Lior Kogut 1 Mem. ASME e-mail: mekogut@tx.tehnion.a.il Izhak Etsion Fellow ASME e-mail: etsion@tx.tehnion.a.il Dept. of Mehanial Engineering, Tehnion, Haifa 32000, Israel A Stati Frition Model for Elasti-Plasti Contating Rough Surfaes A model that predits the stati frition for elasti-plasti ontat of rough surfaes is presented. The model inorporates the results of aurate finite element analyses for the elasti-plasti ontat, adhesion and sliding ineption of a single asperity in a statistial representation of surfae roughness. The model shows strong effet of the external fore and nominal ontat area on the stati frition oeffiient in ontrast to the lassial laws of frition. It also shows that the main dimensionless parameters affeting the stati frition oeffiient are the plastiity index and adhesion parameter. The effet of adhesion on the stati frition is disussed and found to be negligible at plastiity index values larger than 2. It is shown that the lassial laws of frition are a limiting ase of the present more general solution and are adequate only for high plastiity index and negligible adhesion. Some potential limitations of the present model are also disussed pointing to possible improvements. A omparison of the present results with those obtained from an approximate CEB frition model shows substantial differenes, with the latter severely underestimating the stati frition oeffiient. DOI: 10.1115/1.1609488 Keywords: Frition Modeling, Contating Rough Surfaes, Stati Frition, Adhesion Introdution It is well known from everyday experiene that to displae one body relative to another when the bodies are subjeted to a ompressive fore neessitates the appliation of a speifi tangential fore, known as the stati frition fore, and until the required fore is applied the bodies remain at rest. Aurate predition of the stati frition fore may have an enormous impat on a wide range of appliations suh as bolted joint members 1, workpiee-fixture element pairs 2, stati seals 3, luthes 4, ompliant eletrial onnetors 5 magneti hard disks 6,7, and MEMS devies 8,9, to name just a few. Stati frition was onsidered by the pioneers of frition researh: Leonardo da Vini, 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 fore opposing relative motion to the fore holding the bodies together seemed to be onstant over a range of onditions. Amontons, for example, is remembered for his two laws of frition: 1. The fore of frition is diretly proportional to the applied load. 2. The fore of frition is independent of the nominal area of ontat. A ommon method for alulating the stati frition fore Coulomb frition law was drawn from these two basi laws by multipliation of the normal applied load by a proportionality onstant, known as the stati frition oeffiient, taken from engineering handbooks as a funtion of the ontating materials. Stati frition oeffiients are onveniently tabulated and inorporated into engineering handbooks for at least 300 years. However, these tabulated values represent average oeffiients of frition 1 Currently Post-Dotoral Fellow, Dept. of Mehanial Engineering, UC Berkeley, kogut@newton.berkeley.edu Contributed by the Tribology Division of THE AMERICAN SOCIETY OF ME- CHANICAL ENGINEERS for presentation at the STLE/ASME Joint International Tribology Conferene, Ponte Vedra, FL Otober 26 29, 2003. Manusript reeived by the Tribology Division January 24, 2003; revised manusript reeived June 10, 2003. Assoiate Editor: G. G. Adams. determined over a broad spetrum of test onditions. While these numbers provide a general guideline of the sensitivity of the oeffiient of frition to the materials in ontat, they may not neessarily be representative of the oeffiient of frition that will result between atual ontat pairs. The frition oeffiient is presently reognized as both material- and system-dependent 11 and is definitely not an intrinsi property of two ontating materials. Blau 11 in his review paper indiated that the frition oeffiient is an established, but somewhat misunderstood, quantity in the field of siene and engineering. While frition oeffiients are relatively easy to determine in laboratory experiments, the fundamental origins of sliding resistane are not so lear, and hene, it is extremely important to understand the proess involved in frition. Indeed, a great deal of progress has been made sine the pioneering work of Amontons in 1699 and Coulomb in 1785, as is evident from reent works that onsider both atomisti point of view e.g., 12 14 and ontinuum mehanis prinipals e.g., 15 17. Tabor 18 in his general ritial piture of frition understanding pointed out three basi elements that are involved in the frition of dry solids: 1. The true area of ontat between mating rough surfaes. 