9An explanation of the different regimes os friction and wear using asperity deformation models.pdf

7/23/2019 9An explanation of the different regimes os friction and wear using asperity deformation models.pdf

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Wear , 53 1979) 229 - 243

@ Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands

229

AN EXPIATION OF THE DIFFERENT REG~ES OF FRI~~N

AND WEAR USING ASPERITY DEFORMATION MODELS

J. M. CHALLEN and P. L. B. OXLEY

School o Mechanical and I ndust ri al Engi neeri ng, Uni versit y of New South Wales, P.O.B.

1, Remi ngt on, New Soutfi Wales 2033 ~A~st ~l ia~

(Received August 11,1978)

Summary

A slip-line field analysis is given for the deformation of a soft asperity

by a hard one and equations are derived for the corresponding coefficients of

friction and wear rates. Three main models are proposed. For smooth sur-

faces the first model gives low coefficients of friction and shows how plastic

deformation of the asperity can occur without removal of material. The sec-

ond model shows how wear and high coefficients of friction can occur for

such surfaces. For rougher surfaces a cutting model applies with a chip (wear

particle) being produced. In this way an explanation is offered of why lu-

brication is observed to inhibit wear for smooth surfaces and to encourage

it for rougher surfaces. A possible explanation is also given of why the actual

wear for engineering surfaces under normal working conditions is many or-

ders of magnitude less than that calculated by assuming that all of the plas-

tically deformed material is removed.

1. Introduction

When metallic surfaces are pressed together the load is carried by the

load-bearing areas created by the plastic deformation of the tips of con-

tacting asperities and the sum of these areas (real area of contact) is normally

much smaller than that of the surfaces themselves (apparent area of contact).

In early attempts to explain the frictional force which opposes relative

sliding between such surfaces Bowden and Tabor [l] assumed that this re-

sulted mainly from the forces needed to shear the welded junctions

formed by adhesion at the contact areas although for certain conditions, e.g.

when a hard surface slides over a soft one, ploughing of the hard surface

through the soft one can also contribute. With this model the junctions are

assumed to be parallel to the sliding direction and the normal and shear

stresses acting on them are taken to be independent of each other and to be

related respectively to the indentation yield stress (hardness) and shear flow

stress of the contacting materials. It follows that the frictional force is pro-

portional to the load normal to the surfaces and independent of their area,


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thus satisfying the two basic laws of friction. However, the coefficients of

friction estimated in this way are, at most, of the order of 0.2 and this is

much less than the values normally obtained for unlubricated metals. Later

the theory was modified to take account of the interdependence of the nor-

mal and shear stresses which must be related by a yield criterion [ 21 and the

concept of an interfacial film separating the surfaces in the contact regions

[3]

,

with the film having strengths ranging from zero (perfect lubrication) to

the shear strength of the asperities (strong adhesion), was also introduced.

With this model the resolved shear stress at the junctions resulting from an

applied tangential force (equal to the frictional force when sliding occurs)

will only cause relative sliding of the surfaces if it is equal to the shear

strength of the interfacial film. For lower values, deformation of the asperi-

ties will occur under the combined action of the normal and shear stresses

and the real area of contact will increase (junction growth) from its initial

value (resulting from the normal load alone) with this process continuing

until the tangential force is increased sufficiently to shear the interfacial film.

The predictions made in this way of the increase in the real area of contact

have been shown to be in good agreement with experimental results and the

theory, while still satisfying the basic laws of friction, gives more realistic

values for the coefficient of friction than the earlier theory and in particular

shows that for chemically clean surfaces (strong adhesion) this can reach ex-

tremely high values in agreement with experiment. Weaknesses of the theory

appear to be that it cannot account for the influence of surface roughness

unless a separate ploughing term is introduced and also it is not easy to see

how the model could be applied to make estimates of the wear of sliding

surfaces.

