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Roughness, friction and wear: The effect of contact planform
357
Wear, 57 (1979) 357 - 363 @ Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
ROUGHNESS, PLANFORM*
FRICTION
AND WEAR: THE EFFECT
OF CONTACT
K. L. WOO and T. R. THOMAS Department of Mechanical Engineering, (Gt. Britain)
Teesside Polytechnic,
Middlesbrough,
TSl 3BA
(Received May 15,1979)
Summary Recent theoretical and experimental findings suggest that the statistical properties which affect friction vary slowly with the dimensions of the nominal area. This variation is due to the existence of random long spatial fluctuations which are outside the bandwidth of features imposed by the surface preparation process and which longer wavelengths completely dominate contact behaviour. This leads to the suspicion that the second of Amontons’ laws, i.e. that the limiting static friction is independent of the shape of the nominal area, is not absolutely true. A relation is derived which connects the limiting static friction with the statistical contact properties and with a dimension of the nominal area. In an experiment devised to test this relation, the limiting static friction was measured for three rectangular steel specimens whose nominal area and normal load were kept constant and whose length to breadth ratio varied. The limiting static friction was found to be linearly related to the specimen length in agreement with theoretical predictions. Regression analysis of 36 measurements established that the probability of this result being due to chance was less than 1 in 500. Supporting evidence for this hypothesis has also been found from wear experiments reported in the literature.
1. Introduction In 1699, in a paper published in the Proceedings of the French Royal Academy of Sciences, G. Amontons [l] wrote “... the resistance caused by the friction of surfaces rubbing together increases or diminishes according to greater or lesser pressure and not according to the greater or lesser extent of these surfaces”. This statement forms the basis of what are now known as the Laws of Friction or Amontons’ Laws. *Presented at the International Conference on Metrology and Properties of Engineering Surfaces, Leicester, April 18 - 20, 1979.
358
Almost all real surfaces are rough on a microscopic scale, and when two such surfaces are in contact they touch only at tiny discrete areas where their highest asperities are in contact. Thus in general the real area of contact is only a small percentage of the nominal contact area. The local pressure at the contact regions is then high enough to cause plastic deformation of the asperities even at the lightest load. This hypothesis has been used successfully to explain Amontons’ Laws for plastic contact [ 21. However, it was pointed out that contact between run-in surfaces will be elastic [3] so there was an apparent need for a more general approach. It was suggested that Amontons’ Laws were a consequence of the supposed Gaussian distribution of the asperity heights and not the mode of their deformation [4]. It is known that many engineering surfaces have height distributions (though not necessarily summit distributions) which are Gaussian or nearly so [ 51. Recently it has been pointed out that the variance of the height distribution of a surface is proportional to a significant dimension of the nominal area [6] . This variation or non-stationarity is due to the existence of random long spatial fluctuations which are outside the bandwidth of features imposed by the machining process. These longer wavelengths dominate contact behaviour as shown in contact maps where clustering of contact spots can be observed [ 71. Since friction is related to the topography of the surface, which is itself length dependent, friction would be expected to be affected by the dimensions of the nominal contact area, contrary to the second of Amontons’ Laws. In this paper an expression for limiting static friction is derived using a surface topographical approach and tribological interactions between rough surfaces which appear to confirm this relation are described [ 81.
