Static Friction Coefficient Model for Metallic Rough

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which depend on the extent of plastic deformation of the sur- face asperities as will be discussed in the following sections, determine the static friction coefficient ...
W. R. Chang1 I. Etsion2 D. B. Bogy Department of Mechanical Engineering, University of California, Berkeley, CA 94720

Static Friction Coefficient Model for Metallic Rough Surfaces The friction force required to shear interface bonds of contacting metallic rough surfaces is calculated, taking into account the prestress condition of contacting asperities. The surfaces are modeled by a collection of spherical asperities with Gaussian height distribution. Previous analyses for adhesion force and contact load of such surfaces are used to obtain the static friction coefficient. It is shown that this coefficient is affected by material properties and surface topography, and that it actually depends on the external loading contrary to the classical law of friction.

Introduction Friction is inevitably associated with almost any mechanical component. In some cases, e.g. belt drives, brakes and clutches, joints, etc., friction may be desirable. In other cases, e. g. gears, bearings, seals, etc., friction may be destructive. In both cases it is important to understand the process involved in friction. A great deal of progress has been made in this regard since the pioneering work of Amontons in 1699 and Coulomb in 1785, as is evident from the general critical picture of our present understanding of the frictional process that was presented by Tabor (1981). Tabor pointed out three basic elements that are involved in the friction of unlubricated solids: 1 2 3

The true area of contact between mating rough surfaces. The type and strength of bond formed at the interface where contact occurs. The way in which the material in and around the contacting regions is sheared and ruptured during sliding.

The importance of these three elements can be easily understood from the definition of the friction coefficient /t Q_ (1) /* = F ' where Q is the tangential force needed to shear the junctions between the contacting surfaces and F is the external normal force. The actual contact load P i n the true area of contact differs from F by the amount of the intermolecular forces acting between the surfaces in contact. We shall refer to these forces as the adhesion, Fs, and hence, P=F+FS.

(2)

From equations (1) and (2) we have

Current address: Digital Equipment Corporation, Shrewsbury, MA. On sabbatical leave from Technion, Haifa, Israel. Contributed by the Tribology Division of The American Society of Mechanical Engineers and presented at the ASLE/ASME Tribology Conference, San Antonio, Texas, October 5-8, 1987. Manuscript received by the Tribology Division, September 1987. Paper No. 87-Trib-6.

Journal of Tribology

P-F,

(3)

The right-hand side of equation (3) contains all the three elements mentioned above. The contact load, P, is related to the true area of contact through the general problem of contacting rough surfaces and was treated in Chang et al. (1987). The adhesion, Fs, relates to the strength of the bond formed at the interface, and was treated in Chang et al. (1988). This paper addresses the tangential force, Q, which relates to the shearing of the contact and, hence, to the friction force. Most studies of contact and adhesion of rough surfaces which are essential for understanding the mechanism of friction are concerned with static conditions. Yet most of the published literature deals with dynamic rather than static friction. Of the papers available on static friction, very few deal with the effect of surface roughness. Hisaka&o (1970) offered a theoretical model of a static friction coefficient for conicalshape asperities. In his model, surface forces are neglected and the friction coefficient for shearing and ploughing is presented as a function of the ratio between the shear strength and the yield pressure of the material. Experimental results showed that the friction coefficient increases with surface roughness mainly due to ploughing. Another investigation of the surface roughness effect was carried out by Ghabrial and Zaghlool (1974). They found that the friction coefficient is affected by both the surface roughness and the process used to obtain the surface finish. Temperature effects on static friction coefficient were studied by Brailsford (1973) and by Galligan and McCullough (1985). In both cases it was found that the friction coefficient can either increase or decrease with temperature depending on the temperature range and the combination of the materials in contact. Time effect on static friction coefficient was investigated by Kato et al. (1972). It was shown that the static friction coefficient increases with the time of contact and eventually reaches a constant value. Rabinowicz (1971) measured static friction coefficients of 210 combinations of various metals spanning a wide range of surface energies and hardness values. It was found that the