In investigations of particular relevance to the present one, Green [4]

applied plasticity theory to estimate the forces involved in asperity deforma-

tion and obtained slip-line field solutions for both strong and weak junctions

and then used some of these [5] to show, for example, how the coefficient

of friction could in the case of strong junctions be extremely high. In this

work Green pointed out that, while during the initial junction growth period

the two surfaces move closer together, they must under steady state sliding

conditions (neglecting small fluctuations) move parallel to each other and he

showed that the necessary imposition of this condition on each individual

junction determined both its manner of deformation and the forces exerted

through it. To investigate the influence of the strength of adhesion etc. on

the value of the coefficient of friction Green argued that if there are many

junctions at different stages of development (as there normally would be)

then the coefficient of friction for the surfaces as a whole can be estimated

by taking the ratio of the average tangential and normal forces over the life

cycle of a typical junction, which he said consisted of formation, deforma-

tion and fracture. Lacking a method for predicting the change in shape of an

asperity and the corresponding forces during a life cycle (even today this

problem appears prohibitively difficult), Green carried out experiments using

scaled-up asperities made from Plasticine to give some indication of the type


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in that steady state solutions are sought which can be taken to represent in

an average way the very complex deformation processes which occur in the

actual sliding between surfaces and can indicate how surface roughness and

interfacial film strength influence friction and wear.

2.

Asperity deformation models

To make the problem more tractable attention is limited to the case of

sliding of a hard (rigid) surface over a relatively soft one and in order that

slip-line field theory can be used in analysing the deformation it is assumed

(i) that deformation occurs under plane strain conditions with no flow nor-

mal to planes which are parallel to the sliding direction and normal to the

sliding surfaces and (ii) that the material of the softer surface is an idealized

rigid-plastic material which deforms at constant flow stress, which does not

change its density (volume constancy assumption) and which remains iso-

tropic at all stages of deformation. With this method it is the slip-line field

which consists of two orthogonal families of curves representing the direc-

tions of maximum shear stress and m~imum shear strain rate within the de-

forming region which is treated as the unknown to be determined. The gen-

eral approach to a problem is to construct a field that satisfies the stress

boundary conditions and is internally stress consistent and then to check to

see if it satisfies the necessary velocity conditions; if not the field is adjusted

until all conditions are met. For complete acceptance, a slip-line field should

be checked to see that the rate of plastic work is always positive and that the

stresses in the assumed rigid material are below the yield point but frequently

both of these conditions are assumed without rigorous proof. In carrying out

the above procedure the stresses associated with a slip-line field are calculated

from the stress equilibrium equations referred to slip lines (Hencky equa-

tions) which can be expressed in the form

p + 212$ = constant along a I line

p - 2h =

constant along a II line

(1)

where p is the mean compressive (hydros~tic) stress which acts normal to

the slip lines, h is the shear flow stress which acts parallel to the slip lines and

$Y s the anticlockwise angular rotation of the I lines from a fixed reference

axis with the I lines taken as those on which the shear stress exerts a clock-

wise couple. In checking a field for velocity this can either be done numeri-

cally using equations derived from the condition that, for volume constancy,

the rate of extension along slip lines must be zero or by a graphical method

based on the same condition in which a velocity

diagram ~hodograph) is con-

structed, and it is the latter method which is used in the present work. A de-

tailed description of slip-line field theory and its application is given in the

books by Hill [ 93 and Johnson et al. [lo].


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L

R

soft

asperity

. ../_.---

(4

(b)

Fig. 2. The rubbing model: a) slip-line field, b) hodograph.

2.1. Rubbing model

Figure 2 gives a possible steady state slip-line field and the corresponding

hodograph for the plastic deformation of a soft asperity by a hard (rigid)

asperity. It is similar to one proposed by Green [ 51 for a weak junction which

apparently he made little use of. The interface ED between the hard and soft

asperities and the stress free surface AE are both assumed to be straight with

their directions defined by the angles Q and n measured from the sliding di-

rection which for convenience is represented by the velocity U (Fig. 2) of

the soft material, the hard asperity being assumed to be stationary. It should

be noted that the deforming region ABCDE is that existing after initial junc-

tion growth and consequently that the angle rl will in general be far greater

than the initial slope of asperities on the surface of the soft material. The an-

