Hamrock, Simplified Solution for Elliptical Contact Deformation Between Two Elastic Solids, Transactions ASME, Journal of Lubrication Technology, Vol.. Dowson, Isothermal Elastohydrodyna
Trang 1temperature [56] and this characteristic cannot be neglected in any model of EHL with slidingpresent.
EHL Between Meshing Gear Wheels
From the view point of practical engineering an important EHL contact takes place betweenthe lubricated teeth of opposing gears As is the case with rolling bearings, it is essential tomaintain an adequate EHL film thickness to prevent wear and pitting of the gear teeth Thesame fundamental equations for EHL film thickness described for a simple Hertzian contactalso apply for gears However, before applying the formulae for contact parameters andminimum film thickness it is necessary to define reduced radius of curvature, contact loadand surface velocity for a specific gear The contact geometry is illustrated in Figure 7.42
C 2
R B
R A
R A sinψ + S
W
W
h B
h A
FIGURE 7.42 Contact geometry of meshing involute gear teeth
The surface contact velocity is expressed as:
U= U A + U B
2 = ωA R A sinψ + ωB R B sinψ
2
where:
R A , R B are the pitch circle radii of the driver and follower respectively [m];
ψ is the pressure angle (acute angle between contact normal and the common
tangent to the pitch circles);
ωA, ωB are the angular velocities of the driver and follower respectively [rad/s]
Trang 2W is the total load on the tooth [N];
h B is the distance from the centre of the follower to interception of the locus of the
contact with its base circle, i.e h B = R B cosψ [m];
T B is the torque exerted on the follower [Nm]
The torque exerted on the driver and the follower expressed in terms of the transmittedpower is calculated from:
N A , N B are the rotational speeds of the driver and follower respectively [rps];
H is the transmitted power [kW]
Substituting into (7.54) yields the contact load The minimum and central EHL filmthicknesses can then be calculated from formulae (7.26) and (7.27)
The line from ‘C 1 ’ to ‘C 2’ (Figure 7.42) is the locus of the contact and it can be seen that the
distance ‘S’ between the gear teeth contact and the pitch line is continuously changing with
the contact position during the meshing cycle of the gears It is thus possible to model anyspecific contact position on the tooth surface of an involute gear by two rotating circular discs
of radii (R A sin Ψ + S) and (R B sin Ψ - S) as shown in Figure 7.42 This idea is applied in a testing apparatus generally known as a ‘twin disc’ or ‘two disc‘ machine shown schematically
in Figure 7.43 Since the gear tooth contact is closely simulated by the two rotating discs, thesemachines are widely used to model gear lubrication and wear and in selecting lubricants ormaterials for gears It is much cheaper and more convenient experimentally to use metaldiscs instead of actual gears for friction and wear testing The wear testing virtually ensuresthe destruction of the test specimens and it is far easier to inspect and analyse a worn discsurface than the recessed surface of a gear wheel
It may also be apparent that the fixed dimensions of the discs only allow modelling of oneparticular position in the contact cycle Of particular importance to friction and wear studies
is the increasing amount of sliding as the contact between opposing gear teeth moves awayfrom the line of shaft centres The radii of curvature also vary with position of gear teeth sothat the ‘two-disc’ test rig is not entirely satisfactory and another model gear apparatus such
as the ‘Ryder gear tester’ may be necessary for some studies A recently developed apparatus where two contacting discs are supplied with additional movement of theircorresponding shafts allows a much closer, more realistic simulation of the entire gear toothcontact cycle [68]
Trang 3FIGURE 7.43 Schematic diagram of a ‘two disc‘ machine used to simulate rolling/sliding
contact in meshing gears, i.e.: for S = 0 pure rolling and for S ≠ 0 rolling/sliding
in EHL contact; S is the distance between the pitch line and the gear teeth
contact [m]
