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A contribution to the theory of wear — the abrasive wear of a soft surface against a rough hard surface
127
Wear, 50 (1978) 127 - 144
A CONTRIBUTION TO THE THEORY OF WEAR - THE ABRASIVE WEAR OF A SOFT SURFACE AGAINST A ROUGH HARD SURFACE A. BOS* Shell Research Ltd., Thornton (Gt. Britain)
Research Centre, P.O. Box 1, Chester CHl 3SH
(Received January 9, 1978)
Summary A theoretical model has been developed to describe the abrasive wear of a soft surface over which a rough hard surface with longitudinal asperities slides. The model predicts that: (a) the volumetric wear rate is proportional to the roughness of the hard surface, provided that the shape of the asperity peak height distribution of the hard surface remains the same; (b) the wear rate is inversely proportional to the hardness of the softer surface and proportional to the load, provided that the peak height distribution of the asperities on the hard surface is broad; (c) the wear rate is affected very little by the shape of the asperities on the hard surface.
1. Introduction Many theories of wear have been proposed in the literature. Most of them deal with adhesive wear [ 1 - 41 and a few with other types of wear, such as abrasive wear [ 5,6] , surface fatigue wear [ 71 or chemical wear [S] . In the models for adhesive wear the roughness of the wearing surfaces is considered to be irrelevant. This is justified because in practice the asperities are very shallow and therefore that part of the surface of the asperity making contact with the other surface by penetration, for instance, is roughly equal to the projected area of intersection of the asperity with the other surface. However, in a model of abrasive wear the roughness of the surfaces must be taken into account. This was done to some extent in the model of Archard [5] and Rabinowicz [6] for two-body abrasion. In this model it was assumed that a rough hard surface slides over a smooth soft surface. The asperities of the hard surface were considered to be rigid *Present address: Koninklijke/Shell-Lahoratorium, 0043-1648/78/0050-0127$2.25
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The Netherlands.
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128
circular cones with a base angle* 0 and the aim of the model was to calculate the volumetric wear rate V/s caused by one asperity. It was assumed that the load L was supported by that part of the soft surface in contact with the leading half of the asperity. The result obtained was V 2 tan 0 L
-=--
n
S
H
where H is the hardness of the soft surface and V is the volume worn off the soft surface in the sliding distance s. A more rigorous model would take into account such characteristics of the hard surface as the shape of the asperities and their peak height distribution. The objective of this paper is the development of a new statistical model of abrasive wear. 2. Description of the theory
2.1. Assumptions A rough hard surface slides relative to a soft surface and causes wear of the soft surface. The following assumptions are made. (1) The soft surface is and remains smooth throughout the wear process. (2) The hard surface is rough. The asperities on it have the same length and are of constant height in a longitudinal direction (the sliding direction); in the transverse direction they are sinusoidal in shape. The asperities are evenly distributed over the hard surface as shown in Fig. 1. (3) The number of asperities in any transverse cross section through the area of contact between the surfaces is constant and the height distribution of the asperity peaks in every such cross section is the same. (4) The asperities on the hard surface are rigid, i.e. not deformable, and they do not wear. (5) Only the soft surface wears. In general the wear has abrasive, adhesive and chemical components. Hence the volumetric wear V is given by v = f( via, vadh, Vchem) It is assumed that vahr
3
vdh
+ Vchem
(6) The presence of a lubricant is not specifically taken into account. If there is a lubricant, it can assist in carrying the load by a hydrodynamic action. However, when the roughness of the hard surface is high compared with the film thickness the hydrodynamic action can be neglected.
*The model is normally expressed in terms of the vertex angle but in the present paper, as will appear in Section 4 and the Appendix, it is more convenient to use the base angle.
129
+---
In--+
Fig. 1. Distribution of the asperities over the hard surface: -, of contact area; direction of motion, right to left. Fig. 2. The shape of an asperity in the transverse direction:
asperities; - - -, outline
2 = hcosax,
--n/2
< ax <
3rrl2.
2.2. The shape and size of asperities The asperities have the shape of cosine curves in the transverse direction and each is accompanied by a valley of exactly the same form (Fig. 2). The pair (asperity plus valley) can be described by the equation -1712 < ax < 3n/2
z = h cos ax
(1)
where h is the height of the asperity and 27r/u is the .width of the pair (asperity plus valley) at its baseline. A consequence of this particular shape of asperity is that the mean surface of the hard surface coincides with the plane 2 = 0. It will be assumed that the number of asperities per unit of transverse distance with heights between h and h + dh is given by n(h)dh =:
exp 1\
n(h)dh = 0
-ln2Wht)
21n’o
dh
I
O
(24 or h>h,
(2b)
These equations describe a distribution with a cut-off at h = h,, the maxim u m asperity peak height, and this distribution is shown graphically in Fig. 3. In the equations n, is the total number of asperities in a transverse cross section over the area of contact between the soft and hard surfaces, n(h) has a maximum at ,h = h,, u is a measure of the heterodispersity of the distribution and br is a constant given by
It is assumed that there is a relation between the base of the pair (asperity plus valley) and h. This relation can be expressed as a = bzhb3
(4)
130
f
-----
c
5 F 2 z k E D ILL
I
I I
IL---ht PEAK
-I
T 11 I
hrn HEIGHT,
f-
h
Fig. 3. Distribution
12
---+
of the asperity peak heights.
Fig. 4. Modified model for the calculation of the load-carrying capacity of the asperities: -, asperities; - - -, outline of contact area; direction of motion, right to left.
