A Static Friction Model for Elastic-Plastic Contacting Rough Surfaces

Lior Kogut1 Mem. ASME e-mail: [email protected] Izhak Etsion Fellow ASME e-mail: [email protected] Dept. of Mechanical Engineering, Technion, Haifa 32000, Israel A Static Friction Model for Elastic-Plastic Contacting Rough Surfaces A model that predicts the static friction for elastic-plastic contact of rough surfaces is presented. The model incorporates the results of accurate finite element analyses for the elastic-plastic contact, adhesion and sliding inception of a single asperity in a statistical representation of surface roughness. The model shows strong effect of the external force and nominal contact area on the static friction coefficient in contrast to the classical laws of friction. It also shows that the main dimensionless parameters affecting the static friction coefficient are the plasticity index and adhesion parameter. The effect of adhesion on the static friction is discussed and found to be negligible at plasticity index values larger than 2. It is shown that the classical laws of friction are a limiting case of the present more general solution and are adequate only for high plasticity index and negligible adhesion. Some potential limitations of the present model are also discussed pointing to possible improvements. A comparison of the present results with those obtained from an approximate CEB friction model shows substantial differences, with the latter severely underestimating the static friction coefficient. @DOI: 10.1115/1.1609488# Keywords: Friction Modeling, Contacting Rough Surfaces, Static Friction, Adhesion Introduction It is well known from everyday experience that to displace one body relative to another when the bodies are subjected to a compressive force necessitates the application of a specific tangential force, known as the static friction force, and until the required force is applied the bodies remain at rest. Accurate prediction of the static friction force may have an enormous impact on a wide range of applications such as bolted joint members @1#, workpiece-fixture element pairs @2#, static seals @3#, clutches @4#, compliant electrical connectors @5# magnetic hard disks @6,7#, and MEMS devices @8,9#, to name just a few. Static friction was considered by the pioneers of friction research: Leonardo da Vinci, Guillame Amontons, Leonard Euler, Charles Augustin de Coulomb, George Rennie, Arthur-Jules Morin, Robert Hooke and others @10#. In early experimental work it was observed that the proportionality of the force opposing relative motion to the force holding the bodies together seemed to be constant over a range of conditions. Amontons, for example, is remembered for his two laws of friction: 1. The force of friction is directly proportional to the applied load. 2. The force of friction is independent of the nominal area of contact. A common method for calculating the static friction force ~Coulomb friction law! was drawn from these two basic laws by multiplication of the normal applied load by a proportionality constant, known as the static friction coefficient, taken from engineering handbooks as a function of the contacting materials. Static friction coefficients are conveniently tabulated and incorporated into engineering handbooks for at least 300 years. However, these tabulated values represent average coefficients of friction 1 Currently Post-Doctoral Fellow, Dept. of Mechanical Engineering, UC Berkeley, [email protected]! Contributed by the Tribology Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for presentation at the STLE/ASME Joint International Tribology Conference, Ponte Vedra, FL October 26 –29, 2003. Manuscript received by the Tribology Division January 24, 2003; revised manuscript received June 10, 2003. Associate Editor: G. G. Adams. 34 Õ Vol. 126, JANUARY 2004 determined over a broad spectrum of test conditions. While these numbers provide a general guideline of the sensitivity of the coefficient of friction to the materials in contact, they may not necessarily be representative of the coefficient of friction that will result between actual contact pairs. The friction coefficient is presently recognized as both material- and system-dependent @11# and is definitely not an intrinsic property of two contacting materials. Blau @11# in his review paper indicated that the friction coefficient is an established, but somewhat misunderstood, quantity in the field of science and engineering. While friction coefficients are relatively easy to determine in laboratory experiments, the fundamental origins of sliding resistance are not so clear, and hence, it is extremely important to understand the process involved in friction. Indeed, a great deal of progress has been made since the pioneering work of Amontons in 1699 and Coulomb in 1785, as is evident from recent works that consider both atomistic point of view ~e.g., @12–14#! and continuum mechanics principals ~e.g., @15–17#!. Tabor @18# in his general critical picture of friction understanding pointed out three basic elements that are involved in the friction of dry solids: 1. The true area of contact between mating rough surfaces. 