gle (Y,however, will not change and will be relatedto the initial surface rough-

ness of the hard surface. In constructing the slip-line field the independent

variables are taken to be the angle (Yand the strength f of the interfacial film

defined in the usual way as the ratio of the strength of the film to the shear

flow stress h of the soft material with 0 < f < 1. (To determine the scale of

the field it will also be necessary to know the normal load N (Fig. 2) carried

by the asperity.) With this model the deformation is represented as a stand-

ing wave and the straight line joining A and D must be parallel to U to satisfy

volume constancy. Also the shear force on the soft asperity at the interface

must act in the direction DE to oppose motion. These two conditions to-

gether with the further condition that the slip lines must be inclined at an an-

gle of $n to the free surface AE define the slip-line field and it follows from

geometry that


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rl = sin-l

sin LY

(1 - f)12

(3)

where cf, is the angle between CD and U and is measured positive as shown in

Fig. 2. The field, which is made up of regions of straight slip lines (CDE and

ABE) and a centred fan (BCE), is clearly internally consistent for stress, i.e.

in working round any closed slip-line loop using eqns. (1) the change in p is

zero. The hodograph (Fig. 2) shows that the field is acceptable for velocity.

In this there is a discontinuity in tangential velocity along the slip line ABCD

so that material which enters and leaves the field with a velocity U flows in

the directions AE and ED in the regions ABE and CDE respectively; a typical

stream line is given in Fig. 2. The range over which the solution can apply is

determined from the condition that Cpmust be positive, which taken in eon-

junction with eqns. (2) and (3) gives 01G 77< a~. To find the direction of the

resultant force

R

(Fig. 2) acting between the asperities and hence the coeffi-

cient of friction the equilibrium of the triangular element CDE is considered.

The shear stress on the slip lines is k and the hydrostatic stress p in this region

is found by starting at the free surface AE where p =

k

and applying the rele-

vant equation of eqn. (1) along the slip line ABCD (II line) which gives

P=k{l+2($n+(I,-_rl))

wherein + Q, - 77 s the angle subtended at the centre of the fan BCE. By re-

solving forces it can now be shown that the horizontal and vertical compo-

nents of R (per unit width) are given by

F=k[{l+2($n+cP--)}sinol+cos(cr+2@)]ED

(4)

and

N =

k[

{l + 2($.7r + 9 - n)] cos 01f sin (a + 2@)] ED

(5)

where ED (Fig. 2) is the length of the interface. For a given normal load N,

ED can be found from eqn. (5),

i e

the scale of the slip-line field (Fig. 2) is

determined and substituted in eqn. (4) to give the corresponding frictional

force

F.

From eqns. (4) and (5) and substituting for Cpand n from eqns. (2)

and (3) the coefficient of friction or =

F/N

can be expressed as

A sin Q + cos (COS-~ f - a)

i.r=

A cos a +

sin (cos-l

f-a)

where

A=l+~n+co~~f-22cu--2sin-~

sin (Y

(1 -

f)2

(6)

which shows that 0 < ~1< 1; results showing the influence of surface

roughness CY nd interfacial film strength

f

on P are given in Fig. 3.


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0 10 20 30 40 50 60 70 60 90

Hard asperity angle a/degrees

Fig. 3. Variation of p with CY nd f.

In ending this section it is interesting to note that the idea of a standing

wave in metal deformation processes is not new. In drawing it is well known

that a standing wave can occur ahead of the die and Johnson and Rowe [ll]

have given a slip-line field, part of which is similar to the present one, to ex-

plain this. Collins [ 121 also used a similar field in considering rolling contact.

Further evidence of a standing wave effect has been given by Enahoro and

Oxley [ 131 from machining experiments and by Cocks [ 141 from experi-

ments in which a hemispherical rider and a drum were in sliding contact. An

apparent virtue of the model is that it can contribute to friction while at the

same time it does not involve removal of the deformed material and thus in

theory no wear is involved. For this reason it will be termed the rubbing

model.

