1. Introduction

Urban metro transport has been broadly constructed and commissioned in most major cities all around the world in view of its salient advantages in convenience and environmental protection. However, a flow of damage problems occurring in the parts of metro vehicle and track gradually emerge due to the complex operation conditions as the in-service time grows. The deterioration of wheel and rail surface material, such as the fast-growing wear and cracks at wheel flange and gauge face of outer rail on a tight curve, is immediately related to the frequent and large wheel/rail frictional interactions [1,2]. This consequence can deteriorate wheel/rail contact (for example, facilitating a conformal contact [3]) and ride comfort and shorten the lifetime of wheel and rail. Commonly, rail grinding processes are adopted to restore rail profile, however, along with the depletion of financial resources and stoppage of transport [4]. Thus the wheel and rail surface degenerations are challenging problems from the viewpoint of economic benefits and should be carefully managed.

The field wheel flange tends to contact the gauge corner of the outer rail due to large lateral forces when the train travels on a tight curve, and the resulting wheel/rail contact is usually full-slip in this case. Thus the micro-slip of outer wheel/rail contact and frictional wear of rail gauge corner on the tight curve will be particularly severe. Recently, rail lubrication has come into the limelight since it was proved to be able to mitigate radiating squeal and rail surface wear in tight curves thanks to the low-adhesion wheel/rail interactions [5]. There are usually two kinds of lubrication system with different materials and features: one is the top-of-rail friction modifiers (abbreviated as FM, with a controlling friction coefficient range of 0.3-0.4) applied on the top of low rail; and the other is the lubrication grease (with an approximate friction coefficient of 0.1) applied on the rail gauge corner, as shown in Fig. 1. According to Ref. [6], financial costs of rail maintenance without lubrication effect is larger than that with the lubrication for most tight curves because the rail grinding will be more frequent. Gauge corner lubrication device.

To understand how the lubrication affect the wheel/rail contact and wear and fatigue damage of rails, a rolling-sliding testing apparatus combined with the observations of rail microstructures is often used. Lewis et al. [5,7] and Wang et al. [8] reported the different performances of wheel and rail wear and damage with the presence of water and grease lubricant. It was pointed out that the wear and RCF mechanism of wheel and rail materials are considerably different when the “third-body” exists in the wheel/rail interface. This result is fundamental and important for the assessment of the evolution of tread wear and damage based on train-track interaction dynamic simulations. Chen et al. [9] reported the feature of several types of lubricant and according to effects on traction/braking performances, spread extent on the rail surface and wheel/rail interactions. It was further suggested that the water-soluble lubricant (RTRI-WL) exhibits a good property to optimize wheel/rail interactions and braking efficiency.

In practice, the wheel/rail coupling relation is very complex in view of the track irregularities, contaminants adsorbed on the wheel and rail interface, traction loadings, external excitations, etc. Thus implementing a dynamic simulation to study the effect of lubrication on wheel/rail contact, rail surface wear, and damage is essential. Commonly, the prediction model of wear (Archard or USFD theory) and fatigue (Tγ function or shake-down theory) based on the multi-body dynamics (MBD) modeling is often used [10-12]. Arias-Cuevas et al. [13] investigated the curving behaviors with the presence of lubricant on the high rail by performing a co-simulation including vehicle dynamics and an improved lubricant model. More research emphasis and interests are focused on the FM. Alarcón et al. [14] mentioned that a befitting FM could result in the prominent reduction of energy dissipation and wear of wheel/rail contact. Furthermore, Khan et al. [15] pointed out that the conclusion that the FM with a lower friction coefficient is apt to have a favorable effect on the inhibition of rail RCF on a tight curve.

Most publications conducted numerical simulations upon the effect of FM on the wear and RCF of wheel and rail tread. In China, the RGCL devices (see Fig. 1) are generally installed along the outer metro rail on tight curves. The RGCL has quite different properties and acting mechanisms against the FM, whereas the RGCL affects the wheel/rail contact and rail surface RCF on the tight curve is still unclear. This paper intends to determine the effect of RGCL on the wheel/rail contact and rail surface RCF during curving negotiating based on numerical simulations. For this purpose, a fully nonlinear metro vehicle-slab track interaction model is extended, in which a precise wheel/rail non-Hertzian contact model with consideration of wheelset yaw motions is implemented. The rail surface RCF based on the non-Hertzian contact analysis with different lubrication conditions is given and discussed. Finally, the conclusions are drawn.

