A numerical study on tread wear and fatigue damage of railway wheels subjected to anti-slip control

Tread wear and rolling contact fatigue (RCF) damage propagated on railway wheels are the two extremely important focal points as they can tremendously deteriorate wheel/rail interactions and hunting stability and destroy wheel surface materials, and subsequently, cut down the lifetime of the wheels. The on-board anti-slip controllers are of essence aiming to hold back the striking slipping of the powered wheelsets under low-adhesion wheel/rail conditions. This paper intends to investigate the impact of anti-slip control on wheel tread wear and fatigue damage under diverse wheel/rail friction conditions. To this end, a prediction model for wheel wear and fatigue damage evolution on account of a comprehensive vehicle-track interaction model is extended, where the wheel/rail non-Hertzian contact algorithm is used. Furthermore, the effect of frictional wear on the fatigue damage at wheel surface is considered. The simulation results indicate that the wheel/rail contact is full-slip under the low-adhesion conditions with braking effort. The wear amount under the low-adhesion conditions is observably higher than that under the dry condition. It is further suggested that the wheel tread is prone to suffering more serious wear and fatigue damage issues with a higher anti-slip control threshold compared to that with a lower one.


Introduction
Wear and fatigue damage are the two main intractable issues occurring on wheel and rail tread ( Fig. 1), which are induced by the frictional interactions in the wheel/rail contact patches with a variety of contaminants and a high degree of contact stresses [1,2]. In turn, the degraded wheel and rail profiles caused by the frictional wear can exacerbate wheel/rail interactions, riding quality, curving performances, and coupled vibrations of the train-track system. On the other hand, the fatigue damage (mainly manifested as the cracks and spalling) on the wheel and rail tread will not cause the aggravating wheel/rail interactions. But it can extend down to the sub-surface with the consecutive and deteriorating wheel loadings, and eventually, leading to the abrupt fracture failure of the wheel and rail. During the last few decades, the rapid increase of traction tonnage, operating speed, and axle load of railway trains makes the wheel/rail contact worsening and accelerates the deterioration of wheel/rail surface material, and consequently, supplements the excess maintenance work. Restoration of wheel profiles and installing the lubrication devices are the most commonly used countermeasures against the fast-growing wheel wear and fatigue damage, which consume extra manpower and are very expensive. According to the first-hand information from the metro operating companies, more than 70% of metro wheels cannot be used to the scheduled lifetime in light of the severity of damage on wheel profiles.
www.Springer.com/journal/40544 | Friction Anti-slip controllers have been widely installed on modern metro trains purposing to avoid the unwanted wheel/rail slipping phenomenon and along with the resulting wheel and rail surface damage under poor wheel/rail adhesion conditions. An appropriate adjustment on anti-slip control may be of advantage to ameliorate wheel/rail interactions and relieve wheel tread wear and fatigue damage, which is easier and more economical than wheel re-profiling operations or installing the lubrication device. This research work emphasizes on the effect of anti-slip control on tread wear and fatigue damage of metro wheels, which has hardly been discussed in the past.

