Energy Theory of Rubber Abrasion

Tribology International Vol. 30, No. 12, pp. 839843, 1997 1998 Elsevier Science Ltd. All rights reserved

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Energy theory of rubber abrasionby a line contact

S. W. Zhang and Zhaochun Yang

A new energy theory of rubber abrasion by a line contact hasbeen set up through analyzing the energy distribution andconversion during the abrasion process, corresponding theoreticalwear equation of rubber abrasion by a line contact in unsteadystate has been derived. This energy theory, which is animprovement on the previous theory of rubber abrasion by a linecontact, can be used to clarify the phenomena and processes ofthe rubber abrasion. 1998 Elsevier Science Ltd. All rightsreserved.

Keywords: energy theory, rubber abrasion, line contact

Introduction

The pioneer efforts concerning the mechanisms of rub-ber abrasion were made by Schallamach in his earlywork with a needle to scratch the rubber surface undercontrolled conditions. However, the difficulties werethat so little rubber was removed and a complex stressof three dimensions was applied on the needle. There-fore, a line-contact test mode on rubber abrasion waspresented by Champ et al.1, in which a razor bladeinstead of a needle was used. In the following years,although the physical process of rubber abrasion by aline contact has been extensively studied112, only afew wear theories have been proposed1,2,1012.Champ et al. presented a simple theory1,2 relating therubber abrasion to the crack-growth property of therubber. Gent and Pulford have argued that the rubberabrasion does not account for solely crack-growthproperties of material, but involves other failure processas well3. Moreover, the simple theory is only applicablefor the steady state of rubber abrasion.Over 10 years ago, an amended theory of rubberabrasion by a line contact was proposed by the presentauthor4,6, which can be applied to clarify the wholewear process including unsteady state and steady stateof rubber abrasion. However, the mathematic descrip-tion for the wear process is still a quasitheoreticalequation. Therefore, there is thus a clear need to putforward a more perfect theory to be able to clarify thewhole wear process of rubber abrasion theoretically. In

University of Petroleum, Changping, Beijing 102200, PeoplesRepublic of ChinaReceived 29 July 1996; revised 17 May 1997; accepted 30 June 1997

Tribology International Volume 30 Number 12 1997 839

this paper, based upon the experimental observations, anew energy theory of rubber abrasion by a line contacthas been set up through analyzing the energy distri-bution and conversion during the wear process.

Theory

On the basis of the results of experimental studies46which were carried out in a modified blade abraderdescribed by Gent and Pulford3, the physical processof rubber abrasion might be considered as two alterna-tively proceeded processes, i.e. crack growth of surfacelayer (tongue formation) and rupture of tongue tip ofridges (tongue rupture) (Fig 1)4. Moreover, from theprevious experimental observations, it has beendeduced that the rupture of tongue tip resulted fromtensile stress is the primary cause of material lossesalthough the effect of crack growth via mechanicalfatigue on abrasion is taken into account4. Hence, thephysical model of rubber abrasion could be shown inFig 2.From the point of view of energy, a wear process ofmaterial is a process of energy conversion and energydissipation. Fleisher13 has pointed out that, althoughthe frictional work is dissipated mostly into heat duringthe frictional process, there is still a fraction of workdeforming the material of the frictional pairs andaccumulating in the form of potential (deformation)energy. If the potential energy accumulates to a criticalvalue in the material for a certain volume, the weardebris of material will produce. This potential energyis termed accumulated energy.According to the physical model of rubber abrasion(Fig 2) and based on the theory of fracture mechanics,

Energy theory of rubber abrasion by a line contact: S. W. Zhang and Zhaochun Yang

Fig. 1 Formation of ridge from filled NBR, W =15 kJ/m2 (100 ): (top) tongue formed; (bottom)tongue ruptured

L

q

t

F

tongue-tiprupture

crack growthD L

Fig. 2 Physical model of rubber abrasion

the accumulated energy U could be expressed as thework which is done by the frictional force during theabove two alternatively proceeded processes, i.e. tongueformation and tongue rupture:

U = KPfL (1)whereK coefficient of accumulated energy.840 Tribology International Volume 30 Number 12 1997

P normal load.f coefficient of friction.L length of the crack growth.

However, from the point of view of energy, theaccumulated energy U could be considered to consistof two parts: crack-growth energy and rupture energyof the tongue tip, i.e.

U = UA + UB (2)whereUA crack-growth energy.UB rupture energy of the tongue tip.