2.2. ear model

When @ < 0, q > f n, it is no longer possible to construct a steady

state slip-line field (unless (Y> +X when as will be seen later a cutting model

can apply) and the conditions are similar to those described by Green [ 51 for

a strong junction with no sliding at the interface ED (Fig. 2) and with the life

cycle of the junction consisting of formation, deformation and fracture. That

is, in this range the deformed material is removed and a wear particle pro-

duced. Although a theoretical solution for such non-steady flow appears pro-

hibitively difficult it is possible to make some estimates of the forces and cor-

responding coefficients of friction by assuming that the deformation that oc-

curs before fracture can be characterized by an increasing v (V 2 $ n) as shown

experimentally by Green. This can be achieved by using the slip-line field

given in Fig. 4 in which only stresses are considered and no attempt is made

to satisfy velocity. With this model AD is assumed to remain parallel to U

during deformation and the only change in the external shape of the de-

forming region that is taken into account is that resulting from the increase

in 17with AE still assumed to be straight. The slip-line field in Fig. 4 consists


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236

Fig. 4. The wear model.

of two regions of straight slip-lines which can, however, no longer be joined

by a centred fan as with the previous field (Fig. 2) but instead meet at a line of

stress discontinuity EC (Fig. 4) (see for example ref. 9) across which there is

a jump in hydrostatic stress. The free surface AE again determines the direc-

tion of the slip lines in this region but in considering the interface ED the

strength of the interfacial film is no longer the determining factor as there is

no sliding along ED. However, for convenience CYnd f will still be taken as

the independent variables with f now simply representing the ratio of the re-

solved shear stress at ED to the shear flow stress

k*. It

then follows in the

same way as for the rubbing model that

a--

#

=$ cos-l f

(7?

where Q, s the angle between CD and U measured positive as shown in Fig. 4.

Also by noting that for equilibrium the stress discontinuity EC must bisect

the angle n - p (Fig. 4) it can be shown that eqn. (3) again applies. An ex-

pression for the coefficient of friction is found as before by considering the

stresses acting at the interface ED. In this case the hydrostatic stress p in the

region attached to ED is found by starting at the free surface AE and by con-

sidering the jump in stress across EC which gives

p =

k(1 -

2 sin p)

where from geometry

0 = (y --in - _ cos-l f f sin-l

II

(1 - f)l'2

Now by resolving forces and substituting for n and cp rom eqns. (3) and (7)

it can be shown that

*Alternatively the shortening of AD Fig. 4) could replace f as an independent vari-

able with Q calculated from volume constancy considerations. Then, following Green [ 61,

the life cycle of the junction could be considered but this is not done in the present paper

as it would be inconsistent with the approach used.


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-2. o- 1 i ,

10 20 30

I I 045

Hard asperity angle c/degrees

Fig. 5. Variation of vertical load bearing stress with ~1and f for the rubbing and wear

models.

= {1-2sinp+(1-f2)1/2}coso-fsincu

(3)

and values of p calculated from this equation are given in Fig. 3. In this range

the normal load N can for certain conditions go negative as shown in Fig. 5

in which values of the vertical stress found by dividing N by the horizontal

projection of ED are given for both the rubbing and wear models. This would

imply negative values of p and these have not been included in Fig. 3. Of course

when this stress is zero (Fig. 5) then P is infinite.

To calculate the wear rate associated with this model it is assumed that

the plastically deformed material defined for a given Q by the condition @I

0, Q = in is removed which from the geometry of Fig. 4 gives

wear rate =

volume loss in a given sliding distance

normal load

= 1 sin2 (Y+ + sin 2~

G 1 + sin 2a

The curve representing this equation which shows the influence of (IIon the

wear rate is given in Fig. 6.

2.3. Cutting

model

When cx > a7r the steady state slip-line field for cutting with a restricted

contact cutting tool (Fig. 7) proposed by Johnson [ 151 and Usui and Hoshi

[ 161 can be considered as a possible model. This field is clearly similar to

that for the rubbing model (Fig. 2) with AE again a stress free surface and

ABCD a line of velocity discontinuity. However, there are marked differ-

ences in the flow as shown by the hodograph in Fig. 7. The straight line join-

ing A and D is no longer parallel to U and the distance of D below A deter-

mines the thickness tl (Fig. 7) of the layer which is removed as a chip flowing


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06

cutting model

wea

model

x

; i

5 0. 1.

,

, /

o. *,; o

0 10 20 30 LO 50 60 70 60 90

Hard asperity angle a/degrees

Fig. 6. Variation of wear rate with LY nd f.