On the wheel/rail tread wear
From the viewpoint of tribology, wheel/rail wear can be mainly classified as the abrasive wear, adhesive wear, corrosion wear, erosive wear, and rolling contact fatigue (RCF) wear [2,3], among which the adhesive wear due to wheel/rail rolling contact is the major focus in the publications.
Twin-disc test machine has been mostly used to experimentally reveal wheel/rail wear evolution mechanism. Lewis et al. [4,5] conducted wheel wear evolution testing under dry contact condition and divided wheel/rail wear into three wear regimes according to the wear rates (mild, severe, and catastrophic). Since the wheel/rail interface is exposed to an exoteric environment, it is prone to being attached with the contaminant (water, grease, accumulated leaves, etc.). Aiming at this, Hardwick et al. [6], Eadie et al. [7], and Wang et al. [8] examined the effect of dry, water, and greasy conditions on wheel/rail creep curves and wear evolution (wear resistance) and stated that wheel/rail wear is relatively low due to the low-friction interactions under the wet and greasy conditions. Furthermore, the effect of wheel/rail slipping rates was presented and suggested that the wheel wear can transform from type I (mild wear) and type II (severe wear), as referred in Ref. [9]. Thanks to these experimental research works, the numerical models of wheel/rail wear evolution under complex adhesion conditions can be well established and implemented.
In practice, wheel and rail wear is very intricate due to the highly non-linear wheel/rail contact. On the one hand, the spatial motions of the wheelset and track greatly affect wheel/rail contact and wear region, and the curving process can even lead to severe wheel flange wear. On the other hand, the slipping performances of powered wheelset also have a considerable effect on wheel/rail wear, which are likely caused by the hauling effort and variable friction statuses. These two aspects are difficult to replicate on the twin-disc test machines. The following wheel and rail wear prediction models are available: (i) British Railway Research (BRR) model [10], (ii) Zobory model [11], (iii) the KTH model (derived from Archard wear model) [12], and (iv) the USFD model (developed by University of Sheffield) [4,13]. Among Refs. [14][15][16][17], the combined utilization of train-track coupled simulations and wheel/rail on-line wear prediction is widely used.
With an aim to enhance the computation efficiency that affected by wear predictions and profile updating during the loop iterations, some improved and fast wheel wear prediction models have been proposed, e.g., the artificial-neural-network-based wear model [18], quasi-static wear model [19], and physics-based data-driven wear model [20]. However, whether these prediction models are capable of wheel wear evolution under complex operating conditions, especially for the changeable adhesion conditions, is unknown and should be further proved.
A flow of countermeasures against decelerating the development of wheel and rail wear have been proposed recently. Polach [21] and Ye et al. [22] designed a new wheel profile (S1002-A103 profile) with an aim to reduce the wheel flange wear and make the tread wear evenly distributed based on the S1002 profile. Fröhling et al. [23,24] declared that a conformal design of wheel profile can help to make wear evenly distributed across the tread, and then to reduce the wheel tread wear (mainly the false flanges), which however, increases the possibility of RCF initiation at the same time. Fu et al. [25] stated that the active wheelset steering strategy can reduce wheel flange wear compared to the passive one. Moreover, it was proved that the enhanced wheel steel (the R8T railway steel) with favorable abrasive resistance can be helpful to reduce wheel wear [4].

On the wheel/rail tread fatigue damage
The surface-initiated RCF is a very complicated issue, in-depth understanding of which needs the materialogy and tribology. The wheel and rail surface-initiated RCF comes into being as a consequence of the frictional interactions and resulting thermal stress when the material strength limit is exceeded. The surface-initiated RCF will usually not cause drastic wheel/rail interactions or hunting performances. But the surface-initiated cracks tend to develop down to the sub-surface with the presence of the fluid (grease, water, etc.) trapped in cracks and the continual contact stresses, subsequently leading to the pit or spalling [26,27].
Twin-disc testing combined with microstructure observation (the scanning electron microscope (SEM)) has been commonly used to study wheel/rail RCF. Similar to the study of wheel/rail wear, how the wheel/rail slipping [9,28], wheel/rail surface layer materials [29], temperature conditions [30,31], "thirdbodies" (water, oil, grease, friction modifier, etc.) [7,32], and local defect [33,34] affect wheel/rail RCF has been experimentally examined. On the other hand, how the wheel surface RCF forms is also discussed by field observations, such as the effect of wheel radius difference on the abnormal development of wheel surface cracks [35].
In the perspective of numerical simulations, two wheel/rail RCF prediction models, namely the energy dissipation (Tγ, where T and γ stand for wheel/rail creep force and creepage, respectively) function and shakedown theory on the basis of wheel/rail Hertz contact analysis, have been mostly employed to qualitatively assess the surface damage [27]. Karttunen et al. [36] presented wheel/rail RCF related to the track irregularity and suggested that it is primarily caused by the lateral irregularities with a high level. Spangenberg et al. [37] supplied a conformal design of wheel profile, which has a prominent benefit in the reduction of wheel RCF. Khan et al. [38] stated that the initiation of rail RCF can be avoided through the implementation of the friction control, and it can also be replaced by the large frictional wear on a tight curve without the friction control. On the other hand, due to the complex architectonics of the turnout and the corresponding execrable wheel/rail matching relationship, Ma et al. [39] evaluated the rail surface RCF when the wheelset sideways pass through the turnout switch panel by using the shakedown theory, where the non-Hertzian models were applied for the sake of more precise depiction of wheel/rail contact and damage distributions.
Recently, Dirks et al. [40] proposed an innovative wheel/rail wear and crack prediction model, in which the crack prediction model was derived based on the stress and energy dissipation index as well as the Palmgren-Miner theory. Owing to this study, the size of wheel surface cracks can be well predicted.
In reality, wheel and rail wear and surface damage propagate and develop simultaneously. The competitive effect between these two injuries [41], particularly the eliminating effect of frictional wear on the surfaceinitiated cracks, is in existence and should be carefully treated in terms of the numerical simulations. Therefore, it is quite essential to take the frictional wear into account when executing the RCF evolution predictions.