2. Theory When two rough nominally flat Gaussian surfaces are brought normal contact the density D, of contours [9] is given by
into
where u2 is the variance of the combined height distributions of the two surfaces and z is the separation of the mean planes of the surfaces. For moderate loads (z/a > 1) the above expression closely represents the density of contacts. Thus the number N of contacts for a nominal area A, becomes Na
4
-7
exp
-z2 -
i 2a2
1
However, for real surfaces of similar topography u2 is proportional to L which is a significant dimension of the nominal area [6]. For rectangular nominal areas L is simply the length of the longer side. Thus
359
-z2 Na%exp (-2u2 1
(1)
If the average area of each individual area is A,o:
contact
is a then the real contact
A& L exp -z2 ( 2a2 1
Limiting static friction can be regarded as a tangential force required to overcome the adhesion at regions of intimate contact plus the tangential force required to lift the asperities over each other. If one surface is much softer than the other a ploughing term must be included. For two hard surfaces in contact with each other, e.g. steel on steel, ploughing is negligible [ 21. For simplicity the latter case is considered here. Suppose that s is the average shear strength per unit area of the junction formed by adhesion and is assumed to be constant for a given pair of materials with the same degree of contamination, W is the normal load and 0 is the average slope of the asperities measured in the direction in which the tangential force is applied. When gross sliding occurs during unlubricated contact the upper surface has been shown to undergo both normal and horizontal displacement [lo] . This is explained by assuming that when the specimen is displaced from rest it must climb up and pass the forward slopes, i.e. the total measured force F is given by F=A,s+Wtane +wtane where K is a constant dependent only on the surface topography. If the specimens are given the same topography, i.e. K and 8 are constant, and the same area and load, i.e. W and A, are constant, then the dimensionless separation will be the same for elastic contact [ll].The average contact size might also be expected to be constant. Since the coefficient of static friction ps = F/W, we have finally a relation of the form
where Ci and C2 are constants, i.e. the limiting static friction will be the sum of a constant term and a term varying inversely as the specimen length.
3. Experiment Experiments were performed using the apparatus shown schematically in Figs. 1 and 2. The limiting static friction was measured by a half-proving ring which was attached between the specimen and the loading mechanism. In the loading mechanism (Fig. l), a compression spring was loaded by advancing a micrometer until the spring force was slightly greater than the
360
Fig. 1. Schematic diagram of the tangential loading mechanism. Fig. 2. Schematic diagram of the test rig.
expected limiting static friction. A steadily increasing tangential force was applied to the specimen by the spring when the micrometer barrel was moved backwards. When macro-slip was about to occur the spring force was mostly taken up by the specimen and the normal reaction, and hence the friction between the slider and the micrometer became small. The friction between the slider and the walls of the casing was small owing to the low friction coefficient of polytetrafluorethylene and the lightness of the slider centre-piece which was machined from duralumin. Thus friction in the loading mechanism was a minimum when the limiting static friction between the specimen and track was measured. This is highly desirable since stickslip motion between the moving parts will reduce the resolution of the proving ring. The air trapped behind the slider also acted as a damper and so helped the slider to travel smoothly. Three steel specimens of rectangular planform were cut from the same plate which was 2.5 mm thick. Their aspect ratios were respectively 1.75:1, 3.11:1 and 7.OO:l. Their nominal areas were all 180.6 cm2 and their masses were 3.508 kg. The contacting surfaces of all three were parallel ground simultaneously on the same machine with their lays parallel to their longest sides. Hence the theoretical assumptions of constant nominal pressure and surface topography were satisfied. The specimens and track (also ground) were cleaned in acetone before each experiment to remove most of the outer non-uniform layers of contaminants. No special precautions were made to prevent the oxidation and adsorption of atmospheric gases into the surface layers of the specimens as these were assumed to affect all the specimens uniformly. The duration of static contact with the track was kept constant. A tangential force was applied to each specimen and the maximum force achieved when the specimen just began to slide was measured. This force was taken as the limiting static friction. The applied force acts parallel and above the mean contact interface. The resultant couple will tend to cause the leading portion of the specimen to plough into the track. However, since both specimen and track were made from steel of the same hardness and providing the turning moment is small (i.e. no toppling) ploughing will be negligible. From static considerations the normal reaction, and hence the friction, remains unchanged.
361
The tests were conducted in the following sequence: the shortest to the longest specimen and then vice versa. This should compensate for any change in contact properties with the number of traversals and thus provide a basis for the combination of the results from each of the 12 sets of tests. To investigate the variation of height distributions with specimen length 100 height measurements were made at equal horizontal intervals over the length of each specimen with a coordinate measuring machine (Ferranti Mercury).
