of the rail foot where the straight break has nucleated. In fact, cold lap is the main cause of straight breaks initiating at the top of the rail foot particularly for the welded rails with dissimilar heights.

(a) (b)

Figure 2-10: (a) Cold lap formation at the underhead region of a weld; and (b) Fatigue crack emanated from the apex of the cold lap [48]

Cold laps could also occur in locations where the mold fit is perfect; in this case the formation of cold lap is associated with the lack of enough melt-back on the rail end [49]. Accordingly, the cold lap occurs when the amount of liquid penetration in the rail end (melt-back depth) is smaller than the initial length of rail end passed beyond the mould collar edge (stick-out length). Figure 2-11 shows a schematic section view of the cold lap formation during the solidification stage.

A series of thermal finite element simulations has been performed by Chen [49-51] to investigate the formation of cold lap and the effect of different weld process parameters such as preheating time, liquid temperature, weld gap and stick-out (extent of parent rail material projecting into mould cavity) dimensions. The FE model is basically a heat transfer model in which the preheating, tapping and solidification stages are introduced by predefined heat fluxes and temperature boundary conditions measured during laboratory welding. The amount of melt back and heat affected zone are determined

22 through obtaining isotherms of different temperatures in the model. Accordingly, the line of fusion is defined by plot of the solid-liquid interface temperature isotherm and the outermost boundary of the HAZ profile is determined by the isotherm of eutectoid temperature.

Figure 2-11: Schematic view of cold lap formation when no leak of molten steel has occurred [49]

According to the simulation results, by increasing the preheating time, the melt-back depth increases and the possibility of cold lap formation decreases. Similar conclusion can be drawn for the effect of molten metal temperature (Figure 2-12). However, it was surprisingly found that the width of weld gap cannot significantly change the temperature gradient and the melt-back depth. On the other hand, a shorter stick-out could alleviate the requirement for high input of thermal energy and molten metal temperature and accordingly reduce the cold lap probability.

Figure 2-12: Effect of welding parameters on melt-back depth: (a) Preheating time; and (b) Liquid temperature; the horizontal dotted line is the boundary of cold lap and no cold lap regions [49]

2.1.4 Horizontal Split Webs

Horizontal split web (HSW) or big dipper is a common failure mode in both aluminothermic and flash butt welds in Australian heavy haul railways [52-53]. This failure mode is typified by initiation of horizontal cracks at mid to upper web region of the weld collar surface and propagation in a horizontal plane into the heat affected zone. Once the crack tip extends past the weld collar, it can develop vertically either towards the rail head and/or base resulting in removal of a rail section and dramatically increasing the risk of derailment. The main cause of HSW formation is the presence of large surface or subsurface defects such as shrinkage cracks, hot tears, inclusions (in the form of alumina slag or iron oxides) and pores [15, 54]. That is why in most instances the propagation of HSW is rapid in the form of a brittle fracture and no evidence of fatigue crack growth could be seen on the fractured face. Figure 2-13 shows an example of HSW and the associated shrinkage defect at the corner of one of the weld buttresses.

(a) (b)

Figure 2-13: (a) Horizontal split web fracture; and (b) Fracture face of the weld buttress showing an area of a shrinkage defect [55]

The factors which contribute to the formation of HSW failures in aluminothermic and flash butt welds are well understood and comprise the following [53, 56-57]:

• High vertical residual stresses present at the web collar surface. The value of these stresses may vary between 100 to 300 MPa for aluminothermic welds depending on the collar shape and process factors. The corresponding range for

flash butt weld is between 450 to 600 MPa (Figure 2-14). However, heat treatment techniques such as normalizing and stress relieving may be used to alleviate surface residual stress levels, although they are not considered standard procedures due to possible side effects such as reducing hardness levels in the rail head. • Fatigue crack initiation sites such as surface or near surface gross defects e.g. shrinkage defects, hot tears and inclusions. However, flash butt welds are relatively cleaner than aluminothermic welds and are less likely to contain such defects. • Cyclic vertical stresses as a result of eccentric loading of the rail. According to the observations most of HSW failures in both aluminothermic and flash butt welds occur in curves of 600-900 m radius or tangent tracks prone to vehicle hunting. The repetitive lateral displacement of the vertical loading (on the gauge and field side of the rail) due to this steering behaviour can result in significant reverse bending. Based on a strain gauge measurement on the web surface of an aluminothermic weld, maximum stresses can vary from +150 MPa to -130 MPa for a single train passage. However, according to finite element analysis, the shape of the collar and the related buttress could also influence the vertical stress distribution [53, 57]. • Inferior fatigue crack growth characteristics of the weld compared to those of the parent rail. According to a study by Bulloch [58] the crack growth behaviour in aluminothermic weld is usually unstable and the growth rate could be up to 5 times faster than the upper bound values for rail steels. This is mainly attributed to the much coarser microstructure and presence of highly directional columnar grains.

A linear elastic fracture mechanics approach has been used by Dudley [52] (aluminothermic welds) and Marich [56] (flash butt welds) to determine the threshold defect sizes on the web surface and the effect of applied and residual stresses. Dudley proposes the following equation for aluminothermic welds based on the analysis of semi elliptical surface cracks (as substitutes for defects) in a plate:

1⁄ ∆⁄∆ (2-3)

Where is the threshold crack depth, ∆ is the threshold stress intensity factor range, is the crack shape factor which varies in the range 0.75 to 0.93 for typical HSW initiators and ∆ is the stress range (maximum stress due to wheel load).

However, the value of ∆ depends on the stress ratio which is the ratio of minimum to maximum total vertical stress (incorporating the residual stress) and varies according to the following relation:

Figure 2-14: Vertical residual stress distribution: (a) Flash butt; and (b) Aluminothermic weld [14]

∆ ∆1 (2-4)

Where ∆ is the threshold stress intensity factor range at 0 and is a material constant. According to previous measurements, the values of ∆ and are about 10.4 √ and 0.62 respectively. Figure 2-15 shows the effect of residual stress and the applied stress range on the threshold crack depth for compressive and tensile loading of the web collar [53]. The graph indicates that a combination of lower residual stress at the web region and reduced applied vertical stress (which may be achieved by better design of the collar shape and reducing the impact forces induced by the running surface irregularities) could result in a significant increase of tolerance to defects responsible for HSW failures.

Figure 2-15: Predicted threshold crack depth and the results of testing: (a) Compressive loading of the collar web; and (b) Tensile loading; is the vertical residual stress [53] As previously mentioned, HSW cracks initially develop in longitudinal direction almost parallel to the rail running surface and then kink or branch towards the head and or the foot. A good understanding of the HSW propagation mechanism and the evolution of stress intensity factors could be quite helpful for maintenance purposes such as determination of inspection intervals. Perhaps the only study which has considered the crack propagation characteristics of HSW has been performed by Beretta et al. [59-60] in which the mechanism of fatigue crack propagation and the associated crack path are investigated in flash butt welds. In this study, a finite element analysis has been performed in ABAQUS FE package where the stress intensity factor histories (during the wheel passage) for the tip of a modelled HSW crack is determined in three steps of propagation (Figure 2-16). The modelled crack for each step follows the same path observed in a real fracture so that the calculated stress intensity factors (and their range) can be used to identify the fracture mode for the corresponding step.

Figure 2-17 illustrates the value of (stress intensity factor range under mode I growth) and (stress intensity factor∆ range under mode II) for the abovementioned three crack tips∆ along the propagation path. In all graphs α is the angle between the possible kinking direction and the current crack path.

Figure 2-16: (a) Typical HSW crack path and the three points for which stress intensities are calculated; α is the angle of kinking; and (b) Finite element model of the rail and the HSW [60]

Figure 2-17: Stress intensity factor range for possible kinking angles and the real crack propagation angle observed in service: (a) Step 1 showing mode II crack growth, (b) Step 2 showing mode II growth; and (c) Step 3 suggesting a Mode I growth [60]

For the first and second steps where the crack tip is located at point I and II respectively, the propagation angle observed in real fracture corresponds to a maximum value of with having only a little influence. However, at the third step just before kinking,∆ the real∆ crack path is associated to the maximum of . These results are consistent with the SEM fractographic observations and suggest that∆ HSW cracks initially develop under mode II (in plane shearing) propagation and then switch to mode I (opening) once they kink. Similar results have been obtained for the trailing end of the crack.

The propagation of HSW crack under mode I after kinking suggests that the value of longitudinal residual stress as a result of the welding procedure plays an important role in subsequent propagation and final fracture of the weld. According to Skyttebol [61], the calculated residual stresses in flash butt welds using finite element simulations could reach up to 600 MPa extending from the upper foot region to the rail underhead.

2.2 Multi-axial Fatigue

Fatigue is considered a material behaviour in which localized permanent damage occurs as a result of fluctuating or cyclic loading. The fatigue phenomenon is progressive since it occurs in a period of time, localized as the material changes are restricted to specific locations in material and permanent since it does not recover once it starts [62].

The complete mechanism of fatigue in ductile materials is somewhat complicated but it can be summarised as the initiation of micro cracks from persistent material slip bands and consecutive propagation until the structure is eventually fractured. Slip bands are described as patterns of plastic deformation formed on the surface of a material as a result of the dislocation movements along the slip crystallographic planes in a grain (Figure 2-18a). It is worth mentioning that slip bands occur in grains in which the slip planes are oriented in the maximum shear stress direction and hence it can be concluded that fatigue cracks mainly initiate on maximum shear stress planes. According to Forsyth [63] fatigue cracks initially nucleate and grow in the maximum shear stress range plane for a short length generally in the order of several grain sizes. This stage is referred to as stage I crack propagation and has a relatively low rate. On the second

29 stage the rate of propagation increases dramatically and also the crack growth direction changes course to one which is nearly perpendicular to the applied tensile stress (Figure 2-18b). The crack continues to grow with each load cycle until it reaches a critical length after which the sudden fracture takes place.

Figure 2-18: Fatigue crack nucleation and propagation: (a) Formation of material slip bands under cyclic loading; and (b) Schematic view of fatigue crack growth stages [62]

Fatigue is generally categorized into two regimes of high cycle and low cycle. In high cycle fatigue (HCF) large number of cycles (typically between 10 3 to 10 8 cycles) is considered and the material behaviour at the crack nucleation region is deemed to be predominantly elastic. Such conditions usually involve applied stresses lower than the yield strength of material and the loading is of high frequency nature [64]. On the other hand low cycle fatigue (LCF) involves shorter lives typically below 10 3 cycles. This behaviour applies to the locations which experience high stresses in excess of yield strength and as a result the deformation is predominantly plastic and the strains are relatively high.

2.2.1 Approaches to Fatigue Analysis

Stress-life method is the first and widely used approach to quantify the fatigue characteristics of a material in engineering applications. This method is based on the assumption that the behaviour of material is predominantly elastic and the material endures a high number of cycles so its application is restricted to HCF analysis. The standard stress-life method involves S-N curve (Wöhler diagram) which is a plot

30 relating the fatigue strength of material to the number of cycles to fracture according to experimental fatigue tests (e.g. alternate bending). Basquin [65] has proposed a linear log-log approximation for S-N diagram which can be written in the following format:

(2-5) (2)

Where, is the fatigue strength of material at cycles under a reversed ( ) −1 loading, is the true stress amplitude, is a material constant approximately equal to the true fracture strength of material and is the slope of S-N line. It is worth mentioning that some ferrous materials have an endurance limit (fatigue limit) below which the material’s life is virtually infinite. In these cases, the S-N line is horizontal after 10 6 or 10 7 cycles.

However, the S-N graph described above cannot be directly used for design applications since it lacks the important effect of mean stress. A comprehensive graph (referred to as Haigh diagram) which incorporates the results of the non-zero mean stress tests has been proven to be difficult to develop [66]. Accordingly, some researches among which Soderberg, Goodman, Gerber and Morrow tried to find simplified graphs and relations to accommodate for the effect of mean stresses in the fatigue life of materials. Goodman has proposed a relation between alternating applied stress ( ), mean of applied stress ( ) and the fully reversed fatigue strength (limit) of the material ( ):

(2-6) + 1

It was assumed in the stress-life approach that the plastic strains are negligible; however, it is not always the case in reality bearing in mind that the actual fatigue mechanism involves plasticity at the crack initiation location. This particularly applies to LCF regime where the local stresses exceed the yield strength and substantial plastic deformation occurs. Under such conditions strain-life approach based on the direct measurement of strains (under strain-controlled fatigue test of smooth specimens) can better quantify fatigue characteristics of a component. It is worth mentioning that since

31 strain is linearly related to stress for elastic conditions, the strain based approaches can also be used for HCF analysis whereas the opposite does not apply.

Coffin [67] and Manson [68] were the first who found that the relation between plastic strain amplitude ( ) and life to fracture ( ) can be linearized in a logarithmic scale. The relation is similar to what was proposed by Basquin in stress-life approach:

(2-7) (2 ) Where and are material parameters called fatigue ductility coefficient and fatigue ductility exponent. If equation 2-5 is combined with equation 2-7, it forms a general relation which is the basis for the strain-life approach [69]:

(2-8) + (2) + (2) Where and are the total and elastic strain amplitude respectively and is the modulus of elasticity. Equation 2-8 has the benefit of being applicable to both HCF and LCF regimes.

Similar to stress-based approach, the strain-life relation does not include mean stress effects. Several methods have been proposed to account for the mean stress ( ) among which the modified Morrow relation [70] has been shown to be more consistent with the experimental data:

(2-9) + 1 − (2) + (2) The LCF failure occurs when the material ductility is exhausted under plastic shakedown condition in which the strain cycle is closed (Figure 2-19c). However, another type of failure may also be responsible for the rupture of material under extensive cyclic plasticity. This type of failure called ratchetting occurs when the strain cycle is open and the material accumulates plastic strain during each loading cycle

(Figure 2-19d). The material fails under ratchetting mechanism when the total accumulated plastic strain reaches a critical strain which is comparable to the plastic strain at the failure of a monotonic tension test sample. The number of cycles to failure in ratchetting may be determined by dividing the critical strain ( ) by the ratchetting strain accumulated cyclically ( ). While ratchetting is occurring the material also experiences reversing plastic strain∆ which produces LCF. The number of cycles to failure in LCF can be calculated using the aforementioned Coffin-Manson criterion. According to Kapoor [71] the ratchetting mechanism and LCF are competitive independent phenomena and the final rupture is governed by the mechanism which provides the smaller number of cycles to failure. Ratchetting is one of the most important failure mechanisms responsible for the formation of RCF and wear in rolling/sliding contact conditions.

Figure 2-19: Structural response to cyclic loading and the corresponding type of failure: (a) Perfectly elastic (HCF), (b) Elastic shakedown (HCF), (c) Plastic shakedown (LCF); and (d) Ratchetting mechanism (incremental collapse or ratchetting) [72]

2.2.2 Multi-axiality

In real life there are many instances where the engineering structures are under combined loadings which produce complex and multi-axial stress states. The condition becomes even worse if the applied load components have different frequencies and/or

33 phases. Under such conditions, principal stress vectors may rotate during a loading cycle and/or their magnitudes become non-proportional. This particularly applies to locations with high stress concentration such as notch roots and situations with moving contact load such as the wheel-rail contact.

The question arises whether the abovementioned methods based on uni-axial measurements can be directly applied to multi-axial conditions. To overcome the multi- axiality of stress state, early researchers developed equivalent stress (or strain) approaches which were simply the extensions of static yield criteria [62, 66]. These criteria are widely used in engineering applications and are available both in the form of strain-based models for LCF analysis and stress-based models for HCF regime. A common stress-based model is based on the distortion energy (octahedral stress or von Mises) yield criterion through which an equivalent stress amplitude ( ) is defined in terms of the principal stress amplitudes ( ) in the material point during a loading cycle (Equation 2-10). If mean or residual, , stresses are also present, a similar equation (Equation 2-11) can be written to define an equivalent mean stress ( ) based on the principal mean stresses ( ). The values of and can then be used in Equations 2-5, 2-6 or other, similar, stress-life relations to determine the possibility of fatigue failure and the associated life to fracture.

(2-10) 1 ( − ) + ( − ) + ( − ) √2 (2-11) 1 ( − ) + ( − ) + ( − ) √2 Sines [73] reviewed the results on the alternate bending and torsion fatigue tests and found that the alternating shear stress causes the fatigue damage. However, he indicated that the outcome of analysis based on the maximum shear stress and octahedral shear stress criteria give similar results and the use of either of the methods is governed by the convenience of the mathematical expression. Moreover, he realized that the fatigue life of material is mainly influenced by the mean hydrostatic stress with mean shear stress having no effect. Accordingly, he proposed a failure criterion combining the octahedral shear stress and mean hydrostatic stress in the following form:

(2-12) 1 ( − ) + ( − ) + ( − ) + ( + + ) ≤ 3 Where and are material parameters determined using simple fatigue experiments and , , are the mean stresses in , , directions. A modification to the Sines criterion is the Crossland criterion [74] in which the maximum hydrostatic stress substitutes the mean value. This criterion is particularly suitable for conditions where high hydrostatic stresses are involved.

For the case of LCF regime where plastic deformation is dominant, equivalent strain approaches similar to the equivalent stress methods discussed in HCF can be used. One of the most common models is the octahedral shear strain theory (analogous to the octahedral shear stress model) which can be written as follows [62]:

(2-13) 1 ( − ) + ( − ) + ( − ) √2(1 + ) Where, is the Poisson’s ratio and , , are the principal alternating strains. The equivalent alternating strain can then be applied to Coffin-Manson or similar strain- life relation to evaluate the fatigue life.

The main disadvantage of all the mentioned equivalent stress/strain methods is that they are not accurate for the non-proportional loading situations where the principal stresses change their directions during the loading cycle and so their application is very much restricted to simple multi-axial loadings.

2.2.3 Critical Plane Approaches

According to crack observations, fatigue cracks in ductile materials initiate on planes of maximum shear stress amplitudes and their growth is influenced by the surrounding normal stresses. These observations have led to the development of some fatigue crack initiation models considering both the orientation of fatigue crack planes as well as the risk to fatigue failures. In critical plane approaches (criteria), stresses (or strains) during

35 a loading cycle are resolved on various planes in a material and the most severely loaded plane (according to that specific criterion) is determined as the critical plane on which fatigue cracks nucleate. Because of the directional nature of these methods, critical plane approaches can safely be used for non-proportional multi-axial loading situations where the principal stress axes rotate during the cycle.

2.2.3.1 Findley Criterion

In 1959 Findley [75] developed a criterion for high cycle fatigue analysis based on the stress components acting on the critical plane. This criterion relies on the experimental observations which propose that the normal stress on the critical shear plane reduces the allowable alternating shear stress. The criterion implies that fatigue failure occurs in the material if the following inequality is satisfied during a loading cycle:

(2-14) ∆ , > 2 Where, is the shear stress amplitude or half of the maximum range of shear stress ∆ on the shear plane considered and is the maximum normal stress acting on the shear plane. and are material parameters, which are related to the fatigue limit in alternate bending ( ) and fully reversed pure torsion ( ) according to the following relations [76]:

2 − (2-15) 2 − 1 4 − 1 Findley’s inequality should be assessed for all the shear planes passing through the material point. The plane which results in the highest damage parameter (highest value of the left side of the inequality which is also called the Findley’s Parameter) is the critical plane.

2.2.3.2 Brown and Miller Criterion

In 1973 Brown and Miller [77] formulated a similar criterion but based on the shear and normal strains. This criterion is a strain-based method and could be used for both HCF and LCF regimes. A simple form of this criterion is presented as follows [78]:

(2-16) ∆ ∆ 2 is the maximum shear strain amplitude and is the normal strain range on ∆the shear⁄2 plane considered. is a material constant and∆ is a weighting factor which defines the importance of the normal strain (stress) effect on fatigue crack initiation.

2.2.3.3 Fatemi and Socie Criterion

According to Fatemi and Socie [79-81], the growth of small fatigue cracks under only shear stresses can be restricted by the closure phenomenon. Fractographic evidence indicates that crack faces are usually irregular and so due to the friction and interlocking (closure) between the faces, the applied shear stress on crack plane cannot produce enough driving force at the crack tip. However, the application of normal stresses or strains to the crack plane can open up the crack mouth, reduce the friction and promote crack propagation under shear stresses (Figure 2-20). Considering this fact, Fatemi and Socie developed a criterion in the following form:

(2-17) ∆ , 1 2 Where, and are the maximum shear strain amplitude and the maximum normal stress∆ on⁄2 the plane, of maximum shear stress, is the yield strength of material, and are material constants. This criterion is consider ed a strain-based approach and is widely used for LCF analysis.

Figure 2-20: (a) Fatigue crack under pure shear and the resulting interlocking effect; and (b) Effect of normal stress in enhancing fatigue crack growth by reducing the closure effect [62]

2.2.3.4 Liu Criterion

Liu [82-84] has developed a criterion based on the concept of distinction between critical plane and fracture plane. Fatigue fracture plane is defined as the crack plane observed in macroscopic level while critical plane is the plane of fatigue crack initiation in grain size scale. Liu assumes the fracture plane as the plane which experiences the maximum normal stress amplitude. Critical plane however can be the same as fracture plane or inclined from its direction for different materials. In this criterion, the fracture plane is initially determined according to the maximum normal stress amplitude criterion and then the critical plane is evaluated by calculating the angle between the two planes ( ). is predicted as a function of for brittle and ductile materials. Once the critical plane is defined, fatigue damage ⁄ can be calculated based on the nonlinear combination of the normal stress amplitude ( ,), shear stress amplitude ( ,) and hydrostatic stress amplitude ( ,) acting on the critical plane:

1 , , , , (2-18)

Where , is the mean normal stress, and , and are material parameters which can be determined by fatigue experiments. The advantage of Liu criterion is that it can be used for the range of very ductile to extremely brittle materials and is able to define both fracture and critical planes.

2.2.3.5 Dang Van Criterion

Dang Van [85-88] developed a criterion for HCF analysis based on the microscopic observations indicating that cracks initiate in material slip bands under the influence of the local maximum shear stress amplitude and their growth is facilitated by crack opening as a result of the surrounding tensile hydrostatic stress. According to the criterion in its simple linear form fatigue crack initiates at a point of material if the following inequality is fulfilled on a shear plane of that particular point during at least a time portion in the whole stress cycle [89]:

(2-19) () + () > Where is the Dang Van damage parameter which is considered a numerical index for fatigue damage, is the time dependent shear stress amplitude on the considered shear plane which is defined() as the difference between the instantaneous and the mean shear stress of the whole cycle and is the time dependent hydrostatic stress at the point of interest. and are material () parameters which are obtained through two classical fatigue tests such as alternate torsion and alternate bending as follows [90]:

(2-20) 3( − 0.5) So the fatigue damage (or Dang Van parameter) is simply compared with the value of the fatigue strength of material in pure reversed torsion. Dang Van criterion can also be examined through graphical representation of the load path on ( - ) diagram. In this diagram the loading path is defined by time dependent values of and in a loading cycle and the criterion is introduced by a line (representative (of) Equation ( 2-19).) If the loading path remains in the safe region below the criterion line no fatigue failure occurs, otherwise fatigue crack initiation is incipient (Figure 2-21).

The planes on which the abovementioned inequality is satisfied are called critical planes. Finding the critical plane in each material point is part of the fatigue analysis as

39 the critical plane is not obvious at the outset of the process. Hence, the inequality should be assessed in any shear plane passing through the material point on which the fatigue analysis is performed.

Figure 2-21: Graphic representation of Dang Van fatigue criterion in a failure condition showing a sample loading or stress path crossing the criterion line during a cycle

The applicability of the Dang Van criterion is in elastic or elastic shakedown situations where a stabilized state of stress can be developed. Accordingly, the criterion has been successfully applied to several studies on the formation of rolling contact fatigue failures in rail and wheel running surface under a shakedown condition [41-42, 89-94]. The criterion has also been exploited in the analysis of fatigue crack initiation in web and head regions of the flash butt weld under the influence of high residual stresses [95]. Dang Van criterion is also a reliable tool in automotive industry for vehicle component design optimization and damage tolerance investigations [96-98]. More on this criterion and its computer implementation is provided in Chapter 4.

2.3 Damage Tolerance Investigation

It is well known that nearly all fatigue failures in engineering structures initiate from geometrical irregularities with high stress concentrations such as pores, inclusions, notches, cracks and scratches. Accordingly, it is an important engineering challenge to define the tolerance of structures to defects or to find the range of defect sizes and

40 shapes which do not affect the fatigue performance of the material. Several approaches have been proposed to determine the influence of defects in fatigue strength of materials among which the following methods are widely used:

2.3.1 Defects as Notches

In these approaches the defects are considered as notches and the stress state on the surface and at the vicinity of the notch is used to quantify the effect of defects. Perhaps the simplest approach is to relate the fatigue strength of material to maximum stress at the tip of a notch and use stress concentration factor ( ) or fatigue stress concentration factor ( ) as the only parameters characterizing the severity of defects. However, different studies suggested that beside the maximum stress the stress distribution from the surface of the notch to the interior of the material (usually defined as stress gradient) is also a governing factor particularly for large defects with low stress gradients [99- 101]. It is known that a defect with low gradient of stress (gradual decrease of stress) is more damaging than a defect with high gradient (steep decrease of stress) [102].

Maybe a better approach rather than stress analysis of the defect surface is to account for plastic deformations at the vicinity of defects and perform an elastoplastic stress analysis [103]. Such approach has been exploited by Sigl et al. [104] in which the strain amplitude at the surface of the defect is evaluated and the Coffin-Manson relation (Equation 2-7) is used to determine the fatigue life of the defective material. However, the main disadvantage of this approach is the necessity to define an advanced consistent constitutive equation for cyclic plastic deformations.

2.3.2 Multi-axial Fatigue Criteria

According to the observations by Nadot [105-106] the first stage of fatigue crack nucleation in defective materials occurs on the maximum shear stress amplitude plane in the most severely loaded point of the defect. This is similar to what is observed in defect free materials. Accordingly, it seems rational to use the same multi-axial fatigue criteria to investigate crack initiation from the surface of the defects. This idea is more strongly

41 supported by experimental data showing that for defects with low stress concentration fatigue limit is defined by the threshold for crack initiation from the defect surface. The reason is that under a low gradient stress at the periphery of a blunt notch the initiated crack can more readily propagate and lead to a total failure [102].

Fry et al. [107] used Findley critical plane criterion to investigate the effect of pores and inclusions in formation of rail head damages in aluminothermic welds. Several defect geometries (basketball, football and poker chip shapes) were investigated through analytical calculation of the cyclic stress distribution on the periphery of defects and applying critical plane criterion. The methodology is also capable of determining the type of rail head damage (shelling, vertical split head and detail fracture) by examining the direction of fatigue critical plane.

According to the study, pores are more damaging than inclusions and the shape and orientation of pores have a strong effect on the type of rail head damage and the life consumed in stage I of crack propagation. It was also found that the defects located at the highest tensile residual stress depth of the rail head are most damaging. Figure 2-22 illustrates the damage map for different types of pores considered in this study.

Nadot [105] proposed a criterion based on the Crossland multi-axial criterion considering the effect of hydrostatic stress gradient at the vicinity of defects. The criterion is written based on the linear combination of shear stress amplitude , (amplitude of the second invariant of the deviatoric stress tensor) and the maximum value of the hydrostatic stress including gradient effect ( ). ∗

(2-21) ∗ ∗ , + ≤ 1 − Where is the gradient of hydrostatic stress, is the maximum hydrostatic stress in a load cycle, and are material parameters in Crossland’s criterion and is a material parameter describing the influence of defect. The stress distribution of the

42 defect surface required for this criterion is extracted from elastoplastic finite element simulation of predefined defect models.