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friction coefficients obtained with combinations including Indium, which is an extremely soft metal, are distinctly higher than those obtained with the other metals. This was attributed to the higher ratio of surface energy to hardness, characterizing softer metals. The effect of contact load on the static friction coefficient was investigated by Paslay and Plunkett (1953). In this work static friction coefficient was measured for shrink fitted steel cylinders as a function of the interface pressure and the mode of surface preparation. It was found that static friction coefficient is lower for rougher surfaces and that the friction coefficient for a given surface roughness decreases as the contact pressure increases. Nolle and Richardson (1974) suggested that the reduction in friction coefficient at high contact pressures is attributed to plastic deformation of the asperities. This eventually results in lower friction coefficient when the material fails in compression. They also investigated the effect of surface roughness and found that at relatively large roughness the static friction coefficient becomes insensitive to the surface roughness. Buckley (1977) measured friction force between a gold surface in sliding contact with various metals in high vacuum. Reduction in the friction coefficient is seen in some cases as the normal load exceeds a certain value. Broniec and Lankiewicz (1980) measured the effect of normal load and vibration on the static friction coefficient in steel with smooth and rough surfaces. With the smooth surfaces the static friction coefficient decreases with increasing contact pressure. A similar effect was obtained with the rougher surface, but only at high amplitude of lateral vibration. From the discussion above, it is clear that the normal load does affect the friction coefficient contrary to the classical law of friction. The classical friction model arising from interfacial bonds (see Tabor 1981) assumes that the contact produces plastic flow at the tips of the asperities. In this case the true area of contact^, will be

Because of strong bonding at the interface, the force required to shear the junction, which is essentially the friction force, will be Q=AlS = -?-S, ti

(5)

where S is the interfacial shear strength. The coefficient of friction /x is then

Q s H=— = ^ F H

(6)

For most materials, S is of order 0.2H so that from this model ,u = 0.2. This is small compared to values of friction obtained for clean metals. This simple classical model has two serious shortcomings. It neglects the effect of surface forces in obtaining the true area of contact, A,, and it neglects the effect of prestressing due to the contact load, P, on the allowable friction force, Q. In fact, an asperity that yielded because of normal pressure is unable to support additional tangential force. Only asperities that did not reach their elastic limit can support appreciable additional tangential force before they fail, and the amount of this force depends on the prestressed condition of the individual asperity. Clearly, a realistic friction model has to consider surface forces and the degree of prestressing of the contact area.

Background The basic assumptions of the asperity based model of Greenwood and Williamson (1966) (GW model) are adopted: 1 The rough surface is isotropic. 2 Asperities are spherical near their summits. 3 All asperity summits have the same radius R but their heights vary randomly. 4 Asperities are far apart and there is no interaction between them. 5 There is no bulk deformation. Only the asperities deform during contact. Assumptions 1, 2 and 3 above were relaxed by McCool (1986) who treated an anisotropic rough surface with random distribution of elliptical paraboloidal asperities. His results, however, showed very good agreement with those of the simple GW model. Hence, these three assumptions, which greatly simplify the analysis, do not limit appreciably the generality of the model. Furthermore, as shown by Greenwood and Tripp (1971) the contact of two rough surfaces can be modeled by an equivalent single rough surface contacting a smooth plane. When dealing with contacting rough surfaces, two reference planes can be defined. One is the mean of the asperity heights and the other is the mean of the surface heights. The second one is more readily obtained experimentally than the first one and, hence, is more practical for use. Figure 1 shows the basic geometry model of contracting rough surfaces. Let u and d denote the asperity height and separation of the surfaces, respectively, measured from the reference plane defined by the

Nomenclature

A„ = nominal contact area A, = true contact area a = radius of contact area of a deformed asperity a* = a/a d = separation based on asperity heights, h-ys d* = dimensionless mean separation, d/a E l , 2 = Young's moduli E = Hertz elastic modulus F = external force Fs = adhesion force Fs = adhesion force of an individual asperity Fs' = expected adhesion force of a single asperity p* = dimensionless external force, F/An~E p * = dimensionless adhesion force, FS/A„E 1