4. Discussion The maximum coefficient of static friction obtained in the experiment is 0.24. This low value indicates the presence of an outer contaminant film but that should not affect our results except in reducing the absolute value of vs. In the limiting case where a perfect boundary lubricant is present the coefficient of static friction will tend to tan 0. In contrast, when the surfaces are very smooth and clean the effect of surface roughness is negligible [ 121. Others have shown that friction increases with the profile slope [ 131 or with the initial asperity angle [ 141. The probability of the linear relation observed in Fig. 3 being due to chance is less than 1 in 500. According to our theory the intercept at 5.34 f 0.71 N is the value of W tan 8, from which we can deduce that the mean surface slope is 8.8 f 1.2”. This is much steeper than the slopes generally associated with ground surfaces measured along their lay which suggests that the asperities responsible are too small to be measured by conventional stylus instruments. The condition of proportionality between u and L1j2 is obviously met by the results of Fig. 4. The intercept of the regression line is -0.205 + 0.17 E.tmand thus it only fails to pass through the origin by an insignificant margin.
9
a 5 7
L
INI
I$ 6
3 2 1
F
5
E 0
1
2 l/L
3 Im-‘I
L
5
6
!LL 5
10
.JT
15
112 Imm
20
I
Fig. 3. Variation of limiting static friction with reciprocal specimen length. Each circle represents the mean of 12 measurements. The error bars are 50% confidence limits. The solid line is the regression line (p = 0.493, v = 34, p < 0.002, one-sided). Fig. 4. Variation of the root mean square height deviations with the square root of the specimen length. The circles are the experimental results and the solid line is the regression line (p = 0.9994, v = 1, p < 0.025, one-sided).
362
In a survey of wear experiments some additional evidence of this shape criterion was found. Play and Godet [ 151 measured the linear wear rate of rectangular chalk pins of equal nominal area but with varying aspect ratio sliding on a glass disc. They found that an increase in the aspect ratio significantly reduces wear. If it is considered that wear of the chalk pin can only occur when there is intimate contact between the surfaces the wear rate would be related to the number of discrete contacts. Equation (1) predicts that the number of contacts is proportional to the reciprocal of the specimen length. Replotting their results as linear wear rates after first rotation against the reciprocal of the specimen length (Fig. 5) gives good agreement with our hypothesis. Lancaster [ 161, using electrographitic pins sliding on hardened steel, also reported a similar decrease in linear wear rates with increasing aspect ratio. In the experiments described above it is suggested that the observed dependence of the friction and abrasive wear processes on aspect ratio is a result not of material or thermal properties nor of the mode of asperity deformation but of the increasing predominance of long surface wavelengths as the specimen length increases.
15
r
1 3
Load
1 1
=
8 8N
09 07 13
.:
/ -
0
0 1 1 /L
0.2
0.3
Imrti’)
Fig. 5. Variation of linear wear rates of chalk pins on glass with reciprocal of specimen length replotted from Table 3 of ref. 14. The circles are the experimental results and the solid lines are regression lines. Correlations, from top (U = 1, one-sided): p = 0.99998, p < 0.005;~ = 0.9996,p < 0.02;~ = 0.964,p > 0.1;~ = 0.9995,p < 0.02.