2. The type and strength of bond formed at the interface where contact occurs. 3. 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, m: m5 Q max Q max 5 F P2F s (1) where Q max is the tangential force needed to fail the junctions created between the contacting surfaces, and F, the external force, ~see Fig. 1! is the balance between the actual contact load, P, in the true area of contact and the amount of the intermolecular forces or the adhesion, F s , acting between the surfaces in contact. The right hand side of Eq. ~1! contains all the three elements mentioned above. The contact load, P, is related to the true area of Copyright © 2004 by ASME Transactions of the ASME Fig. 1 The forced acting between contacting rough surfaces contact. The adhesion, F s , is related to the strength of the bond formed at the interface. The maximum tangential load, Q max , is related to the failure of the contact. Chang et al. @19# presented a model ~CEB friction model! for predicting the static friction coefficient of rough metallic surfaces based on the three elements indicated by Tabor @18#. The CEB friction model uses a statistical representation of surface roughness @20# and calculates the static friction force that is required to fail all of the contacting asperities, taking into account their normal preloading. This approach is completely different from the classical Coulomb friction law and shows that the latter is a limiting case of a more general behavior where static friction coefficient actually decreases with an increasing applied load or decreasing nominal contact area. The CEB friction model actually treats the static friction as a plastic yield failure mechanism corresponding to the first occurrence of plastic deformation in the contacting asperities. This can severely underestimate friction coefficient values for contacting rough surfaces since it neglects the ability of an elastic-plastic deformed asperity to resist additional loading before failure occurs as was demonstrated recently by Kogut and Etsion @21#. Roy Chowdhury and Ghosh @22# followed the same approach of the CEB friction model with additional adhesion related restriction on the maximum tangential load that can be carried by a single asperity. Etsion and Amit @23# demonstrated experimentally, for small normal loads and relatively smooth surfaces that the static friction coefficient decreases with increasing normal loads as predicted by the CEB friction model. Polycarpou and Etsion @24# extended the original CEB friction model to include the presence of sub-boundary lubrication. In a following paper @25# they compared their model prediction @24# with published experimental results and found good agreement. Liu et al. @26#, in yet another extension of the CEB friction model, developed a static friction model for the case of rough surfaces in the presence of thin metallic films and compared their theoretical results with experimental data in @27#. The original CEB friction model @19# as well as its following extensions @24# and @26# calculate the static friction coefficient by using Eq. ~1! where the contact load, P, and adhesion force, F s , are obtained from previous approximate models of Chang et al. @28# and @29#, respectively. However, as was shown in a series recent works, @21,30,31#, that are based on finite element analysis, the previous approximate models @19,28,29# produce large discrepancies on the single asperity level. As can be seen from the above literature survey, available tabulated values of static friction coefficient do not account for such important parameters as surface roughness, surface energy, mechanical properties and contact load that have strong effect on the friction. An adequate theoretical model will eliminate the current need for extensive empirical work and will shed more light on understanding the dominant parameters affecting the static friction coefficient. The aforementioned approximate models for static friction coefficient assume failure of a contacting asperity as soon as the first plastic point appears, and hence, underestimate the actual friction force. These models also rely on approximate contact and adhesion solutions for a single asperity, that present large discrepancies with respect to recent finite element solutions. The present work relies on these finite element solutions for contact, adhesion and friction, and hence, should improve the accuracy of the original CEB friction model. This remains to be verified by comparison with controlled experiments that will hopefully be presented in subsequent works. Analysis Figure 2 describes schematically the geometry of the contacting rough surfaces. The two rough surfaces of Fig. 