Elastohydrodynamic Lubrication With Herschel-Bulkley ... - CiteSeerX

Elastohydrodynamic Lubrication With Herschel-Bulkley Model Greases JAMES J. KAUZLARICH, ASLE University of Virginia, Charlottesville, Virginia 22901

Downloaded by [75.149.200.233] at 08:32 25 July 2012

J. A. GREENWOOD University of Cambridge, Cambridge CB2 lPZ, England, UK

The analysis of grease-lubricaled rolling element bearings i s presented. Exferimentally determined flow curaes for grease are found to be well correlated by the Herschel-Bdkley model flow eqzration. A theory for predicting roller Jilm thickness based on the assumed flow model i s derived. Experimental results show that grease will deveIop a larger film lhickness lhan the grease base oil at jirst, but the film llzickness falls during rolling until it reaches a steady thickvess usually lower than that of its base oil. This effect i s shown lo depend on the degree of shear degradation of flze grease, its Presented at the 27th ASLE Annual Meeting in Houston, Texas, Moy 1-4, 1972

resulting flow curve, and the temperature rise due to shear in the inlet. The grease yield stress i s fo~cndto have a negligible effect on E H L perfornzance. INTRODUCTION

Grease lubrication of rolling element bearings is still more of an art than a science. Most of the standard tests of grease for bearing applications are believed to have low sensitivity or are of secondary importance. Hutton ( 1 ) states that the tests are not reliable substitutes for trials under service conditions. A similar viewpoint is presented by Harris (2). Theories of lubrication under extreme pressure condi-

NOMENCLATURE a = stress decay constant b = Hertzian half contact width, m D = shear rate, sec-I E' = reduced elastic moclulus, N/m2 EHL = elastohydrodynamic lubrication 11 = roller gap, m = wIIo/(~oUR) 110 = gap a t zero pressure gradient, m lip = plug flow width, m I ( n ) = EHL film thickness integral K = thermal conductivity, Watt/m-C L = roller width, m R2 = 1/11 ?I = power law exponent p = pressure, N/m2 p' = pressure gradient, N/ma PO = maximum Hertz pressure, N/m2 q = stress decay exponent q, = flow per unit roller width, ml/sec R = reduced roller radius, m

-

R I ,= ~ roller radius, m s = surface of roller 1 = time, sec T = temperature, C zc = velocity, m/sec tr, = plug flow velocity, m/sec U = roller surface velocity, m/sec w = load per unit roller width, N/m x,y,z = coordinates, m a = viscosity pressure coefficient, mZ/N a p= ~ Grubin parameter aq, = ( L Y W / R ) ~ ' ~ / ( ( Y ~ ~ U / R ) ~ ' ~ 7 = viscosity temperature coefficient, C-I 7 = Newtonian Viscosity, N-sec/mz 0 = angle, deg A = variable v = Poisson's ratio 7 = shear stress, N/mZ T~ = yield stress, N/m2 $J = plastic viscosity, units vary


tions encountered in rolling element bearings and gears have only emerged since the studies of Ertel and Grubin were published in 1949, e.g., Cameron (3) p. 203. As a result of these studies, it was found that it was necessary to take into account the deformation of the bearing surfaces and the increase in viscosity of the lubricant due to the high pressures, e.g., Dowson and Higginson (4). When this was done, the theory predicted that it was possible to have complete fluid film separation of the bearing surfaces on the order of a micrometer under operating conditions. This area of study is called elastohydrodynamic lubrication, abbreviated EHL, and is now well established by experimental verification, e.g., Dyson, Naylor, and Wilson (5). A literature search was undertaken, but no theoretical EIZL studies of grease-lubricated rollers were found. However, Sasaki, Rfori, and Okino (6) consider a Bingham plastic flow niodel of grease lubrication of undeformed rollers. Their results are questionable since they assume the plug flow velocity constant a t all parts. A more tractable assumption takes the plug flow velocity constant over any cross section but allows it to vary in the flow direction. In this paper the selection of an appropriate flow model for grease is considered, and this flow model is then applied to the analysis for film thickness developed under conditions of elastohydrodynamic lubrication of rollers. Theoretical and experimental results are compared to support the assumptions. RHEOLOGY

