The nature of the wear process in the four-ball lubricant test

Wear, 68 (1981) 109 - 127 0 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands

THE NATURE OF THE WEAR PROCESS LUBRICANT TEST

109

IN THE FOUR-BALL

A. DORINSON Tribology and Lubrication,

3E 825 Indian Trail, De&in, FL 32541

(U.S.A.)

(Received September 10, 1980)

Summary Geometrically the generation of a wear scar in the four-ball test can be divided into a phase in which the surface of the scar is convex and a phase in which it is concave. It is shown how the geometry of the wear scar is fitted to a linear volume-rate model of wear. The course of a real wear experiment of 60 min duration was found to fit the linear model for only the first 10 min of rubbing, after which the wear rate increased. A plausible explanation for this behavior is the increase in the temperature of the material in the vicinity of the rubbing interface due to the accumulation of heat generated by friction. The distinction between the four-ball “wear” and “extreme pressure” tests is ascribed to the way that the accumulation of heat is affected by conditions such as load and rubbing time.

1. Introduction Since it was first described by Boerlage [l] the four-ball machine has acquired the status of an established institution in the fundamental investigation of lubricants as well as in their technological testing. There are a number of procedures by which the machine is used to evaluate the performance of lubricants, but the most frequently used tests fall into two broad categories: methods for assessing antiwear efficacy and methods for rating extreme pressure behavior. Neither catagory has a basic definition other than that of the methodology and the descriptions of the test themselves [ 2, 31. The wear test as specified by ASTM Method D-2266 [ 21 is carried out under an applied load of 399.3 N (40 kgf) at a spindle speed of 20 rev s-l for a period of 3600 s. There are variants of the test conditions, e.g. a load as low as 58.8 N (6 kgf), a spindle speed as low as 10 rev s-l or a test time as short as 900 s. The extreme pressure test consists of a set of runs under a series of graduated loads, each run of 10 s duration at 29 - 30 rev s- ’ in the ASTM procedure [3] but as long as 60 s duration in some variants of the test. The evaluative criterion in the wear test is the average diameter of the circular

110

scars formed on the three stationary balls. For the extreme pressure test the results are reported as the load/wear index, derived from an arbitrary rather involved calculation, which in the last analysis rests on the measured dimensions of the scars worn on the stationary balls. Considerable interpretive significance is attached to the difference between the wear and the extreme pressure tests. That lubricants behave differently in these two types of testing has been demonstrated systematically by Allum and Ford [4] and by Allum and Forbes [ 51 for lubricating fluids containing organic sulfides as additives. Forbes [ 61 , in a publication reviewing the “antiwear” and the “extreme pressure” action of various chemical structures in additive substances, defined antiwear action as occurring when the oil film is not extensively penetrated by contacting asperities, whereas extreme pressure action is observed when rapidly increasing temperature brings about collapse of the oil film, thus bringing the action of the additive to the forefront. This explanation is plausible but not particularly enlightening from a basic mechanistic point of view. Even at the modest applied load of 147 N (15 kgf), which is one of the standard loads at which antiwear action is tested, the average initial elastic deformation pressure per contact spot on each ball is 16.2 GPa (16 500 kgf cm-*). Contact pressures of this magnitude, when they are encountered in engineering practice, are considered to be in the extreme pressure domain. It is thus apparent that, whatever significance the difference between the antiwear and the extreme pressure test has for practical extreme pressure lubrication, it is not simply and directly related to the absolute magnitude of the applied load in the four-ball testing of lubricants. In this paper we propose to demonstrate the complexity of the wear process in the four-ball machine from the initial contact of the rubbing specimens to the termination of the course of wear and to advance some suggestions on the conclusions to be drawn from the observed wear behavior.