On the anti-slip control
It is well known that the poor wheel/rail friction conditions caused by the surface attachments contribute greatly to the reduction of adhesion and wheelset slip www.Springer.com/journal/40544 | Friction with presence of traction/braking effort. The wheel/rail surface damage is highly related to these adverse consequences, such as the wheel flat. Therefore, it is very essential to introduce an effective measure to prevent the excessive wheelset slipping. The PI-or PID-based anti-slip controller is commonly applied to cut short the traction/braking torque when the powered wheelset slips or skids [42,43].
Recently, some advanced anti-slip control strategies have been proposed, where the observation for wheel/rail adhesion conditions (coefficient) and automatic regulation of controlling parameters is requisite, and the optimized adhesion utilization can be acquired [44,45]. However, wheel/rail adhesion is highly non-linear and difficult to be observed due to the high-frequency vibrations and complicated relationship of the wheel/rail system. Therefore, these improved anti-slip controllers just stay at the level of numerical calculations, or at most, laboratory experiments [46].
So far, the anti-slip controllers with constant controlling parameters are commonly employed in the actual railway industry. Only wheel/rail slipping behaviors and traction/braking efficiency are mainly focused among the published literatures. But how the anti-slip control affects wheel profile wear and fatigue damage has been rarely discussed [47], which is the main concern of this paper.

Main work and structure of this paper
The need for remission on the wear and fatigue damage issues of metro wheels that experience excessive and violent wheel loadings is pressing. This paper aims to investigate the effect of the anti-slip control on wheel tread wear and fatigue damage, which has been rarely discussed. What is more, the wheel/rail non-Hertzian contact and the impact of frictional wear on the fatigue damage are carefully considered during the long-term evolution simulations.
The main research work of this paper can be summarized as follows: 1) An elaborate numerical model for metro vehicle-track interactions is performed based on the fundamentals of vehicle-track coupled dynamics, in which an advanced wheel/rail non-Hertzian contact model is applied in terms of better representation of wheel/rail contact.
2) A long-term wheel profile wear and fatigue damage prediction model based on the wheel/rail non-Hertzian contact analysis is proposed, which has been relatively few implemented in Refs. [48,49]. Most importantly, the size of wheel surface crack and the eliminating effect of the frictional wear on it are carefully treated during the loop iteration.
3) The influence of changing wheel/rail adhesion conditions and anti-slip controlling parameters on wheel/rail non-Hertzian contact as well as profile wear and fatigue damage of wheel tread is presented and carefully discussed from the viewpoint of numerical simulations.
The rest of this paper is structured as follows. In Section 2, a vehicle-track coupled dynamics model with consideration of an advanced wheel/rail non-Hertzian contact algorithm is built. In Section 3, a long-term prediction model covering wheel tread wear and fatigue damage based on the wheel/rail non-Hertzian contact is extended. In Section 4, the influence of anti-slip controlling parameters on wheel wear and fatigue damage is presented and particularly discussed. This paper ends with conclusions, as given in Section 5.

Vehicle-track spatial interaction model
Here, a fully non-linear metro vehicle-track spatial interaction model is established based on the fundamental principle of vehicle-track coupled dynamics [50][51][52], which can be mainly classified as the vehicle sub-model, slab track sub-model, wheel/rail interaction sub-model, anti-slip control sub-model, and numerical integrator.

Vehicle sub-model
The metro train is modeled based on the multi-body dynamics principle, in which the two main components are involved: 1) Vehicle structures (including the wheelsets, bogie frames, and carbodys), simulated as rigid bodies with consideration of all six degrees of freedom (DoFs).
2) Suspension components (including the primary and secondary suspension systems and in-train coupler In Eq. (1), M Tv , C Tv , and K Tv represent the mass, damping, and stiffness matrices of the vehicle system, respectively. x Tv denotes the displacement vector of the vehicle system. F tt is the wheelset-rail interaction vector. The spatial vibrations of vehicle system can be solved by the Zhai method [51].

