I. INTRODUCTION
From the initial use of rolling elements to
support and transmit loads and provide alignment
to present rigorous demands on bearing and gear
performance, the failure of these rolling surfaces
to function as designed has presented a wide varie-
ty of problems. Many scientific and engineering
disciplines are involved in the study of these prob-
lems. Surface failures range from abrasive
wear and surface-lubricant interaction phenomena
through pitting, flaking, case crushing, excessive
or variable torque and material dimensional in-
stabilities, and depend on the particular environ-
ments, materials, types of loading and relative
motions. Basic information has come from such
fields as the physics and chemistry of surfaces
and lubricants, metallurgy and mechanics.
High reliability and high temperature appli-
cations have placed greater emphasis on the whole
class of problems, accentuating several and pre-
senting new ones. Under this pressure, research
(primarily in separate industries) has concen-
trated on materials evaluation and, in some in-
stances, on the effect of metallurgical variables
on one or more of the phenomena associated with
a particular failure mode. The use of bench rigs
or simulated testers, for reasons of speed and
economy of evaluation, has increased in popular-
ity.
Some features of this research may be
criticized:
1. Much of the research is scattered, un-
organized, and unreported in the technical litera-
ture. Industries themselves, recognizing the
state of affairs, have made an initial attempt to
correct this by establishing an A. S. M.E. Re-
search Committee on Contact Fatigue of Rolling
Elements. (1
2. Although the place of the bench rig in
accumulating masses of data is important, not
enough careful investigations have been conducted
employing instruments capable of comprehensively
studying the mechanics of the rolling process for
a practical range of loads, speeds, and environ-
ments.
3. Because of a few discrepancies or lack
of direct correlation with standard material tests,
certain established principles of material behavior
are often ignored.
4. The importance of fundamental research
in the rolling contact phenomena is self-evident,
but principles for its rational extension to techno-
logical application are lacking.
It is particularly in this last category that
a contribution out of the discipline of mechanics of
solids may be made. The engineering concept of
material properties, together with geometric and
dynamic principles familiar to the discipline, form
a basis for the rational extension of fundamental
data to various configurations, sizes, and condi-
tions of material elements in particular applica-
tions.
In Appendix A, research limited to those
phenomena or failures that are amenable to the
field of mechanics are classified. In Appendix B
* Superscript numbers in parenthesis refer to
entries in the References Cited section.
past research is critically reviewed according to
this classification. The body of this publication
is devoted to the investigation and analysis of two
important modes of failure (pitting and cumulative
deformation). Considerations necessary to extend
or correlate bench rig data, and to establish the
role of cumulative cycle-dependent plastic strain
as an anticipated failure mode in critical high-
temperature applications, are emphasized. Re-
search results from the literature and some un-
published original data collected in the course of
this investigation are used in the analysis. The
novice to the field of rolling contact is advised to
study Appendices A and B before continuing, while
the more knowledgeable reader will prefer to
proceed, referring to the Appendices as the need
arises.
DEFINITION OF TERMS
A actual contact area
B normal stress influence factor
B10, B50 lives at 0.10 and 0. 50 probability of
10' 50 failure
C quantity used to determine track width
(Fig. 12)
D diameter of a rolling element
E modulus of elasticity
F mean probability of failure
K percentage increase in track width over
theoretical width
L track length
M quantity used to determine track length
(Fig. 13)
N number of stress cycles
load
range of maximum orthogonal shear
stress + maximum contact stress
(2 7 /omax)
minor principle profile radius
major principle profile radius
original profile radius
deformed profile radius
probability of survival
torque
volume of highly stressed material
depth to maximum orthogonal shear
0 stress
a semi-major axis of contact ellipse
a' length of contact ellipse in rolling
direction
b semi-minor axis of contact ellipse,
otherwise indicates Weibull's slope
c material exponent appearing in Lundberg
and Palmgren statistical analysis
d normal force offset in rolling (Fig. 36)
e material exponent appearing in Lundberg
and Palmgren statistical analysis, other-
wise indicates configuration index
h material exponent appearing in Lundberg
and Palmgren statistical analysis, other-
wise indicates chord height relating
radial deformation and track width
k factor relating Radzimovsky's surface
endurance limit to other static strength
properties
k function of bending and torsion fatigue
strength
2 chord length relating radial deformation
and track width
u radial deformation
uE theoretical approach of elastic bodies in
contact
u /uE groove depth after 20 cycles - elastic
approach of a ball and plate
w track width
wE theoretical elastic track width
A quantity used to determine track width
0 characteristic life in Weibull cumulative
distribution function
1 sum of reciprocals of principle profile
radii
a size effect exponent
f3 size effect exponent
p Poisson's ratio
J normal stress
aB bending fatigue strength
a maximum contact (Hertz) stress
max
a' corrected maximum contact stress
max
a complimentary normal stress on critical
shear plane
0o normal stress on plane of maximum
orthogonal shear stress
at maximum surface tensile stress
ac yield strength
T*C critical shear strength in fatigue
TT fatigue strength in torsion
* orthogonal shear stress
7oct octahedral shear stress
T experimentally determined essential
E shear strength in fatigue
w angular velocity
II. ANALYSIS OF THE PITTING MODE OF FAILURE
Synopsis: This section contains informa-
tion from many sources that illustrates the effect
of bench rig design and/or rolling element con-
figuration on pitting endurance. Significant fea-
tures are presented of the stress system created
by the rolling action, and the mechanism of pit-
ting is discussed with respect to both initiation
and propagation. An analysis is given which in-
cludes the effect of size of the critically stressed
region of material on endurance and the effect of
stress level on statistical variability. The
change in stress due to plastic deformation of the
contacting bodies is also discussed.
The normal end of life for most rolling
element mechanisms is by surface pitting or
flaking. Such failures have been associated with
the usual fatigue process discussed in Appendix B.
However, the influence of many features peculiar
to the rolling contact situation complicate this al-
ready complex process. The metallurgist is
deeply involved in discerning what properties of
materials are significant in resisting such failure
and in developing better materials to prevent it.
A wide variety of bench rig tests are used in this
type of research (Figure 1).
The purpose here is to examine what valid
statements can be made about pitting, apart from
the discussion of specific materials, to help elimi-
nate the confusion of specimen configuration and
size effects in such research.
In addition to relative evaluation informa-
tion sought in bench rig testing, there is interest
in a more quantitative relation between bench rig
data on pitting resistance of a material and its
performance in an assembled commercial mecha-
nism. In fact there has been hope expressed
from time to time that a correlation between pit-
ting resistance and standard material tests could
be established. It has been popular to refute this
(2)
hope although as Sachs(2) points out, no systematic
or conclusive argument has been advanced against
the principle of such a correlation.
Complete success in direct correlation,
even when the testing conditions in the bench rig
are as near to the bearing as possible, cannot be
expected at present in the face of such problems as
the lack of full dynamic comparison of the rig and
bearing, or the effect of rolling element shape and
motion on pressure distribution through the usual
lubricant film. However, certain important con-
siderations can be made on the basis of existing
knowledge that will provide a more rational foun-
dation for the correlation.
It is the purpose of this section to demon-
strate the effect of rolling element configuration
and size on pitting strength or life and to analyze
some factors responsible for this effect. This
necessarily involves a proposed mechanism for
the initiation and propagation of the fatigue damage
under normal rolling conditions. Although the
usual inhomogeneities in material contribute to
failure as in other fatigue phenomena with espe-
cially steep stress gradients, the variation of
material properties with depth will not be consid-
ered in the following analysis. Thin cased steel
presents further complications that will not be in-
cluded here.
A. EFFECT OF ROLLING ELEMENT CONFIGU-
RATION ON PITTING
A striking demonstration of the influence of
rolling element configuration on pitting strength is
provided by tests at the U.S. Naval Engineering
Experiment Station at Annapolis. An extensive
series of tests of hardened 52100 steel rollers
with a major radius of 1. 5 inches and a profile
radius of 0. 25 inches mated with a cylinder of
1. 562-inch radius were initially conducted. When
in a later series of tests the toroid geometry was
changed from 1/4-inch to 1/2-inch radius, unex-
pected results were obtained. Since this result
is of particular significance, the original paper is
quoted below: (3)
Rollers with increased radii in the longitudinal
direction were operated in NaK at 2500OF and in
oil at 1100F. In these rollers, the longitu-
dinal radius of the double curvature element
was 0.500" as against 0.250" for standard
rollers. Thus, under equivalent loads the
stresses induced in these rollers were 80% of
those of standard rollers while the theoretical
area of contact was greater by 37%. Because
of the reduced stress in rolling contact, it was
considered that the life of these rollers would
be increased considerably beyond that obtained
for standard rollers. Increasing the longitu-
dinal radius of the contact rollers did not result
in increasing rolling fatigue life. In NaK at
2500F and, more surprising, in oil at 1100F,
lower life or load carrying capacity was indi-
cated for rollers with increased radii. In
bearing studies, reduced load capacity has been
found to result from increasing ball size. Rec-
ognition of this fact is incorporated in empirical
rating formulae for bearings--. Such reduc-
tions are attributed to metallurgical factors
associated with the depth of sections. In these
rollers, however, section change resulting
from the increased radius is slight; as the in-
crease in roller mass was only 0.6%. Thus,
it is not seen how metallurgical factors would
account for these reductions. Further work
on the effect of radius change on rolling fatigue
of metallic materials is planned in future
work.
Data collected from several sources, in-
cluding cylindrical and "point" contact, are re-
ported in Reference 4 and summarized graphically
in Figure 2. With the exception of the specimens
from the General Electric Rolling Contact Rig, all
the specimens are 52100 steel of approximately the
same hardness. The G. E. specimens are MV-1
tool steel of comparable hardness (Rockwell C 64).
In terms of maximum theoretical Hertz stress the
apparent fatigue strength for toroidal rollers is
several times that for cylindrical rollers. The
theoretical contact area shape is also indicated in
Figure 2. The orientation of the contact ellipses
is for rolling in the horizontal direction.
Tests are usually conducted at one stress
level in rolling contact fatigue programs to deter-
mine the effect of material variables. The data
are often reported on Weibull probability paper.
In this way both mean life and variability can be
graphically represented by means of a straight
line. Data collected from many sources for
three popular steels are presented in Weibull
fashion in Figure 3. The ordinate of these plots
1
log log T-- -
where F is the mean (50 percent) probability of
failure (see Johnson (5) for derivation). In order
to compare these data on a fair basis, since the
actual stress level of tests varied, they were ad-
justed in life (stress cycles) by the relation
N max rep) (N
800, 000 psi 800,000 ) (Nreported) (1)
Equation 1 employs the observation men-
tioned in Appendix B: for both ball bearings and
many bench rigs most researchers agree that the
life varies inversely as the ninth or tenth power of
the maximum contact stress.
The conditions for the tests shown in Figure
3, in so far as they are known, are listed in Table
1. The Weibull lines are keyed to the particular
rig and test conditions by the numbers 1 through 12.
A discussion of the divergence of data from various
bench rigs is given in Reference 6.
A convenient parameter of rolling element
configuration is the ratio of the axis of contact
ellipse in the rolling direction to the transverse
axis, e. There are notable exceptions, partic-
ularly the Macks spin rig data, but in general
those data with the largest e have the longest life
(see the Annapolis data for three sets of toroid
geometries, all tested in the same type of rig
Figure 3, lines 1, 2 and 3).
TABLE 1
Summary of Material and Test Conditions for Pitting Endurance Data Presented in Figure 3
Key Mat'l Rig
No. Type Type
Material
Ref. e Modifications
(Rockwell C
Hardness)
Lubricant Temp.
OF
Unadj.
Speed Max.
(SCM) Stress
1 52100 Toroid and
Cylinder
2 52100 Toroid and
Cylinder
3 52100 Toroid and
Cylinder
4 52100 Flat Washer
and Ball
5 52100 Macks Spin
Rig
6 M-1 Nut-Cracker
(G.E. RC Rig)
7 M-1 Flat Washer
and Ball
8 M-1 Macks Spin
Rig
9 M-50 Nut-Cracker
(G.E. RC Rig)
10 M-50 Nut-Cracker
(G.E. RC Rig)
11 M-50 Flat Washer
and Ball
12 M-50 Macks Spin
Rig
0.84 RC 61-63
1.00 RC 61-63
1.33 RC 61-63
1.00 Induction
Vacuum
180 1.13 Commercial
air melt
SAE grade 1
balls
181 0.77 Commercial
air melt
RC 61
43 1.00 Vacuum
Melt
42 1.13 Induction
Vacuum
8-ASTM G.S.
RC 62
0.77 Commercial
air melt 10. 4
ASTM G.S.
RC 62
0.77 Commercial
air melt
3.4 ASTM
G.S. RC 62
43 1.00 SKF Desig-
nation "Ingot
Bar 1A"
42 1.13 Induction
Vacuum
8 ASTM G.S.
RC 62
Navy
designation
2190T oil
Navy
designation
2190T oil
Navy
designation
2190T oil
DTE Med.
Heavy oil
SAE 10
mineral
oil
Mil-L-
7808
DTE Med.
Heavy Oil
Mil-L-
7808
Mil-L-
7808
Mil-L-
7808
DTE Med.
Heavy Oil
Mil-L-
7808
90 1,780 840,000 26
90 1,780 851,500 13
90 1,780 826,000 21
room
2, 625
room 163,000
room 30, 000
room 2,625
room 178,000
room 30,000
room 30,000
room 2,625
room 178,000
731,000 24
750,000 74 balls
27 fail
725,000 4
731,000 22
800,000 17
732,000 4
732,000 4
731,000 14
800,000 12
No.
Spec.
1. Pitting Tests at Illinois
In order to further explore the effect of
rolling element geometry on pitting endurance, a
brief experimental program was initiated in 1958
at the University of Illinois using the apparatus
shown in Figure 4. This apparatus was used in
both the pitting endurance tests and the cumulative
deformation experiments to be described later.
Lubrication was provided by dip-feed from
a constant-level reservoir of non-detergent SAE
10-20 oil. For the pitting tests a 2-hp motor
driving the load roller shaft at 200 rpm was used.
The specimen spindle was supported in tapered
roller bearings and the lower drive shaft was sup-
ported and held in axial position between spherical
ball bearing pillow blocks.
The main feature differentiating this roller
rig from many others of nearly the same design is
the provision for the introduction of any desired
angle of skew between the driver roller and the
specimen roller in the upper level arm. This is
accomplished by pivoting the "foot" supporting the
lever arm directly below the point of contact (see
Figure 4).
Toroid specimens of the two configurations
shown in Figure 5 were used in the pitting endur-
ance tests. When pressed against the load cylin-
der the ellipses of contact have the same eccentric -
ity for both specimens; however, the rolling di-
rection is reversed. The ratio of theoretical
contact ellipse axis in the rolling direction to the
transverse axis, e, is 1.44 for the specimen
having a 0.36-in. profile radius and 0.695 in. for
the specimen with a 1.08-in. profile radius.
The specimens were machined from SAE
4340 steel to within 1/32-in. of finish dimension
and austenitized for one hour in a salt bath at
1525 F. After an oil quench (Rockwell hardness
C 55) they were tempered at 600°F for 1-1/2 hours
in cast iron chips. The final hardness was C 50.
Finish machining of profile radius was accom-
plished with a sharp tool, using a high speed and
low feed. Final polishing was performed with
fine emery and oil to remove all tool marks. Run-
out on some of the specimens amounted to as much
as 0. 003 in. The load rollers for these tests
were 4620 steel, case hardened to Rockwell C 60,
with dimensions as shown in Figure 5.
Observations of the specimen tracks were
made throughout many of the tests with a 40-power
microscope. Radial deformation was determined
on several of the specimens with a supermicrome-
ter. Track widths were obtained by microscope
and shadowgraph techniques. Most of the tests
were audited by a tape recorder which was modi-
fied to sample the noise level every 12 minutes,
to obtain the time and number of cycles to failure.
The data are tabulated in Table 2 and typical fail-
ures are shown in Figure 6.
Since several modifications of the test rig
evolved over a period of three years, three main
machine adjustments have to be taken into account
when considering the data. Specimens designated
P-1, P-2 and P-3 were preliminary tests on the
first setting, after which the machine was aligned
as accurately as possible. However, later ex-
periments on the brass toroids revealed a lack of
parallelism and a third adjustment was made,
after which specimens B-4 and A-6 were run.
Despite the brevity of the program and the
unknown influence of skew adjustment, the results
lend support to the view that even under identical
environmental and test conditions there is an effect
of rolling element geometry that is not accounted
for simply in terms of maximum surface stress.
It is clearly shown in Table 2 that the larger e is,
the longer is the pitting life for the same maximum
contact stress and conditions of testing. Before
attempting to explain this it is appropriate to sum-
marize the significant influence of the stress
system in rolling contact on the fatigue life.
B. ANALYSIS OF CRITICAL STRESS IN PURE
ROLLING
Almost without exception, all the stress
calculations employed in the analysis of the rolling
TABLE 2
Pitting Endurance Tests
Specimen Configuration
Designation Index, e
Load Maximum Stress
(lbs) Theoretical Cycles
Contact Stress (Millions)
max (psi)
Measured Theoretical
Track Width Track Width
w (in.) wE(in.)
688 532,000
562 496,000
688 365,000
1698 500,000
578 500,000
1698 500,000
578 500,000
1698 500,000
578 500,000
1698 500,000
1.00*
0.15
1.28
15.13*
0.12
6.71
7.74*
0. 18*
2.98*
0.005
0. 22*
2.05
0. 77*
8.70*
0.15*
578 500,000 52
1698 500,000 31
Specimens presented in order of testing.
Last two specimens (B-4 and A-6) were tested at adjusted angle of skew (approximately
to produce symmetrical blunting of brass toroid.
1/20 change) so as
* Indicates failure by pitting.
Two pits approximately 0.025-in. in diameter.
At 14x106 cycles oil supply cut off temporarily. Some burnishing occurred. Large pit at failure.
Several small pits (approximately 0.020 in.) began at 3.78x106 cycles and continued until test stopped.
Abrasion due to accidental motion of load roller--repolished--large shallow pit and incipient flake.
Two small pits (less than 0.015 in.).
Period of burnishing and wear early in life.
Vibration and chatter causing wear-repolished. Test stopped after appearance of small pits.
Burnishing as load was increased at start of test.
Radial deformation indicates change from first cycle.
1.440
1.440
P-3c
Radial
Deformation
u (in.)
0.695
A-2d
B-l1e
A-3
B-29
0.062
0.043
0.050
0.057
0.081
0.092
0.065
0.109
0.140
0.109
0.047
0.124
0.050
0.120
0.695
1.440
0.695
1.440
0.695
1.440
0.695
1.440
0.695
0.041
0.035
0.035
0.035
0.072
0.072
0.072
0.097
0.039
0.097
0.097
0.039
0.097
0.039
0.097
0.039
0.097
A-5h
B-4
0.00047
0.00038
0.00052
0.00027
0.00062
p-1 a
contact problem are based on the elasto-static
solution stemming from Hertz's approximation for
the distribution of contact pressure between
elastic bodies in contact. For pure rolling, it is
customary to ignore any tangential traction or
dynamic effects or even the influence of possible
lubricant films. Many other features in which the
ideal differs from the real could be listed, but
important information can still be gleaned from the
elasticity solution, including numerical values of
engineering significance. The "pseudo-effect" of
rolling is determined by solving for the stress at
a point as a function of distance from the contact
area on a plane parallel to the surface and passing
through the point.
1. Maximum Range of Shear Stress
The greatest variation or range of shear
stress on planes of fixed orientation occurs on
planes perpendicular and parallel to the surface.
The location or depth to the maximum range, as
well as the magnitude of this range, is a function
of the ellipticity and orientation of the contact
ellipse with respect to the rolling direction. This
may be described by the parameter e. The range
(2 To) of this orthogonal shear stress expressed
as a fraction of the maximum contact stress is
plotted against e in Figure 7. Other stress com-
ponents may be plotted as a function of e but none
vary as significantly as the orthogonal shear
stress which decreases continuously with increase
in e. For cylinders this range of shear stress
which is created by the action of rolling is 60 per
cent larger than the maximum range of shear
stress under repeated normal loading. It is
significant that tests conducted by Kennedy (7) re-
vealed no evidence of subsurface cracks when hard
steel balls were repeatedly pressed together, in-
dicating that rolling is required to cause cracking
at these stress levels.
2. Surface Tensile Stress
It is interesting to examine the surface
tensile stress at the leading (or trailing) edge of
the contact ellipse (ignoring any dynamic effect) as
a function of e. Contrary to the trend with the
orthogonal shear stress, the tensile stress in-
creases for a significant initial range with increase
in e as shown in Figure 8. Calculations for this
figure were made on the basis of the solution given
by Timoshenko.(8) Additional analysis of stress
as a function of e is given in Reference 4.
C. DISCUSSION OF THE MECHANISM OF
PITTING
Both the stress-life dependence and the
microexamination of failure sites verify that pit-
ting is a result of metal fatigue. As such, pitting
failure may be associated with the generally
accepted body of knowledge in ordinary fatigue.
It is convenient to consider the fatigue process in
two stages, initiation and propagation, for pur-
poses of the discussion which follows.
1. Initiation
In the past, a great deal of argument has
been devoted to the location of the crack nucleation
site, whether surface or subsurface. This will
of course depend on both the type of surface trac-
tion or interaction and the material, if its strength
properties are a significant function of depth. For
nearly pure rolling it is generally accepted(9' 10,
11, 12) that the origin is subsurface. These con-
clusions are based on many metallographic ob-
servations of longitudinal subsurface cracks of
considerable length, the pronounced evidence of
severe subsurface plastic flow, and the influence
of element configuration on pitting. This last
point is discussed more fully in Reference 4 but
briefly may be summarized: there is a significant
effect of configuration, with large e values having
greatest strength for similar test conditions. The
significant fatigue stress is the range of alternating
shear stress due to rolling. The range of alter-
nating shear stress decreases as e increases.
This stress range occurs in the subsurface. The
origin of fatigue is therefore subsurface. This
argument is simple but direct and does not depend
on quantitative unknowns. In fact, the argument
allows some rational connection of fatigue strength
values from rolling contact and a more standard
laboratory test like torsion fatigue.
The possibility of surface initiation under
certain circumstances should not be overlooked,
however. It is significant that the reports of
surface initiation are usually in connection with
gear research. Tests by Dawson (13)with a bench
rig having cylindrical rollers geared to produce
various degrees of rolling and sliding and under
various lubrication conditions demonstrate the
existence of surface initiated pitting.
Many of the discussions of the mechanism
of fatigue on the atomic or crystalline level have
been concerned with the intrusion-extrusion phe-
nomenon observed at free surfaces of pure crys-
tals subjected to fatigue stresses. (14) This
method of crack initiation is similar in some re-
spects to the scheme proposed by Shanley. (15)
However, the fact that most fatigue failures occur
at the surface is probably only evidence that in
most cases the severe stress or chemical condi-
tions exist at the surface, and not proof that all
fatigue failures are surface originated. Obser-
vation of blisters in the rolling track in pure
copper by Eldridge (16) is significant in this
context.
There has been much emphasis placed on
the role of inclusions in the initiation of fatigue
cracks. The role of residual tensile micro
stresses in crack initiation at inclusion sites has
been extensively discussed by Almen. (17 and 18)
Part of the emphasis is due to the fact that it pre-
sents a possible means for manufacturers to do
something about fatigue pitting. The other is the
fundamental difficulty in imagining the opening of
a fatigue crack against a macro-stress field with
a significantly high hydrostatic compressive stress
component. The role of inclusions, though impor-
tant, is not peculiar to the rolling contact situation
nor, for that matter, is fatigue under nominally
high compressive stress fields. Crossland's(19)
torsion fatigue tests under hydrostatic fluid pres-
sure indicate that the fatigue process is inhibited
but not stopped. However, the mechanism of
crack propagation in this unusual applied stress
environment deserves consideration.
2. Propagation
The propagation of fissures or microcracks
through the subsurface band of plastically worked
and slip damaged metal is probably influenced by
the macro-residual radial tensile stress, which
acts when the applied stress is released after
passage of the rolling load. The origin of this
residual tensile stress is discussed by Johnson and
Jefferis (20) and Stulen, (21) although no accurate
estimation of its magnitude for three dimensional
contact has been made. In the advanced stages of
propagation the stress due to the presence of the
crack itself influences propagation, even to the
extent of causing more severe tensile stress at the
surface, as Barwell (22) has conjectured. It is of
interest in this connection to consider an obser-
vation made in the pitting tests reported above.
In Figure 9 a plan and section view of a flaked area
and an incipient flake are shown. The orientation
of the incipient flake is significant in that its
"mouth" is directed away from the approach of the
contact load, thus opposing the hydrodynamic
theory of propagation of initial surface cracks. It
is also of interest to consider a phenomenon re-
ported in personal communication with Dr.
Yoshiaki Masuko of Sumitomo Metal Industries,
Osaka, Japan. Dr. Masuko investigated the
nature of fatigue pitting on hollow steel rolls that
had been press fitted to roll cores. Subsurface
examination revealed a number of radial cracks
near the surface but not reaching it. The cir-
cumferential residual tensile stresses near the
surface due to press fit were about 25 percent of
the fatigue limit of the metal in ordinary fatigue
tests. A reasonable explanation of these obser-
vations of initiation and incomplete propagation on
radial planes can be proposed. Initiation took
place in the subsurface region of high shear re-
versal (shear stress on radial plane = shear stress
on circumferential plane), and under the influence
of the circumferential tensile stress due to press
fit, propagated radially out of this region, being
arrested because of subcritical stress conditions
or even opposing compressive stress due to plastic
deformation.