Figure 2-22: Fatigue damage prediction using Findley criterion (RAHELS model) and comparison with experimental data. Shaded regions show different rail head damages predicted with respect to variable geometrical features of defects [107]

2.3.3 Defects as Cracks

If experimental results for fatigue strength of material based on notches with different root radius and subsequently different stress concentrations are plotted, a similar curve such as the one depicted in Figure 2-23 will be developed. The fatigue limit dramatically decreases with the increase of stress concentration up to a certain stress

43 concentration factor ( ∗) after which the fatigue limit becomes virtually invariable ∗ [108]. According to Smith and Miller [109] below the full effect of stress concentration is felt by the material and the occurrence of fatigue is governed by the ∗ maximum stress at the notch root and hence the geometry of the defect. Above all defects behave similar to a crack of the same length. The reason is that for sharp defects (high stress concentration at the root and high stress gradient) there are some small fatigue cracks at the fatigue limit of material which cannot propagate due to lower stresses far from the defect root. So the fatigue limit is not governed by the initiation of cracks from the root but by the propagation or non-propagation of these peripheral cracks. Hence, it is rational to model high stress concentration defects as cracks and analyze the propagation of the substituting cracks.

Figure 2-23: Schematic of experimental data for notches with different root radius and predictions with notch method (defect modelled as notch) and crack method (defect modelled as crack) [108]

Several studies have been performed on the influence of defects through describing defects as equivalent to sharp cracks [52, 104, 110-112]. The general approach is to find the stress intensity factors at the tip of the modelled crack and then compare them with threshold stress intensity ( ) or its range ( ∆) for possible crack propagation or critical stress intensity factor ( ) as the threshold for fast fracture.

2.3.4 Murakami’s Approach

As previously mentioned, the solution for the effect of small defects (with high stress concentration) is essentially a crack propagation problem due to the presence of non propagating cracks at the periphery of defects in fatigue limit of material. Murakami [113] suggested that a good geometrical parameter governing the stress intensity factor range along a crack front is or the square root of the crack area (in the case of a defect it is the square root of√ the projected area of a defect on the maximum principal stress plane). Accordingly, the stress intensity factor range (for mode I) can be defined by the following relation:

(2-22) 0.65 ∆ ∆ √ Where, is the applied remote stress range. He also proposed that the threshold stress intensity∆ factor range for a small crack (as a substitute to a small defect) can be written as a function of both and the material Vickers hardness based on several experiments on different√ materials and defect sizes [102]:

(2-23) / ∆ 3.3 ∗ 10 ( + 120 )(√ ) Where, is in ( ), in ( ) and in . Combining Equations∆ 2-22 and 2-23, the√ fatigue limit ( ) for a defective material (with surface defects) can be presented in the following form:∆

(2-24) / ∆ 1.43 ( 120 )⁄(√ ) A similar approach can be used to estimate the fatigue limit of defective materials in reversed pure torsion. However, under such conditions the stress intensity factor range cannot be solely defined in terms of the square root of the projected area ( ) but is also dependent on a geometrical parameter , a function of the crack√ aspect ratio. Fatigue limit for pure shear can be determined by the following relation [102]:

.93 120 ∗ (2-25) / √ / ∆ 0

Where, and are dimensions of the semi-elliptical surface crack as illustrated in Figure 2-24. / can be approximated using polynomial function 2-26.

Figure 2-24: A semi-elliptical surface crack under reversed pure torsion [102]

0.0957 2.11 2.26 1.09 0.196 (2-26)

These equations can be used for a wide range of hardness values from 70 to 720. However, their applicability is limited to small cracks with √ less than 1000 [114]. For the case of larger defects, the fatigue limit may be governed by the crack initiation threshold for which the actual geometry of the defect becomes important.

CHAPTER 3 IN-TRACK BENDING BEHAVIOUR

3.1 Finite Element Modeling

The first step to investigate fatigue performance of aluminothermic welds is to study the in-track bending behaviour and the resulting stress distribution arising from contact, thermal (seasonal-induced) and service loadings. Finite element (FE) method if carefully applied has shown to be a reliable tool for such analysis and has been widely used in performance characterizations of railway components. In this chapter the finite element analysis of a specific type of aluminothermic weld widely used in Australian heavy haul railway systems using ANSYS 12.0 FE package is presented. The FE model comprises a specific length of track incorporating concrete sleepers, elastic foundation and seasonal thermal loading effects, hence considered as a thermo-structural model. At the beginning of this chapter, the model construction and load application are described and the reliable mesh type and sizing is determined. Subsequently, the model is validated using experimental measurements and finally the bending behaviour of weld is studied for different operational conditions.

3.1.1 Weld Geometrical Modelling

The aluminothermic weld considered in this study is a Boutet PLK type weld made using consumables from Railtech Ltd. Since the original drawing for the external geometry of this weld was not available, the relevant dimensions were extracted from a scanned geometry of the sample weld using Roland Picza laser Scanner/Digitizer. Eventually, the 3D model of the weld was constructed in Solidworks CAD design software for later implementation in FE simulation. Figure 3-1 shows the actual PLK weld sample, laser scanned geometry and the Solidworks model constructed accordingly. It is worth mentioning that the modelled weld is free from surface defects and other geometrical irregularities which as a result of the cast-like nature of the process are visible on the exterior of the real weld. So, at this stage of the study on-site

47 weld imperfections and other surface features which do not form part of the as-designed collar shape are not considered. Analysis on some type of welding defects and their effect on fatigue behaviour is reported in Chapters 6 and 7.

(a) (b) (c)

Figure 3-1: PLK weld, (a) Original weld, (b) Laser scanned geometry; and (c) Model constructed in Solidworks and used in FE analysis

3.1.2 Track Modelling

To investigate in-track bending behaviour of the aluminothermic weld a specific length of the track was modelled in ANSYS 12.0 package. The model comprises one side of a 3237 mm length of track with six concrete sleepers and vertical and longitudinal bedding stiffness under the sleepers representing the bedding flexibility. The standard AS68 kg/m rail profile (AS 1085.1 [115]) was incorporated into the model, since this is the rail size used in the heavy haul system under investigation. It was also assumed that the weld is located halfway between the middle sleepers. The sleeper cross section was adopted from the work by Kaewunruen [116] and the sleeper spacing was set to 600 mm which is common in heavy haul applications. However, the effective length of the sleeper modelled per one rail was selected as one third of the total sleeper length and the sleeper was centered under the rail [117-118]. Figure 3-2 shows schematic of the model used in this study.

The required vertical stiffness of ballast per sleeper was calculated according to the model suggested by Zhai et al. [119]. In this model the load carrying part of the ballast can be simplified by overlapping cones and it is assumed that outside these cones the normal stress is zero. Under this assumption the stiffness of ballast can be approximated by the equivalent stiffness of the two ballast masses depicted in Figure 3-3 as follows:

Figure 3-2: Schematic of the modelled track with aluminothermic weld at the midway between the two middle sleepers, concrete sleepers, and vertical and horizontal ballast stiffness

Figure 3-3: Model of ballast used in the calculation of ballast stiffness. Ballast stiffness is the equivalent stiffness of the two shaded areas

(3-1) + (3-2) 2( − ) tan ln ()/( + − ) (3-3) ( − + 2 + 2ℎ tan ) tan − + 2ℎ tan

Where, is the sleeper sitting width, is the effective support length of the sleeper per rail (here, one third of the total length), is the sleeper span, is the ballast thickness, is the ballast stress distribution angle and is the ballast modulusℎ of elasticity. The longitudinal stiffness can vary between 1.0e5 to 1.0e7 N/m per metre of track [120]. However, in this study an average value of 5.0e6 N/m per metre of track is used which together with the applied vertical stiffness can result in a 2-3 mm vertical deflection underneath the wheel load in finite element analysis. Such deflection is consistent to the measured values in service under similar loading conditions [36].

Additional to the elastic foundations applied at the underneath and sides of the sleepers, two other boundary conditions were implemented in the track model. A displacement boundary condition is applied on the two rail ends to prevent longitudinal movement of the track and also to enable the development of seasonal thermal stresses in the rail. However, displacements and rotations in other directions are permitted at the rail ends. A sensitivity analysis on the simplification of the boundary condition at the rail ends and the modelled track length is reported later in this chapter. The last displacement boundary condition is applied on the inner cross section of the sleepers to prevent sleeper movements in lateral direction (transverse to the rail length). This is equivalent to the application of a symmetry boundary condition for one side of the track. This type of boundary condition truly applies to the tangent track where the contact patches on both rails are nearly at the centreline of the rails. For the curved track, the actual lateral restraint (fixture or stiffness) on one side of the track cannot be directly defined and depends on both the conditions of track bedding and the service loading on the other side of the track. Figure 3-4 shows the three dimensional model used for FE analysis and the type of displacement boundary conditions applied.

3.1.3 Loading

The axle load used in this study is set to 343.3 kN (which is representative of Australian heavy haul railways) and half of this value (171.7kN) is applied to the rail running surface as the vertical wheel-load contact force. The actual contact region between the rail and wheel may be of Hertzian, multi-Hertzian, semi Hertzian or non-Hertzian type

50 according to the rail-wheel profiles and the location of contact on the rail running surface. However, in this study an elliptical contact patch was assumed for all contact conditions and the vertical load is applied on a Hertzian elliptical patch modelled on the rail running surface. The dimensions of the elliptical patch are calculated from the curvatures of the rail and wheel surface in the contact point using the following relations [121]:

Figure 3-4: Track model used in FE analysis; displacement constraint in Y direction is applied to the red colour shaded areas of the sleepers and constraint in X direction is applied to the rail ends

(3-4) 1 1 1 1 | − | , + , cos 2 2 +

/ (3-5) 3 1 − 1 2 +

/ (3-6) 3 1 − 1 2 + Where, is the vertical applied load, is the rolling radius of the wheel, is the transverse wheel radius and is the transverse radius of the rail running surface (Figure 3-5). and n are non-dimensional coefficients which are tabulated as functions of [121] and and are the ellipse semi axes in x and y directions. The values of and are calculated as 9.12 mm and 5.92 mm respectively considering a tangent track and according to the wheel and rail dimensions used in this study.

Wheel load was applied to the elliptical contact patch as a uniform mean pressure calculated by . As a matter of fact, the Hertzian contact pressure has an ellipsoidal distribution/ and is not constant over the contact patch. However, since only the weld regions below the rail head are considered for fatigue analysis (the current study focuses on the formation of straight break and horizontal split webs) it is unlikely that the simplification on the contact pressure could affect the stress distribution far from the running surface. Nevertheless, the effect of this simplification is studied by comparing three cases of the uniform contact pressure, Hertzian contact pressure and the simulated contact pressure resulting from the wheel rail contact modelling. The finite element simulation and its results will be presented later in this chapter.

Figure 3-5: Schematic of the rail-wheel Hertzian contact and the dimensions used for calculation of the contact patch semi-axes and . (Figure adopted from Iwnicki [121]) Since the main objective of this study is to quantify the fatigue performance, it is necessary to calculate the cyclic stress history at the points of interest in the weld. This requires FE modelling of one load cycle. The load cycle is defined by the passage of the wheel over the weld and this is achieved in the current model by moving the contact pressure from one end of the modelled track to the other end. The contact pressure movement in FE software is applied using multiple load steps through which the contact

52 pressure is sequentially translated from one elliptical patch to the other starting from one end of the track up to the end. In this case the result of the simulation for each load step defines the stress distribution of the weld under the wheel load located at that particular point of the track. Figure 3-6 shows the approach used in FE analysis as a representative of wheel movement and the associated load cycle. According to the figure, in loading step i the contact pressure is applied on patch i and the simulation is performed and the pertaining stress distribution is recorded. In the next step i+1 the contact pressure is applied on patch i+1 and patch i is unloaded. This process continues for the full length of the track. The central distance of the two adjacent contact patches is set to 22 mm.

Figure 3-6: Wheel passage representation in FE model

3.1.4 Seasonal Thermal Load

As previously mentioned, the effect of seasonal thermal stresses as a result of the difference between the stress free temperature or the rail temperature in which the weld is installed and the service temperature or the ambient temperature in which the railway operates is considered in this study. The stress free temperature or the reference temperature (in FE analysis) is set to 35 oC (the nominal stress free temperature for the railway considered may vary between 35 to 40 oC and in practice a greater range may also exist. Hence, a sensitivity analysis on thermal loading will be reported in Chapter 4). The service temperature or the simulation temperature (in FE analysis) is set to 19

53 oC which represents the average ambient temperature during cold months of the year. In this condition the rail is normally under seasonally induced static tensile stress which is believed to facilitate initiation and propagation of straight breaks as previously discussed. The other factor necessary to simulate thermal stresses is the linear coefficient of thermal expansion which is set to 1.2e-5 /oC for rail steel. The mentioned parameters are used in all the simulations in this study unless otherwise specified. Table 3-1 summarizes the model parameters used in FE simulations.

Table 3-1: Summary of the finite element model parameters Rail section (kg/m) 68 Sleeper Poisson’s ratio 0.18

Rail length (mm) 3237 Ballast young’s modulus (MPa) 110

250*237 Sleeper section dimensions (mm) Ballast thickness (mm) 450 *180

Sleeper effective length (mm) 834 Ballast stress distribution angle ( o) 35

Sleeper span (mm) 600 Ballast vertical stiffness (MN/m) 123

Ballast horizontal stiffness per meter Axle load (kN) 343.4 5 of track (MN/m 2)

Rail young’s modulus (GPa) 205 Thermal expansion coefficient (/ oC) 1.2e-5

Rail Poisson’s ratio 0.3 Reference temperature ( oC) 35

Sleeper young’s modulus (GPa) 47.5 Simulation temperature ( oC) 19

3.1.5 Meshing

Two types of 3D continuum finite elements were used in this analysis to mesh the weld, rail and the supporting sleepers. Since only the regions far from the running surface are under investigation, the mechanical behaviour was deemed elastic for all FE elements. The FE mesh for weld region was designed using 10-node tetrahedral structural solid elements SOLID187. This element has a quadratic displacement behaviour and is very suitable for meshing complex geometries like aluminothermic weld exterior shape. For other parts of the model including the rail and sleepers, a more numerically efficient element SOLID185 was used. Solid185 is an 8-node brick structural solid element

54 which has a linear displacement behaviour. This element is widely used for modeling general 3D solid structures. However, for the case of complex geometries it may degenerate into 6-node prism or 4-node tetrahedral elements that have low accuracies particularly at locations with high stress gradients. Figure 3-7 illustrates the two solid structural elements SOLID187 and SOLID185.

Figure 3-7: Finite elements used is study: (a) SOLID185 for sleepers and rail; and (b) SOLID187 for aluminothermic weld [122]

For the contact between the rail and the supporting sleepers or different sections of the rail, a bonded contact algorithm with pure penalty formulation is used. The contact surfaces are represented by the two surface elements CONTA174 and TARGE170 each of which lying on one surface of the contact pair.

The element or mesh sizing is an important part of the model construction since inconsistent element sizes (coarse meshing) and shapes (highly distorted or stretched elements) can both affect the convergence of the FE analysis and/or reduce the accuracy of results. ANSYS 12.1 has provided an advanced meshing tool and quality check to avoid occurrence and early detection of such defective elements. However, the suitable mesh sizing for different regions of the structure depends on the type of loading, related geometry and the required accuracy and still needs user discretion. In this study several sensitivity analyses were performed to determine the suitable mesh sizing for the sleepers, rail and weld regions. The objective is to find a sufficiently fine mesh size below which the element size does not noticeably alter the stress or strain results at the

55 locations of interest in the structure. In the meantime, the constructed mesh should not be too fine so as to extensively increase the number of elements and reduce the efficiency of solution. Of particular importance in mesh sizing is the collar edge of the weld where the geometry is inhomogeneous and stress concentration is relatively high. The stress distribution readings at this region should also be very accurate since the stress history is later used for fatigue analysis and crack initiation studies. A sample of these sensitivity analyses is presented in Table 3-2 where different stress components at two locations of the collar edge root are compared with respect to the element size used.

Table 3-2: Stress values in two locations of the weld collar with respect to element size for the case of a vertical load located at the centreline of the weld.

Underhead Fillet Base Fillet

Element Longitudinal Equivalent Maximum Longitudinal Equivalent Maximum Size Stress Stress Shear Stress Stress Stress Shear Stress (mm) (MPa)S (MPa)S (MPa) (MPa)S (MPa)S (MPa) τ τ 0.05 191.8 196.8 109.8 423.9 421.6 233.2

0.1 191.5 195.4 108.9 423.1 420.7 231.8

0.15 190.1 194.8 107.9 421.7 414.9 228.9

0.2 186.9 192.4 107 414.8 406.2 223.3

0.25 183.9 184.8 102.9 392.6 395.1 215.5

0.3 178.7 173.5 94.8 353.4 374.7 205

It is apparent from the above values that by reducing the element size at the collar edge, the model can better predict the stress concentration. However, more refinement in element size from 0.1 mm to 0.05 mm has little effect (less than 1%) on stresses and so the former size seems both sufficiently accurate and numerically efficient. The finite element mesh of the track resulted from the sensitivity analysis is depicted in Figure 3- 8. The following mesh consists of 371,106 SOLID187 tetrahedral elements mostly used for weld region and the surrounding area, 16,192 SOLID185 brick elements for rail and sleepers and 561,046 nodes in total.

(a)

(b)

(c)

Figure 3-8: Finite element mesh of the structure: (a) Track model, (b) Weld region magnified; and (c) Collar edge magnified

3.1.6 Track Length Sensitivity Analysis

As a matter of fact the railway track is a continuous structure but due to restrictions in finite element analysis, modeling of this structure as in real life is computationally expensive. That is why the continuation of the track was simplified by application of longitudinal boundary conditions (restriction in x displacement) at the two rail ends. Here, the objective is to find a modelled track length in which the simplified applied boundary condition at the rail ends does not significantly affect the stress distribution of the weld region. Accordingly, a sensitivity analysis was performed on the modelled track length. Three track models were constructed each of which with a specific length increasing from 3237 mm (generally used throughout this study) up to 5637 mm (Figure 3-9). The simulation is based on a vertical load located at the centerline of the weld and rail representing the contact condition in a tangent track. Table 3-3 presents stress values evaluated at the underhead radius, base fillet and midway on the web buttress of the weld for the three track models.

Figure 3-9: Track models used for the length sensitivity analysis

Table 3-3: Stress values in three regions of the weld versus the track length modelled Underhead Fillet Base Fillet Web

(Point A) (Point C) (Point B) Track

Length (MPa) (MPa) (MPa) (MPa)S (MPa)S (mm) S S S

5637 192.6 197.3 422.2 419.8 -44.7

4437 191.7 197.1 422.1 419.3 -44.9

3237 191.5 195.4 423.1 420.7 -45.2

As the table suggests, the stress values for the three regions of the weld do not change by more than 1% with respect to the track length modelled. This clearly shows that for the three track lengths the simplified boundary condition applied at the rail ends does not notably affect the stress distribution of the weld and so the use of shorter 3237 mm long track is computationally economical and also leads to satisfactory results.

3.1.7 The Effect of Simplification in Contact Pressure

As previously mentioned, a simplification was made on the contact pressure that is the actual Hertzian contact pressure in the form of an ellipsoidal function (Equation 3-7) was replaced by a uniform contact pressure equivalent to the mean pressure.

(3-7) 3 (, ) 1 − − 2

Where, is the maximum contact pressure, is the vertical applied load, and are the contact patch semi axes and and are the coordinates of any point on the patch with respect to a coordinate system lying on the centre of the patch.

A validation study was performed to compare the effect of this replacement on the stress distribution at the underhead region (below rail head) and to verify the acceptance of this simplification. Three finite element models were studied, the first one with a uniform contact pressure applied on an elliptical patch located at the centerline of the weld and rail and the second with a Hertzian contact pressure distribution applied on the same patch size. The third model consists of the finite element contact simulation between the rail and wheel as shown in Figure 3-10. In this simulation the vertical load is applied on the bearing surface of the wheel bore. A displacement boundary condition was applied on the bore surface to restrain movement in any direction except vertical direction. The element size applied on the contact regions of the wheel and rail was set to 5e-4 m and the minimum time step (in the iterative solver) to 1e-5 seconds to achieve a stable convergence.

Figure 3-10: Finite element simulation of real contact between rail and wheel: Equivalent (von- Mises) stress distribution

Table 3-4 shows some stress values at different regions of the weld evaluated using each simulation. The stress readings for the three regions of the weld clearly show that the substitution of the Hertzian pressure with a uniform mean contact pressure has no significant effect on the bending behaviour of the weld regions below the head. This is expected since the regions of interest are far from the running surface and the stresses observed in these locations arise from the total beam section bending behaviour on the sleepers rather than the contact stresses at and below the running surface.

Table 3-4: Stress values in three regions of the weld versus the contact pressure distribution Underhead Fillet Base Fillet Web

(Point A) (Point C) (Point B) Contact

Pressure (MPa) (MPa) (MPa) (MPa)S (MPa)S Function S S S

Hertzian 193.9 197.6 423.1 420.2 -45.0

Uniform 191.5 195.4 423.1 420.7 -45.2

FE Contact 193.8 197.1 422.1 419.8 -44.7 Model

3.1.8 Model Validation Using Experimental Results

The stress distribution achieved through finite element analysis was compared with results of previous strain gauge measurements (Mutton et al. [43]) on a rail undergoing 3-point bending test. Stresses were measured at locations on the collar surface (for vertical stress measurement) and at the vicinity of the collar edge (for the longitudinal stresses). The support span used for the 3-point bending setup was 600 mm and a vertical load of 200 kN was applied through indentations machined onto the head of the rail at lateral offsets of 0 mm and 22.5 mm from the rail centerline towards the gauge side (Figure 3-11). A similar FE model was also constructed to simulate in-lab experimental setup and conditions. However, no thermal stresses (arising from temperature variations) were involved in both FE model and the experimental test.

Figure 3-11: 3-point bending experimental setup for measurement of longitudinal and vertical stresses on the weld region

Figure 3-12 illustrates the vertical stress on the collar surface, and longitudinal stress at the collar edge, under central loading and eccentric loading (22.5 mm offset from rail centerline) respectively. The consistency between the FE results and the strain gauge measurements is satisfactory for most of the results although the strain gauge readings are sporadic at some locations. The difference between the theoretical and experimental results and the irregularities of the strain gauge measurements are attributed to the following reasons:

• General scatter inherent in the strain gauge measurement technique.

Figure 3-12: Comparison of FE results with strain gauge measurements: (a) Vertical stress on the centerline of the weld collar, (b) Longitudinal stress on the rail surface 5 mm distant from the weld collar edge for central loading, (c) The same stress component for eccentric loading (strain gauge measurements from Mutton et al. [43])

• Stress measurements at locations with high stress gradients (i.e. longitudinal strain measurements close to the collar edge) were sensitive to both small variations in the position of the strain gauge relative to the edge of the collar and collar edge geometrical features. Due to the cast like nature of the process, the collar edge features may be slightly different on the four sides of the weld. • Use of uniaxial strain gauges at locations where the strain behaviour is biaxial, particularly below and above the neutral axis for both longitudinal and vertical strain measurements. • Vertical strain measurements at locations where the surface is not oriented in vertical direction, especially on the weld collar neck. • A slight rotation of the rail at the location of supports was evident in tests particularly under eccentric loading while a fixed support was used in the FE model.

3.2 Free Rolling Condition (Tangent Track)

The stress distribution analysis was primarily performed for the tangent (straight) track in which the contact patch is approximately located at the centerline of the rail. It is assumed that no lateral traction as a result of vehicle steering and no longitudinal force due to acceleration or braking exist at the contact patch so the rail/wheel contact experiences a free rolling condition. It is worth mentioning that in free rolling condition friction forces may still exist at the contact patch however, the net traction force is zero since the contact frictional forces are in equilibrium. This assumption can safely be applied for the normal operation in tangent tracks.

Figure 3-13 shows the contour of equivalent (von Mises) stress at the weld region. Since the load application is symmetric, the same stress distribution can be observed on the other side of the weld. As the figure suggests, high values of equivalent stress is present on the collar edge of the weld due to the effect of stress concentration or the notch effect. The equivalent stress reaches its maximum value at the base region (particularly base fillet) where the normal stress as a result of rail section bending is highest and it reduces from the base towards the mid web where the neutral axis of the section lies.

Figure 3-13: Equivalent stress distribution on the weld exterior surface; High stress concentration is observed throughout the collar edge with maximum values at the base region

As previously mentioned the vertical straight breaks nucleating at the collar edge are driven by the longitudinal tensile stresses and accordingly, the contour of this stress component is of great importance in defining the critical regions and possible failure nucleation sites. Figure 3-14 illustrates the longitudinal stress distribution on the weld exterior surface in a tangent track operation.

Underhead Radius

Base Fillet

Figure 3-14: Longitudinal stress contour under central loading (tangent track)

Similar to the equivalent stress contour, high stress concentration is evident throughout the collar edge with the maximum value at the base region. The total stress distribution is in agreement with the section beam bending having high tensile stresses at the base and high compressive stresses at the rail head. However, the seasonal thermal effect which develops longitudinal tensile stress in the whole rail section has resulted in (through superposition) an increase in the severity of tension at the base and reduction in compressive stress at the rail head.

Beside the severe tension at the base, a local tensile stress region is also observed at the underhead region (depicted by a circle), however with lesser extent. The occurrence of this tensile region at the underhead is somewhat in contradiction to the total section

65 bending behaviour due which a compressive stress similar to the surrounding area is expected. This tensile stress described by Eisenmann [123] and Sugiyama et al. [124] is attributed to the local bending of the rail head on the web. The rail head can be assumed as a standalone beam which is elastically supported by the rail web and so the additional stresses developed in the rail head are governed by the rail head section characteristics such as the rail head neutral axis and moment of inertia (Figure 3-15).

Figure 3-15: Local bending behaviour of rail head on web and the resulting longitudinal stress S X

under a bending moment M Y. The local stress S X is superimposed with the longitudinal stress developed by rail section bending behaviour to form the total longitudinal stress at the rail head

The tensile stress at the underhead region is often referred to as tension spike since it has a narrow width of about 100 to 150 mm occurring as a short duration stress cycle under the wheel. Figure 3-16 illustrates the longitudinal stress variation and the characteristics of the tension spike at the underhead region of a plain rail and an aluminothermic weld with and without the effect of seasonal thermal stresses. For the plain rail and weld without thermal effect the longitudinal stress changes sign from compressive to tensile due to the rail head local bending behaviour. The effect of seasonal stresses is to increase the total longitudinal stress and generally shifting the stresses from compressive to tensile. However, in aluminothermic weld the tension spike does not occur at the centreline of the weld because at this location of rail head the weld is reinforced with the collar and has a high moment of inertia. On the other hand, the spikes occur on the collar edges and have highly localised and intensified characteristics which are in fact due to high stress concentration or the notch effect.

(a)

(b)

Figure 3-16: Variation of longitudinal stress at the rail underhead (on a path located 31 mm from rail centreline) under central loading (tangent track) inclusive and exclusive of the seasonal thermal effects: (a) Rail with no weld installed; and (b) Rail with aluminothermic weld. Shaded area shows the rail head length affected by the local bending of the rail head

Tension spikes could be detrimental both to the fatigue cracking of the underhead region of the weld as will be shown in the next chapter and to the rolling contact fatigue cracks approaching the underhead region from the rail surface. The tensile stress region at the underhead is particularly damaging for the growth of RCF cracks in heavily worn rails where the running surface is quite close to the underhead region and so the extended RCF cracks could more easily grow under mode I crack propagation leading to transverse or detail fractures [36].

