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rail contact
Wear 253 (2002) 498–508
A comparison of analytical and numerical methods for the calculation of temperatures in wheel/rail contact夽 Martin Ertz∗ , Klaus Knothe Technische Universität Berlin, Sekr. F5, Marchstr. 12, D-10587 Berlin, Germany Received 3 January 2002; received in revised form 11 April 2002; accepted 1 May 2002
Abstract The maximum surface temperature during rolling contact of railway wheels with sliding friction can be estimated using Blok’s flash temperature formula. For a more detailed investigation, semi-analytical and numerical methods are available. A survey of various methods is given and an efficient approach is proposed for Hertzian contact. The actual contact temperature is confined to a very thin surface layer. Due to continuous frictional heating, the bulk temperature of the wheel increases with time. For the long-term behaviour of the wheel temperature, not only the convection at the free wheel surfaces but also the heat conduction from the wheel into the colder rail has to be considered. Practical consequences of the theoretical results are discussed. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Wheel/rail contact; Flash temperature; Contact temperature; Convection
1. Introduction The slip between wheel and rail causes frictional heating of both bodies. The resulting high contact temperatures have to be taken into account for microstructural alterations, such as formation of white-etching layers (WEL) on rail surfaces and also for failure of wheel and rail. With a possible relation between contact temperature and coefficient of friction, they can also be responsible for decreasing creep force curves. Since contact temperatures under real railway operating conditions can hardly be measured, temperature calculation is an important part in the investigation of rolling contact. Fundamental work in this area has been done by Blok [1] and Jaeger [2]. In his article The flash temperature concept published in 1963, Blok gave a survey of the state-of-the-art at that time [3]. The methods used are based on the theory of heat conduction with moving heat sources as described by Carslaw and Jaeger [4]. Archard [5] used the theory of Blok and Jaeger to investigate some tribological problems. An approximate solution for rolling contact was given by Tanvir [6]. Knothe and Liebelt [7] presented a numerical method for arbitrarily distributed heat sources. 夽 Dedicated to the memory of Prof. Harmen Blok (1910–2000), the pioneer of flash temperature calculation. ∗ Corresponding author. Tel.: +49-30-314-22142; fax: +49-30-314-22866. E-mail address: [email protected] (M. Ertz).
While railway wheels are heated by friction in the contact patch, there is also heat loss due to conduction through the contact patch into the rail. This effect has rarely been taken into account in previous research. It was mentioned by Moyar and Stone [8], and Gupta et al. [9] used a simple approach for the approximate consideration. In addition, heat flows from the wheel into ambient air due to convection at the free surfaces. Cameron et al. [10] calculated a steady-state temperature based on the equilibrium of frictional heating and heat loss due to convection. Fischer et al. [11] found an analytical solution for the surface temperature with combined frictional heating and convection. All these works are confined to smooth surfaces and most of these are based on the theory of Hertz for the mechanical contact problem. But real surfaces are always rough. Using measured profiles of rough surfaces, even the calculation of stress and strain demands computer simulations [12]. With smooth surfaces, the heat conduction can be treated one-dimensionally, but for rough surfaces, this simplification is not possible, and the investigation of temperatures is also a very complicated numerical problem [13]. On the other hand, Archard [5] has already stated that the influence of surface roughness can be neglected to a first approximation, since the largest temperatures are those deduced for the whole region rather than those deduced for the smaller individual contact areas. In the following sections, methods are presented for temperature calculation with smooth surfaces. After a short
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introduction to the theory of fast moving heat sources, the temperature distribution in wheel/rail contact will be investigated in detail. Convection and heat conduction through the contact patch are also taken into account for the long-term behaviour of the wheel temperature. The possible influence on thermally induced phase transformations will be discussed.
2. Moving heat sources in rolling contact When wheel and rail are brought into contact under the action of the static wheel load, the area of contact and the pressure distribution are usually calculated with the Hertz’s theory. In this case, the area of contact is elliptical and the normal pressure distribution is [14] x2 y2 pz (x, y) = p0 1 − 2 − 2 , (1) a b with the maximum pressure 3N p0 = , (2) 2π ab for the normal load N and the semi-axes a (in rolling direction) and b of the contact ellipse (Fig. 1). In the case of a infinitely long cylinder subjected to the normal load N/b∗ per unit length, the normal pressure distribution is [14] x2 pz (x) = p0 1 − 2 , (3) a with the maximum pressure 2N p0 = . (4) π ab∗ This model is often used for a simplified analysis of three-dimensional contact problems. Comparing Eqs. (2) and (4), we obtain the reference length b∗ for the transition from the three-dimensional to the two-dimensional case as b∗ = 43 b.
(5)
Fig. 2. Co-ordinate system for temperature calculation in wheel/rail contact.
If a tangential force T is transmitted between wheel and rail, there is always a mean relative velocity in the contact point. High contact temperatures are to be expected only with the transmission of tractive or braking forces at high relative velocities. In this case, sliding occurs within the whole contact area and the tangential force is T = µN. The coefficient of friction µ is assumed to be constant in the following. From the point of view of an observer who is fixed to the wheel, the contact patch moves with respect to the wheel surface and the frictional heating within the contact patch is a time-dependent heat source (Fig. 2). During the very short time period that every point on the surface is in contact, the thermal penetration depth a δ=√ , (6) L is very small compared to the size of the contact patch. It depends on the non-dimensional Péclet number av L= , (7) 2κ with the semi-axis length a, the speed v of the moving heat source and the thermal diffusivity λ κ= , (8) c that combines the material properties λ (thermal conductivity), (density) and c (specific heat capacity). Since the velocities of wheel and rail with respect to the contact patch are different, this also has to be taken into account for the Péclet number and the thermal penetration depth. As stated by Johnson, L may be interpreted as the ratio of the surface speed to the rate of diffusion of heat into the solid [14]. If L > 10, heat conduction occurs only perpendicular to the contact plane, i.e. in z-direction [2,5,7]. With the typical values for wheel/rail contact, a ≈ 5 mm, κ = 14.2 × 10−6 m2 /s [9] and v0 = 30 m/s, one gets L = 5300. The longitudinal and lateral heat conduction (x- and y-direction) can, therefore, be neglected and the heat conduction equation is [15]
Fig. 1. Elliptical area of contact.
κ
∂Θ ∂ 2Θ = . 2 ∂t ∂z
(9)
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Since Eq. (9) is linear and contains only derivatives of the temperature, the zero-point of temperature may be conveniently chosen as the ambient temperature. Thus, Θ represents the temperature rise due to the heat supply within the contact patch. We first consider the case that wheel and rail are initially at ambient temperature on coming into contact. Inside the contact patch they are subjected to a heat source at their surfaces due to the frictional heating. Since the heat flow is one-dimensional, this problem is similar to a semi-infinite solid with an arbitrarily distributed heat source q(t) ˙ applied to the surface z = 0 at t ≥ 0. The solution Θ(z, t) has to fulfil the differential Eq. (9), the initial condition Θ(z, t = 0) = 0, ∂Θ (z = 0, t) = q(t). ˙ ∂z
(11)
The general solution of this problem can be found in the book of Carslaw and Jaeger [4], Section 2.9,
t z2 1 dt
q(t ˙ − t ) exp − Θ(z, t) = √ (12) √ , 4κt
β π 0 t
with the thermal penetration coefficient λ β = λc = √ . κ
ξ a q˙r (ξ ) π vr −1 dξ
ζ2 , × exp − √ 2(ξ − ξ ) ξ − ξ
1 Θr (ξ, ζ ) = βr
(17)
for the rail with vr = v0 (Fig. 2). The analytical solution of the integral in Eq. (16) is quite simple if we assume a constant heat flow rate q˙w at the wheel surface within the contact patch. For −1 ≤ ξ ≤ 1, we get
2a 2(ξ + 1) ζ2 exp − vw π 2(ξ + 1) ζ −ζ erfc √ , (18) 2(ξ + 1)
q˙w Θw (ξ, ζ ) = βw
(13)
This is usually referred to as b in the literature [15]. To avoid confusion with the length of the lateral semi-axis of the contact ellipse, we use β instead. The heat conduction Eq. (9) can also be solved by means of the Laplace transform for special boundary conditions (e.g. Tanvir [6], Knothe and Liebelt [7] and Fischer et al. [11]). But when the distribution of q(t) ˙ is a prescribed function of time, the temperature can easily be found by solving the definite integral in Eq. (12) using analytical or numerical methods. Therefore, this method should generally be preferred since the mathematical effort is smaller than with the Laplace transform and the results are equal. Points on the surfaces of wheel and rail pass through the contact patch at different speeds due to the sliding velocity. Since the largest heat flux and the highest temperatures occur along the major axis which is parallel to the rolling direction at y = 0, it is most important to examine this case [6]. It is meaningful to substitute the time t elapsed since entering the contact patch with the current position x in a co-ordinate system fixed to the contact patch (Fig. 2), x = vt − a.
