Differential wear modelling - effect of weld-induced material inhomogeneity on rail surface quality

Matin Sh. Sichani1, Yann BezinInstitute of Railway Research, University of Huddersfield, Huddersfield, UK

AbstractA methodology is introduced to enable prediction of differential wear along the rail due to the presence of geometrical imperfections and microstructural inhomogeneities. The method incorporates vehicle-track dynamic interaction, wheel-rail contact and wear modelling in order to predict long-term longitudinal rail profile evolution. The effect of different contact and wear modelling features on the results is investigated. As a case study, the differential rail wear due to material inhomogeneity caused by rail head weld repairs is considered. It is concluded that, regardless of the magnitude of hardness variation within the heat affected zone, the long-term surface degradation can be minimized by limiting the length over which the variation occurs.

KeywordsDifferential wear, Non-uniform wear, Rail weld, Rail head repair, Aluminothermic welding, Cupping, Railway

1. IntroductionOver the past two decades there has been many publications on uniform wear modelling with emphasis on wheel wear.  A thorough overview of the relevant studies is given in [1]. There are also a number of studies on uniform rail wear modelling (see [2], for instance). In these studies, the assumption of uniformity implies that the rail wears out uniformly along a section of the track and the aim is to predict how the transverse profile is changed after a given amount of traffic. Any variations between the transverse profiles along that section is therefore neglected. This makes it possible to sum up the wear due to the contact cases occurring in that section to obtain the wear depth at each transverse coordinate. In contrast, non-uniform wear modelling, also known as differential wear, concerns the evolution of the rail profile in the longitudinal direction. In this case the variations in loading and contact conditions along the rail are considered. The most common example of this is rail corrugation which is a periodic rail surface irregularity. Over the past 60 years, many studies were published on corrugation. A good summary of rail corrugation modelling may be found in [3].

However, few studies [4, 5] are published on differential wear due to discrete imperfections on the rail. The insulated rail joints, welded rail ends or weld repaired lengths are an example of such imperfections affecting the running surface quality. Presence of rail joints causes impact forces known

1 Corresponding author: matin.s.sichani@gmail.com

as P1 and P2 forces [6] that result in contact condition variation and consequently differential surface evolution. Welding of rails introduces both geometrical imperfections and microstructural inhomogeneities to the running surface and can cause the so-called cupping in both sides of the weld [7]. A summary of rail welding processes and various defects associated with them is published in [8]. The plastic deformation caused by impact forces arising from weld imperfections is studied in [9, 10] to explain cupping formation. However, differential wear as a possible mechanism is not addressed. Muttan et al. [11] has discussed the effect of microstructural inhomogeneities in flash butt welds on rolling contact fatigue behaviour under high axle loading conditions.

To predict the long-term rail profile evolution, a methodology is needed which considers the natural feedback loop between the vehicle-track dynamic response and damage mechanisms causing surface change. In corrugation studies, this combination is referred to as the wavelength-fixing mechanism and damage mechanism [12]. The wavelength-fixing mechanism is governed by the dynamic resonances of the vehicle-track system and the damage mechanism is mainly considered to be wear. However, Böhmer [13] has suggested to consider both plastic deformation and wear in a sense that a mid-term elasto-plastic analysis is conducted to consider the profile evolution due to plastic deformation accompanied by a long-term wear analysis to account for wear of material. When material undergoes plasticity, it is work-hardened and thus the material properties are altered. This includes a change in yield strength as well as hardness which influences the wear resistance of the material. Therefore, the effect of work-hardening on wear resistance may also be incorporated. In the study of the development of wheel out of roundness, Meywerk [14] has proposed the use of reduced wear rates to account for the work hardening effect. A frictional work wear model is used in his study.

Thus, to incorporate all the influencing factors, an approach which combines vehicle-track dynamics, elasto-plastic material response and non-uniform wear is necessary. Figure 1 illustrates the workflow for such approach. The proposed methodology in Figure 1 comprises of three main analysis modules. Each module contains features that influence the final prediction of the profile evolution.

Figure 1 Workflow for simulation of rail profile evolution

To conduct a long-term analysis of the rail profile evolution, the operational conditions and traffic data as well as the initial material and geometrical conditions are needed. A so-called load collective approach may be taken in which a series of passing vehicles with various contact conditions, axle loads, running speeds and occurrence frequency are considered to represent the passing traffic over the rail. Since the dynamic excitations that are of interest lies within the mid- to high-frequency range (50-1000 Hz) a more elaborate dynamic model of track is needed. A discretely-supported beam with two layers of support is usually used. This is because the dynamic impact forces that arise from discrete defects (known as P1 and P2) are governed by the flexibility of the track. Therefore, the dynamic characteristic of the track plays an important role on the simulated worn rail profiles. However, to study the long-term effects, the computational efficiency of the method is a concern. Since the most time-consuming part is the dynamics analysis, to improve the computational efficiency, a simplified two degree of freedom (co-running) track model which considers rail and

sleeper as rigid bodies may be utilized. In that case, tuning of the masses and suspension elements is recommended in accordance to [15] for better correspondence.

In the mid- and high-frequency range the carbody is dynamically isolated due to soft secondary suspensions [16]. Same argument is usually valid for bogie’s rigid body dynamics in frequencies above 20 Hz [17]. Therefore it is mainly the unsprung mass of the vehicle that plays the most important role in damage of the rail surface. However, the coupling between the two wheelsets through bogie frame may play a role [16]. Most of the studies on rail corrugation only consider a single wheelset running over the rails. The coupling between the wheelsets through the bogie frame and wave reflections between the wheels may affect the results in particular in case of soft rail pads [18].

