The theory of partial elastohydrodynamic contacts

49

Wear

Elsevier Sequoia S.A., Lausanne ~ Printed in the Netherlands

THE THEORY

OF PARTIAL

ELASTOHYDRODYNAMIC

CONTACTS

T. E. TALLIAN Research Laboratory. SKF Industries, Inc., Engineering and Research Center, King of Prussia, Pa. (U.S.A.) (Received December 14, 1971; in final form January 31, 1972)

SUMMARY

The concept of partial elastohydrodynamic (EHD) Hertzian contacts is defined between solids with real (rough) surfaces, using the film thickness/r.m.s. roughness ratio h/a as parameter. The influence of the partial EHD condition is described on EHD film thickness, normal load sharing between asperities and EHD film, frictional traction, mild and severe wear, surface fatigue and spalling fatigue. Kinematic effects of the traction us. sliding rate relationship characteristic of partial EHD are identified in rolling bearings. Rough surfaces are treated as two-dimensional random processes, using height, slope, curvature and level-crossing statistics derived, for Gaussian height distributions, with the aid of statistics of the height and its first two derivatives. Scanning electron microscope observations of surfaces are used to visualize asperity geometry, lay and defect population. Normal load and traction force sharing between EHD film and asperities is analyzed for plastic and elastic asperities. Indentation hardness and h/a are parameters of plastic load sharing. Elastic load sharing depends on elastic modulus, h/a, and asperity slope and may depend weakly on the spectral distribution of asperity amplitudes. Traction force sharing depends on similar parameters and on the EHD traction/ sliding rate function. Stresses in elastic asperities and the size of asperity contact areas are expressed in terms of roughness height, its derivatives and h/a. Roughness parameters influencing wear, smearing (galling), and surface fatigue are identified from the asperity contact stresses obtained.

I. INTRODUCTION

A type of Hertzian contact’ characterized by the co-existence of elastohydrodynamic (EHD) lubrication film and interacting asperities and designated partially elastohydrodynamic 1- 5 frequently exists in well lubricated rolling bearings. Several types of phenomena chaiacteristic of such bearings are explicable in terms of partial elastohydrodynamic contact. The most salient of these phenomena are : (a) The degree of asperity interaction is governed by the ratio h/a, where h is the average elastohydrodynamic film thickness (as it would exist over perfe.ct Wear. 2 1 (1972)

50

7‘. I.. TALLIAN

surfaces): and cr is the composite root mean square (r.m.s.) roughness of the surfaces [D=(~:+G:)~] where 0, is the r.m.s. roughness of one surface and C-J?is the r.m.s. roughness of the other. (b) Normal load is shared between the elastohydrodynamic film and interacting asperities. (c) The character of frictional traction between two surfaces in Hertzian contact is dependent on the degree of asperity interactions. The curve of friction coefficient versus sliding speed p==f’(ur -uZ) when the friction arises from interacting asperities, is characteristic of Coulomb-type boundary lubrication conditions. i.e. ,D is essentially independent of U, - u2z 0 (for constant surface temperatures). By contrast, the function p==f’(u, - u2) characteristic of an elastohydrodynamic film is typified by Fig. 1 and shows a linear rise for slow sliding speeds, then a peak. and finally a decrease in p for high sliding velocities. As asperity interactions diminish with increasing h/o, a transition takes place in the ,~=f’(u, -u2) function from the Coulomb type to the elastohydrodynamic type. Mixed characteristic curves prevail in the partial elastohydrodynamic regime.

SLIDING SPEED, u; u2

Fig. 1. General

trend of EHD traction

coeffkient

as a function

of sliding

speed

(d) Mild wear rates are dependent on the degree of asperity interaction, with wear steadily diminishing to insignificance as asperity interactions are reduced by increasing h/o. (e) A surface fatigue phenomenon characteristic of rolling contact, evidenced by plastic flow of surface asperities, and subsequent microcracking and micropitting, is dependent on asperity interaction in the sense that in the absence of interaction, surface fatigue is not observed. In the presence of interaction, surface fatigue depends on several surface and lubrication parameters in addition to h/a, i.e. for fixed h/a, the severity of the fatigue varies with these other parameters. ( f ) One of the two principal modes of spalling fatigue failure, surface originated spalling, is dependent on asperity interaction in that the scale parameter (or any given percentile life, say L 1o) of the ( Weibull) life distribution of spalling fatigue of surface origin increases as asperity interactions are reduced. Generally, the increase of h/o will accordingly lead to longer, surface originated, fatigue life. However, localized imperfections of the surface, which are not properly characterized by the surface’s overall roughness, are important originators of surface initiated spalling, and this type Wear, 21 (1972)

THE

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51

of spalling, while it may be dependent on elastohydrodynamic conditions, is not necessarily a function of h/a. (g) Asperity interaction is required for the onset of severe wear (smearing) in sliding Hertzian contacts, and thus a high maintained value of h/o throughout the entire running period prevents smearing. However, film thickness is unstable in the presence of high sliding speeds. When sliding speed increases, the film tends to become thicker as a result of higher oil entrainment velocity, but thinner due to viscosity drop arising from the increasing heat generated with increasing sliding velocity. If the latter effect prevails, smearing is often initiated through collapse of the elastohydrodynamic film just prior to the smearing occurrence, and this can happen even for high initial h/o values. (h) Due to the shape of the elastohydrodynamic p =f’(u, - UJ curve of Fig. 1, a fully elastohydrodynamic rolling bearing under light load can operate in a stable condition with the rolling elements showing little or no rotation around their own axes, i.e. they glide between the two rings. When load is rapidly increased, the rolling elements are no longer in stable equilibrium when sliding and will accelerate to their rolling angular velocity. This process may create sufficient inertial moments to force momentary sliding under high load, resulting in elastohydrodynamic film collapse, and a type of localized micro-smearing known in high speed bearing technology as skid marking. Owing to the Coulomb nature of the p=f’(ul -uz) curve for asperity contacts, a similar sequence of events will arise only in cases of intermittent total unloading, if hJo is low. The physical description of partially elastohydrodynamic contacts is in need of updating to provide better understanding of the multitude of phenomena which are dependent on the partial elastohydrodynamic condition. This paper attempts to provide such an updating, based on the findings of a number of recent investigations, of which the following are the most important for our purposes. Work performed in the author’s laboratory’-3 bases the description of the partial elastohydrodynamic condition on a theory of rough surface&‘, to which was added a wear mode18. The fatigue model’ utilizes elastic analysis of asperity interactions’. A significant improvement in the statistical description of the geometry of rough surfaces was provided by Nayak” which offers a coherent theory of Gaussian random surface geometry, for the isotropic case and the two-dimensional limiting case (with indications of possible expansion to general anisotropic cases). It correlates surface statistics with the statistics of linear profiles traced across the surfaces, and provides formulae and numerical material on the distribution of the height coordinate of surface asperities, of the asperity gradients, summit heights and summits of given height and curvature, all in terms of the power spectrum of the surface geometry process. The spectrum is characterized for isotropic Gaussian surfaces by only three independent variables, which can be either three moments of the power spectrum or the more readily measured profile characteristics of r.m.s. height, density of zero crossings and density of extrema. The sheer simplicity of a surface geometry model permitting description( in the isotropic case) by only three scalars, makes it doubtful that this model reflects physical realities faithfully, but it contains more than enough flexibility to serve our purposes. Pullen and Williamson” provided a new formula relating normal forces acting on plastically deforming asperities to the contact areas and deflections. Chiu9 Wear, 21 (1972)

-3 3_

I. I:. IAI.L.IAh

provided a similar relationship for elastically deforming asperities of a specialized geometry.* Finally, elastohydrodynamic film thickness and traction through elastohydrodynamic films discussed in many articles ‘2.14~-z3 is the background used in the present paper. It. DESCRIPTION

OF ROUGH

SURFACES

Experimental knowledge regarding the topography of surfaces as used in rolling contacts has, until recently, been essentially limited to tracer type profilometry. Recent observations from scanning electron microscopy will be discussed later. The load carrying surfaces of rolling bearings are made by abrasive finishing and profilometry results show that the profile heights are generally well approximated by a Gaussian distribution. A similar statement holds for the distribution of profile slopes, i.~. the amplitudes of the derivative of the profile process. Direct facts concerning the auto-correlation function or the spectrum of surface roughness profiles are not readily available for these surfaces. (This contrasts to the long-wavelength waviness of the surfaces, the spectrum of which has been well described.) We have, however, data on the( average) density of crossings not only over the centerline( zero line) of the profile, but also over levels above the zero line. These findings6 appear compatible with

showing a Gaussian function as the multiplicative factor relating average density of zero crossings to that of crossings at other levels. Once a profile has crossed from below a given level to above that level, it will dwell above it for a finite length before dipping below again. The average dwell length ah above level h is related to the density LV~of crossings above that level by the equation:

a, =

[l-F(h)]; h

where F (h) is the cumulative height distribution for argument h. Equation (2) relates expected values. For any given level h, the dwell time d, has a distribution around LI, experimentally determined by protilometry and compatible with the following exponential cumulative distribution :

All of the above formulae apply to linear profiles taken by tracing with a stylus instrument. Nayak” supplies the following additional formulae of present interest. Firstly, for profiles, the distribution of heights (amplitudes) _r is Gaussian with the frequency function p(y) given by : * A paper by Greenwood and Tripp13. late for utilization in this work. Weur. 21 (1972)

analyzing

surface asperity

interactions.

became

available

too

THE THEORY OF PARTIAL ELASTOHYDRODYNAMIC

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53

where CJis the r.m.s. roughness and p(y/a) is the standardized Gaussian frequency function. The frequency function for the slope 0 of the profile (the derivative of the profile height record) is Gaussian : (5) where rsBis the r.m.s. slope related to the profile height record by: (70= 3c(T&

6)

The following formulae pertain to peaks( local maxima) of the profile. These must be distinguished from summits (local maxima) of the three-dimensional surface roughness. One defines a density N,, of profile peaks which equals half the density is,(Q) of zero crossings of the slope process and then defines the following auxiliary quantity, tl: (7) Note that 2EP is the total density of peaks and valleys (maxima and minima) taken together. Applying eqn. (6) to the slope process, and designating the r.m.s. value of the second derivative of the profile process as c?: cr” = mr@Ro(0)

Sol&g eqn. (6) for No and eqn. (8) for @,(Q) and substituting into eqn. (7). the following alternative form is obtained for ~1: (9) The distribution of peak heights on a profile is plotted by Nayak” as a function of CL,and the plot is reproduced here as Fig. 2. The parameter a defines the width of the power spectrum of the random process forming the surface from which the profile is taken. If CL=I, the spectrum consists of a single frequency. If CI= CD,the spectrum extends over all frequencies. The standardized cumulative peak height distribution $ =~,/a (corresponding to the standardized profile height distribution y* = y/o) as a function of o1taken from Nayak” (Table I)shows that a profile with a very narrow spectrum has no peaks below the centerline, whereas a “white noise” profile with infinite spectral width has half its peaks below this line. The fraction of peaks above any given level increases as the spectrum becomes broader. Nayak lo has calculated the expected value of the curvature at profile peaks &,. Figure 3 shows a non-dimensional form of this curvature with the following expression as ordinate :

Wear.21(1972)

54

‘1’.1,. TALLIAN

SI,MMIl

HEIGHT,

y,*:

Y, /m

Fig. 2. Probability density for standardized heights of peaks on a profile

TABLE

I

CXJMULATIVE i’ * “P

OF STANDARDIZED

PEAK HEIGHT

DISTRIBUTION

F, (y,*)

F, r=l

0.0000 0.2500 0.5000 0.7500 1.0000 1.2500 1.sxlO I .7500 2.0000 2.2500 2.5000 2.7500 3.0000

FORM

1.5

O.OiIOO 0.0918 0.0308 0.1624 0.1175 0.2590 0.2452 0.3757 0.3934 0.5021 0.5422 0.6254 0.6753 0.7347 0.7837 0.8234 0.8647 0.8895 0.9204 0.9350 0.9561 0.9641 0.9772 0.9814 0.9889 0.9909

2.0

3.0

4.0

5.0

10

50

0.1464 0.2278 0.3288 0.4428 0.5605 0.6719 0.7688 0.8465 0.9041 0.9437 0.9689 0.9839 0.992 1

0.2113 0.3012 0.4047 0.5149 0.6234 0.7226 0.8065 0.8726 0.9209 0.9537 0.9745 0.9868 0.9936

0.2500 0.3435 0.4474 0.5548 0.658 1 0.7505 0.8274 0.8871 0.9303 0.9594 0.9777 0.9885 0.9944

0.2164 0.3178 0.4756 0.5810 0.6807 0.7686 0.8409 0.8965 0.9364 0.963 1 0.9798 0.9896 0.9950

0.3419 0.4405 0.5429 0.6425 0.7331 0.8104 0.8720 0.9182 0.9505 0.9717 0.9847 0.9922 0.9963

0.4293 0.5292 0.6273 0.7177 0.7960 0.8597 0.9083 0.943 1 0.9666 0.9814 0.9902 0.9952 0.9977

0.5000 0.5988 0.6915 0.773 I 0.8413 0.8943 0.9332 0.9599 0.9772 0.987X 0.9938 0.9970 0.9987

The variation with peak height, of the dimensionless expected curvature 5 is a function of the parameter cxonly. For a white noise spectrum (a = co) the expected peak curvatures do not vary with peak height. For all other spectra, peaks become more sharply curved as their height increases. The expected value of the actual curvature 2, is obtained from Z: using eqn. (10). The scale fact& contains only the r.m.s. value of the second derivative of the profile. Wear,21 (1972)

THE

THEORY

OF

PARTIAL

ELASTOHYDRODYNAMIC

PEAK

Fig. 3. Expected

value of dimensionless

tip curvature

HEIGHT,

CONTACTS

y+= P

y

P/C

for peaks on a profile

The next group of formulae from Nayak” pertain to the three-dimensional random process itself and apply to the isotropic case. A height maximum of the threedimensional random process is designated as a summit. The density of summits per unit area is: nT,2 1.2n;

(11)

The frequency function of summits of given non-dimensional height y: = y,/a is shown in Fig. 4, again as a function of the parameter cc For isotropic processes, ~11.5 and c( is again related to the width of the spectrum (the narrowest spectrum corresponding to M= 1.5). An important finding is that the o!obtained from a randomly oriented profile on the isotropic surface according to eqn.(7) serves to characterize the three-dimensional process in Fig. 4. The cumulative probability density of summits not exceeding a fixed non-dimensional height is tabulated in Table II, similar in structure to Table I, and permits a comparison of the distribution of peaks and summits. Summits above a given level y,/o are much more numerous than profile peaks owing to the fact that profiles have no finite probability of crossing summits exactly, but rather traverse the slopes of the asperity hills. Defining the mean curvature at the summit in the usual way as the average of the two principal curvatures, and designating its expected value as Z,,,,a non-dimensional curvature is defined in analogy to eqn. (10) as follows : x* =

JJK a” In

(‘2)

The expected value of the dimensionless mean curvature for summits of given nonWear,

21 (1972)

56

r. I:. TALLIAN

a = m. m2/m4 06

PEAK

Fig. 4. Probability

TABLE

*

WEIGHT

summit

’ y*=P

y

P/T

heights.