3.2.1 Longitudinal Residual Stresses

Strain gauge residual stress measurements have revealed that the longitudinal stress is highly compressive at the underside of the weld foot, whereas significant tensile residual stresses occur at the top of the foot and underhead radius regions. According to measurements by Mutton et al. [43] the longitudinal residual stress at the underside of the foot can vary between about -284 MPa and -183 MPa, at top of the foot between 120 MPa and 298 MPa and at the underhead radius between 121 MPa to 300 MPa. Figure 3-17 shows the total longitudinal stress range resulting from the superposition of finite element stresses (due to service loading) with the residual stress measurements at three location of the collar edge suspected as straight break nucleation sites.

Figure 3-17: Residual stress range and total longitudinal stress range at three locations of the weld suspected for straight break formation

Although, the longitudinal stress at the base region is highest among other locations, the total stress observed at this location is comparatively low due to high compressive residual stress. However, at both underhead and top of the rail foot the total resulting stress is highly tensile which indicate that these locations may be critical in terms of straight break formation.

3.3 Contact Patch Lateral Location

As previously mentioned, the contact location in tangent tracks is generally at the vicinity of the rail centerline. However, under some operational conditions such as steering in curved tracks or straight tracks in which vehicle hunting occurs, the contact patch is displaced. The actual location of the contact depends on many factors among which are the wheel and rail profiles, the radius of track curve and whether the leading or trailing wheelset are under investigation. In shallow to medium curves the leading wheelset moves towards the high rail (outer rail) and the trailing wheelset towards the low rail (inner rail). Under this condition the contact between the leading wheel and high rail is displaced towards the gauge side of the rail. In very sharp curves both the leading and trailing wheelsets move towards the high rail and so all the contacts on the high rail will be displaced towards the gauge side and even flange contact may occur. Figure 3-18 shows the possible contact position on the high and low rail running surface of a leading wheelset during steering in a curved track.

Figure 3-18: Possible contact patch locations during vehicle steering in a medium radius left hand curve (adopted from Telliskivi et al. [125])

The eccentricity of the contact patch and the applied load brings about additional torsion and bending on the rail and weld section and it will definitely change the stress distribution on the weld surface. In this study, the effect of contact patch eccentricity was simulated through moving the contact patch from the rail centreline towards the gauge side in steps of 5 mm distance up to 25 mm distant from the rail centreline (Figure 3-19). The contact patch dimensions and the associated contact pressure may change slightly according to the curvatures of the rail and wheel at the point of contact. However, as previously studied on Section 3.1.6 the pressure distribution on the contact patch has a negligible effect on the stress distribution at the underhead regions of the rail and weld. Accordingly, a similar contact patch and pressure distribution were incorporated for the load eccentricity at the running surface.

Figure 3-19: Contact patch locations on the rail running surface for the study of aluminothermic weld stress distribution under eccentric loading

Figure 3-20 illustrates the longitudinal stress contour on the gauge side of the weld collar under an eccentric load located 20 mm offset from the rail centreline. The contour shows elevated tensile stresses at the underhead region of the weld which is attributed to the same local bending behaviour of the rail head. However, the increased severity has its origin in the developed moment Tx (twisting moment) which results from the eccentricity of the applied load (Figure 3-21). In fact it is not expected that the twisting

70 moment on the rail head directly affects the underhead radius. However, as a result of the local twisting deformation enforced on the rail head and the resistance of the sleeper supports and the rest of the rail length to this deformation (Figure 3-22) a longitudinal strain and its associated stress is developed at the underhead region. This stress is additive to the local tension spike which results from the rail head bending on the web.

Figure 3-20: Longitudinal stress contour at the gauge side of the weld: eccentric loading, contact patch 20 mm offset from rail centerline towards the gauge side

The longitudinal stress at the base region does not change significantly compared to the central loading (tangent track). In fact, the total rail section bending behaviour in XZ plane has not notably changed since the wheel vertical load F z and the associated bending moment M y are constant.

Figure 3-21: Developed twisting moment and the resulting deformation

(a)

(b)

Figure 3-22: Local lateral deformation of the rail head at the location of wheel load induced by Tx: (a) Top view; and (b) Front view

The variation of the longitudinal stress and the associated tension spike with respect to the contact patch lateral location is depicted in Figure 3-23. As the figure suggests the effect of eccentricity is pronounced at the collar edge due to the high stress concentration whereas at centre regions a comparatively small influence is observed mostly attributed to the vertical support provided by the weld buttress.

Figure 3-23: Variation of longitudinal stress at the rail underhead (on a path located 31 mm from rail centreline) under eccentric loading with different eccentricities

For other regions of the rail underhead far from the weld centerline the effect of eccentricity virtually vanishes as the distance increases. Since the effect of rail head local bending behaviour is restricted to a short length (as depicted in Figure 3-16), the influence of load eccentricity at the underhead is also locally restricted. The occurrence of tension spikes in curves can also be observed through strain gauge measurements of the rail and weld under normal service. Although, it is impossible to measure the severity of stress at the collar edge due to the limitations in the strain gauge measurement technique similar (however, with less severity) tension spikes at the vicinity of collar edge are evident (Figure 3-24).

Tension Spike

Figure 3-24: Strain gauge measurement of the longitudinal stress at the gauge side underhead of a wide gap aluminothermic weld; data for a high rail of a 68 kg/m section in a 918 m radius curve subjected to high axle load conditions. Figure shows several wheel passages. [16]

Beside, the underhead region affected by the eccentricity of the applied load, the developed longitudinal bending moment M x (Figure 3-21) induces vertical bending stresses on the web regions of the weld. As a result, a compressive stress develops on the gauge side and a tensile stress at the field side of the rail.

Figure 3-25: Contour of vertical stress on the field side of the rail under an eccentric load located 20 mm from the rail centreline towards the gauge side

Figure 3-25 illustrates the contour of the vertical stress (S z) on the field side of the weld under an eccentric load located 20 mm offset from the rail centreline towards the gauge side. The Figure suggests the maximum vertical stress occurs at the mid web region of the weld where the horizontal split web (HSW) failures usually initiate. The mid web region in a sound weld features a nearly smooth surface so no stress concentration is involved and the stress values may not be high enough to nucleate cracks at this region. Nevertheless, the cyclic vertical stress as will be seen in later chapters has a prominent role in initiation of fatigue cracks from existing defects or cracks on the web of defective welds.

Figure 3-26: Variation of longitudinal stress (L.S.) and vertical stress (V.S.) at different regions of the weld with respect to the load eccentricity

The results of the stress distribution sensitivity to the contact patch lateral location (eccentricity) are summarised in Figure 3-26 for four regions of the weld which are noteworthy in terms of straight break and horizontal split web formation. The underhead region is the most affected in which the longitudinal stress increases dramatically with the contact patch lateral location. Considering the longitudinal residual stress range, the stress can reach up to 700 MPa which is quite close to the yield point of the material

(about 740 MPa for the considered weld material). The web region experiences a linear increase in the vertical stress changing from compressive to tensile as a result of a linear rise in the developed lateral bending. For the base region and the top side of the foot the trend is still linear however, with a slight decrease in longitudinal stresses attributed to the small changes in the total section bending behaviour by the introduction of lateral bending and twisting.

3.4 Contact Tractions

Beside the vertical force resulting from the axle load, there are also longitudinal and lateral forces acting on the surface of the contact patch. The longitudinal forces usually occur as a result of acceleration or braking of the vehicle. However, they will also develop during curving or steering due to the resistance of the bogie’s suspension (yaw suspension) to the rotation and the radial alignment of the leading and trailing axles. Lateral tractions mainly develop in curves and sometimes in tangent tracks prone to vehicle hunting. High lateral traction in curves results from the resistance of the wheelset to the curving and increases by the rise of the wheel-rail angle of attack during the curve negotiation. In extreme conditions (very sharp curves or high speeds) these forces can lead to vehicle derailment. However, based on the type of wheelset (leading or trailing), radius of the curve and whether the rail is the high rail or low rail the lateral traction can be inward (towards the centre of the curve) or outward. Figure 3-27 shows the longitudinal and lateral tractions acting on the leading and trailing wheelsets of a two bogie passenger coach travelling on a 1000 metre radius curve.

Calculation of tractive forces requires analytical or numerical multi-body dynamic analysis and depends on many factors such as wheel-rail profiles, curve radius, vehicle speed, and friction condition of the contact region. Commercial Multi-Body-System (MBS) software such as GENSYS, ADAMS and VAMPIRE are capable of determining contact traction forces in different operational conditions. Although, the calculation of these forces is beyond the scope of this study, it can be assumed that their value is proportional to the applied vertical load. Accordingly, the sensitivity analysis on the

76 effect of the longitudinal and lateral tractions can be performed by varying the ratio of these forces to the vertical load (L/V ratio) defined as the traction coefficient.

Figure 3-27: Longitudinal and lateral tractions acting on the leading and trailing wheelsets of a two-bogie passenger coach in a 1000 m curve. The position of each wheelset is schematically shown with respect to the flange way gap and the value of forces is shown by the length of the corresponding arrows [126]

In this study a range of traction coefficients from 0 (free rolling) to 0.4 were simulated for both lateral and longitudinal tractions. Although, the high rail mostly experiences outward lateral traction (towards the field side), both outward and inward (towards the gauge side) directions were considered in the stress analysis to evaluate and compare the results in each case. Figure 3-28 illustrates the longitudinal stress contour at the field side of the weld for a lateral traction coefficient of 0.4 directed towards the field side and located at 15mm from rail centerline towards the gauge side.

The contour clearly shows the increased value of tensile stress at both underhead radius and base regions of the field side towards which the lateral traction is directed. These observations can also be described by the bending behaviour and deformation pattern of the rail section. As the deformation on Figure 3-29b suggests the lateral force on the contact patch produces a high clockwise twisting moment (Tx) and also a lateral bending moment (M z) on the rail section. It is interesting to note that the clockwise twisting moment produced by the lateral force counteracts the counterclockwise

77 twisting moment resulting from the eccentricity of the applied load. As previously described the total resulting clockwise twisting moment produces a high tensile stress at the field side underhead radius. The lateral bending moment (M z) affects both underhead and the base region by inducing tensile bending stresses.

Figure 3-28: Longitudinal stress contour at the field side of the weld for an eccentric load located 15 mm offset from the rail centerline towards the gauge side with a lateral traction directed towards the field side (outward) with a coefficient of 0.4

Figure 3-30 depicts the longitudinal stress contour at the gauge side of the weld for an eccentric load located 15mm offset from rail centerline towards the gauge side with a lateral traction directed towards the gauge side with a coefficient of 0.4. The severity of tensile stress at the underhead region is significantly higher than the previous case of an

78 outward traction although the base region virtually experiences the same stresses. This indicates that an inward traction is much more detrimental at the underhead region than an outward traction of the same coefficient. There are two main reasons for such observation. In an inward traction both the twisting moments resulting from the eccentricity of the vertical applied load and the inward lateral force are counterclockwise and so the total twisting moment is much higher than the previous case of outward traction. The higher lateral deformation of the rail head under inward traction also proves this finding (Figure 3-29a). In addition, for an inward traction since the gauge side underhead is located under the vertical load, it directly experiences an additional local tension spike arising from the eccentricity of the axle load whereas, the underhead at the field side in an outward traction is not subject to such tension spike.

Similar to the case of outward traction the lateral bending moment (M z) provides an additional tensile bending stress to both underhead and the base regions.

(a) (b)

Figure 3-29: Lateral deformation of the rail section under a tractive load applied 15 mm offset from the rail centreline and L/V ratio of 0.4: (a) Inward traction, and (b) Outward traction

Figure 3-30: Longitudinal stress contour at the gauge side of the weld for an eccentric load located 15 mm offset from the rail centerline towards the gauge side with a lateral traction directed towards the gauge side (inward) with a coefficient of 0.4

Figure 3-31 gives a better insight into the effect of lateral traction and its direction on stress distribution at different regions of the weld. As expected the longitudinal stress at the underhead and base fillet of the field side increase with the rise of the traction towards the field side and this increase is significant at the underhead region showing the great effect of local bending of the head. The graph also shows that an inward lateral traction alleviates the stresses at the two regions of the field side mentioned. However, both the longitudinal stress at the top of the foot and the vertical stress at the mid web of the field side increase by the introduction of an inward traction.

(a)

(b)

Figure 3-31: Variation of longitudinal stress (L.S.) at the underhead, base and upper foot region and vertical stress (V.S) at the mid web with respect to different coefficients of lateral traction and the direction: (a) Field side; and (b) Gauge side of the weld. All the measurement points are on one side and the dimensions are based on Figure 3-26.

The increase in the vertical stress at the field side can be described by the development of a longitudinal bending moment M x and its rise with higher inward tractions. On the gauge side an opposite behaviour is observed for the underhead and the base fillet as previously described. Similar to the field side (however, with more severity due to the additional tension spike described earlier), an abrupt increase can be observed at the underhead radius with the introduction of higher inward tractions. Considering the normal range of the longitudinal residual stress (100 to 300 MPa), the total stress at the underhead radius exceeds the elastic limit of material. Under cyclic loading, this can increase the risk of low cycle fatigue if elastic shakedown does not occur. However, this cannot be directly verified since the plastic deformation can release the stress concentration and alleviate the severity of longitudinal stress at the collar edge. In addition, the occurrence of high inward tractions in service is less likely than high outward tractions. For the base fillet in both sides of the weld, the longitudinal stress is expected to be alleviated for all traction conditions due to the high compressive residual stress (up to -300 MPa) at the underside of the foot.

For the longitudinal stress at the top side of the foot and the vertical stress at the mid web of the gauge side an opposite behaviour to the field side is observed and they decrease with the application of an inward traction. Similar to the field side the development of higher vertical compressive stresses is attributed to the bending moment

Mx which linearly depends on the value of the lateral traction.

The effect of longitudinal traction as a result of vehicle acceleration or braking was also studied through considering longitudinal traction coefficients ranging from 0 to 0.4. Figure 3-32 shows the variation of longitudinal and vertical stress at the underhead, base and mid web region respectively based on different longitudinal tractions for a central load (representing operation in tangent track) with no lateral traction. It is obvious from the results that the longitudinal traction irrespective of its magnitude has virtually no effect on the stress distribution of the weld for regions below the underhead. In fact no specific loading scenario is experienced by the depicted regions under a longitudinal traction. Nevertheless, both the longitudinal and lateral tractions play significant roles in failure modes of the running surface such as RCF and wear.

Figure 3-32: Variation of longitudinal and vertical stresses with respect to the longitudinal traction

The results presented for the effect of contact patch lateral displacement and lateral traction (as a representative of curving and hunting behaviour) combining with the range of longitudinal residual stresses show that the underhead region of the weld may be a critical location in terms of fatigue crack initiation. For both field and gauge side of the weld, the longitudinal stress at top of the foot (incorporating high tensile residual stresses at this region) may have an additive role in development of fatigue failures or growth of the existing cracks. However, despite the presence of high service stresses at the base region the total longitudinal stress (bending stress plus residual stress) is less severe than the underhead due to the effect of high compressive residual stress at the base.

The results seem consistent with the failure modes observed under in-service loading. Fatigue cracking at the underhead radius is a common failure mode in the type of weld considered, and mostly occurs in short radius curves and tangent tracks prone to vehicle hunting. The incidence of fatigue cracking at the top of the foot is also high however all failures at this region have been associated with surface features such as cold laps, which act as crack initiators. On the other hand, few failures have been reported initiating from the underside of the rail foot in normal operations.

3.5 Track Support

Track bedding like any other track component deteriorates over time as a result of repeated and uneven service loading. Deterioration is described as the permanent (sometimes called plastic) deformation of ballast or subgrade which may not be uniform along the track since the service loading is irregular. Settlement of the track bedding may bring about poor support under some sleepers, and ultimately a total lack of support or complete looseness. This phenomenon is considered a closed loop since the lack of support brings about increased service loading and an increase in the loading more readily affect the track support integrity. The extent of this interaction needs careful study of vehicle-track dynamic behaviour under the influence of support voids and bedding settlement as described in the references [127-129] and depends on many factors including the vehicle speed, the number of defective-support sleepers involved and the amount of gap developed underneath each sleeper.

Aluminothermic welds are among the locations of the track where the rate of ballast deterioration is fairly high. The problem is mostly due to the irregularities at the surface of the weld head, which impose high dynamic loads on the track bedding and cause ballast pumping. Weld battering (local plastic deformation and dipping of the weld running surface) due to lack of consistent hardness, and vertical misalignment of the rail heads prior to welding, are among weld surface irregularities. In fact the lack of support at the vicinity of aluminothermic welds affect the bending behaviour and stress distribution in critical locations of the weld and may influence the weld performance. According to Ishida et al. [128] loose sleepers can substantially decrease the fatigue life of welds and so regular track maintenance is necessary for their safe operation.

In this study the effect of deficiency in track support on the weld bending behaviour is investigated through the variation of ballast vertical stiffness under the weld two adjacent sleepers which are subject to the largest effect of impact forces. Figure 3-33 illustrates the variation of longitudinal and vertical stresses at four regions of the weld under different ballast stiffness ranging from complete looseness to full support. Two cases of central loading (tangent track) and eccentric loading (curved track) located 15

84 mm from the rail centerline towards the gauge side with an outward lateral traction coefficient of 0.3 are considered.

(a)

(b)

Figure 3-33: Variation of longitudinal and vertical stresses versus different ballast stiffness under the weld adjacent sleepers: (a) Tangent track (central loading); and (b) Field side in a curved track (15 mm load eccentricity towards the gauge side including an outward traction coefficient of 0.3)

The graph shows an increase in the longitudinal tensile stress at the foot region (both base fillet and top of the foot) and a reduction at the underhead radius as the track stiffness decreases, for both tangent and curved tracks. The change in stress distribution

85 is attributed to an increase in the rail bending moment at the weld location, which results in introduction of more tensile stress at the rail foot and more compressive stress at the underhead. The complete loss of support at the sleepers immediately adjacent to the weld is considered to be the worst case. Field observations support the mentioned results since many of the fatigue failures initiating from the foot region occur at locations were considerable ballast pumping has taken place.

By comparison with the top of the rail foot and the underhead region, the vertical stress at the mid web seems to be virtually unresponsive to any change in the track stiffness since the support stiffness does not have a noticeable effect on lateral or transverse bending of the rail section. Nevertheless, it cannot be directly verified that the support has no effect on the fatigue performance of the web region without a vehicle dynamic analysis to investigate the amplification of the applied load under the simulated support looseness, type of weld surface irregularity and vehicle speed. If the load impact factor dramatically increases under a low support condition, the vertical stresses at the web region may rise accordingly and hence the fatigue performance in curved tracks may largely be affected. However, these analyses are well beyond the scope of this current investigation.

In this chapter the bending behaviour of aluminothermic weld under the influence of several operational parameters such as the contact patch eccentricity, lateral tractions and support condition was investigated using a thermo-structural finite element simulation. The results showed that the underhead region experiences a unique state of stress known as tension spike which results from the local bending of the rail head on the web section. This behaviour which has also been observed during in-track stress measurements is exacerbated with the introduction of contact patch eccentricity. Lateral traction was also proved to intensify the stress state at the underhead due mostly to the twisting of the rail section. In the next chapter the stress state determined during a loading cycle using the finite element method is used to investigate fatigue crack initiation mainly at the weld collar edge. Dang Van multi-axial fatigue criterion based on critical plane approach is utilized and the effect of some operational and welding parameters is investigated.

CHAPTER 4 MULTI-AXIAL FATIGUE ANALYSIS

In Chapter 3, load bearing behaviour of the aluminothermic weld under static load located at the centreline of the rail and the resulting stress distribution was studied subject to different service conditions. It was pointed out that the underhead region, top of the rail foot and rail base can experience high stresses due to bending behaviour of the rail head and rail sections. However, static analysis alone cannot be used to examine the most critical locations of the weld and the effect of service conditions in initiation and propagation of straight breaks and horizontal split webs since these failures are permanent fatigue damage resulting from the cyclic characteristic of the loading. In fact, it is necessary to perform fatigue analysis based on the cyclic loading and the stresses which represent the distress experienced by the weld under consecutive wheel passages. It was pointed out in Chapter 2 that the multi-axial fatigue criteria based on the critical plane approach are among the most reliable approaches incorporating the actual mechanism of crack initiation controlled by the cyclic shear stresses and the role of the surrounding normal or hydrostatic stresses. In this chapter fatigue analysis is performed using the Dang Van critical plane approach which is a high cycle fatigue criterion capable of coping with the problem of out-of-phase loading and rotating principal stresses. At the beginning of the chapter the criterion and its implementation using MATLAB software is extensively described. Following, the approach to determine the fatigue parameters of the material and the residual stress distribution are introduced and eventually the criterion is applied to study the fatigue behaviour of aluminothermic weld under several service conditions.

4.1 Dang Van Original Criterion

It was earlier described that fatigue cracks in metallic materials initiate in intragranular slip bands and so the cyclic shear stress at the grain scale is the governing factor. Accordingly, Dang Van [130] in his original work proposed a criterion considering the inelastic behaviour of material at micro-structural scale (at macro scale the behaviour of

87 material can be completely elastic). Two scales are introduced in this criterion; (a) a macroscopic (or engineering) scale which is characterized by a volume surrounding the point of material where the fatigue analysis is performed and (b) a mesoscopic scale (or the micro-structural) scale defined as the subdivision of the previous scale (Figure 4-1).

Figure 4-1: Definition of macroscopic and mesoscopic scale and the associated macroscopic stress ( ) and the mesoscopic stress ( ) () ()

The stresses at the mesoscopic scale are different from the macroscopic stresses since the material in mesoscopic scale undergoes plastic deformation and hardening. In this case the mesoscopic stress can be derived from the macroscopic stress superimposed by the residual stress developed due to plastic deformation at the mesoscopic scale:

(4-1) ∗ () () +

Where, , and are the deviatoric parts of the mesoscopic stress, ∗ () () macroscopic stress and the residual stress tensor. The condition for avoiding crack initiation is that at the mesoscale the unfavourably oriented grains should reach a shakedown condition [92]. Under such a condition, a stable constant mesoscopic residual stress tensor ( ) can be determined from fulfilment of the following min-max ∗ optimization problem (Equation 4-2) [131]:

(4-2) ∗ min max () +

Where, is the second invariant of the deviatoric stress tensor. The above equation can also be described as finding the smallest hypersphere in the six-dimensional space circumscribing the loading path formed by the deviatoric part of the macroscopic stress ( ). The centre of this hypersphere corresponds to or the constant residual ∗ () − stress tensor which enforces the shakedown at the mesoscopic level. The suitable algorithm to determine the radius and centre of the hypersphere has been described elsewhere [86]. The Dang Van criterion can then be written as:

(4-3) () + () > Where, is the instantaneous value of the maximum mesoscopic Tresca shear stress, is() the instantaneous mesoscopic hydrostatic stress and and are material parameters () calculated from Equation 2-20. In this original criterion the critical plane on which the fatigue crack is supposed to initiate is the plane of maximum mesoscopic Tresca shear stress.

4.2 Minimum Circumscribed Circle (MCC)

Another form of the Dang Van criterion which is used in the current study is described as follows [91, 131-132]. In this criterion the mesoscopic shear stress is replaced by the shear stress amplitude ( ) on the considered shear plane in the following form as described in Chapter 2: ()

(4-4) () + () > In this case, the critical plane is the shear plane in which the above inequality is satisfied at least once in the whole stress cycle. Although, the two criteria 4-3 and 4-4 are described at two different material scales, they result in the same damage value and critical plane. The effect of mean shear stress is truly absent in criterion 4-4 through the implementation of shear stress amplitude (fluctuations around the mean shear stress). In the same manner the definition and existence of a unique mesoscopic residual stress tensor in Equation 4-2 leads to the removal of the mean shear stress from the criterion

4-3 [92, 131]. It is also proved that the mesoscopic and macroscopic hydrostatic stresses in criteria 4-3 and 4-4 are the same [130]. The difference between the two criteria is that in criterion 4-4 all the shear planes passing through the material point should be checked for the fulfilment of the inequality in the whole stress cycle to find the critical plane whereas in criterion 4-3 the critical plane is known as soon as the mesoscopic stresses are defined.

If we consider a point of a structure subjected to cyclic loading and derive the stresses acting on a plane ∆ passing through that material point, the resulting stress vector ( ) can be decomposed into two components; a normal stress vector perpendicular to the plane ( ) and a shear stress vector ( ) lying on the plane (Figure 4-2).

Figure 4-2: The stress components acting on the plane (∆) passing through the material point O subjected to cyclic loading [133]

As Figure 4-2 illustrates, tip of the stress vector ( ) forms a closed spatial curve ( Φ) during the stress cycle. Similarly a closed planar curve ( Ψ) is developed by the tip of the shear stress vector ( ) and lies on the considered plane. In this case, the shear stress vector acting on the shear plane ∆ changes both in magnitude and direction during the loading cycle. The main problem here is how to find the shear stress amplitude ( ) on plane ∆. Since the shear stress is a vectorial function of time the determination () of mean shear stress and shear stress amplitude is not a trivial task [133]. It is also worth mentioning that for other planes passing through point O the evolution of the shear

90 stress vector and the associated planar path ( Ψ) will be different form one plane to another and so the amount of shear stress amplitude should be defined for each plane considered.

Several attempts have been made to define the mean shear stress and the shear stress amplitude vectors among which the methods proposed by Grubisic and Simbürger (the longest projection proposal) [134], Lemaitre and Chaboche (the longest chord proposal) [135] and Papadopoulos (minimum circumscribed circle) [133] are noteworthy. In the longest projection method, the projection of the shear path ( Ψ) on all the lines passing through the origin (O) and lying on the plane ∆ are defined and half the length of the longest projection is determined as the shear stress amplitude (Figure 4-3a). On the other hand, the longest chord proposal describes the shear stress amplitude as half of the maximum distance between any two points on the shear path (Figure 4-3b). Unfortunately there are examples where the two mentioned proposals fail to correctly calculate the shear stress amplitude and so their application is limited to certain types of shear path [136-137].

(a) (b)

Figure 4-3: Definition of the shear stress amplitude ( ) and mean shear stress ( ): (a) The longest projection method; and (b) The longest chord method

The minimum circumscribed circle (MCC) was first introduced by Dang Van [138] although the correct formulation was originally implemented by Papadopoulos [133]. The idea comes from the fact that the smallest circle encompassing a planar curve is

91 unique. The maximum shear stress amplitude in this method is described by the radius of the smallest circle circumscribing the shear path Ψ (Equation 4-5) and the mean shear stress vector is defined as the centre of this circle in plane ∆ (Equation 4-6) (Figure 4- 4). According to the literature the MCC method is currently the only correct method of calculating the amplitude and mean shear stress in any periodic non-proportional or proportional loading [133, 136-137].

(4-5) max ‖() − ‖

(4-6) min max ‖() − ‖

Figure 4-4: Definition of the minimum circumscribed circle (MCC); is the time dependent shear stress amplitude ( ) and in the figure is the maximum() − amplitude () Equation 4-6 implicitly describes how the minimum circumscribed circle is constructed. One can assume a minimum circumscribed circle with the centre the smallest radius of which is described by the longest distance between its centre and any point on the shear path written as . However, an infinite number of these circles max ‖() − ‖ 92 can be found with different location of the centre or different vectors. The actual minimum circumscribed circle which can be used for the determination of shear stress amplitude and the mean shear stress vector is the one with the smallest radius or the minimum value of as defined in Equation 4-6. This is similar to the definition previouslymax introduced ‖() − for ‖the determination of the minimum circumscribed hypersphere (Equation 4-2) in the original Dang Van criterion.

