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Critical stresses in rolling contact fatigue
Wear, 71 (1981) 265 - 282 0 Elsevier Sequoia S.A., Lausanne
265 - Printed
in The Netherlands
CRITICAL STRESSES IN ROLLING CONTACT FATIGUE
N. G. POPINCEANU,
E. DIACONESCU
Machine Design Department, Jassy 6600 (Romania) (Received
July 23,198O;
and S. CREW
Jassy Polytechnic
in revised
institute,
form November
23 August Street No. 11,
24.1980)
Summary An analysis of the main hypotheses concern~g the critical stresses in rolling contact fatigue is presented. It is considered that none of these hypotheses correlates adequately with all the experimental aspects of the phenomenon of rolling contact fatigue. A new hypothesis is proposed; this hypothesis states that the equivalent stress is the critical stress in rolling contact fatigue. By considering the influence of the residual stresses it is shown how the optimum fatigue life in rolling contact can be achieved.
1. Introduction Rolling elements deteriorate mainly by rolling contact fatigue; this is caused both by the action of variable stresses during the rolling process and by the presence of stress concentrators which are due to material or working process defects. According to their nature, the defects can be localized both on the contact surface and within the material, ie. under the contact surface. Failure by rolling contact fatigue is now considered to be a process involving competition between the various types of stress concentrators [l - lo]. The models presented by Tallian et ~2. [3 - 81 exemplify this. In order to calculate both the loading capacity and the fatigue life of rolling contact elements, various theoretical equations have been reported. These equations were obtained using a certain variable stress as the critical stress for rolling contact fatigue, There are many reported experimental data which can confirm or conflict with the hypotheses concerning the critical stress in rolling contact fatigue [ 11 - 241. An analysis of the main hypotheses concerning the critical stress in rolling contact fatigue was carried out. This analysis indicated that none of the existing hypotheses correlates with all the experiments aspects of rolling contact fatigue. The equivalent critical stress hypothesis is proposed as a new hypothesis concerning the critical stress in rolling contact fatigue. The new hypothesis correlates with all experimental aspects of the phenomenon. It is shown how an increased life can be obtained for rolling contact elements.
266
2. The stress state in elastic rolling contact In the general case of elastic rolling contact loading, the stress tensor of the elastic half-space has nine stress components; three of these are the normal stresses u, , uy and u, and six are orthogonal tangential stresses. The orthogonal tangential components are equal with respect to the interchange of suffixes, so that the stress state is characterized by a maximum of six separate stresses, the three normal stresses and the three orthogonal tangential stresses rXY, ryr and 7,, (Fig. 1, positions 1 and 2). The variation in all these components on the contact surface and within the material of the rolling elements can be obtained using the standard equation [25 - 301. The sense and magnitude of the stresses on the surface of the contact ellipse are given in Figs, 1 and 2. The normal stresses reach the normal compression values uX,, uY, and uZO in the centre of the contact ellipse; the greatest of these is the stress uZO = uo, which is also known as the maximum hertzian contact stress (Fig. 1, position 9). On the y axis (Fig. 2(b)) the normal stresses u. and uX are compression stresses; at the ends of the y axis the stress uZ is zero while u, has non-zero values (Fig. 2(a)) at these posi-
Fig. 1. The stresses
at the elastic
contact.
267
0.020 0
(a)
'
' 0.2
I
' 1 0.L
1
I j itI a&
0.6
0.8
(b)
Fig. 2. The distribution of normal stresses on the surface of contact ellipses: (a) normal stresses on the contact surface (@ = b/a = 0.4): -, a, ; ---, a, ; . e =, u, ; (b) normal stresses at the ends of the axes of the contact ellipse [ 271: curve 1, at the ends of the semimajor axis; curve 2, at the ends of the semiminor axis.
tions
although it tends to zero outside the contact ellipse. The stress uY has traction values uYt at the ends of the y axis (Fig. Z(b)) and the stress a, has traction values at the ends of the x axis (Figs. 1 and 2). If the rolling process occurs along the y axis, the stresses uZ and u, will cause a pulsating loading with a frequency v equal to the number of passes per second of the rolling elements. Under the same conditions, the u, stress will produce an asymmetric alternating loading with a frequency of 1.5~. Among the orthogonal tangenti~ stresses, only the stress T,~ has nonzero values on the contact surface (Fig. 1, position 8); these values are much smaller than those of normal stresses. In the half-space, under the contact surface, the normal stresses u,, uy and u, are compression stresses; they have maxima on the z axis which are smaller than those (uxO, uY, and uZO) reached in the centre of the contact surface. The orthogonal tangential stresses T,~, 7yz and T,, reach their maxima under the contact surface; the greatest value r. is achieved at a depth z. by the ryt stress [ 201. During the rolling process along the y axis, the normal stresses ux, (rY and e, and the tangential stress rxY produce pulsating loadings with frequency v ; the tangential stresses 7,Y and 7YZ cause symmetrical alternating loadings with the same frequency. Correlation of both the mode of variation of the above stresses and the points where
1
268
these stresses reach their maxima with the distribution of stress concentrators is the concern of various hypotheses regarding the critical stress in rolling contact fatigue which are reported in the literature.
3. Present hypotheses
regarding the critical stresses in rolling contact
fatigue
3.1. The hypothesis of maximum hertzian stress u. The stress u. acting on the contact surface produces, in the case of rolling along the y axis, a pulsating loading of frequency II. This stress was chosen by Moyer and coworkers [ll, 12 ] as a critical stress for rolling contact fatigue. They recommended an equation for the fatigue life of rolling elements:
LIo = Ku,,-
6.66
(1)
3.2. The hypothesis of maximum tangential stresses r45 It is known from the theory of elasticity that in any point of an elastic space there are three tangential stresses given by the equations: 0.x -0, 745(XY)
= 2
oy 745fY.z)
=
745(x2)
= ____
-02
(2)
2
*x
-0.2
2
There are three different hypotheses which consider the maximum tial stresses as the critical stresses in rolling contact fatigue.
tangen-
3.2.1. The hypothesis of maximum maximorum tangential stress r450 In the case of rolling along the y axis, the maximum tangential stresses will produce pulsating loading of frequency V; from these stresses the stress 7450,rj has its greatest amplitude ?46D at a depth Z,sn. Way [14], Styri [ 151 and Reichard and coworkers [ 16,171 considered that the stress 745D is the critical stress in rolling contact. 3.2.2. The hypothesis of maximum tangential stress on the contact surface 745D6 The sense and the magnitude of the maximum tangential stresses on the contact surface are shown in Figs. 1 and 3; the index s is used for the stresses at the ends of the axes of the contact ellipse and the index 0 is used for the stresses in the centre of the contact ellipse 1271. In the case of rolling along the y axis, the stresses 745(xyf and 7451Xzj will generate ~ternating loading of frequency 1.5~ (Fig. 3(a)). Pineghin [lS]
269
0.2 oh
0.6
0.8
1
(b)
Fig. 3. The distribution of maximum tangential stresses on the contact surface: (a) details of the distribution when /3 = b/a = 0.4; (b) T&TO vs. /? for various stresses 745: curve 1, 746(xy)O;
curve 2,
745(yr)O;
curve 3,
745~0;
curve 4,
745~~
i Curve 5Y T45(xy)sbe
considered that the stress 745(xy) at the end of the minor axis of the contact ellipse is the critical stress in roling contact. Discussing some results obtained by Foord et al. [ 191, Cioclov chose the stress 745(XY)at the end of the major axis as the critical stress. When the rolling is performed along the y axis, this stress produces an alternating loading with a frequency u. 3.3. The hypothesis of maximum traction stress on the contact surface uyt The stress uy has an asymmetrical alternating variation on the contact surface; because of this, Moyer and Morrow [13] considered that its maximal value uyt, reached at the ends of the y axis, is the critical stress in rolling contact. 3.4. The hypothesis of maximum orthogonal tangential stress r. Lundberg and Palmgren [20] proposed that the maximal value r. of the orthogonal tangential stress ryr is the critical stress in rolling contact fatigue. Using this hypothesis they established equations for calculating the dynamic capacities of rolling bearings; these have been accepted by the International Organization for Standardization [ 221. 3.5. The hypothesis of critical tangential stress Ollerton and Morey [ 231 and Stulen and Cummings [ 241 proposed following form for the critical stress in rolling contact fatigue: Tc!