FORM

OF

STANDARDIZED

SUMMIT

HEIGHT

DISTRIBUTION

F,(L.,*)

4 2=

0.0 0.25 0.50 0.75 1.00 I .25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

for standardized

II

CUMULATIVE ?‘.s

density

1.5

0.0 0.0001 0.0037 0.0237 0.0785 0.1787 0.3 176 0.4745 0.6254 0.7525 0.8482 0.9134 0.9539

2

3

4

5

10

x

0.0129 0.0306 0.0706 0.1296 0.2281 0.3381 0.4794 0.6030 0.7293 0.8187 0.8936 0.9375 0.9625

0.0601 0.1103 0.1739 0.2696 0.3701 0.4955 0.6052 0.7199 0.8042 0.8785 0.9247 0.9594 0.9778

0.1026 0.1686 0.2446 0.3496 0.4521 0.5724 0.6724 0.7725 0.8435 0.9044 0.9414 0.9687 0.9830

0.1363 0.2118 0.2943 0.403 1 0.5050 0.6205 0.7134 0.8040 0.8668 0.9196 0.95 12 0.9742 0.9861

0.2327 0.3263 0.4187 0.5296 0.625 1 0.7251 0.8000 0.8689 0.9138 0.9498 0.9704 0.9848 0.9920

0.5 0.5987 0.6915 0.7733 0.8413 0.8943 0.9332 0.9599 0.9772 0.9878 0.9938 0.9970 0.9987

dimensional height is shown in Fig. 5. The mean curvature of the summits is generally not greatly different from the curvature of the peaks. An exception is the case of narrow spectra (~(5 2.4) where the profile peaks have substantially greater curvature than the summits. The frequency distribution for heights of the three-dimensional surface roughness process is, as for profiles: yieur. 21 (1972)

THE

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&

P(Y) =

ELASTOHYDRODYNAMIC

57

CONTACTS

[-t(s)‘]

exp

(13)

with (r the r.m.s. roughness determined from a profile. A quantity analogous to the slope of a profile is the surface gradient y, i.e. the steepest slope proceeding from a given point. The frequency distribution of the surface gradient (always positive) is given by :

p(7) =

$yp[-+(k)2]

(14)

where gOis the r.m.s. profile slope defined in eqn.(6). Note that the distribution of the gradient is not Gaussian, but is a Rayleigh distribution. The expected value of the gradient is : y = (7c/2)f 00

(15)

i.e. it is proportional to the r.m.s. profile slope. In a,degenerate two-dimensional roughness process composed of parallel prismatic ridges, the probability distribution of heights is the same as that for the profile, transverse to the direction of the ridges. The surface gradient becomes equal to the absolute value of the transverse profile slope (The absolute value of the slope is not normally distributed.) The density of summits is the same as that of transverse profile peaks, the mean summit curvature is identical to the profile peak curvature, and the frequency distribution of summits of given curvature and height is the same as the frequency distribution of transverse profile peaks of the same curvature and height. A quantity not discussed by Nayak is the number and area of two-dimensional

sT~NO~H”,ZED

Fig. 5. Expected Wear, 21 (1972)

mean standardized

curvature

SIJMMIT

of summits

HEIGHT,

y,z = Y/m

as a function

of standardized

summit

height.

crossings of the surface over given levels (analog to the number and dwell length of one-dimensional level corssings of the profile). A two-dimensional crossing over level /J is defined’ as the(closed)contour line connecting the points on the surface which lie at level h. It was concluded’ that the density ,,N of such two-dimensional crossings can be obtained on anisotropic surfaces by multiplying the density of one-dimensional profile crossings taken in two perpendicular directions on the surface, and, on an isotropic surface, by squaring the number N,, of one-dimensional crossings. i.c).: hlv = ,y,z

(16)

An analogy to Nayak’s result concerning summits [eqn. (1 I)] would suggest ,,N = 1.2 !Vt. A resolution of this 20’%, difference has not been found to date. Therefore, eqn. (16) will continue to be used. (It is noted7 that eqn. (16) holds only for areas with linear dimensions large by comparison to l/N,, in all directions. For narrow areas, a correction applies7.) No expression for the distribution of individual “hill” areas oi circumscribed by two-dimensional crossing lines has been found in the literature to date.An approximation to the average of an individual hill area is: (17) The area fraction a/a, covered by the totality of “hills” in excess of a given level h equals simply the cumulative probability of heights in excess of level h, i.e. it is, based on eqn. (13) u

= p(.r>h)=l-cf,

(18)

~~0

where G(x) is the cumulative Gaussian distribution function for zero mean and unit variahce, taken for the argument x. This is exactly the same fraction as obtained from eqn. (4) for the total dwell of a profile above the level h. Thus, surface area fractions in excess of given levels are not distorted by calculating them from profiles. In determining asperity interactions, the roughness process of interest is the composite process defined as the sum of the two roughness processes characterizing the surfaces in contact6, obtained by adding together the heights of the two asperities found at identical horizontal coordinates, i.e. J’ = 1’1+ 1’2

(19)

Asperity contacts will occur through an EHD film of thickness h when J 2 h and asperity deflections can be calculated by examining a surface of the composite roughness in contact with a smooth rigid plane. This composite roughness process is certainly Gaussian’ if the constituent processes are, and is closer to Gaussian than the constituent processes if the latter deviate slightly from Gaussian. The local slope of the composite process is equal to the sum of slopes, i.e. 0 = H, +u, The r.m.s. height of the composite process, its auto-correlation function, trum and number of zero crossings are, in turn, obtained as follows: Wear. 21 (1972)

(20) spec-

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r.m.s. height: a2=a*+o;

(21)

1

auto-correlation function :

(22)

R(7)= R,‘,z)+R,(z) power spectrum : SW=

(23)

S,(4+S*@)

zero crossing

density :

These equations state that variances, auto-correlation functions, and power spectra are simply additive. The squared zero crossing density is obtained as the weighted average of squared zero crossing densities for the constituent processes, with variances (squared r.m.s. roughnesses) was weighting factors. From eqn. (22) one can derive for the slopes of the composite process: r.m.s. slope: CJf= CJ;.,+a;.,

(25)

and likewise, by applying eqn. (22) to the derivative process, one obtains for the r.m.s. value of the second derivative: r.m.s. second derivative: 0

If 2

III. VISUAL

=[rl ‘I2 +

(26)

4”

APPEARANCE

OF ROLLING

CONTACT

SURFACES

In fixing ideas about the topography of rolling contact surfaces, their visual appearance would intuitively seem to be a big help. However, optical micrographs of abrasively finished fine surfaces are singularly unrevealing, due to the lack of depth resolution and the visually unhelpful shading of these micrographs. The scanning electron microscope (SEM) has recently made available a wealth of highly descriptive surface micrographs I4 . Figure 6 shows the unrun honed track surface of a bearing ring. Parallel ridges are clearly in evidence and a deep “furrow” was created by an abrasive particle. Figure 7 shows a similar surface, but here, localized defects are in evidence, shown at high magnification in Fig. 8. These defects have “raised edges”, i.e. they provide asperity summits, whereas the furrow in Fig. 6 is a depression. A completely different surface appearance is exhibited in Fig. 9, by nital etch after honing, a practice known to be innocuous regarding fatigue life. A large number of white-appearing elevated carbides have appeared, which partly obliterate the parallel ridges created by honing. Also seen are non-metallic inclusions (elliptical shapes) which may or may not be elevated above the surface. After considerable running( 200 million bearing revolutions) without spalling failure, the etched surface appears as a great profusion of depressed defects and comparatively little is left of the original carbide or ridge structure (Fig. 10). Figure 11 depicts an originally existing pair of Wear, 21 (1972)

60

sutfact:note dark furrow

Fig. 6. Scanmng

electron

mluograph

of hvned

Iring

FIN. 7. Scanning

electron

micrograph

of honed

ring surface

defects on honed

Fig. 9. Nital etched Wear. 21 (1972)

honed

ring surface

ring surface

(SEM).

localized

dcfccts

1000x

JUOOX Fig 8. Localized

showng

(SEM)

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CONTACTS

deep furrows, with the material adjacent to the furrows now having undergone much plastic deformation and cracking. Figure 12 shows an unrun surface of a bearing ball composed of a multitude of depressed scratches in random directions. On a run ring or ball surface occasional

3000x Fig. 10. Etched Fig, 11. Bearing

track track

surface surface

after 200 million after running

3000x Fig. 12. Lapped Wear. 21 (1972

bearing

ball surface

(SEM).

bearing

revolutions

(SEM). Note plastically

(SEM) deformed

furrow

edges

62

I-. I T.Al..I.IAN

clusters of curved “scratches” which form in rolling tracks of ball bearings after running under clean operating conditions are found, (Figs. 25 and 26). Similar scratches have been observed on balls. The kinematics of formation of these curved scratches have been interpreted’” using a personal communication from K.L. Johnson to show that they form, when a point on one of the surfaces rolls with spin on the other surface and acts as a “plow” which gouges a scratch into the opposing surface. The curvature of the scratch is due to the spin, and the length of the scratch is comparable to the dwell time of an asperity in the contact. These scratches continue to form on long running surfaces and are not solely a run-in phenomenon. In interpreting scanning electron micrographs, it must be borne in mind that visually based estimates ofthe depth of features can be highly misleading stereo, micrography is required, and very few stereo-micrographs of bearing surfaces have been published. Also, the horizontal scale of the micrographs, e.y. Fig. 6, is magnified five thousand-fold. The typical mean asperity spacing obtained from profile tracing across a bearing ring surface of this type is of the order of several hundred pin. Taking 300 @r. as an example, this spacing spans much of the picture (see arrows at the bottom of the Figure). The visible ridge structure is much more dense, i.e. the SEM appearance does not reflect the dominant roughness amplitude features, but rather, a “fine structure” superimposed on them. Dominant amplitude features must still be obtained by tracing. What can be learned from SEM micrographs are several things about the fine structure of the surfaces. (a) A honed ring surface is almost totally two-dimensional in its visible roughness. It does not follow that the main amplitudes (hills and valleys) are two-dimensional, since these are not visible in the scanning microscope. Thus, summits can rise along the visible ridges. However, the strong predominance of the visible ridges certainly suggests that the abrasion of the surface has taken place unidirectionally and one would expect all roughness, including the dominant amplitudes, to be much more closely spaced in the direction transverse to the ridges (in the language of roughness standards: across the lay), than they are longitudinally. On the other hand, a ball surface obtained by random lapping, is of isotropic appearance and its roughness is expected to be isotropic. (b) At spacings of the order of l/IO the dominant asperity spacing, ring surfaces show relatively sharp, although perhaps shallow, features which create contrast in the SEM. These can either be depressed scratches or “feather edges” at the crest of ridges. They persist after considerable running suggesting that many are depressions. On ball surfaces, the linear features are randomly oriented depressions. (c) In addition to the systematic ridge pattern, rings show localized defects of at least two kinds: first, roughly equi-axial defects placed randomly which are nicks or pits of various types (predominantly nicks in new surfaces and nicks as well as pits in run surfaces). The other type consists of furrows i.e. finishing scratches running parallel with the main lay. However, these furrows differ from the typical texture in that they are sharp-sided and relatively deep. (d) Scanning microscopic appearance of a surface can be drastically altered, e.g. by etching (see Fig. 9), with no known major effect on the behavior of the surface in service. The alteration is not limited to topography : the etched surface is a mosaic of carbides, presumably of higher elastic modulus and hardness than the matrix, protruding above this matrix to some unknown extent. The unetched surface is covered Wear, 21 (1972)

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63

with matrix material. After considerable running, the etch effect disappears and one can no longer identify the presence or absence of prior etching from the SEM appearance. (e) During running(even under relatively well controlled laboratory conditions) the surface undergoes a great deal of plastic deformation on the SEM scale. There is extensive polishing-over of the ridges, and multitudes of pits and nicks, (sharp-sided depressions of very irregular shape) arise. In the presence of good lubrication, this condition does not cause rapid macroscopic failure. Extensive plastic deformation will not obliterate a sharp-sided furrow or other sharp defect (Fig. 11). Indeed, the plastically deformed surface at the furrow appears more disturbed than the original surface, by cracks, pits and other irregular features. Thus, running appears to smooth over the general ridginess but is ineffective in removing, sharp depressions and may make them more harmful. From visual appearance, one may describe ring surfaces finished by unidirectional honing, as composed of three distinct roughness processes. Firstly, the profilometrically evident gentle sloped “dominant” asperity population which determines the r.m.s. roughness and the density of zero crossings. Statistical data of the type discussed in Section II have been obtained from profile traces and will be supplied in the next section. Overlaying this macro-roughness is a system of SEM-visible ridges, which must have smaller amplitudes than the dominant asperities. Their height characteristics are not adequately known. The ridge and valley lines of this visible texture appear to be narrow, relatively sharp crests and scratches. The crests are probably not shown on profile tracings because the stylus deforms them. The valley bottoms are probably bridged by the stylus radius. Thirdly, there is a localized defect population which consists of furrows, nicks, dents and pits with an occasional raised “feather” edge discernible in profile tracings. Their traced shape, however, is almost certainly distorted, by the blunt tip-angle of all tracing styli and limits the maximum discernible slope to about 45”. An apparently innocuous etching can extensively change the SEM-visiti asperity structure of the surface, although the dominant surface profile changes very little. There is also little change in the performance of the surface under rolling contact. It is necessary to bear in mind both the visual and the tracing description of surfaces in order to discuss their function in partial elastohydrodynamic contact. IV. TRACING

RESULTS

FOR ROLLING

CONTACT

SURFACES

A number of surface tracing analyses have been conducted to determine height and slope distributions as well as crossing densities and dwell times on ring and ball surfaces1.2,4,9,26,27. Table III is a summary of the principal published results and unpublished readings. Most surfaces examined had approximately Gaussian height and slope distribution; lapped ball surfaces, tended to be skewed with a lesser number of high peaks than deep valleys, but were Gaussian above their mid-plane. A direct experiment with analog summing of two profile processes obtained from a pair of surfaces forming a rolling contact has yielded the composite process described in eqns. (21) to(25). It was found that eqn. (21) f0rr.m.s. height summation is Wear, 21 (1972)

4

N

s

N z

2 .-

III

VALUES

shown

ring ring ring ring ring ring ring ring ring brg. ring brg. ring ring ring

Notes: Zero crossings

Lapped ball l/4 in. Lapped ball l/4 in. Lapped ball l/4 in. Ground ball l/4 in. Honed ball bearing Honed ball bearing Honed ball bearing Honed ball bearing Honed ball bearing Honed ball bearing Haned ball bearing Ground ball bearing Ground ball bearing Partially honed ball Partially honed ball Honed ball bearing Honed ball bearing

Surfhce type

EXPERIMENTAL

TABLE

arc calculated

14 1.9 2.7 2.1 3.5 3.3 4 2 7.0 0.5 4.0 1.0 2.5 0.6

1.2 13 1.8 2.2 2.8 8.1 8.8 3.0 1.6 14.2 5.5 6.5 2.0 2.2 1.5

in parentheses

0.4

gll

Angk

r.m.s. slope

ASPERIn

0.4

0.4

kei
r.m.s.