in which ba and ba are constant and bs can be chosen freely. Since the width II of the contact between the two surfaces is given by hm 11 =
s 0
n(h) z
a
dh
(5)
ba is then determined by h,
b2=1 4
1
2xn(h)I+dh
(6)
0
The value of bs can be interpreted physically as follows: b, = 0 implies that the width 2n/a of the sets (asperity plus valley) is constant, irrespective of the value of h; bs = -0.5 implies that all the asperities have the same radius of curvature in the transverse direction (the radius of curvature is l/a2h); ba = -1 implies that all the asperities have the same mean slope in the transverse direction. 2.3. The total load L carried by the asperities as a function of the distance D between the two surfaces Assume the soft surface is at a mean distance D from the hard surface. In order to calculate the total load L carried by the asperities, we replace the model temporarily with a simplified model in which the asperities of the same peak height are aligned to form very long asperities (Fig. 4). This is possible because it was assumed that in any cross section over the contact area the number of asperities is constant and that they have the same peak height distribution. Consider now one such long asperity with peak height h. When h > D, this asperity has sunk into the soft surface and carries a weight AL given by AL = HAAl
(7)
131
where H is the hardness of the soft surface and AAl is the projected contact area between the asperity and the soft material: 212
AA1 = -
arccos
a
(8)
in which l2 is the length of the area of contact between the two surfaces. Consequently, 212H AL = a
(9)
The total load L carried by the asperities is then L =
r”
(10)
ALn(h)dh
2.4. The total ploughing surface A, of the asperities as a function of D We return now to our original model in which the longitudinal asperities are regularly distributed over the hard surface (Fig. 1). The formula for L (eqn. (10)) is valid for this model as well. Consider an asperity with peak height h > D which has sunk into the soft surface. The area AA2 of the cross section of the sunken part in the transverse direction is given by AA2 =z (h2 -D2)1’2 a1
-D
arccos
(11)
The total ploughing surface of all the asperities sunk into the soft metal is then A, = J!
AA2n(h)dh
(12)
D
Finally, in order to characterize the roughness of the hard surface in a simple way, the roughness R of the surface can be calculated from R=;
$ lo
b2
-hn(h)dh a
R is identical to the centre-line average value (c.1.a.) 2.5. The volumetric rate V& of abrasive wear as a function of the load It is &assumedthat the length 1, of the asperities is a fraction b4 of the length of contact area between the two surfaces: 1, = b412 (14)
132
The quantity l/b4 can be considered to be the number of rows of asperities in the contact la. Figure 1 shows three such rows. Therefore, when the contact is slid a distance ls, a volume A212/b4 is displaced plastically in a ploughing action at the leading edges of the asperities and a fraction p is removed as wear particles. It is further assumed that grooves in the soft surface caused by the passage of a row of asperities are partially filled up by material plastically displaced sideways by following asperities running in adjacent tracks so that the asperities never run through the tracks formed by asperities in the immediately preceding row. The volumetric rate V& of abrasive wear, which is defined as the volume of material removed per unit distance slid, is then given by VA = pAdb4
(15)
It is convenient to put Vgbr and other quantities in more general form. We introduce therefore the following dimensionless variables: a = h/h,
(16)
P = h/h,
(17)
Y = D/h,
(18)
6= -
L
(19)
4 l2H
E= b* - Xl, PAIR
(20)
44
n*(h) = -
(21)
nt By substitution it can be shown that n*(ah,)da
=
n*(h)dh hm
= exp - ln2 ($Lfo’ht )/dol [h, 1
/ exp / -1n2~~n~~‘h”/d~]-1 (22)
6 = i arccos(~)n*(olh,)a-b~ Y
da 1n / n*(ah,)aTb3da
1 -I
(23)
133
It can be seen from the equations that, for a given value of y, both 6 (dimensionless load) and E (dimensionless wear rate) are independent of n, R, L, H, 1,, 1s and p, but are still functions of h,/h, or /3/a, bs and u. Of course the factor h, in the denominator of the expression for n*(ah,)da is cancelled in the expressions for 6 and E since n*(olh,)da appears in both the numerator and the denominator of eqns. (23) and (24). It can also be shown by calculating 6 and E as functions of y (using the computer) and plotting curves that the relations between 6 and E are always monotonic. These relations can be considered as general dimensionless forms of the load-wear rate relation and will be used to represent the results of the theory. 3. Calculation of the volumetric rate V& of abrasive wear 3.1. Influence of the roughness R It can be shown as follows that, with certain restrictions, Vlabr is a linear function of the roughness R. From the definition of the dimensionless wear rate E (eqn. (20)) the relation between V& and R is EP4 Vi, = R b4 In this formula /I is determined by the apparatus and is a constant. The fraction p (eqn. (15)) has been assumed to be independent of roughness, and the factor b4 (eqn. (14)) is independent of roughness in the transverse direction. The dimensionless wear rate E, as given by eqn. (24), is independent of roughness for given values of (Y//I, u and bs, i.e. for a given distribution of asperity peak heights and a given relation between the wavelength and amplitude of an asperity. In fact, as shown later, once a/P has been fixed E is not sensitive to changes in bs and hence the restrictions are effectively reduced to a given shape of distribution of the asperity peak heights. Within this restriction the model predicts a linear relation between volumetric wear rate VLbr and roughness R. 3.2. Influence of the number n of asperities in a cross section of the contact area From the analysis presented in the preceding section, it follows that both 6 and E are independent of n,; consequently, VLbr is also independent of n,. 3.3. Influence of the choice of the cut-off ratio in the asperity height distribu tion h, /h, (or p/a) In Fig. 5, E is given as a function of 6 when u = 1.5 and bs = -0.5 (this value of bs implies that all the asperities have the same radius of curvature in the transverse direction) for fl/cu = 2, 3,6 and 10. It appears that E increases for a given value of 6 as P/a is increased until an upper limit is
p’a=6Lg, (I=,. PA 1, 0.001 0.001
0.01
0.1
1.0
6
Fig. 5, Dimensionless wear rate f as a function of dimensionless of the cut-off in the asperity height distribution.
load 6 for various values
reached. This behaviour has been found in all the calculations in which /3/ol was varied. 3.4. Influence of the choice of the asperity width factor bs In Fig. 6, E is given as a function of 6 when bs = -1, -0.5 and 0 with (J = 1.5, for p/al = 2, 3 and 6. It appears that the choice of bs does not influence the dimensionless wear rate significantly for given values of the cut-off ratio h, /h, (or p/a). 3.5. Influence of the degree CTof heterodispersity Figure 7 shows the e-6 characteristic for u = 1.1, 1.2, 1.5 and 2 whilst bs = 0.5. Also the relation for a monodisperse distribution is given. In all these cases h,/ht has been chosen to be large relative to u so that the wear rate is at its highest value for the given value of u (see Section 3.3). It appears that the dimensionless wear rate is strongly dependent on the degree of heterodispersity. 3.6. Influence of the asperity shape To investigate whether the asperity shape in the transverse direction has an influence on the wear rate, the calculations have been repeated but with triangular instead of sinusoidal asperities.
135 I.o 1
0 1.