363
References 1 Translated by D. Dowson, personal communication, 1978. 2 D. Bowden and F. P. Tabor, Friction and Lubrication ofSolids, Oxford Univ. Press: Clarendon Press, Oxford, 1954. 3 J. F. Archard, Elastic deformation and the laws of friction, Proc. R. Sot. London, Ser. A, 243 (1957) 190 - 205. 4 J. A. Greenwood, The area of contact between rough surfaces and flats, Trans. ASME, 89F (1967) 81 - 89. 5 J. B. P. Williamson, Topography of solid surfaces, NASA Spec. Publ. SP-181, 1968, pp. 85 - 141. 6 R. S. Sayles and T. R. Thomas, Surface topography as a non-stationary random process, Nature (London), 271 (1977) 431 - 434. 7 J. Dyson and W. Hirst, The true area of contact between surfaces, Proc. Phys. Sot. London, Sect. B, 67 (1954) 309 - 312. 8 K. L. Woo, R. S. Sayles and T. R. Thomas, An exception to Amontons’ Law, Proc. Euromech. Colloq. No. 110 on Contact Problems and Load Transfer in Mechanical Assemblages, Linkiiping, 1978, Linkiiping Inst. Technol., Linkoping, 1978, pp. 175 - 17’ 9 R. S. Sayles and T. R. Thomas, Computer simulation of the contact of rough surfaces, Wear, 49 (1978) 273 - 296. 10 D. M. Tolstoi, Significance of the normal degree of freedom and natural contact vibrations in contact friction, Wear, 10 (1967) 199 - 213. 11 A. W. Bush, R. D. Gibson and T. R. Thomas, The elastic contact of a rough surface, Wear, 35 (1975) 87 - 111. 12 H. Ernst and M. E. Merchant, Surface friction of clean metals - A basic factor in the metal cutting process, Proc. Conf. on Friction and Surface Finish, MIT Press, Cambridge, Mass., 1940, pp. 76 - 101. 13 N. 0. Myer, Characterisation of surface roughness, Wear, 5 (1962) 182 - 189. 14 S. R. Ghabrial and S. A. Zaghlool, The effect of surface roughness on static friction, Int. J. Mach. Tool Des. Res., 14 (1974) 299 - 309. 15 D. Play and M. Godet, Self protection of high wear materials, ASLE/ASME Lubrication Conf., Kansas City, MO., 1977, ASLE Prepr. (Am. Sot. Lubr. Eng.) 77-Lc-2, 1977. 16 J. K. Lancaster, Geometrical effects of the wear of polymers and carbon, Trans. ASME, 77F (1975) 189 - 194.
Wear, 57 (1979) 357 - 363 @ Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
ROUGHNESS, PLANFORM*
FRICTION
AND WEAR: THE EFFECT
OF CONTACT
K. L. WOO and T. R. THOMAS Department of Mechanical Engineering, (Gt. Britain)
Teesside Polytechnic,
Middlesbrough,
TSl 3BA
(Received May 15,1979)
Summary Recent theoretical and experimental findings suggest that the statistical properties which affect friction vary slowly with the dimensions of the nominal area. This variation is due to the existence of random long spatial fluctuations which are outside the bandwidth of features imposed by the surface preparation process and which longer wavelengths completely dominate contact behaviour. This leads to the suspicion that the second of Amontons’ laws, i.e. that the limiting static friction is independent of the shape of the nominal area, is not absolutely true. A relation is derived which connects the limiting static friction with the statistical contact properties and with a dimension of the nominal area. In an experiment devised to test this relation, the limiting static friction was measured for three rectangular steel specimens whose nominal area and normal load were kept constant and whose length to breadth ratio varied. The limiting static friction was found to be linearly related to the specimen length in agreement with theoretical predictions. Regression analysis of 36 measurements established that the probability of this result being due to chance was less than 1 in 500. Supporting evidence for this hypothesis has also been found from wear experiments reported in the literature.
1. Introduction In 1699, in a paper published in the Proceedings of the French Royal Academy of Sciences, G. Amontons [l] wrote “... the resistance caused by the friction of surfaces rubbing together increases or diminishes according to greater or lesser pressure and not according to the greater or lesser extent of these surfaces”. This statement forms the basis of what are now known as the Laws of Friction or Amontons’ Laws. *Presented at the International Conference on Metrology and Properties of Engineering Surfaces, Leicester, April 18 - 20, 1979.