for the wheel with vw = v0 + vs and
(10)
and the boundary condition −λ
Using Eq. (12) with the respective values of β, v and q, ˙ the temperatures of wheel (w) and rail (r) can be calculated as ξ a 1 q˙w (ξ ) Θw (ξ, ζ ) = βw π vw −1 ζ2 dξ
× exp − , (16) √ 2(ξ − ξ ) ξ − ξ
(14)
with the complement erfc(s) of the error function erf(s) defined as [15] s 2 2 erfc(s) = 1 − erf(s) = 1 − √ e−ω dω. (19) π 0 Outside the contact area there is no frictional heating. Neglecting convection, the heat flow rate is zero here and the analytical solution for ξ > 1 is
(20) With ζ = 0 in Eqs. (18) and (20), the surface temperature is a 2q˙w Θw (ξ ) = ξ + 1 for − 1 ≤ ξ ≤ 1 (21) βw π vw and
For a further simplification, we introduce the dimensionless co-ordinates x z ξ= and ζ = . (15) a δ
2a 2(ξ + 1) ζ2 × exp − vw π 2(ξ + 1) 2(ξ − 1) ζ −ζ erfc √ − π 2(ξ + 1) ζ ζ2 −ζ erfc √ × exp − . 2(ξ −1) 2(ξ − 1)
q˙w Θw (ξ, ζ ) = βw
Θw (ξ ) =
2q˙w βw
a ξ +1− ξ −1 π vw
for ξ > 1. (22)
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Table 1 Comparison of mean and maximum temperatures for different distributions of frictional heating √ Heat flow rate Θmean /(εµvs p0 a/vw /βw )
√ Θmax /(εµvs p0 a/vw /βw )
ξmax
Elliptic (numerical solution) Elliptic (improved approximation) Elliptic (Tanvir’s approximation) Parabolic Constant
1.235 1.240 1.276 1.303 1.253
0.652 0.643 0.673 0.5 1.0
0.851 0.851 0.853 (0.2%) 0.860 (1.1%) 0.836 (−1.8%)
(0.4%) (3.3%) (5.5%) (1.5%)
3. Frictional heating in wheel/rail contact
3.2. Numerical solution for arbitrary heat flow rate
In principle, the integral in Eq. (16) can be solved analytically for any distribution of heat flow rate given as a polynomial. For the elliptical heat flow rate in Hertzian contact, this is not the case. The solution for constant heat flow rate is used as a first-order estimation in Section 3.1. For a more detailed investigation, we will introduce a numerical solution in Section 3.2 and an analytical approximation in Section 3.3. The results of all calculations are summarized in Table 1.
An arbitrarily distributed heat source can be approximated by a step-wise constant function. As a modification of the solution for a constant heat flow rate (Eq. (20)), we get k 1 2a q˙i Θw (ξk , ζ ) = β w vw
3.1. Analytical solution for constant heat flow rate
G(ξ, ζ ) =
With constant values of the coefficient of friction µ and the sliding velocity vs , the frictional power dissipation rate in the contact patch is proportional to the pressure:
(23) q˙friction (ξ ) = µvs pz (ξ ) = µvs p0 1 − ξ 2 .
Setting ζ = 0 in Eq. (28), the surface temperature is [7]
It is generally assumed that all the frictional power dissipation is transformed in heat. The heat generated in the contact patch flows into the material of wheel and rail. With the heat partitioning factor ε, this can be written as q˙w (ξ ) = ε q˙friction (ξ ) and
q˙r (ξ ) = (1 − ε)q˙friction (ξ ).
(24)
i=1
× {G(ξk − ξi−1 , ζ ) − G(ξk − ξi , ζ )}, with
2 2ξ ζ ζ . exp − − ζ erfc √ π 2ξ 2ξ
2 Θw (ξk ) = βw
(28)
(29)
k a q˙i ξk − ξi−1 − ξk − ξi . π vw i=1
(30) Using this method, the surface temperature has been calculated for various distributions of frictional heating (Fig. 3), although the case of parabolic heat flow rate could also be solved analytically. Since the average heat flow rate is equal in all cases, the maximum temperatures are almost the same (Table 1). Outside the contact patch, the differences can be
The surface temperatures of wheel and rail must be equal everywhere in the contact patch. Substituting Eq. (24) in Eqs. (16) and (17), the part of the frictional heating that flows into the wheel is √ β w vw ε= (25) √ √ . β w vw + β r v0 With the average heat flow rate at the surface of the wheel, 1 1 π q˙w = q˙w (ξ ) dξ = εµvs p0 , (26) 2 −1 4 the solutions for constant heat flow rate, Eqs. (18)–(22), can be used as a simple estimate for the friction-induced temperature of wheel and rail. The maximum temperature εµvs p0 a Θmax = 1.253 , (27) βw vw occurs at the trailing edge of the contact patch (Fig. 3). This result has already been given by Blok.
Fig. 3. Surface temperature for various distributions of frictional heat flow rate.
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Fig. 4. Comparison of numerical solution and Tanvir’s approximation for Hertzian contact.
Fig. 5. Temperature distribution in normal direction at the trailing edge (ξ = 1) and before renewed contact after one revolution of the wheel.
neglected. As shown in Fig. 4, the high temperatures are confined to a very thin surface layer. With the parameters given in Table 2, the thermal penetration depth, Eq. (6), is δ = 75 m whilst the semi-axis length in rolling direction is a = 5.88 mm.
shown before, the results for different distributions of frictional heating are nearly similar outside the contact patch. Even so, one gets a discontinuity at the trailing edge. The approximation of the heat flow rate can be improved with a least square fit of the Hertzian pressure distribution. Using again a fourth-order polynomial, this can be written as
3.3. Analytical approximation for Hertzian contact
q˙friction (ξ ) = µvs p0 f (ξ ),
Tanvir proposed an alternative approach for Hertzian contact [6]. He approximated the heat source distribution for Hertzian contact (Eq. (23)) to a fourth-order polynomial and found the analytical solution of Eq. (9) with the Laplace transform. The temperature outside the contact patch was approximated to the solution for the constant average heat flow rate (Eq. (22)). It should be mentioned that some coefficients in the Eqs. (15) and (23) of Tanvir’s paper are wrong. They have been corrected for the results shown in Fig. 5. As
with f (ξ ) =
(31)
π (645 − 210ξ 2 − 315ξ 4 ) 2048
(32)
for −1 ≤ ξ ≤ 1. The surface temperature for the approximated heat flow rate can be calculated from Eq. (16) for ζ = 0 as εµvs p0 a Θw (ξ ) = F (ξ ), (33) βw π vw
Table 2 Reference data for the presented results Symbol Semi-axis of the contact ellipse In rolling direction In lateral direction
a b
Reference length Specific heat capacity [9] Normal force (wheel load) Maximum Hertzian pressure Wheel radius Vehicle speed Sliding velocity (longitudinal) Thermal penetration coefficient Thermal diffusivity Thermal conductivity [9] Coefficient of friction Density
b∗ c N p0 r0 v0 vs β κ λ µ
These parameters hold for all calculations unless other values are given.
Value 5.88 10.54 14.05 450 100 770 0.5 30 1 13290 14.2 × 10−6 50 0.3 7850
Unit
Equation
mm mm mm J/kg K kN MPa m m/s m/s Ws0.5 /K m2 m2 /s W/K m 1 kg/m3
Eq. (5)
Eq. (2)
Eq. (13) Eq. (8)
M. Ertz, K. Knothe / Wear 253 (2002) 498–508
with F (ξ ) =
ξ −1
f (ξ ) √
dξ
. ξ − ξ
(34)
This solution can also be used outside the contact area. We get π F (ξ ) = ξ + 1(71 + 12ξ − 20ξ 2 + 8ξ 3 − 16ξ 4 ) 128 (35) for −1 ≤ ξ ≤ 1 and π F (ξ ) = ξ + 1(71 + 12ξ − 20ξ 2 + 8ξ 3 − 16ξ 4 ) 128 − ξ − 1(71 − 12ξ − 20ξ 2 − 8ξ 3 − 16ξ 4 ) (36) for ξ > 1. The maximum temperature (µvs p0 a Θmax = 1.240 βw vw
(37)
is nearly equal to the result of Eq. (27). The local differences between the improved approximation and the numerical solution for Hertzian contact are everywhere less than 1% (Table 1). Therefore, this efficient approach should be the best choice for the investigation of contact temperatures and thermal stresses in the case of Hertzian contact.
4. Heat conduction from wheel into rail The bulk temperature of the wheel increases with time due to the continuous frictional heating on its surface. Therefore, the temperatures of wheel and rail are different when a point on the surface of the wheel comes into the area of contact again. This gives rise to a considerable heat flow from the hot wheel into the cold rail due to conduction through the contact patch. After one revolution of the wheel, the temperature gradient in z-direction is very small (Fig. 5). Therefore, the next contact of a point on the surface of the wheel with the rail can be treated as the contact of two semi-infinite bodies with different initial temperatures that come into contact at t = 0 [4,15]. With the initial temperatures of wheel and rail, Θw (z, t = 0) = Θw0
and
Θr (z, t = 0) = Θr0 = 0,
(38)
the surface temperature goes instantaneously to a constant value Θm . Without frictional heating at the same time, the wheel temperature is (Carslaw and Jaeger [4], Section 2.5) z Θw (z, t) = Θm + (Θw0 − Θm )erf , (39) √ 2 κw t as long as the prescribed constant surface temperature is Θm . With the dimensionless co-ordinates defined in Eq. (15), the
503
temperature can be written for the wheel as ζ , Θw (ξ, ζ ) = Θm + (Θw0 − Θm )erf √ 2(ξ + 1) and for the rail as Θw (ξ, ζ ) = Θm 1 − erf √
ζ 2(ξ + 1)
(40)
.
(41)
The resulting heat flow rate through the contact patch is vw (42) q˙w (ξ ) = −βw (Θw0 − Θm ) π a(ξ + 1) for the wheel and v0 q˙r (ξ ) = βr Θm π a(ξ + 1)
(43)
for the rail. The value of the constant surface temperature Θm can be calculated from the condition that the heat flow rates at the surfaces of wheel and rail must be equal everywhere in the contact patch. From Eqs. (42) and (43) follows: √ β w vw Θm = (44) √ √ Θw0 = εΘw0 , βw vw + βr v0 with ε as defined in Eq. (25). The heat flow rate through the contact patch from the hot wheel into the cold rail is then v0 q˙w (ξ ) = −εβr Θw0 = −q˙r (ξ ), (45) π a(ξ + 1) and the total heat flow per unit width is 1 8av0 ˙ rail /b∗ = a Q q˙w (ξ ) dξ = −εβr Θw0 . π −1
(46)
Outside the contact patch there is no heat flow from the wheel into the rail. The surface temperature of the wheel is Θw (ξ, ζ = 0) = Θw0
2 2 1 − (1 − ε)arcsin π ξ +1
(47)
for ξ > 1 [4]. In this case, an analytical solution for ζ ≡ 0 is not possible. It is not strictly necessary to solve the problem numerically since the analytical solution is sufficient for the calculation of heat transfer and surface temperature. In order to be able to calculate the temperature field inside the material after the surface is out of contact, the heat flow rate Eq. (45) with a known initial wheel temperature Θw0 can be discretized for the application of the numerical solution given in Section 3.2. We have solved this problem with the assumption of one-dimensional heat flow. This is justified by the considerations in Section 2, except at the leading edge where on gets a discontinuity (Fig. 6). Barber et al. [16] investigated the heat flow at the boundary of contact areas in detail by means of asymptotic methods. For usual operating conditions in wheel/rail contact with very high Péclet numbers, the error using the one-dimensional model may be neglected.
