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Chapter 22 Influence of residual and static stresses
Chapter 22 Influence of residual and static stresses In this chapter a theoretical investigation is described of the effect of raceways indentations on the fatigue life of bearings. The parameters dent-size, dent-depth (and thus dent-slope) were varied as well as the parameters contact-size and maximum contact-pressure (or applied load). Some of the theoretical results were compared against experimental endurance test results of artificially dented raceways, see also [Lubrecht 19901.
22.1
Introduction
For as long as endurance tests of rolling element bearings have been carried out, the dramatic influence of surface defects on the fatigue life of the element has been well known. Irrespective of whether these surface defects were formed by the overrolling of hard or ductile particles (contaminants in the oil), whether they stem from handling damage (e.g. dropping the bearing) or whether they were caused by the manufacturing process (residual grinding grooves, grinding burns), their influence on life is detrimental, as discussed in the previous chapter. The relationship between the occurrence of these surface defects and the fatigue life has up until now only been accessed experimentally, which means running the bearings in certain batches, with a certain load, speed and lubricant supply, until they fail [Lorosch 19851. This procedure, however, being the ultimate test, is very time-consuming, costly and eventually destroys the test element. In the past, due to the lack of a proper theory relating the geometry of the dents and the fatigue life, the time-consuming numerical solution techniques and the high price of powerful computers, this problem has been very hard to solve theoretically. Three major developments have taken place recently that have changed this situation. First of all, in 1985 the New Life Theory was published by Ioannides and Harris [Ioannides 19851. This theory enables us to perform a detailed study of the relation between surface defects, the associated stresses and the fatigue life. Using this theory, observed lives of tested helicopter bearings [Sayles 19821 were predicted [Webster 19851 using traditional contact calculation methods. However, more recently a second development which consists of the extension of the Multilevel techniques to the calculation of integrals and the solution of integral equations [Brandt 1990, Lubrecht 19891 was applied to highly loaded lubricated contacts [Venner 19891. This has
357
358
CHAPTER 22. INFLUENCE OF RESIDUAL AND STATIC STRESSES
resulted in shorter computational times and smaller memory requirements. Last but not least, the rapid increase in speed and memory size of the micro- and (super) mini computers has enabled a theoretical study of the influence of realistic surface features on bearing fatigue life. Of course, the main issue remains whether or not these theoretical calculations correctly predict the fatigue life of the bearings. For this purpose endurance tests were performed. This chapter consists of three main parts; emphasis will be on the first topic: Dry contact calculations of the fatigue life of dented raceways. The dents were modelled by relatively simple mathematical equations, and the influence of the depth and size was investigated. In the second part calculations were performed of a lubricated contact. These computations extend the dry contact calculations and shed some light on the directional preference of spa11 initiation with respect to the overrolling direction. In the third and last part, endurance tests are described with artificially dented raceways. The experimentally established lives are compared with the theoretical predictions. With the present state of bearing steel technology and bearing manufacture it is possible to obtain infinite bearing life under well-controlled endurance test conditions. These conditions include careful handling and mounting of the bearings in a well-designed test rig, and filtering the lubricating oil with the utmost care. Under these conditions lives are infinite on a practical time-scale (> lo9 revolutions). However, these laboratory conditions will not be met in many practical applications, and as a result the bearing will perform less well. It is for this reason that more and more research is devoted to understanding the processes taking place under sub-optimal conditions. These conditions might take dynamic loading into account, as well as starved lubrication, chemical influences (atmosphere) and the influence of hoop stresses and surface roughness of the bearing raceways on the fatigue life. A more complete overview is given in chapter 23. This chapter is devoted to the effect of dents in the raceways on bearing life because they occur frequently in practice and because of the relative ease with which they can be modelled numerically. These defects may originate from the overrolling of contaminants in the oil or may have been caused by handling damage. As a result of these defects the stresses in the subsurface will be substantially higher than in the smooth Hertzian case, and consequently the life will be shorter.
22.2 b C
C e
h
LlO N
P
ph
v,
5,z 2’
Notation Hertzian halfwidth, [m] stress exponent dynamic load capacity, [N] Weibull slope depth exponent life that 90% of a bearing population exceeds number of stress cycles load, [N] maximum Hertzian pressure, [Pa] stressed volume where the fatigue limit is exceeded, [m3] coordinates, [m] stress weighted average depth, [m]
22.3. THEORETICAL MODEL X u 0%
A
ID
S
22.3
359
wavelength used to model dents, [m] stress, [Pa] fatigue limit of the material, [Pa] amplitude used to model dents, [m] shape of the dent, [m] probability of survival
Theoretical model
The model used in the next sections is basically two-dimensional, the so-called line contact model describing the contact between two long parallel cylinders. The main advantage of the model is that the number of nodal points required for a detailed description of the surface features, the contact pressures and the stresses is relatively small. For the contact pressure approximately 1000 points were used, whereas the stresses were calculated on an equivalent 1000 x 500 grid. For a realistic description of a non-smooth point contact, at least 500 x 500 points on the surface would be needed, making this problem very hard to solve with the present techniques and hardware. As a result the present analysis is limited to the line contact approach. The contact pressures and the subsurface stresses are calculated using the plain strain assumptions and assuming no friction at the surface. For an overview of the equations used, the reader is referred to Johnson [Johnson 19851. The fast solution algorithms are based on Multilevel techniques, resulting in a major reduction of the computing time. Employing the New Life Theory formulation, the probability S of a bearing to survive more than N overrollings is given by: (22.1) The stresses are related to life using a criterion similar to Dang Van [Dang Van 19881 where the maximum shear stress over any angle is combined with the hydrostatic pressure [Ioannides 19881. A fatigue limit of the material is taken into account.