Sisko (7) formulated a flow equation for grease of the tinie-independent power-law type that agreed with experimental results better than the Bingham or Ree-Eyring equations. Bauer, Finkelstein, and Wiberley (8) in their investigations into flow properties of lithium stearate grease modified Sisko's equation to a form similar to the Herschel-Bulkley (9) equation, and obtained very good correlation with their experiments. Thus, the Herschel13ulkley flow equation has been selected as the flow model in this paper, and is given by T=

+ 41Dlnl

+bY

[ll

Equation [I] implies that grease is a plastic solid; that it acts as a solid until a critical yield stress is reached. Sisko's esperinients support the concept that some fluid flow takes place in the region where "plug flow" would be encountered. The inaccuracy associated with using Eq. [I] is not very severe, as shown by Atahncke and Tabor ( l o ) ,and the assuniption of plug flow is a convenient mathematical approsinlation well within the accuracy required for most engineering applications. In EHL problems the range of shear stress encountered is so large that the assumption of u yield stress in the flow equation will be shown to be of little consequence to the solution of these problems. Shear rates on the order of los to lo7 sec-' are involved in usual EIfL film thickness calculations. A replot of coaxial cylinder viscometer results for calcium (cup) grease, reported by Vinogradov and Mamakov ( l l ) , is given in Fig. 1. The intercept a t a shear rate of unity for the plot of T - r, gives the value of the plastic

w IP
rU).The plug flow region is of width hp where

In the upper shear flow region

Consider first the case of a Bingham plastic (n = 1). Then, Eq. [12] becomes

Substituting for

7,

from Eq. [4] in the first term only:

Now, clearly hP/h lies between zero and unity, so the term [l - (1/3)(hp/h)2] lies between unity and two-thirds. Thus, h - ho dp-- 12+u2x5 [I51 dx ha h

+

where A lies between 1 and 3/2, giving a modified Reynolds equation, taking into account the grease yield stress. For the general case of n # 1, the analysis is more elaborate, but leads to a similar result:


+

+

l)/(n 1). Omitting where A, lies between 1 and (2n the final term gives the result found by Dyson and Wilson (14) for a power law fluid. Equations [IS] and [16] may readily be integrated providing d and 7, depend on pressure in the same way. Unfortunately, nothing about the pressure dependence of either is known. I t is possible that the yield stress depends much more strongly on pressure than the plastic viscosity. This would tie in with the Johnson and Cameron discussion (15) of a shear plane mode of failure of oils a t very high pressure. In the absence of any data this possibility will not be pursued. CASE OF 9 = dotap AND

T,,

= rUOeap

For this analysis the usual assumptions of Gnibin's theory of EHL are chosen: that the film shape is that given by the Hertzian analysis of the dry contact of rollers, and that the pressure reaches infinitely high values a t the beginning of the parallel section. Crook's (16) approximate form for the film thickness outside the parallel section is taken :

where b is the half width of a Hertzian contact,

where

1 1 1 jj=z+Rz '

2

1-2

E' -=-+El

4

4

1 = ( 3 ~ h ~ ) ~ / ~ b690 - l /U ~ [ - "I2 sin7130 COS"~ 0d0 a 3 h;

Substituting $, = sin2 0 reduces the integrals to Beta functions with values

which, using z! = z(z - 1) !, and z!(-z) ! = ?rz/sin az gives n/92/3

and

?r

/&

[241

[after Jeffreys and Jeffreys (17)l. Therefore,

Substituting for b from Eq. [18] leads to

which is Crook's (16) expression for the film thickness with (3A/2)(7,ho/doU)]. Recalling an additional factor of [I that X lies between 1 and 1.5, the greatest change in the extra factor is given when X = 1.5. For the general case of n # 1, the analysis leads in the same manner to the result given by Dyson and Wilson (14) but with an additional term:

+

where,

for ( n > -

1-g E2

~191

it is possible to obtain

I x 1 - b = +(3Rh0)~/~b-l/~ tan4I30

Pol

T o integrate Eq. [IS], take out a factor e-aP to make the left side e-ap(dp/dx). Then,

-1 (1 - e-a~) = .4- (3Rho)21sb-l13 a!

4)

[281

A plot of I(n) over the range of interest for grease is given in Fig. 3.

Making the substitution h = ho sec2 e

Assuming that e-aP g 0 when 0 --+ 0 gives

3

POWER LAW EXPONENT.

