Friction in wheel–rail contact: A model comprising interfacial fluids, surface roughness and temperature

Wear 271 (2011) 2–12

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Friction in wheel–rail contact: A model comprising interfacial fluids, surface roughness and temperature Christoph Tomberger a,∗ , Peter Dietmaier b , Walter Sextro c , Klaus Six d a

Kompetenzzentrum - Das virtuelle Fahrzeug Forschungsgesellschaft mbH, Inffeldgasse 21/A, 8010 Graz, Austria Technische Universität Graz, Rechbauerstraße 12, 8010 Graz, Austria Universität Paderborn, Fürstenallee 11, 33102 Paderborn, Germany d Siemens Transportation Systems GmbH & Co KG, Eggenberger Straße 31, 8020 Graz, Austria b c

a r t i c l e

i n f o

Article history: Received 30 August 2010 Accepted 3 October 2010 Available online 10 October 2010 Keywords: Wheel–rail contact Rolling contact Friction Interfacial fluid Lubrication Surface roughness Contact temperature

a b s t r a c t A profound description of friction in wheel–rail contact plays an essential role for optimization of traction control strategies, as input quantity for railway simulations in general and for the estimation of wear and rolling contact fatigue. A multitude of wheel–rail contact models exists, however, traction–creepage curves obtained from measurements show quantitative and qualitative deviations. There are several phenomena which influence the traction–creepage characteristics: Mechanisms resulting from surface roughness, frictional heating or the presence of interfacial fluids can have a dominating influence on friction. In this paper, a new wheel–rail contact model, accounting for these influential parameters, will be presented. The presented model accounts for the interaction of an interfacial fluid model for combined boundary and mixed lubrication of rough surfaces with a wheel–rail contact model that additionally accounts for frictional heating. A quantitative comparison with measurements found in the literature is not conducted, since the exact conditions of the measurements are mostly unknown and parameters can easily be adjusted to fit the measurements. Emphasis is placed on the qualitative behavior of the model with respect to the measurements and good agreement is found. The dependence of the maximum traction coefficient on rolling velocity, surface roughness and normal load is studied under dry and water lubricated conditions. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Modeling the tribological processes of the wheel–rail contact demands the application of models and methods of different scientific fields. Since the tribological process comprises mechanisms that are linked from atomic to macroscopic length scales, a complete description, including all mechanisms on a physical basis, is not possible today. Dependent on the field of application, appropriate simplifications and abstractions have to be made. The extent of simplifications for a wheel–rail contact model also depends strongly on the intended use of the model. ‘Classical’ contact models like CONTACT [1], deduced simplifications such as FASTSIM [2] or analytical derivatives of the model [3] are able to describe the principal mechanisms linked to the stick and slip zones within the contact and are therefore capable of describing the characteristic correlation between traction coefficient and creepage. The friction coefficient between wheel and rail

∗ Corresponding author. Tel.: +43 316 873 9081; fax: +43 316 325522 21. E-mail addresses: [email protected], [email protected] (C. Tomberger). 0043-1648/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2010.10.025

has to be provided by the user. But it is the friction coefficient that is highly dependent on parameters such as rolling velocity, normal load, microscopic surface roughness of wheel and rail, a.s.o. Furthermore, these parameter influences are strongly dependent on the lubrication state of the contact, whether the contact is humid, i.e. boundary lubricated, dry or in mixed lubrication state, where enough fluid is present between wheel and rail to build up a positive fluid pressure regime. Some models try to resolve these parameter influences by empirical reduction formulae for the friction coefficient [3–5] but neglect the influence of the lubrication state. Therefore, these models only work for one specific lubrication condition. The goal of the presented model is the description of the resulting friction coefficient between wheel and rail with an extended modeling approach and the incorporation of this concept into a wheel–rail contact framework. This is done in order to obtain a more general applicability to dry, boundary- and mixed lubrication conditions, resolving the discussed parameter influences such as rolling velocity or normal load to the rolling contact as well as the impact of the lubrication state to these parameter influences. The correct description of the resulting traction coefficient between wheel and rail is important for investigations related to wear, rolling contact fatigue, traction control or running stabil-

C. Tomberger et al. / Wear 271 (2011) 2–12

Nomenclature Am Am,z An Ax ,Ay d db f FN FT hx ,hy k kr lc lf lm lx ,ly ˙ m n N pf pm pm pm Ra Vm

vr,x vw,x w ˙ m     zs y  l m ϕ

b

metallic contact area metallic surface area from geometrical intersection nominal contact area cross-sectional areas separation of wheel and rail cell base height global traction coefficient related to the resulting, tangential contact forces global normal force global tangential friction force cell length and -width tangential failure stress reduction factor surface roughness parameter fluid cell height fluid height equivalent metallic volume height local contact length and width fluid mass flow number of contacting asperities total number of asperities fluid pressure metallic normal contact pressure nominal metallic normal contact pressure nominal normal contact pressure radius of spherical asperity tips metallic volume proportion rolling velocity, x-direction rigid wheel velocity, x-direction interference of asperity fluid mass flow source term flow resistance factor (Newtonian) fluid viscosity friction coefficient, related to the local, attainable frictional forces fluid density standard deviation of asperity heights bulk material yield stress nominal tangential contact stress nominal tangential adhesion limit tangential failure stress of metallic contacts probability density function of asperity heights probability of asperity occurrence probability of asperity occurrence at cell base height