Before being able to construct the minimum circumscribed circle in a computer program the continuous curved shear path Ψ should be approximated by an n-sided (or n- vertices) polygon with its vertices lying on the shear path. As a matter of fact this has already been done since finite element analysis has been used for determination of the stress cycle. It was previously motioned that the passage of wheel is represented by moving the axle load step by step along the track. This step wise determination of the stress cycle leads to the development of a polygon shear path with the number of vertices equal to the number of steps taken in finite element analysis.

From a mathematical point of view the smallest circle circumscribing an n-vertices polygon can be defined using the following theorem [133, 139]: “The minimum- circumscribed circle to a plane polygon Ψ is: either one of the circles drawn with a diameter equal to a line segment joining any two vertices of Ψ or one of the circumcircles of all the triangles generated from every three vertices of Ψ”. To find the smallest circle all the triangles formed by the two points and three points on the path should be drawn and checked. The number of circles that can be formed by two vertices ( ) is calculated as the two-combinations from the n element set of vertices:

(4-7) ! 2 2! ( − 2)! Similarly, the number of circumcircles of the triangles formed by three points ( ) on the path is defined as the three-combinations from the n element set of vertices:

(4-8) ! 3 3! ( − 3)!

Accordingly, the algorithm to find the minimum circumscribed circle, time dependent shear stress amplitude and the critical plane for the considered material point can be written as follows:

1) As previously mentioned all the shear planes passing through the material point should be considered for fatigue damage calculation or checking the fulfilment of the criterion 4-4. For each shear plane the time dependent shear stress vector is determined from the transformation of the obtained stress tensor from the finite element analysis. The end points of the shear vectors are the vertices of the loading path polygon Ψ on the plane considered. 2) The set of all line segments formed by connecting all pairs of the vertices on the polygon is considered (Figure 4-5a). They comprise all the sides of the polygon as well as all its chords. Each line segment can be the diameter of a circle and in total circles are drawn; the radius and the centre of each circle recorded. 3) Similarly all the triangles formed by any three vertices of the polygon are considered (Figure 4-5b). The circumcircles of these triangles are drawn, the radius and centre of which is recorded. In total circles will be considered.

(a) (b)

Figure 4-5: Samples of circles for the combination of vertices of the shear polygon Ψ: (a) Two-point circles; and (b) Three-point circles

4) For any of the circles drawn in steps 2 and 3 the circumscribability (whether the circle contains the whole shear path Ψ) is checked. This is performed by finding the distance from any of the polygon vertices to the centre of the circle

considered. If all the distances calculated are smaller or equal to the radius of the circle, the circle is a circumscribing circle. The centre and radius of all the circumscribing circles are recorded. 5) Among all the circumscribing circles, the circle with the smallest radius will be the minimum circumscribing circle for the shear plane considered. The coordinate of the centre is the mean shear stress vector and the distance from each point of the polygon to the centre of the MCC is the shear stress amplitude for the corresponding time in the stress cycle ( ). 6) The shear plane with the maximum value of () is the critical plane. From now on this value is called the DV damage() + parameter () . If this value is greater than fatigue will occur and the mentioned plane is the fatigue crack initiation plane. 7) Steps 1 to 6 are repeated for all material points considered for fatigue analysis.

Due to the directionality of the criterion and the associated repetitive algorithm, the computational time can be fairly high especially if the number of material points and the stress cycle steps are large. For each shear plane two-point circles should be drawn and for each circle the distance of vertices should be checked from the centre of that circle (for circumscribability).( Similarly − 2) three-point circles are formed and for each circle the distance of vertices to the centre is checked. If is the number of material points and is( the − number 3) of planes investigated for each point, the total number of operations ( ) can be approximately calculated using Equation 4-9. As an example, for about 32400 shear planes investigated for one material point in this study and a shear path polygon of 69 vertices the number of operations is about 1.171e11 which takes about 0.4 hours in a 3.2 GHz Quad core Xeon CPU.

(4-9) ! ! ∗ ∗ ( − 2) + ( − 3) 2! ( − 2)! 3! ( − 3)! 4.3 Estimation of Fatigue Parameters

Evaluation of the Dang Van fatigue parameters and in Equation 2-20 requires the two fatigue parameters of the weld material (fatigue limit in reverse bending) 95 and (fatigue limit in alternating pure torsion). The common approach to measure the mentioned fatigue limits is to perform separate reverse bending and alternating torsion tests using a staircase algorithm first introduced by Dixon and Mood [140]. A condition of the staircase analysis is that the number of samples should be large in the order of 40 to 50. According to Collins [141] at least 15 to 30 cylindrical samples are needed for each of the fatigue tests mentioned to have a good estimate of the mean fatigue limits. This number largely depends on the quality of samples and the associated repeatability of the test. A couple of other conditions also exist in the staircase method such as the fatigue limit being a normal distribution and some knowledge of the standard deviation prior to testing [142]. In the case of aluminothermic welds the sampling method is considered highly erratic and costly due to the following factors:

• The relatively large number of samples needed for fatigue limit evaluation necessitates extraction from a number of welds. Since the whole fabrication process of an aluminothermic weld is manual and operator dependent, no two manufactured welds are exactly the same in terms of material properties. In the meantime casting defects such as pores and inclusions are common in weld materials however, the equality of defect contents in different samples cannot be verified. This inevitably induces large errors in fatigue limit measurements.

• Different samples even from the same weld but various locations of the weld may not exhibit identical fatigue properties since the thermal conditions (preheat, cooling and post heating) and the solidification pattern of the weld material may not be uniform throughout the weld section.

• Provision of the required number of samples (30 to 60 samples) with similar surface quality suitable for staircase statistical analysis is costly. And also it cannot be verified that the statistical data maintains a normal distribution.

Hence, a different approach was adopted to estimate the material fatigue parameters based on more reproducible and reliable mechanical parameters such as hardness. Experiments have shown that the ultimate tensile strength of metals correlates well with

96 their hardness values. In this study two experimental relations proposed by Umemoto et al. [143], Equation 4-10 (for single structured pearlitic steel) and Callister [144], Equation 4-11 are used to evaluate the ultimate strength of the weld material.

⁄3.27 (4-10)

3.45 (4-11)

Where, is the ultimate strength of material, and are the Brinell hardness and the Vickers hardness respectively. Schroeder and Poirier [145] made some experiments on the hardness and the ultimate strength relation of aluminothermic weld material (different than the weld considered in this study) on standard and alloy mix portions (Figure 4-6). The mean value of the two predictions 4-10 and 4-11 is found to be consistent to the mean trend line of the measurements and accordingly this mean value is used to estimate the ultimate strength of the current weld under investigation.

Figure 4-6: Tensile properties of weld material with respect to the hardness; red line represents the trend line for the mean values of the ultimate strength [145]

Figure 4-7 shows the variation of Vickers hardness on the rail head, web and the foot regions along the rail for the aluminothermic weld considered in this study. At the edge of the weld collar (20 mm offset from the weld centerline for an overall collar width of 40 mm) which is under investigation for the straight break formation, the hardness value of approximately 330-340 HV is observed at all regions of the weld.

Figure 4-7: Hardness distribution on the central longitudinal plane of the rail in head, web and foot regions of the weld versus longitudinal distance from the weld centreline [19]

According to Lee et al. [146] in mild to high strength steels which have defects approximately smaller than the grain size fatigue cracks initiate within grains and are arrested by the grain boundary. The strength of this boundary is nearly proportional to the hardness and the ultimate strength of the material. So the fatigue limit maintains an approximate linear relation with the ultimate strength of material for strengths below a critical value. The ratio of the reversed bending fatigue limit to the ultimate tensile strength of material ( ) depends on the steel microstructure and varies between 0.25 and 0.6. For pearlitic / microstructure of steel Juvinall [147] proposed the value of 0.38 at 10 6 cycles which is used in this study. According to experiments, fatigue limit in pure torsion is approximately proportional to the fatigue limit in alternated bending. For

98 a defect free (unnotched) material the fatigue limit ratio has the value of about 0.58 [102, 146]. Under the above mentioned consider ations/ the fatigue limit in alternated bending ( ) of the current weld is estimated at 405 MPa and in reversed torsion ( ) at 235 MPa which are used throughout this study unless otherwise stated. 4.4 Residual Stresses

Longitudinal and vertical residual stresses as a result of the welding procedure are incorporated into the fatigue analysis since it was found in Chapter 2 that they have large effects on the stress magnitude at different regions of the weld. The surface residual stress values were obtained using strain gauging and trepanning or ring-core technique [148].

In this method the strain gauge (or rosette) is attached to the surface of the component at the point of interest. Then an annulus with a minimum diameter of 15 mm is machined with a depth of about 10 mm around the strain gauge. Due to the isolation of the formed core island from the surrounding surfaces the stresses on the surface are relaxed. The associated relaxed strains are measured and by applying the generalized Hooke’s law the residual stresses are determined.

For this study the longitudinal residual stresses have been measured at 3 mm offset from the collar edge and the vertical residual stresses measured on collar centerline at the mid web (Figure 4-8) [43]. Measurements were undertaken on a total of six additional test welds; the resulting average residual stress values are superimposed with the stress history at the outset of the multi-axial fatigue analysis through a piece-wise linear model (similar to the method proposed by Josefson and Ringsberg [95] for flash butt welds) based on the strain gauge measurements (Figure 4-9). The scatter in the latter measurements is caused by different factors including the restriction in how close the strain gauges could be attached to the weld collar edge, high gradient of residual stress at the vicinity of the collar edge, the location of strain gauges with respect to the weld fusion boundary and in general inequality of heating and cooling conditions in different welds and even in different sides of the same weld.

L V

Figure 4-8: Trepanning technique for residual stress measurements and the measurement locations [43]

To account for the variation in longitudinal and vertical residual stress among different weld samples, a sensitivity analysis will be performed. The sensitivity analysis is based on the residual stress modes depicted in Figure 4-9. The modes illustrated by piecewise linear representations approximately cover the whole range of residual stress variation in the type of weld considered i.e. between 100 to 300 MPa for the longitudinal residual stress from top of the rail foot to the underhead region and about 100 to 200 MPa for the vertical residual stress on the surface of the mid web. Although, the measured vertical stresses used for the current study does not exhibit such high values, stresses up to 200 MPa has occasionally been reported and so is considered for sensitivity analysis. The results and discussion of the pertaining analysis will be presented later in this chapter.

The minimum circumscribed circle (MCC) algorithm already introduced in Section 4-2 together with the superposition of the residual stresses were implemented in a computer code developed in MATLAB programming software (Appendix A) to perform fatigue analysis. The time dependent stress tensor (cyclic state of stress) obtained in finite element analysis for each material point is exported to the mentioned code where the Dang Van critical plane analysis is performed.

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Figure 4-9: Residual stress measurements using strain gauge method (Mutton et al. [43]) and the piece-wise linear models applied in analysis (Mode 1 horizontal and Mode 2 vertical residual stresses are considered for fatigue analysis and other modes are implemented for residual stress sensitivity analysis as will be described later): (a) Longitudinal residual stress at the collar edge; and (b) Vertical residual stress on the centerline of the weld collar surface

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4.5 Free Rolling Condition (Tangent Track)

Multi-axial fatigue analysis using the Dang Van critical plane approach was performed primarily on the collar edge of the aluminothermic weld to define the critical locations in terms of straight break formation. The first case under investigation is for a contact patch located at the centerline of the weld with no lateral or longitudinal tractions representing the free rolling condition or loading in a tangent track.

Figure 4-10 illustrates the DV damage parameter (the value of the left side of the Dang Van inequality on the most critical shear plane) at the collar edge of the weld versus the height from the rail foot. If the value of the damage parameter in any point of the collar edge exceeds the critical value ( in inequality 4-4) fatigue crack will nucleate at the associated point. In this study the value of is determined at 235 MPa based on the estimation of or the fatigue limit in reversed torsion (Equation 2-20).

Figure 4-10: DV damage parameter on collar edge versus height from rail foot in a tangent track

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The damage figure shows a pronounced damage parameter at the underhead radius and less severe damage parameter at the base region. This is in contrast to what was illustrated in Chapter 3 that the bending stress was much higher at the base fillet than the underhead radius (416 MPa versus 189 MPa). In fact this can be described by considering the effect of residual stresses and the localized tension spike. The longitudinal residual stress is highly compressive at the underside of the foot and tensile throughout the upper foot up to the underhead region, which could alleviate the severity of fatigue damage at the base and increase it at the underhead fillet. In addition to high longitudinal residual stress and the localized tension spike the underhead region is very prone to cold laps, a condition in which the weld metal leaks into the gap between the mould and the parent rail and solidifies forming an unfused region. Figure 4-11 shows a cold lap defect formed at the two sides of the underhead fillet of the considered weld. The increased damage parameter evident in the mid-web in Figure 4-10 is also related to high positive hydrostatic stress due to the increased residual stress in both longitudinal and vertical directions.

Magnified view from this side

Figure 4-11: Formation of cold lap at the underhead radius on both field and gauge sides of an AT weld indicated by the white arrows. The cold lap can sometimes contain pores as magnified at the bottom left figure. Pores at the tip of a cold lap could facilitate fatigue crack initiation [149].

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The damage figure 4-10 also shows lower fatigue damage at the top of the foot compared to the underhead radius and base fillet. The tensile longitudinal residual stress at the top of the foot is similar to that of the underhead region. It was also illustrated in Figure 3-17 that the total longitudinal stress (residual plus bending) at the top of the foot may be higher than those in base fillet and underhead radius. So, what is the reason for lower damage parameter at the top of the foot? The Dang Van criterion is described by the superposition of shear stress amplitude and a small portion (about 24%) of the hydrostatic stress on the most damaging () shear plane. In fact the state of shear stress fluctuation on ()the critical plane is the determining factor in high cycle fatigue mechanism not necessarily the state of total stress at the point of material considered. To find the answer for the question, the variation of shear stress amplitude and the hydrostatic stress on the most damaging plane are considered in Figure 4-12 for both the underhead radius and top of the foot (Point A and B in Figure 3-17 respectively).

Figure 4-12: Variation of shear stress amplitude , hydrostatic stress and the DV damage parameter on the most damaging plane at underhead () and top of the rail foot() versus longitudinal distance of the axle load from the weld centreline; maximum DV damage values at the underhead radius and upperfoot are depicted by small circles

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As the figure suggests the hydrostatic stress is notably higher at the top of the foot compared to that at the underhead radius. However, as described earlier the hydrostatic stress has only a partial effect on the DV damage parameter. The shear stress amplitude which has the biggest influence in damage parameter (see Equation 4-4) is much higher at the underhead radius when the load is located between -50 mm to 150 mm distant from the weld centerline. This will result in (by superposing the maximum shear stress amplitude and the associated hydrostatic stress effect) a more pronounced damage at the underhead region compared to the top side of the foot.

However, according to the Dang Van criterion and based on the damage values depicted in Figure 4-10 no fatigue failure is predicted to initiate at the edge of the weld collar (assuming that the weld geometry is as-designed and the material is defect free) in a tangent track since the DV damage value is generally below the critical damage parameter, . 235 4.6 Contact Patch Lateral Location

Curving or hunting causes the contact patch to move away from the rail centerline towards the gauge side and lateral traction forces to develop at the patch. These two effects are separately considered in Sections 4.6 and 4.7. This section solely discusses the effect of contact patch lateral movement incorporating vertical wheel load at all eccentric locations.

The results of the multi-axial fatigue analysis are depicted in Figure 4-13 for an eccentric load located 25 mm offset from the rail centerline towards the gauge side. The graph exhibits the contribution of high cyclic tensile stresses to fatigue damage, with an abrupt increase at the underhead region. In fact, the displaced contact force induces high local bending moment on the railhead and leads to a pronounced tension spike and higher damage at the underhead radius. The predicted damage parameter even reaches the critical damage, which implies that the fatigue failure is expected at the underhead fillet in accordance to the Dang Van criterion. However, fatigue damage at the lower web and foot regions are predicted to be less severe than under central loading.

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Figure 4-13: DV damage parameter at the gauge side collar edge for both central and eccentric loading located 25 mm distant from rail centerline towards the gauge side with no shear tractions; Fatigue region represents the region in which fatigue crack initiation is expected

Figure 4-14 presents the results of fatigue analysis for three locations of interest on the weld collar edge (underhead radius, top of the foot or web-base fillet and base fillet depicted in Figure 4-13) according to the position of contact patch relative to the rail centerline. The graph shows a near linear increase and a slight decrease in the damage parameter of the underhead fillet and base region respectively. Only a minor change in damage parameter is evident at the web-base fillet, confirming little effect of contact patch lateral location at this region. In summary, all three regions of the weld are predicted to remain in a safe condition in terms of fatigue failure, except for the underhead fillet when the contact patch is displaced laterally by a large extent due to curving or hunting behaviour (see the fatigue prone region in Figure 4-14).

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Figure 4-14: DV damage parameter versus the contact patch lateral displacement for the underhead radius, top of the rail foot and base fillet depicted by circles in Figure 4-13; Shaded area shows the fatigue region where crack initiation is expected

4.7 Contact Tractions

As mentioned in the previous chapter lateral traction is considered in this study as representing the steering or hunting behaviour while longitudinal traction is the result of acceleration and braking of the wheel. A range of traction coefficients from 0 (free rolling) to 0.4 were used for both longitudinal and lateral directions, and the lateral traction was deemed to be outwards (towards the field side). The finite element results presented in Chapter 3 showed that under a tractive force towards the field side, the underhead radius undergoes severe twisting and lateral bending which substantially increases the tensile stress at this region. As a result of the lateral bending moment applied on the rail section, the base region also experienced pronounced tensile stresses even higher than that of the underhead radius. The longitudinal traction was shown to have a negligible effect on bending stresses throughout the collar edge from the underhead radius to the base fillet. However, to have a thorough understanding of the fatigue behaviour, multi-axial fatigue analysis is performed for both the cases of lateral and longitudinal tractions.

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The results of the fatigue analysis at the collar edge of the field side are presented in Figure 4-15 for two cases of no lateral traction and traction coefficient of 0.4 towards the field side (outward) on a contact patch located 15mm offset from the rail centerline towards the gauge side. Lateral traction clearly increases the fatigue damage throughout the collar edge except for a narrow band at the upper foot. DV damage parameter at the underhead radius even exceeds the critical value of 235 MPa indicating that the fatigue failure is highly expected at this region. Although, the base region is subject to higher tensile stresses than the underhead the compressive state of residual stress at the base region has resulted in reduced damage compared to the underhead. For other regions of the weld except the underhead no fatigue failure is expected.

Figure 4-15: DV damage parameter at the collar edge of the field side for an eccentric load located 15 mm offset from the rail centerline with lateral traction coefficients of 0 and 0.4 outwards; shaded area shows the fatigue crack initiation region

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Figure 4-16 illustrates the variation of DV damage parameter at the three locations of interest (underhead radius, web-base fillet and base fillet) for different lateral traction coefficients. The figure shows a linear increase in fatigue damage at the base fillet while the top of the foot remains nearly unaffected by the lateral traction. At the underhead radius the effect of lateral traction is severe particularly for the traction coefficients above 0.1 for which the damage parameter increases dramatically in a near linear trend. For lateral traction coefficients of higher than 0.3 fatigue failure is predicted.

Figure 4-16: DV damage parameter versus the lateral traction coefficient for the underhead radius, top of the rail foot and base fillet depicted by circles in Figure 4-13; contact patch located 15 mm offset from the rail centerline towards the gauge side

4.7.1 Contact Patch Eccentricity, Tractions and In-Service Observations

As described in sections 3.3 and 3.4 steering in curved tracks and hunting during tangent track operation can lead to contact patch displacement and development of additional lateral tractions on the rail head. Although, direct comparison of damage values predicted using this study with what occur in service is not possible the results presented for the effect of contact patch lateral location and lateral traction are consistent with the failure modes observed in-service. Fatigue cracking at the underhead

109 radius is a common failure mode in the type of weld considered, and mostly occurs in short radius curves and tangent tracks prone to vehicle hunting. Figure 4-17 shows a typical straight break failure initiated at the underhead of the studied AT weld type.

Figure 4-17: Straight break in the considered AT weld initiated at the underhead radius [16]

Figure 4-18: DV damage parameter for two longitudinal traction coefficients in a tangent track

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The effect of longitudinal traction is illustrated in Figure 4-18 for the minimum and maximum traction coefficients of 0 (free rolling) and 0.4 under a central loading. As the figure suggests a high longitudinal traction has only a limited influence on the DV damage parameter and the fatigue behaviour by slightly increasing the damage at the underhead region (5% increases at the most critical point).

4.8 Track Support

It was described in chapter three that locations of the track at the vicinity of aluminothermic weld suffer from lack of sufficient support due to the irregularities at the weld surface and the resulting impact forces. Finite element analysis pointed out that by losing substantial support the bending moment applied on the weld section is dramatically increased which results in higher longitudinal stresses at the base region.

In this study the effect of ballast stiffness on the fatigue behaviour of AT welds is investigated through changing the vertical stiffness of the foundation underneath the adjacent sleepers. However, it is assumed in this study that the applied load is constant irrespective of the support condition. Figure 4-19 presents the results of fatigue analysis on the collar edge for four combinations of vertical and horizontal track stiffness in a tangent track. The DV damage parameter as described by Equation 4-4 increases dramatically at the base region when the sleepers loose substantial support stiffness. This is in accordance to the condition of longitudinal stress at this region. Although, the longitudinal tensile stress reaches about 650 MPa for completely loose sleepers, high compressive residual stress at the base has managed to reduce the damage parameter and contain it in the safe region (damage less than ) where fatigue crack initiation is not expected. 235

Nevertheless, the actual mechanism of failure under loosely supported track is more complex. In fact, it depends on the dynamic behaviour of vehicle-track and cannot be fully understood unless a comprehensive multi-body dynamics simulation is performed which can describe the variation of the applied load influenced by the combination of weld running surface irregularities, looseness in support and suspension configuration

111 of vehicle. However, the current analysis shows high sensitivity of the base region to lack of support and indicates that deviation from the ideal condition assumed in this study such as surface defect or roughness at the collar fillet may lead to fatigue failure. It is worth mentioning that for some weld types the collar edge at the base region is prone to cold lap or fin formation which is suspected to be correlated with the nucleation of straight breaks (more on cold lap and its relation with straight break is provided in Chapter 6). Moreover, in reality the surface finish of the weld exterior and the collar edge is not smooth since the process is originally a sand casting process. Field observations support the mentioned results since many of the fatigue failures initiating from the foot region occur at locations were considerable ballast pumping has taken place. At the underhead region the results are mixed; for the region between 120 to 140 mm above the rail foot the same behaviour as the base is observed i.e. the DV damage parameter increases by reducing the track stiffness. However, between 140 to 150 mm, damage parameter slightly decreases which could be attributed to the additional compressive stress induced in the region as a result of the section bending behaviour.

Figure 4-19: DV damage parameter at the collar edge versus ballast horizontal stiffness (HS) and vertical stiffness (VS) in a tangent track

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4.9 Sensitivity to Residual Stresses

It is believed that the residual stresses developed during the welding procedure have great influence in the fatigue life of the welded joints. Skyttebol et al. [61] has investigated the effect of residual stress on the propagation of fatigue cracks in flash butt welds (FBW) through finite element analysis and linear elastic fracture mechanics. However, no quantitative assessment has been performed on the fatigue or fracture behaviour of aluminothermic welds under the influence of residual stresses.

In this study the variation of both longitudinal and vertical residual stress fields is considered for multi-axial fatigue analysis. As previously explained, a piece-wise linear representation is used for several combinations of residual stress modes presented in Figure 4-9. Accordingly, longitudinal residual stress affecting the upper foot to underhead regions changes from 100 to 300 MPa and vertical residual stress from 100 to 200 MPa. Another analysis is performed for the zero longitudinal and vertical residual stresses. This condition does not normally occur in service unless the weld is subjected to stress-relieving treatments. However, inclusion of this case into the analysis gives an insight into the effect of residual stresses. In this case the only influential parameters are service and seasonal-dependent stresses.

The results of the fatigue analysis on the collar edge based on the presumed residual stress modes are illustrated in Figure 4-20 for an eccentric load where the contact patch is located 25 mm from the rail centerline with no lateral traction. The lowest damage parameter throughout the collar edge correlates with zero residual stresses, except for the foot region where the stress-free condition exhibits the highest damage. In fact the absence of a compressive residual stress (which had a preventive role in fatigue crack initiation) at the base leads to pronounced damage at this region. It is observed that the longitudinal residual stress increases the fatigue damage parameter throughout the collar edge of the weld. The damage parameter at the underhead region could even exceed the critical damage value in the presence of high residual stresses pointing out the additive effect of 235 longitudinal residual stress in fatigue crack initiation particularly at the underhead.

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Figure 4-20: DV damage parameter at the collar edge versus different longitudinal (L) and vertical (V) residual stress modes depicted in Figure 4-9 for an eccentric load located 25 mm from rail centerline towards the gauge side

The increase in the fatigue damage at the underhead region is greater than that at the upper foot despite the similarity in the residual stress applied to both regions. The current finding could be explained by inspecting the trend of fatigue damage at the mentioned regions during the loading cycle (Figure 4-21a). As the graph suggests, the underhead region experiences the maximum damage when the load is exactly at the top of the weld while the upper foot exhibits the maximum damage when the load is 132 mm away from the weld centerline. The underhead radius shows much higher peaks in fatigue damage parameter which is attributed to the increase in the shear stress amplitude on the critical plane as the residual stress increases, whereas no evident change of shear stress amplitude is seen at the upper foot region (Figure 4-21b).

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(a)

(b)

Figure 4-21: (a) DV damage parameter on the critical plane at the underhead and upper foot regions versus the relative location of axle load with respect to the weld centerline for different longitudinal residual stress (LRS) values; vertical solid lines define the location of axle load when the damage parameter at the mentioned weld regions achieves its highest value; and (b) Value of shear stress amplitude, hydrostatic stress and damage parameter versus LRS

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The severity of fatigue damage at different regions of the weld could be alleviated through reducing the tensile residual stress or even inducing compressive residual stresses where necessary. The applicable measures include (but not limited to) post- weld thermal treatments such as normalizing or stress relieving and post-weld surface peening [14].

In particular, the post weld surface peening or ultrasonic impact treatment (UIT) has gained a good reputation by increasing the fatigue lives of aluminothermic welds without notable side effects on other characteristics of the weld (the deficiency of some post heat treatment techniques). In this method mechanical impulses generated at an ultrasonic pulse generator are applied using a special hand tool to the toe of the weld collar edge where it develops small plastic deformations (Figure 4-22). The result of this local plastic deformation is a reduction in the severity of stress concentration by re- profiling the toe geometric features and increasing the toe radius and also inducing local compressive residual stress which has been shown to play a preventive role in fatigue crack initiation. According to Gutscher [150] under a laboratory condition this process has the potential of increasing the fatigue life of aluminothermic weld by 50 to 170 percent. The process is also time efficient since a full treatment can be performed in less than ten minutes.