0.001 L 0.001
0.01
1 I .o
0.1 6
Fig. 6. Dimensionless wear rate E as a function of the asperity width factor b,.
of dimensionless
load 6 for various values
If the width of the pair (asperity plus valley) is 2n/a at the base and if a is assumed to’be related to the asperity height h by a = b2hb,
(25)
which is the same relation as used for the sinusoidal asperities (eqn. (14)), it can be shown for the triangular asperities that -1 l F=
s
Y
a---y
-
cCbsn*(ah,)da
a
(?i?!2a-bxnyah
E=
m
n*(ah,)Cb3da
(26)
m
)da
-’
(27)
OL Y
In the above equations the dimensionless groups are defined by eqns. (16) (21) in the same way as for the sinusoidal asperity shape. In Fig. 8 E is plotted against 6 and u = 1.1, 1.2, 1.5 and 2 for b3 = -0.5. Again the E-C?characteristic for a monodisperse distribution is shown.
136 LC
‘1
0.1
ui MONODISPERSE
0.01
0.01
6
0.1
1.0
Fig. 7. Dimensionless wear rate E as a function of dimensionless load 6 for various values of the heterodispersity factor u and large values of P/CL The asperities are sinusoidal in the transverse direction.
Figure 8 shows that e is again strongly dependent on the heterodispersity, although when Figs. 7 and 8 are compared it is seen to be not as strongly dependent as in the case of sinusoidal asperities. For a high degree of heterodispersity the shape of the asperities, whether sinusoidal or triangular, has a negligibly small effect on the wear-load relation. 3.7. Dependence of wear rate on hardness of the softer surface and on load From the graphs shown in Figs. 5 - 8 we can make the approximation that the relations between log e and log 6 are straight lines. Thus f ahb5 (28) where b5 is a constant. From Figs. 7 and 8, b5 can be obtained as a function of u for both sinusoidal and triangular asperity shapes. Figure 9 shows b5 plotted against u. For this purpose b5 has been averaged over the range of e values 0.01 < E < 0.5. It is seen that when the distribution is monodisperse b, is much higher than unity, but at higher heterodispersity b5 becomes about unity which is in agreement with experimental observations and the predictions of other wear models proposed in the literature. So, if the peak height distribution is heterodisperse the wear rate is roughly proportional to
137 IO
0.1
‘y
0.01
0.001
001
0.1
I.0
Fig. 8. Dimensionless wear rate E as a function of dimensionless load 6 for various values of the heterodispersity factor u and large values of P/cu. The asperities are triangular in the transverse direction.
6 and, from the definition of 6 (eqn. (19)), it is inversely proportional to the hardness of the softer surface and is a linear function of the load, whether the asperities are triangular or sinusoidal in the transverse direction. 4. Discussion Although the abrasive wear model has taken into account the statistical nature of the surface asperities, the dependence of the volumetric wear rate on surface parameters is generally simple in form. For instance, the volumetric wear rate is linearly related to the roughness (c.1.a.) of the hard surface, provided that, among other things, the distribution of the peak heights of the asperities on the hard surface remains unaltered. Subsequent to the development of the analysis given in this paper, Hisakado [ 91 published a similar theoretical study in which he considered the abrasive wear of a soft surface caused by the conical asperities of a mating hard surface. In his model the slopes of the cones are randomly distributed and the mean value of the slopes as a function of the roughness is given by an empirical relation. He found that the wear volume per unit of
I
0
I
1.5
2.0
o-
Fig. 9. Variation of the wear rate-load index b5 as a function of heterodispersity E a 6 br ; +, sinusoidal asperities; -X -, triangular asperities.
U:
frictional work increased with the roughness of the hard surface, but that the rate of increase diminished at high roughness. When this relation is transformed into a wear rate-roughness relation, which is possible by making use of Hisakado’s eqns. (9), (12), (21) and (18), the wear rate is still a non linear function of the roughness. Therefore there is a difference between Hisakado’s theory and the theory presented here. There are, of course,
139
differences in the models used, but a major factor in Hisakado’s theory is the form of the empirical relation between the mean cone slope and the roughness (his eqn. (18)). From a calculation presented in the Appendix, it appears that the wear rate is roughly a linear function of the cone slope. Therefore if the cone slope levels off with increasing roughness, as it does by Hisakado’s eqn. (18) (reproduced as eqn. (A4) in the Appendix), the wear rate does the same. Likewise, if the cone slope is independent of roughness the wear rate will increase linearly with roughness and in this respect the two models would predict the same behaviour. In an analogous way a different wear rate-roughness relation would be obtained from our model if some alternative form t,han the one assumed was taken for the effect of roughness on the shape of the asperity peak height distribution; at present we have chosen to assume that the shape of the distribution is independent of roughness. A remarkable outcome of our model is that the wear rate is only a weak function of the relation (eqn. (4)) between the asperity base and the asperity height. Another important result, which it may be possible to explore in an experimental test of the theory, is that the volumetric rate of abrasive wear is proportional to the load only if the peak height distribution of the asperities is heterodisperse. If the distribution is monodisperse it is a stronger function of the load, i.e. the rate of wear increases faster than linearly with load. It would be interesting to know what the heterodispersity of the distribution is in practical cases. However, to obtain the distribution from a real surface is difficult because in practice most asperities are not completely separated from each other. The distribution is therefore not exactly defined and it is necessary to use an arbitrary criterion to decide, for example, whether a hump on the shoulder of an asperity is a separate asperity. A real roughness profile is presented in Fig. 10. It was obtained from an En 31 steel disc that was circumferentially ground to a nominal c.1.a. value
Fig. 10. Talysurf trace of a circumferentially across the direction of grinding.
ground
En 31 steel disc (0.19
pm c.1.a.)
140
of about 0.20 pm (8 pin) and which was used in our three-(bronze)pin-on(steel)disc wear machine for bronze-on-steel wear studies. As a result of the grinding the disc has longitudinal asperities and the profile is taken in the transverse direction. It can be shown by numerical integration that the trace shown has a c.1.a. value of 0.19 pm (7.5 pin) and is therefore considered to be representative for the disc. The asperity peak height distribution is given in Fig. 11.
--
NEGATIVE
- --~.