The rest of this paper is structured as follows. The numerical model, including the vehicle and track dynamics model, an enhanced wheel/rail non-Hertzian contact model and rail surface RCF prediction model, is introduced in Sect. 2. The effect of rail gauge corner lubrication (RGCL) on rail surface RCF curves is numerically studied in Sect. 3. Then, the wheel/rail non-Hertzian contact and rail surface RCF subjected to RGCL is discussed, as introduced in Sect. 4. Finally, the conclusions are drawn based on the experimental and numerical investigations in Sect. 5.

2. Prediction method for dynamic wheel/rail interaction considering RGCL

2.1 Overall architecture for the numerical model

The metro wheel/rail contact and rail surface RCF subjected to RGCL and traction effort on the curve is qualitatively assessed based on the dynamic simulations. Wheel/rail contact is an essential and prior part of rail surface RCF prediction. Therefore, the numerical model includes two main parts: the vehicle-track spatial coupled model and rail surface RCF prediction model, as shown in Fig. 2. Flow diagram of rail surface RCF prediction model.

The vehicle-track coupled model has been mentioned and applied in many references, wherein wheel/rail contact was universally performed by the Hertzian contact theory. However, wheel/rail contact actually emerges as a non-elliptical pattern, which deeply influences the prediction accuracy of rail surface RCF. Thus the wheel/rail contact is characterized by a robust non-Hertzian method in this study. On the other hand, the wheelset yaw motions are quite prominent and have a conspicuous effect on the wheel/rail non-Hertzian contact when the vehicle passes the tight curve. As far as the authors know, only a few references referred to the influence of yaw motions of the wheelset on wheel/rail non-Hertzian contact [12,16-18]. In this study, an improved wheel/rail non-Hertzian contact algorithm with consideration of wheelset yaw motions is implemented, which exerts a good trade-off between calculating efficiency and accuracy in solving wheel/rail contact.

The energy dissipation and shake-down theory are the two commonly used methods to assess the surface-initiated RCF of wheel and rail surfaces. Here, the energy dissipation principle is applied to predict rail surface RCF subjected to RGCL since the effect of wheel/rail frictional wear on the surface RCF is treated. The used contact stress distributions and wheel/rail creepage are delivered from the wheel/rail non-Hertzian contact analysis. Furthermore, the effect of wheel/rail wear on rail surface RCF is treated by introducing the turning points determined by wheel/rail creep and wheel and rail material strength characteristics.

2.2 Dynamic model for metro vehicle and floating slab track

A synthetical metro vehicle-slab track coupled model was performed in Refs. [19-21] is extended in this research work. The modeling for the metro vehicle, floating slab track, wheel/rail rolling contact and prediction of rail surface RCF will be depicted in this section. The parameters of the main interest are listed in Table 1. Key parameters for the dynamics model

The metro vehicle is regarded to be a multi-rigid-body system composed of the carbody, bogie frames and wheelsets, and each vehicle component involves six degrees of freedom (DOFs). The spring-damper elements considering nonlinear characteristics are applied to simulate the primary and secondary suspension elements. The equilibrium equation of the metro vehicle system can be expressed as:

where M_vt is the mass matrix of the vehicle system, K_vt(x_vt, ${\dot{x}}_{vt}$) and C_vt(x_vt, ${\dot{x}}_{vt}$) are the damping and stiffness matrices associated with the relative motions of the vehicle components. x_vt, ${\dot{x}}_{vt}$ and ${\ddot{x}}_{vt}$ denote the displacement, velocity and acceleration vectors of the vehicle system. F_wr stands for the wheel/rail contact forces acting on the wheelsets.

The floating slab track consists of a pair of rails, fasteners, concrete slabs, coil springs and a concrete base. In the track model, the fasteners and coil springs are simulated as linear viscoelastic elements. The two rails are modeled by continuous beam supported by discrete fasteners, while the floating slabs are simulated based on the Mindlin plate theory [19]. The deformations of the concrete base are neglected due to the little effect on wheel/rail interactions.