1 are replaced with a single equivalent rough surface in contact with a flat. The basic assumptions of Greenwood and Williamson @20# regarding the shape and statistical distribution of the asperities along with the transformation to the more practical surface height distribution ~see Nayak @32#! are adopted in the present analysis. R is the uniform asperity radius of curvature, z and d denote the asperity height and separation of the surfaces, respectively, measured from the reference plane defined by the mean of the original asperity heights. The separation h is measured from the reference plane defined by the mean of the original surface heights. f (z) is the asperity height probability density function, assumed to be Gaussian: f~ z !5 1 A2 ps s 2 F S DG exp 20.5 z ss (2) where s s is the standard deviation of asperity heights. The interference is defined as: v 5z2d (3) and only asperities with positive interference are in contact. During loading, the contact load, P̄, adhesion force, F̄ s , and the static friction force, Q̄ max , of each individual asperity depend only on its own interference, v, assuming there is no interaction between asperities. The dependence of P̄, F̄ s and Q̄ max on v must be determined by the asperity mode of deformation, which can be elastic, elastic-plastic or fully plastic. Once these expressions are Fig. 2 Contact model of rough surfaces Journal of Tribology JANUARY 2004, Vol. 126 Õ 35 Table 1 The values of a , b , and c for the various deformation regimes in Eqs. „10… to „12… Eq. ~10! Deformation regime Eq. ~11! Eq. ~12! a b a b c i ai bi Fully elastic, v / v c ,1 1st elastic-plastic, 1< v / v c <6 1 1.03 1.5 1.425 0.98 0.79 0.298 0.356 20.290 20.321 1 1 2 3 4 0.982 4.425 3.425 2.425 1.425 2nd elastic-plastic, 6< v / v c <110 Fully plastic, v / v c .110 1.40 3/K 1.263 1 1.19 0.093 N/A 20.332 0.52 20.01 0.09 20.40 0.85 N/A N/A known, the total contact load, P, adhesion force, F s , and static friction force, Q max , are obtained by summing the individual asperity contributions using a statistical model: E E E ` P5 h A n P̄ ~ z2d ! f ~ z ! dz (4) F̄ s ~ z2d ! f ~ z ! dz (5) d ` F s5 h A n f * (z * ) is the dimensionless asperity heights probability density function obtained from Eq. ~2! by substituting the normalized length dimensions z/ s and s s / s . The dimensionless critical interference, v * c , is another form of 21/2 the plasticity index, C>( v * ) , that was first introduced by c Greenwood and Williamson @20#. It was shown in @35# that C is the most important dimensionless parameter in elastic-plastic contact problems of rough surfaces. It has the form: 2` Q max5 h A n S D d16 v c Q̄ max~ z2d ! f ~ z ! dz C5 v * c (6) s ss 20.5 5 S D 2E s s p KH R 0.5 (9) d where A n is the nominal contact area and h is the area density of the asperities. The integrals in Eqs. ~4!–~6! are solved in parts for the different deformation regimes of the contacting asperities. It should be noted that while the contact load, P, and static friction force, Q max , are calculated for contacting asperities only, the adhesion force, F s , is calculated also for non-contacting asperities, and hence, the difference in the lower limit of the integral in Eq. ~5!. The upper limit of the integral in Eq. ~6! is due to the observation in @21# that preloaded asperities are unable to support additional tangential load if their interference is larger than 6 v c . It should also be noted that Eqs. ~4!–~6! are general in terms of the asperity height probability density function f (z). Other nonGaussian distributions can be used in these equations ~see e.g., @33#!. The critical interference, v c , that marks the transition from elastic to elastic-plastic deformation is given by ~see e.g., Chang et al. @28#! v c5 S D p KH 2E R (7) K50.45410.41n where b is a surface roughness parameter defined as 36 Õ Vol. 126, JANUARY 2004 5a F̄ s0 b S DS D v vc « b (10) vc c ~ contacting asperities, v / v c .0 ! vc (11) F̄ snc 5 4 3 2 8 FS D S D G v/vc v/vc 20.25 Q̄ max where E 1 , E 2 and n 1 , n 2 are Young’s moduli and Poisson’s ratios of the contacting surfaces, respectively. All length dimensions are normalized by the standard deviation of the surface heights, s, and the dimensionless values are denoted by*. Hence, y s* is the difference between h * and d * ~Bush et al. @34#! which, after some algebra becomes: 1 F̄ s S D v «/ v c v/vc ~ non-contacting asperities, v / v c ,0 ! 