2. The four-ball

wear process

2.1. The convex wear scar When two spherical balls are first put together in static contact under an applied load, the dimensions of the circular contact area are governed by the laws of elastic deformation. If these deformations are not too large, the following equation gives the relation between the displacement of the balls from their position of initial unloaded contact and the radius of the circular contact area generated by elastic deformation: z=

r2 -

2R

(14

where z is the displacement of each ball along the line of thrust, r is the radius of the deformed area and R is the radius of curvature of the unloaded scar surface. For unworn balls with radius RO, eqn. (la) takes the form

111

I

(a)

(b)

I

Fig. 1. Initial contact and the incremental wear process: (a) initial contact and the magnitude of elastic compliance under load; (b) incremental wear of the elastically deformed contact surface. 2 a0 .z

(lb)

-O-

2Ro

Figure l(a) shows the geometry from which this relation is derived. Figure l(b) is a representation of the wear of the elastically deformed contact during rubbing. To derive an expression for the profile of the resulting scar it is necessary to have a quantitative expression for the wear. It is known from elasticity theory that the pressure distribution over the contact area is hemispherical; therefore along a diameter the pressure is given by p =pojl

-$,,,,

where po, the maximum pressure at the center of the scar, the normal loading thrust by elasticity theory and x is the radius of the scar. If it is postulated that the depth rate of tional to the pressure distribution, the profile of the worn

(2) is calculated from location along the wear is proporlamina is given by

6h =kp

(3)

where 6h is the depth of wear at the location x for a small interval of time 6t and k is a constant of proportionality. On substitution from eqn. (2) 6h =kpo(l-$)1’2

and since for a fixed load and very small 6h we can regard p. as constant

(da)

112

=K(l-$)“’

6h

(4b)

where the new constant K equals kp,. Let us consider what the scar will look like when the load is removed, assuming that &ho, the depth of wear corresponding to po, is less than zo. The original unworn arc whose chord is 2a, can approximately be represented by h=

$(l-$i”2

The equation

(5)

for the scar profile is written as

h’ = h - 6h

Equation h

(6)

(4b) substituted

-_6h

into eqn. (6) gives

$(I - $.)l” -_K(l - $,‘”

=

(7)

However, eqn. (7) is at variance with two facts about the course of wear (a) that the radius of the wear scar increases progressively with the duration of the wear process; (b) that the radius of curvature of the scar surface does not remain identical with that of the original surface of the ball. Accepting both of these facts, the generalized expression for the profile of the wear scar becomes

!&(1-f!“‘-K(l-;!“’

h-&h=

(8)

Furthermore, it is postulated that a > ao, which is supported by direct measurements of the experimental course of four-ball wear, and that K remains constant for the short time interval 6t. For the special case of h = z. and 6h = 62, x = 0. Equation (7) then gives

(9) From eqn. (8) a2

(10)

z=-

2R and hence a2

2

-=

a0 --

2R

2Ro

K

(11)

Solving for R yields R

=

a2Ro ao2 - 2KRo

(12)

113

Fig. 2. Geometry

of a convex

scar showing

incremental

wear.

On examination of eqn. (7) for its physical meaning, we can demonstrate that K < a02/2R,; hence 2KR0 is less than a02 and does not swamp it in eqn. (12). It follows then that, since a > ao, R > Ro. The geometry of the convex wear scar is shown in Fig. 2. Equations (la) and (lb) apply to the case of two spheres in elastic contact. However, it is apparent from eqns. (5), (7) and (8) that the surface of the elastically relaxed wear scar is not spherical but that it is an ellipsoid of revolution. However, for the small scars with which we are concerned the differences are inconsequential. Therefore we can compute z from eqn. (9) and can substitute the result into eqn. (10) to obtain R. This is the basis of the incremental method described in Appendix A for computing the characteristics of the convex wear scar as wear progresses. From the stochastic application of the incremental wear process to the course of wear we write R, =

an2R,-1 a,_12

- 2K,_,R,_1

(13)

where R, is the radius of curvature of the unloaded scar surface at the end of the nth interval. From this, using the reasoning that was applied to eqn. (12), it can be deduced that R increases with each incremental step of the wear process, i.e. the surface of the wear scar becomes progressively flatter as wear continues. On physical grounds, as will be demonstrated later, the terminal radius aN of the wear scar for the convex phase remains finite and of the same order of magnitude as ao. It is also demonstrable on physical grounds that RN becomes infinite, i.e. the surface of the unloaded wear scar is flat (this determines the terminal condition in the incremental calculation). For RN to become infinite requires a discontinuity in the analysis, whereas physically the wear process is continuous. 2.2. The concave wear scar After the surface of the unstressed wear scar has attained the flat stage, further wear generates a concave scar. This is an experimentally verifiable fact to which the following analysis can be fitted. Let us consider the elastic

