# Fatigue Life Assessment in Bainitic Steels Based on The Cumulative Strain Energy Density
## 4 Conclusions  
The present paper studied cumulative strain energy density behavior in three bainitic rail steels with similar chemical compositions but different Mn contents under a symmetric strain-controlled mode. Based on the high correlation between cumulative total strain energy density and fatigue lifetime, a new predictive model, capable of estimating the durability of high-strength bainitic steels, was developed. Predictions were compared to those of the well-known SWT and Liu criteria. The following conclusions can be drawn:  
* The cyclic stress-strain response was affected remarkably by Mn content. For each steel, the stress ranges associated with a given strain amplitude were significantly different, as was the shape of the hysteresis loops.
* In the absence of Mn, cumulative plastic strain energy density was higher than cumulative elastic strain energy density. Nevertheless, when Mn content was maximum, opposite behavior was observed.
* Regarding cumulative total strain energy density at higher strain amplitudes, the maximum values were found for the steel with the highest Mn content. On the contrary, at lower strain amplitudes, the maximum \(Wt\) values were found for the Mn-free bainitic steel.
* A very good correlation between cumulative total strain energy density and fatigue life was found. These two variables were successfully fitted by a single power function independent of Mn content.
* In the form of a power function, a strong correlation between cumulative total strain energy density and elastic strain energy density at the mid-life was found. Nevertheless, in this case, these relationships depended on Mn content.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **Minimum Ne/Np** & **Maximum Ne/Np** & **Mean** & **Standard Deviation** & **Variance** \\ \hline SWT & \(-0.2729\) & \(0.0700\) & \(-0.1583\) & \(0.1378\) & \(0.0189\) \\ Liu & \(-0.2251\) & \(0.1247\) & \(-0.1068\) & \(0.1428\) & \(0.0204\) \\ CTSED & \(-0.1578\) & \(0.2757\) & \(0.0682\) & \(0.1280\) & \(0.01638\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: Statistical parameters computed from the E\({}_{\text{N}}\) values for the three models.  
Figure 9: Probability density functions of the SWT, Liu and the cumulative total strain energy density (CTSED) model.  
* The new model founded on cumulative total strain energy density led to more accurate fatigue life predictions than those of the well-know SWT and Liu criteria, more specifically, lower standard deviations and mean errors closer to zero.  
Conceptualization, R.F.M., R.B. and X.L.; methodology, R.F.M. and R.B.; writing--original draft preparation, R.F.M.; writing--review and editing, R.B., R.F.M.; and X.L.; funding acquisition, X.L. All authors have read and agreed to the published version of the manuscript.  
The authors acknowledge Fundacao para a Ciencia e a Tecnologia (FCT-MCTES) for its financial support via the project UIDB/00667/2020 (UNIDEMI). The authors also would like to acknowledge the sponsoring under the project number 016713 (PTDC/EMS-PRO/1356/2014) financed by Project 3599 Promover a Producao Cientifica e Desenvolvimento Tecnologico e a Constituicao de Redes Tematicas (3599-PPCDT).  
The authors declare no conflict of interests.