22.4
Dry contact calculations
The simplest way of accessing the stresses in the subsurface is by completely neglecting the lubricant film separating the two solid bodies, which simplifies the problem considerably. The results obtained from the dry contact model can be viewed as a pessimistic prediction, a kind of asymptotical value for very unfavourable lubricating conditions such as: High temperature, low speed, poor lubricant supply (grease lubrication) where the thickness of the film is small. The validity of the dry contact assumption will be discussed in the section on lubricated contact calculations. As mentioned before, the analysis of the dry contact case is relatively easy: An algorithm that computes the contact pressures required to carry the applied load; the deformations are calculated using the semi-infinite half space assumptions. The calculation is purely elastic
360
C H A P T E R 22. INFLUENCE OF RESIDUAL A N D STATIC STRESSES
and no plastic deformation is accounted for. The stress distribution in the subsurface is then computed, assuming zero friction at the interface of the contacting bodies and a semi-infinite half space. Finally, the stress tensor in each point is converted to a life estimate, using the stress-life criterion. In this analysis the influence of the residual stress fields is not taken into account. The calculational details are described in [Brandt 1990, Lubrecht 19891. This chapter is mainly concerned with the tribological aspects of the overrolling of dents. Two different types of analyses have been carried out; first, the static one. A parabolical body is statically loaded against the flat surface which is dented. The pressures are calculated and the stresses are computed, resulting in a life estimate which is then compared with the smooth surface life estimate. In a second type of analysis, the parabolic body is moving stepwise over the flat surface, and the maximum stress at each point of the subsurface is recorded. The disadvantage of this analysis is that it consumes much more computer time than the static one. On the other hand, with the static analysis when the contacting body is located directly over the dent the degree of conformity is increased, resulting in lower stresses and a longer life prediction for the dented surface, which is unrealistic (see figure 22.1 curve X = 0.5 mm for small depths). The dynamic analysis does not suffer from this problem, but as can be seen from figure 22.1 the resulting life reductions for larger amplitudes do not differ much except that the increase in life mentioned is not experienced in the dynamic case. The static life reduction was scaled using the dynamic smooth life in order to be able to compare the effect of the deep dents. Since the dynamic calculations agree with the static ones for the small and deep defects, the analysis of the influence of dent size and depth was carried out using the static programs in consideration of computing time. The shape of the defect is given by:
27rx D ( z ) = A * 10-lo(”/x)zcos(-)
(22.2) X where 2) is the dent shape, A is the amplitude of the dent and X is the wavelength of the dent. This shape was chosen because of its properties: no sharp corners at the edges and bulging out of the shoulders, emulating the physical dents. Figure 22.2 shows the influence of the amplitude and wavelength of the dent on the theoretical life. It can be concluded from this figure that the life reduction is proportional to the depth, and inversely proportional to the wavelength, of the dent, indicating that the slope is the important parameter. In figure 22.3 the influence of the load on life was investigated for three different dent depths. Varying the load means that both the maximum Hertzian pressure Ph and the contact width 2b change. Since the lines of different dent depths are approximately parallel to the smooth curve, it can be concluded that the life reduction is almost a constant factor, irrespective of the load. In figure 22.4 the influence of the contact size is investigated, given a constant dent geometry. As can be seen from this figure, the influence of the dent on the life reduces as the contact size increases. This is easily explained by remembering that with the increase in contact size the volume exposed to fatigue grows, whereas the stressed volume associated with the dent remains constant. As can be seen from a simple dimensional analysis, the influence of the absolute contact size
22.4. DRY CONTACT CALCULATIONS
361
dent depth
10-2
10-1
100
rel. life
Figure 22.1 Relative life as a function of the depth of a dent, for a stationary and moving contact. Hertzian pressure is 3.3 GPa, contact half width is 0.5 mm, X=0.5, 0.125 mm.
10-2
10-1
100
rel. life
Figure 22.2 Relative life as a function o f the depth of a dent, for a stationary contact. Hertzian pressure is 3.3 GPa, contact half width is 0.5 m m ,X=1.0, 0.5, 0.25, 0.125 0.0625 mm.
362
CHAPTER 22. INFLUENCE OF RESIDUAL AND STATIC STRESSES
I"
A-
--
_.
Figure 22.3 Relative life as a function ofthe load for a constant dent, of X=0.125 mm, d=2.5, 5.0, 10.0 pm, note that the vertical axis displays the maximum Hertzian pressure that belongs to a certain load.
Figure 22.4 Relative life as a function of the contact size for a constant dent, of X=0.125 mm, d=5.0, 10.0, 15.0 pm, note the two asymptotes where either the dent-refated stresses or the Hertzian stresses dominate.
