**4.1. Third body transfer model**

The third-body model concept involves mass transfer of the solid film lubricant from its source to the contacting surfaces within the rolling-contact system. For example, if a specimen of silver that is stationary is pressed into contact with a rotating shaft the resulting contact will enable transfer of the soft silver from the specimen to the contact surface of the shaft. The rate of silver transfer is related to: i) the force of the contact, ii) the surface roughness and speed of the shaft, and iii) the rate at which excess silver is pushed out of the contact area. Concerning the RCF tests in Figures 8 through 10, the source of the solid lubricant is the amount of thin-film coating on the balls at the beginning of the test, approximately 200 nm. The film lubricant is transferred to the rod and race contact areas, which are considered the third-body volumes. Depending on surface roughness, more or less lubricant may accumulate in the valleys between surface asperities on the rod and race surfaces. The solid lubricant on the ball surface represents the source or input to the thirdbody concept.

106 Performance Evaluation of Bearings

Baseline M50

Carbon 0.97

Manganese 0.32 Molybdenum 4.45

Vanadium 1.21

chapter 8 of (Bhushan, 1999).

**4.1. Third body transfer model**

Bearing Steel

derived from energy dispersive spectroscopy using a SEM.

**4. Thin film solid lubrication modelling** 

Post-test autopsy and Scanning Electron Microscopy (SEM) results may be used to approximate the amount of silver remaining on the ball surface at the end of each test. Table 4 presents element composition of one non-coated and two silver coated balls after testing. These balls account for three types of SEM test results: before coating, after silver depletion failure, and after early spall failure. The surface composition of each ball was derived from energy spectroscopy analyses attached to the SEM instrument. Concerning Table 4, post-test SEM data from a silver depletion failure is shown in column 3 with no silver present. All of the silver was transferred from the ball surface on to the rod and race surfaces, representing termination of the third body transfer mechanism suggested in (Higgs and Wornyoh, 2008). The results of column 4 however show some silver was still present on the ball surface following early spall failure after 9.1 hours. A theoretical analysis for a third-body transfer model using UHV-RCF data collected from suspended tests is presented in the next section.

> SEM 12.7 mm M50 ball, silver depletion failure.

Chromium 4.18 1.58 2.07 Iron 88.60 97.84 77.83

Silicon 0.27 0.48 0.68

Silver 0.00 16.37

**Table 4.** Element composition of one non-coated ball and two coated balls after testing. Composition

Test life comparison using two modeling approaches is the focus of this section. The first model uses a conservation of mass approach and is based on the work presented in (Higg and Wornyoh, 2008), (Danyluk and Dhingra, 2012a). A third-body mass transfer concept is applied to account for the transport of the film from the ball surface to the rod and race contact surfaces as seen experimentally in Figures 8 and 9. The second modeling approach is similar to the Lundberg-Palmgren model in that the RCF data from the test configurations of Table 1 are used to fit a load-capacity parameter, *C*, to *L10* data similar to that found in

The third-body model concept involves mass transfer of the solid film lubricant from its source to the contacting surfaces within the rolling-contact system. For example, if a specimen of silver that is stationary is pressed into contact with a rotating shaft the resulting contact will enable transfer of the soft silver from the specimen to the contact surface of the

SEM 12.7 mm M50 ball, early spall failure.

The control volume fraction coverage model (CVFC) has been presented and explained in (Higgs and Wornyoh ,2008), however, some parts of that formulation are presented here for clarity. The assumptions of the CVFC model for solid lubrication transfer to the third-body volumes are as follows: i) the ball/rod and ball/race contact surfaces are flat within their contact areas, ii) incipient sliding occurs between surfaces due to elastic deformation, iii) the fractional response and friction of the interfaces is primarily a function of the amount of silver present in the third-body volumes of the race and rod, and on the surface of the ball.