Fig. 3. Geometry of a concave scar showing incremental wear. contact

of the rotating upper ball with radius Ra of curvature and the surface of the wear scar with radius R of curvature. As a first approximation for the displacement of the scar surface under load (Fig. 3) we can write

a2 z=--

(14)

2R where the minus sign arises from the curvature for a concave wear scar, the the stationary ball to its periphery. We now apply the wear postulate and (4b). The analogue of eqn. (9) for

reverse orientation of the radius of sense of z being from the interior of from which we obtained a concave wear scar is

eqns. (4a)

2 z’=z+6z=--+K

(15)

2R where the positive sign for the K term arises because z and 6z are oriented in the same direction. By extension of eqn. (14) a

z’=-_

12

(16)

2R’ a

12 -

2R’

!f+K

2R

Equation

(174

(17b) differs from eqn. (12) in that the numerical value of R is the denominator grows larger as the course of wear progresses. However, a’ is larger than a and thus it cannot be predicted from eqn. (17b) alone that R’ will be smaller than R. negative

and therefore

115

However, it can be demonstrated empirically that if ca, the radius of the concave scar area, increases smoothly and progressively with the course of wear then the numerical value of R, the radius of curvature of the scar surface, will also decrease smoothly and progressively. The following relation can be derived from elasticity theory [ 71 : R=

R,-,a3 3(1 -v2)PRo/2E

-aa3

(18)

where v is Poisson’s ratio, E is Young’s modulus (it is assumed that all four balls are made of the same material) and P is the normal loading thrust on the scar area. The specific application of eqn. (18) to the concave wear scar in the four-ball test is given in Appendix B.

It is difficult to apply direct measurements of the scar diameters to explain the course of four-ball wear in terms of possible wear mechanisms. Models of smooth wear using the adhesive mechanism are based essentially on the statistics of asperity contact and particle detachment; specific models such as the constant volume-rate process proposed by Archard [S] or the laminar depth-rate process of Dorinson and Broman [9] are concerned with the realization of the basic concept in a form applicable to physical experimentation. It is helpful to examine the course of four-ball wear in terms of such simple models; even when the experimental findings do not agree with the predictions of the model, scrutiny of the deviations is informative. However, the geometry of the wear scar on a spherical ball makes the scar diameter a cumbersome and insensitive parameter for such scrutiny. To examine the course of four-ball wear (the material properties and the dimensions of the ball specimens, the applied load and the speed of rotation all being fixed), we write the constant volume-rate postulate in the following form: dV = K,, dt

(‘19)

where K, is the wear rate constant. The computation of the wear volume from measured scar diameters is shown in Appendixes A and B. A plot of wear volume against rubbing time shows directly whether the constant volumerate relation holds or not, whereas no such conclusion is directly apparent when the course of wear is plotted in terms of scar diameter or scar radius. In Fig. 4, wear volumes computed geometrically are shown as two linear plots, the wear rate of one being twice that of the other:

The volume data are given in Tables 1 and 2; the methods of computation are described in Section 3 and in Appendixes A and B. Also shown in Fig. 4 are the corresponding plots of scar radius. Time is plotted on a dimensionless scale, the derivation of which is described in Section 3. To show the amount

116

Relative

Fig. 4. Calculated TABLE

Time

of wear at two levels of volume

rate as a function

of relative time.

