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The breakdown of fluid film lubrication in elastic-isoviscous point contacts
25
Wear, 63 (1980) 25 - 40 0 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
THE BREAKDOWN OF FLUID FILM LUBRICATION ISOVISCOUS POINT CONTACTS*
IN ELASTIC-
J. B. MEDLEY and A. B. STRONG Department (Canada)
of Mechanical
Engineering,
University of Waterloo, Waterloo, Ontario
R. M. PILLIAR and E. W. WONG Department
of Metallurgy, Ontario Research Foundation,
Mississauga, Ontario (Canada)
(Received September 24,1979)
Summary Methods have been developed to determine the initiation of film breakdown experimentally. In these experiments the lubricated sliding of a smooth elastomeric surface over a rigid metal sphere is investigated using friction measurements. Specific friction increases are shown to correspond to fluid film breakdown and are related to film thickness values by considering surface roughness in the contact. Within experimental uncertainty these film thickness estimates agree with recent theoretical and experimental work reported in the literature.
1. Introduction One classical method of using friction measurements to investigate lubrication conditions in a journal bearing involves experimentally determining the relation between the friction coefficient and the Sommerfeld number. Nouh [l] and Moore [2, 31 defined an equivalent Sommerfeld number for point contacts and further modified it to include viscoelastic effects of an elastomeric contact surface to produce an “elastohydrodynamic” number defined as follows: EH =-
E*q,VR qoF
Nouh [l] examined the sliding of metal spheres with specified surface roughness over smooth lubricated rubber. Characteristic curves were produced for a generalized friction coefficient (fa = f/tan 6) uersus the *The work reported here was done as part of J. B. Medley’s M.A.Sc. thesis at the University of Waterloo.
(1)
26
elastohydrodynamic number. The rapid drop of the generalized friction coefficient as the elastohydrodyn~ic number increased beyond a critical value was interpreted as a t~sition from boundary to fluid film lubrication. This critical value of the elastohydrodynamic number was found to depend on the surface roughness of the sphere. Biswas and Snidle [4] developed the following theoretical relation to describe fluid film thickness for point contacts with perfectly smooth surfaces in the elastic-isoviscous lubrication regime* :
This theoretical solution required the assumption of a simplified shape of surface deformation in the contact. Recently Hamrock and Dowson [5] examined the same type of contact without any assumptions concerning deformation. They developed the following ~eoretic~ relation**: ii= = 5.29g,o.6”8
Provided that certain conditions are satisfied these solutions can be used to predict the initiation of fluid film breakdown for Nouh’s experimental conditions. These conditions are as follows: (1) viscoelastic effects on the apparent contact area should be minimal; (2) the distance between surface asperities should be small compared with the apparent contact area; (3) the conditions of elastic-isoviscous lubrication should exist. Physically one would expect that the initiation of fluid film breakdown should occur when the theoretical film thickness approaches the magnitude of the surface asperities. This latter method of analysis predicts the initiation of fluid film breakdown at lower velocities for materials of lower elastic modulus given a specific surface roughness. However, the relation defined by Nouh [l] and Moore [ 2, 31 predicts the opposite effect for materials of lower elastic modulus. The present study uses me~urements of the coefficient of friction to determine the initiation of fluid fiIm breakdown for contacts of the type shown in Fig. 1. A primary objective of the present study is the determination of the effect of elastic modulus on the initiation of fluid film breakdown. In addition the behaviour of the friction coefficient during partial and continuous fluid film lubrication is examined. The study of elastomeric point contacts is relevant to a number of engineering applications. The present work is part of an in vitro testing program of a polyurethane surface for use as a liner for a joint replacement
*Biswas and Snidle [4] solved for the minimum film thickness h,. To obtain a formula for h, the relation h, = 0.78 he from Hamrock and Dowson [ 5 f was used, **The original formulas of B&was and Snidle [43 and Hamrock and Dowson [ 5 ] are in a different form than eqns. (2) and (3). In the original formulas the powers to which terms were raised had two significant digits. The three digits of the exponent in eqns. (2) and (3) allow a more exact correspondence to the original forms.
27
prosthesis employed in orthopaedic surgery [ 61. Point contact models have also been used in the study of microasperity lubrication in face seals [ 7,8] and tyres [3, 9, lo]. It may also be possible to apply studies of this type to the natural lubrication of human synovial joints [ 111.
\
POLYURETHANE OR SILICONE RUBBER
Fig. 1. The experimental configuration. Fig. 2. Schematic diagram of the experimental apparatus.
2. Experimental
investigation
The apparatus used for determination of friction coefficients is shown in Fig. 2. This is a pin-on-disc type of machine. The turntable was driven by a variable speed motor and in the present tests the sliding speeds at the pin/ disc interface were varied from 0.3 to 30 cm s-l. The friction force was measured using a configuration of strain gauges fixed onto a thinned section of the pin as shown in Fig. 2. Four materials were used in the experiments. The spherical tipped pins were made from tungsten carbide or glass and the discs from thick polyurethane or silicone rubber sheets. The surface roughness of the tungsten carbide spheres was measured using a Talysurf IV with a radius follower. The surface of the glass sphere was polished to an optically smooth finish. The elastomeric surfaces placed on the disc were polyurethane [6] prepared by pressure moulding a heated polyurethane powder against polished metal sheets or silicone rubber (General Electric RTV-11) prepared by low pressure moulding against optically smooth glass sheets. The experimental procedure consisted of measuring frictional force for a range of sliding velocities. Normal load variation was achieved by adding the appropriate amount of lead shot to the container shown in Fig. 2. The reduced elastic moduli E’ of the polyurethane and silicone rubber were determined from average values of the measured contact area dimensions shown in Fig. 3 using the following Hertzian formula:
28
E’ = l.SFR/a$
(4)
The dimensions of the average contact area were also used to determine a viscoelastic parameter A, which will be described later. These dimensions were determined by placing printers’ ink onto the rigid sphere and observing the area from which the ink was removed during sliding. This method was previously employed by Gujrati and Ludema [ 91. Water was used as a lubricant for all the experiments with polyurethane. The viscosity of the water was determined by measuring its temperature and consulting standard temperature-viscosity charts. A water and detergent solution (0.01 M sodium lauryl sulphate) was used as a lubricant for all of the experiments with silicone rubber. This lubricant was recommended and used by Roberts in experiments involving both tangential [ 121 and normal [ 131 motions of silicone rubber spheres on glass sheets. Although the addition of the detergent did not change the viscosity from that of pure water, Roberts showed that in thin films there was an effective relative viscosity of 2.5. Thus in the present study the viscosity of water at the measured experimental temperature was multiplied by a factor of 2.5 to obtain the effective viscosity of the solution.