OF SURFACE

values. Groups

0.244 2.033 0.045 0.037 0.061 0.057 0.070 0.035 0.122 0.009 0.070 0.018 0.043 0.010

0.007

STATISTICS

with identical

(2100) (7400) (7ooo) (2700) (510) (3400) (2800) (6300) (2200)

(6800) (4200) (2400)

;~oo, 2-4 x lo3 (6000) (5500)

ZWO crossings (Noiin.)

numbers

Unrun Unrun Run Run Unrun Unrun Run in Unrun Run in Unrun Run in Unrun Unrun Unrun Unrun Unrun Unrun

HistorI.

heavy mineral oil( 1) (2) thin polybutene (2) (3) diester (3)

(I)

(I). (2).or (3) drt 1from the same manufacturing

Random Random Random Random Across lay Across lay Across lay Across lay Across lay Across lay Across lay Across lay Along lay Across lay Along lay Across lay Along lay

Tracing direction

lot

Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. New New New New New NW

3 Y 2 9 Y 14 14 14 14 14 14 readings readings readings readings I-cadings rcndings

THE

THEORY

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65

valid. So is eqn. (24) within the experimental error. The experimental summation also showed that deviations of the constituent surface roughness processes, from a Gaussian amplitudedistribution disappeared into insignificance in the composite as a result of the summation. Equations (22) and (23) have not been directly verified. However, they depend for their validity only on rather broad assumptions of independence and stationarity of the two processes and are believed to be well founded. The set of eqns. (21), (24), and (25) h as important consequences for the behavior of partial elastohydrodynamic contacts. Table III shows that in ball to race contacts, the ball quite generally is smoother than the race by a factor of the order of 5. The r.m.s. slope CJ*of the ball is also smaller by a comparable or greater factor. We have no direct measurement of the crossing density in a ring, but eqn. (6) yields values of 2-5 x 103/in. From eqns. (21) and (25), the height and slope distribution of the composite process is essentially that of the ring. From eqn. (25) the number of zero crossings of the composite roughness in a ball bearing is also virtually the same as that of the ring. Thus, in the partial elastohydrodynamic contact of a ball/ring track pair, the surface texture (including the isotropy properties) of the ring surface dominates. The previous section shows that ring surfaces are quite anisotropic( systems of parallel ridges), while balls are isotropic. Owing to ring effect dominance, the composite roughness is anisotropic with ridges running in the direction of rolling on the ring track. We cannot conclude, however, that this anisotropy applies to the dominant roughness essentially invisible in the SEM to the same extent it applies to the microroughness observed in the SEM. It is awkward to trace circumferentially with a sharp stylus, a surface as anisotropic as a ring surface, as the stylus may follow one valley or weave between valleys. The two actions may give substantially different results. Table III contains some results from tracings of the same ring surface, both transversal and circumferential. Both r.m.s. height and slope are much smaller in the circumferential traces. Since the true r.m.s. height of a surface does not depend on tracing direction, the measured lower value must have arisen from the stylus being guided by a valley. This makes the r.m.s. slopes suspect. While true r.m.s. slopes, are expected to be smaller circumferentially, it is not known how much of the measured reduction is due to anisotropy and how much to stylus guiding. One is left, accordingly, with the qualitative statement that the stylus-traceable macro-roughness of honed ring surfaces is anisotropic but the degree of anisotropy remains to be established. One practically awkward but theoretically correct way not yet undertaken is to trace across the lay at two finite angles. Results from the two traces can then be analyzed to yield longitudinal and transversal profile parameters. In roller bearings, a lesser degree of dominance of ring surface roughness is likely than in ball bearings, since roller surfaces are not as smooth as those of balls. However, the ring surface is still rougher than that of the rollers and both rollers and rings have lay in the rolling direction. Thus, it is reasonable to assume that the composite roughness process governing asperity interactions in all rolling bearings is anisotropic with ridge systems running in the rolling direction (or in the direction of the roughness lay where the lay is not circumferential). The statistical description of the composite roughness process is better approximated by the parameters of a degenerate two-dimensional process having the statistics of profiles taken across the lay, than by an isotropic process. Specifically, peak densities and summit densities are likely to be Wear,21

(1972)

7. t TALLIAN

66 related

by the following

equations:

I?, 2 lV; and not by eqn. (I 1). The average gradient absolute slope : ;’ 2 i? =(2/r+

will be closely approximated

(27) by the average

crB

(2X) The number of summits in excess of a given height may still differ from the number of profile peaks since a profile is still unlikely to hit summits (which will be spaced far apart along the ridges). The expected value of the mean curvatures of the summits will be nearly one-half the expected curvatures of the profile peaks %I=

!Zhp

(29)

since curvatures in the direction of the lay will be very small, and the mean summit curvature is the average of the curvature across and along the lay. Another peculiarity of practical surface profiles is illustrated by the profile trace in Fig. 13 across the track of a bearing ring after running showing major ridges separated by valleys (extending below the mid-line) with a multitude of superimposed small peaks. The existence of these major ridges is important to the model discussed in the next section, describing load carrying by elastic asperity deflection.

RANK

-._ ^

TAYLOR

- HO%SON

Fig. 13. Surface profile trace across run ring surface.

quences proceed.

With these results, discussion of film thickness, forces, stresses and their conseregarding material survival in partial elastohydrodynamic contacts can

V. NORMAL

LOADS

IN PARTIAL

ELASTOHYDRODYNAMIC

CONTACTS

A. Plastic asperity deformation The literature on film thicknesses and pressures in pure elastohydrodynamic contact is extensive. For recent results, pertinent to rolling bearings, see ref. 12 and its bibliography. When there are asperity interactions in addition to the elastohydrodynamic film, the question arises how the normal load is shared. As an upper limit estimate of Wear, 21 (1972)

THE

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CONTACTS

67

the portion of the load carried by asperities, it cannot be greater than the load which plastically deforms the asperities to obtain the actual area of asperity contact’. However, the simple proportionality between area of contact and total load suggested in ref. 1 is no longer claimed. Pullen and Williamson 1‘, show by experiment and calculation that plastic deformation of asperities( against a rigid plane) takes place according to the following two rules :(i) the tips of the asperities are depressed by the rigid plane until the contact pressures reach equilibrium with the external load; and (ii) volume of the plastically deformed material is conserved because all non-contacting elements of the surface rise without distortion of the surface geometry. This deformation corresponds to the following conceptual experiment: (a) remove the material volume contained in the asperities above the level to which they are being depressed; and (b) add this volume “hydrostatically” to the bulk of the subsurface material. As a result of this deformation mechanism, the following formula is derived” for the total load C&P in plastic asperity contact :

Qo,p =

Pa,, *

1 - a/a0

where P

= plastic yield pressure (indentation hardness P z 3~,), OCl = tensile yield strength, a, = nominal (Hertz) contact area, a/q, = the fraction of total Hertz area covered by asperity contacts. The most obvious conclusion from eqn. (30) is that for a value of u/a,=OS, the fraction (a/~,)/(1 - a/a,) is unity and the asperities will carry a total indentation load Pa, equal to the limit bulk plastic deformation (brinelling) load for the area a,. If the total load is further increased, brinelling takes place. a, increases to keep P constant, but a/a, remains at 0.5. Accordingly, in an unconfined (Hertzian) contact, a/a,, can never exceed 0.5 (if the contact materials are homogeneous). As long as the contact is predominantly elastohydrodynamic, i.e. most of the load is carried by the EHD film, the total asperity contact area is not determined by load but by the elastohydrodynamic film thickness which defines the separation between the surfaces. The relation between a/a, and average surface separation is obtained as follows2’. 1. The fraction a/a0 of the Hertz area a,, which is occupied by asperity contacts, equals the probability that at any given point, the composite roughness height y exceeds a fixed value y, above the mean of the composite roughness process, where y, is determined by the film thickness in a manner to be described. 2. The distribution of heights above the centerline is, according to eqns. (4) and (12), Gaussian with standard deviation g, both for an isotropic three -dimensional roughness process and for a two-dimensional ridge process. (This distribution is believed to apply equally to all intermediate degrees of anisotropy.) The cumulative distribution of heights y above the process centerline defines the probability of ordinates y 5 y, and is given in eqn. (18) i.e. the fraction a/a0 is : a/a,=prob(yZy,)= Wear, 21 (1972)

I-4

F 0

(31)

where y,, = contact level above the midplane of the composite roughness, 4 (~~~10)= cumulative standardized Gaussian distribution with argument J’~~.;cJ. As shown in eqn. (23) the average separation, S, i.e., the mean height of empty space (EHD film) between surfaces is not equal to yO, the spacing between the mean planes of the undeformed profiles. The average separation S differs from y. because. at the points ofasperity contact, the local separation is zero, not negutiue, a fact reflected in the following expression for the mean separation S of the deformed composite surfaces

(32) where, from eqn. (4):

and the composite surface height when the undeformed mean planes of the profiles y, and y2 are spaced by yO. Also, CJ= (g: + a:)*, the composite r.m.s. roughness. The lower limit of integration in eqn. (32) is zero because only positive local separation values can exist, except at asperity contact points where local separation is zero. The lower limit of integration would be - x. in the absence of asperity contacts and would then yield y, as the integral. With zero the lower limit of integration, one obtains (33) where cp is the standardized Gaussian frequency function and C$the standardized cumulative Gaussian distribution function. The separation S equals the average EHD film thickness h in the contact assuming, for simplicity, a Grubin model of uniform EHD film thickness throughout the Hertz contact area. Thus, from eqn. (33). after division by CJ:

Equations (31) and (34) taken as a system yield a/a0 as a function of h/o. Numerical values of ~$a, and y,/a as a function of h/a are plotted in Fig. 14. When a more sophisticated film shape model based on detailed film profile determinations is used’*, then a(x, ~)/a,, must be calculated point-wise (for all pairs of coordinates x, z) using the film thickness existing at each location (x, z) within the contact area. With the Grubin assumption, one obtains, by combining eqns. (30) (3 1) and (33) a relationship of the form : (35) for the load carried by asperities. The function t,b(h/o) is also shown in Fig. 14 for values a/u0 2 0.5. Equation (35) yields the upper limit of asperity load, and is reached when all asperity deformations are plastic. For the limiting value of u/u,=OS, eqn. (35)yields h/a = 0.4. Accordingly, actual average film thickness between rough surfaces Wear, 21

(1972)

THE

THEORY

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CONTACTS

69

h

Fig. 14. Plastic asperity contact area fraction a/a,,, standardized separation load Y as function of EHD film thickness /roughness ratio h/u.

y,,/o and standardized

asperity

can never fall below h/o = 0.4. At that film thickness, all the load is carried by asperities. Asperities have been depressed to the original midplane of the composite process. Lubricant left in the valleys is completely unloaded. In the general case of films thicker than the minimum, the total contact load is shared between the asperities and the elastohydrodynamic film according to the following equations :

Q = Q,,,+QEHD (36)

Q EHD=

(1-4aoWao (a,)

where QEHD= elastohydrodynamically carried load, = local elastohydrodynamic pressure. P The formula for the elastohydrodynamic share of the load is general for arbitrary film profile. For the Grubin assumption (i.e. that film thickness over the Hertz contact area is constant), the factor 1 -a/a, is constant and can be moved ahead of the integral sign. For the simple assumption that the elastohydrodynamic pressure distribution is Hertzian (heavy loads):

Q EHD

=

2

;hdO

++O) (37)

Wear, 21 (1972)

70

whereIL,,,, =maximum Qo.r,_fy

EHD pressure.

and

rl/rl,

- 2p,,,., (Cdi,,z Q EllD

where the last fraction on the right depends only on /zio. Table IV, a computer printout calculated from eqns. (35) and (361, shows numerical values of hia, o/q1 and QuJQEHD. Since film thickness in an elastohydrodynamic film is hardly load dependent, p,,,/P can be selected freely and was assumed to be TABLE

IV

NORMAL

LOAD

FACTORS

FOR

PLASTIC

ASPERITIES Asprrity!EHD loud rill io

0.5000000 0.4502617 0.4012937 0.3538303 0.3085375 0.2659855 0.2266273 0.1907869 0.1586553 0.1302946 0.1056498 0.0845658 0.0668072 0.05208 13 0.0400591 0.0303963 0.0227501 0.0167932 0.0122244 0.0087745 0.0062097 0.0043325 0.002979X 0.0020202 0.0013500 0.0008891 0.000577 I 0.0003691 0.0002327 0.0001445 0.0000884 0.0000533 0.00003 17 0.0000185 0.0000107 0.000006 I 0.0000034

0.3989423 0.4645550 0.5363447 0.6141687 0.6977966 0.7X69200 0.88 11670 0.9801164 1.0833155 1.1902953 1.3005868 1.4131343 I .5293068

1.6469062 1.7661739 1.8867932 2.0084908 2.1310353 2.2542341 2.3779326 2.5020041 2.6263514 2.7508991 2.8755900 3.0003819 3. I252438 3.2501535 3.3750953 3.5000583 3.6250352 3.7500209 3.8750123 4.000007 I 4. I250040 4.2500023 4.3750013 4.5000007

* This ratio assumes p,_!P=o.5 Note: This Table is a computer Weur, 21 (1972)

printout.