=
~yr + Go,
the
(3)
where rYz is the maximum value of the orthogonal tangential stress, un is the greatest normal stress at the point where the value 7Yz is reached and K, is a coefficient which depends on the number of loading cycles. Introducing the product K,o,, these workers [ 23, 241 endeavoured to correlate rolling contact fatigue with the action of residual stresses. None of the hypotheses mentioned above takes into account all the components of the stress state and the particular mode of their variation. As shown below, none of these hypotheses correlates with all experimental aspects of the rolling contact fatigue phenomenon.
4. The hypothesis
of the equivalent
critical stress ~xn
In a spatial stress state similar to that produced by rolling contact loading, it is rational to obtain the critical stress as an equivalent stress which includes all the components of the stress tensor. For this purpose, the equation of the equivalent stress eE from the Huber-Mises-Hencky theory [ 26, 27, 291 was used: 1
cx = 2112 __ {(U, -- (5,)’ + (u, - ~7~)~+ (uY - u,)~ + 6(~,,~ + rzy2 + ~,,~)}l’~ (4) During the rolling process the stresses in eqn. (4) have different modes of variation. The equation may be written as follows to obtain the same variation for all the components: 1
ox(h) = 2112 -: (((5, - c$)~ + (a, - CJ,)~ + (UY- Uz)2 + 6X2(~,Y2 + rzY2) + + &
XL
21u2
(5)
where h is a coefficient used to transform the values of the symmetric alternating stresses T_ and rYz into values of equivalent pulsating stresses. Because the shear and torsion stresses are tangential stresses, the following equation may be written: h = (&N)sh -=(7.-1N)sh
(TiCv)t
(6)
(T-1N)t
where (rpN)sh and (7pN)t are the fatigue limit stresses after N loading cycles for pulsating shear loading and pulsating torsion loading respectively and (T_rN)& and (T-1~)~ are the fatigue limit StreSSeS for SyImX?triC alternating loading. The value of the coefficient X can be determined using the Wohler curves of the material of the rolling elements for symmetrical alternating and pulsating torsion loading. Figure 4 shows these curves for rolling bearing steel RUL 1. For substantial loadings, close to the static fracture limit in torsion, (7pN)t = (~-r~)~ = rf and the coefficient X is equal to 1, while at light loadings close to the fatigue limit for symmetrical alternating loadings
271
WI
! 10
102
!
!
103
lob
!
!
11.
10s 106 10'N
Fig. 4. The fatigue limits in torsion for roller bearing steel RUL 1 (0.95% - 1.1% C; 0.2% - 0.4% Mn; 1.3% - 1.65% Cr): curve 1, symmetrical alternating loading cycle; curve 2, pulsating loading cycle.
(T&t = Tpt = 2~_~~ and the coefficient X is equal to 2. The value of the coefficient X depends on the loading intensity. For example, Fig. 5 shows the variation in the maximum equivalent stress with depth for three values of the ratio b/a = p under two loadings (corresponding to X = 1.2 and h = 2). For comparison the largest values of the stresses 7yr and 745(yr) are also presented. 02019606 1P
02 ob 06w 1.0
=cd
=a
(a)
@I
(c)
Fig. 5, The maximum values of the stresses for (a) 0 = 0, (b) fl= 0.2 and (c) fl = 1: curves 1, ryz; curves 2, 745(yz); cum% 3, (JE(1.2); curves 4, oE(2).
5. Correlation of hypotheses experimental results
concerning
the critical stress with the
5.1. Correlation with the origin of fatigue failures Tallian et al. [3 - 81 presented the rolling contact fatigue phenomenon as a competition between different modes of failure caused by the existence of various stress concentrators with surface and subsurface origins. The hypotheses presented in Sections 3.1 - 3.3 considered that stresses with the maxima on the contact surface were critical and consequently these hypotheses cannot explain the subsurface origins. The hypotheses presented in Sections 4 and 5 considered that stresses with maxima below the surface of the loading zone were critical. From Fig. 5 the equivalent stresses from the elastic half-space have maxima ~zn(h) at depths Zzn(k, which depend on the p ratio and the coefficient X. At the same time, the
272
equivalent stresses have significant values on the contact surface; these values are increased by various types of stress concentrators on the contact surface. In this way the critical stress hypothesis can explain failure phenomena with both subsurface and surface origins. 5.2. Correlation with the influence of the ratio 0 = b/a Greenert [ 311, Moyer and Morrow [ 131 and Fessler and Ollerton [ 323 have shown that the fatigue life of rolling elements depends on the contact geometry. Fatigue Iife tests performed with the same value of the maximum hertzian stress u. have achieved longer fatigue lives in cases when the 0 ratio had larger values. Because the m~imum hertzian stress o. was unchanged, it means that this stress does not correlate with the test results obtained. The dependences of the stresses TV, 745D and ~zn(~) on the p ratio are shown in Fig. 6. The stress r45D is almost constant, varying within the narrow limits (0.30 - 0.32)~~ and hence does not correlate with the influence of the p ratio on the fatigue life. When the p ratio increases, the values of the stresses r. and ~zn(h) decrease and consequently these stresses explain the influence of the p ratio on the fatigue life. The influence of the 0 ratio on the equivalent critical stress OzDChjis more marked for larger values of the coefficient X; this corresponds to lighter loads. 1.0 -
1
0,9-?_Q Fo -- p__------:
/
o,e.i
0.7 o,G -
!
1.2
_
1
r_-------___--*--+___..--I--~~---'_:,_
Fig. 6. The critical
stresses
TO,
745~
and UED(~) as functions
of the ratio fl= b/a.
5.3. Correlation with the depth and orientation of structural changes Numerous experimental researches [ 33 - 413 have shown that structural changes in the material of rolling elements loaded at high values of the maximum hertzian stress u. occur in a zone under the contact surface at a depth 2,. With respect to the value of the depth 2, there are different opinions: some studies show that 2, x Z,, which corresponds to the stress TV, while other researches indicate that 2, = Ze5n, which corresponds to the stress
273
Tag,,. Structural changes occur in the most heavily loaded zones, where critical stresses operate, and hence only hypotheses of critical stresses with large values of the depth can explain the appearance of structural changes. The depths Z,, 2,s and ZED(hj at which the stresses ro, 745D and UED(~) respectively occur are shown in Fig. 7 as functions of the /3 ratio. The depth .&n(h) depends on the coefficient X and hence is a function of the loading intensity. The depth &,(k) lies between 2, and 245D: for light loads, when = 2, ; for heavy loads, when h = 1, &D(A) = 245D. These x = 2,ZED(h) findings confirm that the hypothesis of equivalent stress correlates with the appearance of structural changes, both at small depths close to 2, and at greater depths close to 24,. IIT
oi i
agL&3
.
0.L
i 06
i
1
/
1 2
L
6
10
Fig. 7. The depths 20, 245D and ZED(Q, where the stresses~o, 745~ and CJED(A)respectively were obtained, as functions of j3.
