INTRODUCTION
The contact temperature distribution at the interface of rail and wheel materials
has been a topic for along time tribology research. Contact temperature distribution
is critical for thermal stress analysis between two sliding bodies (Johnson
and Greenwood, 1997; Chen and Wang, 2008) and thermal
wear modeling (Bucher et al., 2006; Fischer
et al., 1997). Contact temperature may influence material properties
(Habeeb et al., 2008). Heat is generated due
to frictional heating when a body slides on another body. The heat generated
between two contact bodies is influenced of the thermal properties of the bodies,
the contact geometry and the sliding speed.
One of the first models for estimating contact temperature was proposed by
Blok (1963). The maximum temperature rises at the interface
of two bodies are the same. The maximum temperature is independent of the speed
at low Peclet numbers. The maximum temperature rise at the interface was calculated
using a stationary heat source. Similar with Blok (1963)
analysis, Carslaw and Jaeger (1938) and GallardoHernandez
et al. (2006) presented an analytical integral formula for the temperature
rise on the halfspace surface. However, their method was based on the average
temperature rise instead of the maximum temperature rise.
The main objective of this study is to determine contact temperature on interaction
between rail and wheel materials for various applied load and sliding velocity.
The present model is based on pinondisc method. The model was applied to simulate
the sliding contact of a halfspace wheel material over a stationary sphere
rail material. Results of pressure and contact temperature under different sliding
speeds and applied load were discussed and compared.
MATERIALS AND METHODS
Contact problem: There are two kinds of mechanism contacts between rail
and wheel (Fig. 1). First existing one is rollingsliding
contact which occurs on railhead and wheels tread contact. The second is pure
sliding contact which occurs on rail edge and wheel flange. In this study, the
contact between rail and wheel can be assumed as non conformal contact.

Fig. 1: 
Overview of wheelrail contacts 

Fig. 2(ab): 
Pinondisc scheme 
A non conformal contact of two bodies in revolution is classified in the general
elastic contact. Depending on their modulus of elasticity, the two bodies will
deform to certain degrees. The problem may be reduced to that of a single equivalent
elastic body revolution on a plain infinitely rigid body. This assumption is
widely adopted in the theory and application of Hertzian contacts so that the
conversion formulas of geometry and elasticity are well established (AlBender
and De Moerlooze, 2008; Enblom and Berg, 2008).
When at the top of an elastic pin of radius r and equivalent modulus of elasticity
Ee (Fig. 2) is pressed with a load N against a rigid plane
disc surface, there is a mark on the contact area (Knothe
and Liebelt, 1995; Olofsson and Telliskivi, 2003;
Sundh and Olofsson, 2011). The equivalent modulus elasticity
is given by:
The pin is rotating with a fixed angular speed ω. Due to the normal force,
angular velocity and coefficient of friction μ, a reaction force will occur.
This reaction force is frictional force F_{f}:
The net force F_{Net} is equal applied force F_{app} minus
frictional force F_{f} Eq. 3. The contact patch is
defined as the region A in the xyplane: A = {(x, y): x^{2}+y^{2}
= r^{2}}.
The normal stress P_{z} is given by (AlBender and
De Moerlooze, 2008):
The magnitude of these axes depends on the radii and the radii of curvature
of the contacting systems and on the normal load.
Since, the maximum heat flux occurs in the z direction, the problem can be
simplified into two dimensional contacts. The maximum pressure for this case
is (Jendel, 2002):
The materials used in this research are rail steels. The chemical composition
of material is shown in Table 1.
Pinondisc is commonly used in wear test. The tests use Ducom multi specimen
testing machine designed according to ASTM G99 standards. Rail steel was cut
to form disc specimen which is 42 mm in diameter and 5 mm in width. Pin samples
were prepared as 6 mm in diameter and 12 mm in length. During wear tests, the
normal forces were applied. The normal force of 40 N, 60 N, 80 N and 100 N were
selected. Both pin and disc sample were polished using 120, 220 and 500 grit
abrasive papers and cleaned with alcohol and dried.
Heat transfer equations: The assumption is introduced an energy equation
problem to determine the temperature field in the wheelrail contact. The energy
equation heat transfer equation will be governed based on Fig.
2:
If the density of a solid is constant, cp = cv. Dividing Eq.
6 by k and replacing k/ρcp by α, the thermal diffusivity of the
material:
The differential equation that governs the steady state temperature field is:
Mazidi et al. (2011) proposed that there is
a heat partition at the contact surface of two sliding components, because of
thermal resistance due to accumulation wear particles. If the frictional heat
is the only internal heat generation source (Ertz and Knothe,
2002), then:
Where 
v 
= 
Sliding velocity and within the contact area heat is generated
owing to friction. The following assumptions were made: 
• 
There is pure sliding within the whole contact area 
• 
The effects of elastic deformation are negligible. Consequently, the local
relative velocity equals the global sliding velocity v_{s} 
• 
The heat flow rate q (i.e., the product of the friction coefficient μ,
the normal pressure P(x) and the sliding velocity) is transformed completely
into heat 
Temperature prediction of pin and disc: The schematic diagram of the
temperature at some point (r, θ, z, t) within a pin cylinder of length
2L and radius R, is shown in Fig. 2. The cylinder initially
at temperature Ta, at time zero, it is exposed to a fluid at temperature T∞,
with convection coefficient of h, Temperature within the cylinder is a function
of (r, θ, z, t), for a system with internal heat generation:
The differential equation can be solved by the Adomian Decomposition method,
with the assumption of coefficient of friction is constant. Solution for the
temperature of any point located at the position (r, θ, z, t) of a cylinder
at any time t is T = T (r, θ, z, t) and k is thermal diffusivity. Equation
10 can be rewrite in an operator form by:
where, the differential operators L_{r}, L_{θ} and L_{z}
are defined by (Wazwaz, 2002):
So that the integral operator L_{t}^{1} exist and given by:
Applying L_{t}^{1} to both sides of Eq. 13
and using the initial condition lead to:
The decomposition method defines the solution T (r, θ, z, t) as a series
given by:
Substituting Eq. 14 into both sides of Eq.
15 yields:
The components T_{n} (r, θ, z, t), n≥0 can be completely determined
by using the recursive relationship:
Where:
RESULTS AND DISCUSSION
Figure 3 shows comparisons of pressure distribution for various
applied load. The pressure distribution is calculated based on Hertzian contact
solutions. Pressure distribution obtained from the purely elastic analysis well
matches that of the analytical Hertz solution.

