# Bending Fatigue Behavior of 17-4 PH Gears Produced by Additive Manufacturing
## 4 Determination of the Fatigue Limit  
By applying Equation (4) to the experimental points, reported in Table 1, it is possible to draw the \(\rm\sigma_{F0}\)-N curve (at R = 0.1) shown in Figure 5. The initial points of Table 1 have been used to have an idea about the high cycle behavior of the tested gears. The last four points have been used to calculate the fatigue limit.  
Figure 5: Staircase sequence.  
The standards regarding the statistical elaborations of the fatigue data ([41; 42]) require, even in the case of exploratory research, the use of several points greater than the one presented in Figure 5. However, many approaches to short staircase are present in the literature (e.g., [43; 44; 45]). The short staircase methodologies defined by [PERSON] et al. [45] and [PERSON] [44] are used here to calculate the fatigue limit. Both approaches allow us to determine the fatigue limit \(X_{50\%}\) (considered as a sensitivity analysis) at a given number of cycles (i.e., \(5\times 10^{6}\) cycles) at a failure probability of 50%.  
Beside the formulas, the practical difference between the two proposed methods is the condition considered to increase/decrease the load during the staircase sequence. For the first method, the failure condition is represented by the breakdown (i.e., tooth breakage) of the specimen, while in the second case, the failure condition is considered as the breakdown of one of the two specimens that are tested at the same time under the same test conditions (i.e., applied load).  
In previous works [36; 37], the couple of tested teeth is considered as a unique specimen. Therefore, the fatigue limit is calculated according to the first method. \(X_{50\%}\) can be calculated as:  
\[X_{50\%}=X_{\text{f}}+\text{kd} \tag{5}\]  
\(X_{\text{f}}\), defined as the last test level, is equal to 440 MPa and k, a constant depending on the failure-runout history, is set equal to 0.50; therefore, \(X_{50\%}\), the fatigue limit with 50% reliability, results to be 453 MPa.  
As highlighted previously, in the symmetric test case, both teeth are subject to the same stress level. If we consider the tested teeth as two separate specimens, the choice to increase/decrease the test load (i.e., the failure condition) is done \" as soon as the first of the two specimen (i.e., the tested teeth) fails under simultaneous testing responds\". This situation is the so-called \"specimen in series\" [44].  
In this way it is possible to determine \(X_{50\%}\) also considering the fact that, in reality, only one tooth is broken, while the other one (which is subject to the same stress level) is still intact.  
In this case, \(X_{50\%}\) can be calculated as:  
\[X_{50\%}=X_{0}+\text{d}(\Delta) \tag{6}\]  
where \(X_{0}\) is the initial test level and \(\Delta\) is a coefficient related to the staircase sequence and to the hypothesized statistical distribution. \(X_{0}\) is equal to 446 MPa and, under the assumption of normality, \(\Delta\) is equal to 0.54. Therefore, the fatigue limit \(X_{50\%}\) is evaluated to be equal to 480 MPa.  
For the sake of completeness, in Figure 4, the sloping part of the high-cycle fatigue curve is also shown. This part of the curve has been calculated by fitting the data related to the failed tests [45] using a least squares regression. As it has been performed by considering the couple of tested teeth as a unique specimen, the fitting has been extended until the limit calculated with the same statistical consideration (i.e., Equation (5)).  
Moreover, while tests on the equivalent wrought counterpart were not conducted, the results of monoaxial fatigue tests have shown that the wrought material performs better by approximately 30% [30] (Figure 6).