The dynamic equilibrium equation of the slab track system can be given as:

where M_tt is the mass matrix of the track system, K_tt(x_tt, ${\dot{x}}_{tt}$) and C_tt(x_tt, ${\dot{x}}_{tt}$) are the compositive damping and stiffness matrices associated with the relative motions of the track components. x_tt, ${\dot{x}}_{tt}$ and ${\ddot{x}}_{tt}$ denote the displacement, velocity and acceleration vectors of the track system. F_wr stands for the wheel/rail contact forces acting on the rail top, and F_cb is the interactional forces between the floating slab and concrete base.

2.3 Wheel/rail rolling contact model

The wheel/rail rolling contact modeling is the pivotal part that should be carefully treated in the evolution of wheel/rail interactions and rail surface damage. The Hertzian contact model has been broadly applied in the modeling for wheel/rail contact in train-track coupled dynamics in light of its high efficiency. However, the Hertzian contact model may exhibit a poor description for the wheel/rail contact stress distribution and contact patch in most cases [22], especially when metro vehicle passes through the tight curve with a large yaw angle. Hence the assessment of rail surface RCF based on wheel/rail contact analysis will be more precise by using an improved non-Hertzian contact model with consideration of spatial motions of wheelset and rail, which has been proposed recently [23-26]. In this research work, the normal and homologous tangential wheel/rail contact models with consideration of the apparent yaw motions of the wheelset on a curved line, namely the MKP-YAW [18] and FaStrip-YAW algorithm, are applied to calculate wheel/rail normal and tangential contact forces, respectively.

The solution for wheel/rail contact comprises three components: spatial contact geometrical relationship, normal interactions and tangential interactions. The calculation flow of wheel/rail contact stresses and forces is demonstrated in Fig. 3.Calculation process of wheel/rail contact forces.

2.3.1   Spatial contact geometrical relationship

The wheel/rail spatial contact relation should be cleared before the calculation of contact forces. The “trace line method” [19] was introduced to reveal the nonlinear wheel/rail contact geometric situation comprehensively. This method could take the spatial motions of the wheelset and rails into account. The wheel and rail profiles are the LM type and the CN-60 type commonly used in Chinese metro railways, respectively. The schematic diagram of the spatial wheel/rail contact “trace line” and the initial contact point considering this type of wheel/rail profile and the wheelset yaw angle is depicted in Fig. 4. Spatial trace line and initial contact point of wheel/rail contact under different yaw angle of wheelset: a Ψ = 0 deg and b Ψ = 2 deg.

In the absolute coordinate system (O_A-X_AY_AZ_A), the coordinates of the points attached to the trace line of the wheel profile can be written as:

with

where the superscript “(A)” stands for the absolute coordinate system. s_w represents the abscissa of the wheel profile in the wheel coordinate system (O_W-X_WY_WZ_W). Y_w, φ_w and ψ_w are the lateral displacement, roll angle and yaw angle of the wheelset. R_w and δ_w are the wheels rolling radius and contact angle corresponding to y_w.

With the right-side as an example, the coordinates of the rail profile can be written as:

where the superscript “(R)” stands for the rail coordinate system, denoted as O_R-X_RY_RZ_R here. φ_w is the roll angle of the rail. δ_cant is the rail cant angle. X_r, Y_r and Z_r are the longitudinal, lateral and vertical deformations of the rail, respectively. y_irr and z_irr are the lateral and vertical irregularities of the track, respectively.

The main axis of the normal distance [18,19] in the wheel/rail contact patch coordinate system (O_C-X_CY_CZ_C) can be conveniently calculated as:

where the superscript “(C)” stands for the coordinate system O_C-X_CY_CZ_C, y_c is the initial contact point determined by searching the minimum vertical distance between the undeformed wheel and rail profile. The wheel/rail normal distance can be further expressed as:

2.3.2   Normal contact model

In the MKP-YAW contact model, some assumptions should be first implemented to simplify the calculation for non-Hertzian wheel/rail contact considering the yaw angle of the wheelset [18]. The schematic diagram of wheel/rail non-Hertzian contact is shown in Fig. 5. Wheel/rail normal contact considering the yaw angle [18]: a pressure distribution and b contact patch.

The derivation on the leading and rear edges of the contact patch can be given as:

This formula can be further expressed as:

where x_l(y) and x_r(y) are the leading and rear edges of the contact patch, respectively. C_p is the wheel/rail contact region. a(y) is the longitudinal distance between the leading or rear edges of the contact patch and the main axis of the normal distance. The evaluation for the contact patch can be referred to Refs. [18], and it is omitted hereon.

