10th

International Conference on Contact Mechanics

CM2015, Colorado Springs, Colorado, USA

A Numerical Procedure for Analysis of W/R Contact Using Explicit Finite

Element Methods

Y.Ma V.L.Markine

Section of Road and Railway Engineering, Faculty of Civil Engineering and Geosciences, Delft University of

Technology, Stevinweg 1, NL-2628, CN Delft, The Netherlands

e-mail: Yuewei.Ma@tudelft.nl

ABSTRACT

Since no effective experimental approaches have been proposed to assess wheel and rail (W/R) contact performance

till now, numerical computational analysis is known as an alternative to approximately simulate the W/R interaction. In

this paper, one numerical procedure is proposed on the basis of explicit finite element method to analyse the complex

stress state of W/R contact. Moreover, a novel refining method is developed and utilized to discretise both the W/R

solid model. This model considers a realistic wheel and rail geometry and bilinear kinematic hardening material. The

effects of contact points, friction force and contact stiffness on the dynamic response of wheel-rail interaction are

investigated. Based on these results, it can be noticed that the numerical procedure not only provides much realistic

results but is also flexible and efficient enough for dealing with different wheel-rail operational parameters. Also, it is

worth mentioning that this numerical procedure is not limited to straight tracks, can easily be extended to be used for

curved tracks and turnout as well.

1.INTRODUCTION

Prior to understanding the damage mechanism

involved in the friction between wheel and rail, the

contact has to be analysed, which is highly complex

and characterized by multi-axial and out-of-phase

loading. Finding the contact area and pressure

distribution requires complicated programming and

extensive calculation efforts. Thus, intensive research

efforts have been made on developing efficient and

effective numerical procedures for the solution of

wheel-rail contact problems.

Since the pioneering work on contact by Hertz in 1882,

numerous attempts[1, 2] on close-form analytical

solutions have been reported. Although analytical

solutions for wheel-rail contact problem can be used

efficiently, they are limited to the assumptions of half-

space, linear-elastic material behaviour etc. When these

typical assumptions are dropped off or enhanced

geometries are to be considered, the governing

equations become very complex and cannot be solved

in a closed-form manner. To overcome the limitations

inherent in the analytical approaches, Telliskivi et al[3]

developed a FE-based tool to investigate the contact

conditions for measured wheel-rail profiles of an

ordinary track, in which quasi-static loads were

obtained from multi-body dynamics programs and

elastic-plastic material model was adopted. Besides,

Ringsberg et al. [4] developed a full-scale, three-

dimensional, quasi-static model to investigate the

residual stresses and plastic strains in the rail caused by

global bending and local contact forces. Based on his

proposed model, the fatigue life models integrated with

the specific cyclic-hardening material models were

adopted to predict the crack initiation life[5](see also

Liu et al.[6]). Other static or quasi-static FE-models

could be found in [7-11]. However, in order to simulate

the wheel-rail contact more accurately and realistically,

the dynamic contact behavior should be taken into

account. In[12], a three-dimensional transient finite

element model was presented, which has been

implemented to identifying the rail defects, such as

squats, head checks et al. In this model, a “cylinder”

was adopted to represent the simplified wheel

geometry, which might be limited to applications only

on the straight track. In [13, 14], a specific FE-model

accounting realistic wheel and crossing nose geometry

was utilized to study the dynamic process when a

heavy haul train passed over a specified turnout.

Up to now, although significant progress has been

made over the past decades, it was found that few of

the above mentioned numerical models can incorporate

realistic wheel/rail contact geometry together with non-

linearity of material properties and dynamic behavior.

In addition, the commonly used adaptive refining

method when rails and wheels are discretized through

extruding and rotating the 2D refined cross section,

often results in poor quality elements in the out of

contact region and also generates a large amount of

elements which should be reduced as much as possible

in order to save the calculation expense[3, 12-15].

Despite the fact that “bonded contact” interfacing

between coarse and dense mesh in the contact region

can decrease the amount of elements[4, 7, 16], the

influence of the “boned contact” set-up on the obtained

solution has to be more carefully studied.

For the purpose of overcoming the above problems, a

three-dimensional, transient W/R contact model is

developed by using both implicit (ANSYS APDL) and

explicit finite element package (ANSYS LS-DYNA).

In this model, a novel 3D refining method is proposed

and applied on both wheel and rail contact region.