D. STATISTICAL CONSIDERATIONS IN FATIGUE
PITTING
Since fatigue and other fracture phenomena
in real materials are inherently statistical in
nature, interpretation of test results on the basis
of limited samples cannot be made without re-
course to statistical methods of analysis. In
keeping with the popular Weibull presentation of
bearing fatigue data, a discussion is given below of
the influence of rolling element size and contact
stress level on two significant parameters of the
distribution function: characteristic life, e, and
statistical variability or slope, b.
1. Size Effect
The effect of specimen size on fatigue
strength determined in laboratory testing has been
a subject of wide concern and speculation. A re-
view of much of the literature on size effect and a
rational method for allowing for the apparent
lowering of the fatigue limit due to increase in the
effectively stressed volume of material is given by
Kuguel. (23) Weibull's "weakest link" concept24)
and probability arguments have been adopted and
extended to the rolling contact situation by Lundberg
and Palmgren, who incorporated size effect (track
width x depth to maximum shear x length) into a
complex empirical equation for purposes of bear-
ing rating. In view of the many varieties and
sizes of bench rigs it is appropriate to present a
simple analysis that may be applicable in com-
paring results from such test rigs and that illus -
trates the basic statistical formulation of such
phenomena.
For simplicity assume that the volume of
significantly stressed material is represented by
length of rolling track only. If contact situations
having much different track width and depth to
maximum shear stress are considered, the prob-
lem is theoretically more complex since the indi-
vidual elements on a volume basis do not have
independent survival probabilities.
Consider two rolling elements of different
track lengths. The track length, L1, of one
element can be thought of as an assemblage of
tracks of length L2. Since the probability of
survival of the assemblage, S, is the product of
the component probabilities, S2:
L1/L2
S1 = S2 (2)
Using the Weibull cumulative distribution function,
F(N), the probability of survival is
S = 1 - F(N) = e-(N/0)b
where b is the Weibull slope, e is the character-
istic life, and N is the number of cycles. Equation
2 becomes
1bl ) b2 L
( N N 1
01 62
Since this relation must be valid for all N, if the
Weibull life distribution is to describe the assem-
blage
b = b2 = b.
It then follows that
0 1 2
A change in track length would therefore be ex-
pected to translate the Weibull line in accordance
with Equation 3, the slope remaining constant.
2. Effect of Stress Level
The effect of stress level on life variability
L
(Weibull slope, b) in fatigue tests was treated by
Sinclair and Dolan. (25) The general trend of a
reduction in standard deviation in log of life with
increase in stress level is stated to be a feature
inherent to the phenomenon of fatigue fracture in
polycrystalline metals.* It appears that some
size effect may have entered in their analysis of
bending fatigue strength of aluminum, since
stress gradient (and therefore significantly
stressed volume) increases with stress level.
However, this should enter primarily in shifting
mean life and not changing variability to the extent
reported.
There is not enough data available at this
time in the rolling contact situation to make a
similar analysis. It should be pointed out that
the analysis formulated by Lundberg and
Palmgren (9) does not include such an effect. They
have represented the survival probability by a
product in which the exponents of the factors are
independent of stress.
E. CONSIDERATION OF PLASTIC DEFORMATION
The fatigue process is intimately related to
the mechanism of plastic deformation for even
nominally low stresses such as design stresses in
the "elastic" range. This much is axiomatic.
In the rolling contact case, plastic deformation
also affects the magnitude of applied stress by
changing the geometry of the contacting elements.
In comparing contact fatigue results at high
nominal stress with those at lower stress, some
account of this stress reduction should be made
since it is the purpose of an effective stress index
to indicate the true severity of applied stress. In
this connection, certainly, one definite statement
can be made concerning the maximum contact
pressure on the basis of equilibrium of the con-
tacting bodies alone. That is: the maximum
* It is interesting to note that the standard devi-
ation in fatigue strength is nearly a constant at all
lives.
contact stress is greater than or at least equal to
the total load divided by the actual contact area
> P
max -
Although increase in area of contact may
occur in cases of high loads, at least a lower
bound to the maximum contact pressure can be
given if the contact area can be measured. Loose
statements concerning a reduction in stress may
thus be avoided.
There is evidence that even in the presence
of a plastic zone, the stress distribution is nearly
the same as the elastic one. For a partial uni -
form load on an infinite strip, Mesmer's experi-
ments(26) indicate that the elastic distribution re-
mains valid through a considerable range of plastic
strains. Since bearing steels have no pronounced
yield points, the inhomogeneous yielding charac-
teristics of low carbon mild steel is not expected
or observed. Of course, it is questionable to
extend the results of static tension tests directly
to a discussion of the plastic properties of a rela-
tively small volume of material immersed in a
region of supporting elastic material.
Lundberg and Palmgren cope with the effect
of local plastic flow and inclusions in the state-
ment:
However, the microscopic alterations in the
material do not affect the macroscopic stress
distribution and thus need not be considered
in the determination of the macroscopic stress
magnitudes.
They recognized that when finally a micro-
scopic crack becomes macroscopic in size, the
stress distribution is changed. But this is only in
the advanced fatigue stages where the continuation
of the destructive process swiftly leads to rupture.
A number of investigators have noted in this stage
that the presence of a subsurface crack may make
the surface stress condition much more severe.
With such a crack present, the model of a plate
with built in edges resting on a continuously elastic
foundation is useful in imagining the stress situa-
tion.
In summing up, certainly stresses com-
puted on the basis of ideal elastic action are not
strictly valid, expecially for high loads. Yet,
with auxiliary experimental observation of contact
area, and consideration of the materials used,
these elastic stresses may serve as useful indices
for correlating the existing data.
On the basis of the data at high stress
levels presented in the Annapolis reports, indi-
cating the percentage increase in track width over
the theoretical, and a lower limit of first measur-
able plastic deformation established by Timken
(personal communication with H. R. Neifert), the
relation between theoretical stress and percentage
increase in track width, K, can be estimated.
Figure 10 exhibits a plot of this estimated relation.
As the width of the contact ellipse in-
creases K%, the correction factor for contact
1
stress becomes . A demonstration
(1+ K-O2
of this follows:
In terms of the maximum contact stress,
omax' the total applied load P is given by
2
P = -- r ab max
where
P is the total load
a is the semi-major axis of contact
b is the semi-minor axis of contact
Since the total load acting is not affected
by plastic action at the contact, expressions for P
in terms of corrected and uncorrected maximum
contact stress can be equated.
2 2 K
7r ab max r (1 + - -) a
(1 + - ) b max (4)
where K is the percentage increase in track width
over theoretical, and o' is the corrected maxi-
max
mum contact stress.
It is assumed that the percentage increase
in width of contact ellipse is the same for both
axes.
The ratio of a' to a is then, from
max max
Equation 4
max 1
max K
(1+ -I)
Hence, if K is determined, the uncorrected stress
must be multiplied by this factor to obtain a "true"
value.
If sufficient information is available con-
cerning the deformed geometry, principally profile
radii, corresponding stresses may be calculated
elastically. This method was employed by
Niemann and Kraupner (27) and Zaretsky et al. (28)
Some standard procedure should be adopted
since stress reduction is a function of the number
of cycles because of cumulative plastic deforma-
tion. For convenience, the stress reduction at
10 percent of the mean life would be satisfactory.
However, for comparing contact fatigue results at
nearly the same nominal stress these considerations
are not necessary.
III. CORRELATION OF PITTING FAILURES
Synopsis: As an illustration of some of the
considerations necessary in accounting for roll-
ing element configuration and size effects, the
following examples of data correlation are pre-
sented. In the first example, rolling contact
fatigue data from several bench rigs are corre-
lated with torsion fatigue strength. The second
example is a correlation of pitting life data for
specimens of different geometry tested in the
same rig. Finally, full scale bearing perform-
ance is predicted on the basis of minimal rolling
contact rig data.
A. CORRELATION OF PITTING AND TORSION
FATIGUE STRENGTH
It is possible to treat many rolling contact
fatigue failures on the basis of mechanical con-
siderations relating to the critical stress cycle and
inherent material properties. A numerical de-
monstration of the relation of fatigue to pitting in
rolling contact was presented in 1958. (4 and 29)
Improvements in correlation technique and the
addition of recent information are needed to develop
a unified rolling contact fatigue theory. However,
several of the topics treated in these early reports
provide a basis for future development. To this
end a summary of the correlation procedure as
developed previously is provided here.
The rolling contact and torsion fatigue data
selected for analysis have been collected from
four sources and are described in Table 3.
Geometric definitions are given in Figure 11.
Rolling contact tests were made under nor-
TABLE 3
Description of Specimen Geometry and Material
Rockwell Geometry
Source Material Hardness Description
C Scale
0. 25" e = 1.35
Navy (3 and 33) 52100 Steel 59-61 RI
0.50" e = 0.84
R'= 1.500" R2= 1.562"
G.E. (181) MV-1 Tool Steel 64 R1= 0.25" e = 0.77
R'= 3.75" R2= 0.187"
Bacha (182) 52100 Steel 61 R1= e = 0 w= 0.58"
R = 0.58" R2= 2.57"
Styri (32) 52100 Steel 62 Standard fatigue rotating
beam and torsion specimens
mal environmental conditions of temperature and
non-corrosive light oil lubrication. The data are
displayed in Figure 2, in which maximum theoreti-
cal contact stress is plotted against the log of the
number of cycles to failure. Only two points are
available for e = 0.84, but it is significant that
these fall within the range of the data for similar
geometry, e = 0.77. Despite the number of test
points, the stress range for e = 1.35 is relatively
small and therefore a characteristic point is se-
lected in the data and assigned the same S-N slope
as for the e = 0.77 data in subsequent operations.
Before proceeding to the actual data correlation
the following considerations are pertinent.
1. Plasticity
Plastic increase in contact area will cause
a reduction in actual contact stress. A curve,
using these data and a point of first measurable
plastic deformation, obtained at the Timken Roller
Bearing Company, is shown in Figure 10. In the
absence of more complete data, the same stress
deformation curve may be used for all geometries.
If the plastic action is not so excessive as to cause
a significant change in geometry, then from
Equation 5, the reduced stress is
_ max
max K 2 6)
(1 + - )
This assumes only that the contact ellipse after
plastic action has the same eccentricity as before.
2. Statistical Strength Concepts
Another consideration or correction of
stress stems from the observations that the
strength of a specimen increases with reduced
dimensions.
Weibull's analysis is based on the multi-
plicative law of probabilities and the dispersion of
material strength in the volume under considera-
Ition. It was developed only for conditions of uni-
Iform stress. Lundberg and Palmgren (9) have
modified a similar theory for application to rolling
icontact data. The life formula is given as:
1
log -. =V'F(Tro, N, Z ) (7)
where: S is the probability that the material will
endure N million stress cycles and
V is a volume representative of the
stressed region
F is given as a power function, and
Equation 7 becomes:
'r Ne w -L
l o1 o
log- - h-1
0
The constants c, e, and h are determined by bear-
ing tests.
w is width, Z is the depth to the maximum
orthogonal shear stress, and L is length of track
(circumference of rolling body). If for two
different geometries of rolling contact fatigue
specimens, the same probability of survival (or
failure) is desired for a given number of cycles,
the ratio of orthogonal shear stress will be
(T0)1 IF zh1 w- 2 1/c
( L Z;1 2L2
From this standpoint, cylindrical contact will be
inherently the weakest case since it involves the
largest volume subjected to stress. To eliminate
the statistical size effect and judge the data on a
common basis, the corresponding values for
fatigue strength (in terms of T ) for elliptical con-
tact are multiplied by a factor ( ) /(r )t. Using
the values determined by Lundberg and Palmgren:*
d ) 0.129
Tot t
L 0.097
c c
* These values were obtained for a particular type
of hardened steel, but in the absence of more
direct data they will be used for the 52100 steel
data considered here.
where subscripts t and c refer to toroid and
cylinder, respectively. This ratio may be ex-
pressed as a function of the respective maximum
contact stresses with the use of the following
identities from contact theory:
wt Ce At max t
Z = 0.5wt Me
Z =0.5A a
c c maxc
where: C is determined from Figure 12
M is determined from Figure 13
and the geometry parameter A is:
-2 1-p 2
1 +- - 2
1 2 1 2
where E and p are elastic constants.
With these substitutions, Equation 8 becomes:
(To0c
A
c
Me
0. 129 0. 097
L
t
c c
1C
e t
0. 032]
0.129
max c
0. 032
max t
3. Fatigue Criterion
In developing a fatigue criterion for the
stress condition arising from rolling contact,
account must be taken of both the range of shear
stress and the inhibiting influence on crack propa-
gation of the normal compressive stress acting on
these critical shear planes. Fatigue theory and
test results involving multi-axial compressive
stress are scarce but the few excursions into this
area have accentuated the inhibiting influence of
compressive stress fields.
Especially significant are the torsion
fatigue tests under hydrostatic compression of
Crossland.(19) After the specimen surface was
sealed from the direct action of the pressure
fluid, there was a large increase in life and
fatigue strength.
Usual expressions for fatigue strength are
a modification of onset of plasticity, distortion
energy, or maximum shear theories. In these
expressions the hydrostatic component or normal
stress is neglected. The limitations of such
theories become evident under certain stress
states.
Crossland found that his test results did not
support a simple maximum shear stress or Mises-
Hencky theory of failure. A linear modification
of the Mises function by the volumetric stress was
proposed. Figure 14 is a plot of the semi-range
of Mises stress endured against volumetric stress
for several stress states.
In a paper by Stulen and Cummings (30) in
1954, a theory of fatigue failure under multi-axial
stress was proposed. Failure was assumed to be
caused by alternating slip on certain planes. On
these planes, the critical shear, TC, is a linear
function of the complimentary normal stress, an'
For simplicity, an expression of this fatigue
criterion may be taken as
Tr= 7 + B an
C. e n
In this expression, 7 is an experimentally deter-
mined constant (at N cycles) that may be considered
as the semi-range of shear stress necessary to
cause failure when no normal stress is involved.
For reference purposes and convenience, it will be
called the semi-range of essential shear stress.
The symbol B will be referred to as the normal
stress influence factor. Both T and B are con-
stants for a given number of cycles but may vary
with the number of cycles throughout the test. It
is shown in a more elaborate treatment by
Findley (31) that T7 is not necessarily the fatigue
strength in pure torsion, but is given by:
S= 1+k2 T
where TT is the fatigue strength in torsion and k
is related to the ratio of bending fatigue strength,
TB' to TT at N cycles by
SB
k
E 2B
l-2[- - J
Since hard 52100 steel will be used in the
analysis of this paper, it is necessary to examine
whether or not the fatigue strength in torsion can be
used as a close approximation to the semi-range of
essential shear. The data of Styri32) will be used.
Typical S-N curves for bending and torsion are
given in Figure 15.
By selecting fatigue strengths for bending and
torsion at different numbers of cycles, the values
of k may be determined. The ratios of fatigue
strengths are given in Table 4. Thus the difference
in the critical and essential shear stress in torsion
is negligible.
It is not possible to obtain the value of the
normal stress influence factor, B, directly. How-
ever, some indication of the range of values in-
volved may be inferred from examination of a
particular case in which data are available. For
this purpose, Crossland's data will be used. In
these hydro-torsion tests, the so-called "volu-
metric tensile stress" has the same value as the
normal stress acting on the critical shear planes
(longitudinal and transverse). In this case the
semi-range of the Mises function reduces to the
semi-range of applied (or critical) shear stress.
Figure 14 may be considered as a plot of critical
shear versus complimentary normal stress for
torsion specimens. The slope gives the value of
the influence factor as 0.3. Here, the compli-
mentary normal stress is held constant at a maxi-
mum value. In the case of contact fatigue, the
normal stress varies from zero to its maximum
value during rolling. A value of B, somewhat less
than 0.3, would be expected.
Therefore, the fatigue criterion will be
taken as Equation 10 with B to be determined from
available data on rolling contact fatigue.
4. Correlation Procedure
First the data are corrected for plastic in-
crease in track width by use of Equation 6 and
Figure 10. The data for cylinders are of course
not corrected. In order to facilitate the remain-
ing steps the corrected data are presented using
log-log coordination (Figure 16). In this plot the
S-N line for Styri's torsion data is also given.
Next, the data are corrected for geometry
using the range of orthogonal shear stress. This
is accomplished by multiplying the maximum con-
tact stress by the ordinate of Figure 7, Q, which
corresponds to the given geometry factor, e.
The correction for volume stressed is
TABLE 4
Comparison of Essential and Critical Shear for Torsion using Styri's Data for Hard 52100
No. of TT, psi OB, psi k + k2
Cycles TB e
105 115,000 200,000 0.57 0.14 1.01
106 90,000 170,000 0.53 0.06 1.00
107 70; 000 140,000 0.50 0 1.00
made by use of the semi-range of orthogonal shear
stress and Equation 9. A constant factor of 0.5
is applied for the semi-range transformation.
For e = 1.35, Equation 9 becomes:
0. 129
oc - 0.215 max c
max t
for e = 0.77, Equation 9 becomes:
(r) a °0.129
- =0.236maxc 32
oTYt amax t
Typical values at N = 3 x 106 cycles for these
combined factors are:
S o) 0.40Z for e = 1.35
O'r t 0.368 for e = 0.77
There is a slight decrease in this factor with in-
creasing N. After multiplying by the correction
factor, the data for the two toroids are nearly co-
incident and the maximum percentage difference in
fatigue strength between elliptical and line contact
has been reduced from about 97 per cent to 5 per
cent at 3 x 106 cycles.
The fatigue criterion may now be applied
to the contact fatigue data in order to relate it to
the standard torsion data. The close similarity
in S-N slope is considered to be due to similarity
in mechanism of fatigue. Essentially, this is
the observation made by Styri.
A transformation of critical shear stress
to essential shear stress by use of Equation 10
can be performed. Under this transformation,
the torsion data will not be influenced since it is
demonstrated above that in torsion, the critical
shear stress is the essential shear stress. The
normal stress influence factor B will be selected
so as to cause coincidence of the contact fatigue
data for cylinders and the torsion data. The
essential shear stress for cylinders can be ex-
pressed as:
(E ) cc =(T E - B( T cn
o()c 1 -B( )1
From Figure 17 the ratio of maximum normal
stress to shear stress amplitude on the plane of
maximum orthogonal shear stress is
o = 3.5,
0
at 3 x 106 cycles:
(T ) = 108, 000 psi
and ( )T = 80, 000 psi
where the subscript T refers to torsion fatigue
data. Since
c = (T)T = (TC)T, then B= 0.074
The above operations may be reversed and
combined to provide a means of predicting the
fatigue strength of a toroid of general e:
max
where:
Me 0.129 w cL 0.097
( max cd
0.968
0.032
( e t)
0. 129
In Equation 11, K is a function of a
max
given graphically by Figure 10. Hence, two
simultaneous relationships must be solved. How-
ever, the desired value of amax may be obtained
by a simple iteration procedure.
For example, consider a hypothetical
toroid similar to the Annapolis rollers except
R = 1 inch instead of 1/4 inch or 1/2 inch. This
results in e = 0.53. Equation 11 and Figure 10
yield amax = 570, 000 psi at N = 3 xlO cycles.
max
The slope may be determined by calculating
additional points.
More generally, if only torsion data on a
new material are available, a minimum fatigue
strength in rolling contact can be estimated by
initially ignoring the correction for volume
stressed and plastic increase in track width, thus
obtaining:
2 T
cm ax Q 6
(1-B -.)
'7r
This result can be modified as information con-
cerning the plastic behavior of the metal and
fatigue behavior at one particular geometry is
obtained.
B. CORRELATION OF BENCH RIG LIFE DATA
A less ambitious and more straightforward
problem than the above illustration is the correla-
tion of life data from the same test rig but with
different rolling element geometries. For this
purpose the apparent divergence in life for the
Annapolis (33) data will be examined. The unad-
justed data are presented in Figure 18. Compari-
son of these data after a series of adjustments is
made in Table 5. The volume of significantly
stressed material is taken as the product of theo-
retical track width, depth to maximum shear
stress, and track length. The relative values,
compared to the Annapolis elements of e = 1.00
as base, are also given in Table 6. It should be
noted that the value of B10 and B50 life (life at
which 10 per cent and 50 per cent of specimens
tested may be expected to have failed) in terms of
maximum contact stress have all been adjusted to
800, 000 psi in accordance with Equation 1. If the
range of orthogonal shear is significant then
further adjustment is needed. The orthogonal
shear stress is obtained from Figure 7 and all the
data are again adjusted in accordance with Equation
1 to the orthogonal shear stress level of the
Annapolis element e = 1.00 for illustrative pur-
poses. That is, the adjustment in life is made
according to the equation
N1 (o) 9
1
where To is the amplitude of orthogonal shear from
Figure 7. To account for the size effect,
Equation 3 is used disregarding the theoretical
limitations, and replacing length as follows:
V 1/b
Ni =(-) N2
A Weibull slope of two is used in this cal-
culation, and the results are given in the final
Column of Table 5. All of the adjustments, in
keeping with the above considerations, tend to
cause convergence of the data.
TABLE 5
Successive Adjustments of Pitting Endurance Data for Annapolis 52100
Steel Rolling Elements (Life in Millions)
Same Critical Same Effective
800, 000 psi Shear Range Size
e10 B50 10 B50 B10 50
0.84
1.00
1.33
0.74 1.70
1.72 4.50
3.10 16.50
0.92
1.72
1.60
2.2
4.50
8.60
1.10
1.72
1.16
2.61
4.50
6.24
TABLE 6
Critically Stressed Volume for Annapolis Rolling Elements
Depth to
Maximum Shear
Critically
Stressed Volume
0.84 0.0446 0.0241 0.00107 0.53
1.00 0.0656 0.0308 0.00203 1.00
1.33 0.0812 0.0350 0.00284 1.40
C. CORRELATION OF BENCH RIG AND FULL
SCALE BEARING DATA
In an effort to more easily obtain bearing
design information without the expensive and time
consuming testing of full scale assembled bearings,
the General Electric Company has developed the
rolling contact fatigue machine (RC Rig). Bam-
(34)
berger and Baughman (34) fully described the
machine in 1957 and reported that on the basis of
preliminary data, "The fatigue data indicates a
correlation with actual bearing data may be
possible." Since then Bamberger et al have re-
ported a number of studies concerned with the
rolling contact fatigue behavior of different mate-
rials, variations from heat to heat of the same
material, effect of hardness, grain size, surface
finish, lubricant, and temperature. In the
summer of 1959 the second author of this bulletin
worked with the General Electric group on the
correlation of the RC Rig data with full scale
bearing results. (35)
The problem and approach taken may be
stated as follows:
Given the dimensions, loading and operating
conditions of a bearing, made of a particular
material, determine the life of the bearing
from minimal RC Rig data.*
Following is a partial list of the variables
which are important in the life of bodies in rolling
* This analysis is only concerned with the type of
bearing failure known as pitting. Excessive wear
or deformation or other types of failure will not
be treated here.
contact.
a)
Maximum alternating shear stress
(function of load and radii of the con-
tacting bodies)
b) Plastic increase in track width
c) Volume of material stressed (size effect)
d) Material composition
e) Material hardness
f) Grain size
g) Surface roughness
h) Lubricant
i) Temperature
j) Frequency
It will be presumed that variables "d"
through "j" are controlled in the RC Rig to be the
same as in the bearing of interest. Some of these
variables, such as frequency, only have a measur-
able effect when varied over a wide range. In
these cases it is not necessary to exactly duplicate
the service conditions for the bearing in the RC
Rig.
Variables "a", "b" and "c" are usually not
the same in the RC Rig as in bearings. Each of
these will be discussed below in detail.
a) Maximum Alternating Shear Stress: The signif-
icant stress quantity is the orthogonal shear stress.
For the same Hertz stress the orthogonal shear
stress depends on e. The value of e depends on
the geometry of the contacting bodies. For the
RC Rig e = 0.79. For any practical ball bearing,
e at the inner race is less than 0.79 approaching
e = 0, the value for a roller bearing.
Theoretical
Track Width
Relative
Volume
From Figure 7 it can be seen that the error
in stress would be less than 10% if Hertz stress is
used rather than the orthogonal shear. It will be
shown later that this change in the magnitude of
stress causes a change in the life of only about a
factor of two. The error is always in the same
direction. Other factors being equal, a bearing
will have a slightly shorter life than the RC Rig
for the same maximum Hertz stress. Such varia-
tion in life is well within the scatter normally
found in fatigue life measurements, so the maxi-
mum Hertz stress may be used to compare results
from the RC Rig with results from tests of full
scale bearings.
There is still another factor concerned with
the stresses present in rolling contact which is not
reproduced in the RC Rig. This is the general
level or magnitude of stress. The maximum
Hertz stress in bearings normally is between
200, 000 psi and 500, 000 psi. The stresses in
the RC Rig are normally set between 500, 000 and
800, 000 psi. The reason for this large differ-
ence in the stress levels is that the "idea" of the
RC Rig is to perform a short time test whereas a
bearing is expected to last a long time.