Track sub-model
The slab track is modeled as a layered structure with consideration of its flexible deformations, which involves two rails, discrete-supported rail fasteners, floating slabs, and concrete asphalt (CA) layers. Furthermore, the vertical vibrations of the floating slabs are modeled as the Kirchhoff plate, which can be referred in Ref. [50]. The deformations of the base are neglected in view of the negligible effect of which on wheel/rail interactions. The dynamics equation for the track system can be expressed as

Tt Tt
Tt Tt Tt Tt tt sb where M Tt , C Tt , and K Tt represent the mass, damping, and stiffness matrices of the track system, respectively.
x Tt denotes the displacement vector of the track system. F sb is the slab-base interaction vector. Also, the calculations for the spatial vibrations of the track system are solved by the Zhai method [51].

Anti-slip control sub-model
The on-board anti-slip controller is of importance with presence of the poor adhesion conditions and the resulting noteworthy wheelset slipping. In this paper, each metro wheelset is assumed to be configured with the independent PI-based anti-slip controller [45,46]. The pre-set anti-slip control threshold value (s ref ) is an important parameter that regulates wheel/rail slipping, which is usually constant in practical railway applications. When the observed wheel/rail slipping rate exceeds the pre-set control threshold, the anti-slip controller will be triggered, and the traction/braking torque will be reduced; otherwise, the anti-slip controller will not be launched [46]. The influence of the pre-set anti-slip control threshold on the tread wear and fatigue damage of the metro wheels under braking operation is the main target of attention in this paper.
Since the wheel/rail creepage is hardly possible to be obtained, the anti-slip controller takes the observed wheel/rail slipping rate wj ( ) e t as an input, which can be expressed as where ref s is the pre-set anti-slip control threshold, R is the nominal rolling circle radius of the metro wheel.
Further, the braking torque wj ( ) T t with consideration of the reduced quantity can be given as with the compensation torque cj ( ) T t given as where P and I indicate the proportional and integral constant values, respectively. The time range s e t t  represents the relative slipping velocity exceeding the pre-set threshold value.
T v t is the reference braking torque that determined by the vehicle speed.

Wheel/rail interaction sub-model
Wheel/rail interactions are heavily complicated due to the unstable vibrations of wheelset and track and wheel/rail interface conditions (including the surface roughness and the "third-bodies"). Performing a robust and accurate wheel/rail interaction model is therefore indispensable for wheel profile wear and fatigue damage prediction simulations. Recently, some non-Hertzian wheel/rail contact models have been www.Springer.com/journal/40544 | Friction proposed and implemented in train-track interaction and wheel/rail wear simulations, and the precision of them is proved to be more than that of the Hertzian model [49,[53][54][55]. On the other hand, the multi-point wheel/rail contact conditions are likely to appear with consideration of the wheel/rail frictional wear. In view of this, a non-Hertzian model and related multi-point contact situations are modeled even if they are obviously time-consuming in numerical solutions.
Commonly, the derivation and solution for wheel/rail rolling contact mainly involve the spatial contact geometry and normal and tangential contact. An improved wheel/rail non-Hertzian contact model composed of the modified Kik-Piotrowski (MKP) model and MKP-FaStrip model is employed to calculate wheel/rail normal and tangential interactions in this paper.

Wheel/rail contact geometry model
The employed wheel and rail profiles are displayed in Fig. 2. According to the "trace method" [50], the coordinates of the wheel tread in the absolute coordinate system can be given as x z y x x y l y l R y y R y y y l y y l l y l m y l R y z y l y z l l y l m y l with the symbols l x , l y , l z and s t r ( ) m y given as w w w w w 2 2 s t r w t r cos sin , cos cos , sin where the superscript "(A)" represents the absolute coordinate system. tr y is the lateral position of the wheel profile in the local coordinate system. w t r ( ) R y is the wheel rolling circle radius. w t r ( ) y  is the contact angle. w y and w z are the transverse and vertical coordinates of the rolling circle center in the absolute coordinate system, respectively. w  and w  denote the roll angle and yaw angle of the wheelset, respectively. The coordinates of the rail tread in the absolute coordinate system can be given as where the superscript "(L)" represents the