1 12 n 21 12 n 22 1 5 E E1 E2 A48p b 5a P̄ c F̄ s0 E is the Hertz elastic modulus defined as: b5hRs P̄ 2 where H is the hardness of the softer material and K, the hardness coefficient, is related to the Poisson’s ratio of the softer material by ~see CEB friction model @19#!: y s* 5h * 2d * 5 and as can be seen it depends on surface roughness and material properties. Rougher and softer surfaces have higher plasticity index. Kogut and Etsion @30# found that the entire elastic-plastic contact regime of a single asperity extends over the range 1< v / v c ,110 with a transition at v / v c 56 that divides it into two subregions. Dimensionless contact parameters of a single asperity i.e., P̄/ P̄ c , F̄ s /F̄ s0 and Q̄ max /P̄c were presented in @30,31#, and @21#, respectively, where P̄ c 5(2/3)KH pv c R is the critical contact load at yielding inception ( v 5 v c ), F̄ s0 52 p RD g is the adhesion force at point contact ( v 50) and Dg is the energy of adhesion. These dimensionless contact parameters can be expressed in the general form: (8) P̄ c 5 (a i S D v i vc (11a) bi (12) where « is the intermolecular distance that is typically about 0.3– 0.5 nm. The constants a, b, and c for the elastic, elastic-plastic ~in the two sub-regions!, and plastic regimes are summarized in Table 1. Note that Eq. ~12! is not applicable for v / v c .6 ~see @21#!, and Eq. ~11! is not applicable for v / v c .110 ~see @31#!. The analyses in Refs. @30#, @31#, and @21# are all based on an assumption of elastic perfectly-plastic material behavior and hence, the present model is also adequate for such materials. The dimensionless contact load P * , is obtained from Eqs. ~4! and ~10! ~see @35#! in the form: Transactions of the ASME P *5 P 2 5 pbKv* c A nH 3 11.4 E d * 1110v c* d * 16 v c* SE d * 1 v c* d* 3 K I 1.2631 I 1.511.03 E E ` I1 d * 1110v c* d * 16 v c* I 1.425 d * 1 v c* D (13) S z * 2d * v* c D Fs 52 p bu A nH 10.79 E SE f * ~ z * ! dz * d* J nc 10.98 2` d * 16 v c* d * 1 v c* (14) 0.356 J 20.321 11.19 E E d * 1 v c* d* 4 3 FS «* d * 2z * D 2 20.25 S d * 1110v c* 0.093 J 20.332 D (15) 8 DG f * ~ z * ! dz * (16) and J bc is a general form of the integrands accounting for the contribution of contacting asperities: J bc 5 S z * 2d * v* c b * * Q max Q max 5 F* P * ~ 12F s* / P * ! (19) «* c f * ~ z * ! dz * v* c DS D (16a) (20) It should be noted here that other definitions for the friction coefficient are found, e.g., @36#. However, Eq. ~1! seems to be the more practical definition. Some insight regarding the role of the plasticity index C in the static friction problem can be gained by following the analysis in @35# for the contact problem. With a Gaussian distribution of asperity heights the maximum practical height of a given asperity is z * >3. Therefore, the integrals for the contacting asperities in Eqs. ~13!, ~15! and ~18! are practically zero whenever their lower limit 22 is higher than 3. Using the approximation v * and noting c >C that the relevant limits of integration have the general form d * 1kC 22 the condition for meaningful contribution of any of these integrals is: C. d * 16 v c* «* d * 2z * m5 0.298 J 20.29 where J nc accounts for the contribution of the noncontacting asperities and has the form: J nc 5 F 5 P * 2F s* A nH b The four integrals in Eq. ~13! and their corresponding limits of integration represent the contribution of the asperities in elastic, elastic-plastic ~in the two sub-regions! and fully plastic contact, respectively. This methodology will be maintained in the following. Multiplying Eqs. ~11! and (11a) by F̄ s0 and using the values in Table 1, the adhesion force, F̄ s , of a single contacting asperity in the elastic and elastic-plastic regimes as well as F̄ snc for a single noncontacting asperity, can be obtained. Substituting in Eq. ~5!, and using the dimensionless form of Eq. ~3! one can obtain the dimensionless adhesion force, F s* , between rough surfaces in the form: F s* 5 F *5 The static friction coefficient as defined in Eq. ~1! may be expressed in the form: where I b is a general form of the integrand: I b5 more practical parameters h * and C. Also, the dimensionless external force F * ~see Fig. 1! as a function of these parameters can be obtained in the form: S D k 32d * 1/2 (21) It is clear from Eq. ~13! for example, that the contribution of its last three integrals ~where k>1) vanish for any d * >0 whenever C,1/). Therefore, C50.6 can be defined as the plasticity index value below which the contact problem is fully elastic. Similarly, the last integral in Eq. ~13! ~where k5110) becomes appreciable only if C.6. Hence, as was shown in @35#, C>8 indicates a fully plastic contact. Following the same reasoning the last integral of Eq. ~15! ~where k56) becomes increasingly significant as C becomes larger than &. Since it was found in @31# that the adhesion force of asperities with v / v c .6 is negligible compared to their contact load, it is reasonable to expect negligibly small effect of F s* / P * in Eq. ~20! when C increases above 1.4. The dimensionless adhesion parameter, u, is: u5 Results and Discussion Dg sH (17) Note that contribution of fully plastic asperities ( v / v c >110) was not included in Eq. ~15! in accordance with the observation made in Ref. @31#. Multiplying Eq. ~12! by P̄ c and using the values in Table 1, the static friction force, Q̄ max , of a single asperity in the elastic and first elastic-plastic sub-region can be obtained. Substituting