1

Calculated n

course

of the convex

R (mm)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Radius

course

6.350 6.615 6.902 7.216 7.560 7.937 a.355 8.819 9.338 9.922 10.583 11,339 12.212 13.229 14.432 15.875 17.639 19.844 22.679 26.458 31.750 39.687 52.917 79.375 158.750 a of ball, 6.350 mm; load, infinite here.

aR becomes

phase of wear with a constant

V(X10p3 mm3)

c (mm)

0.000 0.000 0.001 0.002 0.003 0.004 0.004 0.006 0.007 0.008 0.009 0.010 0.012 0.013 0.015 0.016 0.018 0.020 0.022 0.025 0.027 0.030 0.033 0.036 0.040 0.044

0.1091 0.1098 0.1106 0.1114 0.1122 0.1130 0.1139 0.1148 0.1157 0.1166 0.1176 0.1186 0.1196 0.1207 0.1218 -0.1229 0.1241 0.1254 0.1266 0.1280 0.1294 0.1309 0.1324 0.1340 0.1357 0.1375 15 kgf (147

volume

N).

00 72 48 29 14 04 99 01 08 23 45 75 14 63 23 94 78 76 91 22 74 47 45 71 28 17

rate V’ 0.0000 0.0163 0.0336 0.0518 0.0710 0.0914 0.1131 0.1360 0.1604 0.1863 0.2140 0.2434 0.2749 0.3086 0.3447 0.3835 0.4252 0.4701 0.5186 0.5711 0.6280 0.6899 0.7574 0.8312 0.9120 1.0000

117 TABLE 2 Calculated course of the concave phase of wear with a constant volume rate = (mm)

B (mm)

V (X 10e3 mm3)

V’

0.1465 0.1500 0.1600 0.1684 0.1750 0.1876 0.2000 0.2338 0.2500 0.2750 0.3022 0.3250 0.3522 0.3750 0.4000 0.4500 0.4750 0.5000

-36.861 -27.650 -17.385 -13.938 -12.333 -10.475 -9.407 -7.972 -7.618 -7.257 -7.011 -6.870 -6.752 -6.679 -6.619 -6.537 -6.508 -6.485

0.067 0.077 0.111 0.145 0.176 0.246 0.331 0.664 0.886 1.326 1.965 2.655 3.691 4.771 6.204 9.999 12.440 15.300

1.51 1.74 2.50 3.27 3.97 5.56 7.50 15.01 20.03 29.99 44.42 60.04 83.46 107.89 140.29 226.10 281.30 345.97

Radius of balls, 6.350 mm; load, 15 kgf (147 N).

TABLE 3 Dependence of extent of wear as evaluated by scar radius on duration of rubbing in a constant volume-rate process Volume rate Relative time Gross scar radius (mm) Net scar radius (mm)” Ratio of net scar radius for 2K, to net scar radius for K,

KW 5

2Kw 5

0.187 0.078

0.217 0.108

-

1.40

KVJ 180

0.425 0.316 -

2Kw 180

0.503 0.394 1.25

aThe hertzian scar radius 0.109 mm was computed for a load of 15 kgf (147 N) on steel balls 12.70 mm in diameter.

of wear in terms of the scar radius, the measured radius of the scar area must be corrected by subtracting the radius of the hertzian elastic deformation area. Table 3 illustrates the problems associated with making quantitative assessments of the course of wear from scar radius data. The volume ratio remains at 2:l at any given time, but the ratio of the net radii changes from 1.40:1 at relative time 5 to 1.25:1 at relative time 180. 2.4. Change in contact pressure during the course of wear The average contact pressure over the area of the wear scar is given by p= ” 7ra

(20)

118

where P is the normal thrust force acting on this area. Obviously, as a increases through wear, p decreases. Over the full course of wear, from the start at hertzian contact pressure to ~rmination with a well-formed concave scar, the decrease in pressure can be substantial; thus, for an applied load of 147 N, on termination of wear with a scar measuring 0.900 mm in diameter the final. contact pressure is only 5.8% of the initial hertzian pressure. If the true area of asperity contact remains constant even though the macroscopic contact pressure diminishes, the volume rate of wear also remains constant and eqn. (19) correctly describes the wear process. However, if the decrease of contact pressure is associated with drastic changes in the nature of the interfacial contact, then the volume rate of wear will no longer be constant. For example, in lubricated rubbing, better opportunity for access of lubricant as contact pressure decreases theoretically should be manifest as a decrease in wear rate.