Wheel/rail normal and tangential interaction model
The location, shape, and number of wheel/rail contact patches can be solved based on wheel/rail interpenetration ( )  and the normal distance, which can be referred in Refs. [53,54] and omitted here. The normal contact pressure distributions ( ) i p under the case of wheel/rail multi-point contact can be given as (10) where N c is the number of wheel/rail contact points.
where C (i) represents the area of the ith contact patch.
The MKP-FaStrip model referred in Refs. [49,55,56] is introduced to solve the wheel/rail shear stresses. The longitudinal and lateral shear stresses within the adhesion region and slipping region can be given as Eqs. (12a) and (12b), respectively.
with the uncorrected shear stresses within the slipping region given as are dependent on the wheel/rail creep behaviors. The detailed expressions for these employed symbols can be referred in Refs. [56,57] and omitted here.
Furthermore,  is the alterable wheel/rail friction coefficient, which can be given in Ref. [21].
where p A and p B denote the ratio of friction coefficient and the coefficient of exponential friction decrease, respectively. 0  is static friction coefficient. x q x y y q x y x y where x' and y' represent the longitudinal and lateral relative coordinates inside the contact patches, respectively. The wheel/rail tangential contact is demonstrated in Fig. 3. Furthermore, the complex wheel/rail adhesion conditions covering the dry, wet, and greasy conditions are treated in the simulations, and the according parameters are listed in Table 1. The wheel/rail adhesion features with the concerned adhesion conditions are displayed in Fig. 4.

General architecture
The influence of anti-slip control on the profile wear and fatigue damage of metro wheels under changing wheel/rail adhesion conditions is numerically investigated in this paper. To this end, a long-term wheel tread wear and fatigue damage prediction model is built based on the wheel/rail non-Hertzian contact simulations.
The overall computing framework for wheel tread wear and fatigue damage evolution is shown in Fig. 5. Hereon, three sub-modules are mainly contained: (1) a vehicle-track interaction model with consideration of the wheel/rail non-Hertzian rolling contact, (2) a long-term wheel tread wear prediction model, and (3) a long-term wheel tread fatigue damage prediction model with consideration of the frictional wear.
The following steps are involved in this numerical computation: 1) Input the initial wheel and rail profile along with the track irregularity and layouts, vehicle and track designing parameters, and changeable wheel/rail adhesion conditions.
2) Calculate wheel/rail non-Hertzian interactions containing the pressures and tangential contact stresses as well as the wheel/rail local creepage distributions by using the performed vehicle-track coupled model.
3) Calculate wheel tread wear by using the USFD wear model [10,49] based on wheel/rail non-Hertzian contact, and simultaneously, the raw fatigue damage (cracks) of the wheel tread through the "fatigue damage" approach [41]. | https://mc03.manuscriptcentral.com/friction 1) While the wheel tread wear evolves, the rail profile is set to be a constant and unaffected by the wheel/rail frictional wear.
2) The evolved wheel fatigue damage is supposed to have no influence on wheel tread wear.
3) An appropriate multiplying factor is applied to amplify both wheel wear and fatigue damage with an aim to shorten track integrated length and computing time. Moreover, a smoothing method is employed to smooth the simulated wear and fatigue damage profiles for the purpose of avoiding the wheel/rail contact noise interference.

Wheel tread wear prediction model
Among the mentioned methods in Section 1, the Archard wear model and USFD wear model have been the most mainstream methods to predict wheel and rail wear evolution [10,48,49]. The USFD wear model that associates material loss of wheel/rail with the frictional energy dissipation within the contact patch is introduced to assess the material loss of the wheel tread in this paper.
The energy dissipation index based on the wheel/rail non-Hertzian contact analysis can be given as with the local wheel/rail relative velocity distributions in the slip region derived in Ref. [57,58]. According to the USFD principle, the wheel/rail material loss is highly related to the frictional interactions and the resulting energy dissipation. Most importantly, the wheel/rail adhesion status also has a considerable impact on the material loss. In Ref. [8], the relationship between wheel/rail wear rate and Tγ/A value was described and fitted based on the experimental investigations on a twin-disc machine, where the third-body materials (such as water, grease, and friction enhancers) are applied in the wheel/rail interface, and the impact of various contact conditions is determined. In this way, the numerical models of wheel/rail wear evolution under complex adhesion conditions can be well determined.
Given the wheel/rail contact condition as well as energy dissipation distributions obtained by Eq. (16), the material loss distributions in the ith contact patch can be calculated as where w  is the wheel material density.

 
I x y denotes the wear rate that prominently depends on the wheel/rail energy dissipation and adhesion conditions [8], where the superscript "A" denotes wheel/rail adhesion status that covers the dry, wet, and greasy conditions. Then, the wheel tread wear after one iteration computation can be derived by Eq. (19), where the wheel/rail multi-point contact scenario is considered.
with the wheel tread wear of the ith contact patch expressed as where T start and T end represent the beginning and end of the track integration, respectively. ( ) ( ) i R t is the radius of rolling circle corresponding to the ith contact patch.
Finally, after the smoothing and multiplying processes of the material loss of the wheel tread, the updated wheel profile can be given as where the superscripts "sm" and "sc" mean the smoothing and multiplying processes, respectively. Here, a five-point cubic smoothing algorithm is used to smooth wheel tread wear and to eliminate the noise in wheel/rail interaction calculations. The superscripts "O" and "N" mean the original and updated wheel profiles, respectively.