The normal contact pressure based on the MKP-YAW contact model can be derived as:

with

where δ₀ is the interpenetration between wheel and rail tread. y_l and y_r denote the left and right edges of wheel/rail contact patch, respectively.

Thus, the wheel/rail normal contact force can be then calculated as:

2.3.3   Tangential contact model

The wheel flange of the outer side will potentially be in contact with the rail gauge corner when the vehicle travels on the curved section with a small curve radius, and the outer in-contact surface of the wheel and rail will be under the lubricated condition with the RGCL that smeared on the high-rail side. The wheel/rail friction coefficient decreases significantly under lubricated contact conditions compared with dry contact conditions. The gauge corner of the outer rail located on the circular curve and two transition curves is supposed to be installed with the lubrication device. The changing friction coefficient of wheel/rail contact under different friction conditions is considered. The maximum friction coefficient along the transverse direction of the rail profile that considers the lubrication condition is shown in Fig. 6 [27].Friction coefficient along the transverse direction of rail profile [27].

Kalker’s Fastsim has been broadly applied to solve wheel/rail tangential problems covering the shear stress and stick-slip distributions as well as tread wear evolution. Recently, Ye and Sichani et al. [12,28] proposed a novel model for the wheel/rail tangential solutions based on the Hertzian and non-Hertzian contact theory, namely the FaStrip method, which was proved to reveal a better accuracy in the estimation of shear stress distribution against Fastsim. In this paper, the modified FaStrip based on the MKP-YAW contact model considering the yaw angle of the wheelset, namely the FaStrip-YAW model, is developed to solve the tangential contact problem.

The wheel/rail contact patch is divided into adhesion and slip regions in the FaStrip-YAW algorithm, and the wheel/rail shear stress distribution is calculated within these two regions, respectively. Based on the MKP-YAW normal contact model, the demarcation for wheel/rail adhesion and slip regions can be written as:

with

In the adhesion area, the longitudinal and lateral shear stress distribution can be formulated as:

The expression for symbols ξ(y), η(y) and ψ(y) as well as ĸ, ĸʹ, λ and λʹ appeared in the above equations can be referred to Refs. [12,28], and omitted here.

An analogous algorithm as Fastsim is introduced to deal with the stress distributions within the slip region according to the FaStrip-YAW method. The shear stresses in the slip region are first calculated as:

where s_x, s_y and s_ϕ denote the longitudinal, lateral and spin creepage, respectively. Superscript n is the number of the stripes in the x-direction starting from the leading edge of the contact patch to the trailing edge. For non-elliptic algorithm, the flexibility parameters L_x, L_y and L_ϕ employed for this part are defined as:

with

where G is the equivalent shear modulus. S_Cp denotes the area of contact patch. R_c is the modified radius, the solution of which can be referred to Ref. [12,28]. z_wr(x,y) is the corrected separation with the yaw angle of wheelset taken into account.

The total shear stress of each stripe is then compared against the traction bound, which is given as:

And when ${\tilde{q}}_{\left(x,y\right)}^{\left(n\right)}\ge g_f\left(x,y\right)$, the shear stress should be corrected as:

The shear stress distribution in the slip region should be finally modified as:

In the above formulas, μ is the variable wheel/rail friction coefficient considering the relative slipping speed and surface conditions.

In summary, the stress distributions in the adhesion and slip region can be obtained through Eqs. (16) and (22), respectively. The longitudinal and lateral creep forces as well as the spin creep moment can be further evaluated as:

2.4 RCF prediction model

The damage function (also abbreviated as the Tγ theory) and shake-down map based on Hertzian contact analysis are the two commonly used methods to assess the surface-initiated RCF of wheel and rail interface [11,15,29,30]. The Tγ theory is on the strength of the energy dissipation principle that takes the wheel/rail creep force and creepage into account, in which the effect of wheel/rail frictional wear is also treated. For the shake-down diagram, one can assume that the wheel and rail surface will have a risk to suffer RCF problems due to the plastic deformation of surface material once the wheel/rail contact shear stresses outstrip the material yield stress (namely the so-called “shake-down limit”) of the wheel and rail. A dominating shortcoming of the shake-down diagram is that the wheel/rail creepage and effect of wear subjected to frictional relative sliding of wheel/rail contact is not treated in this method, which has a conclusive influence on wheel and rail surface fatigue damage. In terms of this, an improved RCF prediction model based on the non-Hertzian contact is used to qualitatively evaluate the rail surface RCF as the vehicle passes through the curves in this study.