Muro and Tsushima [ 411 reported hardness changes in the material of loaded elements in rolling contact. Similar changes did not occur in the material of some elements under the same load but did occur with pulsating loading (without rolling). This indicates that the hardness changes are due to tangential orthogonal stresses with a symmetrical alternating variation. The equivalent stress Usn(h) also contains the tangential orthogonal stresses and consequently correlates with the hardness changes reported by Muro and Tsushima [41] : 5.4. Correlation with the influence of residual stresses Residual normal stresses may exist in superficial layers of the material of rolling elements. These stresses are initiated by the final cutting process, by heat treatment or by a plastic loading process. Fatigue life tests performed both on test specimens and on rolling bearings which had residual compressive stresses in the shallow layer have achieved lives greater by factors of 1.5 - 4 than those of similar elements without residual compressive stresses [ 32,42 - 481. The presence of residual traction stresses leads to a considerable decrease in the rolling element fatigue life. Theoretical studies which try to explain the influence of residual stresses are based on the critical stresses
214 745D [42] or on the critical stress 7, [ 191. The stresses 745D and 7, are located at certain depths and they cannot be used to explain the influence of residual stresses situated at other depths or of stresses situated on the COntaCt SUrfaCe. Also, the StreSS 745DS on the contact surface cannot explain the influence of internal residual stress. The orthogonal tangential stress TV, which depends only on uo, is unchanged by the presence of residual stresses and consequently the stress r. does not correlate with the influence of 74SD and 7, on the fatigue life. The residual stresses (Ix& oya and oza, which are oriented along the x, y and z axes respectively, change the value of the equivalent stress according to the equation
uER(h)=
&rc-h
+
+ uxR)
{(a, + uxR)
-
-(us
+ u,R)j2
(U, + UZR)}~
+
+ {(u, + UyR)
6h2(7,.2
-(uz
+uzR)j2
+ TzY2) + 6~,,~ll’~
+
(7)
is the value of the equivalent stress that allows for the presence UERW of residual stress. Residual stress measurements performed by Muro and coworkers [45,46] using an X-ray technique and by Pomeroy and Johnson [49, 501 using mechanical methods show that, in the case of elements with axial symmetry, the residual stress state is characterized by two normal stresses uYR and u,a but there is no information regarding the value of the residual stress (J,a in the radial direction. For equilibrium reasons, the value of the stress (s,a should be much smaller than the values of the other two normal residual stresses. The majority of experimental tests [42 - 471 have determined only the distribution of residual stresses along the y axis. However, in the case of residual stress measurements performed on balls, it can be considered that there is a residual stress distribution along the x axis identical with that measured along the y axis. The fatigue life results plotted in Fig. 8 were obtained by Scott et al. [44] using four groups of 50 balls each with the residual stress distributions shown in Fig. 9. The balls from the group A had the highest residual compressive stresses and achieved the longest fatigue lives, while the balls from group C had the lowest residual stresses and achieved the shortest fatigue lives. If we consider a probability S of survival of 0.1, the ratio of the fatigue lives of the two groups is (L1O)A/(L1O)C = 2.5. The tests were carried out at a maximum hertzian stress of u o = 8.428 GPa corresponding to r. = 2.106 GPa for the maximum orthogonal tangential stress. From Fig. 4 the value h = 1 is obtained. Using eqn. (7) and the residual stress distributions presented in Fig. 9, the equivalent stress distributions determined for groups A and C are shown in Fig. 10. The maximum equivalent stresses are (uRRD)A = 0.567~~ for group A and (uERD)C = 0.6150~ for group B. Considering values between 9 and 12 for the exponent of the fatigue life critical stress equations [ 511, the theoretical value of the fatigue lives ratio is where
('hO)A = y=(Llo)c=
2.07 - 2.65
(8)
215
GPa ;
0.033
l.5 0.066 mt - ? 0,l 32 .u 0 0.123 ‘u, U 0,166 a” 0.2 L 6 10
20
Fatigue
LO 60100
life
0 40125 QO2SlWW2S~Q225
200
Depth
( minutes)
below
the
mm
surface
Fig. 8. Fatigue life plots for bails. (Data from Scott et al. [44].) Fig. 9. Residual stress distribution plots for balls. (Data from Scott et al. [44] .)
0.1 0.1. Q2--
OA-
0.2 -__
0.3
04
0.5
0.6
0.7
0.8
-3
:._
~=SE~(A)
A”
’
C
Fig. 10. Distributions as functions of the depth of maximum equivalent stresses for balls of groups A and C from Fig. 9.
There is a good correlation between the theoretical fatigue lives ratio and the experimental ratio. This finding raises the problem of establishing the optimum residual stress distributions.
6. The optimum residual stress distributions The optimum residual stress states will achieve minimal values for the equivalent stresses in eqn. (7). A residual stress state characterized by the nOrId COInpreSSiVe StreSSeS uxR, uyR and (J,n Will be Opthal if it CanCelS the effect of the normal elastic stresses u,, uY and uz respectively; these normal elastic stresses are caused by the contact loading. For the case of point contact, eqn. (7) becomes
276
and, for the case of line contact
where 7,Y = 0 and 7,, = 0, eqn. (9) becomes
UER = 31’2A,zY Equation
(9) is correct
(19) if the residual stresses satisfy the following
equations:
(11) The stress (I,R can be neglected and so eqns. (11) become axR
=
(5,
--ax
in comparison
with the
StreSSeS
aXR and (Jyn
(12)
There is obviously no residual stress distribution able to satisfy eqns. (11) or (12) at all points of the elastic half-space. In order to determine the optimal residual stress distribution the following numerical procedure was used. On each axis of the elastic halfspace, shown in Fig. 1, a number of points were fixed; these were xl, x2, . . . , x, for the x axis, yl, y2,. . . , yn for the y axis and zl, z2,. . . , z, for the z axis. In the plane z1 = constant, the optimum residual stresses were computed using eqns. (12) for the point with the coordinates x1, yl, zl. These residual stresses were used in eqn. (7) and the equivalent stresses were determined for all points of the plane .a1 = constant; the maximal value of these equivalent stresses is (a&. In the next step, again using eqns. (12), the optimal residual stresses for the point with coordinates x2, yl, z1 were computed. These new values for the residual stresses were used in eqn. (7) and new values for the equivalent stresses were determined in the plane .a1 = constant with the maximal value (u~)~. By using this procedure for all points of the plane z1 = constant, a number of maximum equivalent stresses were obtained. The smallest of these maximum equivalent stresses is the most favourable equivalent stress for the plane .a1 = constant. The residual stresses which lead to this equivalent stress will be the optimal residual stresses for the plane zr = constant. This procedure was repeated for all planes z = constant and in each repetition both new optimal residual stresses and corresponding maximal equivalent stresses were found. Figure 11 shows the distribution of the maximum elastic equivalent stress aE, the distribution of the optimal residual stresses cXR and eyR and the distribution of the resultant maximal equivalent stress uER. The maximum maximorum value of the elastic equivalent stress is the equivalent critical stress for .the case of elastic loading without residual stresses, while the maximum maximorum value of the resultant equivalent stress is the equivalent critical stress when the loaded zone contains the optimal residual stress distributions. On the basis of the reasons presented in Section 4 the fatigue life ratio will be (13)
277 0
0.l 0.2 0,3Oj. 05 0.6 0.7
42
0
0.l 0.2 0.3 04 Q5 0,6 0.7
-‘ERFD
0.4
46 0.8
‘ERFD
1
‘VED
‘FED
1.2 14 1.6
(4
1.8 2
(b) 0 0.I 4293G.LO50.6
0 0,l 0.2 02 0.L 0.5 06 0.7 JED
0.2 0.4
‘FERFD
0.6 0.8
970)
-‘ERFD “ED
1.0 1.2 1.2 1.6
(d) 0 0142
1.8 2
43 Oh Q5 46 0,7
I-
OERFDs
d 41 0.2 q
Q6 06 0.7
-‘ERFD
“ERFD
“ED
(e)
,FERFD o/l
42 04 0.6 48 1.0 1.2
03)
-7
0 O.lO.2’03
a4 0.5 ob 0.7
0 0,l02OJO&qSQ6
42 0.1
-‘ERFD’
“ERFD
46 0.8 1.0 1.2
“ED-
1h
1.L
1,6
1.6
1.8 2
0.7
W
‘FED
?