in Eq. ~6!, and following the same procedure that have lead to Eq. ~15!, The dimensionless static friction force is obtained in the form: * 5 Q max F E Q max 2 5 pbKv* c 0.52 A nH 3 1 E d * 16 v c* d * 1 v c* d * 1 v c* I 0.982 d* ~ 20.01I 4.42510.09I 3.42520.4I 2.425 10.85I 1.425! G (18) where I b is defined in Eq. ~14!. Equations ~13!, ~15!, and ~18! can be transformed, by using * as functions of the Eqs. ~8! and ~9!, to present P * , F s* and Q max Journal of Tribology In accordance with the findings of @35# a wide range of plasticity index values from C50.5 to C58, was covered to analyze the effect of surface roughness and material properties on the static friction of contacting rough surfaces. A value of b 50.04 was selected according to the finding of Greenwood and Williamson @20#. A constant value of K50.577 was used corresponding to a typical Poisson’s ratio, n 50.3, for metals. For typical values of adhesion energy, material hardness and surface roughness the range of the adhesion parameter, u, is 1024 < u <0.01 where the upper limit corresponds to very high adhesion energy that can be obtained with very clean surfaces under vacuum conditions. The numerical results of Eqs. ~13!, ~15!, and ~18! to ~20! for any given h * and C, can be cross-plotted to provide a practical presentation of the relevant parameters vs. the known external applied force F * . Following the reasoning of @35# only the results for the range of practical engineering interest namely, 0<h * <3 and F * <0.1, will be presented. Lower h * and higher F * values may invalidate the basic assumptions of no interaction between neighboring asperities and no bulk deformation, respectively ~see @20#!. From Eq. ~20! it is clear that the effect of adhesion on the static friction coefficient depends on the ratio F s* / P * and this effect becomes negligible when F s* / P * !1. JANUARY 2004, Vol. 126 Õ 37 Fig. 3 Dimensionless force ratio, F s* Õ P * , as a function of the dimensionless external force, F * , for various values of the plasticity index, C at u Ä0.003 Figure 3 presents the ratio F s* / P * vs. the dimensionless external force, F * , for the range of the plasticity index, C and for a relatively high value of the adhesion parameter u 50.003. This high value of u was selected to facilitate the distinction of the effect of C at its higher values where F s* / P * may become very small. Note that the ratio F s* / P * depends linearly on u ~see Eq. ~15!! and, hence, it can be easily deduced for values of u different than 0.003 from the results shown in Fig. 3. As can be seen from Fig. 3 the ratio F s* / P * decreases sharply with increasing plasticity index. For C>2, F s* / P * becomes less than 0.11 throughout the range of F * even for the high value of u 50.003. Hence, for C>2 and more practical ~smaller! values of the adhesion parameter u, it can be concluded that P * 5F * is a reasonable approximation ~see Eq. ~19!! and the effect of adhesion on the static friction coefficient is negligible. In contrast, the ratio F s* / P * is significant at low plasticity index, C50.5, over most of the range of F * , and becomes small enough only at the upper limit of F * . For C51 the ratio F s* / P * becomes less than 0.1 for external force higher than a threshold value of F * 50.01. It can, therefore, be concluded that the effect of adhesion is important only in purely elastic contacts where C,0.6, or in lightly loaded contacts with plasticity index up to C51 and high adhesion parameter u .0.001. Hence, whenever u ,0.001 or C.2 the effect of adhesion on the static friction can be safely neglected. * , Figure 4 presents the dimensionless static friction force, Q max versus the dimensionless external force, F * , for various values of the plasticity index, C, when u 50.003. It can be seen that at a given external force, the static friction force decreases with increasing plasticity index. At higher plasticity index the contact is more plastic and a larger population of the contacting asperities undergo interference in the range v / v c >6, where according to the finding in @21# they are unable to support any tangential load and hence, do not contribute to the static friction. Increasing the external force at a given plasticity index also increases the number of such high interference asperities but at the same time brings into contact more asperities that were initially noncontacting. It turns out that the latter effect is more dominant, and, hence, causes an increase of the static friction force with increasing external force. This behavior of the static friction force is different from that in the case of a single asperity @21# where the static friction force first increases with increasing normal load, reaches a maximum and than starts decreasing. Reducing the adhesion parameter u reduces somewhat the static friction at C50.5 and low external force but otherwise has very 38 Õ Vol. 126, JANUARY 2004 * , as