Wheel tread fatigue damage prediction model
During the last few decades, the energy dissipation theory and shake-down theory are the two most popular methods to evaluate the fatigue damage of wheel and rail surfaces, which however, can only perform the qualitative analysis and are not capable of assessing the dimension of the fatigue damage [3,35]. Recently, Dirks et al. [40] conducted the tracking observations upon the wheel/rail surface cracks on a railway over a period of five years. Butini et al. [41] conducted the crack evolution tests on two scaled rolling-sliding test machines (MMSA-2A test-rig and JD-1 test-rig), where the surface cracks were measured by the optical microscope (OM) and SEM. Owing to these terrific investigations, the so-called "fatigue damage approach" can be well implemented in the crack predictions, and the impact of wheel/rail frictional wear on the wheel/rail surface fatigue damage can be considered. Hereon, the above wheel tread fatigue damage prediction model is appropriately modified based on the non-Hertzian contact and applied in this research work. The stress index (SI) and energy dissipation index (EI) can be performed in the "fatigue damage approach". Hereon, the energy dissipation index is selected in wheel crack predictions since it can take wheel/rail relative slipping caused by the low-adhesion wheel/rail conditions into account. The stress magnitude (the maximum SI in the longitudinal direction) can be derived as with the energy dissipation index distributions within the ith contact patch given as Note that a threshold limit of 15 N (according to the T γ model) should be treated when applying the energy dissipation index in assessing the surface fatigue damage.
According to the Palmgren-Miner theory, the dimension of the wheel/rail surface fatigue damage is closely related to the stress magnitude [27,40]. Furthermore, the length of the wheel/rail fatigue damage is defaulted to be proportional to its width. On account of this, the length and depth of the wheel/rail fatigue damage with respect to one wheel/rail loading can be expressed by Eqs. (24a) and (24b), respectively: where α and β are the calibrated material parameters obtained in the crack measurements [27,39]. c f is the scaling factor between the length and depth of wheel/rail fatigue damage.
Similar to the solution of wheel tread wear, the length and depth of wheel tread fatigue damage after one iteration computation can be derived as Eqs. (25a) and (25b) with the length and depth of fatigue damage of the ith contact patch expressed as Finally, the depth of wheel tread fatigue damage with consideration of the eliminative effect of the frictional wear can be given as [sm ,sc] N N tr tr tr Dc tr w tr Dc tr ( ) ( ) ( ) max ,0 y y y c y P y c y Also, the identical five-point cubic smoothing algorithm is used to smooth the wheel tread fatigue damage hereon.

Effect of anti-slip control on wheel tread wear and fatigue damage
Here, wheel/rail creep behaviors as well as wheel wear and fatigue damage evolution subjected to the anti-slip control, primarily the controlling threshold, are numerically assessed based on the model elaborated above. In the simulation, the anti-slip controller with the threshold values of 2%, 4%, and 6% are included. The changing wheel/rail adhesion conditions containing the dry, wet, and greasy statuses are treated. The vehicle is supposed to travel on a straight track with consideration of the long-wave random irregularities (Fig. 6), and the rail cant is set as 1/40. The braking effort delivered to the metro vehicle (including all four wheelsets) is assumed to be 78.5 kN (the upper limit), which initiates to be released when the leading wheelset travels to 80 m. The initial traveling velocity of the metro vehicle is set to be 75 km/h. These dynamics simulations are compiled in FORTRAN environment, and a workstation (Dell, with the processor Intel E5-2698 v3, 2.3 GHz) is employed for the calculations.