Detailed information about the W/R FE model are

outlined as well. Following that, the obtained dynamic

stress/strain responses under a specific loading

condition would be discussed in section 4. Furthermore,

parametric studies on the friction force, contact point

and contact stiffness variation are highlighted in

section 5. In the end, concluding remarks and outlooks

are presented..

2. REFINEMENT METHOD AND DYNAMIC FE-

MODEL

Following the idea of a classical finite element

approach for nonlinear contact problems, the un-

deformed structure is required to be discretized and

further refinement in the vicinity of stress

concentration area has to be performed to achieve the

desired accuracy. In the case of rail and wheel rolling

contact, a couple of difficulties will arise due to the

special contact geometry and rather small resulting

contact patch[12]. First, a long part of wheel and rail

interface needs to be refined in their circumferential

and longitudinal directions respectively. In this region,

the refined element size would be as small as 1000

times over the dimension of wheel and track structure.

Second, the reliability of obtained solutions has to be

checked as well as the accumulated numerical errors.

In order to avoid the above mentioned meshing

drawbacks in chapter 1, one novel refining

methodology, namely nested transition mapped

hexahedral meshing, is developed and used in this

model. The main idea of this novel refining method is

demonstrated in figure 3, in which two six-sided blocks

are generated and overlapped initially. After that, the

overlapped blocks will be segmented into five new

ones through subtracting Boolean operations integrated

with ANSYS. By specifying the line divisions on the

edges of the split blocks, a transition mapped

hexahedral finite element model tends to be obtained

(shown in figure(a-c)). To achieve the nested transition

mapped hexahedral meshing as shown in figure 3(d),

the refining process from (a) to (c) has to be repeated

again on the smallest split block. After finish the nested

transition mapped hexahedral refining process in the

vicinity of contact area, the mesh size in out-of-contact

regions would be much more enlarged, which would be

significantly helpful for reducing the amount of

element.

In order to clearly demonstrated the wheel-rail contact

system, a schematic diagram of wheel-rail dynamic

model was shown in figure 2. The sprung mass is

lumped and supported by a group of springs and

dampers of the primary suspension. The axle load is

chosen as 20t, which means the static load applied to

the wheel is 10t. Two considered counterparts are a

S1002 wheel profile and a 54E1 rail. The wheel has a

nominal radius in rolling direction of 460mm and a

cant angle equivalent to 1/20 was also applied onto the

rail. The inner gauge of the wheelset is 1360mm and

the track gauge is 1435mm.

Fig. 1. Nested transition mapped hexahedral refining procedure

The rolling distance is restrained to be 240mm, while

the length of super refined regions on both wheel and

rail is set to 60mm. In order to reduce the calculation

expense, a 680 mm length of track was selected to

cover the distance between two sleepers[13, 17, 18].

The initial wheel/rail contact position is taken as 0.14m

far away from the origin of the coordinate. For the

model the coordinate directions are : the Z-

axis(longitudinal direction)is parallel to the direction in

which wheel travels, the Y-axis is the vertical direction

and the X-axis is the lateral direction. The wheel is set

to roll along the rail with prescribed angular and

translational velocities. The default translation velocity

of the wheel was held constant throughout the

calculation at 140km/h. Before the application of

velocity, the rotation of the wheel around the Y-axis

and the Z-axis was disabled since it is assumed that

changes in the wheel set’s yaw and roll angles are very

small over a short rolling distance.

Under this condition, the contact patch is located at the

centre of the railhead. A three dimensional model,

composed of a half wheel and a rail is shown in figure

2. The element size is the solution region is set to be

1.0 1.0 1.0 mm using the proposed refining method.

All together the model consists of 110,000 eight-noded

hexahedral solid elements. In the contact region extra

care are taken for the elements to be of good quality,

while less effort was spent outside the contact region

and distorted elements were accepted.

Tab 1. Material and primary suspension parameters[12].

Components Parameters Values

Primary

suspension

Stiffness / K1 1150MN/m

Damping / C1 250KNs/m

Wheel and rail

material

Young’s modulus 210 GPa

Shear modulus 21GPa

Passion ratio 0.3

Density 7900 kg/m3

Yield stress 480MPa

A classical bi-linear elastic-plastic material model was

utilized for wheel and rail materials properties ( as

(d) (c)

(a) (b)

denoted in table 1). The wheels and rails are connected

with “surface-to-surface” contact constraint. The

friction between wheel and rail is effective with an

applied friction coefficient of 0.5, which is used to

handle the stick and slip situation of the contact

surfaces between wheel and rails .