3. The course of wear 3.1. The geometric model Algorithms for computing the wear vofume, the radius and the projected area of the wear scar, and the radius of curvature of the scar surface area are given in Appendixes A and B. These quantities are computed solely on a geometric basis and have no reference per se to a time-governed model of wear. We can elect to normalize the course of wear with respect to the volume at the termination of the phase with the convex scar; the radius R is then equal to infinity and the corresponding wear volume is designated as V,. The expression for the normalized non-dimensional wear volume V’ is V’=

;

(21a)

If, assuming a constant volume-rate mech~ism, V, = I&, then fixing the relative volume scale with respect to Vf is equivalent to fixing the relative time scale with respect to tf : v’=

4 tf

(21b)

Thus in Tables 2 and 3, which were computed for hardened steel balls 12.7 mm in diameter under an applied load of 147 N, the values of V’shown are the measures of non-dimensional relative time for a geometrical constant volume-rate model of wear, and therefore Fig. 4 is geometrical and not necessarily representative of a case of real physical wear. 3.2. The physical course of wear In an actual four-ball wear test the experimentally measured quantities are scar diameter and rubbing time, and the course of the physical wear is usually plotted as scar diameter against time. The geometry of scar radius,

119

wear volume and radius of curvature of the wear scar surface is also valid for physically observed wear, where the time is real and no a priori assumptions are made regarding the mechanism. Plotting the course of observed wear against real time and comparing the results with computed plots against relative time can yield clues about the nature and mechanism of the physical wear process. The constant volume-rate model is the simplest model to use for comparison because deviation of the real course of wear from it is evident.

4. Examination

of physical

wear

4.1. Course of wear with a load of 147 N Table 4 shows the directly measured data (wear scar diameter and real rubbing time) and the wear volume computed from them as obtained in an investigation of four-ball wear carried out under an applied load of 147 N (15 kgf) with a solution of 5.37% di-t-octyl disulfide in light white oil as the lubricant. Wear volume is plotted against real rubbing time in Fig. 5. Observed wear, instead of following the constant volume-rate model, shows an increasing rate as the course of wear progresses. The average rate for the first 300 s is 0.844 X 10e6 mm3 s-l, and the average rate for the terminal 2000 s is 2.19 X lop6 mm3 s-l. Plotted on an expanded scale (Fig. 6), the initial course of wear is seen to consist of an initial phase with a rate of 0.844 X lop6 mm3 s-l, which changes to a progressively increasing rate at about 350 s with an intermediate value of 1.85 X lop6 mm3 6’ at 600 s and TABLE 4 Experimental four-ball wear data

Time (s)

Scar diameter

3 6 10 15 30 45 60 150 300 600 1500 2400 3000 3600

0.2205 0.2289 0.2314 0.2388 0.2604 0.2705 0.2929 0.3368 0.3752 0.4676 0.6043 0.7043 0.7783 0.8108

(mm)

Wear volume

(X

10P3 mm3)

0.0015 0.0060 0.0071 0.0121 0.0292 0.0401 0.0669 0.1448 0.2461 0.6639 1.9646 3.6908 5.5671 6.5485

SteelMls, 12.70 mm L;diameter; applied load, 15 kgf (147 N); rubbing speed, 1800 rev min , i.e. 59.85 cm s ; lubricant, 5.37% di-t-octyl disulfide in white oil of kinematic viscosity 18.14 X lo-+ m2 s at 310.9 K (37.8 “C).

120

12

4

Rubbing

Fig. 5. Experimentally

16 Time,

observed

28

32

seconds

wear volume

as a function

of real rubbing

time,

/,

;

I

24

20

10’

/I -

0.30

-

0.25

E -0.20

_ m 1 $

-0.15

m Fi ::

-0.10

-0.05

300

Rubbing

Fig. 6. Initial

portion

400

Time,

500

600

lo.00 700

seconds

of the experimentally

observed

four-ball

wear process.

rises to the terminal value of 2.19 X lo-6 mm3 se ’ at 1500 s. The curve for wear scar radius in Fig. 6 shows a small upward inflection for the interval 540 - 600 s, but this does not reveal the departure from the constant-rate wear mechanism as dramatically as the volume wear curve. 4.2. Examination of the wear scars The wear scars on the stationary balls were inspected through a 50power microscope and their topography was studied with the aid of a Michelson interferometer attachment. Figure 7 shows details of two scars with

121

m b

mf Fig. 7. Convex wear scars: (a) photomicrograph, after 6 s; (b) interferogram oriented perpendicular to rubbing direction, after 6 s; (c) interferogram oriented parallel to rubbing direction, after 6 s; (d) photomicrograph, after 45 s; (e) perpendicularly oriented interferogram, after 45 s; (f) parallel-oriented interferogram, after 45 s. The leading edge of the wear scar is at the bottom of the photographs here and in Figs. 8 and 9.