I .ooooooo 0.8190475

0.670268 I 0.54758 I I 0.4462 IO1 0.3623709 0.2930376 0.2357684 0.1885734 0.1498146 0.1181303 0.0923778 0.07 I5900 0.0549428 0.0417308 0.03 13492 0.0232797 0.0170801 0.0123757 0.0088521 0.0062485 0.00435 13 0.0029887 0.0020243 0.0013518 0.0008899 0.0005774 0.0003693 0.0002327 0.0001445 0.0000885 0.0000533 0.00003 17 0.0000185 0.0000107 0.000006 I 0.0000034

6.283 1852 4.6806156 3.5170993 2.6622675 2.0273 I23

1.5509524 I. 1903766 0.9153194 0.7041357 0.5411676 0.4149574 0.3 I70226 0.2410075 0.1820913 0.1365722 0.1015739 0.074837X 0.054575

I

0.0393606 0.0280560 0.0197528 0.0137296 0.0094174 0.0063724 0.0042525 0.002798 1

0.0018151 0.0011605 0.00073 13 0.0004541 0.0002779 0.000 I676 0.0000995 0.00005x3 0.0000336 0.0000191 0.0000 107

Use of more than three significant

digits is unjustified

THE

THEORY

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ELASTOHYDRODYNAMIC

CONTACTS

71

0.5. For hardened steel, pi 750 ksi, i.e. in this example, p,,, = 375 ksi,a high but realistic value for rolling bearings. The last column of Table IV shows the ratio of loads carried by plastically deforming asperities and elastohydrodynamic film respectively, for each h/o value. For h/0=0.4, this ratio is calculated to be 6.3. At this h/o value, the assumption of pm,, =constant is merely a limiting assumption. The pressure can be any lesser value, without altering the asperity contact area. Thus, as h/c =0.4 is approached, the film no longer carries load. No further approach of the asperities is possible. Table IV shows the well known fact that for high values of h/a, say h/o z 3, the fraction ofload carried by asperities, even at plastic flow stresses, is small. However, for values of the order of h/o = 1, the asperities carry, even for p,,,/P = 0.5, about 30 y/, of the total load, a substantial fraction. In elastohydrodynamic film thickness calculations, this load sharing by asperities does not introduce substantial errors since the influence of load on film thickness is so small. What it does signify, however, is that the detailed mechanism ofasperity load carrying begins to be important to the load sharing between asperities and film whenever h/a approaches unity, as will be seen in the following sections. VI. NORMAL

LOADS

IN PARTIAL

ELASTOHYDRODYNAMIC

CONTACTS

B. Elastic asperity deformation

All published descriptions of load sharing among elastic asperities contain an element of arbitrariness in the underlying assumptions of asperity geometry, e.g. that the asperities are hemispherical or in the form of an elliptical paraboloid, or straightsided prismatic ridges (two-dimensional roughness) with a small tip radius5v9. The fundamental reason for this arbitrariness is apparent from the following integral equation’ relating the total load Q1 carried by a single two-dimensional asperity ridge in contact with a plane, to the surface profile of the asperity: (39)

where Q1 = load on the single asperity contact, 1 = length of contact strip (large compared to width), A = half-width of contaot strip, 0(t) = slope of asperity profile at abscissa t. This equation is limited to a load distribution where the pressure drops to zero at the contact edges, + A. At these edges of the contact, there is, then, no singularity in the profile. In this integral, all slopes O(t) as well as the strip half-width A, depend on the profile of the asperity. All slopes within the contact strip - As t I A (and the strip width 2A itself)co-determine the load carried by the contact. Accordingly, the complete geometry of the asperities over their load carrying portions must be known to calculate load. Assumptions made about this geometry create the arbitrariness encountered in discussions of asperity load sharing. An approach to finding the total load carried by asperities within a Hertzian contact can be made without deterministic assumptions regarding asperity shape, by considering Qr, the load carried by an individual asperity, a random variable, determined by eqn. (39), in which 0(t) IS a random process and the interval (-A, A) is Wear,21

(1972)

I’ I..

72

T/zLLlAN

a random variable related to the distribution of dwell lengths of asperity height J’ above a level h. Then eqn. (39) is a stochastic integral. From the theory of such integrals one can, at least in principle, derive statements regarding the statistics (mean, perhaps variance) of Q I. and correlations between several Q ,, pertaining to different asperities. Since all parameters of the integral are defined by the surface roughness process, the random variable Q,, is also defined by it without recourse to assumptions as to asperity shape. Over the entire two-dimensional Hertzian area, the load carried by asperities is QE, =C,Q,. with Qi the load on the ith asperity. The summation is over all asperity contacts present. Then, the expected value 6(QE,) of QE, is 6 (QE,I = A {Xi Qi; = Ci {A [QJ;

(40)

The quantities d [Qi] are all identical. Assuming that the number IZ of asperity contacts, while a random variable, is one with very small coefficient of variation vart n/ii one may approximately write

zZnc5' [Qi] fi;Q,,i

:

(41)

From eqn. (39) (42) Equation (42) can be written in a more manageable form by using the conditional mean &(QJA) i.e. the mean of Qi for fixed A. This mean is a function of A, say, !??(A) and can be written (43) Then, the unconditional

mean &(Qi) is obtained

by averaging

over all values

of A :

g(Qi)= 11 9 (df'(A)dA wheref’(A) is the frequency function of A. The quantity of &[[e(t)] is further expressed CY[B(r)] =

\’ Wf[W ,- I

Since the 9 process is stationary, 8[0(t)]

(45)

Al d@(r) the quantities

in eqn. (45) do not depend

= &(0) = {‘I,,, Q’(BIA)dO

wheref’(8 1A) is the conditional frequency function are not known to be independently distributed,

f’@IA) = Wecrr, 21 (1972)

w

as

on t, so that (46)

of 0, given a fixed A. Since 8 and A

(47)

THE

THEORY

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ELASTOHYDRODYNAMIC

CONTACTS

73

with w[A, 01 the joint frequency function of A and 0 andf(4) the unconditional frequency function of A as used in eqn. (44). &[Q,] is uniquely defined if the roughness process and h are known. However, an evaluation of eqns. (42)-(45) has not yet been undertaken. Thus, it is still necessary to explore asperity load carrying by the use of a deterministic asperity geometry model. A particularly simple treatment of the elastic load carrying problem of rough surfaces in a Hertzian contact can be obtained by assuming that the asperity shape is such that the load carried by an asperity is proportional to the asperity contact area 9. For the entire Hertz contact, the proportionality of load to total asperity contact area was first postulated by Archardz9. While Archard required a complex array of hemispherical asperities to achieve this proportionality, it follows directly if the assumption that asperities are second order surfaces is abandoned. For a two-dimensional prismatic asperity ridge, with very small tip radius Y,contacted by a plane perpendicular to its line of symmetry, the total force carried is9 : lim Q1 = EOAl

(48)

r-+0 i.e. this force is, proportional to the asperity contact area and, importantly, also to the slope 6 which, for a symmetrical prismatic asperity, is defined in Fig. 15. For this situation, a maximum shear stress exists in the centerline of the asperity at the following depth:

z,,, = 0.9 (Ab)”

(49)

where b is defined in Fig. 15. The stress is of the magnitude lim zmaX= $8

(50)

r+O

i.e. it is asymptotically constant irrespective of the load on the asperity, and depends only on the slope. It is shown in the Appendix that for an asymmetrical asperity, analogous formulae apply, as follows. Using the nomenclature of Fig. 16, the load carried by an asperity is lim Q1 = :

Al

!5+!3,,,

[!dz;]

*+O

1

2 b t&dALL

RADIUS

-2A,--4 Fig. 15. Symmetrical Fig. 16. General Wear, 21 (1972)

prismatic

prismatic

asperity

asperity

model profile.

model profile.

(51)

74

‘I‘. l- TALLIAN

which reduces to eqn. (48) for 8, = 0, = 8. Designating 8,~ (0, - U,)/(d, + 0,)s 1 as skew ratio of the asperity, the cosine term in eqn. (51) depends as follows on Cl,:

f’s cos

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1.0

I

0.99

0.95

0.89

0.81

0.71

0.59

0.45

0

For skew ratios 0.4 or less, the cosine term exceeds 0.8 and can approximately be set equal to unity. For these cases, a symmetrical asperity with slope O= i(6, + 0,) is a good approximation to the asymmetrical one, for computation of the total load carried. It is also shown in the Appendix that the maximum shear stress under an asymmetrical asperity is lim r,,, Y- 0

E’ O,+& = 71i 3

(52)

which, with 8 =$( 0, + 0,) is formally identical to eqn. (50) for the symmetrical asperity. Consider now the case of two prismatic asperities contacting each other. Figure 17 shows two asperities numbered 1 and 2, with slopes 0, r, 6,, and O,,, Q,, on the two sides of each asperity. The asperities will, in general, contact each other so that the tips do not coincide. Assume the tip of asperity No. 2 contacts the 8ii slope of asperity No. 1. From eqn. (20) the composite slope of the contacting asperities will be: to one side of both peaks (a flii+L between the two peaks (b 1 fj,,+L to the other side of both peaks (c) ()12+&Z where the slopes are added algebraically. For small enough deflections, the slopes in the vicinity of the peak of the composite asperity govern. Thus the composite slopes (a) and(b) will enter eqns. (51) and (52) and the equivalent slope is 8 = 0i 1 +3(0,, + O,,). For larger deflections the simple prismatic model is no longer equivalent to the composite of two prismatic asperities in off-center contact, and eqn. (39) is used for a solution. In the discussions that follow, the symmetrical asperity approximation is used, ix. it is assumed that fl, 2 0.4 for most asperities on the surface, so that in calculating total asperity load within the Hertz area, the cosine term in eqn. (51) can be set equal to unity. Assume further that the complication illustrated in Fig. 17 can be neglected, so that forces and stresses calculated on the basis of simple triangular-section equivalent asperities are acceptable approximations. For symmetrical asperities, typical calculated values of r,,, as a function of I) are given in Table V. The 0 values used are in the range of r.m.s. slopes crBlisted in Table III. For all slopes corresponding to angles 0 > 1.3”, r,_ is in excess of the yield strength T),= 120 ksi of hardened steel (R, ‘v _ 60). (0 = 1.3’ is the Halliday angle characterizing incipient plastic flow.) The variation of z with z for various values of the parameter b is shown in Fig. 18 as calculated from the formulae in ref. 9. The important fact emerging from this figure is that for b/AZ 0.15, i.e. sharp-tipped asperities, the portion of the r(z) curve extending beneath z,,, varies very little. The portion above r lll.lXcontracts vertically as h/A decreases. Thus, the volume of material under substantial r-stress approaches, with decreasing b, an asymptotic value illustrated by the b = 0 curve. The volume at the surface, which is not under high z-stress, diminishes Wear. 21 (1972)

THE

THEORY

OF

PARTIAL

ELASTOHYDRODYNAMIC

75

CONTACTS DIMENSIONLESS 0.0 01 02

Fig. 17. Profile of two general Fig. 18. Depth distribution perity. Tip intercept is b.

TABLE

prismatic

asperities

of dimensionless

in off-center

contact,

STRESS

03 04

05

06

07

and composite

& 09

.09

.I0

profile.

shear stress in the center plane of a symmetrical

elastic as-

V

MAXIMUM

ASYMPTOTIC

SHEAR

STRESS

Slope angle of symmetrical asperity 0” (degrees)

Maximum shear stress

0.1 0.4 0.7 1.0 1.3 1.6 2.2 5.0 30.0

0.092 0.37 0.65 0.92 1.2 1.5 2.0 4.6 28

FOR

VARIOUS

MEAN

ASPERITY

ANGLES

t,,X(105 psi)

without limit as h/A decreases. Sharp tipped asperities, if stressed above the yield strength, will flow plastically at the very surface, whereas rounded ones retain a nonplastic surface layer. From the scanning electron microscopic appearance of ring track surfaces Wear, 21 (1972)

I’. I,. TAI

76

L.IAN

and the fact that their topography dominates the composite roughness, theassumption of essentially prismatic asperity ridges with small tip radii seems reasonable. Each ridge side, having been cut by a single pass of a relatively straight-edged piece 01 grinding or honing grit, should have a unique value of 0. It is not realistic to assume that the two sides coming together in a ridge crest have the same I). Little is known about the realism of the assumption 0,s 0.4 used to justify the use of symmetricaI asperity formulas based on the average slope Cl={(l), +0,). Thus. their use ma) overestimate Q1 for a certain fraction of the asperities in a contact, and this may lead to some overestimation of the total asperity load. Equation (48) represents a linear relationship between contact area 2A1 and load Q1 carried by the asperity, and the asperity slope 0 is a multiplicative factor. Regarding this equation and eqns. (49) and (50). 1. It has been shown5 that the contact of two elastic asperities on surfaces made of the same material is equivalent to the contact of one elastic asperity, with height coordinates equal to the linearly combined height of coincident points on the two contacting asperities, depressed by a rigid half-plane. Equations (48)-(52) are based on this equivalence, also used to define the compbsite roughness process characterized by eqns. (21)-(29). 2. For an elastic asperity, the deflection does not take place in the manner described for plastic asperities29, i.e. by squeezing down the volume above the contact plane and leaving the topography of adjacent surfaces undisturbed. Rather, an elastic asperity undergoes deflections beyond the width of the contact strip. In twodimensional (half-plane) theory, the deflections extend to infinity in both directions from the contact, and the total deflection integrated to infinity fails to converge. (It does converge in three-dimensional contact theory.) The lack of convergence implies that the deformation of any asperity significantly influences the level of the surface everywhere. As a result, one cannot strictly define an average spacing between two surfaces by assuming that their mean distance will remain unchanged at points remote from the asperity in contact. To treat elastic asperity deformations in isolation by twodimensional theory, the assumption has been made that the relutiue displacement of surface points within the Hertz contact area is negligible for all points on the surface except those on the asperity in contact and lying above the midplane of that asperity. even though the total deflection of the surface midplane, integrated to infinity, does not converge. On this basis, one can define the asperity deflection by the equation c=