The energy dissipation (wear index) distribution within the contact patch is derived locally for each element in the non-Hertzian contact patch:

where s_x, s_y and s_ϕ denote the longitudinal, lateral and spin creepage, respectively.

The fatigue index (FI) is employed to assess rail surface RCF in this study. The distributions of fatigue index with consideration of the impact of wheel/rail frictional wear can be expressed as:

where the symbol E_k is the turning point of the energy dissipation related to the material property of the wheel and rail and corresponding creep level, and the first turning point E₁ and the second turning point E₂ equal to 1.98 MPa and 6.61 MPa, respectively. F_EI[WI(x,y), E_k] is the correction factor that derived from the raw energy dissipation distributions and turning points [29,30]. The wheel/rail wear starts to dominate as the energy dissipation WI(x,y) is more than E₁, and the surface fatigue damage will be completely substituted by wear when WI(x,y) exceeds E₂ [29,30].

The total fatigue indexes inside the contact patch can be expressed as:

3.4 Model validation

With the purpose of verifying the accuracy of the metro vehicle-track coupled dynamics model, the field-tested data, including the running velocity and vertical acceleration of the axlebox is used to compare against the simulation results, as shown in Fig. 7. The constitution of the metro vehicle of the simulated scenario is set to be identical to the practical condition. The metro vehicle travels on a straight line with the initial velocity of 72.6 km/h with applied braking loadings. The irregularities of the wheel and rail surface are slight, which is ignored in the simulation. The effect of American Federal Railroad Administration Class 5 track irregularities is considered in the comparison. Comparison results include the running velocity and vertical vibrations of the axlebox.

The comparison of simulation results and the on-track tested data under braking conditions are displayed as follows. It is seen that both the simulated vertical vibrations of the axlebox and traveling speed are basically close to the measured ones in the braking process. But the tested vertical vibrations of the axle-box are slightly greater than that of the simulated ones. The difference should be due to the irregularities that existed on the railhead surface (such as rail corrugation), which are not treated in the simulations. Overall, the errors between the measured results and the calculated ones are acceptable, and the proposed dynamics model is regarded to be reliable and reasonable to investigate the wheel/rail interactions.

3. Simulation results and discussions

The effect of the RGCL on wheel/rail interaction behaviors and rail surface RCF is determined When the metro vehicle passes through a tight curve with the radius of 400 m. According to the practical condition, the vehicle speed is set to be 60 km/h and the superelevation of the curve is set as 105 mm. The traction torque was set as 4200 N m under the traction condition. Furthermore, to keep the vehicle operating at a constant speed, the traveling resistance of the vehicle is assumed to be equal to the total longitudinal hauling loadings under the traction condition and to be 0 under the coasting condition. The wheel and rail profiles are supposed to be unworn, and the track irregularity is not considered in the simulation. The simulation analysis related to the leading wheelset is mainly emphasized and presented hereon.

3.1 Wheel/rail contact and rail surface RCF on a tight curve

3.1.1   Wheel/rail contact

Both the spatial wheel/rail contact locations with different contact curvature and angles and the RGCL contribute deeply to wheel/rail creep behaviors, contact stresses as well as rail surface RCF. The spatial wheel/rail contact points located on the outer and inner rail profiles are firstly presented here, as shown in Fig. 8. It is seen that the contact points of the outer and inner side are locating on the rail top when the vehicle runs on the straight segment. The outer and inner wheel/rail contact points gradually move to the rail gauge corner and rail top respectively as the vehicle enters into the transition curve, and the outer wheel/rail contact points locate at the RGCL region when the vehicle travels on the curve segment with the radius of 400 m. Thus the outer wheel/rail contact is exposed to the RGCL under the lubricated condition. Outer and inner wheel/rail contact points on rail profile under the a coasting condition without RGCL, b coasting condition with RGCL, c traction condition without RGCL and d traction condition with RGCL. .