Figure 4-22: Application of ultrasonic impact treatment on collar edge of an aluminothermic weld using the specialized hand tool [150]

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4.10 Sensitivity to Seasonal Temperature

A sensitivity analysis was performed on the effect of seasonal temperature variations on the fatigue behaviour of aluminothermic weld. Since the weld is generally under tensile stress in temperatures below the weld installation temperature (colder months) and compressive stress in temperatures above the installation temperature (warmer seasons), it is expected that the probability of fatigue failure increases in colder months and decreases in warmer months. A range of rail temperatures varying from 10 to 60 oC was considered and the reference or installation temperature was assumed to be 35 oC. Figure 4-23 presents the extent of fatigue damage parameter on the weld collar edge for three different temperatures representing the mean of colder months, neutral temperature (installation temperature) and mean of warmer months.

Figure 4-23: Damage parameter at the collar edge in a tangent track for three rail temperatures: neutral temp=35 oC, 17 oC above neutral temp=52 oC, and 16 oC below neutral temp=19 oC

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The figure generally supports the preventive role of high temperatures on fatigue crack nucleation. As expected the lowest damage parameter is associated to the temperatures above the neutral temperature which develops longitudinal compressive stress throughout the weld region and the highest damage relates to the temperatures below the neutral temperature which induces longitudinal tensile stress on the collar edge. Figure 4-24 illustrates the variation of shear stress amplitude, hydrostatic stress and the DV damage parameter in two important regions of the weld i.e. underhead radius and upper foot in terms of the rail temperature during service. As the figure suggests the shear stress amplitude on the most damaging plane is constant for all rail temperatures and does not have any effect on variation of damage parameter whereas the governing factor is the hydrostatic stress which increases by reducing the rail temperature.

Figure 4-24: Variation of shear stress amplitude, hydrostatic stress and DV damage parameter on the most damaging shear plane at the underhead and top side of the foot (depicted by circles in Figure 4-23) for different rail temperatures; vertical solid line illustrates the neutral temperature

Therefore, it seems that a good practice for reducing the risk of fatigue failure is to perform welding in lower ambient (neutral) temperatures so that the temperature

118 gradient in cold months is kept as small as possible. However, there are restrictions in defining the optimum welding temperature among which are the risk of buckling during warm months of the year and the maintenance schedule. Moreover, many of the aluminothermic weld installations are performed for removing the defective rails and welds in which case the weld installation temperature is more difficult to control.

In this chapter multi-axial fatigue analysis was performed on the collar edge of an aluminothermic weld to investigate crack initiation under heavy haul conditions. Dang Van critical plane approach was utilized using a developed computer code based on the MCC algorithm for shear stress amplitude calculation. The results show that the underhead region is the most critical location of the weld which experiences severe fatigue damage particularly under the effect of displaced contact patch. It is believed that a combination of factors including the tension spike, the successive change of longitudinal stress from compressive to tensile (which results in increased shear stress amplitude) and high longitudinal residual stress increase the risk of crack initiation at this region. Lateral traction as the other effect of curving and hunting behaviour can also dramatically increase the damage value at the underhead radius. Seasonal temperature also plays an important role in fatigue performance through its effect in hydrostatic stress parameter in Dang Van criterion. As expected lower ambient temperature introduces positive hydrostatic stress throughout the collar edge and increases the related damage index. Since the damage index evaluated in this study critically depends on the stress concentration at the collar edge it is believed that the geometric features of the surface could affect the fatigue behaviour. This parameter is going to be investigated in the next chapter using the fatigue tool developed in this study.

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CHAPTER 5 FATIGUE AND WELD COLLAR DESIGN

Analysis of weld bending and fatigue behaviour performed in Chapters 3 and 4 shows that the stress concentration at the collar edge of the weld plays an important role in increasing the fatigue damage and the likeliness to develop straight break failures. Stress concentration arises from geometric irregularity at the surface of the rail-weld boundary and is mainly related to the radius of the curve at the collar edge and the flank angle or the angle between the collar and the rail surfaces at the edge. However, the stress concentration is also influenced by the surface roughness at the collar edge and the presence or otherwise of casting defects. Several attempts have been made by the weld consumable manufacturers to design weld collar geometries with better features at the collar boundary and lower stress concentrations. However, designing the ideal collar geometry is always restricted by how the collar performs in the casting process. The change in collar geometry may change the amount of heat received by the rail head during the preheating step, the accessibility of molten metal to different parts of the mould, the cooling condition at different regions of the weld and also the formation of casting defects. Accordingly, only a limited number of weld collars are used around the world which can fulfil all the requirements of non-defective production and safe railway operation.

5.1 Design Alternatives and Geometric Features

In Australian heavy haul railway system the type of weld discussed earlier (hereinafter referred to as Type A) is widely used. This weld type has shown acceptable resistance to nucleation of HSW fatigue failures due to a smooth surface finish at the web region of the collar and lower incidence of gross defects. However, as mentioned this type of weld is prone to straight breaks resulting from fatigue cracking at the underhead and upper foot regions. Recently, a further type of AT weld has been developed and this type of weld, hereinafter referred to as Type B, features lower geometric complexities and a smoother collar edge (lower flank angle and larger toe radius) which may relieve

120 the severity of stress and hence sensitivity to straight break initiation. The Type B weld also incorporates the use of selective alloying of the rail head, in order to match the weld hardness to that of the parent rail [151]. Figure 5-1 illustrates the two weld samples and the associated Solidworks models used for bending and fatigue analysis.

(a) (a) (b)

Figure 5-1: Two weld collar designs under investigation: (a) Type A sample weld; (b) Type B sample weld [43]; (c) Type A computer model; (d) Type B computer model; and (e) Section view from top of the rail foot indicating the toe radius and flank angle in the two collar designs

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The purpose of this study is to investigate the effect of geometric features at different regions of the weld collar by comparing the stress distribution and the associated fatigue behaviour in the two collar designs with emphasis on the straight break formation at the collar edge and HSW at the web region.

Similar to the Type A weld, the model used in finite element analysis of Type B has been constructed in Solidworks CAD design using the dimensions extracted from laser scanned geometry of a sample weld. In this case, the weld is assumed to be defect free with ideal exterior geometry and zero surface roughness (both assumptions apply to the Type A weld). Figure 5-2 illustrates the finite element mesh of type B weld incorporated in the analysis of the bending behaviour with high mesh concentration throughout the collar edge where stress concentration and gradient are very high. Similar to the approach described in Chapter 3 element sizing is based on related sensitivity analysis. All other parameters including the track model, foundation stiffness and the temperature settings are the same as Type A weld and summarized in Table 3-1.

Figure 5-2: Finite element mesh of the Type B weld with the collar edge of the upper foot magnified at the bottom right figure

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Based on the stress history obtained in the finite element analysis the Dang Van multi- axial fatigue criterion is utilized to determine the critical locations in the new weld and to evaluate the extent of effect induced by the collar edge geometric features (flank angle and toe radius) of the two welds.

5.2 Residual Stress Distribution

Longitudinal and vertical residual stresses resulting from the Type B process has been measured using the trepanning and strain gauging method similar to that applied to Type A (Figure 5-3). The longitudinal residual stress features high compressive stresses at the base region and tensile stress from top of the foot up to the underhead region resembling the behaviour in Type A weld (Figure 4-9) although the tensile residual stresses are slightly lower in the new weld. However, variable measurements are still evident due to the limitations in the measurement technique and the variability of the welding process. Result of the vertical residual stress measurements is less sporadic and shows high tensile values at the mid web region with a similar range compared to those observed in Type A weld. To incorporate the residual stresses in the multi-axial fatigue analysis a piece-wise linear representation is used.

5.3 Performance in Tangent Tracks

The first comparison on the performance of the two collar designs is performed for the case of a tangent track where the contact patch is located at the centerline of the rail with no lateral or longitudinal tractions representing the free rolling condition. Figure 5- 4 shows the longitudinal bending stress for the two types of weld when the wheel load is located at the centerline of the weld (exactly at the middle of the weld collar). Similar to Type A weld, high stress concentration is visible throughout the collar edge of the Type B weld with pronounced stresses at the base (due to the section bending behaviour) and with lesser extent at the underhead region (tension spike resulting from the rail head local bending). However, the magnitude of the stresses at the collar edge of Type B weld is less pronounced than in Type A weld confirming lower stress concentration for the weld type with larger toe radius and smaller flank angle.

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Figure 5-3: Residual stress measurements using trepanning and strain gauge method (Mutton et al. [43]) and the piece-wise linear model applied in fatigue analysis: (a) Longitudinal residual stress at 3 mm offset from the collar edge; and (b) Vertical residual stress measured on the surface of the collar mid-web

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(a) (b)

Figure 5-4: Longitudinal stress contour for a tangent track when the load is located at the centerline of the weld: (a) Type A weld; and (b) Type B weld

Fatigue analysis is performed using the critical plane searching algorithm previously described in Chapter 4. Figure 5-5 illustrates the DV damage parameter at the collar edge from the base region up to the underhead determined on the most damaging plane using Equation 4-4. The trend of damage parameter variation throughout the collar edge is similar to that observed in Type A weld. Both welds experience lower damage parameter at the base compared with the underhead despite the higher longitudinal bending stress. In fact the beneficial effect of compressive residual stress at the base region is also evident in the new Type B weld. The underhead region in both welds is under a high tensile longitudinal residual stress and a tension spike associated to the rail head bending behaviour and these two induce high damage values.

Nevertheless, it is apparent that the damage parameter at the collar edge of the Type B weld is lower than that in Type A in most of the regions, particularly at the underhead radius. This can be attributed to the better geometric features (lower flank angle and larger toe radius) of the Type B weld and slightly lower longitudinal residual stress developed in the welding process (Figure 5-3). In particular at the underhead region the

125 difference in the geometric features is very much evident (Figure 5-6). Type B weld incorporates a widened collar at the underhead which has enabled a low flank angle and a smoother transition of the rail-weld surfaces. For the standard gap weld, Type A has a constant collar width of 40 mm whereas in Type B the collar width gradually increases reaching a maximum width of about 55 mm. The collar design of the Type A weld at the web region features a bulged geometry compared to the flatter shape of the Type B and so induces higher stress concentration. Perhaps this is also influential in pronounced reduction of damage at the mid web.

In any case no fatigue failures are predicted to initiate at the collar edge of a sound and defect free weld of either types in a tangent track since the DV damage value does not exceed the critical damage parameter ( ). It is however assumed that the fatigue characteristics of the two weld materials 235 are the same.

Figure 5-5: DV damage parameter versus height above the rail foot at the collar edge of the two weld types in a tangent track

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Figure 5-6: Collar shape with the flank angle at the underhead region of the two weld types

5.4 Contact Patch Lateral Location

The displacement of the contact patch under the effect of curving was described in section 3.3 and it was mentioned that as the curve radius decreases the contact band generally moves towards the gauge side and eventually flange contact occurs for very sharp curves. In this section the effect of collar design on the fatigue performance is studied as the contact patch is displaced towards the gauge side. A lateral displacement of up to 25 mm from the rail centerline is considered for this purpose. Figure 5-7 illustrates the longitudinal stress contour for a contact patch located 25 mm from the rail centerline above the weld collar with no lateral or longitudinal tractions. As expected the stresses at the collar edge are less pronounced in Type B weld due to the different geometric features of the collar edge.

Figure 5-8 shows the variation of longitudinal stress at the underhead region on a path located 31 mm from the rail centerline with respect to the longitudinal distance from the weld centerline for both cases of central and eccentric loadings. The wider gap between the two tension spikes in the Type B weld compared to that of Type A weld is due to the wider design of the collar at the underhead of the Type B (Figure 5-6). As expected, the tension spikes in both welds are more severe in curved track (displaced contact patch) than a tangent track due to the twisting moment applied to the rail section. However, tension spikes in Type B weld are lower in both conditions due to the lower stress concentration and mainly the more preferable flank angle and toe radius.

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(a) (b)

Figure 5-7: Longitudinal stress contour on the gauge side for a curved track with contact patch eccentricity of 25 mm with no tractions: (a) Type A weld; and (b) Type B weld

Figure 5-8: Variation of longitudinal stress at the underhead radius (at 31mm from rail centerline) of the gauge side under central loading (tangent track) and eccentric load located 25mm from rail centerline towards the gauge side

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The results of the multi-axial fatigue analysis on the collar edge of both weld types are depicted in Figure 5-9 for a contact patch located 25mm from the rail centerline towards the gauge side. The graph exhibits the contribution of high cyclic tensile stress to the increased damage value at the underhead region. The DV damage parameter at the underhead radius of the Type A weld even exceeds the critical limit according to the Dang Van criterion. However, the damage values for the Type B weld are less pronounced and located in a safe margin provided that the weld material is sound and defect free. For other regions of the collar edge DV damage values are expected to be lower in Type B weld compared to those in Type A as a result of better geometric design and slightly lower longitudinal residual stresses. However, it is important to mention that the in-service weld performance largely depends on the extent by which the as-built weld collar features and the roughness deviates from the ideal conditions assumed in this study and also on the residual stresses observed in that specific weld.

Figure 5-9: DV damage parameter on the collar edge versus height above rail foot for a contact patch located 25mm offset from rail centerline towards the gauge side

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5.5 Contact Lateral Traction

Development of lateral tractions on the contact patch is another effect of steering in curved tracks and hunting behaviour which was discussed in section 3.4. To study the role of collar design under such conditions traction forces with coefficients (lateral to vertical load ratio, L/V) ranging from zero (free rolling) to 0.4 were applied to the rail head towards the field side of the rail. Figure 5-10 illustrates the longitudinal tensile stress contour on the field side of the weld collar for a traction coefficient of 0.4 applied on a contact patch located 15mm from the rail centerline towards the gauge side. The contours show elevated tensile stresses at the underhead region and base of the weld collar in both weld types which are due to the increased lateral bending moment and torsion as the applied load inclines towards the field side. As mentioned in Chapter 3 the trend evident in the longitudinal stress data suggests that the underhead radius is more sensitive to lateral traction than the base, although the magnitude of the longitudinal stress is generally higher at the base region (Figure 5-11).

(a) (b)

Figure 5-10: Longitudinal stress contour at the field side of the weld for an eccentric load located 15 mm from the rail centerline with lateral traction coefficient of 0.4: (a) Type A; and (b) Type B

In any case the stress values obtained are less pronounced in Type B weld which suggests lower stress concentration at the collar edge arising from the more preferable collar edge geometric design.

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Figure 5-11: Variation of longitudinal stress at the under head and base fillet of the two weld types under different lateral traction coefficients

The results of the fatigue analysis at the collar edge of the field side are presented in Figure 5-12 for both Type A and Type B welds for a lateral traction coefficient of 0.4. The damage is severe at the underhead fillet due to the increased longitudinal and high residual stresses at this location. The damage values are lower in Type B weld compared to Type A for most of the collar regions which proves the favorable effect of lower geometric complexities at the collar edge of Type B and the lower longitudinal residual stress. While the state of fatigue at the underhead radius of the Type B weld is still in the safe region, the damage parameter at the underhead of the Type A weld exceeds the critical value, indicating that the fatigue failure is expected at this location. However, a lower damage value is observed at the base fillet compared to the underhead region since the state of the longitudinal residual stress is compressive. The effect of the longitudinal traction on the fatigue behaviour of the weld collar edge was shown in Chapter 4 to be insignificant for various traction values and hence is not considered for the comparison of the two weld types. Summing up the effect of contact eccentricity and the lateral traction, it can be shown that under the simulated conditions, the Type B weld performs more preferably in curved tracks and tracks prone to vehicle hunting.

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Figure 5-12: DV damage parameter at the collar edge of the two weld types for an eccentric load located 15mm from the rail centerline with lateral traction coefficient of 0.4 towards the field side

5.6 Track Support Condition

As part of the comparison of the two weld geometric designs the performance of weld in a deteriorated or ill-supported track is important since this condition occurs for a majority of aluminothermic welds due to the running surface irregularities. In this study the effect of ballast stiffness on the fatigue behaviour of the two welds is investigated through altering the vertical and horizontal stiffness levels of the foundation underneath the weld adjacent sleepers.

Figure 5-13 presents the results of the fatigue analysis on the collar edge for four combinations of vertical and horizontal track stiffness levels in a tangent track. The DV damage parameter increases dramatically at the base region of both AT welds and reduces smoothly at the underhead when the sleepers loose support stiffness.

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Figure 5-13: Damage parameter at the collar edge versus ballast vertical and horizontal stiffness for a tangent track (a) Type A; and (b) Type B

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This fatigue behaviour is attributed to an increase in the rail bending moment at the weld location, which results in an increased tensile stress at the foot and more compressive stress at the rail head. However, the damage parameter at the collar edge of the Type B weld is less pronounced than that of the Type A weld due to the better geometric features at the root of the collar edge. Even though the damage values are below the critical limit, any dynamic event (high impact forces as a result of weld head irregularities and poor support) may cause the stress state to move to fatigue region. A comparison of Type A and Type B welds shows that the Type B weld has better capacity to absorb such dynamic loading events. Also the Type B weld features head alloying in which the hardness characteristic of the running surface is improved through incorporating an alloy material locally at the rail head. This local hardening method does not affect the ductility of the underhead regions which is particularly important in fatigue behaviour of the weld material. The local hardening of the running surface in Type B weld may be able to prevent or reduce cyclic plastic deformation at the rail running surface and so reduce the battering phenomenon. Under such condition the weld will experience lower impact forces and the damage induced in the track foundation will diminish. Consequently, the bending behaviour of the rail is improved and the longitudinal stress and the resulting damage parameter may be decreased.

5.7 Web Fatigue Behaviour

Horizontal split web (HSW) failures mostly occur in short-radius curves and tracks prone to vehicle hunting. Under such conditions the contact patch is displaced towards the gauge side of the rail imposing high vertical bending stresses on the web region of the weld collar. The contribution of bending stress, vertical residual stress and the presence of gross defects has been shown to facilitate the occurrence of fatigue or overload failures in the web region [14]. In this study the fatigue behaviour of the two weld types is investigated on the web collar surface for the case of an eccentric load located 25mm from the rail centerline towards the gauge side with no tractive forces. In this case, vertical residual stresses are implemented in the fatigue code through a quadratic interpolation of the experimental data to better determine the effect of residual stress variation along the centerline of the weld on the collar surface.

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(a) (b)

(c)

Figure 5-14: Vertical stress contour on the field side of the weld for an eccentric load located 25mm from the rail centerline with no lateral traction: (a) Type A weld, (b) Type B weld; and (c) Section view of the collar at the location of maximum bending stress and the effective bending moment

Figure 5-14a and 5-14b show the vertical bending stress contours at the field side of the two welds. The contours show maximum tensile stress at the mid web of the Type A weld and close to web-head fillet of the Type B weld. Vertical stress is also less pronounced and more localized in Type A weld than Type B weld. The difference in the location and magnitude of the maximum stress relates to the collar design of the web region (the collar shapes shown in Figure 5-1). Type A weld features a reinforced collar

135 at the mid to upper web region (compared to Type B welds) which increases the second moment of inertia and reduces the vertical bending stress in this area. However, less stress localization is observed in the Type B weld collar since the collar surface is flatter and the stresses are better distributed compared to the bulged collar design of the Type A (Figure 5-14c).

Figure 5-15 depicts the variation of vertical stress along the web and the contribution of the residual stress to the total vertical stress observed on the web region of the two weld types. The effect of the reinforced collar design is noteworthy particularly at the upper web region in which the total stresses differ by up to 125MPa (at 120 mm above the rail foot) between the two weld types. For both weld types the vertical residual stress variably increases the total vertical stress throughout the region.

Figure 5-15: Vertical stress versus height from rail base and the contribution of residual stresses for an eccentric load located 25 mm from the rail centerline

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The results of the multi-axial fatigue analysis on the web are presented in Figure 5-16. The graph closely follows the trend of vertical bending stress, with residual stresses having only a small effect. Maximum damage value in the Type B weld is around 17% higher than in the Type A weld and occurs around 30 mm above the location of maximum damage in the Type A weld. It may be concluded that the collar in Type B weld is more sensitive to presence of defects in its critical location although it also depends on the type of defect, its geometric features, size and the type of loading. A separate study is necessary to determine the influence of defects in formation of horizontal split webs and the performance of the two weld types. A relevant study is performed and reported in Chapter 7 considering specific defect types using defect surface fatigue analysis and linear elastic fracture mechanics. However, as the figure suggests and also practically observed no fatigue failure develops on the smooth surface of the web under a defect free condition.

Figure 5-16: DV damage parameter on the centerline of the web surface for an eccentric load located 25 mm from the rail centerline

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In this chapter multi-axial fatigue analysis was undertaken to compare the fatigue behaviour of two AT weld types in order to evaluate the effect of geometric features and operational conditions on fatigue damage at the edge of the weld collar and surface of the web. The severity of fatigue damage under all the simulated loading conditions was lower in Type B weld than in Type A weld, which was mainly due to the better geometric features (lower flank angle and larger toe radius) and partly due to the lower longitudinal residual stress. While fatigue crack initiation was expected at the underhead region of the Type A weld under severe curving and hunting behaviour, Type B welds remained safe provided that the weld material is defect free and the surface finish is similar for the two processes. It was also concluded that the collar design of the web region has a substantial effect on the bending and fatigue behaviour of the weld with respect to HSW. However, for a defect-free weld no fatigue failure is expected under normal heavy haul conditions.

The fatigue problem investigated so far primarily focused on crack formation in defect free weld and idealized weld geometry. In the next two chapters effort is made to study the straight break and horizontal split web formation under the influence of certain welding defects using both multi-axial fatigue analysis and fracture mechanics.

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CHAPTER 6 COLD LAP DEFECT

Previous analysis on the fatigue behaviour of the aluminothermic weld showed that the underhead radius is a critical location for the initiation of straight break failures. In particular, under harsh operational conditions such as negotiating short radius curves and severe vehicle hunting where the contact patch is extensively displaced towards either sides of the weld and high lateral traction emerges crack initiation is inevitable. Practical observations are consistent to this fact since many of failure reports for the specific weld type considered relate to straight breaks initiating at the underhead radius.

Nevertheless, failure reports also denote the top of the rail foot as one of the most common straight break initiation sites in this type of weld. Straight breaks may initiate at the collar edge or in its vicinity of one or both sides of the weld and develop in vertical or near vertical direction until they lead to the final brittle fracture of the rail section (Figure 6-1). However, multi-axial fatigue analysis reported in Chapter 4 was unable to detect any severe condition at the upperfoot region in a defect free weld under the simulated loading conditions. The longitudinal residual stress is high at the top of the rail foot and the region occasionally experiences high cyclic longitudinal stresses. However, since the cyclic component of the shear stress on the most damaging plane (shear stress amplitude) is not high enough, the shear based failure according to the Dang Van criterion seems improbable.

So, what is the main reason for possible straight break failure at the upperfoot and the web base fillet? According to site observations and failure reports straight break failures at the upperfoot have usually been accompanied with the presence of a specific welding defect known as heavy flashing or cold lap. This defect has also been reported to be responsible for many straight break failures at the foot regions for aluminothermic weld types in other heavy haul systems overseas [17, 46]. In this chapter effort is made on defining the role of cold lap in the formation of straight break at top of the rail foot considering the effect of some cold lap geometric features and loading conditions.

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(a)

(b) (c)

Figure 6-1: Fatigue crack propagation at top of the rail foot underneath the associated cold lap defects in three weld samples; arrows show the fatigue crack propagation regions [44, 152]

6.1 Cold Lap Defect

Cold lap is a condition which arises from the lack of sufficient melt back on the rail ends subject to welding which eventually leads to lack of fusion between parts of the weld block. Two different cases may be considered: (a) Finning or flashing (EN 14730- 1:2010 [10]) which occurs due to the lack of fitness between the mold and the rail end surfaces. In this case the molten metal seeps out of the resulting gap and solidifies on the rail surface forming an unfused fin (Figure 6-2a). The fin is fused to the rail surface for part of its length beyond which the fusion does not take place due to the lack of melt back or molten metal penetration into the rail ends. Accordingly, the apex of the separation region also corresponds to the location of fusion boundary of the weld section. (b) Cold lap which does not relate to the fitness of the mold. In fact, the mold

140 can be perfectly fit while the cold lap emerges. In this case because of the insufficient heat input, the melt back length of the rail ends cannot extend beyond the edges of the weld collar and so the separation takes place inside the collar (Figure 6-2b). In contrast to the finning condition, cold laps may be invisible and the weld may seem geometrically perfect. However, both these cases are referred to as cold lap due to the nature of the defects which incorporate lack of fusion on the surface of the rail ends.

(a)

Lap Apex

(b)

Lap Apex

Figure 6-2: Section view from the top of the rail foot illustrating two types of cold lap defect: (a) Leaking of weld material and formation of an unfused appendix (finning); and (b) Lack of fusion on the rail surface inside collar boundaries (cold lap)

141

The problem with cold laps is the initiation of fatigue cracks from the apex or tip of the defect where the two unfused surfaces meet and fuse to each other. In aluminothermic welds, the crack extends in a vertical or near vertical direction into the weld (or rail) section forming a straight break failure. As a matter of fact cold lap is not exclusive to aluminothermic welding procedure and any type of welding or casting which incorporates melting and fusion of a third welding agent to the parent components may suffer from this defect. In particular, butt and T-joint arc weldments are susceptible for the occurrence of cold laps [153-154]. Among the contributing factors in formation of cold laps in aluminothermic welds are preheating duration, stick-out length of the rail ends, flank angle of the collar, contamination on the rail surface, sealing of the mold, dissimilarity in rail heights, cooling and solidification pattern.

6.2 Analysis of Cold Lap

Close examination of the cold lap defect shows that this planar defect is very similar to a crack. The unfused surfaces of the rail and the lap act as the two faces of a crack and the cold lap apex as the crack tip since it has an extremely small notch radius with very high stress concentration. Accordingly, it is possible to investigate the effect of cold lap on fatigue behaviour by replacing the defect by a pertinent planar crack. As explained in Section 2.3.3 this standard procedure is widely used in damage tolerance investigations when the defect features high stress concentration under which the fatigue failure is governed by the non propagation of cracks emanating from the defect. Under this assumption the formation or otherwise of a straight break from the tip of a cold lap defect is mainly a fatigue crack propagation problem (Figure 6-3). This would be the basis for the analysis of cold lap and the effect of its geometric features.

For this purpose, it is assumed that the behaviour of material in the vicinity of the crack is predominantly elastic with only limited plasticity confined in the singularity dominated region of the crack tip. In fact this is a true assumption considering the cast- like nature of the process and the noticeably low ductility of the weld material compared to that of the rail. In this case linear elastic fracture mechanics (LEFM) approach can be implemented to investigate the propagation of the presumed crack (cold lap defect).

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Figure 6-3: The unfused surfaces of a cold lap and its apex may resemble the faces and tip of an existing crack which can propagate under fatigue loading and form a straight break fracture

An advantage of the LEFM approach is the ability to characterize the crack tip condition with one single parameter, stress intensity factor ( ) or the energy release rate ( ) both referred to as the driving forces of the crack. Either of these parameters can determine whether the original crack will become destabilized under a static loading and lead to an unconditional rupture. Under a fatigue loading (cyclic loading) the range of the stress intensity factor variation or the stress intensity factor range ( ) can determine the cyclic propagation of the presumed crack. ∆

Only a limited number of structures and crack types have analytical solutions for the calculation of stress intensity factors. For most of the practical and complex structures and crack shapes numerical methods such as finite element (FE), boundary element (BE) or mesh free methods should be used. Among special purpose packages to handle crack analysis are ZENCRACK (FEM), BEASY (BEM) and FRANC2D, 3D (FEM). Crack analysis may also be performed in general purpose packages such as ANSYS or ABAQUS although a special post-processing and manipulation technique may be necessary to achieve stress intensity factors. Some techniques are exclusive to LEFM while others can be used for elastic-plastic and the dynamic fracture problems. Some of the widely used numerical methods are the displacement extrapolation or the crack opening displacement (COD) [155], force method [156], J-integral [157] and virtual crack closure technique (VCCT) [158] which is utilized in this study.