2. The type and strength of bond formed at the interfae where ontat ours. 3. The way in whih the material in and around the ontating regions is sheared and ruptured during sliding. The importane of these three elements an be easily understood from the definition of the frition oeffiient, : Q max F Q max (1) PF s where Q max is the tangential fore needed to fail the juntions reated between the ontating surfaes, and F, the external fore, see Fig. 1 is the balane between the atual ontat load, P, in the true area of ontat and the amount of the intermoleular fores or the adhesion, F s, ating between the surfaes in ontat. The right hand side of Eq. 1 ontains all the three elements mentioned above. The ontat load, P, is related to the true area of 34 Õ Vol. 126, JANUARY 2004 Copyright 2004 by ASME Transations of the ASME
Fig. 1 The fored ating between ontating rough surfaes ontat. The adhesion, F s, is related to the strength of the bond formed at the interfae. The maximum tangential load, Q max,is related to the failure of the ontat. Chang et al. 19 presented a model CEB frition model for prediting the stati frition oeffiient of rough metalli surfaes based on the three elements indiated by Tabor 18. The CEB frition model uses a statistial representation of surfae roughness 20 and alulates the stati frition fore that is required to fail all of the ontating asperities, taking into aount their normal preloading. This approah is ompletely different from the lassial Coulomb frition law and shows that the latter is a limiting ase of a more general behavior where stati frition oeffiient atually dereases with an inreasing applied load or dereasing nominal ontat area. The CEB frition model atually treats the stati frition as a plasti yield failure mehanism orresponding to the first ourrene of plasti deformation in the ontating asperities. This an severely underestimate frition oeffiient values for ontating rough surfaes sine it neglets the ability of an elasti-plasti deformed asperity to resist additional loading before failure ours as was demonstrated reently by Kogut and Etsion 21. Roy Chowdhury and Ghosh 22 followed the same approah of the CEB frition model with additional adhesion related restrition on the maximum tangential load that an be arried by a single asperity. Etsion and Amit 23 demonstrated experimentally, for small normal loads and relatively smooth surfaes that the stati frition oeffiient dereases with inreasing normal loads as predited by the CEB frition model. Polyarpou and Etsion 24 extended the original CEB frition model to inlude the presene of sub-boundary lubriation. In a following paper 25 they ompared their model predition 24 with published experimental results and found good agreement. Liu et al. 26, in yet another extension of the CEB frition model, developed a stati frition model for the ase of rough surfaes in the presene of thin metalli films and ompared their theoretial results with experimental data in 27. The original CEB frition model 19 as well as its following extensions 24 and 26 alulate the stati frition oeffiient by using Eq. 1 where the ontat load, P, and adhesion fore, F s, are obtained from previous approximate models of Chang et al. 28 and 29, respetively. However, as was shown in a series reent works, 21,30,31, that are based on finite element analysis, the previous approximate models 19,28,29 produe large disrepanies on the single asperity level. As an be seen from the above literature survey, available tabulated values of stati frition oeffiient do not aount for suh important parameters as surfae roughness, surfae energy, mehanial properties and ontat load that have strong effet on the frition. An adequate theoretial model will eliminate the urrent need for extensive empirial work and will shed more light on understanding the dominant parameters affeting the stati frition oeffiient. The aforementioned approximate models for stati frition oeffiient assume failure of a ontating asperity as soon as the first plasti point appears, and hene, underestimate the atual frition fore. These models also rely on approximate ontat and adhesion solutions for a single asperity, that present large disrepanies with respet to reent finite element solutions. The present work relies on these