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Wear
Volume 253, Issues 1–2, July 2002, Pages 9–16
CM2000 S.I.

Wear and fatigue in rolling contact * Yoshitsugu Kimuraa, , , * Masami Sekizawab, * Akio Nitanaic * a Kagawa University, 1-1 Saiwai-cho, Takamatsu, Kagawa 760-8521, Japan * b NTN Corporation, 1578 Higashi-Kaizuka, Iwata, Shizuoka 438-8510, Japan * c Tamagawa University, 6-1-1 Tamagawa Gakuen, Machida, Tokyo 194-8610, Japan * http://dx.doi.org/10.1016/S0043-1648(02)00077-7, How to Cite or Link Using DOI * Permissions & Reprints

Abstract
Researches on wear and fatigue in rolling contact from a tribological viewpoint are introduced. Transmission of traction and accompanying microslip in the contact region play critical roles in these phenomena. First, a quantitative analysis of wear in rolling contact with microslip is introduced, and it is shown that a simplified microslip theory can explain its behavior. Second, a marked change in contact fatigue life with traction is demonstrated, and a theory is introduced which relates the fatigue life with cumulative shear strain in the subsurface.
Keywords
* Contact fatigue; * Microslip; * Rolling contact; * Wear

1. Introduction
Reduction of wear and prevention of contact fatigue are important objectives of railway technology and of tribology as well. However, these two engineering fields have been developing as different worlds, and researchers in the two fields seem to have been accustomed to different ways of thinking and different jargons. The present authors are tribologists (one of them is also a railway enthusiast!) and, in this paper, researches on wear and contact fatigue in rolling contact from a tribological viewpoint are introduced. The detailed mechanisms of wear are put aside, and a quantitative theory of wear at rolling contact is first introduced. Then a theory of contact fatigue accompanied by certain plastic flow is described featuring the effect of tangential traction.
2. Wear and contact fatigue
Wear is defined by the progressive loss of substance from the operating surface of a body occurring as a result of relative motion at the surface [1]. The operating surface means the solid surface which makes sliding or rolling contact and, in what follows, it shall generally be termed the contact surface.
Three major modes of wear due to different mechanisms are known, i.e. abrasive wear, corrosive wear and adhesive wear. Abrasive wear is characterized by microscopic cutting of contact surfaces by hard matter, i.e. a hard counterface or hard third bodies present at the interface. Wear of wheels or rails by sand provides an example. Corrosive wear is the removal of reaction products of contact surfaces with environments. One of the important reactive environments is oxygen in air, and rusting of rails leads to this mode of wear.
The most essential mechanism of wear in that it occurs even in the absence of hard matter or reactive environments has been termed adhesive wear. This terminology originated from the adhesive theory of friction first proposed by Holm [2], which claimed that friction was the force necessary to shear adhered junctions formed at real contact points between mating contact surfaces. However, it is the current understanding of the adhesive wear that it is caused by the microscopic fatigue failure at real contact points by the repetitive action of normal and frictional forces.
The process of the so-called adhesive wear is broken down in Fig. 1. Macroscopic operating conditions and the particular microgeometry of contact surfaces determine the real contact points. The actual forces working on them determine the subsurface stress field and the strain distribution, which result in the accumulation of damage leading eventually to the removal of particles.

Fig. 1. Process of adhesive wear.
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It must be noted that the processes along the downward vertical line in Fig. 1 are common to those of contact fatigue. Actually, theories of this mode of wear have made full use of analytical results on contact fatigue. The advantage to the analysis of contact fatigue is that the contact geometry is macroscopically well defined, mostly Hertzian, and the process generally terminates in the removal of fragments. In a wear process, on the other hand, the removal is accompanied with microscopic geometrical changes in the surfaces, so that a feedback loop is provided as shown. Thereafter, the real contact points are formed at the asperities of the modified microirregularities, which makes the phenomena much more complicated.
3. Wear in rolling contact
3.1. Microslip and traction
Most rolling contacts are accompanied by certain slip within the contact region termed microslip, and it has critical roles in wear and contact fatigue. Microslip is defined by the small relative tangential displacement in a contacting area at an interface, when the remainder of the interface in the contacting area is not relatively displaced tangentially [1].
Different mechanisms are responsible for its occurrence. However, the most relevant microslip is one caused by traction. Fig. 2 shows a typical traction curve. With the increase in the slide/roll ratio s, which is defined by the ratio of the speed difference to the average speed of two contact surfaces, traction first increases almost linearly and then gradually levels off. This implies that, when a certain magnitude of traction is to be transmitted, a certain amount of slide/roll ratio is required. Thus microslip exists in most rolling contacts, ones between wheels and rails which are driving or being braked providing typical examples.