3. Results Compared with the metal spheres the moulded silicone rubber surfaces could be considered smooth and the polyurethane surfaces, although not as smooth after moulding, were smooth after a brief running-in period. Thus the combined surface roughness of the metal-elastomer contacts could be defined by the surface roughness of the metal spheres. Measured surface roughness distributions of the metal spheres were plotted and found to be Gaussian in nature. The average standard deviation u of the metal surface or equivalently, for Gaussian surface distributions, the root mean square (r.m.s.) roughness was, chosen as an experimental parameter in this study. The contact area dimensions were measured over the experimental velocity range and remained essentially constant. Thus the elastic and bulk viscoelastic properties of the elastomeric surfaces did not depend on the sliding velocity for this study. Table 1 gives the parameters for each of the fourteen experiments performed*. A plot of typical results for the friction coefficient f uersus sliding velocity V for both the glass and metal spheres on polyurethane and silicone rubber surfaces is shown in Fig. 4. These results show no significant difference between the constant values of the friction coefficient for the smooth glass and rougher metal
*Detailed
tables of the data of this work are available
in ref. 14.
Ez
VELOCITY
(VI
CONTACT
AREA
OF SPHERE
-7 orI--
I 20
I
IO V [cm/set
I 30
]
Fig. 3. Contact area dimensions. Fig. 4. Typical f us. V results: solid curves, metal sphere; broken curves, glass sphere; vertical bars, estimated experimental uncertainty. The numbers on the curves are the experiment numbers.
spheres at the higher sliding velocities. This is expected if continuous fluid films separated the surfaces at the higher velocities. A rapid increase in the friction coefficient for the rougher metal spheres compared with the smooth glass spheres occurs at the lower sliding velocities. This is consistent with a breakdown in continuous fluid film lubrication which would allow surface contact and thus increased friction. It is also important to note the differences between the results for the polyurethane and silicone rubber surfaces. For the polyurethane surfaces the magnitudes of the friction coefficients are always higher. In addition, the rapid increase in friction coefficient for the metal spheres occurs at lower velocity values for the silicone rubber compared with the polyurethane surfaces. If this increase in friction coefficient can be shown to indicate fluid film breakdown then it is apparent that fluid films existed at lower sliding velocities for the silicone rubber contacts.
4. Analysis The correlation between fa and EH developed by Nouh [l] and Moore [ 2,3] is applied to the data of the present study by making the following assumptions. Firstly the contact can be approximated as Hertzian with a Poisson ratio of 0.5 and therefore
Contact surface
Metal/polyurethane Metal/polyurethane Metal/polyurethane Metal/polyurethane Metal/polyurethane Metal/polyurethane Metal/silicone rubber Metal/silicone rubber Metal/silicone rubber Metal/silicone rubber Metal/silicone rubber Glass/polyurethane Glass/silicone rubber Glass/silicone rubber
Experiment number
1 2 3 4 5 6 7 8 9 10 11 12 13 14
The experimental parameters
TABLE 1
Water Water Water Water Water Water Water Water Water Water Water Water Water Water detergent detergent detergent detergent detergent
+ detergent + detergent
+ + + + +
Lubricant
0.635 0.635 0.635 0.635 0.635 0.635 0.635 0.635 0.635 0.635 0.635 0.567 0.567 0.567
R (cm) 0.0086 0.0086 0.0086 0.0086 0.0086 0.0086 0.0086 0.0086 0.0086 0.0086 0.0086 < 0.0086 < 0.0086 < 0.0086
11.1 11.1 11.1 11.1 11.1 11.1 11.1 11.1 11.1 6.3 16.0 11.1 11.1 11.1
29.5 29.5 35.4 35.4 35.4 35.4 5.49 5.49 5.49 5.49 5.49 35.4 5.49 5.49
770
9.8 x 10-4 9.1 x 10-4 8.6 x 10-4 8.6x10 4 8.6 x 10P4 9.1 x lop4 2.3 x 1O-3 2.3 X 10 -3 2.3 x 10P3 2.3 x 1O-3 2.3 x 1O-3 8.6 x lop4 2.3 x 1O-3 2.3 x 1O-3
(Pa s)
31
E* = 0.5625FRla:
(5)
Secondly
a,/ar = (1 + tan2 6 )112 - tan 6
(6)
which was derived by Moore [15] and is used to calculate tan 6. It is also convenient to set a,= 0.962af for the silicone rubber calculations in order to obtain fa values similar to those for the polyurethane. This specification, although not equal to the experimental result, is within the estimated accuracy of the measurements. Figure 5 shows the data of the present study with estimated “transitional” elastohydrodynamic number values (EH)T.Contrary to Nouh’s results the transitional values of the elastohydrodynamic number were apparently not unique for a specific surface roughness. The application of the theoretical film thickness relation assumes certain conditions as discussed earlier. These conditions appear to be satisfied in the present experiments. Since the minimum a,/af values in this study are approximately 0.9 (for the polyurethane contacts), it is assumed that viscoelastic effects on the contact area are minimal (Condition 1). Talysurf traces 4 X 10d4 m in length revealed that distances between surface asperities were of the order of 5 X 10m6 m. Since the radius afof the contact area is always greater than 1.4 X 10h3 m it is assumed that the interasperity distance is small compared with the apparent contact area (Condition 2). The lubrication regime is determined by considering the chart constructed by Hamrock [ 161. A pressure-viscosity coefficient for water is estimated at
(a)0.1L
BOUNDARY y
FLUID
FILM
(hi),
0
EH
Fig. 5. Nouh’s correlation applied to data from the present study: (a) polyurethane (experiments 1 - 6); (b) silicone rubber (experiments 7 - 11).
32
1.5 X 10-l’ Pa-‘. Figure 6 shows that the experimental conditions correspond to elastic-isoviscous lubrication (Condition 3). The conditions for the experimental work of Roberts [12] and Nouh [l] are also shown in Fig. 6 to fall within this regime.
Fig. 6. The regime of elastohydrodynamic
To determine friction coefficient f =fc
the initiation of fluid film breakdown are considered as follows:
+ fb
contact
lubrication.
components
+ ff
bulk
of the
(7)
film
(hysteresis)
For experiments 6 - 14 the friction coefficient is constant at the high velocities. If it is assumed that continuous fluid film lubrication existed at the high velocities f, = 0 and f = fb + ff. Gujrati [ 171 showed that for continuous fluid film lubrication an empirical relation could be described between f, (where f, = fb + ff ) and A, (where A, = (af/R)( 1 - a,/af)) which applied when fb % ff. Figure 7 shows the results of the present study for experiments 6 - 14 included with Gujrati’s. The good agreement with Gujrati’s results indicates that continuous fluid film lubrication existed in our studies for the measured constant friction at high sliding velocities. The following empirical relation results from a least squares curve fit of the points in Fig. 7 corresponding to the high velocity data from the present study and Gujrati’s data: f, =
0.577A,
+ 0.00837
standard
error, 4.69 X lop3
(8)
Included in Fig. 7 are two typical experimental points which are not included in the curve fit and which correspond to low velocity friction coefficient measurements. These points demonstrate the significant increase in friction coefficient at the low velocities.
33
Fig. 7. Friction formula for continuous fluid film lubrication: 0, present study; 0, Gujrati [ 171; w, fat low velocity for experiment 3; 0, fat low velocity for experiment 10.