In the mid-frequency range, the flexibility of the wheelset may also be considered. It is shown in [19] and [20] that, around the resonance frequency of the axle, wheelset flexibility has considerable effect on longitudinal creep force but negligible effect on normal contact force.

The evolution of the running surface may not only be caused by wear. Plastic flow of the material in the normal direction may also contribute to geometrical variations along the rail. However, the profile evolution due to plastic deformation is noticeable within few passages while the evolution due to wear is of a longer time scale [13]. It should be noted that here only plastic deformation of the material on the running surface and in the normal direction is of interest. Material may undergo plastic deformation below the surface which does not contribute to rail profile variation.

The wear depth along the rail may be calculated using either a frictional work method (e.g. T-gamma method [21]) or Archard method [22]. In most corrugation studies (e.g. rate [23, 24, 25, 14, 20]) a frictional work wear model is used. The amount of worn material is related to the work done by the creep forces (stresses) using a wear rate. Usually a single wear rate is used. This single value reflects the hardness and wear resistance of the material microstructure. However, using a wear model based on the Archard method, the hardness is directly incorporated in the wear calculation and the wear resistance of material is represented by the so-called wear coefficients. This facilitates the study of hardness variation effect on differential wear.

The reduction of the worn material from the rail profile may be done in two manners. A less time-consuming and commonly used approach is to calculate the wear amount for the whole contact patch (globally) and subtract the equivalent wear depth from the rail profile at the point of contact or the centre of slip area. In this case, there is no need to know the distribution of the shear stress over the patch and the creep force is used for the wear calculation. A more elaborate approach is to calculate the wear depth at each point within the slip area of the contact patch (locally). For instance in [26, 27], the FASTSIM algorithm [28] is used to find the shear distribution over the patch needed to calculate the frictional-work-based wear depth at each contact discretization. Using Archard method, the distribution of contact pressure and sliding distance over the slip area is needed instead. The accuracy of the wear distribution is therefore dependent on the choice of contact model. A more elaborate contact model than the one used in the dynamics analysis may be used. In this case, however, some incompatibilities between the calculated contact forces may arise.

Various features of contact modelling may affect the predicted results. These features may play a more important role when it comes to shorter wave-length irregularities such as in short-pitch corrugation or rolling noise generation. The elastic wheel-rail contact acts as a filter on contact surface roughness. The contact filtering effect is mainly accounted for in rolling noise studies [29,30], however, it can be used to filter out the non-physical high frequency excitations arising from the calculated overall wear depth distribution.

The use of Hertz contact model is by far the most common approach due to its time efficiency although some of its underlying assumptions may be violated when it comes to irregular running

surfaces. Nielsen [31] shows that, unlike the elliptic Hertzian pressure distribution, the pressure distribution (and consequently the slip distribution) may be asymmetric with respect to the contact point. This is because the Hertzian assumption of representing the rail surface with a second-order polynomial within the contact area may no longer be valid. Moreover, a rapid change of the contact conditions may necessitate the use of non-stationary models for calculation of the slip. Baeza et al. [32] has concluded that as the wavelength of the force variation decreases or its amplitude increases, the non-stationary effects are more pronounced. By combining the non-stationary contact model with an Archard wear model, he showed that the wear is more spread along the rail and has lower peaks. This is partly in-line with the findings in [26] which showed that initial rail roughness is levelled out when using non-Hertzian, non-stationary contact modelling.

For the range of surface irregularities that this paper is addressing (where a discrete irregularity with relatively longer wave-length than surface roughness or short-pitch corrugation is of interest) some of the above features may be discarded. Of course the level of complexity within each of the aforementioned modules in Figure 1 depends on the case study and is dictated by the required time efficiency and computational cost.

2. Case studyThe main focus of this paper is to study the effect of local hardness inhomogeneity and imperfect geometry along the rail surface due to aluminothermic welding repair of rails. Welding process involves high thermal input that results in a so-called heat affected zone (HAZ) on either side of the weld. Within these zones the metallurgical properties vary from the parent rail. This includes a variation in hardness and wear resistance along the running surface. Such variations cause differential wear of the rail surface which in turn results in a geometric irregularity that give rise to variation in contact forces and further differential wear. Figure 2 shows the potential deterioration loop after welding the rail ends or repairing the rail head by welding.

Figure 2 Deterioration loop due to material and geometrical imperfections induced by welding

To limit this vicious cycle, standards address both geometrical and metallurgical imperfections due to welding of rails or weld repairs. There are two European standards that address the criteria for the acceptance of rail welding. EN14730-2 [33] regulates the aluminothermic welding of rails while EN14587-1 [34] standard specifies regulations for flash butt welding. The alignment and flatness of the rail surface within the welded region are specified. The vertical alignment tolerance concerns with aligning of the two rail ends before welding. Flatness is measured using straightedge and feeler gauge over the grinding length which can be interpreted as a measure of vertical deviation of the running surface geometry. The flatness limit is set in different track categories to be specified by the railway authority. The limit is 0.2 mm for most categories with 0.1 mm and 0.15 mm for higher categories of track quality.