ity analyses. Furthermore, information about local mechanical and thermal load distributions within the contact zone are desirable. A profound description of the state of the art of dynamical contact problems with friction can be found in [6]. The fundamentals of friction in rolling contacts are outlined in this book. Within the framework of the presented model, friction will be described by a maximum tangential shear stress, further called tangential failure stress, that the real area of metallic contact can transmit. The presence of interfacial fluids generally shows a friction-decreasing effect by reducing the tangential failure stress at boundary and mixed lubrication conditions. Opposed to the generally well known mechanisms at full hydrodynamic lubrication, where the interfacial fluid fully separates the two contacting bodies, the wheel–rail contact is generally limited to the boundary and mixed lubrication regime. Boundary lubrication addresses lubrication of the contacting, microscopic surface roughness peaks, the metallic asperities, without forming a continuous fluid pressure between the bodies. Mixed lubrication addresses the state where fluid pressure builds

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up between the bodies, causing an increased separation but with the metallic asperities still in contact. Opposed to the ideas of the traditional Stribeck curve, the presented model is also capable of describing boundary lubrication independent of the rolling and relative velocity of wheel and rail, taking the microscopic roughness of the surface into account. This can be of importance when only a limited amount of fluid is present on the rail at, e.g. humid conditions, where the space available between the roughness peaks is sufficient to accommodate the fluid. The friction-reducing effect of interfacial fluids is approximated in computationally fast point contact models [4,5]. These models are based on sound assumptions, however, the simple approach of a friction-reducing factor for the point contact may lead to deteriorating results for, e.g. low or very high rolling velocities. These models predict rising traction coefficients for rougher surfaces even for low lubricated situations at low rolling velocities. The reduction of the traction coefficient is not limited, so it seems that the traction coefficient can become zero or even negative at high rolling velocities. In these models, the influence of surface roughness is modeled for the lubricated contact only, so it may not be appropriate under dry contact conditions. Due to the point contact formulation, it is not possible to calculate local mechanical and thermal loads. They provide a good insight into the principal mechanisms and within their range of application, these models can provide computationally fast and sound results. In the present paper, the range of application will be extended to all rolling velocities, dry and lubricated contacts in the mixed- and boundary lubricated regime, further providing local thermal and mechanical loads within the contact zone. With emphasis on the lubrication only, sophisticated models for elasto-hydrodynamic lubrication (EHL) have been developed and extended to the application to rough surfaces. Precursors of these methods are Patir and Cheng [7], who introduced flow factors to averaged Reynolds equations for an application to rough surfaces. This method is applicable to the full and mixed lubrication regime, where the fluid fully occupies the area between the two contacting bodies, causing hydrodynamic lift. The method was applied to isothermal rolling contact and the adhesion coefficient was approximated [8]. A tangential contact model was not employed. A method to use this model even under boundary lubrication conditions is proposed in [9]. The interfacial fluid model that will be presented in this paper can be used in combined mixed and boundary lubricated contacts. A new approach is the consideration of the metallic volume occupation by the surface roughness peaks, called asperities. This reduces the amount of volume that can be occupied by the fluid as well as the cross-sectional area for the fluid flow. The dependence of the local flow resistance is therefore implicitly accounted for. The interfacial fluid model is based on the continuity equation, a pressure–mass flow relation and an abstraction of the surface geometry. The wheel–rail contact model, where the interfacial fluid model is integrated into, employs two length scales, a macroscopic and a microscopic scale. At the macroscopic scale, mean values of contact temperature, tangential contact stresses and local slip velocities are calculated. On the microscopic cell level, the real area of metallic contact is calculated with a common, single-scale statistical approach for rough surfaces [10]. The combination of a local resolving tangential contact model with an interfacial fluid model results in effects that were not describable until now.

2. Modeling 2.1. Modeling framework Sound models for the description of the principal rolling contact mechanisms do exist, e.g. [1,2]. The values for the friction

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Fig. 1. Overview of the model, input parameters and calculation modules and their interdependencies.