The effect of the RGCL on wheel/rail creep behaviors of the leading wheelset in the coasting and traction process, including the longitudinal creepage and creep forces, is demonstrated in Figs. 9 and 10, respectively. It is seen that the wheel/rail longitudinal creep behaviors of both outer and inner contact are greatly influenced by the RGCL under coasting and traction conditions. The wheel flange of the outer wheel keeps in contact with the gauge corner during the curving negotiating (on the circular curve and two transition curves), and the friction coefficient of the outer wheel/rail contact reduces with the presence of the RGCL in comparison to that without the RGCL. Thus the longitudinal creepage of the outer contact under the lubricated contact condition is higher than that under the non-lubricated contact condition in both coasting and traction conditions. On the other hand, the longitudinal creepage of the outer contact under the traction condition is higher than that under the coasting condition. Dynamic responses of wheel/rail longitudinal creepage under the a coasting condition and b traction condition.Dynamic responses of wheel/rail longitudinal creep force under the a coasting condition and b traction condition.

The wheel/rail longitudinal creep forces of both inner and outer wheel/rail contact reduce due to the falloff of adhesion coefficient with the RGCL. Moreover, the difference of the inner and outer longitudinal creep forces reduces under the traction condition compared with that under the coasting condition. The difference almost equals 0 with the RGCL under the traction condition.

Graphed in Figs. 11-13 are the distributions of contact pressure, shear stresses and direction of the shear stress acting on the rail surface of the outer and inner wheel/rail contact under different conditions, respectively. The focused location is the circular curve (The running distance is 150 m). It is observed that the contact patch of the outer wheel/rail contact is relatively narrow when the wheel flange is in contact with the rail gauge corner on the circular curve. There is little difference existing in the contact patch geometry and amplitude of the contact pressure of both inner and outer wheel/rail contact. Pressure distributions of the outer and inner wheel/rail contact under the a coasting condition without RGCL, b coasting condition with RGCL, c traction condition without RGCL and d traction condition with RGCL.Direction distributions of the shear stresses acting on rail surface of the outer and inner wheel/rail contact under the a coasting condition without RGCL, b coasting condition with RGCL, c traction condition without RGCL and d traction condition with RGCL.

The distribution characteristics of the shear stresses of both inner and outer wheel/rail contact are similar to that of the normal pressures thus the wheel/rail contact is basically full-slip on the tight curve. It is interesting to find that amplitudes of the shear stresses of the outer wheel/rail contact under the lubricated condition are much lower than that under the non-lubricated condition due to the reduction of the friction level. The shear stresses of the outer wheel/rail contact under the traction condition are slightly higher than that under the coasting condition. On the other hand, there is a slight difference lying in the shear stresses of the inner wheel/rail contact with different lubrication conditions. The shear stress directions of the outer wheel/rail contact are mostly in the fourth quadrant (see Fig. 13), and they are not affected by the RGCL. However, the shear stress directions of the inner wheel/rail contact can be altered by the RGCL and traction loadings. Therefore, the RGCL and traction loadings have less influence on the normal pressures, whereas they have a significant effect on the shear stresses, in which the RGCL plays the dominating role.

3.1.2   Rail surface RCF

The effect of RGCL on the global fatigue indexes of the leading wheelset under the coasting and traction conditions is demonstrated in Fig. 14. The distributions of the wear index and fatigue index of outer and inner wheel/rail contact on the circular curve (with the distance of 150 m) are shown in Figs. 15 and 16, respectively. In this research work, the rail surface RCF related to the first wheelset load is focused. Dynamic responses of global fatigue indexes (T_FI) under the a coasting condition and b traction condition.Energy dissipation distributions of the outer and inner wheel/rail contact under the a coasting condition without RGCL, b coasting condition with RGCL, c traction condition without RGCL and d traction condition with RGCL.Fatigue index distributions of the outer and inner wheel/rail contact under the a coasting condition without RGCL, b coasting condition with RGCL, c traction condition without RGCL and d traction condition with RGCL.