finite element solutions for ontat, adhesion and frition, and hene, should improve the auray of the original CEB frition model. This remains to be verified by omparison with ontrolled experiments that will hopefully be presented in subsequent works. Analysis Figure 2 desribes shematially the geometry of the ontating rough surfaes. The two rough surfaes of Fig. 1 are replaed with a single equivalent rough surfae in ontat with a flat. The basi assumptions of Greenwood and Williamson 20 regarding the shape and statistial distribution of the asperities along with the transformation to the more pratial surfae height distribution see Nayak 32 are adopted in the present analysis. R is the uniform asperity radius of urvature, z and d denote the asperity height and separation of the surfaes, respetively, measured from the referene plane defined by the mean of the original asperity heights. The separation h is measured from the referene plane defined by the mean of the original surfae heights. (z) is the asperity height probability density funtion, assumed to be Gaussian: z 1 2 s exp 0.5 z 2 s (2) where s is the standard deviation of asperity heights. The interferene is defined as: zd (3) and only asperities with positive interferene are in ontat. During loading, the ontat load, P, adhesion fore, F s, and the stati frition fore, Q max, of eah individual asperity depend only on its own interferene,, assuming there is no interation between asperities. The dependene of P, F s and Q max on must be determined by the asperity mode of deformation, whih an be elasti, elasti-plasti or fully plasti. One these expressions are Fig. 2 Contat model of rough surfaes Journal of Tribology JANUARY 2004, Vol. 126 Õ 35
Table 1 The values of a, b, and for the various deformation regimes in Eqs. 10 to 12 Eq. 10 Eq. 11 Eq. 12 Deformation regime a b a b i a i b i Fully elasti, / 1 1 1.5 0.98 0.298 0.290 1 0.52 0.982 1st elasti-plasti, 1.03 1.425 0.79 0.356 0.321 1 0.01 4.425 1/ 6 2 0.09 3.425 3 0.40 2.425 4 0.85 1.425 2nd elasti-plasti, 6/ 110 1.40 1.263 1.19 0.093 0.332 N/A Fully plasti, / 110 3/K 1 N/A N/A known, the total ontat load, P, adhesion fore, F s, and stati frition fore, Q max, are obtained by summing the individual asperity ontributions using a statistial model: PA P zdzdz (4) nd F s A F szdzdz (5) n Q max A ndd6 Q max zdzdz (6) where A n is the nominal ontat area and is the area density of the asperities. The integrals in Eqs. 4 6 are solved in parts for the different deformation regimes of the ontating asperities. It should be noted that while the ontat load, P, and stati frition fore, Q max, are alulated for ontating asperities only, the adhesion fore, F s, is alulated also for non-ontating asperities, and hene, the differene 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 interferene is larger than 6.It should also be noted that Eqs. 4 6 are general in terms of the asperity height probability density funtion (z). Other non- Gaussian distributions an be used in these equations see e.g., 33. The ritial interferene,, that marks the transition from elasti to elasti-plasti deformation is given by see e.g., Chang et al. 28 KH 2 2E R (7) where H is the hardness of the softer material and K, the hardness oeffiient, is related to the Poisson s ratio of the softer material by see CEB frition model 19: K0.4540.41 E is the Hertz elasti modulus defined as: 1 E 1 2 1 1 2 2 E 1 where E 1, E 2 and 1, 2 are Young s moduli and Poisson s ratios of the ontating surfaes, respetively. All length dimensions are normalized by the standard deviation of the surfae heights,, and the dimensionless values are denoted by. Hene, y s is the differene between h and d Bush et al. 34 whih, after some algebra beomes: y s hd 1 (8) 48 where is a surfae roughness parameter defined as R E 2 (z) is the dimensionless asperity heights probability density funtion obtained from Eq. 2 by substituting the normalized length dimensions z/ and s /. The dimensionless ritial interferene,, is another form of the plastiity index, ( ) 1/2, that was first introdued by Greenwood and Williamson 20. It was shown in 35 that is the most important dimensionless parameter in elasti-plasti ontat problems of rough surfaes. It has the form: s 0.5 2E KH s 0.5 R and as an be seen it depends on surfae roughness and material properties. Rougher and softer surfaes have higher plastiity index. Kogut and Etsion 30 found that the entire elasti-plasti ontat regime of a single asperity extends over the range 1/ 110 