358
Almost all real surfaces are rough on a microscopic scale, and when two such surfaces are in contact they touch only at tiny discrete areas where their highest asperities are in contact. Thus in general the real area of contact is only a small percentage of the nominal contact area. The local pressure at the contact regions is then high enough to cause plastic deformation of the asperities even at the lightest load. This hypothesis has been used successfully to explain Amontons’ Laws for plastic contact [ 21. However, it was pointed out that contact between run-in surfaces will be elastic [3] so there was an apparent need for a more general approach. It was suggested that Amontons’ Laws were a consequence of the supposed Gaussian distribution of the asperity heights and not the mode of their deformation [4]. It is known that many engineering surfaces have height distributions (though not necessarily summit distributions) which are Gaussian or nearly so [ 51. Recently it has been pointed out that the variance of the height distribution of a surface is proportional to a significant dimension of the nominal area [6] . This variation or non-stationarity is due to the existence of random long spatial fluctuations which are outside the bandwidth of features imposed by the machining process. These longer wavelengths dominate contact behaviour as shown in contact maps where clustering of contact spots can be observed [ 71. Since friction is related to the topography of the surface, which is itself length dependent, friction would be expected to be affected by the dimensions of the nominal contact area, contrary to the second of Amontons’ Laws. In this paper an expression for limiting static friction is derived using a surface topographical approach and tribological interactions between rough surfaces which appear to confirm this relation are described [ 81.
2. Theory When two rough nominally flat Gaussian surfaces are brought normal contact the density D, of contours [9] is given by
into
where u2 is the variance of the combined height distributions of the two surfaces and z is the separation of the mean planes of the surfaces. For moderate loads (z/a > 1) the above expression closely represents the density of contacts. Thus the number N of contacts for a nominal area A, becomes Na
4
-7
exp
-z2 -
i 2a2
1
However, for real surfaces of similar topography u2 is proportional to L which is a significant dimension of the nominal area [6]. For rectangular nominal areas L is simply the length of the longer side. Thus
359
-z2 Na%exp (-2u2 1
(1)
If the average area of each individual area is A,o:
contact
is a then the real contact
A& L exp -z2 ( 2a2 1
Limiting static friction can be regarded as a tangential force required to overcome the adhesion at regions of intimate contact plus the tangential force required to lift the asperities over each other. If one surface is much softer than the other a ploughing term must be included. For two hard surfaces in contact with each other, e.g. steel on steel, ploughing is negligible [ 21. For simplicity the latter case is considered here. Suppose that s is the average shear strength per unit area of the junction formed by adhesion and is assumed to be constant for a given pair of materials with the same degree of contamination, W is the normal load and 0 is the average slope of the asperities measured in the direction in which the tangential force is applied. When gross sliding occurs during unlubricated contact the upper surface has been shown to undergo both normal and horizontal displacement [lo] . This is explained by assuming that when the specimen is displaced from rest it must climb up and pass the forward slopes, i.e. the total measured force F is given by F=A,s+Wtane +wtane where K is a constant dependent only on the surface topography. If the specimens are given the same topography, i.e. K and 8 are constant, and the same area and load, i.e. W and A, are constant, then the dimensionless separation will be the same for elastic contact [ll].The average contact size might also be expected to be constant. Since the coefficient of static friction ps = F/W, we have finally a relation of the form
where Ci and C2 are constants, i.e. the limiting static friction will be the sum of a constant term and a term varying inversely as the specimen length.
3. Experiment Experiments were performed using the apparatus shown schematically in Figs. 1 and 2. The limiting static friction was measured by a half-proving ring which was attached between the specimen and the loading mechanism. In the loading mechanism (Fig. l), a compression spring was loaded by advancing a micrometer until the spring force was slightly greater than the
360
Fig. 1. Schematic diagram of the tangential loading mechanism. Fig. 2. Schematic diagram of the test rig.