Fig. 11. Distributions w functions of the depth of the optimal residual etreeees and the equivalent st=es: cuwe8 1, (IE(h) wing eqn. (5); curve8 2, uER(h) using eqn. (7); curves 3, %R and ‘JyR using eqnt3. (12).
278
For a given rolling element contact geometry (which means a definite value of the 0 ratio) and for a given loading (which means a definite value of the h coefficient) the optimum distribution of the residual stresses can be obtained using the above procedure. If these residual stress distributions are achieved in practice, the longest fatigue life will be obtained. A residual stress state can be optimum only for certain contact geometry tid loading conditions. For a given case, with particular contact geometry and loading, the stresses u. and r. are the first to be determined; the value of the X coefficient is then obtained using the plot presented in Fig. 4. The values of the p ratio and the h coefficient are now known and the optimal residual stress d~tributions are determined using the procedure presented above, The influence of the p ratio and the X coefficient on the critical stresses is shown in Fig. 12 for the critical equivalent stress without residual stresses
I
0.5
.Q3
q2
1
0.2
0.6
od
1
0 (b)
Fig. 12. Distributions of the critical equivalent stresses without residual stresses (am) and with optimum residual stresses (o&z (a) on the rolling contact surface; (b) beneath the rolling contact surface as functions of 0; (c) beneath the rolling contact surface as functionsofh:o,*,pPf;A,A,pIO.
279
and for the critical equivalent tions .
stress with optimum
residual stress distribu-
7. Conclusions (1) A new hypothesis regarding the critical stress in rolling contact fatigue is proposed. Table 1 shows the correlation of the present hypothesis and alternative hypotheses concerning the rolling contact fatigue phenomenon with some experimental aspects. Only the new hypothesis of equivalent critical stresses shows good correlation with all experimental aspects. TABLE 1 Summary of critical stress hypotheses and their correlations with experimental data Critical stress hypothesis
Success of the correlation experimental data
with the following
Origin of failures
Structuml altemtions
b/a mtio
Residual
Partial
No
No
No
Partial
No
No
Partial
Partial
Partial
No
Partial
The maximum tangential stress on the surface
Partial
No
No
Partial
The maximum tangential orthogonal stress ~0
Partial
Partial
Good
No
The critical tangential stress
Partial
Partial
Good
Partial
The equivalent stress
Good
Good
Good
Good
The maximum stress
hertzian
The
maximum traction stress on the surface The maximum tangential stress
stresses
745
(2) The hypothesis of critical equivalent stresses permits, for a certain loading case, the establishment of the optimum residual stress distribution. If these optimum residual stress distributions are achieved in practice, the lowest values of equivalent critical stresses wiIl result and the longest fatigue life will be obtained. The value of the fatigue life increase can be estimated using eqn. (13). The new hypothesis allows an optimum design from which the maximum fatigue life for rolling elements can be obtained.
280
References 1 T. L. Carter, R. H. Butler and R. H. Bear, Investigation of factors governing fatigue life with a rolling contact fatigue spin rig, ASLE Tmns., 1 (1958) 23 - 32. 2 W. E. Littman and R. L. Widner, Propagation of contact fatigue from surface and subsurface origins, J. Basic Eng., 188 (3) (1966) 624 - 636. 3 T. E. Tallian, On competitive failure model in rolling contact, ASLE Trans., 10 (1967) 418 - 435. 4 Y. P. Chiu, T. E. Tallian and J. I. McCool, An engineering model of spalling fatigue failure in rolling contact. I. The subsurface model, Wear, I7 (1971) 433 - 446. 5 T. E. Tallian and J. I. McCool, An engineering model of spalling fatigue failure in rolling contact. II. The surface model, Wear, 17 (1971) 447 - 461. 6 T. E. Tallian, An engineering model of spalling fatigue failure in rolling contact. III. Engineering discussion and illustrative examples, Wear, 17 (1971) 463 - 480. 7 T. E. Tallian, Prediction of rolling contact fatigue life in contaminated lubricants. Part I. Mathematical model, J. Lubr. Technol., 101 (April 1970) 251 - 257; Prediction of rolling contact fatigue life in contaminated lubricants. Part II. Experimental, J. Lubr. Technol., 101 (July 1976) 384 - 392. 8 T. E. Tallian and J. I. McCool, A new model of fatigue life dispersion in rolling contact, Wear, 47 (1978) 147 - 154. 9 C. F. Smith, Fatigue failure distributions for ball bearings, J. Lubr. Technot., 95 (April 1973) 123 - 131. 10 G. S. Reichenbach, W. D. Syniuts and D. Walter, An electron microscope study of rolling contact fatigue, ASLE Trans., 8 (3) (1965) 943 - 948. 11 C. A. Moyer and N. E. Neifert, A first-order solution for the stress concentration present at the end of the roller contact, ASLE Trans., 6 (1963) 324 - 336. 12 R. E. McKelvey and C. A. Moyer, The relation between critical maximum compressive stress and fatigue life under rolling contact, Proc. Symp. on Fatigue in Rolling Contact, London, March 1963. 13 C. J. Moyer and J. Morrow, Relation of fatigue to pitting failure of surfaces in rolling contact, Appl. Mech. Rep. 565, 1958 (University of Illinois). 14 S. Way, Pitting due to rolling contact, J. Appt. Mech., 2 (1) (1935) 46 - 56. 15 M. Styri, Fatigue strength of ball bearing races and heat treated 52100 steel specimens, Proc., ASTM, 51 (1951) 682 - 697. 16 D. W. Reichard, R. J. Parker and E. V. Zaretsky, Residual stresses and subsurface hardness changes induced during rolling contact, NASA Tech. Note 04456, 1968. 17 E. V. Zaretsky, R. J. Parker and W. J. Anderson, A study of residual stress induced during rolling, J. Lubr. Technol., 91 (2) (1969) 314 - 319. 18 S. V. Pineghin, Contactnaia Procinosti i Soprotiulenia Kaceniu, Mashinostroenie, Moscow, 1969. 19 C. A. Foord, C. G. Hinglay and A. Cameron, Pitting of steel under varying speeds and combined stresses, J. Lubr. Technol., 91 (2) (1969) 282 - 293. 20 G. Lundberg and A. Palmgren, Dynamic capacity of rolling bearings, Acta Polytech., 7, Mech. Eng. Ser., 1 (3) (1947) 1 - 80. 21 J. H. Rumbarger, R. C. Herrick and P. R. Eklund, Analysis of the elastic contact of a hollow ball with a flat plate, J. Lubr. Technol., 92 (January 1970) 138 - 144. 22 A. F. B. M. A., Method of evaluating load ratings for ball bearings, Z.S.O. Stand., Section 9, Revision 4, October 1960. 23 E. Ollerton and J. W. Morey, Fatigue strength of rail steel in rolling contact, Proc. Symp. on Fatigue in Rolling Contact, London, March 1963. 24 F. B. Stulen and H. N. Cummings, A failure criterion for multiaxial fatigue stresses, Proc., ASTM, 24 (1954) 82. 25 D. Rothbart, Mechanical Design and Systems Handbook, McGraw-Hill, New York, 1964, pp. 13.1 - 13.76. 26 N. M. Beleaev, Trudi po Teorii Uprugosti i Plastichinosti, Gosoodarstvyenniy Izdatelvsko, Tekhniko-tyeoryetichyeakiy Lityeratoora, Moscow, 1957.