For the low velocity friction coefficient measurements of experiments 6 - 11 f, values are calculated individually using eqn. (7) and setting f, (where f, = fb + ff) equal to the constant high velocity friction coefficients described above. For experiments 1 - 5 in which velocities do not reach high enough values to produce constant friction coefficients, eqn. (8) is used to calculate f, and then eqn. (7) to calculate f,. The contact friction component f, of the total friction coefficient f appears for experiments 1 - 11 as the sliding velocities decrease below their respective critical values. These critical velocity values can be determined in a consistent manner for all the experiments performed. Then using the surface roughness values of the metal spheres the experimental film thickness for this study can be estimated at the initiation of fluid film breakdown and compared with the theoretical and experimental work of other investigators. To obtain a generalized method of specifying the critical velocities mentioned above it is assumed that eqn. (3) due to Hamrock and Dowson [ 51 yields the best estimate of the central film thickness h, in the experimental contacts. Based on the work of Johnson et al. [18] the following parameter is chosen to describe surface separation: h = h,/a
(9)
A plot of log f, uersus h for the region in which f, decreased for increasing A is shown in Fig. 8 for experiments 1 - 11. Although considerable scatter is shown in A for a given value of log f, , the rate of decrease of log f, as a function of h appears to be consistent between experiments. Thus a curve fit of A uemus log f, is made which allows this consistent slope to be retained. The resulting curve is described empirically by the following relation: f, = 0.753 exp(-1.06h)
(10)
0.00
0.c
I 2 (
1
I,
I
I
4
I
I
8
/
1 IO
J
X6
Fig. 8. The fC vs. h correlation for the region in which f, decreased for increasing h. (a) Present study: 1, 2,. . ., 11, experiment numbers; solid curve, curve fit (eqn. (10)); horizontal bars, estimated experimental uncertainty. (b) From Nouh [l] : Nl, N2, . _ ., N9, experiment numbers; solid curve, curve fit (eqn. (11)).
The data of Nouh [l] are also included in Fig. 8. This is done by assuming the following: (1) eqn. (5) can be used to estimate af ; (2) eqn. (6) can then be used to estimate a, ; (3) u = 0.017 pm since the abrasive particles used to polish Nouh’s spheres were twice the diameter of those of the present study. The following relation is obtained by a h uersus log f, curve fit: f, = 15.7 exp(-1.66h)
(11)
By assuming that h, is equal to the mean surface separation for the contact, as was done by Johnson et al. [18], the criterion h, = 30 is set for initial asperity contact as shown in Fig. 9. At this point the measured coefficient of friction f should include a contact friction component f,. The standard error in the f, uersus A, curve fit is 4.69 X 10V3 and it is felt that measured values of f > f, + 4.69 X low3 should therefore correspond to real physical changes in the contact, i.e. asperity contact and the initiation of fluid film breakdown. Using the above concepts each experiment is separately analysed. This is done by fitting the data of each experiment with a X uersus log f, curve of
35
METAL SPHERE
ELASTOMER I
’ INITIAL CONTACT
+hc *
FOR
CASE
SHOWN
hc=
Fig. 9. The initiation
3.0
of fluid film breakdown.
the same slope as given in eqn. (10). The values of X at which f, = 4.69 X 10d3 (corresponding to f = f, + 4.69 X 10m3) are calculated for each experiment using the following equation: hT = 2.17 log f, + i; + 5.06
(12)
The details of the derivation of this equation are given in Appendix A. The critical velocity values at the initiation of fluid film breakdown for each experiment are calculated from the hT evaluations of eqn. (12) using eqns. (3) and (9). Using the calc_ulatedcritical velocity values and the assumption that h, = 30, a plot of H, uersusg, is made as shown in Fig. 10 using the following equations: (13) 213
(14) A curve fit of the data of the present study resulted in the following expression : fiC = 0.39og,o.734
(15)
with a standard error for log I?, ‘versus log g, of 0.145. The following are included in Fig. 10: (1) the data from the direct optical measurements of Roberts [ 121; (2) the theoretical curves of Biswas and Snidle [4] and Hamrock and Dowson [ 51; (3) the data of Nouh [l] which are analysed using eqn. (11) in the same manner as the data of the present study; (4) the following curve fit of the data from the direct optical measurements of Jamison et al. [ 191: I?, = 2.4g,o.72
(16)
36
I o’+
‘._
IO'=
I
Id6
se Fig. 10. The kc us. g, correlation: 1, 2, . . ., 11, present study; Nl, N2, . . ., N9, Nouh [I] ; a, Roberts [ 121; broken curve, present study curve fit (eqn. (15)); vertical bars, estimated experimental uncertainty.
4. Discussion The correlation of Nouh [ 11 and Moore [ 2, 31 does not result in a unique transition value of the elastohydrodynamic number for the data of this study as shown in Fig. 5. In the derivation of the elastohydrodynamic number the elastic deformations must be small enough to allow the use of a squeeze film solution for rigid surfaces. It is very probable that this condition is not satisfied in the experiments of the present study. As shown in Fig. 10 the data of the present study are not in good agreement with the theoretical curve of Biswas and Snidle [4] or the experimental results of Jamison et al. [ 191. The dis~eement with Biswas and Snidle is not surprising since their derivation is based on a simplified surface deformation. The data of Jamison et al. showed increased film thickness values compared with the data of this study. However, their analysis showed that their lub~cation conditions were on the boundary between the elastic variable viscosity and elastic-isoviscous regimes. Thus enhancing viscosity effects may explain the higher film thickness values. Roberts 1121 used direct optical measurements of film thickness for smooth-surfaced contacts which are shown in Fig. 6 to be well within the elastic-isoviscous regime. The resulting data are shown in Fig. 10 to fall on the theoretical curve of Hamrock and Dowson [ 51. This excellent agreement suggests that the theoretical curve of Hamrock and Dowson correctly describes smooth-surfaced elastic-isoviscous point contacts.
37
The data of the present study and that of Nouh [l] are shown in Fig. 10 to fall slightly below the theoretical curve of Hamrock and Dowson [ 51. Whether this is caused by biased errors in some of the experimental measurements or incorrect assumptions in the analysis is not certain. However, the theoretical curve of Hamrock and Dowson does fall within the estimated limits of experimental uncertainty shown on Fig. 10 for the present study. The assumption of continuous fluid film lubrication in the constant friction high velocity region (see Fig. 4) is an essential part of the analysis of the present study which is supported by the experimental work of Gujrati [17] as shown in Fig. 7. Further support is provided by noting that the friction coefficient values at high velocity in Fig. 4 are virtually the same for both the metal and the smooth glass spheres. This is unlikely to be the case if asperity contact is occurring. Based on the agreement with the theory of Hamrock and Dowson [ 51 and the result8 of the friction experiments of this study, the following expression is proposed to give the conditions necessary for continuous fluid film lubrication : R 0.78tvo
u)0.67
O(,y’)0.45~0.22
a
0.902 f 0.335
(17)
The details of the derivation of this equation are given in Appendix B.