22.5. LUBRICATED CALCULATIONS
0.0
I
I
I
I
I
-1.0
-5.0
I
I
I
I
I
I
1.0
-
-1.0
0.0
1.0
I
I
363
I
s
I
I
I
s
Figure 22.5 Comparison of pressure distribution from lubricated and dry contact calculation; note the inlet and pressure spike in the lower pressure profile. Hertzian pressure is 3.3 GPa, half contact size is 0.5 mm, dent: X=0.50 mm, d=3pm. on life can be omitted in the calculations, as long as the ratio of contact size and dent size and the dent slope are kept constant. The stressed volume can be added at the end of the calculation. Thus it is possible to reduce the number of relevant parameters; the remaining ones are: Hertzian pressure, ratio of the contact size and the dent size, and dent slope (and shape, of course). Furthermore, the material parameters such as elasticity modulus and fatigue limit are of interest.
22.5
Lubricated calculations
As an extension of the previous section the influence of the oil film thickness on the stresses will be taken into account in this section. By and large the oil film acts as a damper to the pressures; the thicker the film, the more the large pressure gradients are reduced. The fluid is assumed to behave in a Newtonian way, the viscosity-pressure relation according to Roelands and the density-pressure relation according to Dowson and Higginson. The calculations are described in (Venner 19891. As in the previous section the analysis was carried out statically (i.e. the dent in a constant position sliding over a smooth surface) and dynamically (i.e. time dependent calculations with different positions of the dent relative to the ‘contact’ area). In figure 22.5 the pressure profiles from the static dry contact and the static lubricated contact
364
CHAPTER 22. INFLUENCE OF RESIDUAL AND STATIC STRESSES
Figure 22.6 Photograph of an artificially created dent and spall, overrolling direction is from right t o left. calculations are compared for a certain dent (notice the difference in the inlet region and the pressure spike). The height of the pressure spike is underestimated because the grid is not fine enough, but the exact height is not our aim here. From this figure it can be concluded that for these conditions (high load, thin film) the dry contact assumption is accurate, the pressure profiles are nearly identical and the life predictions differ only by 2%. The lubricated situation is analysed for relatively small dents only. For larger amplitude to wavelength ratios, a cavitated zone will occcur at the dent location. To solve these situations, an extension of the current algorithm is required to calculate the location where the pressure build-up starts. The transient results can be of help in explaining why the spall initiates behind the dent, figure 22.6. The pressures at this spot are higher than at the leading edge of the dent, resulting in higher local stresses. This is caused by the ‘squeeze’effect included in the transient lubricated calculation and cannot be understood by using static or dry contact calculations. Consequently the stresses below the trailing edge of the dent will be significantly higher, as can be seen from figure 22.7. The total risk integral over the trailing (left) part of the contact is almost 50% higher than the same integral over the leading (right) part of the contact. Since the difference comes from the area around the dent, and since this area spans only approximately 1/20 of the total area, it can be concluded that the risk of fatigue at the trailing edge of the dent is much larger than at the leading edge. This effect is assumed to be even more pronounced at greater dent depths than the 3 pm used in this calculation. Comparing the ,510 life obtained with the lubricated transient model to the dry contact transient model (figure 22.1) the difference in life is only 6%, indicating once more the validity of the dry contact assumptions.
22.6. ENDURANCE TESTS
365
F i g u r e 22.7 Maximum subsurface stresses below a dent, employing the time-dependent lubricated calculation, the dent geometryis that offigure 22.5. The timestep is 0.05, the calculation starts with the dent at x/b = -10. The overrolling direction is from right to left.
22.6
Endurance t e s t s
The endurance tests were performed on deep groove ball bearings (6309) using a pure radial load equivalent to C / P = 2.8. At this load the maximum Hertzian pressure is 3.3 GPa and the size of the contact ellipse is 0.76 x 7.8 mm. The indentations were produced on the bearing inner rings with 1 mm hard metal balls, using different loads to produce indentations of different sizes and depths. The bearings were then assembled and run under standard “full film lubrication” conditions, i.e. at 6000 rpm with very clean mineral oil, of a paraffinic type. The inlet temperature was 25 “C, the bearing operating temperature was 58 “C. Four indentations were produced in each inner ring and after a spall was detected the test was stopped and the inner ring was inspected. All bearing failures showed that the spalls were initiated by the indentations. It was found that some of the unspalled indentations exhibited small cracks, showing that they were about to spall, indicating that the spalling time period for the different dents in one inner ring is narrow. The artificial indentations were measured and a theoretical Llo life was computed, as explained in section 22.3. As a next step, these dents were ‘placed’ in a bearing by adding to the risk integral of the dented part of the raceway the risk integral related to the undented raceway. Finally, the ,510 life was calculated for this combination and it was compared to the smooth raceway Llo. The ratios of the theoretical life predictions varied between 1/40 to 1/100 for the 250 pm dents of approximately 10 pm depth. The experimental ratios are < 1/35 to < 1/60, showing that the theoretical predictions are of the proper order of magnitude. The life of the smooth bearings (without indentations) is in excess of 500 Mrevs; the life tests were suspended after this period. As a result the experimental ratio can only be given as an upper limit. The 1510 life of bearings with 250 pm indentations of 10 pm depth was 8.1 Mrevs at C / P = 2.8. Running at lower loads C / P = 20 the experimental life was in excess of 300 Mrevs. Tests to determine the ,510 life at intermediate loads were later carried out to check the theoretical predictions in greater detail.