A fundamental lubrication mechanism involved in highly loaded concentrated contacts wasdiscussed in this chapter The remarkable efficiency of elastohydrodynamic lubrication inpreventing solid to solid contact even under extreme contact stresses prevents the rapiddestruction of many basic mechanical components such as rolling bearings or gears EHL is,however, mostly confined to mineral or synthetic oils since it is essential that the lubricant ispiezo-viscous The mechanism of EHL involves a rapid change in the lubricant from anearly ideal liquid state outside of the contact to an extremely viscous or semi-solid statewithin the contact This transformation allows the lubricant to be drawn into the contact byviscous drag while generating sufficient contact stress within the contact to separate theopposing surfaces If a simple solid, i.e a fine powder, is supplied instead, there is no viscousdrag to entrain the powder and consequently only poor lubrication results A non-piezo-viscous lubricant simply does not achieve the required high viscosity within the contactnecessary for the formation of the lubricating film The formulae for the calculation of theEHL film thickness are relatively simple and are based on load, velocity, dimensions andelastic modulus of the contacting materials As well as providing lubrication of concentratedcontacts, the EHL mechanism can be used to generate traction, i.e where frictional forcesenable power transmission A unique combination of high tractive force with minimal wear,reduced noise levels, infinitely variable output speed and an almost constant torque over thespeed range can be obtained by this means
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Trang 4ELASTOHYDRODYNAMIC LUBRICATION 353
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21 A Dyson, H Naylor and A.R Wilson, The Measurement of Oil Film Thickness in Elastohydrodynamic
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22 L.B Sibley, J.C Bell, F.K Orcutt and C.M Allen, A Study of the Influence of Lubricant Properties on the Performance of Aircraft Gas Engine Rolling Contact Bearings, WADD Technical Report, 1960, pp 60-189.
23 L.B Sibley and A.E Austin, An X-Ray Method for Measuring Thin Lubricant Films Between Rollers, ISA
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24 D.R Meyer and C.C Wilson, Measurement of Elastohydrodynamic Oil Film Thickness and Wear in Ball
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25 A.T Kirk, Hydrodynamic Lubrication of Perspex, Nature, Vol 194, 1962, pp 965-966.
26 A Cameron and R Gohar, Theoretical and Experimental Studies of the Oil Film in Lubricated Point
Contacts, Proc Roy Soc., London, Series A, Vol 291, 1966, pp 520-536.
27 N Thorp and R Gohar, Oil Film Thickness and Shape for Ball Sliding in a Grooved Raceway, Transactions
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28 D Dowson, Recent Developments in Studies of Fluid Film Lubrication, Proc Int Tribology Conference, Melbourne, The Institution of Engineers, Australia, National Conference Publication No 87/18, December,
1987, pp 353-359.
29 T.E Tallian, On Competing Failure Modes in Rolling Contact, ASLE Transactions, Vol 10, 1967, pp 418-439.
30 K.L Johnson, J.A Greenwood and S.Y Poon, A Simple Theory of Asperity Contact in Elastohydrodynamic
Lubrication, Wear, Vol 19, 1972, pp 91-108.
31 J.A Greenwood and J.B.P Williamson, Contact of Nominally Flat Surfaces, Proc Roy Soc., London, Series A,
Vol 295, 1966, pp 300-319.
32 T.E Tallian and J.I McCool, An Engineering Model of Spalling Fatigue Failure in Rolling Contact, II The
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Hertzian Contacts, Tribology International, Vol 14, 1981, pp 315-322.
34 G.M.S De Silva, J.A Leather and R.S Sayles, The Influence of Surface Topography on Lubricant Film Thickness in EHD Point Contact, Proc 12th Leeds-Lyon Symp on Tribology, Mechanisms and Surface Distress:
Global Studies of Mechanisms and Local Analyses of Surface Distress Phenomena, editors: D Dowson, C.M.
Taylor, M Godet and D Berthe, Sept 1985, Inst Mech Engrs Publ., London, 1986, pp 258-272.
35 N Patir and H.S Cheng, Effect of Surface Roughness Orientation on the Central Film Thickness in EHD Contacts, Proc 5th Leeds-Lyon Symp on Tribology, Elastohydrodynamics and Related Topics, editors: D Dowson, C.M Taylor, M Godet and D Berthe, Sept 1978, Inst Mech Engrs Publ., London, 1979, pp 15-21.