The difference in stress level is not such an
impediment to correlation as it might first appear.
It has been shown many times for both bearings
and for short time tests such as the RC Rig that
the following relation between the maximum Hertz
stress and life is reasonably valid for a range of
lives from 105 to 1010 cycles:
2 _ max 1 (13)
1 \max 2
Where N and N2 are the pitting fatigue lives of
two identical rolling assemblages under the maxi-
mum Hertz stress max 1 and a .
max 1 max 2
Another way of stating Equation 13 is that
the S-N curve for bodies in rolling contact is linear
on a log-log plot and has a slope of -1/9 when log
Umax is plotted vertically and log N is plotted
horizontally. Such a plot is shown in Figure 19
for M-50 tested on the RC Rig at room temperature.
The line with a slope of -1/9, (Equation 13) ap-
pears to fit the RC Rig data reasonably well.
If there were no other differences between
the RC Rig and bearings except stress level, the
number of stress cycles a bearing could endure,
N , at a particular Hertz stress, a b would be:
S a maxb
N =N ( max RC
b RC a
max b /
where NRC is the life of the same material tested
in the RC Rig at some convenient maximum Hertz
stress of amax RC
b) Plastic Increase in Track Width: It is possible
to estimate the effect of the plastic increase in
track width, as follows: for amax= 650, 000 psi
(which would be representative of the stresses in
the RC Rig), the value of K (Figure 10) is about
5%. For amax = 300, 000 psi (which would be
typical for a bearing), the value of K would be near
to zero. Thus, if comparison of a bearing and
the RC Rig is made on the basis of theoretical
Hertz stress using Equation 14 there will be a
small error introduced. The stress in the RC
Rig will be less than computed by the Hertz method
due to yielding at high stresses, whereas the
stresses in the bearing will be close to the theo-
retical stress computed. An estimate of the
error vouldbe (Equation 5):a' /(1.05)2 =
max /1.10.
Approximately the same magnitude of error
is introduced by neglecting the plastic increase in
track width that was introduced by using the maxi-
mum Hertz pressure rather than the more appro-
priate orthogonal shear.
c) Volume of Material Stressed (Size Effect): In
general, the volume of material stressed in the
RC Rig is only a small fraction of the volume of
material stressed in a bearing.
From the analysis of Lundberg and Palm-
gren, it is possible to obtain the followingequation:
21 Z I W L1\ (15)
where (T )land (T )2 are the orthogonal shear
stresses in two bodies of different geometry for
the same probability of surviving a given number
of cycles. w1 and w2 are the widths and L1 and
L2 are the lengths of track (circumference of roll-
ing bodies) and Z1 and Z2 are the depths of the
maximum orthogonal shear stress. The exponents
a and 03 are experimentally determined constants.
For our present purpose of relating bear-
ings and the RC Rig,the ratio of the orthogonal
shear stresses of Equation 15 may be replaced by
the ratio of the maximum Hertz stresses.
NC wb LbRC
N b w b L b
(17a)
But from the previous analysis of statistical
size effect the exponent 90 is identical to the re-
ciprocal of the Weibull slope, b. This is based
on the assumption that size effect is effectively
given as the product of track width and length as in
the analysis of the Annapolis bench rig data. A
Weibull slope of 2 is an approximate value from the
RC Rig data.
In Reference 35 an alternative approach is
explained whereby an estimate of P is obtained
using a large number of the same size specimens
of M-50. From this data it was possible to show
that for the same Hertz stress and all other factors
equal, the life depends approximately on the square
root of the track length* or
N1 L 1 1/2
max RC \ RC/ \ b b /
where the subscripts b and RC refer to bearing
and RC Rig.
Lundberg and Palmgren determined from
bearing tests a = 0. 129. Using this value it can
be shown that the influence of the "Z" term in
Equation 16 is small for practical bearings.
Dropping this factor has the opposite effect
on the predicted life as using Hertz stress instead
of orthogonal shear or as neglecting plastic in-
crease in track width. Thus, this simplification
partially compensates for the error introduced by
the two earlier simplifications.
Equation 16 may be reduced to:
max b w RC LRC
amax RC Wb Lb /
In terms of life from Equation 14:
N \LJ
' \ L /
Since life and stress are related by the
ninth power as in Equation 14 the value of 3 in
Equation 17 is approximately 0 = 1/2 x 1/9=0. 056.
It is now possible to make an estimate from
minimal RC data of the number of stress cycles a
bearing will endure. Equation 14 accounts for
difference in maximum Hertz stress. Equation
17 accounts for difference in volume of material
stressed. All other factors are presumed to
have only minor influence or to be controlled the
same in the RC test as in the bearing. Combining
the effect of relative stress level (Equation 14) and
size (Equation 17a);
N = N max RC WRCLRCC
b RCmax Lb
max b b /
* Proportional to the stressed volume.
-- = e-7- -
Where: -
Nb = Life of bearing, in stress cycles on
critical element of bearing (usually
inner race).
a = Maximum Hertz stress on critical
max b element in bearing.
wb L = Width and length of wear track in
bearing.
NRC = Measured life from RC Rig on same
material, lubricant, etc. as bearing.
NRC should be determined statistic-
ally such as B 0 or B50 life from a
Weibull plot. Nb will then be for the
same probability of failure as NRC.
max RC= Maximum Hertz stress at which NRC
was obtaine.d
w RCLRC= Width and length of wear track on RC
specimen.
Equation 19 is intended only for use as an
estimate of the life of M-50 bearings. It is possi-
ble that the "1/2" exponent will also apply to other
materials. It is fairly certain that the exponent
"9" is universal for hardened (RC 60) steels.
The next logical step is to apply Equation
19 to a number of bearings on which life data is
available. Comparison between predicted and
observed lives would suggest in what manner the
concepts involved need to be refined.
Examples of the application of the correla-
tion Equation 19 follow.
1. Analysis of a Large Thrust Ball Bearing
Life data for this bearing were furnished by
Mr. C. C. Moore of the General Electric Co.
The Weibull plot for the bearing with life in hours
is shown in Figure 20 under a thrust load of
20, 000 lbs. Some of the details of the tests are
given in the legend. The bearing consists of 14
balls, 1.531 in. in diameter. It has an initial
contact angle of about 30 . The length of the wear
track, Lb, is about 6.20 in.
The maximum Hertz stress is 290, 000 psi
and the track width is 0. 416 inches. One bearing
revolution produces about eight stress cycles on
any point of the inner race track, which is the
assumed failure location.
Equation 19 is applied to establish the pre-
dicted Weibull plot shown as a dashed line on
Figure 20. The close correlation which was ob-
tained for this bearing cannot be expected in
general. There are several factors in the
analysis which admittedly could account for a
factor of as much as 4 in life.
2. Analysis of GEL 202K Radially Loaded Bearing
Tests on full scale bearings of this size are
reported in Reference 36. A Weibull plot of the
data is shown in Figure 21. The bearings are
rather small, having a ball diameter of D= 0.250
in. as compared to the ball diameter of the previous
bearing, which was over 1 1/2 in. These are the
same bearings which have been cited in earlier
publications concerning the correlation of the RC
Rig with bearings. The calculated maximum
Hertz stress is 440, 000 psi. The calculated value
of track width is 0.113 inches. For simplicity it
is assumed that the radial bearing experiences
effectively one stress cycle per revolution.
The B10 and B50 life of these bearings was
computed using RC Rig data from Reference 34.
The material is M-50, the lubricant is 7808, and
the tests were conducted at room temperature.
The conditions of the RC and the bearing tests are
nearly identical.
The predicted Weibull plot shown as a
dashed line in Figure 21 gives lives about twice
those actually measured. Even this is considered
good correlation for a process as statistical as
fatigue. Also, the presumption that the inner
race experiences only one stress cycle per revo-
lution of the bearing in a radially loaded bearing is
open to question. As the race begins to come
into the load region, it is sure to experience at
least one load nearly as large as the maximum
load used in the calculations. This would have the
effect of reducing the life of the bearing, thus
making correlation even better.
3. Analysis of 207S MRC Radial Bearing
The Weibull plot for these tests which in
TABLE 7
Summary of Bearing Correlation with RC Rig Data
Bearing and Temp. & Load-(lb.) B10(Test) B50 (Test)
Ball Size Lubricant Frequency B-- (C ) B50 (Cac)
Thrust Ball Bearing 400F T = 20, 000 180 hrs 480 hrs
D= 1.531" 7808 7800 rpm = 1.4 46 = 1.0
202K R.T. R = 426 15 hrs 32 hrs 0.4
D =0.250" 7808 50,000 rpm 40 hrs 80 hrs
207S R.T. R = 1750 120 x 106sc =1.0* 210x 106sc 1.0*
D= 0. 437" 6081 11, 250 rpm 120 x 10Osc 240 x 106sc
* Corrected for differences in lubricant.
Figure 22 is plotted in terms of stress cycles
rather than hours, since that is the way it was
reported in Reference 34. The general size of
this bearing is between the two which have just
been analysed. Ball diameter is 7/16 in.
The computed Hertz stress is 450, 000 psi.
The theoretical track width is 0. 184 inches. A-
gain, it is assumed that the inner race experiences
one effective stress cycle per revolution.
There is no available RC Rig data for M-50
at room temperature using 6081 lubricant, which
is the condition under which these bearings were
tested. The closest set of data is that reported
in Reference 34 using 7808 lubricant at room tem-
perature. In a way this lack of complete data is
fortunate because it provides an opportunity to in-
dicate the manner in which other variables which
are not included in the "correlation equation" may
be taken into account. Other RC Rig studies using
M-1 steel have indicated that about twice the life
is obtained using 6081 as is obtained using 7808.
This agreed qualitatively and quantitatively with
independent results obtained on the Mack spin rig.
To "correct" the predicted Weibull plot shown on
Figure 22 using RC data with the 7808 as lubricant
to predict behavior of a bearing which is lubricated
by 6081, the lives should be approximately doubled.
The predicted Weibull plot "corrected" for differ-
ence in lubrication is shown in Figure 22. It
correlates nicely with the 207S bearing tests.
A summary of these examples of bearing
correlations with bench rig results is provided
in Table 7.
It should be possible to use the method
employed above to obtain "correlation equations"
for other bench rigs. In addition, rigs can be
designed which eliminate much of the guess work
in correlation by matching the conditions of rolling
contact as much as possible to that of the bearings
of interest. Variables which cannot be completely
matched can be accounted for as illustrated above.
IV. ANALYSIS OF CUMULATIVE DEFORMATION
Synopsis: Available data on hard steel and
results on brass in rolling contact obtained as
part of this research are analyzed with regard
to both the initiation and accumulation of plastic
deformation. The nature of track strains and
the effect of preload is experimentally studied
using annealed brass rolling elements. Supple-
mentary tests on the brass subject to complex
combined stress cycles were also performed.
The results are analyzed in an effort to relate
the observations to existing theory and to
establish that the magnitude and rate of accumu-
lation of deformation in rolling contact might
reasonably be accounted for by accumulation of
plastic strain due to cycles of complex com-
bined stress.
A few observations of failure due to dimen-
sional change of rolling elements have been made
in full scale tests, and this mode is anticipated as
the dominant cause of failure in unlubricated
bearings at high temperatures. The possible
sources or mechanisms of the dimensional change
are not well defined or understood. The term
"wear" is sometimes used, although it has not
been established to what degree attrition of sur-
faces contributes to this failure. Density in-
crease caused by metallurgical instability is a
possible source in some metals, for instance
transformation of martensite to troostite in some
hard steels. But even in this particular case it
is difficult to account either for the magnitude of
the observed dimensional changes or the influence
of speed.
Under the complex cyclic stress state pre-
sent in rolling contact, pronounced accumulation
of deformation can occur. It is necessary, then,
to establish and explore cumulative plastic strain
under complex combined stress cycles as a major
source of deformation in this mode of failure.
Viscoelastic and anelastic strains will not be
treated here, although these introduce important
effects in cases such as instrument bearings. In
addition, no consideration of speed or temperature
effects will be made. In a particular case (steel
at 10000F) Bamberger (37) has explored this
effect.
The consequences of cyclic accumulation of
plastic strain are a permanent change in geometry
of contacting surfaces, causing a simultaneous
reduction of applied stress, and an intensification
of residual stresses due to the high degree of
elastic restraint. The phenomenon itself is due
to the complex loading cycle induced by rolling
contact as well as the inherent cyclic material
properties. Some of these aspects will be ex-
plored and substantiated in the following sections.
A. PRELIMINARY CONSIDERATION OF
EXTERNALS
Without concern, at the moment, for details
of the internal mechanism responsible for cumu-
lative deformation in rolling contact, certain ex-
ternal aspects may be examined. This will in-
clude a convenient classification as to the appear-
ance of deformation on the basis of limited obser-
vations and range of variables, analysis of contact
stress as a function of deformation for these ideal
types, and establishment of simple relations be-
tween the parameters of the deformed profile
(track width, radial deformation, and deformed
profile radius).
1. Classification of Extreme Cases
In static indentation tests, which always
involve severe localized plastic strains, there is
a natural classification according to the two
general types of imprints. One type exhibits a
raised coronet of metal displaced locally from the
impression (characteristic of cold worked and
high strength metals), and the other is manifested
by a "sinking in" impression, the plastic strain
being distributed and accommodated throughout
the specimen (characteristic of soft annealed
specimen). Similar observations were made in
rolling tests performed as part of this research
when a few cold worked brass toroids (Rockwell
B 70 on the surface) were blunted by rolling them
against a hard steel roller. Compared to the
majority of specimens which were annealed
(Rockwell B 10), the track edge or shoulder build-
up of the cold worked brass was noticeably
sharper. It seems appropriate to make two ex-
treme classifications with regard to severity of
plastic deformation and its proximity to the elastic
strain range. To this end, the two types will be
designated hereafter as pronounced plastic defor-
mation and elastically restrained plastic defor-
mation.
In pronounced plastic deformation, the
loads are so high with respect to the yield strength
of the rolling bodies that extreme and immediate
changes in contact geometry result. The bound-
aries of this deformation, whether of the "raised
coronet" or "sinking-in" type, are sharp and well
defined. Further plastic deformation during
rolling pushes these frontiers aside. The primary
single factor in determining the relations among
the dimensions of the deformed track is the profile
of the harder rolling body either grooving or
blunting this surface. This is true, since such
extreme cases are, in general, deliberately in-
duced, whereas in more practical circumstances a
combination of both blunting and grooving of pro -
filed rolling elements might be expected.
In the elastically restrained case the dis-
tortion of profile geometry that develops with
cycles of rolling is a continuous transition from
the original profile radius, increasing in the case
of blunting and decreasing in the grooving case.
This transition is such that on any cycle the track
width calculated on the basis of the deformed pro-
file radii using the usual elastic theory closely
approximates the measured track width. Niemann
and Kraupner (27) have verified this approximation
for the blunting of steel at room temperature, by
direct measurements of the deformed profiles.
2. Relations Among Parameters of Deformation
For pronounced plastic deformation, rela-
tions among original and deformed track dimen-
sions may be derived from simple trigonometric
considerations. Consider first the blunting of a
convex element by a cylindrical surface. Assum-
ing local redistribution of plastically displaced
material, the area of the original segment shown
shaded in Figure 23 a must equal the strip above
the chord of this segment, bounded by the flattened
track. Mathematically,
R sin-l -(Ro- h) = (h - u) w
0
where R° is the original profile radius, u is the
radial deformation, w is the track width, and h is
the chord height. The angle sin- (w/2R ) is ex-
pressed in radians.
Radial deformation in terms of original
profile radius and track width is
1 2 w o -1 w (20)
u = Ro - (2 - -- sinm (20)
For the redistribution shown in Figure 23 b,
a similar result for grooving of an originally
straight profile may be obtained. Here the area
of a sector of radius equal to the deformed profile
radius must be set equal to a triangle formed by
these rays and the original surface. The result,
expressing radial deformation as a function of de-
formed profile radius and track width is
w-1
sin 2R
u = R (1 ----
p -1 w
[tan sin 2R
p
Without further consideration the deformed
profile radius (R ) is not known. Indeed, even in
the case of pronounced deformation there is a
slight decrease in curvature with cycles. (16)
However, with simplicity consistent with these
approximations this radius can be taken as slightly
greater than the profile of the grooving body, say
10 per cent greater. This approximate degree of
conformity was observed in tests reported here as
well as in Eldredge's tests. The applicability of
these expected relations between radial deforma-
tion and track width to tests performed as part of
this research is shown in Figure 24.
In the blunting tests, annealed brass toroids
of 2.5-in. major diameter and 5/8-in. profile were
loaded and driven by a 2.5-in. steel cylinder. In
the grooving tests annealed brass cylinders of
2.5-in. diameter were loaded and driven by a hard
steel toroid with the same dimensions as the brass
toroids.
Also shown in Figure 24 a is the relation to
be expected if the change in dimension were due
entirely to wear or complete accommodation of the
displaced material throughout the specimen. The
theoretical curves for blunting and grooving form
a lower limit, predicting less radial deformation
-or greater track width than measured. The agree-
_ment is sufficiently close that the difference may be
attributed primarily to the nature of annealed brass
in increased accommodation of elastic strain as
mentioned above. The relation expected if wear
were the mechanism is considerably too high.
This was confirmed by auxiliary weight loss deter-
mination.
For the elastically restrained case, essen-
tially two equations are available for the deter-
mination of the relations among the three param-
eters: radial deformation u, track width w, and
profile radius R. This first equation, in keeping
with the definition of this case, is the elastic rela-
(21) tion between load, principal curvatures, and con-
tact area dimension (track width):
P 1/3
w = constant. ( )
2T
where P is the load and S(1/R) is the sum of the
reciprocals of the radii of curvature. The second
is a trigonometric relation among the three dimen-
sions that was derived ignoring plastic redistribu-
tion for initial simplicity. From the inset of
Figure 25 this relation is
2 122 2
u R -(-) - -( ) - (R -R ) (23)
Sp°P
p op o
or approximately* for R , the deformed profile
radius: 2
_-2
S2(R -u- R - ( ) 2
0 o 2
To examine the range of applicability of
such a relation the data of Niemann and Kraupner
on the blunting of hard steel toroids is examined.
The reported values of track width for various
loads at 30 x 106 stress cycles are used in Equa-
tion 22 to obtain the deformed profile radii. The
trigonometric relation expressed in Equation 23 is
used to determine u. Experimental and predicted
values of u are plotted as a function of maximum
contact stress in Figure 25. There is an increas-
ing divergence of the two for high load values with
the theoretical relation predicting greater defor-
mation. This is to be expected in view of the
simplified trigonometric relation if local redistri-
bution of material at contact is significant, as it
should be for the higher loads. If the profile
crescent in Figure 25 were deformed into a plate
of uniform thickness with the same curvature, the
4
* Ignoring (-) and higher order terms in the
p
binomial expansion of the first radical.
theoretical relation would be approximately
halved.
3. Contact Stress as a Function of Deformation
As deformation accumulates, the area of
contact increases, and as a consequence the con-
tact stress is reduced. A rational estimate of
this stress as a function of track width may be ob-
tained in pronounced blunting by assuming the
toroid is simply deformed into a cylinder of length
equal to the actual track width. For cylindrical
contact,
ma = constant --
max uw
where oa is the maximum contact stress. This
situation is illustrated in Figure 26 where the re-
sult of typical calculation is presented graphically.
It should be noticed that after an initial rapid drop
in contact stress from some theoretical elastic
value there is a flattening of the curve so that the
contact stress drops only about 20 per cent for a
100 per cent increase in track width. The situa-
tion for pronounced grooving is more complex
since no simple contact area is generated. In
fact, for some initial period, perhaps 102 cycles,
the contact area resembles an ellipse, truncated
at the ends of its transverse axis. With soft tin
and other materials Eldredge has concluded that a
complete contact ellipse is eventually established.
Without unwarranted detail, the two situations of
blunting and grooving can be compared as regards
contact stress. For the same track width a
higher contact stress would be expected in the
grooving situation.
The contact stress for the elastically re-
strained case may be determined by using the de-
formed geometry in calculations based on the
elastic solution. Stress and strain indices are
derived on this basis in Reference 38.
B. PLASTIC DEFORMATION IN ROLLING CON-
TACT
Analysis of the phenomena of cumulative
plastic deformation divides naturally into two
categories: initiation and accumulation. Questions
pertinent to this section are: How much plastic de-
formation is to be expected initially in rolling with
a given load, material and element configuration?
What is the effect of number of cycles on this
amount? Can this increase in groove depth or
radial reduction with cycles be rationally attri-
buted to plastic deformation? While these
questions cannot at this time be answered com-
pletely for a full range of variables and conditions
because of scarcity of experimental data, the
foundation for more comprehensive answers may
be constructed.
1. Initiation
From experiments at one extreme (grooving
of very soft metals like tin and lead), Eldredge's
data may be interpreted as predicting the amount
of plastic deformation on the basis of the static
hardness (load divided by projected area). That
is, the initial track width for any size ball groov-
ing a flat surface may be estimated closely by cal-
culating the impression diameter expected on the
basis of the static hardness. For annealed
copper, because of work hardening caused by the
forward displacement due to the rolling motion,
the track width is noticeably smaller than the
static imprint diameter. The effect is much
smaller for cold worked metals.
For hard 52100 steel with various amounts
of retained austenite Drutowski (39) reported track
depth as a function of nominal maximum contact
stress for plate grooving with cemented carbide
balls of two diameters (1/4 in. and 1/2 in.). This
data may be presented all on one plot (Figure 27)
over the full range of contact stress if the radial
deformation is made dimensionless by dividing by
the theoretical elastic approach of the ball and
plate, uE. These data actually represent the
groove depth u after 20 cycles. This method of
presenting data, i.e., normalizing the measured
deformation by dividing by the corresponding theo-
retical elastic deformation, is a convenient way of
taking into account the element size.
Often the high calculated values of maxi-
mum contact stress are defended by the statement
that even though these numbers are much higher
than ordinary yield stress values, such stresses
are possible because of the high degree of con-
straint. As regards initiation of plastic flow,
elastic constraint on gross plastic flow is signifi-
cant in some situations where the plastic nucleus
is small. However, a more rational examination
of the situation reveals that the values of initiation
stress predicted on the basis of uniaxial tests are
not as divergent as has been supposed. For this
purpose consider the uniaxial tension and com-
pression results of Sachs et al (2) for reasonably
similar 52100 steel, and the elastic limits in ten-
sion reported by Drutowski. ('9) The value of
octahedral shear stress at these "elastic limits"
and the maximum octahedral shear stress at the
point of first measurable plastic deformation in
rolling contact for the various hardnesses are
presented in Table 8. There is variation to be
sure, especially with so ill-defined a property as
elastic limit, but the tension and compression values
tend to bracket the rolling contact values. As
stress is increased, the deviation of actual con-
tact stress from theoretical is probably in accord
with the allowances proposed above. This is fur-
ther evidence that the contact situation cannot be
examined wholly on the basis of maximum contact
stress but requires consideration of state of
stress and, as will be established below, the
effect of the rolling action. This is true whether
the phenomenon considered is static indentation,
initiation, or accumulation of plastic deformation
in rolling or fatigue.
2. Accumulation
Tabor (40) has reported that in static in-
dentation tests there is no accumulation of plastic
deformation with repetition of load after the first
(41)
application. In addition Jones (41) reports that
the flattening of the ball in the indentation test is
several orders of magnitude smaller than the plate
indentation. Both of these statements are invalid
when plastic deformation due to rolling contact is
TABLE 8
Octahedral Shear Stress for Initiation of Plastic Deformation of
52100 Steel in Tension, Compression and Rolling Contact
Drutowski
Ret. Aust. Hardness* T **
oct
Tension
0
7.4%
3.9%
58,000
27,500
60, 000
Compression --- -- ---
0 58 115,000
Rolling 7.4% 62 90,000
Contact 3. 9% 64 114,000
Sachs
Ret. Aust. Hardness
0
Trace
Trace
84,500
66,000
47,000
0 58 153,000
Trace 62 145,000
Trace 64 127,000
* Rockwell C hardness
** Maximum octahedral shear stress at reported "elastic limit", psi.
involved. In addition to the experimental evidence
to be presented here from cumulative blunting and
grooving of brass, there is the published data of
Neimann and Kraupner (27) and Zaretsky and
Anderson (42) referred to in the review.
A qualitative verification that plastic de-
formation accumulates, even at room temperature
in hard steels, is furnished by the superimposed
profile traces of a grooved surface (originally
flat) presented in Figure 28. The traces were
made at SKF(43) on a hard steel surface (Rockwell
C 61.3) after various numbers of stress cycles in
rolling. The maximum nominal contact stress
was 730, 000 psi (500 lbs. on 1/2 in. -diameter
ball). Clearly, not only does the depth of the
groove increase but the shoulder of displaced
materials grows with cycles. Similar traces
are presented by Zaretsky and Anderson for
blunting of a ball constrained to roll over the same
path in the Macks spin rig. On the basis of the
limited evidence from the SKF rig there appears
to be considerably less wear than reported by
Zaretsky and Anderson at 750, 000 psi after
10, 000 stress cycles in Figure 9 of their paper.(42)
Wear is determined by comparing the area of the
shoulder bulge to the groove area.