Some other institutions have a different measure for finished weld geometry where instead of vertical deviation, the surface inclination is considered. For instance, ETM-01-01 standard [35] by Australian

infrastructure manager ARTC specifies the limits for finished welds of any kind including head repair welding. The surface deviation of rail is limited in terms of change in weld ramp angle (surface inclination). The limit is 7 mrad over 50 mm base. The studies in [36] show that maximum dynamic force arising due to weld geometrical irregularities is better correlated to the surface gradient than vertical deviation. This criterion is adopted by the Dutch infrastructure manager where the maximum allowable absolute gradient on a 25 mm basis is categorized based on the line speed and ranges from 1 mrad to 3.2 mrad [8]. In comparison to the Australian standard, tighter limits on the surface gradient is enforced. Further study on assessment criteria of rail weld geometry is provided in [37].

In addition to the rail surface geometry, the size of the HAZ and hardness variation along the weld is also regulated for various rail grades. It is mentioned that the visual HAZ should be symmetrical and range between 20 mm to 45 mm (different limits are assigned based on the welding technique). The hardness values across the HAZ shall not be more than 30 HV10 below or 60 HV10 above the average hardness of the parent rail. This applies to certain rail grades including R260. For higher rail grades a maximum and minimum hardness value is set within 10 mm from both sides of the weld line. In case of R350HT, for instance, it should be between 325 HV10 to 410 HV10. EN 15594 standard [38] addresses the restoration of rails by electric arc welding. It specifies the maximum hardness limits both on- and sub-surface for various rail grades and single or multilayer weld deposition. The maximum surface hardness for R260 grade ranges from 300 HV10 to 400 HV10 depending on number of deployed layers.

Although there have been attempts on modelling plastic deformation due to weld-induced material inhomogeneity [9, 10], to the best of the author’s knowledge, there has been no study published on numerical modelling of the differential wear. Therefore, a preliminary study on the topic is provided in this paper. To this aim, some simplifying assumptions are utilized. As a first attempt, we limit our study to two-dimensional vertical dynamics. Thus, the wheelset dynamics and lateral effects are neglected. This assumption limits the applicability of the current work to ideal tangent track. Moreover, the axle flexibility and multiple wheel on track effects are neglected to keep the simulation times reasonable. The main damage mechanism in this specific case is assumed to be wear. To make sure if this assumption is valid, the occurrence of plasticity on the surface is checked within every wear step.

Based on these assumptions a methodology is implemented to predict longitudinal rail profile evolution under the effect of inhomogeneous hardness along the rail surface. Figure 3 shows the detailed workflow of the implementation. A short description of each analysis within it comes as follows.

Figure 3 Workflow of the methodology for calculation of the differential wear due to hardness variation along the rail

2.1.Vertical dynamicsTo incorporate the variation of the contact force, a vertical dynamic model of the unsprung mass rolling on a discretely-supported, Euler-Bernoulli beam model of the track is used. The track model consists of a rail beam and rigid sleepers with rail pad and ballast characterized by spring-damper

elements. The wheel-rail interaction is modelled using a non-linear Hertz spring, accounting for loss of contact. The length of the model and the number of elements per sleeper-spacing is set as recommended in [39]. A schematic of the model is shown in Figure 4.

Figure 4 Schematic of the vehicle/track dynamics model used in this study

2.2.Traction dynamicsTo generate wear, wheel should slip over the rail. For simplicity, lateral vehicle dynamics is discarded and the study is limited to tangent track assuming the contact position does not change considerably. For a driven wheel, the longitudinal creepage due to acceleration or braking is the main source of slip. Assuming a constant tractive torque, the longitudinal creepage changes as the contact conditions vary. Figure 5 depicts the simplified driven wheel model with variable contact force, N, constant tractive torque, T, creep force, Fx, wheel velocity, V, wheel angular velocity, ω and longitudinal creepage, ξ. The wheel velocity, V, is assumed to be constant hence the wheel is in translational equilibrium. Using the rotational equation of motion for the wheel together with the longitudinal creepage relationship, ξ = 1-Rω/V (where R is wheel’s radius) and Carter’s creep force equation for line contact [40] a system of non-linear equations is formed. The longitudinal creepage at each time-step is then calculated using an explicit time integration scheme.

Figure 5 simple driven wheel model.

For a constant contact geometry and friction coefficient, the creep curve changes with reducing normal load, so that a higher creepage is needed to provide the same tractive force. This is shown in Figure 6 with Ftract = T/R. A more realistic approach would be to assume a constant tractive power (P = Tω), as done in [24], instead of a constant tractive torque.

Unlike the driven wheel, for a free wheel running with constant speed on tangent track a constant longitudinal creepage may exist due to lateral wheelset offset. In this case a reduction in the normal load will not affect the creepage but the creep force on the wheel.

Figure 6 Effect of reducing the contact force (N0 to N1) on utilized creepage (ξ0 to ξ1) to provide a constant tractive force (Ftract), having constant friction coefficient (μ).

2.3.Contact mechanicsThe vertical contact force as well as longitudinal creepage are needed in order to calculate the pressure and slip distribution along the contact line. Since the contact problem in the vertical dynamics module is treated as a point contact with Hertzian spring connection, several features of the actual elastic contact between the wheel and rail are lost. For instance, a two-point contact (in the longitudinal direction) may occur if the longitudinal wheel curvature (1/ R) is smaller than the curvature of the rail with a dipped surface. This is checked for each point along the rail to make sure single-point contact assumption holds.