Fig. 3: 
Pressure distribution for various applied load 
Table 2: 
Contact parameters and material properties 

The peak pressure distribution occurs at the centre of contact, x = 0. The
peak pressure distributions increase linearly with the increasing applied load
with an average slope of 0.053. These results are same trend with Windarta
et al. (2011) conclusions. The pressure and wear rate increase linearly
with increasing applied load.
However, the temperature rise due to frictional heating softens the material
(degrades the yield strength) when the thermal softening effect is included
which further decreases the von Mises stress intensity. Table
2 contains the circular contact parameters and material properties for temperature
calculation throughout this study.
Figure 4 presents the comparison of contact temperature profiles
along the x axis for various models. Chen and Wang
(2008) Model assumes purely elastic contact based on Hertzian contact. There
is not any temperature rise outside contact area. The contact pressure is approximately
symmetric with respect to x = 0. Ertz and Knothe (2002)
proposes a model using four degree polynomial. However, frictional heating distorts
the contact pressure distribution under higher sliding speeds and shifts the
pressure profile along the mating surface sliding direction. Proposed model
modified (Ertz and Knothe, 2002) model, using assumption
that frictional heating outside contact produce by conduction and fixed source
from contact area.

Fig. 4: 
Contact temperature rise comparison along the longitudinal
(x) axis with respect to various models 

Fig. 5: 
Peak temperatures rise with respect to sliding velocity 
The maximum temperature rise is 259.98 K at x = 1 mm for 5.3 MPa pressure and
3.14 m sec^{1} sliding speed. The peak temperatures rise with respect
to sliding velocity present in Fig. 5.
The surface temperature rise model developed is evaluated based on a pure sliding
circular Hertzian contact between an adiabatic moving surface and a stationary
surface is considered. In this case, all of the frictional heat is transferred
into the stationary surface.
Comparisons between prediction temperature rise and pinondisc measurement
results with respect to sliding velocity are presented in Table
3. The measurement were taken 6 mm from the centre of pin and disc contact.
Average error between prediction and measurement results is 4.99%. The highest
error occurs at low sliding velocity (0.16 m sec^{1}).
Figure 6 gives the variation of the surface temperature rise
distribution along the xaxis with respect to the increasing sliding speed.

Fig. 6: 
Distributions of surface temperature rise along the x axis
corresponding to various sliding velocities 
Table 3: 
Temperature rise comparison for various sliding speed 

The surface temperature rise increases noticeably as the sliding speed increases
(the maximum temperature increases by about 500% when the sliding speed changes
from 0.163.14 m sec^{1}. The temperature rise distribution becomes
skew under higher sliding speeds (the maximum temperature shifts from the center
to the leading edge of the contact area). This trend of temperature with the
sliding speed is consistent with that by Chen and Wang (2008).
CONCLUSION
The pressure distribution is calculated based on Hertzian contact solutions.
Pressure distribution obtained from the purely elastic analysis well matches
that of the analytical Hertz solution. The peak pressure distributions increase
linearly with the increasing applied load with an average slope of 0.053.
The maximum temperature rise is 259.98 K at x = 1 mm for 5.3 MPa pressure and
3.14 m sec^{1} sliding speed. Prediction temperature rise results were
successfully validated using pinondisc method. Average error between prediction
and measurement results is 4.99%. The highest error occurs at low sliding velocity
(0.16 m sec^{1}).
ACKNOWLEDGMENT
The authors would like to thank Keretapi Tanah Melayu Berhad (KTMB) for providing
railway and wheel materials. This research is under Graduate Assistantship Universiti
Teknologi PETRONAS.
NOMENCLATURE
a_{0}, p_{0} 
= 
Hertz contact radius (mm) and peak pressure (MPa) 
E_{i} 
= 
Young’s modulus two contact bodies (i = 1 and 2) 
E* 
= 
Equivalent Young’s modulus (GPa) 
H 
= 
Hardness (GPa) 
k 
= 
Thermal conductivity (W m^{1} K^{1}) 
l 
= 
Characteristic length (mm) 
p, s 
= 
Pressure and shear traction (MPa) 
q, q1, q2 
= 
Total heat flux and heat flux flowing to two bodies (W m^{2}
) 
ΔT 
= 
Temperature rise (K) 
Tm, Ta 
= 
Material melting point and reference room temperature (K) 
Vs 
= 
Sliding velocity (m sec^{1}) 
W 
= 
Normal load (N) 
x, y, z 
= 
Space coordinates (mm) 
α 
= 
Linier thermal expansion coefficient (mm K^{1} m^{1}) 
δ 
= 
Rigid body approach (mm) 
Δ 
= 
Mesh size (mm) 
μ_{f} 
= 
Coefficient of friction 
υi 
= 
Poisson ratio of two contact bodies 
σij 
= 
Total stress component (MPa) 
σ_{VM} 
= 
Von Mises equivalent stress (MPa) 
σy 
= 
Initial yield strength (MPa) 