coefficients between the two contacting bodies, being crucial to the solution, have to be supplied to the model by the user and are assumed to be constant. This friction coefficient is however strongly dependent on the contact situation, contact load, lubrication, microscopic roughness of wheel and rail, rolling velocity, a.s.o. The presented model aims at calculating the friction coefficient between wheel and rail, allowing it to vary locally within the contact zone. The description of the nominal normal contact is accomplished with the theory of Hertz, the nominal tangential contact is calculated with independent, discrete contact stiffnesses, adapted from [2]. Normal and tangential loads as well as local slip velocities are then used to calculate local friction coefficients based on the idea of adhesion, where the local friction coefficients are proportional to the real area of metallic contact. This real area of metallic contact as well as the proportionality factor are dependent on local contact temperatures, microscopic roughness of the surfaces and the presence of an interfacial fluid. The calculated, local friction coefficients have a strong influence to the calculated tangential loads and slip velocities, from which they are computed from, Fig. 1. This results in an implicit set of equations. The calculation of the nominal contact area and nominal normal contact pressure distribution is not part of the wheel–rail contact framework. The nominal contact area determines the boundary of the contact zone. The nominal normal contact pressure distribution is the normal loads related to the nominal contact area. They have to be supplied to the model. This can be done by the use of, e.g. Finite-Element-Methods or the application of the theory of Hertz, if the contact conditions allow to do so. The contact area itself is divided into n·m elements of equal size, see Fig. 2. The computational domain is limited to the grid points that lie within the contact zone. The four major submodels of the wheel–rail contact framework, ‘contact temperature’, ‘microcontact’, ‘interfacial fluid’ and ‘tangential contact’ show interdependencies The contact temperature model computes the local temperature distribution, resulting in locally varying material parameters. The microcontact model computes the area of metallic contact according to the temperature dependent material parameters. The microcontact model interacts with the interfacial fluid model that provides a fluid pressure distribution, supporting the normal contact. At this point, local friction coefficients, the ratio of frictional force to normal contact force, can be defined: i,j =

mi,j Ami,j pni,j Ani,j

(1)

where mi,j is the temperature dependent, local tangential failure stress of the asperity contact, Ami,j the local metallic area of contact, pni,j the local, nominal normal contact load and Ani,j the local, nominal contact area. The tangential contact model then computes local tangential stresses with the restriction to the local friction coefficient i,j . Local slip velocities are calculated, providing the local heat sources for the contact temperature. This closes the loop for the iterative solution. The resulting global traction coefficient f is expressed as f =

FT FN

(2)

The definition of the tangential failure stress  m is however difficult. In addition to the mechanism of adhesion, plastic asperity deformation, plowing and other dissipative mechanisms, boundary lubrication and influences of solid interfacial layers have an effect on the tangential failure stress of a contact. Furthermore, so-called third bodies, forming a solid interface between the contacting bodies, can alter the friction process [11,12]. Due to the relatively difficult measurability of the third body interface, no comprehensive rheological model exists to the authors knowledge. Therefore, third bodies are not considered in the present model. Due to the manifold of mechanisms that can contribute to the tangential frictional force, the determination of the tangential failure stress is accompanied with a great degree of uncertainty. Therefore, a rather heuristic approach was chosen, as is common in the literature: The failure stress is assumed proportional to the bulk material yield stress  y m = y k

(3)

For dry contact conditions, a value of k = 1 lead to good agreement with measurements found in the literature. When a fluid is present, the mechanism of boundary lubrication reduces the shear stress limit of the contacting asperities [13,14]. This reduction of the shear stress limit is largely independent of the macroscopic properties of the fluid [15]. The usage of a reduction factor is therefore considered a straightforward approach. The proportionality to the bulk material yield stress is furthermore assumed to remain valid since the interface is assumed to vary rapidly due to microwear. Unlubricated local microcontacts are still likely to occur. 2.2. Microcontact model Based on the surface roughness, temperature dependent material properties, nominal normal contact pressure- and fluid pressure distribution, the metallic area of contact and resulting

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Fig. 2. Interfacial fluid-, microcontact-, temperature- and tangential contact-model are solved on a rectangular grid.

metallic normal contact pressures are computed with the microcontact model. It is based on a single-scale statistical method for the surface description initially proposed by Greenwood and Williamson [16]. This method was extended to combined elastic, elasto-plastic and plastic asperity deformation by Zhaoet al. [10]. The asperity tips are assumed to be spherical and of radius Ra . Both rough surfaces are reduced to an equivalent rough surface, contacting with a rigid flat. The separation d fully determines the contact, i.e. contact load, metallic contact area, etc. can be calculated, Fig. 3. The interferences w define if asperities deform elastically, elasto-

Fig. 3. Sketch of a microcontact.

plastically or plastically. In this paper, a base height db is introduced, where the cell height lc is measured from. This is relevant for the interfacial fluid model and will be discussed in Section 2.5. Elastically deforming asperities are calculated according to the theory of Hertz. For plastically deforming asperities, the geometrical intersection and the material hardness are used. To avoid a discontinuity of the contact load at the elastic–plastic transition, an elasto-plastic transitional regime is used. The ratio of plastically deforming asperities to the total number of contacting asperities will further be addressed as ‘plasticity of the contact’. It should be noted at this point, that the plasticity of a contact is mainly determined by the surface roughness. The normal load of a contact has only minor influence on the plasticity of the contact, since a decrease of separation with increasing load leads to new elastic contacts of formerly non-contacting asperities. This will be of relevance for the interpretation of the results. The statistical nature of the model arises from the description of the asperity heights. The heights of the asperities are assumed to follow a Gaussian distribution with probability density function ϕ(z) ϕ(z) =

2 1 √ e−1/2(z/zs ) zs 2

(4)

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Fig. 4. The assumed temperature variation of elastic modulus, hardness and yield stress.