5. Conclusions (1) An analysis has been developed using friction measurements to determine the initiation of fluid film breakdown for elastic-isoviscous point contacts consisting of a rigid surface of random roughness and a smooth lowmodulus surface. (2) Within experimental uncertainty the friction experiments of this study show agreement with the theoretical solution of Hamrock and Dowson [ 51, the experimental work of Roberts [12] and the experimental work of Nouh [l]. (3) The results for different reduced elastic moduli show qualitative agreement with the proposed correlations used in the present analysis. (4) Equation (17) can be used to estimate whether a given contact is boundary or fluid film lubrication. (5) The data of the present study do not show good agreement with the correlation proposed by Nouh [l] and Moore [ 2,3]. (6) The data of the present study support the correlation proposed by Gujrati [17] for the friction coefficient f,., during continuous fluid film lubrication. (7) For the friction experiments a general f, uerSu8 h curve describes the data.
38
Acknowledgments
The authors wish to thank J. L. Tevaarwerk, Assistant Professor, Department of Mechanical Engineering, University of Waterloo, for his many helpful discussions. Financial support from the Atkinson Charitable Foundation and the Natural Sciences and Engineering Research Council of Canada are gratefully acknowledged.
Nomenclature a =f, E+ E’
a,
f fb
k fn F
h, h, qo R u uT
V
01 tan 6 QO (5
Hertzian radius viscoelastic contact area dimensions shown in Fig. 3 viscoelastic Young’s modulus reduced elastic modulus total coefficient of friction coefficient of friction resulting from subsurface bulk (or hysteresis) losses coefficient of friction resulting from surface contact coefficient of friction resulting from fluid flow in a continuous lubrication fb + ff, coefficient of friction during continouus fluid film lubrication normal load film thickness at the centre of the contact minimum film thickness maximum Hertziancontact stress reduced radius of contacting bodies (in this study it is equal to the sphere V/2, mean surface velocity mean surface velocity at the initiation of film breakdown relative surface velocity (in this study it is equal to the elastomer velocity) pressure-viscosity coefficient phase shift between sinusoidal stress and strain in a viscoelastic material absolute viscosity at atmospheric pressure standard deviation of the rigid surface profile
film
radius)
S_uperscripts X average of the values Xi for a single experiment Dimensionless grou$s (af/R)( 1 - a,/af), contact asymmetry parameter 4 E*QVR/~OF, elastohydrodynamic number EH (EH)T
fR ge gv fiC
elastohydrodynamic number at transition to fluid film lubrication f/tan 6, generalized friction coefficient (F4/g3u3E’R5)2/3, elasticity number F%/&R2 , viscosity number h,F2/&%$$uZ, film thickness number he/a, film breakdown number
film breakdown number at the initiation of film breakdown
References 1 S. T. Nouh, Transition to elastohydrodynamic West Virginia University, 1970.
lubrication of rubber, M.Sc. Thesis,
39
6
10 11 12 13 14 15 16
17 18 19
D. F. Moore, The Friction and Lubrication of Elastomers, Pergamon Press, Oxford, 1971. D. F. Moore, The elastohydrodynamic transition speed for spheres sliding on lubricated rubber, Wear, 35 (1975) 159 - 170. S. Biswas and R. W. Snidle, Elastohydrodynamic lubrication of spherical surfaces of low elastic modulus, J. Lubr. Technol., 98 (1976) 524 - 529. B. J. Hamrock and D. Dowson, Elastohydrodynamic lubrication of elliptical contacts for materials of low elastic modulus I - Fully flooded conjunction, J. Lubr. Technol., 100 (1978) 236 - 245. J. B. Medley, R. M. Pilliar, E. W. Wong and A. B. Strong, Hydrophilic polyurethane elastomers for hemiarthroplasty : a preliminary in vitro wear study, Eng. Med., 9 (2) (1980) 59 - 65. C. R. McClune and D. Tabor, An interferometric study of lubricated rotary face seals, Tribal. Int., 11 (4) (1978) 219 - 227. E. T. Jagger and P. S. Walker, Further studies of the lubrication of synthetic rubber rotary shaft seals, Proc. Inst. Mech. Eng., 181 (1) (1966-67) 191 - 204. B. D. Gujrati and K. C. Ludema, Viscoelastohydrodynamics: lubricated elastomeric contacts. In Lieng-Huang Lee (ed.), Advances in Polymer Friction and Wear, Plenum Press, New York, 1974. B. D. Gujrati, The frictional dynamics of lubricated elastomers in contact with regularly spaced sliders, J. Lubr. Technol., 100 (1978) 219 - 225. G. R. Higginson, Elastohydrodynamic lubrication in human joints, Eng. Med., 7 (1) (1978) 35 - 41. A. D. Roberts, The shear of thin liquid films, J. Phys. D, 4 (1971) 433 - 440. A. D. Roberts, Squeeze films between rubber and glass, J. Phys. D, 4 (1971) 423 432. J. B. Medley, The tribology of an elastomeric polyurethane for orthopaedic joint replacement, M.A.Sc. Thesis, Univ. Waterloo, 1979. D. F. Moore, On the decrease in contact area for spheres and cylinders rolling on a viscoelastic plane, Wear, 21 (1972) 179 - 194. B. J. Hamrock, Additional aspects of elastohydrodynamic lubrication, Conf. on the Fundamentals of Tribology, Massachusetts Institute of Technology, June 1978, NASA Tech. Memo. TM-78898, 1978. B. D. Gujrati, Visco-elastohydrodynamics (VEHD) of “soft” polymeric contact(s) under rigid indenter(s), Ph.D. Thesis, University of Michigan, 1974. K. L. Johnson, J. A. Greenwood and S. Y. Poon, A simple theory of asperity contact in elastohydrodynamic lubrication, Wear, 19 (1972) 91 - 108. W. E. Jamison, C. C. Lee and J. J. Kauzlarich, Elasticity effects on lubrication of point contacts, ASLE Trans., 21 (4) (1978) 299 - 306.
Appendix A Equation (12) is derived by considering a general equation for the experiments of the present study which is based on eqn. (10) as follows: f, = cl exp(-1.06h) where c1 is a constant or In f, = -1.06X
+ In c1
To obtain a fitted curve for the data of a single experiment which has the same slope as the above general In f, uersus h curve implies In cl = 1.06); + In f,
40 where i and ln f, are average values for the experiment iment curve is In f, = -1.06h By applying
and the single exper-
+ 1.06x + In f,
the criterion
h = h, when f, = 4.69 X 1O-3 it follows that
In (4.69 X 10-3) = -1.06hr
+ 1.06i
+ In f,
h, = 0.943 In f, + x + 5.06 or h, = 2.17 logf,
+ h + 5.06
which is eqn. (12).