The data of Niemann and Kraupner for the
effect of cycles on the blunting of hard steel may
be condensed and presented as a single graphical
relationship if the reported track width is made
dimensionless by dividing by the nominal elastic
track width. This plot is given in Figure 29 a
and the closeness of fit to the reported data is
illustrated in Figure 29 b. It must be concluded
that the accumulation of plastic deformation into
millions of cycles, even for moderate loads at
room temperature, is appreciably greater than
the initial values. While a careful separation of
wear and plastic deformation for a range of condi-
tions has been achieved, there is indication that
at high temperatures plastic deformation is very
pronounced (Bamberger (44)).
In order to conduct a more complete in-
vestigation in a situation allowing simple measure-
ments, tests were performed on specimens of
annealed brass in both blunting and grooving situa-
tions. Radial deformation and track width were
measured at logarithmic intervals of 1, 10, 102
103 and 10 cycles for five load levels using the
apparatus shown in Figure 4. A 1-1/2 h.p. motor
drove the load roller at 1000 rpm for the plastic
deformation experiments.
Specimens were designed so that a circular
contact area would theoretically develop when they
were mated with a hard steel load roller (either
cylindrical or toroidal depending on the specimen
tested). The dimensions of these specimens are
given in Figure 30. As received, the brass varied
from Rockwell hardness B 80 near the surface to
B 53 in the center. After machining, the speci-
mens were annealed at 9600F for four hours and
furnace cooled. The final hardness was Rockwell
B 12. Several of these specimens, both toroids
and cylinders, had grid lines spaced at 0.020 in.
scribed at two locations on the surface. After
annealing, the specimens were polished with fine
emery and oil, then buffed. The bore of the
specimens was machined so as to provide approxi-
mately 0.001-in. press fit with the spindle.
In both the experiments on blunting and
grooving the brass specimen was placed on the
spindle in the lever arm and driven by the steel
load roller. Before the tests on the annealed
specimen, the effect of skew on the nature of the
plastic deformation was assessed by use of un-
annealed brass toroids. Various amounts of
skew were introduced by adjusting the angle of the
lever foot. It was observed that a change in angle
of skew of less than 30 minutes was sufficient to
cause unsymmetrical deformation to transfer from
one side to the other. The foot was locked in the
position that produced symmetrical blunting.
The lubricant for all tests was a commer-
cial SAE 10-20 motor oil. The first 100 cycles
were turned by hand at a rate of approximately 100
rpm. The load was removed and gradually re-
applied to the lever arm when the motor was
started.
The test was interrupted and measurements
were made at logarithmic intervals. A super-
micrometer was used in measuring diametrical de-
formation while the specimen was still mounted on
the spindle. For the measurement of groove
depth the micrometer was fitted with knife edge
feelers. Track width was measured by use of a
scale and low power magnification.
Results are plotted in Figures 31 and 32.
A dimensionless summary of the track width data
for both blunting and grooving after the manner of
Figure 29a is given in Figure 31c. While the
accumulation of deformation beyond 102 cycles is
greater for grooving, the numerical results on the
whole are comparable, the rate of accumulation
being nearly a linear logarithmic increase for
grooving.
Wear could not possibly account for the
phenomena reported here. Weighing of the speci-
men before, after, and at intervals during the
tests, confirmed that the amount accounted for by
wear, either the total amount or the change between
various intervals, is a small fraction of the total
volume displaced.
The fact that plastic deformation does
accumulate in blunting as well as grooving con-
firms that the growth in track width is not primari-
ly an external configuration effect as proposed by
-Eldredge. It has been shown that there is a de-
finite reduction in contact stress with increase in
track width in the case of pronounced blunting.
There is no density transformation in brass that
would account for such a large deformation.
Accepting that the phenomenon is indeed due to
accumulation of plastic strain by applied cycles of
rolling even though the contact stress is reduced
by the deformation, it is natural to ask the question:
how?
Two possible sources of this phenomenon
are:
1.) Interaction of bulk residual and applied
stresses during the complex loading cycle to pro-
duce cyclic flow even though the material is basi-
cally cycle-independent. The mechanics analysis
of plane strain rolling of an elastic-perfectly
plastic half space by Merwin and Johnson (45) is
such a case.
2.) Inherent cycle-dependent material be-
havior even in the absence of macro-stress or
strain gradients.
These possible sources were proposed in a
discussion of papers presented at the General
Motors Symposium on Rolling Contact Phenomena
in Warren, Michigan, October 1960. (46)
The actual situation may involve both. The
second possibility should be explored before exten-
sive mechanics analysis is devoted to the first.
This is the approach taken below.
The usual assumption of the identity of
current loading and subsequent yielding surface in
classical plasticity theory is reasonably well
founded for simple radial (proportional) loading.
However, there are important experimental quali-
fications even for repeated tensile loading at room
temperature.
The observation of accumulation of plastic
strain by means of repeated stress was made early
in the investigation of behavior of metals under
cyclic stressing. Bauschinger's "natural elastic
limit" was a consequence of this observation.
Bairstow (47) also measured cumulative plastic
strains for repetition of stress.
Yet these deviations from the theory do not
appear sufficiently pronounced to account for such
a large accumulation of plastic strain in rolling
contact where the contact stress actually reduces.
However, for the particular brass used in the
toroid specimens, tensile tests under repeated
load were made to explore this possibility. No
accumulation of plastic strain of the magnitude ob-
served in rolling contact could be measured for
cyclic loading in tension.
The effect of the rolling action on the
stress history of an element in the highly stressed
volume of material in the contact region appears
to be a significant feature in accounting for the
observed accumulation of plastic deformation. A
subsurface element in the track center is first
sheared forward as the rolling load approaches
and then backward after it passes. When the
contact area is in the near proximity of the element
the associated stress deviator has a significant
tensile component transverse to the rolling direc-
tion or track. Subsurface elements at the track
edge also experience a shear reversal, in addition
to the orthogonal shear in the transverse direction
which increases to a maximum and decreases as the
load passed by. Thus, while the element is in
the region of high distortion there is a shear re-
versal, flow being induced transverse to the track.
As an illustration of this history, the stress
variations on a subsurface element for the elastic
case of cylindrical contact is given in Figure 33.
Unlike one directional repeated stressing,
reversal of the shear stress may cause large
changes in the inherent deformation resistance of
metals. Tuler and Morrow (48) have reviewed
the cycle-dependent stress-strain behavior of met-
als and present data for OFHC copper demonstrat-
ing both hardening and softening in the cyclic
state. Smith, Hirschberg, and Manson (49) re-
port on the cyclic stress-strain behavior of
several steels including 52100 at a hardness of
Rockwell C 53. All of the heat treated steels
softened as a result of being cyclically strained.
For the 52100 steel a cyclic strain amplitude of
1% caused a drop in the stress amplitude from
about 280, 000 psi to 200, 000 psi. Such a cycle-
dependent softening would lead to a cycle-depend-
ent increase in contact area. Thus, the geo-
metric changes of elements during rolling contact
may be partially due simply to an inherent cycle-
dependent reduction of deformation resistance of
the metal subjected to repeated reversals of shear
stress.
To explore the possibilities of accumulation
of plastic strain with cycles of complex stress
under controlled conditions, combined tension-
torsion tests were preformed on a double gage
length solid specimen of annealed brass. The
axial load was increased to a maximum value with
a constant torque, unloaded, the torque reversed,
and loaded again. This loading program caused
an accumulation of axial plastic strain.
The nominal tensile stress-plastic strain
path for a series of these cycles is shown in Figure
34. It should be noted that repetition of the load
cycle without reversing the torque did not cause
measurable plastic deformation to accumulate.
The magnitude of the increments in plastic strain,
the tensile yielding at a point well below the maxi-
mum load, and the cyclic work hardening are fea-
tures which are qualitatively in agreement with the
observed cyclic deformation in rolling tests.
Certain features of the phenomenon might
be associated with incremental collapse or "shake-
down" models familiar to structural analysis.
However, it is probable that this cyclic instability
is an inherent feature of metal behavior.
Because the specimens were solid the torque
causes a macro-strain gradient. In order to
eliminate this, steady tension and alternating shear
stress tests were performed on thin wall copper
tubing. Some of the results of these tests are
presented in Figure 35. The shear stress-strain
hysteresis loops are also shown. Therefore, it
appears that cyclic strain accumulation is an
"inherent" material response to such complex stress
cycles, and may be expected to play a significant
role in the cumulative deformation observed in
rolling. A more complete exploration of this
phenomenon is given in Reference 50.
Because of unexplored areas of cyclic de-
formation even in simple radial loading, lengthy
speculation as to the fundamental mechanism is
unwarranted. Nevertheless, it seems reasonable
that after considerable dislocation generation due
to initial plastic deformation, new routes of escape
from dislocation entanglement or other barriers
to slip are provided by the complex stress cycles
used in these tests. In addition, vacancy capture
and mechanically induced climb mechanisms may
be expected to operate under repeated load condi-
tions. This process would be accelerated with
increase in thermal energy provided in high tem-
perature environments.
C. ASSOCIATED PHENOMENA
In addition to observations of the amount
and rate of accumulation of external deformation,
some associated phenomena were studied. The
nature and intensification of distortion of orthogo-
nal grids on the rolling tracks for both blunting
and grooving, the effect of an initial cycle at high
load on subsequent deformation at lower load, and
some remarks about the nature of residual stress
pattern to be expected as a result of plastic defor-
mation in a simple case, complete the treatment
of cumulative deformation in rolling presented
here.
1. Influence of Rolling on Surface Grid Distortion
A phenomenon that has provoked a great
deal of discussion is the surprising forward
(direction of disk rotation) displacement of trans-
verse surface lines and the associated subsurface
displacement due to rolling that Crook (51) and
Welsh (52) observed in mild steel disks. The
nominal maximum contact stress was 240, 000 psi,
and the frequency was 1500 rpm. The contact
was lubricated with a jet of oil of about 50 centi-
stokes viscosity. For pure rolling, originally
straight lines parallel to the cylinder axis appear-
ed to be "pulled" into the contact area. This
occurred on both the driver and driven roller,
thus opposing any tangential friction. As might
be expected the plastic zone is confined to a sub-
surface band because of the depth to the maximum
shear distortion in the contact situation. But this
:and is displaced forward in the rolling direction,
:ausing a distortion of radial lines as shown
ichematically in Figure 36. Crook gives a
rather improbable explanation of this by stating that
the forward shear is a result of the increase in
velocity of material under the contact point, due to
change in effective cross sectional area. Thus,
it appears that this velocity increase must be due
to increments of plastic deformation each cycle
and not elastic compression, and further, that the
intensity of the phenomenon must depend predomi-
nantly on speed of rotation. A number of more
feasible alternative explanations based on the
existence of a hydrodynamic film which modifies
the static Hertzian contact pressure have been
offered in the discussion which accompany Crook's
paper. Milne and Lewicke (51)proposed that the
high local tangential lubricant shear stress in the
outlet direction (because of short outlet area)
produce the plastic tangential "creep. "
Another suggested explanation is that pro-
posed by Christopherson and Johnson. For
simplicity, consider the -identical disks shown in
Figure 36 which are independently driven so that
no resultant friction exists between them. Since
plastic work is being done (as well as "elastic"
hysteresis energy dissipation), there must be an
offset of resultant normal force towards the lead-
ing edge of the contact area. Also, from hydro-
dynamic lubrication theory, there must be a dis-
placement of resultant force from the point of
nearest approach toward the leading edge, and more
important, a gradual build-up of pressure at
entrance and a sharp drop-off at exit.
As the subsurface element approaches the
contact it is sheared backward and then forward
as it passes under. The maximum shear stress
will be expected in the forward direction due to
the abrupt drop-off of contact pressure after the
point of resultant force is reached. To examine
the possible order of magnitude of this effect, a
calculation for the orthogonal shear stress was
carried out in Reference 38 for an extreme
asymmetric pressure distribution. This shear
stress is plotted in Figure 37 at a depth of one-
half the contact length. The increase of forward
over backward shear stress is 18 per cent. Thus
there is a small mean shear stress in the forward
direction which could account for the preference
for distortion in that direction. Benham, (53)for
example, has shown that only a small bias in
stress is required to cause a large accumulation
of strain under cyclic loading. However, the
small bias in stress proposed above depends on
the presence of a lubricant. The validity of the
above concept is clouded by the conflicting ob-
servation made by Hamilton (54) that forward dis-
placement occurs even in the absence of a lubri-
cant.
Perhaps the most probable explanation is
that based on an analysis recently proposed by
Merwin and Johnson.(45) By assuming the elastic
strain cycle and integrating the Prandtl-Reuss
plasticity relations by increments, they determine
the corresponding stress cycle. This procedure
gives residual values of orthogonal shear stress
which because of final equilibrium considerations
are relaxed at zero. This implies a residual
shear strain that accounts for a "forward" shear
displacement. Even though the analysis is
approximate in that it violates equilibrium during
the loading cycle and compatibility when the resid-
ual stresses are relaxed, many essential features
of the phenomenon are dealt with logically.
Using brass toroids and cylinders a similar
study for the three-dimensional situation are con-
ducted as a part of this investigation. Several
factors emerge: For severe grooving, differential
slip had the predominant influence on distortion of
transverse surface lines. In blunting, the original
backward displacement of transverse lines is in
accord with the fact that a bulge or hill of material
is formed on the inlet side, as in plastic reduction
in strip rolling. This pattern is not accentuated
with cycles after the initial severe blunting. The
tangential drag, or possible mechanisms mentioned
in connection with Crook's phenomenon, tends to
straighten the line after this. The specimens
before test are shown in Figure 38, and photographs
of the tracks are displayed in Figure 39. The back-
ward displacement of the side portion of the lines in
grooving is not as pronounced as the forward dis-
placement, possibly because of the net friction
force required to drive the brass cylinder.
2. Effect of Preload
After large numbers of cycles Niemann and
Kraupner (27) observed smaller total accumulation
of deformation in rolling after an initial period of
high load, compared to tests at the lower load
during the same running time. The effect of pre-
vious cold work in reducing the rate of accumula-
tion of deformation even at high temperatures has
been observed by Bamberger. (37)
This preload effect was also observed in the
rolling tests with brass specimens. In both blunt-
ing and grooving only the initial cycle was at the
high load. The results are presented graphically
in Figures 31 and 32. When the cross over occurs
in track width the contact stress on the preloaded
specimen is actually higher than that in a com-
panion toroid at the same load, yet the rate and
total accumulation is slightly lower.
Two factors contribute to the increased
resistance to deformation as a result of preloading.
There is a set of residual stresses created by
preloading which algebraically opposes the applied
stresses at lower loads, and the metal in the con-
tact region is cyclically hardened to a higher level
than possible by continuous operation at the small-
er rolling load.
3. Residual Stress
As a result of cumulative plastic deforma-
tion, residual stresses may be expected to inten-
sify with cycles of rolling. Such observations in
hard steel have been reported by Bush et al. (55)
Analysis of the residual stress distribution in a
perfectly plastic half space due to rolling has been
made by Johnson. (56) Unlike the half space
problem a tensile residual stress normal to the
surface can be developed in a disc or roller.
Consider a solid cylinder subdivided into three
regions: an outer ring, an inner concentric ring,
and a solid core. If the inner ring is plastically
expanded and the assembly is fitted back together,
residual tension between the plastically deformed
inner ring and the elastic core, as well as tangen-
tial compression, develops. Typical calculations
are illustrated in Reference 38.
In the three dimensional situation the
elastic constraint is even higher because of the
adjacent elastic material at the side of the track.
Hence, a more severe residual stress pattern
might be expected.
For more complete information on residual
stresses induced by rolling and the influence of
residual stresses on fatigue, the reader is re-
ferred to a recent book by Almen and Black. (18)
V. SUMMARY AND CONCLUSIONS
This bulletin has two main parts. The
Appendices deal with a review and classification of
research concerning aspects of the failure of sur-
faces in rolling contact. The text contains
analyses of certain features of two modes of
failure (pitting and cumulative deformation) in-
volving original experiments and interpretation of
appropriate experiments from the literature and
examples of correlation of pitting data. A brief
summary and statement of conclusions follows.
A. REVIEW
Research has been classified and identified
according to two indices: mode of failure and phase
of research. Representative research from
areas defined or located by these indices have been
critically reviewed.
1. Pitting
A great deal of research over the years has
been conducted in all phases of pitting failure,
centering heavily in the area of controlled experi-
mentation (material and lubricant evaluation) with
the bench rig tester playing an important role.
Theory and interpretation have not been so well
explored. Effort exists not only to screen and
evaluate, but to extend minimum material data
and fundamental properties to actual component
performance. Principles for this extension or
correlation of data are not well established, the
possibility and extent of the correlation being de-
bated in the literature. Some fundamental re-
search into the nature of lubricant films involved
in rolling contact has been started for the two-
dimensional situation.
2. Cumulative Deformation
Observation of excessive run-out develop-
ing in bearings in the relatively few high-tempera-
ture investigations has alerted researchers to a
new failure mode, but the primary cause has not
been established even in the first efforts at con-
trolled experimentation. Cycle-dependent cumu-
lative plastic deformation has been advanced as a
cause. However, no comprehensive theory or
interpretation is available even from fields of
pure material research concerning the accumula-
tion of plastic strain under repeated stress. A
three-dimensional plasticity solution for one roll-
ing cycle might at least serve as a rational basis
for eliminating size and configuration effect and
estimating plastic strain associated with a given
external deformation. But the complications
existing in the present static indentation ap-
proaches are compounded, and a comprehensive
solution even for simple stress-strain idealizations
appears beyond present expectations. Some
analyses of the two-dimensional case for perfect
plasticity have been developed. As a result of
this sparse research there are yet no well-estab-
lished fundamentals to serve as the basis for pre-
diction or design.
3. Excessive Rolling Resistance
The phenomena associated with this mode
of failure have been subjects of research interest
for perhaps the longest time. Yet the least con-
trolled experimentation of fundamental signifi-
cance in the range of practical interest has been
conducted in this area. This is especially dis-
concerting since such information is important to
a rational mechanics analysis of the two other
failure modes. Rejection or failure of instru-
ment bearings in service due to excessive or
variable torque has led researchers to consider
both surface interaction energy losses and internal
dissipation losses. Data for experimental sepa-
ration of these phenomena are not available.
B. ANALYSIS
Features of two modes of failure (pitting
and cumulative deformation) have been analyzed
with regard to primary material cause and con-
figuration and size effects.
The significance of the action of rolling in
both of these modes is emphasized. The in-
fluence of rolling element configuration on pitting
fatigue life is directly related to the effect of roll-
ing in contrast to repeated contact. The rolling
action reverses a subsurface shear, which is a
function of configuration, producing a critical
range of stress. Regarding cumulative defor-
mation, the complex stress cycling involving shear
reversal in highly distorted regions is responsible
for the continuance of cyclic plastic flow, despite
the resulting contact stress reduction.
1. Pitting
The effect of geometry or rolling element
configuration on pitting fatigue life in bench rig
experiments was established on the basis of the
presentation of data collected from several
sources. The effect of configuration on the
magnitude of maximum reversed subsurface shear-
stress is such as to suggest it to be the critical
stress, and an appropriate strength index for
comparing widely divergent configurations of
rolling elements. It is assumed that comparison
is made in otherwise equivalent circumstances.
Subsurface origin of fatigue cracks responsible
for pitting is indicated, propagation parallel to
the surface being enhanced by the residual radial
tensile stress due to plastic deformation. A
few observations of subsurface structure in pitting
tests conducted here tend to confirm this view.
A simple size effect analysis based on the
usual Weibull distribution function and length of
rolling track is the basis of an expression that
permits extension of pitting life data to various
sizes of rolling elements.
These considerations are applied in the
correlation of
a) torsion fatigue and rolling contact bench
rig data
b) rolling contact bench rig data and full
scale bearing tests.
2. Cumulative Deformation
In an effort to identify the primary cause
of this failure in a pronounced and easily measured
situation, rolling experiments on brass elements
mated with hard rollers, and tests in combined
tension and torsion were conducted. The intensi-
fication in external deformation with cycles of
rolling was attributed primarily to a material
phenomenon of cumulative plastic strain via com-
plex stress cycles for the following reasons:
1. The progressive reduction in diameter
is not due to wear, since the weight reduction of
the specimens account for only a small percentage
of it.
2. The success of the relation among the
parameters of deformation, which assumes re-
distribution of material at contact, in approxi-
mating the actual correlation of track width and
radial deformation, as well as the relatively large
volume involved, rule out simple density trans-
formation in brass.
3. The cyclic intensification of plastic
flow in the track indicated by the forward displace-
ment of orthogonal grid lines scribed on the speci-
men attests to the accumulation of plastic strain.
4. Combined stress tests for complex
loading cycles with the same material resulted in
a cycle-dependent accumulation of plastic strain,
indicating that the rolling contact mode is a mani-
38
festation of the same phenomenon.
Analysis of available data on hard steel
was made in terms of a convenient deformation
index. The index expresses plastic-deformation
(radial or track width) as a fraction of its elastic
counterpart, and its utility was indicated in con-
densing the data of Drutowski and Niemann and
Kraupner.
Three salient features associated with this
mode of failure are accumulation of plastic strain,
reduction in applied stress, and cyclic intensi-
fication of residual stress.
VI. RECOMMENDATIONS
From the above review and analysis a
number of recommendations for both research
aims and technical application follow:
A. PITTING
The effect of alignment of rolling elements
on surface failure, notably pitting, should be in-
vestigated. Skew of less than 1/2 degree has
been found to be sufficient to cause unsymmetrical
deformation in blunting. There was also some
indication of a marked decrease in pitting life for
skew of that order of magnitude.
The effect of stress level on pitting life
variability or Weibull slope should be investigated.
This effect cannot be included in the form of the
equation developed by Lundberg and Palmgren.
The investigation will require collection and
statistical analysis of sufficient contact fatigue
data for a large stress range.
A fundamental correlation equation based
on these findings and the most recent engineering
descriptions of fatigue should be developed. The
influence of lubricant, relative surface motions
and interaction, and transient temperatures on
the two major categories of mechanics analysis,
critical loading cycle and inherent material prop-
erty, must be evaluated.
B. CUMULATIVE DEFORMATION
The extent and intensity of plastic strain
due to three dimensional rolling, even in pro-
nounced cases, needs further experimental in-
vestigation. It is obvious that more controlled
investigations will have to be conducted into effect
of rate, temperature, and material on magnitude
of deformation accumulated. However, as a
corollary to this, further investigation into the
effect of configuration and size is needed for the
extension of these data to practical designs.
The effect of preload or mechanical work-
ing, apart from or in conjunction with heat treating
procedures, should be considered in developing
more cyclically stable materials with a slower
rate of accumulation of plastic deformation in high
temperature applications. Such possibilities have
been pointed out by Niemann and Kraupner (pre-
load) and by Bamberger (previous cold work). It
is possible that a short running period of high' pre-
load together with close fitting reassemblage of
pre-run component parts may increase life to
excessive run-out. At any rate, material suit-
ability in high temperature application is not deter-
mined by constant load creep or hot hardness tests
but by stability under complex cyclic loading.
The basic material phenomenon must be
investigated as it will most certainly become an
important failure mode as higher temperatures
and complex cyclic loading in structures are en-
countered. In this respect the rolling contact
problem is only a single manifestation or forerun-
ner of a broad class of problems. Gross dimen-
sional instability of structures or frames subject
to variable and repeated combined loading cycles
is expected to be due in large part to this mecha-
nism. The usual shakedown analysis based on
ideal plastic action of frame components will not
be directly applicable. Further laboratory tests
of metals under complex stress cycles, extending
the combined tension-torsion tests reported here,
are needed. Thin walled tubular specimens
should be used to obtain macroscopically uniform
stress states, thus eliminating compatibility or
plastic strain gradient effects which give rise to
shakedown phenomena in continuous media.
C. EXCESSIVE ROLLING RESISTANCE
A complete dynamic investigation is needed
of the rolling phenomenon, including measurement
of normal and tangential forces, as well as normal
force offset or coefficient of rolling for a practical
range of loads and speeds. This will necessitate
independently driven surfaces--the torque on the
system will have to be determined in a manner that
eliminates the effect of support bearing friction.
Only in this way can friction and internal hystere-
sis losses be separated for a range of controlled
conditions.