H-y,

(53)

with the quantities shown in Fig. 19. The deflected asperity is no longer straight-sided, and therefore the asperity contact strip width 2A is not the same as the undeformed intercept 2A, at level v, given by 2A, = 2v/tan 8. The ratio A/A, between these two strip widths was calculated5 and is shown as a function of the relative deflection v/H in Fig. 20. For all but the smallest deflections, i.e. for G/H =O.l _ 1, A/A, lies between for purposes of 0.35 and 0.6. (For v/H -+ 0, A/A, -0). Thus as an approximation, asperity load calculation, one can select a central value of the range A/A, 0.35 -0.6, e.g. : AlA 1~0.5

v/H 20.1

with which eqn. (48) becomes: Weur, 21

(1972)

(54)

THE

THEORY

OF

PARTIAL

ELASTOHYDRODYNAMIC

77

CONTACTS

0.6-

05-

04n A, 0.3" 0.2-

0.1-

0, 0

Fig. 19. Elastically Fig. 20. Contact deflection u/H.

deformed

shape of symmetrical

width/intercept

ratio

A/A,

prismatic

04

I 0.6

08

asperity.

of elastic prismatic

Q 1,El +A11

02

asperity

as a function

of dimensionless

(55)

provided that :

From Fig. 19 A,B=v

(56)

so that eqn. (55) becomes

Q 1,El z

(57)

$0

where V,defined in eqn. (53) is the deflection of the asperity( its undeformed elevation above the plane of contact). The total load carried by all asperities in the Hertzian area is the sum of the individual asperity loads given by eqn. (57) and it is obtained by summing over all asperity ridges within the width AH( major axis) of the Hertzian contact area. Thus Qa,~l =

C (AH)

QI,EI

= g C (u/J

(58)

(AN)

The length of the ridges 1,varies across the width of the elliptical contact area, from zero, at the end points to Z,,,,,,which equals the minor axis. The mean value of 1 is 1=+7c l,,,. In case of a rectangular (Zdimensional) contact, 1 is constant and i= 1,,,. Wear,21

(1972)

I 1.0

In both cases, I is independent of the roughness process( assuming uniform roughness across the track). The deflection 1: is a random variable dependent on the peak height yP of each ridge in the contact area, as measured from the composite profile centerline : L!=yp-l‘”

(5’))

with mean F,, dependent on film thickness h. The number IZof ridges over a contact width A,, counting only those with height in excess of y0 is also a random variable, with mean fi which is (from the definition of peak density N,, and of the cumulative distribution function F,,(yP) of peaks no higher than yP): rz= &Np[l

-F,(_r,)]

(60)

Thus, Qm is also a random variable. It will be assumed than fi is large by comparison to varin, i.e.: that n is essentially a constant equal to fi. The probability density of peaks of height between y, and yP + dyp, to be found within the population of peaks higher than ~1~is:

(61) where,& is the frequency function of peak heights. Therefore the mean value of the sum in eqn. (58) is, with i and V denoting the means of 1 and U:

Q l&Elcan, for large ti be considered simplification, this reduces to

a constant

equal to this mean value. After

with (63) and a,=A,l, the Hertz contact area. Recalling eqn. (32) of the plastic case, there is also a distinction in the present, elastic, case between average surface separation S, and y,. Since there is, in the present case, deformation of the surface outside the asperity contacts, S is, exactly, the average of the deformed composite roughness heights, setting y=O in the contact areas. The calculation of this exact average separation requires pointwise knowledge of the deformed profile. As a simplification, one can assume that outside the asperity contacts, the surfaces are undeformed, and then eqn. (33) holds as an approximation. With the Grubin assumption of uniform EHD film thickness, h-S, and thus, eqn. (34) applies as an approximation and is repeated here for clarity :

Wear, 21 (1972

THE THEORY

OF PARTIAL

ELASTOHYDRODYNAMIC

CONTACTS

79

Equation (63) is valid without qualification only if all asperities are elastic according to the inequality attached to eqn. (55). This is never strictly the case as the theoretical distribution of 0 is not bounded from above. Physically, however, the distribution of 8 must be bounded for all real surfaces. Whatever the bound of 8 is for the unrun surface, running-in can be expected to result, after the first few hundreds of thousands of cycles, in a shakedown of cumulative plastic flow. 8 values in excess of the Halliday angle will be flattened to the Halliday value. Thus, for stationary conditions, the load distribution among asperities can be calculated from eqn. (63) using a peak height distribution : &(ya) altered to correspond to the run-in condition. This argument cannot be used when discussing stresses in individual asperities and their fatigue, since micro-plastic occurrences determine fatigue and these do not shake down even after long running as will be discussed later. For the elastohydrodynamic portion of the load carried in the presence of elastic asperities, the formula for elastohydrodynamic load given in eqn. (36) still applies, except that the fraction of the contact area occupied by asperity contacts now is, in view of eqn. (54): i$)L,

= + p-4

(64)

@]

where C$is the cumulative Gaussian distribution function (with zero mean and unit variance). Equation (37) continues to hold if one inserts a/@,),, from eqn. (64) i.e. QEHD= Pmaxao

[I++

@]

(65)

Equations (34)and (64) together yield a relationship between u/(aO)rl and h/o, i.e. (C),,

(66)

= $)

Equations (64) and (65) together yield QEHDas a function of h/a, i.e. QEHD

=

Pmax aO $2

k

(67)

i)

The functions $1 and ti2 are shown in Table VI. Standardizing eqn. (63) by the r.m.s. roughness (T.one obtains I

Qa,E,

=

& (68)

where Zp [$+

= j-1,. [Y: - $‘J f;(y;)dyp*

andf,(y,*) is the standardized peak height density function given in Table I, with tl as a parameter. From eqn. (34) h/a and y,/a are uniquely related so that Z,,(y,/o, ~1)can also be expressed as a function of h/o and LX. Eqns. (68) and (34) show the following. 1. The value of integral Z,(y,/a, LX)depends for given a, only on h/o and deWear,21 (1972

80

T. t. ‘TALLIAY

TABLE

VI

ASPERIn ELASTIC

CONTACT ASPERITIES.

AREA FRACTION AS A FUNCTION

Y’, (h/n) 0.3989423 0.4645550 0.5363447 0.6141687 0.6977966 0.7869200 0.8811670 0.9801164 1.0833155 1.1902953 1.3005868 1.4137343 I .5293068 I .6469062 1.7661739 1.8867932 2.0084908 2.1310353 2.2542347 2.3779326 2.5020041 2.62635 14 2.7508991 2.X755900 3.0003819 3.1252438 3.2501535 3.3750953 3.5000583 3.6250352 3.7500209 3.8750123 4.000007 1 4.1250040 4.2500023 4.3750013 4.5000007 Note:

0.2500000 0.2251309 0.2006469 0.1769151 0.1542688 0.1329927 0.1133136 0.0953935 0.0793276 0.0651473 0.0528249 0.0422829 0.0334036 0.0260406 0.0200296 0.0151981 0.0113750 0.0083966 0.006 1122 0.0043872 0.003 1048 0.0021662 0.0014899 0.0010101 0.0006750 0.0004445 0.0002885 0.0001846 0.0001163 0.0000723 0.0000442 0.0000267 0.0000158 0.0000093 0.0000053 0.0000030 0.0000017

This Table is a computer

‘Y, AND DIMENSIONLESS EHD OF FILM THICKNESS,ROUGHNESS

LOAD ‘yz FOR RATIO /zig

Y’?(h,:rr) 0.5984134 0.6182561 0.6377915 0.6567267 0.6747959 0.6917717 0.7074734 0.7217716 0.7345903 0.7459045 0.7557364 0.7641477 0.7712323 0.7771071 0.78 19033 0.7857582 0.7888086 0.7911850 0.7930077 0.7943841 0.7954073 0.7961561 0.7966958 0.7970786 0.7973460 0.7975299 0.7976543 0.7977373 0.79779 17 0.7978269 0.7978493 0.7978633 0.7978719 0.7978772 0.7978803 0.7978821 0.7978832 printout.

Use of more than three significant

digits is unjustified.

creases as h/a increases. Thus increasing elastohydrodynamic film thickness relieves asperities of load. 2. The Hertzian contact area a, is a constant of proportionality in the load equation, as was the case for plastic asperities. 3. Two roughness parameters are multiplicative factors in the load formula: cr and EP. The presence of 0, the r.m.s. roughness( readily measured)in the formula signifies that for a given h/a,the total load carried by asperities will( for constant Iv,) increase with roughness amplitude. Wear. 21 (1972)

THE

THEORY

OF

PARTIAL

ELASTOHYDRODYNAMIC

CONTACTS

81

NP the density of peaks of all heights is not readily measured from the height profile. However, the derivative of the height proIile( the &prolile)permits its measurement using eqn. (7) which gives NP = $N0(8) with No(e) the zero crossing density of the &profile. From eqn. (7) NP = $xNO with a 2 1. The value CI= 1 is obtained for a singlefrequency protile only. For all finite band width spectra, LY > 1, i.e. the number NP will be larger than l/2 No. There will, on the average, be more than one peak between subsequent zero crossings. In the prismatic asperity model, critical to eqn. (68) exactly one peak in excess of level y, occurs between an upward crossing and subsequent downward crossing of the profile over y,,. Thus for consistency of the model, one should make the substitution [ 1 - FP(y,)] NP -+ +Ny, = +K$ exp [+j2] with & defined by eqn. (1). Also, one should, for consistency, obtain the distribution setting, from eqn. (56): v=A,O

(69)

of deflections u by (70)

and obtaining the distribution of A, from eqns. (2) (3) and (54) and the distribution of the slope 8, typical of the prismatic asperity, by making a suitable assumption, e.g. using a fixed fraction of the slope with which the profile crosses over level y,. It seems, however, that the latter would not add appreciably to the precision of the calculation and would greatly complicate it. The use of eqn. (69) on the other hand does appear to add valuable precision, and thus eqn. (60) will be rewritten : ii,, = *AHNo

exp [-ii:,

‘1

(71)

whereas eqn. (61) is left unchanged. Equation (63) is then replaced by:

Q o,El =

%,No 4X

(72)

with I,(Yo) = O”(YP-YO)fp(Yp)dYp 1’ or after standardization a:alogous to eqn. (68)

Q &El = with

Wear, 21 (1972)

$o~o~

(73)

82

1. I. TALLlAh

(74)

Q

From

R,El

eqn. (6): (75)

From eqn. (34) one can substitute 1 [(h/g) xl= I’[Y,/o, ~1 f or corresponding

values of h/cr for y,Ja so that one can define value of h/a and ~,/a. With this definition :

Q ll.El Table TABLE ELASTIC

(76) VII shows numerical

values of I [(h/a),

cc] for selected c(.

VII ASPERITY 1.0

LOAD 1.5

FACTOR

I (h/o.

2.0

3.0

a) 4.0

5.0

10

50

‘X

1.0515

1.0177

0.9947

0.9379

0.8613

0.7979

0.8919

0.8668

0.8495

0.8053

0.7439

0.6916

0.7131

0.6961

0.6840

0.6522

0.6062

0.5658

0.5383

0.5276

0.5198

0.4985

0.4662

0.4369

h/IO

I=

0.0000

1.2533

1.1596

1.1092

0.2500

1.0059

0.9656

0.9328

0.5000

0.7734

0.7581

0.7393

0.7500

0.568 I

0.5630

0.5535

1.0000

0.3977

0.3962

0.3920

0.3839

0.3778

0.3731

0.3598

0.3386

0.3186

I .2500

0.2648

0.2644

0.2628

0.2589

0.2556

0.2452

0.232 1

0.2192

I .5000

0.1675

0.1674

0.1651

0.1502

0.1424

0.1004

0.1004

0.1634 0.0988

0.1578

1.7500

0.09x I

0.0959

0.09 18

2.0000

0.0570

0.0564

0.0561

0.0529

2.2500

0.0301

0.0570 0.0306

0.1668 0.1002 0.0570

0.2530 0.1621

0.0873 0.0505 0.0276

2.5000

0.0156

0.0156

2.7500

0.0075

3.0000

0.0034

0.0995 0.0657

0.0306

0.0305

0.0304

0.0303

0.055 I 0.0298

0.0155 0.0075

0.0155 0.0074

0.0154

0.0152

0.0074

0.0073

0.0148 0.007 1

0.0142

0.0075

0.0156 0.0075

0.0034

0.0034

0.0034

0.0034

0.0034

0.0033

0.0033

0,003 1

0.0288

0.0069

Equation (76) presents a much simplified statement regarding the influence of roughness on asperity carried load: the load depends, via l[(h/a), x] on the film thickness/roughness ratio h/o and is proportional to cr,,,the r.m.s. slope of the composite roughness. There is an apparent effect, via 1, of the spectrum width measure a, but having postulated a = 1 in calculating iij,, it is questionable whether a more precise selection of IXas a parameter of I will add precision to the load formula. Dividing eqn. (76) by eqn. (67) one has, for elcrstic asperities. the ratio ofasperity load to EHD film load:

THE THEORY

OF PARTIAL

ELASTOHYDRODYNAMIC

83

CONTACTS

QaE1

(77)

Q EHD

Comparison of eqns. (38) and (77) shows (numerical factors aside) that the indentation hardness P entering the plastic asperity load sharing formula corresponds to the elastic and geometrical factor E'a,in the elastic load sharing formula. Functions of h/o control load sharing in both cases. The two h/afunctions, however, are not identical. Table VIII compares the load sharing between asperities and elastohydrodynamic film for elastic and plastic (Table IV) asperity behavior. TABLE VIII ASPERITY

LOAD/EHD

LOAD RATIOS

hlo

Elastic asperities*

Plastic asperities**

Qo.dQ~mCm=1.0) OB= 0.007 (0.4O)

0.0349 (2”)

0.122 (7’)

0.862 (15’)