Fig. 8. Concave wear scars: (a) photomicrograph, after 60 s; (b) perpendicularly oriented interferogram, after 60 s; (c) parallel-oriented interferogram, after 60 s; (d) photomicrograph, after 150 s; (e) perpendicularly oriented interferogram, after 150 s; (f) paralleloriented interferogram, after 150 s.

convex surfaces. The progressive increase in the radius of curvature is readily apparent on comparing the interferograms of the 6 s and the 45 s scars. It is seen in Fig. 8 that the scar surface is substantially flat at the 60 s interval and that the interferogram of the 150 s scar shows a definitely concave radius of curvature.

Fig. 9. 15 s wear showing the complex perpendicularly oriented interferogram;

character of a convex scar: (a) photomicrograph; (c) parallel-oriented interferogram.

(b)

The photomicrographs of the wear scars in Figs. 7 and 8 indicate that the wear process even at the low applied load of 147 N (15 kgf) is not ideally smooth. The 6 s scar has distinct furrows oriented in the rubbing direction; the 45 s scar shows a distinct difference between the topographies of the leading and of the trailing edges; the 150 s scar has a longitudinal groove at least 25 pm deep. Interferograms oriented parallel to the rubbing direction, such as that in Fig. 9 for the 15 s scar, show that during much of the convex wear phase the scar is characterized by a concave depression at the trailing border. For the particular experiment under consideration here, the termination of the convex phase of wear occurred at a real rubbing time of 49 ? 1 s, as nearly as could be estimated from a comparison of the geometrical relative time plots and the experimental real time plots of scar radius. The interferograms of the 45 s and the 60 s wear scars are consistent with this. Detailed plots of the experimental scar radii for the first 150 s of real time rubbing do not show any signs of discontinuity or transition between the convex and the concave phases of wear scar geometry.

5. Discussion The geometrical wear calculations of Section 2 were put into terms of constant volume rate and relative time for comparison with the experimental observations of Section 3 in terms of real time to see whether the latter conformed to or departed from a constant volume rate because wear at constant volume rate can be fitted to models that are attractive from the viewpoint of both a priori logic and real phenomenology. One such model is Archard’s treatment of contact and wear of plastically deformed asperities [ 81, which is an extension of an earlier analysis by Holm [lo] . A key’element of this model is the postulate that the surface density of the asperities is fixed. This means that for rubbing specimens of geometry such that the nominal area of the interface increases with the progress of wear there must be some compensating mechanism operative at the asperities that keeps the wear rate constant. Archard’s original simple treatment comes remarkably close to the results of more sophisticated analyses of the behavior of surfaces carrying