A conservation of mass formulation for the transfer of film lubricant from the ball surface to the wear tracks of the race and rod is as,

$$
\begin{pmatrix} \text{Third Body} \\ \text{Storage Rate} \end{pmatrix} = \begin{pmatrix} \text{Third Body} \\ \text{Input Rate} \end{pmatrix} - \begin{pmatrix} \text{Third Body} \\ \text{Output Rate} \end{pmatrix}. \tag{1}
$$

The output rate in equation (1) is driven by the load between the ball-rod and ball-race that forces some of the solid silver out of the wear track. Examination of the wear tracks on the races and on the rod and race in Figures 8 through 10 illustrate that silver is pushed outside of the CVFC volume over time, and hence removed from the third-body storage volumes. The input rate to the third-body storage volumes of the rod and race contact zones is influenced by the fiction coefficient between the solid lubricant and contact area. Concerning RCF contact and Figures 8 through 10, incipient sliding between the ball-rod and ball-race is assumed throughout this formulation.

Equation (1) may be described as the rate of change of the fractional coverage, *X(t)* of the third-bodies on both the rod and race wear tracks. For the present study, *X(t)* will be normalized to the average surface roughness of the race and rod as presented in Table 3, or approximately 250 nm and represents the maximum asperity height defined as, *h*max. The asperity depth is about the same as the initial silver coating thickness on the balls as well, approximately 200 nm. Following the form of (Higgs and Wornyoh, 2008), and (Danyluk and Dhingra, 2012a) the fractional coverage variable is defined as,

$$X = \frac{h}{h\_{\text{max}}},$$

where *h* is the local height of silver coating in the third-body volumes. Archard's volume wear rate law is used to account for surface wear interactions and is defined as,

$$\frac{dV}{dt} = KF\_N \mathcal{U}\_\prime \tag{3}$$

Rolling Contact Fatigue in Ultra High Vacuum 109

(m2)

**Figure 11.** Fractional coverage of the third body volume calculated using equations (5) and (6) with

K (m2/N) Kbc Kbr KbEc KbEr Test Configuration 3 1.0E-15 2.0E-16 1.0E-15 5.0E-17 Test Configuration 1 1.0E-15 2.0E-15 1.0E-15 2.0E-17

Hertzian Contact Stress (GPa) Fr (N) Fc (N) Third Body Surface Area

2.8 101 56 5.4E-05

2.2 67 37 3.7E-05

Test Configuration 3 3.7 264 145 7.2E-05

Test Configuration 1 3.5 237 130 6.2E-05

Observation of the curves in Figure 11 suggests steady-state third-body coverage after 400 seconds of RCF testing. For comparison, thickness measurements of the silver remaining on the balls of suspended RCF tests reveal a third-body-coverage steady state value, *Xss*, between 0.46 to 0.61 when testing in configuration 3 and, *Xss* between 0.68 and 0.89 when testing in configuration 1. This data was collected from suspended tests similar to Figure 8 and do not include spall failures. These values represent a range of steady state fractional coverage for each configuration and are shown in Figure 11 as dashed lines with solid lines

values from Tables 2, 5, and 6 for two test configurations from Table 1.

**Table 5.** Wear coefficients used in equations (5) and (6).

**Table 6.** Normal forces and third-body contact area calculations.

above and below indicating the range of measured coverage, *Xss*.

where *V*, *K*, *FN*, and *U* are the volume, wear coefficient, normal force, and sliding velocity, respectively. The wear coefficient *K* is the probability that a surface is being worn due to sliding contact, and for this section incipient sliding assumed. Combining equations (1) through (3) gives the following differential equation for *X(t)* as,

$$A\hbar\_{\text{max}} \frac{dX}{dt} = \left(K\_{bc}F\_c\mathbf{U}\_c + K\_{br}F\_r\mathbf{U}\_r\right)\left(\mathbf{1} - \mathbf{X}\right) - \left(K\_{b\to c}F\_c\mathbf{U}\_c + K\_{b\to r}F\_r\mathbf{U}\_r\right)\mathbf{X}\_r\tag{4}$$

where the first term on the right hand side accounts for third body input and the second term for third body removal. The solution of Eq. (4) is given as:

$$X(t) = \frac{K\_{bc}F\_cU\_c + K\_{br}F\_rU\_r}{K\_{bc}F\_cU\_c + K\_{br}F\_rU\_r + K\_{bE}F\_cU\_c + K\_{bE}F\_rU\_r} \left(1 - \exp\left(-\frac{t}{\tau}\right)\right). \tag{5}$$