Fig. 2. wheel-rail contact dynamic FE model. a) Schematic diagram of

FEM; b) FE model – side view; c) Nested mesh at rail refined region;

d) FE model – cross sectional; e) Close-up view in the refined contact

region.

3. RESULTS AND DISCUSSIONS

Using the model presented in section 2, a numerical

study is conducted to investigate the contact behaviour

under operational loading. For the simulation of the

model shown in figure 2, it takes about 4 hours under

the element size of 1.0 1.0 1.0 mm in the refined

contact region with a 3.10 GHz Pentium 16 processors’

workstation to run over a rolling distance of 0.24m.

3.1. RESULTANT INTERFACE FORCES

In order to save the calculation efforts, the wheel will

roll a limited distance from -120mm to 120mm (shown

in figure 2) and the dynamic response over this region

will be investigated in current study. The resultant

contact forces along and perpendicular to the rolling

direction are of maximum interest since they are

related to the axle load and applied traction force. From

figure 3, the resultant reaction force variation on the

rail surface with respect to rolling distance can be

observed. The vertical force FY oscillates distinctly at

the initial period rolling from -120mm to 50mm, which

is caused by the implicit dynamic relaxation process.

Later on, it almost keeps constant and is slightly

varying about the wheel axle load of -100KN. Since

the wheel motion towards to the X direction is

constrained, the lateral reaction force FX is nearly

unchanged at zero. While for the longitudinal resultant

reaction force FZ, it decreases from zero to -25KN

from -120mm to -80mm and then remains constant.

This phenomena can be explained by the fact that a

short response distance is required for the wheel

acceleration varying from 0 to a constant value because

of the initial angular velocity and translational velocity

applied on the wheel. The magnitude and direction of

FZ is related to applied traction, which can be varied

from different operational conditions such as braking

and accelerating.

Fig. 3. Maximum Von Mises stress and contact pressure variation w.r.t

rolling distance.

Besides, aiming to check the surface contact pressure

and sub-surface stress distribution, the result at the

moment when the wheel rolls over the origin will be

extracted and discussed in the following subsection.

3.2. SURFACE PRESSURE

From figure 4, normal contact pressure distribution on

the rail surface is observed with a 3D shaded surface

plot and 2D contour plot at the origin. In order to

demonstrate the surface pressure distribution better, the

compressive normal pressure is treated as positive. Due

to the non-Hertzian contact conditions, a very high

normal contact pressure which amounts to 1010MPa is

obtained. The contact patch is not a standard ellipse

and asymmetric with respect to its central axis. The

length of the contact patch in X and Z direction is

17mm and 14mm respectively. Approximately, the

area of the contact patch is 150mm2. The reason for

the non-elliptical and larger contact patch in

comparison to literature [12] is attributed to changing

radius of curvature of the realistic wheel/rail

geometries being in contact as well as the cant angle

considered in the model.

Y

0 60

0

solution area

dynamic relaxation area

coarse meshed area

further rolling area

(c) 16 0

(a) (b)

(d) (e)

0 30 -30 -1 0 1 0 tart point nd point

Fig. 4. rail surface normal contact pressure distribution at origin Z = 0.

a) 3D shaded surface plot; b) 2D Contour plot.

Accurate estimation of wheel-rail friction levels is of

extreme importance in train simulation, since the

magnitude of the frictional force on the contact

interfaces can determine the crack initiation path and

its propagation as well as the development of wear.

From figure 5, the distribution of surface shear

pressure is observed and it can be seen that the

maximum surface shear pressure amount to as large as

500MPa, which cannot be ignored in comparison to the

normal pressure. The surface shear pressure ( shown in

figure 2(b) contour plot) is mainly distributed at the

trailing edge of the contact patch instead of leading

edge. The surface shear pressure (shown in figure 5(c)

quiver plot) is pointing at the direction of wheel

moving, which is logical and consistent with the

direction of resultant longitudinal force FZ(shown in

figure 4). The criteria for distinguishing the slip and

stick area is the same as the one used in [12]. As can be

seen from figure 5(d), the leading edge of the contact

patch is still in stick, whereas the trailing edge is in

micro-slip. The micro-slip phenomenon discussed in

figure (5) is also consistent with the analytical results

presented in [19].