Appendix
B
Equation (17) is de$ed by combining eqns. (3) and (9) with the appropriate definitions of H, and g, to obtain the following expression : R 0.‘8(?jo z+s ,(E’)e.4aFe.s2
=-
h 5.29
(Bl)
At the initiation of fluid film breakdown h = hr. The arithmetic average for XT from experiments 1 - 11 is hT = 4.77. An estimated lower limit value of hT = 3 can be obtained from the theory as shown in Fig. 9. The difference between the average hT value from the present experiments and the theoretical lower limit value can be used to define the following range for hr : h* = 4.77 * 1.77
This range can be used in eqn. (Bl) to give the following expression ing the conditions necessary for continuous fluid film lubrication: ~0.78(~~~)0.67 +')0.45~0.22
which is eqn. (17).
a
0.902 + 0.335
describ-
Wear, 63 (1980) 25 - 40 0 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
THE BREAKDOWN OF FLUID FILM LUBRICATION ISOVISCOUS POINT CONTACTS*
IN ELASTIC-
J. B. MEDLEY and A. B. STRONG Department (Canada)
of Mechanical
Engineering,
University of Waterloo, Waterloo, Ontario
R. M. PILLIAR and E. W. WONG Department
of Metallurgy, Ontario Research Foundation,
Mississauga, Ontario (Canada)
(Received September 24,1979)
Summary Methods have been developed to determine the initiation of film breakdown experimentally. In these experiments the lubricated sliding of a smooth elastomeric surface over a rigid metal sphere is investigated using friction measurements. Specific friction increases are shown to correspond to fluid film breakdown and are related to film thickness values by considering surface roughness in the contact. Within experimental uncertainty these film thickness estimates agree with recent theoretical and experimental work reported in the literature.
1. Introduction One classical method of using friction measurements to investigate lubrication conditions in a journal bearing involves experimentally determining the relation between the friction coefficient and the Sommerfeld number. Nouh [l] and Moore [2, 31 defined an equivalent Sommerfeld number for point contacts and further modified it to include viscoelastic effects of an elastomeric contact surface to produce an “elastohydrodynamic” number defined as follows: EH =-
E*q,VR qoF
Nouh [l] examined the sliding of metal spheres with specified surface roughness over smooth lubricated rubber. Characteristic curves were produced for a generalized friction coefficient (fa = f/tan 6) uersus the *The work reported here was done as part of J. B. Medley’s M.A.Sc. thesis at the University of Waterloo.
(1)
26
elastohydrodynamic number. The rapid drop of the generalized friction coefficient as the elastohydrodyn~ic number increased beyond a critical value was interpreted as a t~sition from boundary to fluid film lubrication. This critical value of the elastohydrodynamic number was found to depend on the surface roughness of the sphere. Biswas and Snidle [4] developed the following theoretical relation to describe fluid film thickness for point contacts with perfectly smooth surfaces in the elastic-isoviscous lubrication regime* :
This theoretical solution required the assumption of a simplified shape of surface deformation in the contact. Recently Hamrock and Dowson [5] examined the same type of contact without any assumptions concerning deformation. They developed the following ~eoretic~ relation**: ii= = 5.29g,o.6”8
Provided that certain conditions are satisfied these solutions can be used to predict the initiation of fluid film breakdown for Nouh’s experimental conditions. These conditions are as follows: (1) viscoelastic effects on the apparent contact area should be minimal; (2) the distance between surface asperities should be small compared with the apparent contact area; (3) the conditions of elastic-isoviscous lubrication should exist. Physically one would expect that the initiation of fluid film breakdown should occur when the theoretical film thickness approaches the magnitude of the surface asperities. This latter method of analysis predicts the initiation of fluid film breakdown at lower velocities for materials of lower elastic modulus given a specific surface roughness. However, the relation defined by Nouh [l] and Moore [ 2, 31 predicts the opposite effect for materials of lower elastic modulus. The present study uses me~urements of the coefficient of friction to determine the initiation of fluid fiIm breakdown for contacts of the type shown in Fig. 1. A primary objective of the present study is the determination of the effect of elastic modulus on the initiation of fluid film breakdown. In addition the behaviour of the friction coefficient during partial and continuous fluid film lubrication is examined. The study of elastomeric point contacts is relevant to a number of engineering applications. The present work is part of an in vitro testing program of a polyurethane surface for use as a liner for a joint replacement
*Biswas and Snidle [4] solved for the minimum film thickness h,. To obtain a formula for h, the relation h, = 0.78 he from Hamrock and Dowson [ 5 f was used, **The original formulas of B&was and Snidle [43 and Hamrock and Dowson [ 5 ] are in a different form than eqns. (2) and (3). In the original formulas the powers to which terms were raised had two significant digits. The three digits of the exponent in eqns. (2) and (3) allow a more exact correspondence to the original forms.
27
prosthesis employed in orthopaedic surgery [ 61. Point contact models have also been used in the study of microasperity lubrication in face seals [ 7,8] and tyres [3, 9, lo]. It may also be possible to apply studies of this type to the natural lubrication of human synovial joints [ 111.
\
POLYURETHANE OR SILICONE RUBBER
Fig. 1. The experimental configuration. Fig. 2. Schematic diagram of the experimental apparatus.
2. Experimental
investigation
The apparatus used for determination of friction coefficients is shown in Fig. 2. This is a pin-on-disc type of machine. The turntable was driven by a variable speed motor and in the present tests the sliding speeds at the pin/ disc interface were varied from 0.3 to 30 cm s-l. The friction force was measured using a configuration of strain gauges fixed onto a thinned section of the pin as shown in Fig. 2. Four materials were used in the experiments. The spherical tipped pins were made from tungsten carbide or glass and the discs from thick polyurethane or silicone rubber sheets. The surface roughness of the tungsten carbide spheres was measured using a Talysurf IV with a radius follower. The surface of the glass sphere was polished to an optically smooth finish. The elastomeric surfaces placed on the disc were polyurethane [6] prepared by pressure moulding a heated polyurethane powder against polished metal sheets or silicone rubber (General Electric RTV-11) prepared by low pressure moulding against optically smooth glass sheets. The experimental procedure consisted of measuring frictional force for a range of sliding velocities. Normal load variation was achieved by adding the appropriate amount of lead shot to the container shown in Fig. 2. The reduced elastic moduli E’ of the polyurethane and silicone rubber were determined from average values of the measured contact area dimensions shown in Fig. 3 using the following Hertzian formula:
28
E’ = l.SFR/a$
(4)
The dimensions of the average contact area were also used to determine a viscoelastic parameter A, which will be described later. These dimensions were determined by placing printers’ ink onto the rigid sphere and observing the area from which the ink was removed during sliding. This method was previously employed by Gujrati and Ludema [ 91. Water was used as a lubricant for all the experiments with polyurethane. The viscosity of the water was determined by measuring its temperature and consulting standard temperature-viscosity charts. A water and detergent solution (0.01 M sodium lauryl sulphate) was used as a lubricant for all of the experiments with silicone rubber. This lubricant was recommended and used by Roberts in experiments involving both tangential [ 121 and normal [ 131 motions of silicone rubber spheres on glass sheets. Although the addition of the detergent did not change the viscosity from that of pure water, Roberts showed that in thin films there was an effective relative viscosity of 2.5. Thus in the present study the viscosity of water at the measured experimental temperature was multiplied by a factor of 2.5 to obtain the effective viscosity of the solution.