D. GENERAL
In closing, the authors would like to
criticize the current direction of research on
rolling elements. It is our opinion that a dis-
proportionate amount of effort is spent on uncon-
trolled and uncorrelated routine bench testing and
on the lubricant and "metallurgical aspects" of the
problem while the "mechanics" of rolling contact
is slighted. We would encourage researcher and
sponsors of research to reassess the obstacles to
understanding the rolling contact problem and see
if more effort could not be wisely spent in studying
the forces, displacements, stresses and strains
for bodies in rolling and sliding contact.
v/
3
/
/
232 =
* 52/00 steel
.I / /0 /00
Millions of stress cycles
FIGURE 3. EFFECT OF RIG TYPES AND
ROLLING ELEMENT CONFIGURATION ON
PITTING ENDURANCE FOR THREE STEELS
FIGURE 4. ROLLING CONTACT RIG
USED IN CUMULATIVE DEFORMATION
AND PITTING EXPERIMENTS
/000
(a) Pit on specimen A-4 ofter .77x 10 cycles
Load roller / ' Type B
FIGURE 5. ROLLING ELEMENTS USED IN
PITTING ENDURANCE TEST
l/ -'tir on specimen B-J offter 8.7x /0 ' cycles
FIGURE 6. TYPICAL PITTING FAILURES
mm I I
I I TJ
i
^_
/ M-50 steel /2
./ , ,
I-T^-
v W1 piE l ii f i
i /Z ! i
ý41*1.
e="/ =/440
.5 1.0 1.5 2.0 2.5 3.0
e Contact ellipse axis in rolling direction
Axis in transverse direction
FIGURE 7. RANGE OF ORTHOGONAL
SHEAR STRESS AS A FUNCTION OF
ROLLING ELEMENT CONFIGURATION,e
.5
.4
0 .2
e
FIGURE 8. SURFACE TENSILE STRESS AT LEADING
EDGE OF CONTACT ELLIPSE AS A FUNCTION OF
ROLLING ELEMENT CONFIGURATION, e
(a) P/oan view approximately actual size
(b) Enlarged photograph of sectioned specimen showing flake
fragments lifted during grinding, opprox 25x
FIGURE 9. INCIPIENT FLAKE ON
TOROID SPECIMEN A-2
K, percentage increase in track width over theoretical
FIGURE 10. INCREASE IN TRACK WIDTH
o -.0
.0
FIGURE II. GEOMETRIC DEFINITIONS
e
FIGURE 12. FACTOR FOR TRACK WIDTH CALCULATION
FIGURE 13. DEPTH OF MAXIMUM
ORTHOGONAL SHEAR STRESS
.~ *;;;
(j~
-40 -30 -20 -/0 0 /0 20 30 40
Volumetric tensile stress, thousands of psi
FIGURE 14. EFFECT OF HYDROSTATIC
PRESSURE ON FATIGUE (CROSSLAND)
; 300
250
I
0
/10 10o5 106 I07 I0 109
Cycles to failure
. S-N CURVES FOR TORSION AND ROTATING
103
BEAM TES
104 105 106 107 /08
Cycles to failure
TS OF HARD 52100 STEEL (STYRI)
I Maximum contact stress corrected for plastic increase
in track width
U1 Range of orthogonal shear stress, geometry effect
iZ Semi-range of orthogonal shear stress corrected for
volume stressed
17 Torsional fatigue strength or essential shear stress for
contact fatigue, effect of state of stress
106 /0r
N, Cycles to failure
FIGURE 16. SUCCESSIVE STEPS IN
CORRELATION OF DATA
108
FIGURE 17. RATIO OF NORMAL TO SHEAR
STRESS ON PLANE OF MAXIMUM
ORTHOGONAL SHEAR STRESS
. 50
25
S10.0
75
5.0
2.5
./0 .25 .50.75 /.00 10
Millions of stress cycles
FIGURE 18. WEIBULL PLOT OF
UNADJUSTED ANNAPOLIS DATA
S300
S250
S200
150
lO0
50
103
FIGURE 15
_ _ i o-- Did not fail
0
(a) Torsion specimens RC 62.1
I I I
0 0
t09
'.- J.v
106
te = 1.35
/ 3-
17 "- -0" ,,
ELT .77a
to05 1 -. .35
15 , i i i ii i I I I I I0
(b) Rotating beam specimens RC 61.8
I I I I
v( 50
"" '...; c.v
1z
4z
'<:
§
<
s.
g
.
L
Slope = - /9
6.40
Material. M-50 (MV-1) RC 64
Lubricant, Mil - L-7808
Room temp
5.90
5.88
5.86
S5.84
5.82
6.80
720 760
Log life, cycles
FIGURE 19. LOG-LOG PLOT OF RC DATA
SHOWING SLOPE TO BE I/9 TH
Hours to failure
FIGURE 21. GEL 202K RADIAL BEARING DATA
Hours to failure
FIGURE 20. THRUST BALL BEARING DATA
95
75
50
2b
25
7.5
(a) Blunting (b) Grooving
FIGURE 23. IDEAL PRONOUNCED PLASTIC
DEFORMATION
Predicted, Eq. 19 using
data with 7808 as lubricant
Prediction 'corrected" for
difference in lubricant, life
multiplied by factor two -
I
I
IC
II
I
/
/ I
/ Measured
/ o
Material, M- 50
/I P, radial, 1750 lbs
Lubricant, 6081B
lRoom temp 8 /11250 rpm
I
N, Cycles to failure
FIGURE 22. 202S MRC RADIAL BEARING DATA
X50
6.00
- r
5,0n
L
75
50
/0.0
75
5.0
2.5
1
7
25 1
.00O
.003ý
.00
0
6
Track width, thirty-seconds of on inch
FIGURE 24. COMPARISON OF THEORY AND EXPERIMENT FOR PRONOUNCED PLASTIC DEFORMATION
7
6
-5
O 4
2
300 400 500 600
Max contact stress, thousands of psi
FIGURE 25. RADIAL DEFORMATION FOR ELASTICALLY
RESTRAINED PLASTIC BLUNTING
w//wE
FIGURE 26. REDUCTION OF CONTACT STRESS IN
PRONOUNCED PLASTIC BLUNTING
- - Calculated from measured w using elastic
solution, Eq. 22, with actual load to obtain
Rp and Eq. 23 to obtain u
- u at 30 x 106 stress cycles from /
Niemann and Kroupner /
---------------------/ --
Hardness RC 66 R, = .465-
Snnd r .oll'r -G-di,, I QOn
.25
&
I¼'1
t
05
q,
r
200 300 400 500 600 700 800 900
Max contact stress, thousands of psi
FIGURE 27. DIMENSIONLESS PRESENTATION OF
DRUTOWSKI'S GROOVING DATA
.0001
0
(nnni3
FIGURE 29. DIMENSIONLESS SUMMARY
TO ACTUAL EXPERIMENTAL RESULTS
2.2
S1.8
1.4
S1.0
30 0
Millions of stress cycles
OF KRAUPNER AND NIEMANN'S
BLUNTING DATA AND COMPARISON
FIGURE 30. ROLLING ELEMENTS USED IN
CUMULATIVE DEFORMATION TESTS
Type r i
(0) Brass grooving and (b) Companion steel
blunting specimens load rollers
o * 18.4% retained oustenite RC 59 l t
a0 7.4% .. RC58 /
S1 0% .. .. RC62
A
S6.000 stress
I
I
Hardness
Rockwell C 66
I (a)
_ _ _ _ 1._I_ _ _ _ _ I
- - - Predicted from part (ao)
Reported experimental curve
S 595,.000 psi
'4 530,000
'- - -- 480.Zo
- 415,000
367 .000
01 100 316,000
"e (b)
e5fýr-'ý
1340000 stress cycles
I,4,00stres cyles
7
Z
z
.10 .05 0 .05 .10 .15
Inches
Maximum contact stress. 730, 000 psi Hardness, RC 61.3
Stress cycles, 1200 per minute Lubricant, MIL-0-6082A
Average test temperature, 44 C
Grooved by Y/" diameter 'Hofors' ball, Ref 43
FIGURE 28. SUPERIMPOSED TRACES OF FLAT
PLATE GROOVED BY ROLLING BALL
.0
.0
ZZ :Z:
c3 t ,
\'ý11 ----
^.l
cycles
I
L
b
Q)
K
Number of stress cycles
FIGURE 31. CYCLIC GROWTH OF TRACK WIDTH FOR PRONOUNCED PLASTIC DEFORMATION OF BRASS
I- b -- - b'-----
t
A A O
cr, y aza
6 1.1 rxtO y
/.5b .Ob .5b 0 .5b I.Ob 1.5b
FIGURE 33. VARIATION OF DEVIATOR STRESSES
AT DEPTH OF MAXIMUM REVERSING SHEAR STRESS
Number of stress cycles
FIGURE 32. CYCLIC INCREASE OF
RADIAL DEFORMATION FOR PRONOUNCED
PLASTIC DEFORMATION OF BRASS
.004
.003
.002
.001
0
S.004
.003
002
.00,
Specimen subjected to a torque of 62.5 in. - bs ( s 22,000 psi) which was reversed offer each load cycle.
25 -- ^^- - ---
25 --- __
20 --
15 _
/5 min. time delay
-Repeated load without torque reversal - no measurable plastic strain increment
i, I I I I 1
.030
FIGURE 34. AXIAL STRESS - PLASTIC
k
Axial plastic strain accumulated
STRAIN LOCUS FOR COMBINED TENSION TESTS WITH
ANNEALED BRASS
yi,(C3 Vl 1furqu wUwrvlJWu foluu W"rfiva
FIGURE 35. THIN WALL TUBE DATA FIGURE 36. KINETICS OF LUBRICATED ROLLING DISKS
(4
FIGURE 37. EFFECT OF ASYMMETRIC CONTACT STRESS ON ORTHOGONAL SHEAR STRESS
(0) Specimen T-5
FIGURE 38. PHOTOS OF MOUNTED BRASS
ROLLING ELEMENTS BEFORE TESTS
(b) Specimen C-9
r1do =
103
'4
f
/0
I
)
Specimen T-5 Specimen C-9 Specimen C-8
432 lb load 432 lb load 683 lb load
Blunting Grooving Grooving
Relative motion of load over surface
9 Rolling direction (surface velocity)
FIGURE 39. SEQUENCE PHOTOS OF ROLLING TRACK GRID DEFORMATION IN BLUNTING AND GROOVING
/
102
b.~
ýi
VII. APPENDIX A. RESEARCH CLASSIFICATION
Research on the failure of surfaces in
rolling contact has dealt v ith a wide class
of phenomena under a multitude of particular
conditions. The purpose of this classification
is to establish a logical framework for future
reference and to provide the basis for an
organized review.
The major areas of research will be
designated by two principal indices: mode of
failure and phase of research. The research
phases under each mode are drawn from
Tykociner's classification of recurrent
characteristics of any research problem. (57)
Each of the areas located by the two princi-
pal indices may be expanded by the problem-
atology principles enumerated by Tykociner,
thus providing specific topics for research.
A. MODES OF FAILURE
Here failure means an action of the sur-
face resulting from, or developing with,
rolling under load in a particular environ-
ment that leads to unsatisfactory functioning
of the mechanism. Not included in this defi-
nition are failures due to vibration and noise
resulting from original imperfections in, or
misapplication of the mechanism, severe
lubricant deterioration and fouling, (59or
abrasive wear. Surface-lubricant
interaction phenomena are not excluded, but
are subjugated to contributory roles in the
modes treated. Surface includes the body
material at or near the surface and corre-
sponding films and lubricants involved in the
contact. Thus, gross body failure via frac-
ture or elastic deformation is excluded.
Rolling contact is intended to include cycle-
dependent phenomena, but excludes those
failure mechanisms associated with gross
relative sliding. Thus, no extensive con-
sideration of gear failures will be made.
Three modes of failure have been se-
lected for review: 1) pitting or flaking,
2) cumulative deformation of rolling elements,
and 3) excessive or variable rolling resist-
ance. Although these modes are more fully
defined in the review, a few words to justify
their treatment are appropriate.
1. Failure by pitting is a long established
and recognized mode; in fact, pitting failure
is the basis of life rating in bearing design.
2. Cumulative deformation leading to re-
jection of a mechanism due to dimensional
instability of the rolling elements (in the case
of rolling bearings--excessive run-out) is
not a well-established or well-explored mode.
However, its importance in future application
will depend upon research at high temperature.
3. Rolling resistance has been the subject
of curiosity and research for perhaps the
longest time, and its minimization is a con-
sideration in most design. It is included
here as a failure mode, not as an external
manifestation of other causes (for instance
increased friction torque due to lubricant
deterioration or excessive bearing preload,
etc. ), but as an inherent phenomenon of real
deformable bodies in rolling contact. Con-
cern generated because of required predict-
ability of instrument bearings has placed new
emphasis on this mode.
Additional modes of failure could be
added and treated within a similar framework.
B. PHASES OF RESEARCH
The four phases of research (57are:
A. Phenomenological exploration: observa-
tion and classification. As applied to
this research subject it has two sub-
divisions:
1. Description of mode, place, and
range of occurrence of failure.
2. Primary observations and purely
empirical relations among recog-
nized variables.
B. Controlled experimentation: measure-
ment and collection of data. The best
way to catalogue this area is by type of
test. The following are considered here
1. Full scale tests of associated
mechanisms in which failure occurs,
with appropriate instrumentation.
2. Bench rig or simulated testers in
which certain essential features of
the rolling contact conditions are
isolated and explored.
3. Pertinent material tests which ex-
plore material behavior associated
with the failure mode under simpler
conditions ranging from tension tests
to lubricant pressure viscosity tests.
Each of these items may be further
modified by a statement of:
a) Variables treated
b) Measured quantities
c) Factual relations
C. Hypothesis or theory: interpretation of
experimental results. The analysis
implied in this phase of research can be
grouped into three sections:
1. Mechanics analysis. This section
involves contributions from elasticity,
plasticity, fluid mechanics, thermo-
dynamics, etc. , on the basis of
idealizations pertinent to the above
research findings.
2. Material analysis. From funda-
mentals established by pertinent
material tests, mathematical rela-
tions among "purified" variables are
formulated.
3. Synthesis. The findings of material
analysis are extended by means of
the mechanics of the particular situa-
tion to form a "solution" (correlation
of facts).
Research in these sections can be
modified by:
a) Idealization or conditions treated
b) Nature of solution or variables
interrelated
D. General review and prediction. This
area should include:
1. A summary of fundamental results
and predictable features.
2. A critical review in preparation for
further research effort.
A diagrammatic representation of this
classification is given below. It should be
pointed out that some published research pa-
pers may be placed with validity in more than
one area as defined above.
CLASSIFICATION OF RESEARCH
Mode of Failure
Cumulative
Deformation
Phase of Research
Excessive
Rolling
Resistance
I II III
1 1 1 Observation
A &
2 2 2 Classification
1 1 1
2 2 B Controlled
Experimentation
3 3 3
1 1 1
Theory
2 2 2 C &
3 3 3 Interpretation
S1 1 Fundamentals,
D Prediction and
2 2 2 Review
1.
A
2.
1.
B - 2.
3.
I.
C 2.
3.
1.
- 2.
Description of mode, place and range of occurrence.
Primary phenomenological observation.
Full scale tests a) Variables treated
Bench rig tests b) Measured quantities
Pertinent material tests -- -- c) Factual relations
Mechanics analysis a) Idealizations of conditions treated
Materials analysis b) Nature of solution or variables
Synthesis (correlation of facts) interrelated
Results, fundamentals, predictable features
Critical review
Pitting
VIII. APPENDIX B. REVIEW OF RESEARCH ACCORDING TO CLASSIFICATION OUTLINE
A review of research conducted under
the three modes of failure selected earlier is
given below according to the classification
outline. The review of the literature is by no
means comprehensive but the references
selected are representative of research in
the particular phase. However, research in
certain phases is sparse or nonexistent.
These areas are indicated.
A. PITTING
Description of Mode I-A-l: Eventually
the contacting surfaces of rolling elements
experience a failure that is characterized by
a flaking of metal fragments from the surface,
leaving pits of such size as to interfere with
normal operation.
The mode represents the end of useful
life of all ball and roller bearings in the
normal range of load and temperature, and
for that reason it has been the basis of life
and capacity rating. The same name has
been extended to a similar class of failures
found on gear teeth, valve tappets, etc.
This review is limited primarily to English
language publications. A review of contact
fatigue treating many European references is
provided by Schreiber and Ulsenheimer (61)
Primary Phenomenological Observa-
tions I-A-2 : In his experiments at the turn
of the century, Stribeck(62) noticed this type
of failure but did not consider it of funda-
mental importance in determing ball bearing
capacity. This mode of failure was noted
earlier by Crandell and Marston(63) in the
study of the endurance of bridge rollers to
repeated rolling. Load and cycles to failure
are tabulated for 36 tests although no attempt
is made to establish a life formula on the
basis of the limited data. It remained for
Palmgren some thirty years later(64) to pro-
duce an empirical endurance formula, based
on pitting failure, relating life in bearing
revolutions inversely to the cube of the load.
Since that time, many full scale tests have
been conducted for purposes of load rating,
the kinship of the phenomenon to fatigue being
manifest further in the dispersion in life
exhibited in typical bearings. Perhaps the
most far-reaching determination of dynamic
capacity based on ordinary fatigue pitting, as
it came to be called, is the work of Lundberg
and Palmgren reported in 1947,9)
A case study of many gear failures in
service is given by Hoshino. The failure
is classified into two groups according to
appearance: small pitch line pitting (position
of pure rolling) and flaking of other regions
of the tooth face experiencing approach and
recess sliding action. The directional flow of
metal in the subsurface region of relative
sliding and the orientation of the "oyster shell"
flake indicates the influence of sliding on the
nature of the failure.
Primary design data and service con-
ditions, including numerous microphotographs
of pitting failure sites in ball and roller bear-
ings and gears for aircraft service, are sum-
marized by Davies and Day. ( The authors
demonstrate that on the basis of a constant
design life for all examples, except the gear,
the stress frequency is inversely proportional
to the ninth power of the stress. Observation
of a "white phase" structural change in highly
loaded zones and sometimes in association
with subsurface fatigue cracks are reported.
"Spalling" is a failure mode of some con-
cern in steel mill work rolls and backup rolls.
A description of these failures and an analysis
of roll deflections, pressure distribution, and
critical subsurface stresses is given by
Keller.(67) Here "pits" of 20-square-inch
surface area are not uncommon in large (two-
foot diameter or more) rolls.
From full scale tests conducted on rear
axle assemblies, Almen (68)presents S-N
plots of pitting fatigue failures for through
hardened and carburized gears. Based on
nominal maximum contact stress at the pitch
line, the fatigue strength is noticeably less
than for roller bearings of comparable steel.
Full Scale Tests I-B-l: In addition to
testing for basis of load rating and relative
material evaluation, some research of basic
importance has been conducted with full scale
bearings. Jones 69)undertook an extensive
investigation of the metallographic trans-
formations induced in the subsurface of
specially designed ball bearing races at a
nominal contact stress of 800, 000 psi for
room temperature conditions, and in standard
races for loads closer to design values. The
material was a fully hardened 52100 steel.
Interruption at various running times, to-
gether with metallographic observation, re-
vealed subsurface transformation identified
as troostite and slip lines. These regions
intensified with running time. Hardness
surveys indicated the material near the sur-
face to be at a hardness equivalent to Rockwell
C 63 or C 64 while in the center of these trans-
formed and fatigued zones a few thousandths
of an inch below the surface the hardness was
Rockwell C 59. Sections were exhibited show-
ing fatigue pits that corresponded to the depth
of these transformed zones.
Endurance tests of groups of deep
groove ball bearings with inner races (failed
component) made of steel from various air and
vacuum steel making processes were analyzed
by Morrison et al. They report that aver-
age L10 life for all vacuum melted steels is
1. 7 times the corresponding average life for
all air melt steels and that additional vacuum
remelting continues the improvement to some
extent. Data from bearing groups showing
significant differences in trace elements are
examined and evidence of a significant nega-
tive correlation between aluminum, copper
and vanadium is cited.
A number of investigations have been
conducted on full scale bearings to determine
the effect of lubricants or lubricant properties
on pitting. From a series of tests with typical
gear box aircraft roller bearings and lubricants
of various viscosity and composition,
Otterbein concluded that, in general,
lubricants with low bulk viscosity have a detri-
mental effect on pitting life. He established a
linear relation between viscosity and an index
of bearing life. However, other results with
bench rig apparatus have indicated that bulk
viscosity alone is not a sufficient index and
there has been more attention directed toward
pressure viscosity and other lubricant
properties.
In an effort to study the effect of local
contact temperature without significantly
changing lubricant properties, Sternlicht
et al. (72)conducted ball bearing tests at two
outer race temperatures (1000 F and 1250 F)
with two lubricants of the same bulk viscosity
at the respective test temperatures. The
mean life of the lower temperature test
series was eight times longer than the high
temperature series. Taken by themselves,
however, these tests do not establish maxi-
mum contact zone temperature as a criterion
for fatigue.
The nature of rolling bearing lubrica-
tion and its possible influence on pitting fail-
ure has been investigated by attempts to mea-
sure lubricant film thickness in an angular
contact ball bearing while running under load.
A technique involving dielectric oilfilm break-
down voltage measurements was employed by
Sibley etal. (73)to indicate significant lubricant
films under various operating conditions of
speed, load, and temperature for a number of
lubricants. It is difficult to obtain more than
qualitative results from their research at
this time. Assuming a pitch circle of 85 mm
and osculation of ball and inner race equal
0. 96, the thrust loads reported would pro-
duce nominal maximum contact stresses on
the inner race of 185, 000 psi and 266, 000 psi,
respectively, under static conditions. By
way of illustration of results, the maximum
dielectric breakdown voltage was reached at
a shaft speed of 8, 000 rpm for the higher
load (1500 with Mil-L-7808 lubricant), having
increased steadily from a shaft speed of
2, 000 rpm. While boundary lubrication is
undoubtedly important, there is indication
that contact stress is modified significantly
in transmission through hydrodynamic films
of some nature, particularly in light load and
high speed applications.
Bench Rig Test I-B-2: A great many
experiments with bench rig or simulated
tester have been conducted since the early
work of Way(74) and Buckingham. (75) These
have increased both in number and diversity
of design in recent years. The purpose of
this research has been primarily to deter-
mine the influence of material and lubricant
variables on pitting life. These variables
may be grouped as follows:
1. Material types; metallurgical, surface
finish and other material variables.
2. Elevated temperature.
3. Lubricant and lubrication variables.
4. Relative sliding of surfaces.
5. Rolling element hardness compatibility
or pairing, and geometry or configura-
tion effects.
Perhaps the greatest utility of the bench
rig is in selecting or screening materials and
their corresponding treatments. Carter(76)
has reported Weibull lines determined from
pitting tests in the Macks spin rig for ten
promising high temperature steels and the
standard 52100 steel. Apart from ranking of
air melt steels as to B10 and B50 life at a
reported nominal maximum contact stress of
725, 000 psi, he found evidence that vacuum
melting improved fatigue life for AISI M-1
tool steel, although no correlation of rolling
contact fatigue life with cleanliness was ob-
tained.
Baughman(77)has given a statistical
analysis of the effect of various combinations
of material variables on M-50 steel pitting
life in the General Electric rolling contact
rig. Although certain optimum values of
significant variables like surface finish and
grain size are determined, not enough tests
were performed at each combination (four
tests) to justify such elaborate statistical
analysis. Zaretsky and Anderson (42) have
examined the influence of hardness on pitting
life of four steels that range in hardness
from Rockwell C 55 to C 68, finding that the
B10 life increased continuously with increasing
hardness in all cases. It should be noted
that only the B10 life was taken as a charac-
teristic of life. Some inversions occurred at
the B50 life.
Bear and Butler examined 52100 and
AISI M-l tool steels metallographically after
testing in the Macks spin rig first at room
temperature, then at 2000 and 2500 F. They
found that metallurgical structure was stable
at room temperature but was unstable at 2000
F or 2500 F, the decomposition product being
troostite. Fiber direction, inclusions, and
chemical segregation were said to contribute
to fatigue failures.
Various metallurgical factors, particu-
larly nonmetallic inclusions, are studied in
contact fatigue specimens from different steel
making processes by Johnson et al. (78) (79)
The rig resembles a simple thrust bearing and
maximum contact pressures of 563, 000 psi are
developed. The most detrimental inclusions
are the brittle types like alumina and silicates
even in small size ranges, especially if
occurring in "cloud" distributions as observed
in some vacuum induction-melted steels.
Non-uniformity of carbide particles and re-
tained austenite, associated with chromium
segregation, are also held to be detrimental.
It is interesting to note that some general
correlation (ranking) between rolling contact
and rotary bending fatigue tests is established
(79)
in the earlier paper.
Tests performed in the G. E. rolling con-
tact rig 80)on thirty-two heats of M-50 steels
(hardness ranged from Rockwell C 62-C 65) at
500°F may be summarized as regards tem-
perature effect on life by averaging all
reported B10 and B50 lives. These values
are respectively 2. 54 x 10 stress cycles and
7.90 x 106 stress cycles. However, if room
temperature data reported on essentially the
same material is examined it is noticed
that the change in life is relatively small. In
fact the difference in life is less than the
variations due to metallurgical variables at
room temperature. In the same rig under
identical test conditions (700, 000 maximum
contact stress, Mil-L-7808 libricant), except
room temperature, the B and B lives for
the best test were 3.4 x 10 and 10 x 10
respectively (the steel hardness was Rockwell
C 62. 2). Other tests at the same hardness
and higher (Rockwell C 64) gave lives shorter
than the reported 5000F results. At elevated
temperatures the contact stress may have
been reduced by the occurrence of plastic
deformation if the mechanical load cell was
not monitored and adjusted frequently.
As regards lubricant research, apart
from ranking of various oils and synthetic
fluids, the most significant findings are that
bulk lubricant viscosity does not seem to be a
dominant factor in determining pitting life.
However, the test results of several lubri-
cants (including mineral oils) show that with
the same type of lubricant, pitting life is
increased with increasing viscosity. Barwell
and Scott(81) state that compressibility and
variation of viscosity with pressure may be
most significant. Barwell(22) states that
transverse cracks originate in the surface,
being influenced by the chemical action of the
lubricant. He suggests that these cracks
associate with subsurface cracks developed
in metallurgically altered subsurface regions
to produce pits. Loads were usually exceed-
ingly high in these four-ball tests. Nominal
maximum contact stress as high as 1, 000, 000
psi was employed.