expected limiting static friction. A steadily increasing tangential force was applied to the specimen by the spring when the micrometer barrel was moved backwards. When macro-slip was about to occur the spring force was mostly taken up by the specimen and the normal reaction, and hence the friction between the slider and the micrometer became small. The friction between the slider and the walls of the casing was small owing to the low friction coefficient of polytetrafluorethylene and the lightness of the slider centre-piece which was machined from duralumin. Thus friction in the loading mechanism was a minimum when the limiting static friction between the specimen and track was measured. This is highly desirable since stickslip motion between the moving parts will reduce the resolution of the proving ring. The air trapped behind the slider also acted as a damper and so helped the slider to travel smoothly. Three steel specimens of rectangular planform were cut from the same plate which was 2.5 mm thick. Their aspect ratios were respectively 1.75:1, 3.11:1 and 7.OO:l. Their nominal areas were all 180.6 cm2 and their masses were 3.508 kg. The contacting surfaces of all three were parallel ground simultaneously on the same machine with their lays parallel to their longest sides. Hence the theoretical assumptions of constant nominal pressure and surface topography were satisfied. The specimens and track (also ground) were cleaned in acetone before each experiment to remove most of the outer non-uniform layers of contaminants. No special precautions were made to prevent the oxidation and adsorption of atmospheric gases into the surface layers of the specimens as these were assumed to affect all the specimens uniformly. The duration of static contact with the track was kept constant. A tangential force was applied to each specimen and the maximum force achieved when the specimen just began to slide was measured. This force was taken as the limiting static friction. The applied force acts parallel and above the mean contact interface. The resultant couple will tend to cause the leading portion of the specimen to plough into the track. However, since both specimen and track were made from steel of the same hardness and providing the turning moment is small (i.e. no toppling) ploughing will be negligible. From static considerations the normal reaction, and hence the friction, remains unchanged.
361
The tests were conducted in the following sequence: the shortest to the longest specimen and then vice versa. This should compensate for any change in contact properties with the number of traversals and thus provide a basis for the combination of the results from each of the 12 sets of tests. To investigate the variation of height distributions with specimen length 100 height measurements were made at equal horizontal intervals over the length of each specimen with a coordinate measuring machine (Ferranti Mercury).
4. Discussion The maximum coefficient of static friction obtained in the experiment is 0.24. This low value indicates the presence of an outer contaminant film but that should not affect our results except in reducing the absolute value of vs. In the limiting case where a perfect boundary lubricant is present the coefficient of static friction will tend to tan 0. In contrast, when the surfaces are very smooth and clean the effect of surface roughness is negligible [ 121. Others have shown that friction increases with the profile slope [ 131 or with the initial asperity angle [ 141. The probability of the linear relation observed in Fig. 3 being due to chance is less than 1 in 500. According to our theory the intercept at 5.34 f 0.71 N is the value of W tan 8, from which we can deduce that the mean surface slope is 8.8 f 1.2”. This is much steeper than the slopes generally associated with ground surfaces measured along their lay which suggests that the asperities responsible are too small to be measured by conventional stylus instruments. The condition of proportionality between u and L1j2 is obviously met by the results of Fig. 4. The intercept of the regression line is -0.205 + 0.17 E.tmand thus it only fails to pass through the origin by an insignificant margin.
9
a 5 7
L
INI
I$ 6
3 2 1
F
5
E 0
1
2 l/L
3 Im-‘I
L
5
6
!LL 5
10
.JT
15
112 Imm
20
I
Fig. 3. Variation of limiting static friction with reciprocal specimen length. Each circle represents the mean of 12 measurements. The error bars are 50% confidence limits. The solid line is the regression line (p = 0.493, v = 34, p < 0.002, one-sided). Fig. 4. Variation of the root mean square height deviations with the square root of the specimen length. The circles are the experimental results and the solid line is the regression line (p = 0.9994, v = 1, p < 0.025, one-sided).