PEAKS
PEAh
Fig. 11. Distribution
IHEIGHT,
POSITIVE
PEAKS h
h ,/~rn
of the asperity peak heights of the trace shown in Fig. 10
Comparison of Fig. 11 with Fig. 3 shows that in practice the peak height distribution has a different form from that assumed in the model. A real surface shows negative peaks (i.e. peaks with their tops below z = 0, the mean level of the surface). In our model negative peaks have been excluded. However, it is considered that in practice only the highest of the positive peaks contribute to the wear process, so that only the peak height distribution for, say, h > h, is important. From the maximum distribution to larger peak heights, the distributions in Fig. 11 and Fig, 3 are similar, and the distribution in Fig. 11 can be described satisfactorily by eqn. (2a) when u is given the value 2. Hence the disc surface has a heterodisperse asperity peak height distribution and according to the theory the wear rate should be roughly proportional to the load. This is in agreement with experimental observations quoted in the literature [lo] and suggests that many other practical surfaces are heterodisperse in the terms of our model.
141
1 IOOpm -
(b)
(a)
Fig. 12. Comparison of the roughness of (a) the bronze pin surface with the roughness of (b) the steel disc surface after a wear experiment.
An important assumption in our model is that the soft surface remains much smoother than the hard surface throughout the wear process. Figure 12 shows the transverse roughness profiles of the surface of a bronze pin and a rough steel disc (0.4 pm c.1.a.) from a three-pin-on-disc experiment run under lubricated conditions in which abrasive wear was dominant and no wear of the asperity tips of the steel surface took place. From the profiles it can be seen that the pin surface is about five times smoother than the mating steel surface, so the assumption given above seems to be justified. Nomenclature
il A2
bl bzv b3 b, b5
D h 4,
ht
H
h 12 4
L n(h)dh n*(h)
nt
P R s V V’
parameter giving information about the sharpness of the asperity total load-carrying surface total ploughing surface of the hard asperities sunk into the soft surface in a cross section over the contact between the surfaces constant used in expression for n(h) constants used in the relation between h and a the ratio of the asperity length in a longitudinal direction and the length 22 of the contact constant used in the relation between the dimensionless wear rate E and the dimensionless load 6 mean distance between the hard and the soft surfaces peak height of an asperity on the hard surface maximum asperity peak height asperity peak height for which n(h) is maximum hardness of the soft surface width of the contact between the surfaces length of the contact between the surfaces length of the asperities in the longitudinal direction total load carried by the asperities number of asperities with peak height between h and h + dh in a transverse cross section over the contact n(h)lnt number of asperities in a transverse cross section over the contact fraction of ploughed material removed as wear particles roughness of the hard surface (centre-line average of the asperities) sliding distance volumetric wear, volume of material removed by wear V/s, volumetric wear rate, i.e. volume of material removed per unit distance slid
142
h/h, h/h, sy D/h,
L/1112H, dimensionless load bdV’/&R, dimensionless wear rate a measure of the heterodispersity (or width) the asperities
E a
of the peak height distribution
of
Subscripts abr adh them
abrasive adhesive chemical
Acknowledgments The author wishes to thank Mr. A. Dyson and Mr. J. F. Hutton for stimulating discussions and worthwhile suggestions. References 1 J. F. Archard, Contact and rubbing of flat surfaces, J. Appl. Phys., 24 (1953) 981. 2 E. Rabinowicz, Practical use of the energy criterion, Wear, 7 (1964) 9. 3 C. N. Rowe, Some aspects of the heat of adsorption in the function of a boundary lubricant, ASLE Trans., 9 (1966) 101. 4 N. P. Suh, The delamination theory of wear, Wear, 25 (1973) 111. 5 J. F. Archard, Interdisciplinary Approach to Friction and Wear, NASA Symp., Nov. 28 - 30, 1967, San Antonio, Texas, NASA, Washington, D.C., 1968, p. 283. 6 E. Rabinowicz, Friction and Wear of Materials, Wiley, New York, 1965, p. 168. 7 J. Hailing, A contribution to the theory of mechanical wear, Wear, 34 (1975) 239. 8 T. F. J. Quinn, The effect of “hot spot” temperature on the unlubricated wear of steel, ASLE Trans., 10 (1967) 158. 9 T. Hisakado, The influence of surface roughness on abrasive wear, Wear, 41 (1977) 179. 10 B. W. E. Avient, J. Goddard and H. Wilman, An experimental study of friction and wear during abrasion of metals, Proc. R. Sot. London, Ser. A, 255 (1960) 159.
Appendix The transformation
of Hisakado’s
abrasive wear equation
We combine eqns. (9) and (12) of Hisakado’s [9] paper, which are derived for unlubricated sliding, and transform these to lubricated conditions by making use of his eqn. (21). This gives the following expression for the volumetric wear rate, equivalent to our VLbr : o,W(l/tan 19)~ Vb -= (AlI L
p,{(7T/2)(1/tan28
--/tan
e)l,
143
For the limiting case, that tan 6 is constant for a given roughness ((tan fJ)min /(tan 01, = l), the expression transforms into
-=-
Vb L
2 a,Wtanf3 71 Pm
I
1+
z7r
P tan 0 +
i 27r
P tan 0
12
+ ---tan0 ( 271
13
+“’
I
(AZ)
It is justifiable to consider this limiting case because the general shape of the function between wear rate and roughness is independent of the details of the distribution of tan 6. The conditions considered by Hisakado are 0 < P < 1.2 and 0 < tan /3 < 0.31, which from his eqn. (18) is equivalent to 0 < R,,, d 30 pm. It has been found by numerical analysis that eqn. (A2) can be approximated to within 10% accuracy by the following equation which is linear in tan 0 :
‘b -=--
2 a,w
(1 + 0.15 P) tan e 7-l Pm If&b = 1 and /J is small, the expression reduces to the relation following from Archard’s model, which was discussed in the introduction of this paper: L
v,, -=
(A31
2 w --he
L
71 Pm The empirical relation between the asperity slope and roughness is given by Hisakado’s eqn. (18) as follows:
(tan e), = 4 41imax ‘2
t&44)
max + 18.8
in which R is in micrometres. The final result is obtained by eliminating 0 between eqns. (A4) and (A3) to give vb -.---=-
L
Olbw
Pm (' +"'15p)
(,.,,R~~;+
(A5)
18.8)
which expresses an increase in wear rate with surface roughness with a fall in the rate of increase at high roughness. Nomenclature (from ref. 9) L Pm R max s
vb
sliding distance yield pressure of the softer metal maximum height of the asperities shear strength at the contact wear volume of the softer metal under
lubricated
conditions
144 W ab
e IJ
total normal load ratio of the wear volume to the volume of a groove ploughed by a hard asperity under lubricated conditions base angle of a single asperity SIP,, coefficient of friction due to adhesion between the two surfaces in unlubricated conditions mean value of ( ) minimum value of ( )
Wear, 50 (1978) 127 - 144
A CONTRIBUTION TO THE THEORY OF WEAR - THE ABRASIVE WEAR OF A SOFT SURFACE AGAINST A ROUGH HARD SURFACE A. BOS* Shell Research Ltd., Thornton (Gt. Britain)
Research Centre, P.O. Box 1, Chester CHl 3SH
(Received January 9, 1978)
Summary A theoretical model has been developed to describe the abrasive wear of a soft surface over which a rough hard surface with longitudinal asperities slides. The model predicts that: (a) the volumetric wear rate is proportional to the roughness of the hard surface, provided that the shape of the asperity peak height distribution of the hard surface remains the same; (b) the wear rate is inversely proportional to the hardness of the softer surface and proportional to the load, provided that the peak height distribution of the asperities on the hard surface is broad; (c) the wear rate is affected very little by the shape of the asperities on the hard surface.