Fig. 3-EHL

film thickness integral

+I

273

Elastohydrodynamic Lubrication With Herschel-Bulkley Model Greases

this term is never greater than 0.01. Thus, in all practical cases, the yield stress has a negligible effect on the film thick?Less.

Solving for ho from Eq. [27] gives

x

[(4

+ Xn7yoht -

+

LOW YIELD STRESS PRESSURE DEPENDENCE

~291

40U n

Again, the coefficients are of comparable size, so that the effect of the yield stress depends on the value of (r,,lz,"/ C$~U").Inspection of the values in Tables 1 and 2 shows that

If the yield stress does not increase with pressure as quickly as the plastic viscosity, then it can only influence the film thickness if its value a t ambient pressure is much higher than for the case above. This does not seem to occur


TABLE1-ROLLER THEORY AND EXPERIMENT COMPARED FOR A 10% LITHIUM HYDROXYSTEARATE GREASE

No. G I 3 (19) Li-Soap+90% Mineral Oil

35 35 75

0.130 0.130 0.025

2.3 2.3 0.72

2.2 2.2 .. .

(b)900+42D0.78 cc) 48+1.6D0.82 ...

(d)12 1.09

1.6 1.1

.. .

...

-

(8)

(" (0)

w = 2.86 X lo6 N/m, U = 3.68 m/sec, R = 2.2 X (3.57 0.985 log10 X m2/N. Little preshear. Presheared one week in a Klein mill. Neglects shear heating.

+

TABLE2-THEORETICAL

GREASE

m, E' = 2.27 X loL1N/m2, a,,.

ra,il =

PERFORMANCE OF GREASES I N ROLLERS

VISCOSITY FILM,pm TEMP. . 110 THEORY "C N-sec/mz EQ. [31]

KHEOLOGY(~) T , N/m2

No. 1 (12) Na-Complex Soap +86y0 Min. Oil

25 50 75

0.095 0.029 0.013

1.80 0.68 0.3.5

600+1. 5D0.84 420+0. 55D0.a4 350+1 .

NO. 2 (12) Ba-Complex Soap +77% Min. Oil

25 50 75

0.095 0.029 0.013

1.80 0.68 0.35

lOOO+l. 15D0.88 6004- 5. 5D0.86 450+5 .0D0.80

No. 5 (12) Gel. of Bentonite +90% Synth. Oil

25 50 75

0.048 0.019 0.008

1.03 0.48 0.23

-

No. 6 (12) Clay +?% Synthetic Oil -No. 7 (12) Silica+95% Mineral Oil

800+0 .37D1.O0 400+0.37D0~96 500+0.044D~~~0

1.91 1 .OO(c) 0.75Ce) 4.38 6.16 13.6

-

No. 8 (1 1) Calcium Soap +86y0 Spin. Oil +lyo Water

w = 2.86 X 106 N/m, U = 3.68 m/sec, R = 2.2 X 10-2 m, E' = 2.27 x 1O11N/m2, a,,, (3.57 0.985 loglo$0) X 10-8 m2/N. cb) Presheared a t D.= 7.2 X lo4 sec-I, except No. 8. ( 0 ) Presheared a t D = 5 X lo4 sec-1. (d) Small preshear history. ( 0 ) ?(grease) falls below base oil) a t max. shear rate. (")

+

~ a . , i l=

in practice, and no further analysis of this possibility need be considered. FLOW PATTERN I N THE INLET

If (+/rl/) is indc:.xident of pressure, Eq. [12] or [13] can be solved to find the variation of the width of the plug flow region. Results are plotted in Fig. 2 for a Bingham plastic assuming the nontypical value for (+oU/r,,ko) = 10. T h e plug width narrows to very low values immediately before the parallel section and stays small until the inlet has widened to ten times the minimum width. For larger values of (+OU/~,o/tO) the plug region becomes too narrow to show. Tt follows that in Eq. [IS] X must be very close to 3/2, and that the factor giving the effect of the yield stress on film thickness from Eq. [26] is very close to [I 2.25(~,ho/ +oU)I. Ultimately the plug widens to fill the inlet. T h a t is, //,,/I/--t 1; but the width of the shear zone ( h - h,) continues to increase. T o find the velocity of the plug flow region, substitute from Eq. [11] into Eq. [9] and get