The constants *Kbc* and *Kbr* are the wear coefficients for silver between the ball-race and the ball-rod, respectively, and influence how the third body is filled with silver from incipient sliding during the test. The constants *KbEc* and *KbEr* are the wear coefficients for the silver that is pushed out of the wear track between the ball-race and ball-rod. The wear coefficients *KbEc* and *KbEr* influence how much silver is removed from the third-body due to ball sliding with the edge of the wear track during the test as shown in Figures 8 and 9. The time constant in equation 5 is defined as,

$$\tau = \frac{A h\_{\text{max}}}{K\_{bc} F\_c U\_c + K\_{br} F\_r U\_r + K\_{bEc} F\_c U\_c + K\_{bEr} F\_r U\_r} \, \tag{6}$$

and defines the time to steady state third-body thickness. It was found that also correlates with the run-in time of the RCF test configurations in Table 1. The condition *X(t) >* 0 signifies that silver is being transferred from the ball surface to the third-body volumes on the race and rod. When all of the silver has been transferred from the ball, the condition *X(t) =* 1 exists and the third body input rate goes to zero as defined in equation (4). As the thirdbody volume becomes depleted, that is, as silver is pushed out of the wear track as defined in the second term on the right hand side of equation (4), the test results of Figure 9 and Table 4 column 3 begins to occur. As soon as the input to the third-body volumes ceases, the volume coverage *X(t)* diminishes resulting in asperity-to-asperity contact such that friction and vibration increase and the stopping threshold criteria of Section 2.3 is exceeded.

Equation (5) is plotted in Figure 11 using the material properties, wear coefficients, and loads presented Tables 2, 5, and 6. The wear coefficients presented in Table 5 are within the range and order of magnitude of those tested between bearing steels like Rex20 and silver, and those tested between Si3N4 and silver under UHV conditions found in references (Holmberg and Matthews, 2009) and (NASA/TM 1999-209088, 1999). Table 6 contains normal load and contact area calculations from the RCF test rig of Figure 2c and are used in the calculations of equations (5) and (6).

**Figure 11.** Fractional coverage of the third body volume calculated using equations (5) and (6) with values from Tables 2, 5, and 6 for two test configurations from Table 1.



in equation 5 is defined as,

, *<sup>N</sup> dV KF U*

where *V*, *K*, *FN*, and *U* are the volume, wear coefficient, normal force, and sliding velocity, respectively. The wear coefficient *K* is the probability that a surface is being worn due to sliding contact, and for this section incipient sliding assumed. Combining equations (1)

> max 1 , *bc c c br r r bEc c c bEr r r dX Ah K FU K FU X K FU K FU X*

where the first term on the right hand side accounts for third body input and the second

The constants *Kbc* and *Kbr* are the wear coefficients for silver between the ball-race and the ball-rod, respectively, and influence how the third body is filled with silver from incipient sliding during the test. The constants *KbEc* and *KbEr* are the wear coefficients for the silver that is pushed out of the wear track between the ball-race and ball-rod. The wear coefficients *KbEc* and *KbEr* influence how much silver is removed from the third-body due to ball sliding with the edge of the wear track during the test as shown in Figures 8 and 9. The time constant

*bc c c br r r bEc c c bEr r r*

*Ah K FU K FU K FU K FU*

with the run-in time of the RCF test configurations in Table 1. The condition *X(t) >* 0 signifies that silver is being transferred from the ball surface to the third-body volumes on the race and rod. When all of the silver has been transferred from the ball, the condition *X(t) =* 1 exists and the third body input rate goes to zero as defined in equation (4). As the thirdbody volume becomes depleted, that is, as silver is pushed out of the wear track as defined in the second term on the right hand side of equation (4), the test results of Figure 9 and Table 4 column 3 begins to occur. As soon as the input to the third-body volumes ceases, the volume coverage *X(t)* diminishes resulting in asperity-to-asperity contact such that friction

and vibration increase and the stopping threshold criteria of Section 2.3 is exceeded.