362
In a survey of wear experiments some additional evidence of this shape criterion was found. Play and Godet [ 151 measured the linear wear rate of rectangular chalk pins of equal nominal area but with varying aspect ratio sliding on a glass disc. They found that an increase in the aspect ratio significantly reduces wear. If it is considered that wear of the chalk pin can only occur when there is intimate contact between the surfaces the wear rate would be related to the number of discrete contacts. Equation (1) predicts that the number of contacts is proportional to the reciprocal of the specimen length. Replotting their results as linear wear rates after first rotation against the reciprocal of the specimen length (Fig. 5) gives good agreement with our hypothesis. Lancaster [ 161, using electrographitic pins sliding on hardened steel, also reported a similar decrease in linear wear rates with increasing aspect ratio. In the experiments described above it is suggested that the observed dependence of the friction and abrasive wear processes on aspect ratio is a result not of material or thermal properties nor of the mode of asperity deformation but of the increasing predominance of long surface wavelengths as the specimen length increases.
15
r
1 3
Load
1 1
=
8 8N
09 07 13
.:
/ -
0
0 1 1 /L
0.2
0.3
Imrti’)
Fig. 5. Variation of linear wear rates of chalk pins on glass with reciprocal of specimen length replotted from Table 3 of ref. 14. The circles are the experimental results and the solid lines are regression lines. Correlations, from top (U = 1, one-sided): p = 0.99998, p < 0.005;~ = 0.9996,p < 0.02;~ = 0.964,p > 0.1;~ = 0.9995,p < 0.02.
363
References 1 Translated by D. Dowson, personal communication, 1978. 2 D. Bowden and F. P. Tabor, Friction and Lubrication ofSolids, Oxford Univ. Press: Clarendon Press, Oxford, 1954. 3 J. F. Archard, Elastic deformation and the laws of friction, Proc. R. Sot. London, Ser. A, 243 (1957) 190 - 205. 4 J. A. Greenwood, The area of contact between rough surfaces and flats, Trans. ASME, 89F (1967) 81 - 89. 5 J. B. P. Williamson, Topography of solid surfaces, NASA Spec. Publ. SP-181, 1968, pp. 85 - 141. 6 R. S. Sayles and T. R. Thomas, Surface topography as a non-stationary random process, Nature (London), 271 (1977) 431 - 434. 7 J. Dyson and W. Hirst, The true area of contact between surfaces, Proc. Phys. Sot. London, Sect. B, 67 (1954) 309 - 312. 8 K. L. Woo, R. S. Sayles and T. R. Thomas, An exception to Amontons’ Law, Proc. Euromech. Colloq. No. 110 on Contact Problems and Load Transfer in Mechanical Assemblages, Linkiiping, 1978, Linkiiping Inst. Technol., Linkoping, 1978, pp. 175 - 17’ 9 R. S. Sayles and T. R. Thomas, Computer simulation of the contact of rough surfaces, Wear, 49 (1978) 273 - 296. 10 D. M. Tolstoi, Significance of the normal degree of freedom and natural contact vibrations in contact friction, Wear, 10 (1967) 199 - 213. 11 A. W. Bush, R. D. Gibson and T. R. Thomas, The elastic contact of a rough surface, Wear, 35 (1975) 87 - 111. 12 H. Ernst and M. E. Merchant, Surface friction of clean metals - A basic factor in the metal cutting process, Proc. Conf. on Friction and Surface Finish, MIT Press, Cambridge, Mass., 1940, pp. 76 - 101. 13 N. 0. Myer, Characterisation of surface roughness, Wear, 5 (1962) 182 - 189. 14 S. R. Ghabrial and S. A. Zaghlool, The effect of surface roughness on static friction, Int. J. Mach. Tool Des. Res., 14 (1974) 299 - 309. 15 D. Play and M. Godet, Self protection of high wear materials, ASLE/ASME Lubrication Conf., Kansas City, MO., 1977, ASLE Prepr. (Am. Sot. Lubr. Eng.) 77-Lc-2, 1977. 16 J. K. Lancaster, Geometrical effects of the wear of polymers and carbon, Trans. ASME, 77F (1975) 189 - 194.