Anderson and Zaretsky(82) give results
of tests from the fatigue spin rig with lubri-
cants of five different base stock lubricants
that have approximately the same atmospheric
pressure viscosity. A relation of L10 life
with pressure coefficient of viscosity is indi-
cated. On the other hand Rounds(83) tends to
discount the significance of pressure viscosity
coefficient or viscosity index when lubricants
of different chemical class are involved, and
emphasizes other lubricant properties like
molecular shape, reactivity or polarity and
antiwear characteristics. From short time
tests with the four-ball rig at extreme contact
pressures he reports longest lives with the
diesters and polyphenyl ethers and shortest
lives with the fatty acids and halogenated
hydrocarbons. In general, ring structures
or molecular branching usually gave longer
lives, but reactive or polar groups were
detrimental. The pertinence of such short
time tests to the study of time-dependent
phenomena and actual service failure is ques-
(84)
tioned in a discussion by Dolan. Rounds
suggests that the basis for observed correla-
tion between full scale and four-ball tests may
be that the resulting high surface temperatures
accelerate the chemical reactions.
An extensive summary of research at
the National Engineering Laboratory with the
four-ball tester is provided by Scott. In
this context the results of tests with various
lubricant additives and water contamination are
of interest. Increased concentration of the
more active extreme pressure additives re-
sulted in life reduction. Even small amounts
of dissolved water in lubricants are harmful
except in conjunction with stainless steel balls.
This effect of water contamination in mineral
oil lubricants is attributed to hydrogen em-
brittlement.
The influence of viscosity of mineral oil
lubricant on pitting fatigue limit and fatigue
life of lower strength steels (96, 000 psi tensile
strength) is explored by Martin and Cameron(86)
with the same test rig used by Niemann and
Kraupner in their cumulative deformation
experiments. Although it is possible to rank
pitting fatigue limit with viscosity (fatigue
limit varies from 50, 000 psi maximum contact
pressure to 120, 000 psi for variation of
viscosity from 3 to 530 cS) some other prop-
erty is apparently significant in determining
life or the slope of the S-N curve.
Buckingham and Talbourdet(87) give S-N
curves for pitting of steel rollers with various
percentages of sliding showing the decrease in
life with introduction of sliding. Nishihara
and Kobayaski(88) have found that 20 per cent
relative sliding produces the greatest tend-
ency to cause pits in contacting cylinder. At
higher values "the growth of pits can scarcely
be observed. " For relatively low strength
steels (80, 000 psi tensile strength) Dawson(13)
reports pitting life nearly independent of
sliding for smaller slide/sweep ratios (sur-
face velocity of disc - surface velocity of
mating disc/surface velocity of disc) in the
range -0. 0005 to -0. 04. He also reports
that pitting is confined to the disc with the
smaller peripheral velocity. Many of Way's
experiments, indicating surface origin of
pitting, are reproduced.
Further experiments (89)at two slide/
sweep ratios (-0. 046 and -0. 0046) and with
two test conditions giving calculated film
thicknesses at first less than, and secondly
much greater than, the total surface rough-
ness of the discs, indicate that pitting life is
almost independent of load above the fatigue
limit. It should be noted that rather small
pits a few thousandths of an inch across are
the basis of these curves. The authors con-
trast these "inert" pits to the "continuously"
propagating type observed at much higher
stresses. The possibility arises that these
pits may be "premature pin hole" pits. (90)
By increasing the contact angle, and
thereby increasing the spin velocity or rela-
tive sliding, in the NASA five-ball tester at
constant maximum contact pressure,Zaretsky
(Z8)
et al. were able to study the resulting
decrease in fatigue life for hard ball bearing
steels. As an example, a change in relative
spin velocity of a factor of two corresponds
to a change in life of a factor of two. Although
the maximum contact pressures are high
(800, 000 psi), the results are in keeping with
results at lower stress in full scale ball bear-
ing tests but are not predicted by standard
capacity formulae. Surface temperature at the
higest contact angle (400) is 600 F higher than
at the lowest (100).
Investigations of the influence of com-
bined roll and spin on fatigue strength of hard
(91)
steels are reviewed by Wernitz. These
are tests conducted by Maass in Germany that
indicate a relatively rapid drop in strength
with slip until a value of about 4% slip is
reached, whereupon a threshold fatigue limit
appears to be established.
In the pitting fatigue research on 52100
steel conducted at the U. S. Naval Engineering
Experiment Station (33)three geometries of
toroidal test specimens were used. The pro-
file radii were 0. 250 in. , 0. 383 in. , and
0. 500 in. The toroid diameter was 1. 500 in.
and the cylindrical driving load roller was
1. 562 in. in diameter. Tests were conducted
at nearly the same maximum nominal contact
stress for all geometries; however, mean
life increased with decrease in profile radius.
Other tests performed at the Station were in
liquid sodium environments.
The effect of profile radius change or
"crown" of toroids of large radius (near cylin-
der) on fatigue strength of case hardened roller
bearing steels is studied in detail by McKelvey
(92)
and Moyer. In this case, except for the
smallest profile radii (100 inch) used, a com-
plete eliptical contact area is not formed, as
in Greenert's tests. A correlation of S-N
data on the basis of maximum contact stress
is possible, taking into account stress concen-
tration at the end of roller contact. End
failures are shown to predominate except for
the smallest profile radii, where the maxi-
mum stress occurs at the roller center.
Relative hardness or material combina-
tion is shown to be an important factor in
(90)
pitting tests by Chesters. The test rig
consisted of a roller driving a pair of speci-
men discs similar to the Niemann and
Kraupner rig. With the exception of case
hardened rollers, the effect of increasing the
hardness or tensile strength of the driving
roller was to reduce the fatigue resistance of
the driven disc materials. For a given roller
material the fatigue strength of the disc ma-
terial increases directly with tensile strength.
Four different steels were used in the driven
discs and six steels were used in the rollers,
representing materials often employed in
pinion/wheel gear combinations.
Tests with hard (958 D. P. H. ) and soft
(800 D. P. H. ) balls in the four-ball rig (1
rotating -- 3 rolling balls) yielded results
that indicate a marked influence of intrinsic
hardness and relative position (rotating or
rolling). Milne and Nally (93)also associated
the physical appearance of the pitting failure
with kinematic conditions of the rolling ele-
ments (whether rotating ball in four-ball rig,
or cone in cone and three-ball rig). Greater
tangential drag or traction on the balls is said
to cause initiation of surface cracks whereas
the cone failures are subsurface.
Rigs other than those involving rolling
contact have been employed to obtain some
insight into the pitting phenomenon. Burton
et al. (94)have developed a rig for repeated
contact loading of 52100 steel balls and have
reported possible early indications of fatigue
in the observed discontinuity in the specimen
temperature vs. time plots. In addition to
surface damage they report some indications
of subsurface cracks, which Kennedy (7)was
unable to observe in a similar test arrange-
ment.
Pertinent Material Tests I-B-3: Ma-
terial tests, apart from the rolling contact
situation, have been conducted on both steels
and lubricants primarily for the purposes of
screening or ranking for use in rolling ele-
ment designs, evaluating heat treatment
methods and determining the effect of environ-
mental variables on purified material variables
thought significant to the pitting fatigue strength.
Styri(32)made an attempt to develop a
simple fatigue test which would allow study of
the effect of a single factor pertinent to con-
tact fatigue. The stress-life relations from
various fatigue tests on hard bearing 52100
steel were compared to the usual contact
stress-life relation determined from bearing
tests. Torsion fatigue tests gave the closest
correlation. He also noted the wide life scat-
ter of the specimens of vacuum-melted steel
which is relatively free of inclusions.
Because of the scarcity of reliable data
for extremely hard steels such as used in
rolling element bearings, Sachs, Sell, and
Brown(2)undertook an investigation to deter-
mine the relation of tension, compression,
and fatigue properties to steel hardness. The
materials were conventional melt 52100 steel
and three tool steels (Halmo, M-1 and MV-1)
ranging in hardness from Rockwell C 50 to
C 65. Tension and compression test results
indicate an optimum in hardness at about
Rockwell C 60. However, the fatigue strength
determined in rotating beam tests did not de-
crease at high hardnesses. Induction vacuum
melted 52100 steel appeared to have a higher
fatigue strength at 108 cycles than the con-
ventional melt. Fatigue strength (at 108
cycles) values ranked the materials (all
Rockwell C 62) in this order:
1. Halmo (130, 000 psi)
2. M-1 alloy and vacuum melt 52100
(120, 000 psi)
3. MV-1 alloy (110, 000 psi)
4. Conventional 52100 (100,000 psi)
In an appendix the results of full scale
bearing tests with three of the same materials
used in the investigation were presented in
Weibull plots. The same ranking was given
here as in the rotating beam tests.
Hersey and Hopkins (95)have given an
extensive historical review and source of
condensed technical data on the effect of pres-
sure and temperature on lubricant properties
such as pressure coefficient (defined as the
rate of change of viscosity with respect to the
pressure at constant temperature divided by
viscosity). Experiments with a falling body
viscometer illustrate the variation of pres-
sure coefficient with pressure. Non-Newtonian
characteristics such as plasticity and time
effects are said to be conspicuous in solidified
oils at low shear rates. Non-Newtonian be-
havior such as the approach to limiting shear
stress with high rates of shear has been
observed more recently.
Sibley etal.(73)have reviewed the experi-
ments of Charron, which involved impacts
against a small piston which then forced the
lubricant through a capilliary tube. The mea-
sured viscosities were significantly smaller
than would be expected from the magnitude of
the pressures. Further experiments indicated
a decreasing viscosity presumably caused
by lubricant heating due to shearing action.
A falling needle viscometer with exter-
nal electromagnetic indication suitable for
pressures up to 20, 300 psi was used by
Boelhouwer and Toneman 96)in investigating
the variation of pressure coefficient with
pressure. The established pressure-viscosity
relation is only satisfied for one of the fluids
examined.
The fundamentals of boundary lubrication
are treated by Bowden and Tabor. (97) Many
experiments concerning the lubricating prop-
erties of films of various chemical composi-
tion and the associated surfaces are reviewed.
Mechanics Analysis I-C-1: Mechanics
analysis of idealization pertinent to the rolling
contact pitting failure have been directed to-
ward determination of stress induced in con-
tacting elastic bodies due to normal and com-
bined traction, lubricant films, or thermal
gradients caused by friction. The assumption
made is that stress is a quantity significant
to rational life determination. Some work
has included inertial or dynamic effects in
the elastic bodies (moving concentrated load),
but in general all analysis is elasto-static.
Recent analysis has extended into the plastic
range of strain caused by rolling contact.
The classical solution for the pressure
distribution between elastic bodies in contact
(98)
is due to Hertz. A solution for the sub-
surface stresses on a vertical line through the
center of the contact area in a half-space
loaded with a Hertzian ellipsoidal pressure
(99)
was given by Thomas and Hoersch. The
maximum shear stress was discovered to
occur below the surface. Lundberg and
Palmgren and Fessler and Ollerton(100)
solved for the subsurface stresses on orthogo-
nal planes, including the axis of the contact
ellipse. Here the maximum orthogonal shear
stress occurring in the subsurface is accentu-
(101)
ated. For the two-dimensional case Poritsky
and Smith and Liu 102)have solved for the
stresses due to a combination of Hertzian
normal and tangential surface stresses. The
maximum shearing stress moves toward the
surface with increased tangential traction and
the maximum tensile stress is at the "trailing"
edge of the contact area. A solution for the
compliance (displacement divided by load)
for combined surface traction in the more
(103)
general elliptic contact is given by Mindlin.
The combined influence of tangential friction
and twist or spin on surface displacement,
strains, and ball motion has been analyzed by
Hetenyi and McDonald (104)and Johnson. (105)
The effect of curvature and finite size in the
Hertz theory for cylindrical contact is studied
by Loo(106)and discussed by Lubkin. Moyer
and Neifert (107)present a method of calculat-
ing stress concentration at the edge of a
finite length roller in contact with an elastic
half space.
A start on the plastic problem has been
made. For the elastic contact pressure dis-
tribution, and the assumption that subsurface
total strain cycle is essentially the elastic
strain cycle, Merwin and Johnson (45)integrate
the Prandtl-Reuss plasticity relation to obtain
the plastic stress cycle and residual stresses.
The influence of tangential force is also
(20)
treated.
A solution for the viscoelastic problem
has been proposed by Lee and Radok. (108)
The problem treated is that of a rigid sphere
and viscoelastic half space (linear viscoelastic
operators in the time variable replace the
elastic constants) and the cases of prescribed
uniform indentation rate and load history are
illustrated. There is a marked deviation from
the Hertzian distribution, the distribution
being flattened at indentation times equal to
the relaxation time of the material. The
solutions approach each other for very short
indentation times.
Morland (109)solves the viscoelastic
problem for the rigid cylinder rolling with
constant velocity over a linear viscoelastic
half space. The basic integral equations are
solved by series expansion. A numerical
example is presented for the case of contact
time equal to the retardation time of the stand
ard viscoelastic body.
It is apparent from the analysis of Cole
and Huth (110)that only for speeds near the
sonic velocity of the material is the stress
distribution due to a moving concentrated
load significantly different from the corres-
ponding static solution.
Based on the elasticity analysis of Smith
and Liu and the thermal analysis for flash
temperature of H. Blok, Kelley(111)has
solved for the stress distribution due to the
sliding action of bodies in contact. The com-
bination of thermal and contact stresses are
particularly severe on the disc with negative
sliding (slower surface speed).
Photoelastic tests have supported ana-
lytical results in this phase. A dynamic
photoelastic technique was applied by Sternlicht
et al. (72)to compare the stress distribution
under dry and lubricated rolling. Theyconclude:
"For all practical purposes - the dry and
lubricated pressure patterns are the same,
and are not affected by load or speed. "
Greatest emphasis is placed on film tempera-
ture on the basis of their elasto-hydrodynamic
analysis of lubricants with different pressure
and temperature characteristics.
Since considerable attention has recently
been directed to the role of the lubricant in
pitting failure, analyses of the mechanics of
rolling contact lubrication and related experi-
ments have multiplied in number over the
last few years. Some of these will now be
reviewed.
Sibley et al. (73)have given a short but
excellent survey of fluid mechanics and
elasto-hydrodynamics analyses of film thick-
ness and pressure distribution pertinent to
gear and roller bearing lubrication. This has
irncluded Martin's isoviscous hydro-dynamic
theory of 1916, its modifications for variable
viscosity by Gatcombe, Bell, Cameron, etc. ,
and the modification for elastically deformable
surfaces by Lewicki and Dorr. Recentanaly-
ses such as Poritsky's have attempted to take
into account the effect of pressure on both
viscosity and surface deformations. One
incentive for these modifications is the good
performance of roller bearings and gears at
load values above those predicted as critical
by the theories. The subsurface stress dis-
tributions corresponding to pressure distri-
butinns predicted from such two-dimensional
elasto-hydrodynamic analyses are presented
by Dowson et al. in terms of three defining
parameters for speed, load, and pressure
viscosity coefficient. In general they find
that the pressure distribution and resulting
subsurface stresses are near-Hertzian except
at very high speed. The authors consider rise
in temperature and consequent viscosity re-
duction to be insignificant in pure rolling,
although deviation from linearity between
shear stress and shear rate (non-Newtonian
behavior) is thought to be significant in some
lubricants.
Direct measurement of lubricant film
thickness by use of columnated X-ray beams
has indicated thinner films than predicted by
the elasto-hydrodynamic theories mentioned.
Sibley et al. (113)attribute this to non-Newtonian
behavior which leads to a 'spreading" of the
pressure distribution and thinner films. In
these tests maximum contact pressures in the
range of 100, 000 psi to 180, 000 psi are em-
ployed, and film thicknesses in the range of
four to fifty micro-inches are reported. An
empirical correlation of film thickness data
on the basis of dimensionless forms obtained
from examination of Grubin's theory is pre-
sented graphically. In a discussion to the
paper Sternlicht criticizes the neglect of
temperature rise in the film, especially since
the authors show that film thickness is sensi-
tive to temperature and viscosity. Sibley in
turn criticizes Sternlicht's analysis (which
includes the energy equation) for assuming
constant temperature across the oil and no
conduction into the steel. He argues, as does
Crook, (114)that steel temperature or inlet oil
temperature controls film thickness.
The effects of temperature rise and high
rates of shear in the lubricant at the contact
area on lubricant viscosity have not been con-
sidered extensively in analyses. However,
some data of fundamental significance with
regard to rate of shear and ambient tempera-
ture was obtained by Smith 115)with a crossed
axis ball and cylinder machine. In general a
limiting shear stress or coefficient of friction
is approached at high rates of shear for sever-
al fluids. Results are interpreted in terms of
a shear plane model of flow where frictional
force represents the shearing of a thick plas-
tic film of lubricant. Smith remarks that his
findings that the coefficient of friction de-
creases with temperature is contrary to the
data of Misharin for somewhat similar experi-
mental conditions. He suggests that a com-
plete lubricant film was not established in
Misharin's tests.
Other possible idealizations are the
time-dependent Maxwell body, and even plas-
ticity idealizations for semi-solid films.
Many rheological behaviors and effects are
discussed by Smith. (116) He illustrates the
effect of high rates of shear on two lubricants.
In the crossed-axis ball and cylinder rig at a
maximum contact pressure of 155, 000 psi an
ester-based lubricant behaves as a liquid of
great viscosity below film shear stresses of
1,450 psi, but at higher rates of shear the
ester behaves as a plastic solid. A silicone
fluid, on the other hand, begins to lose vis-
cosity and undergo plastic shearing at much
lower shear stresses.
A theory of roller lubrication with the
Bingham rheological idealization is proposed
by Sasaki et al. At high speeds the dis-
tinction between this plastic-viscous model
and the Newtonian ideal diminishes. Criti-
cisms of certain assumptions regarding the
boundary conditions of the solution and
accuracy of some calculations are made in a
discussion by Osterle. The form of funda-
mental equations for a visco-elastic body is
given by Burton. He demonstrates that
the influence of shear elasticity onthe pressure
distribution is one of magnitude and not of kind.
With a Maxwell fluid the elastic component be-
comes increasingly significant at higher rates
of loading.
It is interesting to note that the concept
of full hydrodynamic lubrication with slip-free
rolling is challenged by Blok. (119) Evidence
of complete fluid films concurrent with what
they consider to be negligible slipping is pre-
sented in a discussion by Sibley and Whitney.
Blok concedes that slip may be small but can-
not vanish on the basis of equilibrium considera-
tions.
Material Analysis I-C-2: Material
analyses have been performed on both the solid
rolling element parent metal, the lubricant,
and their associated films. Apart from the
many fundamental studies of the atomic mecha-
nism of fatigue such as those reviewed by
Mott, (120)analysis of the effect of state of
stress on fatigue has been most applicable to
rolling contact conditions. In addition, re-
search into the origin and nature of lubricant
and film properties has contributed to the sum
of knowledge pertinent to this failure mode.
In connection with a discussion of fatigue in
rolling contact, it is appropriate to mention
that observations of subsurface-originated
cracks have been made in basic fatigue studies,
even in pure metals, despite the concentration
on intrusion-extrusion models of fatigue.
Bendler and Wood(121) report subsurface
fatigue fissures in OFHC copper specimens
subjected to steady tension and alternating
torsion.
From analysis of data compiled from
many fatigue tests under conditions of com-
bined stress, Stulen and Cummings (30)have
presented a fatigue criterion expressed in
equation form that modifies the critical
damaging effect of the maximum reversed
shear stress for the inhibiting effect on propa-
gation of the compressive stress normal to
the critical shear plane. Findley (31)has also
proposed a criterion based on essentially the
same grounds.
As regards variation of pressure vis-
cosity in many polymer lubricants, Boelhouwer
and Toneman (96)suggest that it is related to
the "flexibility" of the molecular chains.
Burton (122)gives a brief analysis of the
molecular mechanism of deformation of simple
liquids involving the kinetics of slip and "acti-
vation energy of hole formation. " The mecha-
nisms would explain viscosity increase with
pressure and viscosity decrease with tempera-
ture. It is stated that these two effects are
quite important and may account for more than
a thousandfold variation in viscosity in typical
rolling contact situations.
The nature of surface films composed of
parent metal and environmental contaminants,
like lubricants and their compounds that form
under pressure and temperature extremes,
has been the subject of wide research. Some
of this research is, of course, pertinent to
the comprehensive understanding of rolling
contact failure. Some indication of the nature
of this research is given by Deryaguin etaL,(123)
Davy and Edwards, (124)and Loeser and Twiss.125)
The importance of these films, acting
both as lubricants modifying the friction and
stress systems and their chemical effect on
macroscopic crack initiation and propagation,
emphasizes the need for more pertinent
research.
Synthesis I-C-3: Synthesis of the re-
sults of both mechanics and material analyses
in application to the mode of failure in actual
rolling contact situations is still in the primi-
tive stage, but some representative attempts
have been made.
From an analysis of the effect of rolling
on the range of subsurface shear stress in
cylinders, Radzimovsky took the orthogonal
(occurring on planes parallel and normal to
the surface) shear stress as the most critical.
The strength condition is based on a modified
Huber-Mises condition where the alternating
stresses are reduced to statical form accord-
ing to Goodman's linear relation. To account
for the effect of ranges of compressive stress,
the endurance limit due to completely com-
pressive stress is used in the strength condi-
tion rather than the endurance limit caused by
reversed stress. In this way an expression is
developed for the "surface endurance limit,"
CT = k a
max y
where a is the uniaxial yield stress of the
material and k depends on static material
properties as well as the endurance limit in
torsion. The equation compares favorably
with results of Way and Buckingham for rela-
tively soft steel rollers.
Moyar (4)attempted to explain the effect
of rolling element configuration on contact
fatigue strength for hard steels and showed a
rational numerical relation between torsion
fatigue strength and contact fatigue strength.
The stress analysis was based on the solu-
tion of Fessler and Ollerton. The strength
criterion modified the critical reversed shear
stress in a manner similar to the criterion of
Stulen and Cummings (30)for combined stress.
Account was also taken of the volume of ma-
terial subject to critical stress (statistical
size effect) after the method of Lundberg and
(9)
Palmgren. An application of this approach
was made recently by Ollerton and Morey(127)
to relatively soft rail steels with water lubri-
cation.
The possibility of applying low cycle fa-
tigue data to rolling contact situations involv-
ing appreciable cyclic plastic flow is opened
by the analysis of Johnson and Jefferis. (20)
They illustrate a possible relation between
total strain accumulation and fatigue life.
Although a synthesis of facts to predict
the effect of lubricant on pitting has not been
conducted, application of theory to actual
rolling element mechanisms has been made
related to lubricant film thickness, pressure,
and friction torque. Hopkins and St. John,(128)
under the simplest assumptions of constant
viscosity, density, and negligible surface
deformation, made an analysis of the motion
of bearing elements in a typical roller bearing,
accounting for both hydrodynamic conditions
and metal-to-metal friction. V. Hackewitz(129)
applied hydrodynamic theory based on the
work of Dorr and Kapitsa to a particular
roller bearing as an example of obtaining
"accurate numerical results" for peak lubri-
cant pressure and film thickness. Tempera-
ture and pressure viscosity effects are
excluded. However, in general, oil film
thickness between roller and raceways varies
more with speed than with load. The dynamic
peak pressure in the rotating bearing may be
considerably less than the peak pressure (or
maximum Hertz stress) in the stationary
bearing.
Because the statistical method is a
powerful tool for the correlation of experi-
mental data, some research dealing parti-
cularly with this method as it applies to the
fatigue phenomenon will be briefly reviewed
in this section.
Weibull (130)has reviewed some of his
early work in this area and has examined
several methods for efficiently estimating
the three parameters (location, scale, and
shape) of his widely used distribution function.
He makes an illustrative comparison of full
scale bearing data and data from the Mack's
spin rig. Assuming the location parameter
to be identically zero, he demonstrates the
close coincidence of the remaining parameters
for both sets of data, although material load-
ing and test conditions are not presented in his
paper. He recognizes that the scale parameter
depends on load and hence the close correlation
is described as "accidental. " However, he
implies that shape parameter (an indication
of "scatter") is not load dependent.
An extensive summary of ball bearing
data was standardized by Tallian (131) for ex-
amination of the closeness of fit to the usual
Weibull distribution with zero lower bound.
He examines the deviations from theory at
the lower and higher ends of the cumulative
failure probability region of usual interest
which result in longer lives than expected.
Special regard is given to the postulates for
the existence of the distribution and to the
basic sequence of events in fatigue. The
apparent minimum standardized life is associ-
ated with the short but finite macro-crack
propagation time period. Tallian proposes
that the variable excess experimental life
observed up to about 6% failure probability is
due to the variation of the period of this phase
of the fatigue process with the concurrent
development of "beneficial structural changes."
The deviations from theory at the higher failure
probabilities is attributed to the insufficient
number of independent potential failure sites
in the longer-lived specimens.
