10thInternational 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)
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
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 0.0m m 2.0m m 5.5m m 1393.5 1063.5 962.8 975.6 1066.3 915.9 898.0 867.7 890.6 Maximum normal pressure [MPa]
La te ra l co ord ina te [mm ] -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]
La te ra l co ord ina te [mm ]
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.
0 s l ki i i if li 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 ki for the contact segments is given
in terms of bulk modulus Ki, the volume Vi, and the
face area A of the element that contains master i
segment asfor brick elements. is the interface stiffness scale factor.
2 i i i i K A k 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 d k 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 x x fs .
Introducing the force contact stiffness and segment force into equation(1),
s xd xd
x x f l k n l K x d n ( 4 )
The penetration depth is s xd d x l 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 ) E K ( 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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!["Niemcy-Warmiacy-Polacy 1871-1914. Z dziejów niemieckiego ruchu katolickiego i stosunków polsko-niemieckich", Robert Traba, Olsztyn 1994 : [recenzja]](data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==)