281 27 S. D. Ponomariov et al., Rashetina Prochinosti y Mashinostroyeniye, Gosoodarstvyenniy, Naukina-tekhnichyeskiy Izdatelstvo Mashinostroitelnoi Lityeratoora, MOSCOW, 1958. 28 K. H. K.loss, Werkstoff Paarung und Gleitreibungsverhalten in Fertigung und Konstruktion, Fortschr. Ber., VDI 2. Reihe 2, 25 (September 1972). 29 S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, McGraw-Hill, New York, 1973. 30 M. J. Hartnett, The analysis of contact stresses in rolling element bearings, J. Lubr. Technol., 101 (January 1979) 105 - 109. 31 W. J. Greenert, The toroid contact roller test as applied to the study of bearing materials, J. Basic Eng., 84 (1) (1962) 181 - 191. 32 M. Fessler and E. Ollerton, Contact stresses in toroids under radial loads, Br. J. Appl. Phys., 9 (1957) 387 - 393. 33 J. A. Martin and A. D. Eberhardt, Identification of potential fatigue nuclei in rolling contact fatigue, J. Basic Eng., 89 (3) (1967) 932 - 942. 34 D. Scott, B. Loy and G. H. Mills, Metallurgical aspects of rolling contact fatigue, Proc., Inst. Mech. Eng., London, 181 (30) (1966 - 1967). 35 J. J. Bush, W. L. Grube and G. H. Robinson, Microstructural and residual stress changes in hardened steel due to rolling contact. In J. B. Bidwell (ed.), Rolling Contact Phenomena, Elsevier, Amsterdam, 1962, pp. 365 - 399. 36 G. H. Vasilca and V. A. Raszillier, A study of dark etching area type structure of material and hertzian contact area induced by ball bearing type motion, Wear, 19 (1972) 1 - 15. 37 J. A. Martin, S. Borgese and A. D. Eberhardt, Microstructural alternations of rolling bearing steel undergoing cyclic stressing, J. Basic Eng., 88 (1967) 555 - 567. 38 R. K. Kepple and R. L. Mattson, Rolling element fatigue and microresidual stress, J. Lubr. Technol., 92 (January 1970) 76 - 82. 39 K. Fujita and A. Yoshida, Surface durability of case-hardened nickel-chromium steel rollers under pure rolling and sliding-rolling contacts, Wear, 52 (1975) 37 - 48. 40 K. Fujita and A. Yoshida, Surface failure of soft and surface-hardened steel rollers in rolling contact, Wear, 55 (1979) 27 - 39. 41 H. Muro and N. Tsushima, Microstructural, microhardness and residual stress changes due to rolling contact, Wear, 15 (1970) 309 - 330. 42 I. Sprishevschi, Podsipniki Kachenia, Mashinostroenie, Moscow, 1969. 43 J. 0. Almen, Effects of residual stress on rolling bodies. In J. B. Bidwell (ed.), Rolling Contact Phenomena, Elsevier, Amsterdam, 1962, pp. 400 - 424. 44 R. L. Scott, R. K. Kepple and M. H. Miller, The effect of processing - induced nearsurface residual stress on ball bearing fatigue. In J. B. Bidwell (ed.), Rolling Contact Phenomena, Elsevier, Amsterdam, 1962, pp. 301 - 316. 45 M.Muro, T. Tsushima and K. Nunome, Failure analysis of rolling bearings by X-ray measurement of residual stress, Wear, 25 (1973) 345 - 356. 46 H. Muro, T. Tsushima and N. Nagafuchi, Initiation and propagation of surface cracks in rolling fatigue of high hardness steel, Wear, 35 (2) (1975) 261 - 282. 47 E. V. Zaretzky, R. J. Parker and W. J. Anderson, Effect of component differential hardnesses on residual stress and rolling contact fatigue, NASA Tech. Note 02664, 1965. 48 D. D. Papshev and A. N. Pikovski, Povishnie dolgovecinosti podsipnikov kacenia, Vestn. Mashinostr., 6 (1969) 10 - 12. 49 R. J. Pomeroy and K. L. Johnson, Residual stresses in rolling contact, J. Strain Anal., 4 (3) (1969) 208 - 218. 50 R. J. Pomeroy, Measurement of residual stresses in rolling contact, Wear, 16 (1970) 393 - 412. 51 R. J. Parker and V. E. Zaretsky, Reevaluation of the stress-life relation in rollingelement bearings, NASA Contract E6768, 1972. 52 L. J. Nypan, H. H. Coe and R. J. Parker, Bending stresses in spherically hollow ball bearing and fatigue experiments, J. Lubr. Technol., 98 (July 1976) 472 - 475.
282 53 G. J. Moyar and G. M. Sinclair, Cumulative plastic deformation in rolling contact, J. Basic Eng., 85 (1) (1963) 105 - 110. 54 A. V. Orlov and S. V. Pineghin, Ostatocinie Napriajenie, Deformatii pri Kontakton Nagrujenii, Mashinostroenie, Moscow, 1971. 55 S. Cretu, Ph.D. Thesis, Jassy Polytechnic Institute, 1978. 56 R. K. Kepple and R. L. Mattson, Rolling element fatigue and microresidual stress, J. Lubr. Technol., 93 (1) (1970) 76 - 81. 57 I. Crudu, Ph.D. Thesis, Jassy Polytechnic Institute, 1969. 58 A. I. Birger, Ostatochinie Napriajenia, Gosoodarstvyenniy Naukino-tekhnichyeskiy Izdatelstvo Mashinostroitelnoi, Moscow, 1963.
265 - Printed
in The Netherlands
CRITICAL STRESSES IN ROLLING CONTACT FATIGUE
N. G. POPINCEANU,
E. DIACONESCU
Machine Design Department, Jassy 6600 (Romania) (Received
July 23,198O;
and S. CREW
Jassy Polytechnic
in revised
institute,
form November
23 August Street No. 11,
24.1980)
Summary An analysis of the main hypotheses concern~g the critical stresses in rolling contact fatigue is presented. It is considered that none of these hypotheses correlates adequately with all the experimental aspects of the phenomenon of rolling contact fatigue. A new hypothesis is proposed; this hypothesis states that the equivalent stress is the critical stress in rolling contact fatigue. By considering the influence of the residual stresses it is shown how the optimum fatigue life in rolling contact can be achieved.
1. Introduction Rolling elements deteriorate mainly by rolling contact fatigue; this is caused both by the action of variable stresses during the rolling process and by the presence of stress concentrators which are due to material or working process defects. According to their nature, the defects can be localized both on the contact surface and within the material, ie. under the contact surface. Failure by rolling contact fatigue is now considered to be a process involving competition between the various types of stress concentrators [l - lo]. The models presented by Tallian et ~2. [3 - 81 exemplify this. In order to calculate both the loading capacity and the fatigue life of rolling contact elements, various theoretical equations have been reported. These equations were obtained using a certain variable stress as the critical stress for rolling contact fatigue, There are many reported experimental data which can confirm or conflict with the hypotheses concerning the critical stress in rolling contact fatigue [ 11 - 241. An analysis of the main hypotheses concerning the critical stress in rolling contact fatigue was carried out. This analysis indicated that none of the existing hypotheses correlates with all the experiments aspects of rolling contact fatigue. The equivalent critical stress hypothesis is proposed as a new hypothesis concerning the critical stress in rolling contact fatigue. The new hypothesis correlates with all experimental aspects of the phenomenon. It is shown how an increased life can be obtained for rolling contact elements.
266
2. The stress state in elastic rolling contact In the general case of elastic rolling contact loading, the stress tensor of the elastic half-space has nine stress components; three of these are the normal stresses u, , uy and u, and six are orthogonal tangential stresses. The orthogonal tangential components are equal with respect to the interchange of suffixes, so that the stress state is characterized by a maximum of six separate stresses, the three normal stresses and the three orthogonal tangential stresses rXY, ryr and 7,, (Fig. 1, positions 1 and 2). The variation in all these components on the contact surface and within the material of the rolling elements can be obtained using the standard equation [25 - 301. The sense and magnitude of the stresses on the surface of the contact ellipse are given in Figs, 1 and 2. The normal stresses reach the normal compression values uX,, uY, and uZO in the centre of the contact ellipse; the greatest of these is the stress uZO = uo, which is also known as the maximum hertzian contact stress (Fig. 1, position 9). On the y axis (Fig. 2(b)) the normal stresses u. and uX are compression stresses; at the ends of the y axis the stress uZ is zero while u, has non-zero values (Fig. 2(a)) at these posi-
Fig. 1. The stresses
at the elastic
contact.