Due to the geometrical irregularity of the rail surface in the longitudinal direction, the contact point between wheel and rail may not be located right under the wheel’s rolling axis. Therefore, a geometrical (longitudinal) shift of the contact point location with respect to the wheel centre may exist. An exaggerated illustration of this shift, denoted as S, is shown in Figure 7. This implies a torque on the wheel which is negligible [41]. However, the geometrical shift of the contact patch centre is important when it comes to where on the rail the calculated worn depth should be applied. The following relation holds for the shift,

SR

=sin α= Z ' ( X+S )

√1+( Z ' ( X+S ) )2, (1)

where Z' ( X)=∂ Z (X )/∂ X with Z(X) being the rail longitudinal profile.

Figure 7 Geometrical shift of the contact point from the wheel centre.

Moreover, the irregular rail surface presents a non-zero longitudinal curvature, therefore, rail curvature in the X-Z plane is calculated and taken into account at each point along the rail. This curvature is then used for calculating the contact length and the pressure distribution. Hertz solution is assumed to be valid for pressure distribution. This implies two important underlying assumptions; the

rail longitudinal profile can be approximated as a second-order polynomial within the potential contact zone and the material properties are constant (homogeneous) within this zone. It should be noted that these two assumptions may be violated in case of irregularities with wavelengths close to the contact size or in case of abrupt changes of the material properties.

The contact ellipse is calculated assuming the wheel and rail transverse profiles at a constant wheelset lateral position. However, only central line of the contact patch is considered for the two-dimensional wear studies. To calculate the slip distribution within the contact patch a modified strip theory implementation proposed in [42] is used. Since only longitudinal creepage is considered, an analytical solution exists. The pressure distribution along the contact centreline is,

p ( x )=p0

a0√a0

2−x2, (2)

with p0 and a0 representing the maximum Hertzian contact pressure and contact half-length, respectively, and x is the local longitudinal coordinate within the contact patch. The relative slip velocity distribution within the contact centreline is,

s ( x )={2 (1−ν ) μ p0

cx G a0√ ( x−d0)2−(a0−d0 )2 ,∧d0<a0

|ξ|−2 (1−ν ) μ p0

cxx ,∧d0≥ a0

(3)

where, d0 is half-length of the slip area calculated as,

d0=cx G a0

2 (1−ν ) μ p0∨ξ∨¿, (4)

having Poisson’s ratio, ν, shear modulus, G, friction coefficient, μ and Kalker’s coefficient, cx [28]. Maximum contact pressure, p0, contact half-length, a0 and slip area half-length, d0 are all dependent on the contact force and rail surface geometry at each time-step.

2.4.Wear calculationAn Archard wear model is used to enable accounting for the hardness variation effect on wear. The Archard model is applied locally over the slip area to achieve a more accurate distribution of the wear. Thus, the wear depth, w, at each local coordinate x is calculated as,

w (x)=kp ( x ) sd(x)

H (x) , (5)

where, sd is the sliding distance, H is the Vickers hardness and k is the non-dimensional wear coefficient dependent on the sliding velocity and contact pressure as described in [22] to represents mild, severe or catastrophic wear regimes. The sliding distance, sd, is calculated by multiplying the relative slip velocity, s, calculated from Equation (3) to the contact discretization dx. Figure 8 shows a schematic of the distribution of relevant parameters within a single contact case with respect to its local coordinate.

Figure 8 Schematic of contact pressure (dash doted), sliding distance (dashed) and wear depth (solid) distribution along the contact line for a single contact case

The calculated wear depth within each local contact coordinate is then applied to the (global) longitudinal rail coordinate, X, taking into account the longitudinal shift of the contact centre. The summation of the wear due to each contact case gives the overall wear of the rail profile due to a wheel passage as shown in Figure 9.

Figure 9 Schematic of wear and pressure distribution for contact at each time-step and the summation of wear depth at each global rail coordinate to represent the overall rail wear due to the passage of a wheel.

Although the surface irregularities considered in this study are generally larger than the contact length, a contact filter may still be used to smooth the rail profiles resulted from wear calculations. This is because the wear depth deduction is done at each discretization within the contact line and a discretization error may cause non-physical perturbations in the predicted worn profile. Here, the contact filter is applied as a high-pass filter for wavelengths above the nominal contact length.

After calculation of the wear depth for the passage of the simulated vehicles (unsprung masses), the wear depth is scaled up (or down) to comply with the limit in each wear step and the rail profile is updated to be used in the dynamics simulation in the next wear step. This limit may either be set on the maximum wear depth (usually set to 0.1 mm as suggested in [43] for wheel wear) or it may be set on the amount of gross tonnage passing the rail in each wear step. For the cases studied here, it is concluded that a limit on gross tonnage yields a more realistic and smoother wear pattern. A study is

conducted on what the appropriate million gross tonne (MGT) limit should be. It is concluded that the wear pattern converges for a wear limit of 1 MGT.

2.5.Onset of surface plasticityThe variation in material properties of the rail within the HAZ may result in parts of the rail having lower yield limit and more prone to plastic flow on the surface. To study the possibility of having plastic flow on the surface, the yield limit at each point along the rail is estimated from the given hardness value at that point using the following empirical relationship from [44],

Y lim ¿=−90.7+2.876 H ¿, (6)where, H is the Vickers hardness number and Ylim is the yield limit in MPa. The Von Mises shear strength (κ=Y lim ¿/√3¿) is therefore known along the rail. It is then checked if the contact pressure at each time-step would lead to the onset of plasticity according to the elastic limits for two-dimensional tractive rolling [45]. Figure 10 depicts the elastic limit dependent on friction coefficient and tractive to normal load ratio (adhesion utilization).