It is seen that the T_FI values are very low on the straight segment. The T_FI values increasingly rise when the vehicle travels on the transition curve with the increase of curve curvature. When the vehicle travels on the circular curve, the fatigue levels of the outer side are more serious than that of the inner side without lubrication. The T_FI values of the inner side are slightly higher with RGCL than those without RGCL under traction conditions. For the outer contact, however, the energy dissipation and fatigue indexes are prominently reduced with the presence of the RGCL compared with that without lubrication under both coasting and traction condition. Therefore, both frictional wear and RCF of the outer rail surface can be significantly relieved with the RGCL on the circular curve. Additionally, the T_FI values of the outer contact under traction conditions are higher than that of inner contact for the sake of larger contact stresses (see Fig. 14). The fatigue index of the outer side is reduced compared with the raw energy indexes when the vehicle travels on the tight curve with the lubrication effect, but it is unaltered under the lubricated conditions or with the loose curve. Thus the rail surface cracks maybe take place by the prominent frictional wear under the non-lubricated conditions as the vehicle travels on the tight curve. It is further suggested that the RGCL can effectively reduce both the wear and the surface damage of the outer rail.

3.2 Sensitivity analysis on rail surface RCF

The effect of the RGCL on rail surface RCF of the outer and inner side on the circle curve (with the running distance of 150 m) with different curve radius is displayed in Fig. 17. The related key parameters of the curves and according to passing velocities are listed in Table 2. The traction torque is set as 4200 N m under the traction condition. It is seen that the T_FI values present a general downward trend with the increase of curve radius under both coasting and traction conditions. Thus the rail surface RCF is much more serious on a tight curve than that on a loose curve and straight segment. The T_FI values of the outer rail are higher than that of the inner rail under the non-lubricated condition in terms of the higher contact stresses, especially with the traction loadings. The RGCL has a significant effect on the rail surface RCF of the outer side. The T_FI values of the outer rail surface are greatly mitigated with the presence of RGCL, especially with a smaller curve radius and without traction effort. On the other hand, the fatigue damage of the inner rail surface is less affected by the RGCL. Moreover, the RGCL does not play any role in reducing rail surface RCF of both outer and inner rail when the vehicle travels on a loose curve (with a large curve radius ≥ 1000 m) under the coasting condition, because the outer wheel flange is not in-contact with the related gauge corner with small transition displacement is this case. Effect of RGCL on rail surface RCF with different curve radius: a coasting condition and b traction condition. Key parameters for the curves and curve negotiation velocities

The effect of the RGCL on rail surface RCF of the outer and inner side on the circle curve with different traction loadings is shown in Fig. 18. Hereon, the curve radius is set as 400 m, and the superelevation of the curve is 105 mm. The constant speed is set to be 60 km/h. It is seen that the T_FI values of the outer rail increase as the traction loadings increase without RGCL, and also, the growth trend can be found for both outer and inner side with RGCL. Thus the surface RCF of the outer rail will have a risk to deteriorate with a larger traction effort. Furthermore, the T_FI values of the inner rail decrease with an increase in traction loading and the resulting declining contact stresses on the curve. Effect of RGCL on rail surface RCF with different traction loadings.

According to Spiryagin et al. [27], friction coefficient of the lubrication applied on gauge face has the approximate range of 0.05-0.2. Here, the effect of friction coefficient of RGCL on the rail surface RCF of the outer and inner side on the circle curve (with the running distance of 150 m) under coasting and traction conditions is shown in Fig. 19. The curve radius and superelevation of the curve is set as 400 m and 105 mm, respectively. The traction torque is set as 4200 N m under the traction condition. It is seen that the higher friction coefficient of RGCL is apt to aggravate surface RCF of the outer rail, but it has less effect on the surface RCF of the inner rail. That means the RGCL with lower friction coefficient can ameliorate the surface RCF of outer rail. However, one can expect that the RGCL with lower friction coefficient can worsen the steering capacity of the leading wheelset when passing through a tight curve. Effect of RGCL on rail surface RCF with different friction coefficient: a coasting condition and b traction condition.

4. Discussion

4.1 Wheel/rail contact

When the vehicle travels on a tight curve, the adhesion levels of the outer wheel/rail contact are greatly reduced with the presence of RGCL compared with the dry contact condition, and subsequently, the outer longitudinal creep force decreases and the inner longitudinal creep force increases (alter towards the forward direction). Thus the steering torque formed by the difference between inner and outer longitudinal creep force under the lubricated condition is much lower than that under the non-lubricated condition, especially with the traction loadings. The steering capacity will mainly depend on the leaning force between wheel flange and rail gauge face (namely the transverse component of the normal contact force) in this way.