3. Results Compared with the metal spheres the moulded silicone rubber surfaces could be considered smooth and the polyurethane surfaces, although not as smooth after moulding, were smooth after a brief running-in period. Thus the combined surface roughness of the metal-elastomer contacts could be defined by the surface roughness of the metal spheres. Measured surface roughness distributions of the metal spheres were plotted and found to be Gaussian in nature. The average standard deviation u of the metal surface or equivalently, for Gaussian surface distributions, the root mean square (r.m.s.) roughness was, chosen as an experimental parameter in this study. The contact area dimensions were measured over the experimental velocity range and remained essentially constant. Thus the elastic and bulk viscoelastic properties of the elastomeric surfaces did not depend on the sliding velocity for this study. Table 1 gives the parameters for each of the fourteen experiments performed*. A plot of typical results for the friction coefficient f uersus sliding velocity V for both the glass and metal spheres on polyurethane and silicone rubber surfaces is shown in Fig. 4. These results show no significant difference between the constant values of the friction coefficient for the smooth glass and rougher metal
*Detailed
tables of the data of this work are available
in ref. 14.
Ez
VELOCITY
(VI
CONTACT
AREA
OF SPHERE
-7 orI--
I 20
I
IO V [cm/set
I 30
]
Fig. 3. Contact area dimensions. Fig. 4. Typical f us. V results: solid curves, metal sphere; broken curves, glass sphere; vertical bars, estimated experimental uncertainty. The numbers on the curves are the experiment numbers.
spheres at the higher sliding velocities. This is expected if continuous fluid films separated the surfaces at the higher velocities. A rapid increase in the friction coefficient for the rougher metal spheres compared with the smooth glass spheres occurs at the lower sliding velocities. This is consistent with a breakdown in continuous fluid film lubrication which would allow surface contact and thus increased friction. It is also important to note the differences between the results for the polyurethane and silicone rubber surfaces. For the polyurethane surfaces the magnitudes of the friction coefficients are always higher. In addition, the rapid increase in friction coefficient for the metal spheres occurs at lower velocity values for the silicone rubber compared with the polyurethane surfaces. If this increase in friction coefficient can be shown to indicate fluid film breakdown then it is apparent that fluid films existed at lower sliding velocities for the silicone rubber contacts.
4. Analysis The correlation between fa and EH developed by Nouh [l] and Moore [ 2,3] is applied to the data of the present study by making the following assumptions. Firstly the contact can be approximated as Hertzian with a Poisson ratio of 0.5 and therefore
Contact surface
Metal/polyurethane Metal/polyurethane Metal/polyurethane Metal/polyurethane Metal/polyurethane Metal/polyurethane Metal/silicone rubber Metal/silicone rubber Metal/silicone rubber Metal/silicone rubber Metal/silicone rubber Glass/polyurethane Glass/silicone rubber Glass/silicone rubber
Experiment number
1 2 3 4 5 6 7 8 9 10 11 12 13 14
The experimental parameters
TABLE 1
Water Water Water Water Water Water Water Water Water Water Water Water Water Water detergent detergent detergent detergent detergent
+ detergent + detergent
+ + + + +
Lubricant
0.635 0.635 0.635 0.635 0.635 0.635 0.635 0.635 0.635 0.635 0.635 0.567 0.567 0.567
R (cm) 0.0086 0.0086 0.0086 0.0086 0.0086 0.0086 0.0086 0.0086 0.0086 0.0086 0.0086 < 0.0086 < 0.0086 < 0.0086
11.1 11.1 11.1 11.1 11.1 11.1 11.1 11.1 11.1 6.3 16.0 11.1 11.1 11.1
29.5 29.5 35.4 35.4 35.4 35.4 5.49 5.49 5.49 5.49 5.49 35.4 5.49 5.49
770
9.8 x 10-4 9.1 x 10-4 8.6 x 10-4 8.6x10 4 8.6 x 10P4 9.1 x lop4 2.3 x 1O-3 2.3 X 10 -3 2.3 x 10P3 2.3 x 1O-3 2.3 x 1O-3 8.6 x lop4 2.3 x 1O-3 2.3 x 1O-3
(Pa s)
31
E* = 0.5625FRla:
(5)
Secondly
a,/ar = (1 + tan2 6 )112 - tan 6
(6)
which was derived by Moore [15] and is used to calculate tan 6. It is also convenient to set a,= 0.962af for the silicone rubber calculations in order to obtain fa values similar to those for the polyurethane. This specification, although not equal to the experimental result, is within the estimated accuracy of the measurements. Figure 5 shows the data of the present study with estimated “transitional” elastohydrodynamic number values (EH)T.Contrary to Nouh’s results the transitional values of the elastohydrodynamic number were apparently not unique for a specific surface roughness. The application of the theoretical film thickness relation assumes certain conditions as discussed earlier. These conditions appear to be satisfied in the present experiments. Since the minimum a,/af values in this study are approximately 0.9 (for the polyurethane contacts), it is assumed that viscoelastic effects on the contact area are minimal (Condition 1). Talysurf traces 4 X 10d4 m in length revealed that distances between surface asperities were of the order of 5 X 10m6 m. Since the radius afof the contact area is always greater than 1.4 X 10h3 m it is assumed that the interasperity distance is small compared with the apparent contact area (Condition 2). The lubrication regime is determined by considering the chart constructed by Hamrock [ 161. A pressure-viscosity coefficient for water is estimated at
(a)0.1L
BOUNDARY y
FLUID
FILM
(hi),
0
EH
Fig. 5. Nouh’s correlation applied to data from the present study: (a) polyurethane (experiments 1 - 6); (b) silicone rubber (experiments 7 - 11).
32
1.5 X 10-l’ Pa-‘. Figure 6 shows that the experimental conditions correspond to elastic-isoviscous lubrication (Condition 3). The conditions for the experimental work of Roberts [12] and Nouh [l] are also shown in Fig. 6 to fall within this regime.
Fig. 6. The regime of elastohydrodynamic
To determine friction coefficient f =fc
the initiation of fluid film breakdown are considered as follows:
+ fb
contact
lubrication.
components
+ ff
bulk
of the
(7)
film
(hysteresis)
For experiments 6 - 14 the friction coefficient is constant at the high velocities. If it is assumed that continuous fluid film lubrication existed at the high velocities f, = 0 and f = fb + ff. Gujrati [ 171 showed that for continuous fluid film lubrication an empirical relation could be described between f, (where f, = fb + ff ) and A, (where A, = (af/R)( 1 - a,/af)) which applied when fb % ff. Figure 7 shows the results of the present study for experiments 6 - 14 included with Gujrati’s. The good agreement with Gujrati’s results indicates that continuous fluid film lubrication existed in our studies for the measured constant friction at high sliding velocities. The following empirical relation results from a least squares curve fit of the points in Fig. 7 corresponding to the high velocity data from the present study and Gujrati’s data: f, =
0.577A,
+ 0.00837
standard
error, 4.69 X lop3
(8)
Included in Fig. 7 are two typical experimental points which are not included in the curve fit and which correspond to low velocity friction coefficient measurements. These points demonstrate the significant increase in friction coefficient at the low velocities.
33
Fig. 7. Friction formula for continuous fluid film lubrication: 0, present study; 0, Gujrati [ 171; w, fat low velocity for experiment 3; 0, fat low velocity for experiment 10.
