*3.4. Fatigue Life*

Figure 8 shows the two-parameter Weibull probability plot [35–38] for bearing fatigue-life testing, using maximum likelihood estimation. Here, the failure probability of four series is plotted against the number of load cycles in terms of inner-ring rotations in a double logarithmic diagram. The three production strategies are illustrated, which had a sample size of *n* = 3 each: turning with sharp and rounded cutting edges, respectively, and deep rolling. The tailored forming shafts were compared to *n* = 15 conventionally produced samples with the same geometry made only from SAE5140 from Reference [36], which were ground after heat treatment. Due to restrictions in test bench operation, a reduced speed of 1250 min−<sup>1</sup> at 60 ◦C (κ = 1.6) was used for the reference series. The slope of the regression line is named as shape parameter β of the Weibull distribution. This parameter can be understood as a constant that depends on the damage mechanism or fatigue behavior of a system. The characteristic lifetime, *L*63, also called location parameter, *T* ˆ , of a series is the time period after which approximately 63.2% of the samples have failed.

For the aforementioned load parameters, a bearing life of *L*63 = 12 × 10<sup>6</sup> revolutions could be determined for the reference series made from monolithic SAE5140. The shape parameter of this Weibull distribution is β = 1.39, which indicates that the failure rate slightly increases with time, suggesting rolling contact fatigue as the damage mechanism. As expected, the density function shows negative skew. For the roller bearing life of SAE52100 bearing steel under good lubrication conditions (κ ≥ 2), the size of the Weibull exponent ranges from β = 1 to 1.5 [28,37]. According to Reference [17], a general shape parameter of β = 1.35 for roller bearings is known, which shows that the conventional reference tests correspond well to the expected values, leading to rolling-contact fatigue.

The hybrid samples manufactured with a sharp cutting edge achieved a bearing life of *L*63 = 10.4 × 10<sup>6</sup> revolutions. This is within an 86% margin of the reference series. A shape parameter of β = 1 suggests a constant failure rate, which usually indicates random failures. Gleß [39] also carried

out fatigue tests for bearing surfaces with different mechanical surface treatments. For rough surfaces, he achieved significantly lower lifetimes than for surfaces with subsequent finishing.

**Figure 8.** Weibull plots of fatigue-test failures for different machining processes.

During the test series with a rounded cutting edge, an early failure after only 0.15 × 10<sup>6</sup> revolutions occurred. This results in a bearing fatigue life of *L*63 = 3.62 × 10<sup>6</sup> revolutions, which is only 30% of the reference lifetime. The Weibull slope of β = 0.35 also indicates early failures with a different mechanism for this series. Due to the small sample number of *n* = 3, this early failure has a strong impact, leading to a very wide confidence interval. The sample set manufactured with rounded cutting edge can therefore not be used for further considerations. Due to comparable lubrication conditions (comparable roughness values), as well as the similar residual stress states in relation to the sample set manufactured with sharp cutting edge, a very similar service life was expected for both test series.

The Weibull plot of the deep-rolled samples shows a relatively steep slope of β = 5. The samples fail within a short period, resulting in a relatively narrow confidence interval. The confidence intervals of reference dataset and the deep-rolled samples overlap for an interval width in abscissa direction of 4.21 × 10<sup>6</sup> revolutions at *L*63. A bearing fatigue life of *L*63 = 13.34 × 10<sup>6</sup> revolutions was achieved. This is 28.5% above the sample set manufactured with sharp cutting-edge geometry and 10% above the reference. Reasons for a lifetime-increasing effect of the rolling process are believed to be improved surface properties, such as higher hardness and especially lower roughness. Microcontacts, which lead to local stress maxima and thus influence surface degradation, can then result in early failures. Various studies have also shown that compressive residual stresses induced by the manufacturing process can extend the fatigue life of roller bearings by a factor of 2.5 [30]. Table 3 summarizes the results of bearing-fatigue testing.

**Table 3.** Weibull distribution parameters and relative fatigue life to reference samples.