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6.2.1 Virtual Crack Closure Technique (VCCT)

Among different numerical methods to calculate mixed-mode stress intensity factors and strain energy release rates from finite element simulation results, virtual or modified crack closure technique (VCCT or MCCT) has gained popularity among researchers and scientists due to its simplicity and applicability. The application of virtual crack closure technique to finite element analysis was originally introduced by Rybicki and Kanninen [159]. The method is based on the crack closure integral of Irwin [160] which states that the energy released due to the extension of the crack by a very small length is the same as the work required to close the crack back to its original length. Based on this assumption the technique was initially implemented in a two-step form which is referred to as the crack closure technique (CCT). Accordingly, the energy released during the extension of the crack from length to (Figure 6-4) is equal to the energy required to close the same extension ( ) which + is∆ written in the following from [158]: ∆

(6-1) 1 ∆ . ∆ . ∆ 2 where and are the shear and opening forces at point which are calculated at the first step of the simulation when the crack tip is located at . and are the differences in shear and opening displacements of the two crack faces∆ at point∆ which are obtained at the second simulation when the crack has been extended (from length to ) and the crack tip is located at point . In this case the forces necessary to close the ∆ crack are the same forces available at the crack tip before extension (when the crack is closed). The main problem with this technique is the requirement for simulating two configurations of a crack (original and extended) to be able to calculate the stress intensity factors of the crack tip.

The virtual crack closure technique (VCCT) enjoys the same assumption of the Irwin crack closure algorithm furthermore it overcomes the two-step analysis requirement. In this technique it is assumed that the extension of crack from to does not noticeably change the condition at the crack tip provided that ∆ is a small 2∆ value. ∆ 144

Accordingly, it is assumed that the displacements at node when the crack tip is located at node are approximately the same as the displacements at node when the crack tip is located at node .

(a) (b)

Figure 6-4: Crack closure technique (CCT) using two step finite element simulations: (a) Step one, original crack (separation forces are calculated at ); and (b) Step two, crack is grown and the displacements at are obtained Under such assumption it is possible to obtain energy release rate for a specified crack in one single simulation of the original crack using the separation forces at the crack tip and the displacements behind the crack tip. In the case of the three dimensional cracks where all the crack modes (I, II, III) are present VCCT can be formulated for both 8- node and 20-node brick elements. For 8-node brick elements the separated energy release rates ( , and ) for any point of the crack front can be obtained using Equation set 6-2 [158].

∗ 1 ( − ) 2 . ∗ (6-2) 1 ( − ) 2 . ∗ 1 ( − ) 2 . where , are the shear forces and is the opening force at node . ∗ , ( − )* ∗ and ∗ are the relative displacements of the two nodal points l and l ( − ) ( − )

145 located on the two crack faces, is the element length in the crack front and is the width of the elements adjacent to∆ the crack front (Figure 6-5).

Figure 6-5: Virtual crack closure technique (VCCT or MCCT) for 8-node brick elements

However, there are some conditions to be met before the Equation set 6-2 can safely be implemented without the need to apply any specific corrections:

• The element length ( ) in front of the crack tip and behind the crack tip should be equal. ∆ • The element width ( ) should remain constant along the crack front. • For the case of an irregular crack front all the elements at the crack tip should be perpendicular to the crack front. • If the crack front is not straight a separate coordinate system should be used for any node on the crack front and the displacements and forces at that node should be calculated in the new coordinate system.

One of the VCCT advantages is the determination of separated energy release rates for mixed mode cracks. Subsequently, the stress intensity factor at any point of the crack

146 front can be obtained for each crack loading mode from general LEFM equations relating energy release rates with the associated stress intensity factors (Equations 6-3).

. (6-3) .

2. . where is equal to the modulus of elasticity ( ) for plane stress condition and equal to for plane strain stress state and and are shear modulus and the Poisson’s ratio/(1 respectively. − )

6.2.2 Mixed-Mode Fracture Criteria

In pure mode I the condition for fast fracture is to have the first mode stress intensity factor above or equal to the fracture toughness of the material ( ). In this case the crack generally propagates in the direction where maintains its≥ maximum value. However, most of the engineering structures have multiple loading types imposed on the crack or the crack is oriented in such a direction that two or three loading modes are applied on the crack tip. Similar to the failure criteria (for example Tresca and von Misses yield criteria) which introduce critical conditions for materials under multi-axial loading, suitable criteria should be available to determine the critical conditions for the mixed-mode loadings of cracks.

A number of fracture criteria have been proposed to determine when a presumed mixed- mode crack becomes unstable and to what direction the crack propagates. Among well known criteria are the ones introduced by Erdogan and Sih [161], Schöllmann [162], Hussain at al. [163] and Richard [164]. Richard criterion is based on approximation formulae and has shown to be numerically accurate and practically simple to apply. The criterion has also proved to be in good agreement with the experiments and other accurate criteria such as the Schöllmann maximum principal stress criterion [165-167]. Richard proposes an equivalent stress intensity factor which superposes the effect of all

147 loading modes. According to this criterion, unstable (fast) fracture occurs when becomes equal to the fracture toughness of the material (Equation 6-4) [165]:

1 4 4 (6-4) 2 2

Where / , / and and are the critical stress intensity factors for mode II and III shear loading respectively. By considering the values of

1.155 and 1.0 the Equation 6-4 is in perfect agreement with the equivalent intensity factor determined by Schöllmann [165]. Richard also proposes two functions for the determination of crack growth angles (kinking angle) and (Figure 6-6):

| | | | ∓ (6-5) | | | | | | | |

where is negative for positive values of and positive for negative values of and when 0.

| | | | ∓ (6-6) | | | | | | | |

where is negative for positive values of and positive for negative values of and when 0.

Figure 6-6: Crack deflection angles and for mixed-mode crack problem [166]

148

Considering , , and the Equations 6-5 and 6-6 ° ° ° ° are in good agreement 140 with the −70 direction angles 78 derived −33 by Schöllmann [165].

Under fatigue loading, single mode I crack propagates if the stress intensity factor range exceeds the threshold stress intensity factor range which is a property of the material.∆ According to Richard criterion, in mixed-mode∆ cyclic loading the propagation or otherwise of the crack can be predicted by comparing the equivalent stress intensity factor range (Equation 6-7) with the threshold value . However, fast fracture is still governed∆ by the maximum value of the equivalent ∆stress intensity factor in comparison with the fracture toughness of the material ( ).

(6-7) ∆ 1 ∆ ∆ 4(∆ ) 4(∆ ) 2 2 6.2.3 Local Stress Intensity Factors

Cracks usually propagate and kink in a direction where the resistance is the lowest or the driving force has the maximum value. The stress intensity factors at the tip of an infinitesimal kink just emanated from the crack are different from those of the original crack tip and depend on the direction of the prospective kink (Figure 6-7). For the case of a plane mixed-mode crack the local stress intensity factors are obtained using the following relations [168]:

(6-8) ( ) () Where and are the stress intensity factors at the kink tip and and are the stress (intensity) factors () at the original crack tip. , , and are obtained using the following relations:

(6-9) 3 1 3 cos cos 4 2 4 2 149

3 3 − sin + sin 4 2 2

1 3 sin + sin 4 2 2

1 3 3 cos + cos 4 2 4 2

Figure 6-7: Kinking at a crack tip and the local stress intensity factors (a) Original crack; and (b) Crack with a kink at the tip

The local stress intensity factors may be used to determine the variation of energy release rate with respect to the kink angle and to predict the direction of crack propagation using the maximum energy release rate criterion [163].

6.2.4 Cold Lap Finite Element Model

The modeling of the cold lap as a planar crack is performed in the general finite element package ANSYS 12.0 as was used for the previous investigations. For the sake of simplicity the presence of cold lap on the top of the rail foot is modelled as a rectangular parallelepiped appendix with specific dimensions depicted on Figure 6-8. Lap thickness, unfused length (extension of lap from its tip), width and the edge offset (distance between collar edge and the lap tip) are variable and related sensitivity analyses are performed later in the chapter to determine the influence of the geometric features on the behaviour of cold lap.

As previously explained in section 6.2.1, to determine the stress intensity factors at the crack front (lap tip) separation forces comprising the opening and shear forces of the nodes lying on the crack front should be extracted. For this purpose the interaction between the cold lap surface and the rail surface is modelled as a contact boundary

150 condition (a feature of the software package). For the unfused region, a frictional contact with a coefficient of 0.3 is defined between the two surfaces (crack faces). However, it is worth noting that as will be seen later the crack is generally open during the whole stress cycle and the closure phenomenon never occurs and the variation of the friction coefficient has virtually no effect on the growth characteristics. For the rest of the interaction (where the lap is fully fused to the rail surface) a bonded contact is introduced which virtually bonds the two surfaces into one body. The benefit of this method is that the separation forces can be obtained from the bonded contact reaction forces (as a post-processing feature) at the crack front.

Figure 6-8: Cold lap model as a simple rectangular parallelepiped and its geometric features

To investigate whether fatigue crack propagates from the lap tip, the equivalent stress intensity factor range ( ) is required and to extract this parameter the simulation of at least one load cycle∆ is necessary. To achieve this, the same approach as the one proposed in Chapter 3 for the wheel passage is used where the rail-wheel contact load moves from one end to the other end of the track by sequentially moving the load from one patch to the adjacent patch (Figure 3-6). Thereby, for each load step the force and displacements at the crack tip are extracted and the stress intensity factors are calculated through VCCT. The variation of stress intensity factors during the load passage will determine the stress intensity factor range for the associated crack loading mode and then the equivalent stress intensity factor range is extracted through Richard’s criterion.

151

The finite element mesh applied to the weld and rail sections are similar to the ones used in Chapter 3 with 8-node brick elements for the rail and sleepers and 10-node tetrahedral elements for the weld region. However, no mesh refinement is used at the collar edge of the weld since the stresses at these regions are not of interest in this study and also the computational time is enhanced substantially. For the lap region cubic 8- node brick elements are applied with special refinement at the crack tip. A small block of highly refined mesh is used at the crack tip featuring equal element widths and lengths and perpendicularity to the crack front (Figure 6-9). These conditions guarantee the safe application of the VCCT.

Figure 6-9: Finite element mesh and the refined block at the crack tip

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All the model parameters including the loading, seasonal and reference temperatures, boundary conditions and the track foundation levels are similar to the main model introduced in Chapters 3 and 4. However, in this analysis the effect of rail running surface irregularity is introduced into the model through a contact load magnification factor of 1.3 applied directly to the contact patch located exactly above the weld region. This magnification factor is the average impact factor which aluminothermic welds usually experience during normal operation and varies from 1.1 to about 1.5 (Figure 2- 5). As mentioned in Chapter 4 the longitudinal residual stress at the location of cold lap may vary between 100 to 300 MPa depending on the welding procedure and the thermal conditions. In this study the average value of 200 MPa is considered for fracture analysis. As Figure 4-9 suggests the vertical residual stress at this location is zero. Table 6-1 summarizes important simulation parameters and the lap dimensions used throughout this chapter. Any deviation from the values mentioned will be advised in the related section.

Table 6-1: Summary of the finite element model parameters Track length (mm) 3237 Lap thickness (mm) 2

Axle load (kN) 343.4 Lap width (mm) 20

Reference temperature ( oC) 35 Edge offset (mm) 5

Simulation temperature ( oC) 19 Unfused length (mm) 5

6.2.5 Element Size of the Crack Tip

The main condition in the VCCT is the assumption that the growth of the crack by the size of an element ( ) does not noticeably change the condition at the crack tip. Accordingly, the size ∆of the element used in the crack tip block should be small enough to satisfy such assumption. However, very small element size should also be avoided to limit the increase in simulation time and maintain the solution efficiency. A sensitivity analysis is performed on the element size (length and width are equal) of the crack tip to evaluate the optimum size. Table 6-2 presents the stress intensity factors of the crack tip at the middle of the lap versus the element sizing when the load is located exactly at the centreline of the rail and weld representing free rolling in a tangent track.

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Table 6-2: Crack tip stress intensity factors versus the element size at the crack tip block

Element Size (m) ( √) ( √) ( √) 0.0001 6141493 6774717 506939

0.00025 6146586 6787869 507147

0.0005 6205724 6790984 508060

0.001 6709883 6791567 516342

As the table suggests all the element sizes perform well with low deviation (less than 2%) in determining the mode and mode stress intensity factors. For mode except the 1mm element which has a high error (about 8.5% with respect to the smallest element) all the other three sizes are in good correlation. In fact, the stress intensity factors obtained with the 0.25 mm element show negligible difference with those of the 0.1 mm element while the larger element results in a better computational efficiency. The simulation of a full loading cycle with 0.25 mm element takes 3.1 hours compared to 4.2 hours with 0.1 mm element in a 3.2 GHz Quad core Xeon CPU. Accordingly, the 0.25 mm element size is adopted for the analysis of cold lap throughout this study.

6.3 Influence of Cold Lap on Fatigue Behaviour

Initial analysis is performed on a cold lap located at the top of the rail foot under a tangent track operation where the contact patch is located at the centerline of the rail with no lateral and longitudinal tractions. Figure 6-10 illustrates the equivalent stress contour on the lap and the surrounding region when the wheel load is located exactly at the centerline of the weld. As the figure suggests, the crack mouth opens under the applied load and high stress concentration emerges at the crack tip. In fact the reason for the opening of the crack is the load applied to the left surface or the bearing surface of the lap from the weld collar. This load results from the bending stresses at the top of the weld foot as well as the longitudinal residual and seasonal-dependent tensile stresses. However, the lap itself (depicted by an ellipse) is not subject to any significant stresses since no external load or support exists on the other three surfaces of the lap (top side, bottom side and the right side) and so it lacks the ability to transfer any load.

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It is worth mentioning that the crack tip stresses in LEFM are theoretically infinite and a singularity exists at the tip. However, the stress values predicted by finite element method with the VCCT meshing pattern are limited since no singular element is used in this method. The absence of singular elements in VCCT is an advantage and it does not affect the accuracy of the stress intensity factors derived in this technique.

Figure 6-10: Equivalent (von Mises) stress in the cold lap region under a central load (tangent track) and the lap cross section showing crack mouth opening

As mentioned in the VCCT, the stress intensity factors are calculated from the forces between the two nodes in contact at the crack tip and the opening displacements of the nodes behind the crack tip. Figure 6-11 illustrates the variation of stress intensity factors at the centre of the lap in the three loading modes when the wheel load passes over the rail running surface. As the figure shows the values of the stress intensity factors in opening ( ) and the sliding ( ) modes are much higher than the out of plane tearing ( ) mode suggesting that the propagation of the cold lap defect is mainly driven by the in-plane opening and shear forces. The stress intensity factors of the two prominent

155 fracture modes start and end at a fixed value rather than zero. This is due to the presence of constant longitudinal residual and seasonal-induced thermal stresses during the load cycle. The maximum stress intensity factors in the two in-plane modes correspond to the position of the wheel at the centreline of the weld and the distinguished peak at this point refers to the load amplification factor applied at this region.

Figure 6-11: Variation of stress intensity factors and the resulting stress intensity factor ranges at the middle point of the cold lap during one wheel passage (loading cycle)

As mentioned in Section 6-2-2 crack propagation is governed by the value of the equivalent stress intensity factor range ( ). Based on Richard’s criterion, is written as a function of the stress intensity∆ factor ranges in the three loading∆ modes ( , Equation 6-7). According to the corresponding values depicted in Figure∆, ∆ 6-11, , ∆ the equivalent stress intensity factor range is obtained as 3.01 MPa √m. The threshold stress intensity factor range ( ) or the critical limit for the propagation of a crack under fatigue loading is a material∆ characteristic and is obtained using cyclic loading of notched specimens extracted from the parent material. However, this value also depends on the applied stress ratio ( ) which is the ratio of the minimum stress

156 intensity factor ( ) to the maximum stress intensity factor ( ) during one loading cycle. The correct measurement of for aluminothermic weld is quite difficult due to the variability of material charac∆teristics as well as microstructure and the defect content throughout the weld section and between different weld samples. The problem becomes more complex and even impossible to solve when the measurements need to be performed on the material properties of the cold lap tip region. The cold lap tip is located on the fusion boundary of the weld (the border of the fusion and heat affected zones) where the material properties may change drastically across the border. In addition, material sampling at this narrow band for the purpose of fatigue testing seems unfeasible.

Dudley et al. [52] made limited experiments on fracture properties of aluminothermic weld at the web region for the purpose of determining damage tolerance with respect to the web defects. According to his measurements, the threshold stress intensity factor range can be written as a function of stress ratio ( ) in the following form: (6-10) ∆ ∆(1 − ) where is the threshold stress intensity factor range at and is a material constant.∆ Based on the measurements and are derived at 10.40 MPa √m and 0.62 respectively. Although, the location of∆ measurement s are different from the cold lap region the values estimated using Equation 6-10 are used for the purpose of crack propagation from the lap tip. The stress ratio obtained under the simulated conditions is about 0.75 calculated using the values in Figure 6-11. The high value of the stress ratio results from the large contribution of the longitudinal residual stress and the seasonal- induced thermal stress. The corresponding threshold stress intensity factor range is estimated at 4.4 MPa √m. Comparing the simulated condition ( and the threshold value ( ) it can be concluded that the crack may∆ not propag 3.01)ate from the tip of the cold∆ lap under 4.4 the simulated conditions. However, is sensitive to the stress ratio and the stress ratio is also a function of the constant∆ stress components of residual and thermal stresses. By increasing the longitudinal residual stress or reducing the ambient temperature the stress ratio increases and decreases. In a simple ∆

157 example if the longitudinal residual stress reaches 300 MPa (upper limit of measured values) exceeds 0.8 and drops to 3.8. On the other hand, the crack tip is a suitable location for the congestion∆ of defects including contaminations of the rail surface and porosities entrapped at the crack tip as well as on the contact of the lap and rail surfaces. The presence of defects inevitably reduces the threshold stress intensity factor range and increases the risk of crack propagation. Fracture reports suggest that many of the straight breaks initiate from cold laps having entrapped porosity at the tip.

Since the lap tip is located on the fusion boundary, it may also be suitable to consider the threshold stress intensity factor range of the rail steel (However, the HAZ material fracture toughness may be different from that of the parent rail). According to Skyttebol et al. [61] for very high R s-ratios (0.7-0.9) the value of may be used √ for rail steel. In this case the equivalent intensity factor∆ range 2 of the current problem is beyond the threshold value and the crack will propagate from the lap ∆ = 3.01 tip. However, the correct value of at the crack tip may considerably vary ∆ according to the loading ratio, defect content and the welding procedure and conditions.

Under the assumption of crack propagation it is important to determine the angle of kinking or crack initial growth. Several criteria are available for the estimation of crack growth among which the Richard’s criterion was mentioned earlier. According to Richard’s criterion the predicted kink angle for the current simulation parameters is about -53 o which is obtained through Equation 6-5. For the sake of comparison, two other criteria are explained and used for kink angle determination. According to the maximum circumferential tensile stress criterion [161], the crack propagates in a plane perpendicular to the direction of maximum tensile stress. The kink angle can then be obtained using the following relation:

3 + + 8 (6-11) = − cos + 9

The kink angle predicted by this criterion for the current analysis is about -55 o. Another widely used criterion is the maximum energy release rate (MERR) proposed by Hussain

158 et al. [163]. According to this criterion the crack propagates in the direction where the strain energy release rate achieves its maximum value: ()

(6-11) () () 0 , < 0 Figure 6-12 illustrates the variation of energy release rate versus the direction of kinking. The kink angle corresponding to the maximum energy release rate is about -55 o similar to the one proposed by the maximum circumferential tensile stress criterion.

Figure 6-12: Variation of energy release rate (G) with respect to the kink angle. Maximum energy release rate corresponds to the kink angle of 55 o

As previously mentioned, the stress intensity factors at the kink tip are different from those of the original crack tip and depend on the kink angle. Figure 6-13 illustrates the variation of stress intensity factor ranges versus prospective kink angle. The kink angles predicted by the three criteria are also depicted in the figure. It is important to note that the predictions are consistent to the kink angle which provides the highest and zero . In fact, the crack propagates in such way that the mode II propagation∆ changes into∆ mode I. The kink angle may change further during the propagation phase in order to maintain the mode I growth. The propagation of straight break in real life is also

159 subject to the same condition since it eventually propagates in vertical direction where the crack opening mode I achieves the maximum value.

Figure 6-13: Variation of stress intensity factor range at the tip of a kink with respect to the kink angle. Note that the kink angle predicted using different criteria is consistent to the maximum ∆ 6.3.1 Sensitivity to Lap Thickness

Depending on the size of gap between the mold and the rail surface and the extent of dissimilarity in rail heights, the thickness of the produced cold lap may vary. It is important to determine the effect of lap thickness in order to gain an insight into the damage tolerance of AT weld. For this purpose a range of thickness values between 1 mm to 3 mm is considered while other geometric parameters of the lap are kept unchanged. Figure 6-14 illustrates the variation of stress intensity factors at the middle point of the lap for two lap thickness of 1 mm and 3 mm during one load cycle. For both loading modes I and II the stress intensity factors increase dramatically with the thickening of the lap. In fact as the lap enlarges, the forces are better transferred from the collar to the bearing surface of the lap and the crack opening and sliding increase. Similarly, the stress intensity factor ranges for both prominent modes rise with an

160 increase in the lap thickness. It is also apparent in Figure 6-14 that the mode I stress intensity factor values for the thicker lap are much closer to those of the mode II and it suggests that higher vertical forces (opening forces) are applied to the lap. However, the effect is very limited on the mode III tearing and this mode does not play an important role in the propagation of the crack.

Figure 6-14: Variation of stress intensity factors and the resulting stress intensity factor ranges at the middle point of the cold lap for two lap thickness values

Figure 6-15 depicts the variation of equivalent stress intensity factor range ( obtained by Equation 6-7 with respect to different lap thickness values. A dramatic∆ ) increase in is evident with respect to the cold lap thickens and it suggests that the risk of crack∆ propagation from the lap tip rises for thicker laps. Ross [45] has also reported high dependence of stress concentration at the lap apex on the lap thickness based on a simplified analytical model.

The two threshold stress intensity factor ranges mentioned earlier for the weld web region and the rail steel are also illustrated in the figure by the two dashed lines. For all

161 the thickness values considered is beyond the steel threshold limit of 2 MPa √m and it may imply the risk of crack∆ propagation for all the thickness values considered. However, even for the thickest lap the equivalent stress intensity factor range is still below the threshold limit of the weld material measured at the web region (4.4 MPa √m) and it may suggest a safe operation. Nevertheless, as previously mentioned there are many uncertainties at the lap region which may affect the threshold values as well as the risk of crack development. Based on the experimental data reported by Lawrence et al. [46] even a 1 mm lap can reduce fatigue life of the weldment by 80%.

Figure 6-15: Equivalent stress intensity factor range at the middle point of the lap versus the cold lap thickness. The two threshold values are included in the figure

6.3.2 Sensitivity to Lap Unfused Length and Width

A sensitivity analysis is performed on two geometric features of the lap, unfused length and lap width. The unfused length refers to the extension of the leaked material beyond the fusion boundary of the weld and in a way it may represent the crack length. Figure 6-16 illustrates the variation of with respect to the unfused length ranging from 2 mm to 12 mm. The influence ∆ of the lap length is quite limited with only a slight

162 decrease of for the unfused length of 2 mm to 8 mm and lack of sensitivity to increase of length∆ beyond 8 mm. In fact, the load transferred to the lap by the weld collar only affects a small portion of the lap at the vicinity of the lap tip and the rest of the lap length does not support any load. In this case the unfused length should not be referred to as the crack length since most of the lap length has no role on opening or sliding of the crack mouth. The insensitivity of the cold lap fatigue behaviour to its length has also been reported in the analytical work by Ross [45].

Figure 6-16: Equivalent stress intensity factor range at the middle point of the lap with respect to lap unfused length

The sensitivity analysis on the lap width leads to similar results. Figure 6-17 depicts the variation of for different width values ranging from 5 mm to 30 mm. Figure shows a slight decrease∆ of the equivalent stress intensity factor range from about 3.2 MPa √m for a 5 mm lap down to 3 MPa √m for a 30 mm lap. Insensitivity to the lap width is also observed for laps with widths beyond 20 mm. Nevertheless, the increase in the width of the lap may result in higher probability for the entrapment of contamination, defects and porosities at the lap tip along its long width. This eventually leads to an abrupt decrease in values of the local material and the higher probability of crack propagation. ∆

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Figure 6-17: Equivalent stress intensity factor range at the middle point of the lap with respect to lap width

6.3.3 Sensitivity to Edge Offset

As described in Section 6-1 according to the heat input, rail stick-out length and other welding parameters the fusion boundary of the weld may either form inside or outside of the weld collar. Since the location of the lap apex coincides with that of the fusion boundary, the lap tip can also emerge in or out of the collar forming one of the two cases depicted in Figure 6-2. It is important to determine the effect of lap tip location (distance of the lap tip from the collar edge) on the fatigue behaviour of the crack. This sensitivity analysis is performed for a range of distances from -5 mm (representing an internal lap tip located 5 mm from the collar edge) to +10 mm (external lap tip located 10 mm from the collar edge). Lap thickness is set to 2 mm and the width to 20 mm.

Figure 6-18 illustrates the variation of stress intensity factor ranges in all three loading modes as well as the equivalent stress intensity factor range with respect to different lap tip locations. As the figure suggests dramatically increases with the lap tip approaching the collar edge from outside∆ the weld collar. An increase of more than 40% is evident for the lap tip located at the collar edge compared to a tip having 10 mm distance from the edge. Moving the lap apex from the collar edge location inward (into

164 the weld collar) causes a slight relief of from about 3.74 to 3.58 MPa √m, a decrease of about 4%. The crack tip located∆ on the collar edge has the highest suggesting the highest risk of crack propagation for such laps. This may be attributed∆ to the high stress concentration at the collar edge which affects the loading of the crack at its tip. The figure also suggests that laps with apexes located out of the collar (when the fusion zone is wide and the fusion boundaries are far from the collar edge) are safer for the operation than those in the inside.

The variation of separated stress intensity factor ranges provides some explanation for the current findings. The trend of which prominently governs the value of through Equation 6-7 shows a great∆ increase for the lap tip approaching the collar ∆edge particularly from the inside of the collar. However, the variation of for the crack tips at the left side of the collar edge is not as severe as due to the∆ adverse trend of . In fact, for crack tips located inside the collar the in-plane∆ tearing or sliding mode is∆ the governing factor for crack propagation.

Figure 6-18: Variation of stress intensity factor ranges and the equivalent parameter with respect to the distance of the lap tip from the collar edge. Negative values relate to lap tips inside the collar while positive values imply lap tips located outside

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6.3.4 Contact Patch Displacement and Lateral Traction

Among important parameters to investigate is the fatigue behaviour of cold laps in curved tracks or tangent tracks prone to vehicle hunting. It was earlier described that the effect could be simulated by two distinct representative conditions; contact patch lateral displacement and lateral tractions with various ratio of the lateral load to vertical load. In this study cold lap on both field and gauge sides of the weld is considered and the traction is deemed to be inwards as well as outwards.

Contact patch lateral displacement is considered for a patch moving from the centerline of the weld to 25 mm distant from the centerline towards the gauge side. Figure 6-19 illustrates the variation of separated stress intensity factor ranges as well as the equivalent value at the middle point of the lap located on the gauge side (left side of the graph) and the field side (right side of the graph).