36 H.S Cheng, On Aspects of Microelastohydrodynamic Lubrication, Proc 4th Leeds-Lyon Symp on Tribology, Surface Roughness Effects in Lubrication, editors: D Dowson, C.M Taylor, M Godet and D Berthe, Sept.
1977, Inst Mech Engrs Publ., London, 1978, pp 71-79.
37 X Ai and L Zheng, A General Model for Microelastohydrodynamic Lubrication and its Full Numerical
Solution, Transactions ASME, Journal of Tribology, Vol 111, 1989, pp 569-576.
38 P Goglia, T.F Conry and C Cusano, The Effects of Surface Irregularities on the Elastohydrodynamic
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Part 1, pp 104-112, Part 2, pp 113-119.
39 C.C Kweh, H.P Evans and R.W Snidle, Micro-Elastohydrodynamic Lubrication of an Elliptical Contact
With Transverse and 3-D Sinusoidal Roughness, Transactions ASME, Journal of Tribology, Vol 111, 1989, pp.
577-584.
40 L Chang and M.N Webster, A Study of Elastohydrodynamic Lubrication of Rough Surfaces, Transactions
ASME, Journal of Tribology, Vol 113, 1991, pp 110-115.
41 L.G Houpert and B.J Hamrock, EHD Lubrication Calculation Used as a Tool to Study Scuffing, Proc 12th Leeds-Lyon Symp on Tribology, Mechanisms and Surface Distress: Global Studies of Mechanisms and Local
Analyses of Surface Distress Phenomena, editors: D Dowson, C.M Taylor, M Godet and D Berthe, Sept.
1985, Inst Mech Engrs Publ., London, 1986, pp 146-155.
42 K.P Baglin, EHD Pressure Rippling in Cylinders Finished With a Circumferential Lay, Proc Inst Mech.
Engrs, Vol 200, 1986, pp 335-347.
43 B Michau, D Berthe and M Godet, Influence of Pressure Modulation in Line Hertzian Contact on the Internal
Stress Field, Wear, Vol 28, 1974, pp 187-195.
44 J.F Archard and R.A Rowntree, The Temperature of Rubbing Bodies, Part 2, The Distribution of Temperature,
Wear, Vol 128, 1988, pp 1-17.
45 F.P Bowden and D Tabor, Friction and Lubricating Wear of Solids, Part 1, Oxford: Clarendon Press, 1964.
46 H Blok, Theoretical Study of Temperature Rise at Surfaces of Actual Contact Under Oiliness Lubricating
Conditions, General Discussion on Lubrication, Inst Mech Engrs, London, Vol 2, 1937, pp 222-235.
47 J.C Jaeger, Moving Sources of Heat and the Temperature at Sliding Contacts, Proc Roy Soc., N.S.W., Vol 76,
1943, pp 203-224.
48 J.F Archard, The Temperature of Rubbing Surfaces, Wear, Vol 2, 1958/59, pp 438-455.
49 F.E Kennedy, Thermal and Thermomechanical Effects in Dry Sliding, Wear, Vol 100, 1984, pp 453-476.
50 H Blok, The Postulate About the Constancy of Scoring Temperature, Interdisciplinary Approach to Lubrication of Concentrated Contacts, P.M Ku (ed.), Washington DC, Scientific and Technical Information Division, NASA, 1970, pp 153-248.
51 T.A Stolarski, Tribology in Machine Design, Heineman Newnes, 1990.
52 V.K Ausherman, H.S Nagaraj, D.M Sanborn and W.O Winer, Infrared Temperature Mapping in
Elastohydrodynamic Lubrication, Transactions ASME, Journal of Lubrication Technology, Vol 98, 1976, pp.
236-243.
53 V.W King and J.L Lauer, Temperature Gradients Through EHD Films and Molecular Alignment Evidenced
by Infrared Spectroscopy, Transactions ASME, Journal of Lubrication Technology, Vol 103, 1981, pp 65-73.
54 A.R Wilson, An Experimental Thermal Correction for Predicted Oil Film Thickness in Elastohydrodynamic Contacts, Proc 6th Leeds-Lyon Symp on Tribology, Thermal Effects in Tribology, Sept 1979, editors: D Dowson, C.M Taylor, M Godet and D Berthe, Inst Mech Engrs Publ., London, 1980, pp 179-190.