(b)

Fig. 7. Restricted contact cutting model: (a) slip-line field, (b) hodograph.

in the direction given by the hodograph. The shear force on the soft asperity

at the interface ED must now act in the direction ED to oppose motion,

i e

in the opposite direction to that for the rubbing model, and the equation cor-

responding to eqn. (2) is

(Y $J =+ -cos-1 f)

(10)

where # is again measured positive as shown in Fig. 7. A limitation of the re-

stricted contact cutting model in relation to the present problem is that the

hard asperity will usually extend beyond E and the chip will be constrained

to move parallel to ED. If the interface is sufficiently long then the distance

ED over which contacts occur can be termed the natural contact length and

is determined as part of the solution. For such conditions Lee and Shaffer


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[ 171 have proposed the slip-line field given in Fig. 8 in which the velocity

discontinuity ABCD is straight and ED is determined by the values of 4 and

$. With this model AE is again assumed to be stress free and it follows that

along ABCD

p = k

and therefore that the resultant force transmitted by ED

and ABCD will be inclined at an angle of $7r to ABCD. Noting that eqn. (10)

still applies it can now be shown from geometry that the coefficient of fric-

tion associated with this model is given by

/J=mn(a--

an++

cos-l

f)

11)

and results calculated from this equation are given in Fig. 3. In the same way

as with the rubbing model the scale of the slip-line field, represented by say

t1 (Fig. 8), is determined from the normal load and it can be shown by con-

sidering the corresponding volume of chip removed over a given sliding dis-

tance that the wear rate for the cutting model is given by

1

wear rate = -

cos (OL $ cos-1 f)

dzk cos {n + a -+ + + cos-1 )}

(12)

Curves calculated from eqn. (12) showing the influence of LY nd

f

on the

wear rate are given in Fig. 6.

3. Discussion

The models of asperity deformation proposed give results which are

consistent with the two basic laws of friction with the frictional force pro-

portional to the load normal to the contacting surfaces and independent of

their area. Also, in agreement with the trends usually observed in experi-

ments, it can be seen (Fig. 3) that for a given interfacial film strength

f

the

coefficient of friction 12 is predicted to increase with increase in surface

roughness (Yover the entire range of conditions considered while for a given

Q a decrease in f is predicted to decrease P in the rubbing model range and to

increase P in the cutting model range. It is difficult to check the accuracy of

the actual numerical values of 1-1Fig. 3) because of the problems in relating

the idealized models of asperity deformation used in the analysis to the ac-

tual surface conditions in friction experiments. However, for engineering sur-

faces under normal working conditions OLmight be expected to have average

values in the range 0 - 10 with

f 0.5. The coefficient of friction ,u can

also become extremely large in the cutting range and, for example, for f = 0

P + 00 as (11 90. A further observation made by Bowden and Young in

their experiments was that for chemically clean surfaces junction growth

continued until the real and apparent areas of contact became equal and gross

seizure occurred. In contrast, for contaminated surfaces junction growth was

soon ended. An estimate of the junction growth associated with the rubbing

and wear models for comparison with these findings can be made from the

values of vertical stress given in Fig. 5. For a given Q he initial area when only

the normal load is applied can be assumed to be inversely proportional to the

stress corresponding to f = 0 (F = 0 for small 0) and the final area will be in-

versely proportional to the stress corresponding to the actual value of f.

Therefore the ratio of these two stresses can be taken as the ratio of the areas.

In this way it can be seen (Fig. 5) in agreement with Bowden and Youngs re-

sults that as f is increased for a given cr the growth in area increases and that

for sufficiently large values of f the real and apparent areas will be equal.

Two wear processes have been suggested both of which give wear rates

which are inversely proportional to the shear flow stress (hardness) of the

softer material and thus satisfy at least one of the basic laws of wear. For

1y< in a wear particle is produced by shearing off the deformed part of the

asperity when, for a given cr, f is sufficiently large to make the wear model

applicable. In this range the wear rate results (Fig. 6) plot as a single curve?

and as can be seen the larger the value of 01 he higher is the wear rate and the

smaller is the value of frequired to give wear. The role of contaminant films

(lubricants) in this range is therefore to reduce f so that rubbing conditions

are maintained and clearly the rougher the surface the more effective must

be the lubrication. In passing it should be noted that although with the rub-

bing model there is theoretically no wear it is possible that wear could occur

on a much reduced scale at the interface ED (Fig. 2). For 1y> in wear is pro-

duced by a cutting action and in this case, as shown by the family of curves

in Fig. 6, a reduction in f for a given 01 ncreases the wear rate which is con-

sistent with experience - see for example the results of Mulhearn and Samuels

[ 191 for abrasive wear. Also an increase in sharpness (Y or a given f in-

creases the wear rate as would be expected.