Fig. 5. rail surface shear pressure distribution and slip-stick area

distribution at origin Z = 0mm. a) 3D shaded surface plot ; b) 2D

Contour plot; c) Quiver plot; d) Slip-stick area plot.

3.3. SUB-SURFACE PRESSURE

Generally, Von-Mises stress is adopted as a measure[6]

of material performance assessment under specific

contact loading conditions for elastic-plastic material.

From figure 6 (a) and (d), the distribution of Von-

Mises stress inside the rail contact patch at the moment

of wheel passing through the origin Z = 0mm is

observed. The shape of the Von Mises stress is similar

to the normal surface pressure contour plot shown in

figure 4(b).

In order to check the sub-surface stress response, two

cutting surfaces, namely A-A in lateral-vertical plane

and B-B in longitudinal-vertical plane, are created. The

stresses mapped on the two cutting surfaces are shown

in figure (6) (b-c, e-f). For the stress distribution on the

A-A cutting surface, the maximum Von Mises stress

occurs at the rail top surface and the shear stress

exhibits two equal size compressive and tensile

components. This phenomenon can be attributed to the

contact angle between interfaces, which will lead to a

tendency for preventing the wheel sliding away from

the rail surface. Since the magnitude and area of the

tensile and compressive shear components are equal, it

is logistical with the FX lateral force variation shown

in figure 3.

While for the stress results on B-B cutting surface, the

maximum Von Mises stress moves from sub-surface to

the surface caused by the large shear stress at the

trailing edge of the contact patch. The tensile and

compressive shear stress components are different from

the one on A-A cutting surface, which is resulted from

the traction moment applied on the wheel axle. The

results obtained in this paper is also comparable with

reference [6].

Fig. 6. Stress distribution in longitudinal-vertical plane at origin Z =

0mm. a) Cutting surface on X-Y Plane; b) Von-Mises stress on A-A

Cutting plane; c) Shear stress on A-A cutting plane; d) Cutting surface

on Y-Z Plane; e) Von-Mises stress on B-B Cutting plane; f) Shear

stress on B-B cutting plane

4. PARAMETRIC STUDIES

(a) (b)

(b) (a)

(c) (d)

A

A B B

(d) (a)

(b) (e)

(c) (f)

In order to check the validity and flexibility of the

model, parametric studies related to contact point ,

frictional force and contact stiffness will be conducted

in this chapter.

4.1. INFLUENCE OF CONTACT POINT

Generally, the wheelsetis not fixed to the central line

for enhancing its track negotiating capability under

different track conditions such as curves, turnouts[20].

Because of lateral shift of the wheelset relative to the

rail, the contact conditions in both wheel and rail

coordinates will be changed. Considering the fact that

contact conditions of wheel and rail interface are

characterized by the geometry at the contact point,

itsdistribution under different lateral shifts of

wheelsetwas determined using the developed

geometrical contact model{Ma, 2012 #2223}by

calculating the minimum distance between the profiles

in vertical direction (as shown in figure 16). The other

parameters were kept the same.

Fig. 7 Contact points distribution. a) 5.5mm; b) 2.0mm; c) -3.5mm; d)

-5.5mm.

Moreover, based on the developed numerical models,

the influence of the contact point variation under

different lateral wheelset shifts (-5.5, -3.5, -2, -1, 0, 1, 2,

3.5, 5.5mm ) on wheel-rail interaction would be

assessed in this section. The resultant interface force

variation with respect to rolling distance are shown in

figure 8. It can be noticed that between a lateral shift of

-5.5mm and -3.5mm the vertical resultant force shows

a larger deviation in comparison to others. Besides, the

magnitude of resultant lateral force is higher than the

ones under other lateral shifts.

Fig. 8. Resultant interface force variation w.r.t rolling distance. (Solid

line: FY; Dash-dot line: FX; Dashed line FZ).

Furthermore, the maximum normal and shear contact

pressure between the wheel and rail interface are

shown in figure 9. It has to be mentioned that both the

contact area and pressure are significantly different.

This can be explained by the geometrical change of

contact angles and the radii of curvature at different

contact points.

Fig. 9. Comparison of the position of contact patches and pressure

distribution of left wheel-rail pair on the variation of wheelset lateral

shift.

For a lateral shift of -5.5mm, contact occurs between

the wheel flange root and rail gauge corner. In this area,

the wheel-rail contact angle is significant with respect

to the horizontal plane. Consequently, a larger lateral

force will be generated which plays a more important

role in wheel-rail interaction. This explanation can also

be verified by the surface shear pressure quiver plot in

figure 10, in which a large shear pressure components

towards to X lateral direction is observed.