1. Introduction Many theories of wear have been proposed in the literature. Most of them deal with adhesive wear [ 1 - 41 and a few with other types of wear, such as abrasive wear [ 5,6] , surface fatigue wear [ 71 or chemical wear [S] . In the models for adhesive wear the roughness of the wearing surfaces is considered to be irrelevant. This is justified because in practice the asperities are very shallow and therefore that part of the surface of the asperity making contact with the other surface by penetration, for instance, is roughly equal to the projected area of intersection of the asperity with the other surface. However, in a model of abrasive wear the roughness of the surfaces must be taken into account. This was done to some extent in the model of Archard [5] and Rabinowicz [6] for two-body abrasion. In this model it was assumed that a rough hard surface slides over a smooth soft surface. The asperities of the hard surface were considered to be rigid *Present address: Koninklijke/Shell-Lahoratorium, 0043-1648/78/0050-0127$2.25
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in the Netherlands
128
circular cones with a base angle* 0 and the aim of the model was to calculate the volumetric wear rate V/s caused by one asperity. It was assumed that the load L was supported by that part of the soft surface in contact with the leading half of the asperity. The result obtained was V 2 tan 0 L
-=--
n
S
H
where H is the hardness of the soft surface and V is the volume worn off the soft surface in the sliding distance s. A more rigorous model would take into account such characteristics of the hard surface as the shape of the asperities and their peak height distribution. The objective of this paper is the development of a new statistical model of abrasive wear. 2. Description of the theory
2.1. Assumptions A rough hard surface slides relative to a soft surface and causes wear of the soft surface. The following assumptions are made. (1) The soft surface is and remains smooth throughout the wear process. (2) The hard surface is rough. The asperities on it have the same length and are of constant height in a longitudinal direction (the sliding direction); in the transverse direction they are sinusoidal in shape. The asperities are evenly distributed over the hard surface as shown in Fig. 1. (3) The number of asperities in any transverse cross section through the area of contact between the surfaces is constant and the height distribution of the asperity peaks in every such cross section is the same. (4) The asperities on the hard surface are rigid, i.e. not deformable, and they do not wear. (5) Only the soft surface wears. In general the wear has abrasive, adhesive and chemical components. Hence the volumetric wear V is given by v = f( via, vadh, Vchem) It is assumed that vahr
3
vdh
+ Vchem
(6) The presence of a lubricant is not specifically taken into account. If there is a lubricant, it can assist in carrying the load by a hydrodynamic action. However, when the roughness of the hard surface is high compared with the film thickness the hydrodynamic action can be neglected.
*The model is normally expressed in terms of the vertex angle but in the present paper, as will appear in Section 4 and the Appendix, it is more convenient to use the base angle.
129
+---
In--+
Fig. 1. Distribution of the asperities over the hard surface: -, of contact area; direction of motion, right to left. Fig. 2. The shape of an asperity in the transverse direction:
asperities; - - -, outline
2 = hcosax,
--n/2
< ax <
3rrl2.
2.2. The shape and size of asperities The asperities have the shape of cosine curves in the transverse direction and each is accompanied by a valley of exactly the same form (Fig. 2). The pair (asperity plus valley) can be described by the equation -1712 < ax < 3n/2
z = h cos ax
(1)
where h is the height of the asperity and 27r/u is the .width of the pair (asperity plus valley) at its baseline. A consequence of this particular shape of asperity is that the mean surface of the hard surface coincides with the plane 2 = 0. It will be assumed that the number of asperities per unit of transverse distance with heights between h and h + dh is given by n(h)dh =:
exp 1\
n(h)dh = 0
-ln2Wht)
21n’o
dh
I
O
(24 or h>h,
(2b)
These equations describe a distribution with a cut-off at h = h,, the maxim u m asperity peak height, and this distribution is shown graphically in Fig. 3. In the equations n, is the total number of asperities in a transverse cross section over the area of contact between the soft and hard surfaces, n(h) has a maximum at ,h = h,, u is a measure of the heterodispersity of the distribution and br is a constant given by
It is assumed that there is a relation between the base of the pair (asperity plus valley) and h. This relation can be expressed as a = bzhb3
(4)
130
f
-----
c
5 F 2 z k E D ILL
I
I I
IL---ht PEAK
-I
T 11 I
hrn HEIGHT,
f-
h
Fig. 3. Distribution
12
---+
of the asperity peak heights.
Fig. 4. Modified model for the calculation of the load-carrying capacity of the asperities: -, asperities; - - -, outline of contact area; direction of motion, right to left.