For a given separation d of the contacting bodies, the probability of asperity occurrence at separation d, (z), equals the relative number of contacting asperities n

(d) = = N



∞

ϕ(z)dz

(5)

d

With the chosen formulation, 50% of the asperities are in contact at a separation value of zero. The absolute value of separation d is negative if more than 50% of the asperities are contacting, respectively positive if less than 50% of the asperities are contacting. The modeling of the rough surface will be of particular importance for the interfacial fluid model. In order to represent typical ‘rough’ and ‘smooth’ surfaces, the roughness-parameter kr is introduced, ranging from 0 to 1. A value of 0 represents very ‘smooth’ and a value of 1 ‘rough’ surfaces. ‘Smooth’ is linked to large asperity radii and small standard deviations of asperity heights and ‘rough’ vice versa. A linear variation of zs /Ra with kr is assumed. For the calculation of metallic contact pressure and metallic contact area, the elastic modulus and the asperity hardness have to be defined. The assumed temperature dependence of the elastic modulus and yield stress is gained from the bulk material behavior, the hardness is assumed to vary linearly with temperature, see Fig. 4. The high value of the asperity hardness is argumented by the fact that the contact is supposed to be work-hardened. Furthermore, the size of the asperities can be in the order of magnitude of single grains. Since the strain mechanism of grain boundary sliding is eliminated for single grains, a lower ductility and higher strength can be assumed. The local anomaly of the yield stress between approximately 300 and 600 ◦ C can be found for typical rail steels. This behavior is attributed to the interstitially dissolved C and N atoms that block the mobility of the dislocations. For temperatures above 250 ◦ C, the rate of diffusion of the C and N atoms is sufficiently high to slow down moving dislocations that would have already detached at lower temperatures [17]. Fig. 5 shows the dependency of the metallic contact area and the separation of the wheel and rail on the roughness parameter kr , at ambient temperature and constant load. As to be expected, the metallic contact area reduces for ‘rougher’ surfaces, since the number of plastically contacting asperities increases, the contact becomes more plastic. Plastically contacting asperities have reached the normal stress limit, the asperity hardness. For the same reason, the separation d also increases with increasing roughness parameter creasing roughness parameter. This increased cell height has an important influence on the roughness-dependency of lubricated contacts, as to be discussed in Section 3.

Fig. 5. Dependency of roughness parameter kr on the relative real area of metallic contact Am /An and relative contact separation d/zs for a constant nominal load.

2.3. Contact temperature model The calculation of the contact temperature is linked to the global, macroscopic level. Flash temperatures at the asperity contacts can be approximated by a method given in [18]. However, due to the assumed rapid change in the microcontact where asperity-tips, showing excessive flash temperatures, can loose and regain contact many times during the passage of the contact zone, the mean temperature distribution will be used. The computed temperature distribution changes the material parameters locally. The heat source distribution is gained from the frictional power, i.e. from local normal loads, local friction coefficients and local slip velocities. It is assumed, that wheel and rail can be treated as a half-space. Therefore, analytical solutions to the temperature distribution can be found for point-type heat sources [19]. The singular solution to a point source can be integrated over the discretised element i,j of the contact zone. With integration, a finite value for the element, where the heat source is located, can be calculated. Therefore, n · m temperature distributions for every discrete source term can be found on a n · m grid. The linearity of the heat equation allows the superposition of each solution to an overall solution. Since the heat flow partitioning into wheel and rail is not a-priori known, a system of n · m equations can be set up with the condition that the interface temperatures have to be equal and the sum of heat flow into wheel and rail has to equal the heat source. The advantage of this method is the possibility of describing arbitrary heat source distributions within the contact zone. With the price of higher computation times, the restrictions to, e.g. perfectly elliptical heat source distributions, sufficient high rolling speeds and the mandatory presence of heat sources, where analytical solutions can be found [4,20], are eliminated. 2.4. Tangential contact model In order to describe the global tangential contact and local stick and slip zones between wheel and rail, the elasticity of the contacting bodies has to be taken into account. The discretisation of the contact zone and division into independent, discrete contact stiffnesses is adapted from [6]. For metallic contacts, damping is not considered. In the present paper, the discrete contact stiffnesses are adapted from [2]. The modeling is based on the FASTSIM method, but it allows for varying friction coefficients within the contact zone. In the stick region, the deformations of the discrete stiffnesses are determined by the kinematics of the relative wheel and rail motion. The tangential contact stresses cannot supersede

C. Tomberger et al. / Wear 271 (2011) 2–12

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the tangential adhesion limit  li,j  li,j =

mi,j Ami,j Ani,j

= i,j pni,j

(6)