+

so th:~t once the variation of It, has been found, the local plug velocity is determined and the complete velocity prolile easily follows. These are shown in Fig. 2. rt is clear that the plug is not a solid, moving as a rigid body. Although it moves through the parallel section with the velocity of the rollers, further out it moves more slowly, and a t large distances it moves backwards out of the inlet. 'I'his, of course, esactly parallels the flow field for oils. An interesting difference is that instead of the backward velocity steadily increasing to U/2 a t greater distances, the plug slows down again and ultimately comes to rest-a more pl:wsible behavior than that of oil. SOLUTIONS COMPARED

Equation (251 can be conipared with the known nun~erical solutions for a Newtonian lubricant as follows. T h e solutions obtained by Dowson and Higginson (4) follow the relation :

where (1, = W ~ I ~ / ( ~ ~ is U p) r~o ~ I~ o ~Rt i o n to a l the fluid pressure which would occur in the absence of elastohydrodynaniic effects, e.g., Greenwood (18). Setting T,, = 0 and l z = 1 in Eq. [29] there results

];or comparison with Eq. [31], Eq. [32] may be written as who UR

-= $0

Thus,

[331

For the conditions of Poon's (19) experiments (Table 1) aq, = 273 and ho[33]/hm;,.[31] = 1.57. Much of this discrepancy is due to comparing the parallel section film thickness with the minimum film thickness, which introduces a factor of about 1.3. Thus, Eq. [29] eives reasonable results for this case, and i t appears reasonable to use it to compare the film thicknesses for oil and grease.

ape = 18.7;

EXPERIMENTAL RESULTS

Figure 4 presents roller experiments for a lithium hydrosystearate grease tested by Poon (19). These tests a s well a s those of Dyson and Wilson (20) generally show that grease will develop a thicker film than its base oil for the same roller conditions a t first, but after a short time, the performance of the grease becomes poorer than its base oil. Only grease G1 tested by Dyson and Wilson (20) showed a small increase in film thickness with time, and this was attributed to an increase in viscosity of the grease due to selective evaporation of lighter fractions. In addition to these roller experiments, microscopic examination of greases confirms the observation that the soap structure is degraded by shear in the rollers such that the original fibrous structure is eventually reduced to small spherical particles of soap in the oil medium. T h e effect of shear degradation of calcium grease is clearly evident in Fig. 1. However, the preshear rate of 5 X lo4 sec-' for curve R is far below the maximum shear rate of lo6 to lo7 sec-I encountered in rollers. Thus, viscometer and roller experiments make it clear that the length oi time of application of shear is important to grease performance, a s well as the rate of shear a t which shearing occurs. An estimate of the equivalent percentage of shear degradation for different shear rates can be made using a shear stress decay function proposed by Bauer, Finkelstein, and Wiberley (8), of

T h e values of a and q did not vary greatly for the greases tested ,)'A( with q ranging from 0.39 to 0.51 a t 37.8 C. Extrapolating the flow curve for the 12 percent lithium grease a t 37.8 C in Fig. 11 (8), and assuming q = 0.39 in Eq. [35], gives

Equation [36] predicts that one hour a t D2 = lo5 set-I is the same as 1.3 minutes at Dl = lo6 sec-I. This result is in accordance with the roller and Klein-milled curves shown in Fig. 4, where one week in a Klein mill gives the same results as about 7 minutes in rollers. Viscometer data presented in Fig. 5 was used to estimate the flow curves for Poon's (19) GL3 grease. This d a t a was obtained on a Weissenberg cone and plate rheometer taking test points beginning a t low shear rates and being careful to avoid grease estrusion effects. T h e dashed portions represent curve extrapolation. In Fig. 6, concentric cylinder viscometer d a t a by Schol-

275


ment between theory and test for the grease to oil film ratio is good for the Klein-milled grease, but clearly the isothermal theory is incorrect for virgin grease. Shear rate and temperature rise are examined in the next sections as a possible explanation for the experimental results.

GREASE G L 3 ( 1 9 ) Li-SOAP 9 0 y 0 MINERAL O I L T 3 5 ' ~ . OIL F I L M = 2 . 2 p m

-

+

.

SHEAR RATE AND PRESSURE DISTRIBUTION

0 W

V)

AFTER I WEEK I N

w

A KLElN M I L L .

a

od

I

I

10

20

Fig. 4-EHL

I

" 90 I

I

I

30 4 0 TIME, min.

