# Assessment of Fatigue Life and Failure Criteria in Ultrasonic Testing Through Thermal Analyses
## 4 Results
### Model Evaluation  
The main goal of this topic is to assess the models, described in Table 1, and identify which failure criteria provided a better agreement with ultrasonic fatigue results.  
[PERSON] and [PERSON]'s models are inapt to apply in some cases. Most of the T-N curves did not present phase II, and using these models requires steady-state temperature. In light of this, these models are inappropriate to apply for the obtained T-N curves in this study.  
With the results of the tested specimens failing, the heat dissipation (Q\({}_{\text{cyc}}\)) was calculated based on the equation provided in Table 1 and associated with the number of cycles to failure (N\({}_{f}\)) to obtain a more general failure criterion. As mentioned before, energy heat dissipation according to [PERSON] [15; 17] is independent of the test frequency and the geometry of the specimen.  
Figure 14 demonstrates the estimated Q\({}_{\text{cyc}}\) related to the N\({}_{f}\). The results reveal that N\({}_{f}\) decreases linearly with increasing Q\({}_{\text{cyc}}\). The fatigue function \(Q_{\text{cyc}}=C\left(N_{f}\right)^{m}\) was obtained, where C and m are the constants of the material. The material in this study encountered values of C and m equal to 4.52 and \(-\)0.489, respectively.  
Figure 13: Experimental life vs. predicted life based on S–N curve.  
With the purpose of verifying the accuracy of the predicted life results, they were evaluated and compared with the experimental results. Figure 15 presents this comparison and analysis for 1:5 and 1:2 scatter bands. According to Figure 14, the fatigue life estimated by the energy dissipation presented good accuracy. Around 90% of the fatigue data are within the scatter band and in the conservative region.  
Although the specific heat capacity of steel varies with temperature, a constant value was used in this study to simplify the analysis. Future studies could benefit from incor  
Figure 14: Q\({}_{\rm cyc}\) related to the experimental fatigue life.  
Figure 15: Experimental life vs. predicted life based on \(Q_{\rm cyc}\)-\(N\) curve.  
porating temperature-dependent thermodynamic parameters to provide a more detailed understanding of their influence on VHCF performance [5; 24; 25; 26; 27].  
The latest model by [PERSON] and [PERSON] [16; 20] was assessed by relating the R\({}_{\theta}\) with the N\({}_{\rm f}\). R\({}_{\theta}\) data were estimated by the T-N curves and evaluated in Section 4.1. Figure 16 shows the R\({}_{\theta}\) vs. N\({}_{\rm f}\). The constant parameters, c1 and c2, were attained by the fitted curve and have values of 1.21 and 0.489, respectively. It can be seen that the slope of the curve obtained by the fitted curve in Figures 14 and 16 is the same. This is expected, given that Q\({}_{\rm cyc}\) is dependent on R\({}_{\theta}\).  
Similarly, in Figure 14, the predicted life estimated by R\({}_{\theta}\) presented in Figure 17 demonstrates great reliability. Most of the data are in the 1:2 and 1:5 boundaries.  
Figure 16: R\({}_{\theta}\) related to the experimental fatigue life.  
Figure 17: Experimental life vs. predicted life based on \(R_{\theta}\)-\(N\) curve.