When the tangential contact stresses reach the tangential adhesion limit, the deformations are prescribed by this limit. Therefore, local offsets between wheel and rail are determined in the contact zone and local slip velocities can be calculated. The total frictional force can be gained by integration of the tangential contact stresses over the contact zone. 2.5. Interfacial fluid model The interfacial fluid model is based on the idea of a squeezing motion of the interfacial fluid through the rough surface structure of the solids. Due to the small timescales of the fluid passing the contact zone within one millisecond or below and the resulting, relatively low contact temperatures at lubricated contact conditions, the influence of the contact temperature on the fluid temperature and therefore viscosity is neglected. The contact zone is viewed as a set of discrete cells, partly occupied by the asperities of the contacting bodies, see Fig. 2. The dependency of the flow resistance on the cell height and roughness is therefore implicitly accounted for. In order to define a cell height as well as a qualitative measure for the metallic volume occupation of each cell, the information about statistical asperity occurrence from the microcontact will be used. In order to be able to define a fluid height, a cell base height has to be defined. Due to the statistical nature of the microcontact model, it is straightforward to define the base height db by a probability of asperity occurrence b , Fig. 3. With regard to Eq. (5) one can write √ (7) db = 2zs erf −1 (1 − 2 b ) The height of each cell is then defined by lci,j = di,j − db

Ani,j

(di,j )∼ (di,j )

(8)

(9)

Surface fraction curves can be deduced for typically ‘smooth’ and ‘rough’ surfaces, Fig. 7.

Fig. 6. Sketch of metallic surface area from geometrical intersection, Am,z .

With integration of the metallic surface, the metallic volume proportion can be calculated:



Vmi,j (d) =

di,j

db

Am,zi,j (z)dz

(10)

Since the metallic volume proportion is only a qualitative measure and the exact geometrical shape of the surface is not known, the metallic volume proportion is abstracted to a simple volume reduction, Fig. 8. Cell height lc and equivalent metallic volume height lm are therefore defined by √ lci,j = di,j − 2zs erf −1 (1 − 2 b ) lmi,j

Now, a qualitative value of metallic volume occupation of each cell is required. Metallic volume reduces the volume that can be occupied by the fluid. This metallic volume also reduces the crosssectional areas, increasing fluid flow resistance. The microcontact model does not contain a conception of the geometric shape below the spherical asperity tips. Therefore, a simple abstraction is proposed: The probability of locating an asperity at separation d or more is interpreted as metallic surface fraction, see Fig. 6. Am,zi,j

Fig. 7. Surface fraction Am,z /An curves for a ‘smooth’ and a ‘rough’ surface.

1 = √ zs 2



db +lc

db

i,j



∞



2

e−1/2(z /zs ) dz  dz 

(11)

z 

The fluid is assumed to be motionless on the rail. The rolling motion of the wheel is modeled as a set of variable cell volumes passing the motionless fluid, forcing local fluid expulsion due to decreasing cell volumes downstream. The rolling motion is limited to the x-direction. The governing equations are based on the principle of mass conservation and a pressure–mass flow relationship. It is assumed that the fluid velocity in one Cartesian direction is only dependent on the pressure gradient in the same direction. Inertia forces of the fluid are neglected. These assumptions can be verified for Newtonian fluids by reducing the Navier–Stokes equation by an order-of-magnitude analysis [15]. The pressure–mass flow relationship is deduced from a dimensional analysis. A flow-resistance factor  [Pa s] is introduced, accounting for the shear resistance

Fig. 8. Abstraction of the metallic volume proportion Vm to a simple geometrical reduction of cell volume and cross-sectional areas Ax and Ay .

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of the fluid. The flow resistance factor can be replaced by the equivalent viscous term of the Reynolds-equations to account for perfectly Newtonian fluids. The fluid does not necessarily occupy the complete cell, contacting both wheel and rail. Therefore, fluid drag motions, resulting from different wheel- and rail-velocities, are neglected. ∂pf

˙x= m

∂x

Ax 

lf2

(12)

lf2

∂pf ˙y= Ay  m ∂y

with



=

empirical non-Newtonian fluids 12 unpolluted, Newtonian fluid

(13)

Using the fundamental principle of mass conservation and applying finite differences, the governing equations are defined. ˙ ij(E) + m ˙ ij(S) − m ˙ ij(N) + m ˙ ij = 0 ˙ ij(w) = m m

(14)

where the cross-sectional areas Ax and Ay are defined as Axi,j = (lci,j − lmi,j )hy

(15)

Ayi,j = (lci,j − lmi,j )hx

An anisotropic roughness orientation like longitudinally grooves can be approximated by a weighting function for Ax and Ay . For the present study, isotropic surfaces are considered. The source term ˙ i,j arises from variable cell volumes within the contact zone. m Decreasing cell volumes downstream are forcing fluid expulsion. Increasing cell volumes in the outlet region allow the intake of fluid ˙ i,j is modeled by expelled from other regions. The source term m a mass flow balance of fluid flow rate to free cell volume flow rate from wheel and rail, using the rigid wheel and rail movements in the x-direction vr,x + vw,x ˙ i,j = lfi,j±1 hy vr,x i,j±1 − (lci,j − lmi,j )hy (16) i,j m 2   