Results I-D-1: As a result of the intensi-
fied research effort, particularly with the
rolling contact bench rig, much data has
accumulated that allows relative ranking of
materials and lubricants with regard to in-
fluence on pitting life. The observed relation
between life and load is perhaps the only direct
correlation between the bench rig and the
bearing, with the exception of the correlation
of rolling contact rig data and the results of
full scale bearing tests in a particular circum-
stance made by Morrow. (35)
Certain aspects of this research may be
summarized. It appears that increase in lubri-
cant pressure viscosity and material hardness
increase pitting life. For normal conditions
approaching pure rolling, the origin of the
fatigue crack is subsurface in the transformed
or damaged region, subject to maximum range
of reversed shear stress due to the rolling
action. The particular location may depend on
local stress raisers or variation of material
properties with depth. With increasing tan-
gential traction due to relative sliding or
mutual surface interaction due to severe
environmental conditions, the origin of the
fatigue crack may be in the surface, thus being
directly influenced by the chemical attack of
surface-lubricant films. Both of these mecha-
nisms may occur simultaneously. The influ-
ence of lubricant film as chemical agent, heat
conductor, and pressure and shear transmitter
are all of significance in this respect.
Several complementary avenues of re-
search are indicated. Among these would be
a photoelastic study of the influence of inclu-
sion shape, hardness, and location on distri-
bution of contact stress. To make this fea-
sible experimentally, the area of contact
would have to be exaggerated. Also subsur-
face stress calculation can and should be made
from boundary stress conditions other than
Hertzian (for example see Dowson et al. (112)).
Experimentally, the nature of chemical attack
or stress corrosion on pitting failure in vari-
ous controlled kinematic and thermal condi-
tions is not well explored.
Critical Review I-D-2: Because of the
emphasis on speedy collection of masses of
data with the diverse bench rig types, there
is no comprehensive kinematic or dynamic
information as to the conditions of tests. This
prohibits anything like a standardization of
rolling contact fatigue as a property, limiting
comparison of data from various rigs or com-
plete rational extension to various design con-
figurations. While it is recognized that ideali-
zations for the purposes of mechanics analysis
must by definition differ from all the com-
plexities of the real situation, results of use-
ful engineering significance can be obtained
if a more realistic knowledge of even the
mechanical forces involved is established
experimentally.
B. CUMULATIVE DEFORMATION
Description of Mode II-A-1: The term
"cumulative deformation" is used to designate
failures of rolling element bearings caused by
decrease in diameter of the rolling elements
and grooving of raceways which accumulates
with stress cycles, leading to excessive "run-
out. " The result may be unacceptable vibra-
tion, noise, friction, torque, or the precipi-
tation of a more catastrophic failure. It is
anticipated as a dominant mode of failure in
elevated temperature applications and may be
observed as a phenomenon and perhaps a
basis of failure at room temperatures in cer-
tain high-precision situations such as machine
tool applications.
Primary Phenomenological Observations
II-A-2: From their tests with bridge rollers
in the 1890's, Crandall and Marston(63)"
reported observation of "a flow of metal at the
ends of the rollers and around the edge of the
contact portion of the plate" after repeated
rolling. Such cumulative deformation is
particularly pronounced in heavily loaded
rail heads in association with "shelly" fail-
ures. (132) Glaeser etal. 133)have examined
the failure modes of many air frame oscil-
lating roller bearings tested in a tempera-
ture range between 3000F and 6000F, and
conclude that the predominate failure mode
is due to plastic deformation. In these oscil-
lating bearings not only is the spherical inner
race blunted, but the "hourglass" rollers are
reduced in diameter. This reduction becomes
so pronounced that in some cases the roller
separator is allowed to ride on the inner race.
Also, there is evidence of "mounding" at
periodic locations along the inner race caused
by the oscillating action. This leads to more
catastrophic failure due to stress concentra-
tion and localized material work hardening.
The results of subsurface micro-hardness
surveys are reported on three steels (52100,
M-2, and type 440C stainless) showing evi-
dence of considerable subsurface plastic flow.
From these tests load-life diagrams are re-
ported.
In another exploratory high-temperature
rolling contact investigation, unlubricated
needle bearings (0.051-in. roller diameter,
0. 540-in. inner race diameter) with various
aging treatments were tested at 12000F. Con-
stant loads were employed which resulted in
nominal contact stresses of 234, 000, 281, 000,
and 456, 000 psi. The approach of inner and
outer race under load as a function of running
time was recorded, 0.001 in. being defined as
failure. This phenomenon was termed "wear,"
but no determination of the amount of abrasive
wear was made, although evidence of plastic
deformation of the rollers and race impressions
were cited. It should be noted that rolling
contact rig tests at the same company(44)
have led to the statement that the mode of
failure in high-temperature bearings is pre-
dominately due to plastic deformation.
Full Scale Tests II-B-1: Apart from a
few exploratory tests such as those mentioned
above, no controlled experiments with full
scale bearings to investigate a range of vari-
ables such as temperature, speed, bearing
design, and material variables have been
found in the literature.
Detailed observations of associated
phenomena -- subsurface microplastic defor-
mation and the development of residual
stress -- in full scale bearings have been
reported, however. Bush et al. (55)identified
from electron micrographs the "gray lines"
observed previously by Jones (69)as alterna-
ting fine slip. A threshold maximum contact
stress in the range of 422, 000 to 480, 000 psi
for the hardened SAE 52100 steel ball bear-
ings was indicated on the basis of the initiation
of an altered microstructure. As contact
stress is increased beyond this range, this
microstructure and the residual stress pat-
tern develop to a saturation value more
rapidly.
Bench Rig Tests II-B-2: Research
efforts utilizing rolling contact bench rigs
apparently have centered about two purposes.
The first type of research measures the
amount of plastic deformation after some
number of cycles as a method of ranking
materials and treatments, reminiscent of
identation tests performed for static load
capacity determination. The second recog-
nizes the cumulative nature of the phenomenon
and its significance as a possible failure mode
in itself. The first type is represented by the
work of Drutowski,39) Zaretsky and Anderson,(42)
and Akaoka; (135) the second by the research of
Niemann and Kraupner and Bamberger. (37)
Rolling cemented carbide balls between
steel plates with various percentages of re-
tained austenite, Drutowski measured groove
depth after 20 cycles. The nominal contact
stress ranged from 150, 000 psi to over
900, 000 psi. He determined that for low
stresses (below 500, 000 psi), steels with the
lowest amount of retained austenite exhibit
the smallest plastic deformation and highest
initiation stress for measurable plastic defor-
mation. At higher stress levels they were
ranked roughly by hardness.
The first measurable plastic deformation
depended on percentage of retained austenite,
but was in the range 160, 000 to 377, 000 psi
maximum contact stress.
Incidental to a series of fatigue pitting
tests on both the Macks spin rig and a five-
ball tester, Zaretsky made traces of ball
track profiles to determine the amount of
plastic deformation and wear as a function
of hardness for five steels. No explicit nu-
merical results are reported, but a relative
scale indicates continuous decrease in defor-
mation with increased hardness. Plastic
deformation was from 12 per cent to 90 per
cent greater than wear for various steels and
hardnesses at 10,000 stress cycles and
750, 000 psi in the spin rig. The data reported
from the five-ball rig on M-1 steel after 1. 7 x
10 cycles at 800, 000 psi indicates wear 40
per cent to 60 per cent greater than plastic
deformation. Emphasis is placed on hardness,
closely controlled at the highest value con-
sistent with other metallurgical factors,
despite the inverse effect at lower stress
reported by Drutowski and the lower opera-
ting stress in many practical bearings.
Akaoka (135)replaced the inner race of
standard radial ball bearings with hardened
steel cylinders equivalent to SAE 52100 with
various forging ratios and melt compositions
and tested them in a dynamic rotating unbal-
ance machine at high stress levels. From
dimensions and load given in the paper a
theoretical maximum contact stress of
1,210,000 psi may be calculated. He reports
plastic deformation (grooving) at the end of
fatigue life, and correlates long lives with
those specimens exhibiting greatest plastic
deformation. This correlation is not a simple
consequence of accumulation of plastic defor-
mation with longer running time, he argues,
because almost all the deformation occurs
very early in life -- as might be expected at
these high stresses from the findings of Bush
et al. Although the four materials had nearly
equivalent chemical composition, macro and
microstructure, and mechanical properties,
the materials with the higher forging ratio
had the greater plastic deformation (groove
depth and width) and longer lives. Many sub-
jects are treated in this paper, and some
important questions raised.
The rig used by Niemann and Kraupner(27)
consisted of a test bar with various profile
radii, driven and loaded between cylindrical
rollers. The variables were hardness, pro-
file radius, and load (maximum contact stress
range from 31, 800 psi to 596, 000 psi), and
the quantities measured were radial deforma-
tion, track width, and deformed profile radius.
A major part of the deformation occurred be-
tween 2 million and 30 million stress cycles.
The "initiation stress" was determined by
extension of the relation between deformed
radius and nominal contact stress to the value
of original profile radius. This stress value
was expressed empirically as a function of
hardness. Specific plastic deformation deter-
mined in static tests was much smaller than
that determined on the basis of permanent
approach at 30 x 10 cycles. For hardness
equivalent to Rockwell C 61 the initiation
stress is 164, 000 psi.
(37)
Bamberger explored the effect of
speed, temperature, and previous cold work on
cumulative deformation of a cobalt base alloy
using a rolling contact rig consisting of a
3/8-in. -diameter test bar driven between two
toroid load rollers. Radial deformation was
given as a function of running time for a nomi-
nal contact stress of 500, 000 psi at 12000 F
and 25, 000 stress cycles per min. The rate
of accumulation of radial deformation is rough-
ly constant under these conditions, the most
severely cold worked steel having the smallest
rate. It was stated that the process is akin to
creep. However, in high temperature creep
tests there is usually an increase in creep
rate for material with the most extensive grain
boundaries such as highly cold worked materials.
Although the process may resemble creep,
important differences in the nature of the plastic
slip process are indicated. Life (N, number
of cycles) to some prescribed deformation was
in accord with the equation N = C a m, where
max
C is a constant that depends on amount of
deformation defined, ( is the stress level,
max
and m is an empirical constant with a value
of 0. 7 for these conditions.
In an investigation treated more com-
pletely elsewhere in this publication, Moyar
and Sinclair (136)studied the cumulative blunt-
ing and grooving of annealed brass rollers at
room temperature.
Other manifestations of cumulative
plastic deformation have been studied by
(54)
Hamilton. (54) The forward (into the leading
contact edge) shear displacement of a sub-
surface layer of material due to rolling (first
observed in mild steel discs by Crook in 1957)
is examined in cold worked copper by
Hamilton. The shear displacement per cycle
is nearly constant for fixed test conditions
and is not dependent on the presence of a
lubricant or strongly influenced by speed.
Corrugation of the surface, another interest-
ing mode of deformation, was observed at
slow speeds and high loads in the driven disc
specimens.
Pertinent Material Tests II-B-3: Ma-
terial tests have been conducted with a view
toward selecting materials suitable for high
temperature rolling bearing applications.
These have included hot hardness, dimensional
stability, and yield strength determination.
Dimensional stability tests involve measure-
ments of expansion (retained austenite trans -
formation) or contraction (martensite tem-
pering) of alloy steels held for specified
periods of time at elevated temperatures.
For example, a typical heat treated 52100
steel contracts 0. 001030 in. /in/at 8000F for
1000 hours while M-50 steel expands 0. 000027
in. under the same conditions. (137)
These determinations of material prop-
erties are convenient and probably serve a
purpose in qualitative ranking, but regarding
more explicit design information related to
failure due to cumulative deformation they
offer no help. Indeed, even pure research
into the cyclic plastic strain behavior of ma-
terials subject to simple repeated loads and
range of temperature is relatively unexplored.
Under conditions of controlled strain Morrow
and Sinclair (138)have examined the effect of
cyclic accumulation of microplastic strain on
the relaxation of the mean stress for various
steel hardnesses. Coffin (139)has observed
the remarkable effect of a small torque on
cumulative permanent twist of a double-gage-
length specimen subject to repeated uniaxial
strain. Some of the complexity of the phe-
nomenon associated with macro strain gradi-
ents was avoided in tests of thin wall annealed
copper tubing by Moyar and Sinclair. (50)
Inherent cycle-dependent accumulation of
plastic strain is observed under steady ten-
sion and alternating torsion. The effect of
periods of cyclic stress on subsequent recov-
ery and creep rate in lead has been explored
by Kennedy. Periods of cyclic loading at
low stress values increase the creep rate
while periods at high stress decelerate sub-
sequent creep. Also in his tests the total
plastic strain accumulated in a creep test
under fluctuating loads was much greater
than in a creep test conducted at the highest
load for the same time.
The fact that either hardening or soft-
ening can occur under cyclic conditions is
reflected by cycle-dependent changes in a host
of bulk material properties including indenta-
tion hardness and the mechanical hystersis
(141, 142, 143, 144, 145)
loop. Every metal
has a range of potential strength or hardness
which can be achieved by cold working,
annealing, and so on. Metals initially on
the low end of this potential hardness spec-
trum cyclically harden: those on the high
(146, 147)
end soften 147) In each case, interme-
diate states are approached, representing a
stable condition for the particular metal and
imposed cycling conditions.
The initial cyclic rate of change in prop-
erties is greatly influenced by the cyclic
strain range but rapidly diminishes with re-
peated cycling -- the major changes occurring
in the first few per cent of the fatigue life.
Materials quickly adjust to a nearly stable
steady state condition which is reflected by a
constant terminal hardness (48)and a stable
(53, 149)
mechanical hysteresis loop. (53, 149) Tuler
and Morrow 48)have recently shown that the
stable intermediate state can also be achieved
by partially annealing cold worked OFHC
copper.
A number of researchers have attempted
to treat mechanistically the phenomenon of
cycle-dependent hardening and softening.
(141)
Notable among these is Thompson, who
uses transmission electron microscopy ob-
servations of dislocations and other lattice
defects (principally dislocation loops) to dis-
cuss the mechanism of fatigue hardening.
Others have invoked annihilation of disloca-
tions of opposite sign 145)and cyclic-induced
polygonization (150)as the mechanism of fa-
tigue softening. The subject has also been
treated as a generalized Bauschinger
effect(147, 47, 151)involving the introduction,
relaxation, and intensification of internal
stresses in localized weak zones.
Mechanics Analysis II-C-1: Contribu-
tion from the theory of plasticity as regards
the effect of rolling on cumulative deformation
is limited. There have been combined analyti-
cal and experimental investigations into the
two-dimensional strip rolling process that
were essentially for purposes of power cal-
culations. However, some idea of the nature
of plastic strain may be obtained from
Reference 152. Actually, analytical plasticity
research into contact problems is still con-
cerned with treating the static indentation
problem in a more realistic way. The com-
plex stress and strain cycle induced during
the passage of a rolling load, the change in
surface geometry in the general three-
dimensional case, and a non-Hertzian pres-
sure distribution all combine to defy a com-
prehensive solution.
A contribution to the two-dimensional
problem of the rolling of a rigid cylinder on
an elastic-perfectly plastic half space was
(56)
made by Johnson. From an examination
of the residual stress distribution he found
that the shakedown limit, irrespective of
yield criterion, occurs when the maximum
contact stress equals four times the yield
stress in pure shear. Below this limit entire-
ly elastic behavior will eventually be estab-
lished. The limit corresponds to a load 70%
greater than the load necessary to initiate
yielding.
However, in a particularly simple case
(16)
Eldredge has made an analysis that demon-
strates certain desirable features of an
analytical solution in this area. Therefore,
his solution will be presented in condensed
form by use of difference equation notation
(subscripts refer to stress cycle). The case
of a rigid ball grooving a perfectly plastic
plane is treated. Perfect plasticity is de-
fined to mean that the projected contact area
for a subsequent cycle is equal to the original
projected area. It should be noted that this
assumption provides a means of including
size effect since the "yield pressure" based
on projected area can be determined as a
property independent of ball size. It is
assumed that in forming the groove the con-
tact area occurs on the front half of the ball
(Figure 40). The original projected area is
bounded by a semicircle of diameter equal to
the track width (w ). The subsequent shape
of the area is determined empirically. This
gives rise to the relation
I w = 0.7 w2 (26)
n+1 n 1
Considering the transverse profile (Figure 16)
of the ball, the trigonometric relation between
the track width w (chord length) and track
n
depth u (chord height) any cycle, n, yields:
2
w = 4D u (27)
n n
Likewise, considering the longitudinal ball
profile (Figure 40c) the trigonometric relation
between the length of the contact area ( ) and
the increment in track depth is
12
2n+1
Au = u - u - (28)
n n+1 n 4D
From Equations 26, 27, and 28 the
fundamental cyclic difference equation for
track width as a function of number of cycles
is
4 2 2 4
w - w w w + 0. 49 wI =0 (29)
n n+1 n 1
Thus with w1 known as a function of ball size,
load and "yield pressure, " the track width at
any cycle may be determined.
A similar analysis for the cycle-depend-
ent blunting of a perfectly plastic sphere roll-
ing between rigid platens has been performed
(153)
by Reis. The corresponding cyclic dif-
ference equation for track width is
4 3 3 4
w 4 + 2w3 w - 2w w w -w
n+1 n+1 n n n+1 n
2
7r 4
- = 0
2 1
(30)
Materials Analysis II-C-2: In view of
the scant experimental data, formulation of
pertinent variables into mathematical equa-
tions for a range of conditions is not available.
That is, analysis of material behavior as
regards rate of accumulation of plastic strain
for ideal conditions has not been attempted.
However, an indication of this type of analy-
sis is given by Morrow and Sinclair in the
analysis of cycle-dependent stress relaxation
(138)
phenomenon.
An atomistic mechanism for the effects
of combined fatigue-creep stressing has been
(140)
proposed by Kennedy. The intersection
of screw dislocations to form a "jog" provides
a source of vacancies or interstitials when
moved back and forth through the crystal
lattice by repeated stress, these in turn
accelerating the climb process of dislocations
to facilitate further polygonization (recovery)
and creep.
Synthesis II-C-3: A synthesis of ma-
terial behavior with regard to the mechanics
of the rolling contact situation to provide a
rational basis for design or estimation of
service behavior is not available.
Results II-D-1: The result of this
research has been isolated design information
obtained directly in rolling contact tests on a
few particular materials or bearing designs.
Only limited indication of the mechanism re-
sponsible for failure mode has been given.
Critical Review II-D-2: Since interest
in high temperature service is relatively new,
there is even a lack of clear phenomenological
information from service failures. Contribu-
tions from wear and plastic deformation have
not been fully separated for any known range
of variables or conditions. Stability to cyclic
stress does not of necessity follow results of
static tests for the purposes of material rank-
ing. Purified material tests into the effect of
cyclic stress and temperature on cumulative
plastic deformation are lacking even for the
"ideal" laboratory materials, much less the
complex bearing steels. Combined stress
tests also need to be conducted. Although con-
tribution from the mechanics of solids is, of
course, limited, some means is needed of
extending minimum data to various rolling
element configurations and of interrelating
geometric variables.
C. EXCESSIVE ROLLING RESISTANCE
Description of Mode III-A-1: Usually,
minimum rolling resistance is a desirable
feature of all rolling contact bearings. In the
case of instrument ball bearings such as those
used in gyro-guidance systems, resistance to
rolling is a major concern. Failure occurs
when the magnitude or variation of rolling
resistance produces a friction torque that
interferes in an unpredictable way with instru-
ment response. Attention is directed to the
inherent features of rolling resistance, not to
other catastrophic failure types that may be
manifest in extreme friction torque.
Primary Phenomenological Observations
III-A-2: Serious scientific interest in the na-
ture of rolling friction as a phenomenon dates
back at least as far as 1785 to the experiments
of Coulomb. Tabor 154)summarizes the
experiments of Morin and Dupuit in France
around 1840, and the interesting controversy
between the two over the dependence of rolling
friction on wheel radius. In 1876 Reynolds(155)
attributed the source of rolling friction to
interfacial slip between elastic surfaces due
to differential deformation. He deduced this
from his observation that a metal cylinder
moved forward a distance less than its circum-
ference when rolled over a rubber surface.
The experiments of Crandall(63) in 1890, which
involved measurement of the force necessary
to pull a plate from between loaded rollers,
substantially verified Dupuit's finding that
friction force is inversely proportional to the
square root of roller radius. Since that time,
bearing manufacturers with more practical
concern have established a so-called "friction
coefficient" for a wide variety of particular
rolling bearings for convenience of engineering
design and selection. The coefficient of fric-
tion does vary appreciably with load, however.
The effect of rotational speed is negligible for
most design purposes. Although there is con-
flict in values, empirical results of this type
of research are summarized in many of the
recent texts treating rolling contact bear-
ings. (156, 157,11, 158) The emphasis placed
on instrument bearing friction torque has given
rise to an effort to standardize methods and
instruments to study friction phenomena,(159)
including significant changes in rolling resist-
ance with running time and test conditions.
Full Scale Tests III-B-1: Many full
scale tests have been performed on a variety
of bearings under various conditions of speed,
load, and lubricant to determine friction
torque or coefficient of friction, but few
experiments of basic significance relating to
contact geometry or rolling element dynamics
have been undertaken.
The research of Palmgren and Snare(
seems to confirm that lubrication of rolling
bearings is hydrodynamic at high speed and
light load, turning into boundary lubrication
at lower speed and heavier load. In an exten-
sive series of tests with various types of ball
and roller bearings, friction torque due to
hydrodynamic lubrication shear, differential
slip (Heathcote) friction and internal hysteresis
were separated. For unloaded bearings, re-
sistance moment increased with viscosity and
speed of rotation to the two-thirds power.
This is in keeping with hydrodynamic theory.
In thrust ball bearings, friction due both to
differential sliding and hysteresis depends on
load to the four-thirds power; hence their
effect on friction torque may be separated
since sliding resistance is dependent on de-
gree of osculation. Resistance moment in
heavily loaded bearings is essentially inde-
pendent of speed and lubricant as opposed to
those operating in the region of hydrodynamic
lubrication. In most bearings, there is a com-
bination of both types. It is assumed in the
analysis of hysteresis losses that the percent-
age of energy dissipated is independent of
stress, but for the higher loads this does not
appear to be well founded on the basis of other
material tests.
Coefficient of friction for a range of
straight mineral oils, synthetic oils, and
mineral oil-additive blends lubricating thrust
ball bearings have been measured by Rounds(161)
for ball velocity speeds up to 600 ft. /min. ;
maximum contact pressures of 300,000 psi,
400, 000 psi, and 500, 000 psi; and oil temp-
eratures of 1000 F, 200 F, and 3000 F. In
general the coefficient of friction decreases
as oil temperature, ball velocity, or load
increases and is independent of oil viscosity
in contrast to hydrodynamic theory, indicating
boundary lubrication and the importance of
surface films.
Bench Rig Tests III-B-2: Bench rig
tests of fundamental importance in rolling
contact have been reported which explored
the effect of surface roughness, established
the role of internal hysteresis losses, and
examined the influence of relative surface
velocity on the point of application and angle
of inclination of resultant contact force for
very light loads. Barwell (162)has reviewed
the results of a number of bench rig tests.
An interesting theory of rolling friction based
on the attractive and repulsive atomic forces
of surfaces in intimate contact is proposed by
Tomlinson and is shown to be consistent with
his measurements of coefficient of friction for
hard steel rollers oscillating on steel plates.
Barwell points out limitations of the theory,
especially the complications introduced by
oxide films.
Tabor(163)performed a series of critical
laboratory experiments demonstrating the
significance of energy losses in rolling due to
internal material hysteresis as opposed to
interfacial slip mechanisms. For rolling of
a ball in a deep groove, differential slip was
a dominant source and could be influenced by
the presence of a lubricant, but not when
rolling on a flat surface. These tests involved
rolling a steel ball on either flat or pre-
grooved rubber. Also, identical copper balls
were rolled back and forth over each other in
a special pendulum apparatus and the decre-
ment in amplitude measured. The symmetry
conditions ruled out interfacial slip, estab-
lishing material hysteresis as the major
source of rolling resistance.
With a rolling friction apparatus de-
signed to measure the force necessary to roll
a ball between flat plates, Drutowski(164)
determined the effect of contact stress and
material on average friction force. There
was, however, a great variation in friction
force, with instantaneous values many times
the average. Roughness of finish or orienta-
tion of lay have only a small effect on average
friction force, but a more marked effect on
peak values. The appreciable number of
friction peaks for even the smoothest steel
surfaces (1 micro-inch) seemed to indicate
that the peaks were due to material inhomo-
geniety as well as geometrical irregularities.
This was confirmed when friction force his-
tory for rolling of steel and synthetic sapphire
were compared. The friction peaks for steel
were still marked, but they were essentially
unmeasurable for the homogeneous sapphire.
Rolling force increased with the 1. 2 power of
load for the initial elastic range. It increased
with the 2.4 power for loads above that caus-
ing observable plastic deformation. For 52100
steel hardened to Rockwell C60, calculation
based on reported data shows maximum con-
tact stress to be approximately 350, 000 psi
at this transition point. Drutowski (39)also
reports the reduction in average rolling fric-
tion force with decrease in retained austenite
in a 52100 steel.
For elastomers, like Neoprene, and
thermoplastics, like Plexiglas, Flom(165)
has measured dynamic mechanical losses
(rebound of a steel ball from the specimen
surface) and correlated them with measure-
ments of rolling friction (three steel balls
rolling on a plastic surface) at various speeds
and temperatures.
The rolling friction of a sapphire ball on
single crystals of copper was shown to be
strongly dependent on crystallographic direc-
tion of rolling in the experiments of Dyer(166)
with the rig developed by Drutowski. The
anisotropic hardening and shape of the ball
tracks are studied and discussed in terms of
slip systems activated and dislocation inter-
actions.