The assumption that the crack angle is constant duringthe wear process is implicit in the following analysis.The crack growth per stress cycle r was given pre-viously6:

r > B(2F)a (3)whereB constant.F frictional force per width.a constant.

As seen, the crack growth per stress cycle is a constantfor the same material under identical operating con-ditions. In this situation, the corresponding crack-growth energy is also kept constant.From the point of view of energy, similar to theconcept of energy density introduced by Fleisher13, anew concept, wear energy density e*, is presented:

e* = UB/VB (4)where VB is the rupture volume per stress cycle, i.e.

VB = SDL (5)whereS cross section of the tongue tip.DL rupture length of the tongue tip per stress

cycle.

From Equations (1) and (2), we haveUA = U 2 UB = KPfL 2 UB (6)

Inserting Equations (4) and (5) into the aboveequation, gives

UA = KPfL 2 e*SDL (7)Since the wear rate is very low in the unsteady state,the wear process would be considered as a continuousprocess. Moreover, in light of that the crack-growthenergy is constant under the same operating con-ditions, then

UA = KPfLi 2 e*SDLi (8)= KPfLi + 1 2 e*SDLi + 1 =

where Li and Li + 1 are the length of crack growth atthe ith and (i + 1)th stress cycle, respectively, DLi and


DLi + 1 are the rupture length of tongue tip at the ithand (i + 1)th stress cycle, respectively.Thus,

KPf(Li + 1 2 Li) = e*S(DLi + 1 2 DLi) (9)Based on the experimental observation, it has beenassumed that a rupture fragment is made for eachstress cycle as stated previously4. Then

Li + 1 2 Li = r 2 DLi (10)From Equations (9) and (10), we have

DLi + 1 2 DLi =KPf(r 2 DLi)

e*S (11)

It can be rewritten asdDLi

di = DLi + 1 2 DLi =KPf(r 2 DLi)

e*S (12)

where i is the number of stress cycles.With the conditions i = 0 and DLi = 0, the rupturelength of the tongue tip per stress cycle DLi is obtainedby integration:

DLi = r 2 r exp S 2 KPfie*S D (13)

The linear wear-rate in unsteady state can be givenaccording to its definition:

Ru = SDLDS

1Da

(14)

whereD distance of friction per revolution.S average spacing of ridge.a contact width.

S = ta (15)where t is the height of tongue.

t > Ssin u (16)where u is the crack-growth angle.Inserting Equations (13), (15) and (16) into Equation(14) gives

Ru = Fr 2 rexp S 2 KPfie*S DGsin u (17)

Then

Ru = rsin u 2 rexp S 2 KPfie*S Dsin u (18)

or

Ru = rsin uF1 2 exp S 2 KPfie*S DG (19)

However, the term (rsin u) in the above equation isjust the theoretical linear wear rate in steady state, Rs,as proposed by Southern and Thomas2, and the presentauthor6 on the assumption that crack growth plays animportant part in the abrasion process when abrasionpatterns are produced, i.e.


Rs = rsin u (20)and

Rs = B(2F)a sin u (21)Inserting Equation (20) into Equations (18) and (19),respectively, the final expressions are given

Rs 2 Ru = rexp S 2 KPfie*S Dsin u (22)

andRuRs

= 1 2 exp S 2 KPfie*S D (23)

As seen from Equation (22), the logarithms of thedifference between the wear rate in unsteady state andthat in steady state is inversely proportional to thenumber of stress cycles, the normal load and thecoefficient of friction, moreover, is proportional tothe wear energy density and the cross-section of thetongue tip.

Experimental

Rubber abrasion tests were carried out in an arrange-ment shown schematically in Fig 3. A steel blade(10 mm wide) with 45 inclined angle was held atright angles pressing on the surface of a rubber disk of48 mm in diameter and rotated around the central axis.The rubber disks were made of natural rubber (NR)and nitrile rubber (NBR), respectively. All of theexperiments were carried out at room temperature, 22 3 C, and mostly with the normal load P at 8 N or10 N, and with the rotating speed of the driving shaftat 50 rev/min, corresponding to a sliding speed v of010 m/s at the rubbing track, under these conditionsno significant temperature rise on the material surfacewas noted.The wear rates were obtained by measuring the lossweight of the rubber disks.