Figure 10 Elastic limit in two-dimensional tractive rolling for different friction coefficients as a function of adhesion utilization (Q/N) where κ is the Von Mises shear strength. Adapted from [45].

In frictionless contacts, initially the material yields under the surface and the plastic deformation is limited to a small zone below the surface. Only after an appreciable load increase the plastic zone reaches the surface. However in presence of friction, yield occurs at all points on the surface if the friction coefficient is above 0.3 [46].

3. Results and discussion

3.1.Effect of various modelling featuresBefore applying the proposed methodology to rail head weld repairs, a brief study is conducted to show the effect of modelling features on the wear results. To this end a sinusoidal irregularity with 200 mm length and 0.1 mm height on a homogeneous R260 rail surface is considered. The irregularity centre is located on middle of the track at the midspan. Track and vehicle data are listed in Table 1. Wear calculation is conducted for 50 MGT.

Table 1 Track and vehicle input data.

Track data:Sleeper spacing 0.65 mRail pad stiffness 195 MN/mRail pad damping ratio 0.1Sleeper mass 325 kgBallast stiffness 65 MN/mBallast damping ratio 0.5Simulated distance 1 mRail data:Rail grade R260Density 7850 kg/m3

Cross-section area 76.7 cm2

Second moment of inertia 3.038 m4

Rail profile 60E1General properties:Young’s modulus 210 MPaPoisson ratio 0.3Friction coefficient 0.3

Vehicle data:Passenger loco:Speed 120 km/hAxle load 20 tonneUnsprung mass 1800 kgTractive torque 4500 NmWheel moment of inertia 120 kgm2

Wheel radius 460 mmWheel profile P8Freight loco:Speed 80 km/hAxle load 22 tonneUnsprung mass 2500 kgTractive torque 6500 NmWheel moment of inertia 250 kgm2

Wheel radius 600 mmWheel profile P10

Figure 11 (a) illustrates the profile evolution incorporating all the modelling features discussed in the previous section and is taken as reference. Figure 11 (b) shows the results for the same case, however, with assuming a constant equivalent longitudinal creepage and hence discarding the traction dynamics module presented in Section 2.2. A clear difference is seen between the two cases. For the case with constant creepage, the only varying parameter that causes differential wear is the contact force variation. As the wheel passes over the irregularity, the contact force rises until it reaches the crest and then falls after the crest. This causes a higher wear before the crest and a lower wear after it. Therefore, the rail profile after 50 MGT is seems to keep a similar shape with lower crest height and a forward-shifted crest location. However, considering the traction dynamics of a driven wheel, a change in contact force also affects the longitudinal creepage, as illustrated in Figure 6. A reduction of the contact force influences wear in two opposing ways; a reduction in wear depth due to a reduction in the contact pressure, and an increase in wear depth due to an increase in the utilized longitudinal creepage. For the studied case, the latter outbalances the former. Thus a different wear pattern is predicted when it is assumed to have constant torque than constant creepage. Possible differences on corrugation growth associated with the assumption of constant or variable longitudinal creepage is also addressed in [47].

(a) (b)

(c) (d)Figure 11 Simulated rail profile evolution after 50 MGT for (a) considering driven wheels (reference case), (b) considering constant creepage (c) not considering the geometrical contact shift (d) deduction of wear depth only at the point of contact.

Each line in a plot indicates the updated rail surface after every wear step of 1 MGT.

Figure 11 (c) shows the results when the geometrical contact shift explained in Figure 7 is not accounted for. A considerable overestimation of the wear in the troughs and underestimation of it in the crests is evident. This is because, considering the geometrical effects, the contact points are shifted forwards before the crest and backwards after the crest. Therefore, more contacts are concentrated near the vicinity of the crest which results in more wear. A similar but opposite phenomenon happens in troughs. Therefore, lower wear in troughs is expected if contact shift is taken into account.

The effect of how the calculated worn material should be deducted from the rail profile is studied too. For the reference case in Figure 11 (a), the calculated wear for each contact case is distributed over the slip area and located on the rail as illustrated in Figure 9. Figure 11 (d) shows the results for the case where the worn material is calculated for the whole patch instead and is deducted from the rail profile at the point of contact. A non-smooth wear pattern is achieved which is more sensitive to the low amplitude fluctuations of the contact conditions (contact force and creepage).

3.2.Differential wear of aluminothermic rail-head repairA case study is conducted on differential wear due to hardness variation caused by aluminothermic rail-head repair. It is assumed that the rail surface is repaired and ground perfectly so that the only source of imperfection is the non-homogeneous rail hardness along the rail. Figure 12 shows measured hardness profiles for two aluminothermic rail-head repairs [48] as well as a general representation used in the simulation. Using this simplified representation, the hardness profile can be categorized by variation length, λ, and peak-to-peak magnitude, Δ. For the sake of simplicity it is assumed that the hardness of the weld material is identical to the parent rail, however, a further parameter could be added to account for a possible deviation.

Figure 12 Measured hardness profiles along the HAZ [48] and a schematic generalization of the hardness profile.

The aim is to study the effect of variation length (λ) and magnitude (Δ) on rail profile evolution in long-term. The wear due to the mixed traffic of typical passenger and freight locos are considered. The characteristics of the track as well as the locos are listed in Table 1. The loco wheels are driven. Since the longitudinal creepage of the free wheels in the wagons and trailer cars are minimal for wheelset central position in tangent track, the wear due to them is negligible and thus discarded.