267
0.020 0
(a)
'
' 0.2
I
' 1 0.L
1
I j itI a&
0.6
0.8
(b)
Fig. 2. The distribution of normal stresses on the surface of contact ellipses: (a) normal stresses on the contact surface (@ = b/a = 0.4): -, a, ; ---, a, ; . e =, u, ; (b) normal stresses at the ends of the axes of the contact ellipse [ 271: curve 1, at the ends of the semimajor axis; curve 2, at the ends of the semiminor axis.
tions
although it tends to zero outside the contact ellipse. The stress uY has traction values uYt at the ends of the y axis (Fig. Z(b)) and the stress a, has traction values at the ends of the x axis (Figs. 1 and 2). If the rolling process occurs along the y axis, the stresses uZ and u, will cause a pulsating loading with a frequency v equal to the number of passes per second of the rolling elements. Under the same conditions, the u, stress will produce an asymmetric alternating loading with a frequency of 1.5~. Among the orthogonal tangenti~ stresses, only the stress T,~ has nonzero values on the contact surface (Fig. 1, position 8); these values are much smaller than those of normal stresses. In the half-space, under the contact surface, the normal stresses u,, uy and u, are compression stresses; they have maxima on the z axis which are smaller than those (uxO, uY, and uZO) reached in the centre of the contact surface. The orthogonal tangential stresses T,~, 7yz and T,, reach their maxima under the contact surface; the greatest value r. is achieved at a depth z. by the ryt stress [ 201. During the rolling process along the y axis, the normal stresses ux, (rY and e, and the tangential stress rxY produce pulsating loadings with frequency v ; the tangential stresses 7,Y and 7YZ cause symmetrical alternating loadings with the same frequency. Correlation of both the mode of variation of the above stresses and the points where
1
268
these stresses reach their maxima with the distribution of stress concentrators is the concern of various hypotheses regarding the critical stress in rolling contact fatigue which are reported in the literature.
3. Present hypotheses
regarding the critical stresses in rolling contact
fatigue
3.1. The hypothesis of maximum hertzian stress u. The stress u. acting on the contact surface produces, in the case of rolling along the y axis, a pulsating loading of frequency II. This stress was chosen by Moyer and coworkers [ll, 12 ] as a critical stress for rolling contact fatigue. They recommended an equation for the fatigue life of rolling elements:
LIo = Ku,,-
6.66
(1)
3.2. The hypothesis of maximum tangential stresses r45 It is known from the theory of elasticity that in any point of an elastic space there are three tangential stresses given by the equations: 0.x -0, 745(XY)
= 2
oy 745fY.z)
=
745(x2)
= ____
-02
(2)
2
*x
-0.2
2
There are three different hypotheses which consider the maximum tial stresses as the critical stresses in rolling contact fatigue.
tangen-
3.2.1. The hypothesis of maximum maximorum tangential stress r450 In the case of rolling along the y axis, the maximum tangential stresses will produce pulsating loading of frequency V; from these stresses the stress 7450,rj has its greatest amplitude ?46D at a depth Z,sn. Way [14], Styri [ 151 and Reichard and coworkers [ 16,171 considered that the stress 745D is the critical stress in rolling contact. 3.2.2. The hypothesis of maximum tangential stress on the contact surface 745D6 The sense and the magnitude of the maximum tangential stresses on the contact surface are shown in Figs. 1 and 3; the index s is used for the stresses at the ends of the axes of the contact ellipse and the index 0 is used for the stresses in the centre of the contact ellipse 1271. In the case of rolling along the y axis, the stresses 745(xyf and 7451Xzj will generate ~ternating loading of frequency 1.5~ (Fig. 3(a)). Pineghin [lS]
269
0.2 oh
0.6
0.8
1
(b)
Fig. 3. The distribution of maximum tangential stresses on the contact surface: (a) details of the distribution when /3 = b/a = 0.4; (b) T&TO vs. /? for various stresses 745: curve 1, 746(xy)O;
curve 2,
745(yr)O;
curve 3,
745~0;
curve 4,
745~~
i Curve 5Y T45(xy)sbe
considered that the stress 745(xy) at the end of the minor axis of the contact ellipse is the critical stress in roling contact. Discussing some results obtained by Foord et al. [ 191, Cioclov chose the stress 745(XY)at the end of the major axis as the critical stress. When the rolling is performed along the y axis, this stress produces an alternating loading with a frequency u. 3.3. The hypothesis of maximum traction stress on the contact surface uyt The stress uy has an asymmetrical alternating variation on the contact surface; because of this, Moyer and Morrow [13] considered that its maximal value uyt, reached at the ends of the y axis, is the critical stress in rolling contact. 3.4. The hypothesis of maximum orthogonal tangential stress r. Lundberg and Palmgren [20] proposed that the maximal value r. of the orthogonal tangential stress ryr is the critical stress in rolling contact fatigue. Using this hypothesis they established equations for calculating the dynamic capacities of rolling bearings; these have been accepted by the International Organization for Standardization [ 221. 3.5. The hypothesis of critical tangential stress Ollerton and Morey [ 231 and Stulen and Cummings [ 241 proposed following form for the critical stress in rolling contact fatigue: Tc!
=
~yr + Go,
the
(3)
where rYz is the maximum value of the orthogonal tangential stress, un is the greatest normal stress at the point where the value 7Yz is reached and K, is a coefficient which depends on the number of loading cycles. Introducing the product K,o,, these workers [ 23, 241 endeavoured to correlate rolling contact fatigue with the action of residual stresses. None of the hypotheses mentioned above takes into account all the components of the stress state and the particular mode of their variation. As shown below, none of these hypotheses correlates with all experimental aspects of the rolling contact fatigue phenomenon.
4. The hypothesis
of the equivalent
critical stress ~xn
In a spatial stress state similar to that produced by rolling contact loading, it is rational to obtain the critical stress as an equivalent stress which includes all the components of the stress tensor. For this purpose, the equation of the equivalent stress eE from the Huber-Mises-Hencky theory [ 26, 27, 291 was used: 1
cx = 2112 __ {(U, -- (5,)’ + (u, - ~7~)~+ (uY - u,)~ + 6(~,,~ + rzy2 + ~,,~)}l’~ (4) During the rolling process the stresses in eqn. (4) have different modes of variation. The equation may be written as follows to obtain the same variation for all the components: 1
ox(h) = 2112 -: (((5, - c$)~ + (a, - CJ,)~ + (UY- Uz)2 + 6X2(~,Y2 + rzY2) + + &
XL
21u2
(5)
where h is a coefficient used to transform the values of the symmetric alternating stresses T_ and rYz into values of equivalent pulsating stresses. Because the shear and torsion stresses are tangential stresses, the following equation may be written: h = (&N)sh -=(7.-1N)sh
(TiCv)t
(6)
(T-1N)t
where (rpN)sh and (7pN)t are the fatigue limit stresses after N loading cycles for pulsating shear loading and pulsating torsion loading respectively and (T_rN)& and (T-1~)~ are the fatigue limit StreSSeS for SyImX?triC alternating loading. The value of the coefficient X can be determined using the Wohler curves of the material of the rolling elements for symmetrical alternating and pulsating torsion loading. Figure 4 shows these curves for rolling bearing steel RUL 1. For substantial loadings, close to the static fracture limit in torsion, (7pN)t = (~-r~)~ = rf and the coefficient X is equal to 1, while at light loadings close to the fatigue limit for symmetrical alternating loadings
271
WI
! 10
102
!
!
103
lob
!
!
11.