123

asperities: for example, Archard’s [ll] later examination of the elastic deformation of surfaces composed of asperities with multiple orders of sizes or Greenwood and Williamson’s [ 121 statistical treatment of the contact of surfaces. The plastic deformation postulate requires that the real area of contact remains constant during the course of wear; hence, if the total number of asperities increases because of enlargement of the nominal interface area, then the average real contact area per asperity must decrease. Given that the ratio of wear-effective asperity contacts to the total number of contacts does not change during the wear process, the size of the wear particle per effective contact must therefore decrease if the wear rate is to remain constant. It follows that, if in fact the observed course of wear shows an increasing rate, either the average size of the particle removed per wear-effective asperity encounter or the ratio of effective encounters to total encounters must increase; perhaps both may increase simultaneously. However, a priori analytical reasoning based on asperity geometry or on the plastic deformation postulate does not convincingly explain why such behavior occurs. The cause must lie with some other aspect of the wear process. The physical phenomenon that comes to mind most readily is the rise in the temperature of the surface lamina of the wear scar as the frictional heat generated by rubbing accumulates in the ball. This lowers the yield strength of the metal, which in turn increases the true area of asperity contact and probably increases the ratio of wear-effective contacts as well. Fein [ 131 has reported the influence of bulk temperature on wear in four-ball tests of 10 s rubbing duration. The major effect for a test lubricated with 0.43% stearic acid in cetane under a load of 265 N (27 kgf) at 21.7 cm 5-l was a lo-fold increase in wear when the bulk temperature was raised to the critical value of 347 K (74 “C). There was also a less spectacular increase of 135% in wear as the temperature rose from 304 to 347 K. Both of these increases are significant in understanding the nature of four-ball wear. Viewed on the scale of Fig. 5, the course of wear observed over the time interval of the present investigation consists of two segments of constant volume rate with a transition between them; the initial segment and the transition are shown in greater detail in Fig. 6. Another aspect of the transition is shown in Fig. 10, where the wear rate is plotted against the average loading pressure exerted on the area of the wear scar. Neither the geometry of the scar surface nor the mechanics of the asperity contact model predicts the 213% increase in the wear rate between 450 and 1500 s of rubbing; on the contrary, the substantial relaxation of pressure due to enlargement of the scar area by wear should favor a decrease in wear rate by allowing more opportunity for quasi-hydrodynamic lubrication. The most reasonable explanation for the wear rate transition lies in a critical build-up of temperature, not to the levels required for the drastic jump in wear rate observed by Fein [13] but rather to levels at which the wear transition is just beginning (ref. 13, Fig. 2). It should be remembered that, although in Fig. 10 the wear rate is plotted against pressure, the experimental variable is really rubbing

124

4

I2

8

Contact

16

Pressure, 10’ kg/cm*

Fig. 10. Wear rate as a function

of contact

pressure.

time, as shown on the irregular scale at the top of the figure. It is thus apparent that the individualistic features of the course of wear in the fourbail test basically reflect the rate at which heat is generated at the rubbing interface and conducted into the material of the balls. From the tribological point of view the operative physical parameters for a given combination of specimen material and lubricant are load and rubbing speed. The portion of the total kinetic energy put into the system which is converted by friction into heat is manifest as a temperature rise. An equivalent effect is obtained by using external heating to increase the temperature, as Fein [ 131 did. The two variations of the four-ball test, the wear and extreme pressure tests, are not two separate entities; in both cases the phenomenology of the tests rests on the course of wear in response to the influences of load and rubbing speed which, as we have seen, become the influence of temperature. Under the relatively mild load and speed conditions of the ASTM wear procedure the initial wear rate has a moderate value and the rate of heat generation is correspondingly moderate; consequently the interfacial temperature increase is likewise moderate. Eventually the interfacial temperature builds up to a level at which a critical change (such as a pronounced decrease in the efficacy of the lubricant or a marked lowering in the shear strength of the material of the rubbing interface) occurs in the system and the wear scar enlarges rapidly. This introduces such counteracting effects as a decrease in the heat flux through the enlarged interfacial area, quasi-hydrodynamic lubrication, additive activation etc. Eventually the increase in wear rate slows down and the wear may even assume a constant rate again at a higher level. At the low loads and rubbing speeds of the wear test the transitional increase in wear rate is moderate. The larger the load and the higher the rubbing speed, the more pronounced is the wear rate transition and the shorter the

125

rubbing time required to achieve it. This is the real basis of the distinction between four-ball wear and extreme pressure testing.