The wear calculations are done over 1 m of the track with the centre of weld repair located at 50 cm in the middle of two sleepers. The hardness of the rail and weld material are 274 HV representative of R260 grade.

A set of 20 cases are simulated for combinations of λ (ranging between 10-70 mm) and Δ (ranging between 60-140 HV). Wear calculations are conducted for 50 MGT which is a reasonable grinding interval for straight tracks. Wheels are considered to be driven and geometrical contact shift as well as wear distribution along the slip area are accounted for. For the studied cases, no two-point contact in longitudinal direction are detected. The plastic onset analysis shows no plastic flow on the surface for any of the studied cases. No loss of contact is detected either.

The surface quality of the rail after 50 MGT is then quantified by calculating the maximum vertical deviation of the rail profile (indicative of rail flatness) and maximum absolute gradient. The profile gradient is calculated as advised in [49]. Figure 13 shows contour lines of these two surface quality indicators for the range of simulated hardness profiles.

(a) (b)Figure 13 Contour plots of simulated (a) maximum vertical deviation and (b) maximum absolute gradient of the rail surface after 50 MGT with respect to variation length and magnitude of the hardness profile

The results show that the rail quality in the long term is highly affected not just by the hardness magnitude variation within the HAZ but also by the variation length which is indicative of HAZ length. The profile evolution for two cases with similar hardness variation magnitude but different variation length is shown in Figure 14. The case of 10 mm variation length shows almost no form of differential wear. It is as if the wheel does not “experience” any inhomogeneity as it travels along the rail. The hardness variation length is, in this case, close to the contact length. A completely different rail profile is achieved for the same hardness variation magnitudes but longer variation length. Extended simulation of this case for higher MGTs show that a corrugation pattern initiates from the rail weld onward.

(a) (b)Figure 14 Simulated rail profile evolution after 50 MGT for a perfect starting geometry but hardness inhomogeneity according to Figure 12 with hardness variation magnitude of Δ=140HV and variation length of (a) λ =10 mm and (b) λ =70 mm. Each line in a plot indicates the updated rail surface after every wear step of 1 MGT.

The results suggest that the negative effect of weld-induced material inhomogeneity on running surface quality is diminished by minimizing the length of the zone within which it occurs. However, one should bear in mind that the Hertz contact model (which relies on homogeneity assumption) may not be considered as a valid approximation for abrupt changes in material properties within the contact length.

Although similar trends are predicted for other combinations of vehicle speeds and axle loads, the magnitude of the contour lines in Figure 13 depends on the track and vehicle characteristics. Furthermore, the simulated traffic load of 50 MGT in this study consists of only locos without

accounting for the trailing wagons. Therefore, to set realistic limits on welding process using the proposed methodology, a more generic operating condition including a wider range of vehicle fleet and various track support conditions should be considered.

It has been shown in many studies [6, 36, 37] that an increase in the vehicle speed has a significant effect on the arising dynamic loads due to discrete geometry defects. In the cases reported in this paper, moderately low speeds are considered. A brief test using higher loco speeds shows that although same contour line trends (as in Figure 13) is obtained, the absolute magnitudes as well as the worn patterns are different. Thus, a suitable relevant choice of vehicle fleet and speed should be used in the simulations for high speed lines.

In addition to variation in hardness, a difference between the metallurgical microstructure of the welding material and parent rail may imply a difference in wear resistance. This effect can be accounted for by using different wear coefficients for the weld material. This effect is more pronounced for head repair techniques such as manual metal arcing where the deposited weld material may have a bainitic microstructure different to the pearlitic rail.

The empirical equation (Equation (4)) taken from [44] to relate hardness to yield strength is reported to have a standard error of 102 MPa. Using this equation, the yield limit for R260 rail is calculated to be about 690 MPa. This value is more than what is reported for the same rail grade by others. In general, there is a large scatter in the reported measured material properties (see Table 1 in [13]) with the yield strength ranging between 440-654 MPa. An overestimation of the yield strength may be a reason why no plastic deformation on the surface is detected for the studied cases.

4. ConclusionIn this paper, a general methodology is proposed for simulating the long-term rail surface evolution due to discrete geometrical imperfections and microstructural inhomogeneities, typical of rail welds and repairs. Different aspects and modelling features within the approach are first discussed and possible variations such as the effect of traction and contact longitudinal shift are addressed. To test the methodology, the specific case of inhomogeneous hardness along the rail resulting from aluminothermic weld repairs is studied. As a preliminary step, a simplified two-dimensional version of the methodology is implemented, representing symmetrical loading on tangent track. The simulated long-term rail profile evolution gives some insight on how welding process should be regulated with respect to length and magnitude of variations in hardness, to limit long-term running surface quality degradation. Since a hardness traverse across welds and weld repairs is process approval criterion, it is prudent to establish the dependence of wear resistance on this parameter. In particular, the case presented in Figure 14 show that geometrical degradation can be effectively controlled by limiting the length over which the hardness variation occurs, without the need for controlling hardness variation. In practice, this can prove difficult to achieve, however, lowering the number of preheats and the magnitude of heat input may help. Alternatively, using Flash-butt welding or innovative welding techniques such as orbital friction welding [50] may contribute to shortening the length of the HAZ.

Finally, this preliminary study is limited to two-dimensional cases neglecting the lateral dynamics. However, the proposed methodology is essentially applicable to three-dimensional cases as well. In addition, the lack of data on wear resistance of cast microstructures that characterise aluminothermic welds and the non-pearlitic (often bainitic) microstructures produced by the consumables used in weld restoration of running surface defects may be a restriction of the current work. Thus, more tests on wear resistance of these materials are needed.