Figure 6-19: Variation of separated stress intensity factor ranges as well as the equivalent value with respect to the contact patch eccentricity from the centerline towards the gauge side for the lap located on the field and gauge side of the weld

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A linear trend for all the crack modes and the equivalent stress intensity factor range is evident. The value of increases dramatically as the patch makes distance from the weld centreline for the∆ laps located on the field side. This is in fact due to the linear increase of in-plane opening and sliding driving forces; and respectively. For the laps located on the gauge side an opposite behaviour∆ is observed.∆ By moving the contact patch towards the gauge side , and decrease linearly. The most critical condition ( ∆) is∆ observed∆ for a lap located on the field side √ when the patch is ∆displaced 3.74 25 mm from the centreline and the least critical condition ( ) for a patch located on the gauge side with the same eccentricity ∆ = 2.31 √ of the loading. This suggests that the highest driving forces are applied to the laps on the field side. However, the contact patch eccentricity has virtually no effect on the out of plane tearing mode and maintains a constant value. ∆

Similar to the approach in the load eccentricity the effect of lateral traction is considered for laps on both field and gauge sides. Lateral traction coefficients of 0 to 0.4 is applied on a contact patch located 15 mm from the weld centreline towards the gauge side and both inwards (towards the gauge side) and outwards (towards the field side) directions are incorporated in the model. Figure 6-20 depicts the variation of separated stress intensity factor ranges and the equivalent value at the midpoint of the lap located on the field side of the weld for different load ratios. The figure generally shows a near linear dramatic increase in when the total applied load is inclined more towards the ∆ gauge side. From traction coefficient of 0.2 to -0.4 the value of changes from 3.19 ∆ to 4.38 MPa √m which is a 37% increase. This value is also very close to the threshold reported for the weld material sampled form the web region ( MPa √m). The ∆ = 4.4 main reason for this notable increase is the linear growth of and which is ∆ ∆ consistent to the variation of longitudinal stress at the top of the foot previously depicted in Figure 3-31. For traction coefficients of 0.2 to 0.4 (towards the field side) a slight increase in is observed mainly attributed to the dramatic growth of ∆ ∆ while and maintain a near constant value. At traction ratio of 0.4 all the crack ∆ ∆ loading modes (I, II, III) have comparable stress intensity factor range values suggesting a multi-axial crack loading.

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Figure 6-20: Variation of , , and with respect to lateral traction coefficient of both inward and outward∆ ∆ directions∆ for∆ a lap located on the field side of the weld

Figure 6-21: Variation of , , and with respect to lateral traction coefficient of both inward and outward∆ ∆ directions ∆ for ∆a lap located on the gauge side of the weld

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Figure 6-21 illustrates the same parameters for the lap located on the gauge side of the weld. In contrast to the trend observed in the previous section, the main increasing trend in relates to the tractions more inclined towards the field side (opposite to the location∆ of lap). From traction coefficient of -0.1 to 0.4 varies between 2.51 to 3.44 MPa √m, an increase of 37%. Similarly, this trend is mai∆nly attributed to the linear increase of and which is also consistent to the increasing trend of the longitudinal stress∆ depicted∆ in Figure 3-31. Although, the values of and do not notably change for inward traction ratios of -0.1 to -0.4 a slight ∆ increase ∆ of is observed which is related to the increasing trend of . For high inward tractions∆ (- 0.3 to -0.4) all the crack loading modes have similar∆ values of stress intensity factor range implying a multi-axial crack loading.

Summing up the sensitivity analysis on the patch eccentricity and tractions it can be concluded that the laps on the field side of the weld are under a more critical condition than those on the gauge side with similar loading scenarios. The most damaging loading is attributed to the extensive contact patch eccentricity with high lateral tractions which are inclined towards the rail side which is opposite to the side of the lap.

6.3.5 Track Support

Degradation of ballast and lack of consistent track support in the vicinity of AT welds was shown to reduce the fatigue performance of the base region using multi-axial fatigue analysis. In this section the effect of track support on the fatigue behaviour of cold lap is investigated through the variation of ballast stiffness supporting the two weld adjacent sleepers.

Figure 6-22 illustrates the variation of , , and with respect to the ballast vertical stiffness in a tangent track.∆ As ∆the figure∆ suggests∆ the reduction of ballast stiffness due to the degradation increases the value of which indeed increases the chance of crack propagation from the lap tip. This is ∆consistent with the trend of the longitudinal stress depicted in Figure 3-33. The higher bending moment and the longitudinal stresses in ill-supported track results in the higher in-plane opening ( ) ∆ 169 and shear ( ) driving forces and consequently larger values. However, as expected the∆ out of plane tearing mode is not affected by the∆ condition of track support since no related loading scenario is altered.

Figure 6-22: Variation of , , and with respect to ballast vertical stiffness in a ∆ ∆tangent ∆ track (central∆ loading)

6.3.6 Crack Kinking

It was assumed throughout the study that the lap apex (crack front) is free from any type of defect or impurities. However, in some instances the material at the tip of the lap contains traces of casting defects such as pores or contamination. The presence of defects can lead to the reduction of and possible formation of a kink (small crack extension) at the lap tip. The formed∆ kink may or may not propagate according to its effect on the crack loading and the final value of at the tip of the kink. In this section, the effect of a presumed kink and its initial∆ length on the fatigue behaviour of the original lap is of interest. The direction of the kink considered in this study is set to 55 o which is the crack propagation angle predicted by Richard’s or the similar criteria such as the maximum circumferential stress.

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Figure 6-23 presents the variation of separated stress intensity factor ranges at the tip of a presumed kink for kink lengths varying from 0.5 mm to 3 mm. Since the kink direction corresponds to the plane of the maximum circumferential stress the largest load applied on the crack faces is the opening force. Accordingly, the value of has an absolute prominence over the in-plane sliding mode and so it is the∆ main parameter governing the equivalent stress intensity factor range∆ . As expected the trend of and consequently shows a notable increase with∆ the extension of the kink. An∆ increase of the kink length∆ from 0.5 mm to 3 mm leads to a 56% rise in the value of from 5.02 to 7.82 MPa √m. ∆ The important finding is the substantial effect of kinking on the fatigue behaviour of the lap. Even a very short 0.5 mm kink can increase from 3.01 MPa √m (at the original crack tip) to 5.02 MPa √m which is beyond the∆ threshold value of the weld sample material (4.4 MPa √m). This can be considered a good indication for the strong effect of casting defects on the initiation of fatigue failures from the tip of cold laps.

Figure 6-23: Variation of , , and at the tip of an existing kink with respect to ∆ ∆ ∆the kink∆ length

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Site observations support the idea since the majority of failures associated with the cold lap defect relate to entrapment of pores at its apex. Figure 6-24 illustrates a straight break failure at the top of the rail foot underneath a cold lap defect. Straight break has initiated from the apex of the lap where large area of porosity is evident.

Cold Lap

Porosity

Figure 6-24: Straight break at the top of the rail foot associated with a cold lap defect with high content of porosity at its apex [169]

In this chapter the formation of straight break under the influence of cold lap defect located at the top of the rail foot was investigated through modelling the defect as an equivalent planar crack and the utilisation of linear elastic fracture mechanics. It was shown that the equivalent stress intensity factor range can reach the threshold value suggesting possible crack initiation from the apex of the cold lap. The study on the lap geometric features revealed that the lap thickness has a great effect on . In fact thicker laps are more influential in the formation of straight breaks. It was∆ also shown that as the fusion boundary or the lap apex moves further from the collar edge the possibility of crack initiation is reduced. According to the analysis of the contact conditions cold laps on the field side are more damaging than those on the gauge side. Inward (towards the gauge side) tractions also apply high driving forces on the field- side located laps whereas outward tractions are effective on the laps located on the gauge-side of the rail.

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CHAPTER 7 TOLERANCE TO WEB DEFECTS

Practical observations show that the occurrence of horizontal split web failures in aluminothermic welds is mainly attributed to the presence of large surface or near- surface defects at the web region of the weld collar. High vertical service and residual stresses usually play an important role in developing fatigue or overload failures at the surface of these large defects. Hence, it is crucial to characterize the sensitivity of the collar web design to different defect geometries and sizes and determine a collar geometry having the highest tolerance to casting defects. It is however important to note that a suitable collar design in terms of damage tolerance does not necessarily provide the highest fatigue performance. The collar geometric design also affects the solidification characteristics, defect formation and the cooling behaviour which determines the distribution of vertical residual stresses on the web region.

As described in Chapter 2 a number of techniques have been developed to investigate the effect of specific defects on fatigue strength and determine the damage tolerance of the material. The selection of a suitable approach mainly depends on the size of the defect and its geometric characteristics which defines its stress concentration factor. The large defects responsible for HSW formation may be categorized into large defects with smooth surface and low stress concentration factors and defects with high surface irregularities and high stress concentration factors. In the low stress concentration regime, stress gradient (variation of stress from the defect surface to the interior of material) is relatively low and the fatigue limit may be described by the initiation of fatigue cracks from the defect surface. In such cases the defect can be considered as a notch and the multi-axial fatigue analysis may be performed on the surface of the defect to determine the initiation of fatigue cracks.

Defects with high stress concentration factors usually feature high stress gradients and so the fatigue cracks formed on the surface of the defect may not propagate further when the crack tip approaches the low stress region at the vicinity of the defect. The

173 fatigue failure and strength of material in this case depends on the propagation or non- propagation of these peripheral cracks and so the initiation of the crack at the surface of the defect may not solely provide the failure condition for the structure. For such defects a widely used approach is to substitute the defect with a crack of the same size and perform fracture mechanics analysis.

In Chapter 5 the fatigue performance of two weld collar designs in absence of casting defects was investigated. It was shown that under similar conditions, Type A collar performs better since the web is reinforced by a buttress which reduces the cyclic vertical stress and damage parameter at the critical point of the collar. The main purpose of this chapter is to analyze the reduction of fatigue performance due to the presence of surface defects with specific predefined geometric shapes and sizes. Both collar designs Type A and Type B are considered to provide a better insight into the effect of collar design with respect to the HSW formation. The difference in collar shape may also affect the propensity of the collar to develop surface defects such as shrinkage cracks, hot tears and inclusions. However, this aspect is not considered in this study.

7.1 Defects as Notches (Multi-axial Fatigue)

In this approach the crack initiation at the surface of the defect is investigated using multi-axial fatigue analysis similar to the analysis performed on the collar edge for straight break formation. The first step is to model the defect on the surface of the weld collar and perform a stress distribution analysis on the surface of the defect to achieve the stress state during one loading cycle. The stress history at the critical points of the defect is inserted into the Dang Van multi-axial fatigue code and the damage parameter on the most critical shear plane of the corresponding point is determined. Fry et al. [107] has also applied a similar approach (using Findley’s criterion) on the casting defects responsible for different failure modes of rail head in aluminothermic welds.

The focus of this section is on pore defects with three different geometric shapes: spherical, ellipsoidal and coin-shape. To investigate the most critical condition, the defects are modelled at the location of the web which experiences the highest fatigue

174 damage or DV damage parameter. Previous analysis in Chapter 5 revealed that this location in Type A weld is about 90 mm and in Type B about 120 mm above the rail foot (Figure 7-1). Throughout the analysis in this chapter the contact load is applied at 25 mm offset from the rail centerline towards the gauge side in order to provide the highest vertical stress on the field side where the defects are located. No lateral or longitudinal tractive forces are considered on the wheel-rail contact patch. The vertical residual stress is also derived from the measurement trend lines depicted in Figure 5-15. All other simulation parameters including the track stiffness, axle load and boundary conditions are the same as previous analysis in Chapter 3.

Figure 7-1: Variation of the DV damage parameter on the web centerline of the Type A and Type B welds and the corresponding locations of the maximum damage where the defects are modelled

(a) (b) (c)

Figure 7-2: Defect models used in the damage tolerance analysis: (a) Spherical defect; (b) Ellipsoidal defect; and (c) Coin-shape defect

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Figure 7-2 illustrates the ideal geometries considered for modeling the pore defects. Since surface defects are the focus of this study only half of the depicted geometries are engraved on the web surface.

7.1.1 Spherical Defects

A set of spherical defects with radius ranging from 0.5 mm to 4 mm was modelled on the surface of the two collar shapes. To correctly simulate the stress concentration and extract the stress history a fine mesh is applied to the surface of the defect. The element size on the defect surface varies between 3.5e-5 m and 2.5e-4 m according to the defect size. Figure 7-3 depicts the finite element mesh for a 3 mm radius defect.

Figure 7-3: Finite element mesh of the 3 mm radius spherical defect

Due to the important role of the vertical stress distribution in initiation of fatigue cracks, the corresponding stress component on the surface of a 0.5 mm and 4 mm radius defects are compared in Figure 7-4 for both weld types. The contours generally show high stress concentration on a horizontal band around the spherical defect due to the notch effect. For the 0.5 mm defect located on Type B collar higher value of stress is observed compared to that of Type A which is in fact attributed to the larger cyclic vertical stress

176 on the web region of the Type B compared to Type A which has reinforcement in the collar (Figure 5-15). The large 4 mm radius defect generally shows lower stress concentration compared to the smaller defect except at the region close to the collar web surface in Type A weld. In fact, a larger radius at the notch root decreases the notch effect and the corresponding stress concentration factor. However, this effect is overridden in Type A weld by the increase of the local stress as the defect surface approaches the boundaries of the collar.

The variation of vertical stress with respect to the defect radius and the type of collar is more clearly illustrated in Figure 7-5. Two important points on the defect are investigated; point D at the deepest location on the defect surface and point S on the intersection of the defect surface and the collar exterior surface. The highest vertical stress corresponds to the smallest defect with the minimum radius and maximum notch effect. As expected the vertical stress decreases with the enlargement of the defect at both locations on the defect surface. For small to medium sized defects the severity of stress concentration is more pronounced in Type B weld. However, this trend is reversed for larger defects implying the increasing effect of the collar shape on the stress distribution of the defect surface. In particular, vertical stress at point S of the defect on Type A weld increases with an increase in the defect size which is unexpected (based on the relation between notch radius and stress concentration). The reason for this unexpected increase is the interaction of the defect surface and the collar boundaries as shown in Figure 7-6 which provides a geometric complexity at the region and accordingly a high stress concentration. Due to the localized bulge shape of the Type A collar a large defect causes the internal force lines to become very dense at the surface and this results in an increased stress concentration. In contrast, the Type B weld features a flatter collar design which provides lower geometric complexity, a better redistribution of the internal force lines and lower stress concentration.

Figure 7-7 depicts the results of multi-axial fatigue analysis on the two points of the defect surface. The DV damage parameter for both welds nearly follows the same trend observed in the vertical stress variation. The highest damage relates to the deepest point of the smallest defect on Type B weld collar. However, opposite to the behaviour in

177

Type B weld the fatigue damage parameter increases at Point S of the defect on Type A weld as the defect radius increases (due to the mentioned effect of the collar design). It can be concluded that for a smooth spherical defect, Type A weld is sensitive to larger defects while Type B is more sensitive to smaller defects.

(a) (b)

Figure 7-4: Vertical stress contour on a spherical defect when the contact load is located exactly at the top of the weld: (a) R=0.5 mm Type A weld, (b) R=0.5 mm Type B, (c) R=4 mm Type A; and (d) R=4 mm Type B

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Figure 7-5: Variation of vertical stress ( σz) at two points on the surface of the defect with respect to the defect radius: point D at the deepest point of the defect and Point S at the intersection of the defect and collar web surfaces

(a) (b)

Geometric complexity causing the stress concentration

Figure 7-6: Lines of force on the surface of the collar around a spherical defect and the location of stress concentration: (a) Type A weld; and (b) Type B weld

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Figure 7-7: Variation of DV damage parameter at two points on the surface of the spherical defect with respect to the defect radius

In any case, the initiation of fatigue cracks on the surface of the defect may not necessarily imply crack propagation and total failure of the web region. In fact, fatigue cracks emanated from small defects are less likely to propagate than cracks on the surface of large defects and this is due to the lower driving force on smaller cracks. In this case fracture mechanics and crack analysis (reported in the last section of this chapter) can provide a better understanding on the tolerance of each collar design to defect sizes. It is worth mentioning that multi-axial fatigue analysis mentioned earlier does not predict fatigue crack initiation from any size of a smooth spherical defect since the damage parameter is generally below the threshold value of bDV =235 MPa . However, real casting defects, particularly inclusions, shrinkage defects and hot tears, may be far from ideal geometric shapes. Defect surface roughness, irregularities and existence of micro cracks and microstructure impurities on the periphery of the defect have also great influence on the nucleation of fatigue cracks. Nevertheless, the above analysis provides an insight into the interaction of the defect and collar geometry and the comparative damage tolerance with respect to the collar design.

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7.1.2 Ellipsoidal Defects

The ellipsoidal defect considered in this study follows the general Equation in Figure 7- 8 with the condition that semi-principal axes b and c are equal and semi-principal axis a is fixed during the analysis. Similar to spherical defects, half of the ellipsoid is engraved on the surface of the collar. The value of a is set to 2 mm while b and c are variable changing from 0.1 mm to 1.25 mm.

Figure 7-8: Equation of the considered ellipsoidal defect with conditions: b=c and a=cte

The vertical stress contour on the surface of the defect is depicted in Figure 7-9 for two semi-axis values of 0.25 mm and 1.25mm. Similar to the spherical defect the stress is concentrated on a horizontal band around the defect surface. However, for the small 0.25 mm defect the stress concentration reduces from its maximum value at the deepest point of the defect towards the two ends on the right and left side. Since, for the small defect the semi-axis b is much smaller than a the defect is very much stretched and the two end points on the x-axis (Figure 7-8) are very shallow on the collar surface and so they impose a very small stress concentration. For large semi-axis b the vertical stress component seems to be similar on the deepest point and the two ends close to the collar surface. In any case the stress values are higher on the Type B collar compared to Type A and this is mainly attributed to the higher cyclic vertical stress observed on the collar Type B (Figure 5-15). Figure 7-10 illustrates the trend of cyclic vertical stress at two points of the defect with respect to the variation of semi-minor axis b. For both the collar designs the normal stress at the deepest point of the defect decreases in a near linear way as the defect enlarges. This is similar to the behaviour observed in spherical defects and is associated to the mitigation of stress concentration (and the notch effect) for larger radius notch roots. For point S at the surface of the collar a mixed behaviour

181 influenced by the effect of depth and radius is observed. As the semi-axis b increases the depth of the defect at point S increases and so the stress becomes more concentrated. This behaviour continues up to a certain defect size ( b≈0.6 mm) for which the effect of defect radius (which is a decrease in stress concentration) at point S overrides the effect of depth and so the notch effect starts to diminish.

(a) (b)

Figure 7-9: Vertical stress contour on an ellipsoidal defect (a) b=0.25 mm Type A weld, (b) b=0.25 mm Type B, (c) b=1.25 mm Type A; and (d) b=1.25 mm Type B

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Figure 7-10: Variation of vertical stress ( σz) at two points on the surface of the defect with respect to the semi-minor axis b

Figure 7-11: Variation of DV damage parameter at two points on the surface of the defect with respect to the semi-minor axis b

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The results of the multi-axial fatigue analysis is reported on Figure 7-11 with very much similar trends as the ones depicted in vertical stress component in Figure 7-10. The highest damage parameter relates to the deepest point of the smallest defect ( b=0.1 mm ) located on collar Type B. However, damage values are below the threshold fatigue damage parameter of 235 MPa required for crack initiation. It is worth mentioning that the interaction of collar border and the defect described for large spherical defects is not notably observed for the ellipsoidal defects investigated since the defect size (semi- major axis a) is too small for such interaction.

7.1.3 Coin-Shape Defects

For analysis of the coin-shape defect specific type of geometry with the cross section shown in Figure 7-12 is considered. The following geometry has the advantage that the defect notch radius remains constant regardless of the defect size and the coin-shape form of the section is maintained. In this case the effect of defect size and its depth can solely be investigated. Defect widths ( w) varying from 1 mm to 6 mm are considered and like other surface defects, half of the geometry is engraved on the collar surface.

Figure 7-12: Geometric parameters of the coin-shape defect considered in this study ( w is variable)

Figure 7-13 illustrates the vertical stress contour on the surface of 1 mm and 6 mm wide defects when the contact load is exactly at the top of the weld. Very high stress concentration factors (between 4 and 9) are visible for all the defect sizes and on both collar designs which is mainly attributed to the small notch radius. For both small and large defects the stress concentration is more notable at the shallow points of the defect close to the surface of the collar and in this region Type B weld exhibits higher stresses

184 than Type A. The stress concentration observed on the large defect is much higher than on the small one and this solely shows the influence of defect size (depth) on the stress values keeping in mind that the defect notch radius is constant. This is different to what was observed in spherical defects where the increase of the defect (sphere) depth also increases the defect radius which has a relieving effect on the stress concentration.

(a) (b)

Figure 7-13: Vertical stress contour on a coin-shape defect: (a) w=1 mm Type A weld, (b) w=1 mm Type B, (c) w=6 mm Type A; and (d) w=6 mm Type B weld

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The trend of equivalent stress (von Mises stress) variation on the two points of the defect surface is presented in Figure 7-14. It is clear that by increasing the defect size (depth) the stress on both regions of the defect surface increases and in occasions the stress exceeds the yield strength of the material. The yield strength of the current weld material is estimated according to the previous experiments on relationship between hardness and 0.2% proof stress for different regions of the weld presented by Mutton [34] (Figure 7-15). For hardness value of 340 HV the proof stress is estimated at about 740 MPa. Accordingly, signs of plastic deformation are observed in some regions of the defect surface as the defect enlarges. The most critical locations are the defect surface on Type B, defect surface on Type A, deepest points on Type A and deepest point on Type B respectively.

Figure 7-14: Variation of the equivalent (von Mises) stress at two points on the surface of the coin- shape defect with respect to the defect width ( w)

The results of the multi-axial fatigue analysis are illustrated in Figure 7-16. As part of the conditions in the safe usage of the Dan Van multi-axial fatigue criterion is the assumption of elastic shakedown. Since the current analysis does not provide enough basis for the investigation of shakedown on the surface of the defect after initial yielding

186 and also due to the lack of information on the constitutive behaviour of the weld material it may be unsuitable to apply the criterion to regions with plastic deformation. Accordingly, the mentioned regions have been shown on Figure 7-16 by dashed lines and although the damage values have been calculated for the whole range of defect sizes they cannot be referred to for fatigue analysis purposes. In this case the damage graph is only applicable to defect sizes below 2 mm width on the surface of Type B and 2.4 mm on the surface of the Type A collar. To be able to use the multi-axial fatigue criterion, the stress state after shakedown should be considered and if elastic shakedown does not occur other approaches applicable to fatigue under ratchetting or plastic shakedown conditions should be applied. However, in Section 2.3.3 it was mentioned that defects with high stress concentration behave in a similar way as cracks with the same size. In this condition the methods based on fracture mechanics can better predict the failure of the structure compared to the methods based on the notch effect and stress state on the surface of the defect (Figure 2-23). This is in fact the reason for the wide usage of fracture mechanics technique to characterize defects and damage tolerance of structures. In the following section the fracture mechanics approach is exploited to determine the severity of high stress concentration defects such as the coin-shape defects and also the propagation of existing cracks on the surface of the web.

Figure 7-15: Determination of relationship between hardness and yield strength of rail welds [34]

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Figure 7-16: Variation of the DV damage parameter at two points on the surface of the coin-shape defect with respect to the defect width ( w)

7.2 Fracture Mechanics Approach

It was earlier described that even if the multi-axial fatigue analysis predicts crack initiation on the surface of the defect it does not necessarily mean a failure particularly for very small defects. In fact most of the defects with very high stress concentration at their periphery include cracks at or close to the fatigue limit of the material but the failure depends on the propagation of these existing cracks. The application of fracture mechanics has the advantage that it can characterize both high stress concentration defects (large defects with severe irregularities on the surface and very small defects) and also cases where cracks themselves are embedded in the structure.

For the purpose of this study semi-circular cracks with radius ranging from 1 mm to 4 mm are modelled on the surface of the two collars at the most critical location of the web. These semi-circular cracks may also represent high stress concentration coin-shape defects and also spherical defects with peripheral cracks with the same total projection area. Similar to what was applied for the analysis of cold lap defect linear elastic

188 fracture mechanics is exploited and the values of the separated stress intensity factors during a loading cycle is extracted using the virtual crack closure technique (VCCT). As a condition for the accurate usage of the VCCT brick elements should be applied all the way perpendicular to the crack front. For this purpose a semi-circular crack block as depicted in Figure 7-17 is embedded on the surface of the web. In contrast to the meshing of the surrounding areas on the weld collar, the crack block features radial meshing using 8-node brick elements which can satisfy the requirements of the VCCT.

Figure 7-17: Finite element mesh of the crack block and the application of radial meshing suitable for virtual crack closure technique

Since the vertical stress acts as the main driving force on the crack, the contour of vertical stress on a 5 mm radius crack block and the lower crack face is depicted in Figure 7-18 for both collar designs. The figure suggests a stress singularity at the tip of the crack throughout the crack front which is more pronounced on the region close to the surface of the collar. It is also apparent that the severity of stress intensity is higher in the Type A collar compared to that of Type B. However, to investigate whether the existing crack can develop into a horizontal split web failure, the chance of crack propagation should be estimated by comparing the equivalent stress intensity factor range ( ) with the threshold stress intensity factor range ( ) which is a material property.∆ The value of also depends on the stress ratio∆ and as suggested by Equation 6-10 its value ∆increases by a decrease in stress ratio. According to the current

189 loading parameters the stress ratio at the crack front is estimated at about 0.5 and values at 6.77 MPa √m. ∆

(a) (b)

Figure 7-18: Vertical stress contour on the crack block and the lower crack face when contact load is located exactly at the top of the weld: (a) R=5 mm Type A weld, (b) R=5 mm Type B weld

The trend of the equivalent stress intensity factor range at two limit points on the crack front with respect to the crack radius is illustrated in Figure 7-19. Similar to what is observed on the vertical stress contour is more pronounced on the regions of the crack front close to the collar surface. The∆ collar type B is more sensitive to small to medium sized cracks while for larger cracks (radius bigger than 3 mm) Type A weld shows higher values which can be attributed to the interaction between the crack front and the collar∆ boundaries previously described in section 7.1.1. According to the current loading parameters the damage tolerance of Type A weld is defined as crack radius below 1.69 mm and for Type B as crack radius below 1.36 mm. So even if fatigue cracks initiate on the surface of a 1 mm radius spherical or coin-shape defect it is less likely that the crack could propagate and lead to a total failure.

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Figure 7-19: Variation of equivalent stress intensity factor range with respect to the crack radius: point D at the deepest point of the crack and Point S at the intersection of the crack front and the collar surface

In order to determine the collar performance in terms of the fast or brittle fracture, mode I stress intensity factor ( ) can be studied. In fact most of the HSW failures observed in Type A weld develop under an overload condition with no sign of fatigue crack growth. Under such condition the fracture toughness of the material ( ) becomes very important. According to the data reported by Mutton and Alvarez [15] in different regions of the weld can vary between 25-35 MPa √m and based on measurements by Dudley et al. [52] the dynamic fracture toughness more preferably used for fast fracture under high strain conditions ( ) ranges between 24-36 MPa √m. Figure 7-20 illustrates the variation of on two points of the crack front with respect to the crack radius for both collar designs. As expected, by increasing the crack size the value of for both points on the crack front increases. However, the severity of is more pronounced on the regions close to the collar surface which implies that these regions undergo fast fracture sooner than the deeper points on the crack front. For most of the crack sizes, the condition in Type A weld is more critical since the slope of the increase of is higher

191 compared to that of Type B particularly at the regions of the crack front close to the collar surface. This is in fact a sign of interaction between the crack front close to the collar surface and the boundaries of the collar design Type A which introduces higher stress concentration at this region. Figure 7-20 suggests that larger cracks on collar Type A can more readily cause overload fractures than those on Type B and this is of great importance since the main cause of the HSW failure in Type A are overload fractures initiated from large defects including inclusions, shrinkage defects and hot tears.