10s 106 10'N
Fig. 4. The fatigue limits in torsion for roller bearing steel RUL 1 (0.95% - 1.1% C; 0.2% - 0.4% Mn; 1.3% - 1.65% Cr): curve 1, symmetrical alternating loading cycle; curve 2, pulsating loading cycle.
(T&t = Tpt = 2~_~~ and the coefficient X is equal to 2. The value of the coefficient X depends on the loading intensity. For example, Fig. 5 shows the variation in the maximum equivalent stress with depth for three values of the ratio b/a = p under two loadings (corresponding to X = 1.2 and h = 2). For comparison the largest values of the stresses 7yr and 745(yr) are also presented. 02019606 1P
02 ob 06w 1.0
=cd
=a
(a)
@I
(c)
Fig. 5, The maximum values of the stresses for (a) 0 = 0, (b) fl= 0.2 and (c) fl = 1: curves 1, ryz; curves 2, 745(yz); cum% 3, (JE(1.2); curves 4, oE(2).
5. Correlation of hypotheses experimental results
concerning
the critical stress with the
5.1. Correlation with the origin of fatigue failures Tallian et al. [3 - 81 presented the rolling contact fatigue phenomenon as a competition between different modes of failure caused by the existence of various stress concentrators with surface and subsurface origins. The hypotheses presented in Sections 3.1 - 3.3 considered that stresses with the maxima on the contact surface were critical and consequently these hypotheses cannot explain the subsurface origins. The hypotheses presented in Sections 4 and 5 considered that stresses with maxima below the surface of the loading zone were critical. From Fig. 5 the equivalent stresses from the elastic half-space have maxima ~zn(h) at depths Zzn(k, which depend on the p ratio and the coefficient X. At the same time, the
272
equivalent stresses have significant values on the contact surface; these values are increased by various types of stress concentrators on the contact surface. In this way the critical stress hypothesis can explain failure phenomena with both subsurface and surface origins. 5.2. Correlation with the influence of the ratio 0 = b/a Greenert [ 311, Moyer and Morrow [ 131 and Fessler and Ollerton [ 323 have shown that the fatigue life of rolling elements depends on the contact geometry. Fatigue Iife tests performed with the same value of the maximum hertzian stress u. have achieved longer fatigue lives in cases when the 0 ratio had larger values. Because the m~imum hertzian stress o. was unchanged, it means that this stress does not correlate with the test results obtained. The dependences of the stresses TV, 745D and ~zn(~) on the p ratio are shown in Fig. 6. The stress r45D is almost constant, varying within the narrow limits (0.30 - 0.32)~~ and hence does not correlate with the influence of the p ratio on the fatigue life. When the p ratio increases, the values of the stresses r. and ~zn(h) decrease and consequently these stresses explain the influence of the p ratio on the fatigue life. The influence of the 0 ratio on the equivalent critical stress OzDChjis more marked for larger values of the coefficient X; this corresponds to lighter loads. 1.0 -
1
0,9-?_Q Fo -- p__------:
/
o,e.i
0.7 o,G -
!
1.2
_
1
r_-------___--*--+___..--I--~~---'_:,_
Fig. 6. The critical
stresses
TO,
745~
and UED(~) as functions
of the ratio fl= b/a.
5.3. Correlation with the depth and orientation of structural changes Numerous experimental researches [ 33 - 413 have shown that structural changes in the material of rolling elements loaded at high values of the maximum hertzian stress u. occur in a zone under the contact surface at a depth 2,. With respect to the value of the depth 2, there are different opinions: some studies show that 2, x Z,, which corresponds to the stress TV, while other researches indicate that 2, = Ze5n, which corresponds to the stress
273
Tag,,. Structural changes occur in the most heavily loaded zones, where critical stresses operate, and hence only hypotheses of critical stresses with large values of the depth can explain the appearance of structural changes. The depths Z,, 2,s and ZED(hj at which the stresses ro, 745D and UED(~) respectively occur are shown in Fig. 7 as functions of the /3 ratio. The depth .&n(h) depends on the coefficient X and hence is a function of the loading intensity. The depth &,(k) lies between 2, and 245D: for light loads, when = 2, ; for heavy loads, when h = 1, &D(A) = 245D. These x = 2,ZED(h) findings confirm that the hypothesis of equivalent stress correlates with the appearance of structural changes, both at small depths close to 2, and at greater depths close to 24,. IIT
oi i
agL&3
.
0.L
i 06
i
1
/
1 2
L
6
10
Fig. 7. The depths 20, 245D and ZED(Q, where the stresses~o, 745~ and CJED(A)respectively were obtained, as functions of j3.
Muro and Tsushima [ 411 reported hardness changes in the material of loaded elements in rolling contact. Similar changes did not occur in the material of some elements under the same load but did occur with pulsating loading (without rolling). This indicates that the hardness changes are due to tangential orthogonal stresses with a symmetrical alternating variation. The equivalent stress Usn(h) also contains the tangential orthogonal stresses and consequently correlates with the hardness changes reported by Muro and Tsushima [41] : 5.4. Correlation with the influence of residual stresses Residual normal stresses may exist in superficial layers of the material of rolling elements. These stresses are initiated by the final cutting process, by heat treatment or by a plastic loading process. Fatigue life tests performed both on test specimens and on rolling bearings which had residual compressive stresses in the shallow layer have achieved lives greater by factors of 1.5 - 4 than those of similar elements without residual compressive stresses [ 32,42 - 481. The presence of residual traction stresses leads to a considerable decrease in the rolling element fatigue life. Theoretical studies which try to explain the influence of residual stresses are based on the critical stresses
214 745D [42] or on the critical stress 7, [ 191. The stresses 745D and 7, are located at certain depths and they cannot be used to explain the influence of residual stresses situated at other depths or of stresses situated on the COntaCt SUrfaCe. Also, the StreSS 745DS on the contact surface cannot explain the influence of internal residual stress. The orthogonal tangential stress TV, which depends only on uo, is unchanged by the presence of residual stresses and consequently the stress r. does not correlate with the influence of 74SD and 7, on the fatigue life. The residual stresses (Ix& oya and oza, which are oriented along the x, y and z axes respectively, change the value of the equivalent stress according to the equation
uER(h)=
&rc-h
+
+ uxR)
{(a, + uxR)
-
-(us
+ u,R)j2
(U, + UZR)}~
+
+ {(u, + UyR)
6h2(7,.2
-(uz
+uzR)j2
+ TzY2) + 6~,,~ll’~
+
(7)
is the value of the equivalent stress that allows for the presence UERW of residual stress. Residual stress measurements performed by Muro and coworkers [45,46] using an X-ray technique and by Pomeroy and Johnson [49, 501 using mechanical methods show that, in the case of elements with axial symmetry, the residual stress state is characterized by two normal stresses uYR and u,a but there is no information regarding the value of the residual stress (J,a in the radial direction. For equilibrium reasons, the value of the stress (s,a should be much smaller than the values of the other two normal residual stresses. The majority of experimental tests [42 - 471 have determined only the distribution of residual stresses along the y axis. However, in the case of residual stress measurements performed on balls, it can be considered that there is a residual stress distribution along the x axis identical with that measured along the y axis. The fatigue life results plotted in Fig. 8 were obtained by Scott et al. [44] using four groups of 50 balls each with the residual stress distributions shown in Fig. 9. The balls from the group A had the highest residual compressive stresses and achieved the longest fatigue lives, while the balls from group C had the lowest residual stresses and achieved the shortest fatigue lives. If we consider a probability S of survival of 0.1, the ratio of the fatigue lives of the two groups is (L1O)A/(L1O)C = 2.5. The tests were carried out at a maximum hertzian stress of u o = 8.428 GPa corresponding to r. = 2.106 GPa for the maximum orthogonal tangential stress. From Fig. 4 the value h = 1 is obtained. Using eqn. (7) and the residual stress distributions presented in Fig. 9, the equivalent stress distributions determined for groups A and C are shown in Fig. 10. The maximum equivalent stresses are (uRRD)A = 0.567~~ for group A and (uERD)C = 0.6150~ for group B. Considering values between 9 and 12 for the exponent of the fatigue life critical stress equations [ 511, the theoretical value of the fatigue lives ratio is where
('hO)A = y=(Llo)c=
2.07 - 2.65
(8)
215
GPa ;
0.033
l.5 0.066 mt - ? 0,l 32 .u 0 0.123 ‘u, U 0,166 a” 0.2 L 6 10
20
Fatigue
LO 60100
life
0 40125 QO2SlWW2S~Q225
200
Depth
( minutes)
below
the
mm
surface
Fig. 8. Fatigue life plots for bails. (Data from Scott et al. [44].) Fig. 9. Residual stress distribution plots for balls. (Data from Scott et al. [44] .)