Fluid flow rate into cell





Wheel and rail volume flow rate

Eqs. (11), (12) and (14)–(16) result in a set of linear equations that can be solved for fluid pressure pf . In the present study, a Gauss-Seidel relaxation procedure is employed. The resulting fluid pressure supports the normal contact, therefore the compatibility condition (17) has to be fulfilled, Fig. 9. This results in increased separations for regions of non-zero fluid pressure.



pfi,j

1−

Ami,j Ani,j



+ pmi,j

Ami,j Ani,j

= pni,j

Table 1 Coefficients and boundary conditions used for the calculations, unless otherwise stated. Nominal contact

UIC60/S1002, running surface

Fluid Nominal contact method Normal load Rolling velocity, vrx Surface roughness parameter, kr Flow resistance factor,  Tangential failure stress reduction factor, k Probability of asperity occurrence, b

Water Hertz 100 kN 30 m/s 0.5 0.012 Pas 0.4 0.9995

Within the presented model, the presence of an interfacial fluid effects the resulting friction in two ways: The achievable shear stresses at the asperity contacts are reduced and positive fluid pressure regimes increase the separation between wheel and rail, resulting in lowered friction coefficients. 3. Results and discussion The model developed in this paper is now used under wheel–rail rolling contact conditions. The contact conditions were chosen to be a UIC60/S1002 contact on the running surface. The parameters for all calculations, unless otherwise stated, can be seen in Table 1. In order to be consistent with the tangential contact model, the nominal normal contact pressure distribution is mapped to a parabolic distribution [2]. The first results show pure rolling under water lubricated conditions. In the inlet region, the decreasing cell heights in the downstream direction are establishing a fluid pressure regime in the front part of the contact. Due to the compatibility condition (17), the local contact separation increases, lowering the fluid pressure peaks and distributing the fluid pressure in the downstream direction, Figs. 10–12. Note that the fluid pressure can exceed the nominal contact pressure, depending on the ratio Am /An . The exact material behaviour in the valleys between the asperity peaks is not modeled, it is however only the projected surface area that is relevant for the fluid normal forces. Due to the pressure-induced flow, fluid is pressed into the regions where the available cell heights are increasing, filling these cells with fluid. At some position further downstream, the pressure induced flow is too weak to fill the cells, fluid pressure drops to zero. The parts of the contact zone experiencing zero fluid pressure are in the boundary lubrication regime. Therefore, the failure stress reduction is still valid, but the fluid does not reduce the metallic contact area by normal contact support. Under most contact con-

(17)

Fig. 9. Sketch of the microcontact. Nominal pressure pn acting on nominal contact area An , has to be supported by metallic contact pressure pm and fluid pressure pf .

Fig. 10. Available cell height lc –lm and fluid height lf at the center of the contact, y = 0. BL denotes the boundary lubrication region.

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Fig. 11. Computed fluid pressure pf for pure rolling conditions.

ditions, boundary lubrication is to be expected in the outlet region of the contact. This is unless the initial fluid height is lower than the initial cell height at the inlet, causing an additional boundary lubrication regime in the inlet part of the contact zone. In the mixed lubrication regime, the nominal metallic contact pressure decreases accordingly. The resulting, local distribution of metallic contact area and local friction coefficients can be seen in Fig. 13. It qualitatively follows the nominal metallic contact pressure distribution. The resulting local friction coefficients follow from (1), where mi,j = const for pure rolling conditions. The local variation of the friction modifies the tangential adhesion limit (6), Fig. 14. Local tangential stresses that are below the tangential adhesion limit are transmitted without slip. The curve for almost zero rolling velocity shows the well-known parabolic type. For the fast rolling case, a distinct decrease in the mixed lubrication regime can be seen. This is due to the decreased metallic contact area. Deduced traction–creep curves show significant deviations from dry traction–creep curves, Fig. 15. The sharp decrease of the traction coefficient under dry conditions for the higher rolling speed is caused by the weakening of the failure stress due to high contact temperatures. The local peak for the fast rolling case is due to the discussed anomaly of the yield stress and is often reported in field measurements. The water lubricated contact does not show this behavior, the contact temperatures are low. Measurements report this behavior for dry and water lubricated conditions [21].

Fig. 12. Computed, metallic contact pressure pm for pure rolling conditions.

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Fig. 13. Computed local variation of the relative metallic contact area Am /An and the resulting friction coefficient distribution  at the center of the contact, y = 0.

Fig. 14. Computed distribution of the tangential adhesion limit l at y = 0. Note that the indicated transition from boundary lubrication to mixed lubrication is valid for the vr,x = 30[m/s] curve only. For vr,x ∼ 0, the entire contact is in the boundary lubrication state.