50

grease performance in rollers

G L - 3 (19 1 LITHIUM HYDROXYSTEARATE + 9 0 % MINERAL OIL

/'

An analysis for the shear rate encountered in the entrance region of rollers with the Herschel-Bulkley flow model of Eq. [I], neglecting the yield stress, is carried out following a similar analysis presented by Dyson and Wilson (14) for a Newtonian fluid. I t can be seen from the flow field sketched in Fig. 2 and from Eq. [3] that the maximum shear rate occurs a t the wall of the rollers. Thus, setting y = h/2 and 7, = 0 in Eq. [5] and rearranging gives


/

Substituting for p'/+ from Eq. [16], with

T,

= 0, results in

On maximizing Eq. [38], it is found that T(A)

-

D

=

Dm,

at

h

=

2ho

[391

Or.

9 0 0 + 42~''"

T(B1 - 4 8 + I . ~ D O ' ~ ~ KLElN MILLED I WEEK

I '06 SHEAR STRESS. NlmZ Fig. 5-Rheological

characteristics af a lithium hydroxystearate grease

A plot of D/D,,,,. is given in Fig. 7. The resulting curve is independent of the power law exponent n. Application of Eq. [40] to the lithium hydroxystearate grease in Table 1, line 1, results in Dm,,.= 1.7 X lo6sec-'. The pressure in the inlet of rollers can be estimated from Eq. [21], with 7, = 0. The result is ap =

-

BA COMPLEX SOAP

+ 77% MINERAL OIL(12) PRESHEARED AT 7.2x10'xc-1 T ~ 1000 ~ + 1~. 2 5 ~- ' " ~

- ln [ I

~ i ' ~ ( n= ) ] In I:" ( n ) z;12( n ) I:(.>

The pressure distribution in the inlet is plotted in Fig. 7 along with the shear rate a t the wall. The power law esponent has little effect on the pressure distribution, and Fig. 7 shows that the pressure rise is concentrated over a very small distance in the rollers inlet. These curves are

-

'Tdc 6 0 0 + 5 . 0 0 ~ ~ ' ~ ~ 'T.,5ec- 4 5 0 + 2 . 4 0 ~ ~ " ~

SHEAR STRESS. Fig. 6-Rheological

~lm'

characteristics for barium grease

ten (12) is plotted for a barium grease, with a maximum shear rate of D = 7.2 X lo4 sec-'. A plot of Eq. [I] in Fig. 6 shows reasonable agreement with the test data. Table 1 presents a summary based on results from Fig. 5 with theoretical results calculated using Eq. [31] and [29], with test results taken a t time zero from Fig. 4. The agree-

[411

Fig. 7-Wall

shear rate and pressure rise in roller inlet

not exact, due to the assumptions leading to Eq. [21], but the trends are in good agreement with more exact calculations (4). A computer solution for I:(n) was found necessary, as the function has not been tabulated in the range of interest. The values for k/ho and x/b are calculated using ISq. 1191 and [20] respectively.

The analysis for grease follows precisely that given above (the yield stress 7, is neglected) and leads to

ROLLER INLET TEMPERATURE

The dependence on n in Eq. (471 is not strong enough for the small deviations from unity found with greases to be significant. But the very much higher values of tbe plastic viscosity of grease compared to the viscosity of oils means that according to Eq. [47] and data in Tables 1 and 2, temperature rises of 20 to 50 C, and higher, are found in the inlet of rollers using grease.

The fuid passing through the inlet of rollers will experience very high viscous shear, as shown in Fig. 7. Crook (21) shows that heat transfer by convection is small compared to that by conduction, so that to a good approximation the heat produced in each element of fluid can be equated to the heat conducted away from the element, and for a Newtonian fluid


THEORY APPLIED

Crook solves this equation approximately to obtain an upper limit to the temperature rise. For his condition for oil thc values found were only 4 to 1 degree C. For experimental results Crook states, "The temperatures due to rolling friction were not influencing the film thickness." When grease is considered, the flow curves shown in Fig. 1 and 5 indicate that there may be considerable heating in the inlet since the shear stress is everywhere greater than for the base oil a t the same temperature. For the properties associated with greases, Crook's analysis leads to predictions of teniperature rise which are unduly high. A sinipler and better upper limit for temperature rise can be obtained by noting that since it is motion, not force, which is imposed on the grease, the effect of a reduction in viscosity must be to reduce the temperature rise. Thus, the teniperature rise for the case of constant viscosity is an overestimate when the viscosity falls with temperature. Taking the assunlption of constant viscosity a t any cross section, Eq. [42] is integrated twice, noting that dT/dy = 0 a t y = 0, and the result is