with a transition at / 6 that divides it into two subregions. Dimensionless ontat parameters of a single asperity i.e., P /P, F s /F s0 and Q max /P were presented in 30,31, and 21, respetively, where P (2/3)KH R is the ritial ontat load at yielding ineption ( ), F s0 2R is the adhesion fore at point ontat (0) and is the energy of adhesion. These dimensionless ontat parameters an be expressed in the general form: F s a b F s0 P P a b F sn 4 / 2 3 / F s0 0.25 / 8 / ontating asperities, / 0 non-ontating asperities, / 0 Q max i P (9) (10) (11) (11a) a i b i (12) where is the intermoleular distane that is typially about 0.3 0.5 nm. The onstants a, b, and for the elasti, elasti-plasti in the two sub-regions, and plasti regimes are summarized in Table 1. Note that Eq. 12 is not appliable for / 6 see 21, and Eq. 11 is not appliable for / 110 see 31. The analyses in Refs. 30, 31, and 21 are all based on an assumption of elasti perfetly-plasti material behavior and hene, the present model is also adequate for suh materials. The dimensionless ontat load P, is obtained from Eqs. 4 and 10 see 35 in the form: 36 Õ Vol. 126, JANUARY 2004 Transations of the ASME
P P A n H 2 3 K d I d 1.5 d6 I 1.425 1.03d 1 d110 I 1.4d6 1.263 3 I (13) K d110 where I b is a general form of the integrand: I zd b b zdz (14) The four integrals in Eq. 13 and their orresponding limits of integration represent the ontribution of the asperities in elasti, elasti-plasti in the two sub-regions and fully plasti ontat, respetively. This methodology will be maintained in the following. Multiplying Eqs. 11 and (11a) byf s0 and using the values in Table 1, the adhesion fore, F s, of a single ontating asperity in the elasti and elasti-plasti regimes as well as F sn for a single nonontating asperity, an be obtained. Substituting in Eq. 5, and using the dimensionless form of Eq. 3 one an obtain the dimensionless adhesion fore, F s, between rough surfaes in the form: F s F s A n H 2 d d 0.298 Jn J0.29 0.98d d6 0.79d 0.356 d110 0.093 J0.321 J0.332 (15) 1.19d6 where J n aounts for the ontribution of the nonontating asperities and has the form: J n 4 8 3 dz 20.25 dz zdz (16) and J b is a general form of the integrands aounting for the ontribution of ontating asperities: J b zd b zdz (16a) The dimensionless adhesion parameter,, is: (17) H Note that ontribution of fully plasti asperities (/ 110) was not inluded in Eq. 15 in aordane with the observation made in Ref. 31. Multiplying Eq. 12 by P and using the values in Table 1, the stati frition fore, Q max, of a single asperity in the elasti and first elasti-plasti sub-region an be obtained. Substituting in Eq. 6, and following the same proedure that have lead to Eq. 15, The dimensionless stati frition fore is obtained in the form: Q max Q max A n H 2 3 K 0.52 d I 0.982 d d d6 0.01I 4.425 0.09I 3.425 0.4I 2.425 0.85I 1.425 (18) where I b is defined in Eq. 14. Equations 13, 15, and 18 an be transformed, by using Eqs. 8 and 9, to present P, F s and Q max as funtions of the more pratial parameters h and. Also, the dimensionless external fore F see Fig. 1 as a funtion of these parameters an be obtained in the form: F F A n H PF s (19) The stati frition oeffiient as defined in Eq. 1 maybeexpressed in the form: Q max F Q max (20) P1F s /P It should be noted here that other definitions for the frition oeffiient are found, e.g., 36. However, Eq. 1 seems to be the more pratial definition. Some insight regarding the role of the plastiity index in the stati frition problem an be gained by following the analysis in 35 for the ontat problem. With a Gaussian distribution of asperity heights the maximum pratial height of a given asperity is z3. Therefore, the integrals for the ontating asperities in Eqs. 13, 15 and 18 are pratially zero whenever their lower limit is higher than 3. Using the approximation 2 and noting that the relevant limits of integration have the general form d k 2 the ondition for meaningful ontribution of any of these integrals is: 1/2 k (21) 3d It is lear from Eq. 13 for example, that the ontribution of its last three integrals where k1) vanish for any d0 whenever 1/). Therefore, 0.6 an be defined as the plastiity index value below whih the ontat problem is fully elasti. Similarly, the last integral in Eq. 13 where k110) beomes appreiable only if 6. Hene, as was shown in 35, 8 indiates a fully plasti ontat. Following the same reasoning the last integral of Eq. 15 where k6) beomes inreasingly signifiant as beomes larger than &. Sine it was found in 31 that the adhesion fore of asperities with / 6 is negligible ompared to their ontat load, it is reasonable to expet negligibly small effet of F s /P in Eq. 20 when inreases above 1.4. Results and Disussion In aordane with the findings of 35 a wide range of plastiity index values from 0.5 to 8, was overed to analyze the effet of surfae roughness and material properties on the stati frition of ontating rough surfaes. A value of 0.04 was seleted aording to the finding of Greenwood and Williamson 20. A onstant value of K0.577 was used orresponding to a typial Poisson s ratio, 0.3, for metals. For typial values of adhesion energy, material hardness and surfae roughness the range of the adhesion parameter,, is10 4 0.01 where the upper limit orresponds to very high adhesion energy that an be obtained with very lean surfaes under vauum onditions. The numerial results of Eqs. 13, 15, and 18 to 20 for any given h and, an be ross-plotted to provide a pratial presentation of the relevant parameters vs. the known external applied fore F. Following the reasoning of 35 only the results for the range of pratial engineering interest namely, 0h3 and F0.1, will be presented. Lower h and higher F values may invalidate the basi assumptions of no interation between neighboring asperities and no bulk deformation, respetively see 20. From Eq. 20 it is lear that the effet of adhesion on the stati frition oeffiient depends on the ratio F s /P and this effet beomes negligible when F s /P1. Journal of Tribology JANUARY 2004, Vol. 126 Õ 37
Fig. 3 Dimensionless fore ratio, FÕP, s asafuntionofthe dimensionless external fore, F, for various values of the plastiity index, at Ä0.003 Fig. 4 Dimensionless stati frition fore, Q max,asafuntion of the dimensionless external fore, F, for various values of the plastiity index, at Ä0.003 Figure 3 presents the ratio F s /P vs. the dimensionless external fore, F, for the range of the plastiity index, and for a relatively high value of the adhesion parameter 0.003. This high value of was seleted to failitate the distintion of the effet of at its higher values where F s /P may beome very small. Note that the ratio F s /P depends linearly on see Eq. 15 and, hene, it an be easily dedued for values of different than 0.003 from the results shown in Fig. 3. As an be seen from Fig. 3 the ratio F s /P dereases sharply with inreasing plastiity index. For 2, F s /P beomes less than 0.11 throughout the range of F even for the high value of 0.003. Hene, for 2 and more pratial smaller values of the adhesion parameter, it an be onluded that PF is a reasonable approximation see Eq. 19 and the effet of adhesion on the stati frition oeffiient is negligible. In ontrast, the ratio F s /P is signifiant at low plastiity index, 0.5, over most of the range of F, and beomes small enough only at the upper limit of F. For 1 the ratio F s /P beomes less than 0.1 for external fore higher than a threshold value of F0.01. It an, therefore, be onluded that the effet of adhesion is important only in purely elasti ontats where 0.6, or in lightly loaded ontats with plastiity index up to 1 and high adhesion parameter 0.001. Hene, whenever 0.001 or 2 the effet of adhesion on the stati frition an be safely negleted. Figure 4 presents the dimensionless stati frition fore, Q max, versus the dimensionless external fore, F, for various values of the plastiity index,, when 0.003. It an be seen that at a given external fore, the stati frition fore dereases with inreasing plastiity index. At higher plastiity index the ontat is more plasti and a larger population of the ontating asperities undergo interferene in the range / 6, where aording to the finding in 21 they are unable to support any tangential load and hene, do not ontribute to the stati frition. Inreasing the external fore at a given plastiity index also inreases the number of suh high interferene asperities but at the same time brings into ontat more asperities that were initially nonontating. It turns out that the latter effet is more dominant, and, hene, auses an inrease of the stati frition fore with inreasing external fore. This behavior of the stati frition fore is different from that in the ase of a single asperity 21 where the stati frition fore first inreases with inreasing normal load, reahes a maximum and than starts dereasing. Reduing the adhesion parameter redues somewhat the stati frition at 0.5 and low external fore but otherwise has very little effet on the results