Wheel/rail creep behaviors
The wheel/rail creep behaviors under the changing wheel/rail friction conditions are determined hereon, which are fundamental to present an insight into wheel wear and fatigue damage evolution subjected to the anti-slip control.
The impact of anti-slip control threshold on wheel/rail longitudinal creepages of the leading and fourth wheelsets is displayed in Fig. 7. It is seen that the wheel/rail longitudinal slipping of both the leading and fourth wheelsets aggrandize rapidly once the powered wheelsets enter into the low-adhesion region (including the wet and greasy conditions), and simultaneously, the anti-slip controllers are instantly triggered. The wheel/rail longitudinal creepages of both leading and fourth wheelsets under the low-adhesion conditions are maintained around the pre-set threshold values, which are significantly higher than those under the dry condition. Clearly, the wheel/rail longitudinal creepage with respect to a larger controlling threshold value is higher than that to a low one.
The wheel/rail adhesion-slip distributions (with the anti-slip controlling threshold of 2%) under different braking and adhesion conditions are shown in Fig. 8. It is found that the adhesion region locates at the front of wheel/rail contact patch under the dry condition. The area of the slipping region with braking effort is larger than that under the coasting situation. Moreover, the wheel/rail contact patches of both leading and fourth wheelsets are completely full-slip under both wet and greasy conditions due to the pronounced wheel/rail relative slipping.  The wheel/rail pressure distributions under the dry condition are displayed in Fig. 9. Furthermore, the wheel/rail pressure distributions under the wet and greasy conditions are demonstrated in Figs. 10 and 11, respectively. It is seen that the pressure amplitudes of the leading wheelset are slightly higher than those of the fourth wheelset under wet condition, but the differences existing in the pressures of these two wheelsets are minute under the greasy condition. To sum up, both the wheel/rail friction conditions and anti-slip controlling threshold values affect the wheel/rail pressures, at least the pressure amplitude, very slightly.
Wheel/rail tangential interactions contribute vastly to the frictional wear and fatigue damage of wheel and rail surfaces. Therefore, the wheel/rail shear stresses subjected to the complex wheel/rail friction statuses, coasting/braking operations, and controlling threshold values need to be followed with interest. The wheel/rail shear stress distributions under the dry condition are demonstrated in Fig. 12. The wheel/rail  The shear stresses located at the rear of the wheel/rail contact patch are larger than those of the front under dry contact condition, where the rear and front of wheel/rail contact patch correspond to the slipping and adhesion regions, respectively (Fig. 8). The shear stresses with braking effort are obviously higher than those under the coasting operation. In the low-adhesion conditions and with the implementation of braking effort, the shear stress distributions of front of wheel/rail contact patch are symmetrical to that of the rear one (without consideration of wheelset yaw angle) in view of the low-friction interactions and full-slip contact. The amplitudes of the shear stresses under the low-adhesion conditions are obviously lower than those under the dry condition, where the shear stresses under the greasy condition are lower than those under the wet condition due to the lower degree of frictional contact. Furthermore, the shear     www.Springer.com/journal/40544 | Friction stresses subjected to a higher controlling threshold are lower than those to a relatively lower one under the wet contact condition in light of the decreasing friction level with the increasing wheel/rail slipping.
All in all, the braking loadings and adhesion conditions have a minor impact on the wheel/rail normal pressures, whereas they affect wheel/rail tangential interactions to a great extent.