Fig. 2. Typical traction curve.
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Analyses are available to quantitatively estimate microslip due to traction. A comprehensive description of a three-dimensional elasticity analysis was given by Halling [3]. Fig. 3 illustrates an elastic contact between like spheres and which are rotating at slightly different angular speeds ω1 and ω2. At the entry to the contact area, at the right in the figure, a stick area exists in which the surface of the two spheres sticks without making relative slip. The ‘strip theory’ is introduced which considers the spheres to be represented by a row of elemental cylinders of axial length δy embracing the Hertzian contact circle with radius a.

Fig. 3. Stick and slip areas in a point contact [3].
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A more simplified, two-dimensional theory was developed by Soda [4]and Soda et al. [5], which can be applied to each elemental cylinder in Fig. 3. The entrance zone of a two-dimensional Hertzian contact is idealized in Fig. 4, where real contact takes place only at discrete asperities. Two cylinders come to contact at O, which is taken as the origin of the coordinate x along the common tangent of the contact surfaces. If the rigid-body peripheral speed of the top cylinder , v1, is larger than that of the bottom cylinder , v2, the cylinder will tend to slide over the cylinder . However, friction working at real contact points resists the sliding, and tangential elastic deformation ε at both surfaces would accommodate the relative displacement. Then actual slip is prevented if the tangential force to shear a contact point δT is smaller than friction working at the contact point, i.e. the product of the normal force δP on it and the coefficient of friction in sliding μ0. This means that the point falls in the stick area and, since δT can be assumed to be proportional to the elastic deformation εwhich increases linearly with the distance x from the origin O, the stick area is located at the entrance edge of the Hertzian contact area. Since the normal force δP is reasonably assumed constant, δT will exceedμ0δP at a certain point and actual slip is initiated to form the slip area.

Fig. 4. Entrance zone of two-dimensional Hertzian contact [5].
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The extent of the stick area and the slip area is shown in Fig. 5 as a function of the slide/roll ratio s and the total normal load P. The curved surface α represents the elliptical increase in the Hertzian contact width withP, and the surface β represents the limit of the stick area which decreases in inverse proportion to s. The hatched volume between the surface α and the surface β gives the slip area.

Fig. 5. Extent of stick and slip areas [5].
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It should be noted that the projection of the intersection of the surfaces α and β on the s–P plane represents the limiting condition of the existence of the slip area in the Hertzian zone, being given by a relation sP1/2=(P∗)1/2 where P∗ is a constant; if sP1/2 is smaller than (P∗)1/2 no actual slip occurs even though a difference in the rigid-body speed is present.
A traction curve as shown in Fig. 2 is obtained from this theory if the coefficient of friction μ0 is assumed constant. With the increase in the slide/roll ratio s, the coefficient of traction μ increases linearly until (1/2)μ0is reached; μ then asymptotically approaches to μ0 for further increase in s. However, this idealized behavior may be modified by the change in μ0 with sliding speed.
3.2. Wear experiments
A series of experiments was conducted on a four roller machine, Fig. 6, in which an inner roller made rolling contacts with three outer rollers arranged at 120° apart [5]. The top outer roller was mounted on a shaft which was supported by vertically movable bearings and a static load was applied to this shaft by weights. The shafts of the two bottom outer rollers were supported by bearings fixed to the machine frame. Then the inner roller was supported centerlessly by the three outer rollers, and identical normal loads worked on the three contacts. The inner roller and the set of the outer rollers had independent driving systems, and their rotational speeds were controlled separately to give a desired slide/roll ratio.