in which ba and ba are constant and bs can be chosen freely. Since the width II of the contact between the two surfaces is given by hm 11 =
s 0
n(h) z
a
dh
(5)
ba is then determined by h,
b2=1 4
1
2xn(h)I+dh
(6)
0
The value of bs can be interpreted physically as follows: b, = 0 implies that the width 2n/a of the sets (asperity plus valley) is constant, irrespective of the value of h; bs = -0.5 implies that all the asperities have the same radius of curvature in the transverse direction (the radius of curvature is l/a2h); ba = -1 implies that all the asperities have the same mean slope in the transverse direction. 2.3. The total load L carried by the asperities as a function of the distance D between the two surfaces Assume the soft surface is at a mean distance D from the hard surface. In order to calculate the total load L carried by the asperities, we replace the model temporarily with a simplified model in which the asperities of the same peak height are aligned to form very long asperities (Fig. 4). This is possible because it was assumed that in any cross section over the contact area the number of asperities is constant and that they have the same peak height distribution. Consider now one such long asperity with peak height h. When h > D, this asperity has sunk into the soft surface and carries a weight AL given by AL = HAAl
(7)
131
where H is the hardness of the soft surface and AAl is the projected contact area between the asperity and the soft material: 212
AA1 = -
arccos
a
(8)
in which l2 is the length of the area of contact between the two surfaces. Consequently, 212H AL = a
(9)
The total load L carried by the asperities is then L =
r”
(10)
ALn(h)dh
2.4. The total ploughing surface A, of the asperities as a function of D We return now to our original model in which the longitudinal asperities are regularly distributed over the hard surface (Fig. 1). The formula for L (eqn. (10)) is valid for this model as well. Consider an asperity with peak height h > D which has sunk into the soft surface. The area AA2 of the cross section of the sunken part in the transverse direction is given by AA2 =z (h2 -D2)1’2 a1
-D
arccos
(11)
The total ploughing surface of all the asperities sunk into the soft metal is then A, = J!
AA2n(h)dh
(12)
D
Finally, in order to characterize the roughness of the hard surface in a simple way, the roughness R of the surface can be calculated from R=;
$ lo
b2
-hn(h)dh a
R is identical to the centre-line average value (c.1.a.) 2.5. The volumetric rate V& of abrasive wear as a function of the load It is &assumedthat the length 1, of the asperities is a fraction b4 of the length of contact area between the two surfaces: 1, = b412 (14)
132
The quantity l/b4 can be considered to be the number of rows of asperities in the contact la. Figure 1 shows three such rows. Therefore, when the contact is slid a distance ls, a volume A212/b4 is displaced plastically in a ploughing action at the leading edges of the asperities and a fraction p is removed as wear particles. It is further assumed that grooves in the soft surface caused by the passage of a row of asperities are partially filled up by material plastically displaced sideways by following asperities running in adjacent tracks so that the asperities never run through the tracks formed by asperities in the immediately preceding row. The volumetric rate V& of abrasive wear, which is defined as the volume of material removed per unit distance slid, is then given by VA = pAdb4
(15)
It is convenient to put Vgbr and other quantities in more general form. We introduce therefore the following dimensionless variables: a = h/h,
(16)
P = h/h,
(17)
Y = D/h,
(18)
6= -
L
(19)
4 l2H
E= b* - Xl, PAIR
(20)
44
n*(h) = -
(21)
nt By substitution it can be shown that n*(ah,)da
=
n*(h)dh hm
= exp - ln2 ($Lfo’ht )/dol [h, 1
/ exp / -1n2~~n~~‘h”/d~]-1 (22)
6 = i arccos(~)n*(olh,)a-b~ Y
da 1n / n*(ah,)aTb3da
1 -I
(23)
133
It can be seen from the equations that, for a given value of y, both 6 (dimensionless load) and E (dimensionless wear rate) are independent of n, R, L, H, 1,, 1s and p, but are still functions of h,/h, or /3/a, bs and u. Of course the factor h, in the denominator of the expression for n*(ah,)da is cancelled in the expressions for 6 and E since n*(olh,)da appears in both the numerator and the denominator of eqns. (23) and (24). It can also be shown by calculating 6 and E as functions of y (using the computer) and plotting curves that the relations between 6 and E are always monotonic. These relations can be considered as general dimensionless forms of the load-wear rate relation and will be used to represent the results of the theory. 3. Calculation of the volumetric rate V& of abrasive wear 3.1. Influence of the roughness R It can be shown as follows that, with certain restrictions, Vlabr is a linear function of the roughness R. From the definition of the dimensionless wear rate E (eqn. (20)) the relation between V& and R is EP4 Vi, = R b4 In this formula /I is determined by the apparatus and is a constant. The fraction p (eqn. (15)) has been assumed to be independent of roughness, and the factor b4 (eqn. (14)) is independent of roughness in the transverse direction. The dimensionless wear rate E, as given by eqn. (24), is independent of roughness for given values of (Y//I, u and bs, i.e. for a given distribution of asperity peak heights and a given relation between the wavelength and amplitude of an asperity. In fact, as shown later, once a/P has been fixed E is not sensitive to changes in bs and hence the restrictions are effectively reduced to a given shape of distribution of the asperity peak heights. Within this restriction the model predicts a linear relation between volumetric wear rate VLbr and roughness R. 3.2. Influence of the number n of asperities in a cross section of the contact area From the analysis presented in the preceding section, it follows that both 6 and E are independent of n,; consequently, VLbr is also independent of n,. 3.3. Influence of the choice of the cut-off ratio in the asperity height distribu tion h, /h, (or p/a) In Fig. 5, E is given as a function of 6 when u = 1.5 and bs = -0.5 (this value of bs implies that all the asperities have the same radius of curvature in the transverse direction) for fl/cu = 2, 3,6 and 10. It appears that E increases for a given value of 6 as P/a is increased until an upper limit is
p’a=6Lg, (I=,. PA 1, 0.001 0.001
0.01
0.1
1.0
6
Fig. 5, Dimensionless wear rate f as a function of dimensionless of the cut-off in the asperity height distribution.
load 6 for various values
reached. This behaviour has been found in all the calculations in which /3/ol was varied. 3.4. Influence of the choice of the asperity width factor bs In Fig. 6, E is given as a function of 6 when bs = -1, -0.5 and 0 with (J = 1.5, for p/al = 2, 3 and 6. It appears that the choice of bs does not influence the dimensionless wear rate significantly for given values of the cut-off ratio h, /h, (or p/a). 3.5. Influence of the degree CTof heterodispersity Figure 7 shows the e-6 characteristic for u = 1.1, 1.2, 1.5 and 2 whilst bs = 0.5. Also the relation for a monodisperse distribution is given. In all these cases h,/ht has been chosen to be large relative to u so that the wear rate is at its highest value for the given value of u (see Section 3.3). It appears that the dimensionless wear rate is strongly dependent on the degree of heterodispersity. 3.6. Influence of the asperity shape To investigate whether the asperity shape in the transverse direction has an influence on the wear rate, the calculations have been repeated but with triangular instead of sinusoidal asperities.
135 I.o 1
0 1.