AcknowledgementsThe support from European Commission H2020 funding of project In2Rail (grant agreement n. 635900) is gratefully acknowledged. The authors would also like to thank Ilaria Grossoni for providing her beam model of the rail on which the vehicle-track dynamics model in this work is based, and Jay Jaiswal (ARR Rail Solutions Ltd) for his advice on metallurgical and welding technology aspects of the work.

References

[1] R. Enblom, “On Simulation of uniform wear and profile evolution in the wheel-rail contact,” KTH Royal Institute of Technology, PhD thesis, Stockholm, 2006.

[2] R. Enblom and M. Berg, “Proposed procedure and trial simulation of rail profile evolution due to uniform wear,” Proceedings of the Institution of mechanical engineers. Part F, journal of rail and rapid transit, vol. 222, no. 1, pp. 15-25, 2008.

[3] P. Torstensson, “Rail corrugation growth on curves,” Chalmers University of Technology, PhD thesis, Gothenberg, 2012.

[4] Z. Li, X. Zhao, R. Dollevoet and M. Molodova, “Differential wear and plastic deformation as causes of squat at track local stiffness change combined with other track short defects,” Vehicle System Dynamics, vol. 46, no. S1, pp. 237-246, 2008.

[5] Z. Li, X. Zhao, C. Esveld, R. Dollevoet and M. Molodova, “An investigation into the causes of squats—Correlation analysis and numerical modeling,” Wear, vol. 265, pp. 1349-1355, 2008.

[6] H. H. Jenkins, J. E. Stephenson, G. Clayton, G. Morland and D. Lyon, “The effect of track and vehicle parameters on wheel/rail vertical dynamic forces,” Journal of Railway Engineering Society, vol. 3, no. 1, pp. 2-16, 1974.

[7] P. J. Mutton and E. F. Alvarez, “Failure modes in aluminothermic rail welds under high axle load conditions,” Engineering Failure Analysis, vol. 11, pp. 151-166, 2004.

[8] M. J. M. M. Steenbergen, “Rail welds,” in Wheel-rail interface handbook, Cambridge, UK, Woodhead Publishing Limited, 2009, pp. 377-408.

[9] Z. Wen, G. Xiao, X. Xiao, X. Jin and M. Zhu, “Dynamic vehicle–track interaction and plastic deformation of rail at rail welds,” Engineering Failure Analysis, vol. 16, no. 4, pp. 1221-1237, 2008.

[10] W. Li, G. Xiao, Z. Wen, X. Xiao and X. Jin, “Plastic deformation of curved rail at rail weld caused by train–track dynamic interaction,” Wear, vol. 271, no. 1-2, pp. 311-318, 2011.

[11] P. Mutton, J. Cookson, C. Qiu and D. Welsby, “Microstructural characterisation of rolling contact fatigue damage in flashbutt welds,” Wear, vol. 366, pp. 368-377, 2016.

[12] S. L. Grassie and J. Kalousek, “Rail corrugation: characteristics, causes and treatments,” Proc. IMechE, Part F: J. Rail and Rapid Transit, vol. 207, no. F, pp. 57-68, 1993.

[13] A. Böhmer and T. Klimpel, “Plastic deformation of corrugated rails—a numerical approach using material data of rail steel,” Wear, vol. 253, pp. 150-161, 2002.

[14] M. Meywerk, “Polygonalization of railway wheels,” Archive of Applied Mechanics, vol. 69, pp. 105-120, 1999.

[15] J. Yang, D. J. Thompson and Y. Takano, “Characterizing wheel flat impact noise with an efficient time domain model,” in Noise and Vibration Mitigation for Rail Transportation Systems, Berlin, Springer-Verlag, 2015, pp. 109-116.

[16] K. Popp, K. Holger and I. Kaiser, “Vehicle-Track Dynamics in the Mid-Frequency Range,” Vehicle System Dynamics, vol. 31, no. 5-6, pp. 423-464, 1999.

[17] J. C. O. Nielsen, R. Lundén, A. Johansson and T. Vernersson, “Train-Track Interaction and Mechanisms of Irregular Wear on Wheel and Rail Surfaces,” Vehicle System Dynamics, vol. 40, no. 1-3, pp. 3-54, 2003.

[18] T. X. Wu and D. J. Thompson, “Behaviour of the Normal Contact Force Under Multiple Wheel/Rail Interaction,” Vehicle System Dynamics, vol. 37, no. 3, pp. 157-174, 2002.

[19] C. Andersson and T. Abrahamsson, “Simulation of interaction between a train in general motion and a track,” Vehicle System Dynamics, vol. 38, no. 6, pp. 433-455, 20025.

[20] L. Baeza, P. Vila, G. Xie and D. Iwnicki, “Prediction of rail corrugation using a rotating flexible wheelset coupled with a flexible tarck model and a non-Hertzian/non-steady contact model,” Journal of Sound and Vibration, vol. 330, pp. 4493-4507, 2011.

[21] H. Krause and G. Poll, “Wear of wheel-rail surfaces,” Wear, vol. 113, pp. 103-122, 1986.

[22] T. Jendel, “Prediction of wheel wear profile - comparison with field measurements,” Wear, vol. 253, pp. 89-99, 2002.

[23] M. Hiensch, J. C. Nielsen and E. Verheijen, “Rail corrugation in The Netherlands—measurements and simulations,” Wear, vol. 253, pp. 140-149, 2002.