Hl2

hardness of contacting surfaces separation based on surface heights dimensionless mean separation, h/a hardness coefficient total number of asperities number of asperities in contact total contact load contact load of an individual asperity expected contact load of a single asperity dimensionless contact load, P/A„E tangential shear force allowable tangential force of an individual asperity Q' = expected allowable tangential force of a single asperity Qi,2 = possible allowable tangential force of a single asperity h h* K N Nc P P P' P* Q Q

= = = = = = = = = = = =

s

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to the fact that only asperities in contact contribute to the expected contact load and shear force, while both contacting and noncontacting asperities contribute to the expected adhesion force. The total contact load P, adhesion force Fs and shear force Q are obtained by summing the individual asperity contributions. Hence, from (7) and (11) Fig. 1

P(d) = 7]A„[ P(u-d)4>(u)du,

Contacting rough surfaces

mean of the original asperity heights. R is the asperity radius of curvature and h is the separation of the surfaces measured from the reference plane defined by the mean of the original surface heights. The total number of asperities is N=VA„

(7)

Fs{.u-d)^u)du,

(12) (13)

— 00

Q(d) = r,AnrQ(u-d)d,(u)du.

(14)

Jd

and the number of asperities in contact is

For the calculation of the friction coefficient, the external normal load, F, rather than the contact load, P, is required. The latter is the resultant of the external applied load F and the (u)du, (8) ashesion force, Fs as indicated in equation (2). The shear force, Q, contact load, P, and adhesion force, Fs, where -q is the areal density of asperities, A„ is the nominal contact area and {u) is the asperity height distribution which depend on the extent of plastic deformation of the surfunction. face asperities as will be discussed in the following sections, Interference is defined as determine the static friction coefficient according to equations u = u-d (9) (l)-(3).

S

ao

and only those asperities with positive interference have contact. During loading, the contact load P, the adhesion force Fs and the shear force Q of each individual asperity will depend only on its own interference co since we assume that there is no interaction between asperities, i. e.,

Static Friction of Rough Surfaces

As indicated in the Introduction, the classical friction model assumes that the friction force, which is the force required to shear the junctions of contacting asperities, is given by the product of the true area of contact, A,, and the shear strength, S, of the weaker material. This, of course, neglects the fact The dependence of P, Fs and Q on oo must be determined by that contacting asperities are already prestressed even before the asperity deformation model which is elastic-plastic. Once any tangential force is applied. These asperities will fail with these expressions are known, the expected contact load, ex- the application of a smaller tangential force than the one corpected adhesion force and expected shear force of each asperi- responding to the full shear strength of unloaded asperities. It ty can be calculated by is reasonable to assume that plastically deformed asperities, which have already failed due to high local contact pressure, are unable to support any additional tangential load if strain P' = \ P(u-d)4>(u)du, hardening is neglected. Hence, only the contacting asperities that have not yet reached their elastic limit can contribute significantly to the strength of the bond and resist sliding. As F's= \_aFs(u-d)(u)du, (11) the tangential force applied to the contacting rough surfaces is gradually increased these asperities approach a point of dQ{u-d)(u)du. failure. At the instant when relative sliding of the surface begins, all contacting asperities must have failed. The tangenThe different limits of integration for F!. in (11) is due tial force corresponding to this limiting condition is called the Nomenclature (cont.)

Q* = dimensionless tangential shear force, Q/A„E R = asperity radius of curvature S = interfacial shear strength u = height of asperity measured from the mean of asperity heights u* = dimensionless height of asperity, u/a x,y, z = coordinate axes ys = distance between the mean of asperity heights and that of surface height y= distribution function of asperity heights = dimensionless distribution function = plasticity index 0) = interference CO* = dimensionless interference, co/a C0C = critical interference at the inception of plastic deformation CO 7- = transition interference, equation (26) « c * = dimensionless critical interference, o}c/a