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Fig. 6. Surface temperature of the wheel due to contact with the cold rail, initial temperature 150◦ C, without frictional heating.
5. Convection Outside the area of contact, heat flows from the wheel into ambient air by convection at the free surfaces. With the heat transfer coefficient α, the boundary condition is λw
∂Θw (x, z = 0) = αΘw (x, z = 0) ∂z
(48)
for x > a. In the case of the wheel, there is forced convection due to the rotation. In order to find the heat transfer coefficient α, Fischer et al. used the empirical formula α ≈ 3.6 v with α in W/K m2 and v in m/s [11]. At v0 = 30 m/s, one obtains α = 108 W/K m2 . Another approach is possible if the wheel is considered as a cylinder in cross flow with the fluid velocity v0 . Using a formula from Baehr and Stephan [15], we get α = 9 W/K m2 for the parameters given in Table 2. As a general rule, typical values of α for forced convection with air or gas are in the range of 10–100 W/K m2 [17]. For the investigation of the convective heat transfer from a solid into a fluid, it is meaningful to consider the Biot number Bi =
αd . λw
(49)
The main characteristic length d in wheel/rail contact is the diameter 2a of the contact patch. With the typical values d = 2a ≈ 10 mm, α = 10 W/K m2 and λw = 50 W/K m, we get the Biot number Bi = 2 × 10−3 . At small values of the Biot number, the temperature gradient inside the solid is small compared with the difference of temperature between the fluid and the surface of the solid. If Bi < 0.1, the temperature gradient in the solid can be neglected for the calculation of the convective heat transfer [15]. The conductive heat transfer at the surface of the wheel can, therefore, easily be estimated with the assumption that the average wheel
Fig. 7. Surface temperature with frictional heating and convection outside the contact patch, no initial wheel temperature, different values of heat transfer coefficient α.
temperature is constant at Θw0 . This gives the heat flow rate as q˙convection = −αΘw0
(50)
at the surface and the total heat flow per unit width is ˙ convection /b∗ = −2π r0 αΘw0 . (51) Q With Eqs. (46) and (51)) two heat sinks exist that reduce the temperature of the wheel. Since both terms are proportional to the surface temperature of the wheel, their orders of magnitude can easily be compared: ˙ convection αr0 π3 Q = . (52) ˙ rail εβr 2av0 Q ˙ convection /Q ˙ rail < With the parameters used here, we get Q 1%. Convection is only taken into account on the wheel tread, i.e. on the area that is subjected to frictional heating. This has also been assumed by Fischer et al. [11]. The result of Eq. (52) shows that convection can be neglected in this case. This is confirmed by numerical calculations (Fig. 7). It is evident that convective heat transfer can be neglected except for values of the heat transfer coefficient α that are far from realistic. Even without convection, the surface temperature outside the contact patch decreases very quickly due to conduction from the surface into the bulk material (Fig. 3). Convection on the sides of the wheel cannot be taken into account with the two-dimensional model of an infinitely long cylinder. In Section 7.3, convection on the whole surface of the real wheel will be considered approximately.
6. Steady-state wheel temperature Since the heat loss due to conduction into the rail is proportional to the initial surface temperature of the wheel, it
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505
will be equal to the frictional heating at a value of the initial temperature that can be calculated from the energy balance ˙ rail εPfriction Q + = 0, b∗ b∗
(53)
with εPfriction being the fraction of the frictional power dissipation Pfriction = µvs N =
π µp0 ab∗ vs , 2
(54)
that flows into the wheel. Using Eq. (46), this can be solved for the steady-state wheel temperature µvs p0 π 3 a Θ∞ = . (55) βr 32v0 The value of this temperature depends on the thermal penetration coefficient of the rail only. The subscript ∞ has been chosen to indicate that this temperature will only be reached after a very long time at constant operating conditions. While Θ∞ can be calculated with little effort, the transient calculation is much more difficult. Using a finite element model of the wheel, the surface temperature comes near a steady-state after 30–120 min, depending on operating conditions [18]. The ratio of Θ∞ to Θmean (Table 1) is 1 π3 Θ∞ = , (56) Θmean 0.836(1 − ε) 32 for the assumption of constant heat flow rate. The average contact temperature results from the current frictional heating (Table 1) and from the average wheel temperature in steady-state (Eq. (55)). With Eq. (44), one gets Θmean,∞ = Θmean + εΘ∞ .
Fig. 8. Temperature under the surface for two-dimensional model, Hertzian contact, initial wheel temperature for equilibrium without convection, Eq. (55).
7. Application to elliptical contact areas 7.1. Frictional heating The temperature distribution for the elliptical area of contact can be calculated with the polynomial approach given in Section 3.3. As already mentioned, the transverse heat conduction can be neglected and the temperature field within any strip parallel to the rolling direction (x-direction) can be treated independently (Fig. 1). Since the pressure distribution within every strip is similar to that of Eq. (23), the polynomial approximation Eq. (33)can be used for every strip at
ηi = yi /b. One only has to replace a by ai = a 1 − ηi2 and
p0 by pi = p0 1 − ηi2 . The surface temperature is then
(57)
If the heat partitioning factor ε is near 0.5 in Eq. (56), the average contact temperature Θmean,∞ in steady-state is approximately twice as high as the average contact temperature Θmean for the first contact of the cold wheel. This is not at all surprising. It corresponds to the commonly used assumption that the body in continuous contact (in this case the wheel) is an insulator [5]. Thus, all the frictional heating flows into the rail and, therefore, doubles the contact temperature. If we calculate the temperature field in the wheel with the initial temperature Θ∞ from Eq. (55), it is obvious that the temperature of the wheel remains constant at this value. Since there is no resulting heat flow, the average bulk temperature of the wheel does not change. The oscillating surface temperature is due to the different distributions of the heat flow rates resulting from frictional heating and rail contact. But within a very short distance after the trailing edge of the contact patch, the gradient in z-direction has disappeared and the temperature in the wheel will be constant again (Fig. 8).
εµvs p0 Θw (ξ, η) = βw
a (1 − η2 )0.75 F π vw
ξ
1 − η2
,
(58) for ξ > − 1 − η2 and |η| < 1 (Fig. 9). The average surface temperature for the elliptical area of contact can also be calculated with this approach. Using the solution within a single strip (Table 1), and integrating this result over the width 2b of the contact ellipse, one gets an integral that can only be solved numerically. The accurate solution is 3D Θmean = 0.323
εPfriction . √ bβw avw
(59)
In the special case of a circular area of contact (a = b), Archard [5] got nearly the same result with a factor of 0.31 instead. For practical applications, this result can be approximated to 4 Pfriction 3D Θmean ≈ , (60) √ 25 bβ av0
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the total heat loss due to convection is ˙ 3D Q convection = −αAwheel Θw0 ,
(63)
over the total wheel surface Awheel . Taking only the sides of the wheel into account, one gets Awheel ≈ 2π r02 . Thus, the steady-state wheel temperature Θ∞ can be calculated from the energy balance ˙ 3D + q˙ 3D εPfriction + Q rail convection = 0.
(64)
With Eqs. (61) and (63), the result is 3D Θ∞ =
Fig. 9. Surface temperature for elliptical area of contact.
Pfriction , √ 7bβr av0 /2π + αAwheel /ε
taking into account convection as well as heat conduction from wheel into rail. Similar to Eq. (59), the average contact temperature in steady-state is 3D 3D 3D Θmean,∞ = Θmean + εΘ∞ .
with the heat partitioning factor ε ≈ 0.5 for small sliding velocities, i.e. vs v0 , and equal thermal penetration coefficients β of wheel and rail. 7.2. Heat conduction from wheel into rail The heat conduction from wheel to rail due to an initial temperature of the wheel (Section 4) can also be investigated independently within any strip. But the calculation of the steady-state wheel temperature is not as easy as in the twodimensional case (Section 6). If one assumes that the initial temperature is constant over the whole width of the contact patch, the heat flow per unit width within one strip, Eq. (46), can be integrated over the width 2b of the contact ellipse in order to get the total heat flow. This gives approximately av0 3D ˙ Q . (61) rail = −7εbβr Θw0 2π The steady-state wheel temperature Θ∞ without convection can be calculated from the energy balance of frictional heating and heat loss as P 2π Pfriction friction 3D Θ∞ = = 0.358 √ . (62) 7bβr av0 bβr av0 Another approach is possible if heat conduction between strips is neglected. Thus, one has to calculate a steady-state temperature for every strip as in Section 6 and take the average value over the whole contact ellipse. This result differs by only 4% from Eq. (62).