For the low velocity friction coefficient measurements of experiments 6 - 11 f, values are calculated individually using eqn. (7) and setting f, (where f, = fb + ff) equal to the constant high velocity friction coefficients described above. For experiments 1 - 5 in which velocities do not reach high enough values to produce constant friction coefficients, eqn. (8) is used to calculate f, and then eqn. (7) to calculate f,. The contact friction component f, of the total friction coefficient f appears for experiments 1 - 11 as the sliding velocities decrease below their respective critical values. These critical velocity values can be determined in a consistent manner for all the experiments performed. Then using the surface roughness values of the metal spheres the experimental film thickness for this study can be estimated at the initiation of fluid film breakdown and compared with the theoretical and experimental work of other investigators. To obtain a generalized method of specifying the critical velocities mentioned above it is assumed that eqn. (3) due to Hamrock and Dowson [ 51 yields the best estimate of the central film thickness h, in the experimental contacts. Based on the work of Johnson et al. [18] the following parameter is chosen to describe surface separation: h = h,/a
(9)
A plot of log f, uersus h for the region in which f, decreased for increasing A is shown in Fig. 8 for experiments 1 - 11. Although considerable scatter is shown in A for a given value of log f, , the rate of decrease of log f, as a function of h appears to be consistent between experiments. Thus a curve fit of A uemus log f, is made which allows this consistent slope to be retained. The resulting curve is described empirically by the following relation: f, = 0.753 exp(-1.06h)
(10)
0.00
0.c
I 2 (
1
I,
I
I
4
I
I
8
/
1 IO
J
X6
Fig. 8. The fC vs. h correlation for the region in which f, decreased for increasing h. (a) Present study: 1, 2,. . ., 11, experiment numbers; solid curve, curve fit (eqn. (10)); horizontal bars, estimated experimental uncertainty. (b) From Nouh [l] : Nl, N2, . _ ., N9, experiment numbers; solid curve, curve fit (eqn. (11)).
The data of Nouh [l] are also included in Fig. 8. This is done by assuming the following: (1) eqn. (5) can be used to estimate af ; (2) eqn. (6) can then be used to estimate a, ; (3) u = 0.017 pm since the abrasive particles used to polish Nouh’s spheres were twice the diameter of those of the present study. The following relation is obtained by a h uersus log f, curve fit: f, = 15.7 exp(-1.66h)
(11)
By assuming that h, is equal to the mean surface separation for the contact, as was done by Johnson et al. [18], the criterion h, = 30 is set for initial asperity contact as shown in Fig. 9. At this point the measured coefficient of friction f should include a contact friction component f,. The standard error in the f, uersus A, curve fit is 4.69 X 10V3 and it is felt that measured values of f > f, + 4.69 X low3 should therefore correspond to real physical changes in the contact, i.e. asperity contact and the initiation of fluid film breakdown. Using the above concepts each experiment is separately analysed. This is done by fitting the data of each experiment with a X uersus log f, curve of
35
METAL SPHERE
ELASTOMER I
’ INITIAL CONTACT
+hc *
FOR
CASE
SHOWN
hc=
Fig. 9. The initiation
3.0
of fluid film breakdown.
the same slope as given in eqn. (10). The values of X at which f, = 4.69 X 10d3 (corresponding to f = f, + 4.69 X 10m3) are calculated for each experiment using the following equation: hT = 2.17 log f, + i; + 5.06
(12)
The details of the derivation of this equation are given in Appendix A. The critical velocity values at the initiation of fluid film breakdown for each experiment are calculated from the hT evaluations of eqn. (12) using eqns. (3) and (9). Using the calc_ulatedcritical velocity values and the assumption that h, = 30, a plot of H, uersusg, is made as shown in Fig. 10 using the following equations: (13) 213
(14) A curve fit of the data of the present study resulted in the following expression : fiC = 0.39og,o.734
(15)
with a standard error for log I?, ‘versus log g, of 0.145. The following are included in Fig. 10: (1) the data from the direct optical measurements of Roberts [ 121; (2) the theoretical curves of Biswas and Snidle [4] and Hamrock and Dowson [ 51; (3) the data of Nouh [l] which are analysed using eqn. (11) in the same manner as the data of the present study; (4) the following curve fit of the data from the direct optical measurements of Jamison et al. [ 191: I?, = 2.4g,o.72
(16)
36
I o’+
‘._
IO'=
I
Id6
se Fig. 10. The kc us. g, correlation: 1, 2, . . ., 11, present study; Nl, N2, . . ., N9, Nouh [I] ; a, Roberts [ 121; broken curve, present study curve fit (eqn. (15)); vertical bars, estimated experimental uncertainty.
4. Discussion The correlation of Nouh [ 11 and Moore [ 2, 31 does not result in a unique transition value of the elastohydrodynamic number for the data of this study as shown in Fig. 5. In the derivation of the elastohydrodynamic number the elastic deformations must be small enough to allow the use of a squeeze film solution for rigid surfaces. It is very probable that this condition is not satisfied in the experiments of the present study. As shown in Fig. 10 the data of the present study are not in good agreement with the theoretical curve of Biswas and Snidle [4] or the experimental results of Jamison et al. [ 191. The dis~eement with Biswas and Snidle is not surprising since their derivation is based on a simplified surface deformation. The data of Jamison et al. showed increased film thickness values compared with the data of this study. However, their analysis showed that their lub~cation conditions were on the boundary between the elastic variable viscosity and elastic-isoviscous regimes. Thus enhancing viscosity effects may explain the higher film thickness values. Roberts 1121 used direct optical measurements of film thickness for smooth-surfaced contacts which are shown in Fig. 6 to be well within the elastic-isoviscous regime. The resulting data are shown in Fig. 10 to fall on the theoretical curve of Hamrock and Dowson [ 51. This excellent agreement suggests that the theoretical curve of Hamrock and Dowson correctly describes smooth-surfaced elastic-isoviscous point contacts.
37
The data of the present study and that of Nouh [l] are shown in Fig. 10 to fall slightly below the theoretical curve of Hamrock and Dowson [ 51. Whether this is caused by biased errors in some of the experimental measurements or incorrect assumptions in the analysis is not certain. However, the theoretical curve of Hamrock and Dowson does fall within the estimated limits of experimental uncertainty shown on Fig. 10 for the present study. The assumption of continuous fluid film lubrication in the constant friction high velocity region (see Fig. 4) is an essential part of the analysis of the present study which is supported by the experimental work of Gujrati [17] as shown in Fig. 7. Further support is provided by noting that the friction coefficient values at high velocity in Fig. 4 are virtually the same for both the metal and the smooth glass spheres. This is unlikely to be the case if asperity contact is occurring. Based on the agreement with the theory of Hamrock and Dowson [ 51 and the result8 of the friction experiments of this study, the following expression is proposed to give the conditions necessary for continuous fluid film lubrication : R 0.78tvo
u)0.67
O(,y’)0.45~0.22
a
0.902 f 0.335
(17)
The details of the derivation of this equation are given in Appendix B.