Equation (5) is plotted in Figure 11 using the material properties, wear coefficients, and loads presented Tables 2, 5, and 6. The wear coefficients presented in Table 5 are within the range and order of magnitude of those tested between bearing steels like Rex20 and silver, and those tested between Si3N4 and silver under UHV conditions found in references (Holmberg and Matthews, 2009) and (NASA/TM 1999-209088, 1999). Table 6 contains normal load and contact area calculations from the RCF test rig of Figure 2c and are used in

and defines the time to steady state third-body thickness. It was found that

(4)

(5)

max ,

through (3) gives the following differential equation for *X(t)* as,

term for third body removal. The solution of Eq. (4) is given as:

 ( ) 1 exp . *bc c c br r r bc c c br r r bEc c c bEr r r K FU K FU <sup>t</sup> X t K FU K FU K FU K FU*

*dt*

the calculations of equations (5) and (6).

*dt* (3)

(6)

also correlates


**Table 6.** Normal forces and third-body contact area calculations.

Observation of the curves in Figure 11 suggests steady-state third-body coverage after 400 seconds of RCF testing. For comparison, thickness measurements of the silver remaining on the balls of suspended RCF tests reveal a third-body-coverage steady state value, *Xss*, between 0.46 to 0.61 when testing in configuration 3 and, *Xss* between 0.68 and 0.89 when testing in configuration 1. This data was collected from suspended tests similar to Figure 8 and do not include spall failures. These values represent a range of steady state fractional coverage for each configuration and are shown in Figure 11 as dashed lines with solid lines above and below indicating the range of measured coverage, *Xss*.

Comparison of the measured steady state coverage, *Xss* with that calculated from solution of equation (4) shows good agreement between measured and predicted third body fractional coverage using the wear coefficient values presented in Table 5. The trending of coverage, *Xss*, to the same steady state values for each of the configurations 1 and 3 is due to the material type and loading conditions related to the RCF test setup. The run-in time for each of the test configurations 1 and 3 is comparable with the transient portion of the curves in Figure 11, suggesting that the volumes between asperities on the rod and races fill-up within the first 10 minutes of the test rotating at 130 Hz.

A steady state wear factor for the depletion of silver from the ball may be calculated using Archard's wear equation integrated over time as,

$$\mathbf{V}\_{ball} = \int\_{0}^{t\_f} \mathbf{K}\_{ball} F\_{ball} \mathbf{L} I\_{ball} \left(\mathbf{1} - \mathbf{X}(t)\right) dt. \tag{7}$$

Solution of equation (7) and application of equation (5), the ball steady state wear factor may be expressed as,

$$\varphi = \frac{V\_{ball}}{F\_{ball}t\_f \mathcal{U}\_{ball}} \mathcal{g}\_{\prime} \tag{8}$$

Rolling Contact Fatigue in Ultra High Vacuum 111

(9)

coating-life within the range of the stresses tested. In this section a Lundberg-Palmgren model is used to back-calculate a basic load capacity parameter, *C*, for each of the test configurations of Table 1. The load capacity parameter may then be used to plan the length of any UHV-RCF test based on the test-load for each configuration. The load capacity calculation follows (Bhushan, 1999) chapter 8. The stress cycles corresponding to 10% failure

> <sup>10</sup> , *<sup>C</sup> <sup>L</sup> W*

where the variable *W* corresponds to the radial load applied to the ball, and *C* is the basic load capacity of the test configuration with respect to a ball-bearing type system. The basic load capacity parameter, *C* may be calculated using the RCF cycles-to-failure results similar to those presented in Figures 5 through 7. The values of *L*10 and *W* were measured for each of the test configurations shown in Table 1. Using data from Table 6 the *L10* life for each test configuration and loading is plotted as a function of load capacity in Figure 12. The measured *L*10 life for different contact stresses is also plotted in Figure 12 as well,

represented as vertical lines (large and small dashed lines, and one dash-dot line).