0.40 0.54 0.70 0.88 1.08 1.30 1.53 1.77 2.01 2.25 2.50 2.75 3.00

0.0513 0.0386 0.0281 0.0197 0.1323 0.0086 0.0053 0.003 1 0.0018 0.009 0.0005 0.0002 0.0001

0.2558 0.1926 0.1400 0.098 1 0.0661 0.0428 0.0265 0.0157 0.0088 0.0046 0.0024 0.0011 0.0005

0.8940 0.6732 0.4893 0.3428 0.2311 0.1496 0.0927 0.0548 0.0308 0.0162 0.0084 0.0040 0.0018

1.9200 1.4458 1.0507 0.7361 0.4963 0.3212 0.1991 0.1177 0.0662 0.0348 0.0180 0.0086 0.0039

hlcr

Q~.EL/QEHD(U=CO)

-~ 0.40 0.54 0.70 0.88 1.08 1.30 1.53 1.77 2.01 2.25 2.50 2.75 3.00

0.0349 (2”)

0.122 (70)

0.862 (15”)

0.0327 0.0266 0.0205 0.0151 0.0107 0.0071 0.0045 0.0027 0.0016 0.0008 0.0004 0.0002 0.0001

0.1628 0.1324 0.1024 0.0754 0.0530 0.0354 0.0225 0.0137 0.0078 0.0043 0.0022 0.0011 0.0005

0.5692 0.4629 0.3579 0.2636 0.1851 0.1239 0.0788 0.0477 0.0273 0.0149 0.0076 0.0037 0.0017

1.2223 0.9941 0.7687 0.5661 0.3976 0.2659 0.1693 0.1024 0.0587 0.0320 0.0164 0.0080 0.0036

Wear, 21 (1972)

p,,, = 375 ksi

6.283 3.517 2.027 1.190 0.7041 0.4150 0.2410 0.1366 0.0748 0.0394 0.0196 0.0094 0.0043

Q..~QLwo

bg= 0.007 (0.40)

* Max Hertz pressure

Q..,/QEHD

** P,,, = 0.5

6.283 3.517 2.027 1.190 0.7041 0.4150 0.2410 0.1366 0.0748 0.0394 0.0196 0.0094 0.0043

84

‘I. I. TALLIAU

Another method of asperity load calculation uses the mean curvature of asperity summits as the relevant geometry parameter”‘. In view of the detailed information provided by Nayak”’ regarding the distribution of summit curvatures, this is a tempting approach. It is doubtful. however. that it would be more productive than the one used. as is evident from eqn. (39). Unless a Hertzian assumption is made. Cz. that the curvature found at the summit persists throughout the contacting asperity eqn. (39) cannot be solved based on summit curvature alone. If the Hertzian assumption is made, the usual difficulty arises that individual asperity loads are no longer linearwith thearea ofcontact ofeach asperity.An additional deterrent is thecomplexity of formulations involving curvatures which depend on the second derivative of the profile. In the present approach, only the r.m.s. value of the first derivative is used [eqn. (76)]. Accordingly, no asperity load formulae are derived for second-order asperity profiles. However. a relationship between asperity peak curvatures and the slopes of “best fitted” prismatic equivalent asperities will be contrived. since it is needed to permit statements regarding ,stvr.ssc~sin asperities. Slwrr

stresses

in elrrstic. trsprritirs

The distribution of maximum shear stresses existing in elastic asperities can, in principle, be obtained by noting the relationship between maximum shear stress T and 0, given in eqn. (50). However, the distribution off) values found from the derivative profile cannot be substituted into this equation since we require a “typical” (I,. to be associated with each asperity summit which characterizes the “best fit” prismatic asperity substituted for one of random shape. A method for deriving such a typical Or is illustrated in Fig. 21. In this Figure. a randomly shaped asperity ridge is replaced first, by a parabolic cylinder and then by a prism.

PROFILE _I__c_ENTERLINE

x t-Y Fig. 21. Geometrical

--

approximations

o to a random

asperity.

The parabolic shape can e.g. be a mean square fit. The prism is selected such that it, and the parabolic shape, contact each other at the tip and at the mid-line of the asperity. Figure 21 shows that the tip curvatures of the parabolic cylinder, and its height y, above the mid-line, multiplied together, equal twice the square of the slope 07. of the prism, i.e. p!‘y. = Weur, 21 (1972)

20z,

(7x1

THE

THEORY

OF

PARTIAL

ELASTOHYDRODYNAMIC

85

CONTACTS

Nayak” gives the joint distribution of summit curvatures and heights in an isotropic roughness process. In a similar way, the joint distribution of peak curvatures and heights in a profile (two-dimensional ridge process) can be found from which is obtained a distribution of “typical” 8, associated with all peaks, by integrating the joint distribution for selected values of pyo = const., and substituting 8, from eqn. (78). The distribution of 8, is, according to eqn. (50) the same as that of z,,,, except for a scale factor. Asperity loads in the isotropic case

Discussion of asperity loads up to this point covered a two-dimensional roughness topography dominated by prismatic ridges. A similar model for the isotropic case can be sketched on the basis of the elasticity theory of isotropic (axi-symmetrical) asperities given in the Appendix. Consider a conical elastic asperity depressed by a rigid plane. The load is Q1 = E’y A2

(791

where y is the gradient of the cone; A is the radius of the contact area. The load is, as for prismatic asperities, proportional to the contact area (in this case ~cA’). Or, setting v=c, the asperity deflection defined in the Appendix, Q1 = E’ ; v2

(80)

Using eqn. (79), one obtains for the total load carried by all asperities in the Hertz contact :

Qam = c Q1 = 1.57 ; & yA2 = ; (;,; (ao)

0”

(81)

(I

From this equation, it is possible to proceed in analogy to eqn. (68) to obtain an expression for the total asperity load based on the joint distribution of gradients y and contact radii A, or asperity heights v above level h. This analysis has not been undertaken. Even on an isotropic surface, asperities are not generally bodies of rotation. Two conical asperities in off-center contact do not yield a composite asperity with rotational symmetry. Thus, the three-dimensional model is rather a crude approximation of reality. However, the complexities of a more faithful representation of the isotropic case were viewed as unmanageable at the time of writing*. Size and number of individual asperity contacts

As previously stated, the number of asperity contacts for film thickness h is, for two-dimensional roughness : iRhao where is, varies with h per eqn. (1). For a three-dimensional process, the number of contacts is ),,iV_ with ,R related to Rh by eqn. (16), i.e. ,,m=ml. The approximate average size (length) of a two-dimensional contact is, using eqn. (64) * Greenwood and TrippI offer an elegant derive load/deflection relationships. Wear. 21 (1972)

description

of isotropic

surfaces

in elastic contact

and

1. t.

86

and a plausible size distribution For a three-dimensional

is given by eqn. (3). contact. the average clreu of an asperity

_ l--V(;)

ah__Ni exp and a plausible at-d,:

contact

is:

(83)

F-

00

distribution

G’(a,) = 1 -exp VII. TRACTIONS

TALLIAN

of these areas is obtained

[-

from eqn. 3 by the substitution

c::,+]

IN PARTIAL

ELASTOHYDRODYNAMIC

CONTACT

Asperity

tractions The frictional traction transmitted by a partial elastohydrodynamic contact has two components: the traction transmitted by asperity interactions and the traction transmitted by the elastohydrodynamic film, i.e. (85)

T=T,+TFHD The simplest assumption Coulomb law, i.e.

for the traction

transmitted

by asperities

is that it obeys a

(86)

T, = PL,Q,

with the coefficient of sliding friction ,LL~ dependent on boundary lubrication conditions. but identical for all asperities in a contact. This last assumption can be questioned since a distinction exists between elastic asperity contacts in which the boundary film has presumably not been disrupted and plastic contacts which may weld together with obvious destruction of the boundary film. In the first case, one expects values p, < 0.3. In the second case, one expects K -m 1. However, for steady-state traction calculations after run-in, in the absence of smearing, the contribution to traction of contacts which are plastic and weld, will be small. The dominant contribution to asperity traction arises from elastic contacts for which a pa value representative of boundary lubricated friction can be used. On this basis, 11,= constant will be assumed over all asperities. The elastohydrodynamic component of traction T&n depends on many variables, most significantly the sliding speed (see Fig. 1) as will be shown later. HOWever, formally TEm=

PEHD

QEHD

in which pEHn is a variable Defining the normal

Wear.21 (1972)

(87) quantity. load ratios 4, and qEHD

THE

THEORY

4,

OF

=

PARTIAL

ELASTOHYDRODYNAMIC

87

CONTACTS

Q,/Q (88)

QEHDIQ

qEHD=

4cz+%m=1 one can write: T p = - =

Q

/&&+~EHDqEHD

The constant ,u. is available experimentally. The variable &uu will be discussed later, and the factors q., qEHDcan be calculated as functions of h/o from eqns. (88) and (76) or (81). ,Ufrom eqn. (89) is illustrated as a function of sliding speed in Fig. 22.An increase of h/a will reduce qa and increase qEHDand thereby reduce the width of the band representing T,. For A and PEHD comparable (a case representative of heavily loaded, well lubricated contacts) the relative influence of the two terms in eqn. (89) depends primari1Yon qa and (IEHD For this case, Table VIII suggests that qa will be significant for h/as 1.5. In more lightly loaded contacts, the load QEHDcarried by the elastohydrodynamic film drops much faster for fixed h/a than the load Q, carried by the asperities. (Film thickness varies very little with load.) pEHDalso is reduced, whereas p0 is not. Thus, in lightly loaded contacts (e.g. high speed bearings) the contribution of asperities to traction may be significant up to h/&%2. Elastohydrodynamic traction From the principal references for elastohydrodynamic traction behavior’4-‘8,21 L23, the following simplified description of elastohydrodynamic traction can be gleaned. The traction coefficient pEnD as a function of slide to roll ratio s = 2(u, - u2)/ (ur + u,)followsacurveofthetypeshownin Fig.23.Threeregimescanbedistinguished: regime I extends from s=O to 0.2 x 10m2 or 0.5 x 10e2. In this regime, the traction

e

k SLIDE TO ROLL RATIO

Fig. 22. Resultant

traction

coefficient

SLIDE TO ROLL RATIO

p in partial

Fig. 23. Three regions of the EHD traction Wear, 21 (1972)

EHD contact

coeffkient

as a function

paHD as a function

of slide/roll

of slide/roll

ratio s.

s

ratios

88

I

II. I‘ALI-IAK

coefficient is proportional to s. The characteristic parameter is the slope 0 = ~~ni,,,/.s. Regime II extends from the end of regime I roughly to s= 10. ’ or s= lo- ‘, Here, ,u~~,,, first rises roughly as a logarithmic function of s, then reaches a maximum at the boundary between regimes II and III. In regime III, covering all values of s beyond regime II, the traction gradually decreases. For regimes II and III the characteristic parameters are: the value and location of the peak traction and some measure of the rate of decrease of pEHD past its peak value. Recent publications ’ sq23have attempted to define numerically, similar parameters. Both achieve fits to experimental data over limited ranges of variables, but neither is fully satisfactory as a general treatment. The development of a more general treatment, under intensive study by several workers, is beyond the scope of this paper. However, it may be of value to suggest an approach to such a treatment as follows. In the present state of rheological knowledge under elastohydrodynamic conditions, a quantitative theoretical solution covering all practical operating conditions appears remote. To serve the practicing engineer, the treatment should therefore be fitted to experimental data of a type that can be practically measured under EHD conditions. It seems reasonable to require that the user select a luhricunt and specify total contact loud, relative speed between bearing rings and ambient temperature. He cannot specify the temperature or pressure profile in the contact, the film thickness or the sliding and rolling velocities, since all these, in a bearing, depend on the kinematics of rolling body motion between the rings, and these in turn are determined by the traction equilibrium of the rolling bodies with rings and cage. Knowing the selected external parameters, it seems feasible to conduct singlecontact (disk, ball or roller/flat) experiments to measure total tractive force and to ascertain the topography of the film so as to permit a reasonable approximation (Grubin or other) to the average film thickness. These data can be obtained in the form of a traction US. slide/roll ratio curve for selected loads, rolling speeds, a few in partial elastohydrodynamic contact. These forces then serve to calculate the to be used. The model must, as a minimum, yield the initial slope p/s in regime 1, the location and value of pmax and, if possible, some measure of p for the highest values of s to be expected, generally being simple sliding (s = 2). These outputs must be available by interpolation. for loads, rolling speeds and temperatures within the range expected in service (this generally includes zero load) and must be capable of being scaled to contacts of similar shape but different absolute size. Such a model, when available, or its approximation15.23 eventually yields /+nn, Combining this with c(,~ according to eqn. (89) yields the total tractive forces in partial elastohydrodynamic contact. These forces then serve to calculate the equilibrium of rolling body forces in bearings if cage tractions and bulk lubricant losses are known. In the presence of spin or point-wise variable slip, it is necessary to calculate total tractive forces and moments by subdividing the contact area into suitable elements, each of which shows constant sliding velocity. Micro-elustohydrodynamic @cts Christensen3i, Fowles 32, Dowson and Whomes18, and Walters et ~1.~~have called attention to the fact that the existence of asperities (generally represented by ridge systems) influence EHD film thickness and traction even when they do not make contact, by modifying the hydrodynamic forces arising in the film. Wear. 21 (1972)

THE

THEORY

OF

PARTIAL

ELASTOHYDRODYNAMIC

CONTACTS

89

The effects on film thickness, while not negligible, are likely to be no more significant than other neglected effects of thermal, non-Newtonian etc. nature. The effect on traction is important, at least conceptually, since it interferes with the separation of T, and TEHD.Figure 26 shows experimental results33 suggesting that, for circumferential lay, traction can decrease by a factor of two with roughness. For transverse lay, traction can increase or decrease by a factor of two. Enough understanding of the EHD effects of roughness is not available to combine them with asperity contact effects. The deformation of asperities can no longer be described as a dry contact problem when EHD pressures vary over asperity dimensions, as they must in order to create micro-elastohydrodynamic effects. On the other hand, the microelastohydrodynamic deformation has been neither observed nor calculated (for rolling contacts) in enough detail to use it in predicting asperity interactions. The incorporation of micro-elastohydrodynamic effects into the theory of partial elastohydrodynamic contacts must accordingly await further study.

Fig. 24. Clusters

of wear scratches

Fig. 25. Wear scratch VIII. WEAR

on bearing

on bearing track

track

(SEM).

(SEM).