Fig. 6. Four roller machine [5].
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The roller specimens were made from 0.45%C carbon steel heat-treated to HV240. They were 40 mm in diameter and had 9 mm contact length; their contact surface was ground to 6 μm Ry.
Experiments were made under normal loads ranging 27–111 kgf (265–1089 N). The rotational speed of the outer rollers was varied from 800 to 2000 rpm, and the inner roller was driven at speeds 10 rpm lower than that of the outer rollers, thereby giving a constant speed difference of 2.1 cm/s. Consequently, the slide/roll ratio ranged 0.0050–0.0125. All experiments were made without lubricant in the laboratory air.
3.3. Results and discussion
Apparent wear rates dW/dl of the rollers are plotted in Fig. 7 against the slide/roll ratio. Here, the apparent wear rate is the worn mass W during sliding for the apparent sliding distance l; the apparent sliding distance is the product of the rigid-body speed difference between the inner and the outer rollers, 2.1 cm/s in all cases, and the running duration. Of course, the difference in the number of contacts per revolution of the inner and the outer rollers has been taken into account.

Fig. 7. Change in apparent wear rate with s[5].
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The apparent wear rate varies markedly with the slide/roll ratio. In particular, it is recognized that the apparent wear rate remains low up to a certain slide/roll ratio and then shows an abrupt increase. When this result is compared with Fig. 5, it is likely that the slip area comes to being at this limiting slide/roll ratio. Although small amount of wear occurs even in the contact where all Hertzian zone is in the stick area, much more wear is caused by the actual slip.
A mathematical expression for the apparent wear rate was derived by assuming the linear wear law, i.e. the wear amount is in direct proportion to the normal load and the sliding distance. In the present case, the wear amount dw in a small area dA in the Hertzian zone during a time duration dt is assumed to be in direct proportion to dA, dt, the local contact pressure p and the local actual slip speed Δv, such that

Then, from the theory of microslip outlined above, the apparent wear rate dW/dl is given by

When the value of P∗ is assumed to give the best fit with the experimental results, this formula presents the curved surface in Fig. 8 on the s–P plane. It rises from the horizon at the boundary sP1/2=(P∗)1/2 and goes up with the increase in s and P. The straight chain line at the right extreme represents the limiting wear amount for the case of perfect slip. Since the apparent wear rate is normalized by kw, experimental results for the inner rollers, the solid circles, and those for the outer rollers, the open circles, almost coincide and show agreement with the theory.

Fig. 8. Plot of apparent wear rate on s–P plane [5].
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Another point of interest in Fig. 7 is the difference in the apparent wear rates between the inner and the outer rollers. That is, the wear rate of the outer rollers is about 2.5 times as large as that of the inner rollers. A different condition between them exists that the contact surface of an outer roller passes through the contact region at a higher speed than that of an inner roller. In this sense the outer roller is called the driver and the inner roller the follower. More exactly, however, the driver is defined as the rolling surface on which the direction of traction and the direction of travel of the contact region coincide; the follower is defined on which the two directions are opposite. Thus, it is concluded that a driver surface wears more heavily than a follower surface.
4. Rolling contact fatigue
4.1. Fatigue life of the driver/follower
The life due to rolling contact fatigue has been studied with different machine elements like rolling-contact bearings, gears, cams etc. and different names like flaking, pitting or spalling have been given to the fatigue damage on their contact surfaces.
Typically with involute gears, difference in the contact fatigue life between the driver and the follower region was empirically known, i.e. the life is much shorter with the follower. In meshing of involute gears, only the points on their pitch circle make pure rolling contact. With the distance from it, slip of opposite sign increases in addendum and dedendum, and the addendum becomes the driver and the dedendum becomes the follower in the exact terminology given above. Then, contact fatigue or pitting first appeared always on the dedendum.
A classic explanation was given by Way [6]. It was postulated that cracks which led to eventual detachment of flakes started at the contact surface, not in the subsurface layer, and the role of hydrostatic pressure in their growth was noted.
Consider a lubricated rolling contact system in which a surface crack is coming close to a contact region. If the crack exists on a follower making an angle in a certain range to the surface, the traction working at the contact region exerts a tensile stress to the crack, and the crack enters the contact region with its mouth open. At the entrance to the contact region, considerable hydrodynamic pressure is established and the lubricant fluid penetrates the crack. Under the high contact pressure, the mouth is closed and hydrostatic pressure is built up in the fluid entrapped within the clack. It gives a concentrated tensile stress at its tip and causes the crack to grow. Such a phenomenon never happens on a driver, since a surface crack approaches the contact region with its mouth closed by the compressive stress due to traction.
This was the only available theory to explain the difference in the contact fatigue life between the follower and driver surfaces. Although the proposed mechanism seems reasonable, problems have been pointed out. First, the theory predicts the qualitative difference in the fatigue life of the follower and the driver. However, continuous decrease in the fatigue life was reported when traction was increased from the negative range (driver) to the positive range (follower), Fig. 9[7]. Secondly, the theory is applicable only to lubricated systems, but similar behavior has also been found in marginally lubricated or unlubricated systems. Thirdly, it may be reasonable to expect that a further increase in the absolute value of traction in its negative region beyond the range studied in Fig. 9 would result in a decrease in the life because of the increased combined stresses.