5. Conclusions (1) An analysis has been developed using friction measurements to determine the initiation of fluid film breakdown for elastic-isoviscous point contacts consisting of a rigid surface of random roughness and a smooth lowmodulus surface. (2) Within experimental uncertainty the friction experiments of this study show agreement with the theoretical solution of Hamrock and Dowson [ 51, the experimental work of Roberts [12] and the experimental work of Nouh [l]. (3) The results for different reduced elastic moduli show qualitative agreement with the proposed correlations used in the present analysis. (4) Equation (17) can be used to estimate whether a given contact is boundary or fluid film lubrication. (5) The data of the present study do not show good agreement with the correlation proposed by Nouh [l] and Moore [ 2,3]. (6) The data of the present study support the correlation proposed by Gujrati [17] for the friction coefficient f,., during continuous fluid film lubrication. (7) For the friction experiments a general f, uerSu8 h curve describes the data.
38
Acknowledgments
The authors wish to thank J. L. Tevaarwerk, Assistant Professor, Department of Mechanical Engineering, University of Waterloo, for his many helpful discussions. Financial support from the Atkinson Charitable Foundation and the Natural Sciences and Engineering Research Council of Canada are gratefully acknowledged.
Nomenclature a =f, E+ E’
a,
f fb
k fn F
h, h, qo R u uT
V
01 tan 6 QO (5
Hertzian radius viscoelastic contact area dimensions shown in Fig. 3 viscoelastic Young’s modulus reduced elastic modulus total coefficient of friction coefficient of friction resulting from subsurface bulk (or hysteresis) losses coefficient of friction resulting from surface contact coefficient of friction resulting from fluid flow in a continuous lubrication fb + ff, coefficient of friction during continouus fluid film lubrication normal load film thickness at the centre of the contact minimum film thickness maximum Hertziancontact stress reduced radius of contacting bodies (in this study it is equal to the sphere V/2, mean surface velocity mean surface velocity at the initiation of film breakdown relative surface velocity (in this study it is equal to the elastomer velocity) pressure-viscosity coefficient phase shift between sinusoidal stress and strain in a viscoelastic material absolute viscosity at atmospheric pressure standard deviation of the rigid surface profile
film
radius)
S_uperscripts X average of the values Xi for a single experiment Dimensionless grou$s (af/R)( 1 - a,/af), contact asymmetry parameter 4 E*QVR/~OF, elastohydrodynamic number EH (EH)T
fR ge gv fiC
elastohydrodynamic number at transition to fluid film lubrication f/tan 6, generalized friction coefficient (F4/g3u3E’R5)2/3, elasticity number F%/&R2 , viscosity number h,F2/&%$$uZ, film thickness number he/a, film breakdown number
film breakdown number at the initiation of film breakdown
References 1 S. T. Nouh, Transition to elastohydrodynamic West Virginia University, 1970.
lubrication of rubber, M.Sc. Thesis,
39
6
10 11 12 13 14 15 16
17 18 19
D. F. Moore, The Friction and Lubrication of Elastomers, Pergamon Press, Oxford, 1971. D. F. Moore, The elastohydrodynamic transition speed for spheres sliding on lubricated rubber, Wear, 35 (1975) 159 - 170. S. Biswas and R. W. Snidle, Elastohydrodynamic lubrication of spherical surfaces of low elastic modulus, J. Lubr. Technol., 98 (1976) 524 - 529. B. J. Hamrock and D. Dowson, Elastohydrodynamic lubrication of elliptical contacts for materials of low elastic modulus I - Fully flooded conjunction, J. Lubr. Technol., 100 (1978) 236 - 245. J. B. Medley, R. M. Pilliar, E. W. Wong and A. B. Strong, Hydrophilic polyurethane elastomers for hemiarthroplasty : a preliminary in vitro wear study, Eng. Med., 9 (2) (1980) 59 - 65. C. R. McClune and D. Tabor, An interferometric study of lubricated rotary face seals, Tribal. Int., 11 (4) (1978) 219 - 227. E. T. Jagger and P. S. Walker, Further studies of the lubrication of synthetic rubber rotary shaft seals, Proc. Inst. Mech. Eng., 181 (1) (1966-67) 191 - 204. B. D. Gujrati and K. C. Ludema, Viscoelastohydrodynamics: lubricated elastomeric contacts. In Lieng-Huang Lee (ed.), Advances in Polymer Friction and Wear, Plenum Press, New York, 1974. B. D. Gujrati, The frictional dynamics of lubricated elastomers in contact with regularly spaced sliders, J. Lubr. Technol., 100 (1978) 219 - 225. G. R. Higginson, Elastohydrodynamic lubrication in human joints, Eng. Med., 7 (1) (1978) 35 - 41. A. D. Roberts, The shear of thin liquid films, J. Phys. D, 4 (1971) 433 - 440. A. D. Roberts, Squeeze films between rubber and glass, J. Phys. D, 4 (1971) 423 432. J. B. Medley, The tribology of an elastomeric polyurethane for orthopaedic joint replacement, M.A.Sc. Thesis, Univ. Waterloo, 1979. D. F. Moore, On the decrease in contact area for spheres and cylinders rolling on a viscoelastic plane, Wear, 21 (1972) 179 - 194. B. J. Hamrock, Additional aspects of elastohydrodynamic lubrication, Conf. on the Fundamentals of Tribology, Massachusetts Institute of Technology, June 1978, NASA Tech. Memo. TM-78898, 1978. B. D. Gujrati, Visco-elastohydrodynamics (VEHD) of “soft” polymeric contact(s) under rigid indenter(s), Ph.D. Thesis, University of Michigan, 1974. K. L. Johnson, J. A. Greenwood and S. Y. Poon, A simple theory of asperity contact in elastohydrodynamic lubrication, Wear, 19 (1972) 91 - 108. W. E. Jamison, C. C. Lee and J. J. Kauzlarich, Elasticity effects on lubrication of point contacts, ASLE Trans., 21 (4) (1978) 299 - 306.
Appendix A Equation (12) is derived by considering a general equation for the experiments of the present study which is based on eqn. (10) as follows: f, = cl exp(-1.06h) where c1 is a constant or In f, = -1.06X
+ In c1
To obtain a fitted curve for the data of a single experiment which has the same slope as the above general In f, uersus h curve implies In cl = 1.06); + In f,
40 where i and ln f, are average values for the experiment iment curve is In f, = -1.06h By applying
and the single exper-
+ 1.06x + In f,
the criterion
h = h, when f, = 4.69 X 1O-3 it follows that
In (4.69 X 10-3) = -1.06hr
+ 1.06i
+ In f,
h, = 0.943 In f, + x + 5.06 or h, = 2.17 logf,
+ h + 5.06
which is eqn. (12).
Appendix
B
Equation (17) is de$ed by combining eqns. (3) and (9) with the appropriate definitions of H, and g, to obtain the following expression : R 0.‘8(?jo z+s ,(E’)e.4aFe.s2
=-
h 5.29
(Bl)
At the initiation of fluid film breakdown h = hr. The arithmetic average for XT from experiments 1 - 11 is hT = 4.77. An estimated lower limit value of hT = 3 can be obtained from the theory as shown in Fig. 9. The difference between the average hT value from the present experiments and the theoretical lower limit value can be used to define the following range for hr : h* = 4.77 * 1.77
This range can be used in eqn. (Bl) to give the following expression ing the conditions necessary for continuous fluid film lubrication: ~0.78(~~~)0.67 +')0.45~0.22
which is eqn. (17).
a
0.902 + 0.335
describ-