0.001 L 0.001
0.01
1 I .o
0.1 6
Fig. 6. Dimensionless wear rate E as a function of the asperity width factor b,.
of dimensionless
load 6 for various values
If the width of the pair (asperity plus valley) is 2n/a at the base and if a is assumed to’be related to the asperity height h by a = b2hb,
(25)
which is the same relation as used for the sinusoidal asperities (eqn. (14)), it can be shown for the triangular asperities that -1 l F=
s
Y
a---y
-
cCbsn*(ah,)da
a
(?i?!2a-bxnyah
E=
m
n*(ah,)Cb3da
(26)
m
)da
-’
(27)
OL Y
In the above equations the dimensionless groups are defined by eqns. (16) (21) in the same way as for the sinusoidal asperity shape. In Fig. 8 E is plotted against 6 and u = 1.1, 1.2, 1.5 and 2 for b3 = -0.5. Again the E-C?characteristic for a monodisperse distribution is shown.
136 LC
‘1
0.1
ui MONODISPERSE
0.01
0.01
6
0.1
1.0
Fig. 7. Dimensionless wear rate E as a function of dimensionless load 6 for various values of the heterodispersity factor u and large values of P/CL The asperities are sinusoidal in the transverse direction.
Figure 8 shows that e is again strongly dependent on the heterodispersity, although when Figs. 7 and 8 are compared it is seen to be not as strongly dependent as in the case of sinusoidal asperities. For a high degree of heterodispersity the shape of the asperities, whether sinusoidal or triangular, has a negligibly small effect on the wear-load relation. 3.7. Dependence of wear rate on hardness of the softer surface and on load From the graphs shown in Figs. 5 - 8 we can make the approximation that the relations between log e and log 6 are straight lines. Thus f ahb5 (28) where b5 is a constant. From Figs. 7 and 8, b5 can be obtained as a function of u for both sinusoidal and triangular asperity shapes. Figure 9 shows b5 plotted against u. For this purpose b5 has been averaged over the range of e values 0.01 < E < 0.5. It is seen that when the distribution is monodisperse b, is much higher than unity, but at higher heterodispersity b5 becomes about unity which is in agreement with experimental observations and the predictions of other wear models proposed in the literature. So, if the peak height distribution is heterodisperse the wear rate is roughly proportional to
137 IO
0.1
‘y
0.01
0.001
001
0.1
I.0
Fig. 8. Dimensionless wear rate E as a function of dimensionless load 6 for various values of the heterodispersity factor u and large values of P/cu. The asperities are triangular in the transverse direction.
6 and, from the definition of 6 (eqn. (19)), it is inversely proportional to the hardness of the softer surface and is a linear function of the load, whether the asperities are triangular or sinusoidal in the transverse direction. 4. Discussion Although the abrasive wear model has taken into account the statistical nature of the surface asperities, the dependence of the volumetric wear rate on surface parameters is generally simple in form. For instance, the volumetric wear rate is linearly related to the roughness (c.1.a.) of the hard surface, provided that, among other things, the distribution of the peak heights of the asperities on the hard surface remains unaltered. Subsequent to the development of the analysis given in this paper, Hisakado [ 91 published a similar theoretical study in which he considered the abrasive wear of a soft surface caused by the conical asperities of a mating hard surface. In his model the slopes of the cones are randomly distributed and the mean value of the slopes as a function of the roughness is given by an empirical relation. He found that the wear volume per unit of
I
0
I
1.5
2.0
o-
Fig. 9. Variation of the wear rate-load index b5 as a function of heterodispersity E a 6 br ; +, sinusoidal asperities; -X -, triangular asperities.
U:
frictional work increased with the roughness of the hard surface, but that the rate of increase diminished at high roughness. When this relation is transformed into a wear rate-roughness relation, which is possible by making use of Hisakado’s eqns. (9), (12), (21) and (18), the wear rate is still a non linear function of the roughness. Therefore there is a difference between Hisakado’s theory and the theory presented here. There are, of course,
139
differences in the models used, but a major factor in Hisakado’s theory is the form of the empirical relation between the mean cone slope and the roughness (his eqn. (18)). From a calculation presented in the Appendix, it appears that the wear rate is roughly a linear function of the cone slope. Therefore if the cone slope levels off with increasing roughness, as it does by Hisakado’s eqn. (18) (reproduced as eqn. (A4) in the Appendix), the wear rate does the same. Likewise, if the cone slope is independent of roughness the wear rate will increase linearly with roughness and in this respect the two models would predict the same behaviour. In an analogous way a different wear rate-roughness relation would be obtained from our model if some alternative form t,han the one assumed was taken for the effect of roughness on the shape of the asperity peak height distribution; at present we have chosen to assume that the shape of the distribution is independent of roughness. A remarkable outcome of our model is that the wear rate is only a weak function of the relation (eqn. (4)) between the asperity base and the asperity height. Another important result, which it may be possible to explore in an experimental test of the theory, is that the volumetric rate of abrasive wear is proportional to the load only if the peak height distribution of the asperities is heterodisperse. If the distribution is monodisperse it is a stronger function of the load, i.e. the rate of wear increases faster than linearly with load. It would be interesting to know what the heterodispersity of the distribution is in practical cases. However, to obtain the distribution from a real surface is difficult because in practice most asperities are not completely separated from each other. The distribution is therefore not exactly defined and it is necessary to use an arbitrary criterion to decide, for example, whether a hump on the shoulder of an asperity is a separate asperity. A real roughness profile is presented in Fig. 10. It was obtained from an En 31 steel disc that was circumferentially ground to a nominal c.1.a. value
Fig. 10. Talysurf trace of a circumferentially across the direction of grinding.
ground
En 31 steel disc (0.19
pm c.1.a.)
140
of about 0.20 pm (8 pin) and which was used in our three-(bronze)pin-on(steel)disc wear machine for bronze-on-steel wear studies. As a result of the grinding the disc has longitudinal asperities and the profile is taken in the transverse direction. It can be shown by numerical integration that the trace shown has a c.1.a. value of 0.19 pm (7.5 pin) and is therefore considered to be representative for the disc. The asperity peak height distribution is given in Fig. 11.
--
NEGATIVE
- --~.