366
CHAPTER 22. INFLUENCE OF RESIDUAL A N D STATIC STRESSES
Comparing our theoretical and experimental results with the work of Lorosch, the prime difference is that the dents produced in our investigation are relatively shallow (15 pm depth) compared to the dents in [Lorosch 19851 (up to 40 pm depth), for a 1 mm ball and comparable dent sizes. This difference in depth cannot be explained at the moment. The life reductions reported in [Lorosch 19851 as a function of the dent size are of the same order of magnitude as the ones we obtained. The influence of the contact size on the life reduction in [Lorijsch 19851, figure 8 and figure 22.4, is very similar. Whereas the life of the dented surface (10 or 15 pm depth) remains almost constant, decreasing the contact size from 8 mm to 2 mm increases the smooth life by a factor of 12. Thus the ratio of the life reduction between 2 mm and 8 mm is approximately 12. A very similar life reduction ratio can be observed in [Lorosch 19851, figure 8.
22.7
Conclusions
Using the dry contact model, a parametric study of the influence of dents on the life has been carried out for various dent sizes and depths, and the influence of contact pressure and contact size has been investigated. It was found that the dent slope is the main parameter when studying the effect on life. As a result of the volume subjected to fatigue, the influence on life of a certain dent in a large bearing is much less than if the same dent were ‘placed’ in a small bearing. The lubricated contact calculations were then used to validate the dry contact pressure calculations; differences between dry- and lubricated life predictions were small. The lubricated time-dependent calculations explain why the spa11 is always located at the rear of the dent. Comparing the theoretical and experimental results with experimental results from [Lorosch 19851, it can be concluded that although the depth of the dents is larger the life reductions are comparable to the ones found in our analysis; the influence of the contact size on life is very similar. Of course, the (2d line contact) model used here, differs from the contact between a ball and a raceway. The contact itself is an elongated ellipse and therefore is fairly well approximated by the line contact; however, the spherical indentation is modelled as if it were a transverse trench. This means that the calculation will exaggerate the influence of the dent and the predictions can be too pessimistic. However, because the area of high risk will be concentrated in the centre of the raceway (where the highest pressures occur), the line contact analysis concentrates on the right contact patch. Therefore, it is not so surprising that the theoretical results agree so closely with the experiments, in spite of the above-mentioned differences.
Bibliography [Brandt 19901
Brandt, A., and Lubrecht, A.A., “Multilevel Matrix Multiplication and Fast Solution of Integral Equations”, 1990, to be published in the Journal of Computational Physics.
[Dang Van 19881 Dang Van, K., Griveau, B., and Message, O., “On a New Multiaxial Fatigue Limit Criterion: Theory and Application”, in “Biaxial and Multiaxial Fatigue”, editors M.W. Brown, K.J. Miller, Cambridge University Press, 1989, pp. 479-496. [Ioannides 19851 Ioannides, E., and Harris, T.A., “A New Fatigue Life Model for Rolling Bearings”, Trans. ASME JOLT, 107, 1985, pp. 367-378. [Ioannides 19881 Ioannides, E., Jacobson, B.O., and Tripp, J.H., “Prediction of Rolling Bearing Life under Practical Operating Conditions”, Proc. 15th Leeds-Lyon Symposium on Tribology, Elsevier, 1988, pp. 181-187. [Johnson 19851
Johnson, K.L., “Contact Mechanics”, Cambridge University Press, 1985.
[Lorosch 1985)
Lorosch, H.K., “Research on Longer Life for Rolling-Element Bearings”, Trans. ASLE, 41, 1985, pp. 37-43.
[Lubrecht 19891 Lubrecht, A.A., and Ioannides, E., “A Fast Solution of the Dry Contact Problem and the Associated Sub-surface Stress field using Multilevel Techniques”, presented at the 1989 ASME/STLE conference in Fort Lauderdale. [Lubrecht 19901 Lubrecht, A.A., Venner, C.H., Lane, S., Jacobson, B.O., and Ioannides, E., “Surface Damage - Comparison of Theoretical and Axperimental Endurance Lives of Rolling Bearings”, presented at the Japan International Tribology Conference, Nagoya, 1990. [Sayles 19821
Sayles, R.S., and Macpherson, P.B., “Influence of Wear Debris on Rolling Contact Fatigue”, in “Rolling Contact Fatigue Testing of Bearing Steels”, editor J.J.C. Hoo, ASTM STP 771, 1982, pp. 255-274.