Fig. 10. Surface shear pressure quiver plot at lateral shift = -5.5mm.

Furthermore, the distribution of VMS stress and shear

stress in A-A and B-B cutting surfaces are given for the

lateral shifts of -5.5, 0.0 and 5.5mm. The stress

distribution of lateral shift 0.0mm is used as a reference.

As expected from the surface shear pressure

distribution in figure 9 and10,when the lateral shift

reaches to -5.5mm, both the maximum shear stress and

VMS stress on the A-A plane occurs at the rail top

surface because of the large resultant lateral force.

While for the lateral shift of 5.5mm, the resultant

longitudinal force will become dominant, the

maximum shear stress and VMS stress on the B-B-

plane change from sub-surface to the top surface.

(b) (a) (c) (d)

-5.5mm

-3.5mm

0.0mm 2.0mm 5.5mm

1393.5 1063.5 962.8 975.6 1066.3 915.9 898.0 867.7 890.6

Maximum normal pressure [MPa]

Late

ral co

ord

inate

[m

m]

-5.5 -3.5 -2.0 -1.0 0.0 1.0 2.0 3.5 5.5

Lateral displacement [mm]

709.0 525.4 423.3 432.4 491.9 421.7 385.6 372.4 363.8

Maximum shear pressure [MPa]

Late

ral co

ord

inate

[m

m]

Fig. 11. Sub-surface stress variation w.r.t wheelset lateral shifts.

It can be concluded that lateral shift of wheelset plays

as an important role in wheel-rail interaction analysis

and its change will have a significant impact on the

surface pressure and sub-surface stress distribution.

4.2. INFLUENCE OF FRICTION FORCE

Friction between wheel and rail rolling contact has a

major impact on maintenance and logistics because it

determines the degradation process of wear and rolling

contact fatigue(RCF) damage and the capability for

braking and accelerating operations of railway vehicles.

Recently, sand-based friction modifiers has been

widely used to increase the traction forces under low

adhesion conditions or during braking and accelerating

operations. Meanwhile, lubricant has been chosen to

release the wear and RCF damage for further extending

the track and wheels’service life. In this section, the

influence of both the friction coefficient and traction

loads on the surface pressure and sub-surface stress

response will be investigated.

4.2.1. DRY & SLIPPERY TRACK

A statistical study has been done to investigate the

friction coefficient variation in the Dutch railway

network{Popovici, 2010 #2930} and the results show

that approximately 80% of the values are between 0.05

and 0.2 in autumn, which means severe delays or

sometimes even accidents may be caused by the

insufficient traction loads. While in the specific regions

close to train stations, higher friction coefficient is

required to make use of its maximum power for

accelerating and braking operations. Thus, the

difference in stress states and contact patch properties

under low/high frictional levels ( four friction

coefficients varying from 0.2 to 0.7 ) will be

investigated and discussed in this sub-section. All other

parameters are kept the same as preceding sections.

The resultant interface force over the rolling distance is

illustrated in figure 12. It should be noticed that the

vertical forces almost remain the same, while the

longitudinal forces gradually arise with the increment

of friction coefficients up to 0.5. When the friction

coefficient reaches to be as high as 0.7, the longitudinal

force stays the same level as friction coefficient 0.5. In

contrast with the longitudinal force variation, the

lateral forces tend to decrease.

Fig. 12. Resultantinterface force variation w.r.t rolling distance(Solid

line: FY; Dash-dot line: FX; Dashed line FZ).

Since the vertical and lateral resultant forces almost

remain the same under the variation of friction

coefficients, only the surface and sub-surface shear

pressure will be discussed and presented. It can be

noted from figure 13 that varying the value of friction

coefficient whilst keeping the traction coefficient

constant causes the micro-slip zone to change in size.

The pressure response shown above can be explained

by slip-adhesion relationship. When the friction

coefficient is 0.2, saturation is reached in which full

slip almost covers the whole contact patch. With the

increase of friction coefficients, the area of micro-slip

decreases in size and thus increasing the stick area.

Fig. 13. Surface shear pressure distribution & slip-stick area.

Figure 14 shows distribution of Von-Mises and

shearstress on B-B plane. It is interesting to see that the

maximum Von-Mises stress will shift from sub-surface

to surface with the increment of friction coefficient.