r +

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static friction force. Hence, the static friction force in contacting rough surfaces is the sum of all the tangential forces causing plastic flow of the individual prestressed asperities. No complete solution for the stress field and area of contact for combined normal and tangential forces in contacting bodies is found in the literature. The few papers addressing the problem, e.g. Mindlin (1949), Hamilton and Goodman (1966), Golden (1977), and Hamilton (1983) are based on some simplifying assumptions. In the following we shall adopt the approach presented by Hamilton (1983). In this approach it is assumed that the shape of the contact region as obtained from the Hertz solution for contacting bodies under a normal loading P is unaffected by an additional applied tangential force Q. As shown by Hamilton (1983), if the ratio Q/P is less than 0.3 then failure will first occur below the surface, slightly off the point of failure corresponding to the Hertz solution when Q = 0. Immediately after the ratio Q/P exceeds the value of 0.3, the point of failure moves to the surface and failure occurs on the back edge of the contact boundary. We shall neglect the small effect of the tangential force on the exact location of failure below the surface and will assume that in this case it occurs at the point given by the Hertz solution. If we define a rectangular coordinate system with the z axis perpendicular to the contact region at its center and the x and y axes located on the contact surface, then the dimensionless location %=z/a for the inception of plastic deformation is given by the implicit equation (Chang 1986)

-L-l. 2

{\+v)")[itan " ' ( I / O -

+ ff2 JJ

3r -=o,

(1 + f 2 ) 2

(15)

where v is the Poisson ratio. The stress field at this point of failure inception is obtained from the work of Hamilton (1983) for a combined effect of a normal force P and a tangential force Q in the form 3P

[(! + »-) [ft.an-'a/f)-! J + 2(i + r 2 )i'

2«7 2

into equation (18) and solving for Q,, i.e., 9 p2 Ql

~

=

4^V/'

(19)

where

f2 2(1+ r2)

• 1 + — ftan-'(l/f)and

c2 = a + " ) [ f t a n - , ( l / r ) - l ] + -

3

2(1+ f 2 ) ' The constant c2 can also be related to other material properties (see Chang 1986) so that Q, can be obtained in the form 0.2045 (20) P, I where Kiss, hardness coefficient related to the Poisson ratio, v, by the approximate expression

£-0

a = K\c,

# = 0 . 4 5 4 + 0.41)'

(21)

and coc is a critical interference corresponding to the inception of plastic deformation and is given by (22) -)2R. V 2E The parameter E is the Hertz elastic modulus defined in the form

1 \-v\ \-v\ (23) ~E In equation (20) the absolute value of c, is used since only the positive real root of Qi is desired. Similarly the tangential force that causes yield on the surface, Q2, can be obtained by substituting the stress state of equation (17) into equation (18) which results in a quadratic equation c

ay

C2

27c? V

3 ( - ^ )

2

+ ^ + c 5 =0,

(24)

ax,

where 3P 2«? 2 (i + r 2 ) '

36 2ira2 T 1

= T yz

' xy

fl+4nan-Wf)-^J,

3 , c5=—-(l-2e)2

»

where x is in the direction of the tangential force Q and a is the radius of the contact area. The stress field on the surface at the back edge of the contact boundary is: {\-2»)P 27rfl2

3Q - + -4a 2 t

9vQ

T

xy

(-f*0-

(1-2P)P

2ira 2

16a 2 °z

('--r-rO-

9x c4 = — ( 1 - 2 ^ ( 1 - ^ / 2 ) ,

=0 u

9ir2 16

(16)

T

xz

r

yz

(17)

^'

The von Mises failure criterion with nonzero shear stresses has the form (Mendelson 1968) J = (ox- values used. The value ijoJ? = 0.04 is chosen according to the findings of Greenwood and Williamson (1966). The contact load, P, and the adhesion force, Fs, were calculated by the procedures outlined in Chang et al. (1987) and (1988) respectively. The tangential shear force, Q, was obtained from equation (30) for a Gaussian height distribution («). The static friction coefficient can, thus, be obtained from equation (3). Figures 3,4, and 5 show the static friction coefficient vs. the

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Table 1 Plasticity index, surface topography, and adhesion index for rjRo = 0.04 •A

0.5 0.6 0.7 0.8 0.9 1.0 2.5

a/R

9.77Xl0~ 4 1.41 X l O " 3 1.92X10" 3 2.5 X l O " 3 3.17x10-' 3.91 x l O " 3 2.44X10" 2