a function Fig. 4 Dimensionless static friction force, Q max of the dimensionless external force, F * , for various values of the plasticity index, C at u Ä0.003 little effect on the results shown in Fig. 4, in accordance with the negligible effect of the adhesion over most of the practical range of F * as was shown in Fig. 3. Note the log/log scale used in Fig. 4 showing almost linear relation having the general form: * 5C ~ F * ! m Q max (22) * This relation differs from the classical law of friction, Q max 5mF*, whenever mÞ1. Indeed in Fig. 4 the power m is less than 1 and varies from m50.82 at C52 to m50.86 at C50.5, indicating a smaller rate of increase of the friction force compared to that of the external force as more asperities are brought into contact. As shown in Fig. 4 at the highest plasticity index, C58, the static friction force is extremely small being between 3 to 4 orders of magnitude smaller than the external force. This is a result of the contact becoming fully plastic, see @35#, where large percentages of the contacting asperities undergo interferences much higher than v / v c 56. Such small friction force at high plasticity index seems unreasonable. It may be attributed to some of the simplifying assumptions made in Ref. @30# namely, an elastic perfectlyplastic behavior of the materials that neglects more realistic strain hardening effects. In addition, Mesarovic and Fleck @37# presented a finite element analysis that shows a decrease in the mean contact pressure of a single asperity under very high normal loads and extreme interference deep into the fully plastic regime. As a result such an asperity may regain its ability to resist a finite tangential load and thereby contribute to the static friction of highly plastic contacting rough surfaces. The present model, not showing these effects, may be invalid at large plasticity index values. Figure 5 presents the static friction coefficient, m, ~see Eq. ~20!! versus the dimensionless external force, F * , for low and medium values of the plasticity index, C, and u 50.003. Increasing the plasticity index, at a given external force, decreases the friction coefficient, similar to the behavior of the friction force as shown in Fig. 4. However, in contrast to the behavior of the friction force, increasing the external force, at a given plasticity index, decreases the static friction coefficient. This can be easily understood from substituting Eq. ~22! in the expression for the static * /F*, which results in: friction coefficient m 5Q max m 5C ~ F * ! m21 5C ~ F * ! 2n (23) Since m is less than 1 we can see from Eq. ~23! that m decreases with increasing external force. Etsion and Amit @23# investigated Transactions of the ASME Fig. 5 Static friction coefficient, m, as a function of the dimensionless external force, F * , for various values of the plasticity index, C at u Ä0.003 experimentally the effect of external load on the static friction coefficient between aluminum alloy pins and a nickel coated disk. They found for plasticity index values ranging from 0.67 to 1.01 that the power n in Eq. ~23! has values between 0.102 and 0.130, respectively. The corresponding values of n obtained from Fig. 5 for the range of plasticity index values between 0.5 and 2 are between 0.09 to 0.13 for u 50.001. This is a fair agreement considering the unknown exact u value in the experiment. It should be noted that as F * approaches zero the static friction coefficient may become very large and this too was observed in @23#. Also shown in Fig. 5 in dashed lines are the results obtained from the original CEB friction model @19#. As can be seen this approximate model substantially underestimates the static friction coefficient and already at C52 predicts unrealistic small values. This is due to a restrictive assumption made in @19#, that asperities with v / v c >1 are unable to support any tangential load, causing severe underestimation of the static friction coefficient at plasticity index values above 0.6. Another assumption that was made in the CEB friction model @19# is that the elastically preloaded asperities having v / v c ,1 cannot support tangential loads higher than that causing the onset of plasticity. This assumption can severely underestimate the maximum tangential load that can be supported by these asperities as was demonstrated in @21#, and is responsible for the lower static friction coefficient that is predicted by the CEB model even at C50.5 Figure 6 shows the effect of the adhesion parameter, u, on the static friction coefficient, m. It can be seen that for a low plasticity index, C50.5, reducing u from the high value of 0.003 to 0.001 ~a three fold reduction of the adhesion force! reduces substantially the static friction coefficient at a given external force at the lower end of that force. This effect diminishes as the external force increases and becomes negligibly small at the upper limit of the