For the shear stress distribution, the tensile stress

moves upward as well. This phenomena is

corresponding very well to the pressuredistribution in

figure13, in which higher friction coefficient will lead

to smaller slip area and larger surface pressure, thus

larger shear stress and VMS stress shift upward to the

surface.

Fig. 14. sub-surface stress distribution on YZ-plane.

B-B – Plane VMS stress

B-B – Plane Shear stress

A-A – Plane VMS stress

A-A -Plane Shear stress

-5.5 0.0 5.5

Lateral shift [mm]

0.2

0.4

0.5

0.7

0. 0. 0. 0.

Shear pressure

Slip-stick area

0. 0. 0. 0.

Friction coefficient

YZ – Plane

Shear stress

YZ – Plane

VMS stress

4.2.2 THE INFLUENCE OF BRAKING &

ACCELERATING

When the vehicle travels along the track, three typical

operational conditions are distinguished, namely

accelerating, free rolling and braking operations, as

shown in figure 15. Under the free rolling condition, no

traction loads will be applied so that the translational

velocityis proportional to the angular velocity. While

for the braking and accelerating operations, the

magnitude of the applied tractions remains the same

but the directions of them are in opposite.

Fig. 15.Schematic graph of accelerating, free rolling & braking

operation

Figure 16 shows the longitudinal resultant force

variations when the train pass over the track for the

three operational conditions. As we can see, with the

change from accelerating to braking, the resultant

longitudinal force will alternate from positive to

negative which is the same as the change of traction

moments applied in figure 15.

Fig. 16. Resultant interface force variation w.r.t different traction

conditions.

Figure 17 shows the surface shear pressure distribution

under three operational conditions. Under free rolling,

the shear pressure is almost equal to zero and all the

contact patch is covered by adhesion. While for the

accelerating and braking operation, they are equal but

the opposite of each other.

Fig. 17. Surface shear pressure distribution & slip-stick areaw.r.t

different traction conditions.

Figure 18 shows the sub-surface shear stress and VMS

stress distribution under three operational conditions.In

the free rolling process, the shear stress in B-B plane

are distributed equally from left to right in the rolling

direction. Besides for the acceleratingprocess, the VMS

stress was tracked to the right direction and tensile

stress was dominate on the top rail surface. While for

the braking operation, the VMS stress was tracked to

the left direction and compressive was dominate on the

top rail surface.

Fig. 18. Sub-surface stress distribution w.r.t different traction

conditions.

4.3. THE INFLUENCE OF CONTACT

STIFFNESS

Although considerable FE simulations have been done

in the field of wheel/rail frictional rolling contact

interaction, there are certain aspects of their dynamics

such as sensitivity to interfacial parameters(e.g.,

contact stiffness, contact damping, which are of great

importance to contact dynamics and interface

modelling) that are not fully understood and modelled.

The variation of interfacial parameters may cause

uncertainty in dynamic response and reliability

simulations[23]. Thus, the influence of contact stiffness

on wheel/rail interface interaction will be studied in

this section.

Figure 19 shows a schematic graph of the two contact

bodies. A stiffness relationship between two contact

surfaces must be established for contact to occur. The

relationship is generated through an “linear-elastic

spring” that is put between the two contact segments,

where the contact force is equal to the product of

contact stiffness , the penetration and is the normal

vector on the contact surface.

0s i i i il k if l f n ( 1 )

(b) Free rolling

Y

(a) Accelerating

Y

Y

(c) Braking

ccelerating ree rolling ra ing

Shear pressure

Slip-stick area

ree rolling ccelerating ra ing

YZ – Plane

VMS stress

YZ – Plane

Shear stress

Physically no interpenetration are permitted between

two contacting surfaces, which is called “contact

compatibility”, which is not numerically possiblebut as

long as penetration is small enough or negligible,

accurate solution could beguaranteed.

Fig. 19. schematic graph of contact interaction

The stiffness factor ik for the contact segments is given

in terms of bulk modulus iK , the volume

iV , and the

face area iA of the element that contains master

segment asfor brick elements. is the interface

stiffness scale factor.

2

i i

i

i

K Ak

V

( 2 )

From equation (2), it can be notice that the variation of

three factors, namely mesh size, interface stiffness

scale factor and material properties related to contact

stiffness definition, might have an impact on the

contact interaction. Thus, parametric studies on the

three factors will be conducted in this section.