**Figure 12.** Plot of the natural log of *L10* stress cycles verses load capacity parameter, *C*, using equation (9) for three test configurations in Table 1. Vertical lines represent measured *L10* life for three test

The Lundberg-Palmgren model in equation (9) may require life-adjustment factors to fit the model to experimental data. Concerning Figure 12, it is to be expected that the load capacity parameter, *C*, will be the same for each test configuration independent of loading.

3

maybe calculated as,

configurations and three loads (GPa).

where *g* is the gravitational constant and *tf* is the time to failure based on the stopping threshold criteria 0.35g. Table 7 contains evaluation of Equation (8) using RCF depletionfailure data from Figures 6 and 7. Spall failures were not included in the wear factor calculations of Table 7. Configuration 2 shows the smallest wear factor and had the longest RCF test life. The wear factors of configurations 1 and 3 are about the same suggesting similar test-time results using either the Rex20 rod or the Si3N4 rod with 12.7 mm balls. The result that wear factors for configurations 1 and 3 are similar regardless of rod type suggests that most of the third-body storage volume resides on the race. This is confirmed from the autopsy results of Figures 9 and 10 in that silver has been pushed out of the wear track on the race.


**Table 7.** Steady state wear factor of the ball, calculated using data from all non-spall RCF tests.

### **4.2. Lundberg-Palmgren emperical model**

Empirical modeling with ex-situ data allows coating life prediction based on past performance. RCF data collected over a range of contact stresses may be used to extrapolate coating-life within the range of the stresses tested. In this section a Lundberg-Palmgren model is used to back-calculate a basic load capacity parameter, *C*, for each of the test configurations of Table 1. The load capacity parameter may then be used to plan the length of any UHV-RCF test based on the test-load for each configuration. The load capacity calculation follows (Bhushan, 1999) chapter 8. The stress cycles corresponding to 10% failure maybe calculated as,

110 Performance Evaluation of Bearings

be expressed as,

the race.

Table 1 Test

Wear Factor

**4.2. Lundberg-Palmgren emperical model**

the first 10 minutes of the test rotating at 130 Hz.

Archard's wear equation integrated over time as,

Comparison of the measured steady state coverage, *Xss* with that calculated from solution of equation (4) shows good agreement between measured and predicted third body fractional coverage using the wear coefficient values presented in Table 5. The trending of coverage, *Xss*, to the same steady state values for each of the configurations 1 and 3 is due to the material type and loading conditions related to the RCF test setup. The run-in time for each of the test configurations 1 and 3 is comparable with the transient portion of the curves in Figure 11, suggesting that the volumes between asperities on the rod and races fill-up within

A steady state wear factor for the depletion of silver from the ball may be calculated using

Solution of equation (7) and application of equation (5), the ball steady state wear factor may

where *g* is the gravitational constant and *tf* is the time to failure based on the stopping threshold criteria 0.35g. Table 7 contains evaluation of Equation (8) using RCF depletionfailure data from Figures 6 and 7. Spall failures were not included in the wear factor calculations of Table 7. Configuration 2 shows the smallest wear factor and had the longest RCF test life. The wear factors of configurations 1 and 3 are about the same suggesting similar test-time results using either the Rex20 rod or the Si3N4 rod with 12.7 mm balls. The result that wear factors for configurations 1 and 3 are similar regardless of rod type suggests that most of the third-body storage volume resides on the race. This is confirmed from the autopsy results of Figures 9 and 10 in that silver has been pushed out of the wear track on