'l'he teniperature difference between the center line a t y = 0 and the surface of the roller a t y = k/2 is, therefore,

Substituting for dp/dx from the isothermal Reynolds equation,

gives the result wanted:

This differs from Crook's result in the omission of a factor e-7"' from the left side, where y is the temperature coeficient of viscosity.

Scholten (12) presents flow curves for five greases measured with a concentric cylinder viscometer. The greases were tested after being sheared a t D = 7.2 X lo4 sec-I until no apparent change in shear stress could be determined. His paper suggests that about an hour was the longest preshear time allowed. Equation [36] would indicate that an hour a t a shear rate of lo5 sec-' would correspond to only a few minutes in roliers where the maximum shear rate will be greater than lo6 sec-l. An example of Scholten's data for a barium-complex scap with 77 percent mineral oil is shown in Fig. 6 on an In-ln plot. The apparent yield stress determined from Fig. 6 is larger than Scholten's reported yield stress, but his discrepancy is not important since the effect of yield stress on EHL calculations is insignificant, as shown previously. A summary of flow curve equations and theoretical predictions for a number of greases is presented in Table 2. The theoretical EHL grease/oil film thickness ratio for calcium grease gives an opposite trend from that of the lithium grease in Table 1. The predicted values show that the calcium grease should increase its film thickness with time. I t is also interesting to note that although $0 decreases to 1/3 its initial value due to preshear, n increases from 0.63 to 0.77 to produce a thicker film. Unfortunately, no visconieter shear rate trends are available for the other greases listed in Table 2. The bentonite grease in Table 2 looks significantly better than the other greases as far as grease/oil film thickness ratio, but based on the results shown in Fig. 4 for the lithium grease, it must be concluded that roller tests are necessary before any steady-state performance expectations could be predicted. Table 2 includes results for several greases where, a t the maximum shear rate value in the inlet of rollers, the extrapolated viscometer curve for the grease crosses the curve for the base oil, as indicated by note (5). Dyson and Wilson (20) consider the thickening effect by the suspension of soap in the oil and find that the viscosity of the grease should always be greater than its base oil a t the same temperature. With the much greater shear heating of the grease, it is not inconceivable that the viscometer data in these cases is in error. On the other hand, shear



heating in the inlet of rollers has not been included in the theoretical film thickness calculation, so that this data may point to the explanation for the roller results where the grease/oil film thickness ratio falls below unity. Based on the general correlation between Poon's (19) roller esperiments (Fig. 4) and the theoretical predictions (Table 1) using viscometer data (Fig. S), it is suggested that theory based on viscometer data alone will predict grease performance in rollers. However, further experimental work is needed to establish confidence in applying this suggestion to the many types of greases available. The results shown in the section on temperature rise due to shear indicate that the film thickness theory can be improved considerably if the isothermal assumption is discarded in favor of a more realistic assunlption for the temperature distribution in the inlet of rollers.

277

versity of Virginia for its support of the senior author via an appointment as Sesquicentennial Associate of the Center for Advanced Study during 1970-1971. REFERENCES