[Venner 19891
Venner, C.H., ten Napel, W.E., and Bosma, R., “Advanced Multilevel Solution of the EHL Line Contact Problem”, presented at the 1989 ASME/STLE conference in Fort Lauderdale. 367
368 [Webster 19851
BIBLIOGRAPHY Webster, M.N., Ioannides, E., and Sayles, R.S., “The Effect of Topographical Defects on the Contact Stress and Fatigue Life in Rolling Element Bearings”, Proc. 12th Leeds-Lyon Symposium on Tribology, Butterworths, 1985, pp. 207-221.
22.1
Introduction
For as long as endurance tests of rolling element bearings have been carried out, the dramatic influence of surface defects on the fatigue life of the element has been well known. Irrespective of whether these surface defects were formed by the overrolling of hard or ductile particles (contaminants in the oil), whether they stem from handling damage (e.g. dropping the bearing) or whether they were caused by the manufacturing process (residual grinding grooves, grinding burns), their influence on life is detrimental, as discussed in the previous chapter. The relationship between the occurrence of these surface defects and the fatigue life has up until now only been accessed experimentally, which means running the bearings in certain batches, with a certain load, speed and lubricant supply, until they fail [Lorosch 19851. This procedure, however, being the ultimate test, is very time-consuming, costly and eventually destroys the test element. In the past, due to the lack of a proper theory relating the geometry of the dents and the fatigue life, the time-consuming numerical solution techniques and the high price of powerful computers, this problem has been very hard to solve theoretically. Three major developments have taken place recently that have changed this situation. First of all, in 1985 the New Life Theory was published by Ioannides and Harris [Ioannides 19851. This theory enables us to perform a detailed study of the relation between surface defects, the associated stresses and the fatigue life. Using this theory, observed lives of tested helicopter bearings [Sayles 19821 were predicted [Webster 19851 using traditional contact calculation methods. However, more recently a second development which consists of the extension of the Multilevel techniques to the calculation of integrals and the solution of integral equations [Brandt 1990, Lubrecht 19891 was applied to highly loaded lubricated contacts [Venner 19891. This has
357
358
CHAPTER 22. INFLUENCE OF RESIDUAL AND STATIC STRESSES
resulted in shorter computational times and smaller memory requirements. Last but not least, the rapid increase in speed and memory size of the micro- and (super) mini computers has enabled a theoretical study of the influence of realistic surface features on bearing fatigue life. Of course, the main issue remains whether or not these theoretical calculations correctly predict the fatigue life of the bearings. For this purpose endurance tests were performed. This chapter consists of three main parts; emphasis will be on the first topic: Dry contact calculations of the fatigue life of dented raceways. The dents were modelled by relatively simple mathematical equations, and the influence of the depth and size was investigated. In the second part calculations were performed of a lubricated contact. These computations extend the dry contact calculations and shed some light on the directional preference of spa11 initiation with respect to the overrolling direction. In the third and last part, endurance tests are described with artificially dented raceways. The experimentally established lives are compared with the theoretical predictions. With the present state of bearing steel technology and bearing manufacture it is possible to obtain infinite bearing life under well-controlled endurance test conditions. These conditions include careful handling and mounting of the bearings in a well-designed test rig, and filtering the lubricating oil with the utmost care. Under these conditions lives are infinite on a practical time-scale (> lo9 revolutions). However, these laboratory conditions will not be met in many practical applications, and as a result the bearing will perform less well. It is for this reason that more and more research is devoted to understanding the processes taking place under sub-optimal conditions. These conditions might take dynamic loading into account, as well as starved lubrication, chemical influences (atmosphere) and the influence of hoop stresses and surface roughness of the bearing raceways on the fatigue life. A more complete overview is given in chapter 23. This chapter is devoted to the effect of dents in the raceways on bearing life because they occur frequently in practice and because of the relative ease with which they can be modelled numerically. These defects may originate from the overrolling of contaminants in the oil or may have been caused by handling damage. As a result of these defects the stresses in the subsurface will be substantially higher than in the smooth Hertzian case, and consequently the life will be shorter.
22.2 b C
C e
h
LlO N
P
ph
v,
5,z 2’
Notation Hertzian halfwidth, [m] stress exponent dynamic load capacity, [N] Weibull slope depth exponent life that 90% of a bearing population exceeds number of stress cycles load, [N] maximum Hertzian pressure, [Pa] stressed volume where the fatigue limit is exceeded, [m3] coordinates, [m] stress weighted average depth, [m]
22.3. THEORETICAL MODEL X u 0%
A
ID
S
22.3
359
wavelength used to model dents, [m] stress, [Pa] fatigue limit of the material, [Pa] amplitude used to model dents, [m] shape of the dent, [m] probability of survival
Theoretical model
The model used in the next sections is basically two-dimensional, the so-called line contact model describing the contact between two long parallel cylinders. The main advantage of the model is that the number of nodal points required for a detailed description of the surface features, the contact pressures and the stresses is relatively small. For the contact pressure approximately 1000 points were used, whereas the stresses were calculated on an equivalent 1000 x 500 grid. For a realistic description of a non-smooth point contact, at least 500 x 500 points on the surface would be needed, making this problem very hard to solve with the present techniques and hardware. As a result the present analysis is limited to the line contact approach. The contact pressures and the subsurface stresses are calculated using the plain strain assumptions and assuming no friction at the surface. For an overview of the equations used, the reader is referred to Johnson [Johnson 19851. The fast solution algorithms are based on Multilevel techniques, resulting in a major reduction of the computing time. Employing the New Life Theory formulation, the probability S of a bearing to survive more than N overrollings is given by: (22.1) The stresses are related to life using a criterion similar to Dang Van [Dang Van 19881 where the maximum shear stress over any angle is combined with the hydrostatic pressure [Ioannides 19881. A fatigue limit of the material is taken into account.