55 J.L Tevaarwerk, Traction Calculations Using the Shear Plane Hypothesis, Proc 6th Leeds-Lyon Symp on Tribology, Thermal Effects in Tribology, Sept 1979, editors: D Dowson, C.M Taylor, M Godet and D Berthe, Inst Mech Engrs Publ., London, 1980, pp 201-215.
Trang 6ELASTOHYDRODYNAMIC LUBRICATION 355
56 H.A Spikes and P.M Cann, The Influence of Sliding Speed and Lubricant Shear Stress on EHD Contact
Temperatures, Tribology Transactions, Vol 33, 1990, pp 355-362.
57 W.O Winer and E.H Kool, Simultaneous Temperature Mapping and Traction Measurements in EHD Contacts, Proc 6th Leeds-Lyon Symp on Tribology, Thermal Effects in Tribology, Sept 1979, editors: D Dowson, C.M Taylor, M Godet and D Berthe, Inst Mech Engrs Publ., London, 1980, pp 191-200.
58 T.A Dow and W Kannel, Evaluation of Rolling/Sliding EHD Temperatures, Proc 6th Leeds-Lyon Symp on Tribology, Thermal Effects in Tribology, Sept 1979, editors: D Dowson, C.M Taylor, M Godet and D Berthe, Inst Mech Engrs Publ., London, 1980, pp 228-240.
59 K.L Johnson and J.A Greenwood, Thermal Analysis of an Eyring Fluid in Elastohydrodynamic Traction,
Wear, Vol 61, 1980, pp 353-374.
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Conditions by Optical and Infra Red Spectroscopic Methods, Tribology Transactions, Vol 31, 1988, pp
120-127.
61 P.M Cann and H.A Spikes, In Lubro Studies of Lubricants in EHD Contacts Using FITR Absorption
Spectroscopy, Tribology Transactions, Vol 34, 1991, pp 248-256.
62 F.L Snyder, J L Tevaarwerk and J A Schey, Effects of Oil Additives on Lubricant Film Thickness and Traction, SAE Tech Paper No 840263, 1984.
64 M Alsaad, S Bair, D.M Sanborn and W.O Winer, Glass Transitions in Lubricants: Its Relation to
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63 S Bair and W.O Winer, Some Observations in High Pressure Rheology of Lubricants, Transactions ASME,
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65 M Kaneta, H Nishikawa and K Kameishi, Observation of Wall Slip in Elastohydrodynamic Lubrication,
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249-264.
67 K.L Johnson and J.L Tevaarwerk, Shear Behaviour of Elastohydrodynamic Oil Films, Proc Roy Soc.,
London, Series A, Vol 356, 1977, pp 215-236.
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69 C.A Foord, W.C Hammann and A Cameron, Evaluation of Lubricants Using Optical Elastohydrodynamics,
ASLE Transactions, Vol 11, 1968, pp 31-43.
70 P.L Wong, P.Huang, W Wang and Z Zhang, Effect of Geometry Change of Rough Point Contact Due to
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J, Journal of Engineering Tribology, Vol 213, 1999, pp 417-426.
Trang 8discussed in this chapter The traditional name for this type of lubrication is ‘boundary lubrication’ or ‘boundary and extreme-pressure lubrication’ Neither of these terms describe
accurately the processes at work since they were conceived long before any fundamentalunderstanding of the mechanisms was available Several specialized modes of lubricationsuch as adsorption, surface localized viscosity enhancement, amorphous layers and sacrificialfilms are commonly involved in this lubrication regime to ensure the smooth-functioningand reliability of machinery The imprecise nature of present knowledge about these modes
or mechanisms of lubrication contrasts with their practical importance Many vital items ofengineering equipment such as steel gears, piston-rings and metal-working tools depend onone or more of these lubrication modes to prevent severe wear or high coefficients of frictionand seizure
Boundary and E.P lubrication is a complex phenomenon The lubrication mechanismsinvolved can be classified in terms of relative load capacity and limiting frictionaltemperature as shown in Table 8.1, and they will be described in this chapter
These lubrication mechanisms are usually controlled by additives present in the oil Sincethe cost of a lubricant additive is usually negligible compared to the value of the mechanicalequipment, the commercial benefits involved in this type of lubrication can be quite large
In general, boundary and E.P lubrication involves the formation of low friction, protectivelayers on the wearing surfaces One exception is when the surface-localized viscosityenhancement takes place The occurrence of surface-localized viscosity enhancement,however, is extremely limited as is explained in the next section
The operating principle of the boundary lubrication regime can perhaps be best illustrated by
considering the coefficient of friction In simple terms the coefficient of friction ‘µ’ is defined
as the ratio of frictional force ‘F’ and the load applied normal to the surface ‘W’, i.e.:
Trang 9TABLE 8.1 Categories of boundary and E.P lubrication.
