# Fatigue Life Assessment in Bainitic Steels Based on The Cumulative Strain Energy Density
## 3 Results
### Fatigue Life Prediction  
Numerical fatigue lives were calculated for the three steels under study and the four strain amplitudes applied in strain-controlled fatigue tests using the well-known SWT model (Equation (1)) and the Liu model (Equation (2)). These calculations were performed with the stress and strain values of the mid-life cycle (see Table 5). After that, numerical results were compared with the obtained experimental data (\(N_{f}\)) listed in Table 5. For the sake of clarity, a graphical comparison between the predicted and experimental lives is displayed in Figure 7. As can be seen, numerical predictions were almost exact for steel B and resulted in nonconservative values for steel A and steel C. Furthermore, we could conclude that the Liu criterion led to predictions tendentially safer than the SWT criterion.  
Figure 5: Cumulative total strain energy density (Wt) versus fatigue life (\(N_{f}\)) for the three high-strength bainitic steels under study.  
Figure 6: Elastic strain energy density of the mid-life cycle (\(\Delta We_{\rm{,ML}}\)) against cumulative total strain energy _density (Wt)_ for the three high-strength bainitic steels under study.  
where \(k^{\prime}\) is the cyclic hardening coefficient and \(n^{\prime}\) is the cyclic hardening exponent (see Table 3). Once the applied stress was known, the associated elastic strain energy density per cycle could be computed from Equation (3). Thus, based on the energy relationships of Figure 6, we could compute the cumulative total strain energy densities, which allowed the calculation of the corresponding fatigue life from the power function of Figure 5. The predicted values (N\({}_{\text{CTSED}}\)) for different strain amplitudes of the three high-strength bainitic steels are presented in Table 5. Figure 8 compares the predicted lives with those determined in the experiments. As can be seen, the N\({}_{\text{CTSED}}\) values were more accurate than those computed through SWT and Liu criteria (see Figure 7).  
In order to analyze the predictive capabilities of the proposed model in a more systematic manner, probability density functions were used, defined from the following error parameter:  
\[E_{N}=\log_{10}\left(\frac{N_{f}}{N_{p}}\right) \tag{5}\]  
where \(N_{p}\) is the predicted lifetime, and \(N_{f}\) is the experimental life. In theory, more accurate models are those with lower standard deviations and mean errors close to zero. As shown in Figure 9, the proposed model combined these two features, since it had a mean error closer to zero and a lower standard deviation than the other two. Furthermore, it led to conservative predictions, which was another important outcome. Table 6 presents several statistical parameters computed from the \(N\)/\(N_{p}\) ratios obtained for each model, namely minimum error, maximum error, mean, standard deviation and variance. In fact, the cumulative total strain energy density (CTSED) approach led to more accurate results, since its mean deviation was closer to zero and its standard deviation was smaller.  
Figure 8: Comparison between numerical and experimental fatigue lives of the proposed model based on the cumulative total strain energy density (N\({}_{\text{CTSED}}\)).