(I) Hutton, J. F., "Greases," Chap. of: Cameron, A., "The Principles of Lubrication," John Wiley & Sons Inc., New York City, 1966, pp. 520-541. (2) Harris, J. H., "Lubricating Greases," Chap. of: Braithwaite, E. R., Ed., "Lubrication and Lubricants," Elsevier Pub. Co., New York City, 1967, pp. 197-268. (3) Cameron, A,, "The Principles of Lubrication," John Wiley & Sons Inc., New York City, 1966. (4) Dowson, D., and Higginson, G . R., "Elasto-Hydrodynamic Lubrication," Pergamon Press, N.Y., 1966. (5) Dyson, A., Naylor, H., and Wilson, A. R., "The Measurement of Oil-Film Thickness in Elastohydrodynamic Contacts," Proc. I . Meclt. E. 180. 3B, 119-134 (1965-66). (6) Sasaki, T., Mori, H., and Okino, N., "Theory of Grease Lubrication of Cylindrical Roller Bearings," Blrll. J S M E 3 , 212-219 (1960). (7) Sisko, A. W., "The Flow of Lubricating Greases," Ind. & Eng. Clte?n. 50, 1789-1 792 (1958). (8) Bauer, W. H., Finkelstein, A. D., and M'iberley, S. E., "Flow T h e derived theory presented in this paper may be used Properties of Lithium Stearate-Oil hlodel Greases as Functions for predicting grease elastohydrodynamic film thickness in of Soap Concentration and Temperature," Proc. ASLE 3, 215rollers. Within the limitations of the assumptions, the re224 (1960). (9) Herschel, W. H., and Bulkley, R., "hleasurement of Consistency sults appear to be supported by the available experimental as Applied to Rubber-Benzene Solutions," Proc. A S T M 26, evidence. 621-633 (1926). It is found that most greases behave pseudoplastically, (10) Mahncke, H. E., and Tabor, W., "A Simple Demonstration of Flow Type in Greases," Lzib. Evg. 11, 22-28 (19.55). and although their shear stress a t a given shear rate does (11) Vinogradov, G. V., and Mamakov, A. A., "Flow of Greases not fall below the shear stress of their corresponding base Under the Action of Complex Shear," Trans. ASME 90, F , oil a t the same teniperature, it may be possible to have a 604-607 (1968). smaller film thickness than the base oil due to the noniso(12) Scholten, G . J., "Apparent (Dynamic) Viscosity and Yield Strength of Greases after Prolonged Shearing a t High Shear thermal fluid mechanics of the grease in the inlet of the Rates," Proc. I . Meclt. E. 184, 3F, 32-39 (1969-70). rollers. (13) Pavlov, V. P., and Vinogradov, G. V., "Generalized CharacterShear thinning of a grease at a given shear rate in a istics of Rheological Properties of Greases," L~rb.Eng. 21, 479484 (1965). viscometer can be misleading, as in the case of calciuni (14) Dyson, A,, and Wilson, A. R., "Film Thicknesses in Elastohygrease, since some greases will shear thicken or shear thin drodynamic Lubrication by Silicone Fluids," Proc. I . Meclr. E . depending upon the shear rate. 180, 3K, 97-1 12 (1965-66). (15) Johnson, K . L., and Cameron, R., "Shear Behavior of ElastoThe available grease esperiments with rollers tend to hydrodynamic Oil Films a t High Rolling Contact Pressures," support the conclusion that the theory plus viscometer Proc. I . Afeclt. E . 182, 307-319 (1967-68). (16) Crook, A. W., "The Lubrication of Rollers TI. Film Thickness data may be used to predict grease performance. However, with Relation to Viscosity and Speed," Phil. Trapzs. London visconleter data a t higher shear rates than currently re254, A, 223-236 (1961). ported in the literature are needed for E H L predi.. r t' ~ ~ n ~ . (17) Jeffreys, H., and Jeffreys, B. A., "Methods of Mathematical Physics," 3rd Ed., Cambridge Univ. Press, Cambridge, England, A nonisothernial solution for film thickness analysis ap1962, pp. 462-463. pears to be required in order to esplain grease EHL per(16) Greenwood, J. A., "Presentation of Elastohydrodynamic FilmThickness Results," J. Mech. Eng. Sci. 11, 128-232 (1969). for~naticewith greater accuracy. (19) Poon, S. Y., "An Experimental Study of Grease in Elastohydrodynamic Lubrication," Report CUEDIC-hlechlTR3, Cambridge University Eng. Dept., 1969; In abstract, Proc. I . Afeclt. ACKNOWLEDGMENTS E. 184,3F, 97-99 (1969-70). (20) Dyson, A,, and Wilson, A. R., "Film Thicknesses in ElastohyT h e authors wish to express their sincere appreciation drodynamic Lubrication of Rollers by Greases," Proc. I. Meclt. E. 184, 3F, 1-11 (1969-70). to K. L. Johnson, Cambridge University, and A. Dyson, (21) Crook, A. W., "The Lubrication of Rollers 111. A Theoretical Shell Research Ltd., for their helpful discussions and sugDiscussion of Friction and the Temperatures in the Oil Film," gestions on grease E H L . Also, thanks are due to the Uni254, 237-258, (1961).