- 0for various A7 5

A-KJ/m 2 ) 2.5

0.5

15.9 27.4 43.6 64.9 92.7 127 788

31.8 54.9 87.3 130 185 254 1580

159 274 436 649 927 1270 7880

external force for the range of plasticity index \p from 0.5 to 2.5 and surface energies Ay of 5, 2.5 and, 0.5 J/m 2 , respectively. These results show that for a given external force the static friction coefficient increases as the plasticity index decreases (hard and smooth surfaces), and as the surface energy increases (clean surfaces). For a given plasticity index the static friction coefficient decreases as the external force increases. As an example an external force of ION and a nominal contact area of 1 cm2 result in static friction coefficient of about 0.5 for steel on steel when the plasticity index \j/= 1 and the surface energy is 2.5. Increasing the external force to 100N reduces the friction coefficient to 0.38. As can be seen from equation (31) low plasticity index means a smoother surface with smaller standard deviation of asperity heights, as, and a larger asperity radius, R. For such a surface a higher percentage of asperities are in elastic contact (see Chang et al., 1987), therefore the allowable tangential force, Q, is higher and the adhesion force is also higher due to a larger asperity radius. Higher tangential force and adhesion force lead to an appreciable increase in the static friction coefficient (see equation (3)). The increase in surface energy also leads to an increase in the adhesion force, and, hence, the static friction coefficient. On the other hand, higher plasticity index (soft and rough surfaces) means a larger standard deviation of asperity heights and a smaller asperity radius. For high plasticity index most asperities are plastically deformed (see Chang et al. 1987), so less tangential force can be supported before the onset of sliding. Smaller asperity radius and/or lower surface energy lead to a decrease in the adhesion force. Hence, the static friction coefficient decreases. At the equilibrium separation when the contact load and adhesion force are equal, no external force is needed to maintain the contact so the static friction coefficient becomes infinite. As the separation decreases due to the application of an external force, the ratio of the number of plastically deformed asperities to that of elastically deformed asperities increases. The contact load increases faster than the tangential force and the adhesion force becomes negligible compared with the contact load. Hence, the static friction coefficient decreases as the external force increases. When the surfaces are heavily loaded, most asperities are in plastic contact and the allowable tangential force is very small compared with the contact load. Hence, the static friction coefficient becomes very small. The earliest results on friction are due to Amonton in 1699. He proposed the first friction model in which the frictional force is independent of the apparent area of the sliding bodies and directly proportional to the normal load (Bowden and Tabor 1950). As indicated in the Introduction, according to this classical model the coefficient of friction is about 0.2. Nolle and Richardson (1974) reviewed the available published data on static friction of steel. From this review it is evident that the static friction coefficient depends on the external normal load. At low external loads the static friction coefficient increases with external load whereas at high external loads the friction coefficient decreases as the external load increases. Nolle and Richardson concluded that the surface conditions