Figure 7-20: Variation of mode I stress intensity factor (KI) with respect to the crack radius

Nevertheless, a comprehensive damage tolerance comparison between different weld collar designs needs to consider other parameters including the likelihood of each collar design and the welding process to develop large defects suitable for HSW initiation, the strength of the microstructure at the regions of interest on the web and the correct values of and in each structure. The vertical residual stress and the microstructure defect∆ content can also be very variable from one manufactured weld to the other and all of these arise from the operator dependability of the process.

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In this chapter the fatigue performance of weld in terms of HSW formation was investigated under the influence of several predefined web defect geometries. The two collar designs studied in chapter 5 were also included in the analysis to further clarify the effect of collar design of the web. The results shows that for small to medium sized defects the stress concentration and fatigue damage are mainly controlled by the defect surface features and the remote stress on the web surface. For large defects the interaction of the defect surface and collar boundaries particularly for weld Type A increases the local stress and the resulting fatigue damage index. To investigate the effect of planar defects such as coin shape defects linear elastic fracture mechanics was utilised and a sensitivity analysis was performed on the simulated semicircular crack radius. It was shown that the collar Type B is more sensitive to small to medium sized cracks while collar Type A shows higher values of equivalent stress intensity factor range and the mode I stress intensity factor for larger defects suggesting higher probability of crack propagation and earlier overload fracture in this collar design.

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CHAPTER 8 CONCLUSIONS AND FUTURE WORK

8.1 Conclusions

The purpose of this study was to investigate the fatigue behaviour of aluminothermic weld with respect to two major failure modes observed in Australian heavy haul railway systems: straight breaks and horizontal split webs (HSW). The risk of fatigue crack initiation and the associated critical locations on the weld surface were determined through stress and multi-axial fatigue analysis. The effect of several operational and loading parameters including the residual stress distribution arising from the welding process, rail-wheel contact patch location, lateral tractive forces on the contact patch and the track support conditions were evaluated. An attempt was made to characterize the influence of the weld collar design on the risk of fatigue crack formation particularly in relation to straight breaks. The connection between the existence of cold lap or finning defects and the nucleation of straight breaks at the collar edge was demonstrated using linear elastic fracture mechanics. The sensitivity of two different collar designs to specific defect geometries and predefined cracks on the surface of the collar web was evaluated using both notch method and fracture mechanics.

8.1.1 In-Track Bending Behaviour

As the first step in fatigue analysis of the weld, the stress distribution on a specific type of weld widely used in Australian heavy haul railway systems was studied using finite element method. The FE model was generated from the geometric data of the weld collar which was obtained by laser scanning of a representative weld sample and incorporated into a modelled track including sleepers and elastic foundation. The stress analysis shows high stress concentration throughout the collar edge with pronounced stress values at the base and underhead region. High longitudinal stress at the base results from the rail section bending behaviour. However, it was found that the unexpected localized tensile stress (tension spike) at the underhead relates to the local

194 bending of the rail head on the web which develops tensile stress in this region. The presence of tension spike at the underhead has been shown to be associated with greater risk of RCF cracks growing downward and leading to detail fracture. However, to date no analysis on its effect on the performance of welds has been published. The longitudinal tensile stress as a result of the seasonal temperature variation in winter seasons was shown to greatly intensify this tension spike.

The effect of curving and hunting behaviour was investigated through contact patch lateral displacement and tractions applied on the patch. Results of the analysis suggest that the lateral displacement of the patch greatly intensifies the severity of tension spike at the underhead due to the development of a local torsion on the weld cross section. The lateral bending of the rail section due to the eccentricity of the applied load also increases the cyclic vertical stress on the web region which plays an important role in occurrence of the horizontal split web fatigue mode. Tractions towards the field side of the weld increase the severity of tension spike at the underhead region and longitudinal stress of the field side. Inward tractions (towards the gauge side) were shown to be highly detrimental due to which the longitudinal stress at the underhead of the gauge side may even exceed the yield strength of the material. This is in fact attributed to the combination of the local rail head bending and the torsion of the rail section. Lack of vertical support on the sleepers adjacent to the aluminothermic weld affects the base region of the weld by increasing the longitudinal tensile stress.

8.1.2 Fatigue Behaviour

To investigate the fatigue behaviour the Dang Van multi-axial fatigue criterion based on the critical plane concept was exploited. In this method, the shear stress amplitude on the investigated shear plane passing through the material point is evaluated as a vector parameter based on the minimum circumscribed circle (MCC) approach. The shear plane with the highest damage value (linear combination of shear stress amplitude and the hydrostatic stress) is defined as the critical plane. For this analysis the stress history during one loading cycle at the point of interest on the surface of the structure is inserted into the developed MATLAB computer code where critical plane analysis is performed.

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The fatigue damage analysis was mainly performed on the collar edge of the weld with respect to the initiation of straight breaks. The results suggest a pronounced damage value at the underhead region and less severe damage at the underside of the foot. In fact, the high damage value at the underhead relates to the tension spike in this region and the high tensile longitudinal residual stress. The highly compressive residual stress at the base region was shown to have a mitigating effect on the damage parameter observed. Lateral displacement of the contact patch increases the damage at the underhead and for highly displaced contact patch which may happen in short radius curves the damage even exceeds the critical limit indicating a high risk of failure. The effect of lateral traction on the underhead region is noteworthy. Under an outward traction (towards the field side) with L/V ratio (ratio of lateral to vertical force) of more than 0.3 the damage value exceeds the critical limit. The results presented for the effect of contact patch eccentricity and tractions are consistent with the straight break failures observed in service. In fact most of straight breaks initiating at the underhead occur in short radius curves or hunting-prone tangent tracks.

The results of the sensitivity analysis on residual stress indicate that high longitudinal residual stress can provide suitable condition for straight break failure at the underhead while a compressive residual stress has a favourable effect on the base region. High longitudinal residual stress also increases the stress ratio ( ) and reduces the threshold stress intensity factor range of the material (ability to tolerate cracks and defects) as shown in the analysis of cold laps. Ultrasonic impact treatment has recently been proposed to apply post-weld compressive residual stresses to mitigate the risk of crack initiation. Lack of support was shown to increase the fatigue damage parameter at the base region. Rail failure reports from the service conditions of relevance to this study also suggest high failure probability at locations where extensive ballast deterioration has taken place.

8.1.3 Straight Break at Top of the Rail Foot

Besides at the underhead region, the straight break on the investigated weld has been widely reported at the top of the rail foot. However, the multi-axial fatigue analysis of

196 the sound and defect-free weld did not predict any sign of failure in this region. Examination of the fractured regions of failed welds suggests that the failure at the top of the foot is often accompanied by presence of a welding defect called cold lap. To investigate the effect of cold lap on formation of straight break linear elastic fracture mechanics using the virtual crack closure technique (VCCT) was exploited. Analyses show that due mostly to the in-plane shearing and opening force on the lap (which is studied as a planar crack with a straight front) the equivalent stress intensity factor range can reach the threshold value with possible crack initiation from the apex of the cold lap. Based on all the investigated propagation criteria the initiated crack quickly turns down which is also a sign of possible straight break failure. The study on the lap geometric features reveals that among all the dimensions, the lap thickness has the greatest effect on . In fact thicker laps are more influential in the formation of straight breaks. Rise∆ of thickness from 1 to 3 mm can increase the equivalent stress intensity factor range by 65%. It was also shown that as the fusion boundary or the lap apex moves further from the collar edge the possibility of crack initiation is reduced.

It is found that cold laps on the field side are more damaging than those on the gauge side. The highest damage relates to the lap on the field side when the contact patch is severely displaced towards the gauge side. Inward (towards the gauge side) tractions also apply high driving forces on the field-side located laps whereas outward tractions are effective on the gauge-side located ones. In addition, lack of support increases the driving force on the lap and the chance of lap-related straight breaks.

8.1.4 Collar Design

The collar geometric design of the weld is among important parameters which can influence the fatigue behaviour. However, the variability of designs is limited by how the proposed collar performs in terms of the molten metal accessibility, defect formation, post-weld cooling and residual stress distribution. Beside the Type A which is currently the most widely used collar in Australian heavy haul railway system, a further design, Type B is under investigation for possible introduction to the industry. The Type B weld features lower flank angle and bigger toe radius on the collar edge. It

197 also incorporates a flatter collar design on the web and lacks the buttress used in Type A. The multi-axial fatigue analysis performed on the collar edge of both welds reveals that the Type B presents lower damage values in this location which is mainly due to the lower stress concentration throughout the collar edge and partly resulting from the lower longitudinal residual stress measured in its vicinity. In spite of what is observed in Type A weld, Type B weld is less sensitive to the displacement of the contact patch and the transverse forces and no failure is predicted in any of the loading conditions investigated. For a very high lateral traction coefficient of 0.4 adoption of Type B design can decrease the fatigue damage value at the underhead by up to 30%. This results from the tension spike on Type B being much less severe than on Type A which is in fact due to the better geometric features at the collar edge of Type B. The Type B weld also performs better under a defective support by showing less damage value on the base region compared to Type A.

Nevertheless, the Type A weld features a buttress at the web region which increases the moment of area and lowers the vertical stress. Stress analysis of the web surface indicates that the maximum value of the cyclic vertical stress is higher in Type B weld and accordingly the Dang Van damage parameter has a higher value compared to that of Type A. However, no fatigue failure is expected since the web surface studied is defect free and the damage parameter is too low. In fact presence of a large defect is necessary for HSW to initiate and grow and this necessitates further analysis with defects involved.

8.1.5 HSW and Web Defects

A study on the damage tolerance analysis of the web region was conducted based on three large pore defects of spherical, ellipsoidal and coin-shape. Initiation of fatigue cracks on the surface of the defects was investigated through multi-axial fatigue analysis and the Dang Van criterion. For small to medium sized defects on Type A weld vertical stress and damage is lower than those on Type B due to the use of buttress and hence lower bending stresses on Type A collar. However, as the defect enlarges and the defect surface approaches the collar boundary in Type A damage values increase and exceed

198 those of the Type B. The fatigue criterion used fails to solve the crack initiation problem for coin-shape defects as the stress concentration at the notch root is too high and plasticity emerges. For such defects and for defects with peripheral cracks the fracture mechanics approach was applied in which the defect is represented by a crack with the same size (projected area). The results generally support the idea that as the crack enlarges the tolerance of Type A weld is reduced more progressively than the Type B. Apparently, large defects can more readily cause fast fractures in Type A weld and this relates to the interaction of the crack front and the collar boundary which intensifies the stress concentration.

Based on the analysis reported in this thesis the following measures can be proposed to improve the fatigue performance of aluminothermic welds:

• It is favourable to reduce the lateral displacement of the rail-wheel contact patch and the transverse forces. This may be achieved by selection of a suitable wheel profile, re-profiling and timely grinding of the rail head and also through application of lubricant or other friction management products to the rail-wheel interface particularly in short radius curve tracks. • Collar edge features particularly at the underhead region have strong influence in formation of straight breaks. Lower flank angle and larger toe radius provide better fatigue performance. • Tensile residual stress should be kept as low as possible through redesigning the weld process, cooling and post-cooling heat treatment or by manually imposing compressive residual stresses to the collar edge. • Cold laps should be avoided by using better sealing methods and materials. The amount of melt-back on the rail heads should be enough to provide wider fusion regions and to move the lap apex farther from the weld collar edge. This may be achieved by widening the gap between the rail heads and providing longer preheats. However, this may affect the hardness distribution of the rail head and increase the chance of battering and high load amplification factors. • It is favourable to have a collar design with a high moment of area to reduce lateral bending stress on the web region and the risk of HSW formation.

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However, a flatter design in the web region is proposed to reduce the stress concentration at the periphery of large defects. This may delay or even prevent the occurrence of fast and overload fractures.

8.2 Future Work

Several aspects of the aluminothermic weld fatigue performance were studied in this research. However, there are certain topics in which further investigation seems necessary and are pointed out in this section.

The fatigue strength of the material required for the multi-axial fatigue analysis were estimated based on the material hardness and this may not precisely reflect the actual values. At least two fatigue tests such as the reversed bending and alternative torsion are necessary to evaluate the two required fatigue parameters. However, the feasibility of testing is questionable considering the uncertainties and variability of the manufactured welds and the test samples extracted from different regions of the weld. If the testing can be performed a consistent S-N curve could also provide the possibility to calculate the life to crack initiation at the critical locations of the weld. Although, similar fatigue strength was assumed in this study, it is suggested that the tests are performed for both Type A and Type B welds.

The fracture analysis performed on the cold lap and crack like defects incorporated threshold stress intensity factor range and the fracture toughness based on previous limited measurements on different types of aluminothermic weld. Further work may include the measurement of these parameters for the two welding processes studied. In fact, if a crack growth law such as the Paris law can be developed for the weld material it will be possible to investigate the crack propagation and the life to fracture for cold lap defects and the cracks responsible for HSW fractures. However, similar restrictions as mentioned above may apply.

The analysis on the damage tolerance of the collar with respect to HSW formation provided an insight into the fatigue behaviour of the web. Further work is required to

200 investigate the findings through laboratory experiments. The crack growth behaviour can be studied through 3-point bending fatigue test. A high stress concentration notch can be machined on the web surface and the structure cyclically loaded until a pre-crack emerges at the periphery of the notch. A comparison between the crack growth behaviour on the two collar designs can then be performed.

The numerical study can also be further developed by considering different crack shapes and sizes. Since the defects on the web region feature complex geometries it is also of interest to investigate defects with some type of geometric irregularities. However, the current multi-axial analysis cannot be directly used since plasticity forms at the surface of the defect. In this case finite element analysis should be extended to simulate several load cycles in order to achieve an elastic shakedown condition after which the multi- axial criterion can be applied. This needs a fairly accurate constitutive equation incorporating the hardening behaviour. Alternatively, other damage prediction models using the strain based approach such as the Coffin-Manson can be exploited.

It is also proposed that a comprehensive analysis is performed on the effect of track support considering the vehicle/track dynamics and hence the variation of wheel/rail contact position and load. The lack of support in presence of running surface irregularities may amplify the contact load and this has a direct effect on the fatigue behaviour with respect to both straight break and horizontal split web formation. Another important issue which can be considered with or without the lack of track support is the difference in rail height across the weld (arising from welding two rails with different heights). The difference in rail height which is quite a common issue can exacerbate stress concentration at the collar edge and may also develop thicker cold laps at regions of the weld.

Current study primarily discusses fatigue behaviour of welds installed in rails without any wear. It is also important to examine the behaviour as rail wears or is ground (progressive removal of material from the rail head) until such time as it is replaced. Such an approach would provide valuable information on which to assess or review rail wear limits under the relevant service conditions.

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APPENDIX: Implementation of the Multi-axial Dang Van Criterion in a MATLAB Program

% Multi-axial fatigue analysis using Dang Van criterion implemented in MATLAB programming language

% Build the stress matrix from the finite element input data % % data_mat=three dimensional matrix comprising stress tensor at each investigated material point in all load steps which form one stress cycle % X=number of load steps % Z=number of material points (nodes) investigated data_mat=zeros(X,9,Z); % nnodes=size(data_mat,3); % Number of nodes ntimes=size(data_mat,1); % Number of load steps % % Fill data_mat with finite element data da=0; for ii=1:nnodes data_mat(:,:,ii)=x1(:,1+da:9+da); % x1=stress/coordinate matrix from finite element software da=da+9; end % % End of building the stress matrix

% Build the total stress matrix including the residual stresses % % Stressmatrix=same as data_mat with residual stresses superimposed through linear representation stressmatrix=data_mat; for jj=1:nnodes

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if data_mat(1,9,jj)<=24 % 3 sets of linear equations have been specified for longitudinal residual stress based on vertical distance from the rail foot

stressmatrix(:,1,jj)=stressmatrix(:,1,jj)+(17.083*d ata_mat(1,9,jj)-230)*(10^6); else

stressmatrix(:,1,jj)=stressmatrix(:,1,jj)+180*(10^6 ); end

if data_mat(1,9,jj)<=40 % 3 sets of linear equations have been specified for vertical residual stress based on vertical distance from the rail foot stressmatrix(:,3,jj)=stressmatrix(:,3,jj)+0; elseif data_mat(1,9,jj)<=70

stressmatrix(:,3,jj)=stressmatrix(:,3,jj)+(5*data_m at(1,9,jj)-200)*(10^6); elseif data_mat(1,9,jj)<=90

stressmatrix(:,3,jj)=stressmatrix(:,3,jj)+150*(10^6 ); elseif data_mat(1,9,jj)<=120 stressmatrix(:,3,jj)=stressmatrix(:,3,jj)+(- 5*data_mat(1,9,jj)+600)*(10^6); else stressmatrix(:,3,jj)=stressmatrix(:,3,jj)+0; end end % % End of building the total stress matrix

% Calculate the Dang Van fatigue parameters (aDV and bDV) % Sfl=input( 'Enter the fatigue limit for tension-compression test:' ); Tfl=input( 'Enter the fatigue limit for torsion test:' ); aDV=3*((Tfl/Sfl)-0.5); bDV=Tfl;

% Definition of parameters for stress transformation of the considered shear plane and construction of the shear path %

221 transform=zeros(3,3); % Transformation matrix to find the coordinate system of the considered shear plane % Unit vectors of the global coordinate system oldx=[1;0;0]; oldy=[0;1;0]; oldz=[0;0;1]; % Unit vectors of the coordinate system of the considered shear plane newx=zeros(3,1); newy=zeros(3,1); newz=zeros(3,1); newstress=zeros(ntimes,6); % New stress tensor in the coordinate system of the considered shear plane pathpoints=zeros(ntimes,2); % Coordinates of the end point of the shear stress vector on the considered shear plane (these points define the shear path which is used to define the minimum circumscribed circle)

% Definition of parameters for minimum circumscribed circle % centre2=zeros(2,1); % Centre of the constructed two point circle by any two point on the shear path centre3=zeros(2,1); % Centre of the constructed three point circle by any three point on the shear path centre2p=zeros(2,1); % Centre of the two point circle which satisfies the minimum circumscribed circle condition centre3p=zeros(2,1); % Centre of the three point circle which satisfies the minimum circumscribed circle condition centre2p3p=zeros(2,2); % Both centres in one matrix mincentre=zeros(2,1); % Centre of the smallest circle between the 2-point and 3-point circles

% Definition of parameters for hydrostatic stress, shear stress amplitude and damage parameter on the considered shear plane % Shyd=zeros(ntimes,1); % Time dependent hydrostatic stress

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Ta=zeros(ntimes,1); % Time dependent shear stress amplitude on the considered shear plane (value of the shear stress amplitude vector) Dv=zeros(ntimes,1); % Time dependent damage parameter for the shear plane considered Shydcrit=zeros(ntimes,nnodes); % Hydrostatic stress on the critical plane in all load steps and in all nodes Tacrit=zeros(ntimes,nnodes); % Shear stress amplitude on the critical plane in all load steps and in all nodes Dvcrit=zeros(ntimes,nnodes); % Damage on the critical plane in all load steps and in all nodes Anglescrit=zeros(3,nnodes); % Unit vector of normal to the critical plane in all nodes DVpoints=zeros(nnodes,1); % Maximum damage parameter in all nodes considered % Shear path on the critical plane for all nodes pathcritx=zeros(ntimes,nnodes); pathcrity=zeros(ntimes,nnodes); hnn=zeros(nnodes,1); % Parameter to define weather the minimum circumscribed circle on the critical plane has been a 2-ponit circle (hnn=1) or a 3-point circle(hnn=2) mincentrecrit=zeros(2,nnodes); % Centre of the smallest circle on the critical plane minradiuscrit=zeros(nnodes,1); % Radius of the smallest circle on the critical plane

% Critical plane analysis for all nodes: the aim is to consider all the shear planes passing through the considered node % for nn=1:nnodes % Critical plane analysis is performed for all nodes for alpha=0:359 % Angle of rotation (around x-axis) of the coordinate system (each investigated shear plane is the same as the xy plane of the new rotated coordinate system and the normal to the shear plane is the z-axis of the new coordinate system) for beta=0:90 % Angle of rotation (around y-axis) of the coordinate system radius2p=1e30; % A very large number for the radius of the two point circle (for the sake of comparison and finding the smallest 2- point circle) radius3p=1e30; % A very large number for the radius of the three point circle (for

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the sake of comparison and finding the smallest 3-point circle) transform=[cosd(beta) sind(beta)*sind(alpha) sind(beta)*cosd(alpha);0 cosd(alpha) - sind(alpha);-sind(beta) cosd(beta)*sind(alpha) cosd(beta)*cosd(alpha)]; % New coordinate system after rotation newx=transform*oldx; newy=transform*oldy; newz=transform*oldz; a11=newx(1); a12=newx(2); a13=newx(3); a21=newy(1); a22=newy(2); a23=newy(3); a31=newz(1); a32=newz(2); a33=newz(3); % New stress tensor in the rotated coordinate system for nt=1:ntimes s11=stressmatrix(nt,1,nn); s22=stressmatrix(nt,2,nn); s33=stressmatrix(nt,3,nn); s12=stressmatrix(nt,4,nn); s23=stressmatrix(nt,5,nn); s13=stressmatrix(nt,6,nn);

newstress(nt,1)=a11^2*s11+a12^2*s22+a13^2*s 33+2*a11*a12*s12+2*a11*a13*s13+2*a12*a13*s2 3;

newstress(nt,2)=a21^2*s11+a22^2*s22+a23^2*s 33+2*a21*a22*s12+2*a21*a23*s13+2*a22*a23*s2 3;

newstress(nt,3)=a31^2*s11+a32^2*s22+a33^2*s 33+2*a31*a32*s12+2*a31*a33*s13+2*a32*a33*s2 3;

newstress(nt,4)=a11*a21*s11+a12*a22*s22+a13 *a23*s33+(a11*a22+a12*a21)*s12+(a12*a23+a13 *a22)*s23+(a11*a23+a13*a21)*s13;

newstress(nt,5)=a21*a31*s11+a22*a32*s22+a23 *a33*s33+(a21*a32+a22*a31)*s12+(a22*a33+a23 *a32)*s23+(a21*a33+a23*a31)*s13;

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newstress(nt,6)=a11*a31*s11+a12*a32*s22+a13 *a33*s33+(a11*a32+a12*a31)*s12+(a12*a33+a13 *a32)*s23+(a11*a33+a13*a31)*s13; end

% Coordinate of the end point of the shear vector on the shear plane or the xy plane of the rotated coordinate system (these points form the shear path) % pathpoints(:,1)=newstress(:,6); pathpoints(:,2)=newstress(:,5);

% Find the smallest 2-point circle % for hh=1:(ntimes-1) for gg=(hh+1):ntimes out2=0; % Construct the 2-point circle radius2=0.5*sqrt((pathpoints(hh,1)- pathpoints(gg,1))^2+(pathpoints(hh,2)- pathpoints(gg,2))^2);

centre2=[(pathpoints(hh,1)+pathpoints(g g,1))/2;(pathpoints(hh,2)+pathpoints(gg ,2))/2]; for zz=1:ntimes % Check to determine if the 2-point circle contains all the points of the shear path

ch2p=sqrt((pathpoints(zz,1)- centre2(1,1))^2+(pathpoints(zz,2)- centre2(2,1))^2); if (ch2p)>(radius2+100) out2=1; break end end if (out2==0)&&(radius2

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centre2p=centre2; end

end end

% Find the smallest 3-point circle % for kk=1:(ntimes-2) for ll=(kk+1):(ntimes-1) for mm=(ll+1):ntimes % a,b,c are the sides of the triangle formed by the considered three points

a=sqrt((pathpoints(kk,1)- pathpoints(ll,1))^2+(pathpoints(kk, 2)-pathpoints(ll,2))^2); b=sqrt((pathpoints(ll,1)- pathpoints(mm,1))^2+(pathpoints(ll, 2)-pathpoints(mm,2))^2); c=sqrt((pathpoints(mm,1)- pathpoints(kk,1))^2+(pathpoints(mm, 2)-pathpoints(kk,2))^2); if (-a+b+c)*(a-b+c)*(a+b-c)~=0 % Check to determine if the three considered points of the shear path form a circle out3=0; % Construct the 3-point circle radius3=a*b*c/(sqrt((a+b+c)*(- a+b+c)*(a-b+c)*(a+b-c))); ax=pathpoints(kk,1); ay=pathpoints(kk,2); bx=pathpoints(ll,1); by=pathpoints(ll,2); cx=pathpoints(mm,1); cy=pathpoints(mm,2);

centre3(1,1)=((ay^2+ax^2)*(by- cy)+(by^2+bx^2)*(cy- ay)+(cy^2+cx^2)*(ay- by))/(2*(ax*(by-cy)+bx*(cy- ay)+cx*(ay-by)));

centre3(2,1)=((ay^2+ax^2)*(cx-

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bx)+(by^2+bx^2)*(ax- cx)+(cy^2+cx^2)*(bx- ax))/(2*(ax*(by-cy)+bx*(cy- ay)+cx*(ay-by))); for pp=1:ntimes % Check to determine if the 3- point circle contains all the points of the shear path

ch3p=sqrt((pathpoints(pp,1)- centre3(1))^2+(pathpoints(pp,2) -centre3(2))^2); if ch3p>(radius3+100) out3=1; break end end else out3=1; end if (out3==0)&&(radius3

end

% Find the minimum circumscribed circle among the smallest circumscribing 2-point and 3-point circles % centre2p3p=[centre2p centre3p]; [minradius,h]=min([radius2p radius3p]); % Radius of the minimum circumscribed circle defined mincentre=centre2p3p(:,h); % Centre of the minimum circumscribed circle defined

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% Calculate the time dependent damage parameter for the considered shear plane % for bb=1:ntimes

Shyd(bb,1)=(newstress(bb,1)+newstress(bb,2) +newstress(bb,3))/3; % Time dependent hydrostatic stress for the considered shear plane

Ta(bb,1)=sqrt((pathpoints(bb,1)- mincentre(1,1))^2+(pathpoints(bb,2)- mincentre(2,1))^2); % Time dependent shear stress amplitude for the considered shear plane end Dv=Ta+aDV*Shyd; % Time dependent damage parameter for the considered shear plane

% Determine the maximum damage on the considered shear plane and compare with the maximum damage of other shear planes passing through that point to extract the maximum damage value at the considered point and the associated critical plane % if (max(Dv)>max(Dvcrit(:,nn))) Dvcrit(:,nn)=Dv; Tacrit(:,nn)=Ta; Shydcrit(:,nn)=Shyd; Anglescrit(:,nn)=newz; pathcritx(:,nn)=pathpoints(:,1); pathcrity(:,nn)=pathpoints(:,2); hnn(nn)=h; mincentrecrit(:,nn)=mincentre(:,1); minradiuscrit(nn)=minradius; end end end DVpoints(nn,1)=max(Dvcrit(:,nn)); % The final matrix showing the maximum damage on the critical plane of each considered material point disp(nn); end

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