22.4
Dry contact calculations
The simplest way of accessing the stresses in the subsurface is by completely neglecting the lubricant film separating the two solid bodies, which simplifies the problem considerably. The results obtained from the dry contact model can be viewed as a pessimistic prediction, a kind of asymptotical value for very unfavourable lubricating conditions such as: High temperature, low speed, poor lubricant supply (grease lubrication) where the thickness of the film is small. The validity of the dry contact assumption will be discussed in the section on lubricated contact calculations. As mentioned before, the analysis of the dry contact case is relatively easy: An algorithm that computes the contact pressures required to carry the applied load; the deformations are calculated using the semi-infinite half space assumptions. The calculation is purely elastic
360
C H A P T E R 22. INFLUENCE OF RESIDUAL A N D STATIC STRESSES
and no plastic deformation is accounted for. The stress distribution in the subsurface is then computed, assuming zero friction at the interface of the contacting bodies and a semi-infinite half space. Finally, the stress tensor in each point is converted to a life estimate, using the stress-life criterion. In this analysis the influence of the residual stress fields is not taken into account. The calculational details are described in [Brandt 1990, Lubrecht 19891. This chapter is mainly concerned with the tribological aspects of the overrolling of dents. Two different types of analyses have been carried out; first, the static one. A parabolical body is statically loaded against the flat surface which is dented. The pressures are calculated and the stresses are computed, resulting in a life estimate which is then compared with the smooth surface life estimate. In a second type of analysis, the parabolic body is moving stepwise over the flat surface, and the maximum stress at each point of the subsurface is recorded. The disadvantage of this analysis is that it consumes much more computer time than the static one. On the other hand, with the static analysis when the contacting body is located directly over the dent the degree of conformity is increased, resulting in lower stresses and a longer life prediction for the dented surface, which is unrealistic (see figure 22.1 curve X = 0.5 mm for small depths). The dynamic analysis does not suffer from this problem, but as can be seen from figure 22.1 the resulting life reductions for larger amplitudes do not differ much except that the increase in life mentioned is not experienced in the dynamic case. The static life reduction was scaled using the dynamic smooth life in order to be able to compare the effect of the deep dents. Since the dynamic calculations agree with the static ones for the small and deep defects, the analysis of the influence of dent size and depth was carried out using the static programs in consideration of computing time. The shape of the defect is given by:
27rx D ( z ) = A * 10-lo(”/x)zcos(-)
(22.2) X where 2) is the dent shape, A is the amplitude of the dent and X is the wavelength of the dent. This shape was chosen because of its properties: no sharp corners at the edges and bulging out of the shoulders, emulating the physical dents. Figure 22.2 shows the influence of the amplitude and wavelength of the dent on the theoretical life. It can be concluded from this figure that the life reduction is proportional to the depth, and inversely proportional to the wavelength, of the dent, indicating that the slope is the important parameter. In figure 22.3 the influence of the load on life was investigated for three different dent depths. Varying the load means that both the maximum Hertzian pressure Ph and the contact width 2b change. Since the lines of different dent depths are approximately parallel to the smooth curve, it can be concluded that the life reduction is almost a constant factor, irrespective of the load. In figure 22.4 the influence of the contact size is investigated, given a constant dent geometry. As can be seen from this figure, the influence of the dent on the life reduces as the contact size increases. This is easily explained by remembering that with the increase in contact size the volume exposed to fatigue grows, whereas the stressed volume associated with the dent remains constant. As can be seen from a simple dimensional analysis, the influence of the absolute contact size
22.4. DRY CONTACT CALCULATIONS
361
dent depth
10-2
10-1
100
rel. life
Figure 22.1 Relative life as a function of the depth of a dent, for a stationary and moving contact. Hertzian pressure is 3.3 GPa, contact half width is 0.5 mm, X=0.5, 0.125 mm.
10-2
10-1
100
rel. life
Figure 22.2 Relative life as a function o f the depth of a dent, for a stationary contact. Hertzian pressure is 3.3 GPa, contact half width is 0.5 m m ,X=1.0, 0.5, 0.25, 0.125 0.0625 mm.
362
CHAPTER 22. INFLUENCE OF RESIDUAL AND STATIC STRESSES
I"
A-
--
_.
Figure 22.3 Relative life as a function ofthe load for a constant dent, of X=0.125 mm, d=2.5, 5.0, 10.0 pm, note that the vertical axis displays the maximum Hertzian pressure that belongs to a certain load.
Figure 22.4 Relative life as a function of the contact size for a constant dent, of X=0.125 mm, d=5.0, 10.0, 15.0 pm, note the two asymptotes where either the dent-refated stresses or the Hertzian stresses dominate.