Wheel tread wear and fatigue damage evolution
The wear amount and fatigue damage of wheel tread subjected to braking loadings and anti-slip control threshold are firstly presented in Section 4.2.
The wheel tread wear amount distributions (namely the material loss obtained through Eq. (18) respectively. It is found that there is no wear located in front of the wheel/rail contact patch for the reason of the absence of relative slipping in the adhesion region under dry contact condition. The wheel wear amount with braking loadings is higher than that without the braking loadings, the reason for which could come to the larger wheel/rail relative slipping and shear stresses. The wear amount under low-adhesion conditions is evidently higher than that under the dry condition due to the conspicuously larger relative slipping, the distributions of which occupy the entire contact patch in view of the full-slip contact. On the other hand, the wear amount under the greasy condition is lower than that under wet condition due to the lower shear stresses. It is interesting to find that the wear amount with a higher anti-slip controlling threshold is much higher than that with a relatively lower one under the low-adhesion   The wear evolution of wheel tread subjected to the anti-slip controlling threshold is displayed in Fig. 18, and the comparison of final worn wheel profiles with different anti-slip control thresholds is shown in Fig. 19. In this simulation, fifteen wear iterations are executed to examine the wheel wear evolution subjected to the anti-slip control threshold. It can be seen that the depth and lateral region of wheel wear gradually expand with the increasing wear iteration (namely the traveling mileage). The wheel tread is inclined to suffer more serious wear with a larger transverse range and deeper wear depth with a higher anti-slip control threshold. Therefore, one can observe that a higher anti-slip control threshold can cause more pronounced wheel/rail tangential interactions (Figs. 13 and 14) under low-adhesion conditions and the resulting worse wheel wear. It is also seen that the propagation of frictional wear in the initial is faster than that in the late evolution, which may be due to the spread wheel/rail contact caused by the wheel wear and the related lower contact stresses.
The effect of frictional wear on fatigue damage subjected to the anti-slip controlling threshold with the new wheel profile is shown in Fig. 20. It is found that a higher controlling threshold can lead to a more serious surface material loss. The higher controlling threshold can also lead to more serious energy   dissipation and the resulting fatigue damage. The propagation of fatigue damage is more salient than that of the frictional wear. Therefore, the residual fatigue damage can be observed in spite of the frictional wear, where the existed fatigue damage with a higher controlling threshold is obviously severer than that with a relatively lower one.
The fatigue damage evolution of wheel tread subjected to the anti-slip controlling threshold is presented in Fig. 21, and the comparison of final fatigue damage with different anti-slip control thresholds is shown in Fig. 22. Also, it is observed that the depth and lateral region of wheel tread fatigue damage gradually aggrandize with the increasing running mileage. One can conclude that a higher anti-slip control threshold can lead to more pronounced wheel/rail relative slipping and energy dissipation levels in the contact patch with the low-adhesion wheel/rail contact and braking effort, and further, the resulting more execrable fatigue damage. On the other hand, it is found that the propagation of fatigue  damage is faster than that of the wheel wear in terms of the high energy dissipation, especially under the low-adhesion conditions. | https://mc03.manuscriptcentral.com/friction It is worth noting that the memory consumption of one iterative calculation approximately takes up to 122.2-124.4 MB. On the other hand, the wheel/rail non-Hertzian contact analysis consumes a large part of the computational time, especially when the multipoint wheel/rail contact caused by the worn wheel profile appear. The computational times related to different iterative simulation conditions are listed in Table 2.

Conclusions
In this paper, the authors present a numerical investigation into tread wear and fatigue damage evolution of railway wheels subjected to anti-slip control under the changing adhesion conditions, in which the vehicle-track interaction sub-module and according wheel wear and fatigue damage prediction sub-module are covered. A robust wheel/rail non-Hertzian model, namely the MKP-FaStrip model, is applied with an aim to accurately describe the complex wheel/rail frictional interactions and the resulting wheel tread wear and fatigue damage. The following conclusions can be obtained from this research. The low-adhesion wheel/rail conditions can lead to the full-slip contact patch, and furthermore, the shear stresses with respect to them are significantly lower than those under dry conditions with braking loadings due to the lower friction coefficients. The shear stresses with the greasy condition are even inferior compared to those with the wet condition; and under the wet condition, the shear stresses with a relatively higher anti-slip controlling threshold are marginally lower than those with a lower one. However, the adhesion conditions and anti-slip controlling threshold have little influence on wheel/rail pressures.
The wheel tread verges to suffer more serious frictional wear under the low-adhesion conditions in comparison to those under the dry contact condition with braking loadings, which is ascribed to the pronounced wheel/rail slipping. The wheel wear under the wet condition is more significant than that under the greasy condition due to the superior tangential interactions. Furthermore, a higher anti-slip controlling threshold can cause the severer wheel material loss under the low-adhesion conditions and a faster wear evolution. On the other hand, remnant fatigue damage can be observed on wheel tread with consideration of the frictional wear. Similarly, a higher anti-slip threshold can cause the faster fatigue damage evolution in comparison to that with a lower one. It is further suggested that the lower anti-slip controlling threshold is of benefit to relieve the wear and fatigue damage on the wheel tread.
The assessment of wear and fatigue damage of wheel and rail tread under complicated contact conditions is extremely complex, especially for the fatigue damage issues. An inappropriate use of the RCF evaluation method can even lead to the totally wrong conclusion. For instance, the shear stresses under the low-adhesion wheel/rail conditions are obviously lower than those under normal contact conditions (Figs. [12][13][14], and accordingly, the fatigue damage is lower under this case according to the shakedown theory. However, it is demonstrated that the fatigue damage of wheel and rail surface under the low-adhesion conditions is significantly severer than that under the normal condition, which is ascribed to the "oil wedge effect" with presence of the "third-bodies" between the wheel and rail interfaces even with a lower shear stress level [26,27,59]. On the other hand, whether the method proposed in Refs. [40,41] is capable of the complicated adhesion conditions should be further authenticated. Also, this research work should be verified by the field measurements, which are currently being executed by the authors.