PHENOMENA

A rolling wear law has been suggested’,2,33 for partial elastohydrodynamic contacts which, based on Archard’s wear theory34, postulates wear rates for a given configuration (given surface material and microgeometry, given spin-to-roll ratio and lubricant). The wear rate is proportional to the combined contact area of all asperity contacts. From the scanning electron micrographs one concludes that each of the wear occurrences exemplified in Figs. 24 and 25 (sharp-sided gouges) must occur in a single asperity encounter. This, accordingly, is adhesive wear. Wear. 21 (1972)

90

i_ 0

/

10

20

30

40

50

/

60

70

10

h,,I-“” 0

Clrcumfewnt~nl

Fig. 26. Effects of roughness

,

/_

0

20

40

30

50

/

1

60

70

h, > p r lay

on traction

h

Transverse

as a function

lay

of film thickness.

Figure 10, shows a multitude of micropits on a run surface, indicating another way in which “wear” can proceed. The type of pits are unlikely to come about by an individual gouging event. They are attributed to surface fatigue, i.e. to a multitude of roll-overs which gradually exhaust the ductility of the material by small increments of plastic flow. Thus, there are at least two progressive wear phenomena in rolling contact. A third abrasive wear will take place when hard contaminant particles are present in the contact. For adhesive wear occurrences, which appear to be the predominant type during run-in and in operation at moderate to high h/o values, the model describing this type of wear is as follows : two points at the opposing surfaces meet. One surface acts as a tool, gouging a scratch into the other surface. The “tool surface” is a sharp asperity which, due to previous roll-overs, has been work hardened and can gouge the second surface without itself being removed. Or, it can take place by the entrainment of a piece of loose steel debris from external sources, or from previous wear or surface fatigue occurrences within the bearing. However, a model involving debris entrapment cannot account for the good experimental confirmation of the wear rate dependence on h/a reported’. Thus, it is assumed that most of the adhesive wear occurrences are due to plowing by an asperity. The ridge pattern of a new ring track surface shown in the scanning electron micrographs, suggests that many of the asperities on an unrun surface are initially plastically deformed at the tip of the ridges. After initial plastic deformation, the remaining ridges are less sharply crested, generally have slopes less than the Halliday value, and therefore can generally be rolled over without further plastic flow. However: (1) there will be further plastic flow any time the elastohydrodynamic film thickness is reduced; (2) an asperity which does not flow plastically in pure rolling, may do so in the presence of sliding because of the macroscopic shear stresses generated near the surface in a Hertz contact in which interfacial tractions arise; (3) the slope of interest in determining plastic flow is that of the composite roughness. Since surfaces do not meet at identical points during subsequent roll-overs, plastic flow will arise at any point on a surface which encounters an opposing asperity of higher slope angle than those it has previously encountered. Consider an asperity on one of two rolling surfaces designated surface “l”, for which the height y, =y,, is a high value. Let the slope of that asperity be 8, = (I,,. The opposing surface “2” will contact this point with profile features characterized Wear, 21 (1972)

THE

THEORY

OF

PARTIAL

ELASTOHYDRODYNAMIC

CONTACTS

91

by values yz and e2, which vary from cycle to cycle and which are distributed with zero mean and variances of and cri2 per eqns. (4) and (5). Thus, the variables y and 8 of the composite roughness at point “1” will be distributed as follows: Composite Point on surface 1 : Yl=Ylo

01=~,0

Y-~(Y,o,

d)

Point on surface 2 :

Y,-N(O,

(90)

&

&-N(O, &) Q-N@,,, of,) of y and 0 at point “1” will have mean values of yi,, and 8,, and variances 0; and (T&for the height and for the slope respectively. In some of the encounters, there will be a summit over point “l”, i.e. y will be greater at that point than at all adjacent points. Among the sub-set of encounters during which there is a summit at point “l”, a distribution of values 8 exists. There is a finite probability that any given 0 values exceeds the 8, of the Halliday criterion. The probability of a plastic contact at point “l”, given that “1” is a summit, and for film thickness h, is governed by the following two probabilities : . .contact occurs I. PS (Y>hlY, =Y1o) 2.PS(8>e,le,=e,,, . . .plastic flow occurs where 9, designates the probabilities considering only events in which point “1” is a summit. Thus, among occurrences yielding a summit at point “1” the probability of a plastic contact is: i.e. the distributions

P, (plastic occurrence) = P, (J > h 1y I = y, o) . P, I3> 0, i I;l:Q:oi

(91)

According to eqn. (91), at any point of surface “l”, there is a finite probability of a plastic occurrence during the next roll-over. The height and the slope distributions are conditional on the values of y,, and 8io for the fixed point “1”. The probability of a plastic occurrence will be higher if the values of y,, and 19,~are greater. On the basis of using Archard’s wear theory’, the wear rate is proportional to the total area of asperity contact, elastic and plastic. It seems physically plausible that the area of contact governing the wear rate should only be the area covered by plastic contact occurrences. The probability of one of these occurrences is given in eqn. (91). The probability of a contact occurrence regardless of whether it is plastic, is: P, (contact) = P,(y >hly,

=ylo)

(92)

since there is no constraint on the value of 8. The fraction of contact occurrences which will be plastic is governed by P, (plastic) P, (contact)

= P+$:;J

(93)

Bearing in mind the definition of 6’as the sum of 8, and e2, the following holds : 8=0,+8,

>eff;

t91’8,0

or Q, >%-~10 Likewise : Y=Y,+Y, Wear, 21 (1972)

>h;

Y,=Y,,

I. E. IALLIAN

92 or !‘2 >h-Y,,, Thus :

i.e., the fraction of plastic occurrences 0, =8i0 is simply the cumulative probability 8i0 and Bzo positive in all cases). The fraction P,,(h) of plastic contact the value of 8i and y,) is, then: P,,(h)

=

j-

f

&VI,

>Q,-o,ly,

(J.1) ,&

over points on surface “1” for which of 0, exceeding &-OiO (considering occurrences

>h-~Jf,(y,,

anywhere

OMy,dQ,

(irrespective

of

(95)

wheref,(y,, (3,) is the joint distribution of the heights and slopes of summits. The fraction in eqn. (95) is obtained by letting 8, and y, proceed throughout their ranges and computing the weighted average of the probabilities P,. It depends on the contact level (film thickness h) if and only if the distribution of asperity slopes is height dependent. The distribution of slopes in general (as distinct from those associated with peaks according to Fig. 21) is not height dependent”. However, as seen from Fig. 5. summit curvatures are height dependent and increase as the summit height increases. Equation (78) suggests, therefore, that the summit-associated slopes will increase sharply as summit height increases. Since Archard’s law (wear rates proportional to total asperity contact area) requires eqn. (95) to be height independent and experiments approximately confirm this law’, one concludes that the height dependence of eqn. (95) must be small by comparison to the variability of total contact area with h/a. Curvatures for summit heights ranging from h/a = 1 to 3, vary, for realistic x (in excess of 2) by a factor of 3 (Fig. 5.) The product of height and curvature varies, accordingly, by a factor of 9, and 8,, the square root of the product, by a factor of 3. Over the same range of h/a, the total asperity contact area varies by a factor of 20. Thus, the variation of 8, is indeed much the smaller of the two and can be neglected. Experiments suggest’ that the fraction of asperity interactions causing a wear particle increases with test speed, i.e. with h/a. This should result in a wear rate which decreases less than proportionally with h/o. Wear data 35 show a deviation in that direction for numerous cases. In other cases there is no deviation. No case is noted of a deviation in the opposite direction. The preponderance of high-slope contacts at high h/o levels is a likely explanation for these deviations from proportionality of wear versus total asperity contact area. A question of interest is, what influence the parameters of the surface roughness have on the fraction of plastic contacts P,, given in eqn. (95). Consider eqns. (4) and (lo), showing the scale factor for peak curvature. Using eqns. (6) and (7), the scale factor of 6, associated with a peak is given by the following proportionality : var+ 0, = varf (~JI~)+_ go a+

(96)

Aside from a low positive power of a, which can be assumed to vary little among surfaces of similar character, the scale factor is 08, i.e. it is the same for 8, as for the u’ecrr,21 (1972)

THE

THEORY

OF

PARTIAL

ELASTOHYDRO.DYNAMIC

CONTACTS

93

slope 8 at arbitrary points. This suggests that for constant h/o, the fraction of asperity contacts undergoing plastic flow, and accordingly contributing to wear, depends primarily on oB. IX. SMEARING

Smearing phenomena, defined as severe wear occurrences characterized by formation of a wedge of transferred material, of macroscopic magnitude on the surface, require sliding. Under conditions where rolling accompanies sliding, the duration of any asperity contact occurrence is limited to the length of time during which opposing surface elements remain in contact6, and corresponds to different lengths of travel on the surface depending on the degree of sliding. In all cases other than simple sliding, the duration is finite. The wedge-forming process may not continue for the entire duration of the contact, however, this duration is an upper bound. Assuming that a macroscopic wedge can form only if a number of plastic contact occurrences coalesce6, then a condition of smearing is, that the duration of an elemental smearing occurrence be sufficient to reach an adjacent plastically deforming asperity. Traction forces acting in the surface (sliding) increase plastic flow in asperities so that a calculation of plasticity based solely on normal pressures understates the number of plastic contacts. However, for simplicity, only normal pressures will now be considered. The plastically flowing asperities are those for which eqn. (91) holds. Using, again, the simple concept that the density of summits Ns(h) in excess of level h, is identical to the density of contact areas ,N at level h, then the average spacing of summits on an isotropic surface is 2/(,,m)*=2/& N,, is related to m, by eqn. (1). The probabi!ity that any one of the summits of height in excess of h undergoes plastic flow is given by eqn. (95). Therefore, the expected number of plastic occurrences at level h is: N,, (plastic) = ~P,,(h)N,,

(97)

(Nh comprises crossings over h in either direction and this explains the factor l/2). The expected spacing of plastic occurrences is: 2 dpi(h) =@, Pp, (h)

(98)

A smearing occurrence will not take place if the following inequality holds: d,,(h) > S

(99)

where S is the length of a contact occurrence or the length over which the wedge forming process continues, whichever is less. Designate the slide/roll ratio as s and the length of the contact in the rolling direction as 1. Then the maximum available length of lateral travel of two asperities on each other in the process of forming a wedge is 2.~1.Accordingly, eqn. (99) can be written d,,(h) > 2sl

(100)

In eqns. (99) and (100) d,, is the random variable spacing of plastic occurrences, not their expected spacing. As stated in eqn. (90), var* 8, -oB. Thus, P,,(h) is expected to increase as o,, increases. N,, strongly decreases as h/a increases (see eqn. (1.)). An increase of h/a or a decrease of rsewill increase lip, and hence reduce the probability of Wear,21

(1972)

T. k. TALLIAN

random values d,, small enough to violate the inequality (100). Thus the probability of smearing will be reduced. These statements identify three controllable factors as helpful in reducing smearing : increase in elastohydrodynamic film thickness, decrease in roughness amplitude and decrease in roughness slope. X. SURFACE

FATIGUE

PHENOMENA

A model for surface initiated rolling contact fatigue5 assumes that surface fatigue spalls initiate either from contacts between asperities of the general roughness or from stresses arising at imperfections (defects). In the case of roughness-initiated spalling, a two-phase phenomenon requires consideration. In the first phase, micropits form as a result of the fatigue of interacting asperities. In the second phase, spalls form from the micropits, acting as defects. Film thickness h, asperity height y,, and asperity slope 8 are the parameters influencing the plastic strain y, at asperities5. Of the material parameters, a (reduced) elastic modulus E',the Neuber length A,, and a function of the ultimate strength (T, enter the formulae, as follows: Yp = fi

(0,) lo!&,’

17.1 clog*&)

o.659-0.5)

(101)

with r* = E'd

(102)

The severity parameter determining fatigue life is yp/Dwhere D=ductility depends on the hydrostatic stress and on the inherent ductility of the material (neither one of which are dependent on asperity geometry). Asperity geometry enters the equations only through eqn. (102). In this, the two quantities y, and (3 are asperity parameters. y, is the height measured from the mid-line of the composite profile. For 8 in eqn. (102) it is convenient to use the definition of 8, given in eqn. (78) in terms of y, and of the tip curvature p. To obtain the distribution of yP, one needs both a distribution of (yop)“=O, and a distribution of y, alone. The calculation of the distribution of yP is complex. As a simplification, the conversion of eqns. (101) and (102) into power functions was suggested5 but has not been carried out in detail. The general form of the power function proposed is as follows:

where A = the radius of contact (or the half width of the contact strip), depends on the deflection of the asperity and on its slope as shown in eqn. (56). Substituting from eqn. (56) eqn. (103) becomes, for ridge type asperity geometry

To obtain the distribution of y,/Done requires accordingly, the joint distribution of 0, and v or, by substituting for 0, from eqn. (78), the joint distribution of p and y, lo, and, in principle, the necessary integrations to obtain the distribution of y,JDcan be performed. wear.

21 (1972)

THE

THEORY

OF PARTIAL

The probability

ELASTOHYDRODYNAMIC

CONTACTS

95

of forming a micropit, is defined5 as:

i.e. as a function of yp/D and of a factor k’ which describes the boundary lubrication conditions in the contact. Micropit probabilities are assigned to asperity interactions by eqns. (103) and (105). The total number of micropits is calculated as the number of asperity interactions multiplied by the probability that there will be a micropit at a given interaction point. In a two-dimensional roughness process, the number of interactions per unit length is half the density N,, of profile crossings over the level h as defined in eqn. (1). For a three-dimensional roughness process, the number of interactions per unit area is half the density hN of two-dimensional crossings, with t&r = Z;

(106)

The distribution of the density of micropits as a function of their diameter is the determinant of the spalling fatigue life of a surface failing from spalls originating at asperity interactions5. In the absence of an expression for the distribution of the (V -a1 o’I1)product in eqn. (104) one can only state that this number will vary as a function of h, approximately according to eqn. (106)5. This statement can be refined only after determining the distribution of the (Oa*-al ual) product. Regarding spalling failure originating from localized defects (furrows), life can be related to the density and size distribution of the furrows5. There the matter rests until more is learned about localized defect distribution. ACKNOWLEDGEMENTS

This paper consolidates into a unified structure, a multitude of distinct contributions. Those from public literature are referenced. Many more, made by members of the staff in the author’s laboratory, are as yet unpublished. Grateful credit is herewith given to all these contributors, in particular to Dr. Y. P. Chiu, who performed all new elasticity analyses and wrote the Appendix, and to Messrs. J. I. McCool and J. C. Shoemaker, who contributed much to the statistical and numerical treatment. The author is indebted to SKF Industries, Inc., in the Research Division of which all the original work was done, for permission to publish, and to the U.S. Navy and Air Force for support of studies providing background. NOTATION

Half-width of contact strip, Major axis of Hertzian area, Neuber length (notch sensitivity), Undeformed asperity intercept, Total area of “hills” ; Total area of asperity contact, Area fraction of “hills” within a contact area a,, “Hill” area within a two-dimensional crossing, Numerical exponent,

A

AH A, A1

a

ala, ai

al Wear, 21

(1972)

r.