external force. A further reduction of the adhesion parameter to u 50.0001 has a much smaller effect since the adhesion becomes negligible anyway ~see discussion of Fig. 3!. The effect of u on m disappears at the higher plasticity index C52, since in this case too the adhesion force is negligible. High adhesion force decreases the separation, h * , at a given external force and brings more asperities into contact especially when the external force is small, thus enabling to support larger tangential force, and, hence, the static friction force and friction coefficient increase with increasing u. From Figs. 5 and 6 it can be seen that the friction coefficient depends on the dimensionless external force, F * , i.e. on the exJournal of Tribology Fig. 6 Static friction coefficient, m, as a function of the dimensionless external force, F * , for various values of the plasticity index, C, and the dimensionless adhesion parameter, u ternal force as well as on the nominal contact area ~see Eq. ~19!!. This later dependency is due to the effect of A n on the separation d * ~see @35#! that appears in the integrals of Eqs. ~13!, ~15!, and ~18!, which are then substituted in Eq. ~20!. Additionally, the static friction coefficient depends on mechanical properties and surface roughness ~through C! and on the adhesion energy ~through u!. This is quite different from the classical laws of friction. As the plasticity index increases the static friction coefficient becomes much less sensitive to these parameters, similar to the teaching of the classical laws of friction. Hence, the classical Coulomb friction law ~which was obtained some 300 years ago presumably for high C and low u values! can be regarded as a limiting case of the more general model presented in this work. Conclusion A model that predicts the static friction for elastic-plastic contact of rough surfaces was presented. It incorporates the results of accurate finite element analyses for the elastic-plastic contact, adhesion and sliding inception of a single asperity in a statistical representation of surface roughness. Strong effect of the external force and nominal contact area on the static friction coefficient was found in contrast to the classical laws of friction. The main dimensionless parameters affecting the static friction coefficient are the plasticity index C and adhesion parameter u. The effect of adhesion on the static friction was found to be negligible at plasticity index values larger than 2 throughout the practical external force range that was investigated regardless of u. At plasticity index values lower than 1 adhesion may be important if u .0.001 and the external force is not too large. The present model that assumes elastic perfectly-plastic material behavior may be invalid at high plasticity index values where the contact approaches fully plastic state. Unreasonably small static friction was found under this condition and an improved model that considers strain hardening effects and possible junction growths may be required. It was shown that the classical laws of friction are a limiting case of the present more general solution and are adequate only for high plasticity index and negligible adhesion. A comparison of the present results with those obtained from an approximate CEB friction model showed substantial differences with the latter severely underestimating the static friction coefficient. JANUARY 2004, Vol. 126 Õ 39 Acknowledgment This research was supported in parts by the Fund for the Promotion of Research at the Technion, by the J. and S. Frankel Research Fund and by the German-Israeli Project Cooperation ~DIP!. @6# @7# @8# Nomenclature d d* E F F* Fs F s* H h h* K P P* Q Q* R y s* z 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 z* b Dg f* 5 5 5 5 5 5 5 5 5 5 5 5 h m n u s ss v* vc v* c 5 C 5 separation based on asperity heights dimensionless separation, d/ s Hertz elastic modulus external force dimensionless external force, F/A n H adhesion force dimensionless adhesion force, F s /A n H hardness of the softer material separation based on surface heights dimensionless separation, h/ s hardness factor, 0.45410.41 n contact load dimensionless contact load, P/A n H friction force dimensionless friction force, Q/A n H asperity radius of curvature h * 2d * height of an asperity measured from the mean of asperity heights dimensionless height of an asperity, z/ s surface roughness parameter, h R s energy of adhesion dimensionless distribution function of asperity heights area density of asperities static friction coefficient Poisson’s ratio of the softer material dimensionless adhesion parameter, D g / s H standard deviation of surface heights standard deviation of asperity heights dimensionless interference critical interference at the inception of plastic deformation dimensionless critical interference, v c / s plasticity index, Eq. ~9! 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