Temperature Load Lubrication mechanisms
Low
Low High
Surface-localized viscosity enhancement specific to lubricant additive and basestock.
Formation of amorphous layers of finely divided debris from reaction between additives and substrate metal surface.
Reaction between lubricant additives and metal surface.
Formation of sacrificial films of inorganic material on the worn surface preventing metallic contact and severe wear.
Since the contacting surfaces are covered by asperities, ‘dry’ contact is established between theindividual asperities and the ‘true’ total contact area is the sum of the individual contactareas between the asperities Assuming that the major component of the frictional force isdue to adhesion between the asperities (other effects, e.g ploughing, are negligible), then the
expression for frictional force ‘F’ can be written as:
F = A tτ
where:
F is the frictional force [N];
A t is the true contact area [m2];
τ is the effective shear stress of the material [Pa]
Applied load can be expressed in terms of contact area, i.e.:
‘τ’ and ‘py’ does not vary greatly, changing the material type has little effect on friction
Trang 10BOUNDARY AND EXTREME PRESSURE LUBRICATION 359
8.2 LOW TEMPERATURE - LOW LOAD LUBRICATION MECHANISMS
For a very large range of sliding speeds and loads, classical hydrodynamic lubrication prevails
in a lubricated contact As the sliding speed is reduced, hydrodynamic lubrication reaches itslimit where the hydrodynamic film thickness declines until eventually the asperities of theopposing surfaces interact This process was originally investigated by Stribeck and hasalready been discussed in Chapter 4
At low speeds, under certain conditions, contact between opposing surfaces can be prevented
by the mechanism involving surface-localized viscosity enhancement In other words, a thinlayer of liquid with an anomalously high viscosity can form on the contacting surfaces.Hydrodynamic lubrication or quasi-hydrodynamic lubrication then persists to prevent solidcontact and severe wear In such cases linear molecules of a hydrocarbon align themselvesnormally to the contacting surfaces to form a lubricating, protective layer as shown in Figure8.1 Since the molecules are polar the opposite ends are attracted to form pairs of moleculeswhich are subsequently incorporated into the viscous surface layer At the interface with themetallic substrate the attractive force of the free end of the molecules to the substrate issufficient to firmly bond the entire layer
FIGURE 8.1 Low-temperature, low-load mechanism of lubrication [1]
It has been found that linear molecules are more effective than other hydrocarbons inpreventing solid contact The variation in film thickness between parallel discs as a function
of the square root of squeeze time for paraffinic oil and cyclohexane is shown in Figure 8.2 [2].According to the theory of hydrodynamic lubrication described in Chapter 4, there is a lineardecline in film thickness with square root of squeeze time but as can be seen from Figure 8.2this linearity is soon lost The MS-20 oil contains paraffinic molecules which areapproximately linear and this allows for the formation and persistence of a thicker film thanfor cyclohexane Cyclohexane is a non-linear molecule which impedes the linear alignment
of molecules and therefore the resulting film is less effective in preventing solid contact.The effectiveness of this mechanism of lubrication is limited to low temperatures and lowloads The data shown in Figure 8.2 was obtained at contact pressures of 0.4 [MPa], and furtherwork revealed that at contact pressures beyond 2 [MPa] the residual film thickness is verysmall [2] Since in many contacts pressures in the range of 1 [GPa] are quite common, the
Trang 11disadvantages of this lubrication mechanism are obvious The temperature also has apronounced effect on these films It was found that even relatively low operatingtemperatures of about 50°C can result in severe decline in the film thickness [2] Since thepractical applications in which this mode of boundary lubrication occurs are extremelylimited, the topic does not incite much technological interest and has consequently beenneglected by most researchers.