(65)
(66)
It should be mentioned that the heat conduction from wheel into rail can be calculated quite accurately compared with convection. Since the air flow around the wheel is very complicated and depends on the aerodynamics of the train, the average heat transfer coefficient α can only be estimated roughly. 7.4. Transient behaviour The transient thermal analysis is only possible with a finite element model of the wheel. This has been done by Gupta et al., but only for the frictional heating [9]. In their work, the intermittent heat flow due to the consecutive contacts has been replaced with an equivalent reduced heat flow rate applied continuously to the entire circumference. The effect of heat conduction from wheel to rail has been considered by a modified heat partitioning factor ε. With the results of Section 7.2, this approach can be improved. Substituting the intermittent heat transfer from wheel to rail by a continuous flow, this can be included as a sort of convection, with the heat transfer coefficient 7εβr av0 αrail = , (67) r0 32π 3 applied to the width 2b of the contact patch over the entire circumference 2π r0 of the wheel. Neglecting first the convection into ambient air, this gives the same value for the steady-state wheel temperature as in Eq. (62). On the other hand, the transient behaviour can be calculated at any time. Convection into ambient air can be taken into account additionally. More details and results will be presented in a separate paper [18].
7.3. Convection 7.5. Application to tread-braked wheels Convection occurs over the whole surface of the wheel. Based on the consideration of the Biot number in Section 5, a first-order estimation of the heat flow due to convection is possible. If the wheel temperature everywhere is Θw0 , then
The focus of this paper is on the contact temperatures due to slip between wheel and rail. High temperatures and severe thermal stresses in wheels are also caused by the
M. Ertz, K. Knothe / Wear 253 (2002) 498–508
507
Table 3 Summary of results for elliptical contact area (all temperatures in ◦ C) Units
Low speed
High speed
N (kN) v0 (m/s) vs (m/s) µ Pfriction (kW)
100 30 1 0.3 30
100 90 3 0.1 30
Instantaneous contact temperature Maximal temperature
Θmax
149.7
86.4
Eq. (37)
Average contact temperature
3D Θmean
81.7
47.1
Eq. (59)
182.6 173.7
105.4 100.3
Eq. (62) Eq. (66)
153.5 159.0
95.0 95.1
Eq. (65) Eq. (66)
3D Θ∞ 3D Θmean,∞
116.4
79.4
Eq. (65)
140.3
87.2
Eq. (66)
3D Θ∞ 3D Θmean,∞
63.0 113.4
50.3 72.5
Eq. (65) Eq. (66)
3D Θ∞ 3D Θmean,∞
9.1
8.8
Eq. (65)
86.3
51.6
Eq. (66)
Normal load Vehicle speed Sliding velocity (longitudinal) Coefficient of friction Frictional power dissipation
Steady-state: wheel temperature and average contact temperature 3D Without convection Θ∞ α = 10 W/K m2
3D Θmean,∞ 3D Θ∞ 3D Θmean,∞
α = 30 W/K m2 α = 100 W/K m2 α=
1000 W/K m2
non-uniform heating which results from tread braking [8,19]. This is not a rolling contact problem since the brake shoe stays always in contact with the moving wheel tread and the sliding velocity is equal to the circumferential speed of the wheel. It can, therefore, be assumed that nearly all the heat flows into the wheel [5]. Thus, the average contact temperature between brake shoe and wheel from the current frictional heating can be estimated with the equations presented in this paper if the heat partitioning factor ε (Eq. (25)) is set to unity in this case. Some investigations of this problem neglect the periodic contact of the wheel tread with the brake shoe and take only the continuously increasing bulk temperature of the wheel into account [19]. However, after long periods of braking, the steady-state temperature of the wheel will again depend on convection as well as on conduction from wheel to rail through the contact patch. This can also be calculated with Eq. (65), the frictional power now resulting from braking and wheel/rail contact at the same time.
8. Discussion For constant operating conditions in steady-state, the lower limit of the average contact temperature in wheel/rail contact is the instantaneous peak due to the current frictional heating, i.e. without an initially increased wheel temperature. Due to heat conduction from wheel to rail, the upper limit is approximately twice as high if convection is neglected. Taking convection into account, the average contact temperature will always be between these limit values.
Eq. (54)
In Table 3, the instantaneous contact temperature and the steady-state wheel temperature are given for two different vehicle speeds with equal values of the frictional power dissipation. All other parameters are also equal (Table 2). The results show that the temperature decreases with increasing vehicle speed. The steady-state temperature Θ∞ depends mainly on heat conduction from wheel into rail. It is even lower if convection is also taken into account. If the heat transfer coefficient α is in the range of 50–100 W/K m2 , the heat flow into ambient air is nearly equal to the heat conduction from wheel into rail. Thus, Θ∞ is only half as high as without convection. With the overestimation α = 1000 W/K m2 , the average wheel temperature would be nearly equal to ambient temperature. The data in Tables 2 and 3 are usual operating conditions of European locomotives. Maximum contact temperatures in these cases are not high enough to explain thermally induced phase transformations. This may only be the case with extreme conditions, e.g. blocking wheels where the sliding velocity is equal to the vehicle speed. On the other hand, contact temperatures give rise to severe thermal stresses. It has been shown in a number of investigations that thermal stresses in railway wheels and rails can be in the same order of magnitude as the stresses due to mechanical loading, e.g. [8,19,20]. A surface temperature change of 200 ◦ C would result in thermal compressive stresses σx = σy ≈ −700 MPa. This may cause plastic deformation, residual stresses and work hardening at the surfaces of wheel and rail. Since the thermal penetration depth δ is very small, thermally induced plastic deformations are restricted to a very thin surface layer [7,21].
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9. Conclusions The heat flow in wheel/rail contact with smooth surfaces is one-dimensional. Some methods for the calculation of contact temperatures with this model are presented. The maximum surface temperature can be estimated with the flash temperature formula of Blok. Semi-analytical and numerical methods are available for a more detailed investigation. With a polynomial approximation, the case of Hertzian contact can be investigated very efficiently. While the contact temperatures are confined to a very thin surface layer, the bulk temperature of the wheel also increases with time by continuous frictional heating. It can be shown that the wheel temperature for constant operating conditions cannot be more than twice the average temperature for the first contact of the cold wheel. This limit is due to heat conduction from the hot wheel into the cold rail. It corresponds to the usual assumption that the wheel is an insulator and all the frictional heating flows into the rail. If convection is only taken into account on the wheel tread, i.e. on the area that is subjected to frictional heating, it can be included into the usual model for contact temperature calculations [11]. But in this case, its influence is small enough to be neglected. On the other hand, real convection occurs on the whole surface of the wheel. This can only be considered approximately. Therefore, the steady-state temperature of the wheel is even lower. Structural changes in the rail material such as formation of WEL are unlikely to occur due to high contact temperatures only. They are probably the consequence of high mechanical stresses and moderate contact temperatures that result in high thermal stresses. References [1] H. Blok, Theoretical study of temperature rise at surfaces of actual contact under oiliness lubricating conditions, Proc. Gen. Discuss. Lubricat. Inst. Mech. Eng. London 2 (1937) 222–235. [2] J.C. Jaeger, Moving sources of heat and the temperature at sliding contacts, Proc. R. Soc. NSW 56 (1942) 203–224. [3] H. Blok, The flash temperature concept, Wear 6 (1963) 483–494.
[4] H.C. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, 2nd Edition, Oxford University Press, Oxford, 1959. [5] J.F. Archard, The temperature of rubbing surfaces, Wear 2 (1958–1959) 438–455. [6] M.A. Tanvir, Temperature rise due to slip between wheel and rail— an analytical solution for Hertzian contact, Wear 61 (1980) 295–308. [7] K. Knothe, S. Liebelt, Determination of temperatures for sliding contact with applications for wheel–rail systems, Wear 189 (1995) 91–99. [8] G.J. Moyar, D.H. Stone, An analysis of the thermal contributions to railway wheel shelling, Wear 144 (1991) 117–138. [9] V. Gupta, G.T. Hahn, P.C. Bastias, C.A. Rubin, Calculations of the frictional heating of a locomotive wheel attending rolling plus sliding, Wear 191 (1996) 237–241. [10] A. Cameron, A.N. Gordon, G.T. Symm, Contact temperatures in rolling/sliding surfaces, Proc. R. Soc. A 286 (1965) 45–61. [11] F.D. Fischer, E. Werner, K. Knothe, The surface temperature of a halfplane heated by friction and cooled by convection, Z. Angew. Math. Mech. 81 (2001) 75–81. [12] F. Bucher, A. Theiler, K. Knothe, Normal and tangential contact problem of surfaces with measured roughness, Wear 253 (2002) 206–220. [13] J. Gao, S.C. Lee, X. Ai, H. Nixon, An fft-based transient flash temperature model for general three-dimensional rough surface contacts, J. Tribol. 122 (2000) 519–523. [14] K.L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, 1985. [15] H.D. Baehr, K. Stephan, Heat and Mass Transfer, Springer, Berlin, 1998. [16] J.R. Barber, F.B. Ammar, H.G. Georgiadis, Conductive heat exchange between bodies which are in contact for a very short period of time, Collected Papers in Heat Transfer, Vol. 123, ASME, New York, 1989, pp. 101–106. [17] W. Beitz, K.H. Küttner (Eds.), Dubbel–Taschenbuch für den Maschinenbau, 17th Edition, Springer, Berlin, 1990. [18] J. Dohrmann, M. Ertz, K. Knothe, Transient temperature calculation for locomotive wheels, in preparation. [19] M.R. Johnson, R.E. Welch, K.S. Yeung, Analysis of thermal stresses and residual stress changes in railroad wheels caused by severe drag braking, J. Eng. Ind. 99 (1977) 18–23. [20] V. Gupta, G.T. Hahn, P.C. Bastias, C.A. Rubin, Thermal-mechanical modelling of the rolling-plus-sliding with frictional heating of a locomotive wheel, J. Eng. Ind. 117 (1995) 418–422. [21] G. Baumann, Untersuchungen zu Gefügestrukturen und Eigenschaften der “Weißen Schichten” auf verriffelten Schienenlaufflächen (Investigations on structures and properties of white-etching layers on corrugated rail treads), PhD Thesis, Technische Universität, Berlin, 1998.