Not only the lateral displacement but also the angle of attack of the wheelset contributes greatly to wheel/rail non-Hertzian contact (see Figs. 11-13), especially when the vehicle travels on a tight curve. Thus to better predict wheel/rail contact stresses as well as the surface RCF, it is crucial to select a robust wheel/rail non-Hertzian contact model with consideration of wheelset yaw motions than that to select a Hertzian contact model, even though it is more time-consuming.

4.2 Rail surface RCF

The rail surface RCF is evaluated on the strength of Tγ theory in this paper, and it is suggested that the outer rail surface RCF can be greatly reduced with the presence of RGCL, especially with a smaller curve radius and without traction effort. The damage index (DI) based on the shake-down diagram and wheel/rail non-Hertzian contact can be given as:

where k_Rail is the yield stress limit of the rail. It can also be drawn that the outer rail surface RCF can be reduced with RGCL in light of the lower tangential stresses on the circle segment (see Fig. 12). Shear stress distributions of the outer and inner wheel/rail contact under the a coasting condition without RGCL, b coasting condition with RGCL, c traction condition without RGCL and d traction condition with RGCL.

Field observations of rail surface were carried out to conduct an experimental investigation into the effect of RGCL on the rail surface RCF. The in-service outer and inner rail profiles belonging to the same circle segment were observed on a metro line. These rail profiles have been re-profiled with the same interval since the last re-profiling operation. Schematic diagram of the curve conditions, lubricated section and observed locations is shown in Fig. 20. Schematic diagram of curve conditions and observed locations.

The effect of the RGCL on rail surface RCF is determined by comparing the surface status of both outer and inner rails under different lubricated conditions, as shown in Fig. 21a, b. It is seen that the primary wheel/rail contact regions of the outer and inner side locate on the gauge corner and rail top on the transition and curve section, respectively. Obvious RCF damage (manifested as continuous oblique cracks) can be detected near the gauge corner of the outer rail on the curve segment (see Fig. 21a) due to the drastic wheel/rail tangential interactions under the non-lubricated state, and the length of the rail surface inclined cracks is approximately 20-30 mm. The surface damage level of the lubricated rail (outer rail) is much slighter compared with that without the lubrication. On the other hand, little fatigue damage can be observed on the inner rail on the curve segment without the RGCL, but minor surface damage can be discovered on the inner rail top in the curve section with the lubrication existing on the gauge corner of outer rail (see Fig. 21b). This means that the RGCL can effectually mitigate the surface RCF of the outer rail on the tight curve segment. Comparison of rail surface RCF with different lubrication conditions: a outer rail and b inner rail.

5. Conclusions

This paper firstly performed a metro vehicle-slab track interactional model with consideration of the wheel/rail non-Hertzian contact characteristics and a RCF prediction model for rail surface associated with wheel/rail non-Hertzian contact analysis. Then, the influence of RGCL on wheel/rail creep behavior, contact stresses and fatigue damage of rail surface was assessed. The following conclusions can be obtained from this study.

(1) The wheel flange and rail gauge corner are in contact during vehicle curving negotiating, and the friction coefficient of the outer wheel/rail contact drops with the presence of RGCL. The wheel/rail longitudinal creep forces decrease due to the fall off of friction level on tight curves, which reduces the steering capacity of the leading wheelset on a tight curve

(2) The wheel/rail contact patches of the outer rail side are basically full-slip under both lubricated and non-lubricated conditions on curved tracks. The shear stresses of the outer rail side under the lubricated condition are lower than that under the non-lubricated condition. But the dynamic performance of the inner wheel/rail contact is less affected by the RGCL.

(3) The fatigue damage of the outer wheel/rail contact of the metro vehicle is more serious than that of the inner one without RGCL, and it is significantly reduced with the presence of the RGCL compared with that without lubrication under the coasting and traction conditions, especially for the cases with a smaller curve radius. On the other hand, the frictional wear of the outer rail can be effectively reduced by the RGCL.

(4) The RGCL with a lower friction coefficient can reduce contact stresses and ameliorate the surface RCF of outer rail through low-adhesion wheel/rail interactions. But one can expect that the steering capacity of the leading wheelset will be reduced with the presence of the low-friction RGCL on a tight curve.

(5) The rail surface fatigue life is vastly affected by the surface damage. More extensive field and numerical investigations should be conducted to determine an effective way to alleviate this issue, which is still an on-going task.