Fig. 9. Change in contact fatigue life with traction [7]. Three curves represent data for different roller roughness. A: 1.0, B: 0.5, C: 0.1 μm Ry.
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Then another explanation has been tried in an unpublished work by Nitanai [8].
4.2. Contact fatigue experiments
Contact fatigue life was determined on an Amsler friction machine, in which rolling contact was made between two rollers. A top roller of 42.03 mm diameter was driven at a constant speed of 380 rpm; a bottom roller was driven at a speed of 420 rpm, and its diameter was changed in a range 34.55–44.73 mm to give different slide/roll ratios. The contact length was 3 mm.
The top rollers were made of chromium bearing steel JIS SUJ2 (AISI 52100) heat treated to HV800, and their contact surface was ground to 0.4 μm Ry. The materials for the bottom rollers were 0.48%C carbon steel differently heat treated to give HV200, 400 and 600 hardness values, and their contact surface was ground to 3.0 μm Ry.
The top roller was loaded by a spring up to 2940 kN against the bottom roller. Runs were made lubricated with ISO VG7 spindle oil. It formed an elastohydrodynamic film at the contact region, but only insufficiently. The film parameter, i.e. the ratio of the minimum film thickness to the initial composite surface roughness, ranged 0.06–0.10, and only marginal lubrication conditions were established.
Traction was continuously monitored and recorded; the onset of contact fatigue damage, which always occurred on the bottom roller, was detected with a vibration sensor. The forward movement i.e. plastic flow in the surface layer took place with the repetition of contact, and its amount at the contact surface of the bottom roller was determined by measuring the displacement of small Vickers indentations previously formed on the surface. Some rollers were sectioned after a run to determine strain hardening in the subsurface layer.
4.3. Results and discussion
The traction curves are presented in Fig. 10, in which the positive and the negative value of the slide/roll ratio represents that for the follower and the driver. The slight asymmetry of the curves is caused by the presence of the coefficient of rolling friction; it ranges 0.009–0.016 and is reasonably ignored. Since the contact is lubricated, a certain elastohydrodynamic effect can be present in the result. However, elastohydrodynamic traction shows similar behavior to one in solid contact, except for a decrease at higher slide/roll ratios.

Fig. 10. Traction curves over positive and negative regions of s[8].
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The contact fatigue life is plotted against the coefficient of traction in Fig. 11. Although a harder material tends to show a longer life, the characteristic dependence on traction is common to all materials tested. That is, with the decrease in the coefficient of traction from the positive range to the negative range, i.e. from the follower to the driver range, the life shows a continuous increase. It seems to take a maximum when the coefficient of traction is −0.06, and the life tends to decrease with the further decrease in the coefficient of traction, or increase in its absolute value in the negative range. This result is in agreement with the previous one, Fig. 9[7], which was obtained over a range for the coefficient of traction, 0±0.03.

Fig. 11. Change in contact fatigue life for a wider range of the coefficient of traction [8].
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In rolling contact of ductile materials, it has been noted that repetition of the Hertzian contact pressure causes plastic flow of surface layer in the rolling direction, which is called the forward movement. The maximum of the shear strain occurs at certain distance from the surface, typically one third to a half of the Hertzian contact width, and it diminishes near the surface and in the depth. However, traction causes another plastic flow just below the surface, and this enhances the forward movement in the follower while retards it in the driver.
The maxima of the plastic flow observed at the center of the contact surface are plotted in Fig. 12 for different slide/roll ratios. The magnitude of the flow increases with the repetition of contact but not linearly. That is, the rate of the increase is gradually reduced, because strain hardening induced by the plastic flow suppresses further flow. The effect of the slide/roll ratio is evident. The flow is much more marked with the follower indicated by the positive values of the slide/roll ratio, and seems to be minimized when the slide/roll ratio becomes –0.01 where the coefficient of traction takes a value around –0.07. It should be noted that this coefficient of traction is close to the value at which the maximum of the fatigue life appears.