A comparison of analytical and numerical methods for the calculation of temperatures in wheel/rail contact夽 Martin Ertz∗ , Klaus Knothe Technische Universität Berlin, Sekr. F5, Marchstr. 12, D-10587 Berlin, Germany Received 3 January 2002; received in revised form 11 April 2002; accepted 1 May 2002
Abstract The maximum surface temperature during rolling contact of railway wheels with sliding friction can be estimated using Blok’s flash temperature formula. For a more detailed investigation, semi-analytical and numerical methods are available. A survey of various methods is given and an efficient approach is proposed for Hertzian contact. The actual contact temperature is confined to a very thin surface layer. Due to continuous frictional heating, the bulk temperature of the wheel increases with time. For the long-term behaviour of the wheel temperature, not only the convection at the free wheel surfaces but also the heat conduction from the wheel into the colder rail has to be considered. Practical consequences of the theoretical results are discussed. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Wheel/rail contact; Flash temperature; Contact temperature; Convection
1. Introduction The slip between wheel and rail causes frictional heating of both bodies. The resulting high contact temperatures have to be taken into account for microstructural alterations, such as formation of white-etching layers (WEL) on rail surfaces and also for failure of wheel and rail. With a possible relation between contact temperature and coefficient of friction, they can also be responsible for decreasing creep force curves. Since contact temperatures under real railway operating conditions can hardly be measured, temperature calculation is an important part in the investigation of rolling contact. Fundamental work in this area has been done by Blok [1] and Jaeger [2]. In his article The flash temperature concept published in 1963, Blok gave a survey of the state-of-the-art at that time [3]. The methods used are based on the theory of heat conduction with moving heat sources as described by Carslaw and Jaeger [4]. Archard [5] used the theory of Blok and Jaeger to investigate some tribological problems. An approximate solution for rolling contact was given by Tanvir [6]. Knothe and Liebelt [7] presented a numerical method for arbitrarily distributed heat sources. 夽 Dedicated to the memory of Prof. Harmen Blok (1910–2000), the pioneer of flash temperature calculation. ∗ Corresponding author. Tel.: +49-30-314-22142; fax: +49-30-314-22866. E-mail address: [email protected] (M. Ertz).
While railway wheels are heated by friction in the contact patch, there is also heat loss due to conduction through the contact patch into the rail. This effect has rarely been taken into account in previous research. It was mentioned by Moyar and Stone [8], and Gupta et al. [9] used a simple approach for the approximate consideration. In addition, heat flows from the wheel into ambient air due to convection at the free surfaces. Cameron et al. [10] calculated a steady-state temperature based on the equilibrium of frictional heating and heat loss due to convection. Fischer et al. [11] found an analytical solution for the surface temperature with combined frictional heating and convection. All these works are confined to smooth surfaces and most of these are based on the theory of Hertz for the mechanical contact problem. But real surfaces are always rough. Using measured profiles of rough surfaces, even the calculation of stress and strain demands computer simulations [12]. With smooth surfaces, the heat conduction can be treated one-dimensionally, but for rough surfaces, this simplification is not possible, and the investigation of temperatures is also a very complicated numerical problem [13]. On the other hand, Archard [5] has already stated that the influence of surface roughness can be neglected to a first approximation, since the largest temperatures are those deduced for the whole region rather than those deduced for the smaller individual contact areas. In the following sections, methods are presented for temperature calculation with smooth surfaces. After a short
0043-1648/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 3 - 1 6 4 8 ( 0 2 ) 0 0 1 2 0 - 5
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499
introduction to the theory of fast moving heat sources, the temperature distribution in wheel/rail contact will be investigated in detail. Convection and heat conduction through the contact patch are also taken into account for the long-term behaviour of the wheel temperature. The possible influence on thermally induced phase transformations will be discussed.
2. Moving heat sources in rolling contact When wheel and rail are brought into contact under the action of the static wheel load, the area of contact and the pressure distribution are usually calculated with the Hertz’s theory. In this case, the area of contact is elliptical and the normal pressure distribution is [14] x2 y2 pz (x, y) = p0 1 − 2 − 2 , (1) a b with the maximum pressure 3N p0 = , (2) 2π ab for the normal load N and the semi-axes a (in rolling direction) and b of the contact ellipse (Fig. 1). In the case of a infinitely long cylinder subjected to the normal load N/b∗ per unit length, the normal pressure distribution is [14] x2 pz (x) = p0 1 − 2 , (3) a with the maximum pressure 2N p0 = . (4) π ab∗ This model is often used for a simplified analysis of three-dimensional contact problems. Comparing Eqs. (2) and (4), we obtain the reference length b∗ for the transition from the three-dimensional to the two-dimensional case as b∗ = 43 b.
(5)
Fig. 2. Co-ordinate system for temperature calculation in wheel/rail contact.
If a tangential force T is transmitted between wheel and rail, there is always a mean relative velocity in the contact point. High contact temperatures are to be expected only with the transmission of tractive or braking forces at high relative velocities. In this case, sliding occurs within the whole contact area and the tangential force is T = µN. The coefficient of friction µ is assumed to be constant in the following. From the point of view of an observer who is fixed to the wheel, the contact patch moves with respect to the wheel surface and the frictional heating within the contact patch is a time-dependent heat source (Fig. 2). During the very short time period that every point on the surface is in contact, the thermal penetration depth a δ=√ , (6) L is very small compared to the size of the contact patch. It depends on the non-dimensional Péclet number av L= , (7) 2κ with the semi-axis length a, the speed v of the moving heat source and the thermal diffusivity λ κ= , (8) c that combines the material properties λ (thermal conductivity), (density) and c (specific heat capacity). Since the velocities of wheel and rail with respect to the contact patch are different, this also has to be taken into account for the Péclet number and the thermal penetration depth. As stated by Johnson, L may be interpreted as the ratio of the surface speed to the rate of diffusion of heat into the solid [14]. If L > 10, heat conduction occurs only perpendicular to the contact plane, i.e. in z-direction [2,5,7]. With the typical values for wheel/rail contact, a ≈ 5 mm, κ = 14.2 × 10−6 m2 /s [9] and v0 = 30 m/s, one gets L = 5300. The longitudinal and lateral heat conduction (x- and y-direction) can, therefore, be neglected and the heat conduction equation is [15]
Fig. 1. Elliptical area of contact.
κ
∂Θ ∂ 2Θ = . 2 ∂t ∂z
(9)
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Since Eq. (9) is linear and contains only derivatives of the temperature, the zero-point of temperature may be conveniently chosen as the ambient temperature. Thus, Θ represents the temperature rise due to the heat supply within the contact patch. We first consider the case that wheel and rail are initially at ambient temperature on coming into contact. Inside the contact patch they are subjected to a heat source at their surfaces due to the frictional heating. Since the heat flow is one-dimensional, this problem is similar to a semi-infinite solid with an arbitrarily distributed heat source q(t) ˙ applied to the surface z = 0 at t ≥ 0. The solution Θ(z, t) has to fulfil the differential Eq. (9), the initial condition Θ(z, t = 0) = 0, ∂Θ (z = 0, t) = q(t). ˙ ∂z
(11)
The general solution of this problem can be found in the book of Carslaw and Jaeger [4], Section 2.9,
t z2 1 dt
q(t ˙ − t ) exp − Θ(z, t) = √ (12) √ , 4κt
β π 0 t
with the thermal penetration coefficient λ β = λc = √ . κ
ξ a q˙r (ξ ) π vr −1 dξ
ζ2 , × exp − √ 2(ξ − ξ ) ξ − ξ
1 Θr (ξ, ζ ) = βr
(17)
for the rail with vr = v0 (Fig. 2). The analytical solution of the integral in Eq. (16) is quite simple if we assume a constant heat flow rate q˙w at the wheel surface within the contact patch. For −1 ≤ ξ ≤ 1, we get
2a 2(ξ + 1) ζ2 exp − vw π 2(ξ + 1) ζ −ζ erfc √ , (18) 2(ξ + 1)
q˙w Θw (ξ, ζ ) = βw
(13)
This is usually referred to as b in the literature [15]. To avoid confusion with the length of the lateral semi-axis of the contact ellipse, we use β instead. The heat conduction Eq. (9) can also be solved by means of the Laplace transform for special boundary conditions (e.g. Tanvir [6], Knothe and Liebelt [7] and Fischer et al. [11]). But when the distribution of q(t) ˙ is a prescribed function of time, the temperature can easily be found by solving the definite integral in Eq. (12) using analytical or numerical methods. Therefore, this method should generally be preferred since the mathematical effort is smaller than with the Laplace transform and the results are equal. Points on the surfaces of wheel and rail pass through the contact patch at different speeds due to the sliding velocity. Since the largest heat flux and the highest temperatures occur along the major axis which is parallel to the rolling direction at y = 0, it is most important to examine this case [6]. It is meaningful to substitute the time t elapsed since entering the contact patch with the current position x in a co-ordinate system fixed to the contact patch (Fig. 2), x = vt − a.