shown in Fig. 4, in aordane with the negligible effet of the adhesion over most of the pratial range of F as was shown in Fig. 3. Note the log/log sale used in Fig. 4 showing almost linear relation having the general form: Q max CF m (22) This relation differs from the lassial law of frition, Q max F, whenever m1. Indeed in Fig. 4 the power m is less than 1 and varies from m0.82 at 2 tom0.86 at 0.5, indiating a smaller rate of inrease of the frition fore ompared to that of the external fore as more asperities are brought into ontat. As shown in Fig. 4 at the highest plastiity index, 8, the stati frition fore is extremely small being between 3 to 4 orders of magnitude smaller than the external fore. This is a result of the ontat beoming fully plasti, see 35, where large perentages of the ontating asperities undergo interferenes muh higher than / 6. Suh small frition fore at high plastiity index seems unreasonable. It may be attributed to some of the simplifying assumptions made in Ref. 30 namely, an elasti perfetlyplasti behavior of the materials that neglets more realisti strain hardening effets. In addition, Mesarovi and Flek 37 presented a finite element analysis that shows a derease in the mean ontat pressure of a single asperity under very high normal loads and extreme interferene deep into the fully plasti regime. As a result suh an asperity may regain its ability to resist a finite tangential load and thereby ontribute to the stati frition of highly plasti ontating rough surfaes. The present model, not showing these effets, may be invalid at large plastiity index values. Figure 5 presents the stati frition oeffiient,, see Eq. 20 versus the dimensionless external fore, F, for low and medium values of the plastiity index,, and 0.003. Inreasing the plastiity index, at a given external fore, dereases the frition oeffiient, similar to the behavior of the frition fore as shown in Fig. 4. However, in ontrast to the behavior of the frition fore, inreasing the external fore, at a given plastiity index, dereases the stati frition oeffiient. This an be easily understood from substituting Eq. 22 in the expression for the stati frition oeffiient Q max /F, whih results in: CF m1 CF n (23) Sine m is less than 1 we an see from Eq. 23 that dereases with inreasing external fore. Etsion and Amit 23 investigated 38 Õ Vol. 126, JANUARY 2004 Transations of the ASME
Fig. 5 Stati frition oeffiient,, as a funtion of the dimensionless external fore, F, for various values of the plastiity index, at Ä0.003 Fig. 6 Stati frition oeffiient,, as a funtion of the dimensionless external fore, F, for various values of the plastiity index,, and the dimensionless adhesion parameter, experimentally the effet of external load on the stati frition oeffiient between aluminum alloy pins and a nikel oated disk. They found for plastiity index values ranging from 0.67 to 1.01 that the power n in Eq. 23 has values between 0.102 and 0.130, respetively. The orresponding values of n obtained from Fig. 5 for the range of plastiity index values between 0.5 and 2 are between 0.09 to 0.13 for 0.001. This is a fair agreement onsidering the unknown exat value in the experiment. It should be noted that as F approahes zero the stati frition oeffiient may beome 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 frition model 19. As an be seen this approximate model substantially underestimates the stati frition oeffiient and already at 2 predits unrealisti small values. This is due to a restritive assumption made in 19, that asperities with / 1 are unable to support any tangential load, ausing severe underestimation of the stati frition oeffiient at plastiity index values above 0.6. Another assumption that was made in the CEB frition model 19 is that the elastially preloaded asperities having / 1 annot support tangential loads higher than that ausing the onset of plastiity. This assumption an severely underestimate the maximum tangential load that an be supported by these asperities as was demonstrated in 21, and is responsible for the lower stati frition oeffiient that is predited by the CEB model even at 0.5 Figure 6 shows the effet of the adhesion parameter,, onthe stati frition oeffiient,. It an be seen that for a low plastiity index, 0.5, reduing from the high value of 0.003 to 0.001 a three fold redution of the adhesion fore redues substantially the stati frition oeffiient at a given external fore at the lower end of that fore. This effet diminishes as the external fore