Fig. 12. Maximum plastic flow on the surface [8].
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Distribution in the subsurface of the shear strain γxz caused by a single contact is calculated after Merwin and Johnson [9], Fig. 13; the coordinate x is lying on the contact surface and z is perpendicular to it. The ordinate is the depth from the surface z normalized by the half Hertzian width b; the abscissa is γxz multiplied by G/pmax, the ratio of the modulus of rigidity to the maximum Hertzian pressure. For the driver characterized by the positive values of the coefficient of traction, the distribution is simple having a peak marked at around z/b=0.5, which is higher for larger slide/roll ratios. With the decrease in the coefficient of traction beyond 0 to the follower range, the peak diminishes and other peaks and of the opposite sign appear at z/b=0–0.5 and at around 1.0, respectively, and they grow to exceed the peak in the absolute value.

Fig. 13. Shear strain in the substrate [8].
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With the repetitive contact, such shear strain accumulates to cause the macroscopic flow. Since this process intensifies damage in the material, a linear cumulative damage rule is hypothesized to predict the fatigue life. Assumptions are made to take strain hardening into account which decreases the subsurface strain in the successive contacts, to express the stress-strain curve obtained for the materials by a simple power law, to use relations developed for normal strain for shear strain, etc. and a result shown in Fig. 14 is obtained.

Fig. 14. Experimental result and prediction [8].
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The solid symbols and the dashed curve represent the experimental results for HV200 steel reproduced fromFig. 11; the solid curve is the theoretical prediction which is the lower envelope of a family of curves giving the number of contact cycles to failure for the material in different depths. Although the two curves show quantitative discrepancy, qualitatively the theory explains the characteristic feature of the change in the contact fatigue life with the coefficient of traction.
5. Concluding remarks
It is first pointed out that wear and contact fatigue have elemental processes in common. Then, it may be interesting to note that wear is heavier on the driver surfaces while fatigue life is shorter with the follower surfaces. It is reasoned that damage is concentrated in thin subsurface layer with the driver while rather thick layer is damaged in the follower. This might serve to a unified understanding of the two major modes of tribological failure in the wheel/rail systems.
Although the materials and the conditions described here must be quite different from those in the railway practice, it is expected that this article would help for railway researchers to understand the way tribologists are treating the problems of common interests. It must contribute to future cooperation of railway researchers and tribologists for developing more comfortable railway systems, which will fascinate railway enthusiasts like one of the present authors.
References
1. * [1] * Research Group on Wear of Engineering Materials, Glossary of Terms and Definitions in the Field of Friction, Wear and Lubrication, Tribology, OECD, 1969. * 2. * [2] * R. Holm, Electric Contacts, H. Gebers Förlag, 1946, p. 202. * 3. * [3] * J. Halling (Ed.), Principles of Tribology, The Macmillan, New York, 1975, pp. 174–201. * 4. * [4] * N. Soda, Jikuuke (Bearings), Iwanami Shoten, 1964, pp. 99–107 (in Japanese). * 5. * [5] * N. Soda, Y. Kimura, M. Sekizawa * Wear of steel rollers under rolling-sliding contact * Bull. JSME, 15 (85) (1972), pp. 866–876 * Full Text via CrossRef 6. * [6] * S. Way * Pitting due to rolling contact * J. Appl. Mech., 2 (2) (1935), pp. 49–58 * 7. * [7] * N. Soda, T. Yamamoto * The role of surface traction for the pitting on gear teeth * J. Japan Soc. Lub. Eng., 20 (4) (1975), pp. 268–275 (in Japanese) * 8. * [8] * A. Nitanai, Effects of Tangential Traction on the Rolling Contact Fatigue, Ph.D. Dissertation, The University of Tokyo, 1996 (in Japanese). * 9. * [9] * J.E. Merwin, K.L. Johnson * An analysis of plastic deformation in rolling contact * Proc. IMechE, 177 (25) (1963), pp. 676–690 *

Corresponding author. Present address: 5-21-10-4 Nagayama, Tama-Shi, Tokyo 206-0025, Japan. Tel.: +81-42-371-0911; fax: +81-42-371-0911.
Copyright © 2002 Elsevier Science B.V. All rights reserved.
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