22.5. LUBRICATED CALCULATIONS
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Figure 22.5 Comparison of pressure distribution from lubricated and dry contact calculation; note the inlet and pressure spike in the lower pressure profile. Hertzian pressure is 3.3 GPa, half contact size is 0.5 mm, dent: X=0.50 mm, d=3pm. on life can be omitted in the calculations, as long as the ratio of contact size and dent size and the dent slope are kept constant. The stressed volume can be added at the end of the calculation. Thus it is possible to reduce the number of relevant parameters; the remaining ones are: Hertzian pressure, ratio of the contact size and the dent size, and dent slope (and shape, of course). Furthermore, the material parameters such as elasticity modulus and fatigue limit are of interest.
22.5
Lubricated calculations
As an extension of the previous section the influence of the oil film thickness on the stresses will be taken into account in this section. By and large the oil film acts as a damper to the pressures; the thicker the film, the more the large pressure gradients are reduced. The fluid is assumed to behave in a Newtonian way, the viscosity-pressure relation according to Roelands and the density-pressure relation according to Dowson and Higginson. The calculations are described in (Venner 19891. As in the previous section the analysis was carried out statically (i.e. the dent in a constant position sliding over a smooth surface) and dynamically (i.e. time dependent calculations with different positions of the dent relative to the ‘contact’ area). In figure 22.5 the pressure profiles from the static dry contact and the static lubricated contact
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CHAPTER 22. INFLUENCE OF RESIDUAL AND STATIC STRESSES
Figure 22.6 Photograph of an artificially created dent and spall, overrolling direction is from right t o left. calculations are compared for a certain dent (notice the difference in the inlet region and the pressure spike). The height of the pressure spike is underestimated because the grid is not fine enough, but the exact height is not our aim here. From this figure it can be concluded that for these conditions (high load, thin film) the dry contact assumption is accurate, the pressure profiles are nearly identical and the life predictions differ only by 2%. The lubricated situation is analysed for relatively small dents only. For larger amplitude to wavelength ratios, a cavitated zone will occcur at the dent location. To solve these situations, an extension of the current algorithm is required to calculate the location where the pressure build-up starts. The transient results can be of help in explaining why the spall initiates behind the dent, figure 22.6. The pressures at this spot are higher than at the leading edge of the dent, resulting in higher local stresses. This is caused by the ‘squeeze’effect included in the transient lubricated calculation and cannot be understood by using static or dry contact calculations. Consequently the stresses below the trailing edge of the dent will be significantly higher, as can be seen from figure 22.7. The total risk integral over the trailing (left) part of the contact is almost 50% higher than the same integral over the leading (right) part of the contact. Since the difference comes from the area around the dent, and since this area spans only approximately 1/20 of the total area, it can be concluded that the risk of fatigue at the trailing edge of the dent is much larger than at the leading edge. This effect is assumed to be even more pronounced at greater dent depths than the 3 pm used in this calculation. Comparing the ,510 life obtained with the lubricated transient model to the dry contact transient model (figure 22.1) the difference in life is only 6%, indicating once more the validity of the dry contact assumptions.
22.6. ENDURANCE TESTS
365
F i g u r e 22.7 Maximum subsurface stresses below a dent, employing the time-dependent lubricated calculation, the dent geometryis that offigure 22.5. The timestep is 0.05, the calculation starts with the dent at x/b = -10. The overrolling direction is from right to left.
22.6
Endurance t e s t s
The endurance tests were performed on deep groove ball bearings (6309) using a pure radial load equivalent to C / P = 2.8. At this load the maximum Hertzian pressure is 3.3 GPa and the size of the contact ellipse is 0.76 x 7.8 mm. The indentations were produced on the bearing inner rings with 1 mm hard metal balls, using different loads to produce indentations of different sizes and depths. The bearings were then assembled and run under standard “full film lubrication” conditions, i.e. at 6000 rpm with very clean mineral oil, of a paraffinic type. The inlet temperature was 25 “C, the bearing operating temperature was 58 “C. Four indentations were produced in each inner ring and after a spall was detected the test was stopped and the inner ring was inspected. All bearing failures showed that the spalls were initiated by the indentations. It was found that some of the unspalled indentations exhibited small cracks, showing that they were about to spall, indicating that the spalling time period for the different dents in one inner ring is narrow. The artificial indentations were measured and a theoretical Llo life was computed, as explained in section 22.3. As a next step, these dents were ‘placed’ in a bearing by adding to the risk integral of the dented part of the raceway the risk integral related to the undented raceway. Finally, the ,510 life was calculated for this combination and it was compared to the smooth raceway Llo. The ratios of the theoretical life predictions varied between 1/40 to 1/100 for the 250 pm dents of approximately 10 pm depth. The experimental ratios are < 1/35 to < 1/60, showing that the theoretical predictions are of the proper order of magnitude. The life of the smooth bearings (without indentations) is in excess of 500 Mrevs; the life tests were suspended after this period. As a result the experimental ratio can only be given as an upper limit. The 1510 life of bearings with 250 pm indentations of 10 pm depth was 8.1 Mrevs at C / P = 2.8. Running at lower loads C / P = 20 the experimental life was in excess of 300 Mrevs. Tests to determine the ,510 life at intermediate loads were later carried out to check the theoretical predictions in greater detail.