Fig. 15. Computed traction–creepage curves for different rolling velocities under dry und water lubricated conditions.

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attributed to the increase in fluid pressure. The contact separation increases with rolling velocity, therefore the metallic contact area decreases. The fluid pressure also distributes more into the downstream direction, approaching a more symmetric distribution. As it can be seen, a rougher surface shows a declining velocity dependence on the maximum traction coefficient. This is because smooth surfaces reduce metallic contact more quickly with increasing separation, as can be seen in Fig. 7. The wider distribution of asperity heights of rougher surfaces reduces the loss of metallic contact with increasing separation. Under boundary lubricated conditions, almost no velocity dependence can be observed, because the mechanism of normal contact support of fluid pressure is eliminated and the second mechanism, the increase in contact temperature, is significantly reduced. The least velocity dependence is therefore expected to occur under boundary lubricated conditions. 3.2. Influence of surface roughness parameter Fig. 16. Computed dependence of maximum traction coefficient fmax on rolling velocity vrx for different roughness parameters under dry contact conditions.

3.1. Influence of rolling velocity The influence of the rolling velocity on the maximum traction coefficient is studied. Measurements with railway vehicles on track, or twin disc machines in the laboratory, report no distinct velocity dependence of the maximum traction coefficient under dry contact conditions [21–26]. The present model does show a slight decrease of the maximum traction coefficient for rising rolling velocities, Fig. 16. The decrease is however very small compared to the lubricated condition. The explanation is straightforward: Higher temperatures occur within the contact zone. This is due to the fact that the maximum traction coefficient usually occurs at about 1% relative longitudinal creepage for dry contact conditions. Higher rolling velocities therefore cause increased slip velocities within the contact zone. The roughness shows no qualitative influence on the dependence of maximum traction coefficient on rolling velocity. Measurements under water lubricated conditions report a distinct decrease of the maximum traction coefficient with increasing rolling velocities [21,22,24,26–28]. In [29], no velocity dependence is reported for a boundary lubricated contact, using paraffinic-oil. Fig. 17 shows the calculated influence of the rolling velocity under mixed and boundary lubricated conditions with water for different roughnesses. Under mixed lubrication conditions, the dependence of the maximum traction coefficient on the rolling velocity is

Fig. 17. Computed dependence of maximum traction coefficient fmax on rolling velocity vrx for different roughness parameters under water lubricated conditions. ‘BL’ indicates boundary lubricated conditions for the whole contact zone.

The influence of the surface roughness parameter on the maximum traction coefficient is studied. Measurements examining the influence of the surface roughness on the maximum traction coefficient on a twin-disc machine report no distinct tendency under dry contact conditions [24]. The present model predicts falling traction coefficient maxima for increasing surface roughnesses under dry contact conditions, Fig. 18. This is due to the fact, that the plasticity of the contact is mostly determined by the roughness of the contacting surfaces. The contact of ‘rougher’ surfaces therefore implies an increased proportion of plastically contacting asperities, rising the mean asperity normal contact stresses, reducing the metallic contact area. As it was to be expected, rolling velocity has no qualitative influence on the dependency. The difference to the measurement may be explained in two ways: In the measurement, the surface roughness was applied before the test by the usage of abrasive paper of different grain size. Especially under dry contact conditions, microwear could alter the surface during the test runs. Therefore the difference in surface roughness was probably not so significant during the test runs. A difference however existed, what is shown by sketches of the surface roughness profiles after the test runs. On the other hand, plastic dissipation by lateral shearing of the asperities can increase the total frictional force. An empirical equation for this dissipative friction component is proposed in [30]. This mechanism is not accounted for in the present model. However, using asperity interferences gained from the present model, the dissipative frictional forces gained from the empirical equations are many orders of magnitudes smaller than 1, not changing the trend.

Fig. 18. Computed dependence of maximum traction coefficient fmax on the roughness parameter kr for different velocities under dry contact conditions.

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Fig. 19. Computed dependence of maximum traction coefficient kr on the roughness parameter fmax for different velocities under water lubricated conditions.

Under lubricated conditions, the decreasing trend can reverse. Measurements with twin disc machines report rising traction coefficient maxima for rougher surfaces, if the contact is water lubricated [24,28]. Within this model, the increase in standard deviation of asperity heights with roughness, causes more asperities to come into contact. The resulting increase of separation reduces the flow resistance, lowering the fluid pressure. In this case, the effect of loosing normal contact support of the fluid pressure due to the higher separation outweighs the increased plasticity of the contact. This is of course only true up to a certain ‘operating point’, where the fluid is too easily expelled to support the normal contact sufficiently. For lower rolling velocities and thinner fluids, this ‘point’ of maximum traction coefficient is therefore reached at lower surface roughness parameter values, Fig. 19. 3.3. Influence of normal load The influence of normal load FN on the maximum traction coefficient is studied. Measurements found in the literature show a decrease of the maximum traction coefficient with rising normal load under dry contact conditions [21,23,24]. Fig. 20 shows the dependency of the maximum traction coefficient on the normal load for dry contact conditions. Rising contact temperatures due to the rising normal contact pressures are decreasing the failure stress, lowering the maximum traction coefficient. This is the main mechanism for dry contact conditions. The temperature induced