96

P,l 9, P PInax

Q LIEI

EHD ii,,

Ql' 9a qEHD R (7)

Numerical exponent, Tip width of prismatic asperity. Deflection of crucial asperity, Ductility Dwell length of a profile, above level h, Average dwell length of a profile, Expected value operator, (Reduced) elastic modulus, Cumulative distribution function, Cumulative distribution of peak heights, Function sign ; probability density function, Frequency function of peak heights, Joint distribution, height and slopes, at peaks, Cumulative distribution function, Conditional mean load, as a function of A Cumulative distribution of asperity contact areas, (Average) elastohydrodynamic film thickness, Two-dimensional crossing density, at level h, Elastic asperity load integral, Elastic asperity load integral, Elastic asperity load integral, Boundary lubrication constant, Fatigue life, Lower 10th percentile fatigue life, Length of contact strip, Minor axis of Hertzian area, Density of level crossings (surface profile), Profile peak density, Density of crossings over level h (surface profile), Density of zero crossings (surface profile), Summit density, Plastic yield pressure ; Probability, Fraction of plastic occurrences, Probability of (plastic) contacts at summits, Frequency function ; Local EHD pressure, Maximum EHD pressure, Elastic asperity load, Elastohydrodynamically carried normal load, Total normal load in plastic asperity contact, Load at a single asperity, Normal load ratio, asperities, Normal load ratio, EHD, Auto-correlation function, argument z, Asperity tip radius, Average surface separation,

Wear, 21 (1972)

I-. 1.ALLlAh

THE

THEORY

OF

PARTIAL

ELASTOHYDRODYNAMIC

CONTACTS

97

Power spectral density, circular frequency w Slide/roll (velocity) ratio, Traction force, Asperity traction, EHD traction, Abscissa along a, profile (integration variable), Velocity, Asperity deflection, Mean asperity deflection (for film thickness h), Joint frequency function, (A, O), Surface length coordinate, Height (amplitude) coordinate, Standardized height, Peak height, Standardized peak height, Depth of maximum shear stress, Spectrum width measure, Surface gradient, Traction curve, initial slope, Typical asperity slope, Prismatic asperity side slopes, Skew ratio of asperity, Angle corresponding to asperity slope, Profile slope, Curvature, Average profile peak curvature, Standardized average profile peak curvature, Mean summit curvature, Standardized summit curvature, Probability of micropit, Friction coefficient, EHD, Friction coefficient, asperities, Parabolic asperity tip curvature, r.m.s. surface roughness (composite r.m.s. roughness for two surfaces in contact), r.m.s. slope, r.m.s. of the second profile derivative, Tensile yield strength, Ultimate strength, Maximum (asperity) shear stress, Critical shear stress (surface fatigue), Cumulative standardized Gaussian distribution function, argument, Standardized Gaussian frequency function, Spacing of plastic occurrences. NOTATION

A

FOR

APPENDIX

Half of the asperity contact width,

Wear,21

(1972)

98 c E fib.; P(X)

Q I' x fl 1.2 v1.2 ~rTl,x

T. II. TALLIAK

Location coordinate of plane asperity tip. Reduced Young’s modulus. Young’s modulus of the two contacting bodies, Profile of an axial symmetric asperity contact, Contact pressure, Load in an asperity contact, Normal deformation on asperity surface, Coordinate parallel to the asperity surface, Slopes of prismatic plane asperities, Poisson’s ratio of the two contacting bodies, Maximum shear stress in asperity tip.

APPENDIX:

ELASTIC

1. Non-symmetric

MODELS

plane

OF ASPERITY

usprrity

CONTACT

contact

Consider a prismatic asperity with different slopes (- 0, and 8,) under a normal load P in contact with a half plane as shown in Fig. 27. This problem can be solved using complex variable theoryA6. If the contact boundary is (-A, A). the tip will be located at .x = c where c is determined from the following formula :

The ratio of the vertical the tip is given by: iiz ‘S

_

02 8, 8 -! tan2 i 2+

=

AE’ (0, 2rr--.--

+

02)

where

Fig. 27. Asymmetrical Wear, 21 (1972)

of the contact

J

4 71

edges .X= - LIand ,Y= LIfrom

PA,

load P is given by:

The total normal Q,

distances

asperity

cos

H

,

(3A)

THE

THEORY

OF

PARTIAL

ELASTOHYDRODYNAMIC

The maximum shear stress underneath

CONTACTS

99

the tip is:

zmax= E’(8, + 8,)/27c*

(44

The expression for the contact pressure is P(x) =

$

(AZ-x*p I,

(54

where

The expression for the slope of the profile outside the contact area is given for 1x1> A by :

Note that in eqns. (SA) and (6A) the integral I, can be integrated in closed form after using the transformation t = sin 8. Knowing du/dx, the displacement u can be obtained through a numerical integration. II. Axi-symmetrical

contact problem (Asperity (a) Load and deflection

with rotational

symmetry)

Consider a general shape as shown in Fig. 28. y =.f’(r) = Br” where r is the radius. The following results are given by Galin37. The expression for the load is:

?!&+ r’F ’ “)

(a’-r’)+dr

or

(7A)

Y-m”

-\

d 0

Fig. 28. Rotationally Wear, 21 (1972)

r-A

symmetric

i-

asperity

rndX.

100

-1‘. I+.. TALLIAN

where 2(n). q(n) arc functions II

=1

i(n) = 0.636 (p(H) = 1.57 and c is the deflection

of II given by:

3

3

4

0.942 2.67

1.127 3.54

I.253 4.26

at the center of contact

given by :

64) or Qr _A”+l B where .4 =constant radius. P(x) is the gamma function. It may be seen that except for II =O (for a flat ended punch). there is no profile in this category that yields Q1 _ A. The special case II = 2 corresponds to the Hertz problem which yields B= 1/2R. Q, = (2.67E’A3)/(kcR) and C _ Q”‘“. This is a classical result. A conical punch (n = 1) yields Qr = (1.57 E’)/(E). BA2 i.e. the load is proportional to the area of contact, the same as for a prismatic asperity. For a cone B=y, the gradient of the conical surface, i.e. the load is proportional to the gradient as it is for a prismatic asperity for which slope and gradient are the same. The deflection C is proportional to the contact diameter or to the square root of the force, and again contains the gradient as a linear factor: C=;BA

so that Q,

_

“‘“,“”

B-

‘C”

REFERENCES T. E. Tallian, 3. 1. McCool and L. B. Sibley, Partial elastohydrodynamic lubrication in rolling contact. Proc. Inst. Mech. Engrs.. Paper 14, (196566) 169. T. E. Tallian. Y. P. Chiu. D. F. Huttenlocher, et al.. Lubricant films in rolling contact of rough surfaces. ASLE Trans., 7 (2) (1964) 109. R. R. Valori. T. Tallian and L. B. Sibley, Elastohydrodynamic film effects on the load-life behavior of rolling contacts, ASME Paper No 65LUBS-11, 1965. T. E. Tallian and J. I. McCool, The observation of individual asperity interactions in lubricated point contact, ASLE Truns.. 11 (2) (1968) 176. T. E. Tallian and J. I. McCool, An engineering model of spalling fatigue failure in rolling contact--~the surface model, ASLE Preprint 70X-2, 1970; Wear, 17 (1971) 447. N. E. Sindlinger, .I. A. Martin and D. F. Huttenlocher, Progress Report No. 3 on Influence of lubrication on endurance of rolling contacts, U.S. Dept. of the Navy. Bureau of Naval Weapons. Contract NOW-61. 0716-C, SKF Ind. Rept. No. AL62T013, April 1962. W. E. Schmidt, J. I. McCool, et al., Progress Report No. 6 on Influence of lubrication on endurance of rolling contacts, US. Dept. of the Navy, Bureau of Naval Weapons, Contract No. NOW-61-0716-C. SKF Ind. Rept. No. AL63T016, March 1963. E. F. Brady, J. A. Martin, er al., Progress Report No. 9 on Influence of hibricdtion on endurance of rolling contacts, U.S. Dept. of the Navy, Bureau of Naval Weapons. Contract No. NOW-61-0716-C. SKF Ind. Rept. No. AL64T014, December, 1963. Y. P. Chiu. J. A. Martin. J. I. McCool, et al., Development of a mathematical model for predicting life of rolling bearings, Rome Air Development Center. Tech. Rept. RADC-TR-68-54. Wear. 21 (1972)

THE

THEORY

OF

PARTIAL

ELASTOHYDRODYNAMIC

CONTACTS

101

10 P. R. Nayak, Random process model of rough surfaces, JOL7; Ser. F, 93 (3) (1971) 398. 11 J. Pullen, J. B. P. Williamson and R. Hunt, On the plastic contact of rough surfaces, Proc. R. Sot. (London), (1970) submitted. 12 L. B. Sibley and Y. P. Chiu, Contact shape and non-Newtonian effects in elastohydrodynamic point contacts. ASLE Annual Meeting, paper No. ASLE 71-AM-2C-3, 1971. 13 J. A. Greenwood and J. H. Tripp, The contact of two nominally flat rough surfaces, Inst. Mech. Engrs., Tribology Group, 185 (1970-71) 48/71. 14 J. A. Jefferis and K. L. Johnson, Traction in elastohydrodynamic contacts, Proc. Inst. Mech. Engrs., 182 (1967-68) Paper 1. 15 J. M. McGrew, A. Gu, H. S. Cheng and S. F. Murray, Elastohydrodynamic lubrication-preliminary design manual, Wright Patterson Air Force Base, Air Force Systems Command, Tech. Rept. AFAPL-TR70-27. November 1970. 16 K. L. Johnson and R. Cameron, Shear behavior of elastohydrodynamic oil films at high rolling contact pressures, Proc. Inst. Mech. Engrs., 182 (1967-68) Paper 2. 17 M. A. Plint, Traction in elastohydrodynamic contacts, Proc. Inst. Mech. Engrs., 182, (1967-68) Paper 3. 18 D. Dowson and T. L. Whomes, Effect of surface quality upon the traction characteristics of lubricated cylindrical contacts, Proc. Inst. Mech. Engrs., 182 (1967-68) Paper 4. 19 D. Dowson and G. R. Higginson, Elastohydrodynamic Lubrication-The Fundamentals of Roller and Gear Lubrication, Pergamon Press, London, 1966. 20 E. Schoeler, C. G. Hingley, J. W. Caldwell, et al., Final Summary Report on Elemental contact occurrences in rolling and sliding U.S. Dept. of the Navy, ONR Contract Nonr 4895(00), SKF Ind. Rept. No. A L68POO2, April 1968. 21 A. Dyson, Frictional traction and lubricant rheology in elastohydrodynamic lubrication, Phil. Trans. R. Sot., (London), A 266 (1970) 1170. 22 J. C. Bell, J. W. Kannel and C. M. Allen, The rheological behavior of the lubricant in the contact zone of a rolling contact system, ASLE-ASME Lubrication Conf, Paper No. 63-LUB-8, October 1963. 23 C. T. Walters. The dynamics of ball bearings, Trans. ASME, Ser. F, 93, (1) (1971). 24 L. Leonard, G. Cotellesse. Interim Report on Scanning electron microscopy of 6309 size bearings, U.S. Dept. of the Navy, Office of Naval Research, Contract NOOO14-70-C-0229, SKF Ind. Rept. No. AL70COO4, July 1970. 25 N. Sindlinger, et al., Progress Report No. 4 on Influence of lubrication on endurance of rolling contacts, U.S. Dept. of the Navy, Contract No. NOW-61-0716-C, SKF Ind. Rept. No. AL62T015, 1962. 26 E. Schoeler, et al., Final Summary Report on Project III of A study of the geometry of elastohydrodynamic films in point contact, Department of the Navy, Naval Air Systems Command, Contract NOOO1968-C-0310, SKF Ind. Rept. No. AL69P012 1969. 27 J. I. McCool, A. Elshinnawy, et al., Final Summary Report on Elemental contact occurrences in rolling

28 29 30 31 32 33 34 35 36 37

and sliding. US. Dept. ofthe Navy. ONR Contract No. Nonr 4895(00). SKF Ind. Rept. No. AL66L039, August 1966. T. E. Tallian, Discussion to the paper by J. Pullen, et al., On the plastic contact of rough surfaces, SKF Ind. Rept. No. AL70Q013, June 1970. J. F. Archard, Single contacts and multiple encounters. J. Appl. Phys., 32 (1961) 1420. J. A. Greenwood and J. B. P. Williamson, Contact of nominally flat surfaces, Proc. R. Sot. (A), (295) (1966) 300. H. Christensen, Stochastic models for hydrodynamic lubrication of rough surfaces, Inst. Mech. Engrs., Tribology Group, 184 (55) (1969-70) Part 1. P. E. Fowles, A thermal elastohydrodynamic theory for individual asperity-asperity collisions, ASME Paper 70-LUB-25, 1970. C. T. Walters, et al., Study of the behavior of high-speed angular contact ball bearings under dynamic load, Final Rept. on Contract NAS8-21255, National Aeronautics and Space Administration, May 1969. J. Archard and J. Kirk, Lubrication at point contacts, Proc. R. Sot., A261 (1961) 532. T. E. Tallian, E. F. Brady, J. L. McCool and L. B. Sibley, Lubricant film thickness and wear in rolling point contact, ASLE Trans., 8 (4) (1965). N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of’ Statistics, Noordhoff Ltd., Holland, 1959. L. A. Galin, Contact Problems in the Theory of’ Elasticity, English Translation published by North Carolina State University, Dept. of Math. and Eng., 1961.

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