for the wheel with vw = v0 + vs and
(10)
and the boundary condition −λ
Using Eq. (12) with the respective values of β, v and q, ˙ the temperatures of wheel (w) and rail (r) can be calculated as ξ a 1 q˙w (ξ ) Θw (ξ, ζ ) = βw π vw −1 ζ2 dξ
× exp − , (16) √ 2(ξ − ξ ) ξ − ξ
(14)
with the complement erfc(s) of the error function erf(s) defined as [15] s 2 2 erfc(s) = 1 − erf(s) = 1 − √ e−ω dω. (19) π 0 Outside the contact area there is no frictional heating. Neglecting convection, the heat flow rate is zero here and the analytical solution for ξ > 1 is
(20) With ζ = 0 in Eqs. (18) and (20), the surface temperature is a 2q˙w Θw (ξ ) = ξ + 1 for − 1 ≤ ξ ≤ 1 (21) βw π vw and
For a further simplification, we introduce the dimensionless co-ordinates x z ξ= and ζ = . (15) a δ
2a 2(ξ + 1) ζ2 × exp − vw π 2(ξ + 1) 2(ξ − 1) ζ −ζ erfc √ − π 2(ξ + 1) ζ ζ2 −ζ erfc √ × exp − . 2(ξ −1) 2(ξ − 1)
q˙w Θw (ξ, ζ ) = βw
Θw (ξ ) =
2q˙w βw
a ξ +1− ξ −1 π vw
for ξ > 1. (22)
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501
Table 1 Comparison of mean and maximum temperatures for different distributions of frictional heating √ Heat flow rate Θmean /(εµvs p0 a/vw /βw )
√ Θmax /(εµvs p0 a/vw /βw )
ξmax
Elliptic (numerical solution) Elliptic (improved approximation) Elliptic (Tanvir’s approximation) Parabolic Constant
1.235 1.240 1.276 1.303 1.253
0.652 0.643 0.673 0.5 1.0
0.851 0.851 0.853 (0.2%) 0.860 (1.1%) 0.836 (−1.8%)
(0.4%) (3.3%) (5.5%) (1.5%)
3. Frictional heating in wheel/rail contact
3.2. Numerical solution for arbitrary heat flow rate
In principle, the integral in Eq. (16) can be solved analytically for any distribution of heat flow rate given as a polynomial. For the elliptical heat flow rate in Hertzian contact, this is not the case. The solution for constant heat flow rate is used as a first-order estimation in Section 3.1. For a more detailed investigation, we will introduce a numerical solution in Section 3.2 and an analytical approximation in Section 3.3. The results of all calculations are summarized in Table 1.
An arbitrarily distributed heat source can be approximated by a step-wise constant function. As a modification of the solution for a constant heat flow rate (Eq. (20)), we get k 1 2a q˙i Θw (ξk , ζ ) = β w vw
3.1. Analytical solution for constant heat flow rate
G(ξ, ζ ) =
With constant values of the coefficient of friction µ and the sliding velocity vs , the frictional power dissipation rate in the contact patch is proportional to the pressure:
(23) q˙friction (ξ ) = µvs pz (ξ ) = µvs p0 1 − ξ 2 .
Setting ζ = 0 in Eq. (28), the surface temperature is [7]
It is generally assumed that all the frictional power dissipation is transformed in heat. The heat generated in the contact patch flows into the material of wheel and rail. With the heat partitioning factor ε, this can be written as q˙w (ξ ) = ε q˙friction (ξ ) and
q˙r (ξ ) = (1 − ε)q˙friction (ξ ).
(24)
i=1
× {G(ξk − ξi−1 , ζ ) − G(ξk − ξi , ζ )}, with
2 2ξ ζ ζ . exp − − ζ erfc √ π 2ξ 2ξ
2 Θw (ξk ) = βw
(28)
(29)
k a q˙i ξk − ξi−1 − ξk − ξi . π vw i=1
(30) Using this method, the surface temperature has been calculated for various distributions of frictional heating (Fig. 3), although the case of parabolic heat flow rate could also be solved analytically. Since the average heat flow rate is equal in all cases, the maximum temperatures are almost the same (Table 1). Outside the contact patch, the differences can be
The surface temperatures of wheel and rail must be equal everywhere in the contact patch. Substituting Eq. (24) in Eqs. (16) and (17), the part of the frictional heating that flows into the wheel is √ β w vw ε= (25) √ √ . β w vw + β r v0 With the average heat flow rate at the surface of the wheel, 1 1 π q˙w = q˙w (ξ ) dξ = εµvs p0 , (26) 2 −1 4 the solutions for constant heat flow rate, Eqs. (18)–(22), can be used as a simple estimate for the friction-induced temperature of wheel and rail. The maximum temperature εµvs p0 a Θmax = 1.253 , (27) βw vw occurs at the trailing edge of the contact patch (Fig. 3). This result has already been given by Blok.
Fig. 3. Surface temperature for various distributions of frictional heat flow rate.
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Fig. 4. Comparison of numerical solution and Tanvir’s approximation for Hertzian contact.
Fig. 5. Temperature distribution in normal direction at the trailing edge (ξ = 1) and before renewed contact after one revolution of the wheel.
neglected. As shown in Fig. 4, the high temperatures are confined to a very thin surface layer. With the parameters given in Table 2, the thermal penetration depth, Eq. (6), is δ = 75 m whilst the semi-axis length in rolling direction is a = 5.88 mm.
shown before, the results for different distributions of frictional heating are nearly similar outside the contact patch. Even so, one gets a discontinuity at the trailing edge. The approximation of the heat flow rate can be improved with a least square fit of the Hertzian pressure distribution. Using again a fourth-order polynomial, this can be written as
3.3. Analytical approximation for Hertzian contact
q˙friction (ξ ) = µvs p0 f (ξ ),
Tanvir proposed an alternative approach for Hertzian contact [6]. He approximated the heat source distribution for Hertzian contact (Eq. (23)) to a fourth-order polynomial and found the analytical solution of Eq. (9) with the Laplace transform. The temperature outside the contact patch was approximated to the solution for the constant average heat flow rate (Eq. (22)). It should be mentioned that some coefficients in the Eqs. (15) and (23) of Tanvir’s paper are wrong. They have been corrected for the results shown in Fig. 5. As
with f (ξ ) =
(31)
π (645 − 210ξ 2 − 315ξ 4 ) 2048
(32)
for −1 ≤ ξ ≤ 1. The surface temperature for the approximated heat flow rate can be calculated from Eq. (16) for ζ = 0 as εµvs p0 a Θw (ξ ) = F (ξ ), (33) βw π vw
Table 2 Reference data for the presented results Symbol Semi-axis of the contact ellipse In rolling direction In lateral direction
a b
Reference length Specific heat capacity [9] Normal force (wheel load) Maximum Hertzian pressure Wheel radius Vehicle speed Sliding velocity (longitudinal) Thermal penetration coefficient Thermal diffusivity Thermal conductivity [9] Coefficient of friction Density
b∗ c N p0 r0 v0 vs β κ λ µ
These parameters hold for all calculations unless other values are given.
Value 5.88 10.54 14.05 450 100 770 0.5 30 1 13290 14.2 × 10−6 50 0.3 7850
Unit
Equation
mm mm mm J/kg K kN MPa m m/s m/s Ws0.5 /K m2 m2 /s W/K m 1 kg/m3
Eq. (5)
Eq. (2)
Eq. (13) Eq. (8)
M. Ertz, K. Knothe / Wear 253 (2002) 498–508
with F (ξ ) =
ξ −1
f (ξ ) √
dξ
. ξ − ξ
(34)
This solution can also be used outside the contact area. We get π F (ξ ) = ξ + 1(71 + 12ξ − 20ξ 2 + 8ξ 3 − 16ξ 4 ) 128 (35) for −1 ≤ ξ ≤ 1 and π F (ξ ) = ξ + 1(71 + 12ξ − 20ξ 2 + 8ξ 3 − 16ξ 4 ) 128 − ξ − 1(71 − 12ξ − 20ξ 2 − 8ξ 3 − 16ξ 4 ) (36) for ξ > 1. The maximum temperature (µvs p0 a Θmax = 1.240 βw vw
(37)
is nearly equal to the result of Eq. (27). The local differences between the improved approximation and the numerical solution for Hertzian contact are everywhere less than 1% (Table 1). Therefore, this efficient approach should be the best choice for the investigation of contact temperatures and thermal stresses in the case of Hertzian contact.
4. Heat conduction from wheel into rail The bulk temperature of the wheel increases with time due to the continuous frictional heating on its surface. Therefore, the temperatures of wheel and rail are different when a point on the surface of the wheel comes into the area of contact again. This gives rise to a considerable heat flow from the hot wheel into the cold rail due to conduction through the contact patch. After one revolution of the wheel, the temperature gradient in z-direction is very small (Fig. 5). Therefore, the next contact of a point on the surface of the wheel with the rail can be treated as the contact of two semi-infinite bodies with different initial temperatures that come into contact at t = 0 [4,15]. With the initial temperatures of wheel and rail, Θw (z, t = 0) = Θw0
and
Θr (z, t = 0) = Θr0 = 0,
(38)
the surface temperature goes instantaneously to a constant value Θm . Without frictional heating at the same time, the wheel temperature is (Carslaw and Jaeger [4], Section 2.5) z Θw (z, t) = Θm + (Θw0 − Θm )erf , (39) √ 2 κw t as long as the prescribed constant surface temperature is Θm . With the dimensionless co-ordinates defined in Eq. (15), the
503
temperature can be written for the wheel as ζ , Θw (ξ, ζ ) = Θm + (Θw0 − Θm )erf √ 2(ξ + 1) and for the rail as Θw (ξ, ζ ) = Θm 1 − erf √
ζ 2(ξ + 1)
(40)
.