0.1 0.1. Q2--
OA-
0.2 -__
0.3
04
0.5
0.6
0.7
0.8
-3
:._
~=SE~(A)
A”
’
C
Fig. 10. Distributions as functions of the depth of maximum equivalent stresses for balls of groups A and C from Fig. 9.
There is a good correlation between the theoretical fatigue lives ratio and the experimental ratio. This finding raises the problem of establishing the optimum residual stress distributions.
6. The optimum residual stress distributions The optimum residual stress states will achieve minimal values for the equivalent stresses in eqn. (7). A residual stress state characterized by the nOrId COInpreSSiVe StreSSeS uxR, uyR and (J,n Will be Opthal if it CanCelS the effect of the normal elastic stresses u,, uY and uz respectively; these normal elastic stresses are caused by the contact loading. For the case of point contact, eqn. (7) becomes
276
and, for the case of line contact
where 7,Y = 0 and 7,, = 0, eqn. (9) becomes
UER = 31’2A,zY Equation
(9) is correct
(19) if the residual stresses satisfy the following
equations:
(11) The stress (I,R can be neglected and so eqns. (11) become axR
=
(5,
--ax
in comparison
with the
StreSSeS
aXR and (Jyn
(12)
There is obviously no residual stress distribution able to satisfy eqns. (11) or (12) at all points of the elastic half-space. In order to determine the optimal residual stress distribution the following numerical procedure was used. On each axis of the elastic halfspace, shown in Fig. 1, a number of points were fixed; these were xl, x2, . . . , x, for the x axis, yl, y2,. . . , yn for the y axis and zl, z2,. . . , z, for the z axis. In the plane z1 = constant, the optimum residual stresses were computed using eqns. (12) for the point with the coordinates x1, yl, zl. These residual stresses were used in eqn. (7) and the equivalent stresses were determined for all points of the plane .a1 = constant; the maximal value of these equivalent stresses is (a&. In the next step, again using eqns. (12), the optimal residual stresses for the point with coordinates x2, yl, z1 were computed. These new values for the residual stresses were used in eqn. (7) and new values for the equivalent stresses were determined in the plane .a1 = constant with the maximal value (u~)~. By using this procedure for all points of the plane z1 = constant, a number of maximum equivalent stresses were obtained. The smallest of these maximum equivalent stresses is the most favourable equivalent stress for the plane .a1 = constant. The residual stresses which lead to this equivalent stress will be the optimal residual stresses for the plane zr = constant. This procedure was repeated for all planes z = constant and in each repetition both new optimal residual stresses and corresponding maximal equivalent stresses were found. Figure 11 shows the distribution of the maximum elastic equivalent stress aE, the distribution of the optimal residual stresses cXR and eyR and the distribution of the resultant maximal equivalent stress uER. The maximum maximorum value of the elastic equivalent stress is the equivalent critical stress for .the case of elastic loading without residual stresses, while the maximum maximorum value of the resultant equivalent stress is the equivalent critical stress when the loaded zone contains the optimal residual stress distributions. On the basis of the reasons presented in Section 4 the fatigue life ratio will be (13)
277 0
0.l 0.2 0,3Oj. 05 0.6 0.7
42
0
0.l 0.2 0.3 04 Q5 0,6 0.7
-‘ERFD
0.4
46 0.8
‘ERFD
1
‘VED
‘FED
1.2 14 1.6
(4
1.8 2
(b) 0 0.I 4293G.LO50.6
0 0,l 0.2 02 0.L 0.5 06 0.7 JED
0.2 0.4
‘FERFD
0.6 0.8
970)
-‘ERFD “ED
1.0 1.2 1.2 1.6
(d) 0 0142
1.8 2
43 Oh Q5 46 0,7
I-
OERFDs
d 41 0.2 q
Q6 06 0.7
-‘ERFD
“ERFD
“ED
(e)
,FERFD o/l
42 04 0.6 48 1.0 1.2
03)
-7
0 O.lO.2’03
a4 0.5 ob 0.7
0 0,l02OJO&qSQ6
42 0.1
-‘ERFD’
“ERFD
46 0.8 1.0 1.2
“ED-
1h
1.L
1,6
1.6
1.8 2
0.7
W
‘FED
?
Fig. 11. Distributions w functions of the depth of the optimal residual etreeees and the equivalent st=es: cuwe8 1, (IE(h) wing eqn. (5); curve8 2, uER(h) using eqn. (7); curves 3, %R and ‘JyR using eqnt3. (12).
278
For a given rolling element contact geometry (which means a definite value of the 0 ratio) and for a given loading (which means a definite value of the h coefficient) the optimum distribution of the residual stresses can be obtained using the above procedure. If these residual stress distributions are achieved in practice, the longest fatigue life will be obtained. A residual stress state can be optimum only for certain contact geometry tid loading conditions. For a given case, with particular contact geometry and loading, the stresses u. and r. are the first to be determined; the value of the X coefficient is then obtained using the plot presented in Fig. 4. The values of the p ratio and the h coefficient are now known and the optimal residual stress d~tributions are determined using the procedure presented above, The influence of the p ratio and the X coefficient on the critical stresses is shown in Fig. 12 for the critical equivalent stress without residual stresses
I
0.5
.Q3
q2
1
0.2
0.6
od
1
0 (b)
Fig. 12. Distributions of the critical equivalent stresses without residual stresses (am) and with optimum residual stresses (o&z (a) on the rolling contact surface; (b) beneath the rolling contact surface as functions of 0; (c) beneath the rolling contact surface as functionsofh:o,*,pPf;A,A,pIO.
279
and for the critical equivalent tions .
stress with optimum
residual stress distribu-
7. Conclusions (1) A new hypothesis regarding the critical stress in rolling contact fatigue is proposed. Table 1 shows the correlation of the present hypothesis and alternative hypotheses concerning the rolling contact fatigue phenomenon with some experimental aspects. Only the new hypothesis of equivalent critical stresses shows good correlation with all experimental aspects. TABLE 1 Summary of critical stress hypotheses and their correlations with experimental data Critical stress hypothesis
Success of the correlation experimental data
with the following
Origin of failures
Structuml altemtions
b/a mtio
Residual
Partial
No
No
No
Partial
No
No
Partial
Partial
Partial
No
Partial
The maximum tangential stress on the surface
Partial
No
No
Partial
The maximum tangential orthogonal stress ~0
Partial
Partial
Good
No
The critical tangential stress
Partial
Partial
Good
Partial
The equivalent stress
Good
Good
Good
Good
The maximum stress
hertzian
The
maximum traction stress on the surface The maximum tangential stress
stresses
745
(2) The hypothesis of critical equivalent stresses permits, for a certain loading case, the establishment of the optimum residual stress distribution. If these optimum residual stress distributions are achieved in practice, the lowest values of equivalent critical stresses wiIl result and the longest fatigue life will be obtained. The value of the fatigue life increase can be estimated using eqn. (13). The new hypothesis allows an optimum design from which the maximum fatigue life for rolling elements can be obtained.
280
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