0 0.1 0.2 0.3 0.4
FIGURE 8.2 Detection of permanent films formation as evidence of a surface-proximal layer
of aligned molecules [2]
8.3 LOW TEMPERATURE - HIGH LOAD LUBRICATION MECHANISMS
The lubrication mechanism acting at low temperature and high load is of considerable
practical importance It is generally known as ‘adsorption lubrication’ This mechanism of lubrication is quite effective with contact pressures up to 1 [GPa] and relatively low surface temperatures between 100 - 150°C Adsorption lubrication is different from either
hydrodynamic, EHL or even the viscous layer described in the previous section in that theopposing contact surfaces are not separated by a thick layer of fluid A mono-molecular layerseparates the contacting surfaces and this layer is so thin that the mechanics of asperitycontact are identical to that of dry surfaces in contact This mono-molecular layer is formed
by adsorption of the lubricant or, more precisely, lubricant additives on the worn surface Thelubricating effect or friction reduction is caused by the formation of a low shear strengthinterface between the opposing surfaces
It can be seen from equation (8.2) that the role of adsorption lubrication is to reduce theeffective shear stress ‘τ’ at the interface without affecting the plastic flow stress ‘py’ of thesubstrate This is achieved by the formation of an adsorbed film on the surface whichintroduces a plane of weakness parallel to the plane of sliding This principle is illustrated inFigure 8.3 which shows a schematic comparison of contact between dry unlubricated solidsand solids with a lubricant film on asperity peaks
If the film is thin, then any structural weakness in the direction of the contact load will becompensated by the substrate The shear stress anisotropy or low shear stress in the plane ofsliding is obtained by inducing a discontinuity in intermolecular bonding between opposite
Trang 12BOUNDARY AND EXTREME PRESSURE LUBRICATION 361
sides of the sliding interface At all locations other than the interface, bonding between atomseven of different substances, e.g film material and substrate, is relatively strong Thecharacteristics of adsorbed layers, in particular of polar organic substances, allow this system
to form on metallic surfaces The reasons for this are discussed next
Ιnterfacial shear strength
of the surface layer
Small shear stress at interfaces
Contact stresses remain unaffected
Interfacial shear strength
same as substrate
material
Direct contact between
clean (dry) surfaces
Surfaces with low interfacial shear stress
FIGURE 8.3 Lubrication by a low shear strength layer formed at asperity peaks
Model of Adsorption on Sliding Surfaces
Organic polar molecules such as fatty acids and alcohols adsorb on to metallic surfaces andare not easily removed Speculation about the role of these substances in lubrication has along history [3] Effective adsorption is the reason why a metallic surface still feels greasy orslippery after being wetted by a fatty substance and will remain greasy even after vigorouswiping of the surface by a dry cloth Adsorption on a metallic surface of organic polarmolecules produces a low friction, mono-molecular layer on the surface as shown in Figure8.4 The polarity of the adsorbate is essential to the lubrication mechanism Polarity meansthat a molecule is asymmetrical with a different chemical affinity at either end of themolecule For example, one end of a molecule which is the carboxyl group of a fatty acid,
‘-COOH’, is strongly attracted to the metallic surface while the other end which is an alkyl group, ‘-CH 3’, is repellant to almost any other substance
Strong adsorption ensures that almost every available surface site is occupied by the fatty acid
to produce a dense and robust film The repulsion or weak bonding between the contactingalkyl groups ensures that the shear strength of the interface is relatively low The ratio of τ/pyand therefore the friction coefficient is low compared to bare metallic surfaces in contact This
is the adsorption model of lubrication first postulated by Hardy and Doubleday [4,5] and laterdeveloped by Bowden and Tabor [6] The fatty acids are particularly effective because of theirstrong polarity, but other organic compounds such as alcohols and amines have sufficientpolarity to be of practical use