Configuration Configuration 1 Configuration 2 Configuration 3 Contact stress 3.5 GPa 2.2 GPa 4.1 GPa 3.5 GPa 3.7 GPa 2.8 GPa

3 11 *cm cm kg* 3.47E-10 3.23E-10 7.49E-11 3.37E-11 3.12E-10 2.34E-10

Empirical modeling with ex-situ data allows coating life prediction based on past performance. RCF data collected over a range of contact stresses may be used to extrapolate

**Table 7.** Steady state wear factor of the ball, calculated using data from all non-spall RCF tests.

, *ball ball f ball V <sup>g</sup> F tU*

0

*f t*

1 () .

*V K F U X t dt ball ball ball ball* (7)

(8)

$$L\_{10} = \left(\frac{C}{W}\right)^3,\tag{9}$$

where the variable *W* corresponds to the radial load applied to the ball, and *C* is the basic load capacity of the test configuration with respect to a ball-bearing type system. The basic load capacity parameter, *C* may be calculated using the RCF cycles-to-failure results similar to those presented in Figures 5 through 7. The values of *L*10 and *W* were measured for each of the test configurations shown in Table 1. Using data from Table 6 the *L10* life for each test configuration and loading is plotted as a function of load capacity in Figure 12. The measured *L*10 life for different contact stresses is also plotted in Figure 12 as well, represented as vertical lines (large and small dashed lines, and one dash-dot line).

**Figure 12.** Plot of the natural log of *L10* stress cycles verses load capacity parameter, *C*, using equation (9) for three test configurations in Table 1. Vertical lines represent measured *L10* life for three test configurations and three loads (GPa).

The Lundberg-Palmgren model in equation (9) may require life-adjustment factors to fit the model to experimental data. Concerning Figure 12, it is to be expected that the load capacity parameter, *C*, will be the same for each test configuration independent of loading. Configuration 2 demonstrates this attribute. The dash-dotted vertical lines related to configuration 2 are very near to each other, suggesting that life-adjustment factors are not needed to fit the data related to configuration 2. Equation (9) alone may be used to predict RCF life based on *W* and *C* when testing in configuration 2. In contrast, configurations 1 and 3 give different load capacity parameters for the same configurations as shown in Figure 12. The large and small dashed-lines related to Configurations 1 and 3 do not line up on the same load capacity parameter, *C*, suggesting that life-adjustment factors will be needed to accurately calculate *L10* life for these test element combinations.

Rolling Contact Fatigue in Ultra High Vacuum 113

configurations involving 7.94 mm T5 balls with nickel-copper-silver film are not required. Future work with the UHV-RCF platform will involve continued testing for influence of process parameters and deposition methods, such as magnetron, on RCF test life similar to Danyluk and Dhingra (2011, 2012a). Expansion of the platform for study of surface

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mechanisms underlying coating effect on the fatigue life, *Wear* 215: 191-204.

Nickel-Copper-Silver Solid Lubrication, *J. Vac. Sci. Technol. A*, 29: 011005.

and Applications in Surface Engineering, Volume 10: 202-208, Elsevier.

interaction with heat and fatigue under high vacuum will also be explored.

*Mechanical Engineering Department, University of Wisconsin Milwaukee,* 

films: Experiments and modeling, *Wear* 264: 131-138.

*Journal of Tribology*, Vol. 122, Issue 1, pp.1-9.

selection, *J. Phys. D: Appl. Phys* 40: 5463-5475.

and Ion-Plated Silver Films.

and Anoop Dhingra

**Author details** 

*Milwaukee, Wisconsin USA* 

Chapters 4 and 8.

*Transactions*, 32: 490-496.

*Wear* 274-275: 368 - 376.

Mike Danyluk\*

**6. References** 

NASA.

Corresponding Author

 \*