The rolling resistance of copper discs
loaded in the plastic range to produce a for-
ward shear strain accumulation has been
(54)
measured by Hamilton. Although the
sensitivity of the apparatus was limited,
measurable rolling resistance was observed
before the onset of forward flow. It increased
rapidly with load beyond this point.
Dunk and Hall 167)have investigated the
dynamics of rolling and sliding contact con-
ditions for light loads (nominal maximum con-
tact stress = 4, 200 psi) and hydrodynamic
lubrication. A single "coefficient of friction"
is not sufficient to describe the situation fully,
since the resultant force at contact is only
given completely by magnitude, direction
(inclination angle), and point of application
(offset). The apparatus consisted of a pair
of disks separately mounted on low-friction
bearings and independently motor driven
through calibrated spiral springs. The slid-
ing velocity was of primary importance in
determining inclination, while the sum of the
sweep velocities determined the offset. At
higher loads (nominal maximum contact
pressure = 88, 500 psi) in the elasto-
hydrodynamic regime, measurements of sur-
face traction, film thickness (capacitance
technique) and surface temperature at various
speeds and loads are made by Crook 168) in
two ingenious test rigs. The first rig involves
a freely running roller between three driving
peripheral rollers of equal diameter. Known
amounts of braking torque may be applied to
the center roller to control sliding speed.
The portion of the surface traction due to
rolling (determined from extrapolation of
data to pure rolling) is independent of load
and proportional to film thickness in this
regime. As rolling speed is increased the
effect of pressure and temperature on apparent
viscosity diminish. This is attributed to
viscoelastic effects since the oil is under
stress for so brief a period of time. In the
second two-roller rig relative sliding speeds
are considerably increased so that an intrinsic
effect of rate of shear strain upon effective
viscosity (non-Newtonian behavior) occurs.
Pertinent Material Tests III-B-3: While
research on phenomena associated with this
failure mode (material hysteresis and surface
friction phenomena) is extensive, its pertinence
has not been explored. To review even
representative work on the effect of frequency,
stress level, temperature, and material on
mechanical hysteresis or damping capacity
would lead this discussion far afield. A
summary of research (169)shows that for high
stress the specific damping capacity is highly
dependent on stress level, while for low
stresses specific damping capacity may be
independent of amplitudes. Variations in
energy absorbed per cycle stems from grad-
ual material transformation or change in
frequency among many other variables. The
significance of stress gradients and provision
for modifying specific damping energy are
treated by Podnieks and Lazan. For
moderate to high stress levels the change in
energy absorbed per cycle, or damping, is a
manifestation of the cycle dependent hardening
or softening discussed above (II-B-3).
The work of Bowden and Leben (171)is
representative of the basic material and lubri-
cant tests to establish laws of friction and ex-
plore the intimate surface asperity and lubri-
cant film interactions.
Mechanics Analysis III-C-1: Mechanics
analysis has been undertaken which is perti-
nent to the hysteresis phenomenon and hydro-
dynamic film behavior.
Tabor 172)and Drutowski 164)have cal-
culated the elastic strain energy involved in
pressing a ball into a plane. The maximum
elastic energy is proportional to the load
raised to four-thirds power. Drutowski(173,174)
has also defined the volume of significantly
stressed material involved in the rolling of a
ball with regard to an empirically determined
value of strain energy density which bounds
this volume. It is experimentally verified
that the constant of proportionality between
rolling force and stressed volume is inde-
pendent of ball diameter.
In several calculations for dynamic
mechanical losses in rolling contact, such
as those of Flom, only direct compression-
release cycles of elements in the rolling
track are considered for simplicity. In a
discussion Tabor (165)emphasizes the distinc-
tion between energy dissipation under the
actual complex loading cycle involving re-
versal of shear distortion to that under such
simple cycles, concluding that actual losses
may be greater by a factor of three. Flom's
data supports this conclusion.
Rolling friction arising from microslip
phenomena due to elastic surface compliance
may be calculated from solutions such as
those of Johnson (175and dePater. (176) For
a ball rolling in a straight groove Johnson
compares coefficient of rolling resistance
(tangential force/normal force) calculated
according to theories associated with three
sources of rolling resistance: microslip,
Heathcote differential slip, and material
hysteresis. He demonstrates that for apar-
ticular case of close ball and groove conformity,
theoretical rolling resistance due to microslip
is larger than that due to hysteresis.
Calculations for coefficient of friction or
rolling resistance due to lubricant "pumping"
and viscous shear are illustrated by the hydro-
dynamic theory of Kapitsa. In this two-
dimensional laminar flow theory the shape of
the elastically deformed rollers at contact is
prescribed as parabolic.
The source of rolling resistance in
elasto-hydrodynamic lubrication arises from
the curvature forced upon the velocity profiles
by the convergence of flow at contact entry,
according to Crook. He calculates this
component of frictional traction for a Newtonian
fluid whose viscosity varies with pressure and
demonstrates that it is dependent only on film
thickness. The general agreement with ex-
periments indicates that in pure rolling the
temperature due to friction is too small to
significantly change viscosity.
Materials Analysis III-C-2: Mathemati-
cal and graphical description is available to
the relations among variables involved in
hysteresis energy dissipation in simple re-
versed stress states. Demer (178)has com-
piled a list of sources. The atomic mecha-
nisms responsible for these phenomena have
been analyzed and range from diffusion pro-
cesses of various activation energies at low
stress to thermoelastic coupling or simply
non-reversible heat flow processes induced
by cycles of mechanical work.
The work already cited concerning the
effect of pressure and temperature variables
on lubricant properties is pertinent in this
area.
Synthesis III-C-3: Synthesis of results
from the above phases of research in appli-
cation to rolling contact situations is incom-
plete. However, the possibilities of such
synthesis has been illustrated by Drutowski.(164)
On the basis of measured rolling friction and
elastic strain energy calculations he calculated
a specific damping capacity in rolling which
agreed roughly with values determined in
standard tests.
Results III-D-1: In summary, resistance
to rolling may arise from a number of
sources:
1. Fluid resistance offered to the rolling
elements due to the presence of lubri-
cant films.
2. Direct interaction of the rolling sur-
faces and sliding due to elastic defor-
mation (interfacial slip).
3. Hysteresis energy dissipation in the
rolling element material.
4. Effect of rolling in producing an elastic
mound ahead of a grooving element.
This phenomenon is mentioned by
Palmgren 156)and discussed more fully
by Dunk. (179) The resultant effect may
vary considerably with different direc-
tions of tangential force with respect to
rolling direction.
Contrary to the impression perhaps
created by widespread use of a constant co-
efficient of friction in bearing design and power
loss calculation, the variation of rolling
resistance with time as well as its magni-
tude poses a practical problem in instrument
and precision rolling bearings. Small gyro
bearings operate in the speed range of 10, 000
to 20, 000 rpm and may be subject to signifi-
cant loads due to preloading in order to estab-
lish isoelastic conditions in the instrument
assembly. Variation in friction torque, which
is in a range less than 70 dyne-centimenters
may cause serious "drift" in the instrument
even without catastrophic bearing failure.
This variation may come about because of
transformation with time or cycles of the
rolling element material, lubricant or their
films, or it may develop due to changing
operating conditions of speed, load, tempera-
ture, or intermittent periods of service.
Several sources of these phenomena have been
explored in standard laboratory tests: varia-
tion of film thickness with time, effect of a
range of variables on energy dissipation in
materials, etc. Also, various rolling contact
rigs have been used in determining the effect
on rolling resistance of a limited range of
material, speed, and load variables. There
have been some efforts to separate experi-
mentally the various sources of rolling resist-
ance. Instruments have been recently devel-
oped to accurately measure friction torque of
full scale instrument bearings.
Critical Review III-D-2: The greatest
single obstacle to analysis of the phenomena
and failure due to excessive or variable roll-
ing resistance, however, is still the lack of
experimental data in rolling contact in which
the complete force system and kinematic
variables are known and controlled. This has
been attempted only for very low loads.

IX. REFERENCES CITED
1. Arthur D. Little, Inc., "Yearly.Progress
Report on a Study of Contact Fatigue, "
Sponsored by the ASME Research Com-
mittee on Contact Fatigue of Rolling
Elements, 1 October 1961 - 1 October
1962.
2. G. Sachs, R. Sell, and W. F. Brown, Jr.,
"Tension, Compression, and Fatigue
Properties of Several Steels for Air-
craft Bearing Applications, "Proceed-
ings, American Society for Testing and
Materials, Vol. 59(1959), p. 635.
3. W. J. Greenert and Rawlings, "Basic
Information on the Bearing Properties
of Various Materials in Oil and Sodium
Potassium Alloy, " U. S. Naval Engineer-
ing Experiment Station Report 090014D,
April, 1956.
4. G. J. Moyar, "An Analysis of the Fatigue
Strength of Surfaces in Rolling Contact,"
M. S. Thesis, Department of Theoretical
and Applied Mechanics, University of
Illinois, Urbana, Illinois, 1958.
5. L. G. Johnson, "The Median Ranks of
Sample Values in their Population with
an Application to Certain Fatigue
Studies, " Industrial Mathematics, Vol.
2(1951), p. 1.
6. G. J. Moyar and J. Morrow, Discussion
of E. V. Zaretsky and W. J. Anderson,
"Relation Between Rolling-Contact
Fatigue Life and Mechanical Properties
for Several Aircraft Bearing Steels,"
Proceedings, American Society for
Testing and Materials, Vol 60 (1960),
p. 644.
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X. INDEX
Al uminum
bending fatigue strength of, 12
trace element in hearings, 45
Anisotropy, copper single crystal rolling, 66
Bauscninger effect related to cyclic hardening and soft-
ening, 62
Bauschinger natural elastic limit, 31
Bearings
ball, 5,20,23,44,57,60,65
instrument, 42,64,69
needle, 58
roller, 44,53,56,58,65
thrust ball, 23,65
Bench rig, 46-50
divergence of data from, 5
tests of rolling resistance, 65-67
types: cone and three ball, 50; five ball, 49,59; four
ball, 48; G.E.rolling contact, 5, 47; geared roller,
9; Macks spin, 5, 47, 48; repeated contact, 50; U.S.
Naval Eng. Exp. Sta. (Annapolis), 5; Univ. of Ill., 7
Blunting
analysis of perfectly plastic, 63
brass toroids, 27
inner race in "hourglass" roller bearings, 58
profile in hard steel, 27,30
Brass, tensile and tension-torsion tests, 32
Brass toroid rolling element, 7,26,27
Bridge rollers, 44, 58
Carbide cemented balls, 28, 59
Carbide particles in bearing steels, 47
Case hardened steel, 4,7,50
Characteristic life, influence of contact stress and size
on, 11
Chemical effects: in fatigue, 10; lubricant on surface
crack initiation, 48, 55
Chromium segregation in bearing steels, 47
Coefficient of friction in rolling, 64-66
Coefficient of pressure viscosity, 48,51,53,65,66
Cold worked metals
effect of cyclic loading on, 62
grooving of, 28
rate of cumulative deformation in rolling contact, 34,60
Complex loading
cycle in rolling contact, 25,31,39
energy losses under, 68
tension-torsion tests, 32,55
Conformity influence on rolling resistance, 61
Conformity of ball and groove, 27
Contact
angle, effect on pitting life, 49
area, 5,12,15,28,63
ellipse: eccentricity, 7,15; orientation, 5; width
increase, 22,63
stress reduction due to plastic deformation, 13, 28
Copper
balls, rolling resistance of, 66
blister on rolling track in, 10
one-directional loading of, 32
plastic grooving of, 28,66
single crystal, 66
stable cyclic state in OFHC, 62
tension-torsion cyclic tests of, 55,61
trace element in bearing steels, 45
Correlation equation, 22, 24
Correlation of data
bench rig and bearing failures, 4,20-24,48,56,57
elastic limit for tension, compression, and rolling
contact, 29
on basis of maximum contact stress, 50
pitting and standard material tests, 4,10,17,47,49,50,51
pitting life for various rolling element geometries. 19
statistical method, 56
Crack
due to repeated contact, 9
initiation, 9,10
longitudinal subsurface, 9
propagation, 10
radial subsurface in steel rolls, 10
subsurface, effect on stress distribution, 12
surface vs. subsurface, 50,55,57
Creep
high temperature, in rolling contact, 60
subsurface tangential, due to rolling, 33
under fluctuating loads, 61,63
Crown of rollers, effect on pitting strength, 50
Cumulative plastic deformation in rolling
analysis summary, 37
possible sources of, 31
literature review summary, 36
Cycle-dependent
hardening and softening, 62
mechanism, 62
reduction of deformation resistance in rolling contact,32
related to rolling resistance, 67
softening of hard steels, 32
stress relaxation, 63
Cyclic
difference equation for track width, 63
material properties, 31
stress-strain behavior, 32, 61
Cylinders, pitting strength of, 15,17,18
Cylinders, residual stresses in, 10,34,35
Damping capacity, relation to rolling friction, 68
Damping capacity, dependence on stress level, 67
Dielectric oil film measurements, 46
Dimensional instability: metallurgical, 25; tests in
steels, 61
Dimensionless forms for grooving of hard steel, 28
Dimensionless forms from Grubin's lubrication theory, 53
Dislocation interactions in fatigue, 62
Dislocation interactions in rolling, 32,66
Dynamic
capacity of bearings, 44
information, need for in bench rig tests, 58
peak pressure in rotating bearing, 56
photoelastic tests, 57
Elasticity, crack stress model, 12
Elasticity, solution for contact stresses, 7, 52
Elastohydrodynamic
analysis of lubricated rolling contact, 53
source of rolling resistance, 68
Elastomers, 66
Electron micrographs, 59
Endurance formula for ball bearings, 44
Endurance limit, surface, 55
Endurance tests of deep groove ball bearings, 45
Energy
dissipation associated with offset of normal force in
rolling, 33,65
elastic, in rolling and identation, 67
equation in lubrication theory, 54
losses (hysteresis) in rolling contact, 37,65
Failure, definition of modes in rolling contact, 41
Failure, noise level at, 7
Fatigue
creep mechanism, 63
criteria, 16,18,55
low-cycle data, 56
mechanism of, 10,18,54
process, associated with statistical parameters, 57
Fatigue strength
effect of specimen size on, 11
relation of bending and torsion, 17
Flaking, incipient, 10
Fluid mechanics survey, pertinent to bearing lubrication,53
Forging ratio influence on plastic deformation and pitting
life, 60
Friction
coefficient in rolling bearings, 64
force, influence on surface distortion in grooving, 34
laws of, 67
torque in bearings: analysis, 56; due excessive defor-
mation, 58; due lubricant deterioration, 42;
measurement, 65
Gears
comparison of pitting strength to roller bearings, 45
failures in service, 44
pitting tests of materials used in, 50
surface crack initiation in, 10
tests of through hardened and carburized, 45
General Electric rolling contact rig, 5
Grain size, 20
Grooving
analysis of perfectly plastic, 63
of ball bearing races, 60
profile in hard steel, 30
trigonometric relations for, 27
Hardness
hot, 61
influence on pitting, 47
relation to plastic deformation in rolling contact, 28,
59
relation to tensile, compressive and fatigue properties,
51
relative, of roller pairs in pitting tests, 50
surveys in ball bearing races, 45
Hertz contact pressure modification
due finite size bodies, 52
due hydrodynamic film, 33,53
due viscoelastic effects, 52
Hertz pressure distribution, 9,52
High temperature
acceleration of lubricant-surface reactions, 48
effect on cumulative plastic deformation, 25,33,34,
39,58
effect on pitting, 20,46,47
in contact zone as a pitting criterion, 46
steels, 47
Hydrodynami c
film, modification of Hertzian pressure and influence
on plastic deformation, 33,46
propagation of surface cracks, 10
theory applied to roller bearing lubrication, 56,65
Hydrogen embrittlement due to water in lubricants, 48
Hydrostatic compressive stress
influence on crack initiation, 10
influence on fatigue strength, 16
torsion fatigue tests under, 10,16
Hysteresis
atomic mechanisms of, 68
mechanical, 61
losses and rolling friction, 65
Inclusions
detrimental types, 47
in vacuum melted steel, 50
photoelastic study needed, 57
role in fatigue crack initiation, 10
Indentation
repeated, 29
tests for static load capacity, 59
types of imprints, 26
Index
of bearing life, 46
plastic deformation, 38
pressure viscosity, 48
research classification, 41
stress and strain, 28
Induction vacuum melting, 51
Kinematic conditions of rolling elements in bench rig, 50
Kinematic variables lacking in rolling contact tests, 69
Lead, cyclic recovery and creep rate in, 61
Lead, grooving of, 28
Lubricant
contamination by water, 48
effect of rate of shear on viscosity, 51
effect on contact stress distribution, 53
effect on pitting, 20,24,45,48,49
effect on rolling resistance, 68
environmental contaminants, 55
film temperature, 53
film thickness measurements, 46,53
types: ester-based, 54; extreme pressure additives,48;
mineral oil, 65; MIL-L-7808, 24,46,47; MIL-L-6081,24;
of various chemical classes or composition, 48,51;
polymer, 55; sodium potassium (NaK), 5,49; synthetic
oil, 65; water, 56;
Lubrication
boundary, 51,65
dimensionless forms in Grubin's theory, 53
theory of roller, 54
Martensite tempering, 61
Materials, screening of, in bench rig tests, 46,50
Mechanics analyses of pitting failure, 55-56
Metallographic examination of pitting failure, 45,47
Metallurgical structure
change with temperature, 47
transformation due rolling, 45,59
Microplastic deformation, 59
Microresidual stress,
role in crack initiation at inclusions, 10
Microslip, 68
Microstructure, 60; altered by rolling, 59
N.A.S.A. five ball tester, 49
Neoprene, dynamic mechanical losses in, 66
Noise due bearing imperfections, 41
Noise level at pitting failure, 7
Non-Newtonian effects in lubricants, 51,53
Normal stress influence factor, 17
Octahedral shear stress
for initiation of plastic deformation, 29
Orthogonal shear stress
dependence on contact geometry, 9,15,20
depth to maximum, 15
elastic solution for, 52
for asymmetrical pressure distribution, 33
role in cumulative plastic deformation, 32
Photoelastic tests, 53,57
Pitting failures, 4
definition of, 44
initiation and propagation of, 9
list of variables influencing, 20
mechanism of, 9
metallographic observation of, 9
S-N curves, 21
summary: of analysis, 37; of literature review, 36
Pitting life
adjustment for stress level, 5
correlation with plastic deformation, 60
dependence on stress level, 5,9,21,22,45,50
effect of: rolling element geometry or configuration,
4,7,19; rolling element size, 4; sliding, 49;
temperature, 47
list of variables affecting, 20,46
scatter or dispersion in, 21,57
Pitting strength, affected by
relative hardness of roller pairs, 50
rolling element configuration and size, 4,5,9
Plastic shearing of lubricants, 54
Plastic nucleus, 29
Plastic deformation in rolling contact
contact stress reduction due to, 12,13,15,17,21,28
elastically restrained, 26
increase in track width, 21,22
index of, 38
initiation, 12, 28-29, 59,60
mechanism, 32
pronounced, 26
residual stresses due to, 10
rolling element geometry changes due to, 12
Plasticity
effects in solidified oils, 51
loading and yielding surface, 31
Prandtl-Reuss relations, 34
three-dimensional -solution for rolling, 36
two-dimensional solution, 34,36,62
Preload, 34
Probability: multiplicative law of, 15; of failure, 5,57;
of survival, 11,15; paper, 5
Profile radius
changes due: blunting, 60; grooving, 26
effect on pitting, 50
elastic relations based on deformed, 13,26
Rail heads, cumulative plastic deformation, 58
Rail steels, pitting tests of, 56
Rate of cumulative deformation in rolling, 31,33,39,60
Rate of strain accumulation, 63
Repeated contact: compared to rolling, 9; tests, 50
Repeated tensile tests, 31,32
Research classification, 41-43
Residual stress
circumferential, in steel rolls, 10
effect of preload on, 34
in three-dimensional contact, 10,35
influence on crack propagation, 10
intensification due cumulative plastic strain, 25,34,59
measurement of, 59
micro-, at inclusions, 10
tensile, in a cylindrical roller, 34
Retained austenite
effect on initiation of plastic deformation, 28,59
effect on pitting, 47
transformation of, 61
Reversal of shear stress, relation to: rolling contact
energy losses, 68; pitting and cumulative plastic
deformation, 32,37
Rheological behavior of lubricants in rolling and sliding,
54
Rheological models in lubrication theory: Bingham, 54;
Maxwell, 54
Rolling element configuration (geometry) effect
pitting, 4-7,9,19,37; plastic deformation, 28,31,36
Rolling resistance (friction)
comparison of theories, 68
due to lubricant "pumping", 68
experiments, 64
manifestation of cycle-dependent hardening or soften-
ing, 67
needed investigations, 40
of copper discs in plastic range, 66
summary of literature review, 36
theory based on atomic forces in surface, 65
Sapphire balls in rolling, 66
Shakedown applicability to cumulative plastic deformation
in rolling contact, 32
Shakedown limit for two dimensional rolling, 62
Size effect
in cumulative deformation, 36
on fatigue strength, 11,15,56
volume of significantly stressed material, 11,15,17,19,
21,67
Skew angle in rolling contact, 7,39
Skew effect on asymmetrical plastic deformation, 30
Sliding, influence on
formation of "oyster shell" flake on gears, 45; pitting
life, 49; surface initiated pitting, 10
Sliding, resistance losses in rolling, 65,67
Sliding, thermal stresses caused by, 53
Slip
differential, in deep grooves, 34,66,68
due to combined roll and spin, 49
effect on pitting strength, 49
free rolling, 54
in single crystal copper, 66
interfacial, in rolling, 66
lines in subsurface material, 45
Spalling of steel mill rolls, 45
Specific damping capacity, 68
Speed effects
in hydrodynamic lubrication, 46,67
in plastic deformation, 25,61
on elastic stress distribution, 52
on friction coefficient in rolling bearings, 64,65,67
on lubricant film thickness, 56,67
Spin velocity effect
on pitting life in five ball bench rig, 49
on pitting strength (with combined roll), 49
on surface strains and ball motion, 52
Statistical life variability, 5
distribution function for, in ball bearings, 57
in vacuum melted steels, 50
influence of specimen size on, 11
influence of stress level on, 11,39,57
Steel
air melt, 45
making process, effect on pitting, 47
mild low carbon, 12
rail, 56
stainless bearing balls, 48
types: 52100-5,14,15,17,45,47,51,60,61,66; 4620-7;
4340-7; Halmo-51; M-50-21,22,47,61; M-1 -24,44,51,
59; MV-1-5,14; 440 C stainless-58; M-2-58
Strain
accumulation: under cyclic loading, 34; related to
fatigue life, 56
cyclic plastic, 61
nature of distribution in rolling, 62
rate of accumulation, 63
residual shear, 34
uniaxial repeated, 61
Stress
at surface due subsurface crack, 10,12
complimentary normal, 16,17
concentration at roller ends, 50,52
critical in rolling contact, 7,16,17,18,45
cycles per bearing revolution, 23
essential shear, 16,18
maximum contact, 7,13,16,19
state of, 29
Stress distribution: due partial load on infinite strip,
12; effect of plastic flow on, 12
Subsurface
cracks, 9,10,45,50,55
distortion and plastic flow, 9,58,33,61
metallurgical transformation, 45
microhardness surveys, 58
Surface
asperity, 67
compliance, 68
corrugation, 61
cracks, 9,10,49
distortion due to rolling, 33
finish, 20,66
tensile stress, 9
Tangential traction or friction
effect on cumulative plastic deformation, 52
Thermal stress, 53
Thermoplastics, 66
Timken plastic deformation data, 13,15
Tin, 28
Toroid rollers, 5,7,16,18
Torsion fatigue data, 10,18,19,50; tests under hydrostatic
fluid pressure, 10,16,17
Trace elements, 45
Trigonometric relations
original and deformed dimensions for elastically
restrained- plastic deformation, 27
perfectly plastic grooving, 59
pronounced plastic deformation, 26
U.S.Naval Eng. Exp. Sta. (Annapolis)
specimens and materials, 14
toroid and cylinder bench rig tests, 4,18,19,22
track width data, 13
Vacuum melting, 50
Vanadium trace element, 45
Velocity profile in lubricant film, 68
Vibration: due "run-out", 58; from original imperfections,
41
Viscosity
effect on: coefficient of friction (rolling), 65;
pitting fatigue limit, 49; pitting life, 46,48
index, 48
variation with: pressure, 48,55; with temperature, 55,
67
Viscoelastic
body, fundamental equations for, 54
effects in lubricating oils, 67
indentation solution, 52
two-dimensional rolling solution, 52
Viscometer, 51
Water contamination in mineral oil lubricants, 48
Water lubrication of rail steels, 56
Wear, 25; contrasted to plastic deformation, 27,30,31,
58,59; effect of hardness on, 59
Weibull
analysis of material strength, 15
cumulative distribution function, 11
deviations from, 57
presentation of bearing pitting data, 11,23,47,51
probability paper, 5
slope, 11,19,22
"weakest link" concept, 11
Work hardening in copper by rolling, 28
X-ray measurement of film thickness, 53
Yield criterion, 62
Yield strength of rolling bodies, 26
Yield stress, 29
Yielding at high contact stress, 21