## 5 Results and Discussion
### Strain Energy Density Curves  
The fatigue analysis method based on the strain energy density is another way to predict fatigue life. Using the total strain energy density parameters it is possible to assess fatigue resistance in longer lives. Figure 8 presents the fatigue-life curve considering the total strain energy density, and the plastic strain energy density as a function of the number of reversals. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{Basquin} & \(MSE\) & \(\sigma^{\prime}_{f}\) [MPa] & \(b^{\prime}_{f}\) & \(\varepsilon_{a,t}(2N_{f}=2)\) & \(2N_{f,t}\) & \(\varepsilon_{a,t}(2N_{f,t})\) & \(2N_{f,lc\to hc}\) \\  & 0.172 & 1601.15 & \(-\)0.093 & [\%] & [reversals] & [\%] & [reversals] \\ \hline \multirow{2}{*}{Coffin-Manson} & \(MSE\) & \(\varepsilon^{\prime}_{f}\) [\%] & \(c^{\prime}_{f}\) & & & & \\  & 0.010 & 47.80 & \(-\)0.684 & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters of Coffin–Manson and Basquin models (Equation (2)) obtained from the linear regression model. Figure 7: Coffin–Manson and Basquin fatigue models for uniaxial strain-controlled conditions of 51 CrV4 steel. From a qualitative point of view, the slopes of both models were very similar, which indicates a strong influence of the dissipative energy from plastic deformation. As the fatigue life was increased, the gradient of the models tended to diverge due to the influence of the parameter \(\Delta W_{0+}\). \(\Delta W_{0+}\) induces a residual resistance associated with the fatigue limit of the material, and it was determined using the elastic strain energy computed for tensile stress. In the case of steel 51 CrV4, the fatigue limit was estimated by considering 0.5% of the ultimate tensile strength, corrected using a load-type factor of 0.85. From a quantitative analysis of the regression parameters, the similarity of the exponents \(\alpha_{p}=-0.391\) and \(\alpha_{t}=-0.392\) was expected. The differences in the coefficients \(\kappa_{p}\) and \(\kappa_{t}\) were of little significance, around 6.07%; that is, only 6.07% of the strain energy density was from the elastic strain energy density. For both models, a good fit was exhibited using the coefficient of determination, \(R^{2}=0.945\) and \(R^{2}=0.940\). For a fatigue limit, \(\sigma_{a,lim}=446.95\) MPa, a elastic strain energy density, \(\Delta W_{0+}\), of 0.49 was obtained MJ/m\({}^{3}\). For a value 10% superior than \(\Delta W_{0+}\), an infinite fatigue life, \(2N_{f,90\%}\), of 1.09 \(\times\) 10\({}^{7}\) reversals was found. For this fatigue life, the dissipated plastic energy, \(\Delta W_{p}\), was 0.421 MJ/m\({}^{3}\), which corresponds to a value of about 99.83% less than \(\kappa_{p}\). Table 6 summarizes the regression parameters obtained using least-squares methods for both the plastic strain energy density and total strain energy density models. It is interesting to mention the effect of the chemical composition on the strain-life fatigue behavior. For example, with the presence of a spring steel with a higher carbon content, the yield strength, \(\sigma_{02}\), was typically higher, and hence a longer life was associated with the elastic strain component, and vice versa [19]. Quantitatively, this phenomenon was visible in the transition lifetime value, \(2N_{f,t}\), which was normally higher for lower values of \(\sigma_{02}\). Additionally, as \(\sigma_{02}\) decreased, the plastic strain density, \(\Delta W_{p}\), tended to increase. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\kappa_{p}\) (MJ/m\({}^{3}\)) & \(\alpha_{p}\) & \(\kappa_{t}\) (MJ/m\({}^{3}\)) & \(\alpha_{t}\) & \(\Delta W_{0+}\) & \(\alpha_{t,lim}\) (MPa) & \(2N_{f,90\%}\) & \(\Delta W_{p}\) (\(2N_{f,90\%}\)) \\ \hline
237.46 & \(-0.391\) & 252.81 & \(-0.392\) & 0.490 & 446.95 & \(1.09\times 10^{7}\) & 0.421 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameters of Equations (4) and (5) obtained from the linear regression model.