PEAKS
PEAh
Fig. 11. Distribution
IHEIGHT,
POSITIVE
PEAKS h
h ,/~rn
of the asperity peak heights of the trace shown in Fig. 10
Comparison of Fig. 11 with Fig. 3 shows that in practice the peak height distribution has a different form from that assumed in the model. A real surface shows negative peaks (i.e. peaks with their tops below z = 0, the mean level of the surface). In our model negative peaks have been excluded. However, it is considered that in practice only the highest of the positive peaks contribute to the wear process, so that only the peak height distribution for, say, h > h, is important. From the maximum distribution to larger peak heights, the distributions in Fig. 11 and Fig, 3 are similar, and the distribution in Fig. 11 can be described satisfactorily by eqn. (2a) when u is given the value 2. Hence the disc surface has a heterodisperse asperity peak height distribution and according to the theory the wear rate should be roughly proportional to the load. This is in agreement with experimental observations quoted in the literature [lo] and suggests that many other practical surfaces are heterodisperse in the terms of our model.
141
1 IOOpm -
(b)
(a)
Fig. 12. Comparison of the roughness of (a) the bronze pin surface with the roughness of (b) the steel disc surface after a wear experiment.
An important assumption in our model is that the soft surface remains much smoother than the hard surface throughout the wear process. Figure 12 shows the transverse roughness profiles of the surface of a bronze pin and a rough steel disc (0.4 pm c.1.a.) from a three-pin-on-disc experiment run under lubricated conditions in which abrasive wear was dominant and no wear of the asperity tips of the steel surface took place. From the profiles it can be seen that the pin surface is about five times smoother than the mating steel surface, so the assumption given above seems to be justified. Nomenclature
il A2
bl bzv b3 b, b5
D h 4,
ht
H
h 12 4
L n(h)dh n*(h)
nt
P R s V V’
parameter giving information about the sharpness of the asperity total load-carrying surface total ploughing surface of the hard asperities sunk into the soft surface in a cross section over the contact between the surfaces constant used in expression for n(h) constants used in the relation between h and a the ratio of the asperity length in a longitudinal direction and the length 22 of the contact constant used in the relation between the dimensionless wear rate E and the dimensionless load 6 mean distance between the hard and the soft surfaces peak height of an asperity on the hard surface maximum asperity peak height asperity peak height for which n(h) is maximum hardness of the soft surface width of the contact between the surfaces length of the contact between the surfaces length of the asperities in the longitudinal direction total load carried by the asperities number of asperities with peak height between h and h + dh in a transverse cross section over the contact n(h)lnt number of asperities in a transverse cross section over the contact fraction of ploughed material removed as wear particles roughness of the hard surface (centre-line average of the asperities) sliding distance volumetric wear, volume of material removed by wear V/s, volumetric wear rate, i.e. volume of material removed per unit distance slid
142
h/h, h/h, sy D/h,
L/1112H, dimensionless load bdV’/&R, dimensionless wear rate a measure of the heterodispersity (or width) the asperities
E a
of the peak height distribution
of
Subscripts abr adh them
abrasive adhesive chemical
Acknowledgments The author wishes to thank Mr. A. Dyson and Mr. J. F. Hutton for stimulating discussions and worthwhile suggestions. References 1 J. F. Archard, Contact and rubbing of flat surfaces, J. Appl. Phys., 24 (1953) 981. 2 E. Rabinowicz, Practical use of the energy criterion, Wear, 7 (1964) 9. 3 C. N. Rowe, Some aspects of the heat of adsorption in the function of a boundary lubricant, ASLE Trans., 9 (1966) 101. 4 N. P. Suh, The delamination theory of wear, Wear, 25 (1973) 111. 5 J. F. Archard, Interdisciplinary Approach to Friction and Wear, NASA Symp., Nov. 28 - 30, 1967, San Antonio, Texas, NASA, Washington, D.C., 1968, p. 283. 6 E. Rabinowicz, Friction and Wear of Materials, Wiley, New York, 1965, p. 168. 7 J. Hailing, A contribution to the theory of mechanical wear, Wear, 34 (1975) 239. 8 T. F. J. Quinn, The effect of “hot spot” temperature on the unlubricated wear of steel, ASLE Trans., 10 (1967) 158. 9 T. Hisakado, The influence of surface roughness on abrasive wear, Wear, 41 (1977) 179. 10 B. W. E. Avient, J. Goddard and H. Wilman, An experimental study of friction and wear during abrasion of metals, Proc. R. Sot. London, Ser. A, 255 (1960) 159.
Appendix The transformation
of Hisakado’s
abrasive wear equation
We combine eqns. (9) and (12) of Hisakado’s [9] paper, which are derived for unlubricated sliding, and transform these to lubricated conditions by making use of his eqn. (21). This gives the following expression for the volumetric wear rate, equivalent to our VLbr : o,W(l/tan 19)~ Vb -= (AlI L
p,{(7T/2)(1/tan28
--/tan
e)l,
143
For the limiting case, that tan 6 is constant for a given roughness ((tan fJ)min /(tan 01, = l), the expression transforms into
-=-
Vb L
2 a,Wtanf3 71 Pm
I
1+
z7r
P tan 0 +
i 27r
P tan 0
12
+ ---tan0 ( 271
13
+“’
I
(AZ)
It is justifiable to consider this limiting case because the general shape of the function between wear rate and roughness is independent of the details of the distribution of tan 6. The conditions considered by Hisakado are 0 < P < 1.2 and 0 < tan /3 < 0.31, which from his eqn. (18) is equivalent to 0 < R,,, d 30 pm. It has been found by numerical analysis that eqn. (A2) can be approximated to within 10% accuracy by the following equation which is linear in tan 0 :
‘b -=--
2 a,w
(1 + 0.15 P) tan e 7-l Pm If&b = 1 and /J is small, the expression reduces to the relation following from Archard’s model, which was discussed in the introduction of this paper: L
v,, -=
(A31
2 w --he
L
71 Pm The empirical relation between the asperity slope and roughness is given by Hisakado’s eqn. (18) as follows:
(tan e), = 4 41imax ‘2
t&44)
max + 18.8
in which R is in micrometres. The final result is obtained by eliminating 0 between eqns. (A4) and (A3) to give vb -.---=-
L
Olbw
Pm (' +"'15p)
(,.,,R~~;+
(A5)
18.8)
which expresses an increase in wear rate with surface roughness with a fall in the rate of increase at high roughness. Nomenclature (from ref. 9) L Pm R max s
vb
sliding distance yield pressure of the softer metal maximum height of the asperities shear strength at the contact wear volume of the softer metal under
lubricated
conditions
144 W ab
e IJ
total normal load ratio of the wear volume to the volume of a groove ploughed by a hard asperity under lubricated conditions base angle of a single asperity SIP,, coefficient of friction due to adhesion between the two surfaces in unlubricated conditions mean value of ( ) minimum value of ( )