From Figs. 3 and 6 it can be seen that for (II> a~ there is a region in

which for the cutting model used

[

171 there are no solutions for I_(or wear

rate. Also with this model cutting cannot occur and a chip cannot be formed

unless (Y> $n when it is well known from experiments [20] that chips are

produced for much smaller values of (Y.Both of these limitations can be over-

*Note that for f = 0 the resultant force is normal to the interface between the hard

and soft asperities and /J = tan CYor all cases.

The possibility of gross seizure occurring has not been considered.


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Fig. 8. The natural contact cutting model: a) slip-line field, b) hodograph.

come in the following way. With the Lee and Shaffer model it is assumed

that a plastic state of stress exists in the region ABCDE (Fig.8). In an altema-

tive model which has been widely used in machining research the chip is still

assumed to be formed by shearing across a single shear plane, equivalent to

ABCD (Fig. 8), but the material between this plane and the interface is taken

to be rigid. With this model the comer D becomes a stress singularity and

eqn. (10) no longer applies. Working in much the same way as with the Lee

and Shaffer model, but with the hydrostatic stress along ABCD now calcu-

lated from the free surface just ahead of A (Fig, 8), expressions for ~1 nd

wear rate have been obtained and results calculated in this way are given in

Figs. 9 and 10. (A full description of the method is given by Challen [21] .)

It can be seen that values of p and wear rate now exist for all considered

combinations of (Y nd f and that cutting can occur for (Y alues down to

about 20. Over a certain range (a =

20

-

45) the new results (Figs. 9 and

10) have introduced some overlapping of results and for certain values of (Y

and f in this range both the rubbing (or wear) and cutting models give possible

solutions. The values of ~1 or such conditions are always less for the cutting

model than for the other two models and hence the cutting model gives a

smaller frictional force for a given normal load. It might therefore be reasoned

that in these cases the values for the cutting model will apply. This can lead

to some unexpected results and for example in this range an increase in sur-

face roughness 01can for certain values of f cause the predicted values of cc

and wear rate suddenly to decrease. It is not known whether there are any

experimental results showing such effects.


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242

0 10 20 30

LO 50 60

70 80 90

Hard aspmy angl e a/ degrees

Fig. 9. Variation of p with LY nd f (modified cutting model).

06.

0 10 XI 30

40 50 60

70 60 90

Hard asperi ty angle al dcgres

Fig. 10. Variation of wear rate with 01and f (modified cutting model).

In conclusion it is of interest to consider, in the light of the present

work, the observation that the wear calculated by assuming that all of the

plastically deformed material is removed is many orders of magnitude greater

than the actual wear. Actual surfaces will have a distribution of asperities of

varying geometry, and asperity deformation will in general be three dimen-

sional. It might still be expected, however, that the three deformation models

proposed, namely rubbing, wear and cutting, will have their three-dimen-

sional counterparts which will exhibit similar characteristics to those de-

scribed. For engineering surfaces under normal working conditions it would

seem reasonable to suppose that most asperity contacts will have conditions

appropriate to the rubbing model and that these will carry most of the nor-

mal load. Wear will then only occur for the relatively small number of asper-

ities which are sufficiently sharp considering the local interfacial film strength

to give either a cutting or a wear mechanism. This therefore offers a possible


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243

alternative explanation to that, for example, of Kraghelsky [8] of why the

actual wear is so much less than might be expected.

In the present analysis the deformation has been assumed to occur at a

constant flow stress and in future work it is hoped to consider possible varia-

tions in flow stress with strain, strain rate and temperature (as has already

been done for metal cutting [22] ) and in this way to allow for the influence

of such factors as the speed of sliding.

Acknowledgment

The authors wish to thank Professor D. Dowson for many stimulating

discussions of the problems associated with asperity deformation.

References

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-

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