(41)
The resulting heat flow rate through the contact patch is vw (42) q˙w (ξ ) = −βw (Θw0 − Θm ) π a(ξ + 1) for the wheel and v0 q˙r (ξ ) = βr Θm π a(ξ + 1)
(43)
for the rail. The value of the constant surface temperature Θm can be calculated from the condition that the heat flow rates at the surfaces of wheel and rail must be equal everywhere in the contact patch. From Eqs. (42) and (43) follows: √ β w vw Θm = (44) √ √ Θw0 = εΘw0 , βw vw + βr v0 with ε as defined in Eq. (25). The heat flow rate through the contact patch from the hot wheel into the cold rail is then v0 q˙w (ξ ) = −εβr Θw0 = −q˙r (ξ ), (45) π a(ξ + 1) and the total heat flow per unit width is 1 8av0 ˙ rail /b∗ = a Q q˙w (ξ ) dξ = −εβr Θw0 . π −1
(46)
Outside the contact patch there is no heat flow from the wheel into the rail. The surface temperature of the wheel is Θw (ξ, ζ = 0) = Θw0
2 2 1 − (1 − ε)arcsin π ξ +1
(47)
for ξ > 1 [4]. In this case, an analytical solution for ζ ≡ 0 is not possible. It is not strictly necessary to solve the problem numerically since the analytical solution is sufficient for the calculation of heat transfer and surface temperature. In order to be able to calculate the temperature field inside the material after the surface is out of contact, the heat flow rate Eq. (45) with a known initial wheel temperature Θw0 can be discretized for the application of the numerical solution given in Section 3.2. We have solved this problem with the assumption of one-dimensional heat flow. This is justified by the considerations in Section 2, except at the leading edge where on gets a discontinuity (Fig. 6). Barber et al. [16] investigated the heat flow at the boundary of contact areas in detail by means of asymptotic methods. For usual operating conditions in wheel/rail contact with very high Péclet numbers, the error using the one-dimensional model may be neglected.
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Fig. 6. Surface temperature of the wheel due to contact with the cold rail, initial temperature 150◦ C, without frictional heating.
5. Convection Outside the area of contact, heat flows from the wheel into ambient air by convection at the free surfaces. With the heat transfer coefficient α, the boundary condition is λw
∂Θw (x, z = 0) = αΘw (x, z = 0) ∂z
(48)
for x > a. In the case of the wheel, there is forced convection due to the rotation. In order to find the heat transfer coefficient α, Fischer et al. used the empirical formula α ≈ 3.6 v with α in W/K m2 and v in m/s [11]. At v0 = 30 m/s, one obtains α = 108 W/K m2 . Another approach is possible if the wheel is considered as a cylinder in cross flow with the fluid velocity v0 . Using a formula from Baehr and Stephan [15], we get α = 9 W/K m2 for the parameters given in Table 2. As a general rule, typical values of α for forced convection with air or gas are in the range of 10–100 W/K m2 [17]. For the investigation of the convective heat transfer from a solid into a fluid, it is meaningful to consider the Biot number Bi =
αd . λw
(49)
The main characteristic length d in wheel/rail contact is the diameter 2a of the contact patch. With the typical values d = 2a ≈ 10 mm, α = 10 W/K m2 and λw = 50 W/K m, we get the Biot number Bi = 2 × 10−3 . At small values of the Biot number, the temperature gradient inside the solid is small compared with the difference of temperature between the fluid and the surface of the solid. If Bi < 0.1, the temperature gradient in the solid can be neglected for the calculation of the convective heat transfer [15]. The conductive heat transfer at the surface of the wheel can, therefore, easily be estimated with the assumption that the average wheel
Fig. 7. Surface temperature with frictional heating and convection outside the contact patch, no initial wheel temperature, different values of heat transfer coefficient α.
temperature is constant at Θw0 . This gives the heat flow rate as q˙convection = −αΘw0
(50)
at the surface and the total heat flow per unit width is ˙ convection /b∗ = −2π r0 αΘw0 . (51) Q With Eqs. (46) and (51)) two heat sinks exist that reduce the temperature of the wheel. Since both terms are proportional to the surface temperature of the wheel, their orders of magnitude can easily be compared: ˙ convection αr0 π3 Q = . (52) ˙ rail εβr 2av0 Q ˙ convection /Q ˙ rail < With the parameters used here, we get Q 1%. Convection is only taken into account on the wheel tread, i.e. on the area that is subjected to frictional heating. This has also been assumed by Fischer et al. [11]. The result of Eq. (52) shows that convection can be neglected in this case. This is confirmed by numerical calculations (Fig. 7). It is evident that convective heat transfer can be neglected except for values of the heat transfer coefficient α that are far from realistic. Even without convection, the surface temperature outside the contact patch decreases very quickly due to conduction from the surface into the bulk material (Fig. 3). Convection on the sides of the wheel cannot be taken into account with the two-dimensional model of an infinitely long cylinder. In Section 7.3, convection on the whole surface of the real wheel will be considered approximately.
6. Steady-state wheel temperature Since the heat loss due to conduction into the rail is proportional to the initial surface temperature of the wheel, it
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will be equal to the frictional heating at a value of the initial temperature that can be calculated from the energy balance ˙ rail εPfriction Q + = 0, b∗ b∗
(53)
with εPfriction being the fraction of the frictional power dissipation Pfriction = µvs N =
π µp0 ab∗ vs , 2
(54)
that flows into the wheel. Using Eq. (46), this can be solved for the steady-state wheel temperature µvs p0 π 3 a Θ∞ = . (55) βr 32v0 The value of this temperature depends on the thermal penetration coefficient of the rail only. The subscript ∞ has been chosen to indicate that this temperature will only be reached after a very long time at constant operating conditions. While Θ∞ can be calculated with little effort, the transient calculation is much more difficult. Using a finite element model of the wheel, the surface temperature comes near a steady-state after 30–120 min, depending on operating conditions [18]. The ratio of Θ∞ to Θmean (Table 1) is 1 π3 Θ∞ = , (56) Θmean 0.836(1 − ε) 32 for the assumption of constant heat flow rate. The average contact temperature results from the current frictional heating (Table 1) and from the average wheel temperature in steady-state (Eq. (55)). With Eq. (44), one gets Θmean,∞ = Θmean + εΘ∞ .
Fig. 8. Temperature under the surface for two-dimensional model, Hertzian contact, initial wheel temperature for equilibrium without convection, Eq. (55).
7. Application to elliptical contact areas 7.1. Frictional heating The temperature distribution for the elliptical area of contact can be calculated with the polynomial approach given in Section 3.3. As already mentioned, the transverse heat conduction can be neglected and the temperature field within any strip parallel to the rolling direction (x-direction) can be treated independently (Fig. 1). Since the pressure distribution within every strip is similar to that of Eq. (23), the polynomial approximation Eq. (33)can be used for every strip at
ηi = yi /b. One only has to replace a by ai = a 1 − ηi2 and
p0 by pi = p0 1 − ηi2 . The surface temperature is then
(57)
If the heat partitioning factor ε is near 0.5 in Eq. (56), the average contact temperature Θmean,∞ in steady-state is approximately twice as high as the average contact temperature Θmean for the first contact of the cold wheel. This is not at all surprising. It corresponds to the commonly used assumption that the body in continuous contact (in this case the wheel) is an insulator [5]. Thus, all the frictional heating flows into the rail and, therefore, doubles the contact temperature. If we calculate the temperature field in the wheel with the initial temperature Θ∞ from Eq. (55), it is obvious that the temperature of the wheel remains constant at this value. Since there is no resulting heat flow, the average bulk temperature of the wheel does not change. The oscillating surface temperature is due to the different distributions of the heat flow rates resulting from frictional heating and rail contact. But within a very short distance after the trailing edge of the contact patch, the gradient in z-direction has disappeared and the temperature in the wheel will be constant again (Fig. 8).
εµvs p0 Θw (ξ, η) = βw
a (1 − η2 )0.75 F π vw
ξ
1 − η2
,
(58) for ξ > − 1 − η2 and |η| < 1 (Fig. 9). The average surface temperature for the elliptical area of contact can also be calculated with this approach. Using the solution within a single strip (Table 1), and integrating this result over the width 2b of the contact ellipse, one gets an integral that can only be solved numerically. The accurate solution is 3D Θmean = 0.323
εPfriction . √ bβw avw
(59)
In the special case of a circular area of contact (a = b), Archard [5] got nearly the same result with a factor of 0.31 instead. For practical applications, this result can be approximated to 4 Pfriction 3D Θmean ≈ , (60) √ 25 bβ av0
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the total heat loss due to convection is ˙ 3D Q convection = −αAwheel Θw0 ,
(63)
over the total wheel surface Awheel . Taking only the sides of the wheel into account, one gets Awheel ≈ 2π r02 . Thus, the steady-state wheel temperature Θ∞ can be calculated from the energy balance ˙ 3D + q˙ 3D εPfriction + Q rail convection = 0.
(64)
With Eqs. (61) and (63), the result is 3D Θ∞ =
Fig. 9. Surface temperature for elliptical area of contact.
Pfriction , √ 7bβr av0 /2π + αAwheel /ε
taking into account convection as well as heat conduction from wheel into rail. Similar to Eq. (59), the average contact temperature in steady-state is 3D 3D 3D Θmean,∞ = Θmean + εΘ∞ .
with the heat partitioning factor ε ≈ 0.5 for small sliding velocities, i.e. vs v0 , and equal thermal penetration coefficients β of wheel and rail. 7.2. Heat conduction from wheel into rail The heat conduction from wheel to rail due to an initial temperature of the wheel (Section 4) can also be investigated independently within any strip. But the calculation of the steady-state wheel temperature is not as easy as in the twodimensional case (Section 6). If one assumes that the initial temperature is constant over the whole width of the contact patch, the heat flow per unit width within one strip, Eq. (46), can be integrated over the width 2b of the contact ellipse in order to get the total heat flow. This gives approximately av0 3D ˙ Q . (61) rail = −7εbβr Θw0 2π The steady-state wheel temperature Θ∞ without convection can be calculated from the energy balance of frictional heating and heat loss as P 2π Pfriction friction 3D Θ∞ = = 0.358 √ . (62) 7bβr av0 bβr av0 Another approach is possible if heat conduction between strips is neglected. Thus, one has to calculate a steady-state temperature for every strip as in Section 6 and take the average value over the whole contact ellipse. This result differs by only 4% from Eq. (62).