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Fig. 21. Computed dependence of maximum traction coefficient fmax on normal load FN under water lubricated conditions.

reduction of the maximum traction coefficient can clearly be seen. The qualitative difference for the different roughness parameters can be explained by the different temperature slopes of elastic modulus and hardness. For low roughness parameters, the contact is less plastic, with a stronger influence of the elastic modulus that decreases sharply with temperature, Fig. 4. Under water lubricated conditions, measurements show a decrease of the maximum traction coefficient with rising normal load [21]. Ref. [24] reports a dependency of this trend on the surface roughness under water lubricated conditions, using a twindisc machine. For smooth surfaces, increasing traction coefficient maxima are reported for rising normal loads, whereas decreasing traction coefficient maxima are reported for rougher surfaces. Fig. 21 shows the calculated dependency of the maximum traction coefficient on the normal load under water lubricated conditions. For the boundary lubrication condition, where the whole contact area was in the state of boundary lubrication, only a slight decrease can be observed. The mechanism ‘contact temperature’ is significantly reduced, but a slight increased plastification of the contact leads to lower ratios of metallic contact area to nominal contact pressure, reducing the traction coefficient. For the mixed lubrication condition, where the fluid pressure supports the normal contact, the decrease is more significant. The decrease of separation with increased normal load leads to a strong increase in fluid pressure, since cell heights and associated cross-sectional areas decrease drastically. This in turn leads to a non-linear relationship of nominal contact pressure to metallic contact area, decreasing the traction coefficient. The model predicts falling traction coefficient maxima for all ranges of surface roughness. The dependency of the trend on the surface roughness, as reported in [24], cannot be reproduced. 3.4. Limitations of the model

Fig. 20. Computed dependence of maximum traction coefficient fmax on normal load FN under dry conditions.

Under dry contact conditions, the resulting traction coefficient is largely dependent on the resulting contact temperatures. The contact temperatures are determined based on the local slip velocities that are computed with a derivation of FASTSIM. Due to the usage of discrete tangential contact stiffnesses errors are introduced, especially at the edge within the contact zone where local slip initiates. For the computation of the global traction coefficient, this error may not be too significant, since the mean temperature distribution within the contact zone is more relevant to the resulting traction coefficient than the exact distribution of the temperature. Furthermore, contact temperatures become significant above the saturation point, where the complete contact zone slips

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with an average slip velocity equal to the creepage between wheel and rail. However, for a detailed analysis of contact temperature phenomena within the contact zone, this limitation has to be considered. The surface topographical parameters that are set with the ‘roughness parameter’ remain to be calibrated by measurements. Heuristic assumptions have to be made at the microscopic level: The hardness of the microscopic asperities has to be assumed, since exact material parameters are not available in the microscopic range. The definition of the failure stress of the metallic interfaces also follows pragmatic assumptions due to the manifold of mechanisms that contribute to the local frictional forces. Due to the level of detail of the proposed method, it is not possible to perform real-time calculations with the complete set of equations. Some parameters of the model, such as b , cannot be measured directly and have to be determined indirectly. 4. Conclusion An innovative wheel–rail contact model is presented in the present paper. Based on the nominal normal contact, that has to be supplied to the model, it computes local mechanical and thermal stress distributions and resulting traction–creep curves under consideration of interfacial fluids, surface roughness and contact temperatures. The model shows good agreements with measurements from the literature regarding the qualitative behaviour of the traction coefficient between wheel and rail under dry and lubricated contact conditions. The holistic approach of the model is in particular beneficial, since the effects of rolling velocity, lubrication state, normal load, etc. to the resulting traction show distinct interdependencies. Qualitative comparisons with measurements have shown that the model is capable of reproducing these interdependencies correctly. The basic modeling framework of the model has proven to be reasonable. However, the sub-models used for microcontact, temperature, tangential contact and lubrication have to be reviewed critically in future in order to reduce necessary parameter inputs, increase the accuracy of the computed values and reduce computation times. At the current stage of development, the model can be used for, e.g. investigations relating the development of traction control strategies, wear or rolling contact fatigue. Future activities include the deduction of a faster model for usage with vehicle-track MBS simulations. Acknowledgements The authors wish to thank the “Kplus KompetenzzentrenProgramm” of the Austrian Federal Ministry for Transport, Innovation and Technology (BMVIT), Österreichische Forschungsförderungsgesellschaft mbH (FFG), Das Land Steiermark, and Steirische Wirtschaftsförderung (SFG) for their financial support. Additionally, we would like to thank the supporting companies and project partners Siemens Transportation Systems, Graz, Voestalpine Schienen GmbH, Leoben, VAE Gmbh, Zeltweg, and ÖBB-Infrastruktur Betrieb AG as well as Graz University of Technology.

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