inreases and beomes negligibly small at the upper limit of the external fore. A further redution of the adhesion parameter to 0.0001 has a muh smaller effet sine the adhesion beomes negligible anyway see disussion of Fig. 3. The effet of on disappears at the higher plastiity index 2, sine in this ase too the adhesion fore is negligible. High adhesion fore dereases the separation, h, at a given external fore and brings more asperities into ontat espeially when the external fore is small, thus enabling to support larger tangential fore, and, hene, the stati frition fore and frition oeffiient inrease with inreasing. From Figs. 5 and 6 it an be seen that the frition oeffiient depends on the dimensionless external fore, F, i.e. on the external fore as well as on the nominal ontat area see Eq. 19. This later dependeny is due to the effet of A n on the separation d see 35 that appears in the integrals of Eqs. 13, 15, and 18, whih are then substituted in Eq. 20. Additionally, the stati frition oeffiient depends on mehanial properties and surfae roughness through and on the adhesion energy through. This is quite different from the lassial laws of frition. As the plastiity index inreases the stati frition oeffiient beomes muh less sensitive to these parameters, similar to the teahing of the lassial laws of frition. Hene, the lassial Coulomb frition law whih was obtained some 300 years ago presumably for high and low values an be regarded as a limiting ase of the more general model presented in this work. Conlusion A model that predits the stati frition for elasti-plasti ontat of rough surfaes was presented. It inorporates the results of aurate finite element analyses for the elasti-plasti ontat, adhesion and sliding ineption of a single asperity in a statistial representation of surfae roughness. Strong effet of the external fore and nominal ontat area on the stati frition oeffiient was found in ontrast to the lassial laws of frition. The main dimensionless parameters affeting the stati frition oeffiient are the plastiity index and adhesion parameter. The effet of adhesion on the stati frition was found to be negligible at plastiity index values larger than 2 throughout the pratial external fore range that was investigated regardless of. At plastiity index values lower than 1 adhesion may be important if 0.001 and the external fore is not too large. The present model that assumes elasti perfetly-plasti material behavior may be invalid at high plastiity index values where the ontat approahes fully plasti state. Unreasonably small stati frition was found under this ondition and an improved model that onsiders strain hardening effets and possible juntion growths may be required. It was shown that the lassial laws of frition are a limiting ase of the present more general solution and are adequate only for high plastiity index and negligible adhesion. A omparison of the present results with those obtained from an approximate CEB frition model showed substantial differenes with the latter severely underestimating the stati frition oeffiient. Journal of Tribology JANUARY 2004, Vol. 126 Õ 39
Aknowledgment This researh was supported in parts by the Fund for the Promotion of Researh at the Tehnion, by the J. and S. Frankel Researh Fund and by the German-Israeli Projet Cooperation DIP. Nomenlature d separation based on asperity heights d dimensionless separation, d/ E Hertz elasti modulus F external fore F dimensionless external fore, F/A n H F s adhesion fore F s dimensionless adhesion fore, F s /A n H H hardness of the softer material h separation based on surfae heights h dimensionless separation, h/ K hardness fator, 0.4540.41 P ontat load P dimensionless ontat load, P/A n H Q frition fore Q dimensionless frition fore, Q/A n H R asperity radius of urvature y s hd z height of an asperity measured from the mean of asperity heights z dimensionless height of an asperity, z/ surfae roughness parameter, R energy of adhesion dimensionless distribution funtion of asperity heights area density of asperities stati frition oeffiient Poisson s ratio of the softer material dimensionless adhesion parameter, /H standard deviation of surfae heights s standard deviation of asperity heights dimensionless interferene ritial interferene at the ineption of plasti deformation dimensionless ritial interferene, / plastiity index, Eq. 9 Subsripts yielding ineption Supersripts single asperity Referenes 1 Karamis, M. 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