4.3.1. INTERFACE STIFFNESS SCALE FACTOR

Interface stiffness scale factor represents the product

of penalty scale factor and scale factor on master/slave

penalty stiffness, which is scale factor for the interface

stiffness and is set to 0.10 by default. The default

setting of interface stiffness scale factor (0.1) will be

used as a starting point to investigate its influence on

the contact interaction. All the other parameters will be

kept the same as preceding sessions.

Fig. 20. Reaction force variation under different interface scale factor.

From figure 20, it can be found that the resultant force

oscillation are significantly reduced with the increase

of interface stiffness scale factor. This can be explained

by the Hoo s’ law linear spring that the larger the

contact spring stiffness (shown in figure 19), the

smaller the oscillation amplitude of the resultant force.

4.3.2 ELEMENT SIZE

Figure 21 schematically shows the element size

variation under the dimension of d d d mm and

x d x d x d mm . Substitute the element size

x d x d x d mm into stiffness equation (2), the contact

stiffness on the small segment is,

2 2

3

(( ) )(0 1)

( )

K x dk x

x d

( 3 )

Since the element size is reduced from d d d mm to

x d x d x d mm , the segment force applied on the

contact surface will be decreased to be sx x f .

Introducing the force contact stiffness and segment

force into equation(1),

s xd xdx x l k l K x d f n n ( 4 )

The penetration depth is

s

xd d

xl x l

K d

f

n

( 5 )

Which means that the smaller element size, the smaller

penetration depth. Globally, the contact surfaces will

be stiffer.

Fig. 21. Schematic graph of element size variation.a) Element size = d ;

b) Element size = x d

In order to assess the influence of element size on the

wheel-rail interaction, six mesh sizes involving

0.5 0.5 0.5 , 0.75 0.75 0.75 , 1.0 1.0 1.0 ,

1.25 1.25 1.25 , 1.5 1.5 1.5 , 1.7 1.7 1.7 mm of

the same wheel/rail solid model are implemented and

simulated with the developed model.

Fig. 22. Reaction force variation under different element size.

a) 1.5*1.5*1.5mm; b) 1.0*1.0*1.0mm; c) 0.5*0.5*0.5mm.

(a)

(b)

(a)

(b)

(c)

Figure 22 shows the resultant reaction forces variation

in terms of element size variation in vertical direction.

With the decrease of the element size, the force

variation is gradually decreased.

4.3.3. MATERIAL PROPERTY

From equation (2), it can be observed that the contact

stiffness is related to bulk modulus K . While the bulk

modulus can be expressed as

3(1 2 )

EK

( 6 )

Where E is Young modulus, is poison ratio.

For rolling contact problem, hardness is a very

important material property, which will affects the

interface deterioration process and dynamic

performance[6]. Thus, three values of young modulus

as shown in figure 23 are adopted and simulated.

Fig. 23. Reaction force variation under different young modulus

From figure 23, it can be seen that the Young modulus

variation can have a slight influence on the dynamic

response in comparison with element size and interface

stiffness scale factor. But it can also be observed that

oscillation amplitude of the dynamic forces is reducing

with the increment of Young modulus. It is thus clear

that the material hardness plays an important role in the

stress/strain responses of wheel/rail interaction.

5. CONCLUSIONS

A dynamic numerical procedure for the wheel and rail

rolling contact stress/strain analysis is developed and

characterized by a clear explanation of solving

procedure, a novel mesh refinement technique on the

wheel/rail contact region. The obtained results,

involving stress/strain response and contact pressure

distribution as well as contact area, are analysed and

discussed. The main aim of the work is to enhance

knowledge of the contact pressure and maximum

stresses in bulk material. This should provide an

appreciate basis for the study of the degradation

mechanism and wear simulation.

Based on the developed FE model, parametric studies

on contact point, friction force, contact stiffness

variations are investigated. The conclusions can be

drawn as:

1). The lateral shifts of wheel-set on the track will

induce great contact pressure and severe dynamic

contact conditions.

2). The friction coefficient variation and braking &

accelerating operations will have a great influence on

the material response on both surface and subsurface.

3). The mesh size and material properties such as

young modulus can have an impact on the contact

stiffness variation.

In the future work, the proposed numerical model

might be further extended to research on fatigue and

wear behaviours of railway components and structures

to help making a better prediction of damage that can

be used for better maintenance scheduling. It can also

be useful in the optimization of rail and wheel profiles.

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