2

3

1

Fig. 3 Schematic drawing of blade abrader. 1rubbersample (disk), 2steel blade, 3driving shaft


80

60

40

20

00 50 1000 1500 2000 2500

Number of Revolutions i (rev)

Wear

Rat

e (10

5 g

/rev)

NRP = 10Nv = 0.10m/s

Fig. 4 Linear rates of wear of NR material plottedagainst the number of revolutions in unsteady state (P= 10 N, V = 010 m/s)

Results and discussion

The experimental correlation of the wear rate of rubbermaterials (NR and NBR) and the number of revolutions(stress cycle) are shown in Figs 4 and 5, respectively.It has been found that the wear rate of the rubberabrasion in unsteady state is increased with the numberof revolutions, which is the same as observed pre-viously3,4 and is accordant to the wear Equation (17).Moreover, the wear rate is kept unchangeable if oncethe number of revolutions rises to a certain value, itmeans that the wear process at this time reaches thesteady state. In the steady state, the wear rate of NBRmaterials is much lower than that of NR materials. Itwould be ascribed that the wear energy density ofNBR materials is higher than that of NR materials. Adetailed discussion will be given in another paper.In order to compare the experimental results with thetheoretical results as stated above, the differences

30

25

20

15

10

5

00 500 1000 1500 2000 2500


Wear

Rat

e (10

5g/

rev)

NBRP = 8Nv = 0.10 m/s

Fig. 5 Linear rates of wear of NBR material plottedagainst the number of revolutions in unsteady state (P= 8 N, V = 010 m/s)842 Tribology International Volume 30 Number 12 1997

10.00

1.00

0.100 500 1000 1500 2000


Rs

- Ru

Fig. 6 The logarithm of differences (Rs 2 Ru) of NRmaterial plotted against the number of revolutions inunsteady state

between the measured wear rates in steady state andthat in unsteady state of NR and NBR materials aretaken as a logarithm and shown in Figs 6 and 7,respectively. As seen, the differences of experimentallydetermined wear rates between the two kinds of abra-sion state are inversely proportional to the number ofrevolutions, which is in accord with the theoreticalresult, Equation (22). Hence, the wear energy theorycan be used to clarify the phenomena and processesof rubber abrasion. Certainly, what the further workshould do is make a critical comparison between themeasured and calculated value of wear rates in orderto prove the energy theory effective for wear-rate predi-cation.

Conclusions

The energy theory of rubber abrasion proposed, whichis supported qualitatively by evidence from the wear

10.00

1.00

0.100 300 600 900 1200 1500


Rs

- Ru

NBR

Fig. 7 The logarithm of differences (Rs 2 Ru) of NBRmaterial plotted against the number of revolutions inunsteady state


tests of NR and NBR materials, can be used to clarifythe phenomena and processes of the rubber abrasion.

References

1. Champ, D. H., Southern, E. and Thomas, A. C., Fracture mech-anics applied to rubber abrasion. In Advances in Polymer Frictionand Wear, ed. L. H. Lee. Plenum Press, New York, 1974, pp.133140.

2. Southern E. and Thomas A. G. Studies of rubber abrasion,Rubber Chem. Technol. 1979, 52, 10081018

3. Gent A. N. and Pulford C. T. R. Mechanisms of rubber abrasion,Appl. Polym. Sci. 1983, 28, 943960

4. Zhang S. W. Mechanisms of rubber abrasion in unsteady state,Rubber Chem. Technol. 1984, 57, 755768

5. Zhang S. W. Investigation of abrasion of nitrile rubber, RubberChem. Technol. 1984, 57, 769778


6. Zhang, S. W., Theory of rubber abrasion by a line contact. InPolymer Wear and its Control, ed. L. H. Lee. ACS, Washington,DC, 1985, pp. 189196.

7. Pulford C. T. R. Failure of rubber by abrasion, Rubber Chem.Technol. 1985, 58, 653660

8. Gent A. N. A hypothetical mechanism for rubber abrasion,Rubber Chem. Technol. 1989, 62, 750758

9. Uchiyama Y. and Ishino Y. Pattern abrasion mechanism ofrubber, Wear 1992, 158, 141155

10. Fukahori Y. and Yamazaki H. Mechanism of rubber abrasion,Part I: Abrasion pattern formation in natural rubber vulcanizate,Wear 1994, 171, 195202

11. Fukahori Y. and Yamazaki H. Mechanism of rubber abrasion,Part II: General rule abrasion pattern formation in rubber-likematerials, Wear 1994, 178, 109116

12. Fukahori Y. and Yamazaki H. Mechanism of rubber abrasion,Part III: How is friction linked to fracture in rubber abrasion?,Wear 1995, 188, 1926

13. Fleisher G. Energetische methode der bestimmung des ver-schleibes, Schmierungstechnik 1973, 4, 922

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Energy Theory of Rubber Abrasion