I . E-6

I .E-S

I . E-4

DIMENSIONLESS EXTERNAL FORCE, F/A n E

Fig. 5

Static friction coefficient vs. external force

are an important factor in friction and proposed a model for contaminated surfaces. According to this model both surfaces are covered by contaminant layers. At low loads the measured friction is mainly due to the interaction between these two layers which reduce the interaction between the surfaces and, therefore, reduce the friction. As the load increases, the contaminant layers are penetrated, pure metal-to-metal contact is formed and the friction coefficient increases. After full metalto-metal contact is established, the friction coefficient becomes almost constant. At very high loads when most of the asperities are plastically deformed, the friction coefficient decreases as the load increases. This model predicts results that are completely different from Amontons law which does not account for surface contamination and deformation of asperities. Paslay and Plunkett (1953) measured the static friction coefficient at high contact pressures. Their results showed that static friction coefficient is lower for rougher surfaces and that for a given surface roughness the static friction coefficient decreases as the contact pressure increases. This is in agreement with our theoretical results as shown in Figs. 3, 4, and 5. Other experimental results by Nolle and Richardson (1974), Buckley (1976), Bay and Wanheim (1976), and Broniec and Lankiewicz (1982) also showed that at high normal loads the static friction coefficient decreases as the external force increases. Nolle and Richardson (1974) and also Broniec and Lankiewicz (1982) noticed that as the surface roughness increases the static friction coefficient becomes less sensitive to changes in the surface roughness, and that at high surface roughness the effect of the external force on the static friction coefficient diminishes. This behavior can be seen clearly in Figs. 3 to 5. At a given external force the static friction coefficient, 11, decreases rapidly as the plasticity index ip increases from V/ = 0.5 to i/-= 0.9. Thereafter the decrease of \x becomes much more moderate. Also, at low plasticity index there is a strong effect of the external force on /* whereas at \p = 2.5 the static friction coefficient is almost unaffected by the external force. The classical experiments by Amonton were carried out in air, meaning that his specimens were contaminated. Amonton's conclusion that the frictional force is directly proportional to the normal load can be explained by the results for low surface energy in Fig. 5. For the moderate range of dimensionless external force the static friction coefficient at a given plasticity index \p reduces by only 50 percent while the dimensionless external force is increasing by four orders on magnitude from 10~8 to 10" 4 . Hence, in this load range the static friction coefficient can be regarded as independent of the external force. An attempt was made to consider the change in the adhesion due to breaking through an oxide layer as postulated by Nolle

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and Richardson (1974). This was done by assigning low sur- Acknowledgment face energy to asperities in elastic contact and high surface This research was supported in part by the MICRO program energy to the plastically deformed ones. This was expected to show the increase in static friction coefficient with external of the University of California, with industrial sponsorship force which is normally observed at low external forces. from Ampex Corporation, Control Data Corporation, However, the results turned out to be very similar to those ob- Eastman Kodak Company, Memorex Corporation, and tained for the case in which all the asperities have the same low Temescal, a Division of British Oxygen Corporation. It was surface energy. The reasons for this are the following: for also supported in part by the National Science Foundation rough surfaces the static friction coefficient is insensitive to through grant number MSM-8511963. the surface energy as can be seen from the results for xp = 2.5 in References Figs. 3, 4, and 5. For smooth surfaces with low plasticity index Bay, N., and Wanheim, T., 1976, "Real Area of Contact and Friction Stress the percentage of asperities in plastic contact is very small at light loads. Hence, the increase in the static friction due to the at High Pressure Sliding Contact," Wear, Vol. 38, pp. 201-209. Bowden, F. P., and Tabor, T., 1950, "The Friction and Lubrication of change in the surface energy is almost negligible. At heavy Solids," Oxford University Press. loads most of the asperities are in plastic contact. The contact Brailsford, J. R., 1973, "Influence of Electric Current on the Static Friction load is much higher than the adhesion force and hence, the ef- of Metal Surfaces in Air," Wear, Vol. 25, pp. 85-97. Broniec, Z., and Lenkeiwicz, W., 1980, "Static Friction Processes Under fect of adhesion which depends on the surface energy is Dynamic Loads and Vibration," Wear, Vol. 80, pp. 261-271. negligible altogether (see Chang et al. 1988). It can be conBuckley, D., 1977, "The Metal-to-Metal Interface and Its Effect on Adhesion cluded, therefore, that the change in surface energy is not the and Friction," J. of Colloid and Interface Science, Vol. 58, pp. 36-53. Chang, W. R., 1986, "Contact, Adhesion and Static Friction of Metallic main mechanism for the increase of the static friction coefficient with the external force in the low-load range. A possible Rough Surfaces," Ph.D. thesis, University of California, Berkeley. W. R., Etsion, I., and Bogy, D. B., 1987, "Elastic-Plastic Model for different mechanism could be associated with strain hardening theChang, Contact of Rough Surfaces," ASME JOURNAL OF TRIBOLOGY, Vol. 109, pp. and the change in other material properties which would allow 257-263. higher tangential forces to be supported by elastically deformChang, W. R., Etsion, I., and Bogy, D. B., 1988, "Adhesion Model for ed asperities after breaking through oxide layers and prior to Metallic Rough Surfaces," ASME JOURNAL OF TRIBOLOGY, published in this issue, pp. 50-56. the onset of failure. Fuller, K. N. G., and Tabor, D., 1975, "The Effect of Surface Roughness on Summary and Conclusions The classical frictional model assumes that the shear strength of the interface junctions between the contacting surfaces is equal to the shear strength of the weaker material and also neglects the adhesion effect. Hence, the classical model overestimates the static friction coefficient at high external force and underestimates it at low external force. In the present analysis, the effect of the normal contact load on the shear strength of the interface junctions is taken into account. The von Mises yield criterion is used to calculate the tangential force which causes failure of contacting asperities and also to find the location of yield inception. The results show that rougher surfaces and softer materials support less tangential shear force. The static friction coefficient is calculated according to equation (3). The results show that for a given external force the static friction coefficient decreases as the plasticity index increases and as the surface energy decreases. For a given plasticity index the static friction coefficient decreases as the external force increases. The effect of surface roughness and surface energy of adhesion on the static friction coefficient as predicted by the present model is in agreement with experimental results available in the literature. The effect of external force on the static friction coefficient is also in agreement with these experiments for the range of high external loading. More work is needed to understand the effect of external force in the low loading range.