References 1 G. D. Boerlage, Four-ball testing apparatus for extreme-pressure ing, 136 (July 14, 1933) 46 - 47. 2 Wear preventive characteristics of lubricating grease (four-ball 3 4

5

6 7 8 9 10 11 12 13

lubricants, method),

Engineer-

ASTM

Method

D 2266-67, ASTM Standards Book, Part 17 -Petroleum Products. Measurement of extreme-pressure properties of lubricating grease (four-ball method), ASTM Method D 2596-67T, ASTM Standards Book, Part 17 -Petroleum Products. K. G. Allum and J. F. Ford, The influence of chemical structure on the load-carrying properties of certain organo-sulfur compounds, J. Inst. Pet., London, 51 (1965) 145 161. K. G. Allum and E. S. Forbes, The load-carrying properties of organic sulfur compounds - II - The influence of chemical structure on the anti-wear properties of organic disulfides, J. Inst. Pet., London, 53 (1967) 173 - 185. E. S. Forbes, Antiwear and extreme-pressure additives for lubricants, Tribology, 3 (1970) 145 - 152. S. Timoshenko and S. N. Goodier, Theory of Elasticity, McGraw-Hill, New York, 2nd edn., 1951, pp. 372 - 377. J. F. Archard, Contact and rubbing of flat surfaces, J. Appl. Phys., 24 (1953) 981 988. A. Dorinson and V. E. Broman, Contact stress and load as parameters in metallic wear, Wear, 4 (1961) 93 - 110. R. Holm, Electric Contacts Handbook, Geber, Stockholm, 1946, Section 20, p. 214; Berlin, 3rd edn., 1958, p. 248. J. F. Archard, Elastic deformation and the laws of friction, Proc. R. Sot. London, Ser. A, 243 (1957) 190 - 205. J. A. Greenwood and J. B. P. Williamson, Contact of nominally flat surfaces, Proc. R. Sot. London, Ser. A, 295 (1966) 300 - 319. R. S. Fein, Transition temperatures with the four-ball machine, ASLE Trans., 3 (1960) 34 - 39.

Appendix

A

Computation of the course of wear f r a convex scar by successive incremen ts The course of wear for a convex scar is delimited by the value of R,, at the beginning and by R = m at the end. A convenient basis for the computation is to divide the course of wear into N successive increments so that

R, = -

N

N-n

n=0,1,2

Ro

Then the cumulative

wear volume at the nth increment

420)n2{3R, v, = ___ 3

,...,

- (z~)~} - %

(3%

-4

N-l

(AlI

is

(-1

126

where the first term on the right-hand side of the equation is the volume of the spherical segment under the surface of radius R0 of curvature and the second term is the volume of the segment under the surface of radius R, of curvature. The elastic compliances (~e)~ and z, are given by

%I2

(AW

@oh = 2Ro

an2

(A3b)

z,, = -

24, where a, is the radius of the wear scar at the nth interval. theory [Al] a, =

3(1 - v2) p 2E

RoR, Ro +R,

From elasticity

1’3 (A41

I

For steel balls of radius 6.350 mm and a vertical load of 15 kgf (147 N) on the ball assembly a,, = 0.07426 gives the radius of the elastically in Table 1.

recovered

wear scar in millimeters,

as shown

Reference for Appendix A Al

S. Timoshenko and S. N. Goodier, 2nd edn., 1951, pp. 372 - 317.

Appendix

Theory

of Elasticity,

McGraw-Hill,

New York,

B

Computation of the course of wear for a concave scar The wear volume is divided into two parts (Fig. 3). Volume VI, the part under the convex outer surface of the ball, is calculated by the formula for a spherical segment of radius R. : V, = ‘$ a2 zi=

2R,

(3Ro -z,)

(Bl) 032)

U33)

127

The second term of eqn. (B3) can be neglected. The other part V,, of the wear volume, which is bounded by the concave surface of the wear scar, is computed in a similar way: vrr=

Ire* 4R

U34)

where R is the radius of curvature of the relaxed surface of the scar after removal of the load, as given by eqn, (18). V,, can be expressed in terms of the radius of the wear scar as follows: 7fa4 U35) 4Ro The total wear volume is IV, I + IV,, 1.The numerical computation with eqn. (18) gives a negative value for R which, of course, is consistent with the geometrical conventions of the analysis. By this convention the sign of V,, will also be negative. To simplify matters the two volumes are added without regard to sign. For steel balls of 6.350 mm radius under an applied load of 15 kgf the computational algorithms are 3lr P VII = --(1 8E

-v2)a-

V, = 0.1237a4 v,, = 0.7854 ; R=

6.350a3 0.002 654 - a3

Values computed

from these formulae

are shown in Table 2.