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CHAPTER 22. INFLUENCE OF RESIDUAL A N D STATIC STRESSES
Comparing our theoretical and experimental results with the work of Lorosch, the prime difference is that the dents produced in our investigation are relatively shallow (15 pm depth) compared to the dents in [Lorosch 19851 (up to 40 pm depth), for a 1 mm ball and comparable dent sizes. This difference in depth cannot be explained at the moment. The life reductions reported in [Lorosch 19851 as a function of the dent size are of the same order of magnitude as the ones we obtained. The influence of the contact size on the life reduction in [Lorijsch 19851, figure 8 and figure 22.4, is very similar. Whereas the life of the dented surface (10 or 15 pm depth) remains almost constant, decreasing the contact size from 8 mm to 2 mm increases the smooth life by a factor of 12. Thus the ratio of the life reduction between 2 mm and 8 mm is approximately 12. A very similar life reduction ratio can be observed in [Lorosch 19851, figure 8.
22.7
Conclusions
Using the dry contact model, a parametric study of the influence of dents on the life has been carried out for various dent sizes and depths, and the influence of contact pressure and contact size has been investigated. It was found that the dent slope is the main parameter when studying the effect on life. As a result of the volume subjected to fatigue, the influence on life of a certain dent in a large bearing is much less than if the same dent were ‘placed’ in a small bearing. The lubricated contact calculations were then used to validate the dry contact pressure calculations; differences between dry- and lubricated life predictions were small. The lubricated time-dependent calculations explain why the spa11 is always located at the rear of the dent. Comparing the theoretical and experimental results with experimental results from [Lorosch 19851, it can be concluded that although the depth of the dents is larger the life reductions are comparable to the ones found in our analysis; the influence of the contact size on life is very similar. Of course, the (2d line contact) model used here, differs from the contact between a ball and a raceway. The contact itself is an elongated ellipse and therefore is fairly well approximated by the line contact; however, the spherical indentation is modelled as if it were a transverse trench. This means that the calculation will exaggerate the influence of the dent and the predictions can be too pessimistic. However, because the area of high risk will be concentrated in the centre of the raceway (where the highest pressures occur), the line contact analysis concentrates on the right contact patch. Therefore, it is not so surprising that the theoretical results agree so closely with the experiments, in spite of the above-mentioned differences.
Bibliography [Brandt 19901
Brandt, A., and Lubrecht, A.A., “Multilevel Matrix Multiplication and Fast Solution of Integral Equations”, 1990, to be published in the Journal of Computational Physics.
[Dang Van 19881 Dang Van, K., Griveau, B., and Message, O., “On a New Multiaxial Fatigue Limit Criterion: Theory and Application”, in “Biaxial and Multiaxial Fatigue”, editors M.W. Brown, K.J. Miller, Cambridge University Press, 1989, pp. 479-496. [Ioannides 19851 Ioannides, E., and Harris, T.A., “A New Fatigue Life Model for Rolling Bearings”, Trans. ASME JOLT, 107, 1985, pp. 367-378. [Ioannides 19881 Ioannides, E., Jacobson, B.O., and Tripp, J.H., “Prediction of Rolling Bearing Life under Practical Operating Conditions”, Proc. 15th Leeds-Lyon Symposium on Tribology, Elsevier, 1988, pp. 181-187. [Johnson 19851
Johnson, K.L., “Contact Mechanics”, Cambridge University Press, 1985.
[Lorosch 1985)
Lorosch, H.K., “Research on Longer Life for Rolling-Element Bearings”, Trans. ASLE, 41, 1985, pp. 37-43.
[Lubrecht 19891 Lubrecht, A.A., and Ioannides, E., “A Fast Solution of the Dry Contact Problem and the Associated Sub-surface Stress field using Multilevel Techniques”, presented at the 1989 ASME/STLE conference in Fort Lauderdale. [Lubrecht 19901 Lubrecht, A.A., Venner, C.H., Lane, S., Jacobson, B.O., and Ioannides, E., “Surface Damage - Comparison of Theoretical and Axperimental Endurance Lives of Rolling Bearings”, presented at the Japan International Tribology Conference, Nagoya, 1990. [Sayles 19821
Sayles, R.S., and Macpherson, P.B., “Influence of Wear Debris on Rolling Contact Fatigue”, in “Rolling Contact Fatigue Testing of Bearing Steels”, editor J.J.C. Hoo, ASTM STP 771, 1982, pp. 255-274.
[Venner 19891
Venner, C.H., ten Napel, W.E., and Bosma, R., “Advanced Multilevel Solution of the EHL Line Contact Problem”, presented at the 1989 ASME/STLE conference in Fort Lauderdale. 367
368 [Webster 19851
BIBLIOGRAPHY Webster, M.N., Ioannides, E., and Sayles, R.S., “The Effect of Topographical Defects on the Contact Stress and Fatigue Life in Rolling Element Bearings”, Proc. 12th Leeds-Lyon Symposium on Tribology, Butterworths, 1985, pp. 207-221.