the Adhesion of Elastic Solids," Proc. Roy. Soc. (London), A345, pp. 327-342. Galligan, J. M., and McCullough, P., 1985, "On the Nature of Static Friction," Wear, Vol. 105, pp. 337-340. Ghabrial, S. R., and Zaghlool, S. A., 1974, "The Effect of Surface Roughness on Static Friction," Int. J. Mach. Tool Des. Res., Vol. 14, pp. 299-309. Golden, J. M., 1977, "The Actual Contact Area of Moving Surfaces," Wear, Vol. 42, pp. 157-162. Greenwood, J. A., and Tripp, J. H., 1971, "The Contact of Two Nominally Flat Rough Surfaces," Proc. Instn. Mech. Engrs., Vol. 185, pp. 625-633. Greenwood, J. A., and Williamson, J. B. P., 1966, "Contact of Nominally Flat Surfaces," Proc. Roy. Soc. (London), A295, pp. 300-319. Hamilton, G. M., 1983, "Explicit Equations for the Stresses Beneath a Sliding Spherical Contact," Proc. Instn. Mech. Engrs., 1970, pp. 53-59. Hamilton, G. M., and Goodman, L. E., 1966, "The Stress Field Created by a Circular Sliding Contact," ASME Journal of Applied Mechanics, Vol. 88, pp. 371-376. Hisakado, T., 1970, "On the Mechanism of Contact Between Solid Surfaces," Bull, of JSME, Vol. 13, pp. 129-139. Kato, S., Sato, N., and Matsubayashi, T., 1972, "Some Considerations on Characteristics of Static Friction of Machine Tool Slideway," ASME JOURNAL OF LUBRICATION TECHNOLOGY, Vol. 94, pp. 234-247.

McCool, J. I., 1986, "Predicting Microfracture in Ceramics Via a MicroContact Model," ASME JOURNAL OF TRIBOLOGY, Vol. 108, pp. 380-386. Mendelson, A., 1968, Plasticity: Theory and Applications, Macmillan, New York. Mindlin, R. D., 1949, "Compliance of Elastic Bodies in Contact," ASME JOURNAL OF APPLIED MECHANICS, Vol. 16, pp. 259-268.

Nolle, H., and Richardson, R. S. H., 1974, "Static Friction Coefficients for Mechanical and Structural Joints," Wear, Vol. 28, pp. 1-13. Paslay, P. R., and Plunkett, R., 1953, "Design of Shrink-Fits," Trans. ASME, Vol. 75, pp. 1199-1202. Rabinowicz, E., 1971, "The Determination of the Compatibility of Metals Through Static Friction Tests," ASLE Trans., Vol. 14, pp. 198-205. Tabor, D., 1981, "Friction—The Present State of Our Understanding," ASME JOURNAL OF LUBRICATION TECHNOLOGY, Vol. 103, pp. 169-179.

Timoshenko, S., and Goodier, J. N., 1951, Theory of Elasticity, McGrawHill, New York, 2nd Edition.

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