(65)
(66)
It should be mentioned that the heat conduction from wheel into rail can be calculated quite accurately compared with convection. Since the air flow around the wheel is very complicated and depends on the aerodynamics of the train, the average heat transfer coefficient α can only be estimated roughly. 7.4. Transient behaviour The transient thermal analysis is only possible with a finite element model of the wheel. This has been done by Gupta et al., but only for the frictional heating [9]. In their work, the intermittent heat flow due to the consecutive contacts has been replaced with an equivalent reduced heat flow rate applied continuously to the entire circumference. The effect of heat conduction from wheel to rail has been considered by a modified heat partitioning factor ε. With the results of Section 7.2, this approach can be improved. Substituting the intermittent heat transfer from wheel to rail by a continuous flow, this can be included as a sort of convection, with the heat transfer coefficient 7εβr av0 αrail = , (67) r0 32π 3 applied to the width 2b of the contact patch over the entire circumference 2π r0 of the wheel. Neglecting first the convection into ambient air, this gives the same value for the steady-state wheel temperature as in Eq. (62). On the other hand, the transient behaviour can be calculated at any time. Convection into ambient air can be taken into account additionally. More details and results will be presented in a separate paper [18].
7.3. Convection 7.5. Application to tread-braked wheels Convection occurs over the whole surface of the wheel. Based on the consideration of the Biot number in Section 5, a first-order estimation of the heat flow due to convection is possible. If the wheel temperature everywhere is Θw0 , then
The focus of this paper is on the contact temperatures due to slip between wheel and rail. High temperatures and severe thermal stresses in wheels are also caused by the
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Table 3 Summary of results for elliptical contact area (all temperatures in ◦ C) Units
Low speed
High speed
N (kN) v0 (m/s) vs (m/s) µ Pfriction (kW)
100 30 1 0.3 30
100 90 3 0.1 30
Instantaneous contact temperature Maximal temperature
Θmax
149.7
86.4
Eq. (37)
Average contact temperature
3D Θmean
81.7
47.1
Eq. (59)
182.6 173.7
105.4 100.3
Eq. (62) Eq. (66)
153.5 159.0
95.0 95.1
Eq. (65) Eq. (66)
3D Θ∞ 3D Θmean,∞
116.4
79.4
Eq. (65)
140.3
87.2
Eq. (66)
3D Θ∞ 3D Θmean,∞
63.0 113.4
50.3 72.5
Eq. (65) Eq. (66)
3D Θ∞ 3D Θmean,∞
9.1
8.8
Eq. (65)
86.3
51.6
Eq. (66)
Normal load Vehicle speed Sliding velocity (longitudinal) Coefficient of friction Frictional power dissipation
Steady-state: wheel temperature and average contact temperature 3D Without convection Θ∞ α = 10 W/K m2
3D Θmean,∞ 3D Θ∞ 3D Θmean,∞
α = 30 W/K m2 α = 100 W/K m2 α=
1000 W/K m2
non-uniform heating which results from tread braking [8,19]. This is not a rolling contact problem since the brake shoe stays always in contact with the moving wheel tread and the sliding velocity is equal to the circumferential speed of the wheel. It can, therefore, be assumed that nearly all the heat flows into the wheel [5]. Thus, the average contact temperature between brake shoe and wheel from the current frictional heating can be estimated with the equations presented in this paper if the heat partitioning factor ε (Eq. (25)) is set to unity in this case. Some investigations of this problem neglect the periodic contact of the wheel tread with the brake shoe and take only the continuously increasing bulk temperature of the wheel into account [19]. However, after long periods of braking, the steady-state temperature of the wheel will again depend on convection as well as on conduction from wheel to rail through the contact patch. This can also be calculated with Eq. (65), the frictional power now resulting from braking and wheel/rail contact at the same time.
8. Discussion For constant operating conditions in steady-state, the lower limit of the average contact temperature in wheel/rail contact is the instantaneous peak due to the current frictional heating, i.e. without an initially increased wheel temperature. Due to heat conduction from wheel to rail, the upper limit is approximately twice as high if convection is neglected. Taking convection into account, the average contact temperature will always be between these limit values.
Eq. (54)
In Table 3, the instantaneous contact temperature and the steady-state wheel temperature are given for two different vehicle speeds with equal values of the frictional power dissipation. All other parameters are also equal (Table 2). The results show that the temperature decreases with increasing vehicle speed. The steady-state temperature Θ∞ depends mainly on heat conduction from wheel into rail. It is even lower if convection is also taken into account. If the heat transfer coefficient α is in the range of 50–100 W/K m2 , the heat flow into ambient air is nearly equal to the heat conduction from wheel into rail. Thus, Θ∞ is only half as high as without convection. With the overestimation α = 1000 W/K m2 , the average wheel temperature would be nearly equal to ambient temperature. The data in Tables 2 and 3 are usual operating conditions of European locomotives. Maximum contact temperatures in these cases are not high enough to explain thermally induced phase transformations. This may only be the case with extreme conditions, e.g. blocking wheels where the sliding velocity is equal to the vehicle speed. On the other hand, contact temperatures give rise to severe thermal stresses. It has been shown in a number of investigations that thermal stresses in railway wheels and rails can be in the same order of magnitude as the stresses due to mechanical loading, e.g. [8,19,20]. A surface temperature change of 200 ◦ C would result in thermal compressive stresses σx = σy ≈ −700 MPa. This may cause plastic deformation, residual stresses and work hardening at the surfaces of wheel and rail. Since the thermal penetration depth δ is very small, thermally induced plastic deformations are restricted to a very thin surface layer [7,21].
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9. Conclusions The heat flow in wheel/rail contact with smooth surfaces is one-dimensional. Some methods for the calculation of contact temperatures with this model are presented. The maximum surface temperature can be estimated with the flash temperature formula of Blok. Semi-analytical and numerical methods are available for a more detailed investigation. With a polynomial approximation, the case of Hertzian contact can be investigated very efficiently. While the contact temperatures are confined to a very thin surface layer, the bulk temperature of the wheel also increases with time by continuous frictional heating. It can be shown that the wheel temperature for constant operating conditions cannot be more than twice the average temperature for the first contact of the cold wheel. This limit is due to heat conduction from the hot wheel into the cold rail. It corresponds to the usual assumption that the wheel is an insulator and all the frictional heating flows into the rail. If convection is only taken into account on the wheel tread, i.e. on the area that is subjected to frictional heating, it can be included into the usual model for contact temperature calculations [11]. But in this case, its influence is small enough to be neglected. On the other hand, real convection occurs on the whole surface of the wheel. This can only be considered approximately. Therefore, the steady-state temperature of the wheel is even lower. Structural changes in the rail material such as formation of WEL are unlikely to occur due to high contact temperatures only. They are probably the consequence of high mechanical stresses and moderate contact temperatures that result in high thermal stresses. References [1] H. Blok, Theoretical study of temperature rise at surfaces of actual contact under oiliness lubricating conditions, Proc. Gen. Discuss. Lubricat. Inst. Mech. Eng. London 2 (1937) 222–235. [2] J.C. Jaeger, Moving sources of heat and the temperature at sliding contacts, Proc. R. Soc. NSW 56 (1942) 203–224. [3] H. Blok, The flash temperature concept, Wear 6 (1963) 483–494.
[4] H.C. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, 2nd Edition, Oxford University Press, Oxford, 1959. [5] J.F. Archard, The temperature of rubbing surfaces, Wear 2 (1958–1959) 438–455. [6] M.A. Tanvir, Temperature rise due to slip between wheel and rail— an analytical solution for Hertzian contact, Wear 61 (1980) 295–308. [7] K. Knothe, S. Liebelt, Determination of temperatures for sliding contact with applications for wheel–rail systems, Wear 189 (1995) 91–99. [8] G.J. Moyar, D.H. Stone, An analysis of the thermal contributions to railway wheel shelling, Wear 144 (1991) 117–138. [9] V. Gupta, G.T. Hahn, P.C. Bastias, C.A. Rubin, Calculations of the frictional heating of a locomotive wheel attending rolling plus sliding, Wear 191 (1996) 237–241. [10] A. Cameron, A.N. Gordon, G.T. Symm, Contact temperatures in rolling/sliding surfaces, Proc. R. Soc. A 286 (1965) 45–61. [11] F.D. Fischer, E. Werner, K. Knothe, The surface temperature of a halfplane heated by friction and cooled by convection, Z. Angew. Math. Mech. 81 (2001) 75–81. [12] F. Bucher, A. Theiler, K. Knothe, Normal and tangential contact problem of surfaces with measured roughness, Wear 253 (2002) 206–220. [13] J. Gao, S.C. Lee, X. Ai, H. Nixon, An fft-based transient flash temperature model for general three-dimensional rough surface contacts, J. Tribol. 122 (2000) 519–523. [14] K.L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, 1985. [15] H.D. Baehr, K. Stephan, Heat and Mass Transfer, Springer, Berlin, 1998. [16] J.R. Barber, F.B. Ammar, H.G. Georgiadis, Conductive heat exchange between bodies which are in contact for a very short period of time, Collected Papers in Heat Transfer, Vol. 123, ASME, New York, 1989, pp. 101–106. [17] W. Beitz, K.H. Küttner (Eds.), Dubbel–Taschenbuch für den Maschinenbau, 17th Edition, Springer, Berlin, 1990. [18] J. Dohrmann, M. Ertz, K. Knothe, Transient temperature calculation for locomotive wheels, in preparation. [19] M.R. Johnson, R.E. Welch, K.S. Yeung, Analysis of thermal stresses and residual stress changes in railroad wheels caused by severe drag braking, J. Eng. Ind. 99 (1977) 18–23. [20] V. Gupta, G.T. Hahn, P.C. Bastias, C.A. Rubin, Thermal-mechanical modelling of the rolling-plus-sliding with frictional heating of a locomotive wheel, J. Eng. Ind. 117 (1995) 418–422. [21] G. Baumann, Untersuchungen zu Gefügestrukturen und Eigenschaften der “Weißen Schichten” auf verriffelten Schienenlaufflächen (Investigations on structures and properties of white-etching layers on corrugated rail treads), PhD Thesis, Technische Universität, Berlin, 1998.