[24] A. Igeland, “Railhead corrugation growth explained by dynamic interaction between track and bogie wheelsets,” Part F: Journal of rail and rapid transit, vol. 210, pp. 11-20, 1996.

[25] H. Ilias, “The influence of railpad stiffness on wheelset/track interaction and corrugation growth,” Journal of Sound and Vibration, vol. 227, no. 5, pp. 935-948, 1998.

[26] G. Xie and S. Iwnicki, “Simulation of wear on a rough rail using a time-domain wheel–track interaction model,” Wear, vol. 265, pp. 1572-1583, 2008.

[27] X. Sheng, D. J. Thompson, C. J. C. Jones, G. Xie, S. D. Iwnicki, P. Allen and S. S. Hsu, “Simulations of roughness initiation and growth on railway rails,” Journal of Sound and Vibration, vol. 293, pp. 819-829, 2006.

[28] J. J. Kalker, “Rolling contact phenomena - linear elasticity,” in Rolling contact phenomena, Wien, Springer-Verlag, 2000, pp. 1-84.

[29] P. Remington and J. Webb, “Estimation of wheel/rail interaction forces in the contact area due to roughness,” Journal of Sound and Vibration, vol. 193, no. 1, pp. 83-102, 1996.

[30] A. Pieringer, W. Kropp and D. Thompson, “Investigation of the dynamic contact filter effect in vertical wheel/rail interaction using a 2D and a 3D non-Hertzian contact model,” Wear, vol. 271, pp. 328-338, 2011.

[31] J. B. Nielsen, “Evolution of rail corrugation predicted with a non-linear wear model,” Journal of Sound and Vibration, vol. 227, no. 5, pp. 915-933, 1999.

[32] L. Baeza, P. Vila, A. Roda and J. Fayos, “Prediction of corrugation in rails using a non-

stationary wheel-rail contact model,” Wear, vol. 265, pp. 1156-1162, 2008.

[33] “EN 14730-2 Railway applications - Track - Aluminothermic welding of rails - Part 2: Qualification of aluminothermic welders, approval of contractors and acceptance of weld,” European Committee For Standardization, Brussels, 2006.

[34] “Railway applications - Infrastructure - Flash butt welding of rails - Part 1: New R220, R260, R260Mn, R320Cr, R350HT, R370LHT and R400HT grade rails in a fixed plant,” European Committee For Standardization, Brussels, 2016.

[35] “ETM-01-01 Rail weld geometry standard,” Australian Rail Track Corporation, 2009.

[36] M. J. M. M. Steenbergen and C. Esveld, “Relation between the geometry of rail welds and the dynamic wheel–rail response; numerical simulations for measured welds,” Proc IMechE, Part F: Journal of Rail and Rapid Transit, vol. 220, pp. 409-424, 2006.

[37] I. Grossoni, P. Shackleton and Y. Bezin, “Longitudinal rail weld geometry control and assessment criteria,” Failure Engineering Analysis, vol. 80, pp. 352-367, 2017.

[38] “Railway applications - Track - Restoration of rails by electric arc welding,” European Committee For Standardization, Brussels, 2009.

[39] J. Y. Shih, D. Kostovasilis, Y. Bezin and D. J. Thompson, “Modelling options for ballast track dynamics,” in 24th International Congress on Sound and Vibration, London, 2017.

[40] F. Carter, “On the action of locomotive driving wheel,” Proc. R. Soc. London, vol. 112, pp. 151-157, 1926.

[41] J. Piotrowski and J. Kalker, “A non-linear mathematical model for finite, periodic rail corrugations,” Transactions on Engineering Sciences, vol. 1, pp. 413-424, 1993.

[42] M. S. Sichani, R. Enblom and M. Berg, “An alternative to FASTSIM for tangential solution of the wheel-rail contact,” Vehicle System Dynamics, vol. 54, no. 6, pp. 748-764, 2016.

[43] T. Jendel, “Prediction of wheel profile wear - Methodology and verification,” KTH Royal Institure of Technology, Licentiate thesis, Stockholm, 2000.

[44] E. J. Pavlina and C. J. Van Tyne, “Correlation of Yield Strength and Tensile Strength with hardness for Steels,” Journal of Materials Engineering and Performance, vol. 17, no. 6, pp. 888-893, 2008.

[45] D. A. Hills, D. Nowell and A. Sackfield, Mechanics of Elastic Contacts, Oxford: Butterworth–Heinemann, 1993.

[46] K. L. Johnson, “Plastic Deformation in Rolling Contact,” in Rolling contact phenomena, Wien, Springer-Verlag, 2000, pp. 163-201.

[47] L. Afferrante and M. Ciavarella, “Short-pitch rail corrugation: A possible resonance-free regime as a step forward to explain the “enigma”?,” Wear, vol. 266, pp. 934-944, 2009.

[48] S. Fretwell-Smith and C. Zhu, “Investigation of repair methods and welding techniques, Evaluation of welding feasibility of head repair welds,” British Steel, 2017.

[49] M. J. M. M. Steenbergen and C. Esveld, “Rail weld geometry and assessment conscepts,” Proc. IMechE, Part F: Journal of Rail and Rapid Transit, vol. 220, pp. 257-271, 2006.

[50] “WRIST-project,” [Online]. Available: http://www.wrist- project.eu/page/en/about- wrist/objectives- andtargeted-results.php.. [Accessed 7 November 2017].