# Cyclic Elasto-Plastic Behaviour of 51 CrV4 Steel for Railway Parabolic Leaf Spring Design
## 4 Results and Discussion
### Cyclic Elasto-Plastic Response Using a Computational Approach  
After determining the best fitting for the kinematic and isotropic hardening parameters, the numerical model following the finite element method approach as described previously, with the respective elasto-plastic mechanical properties presented in Table 5, was considered and compared with the experimental results to validate the cyclic plasticity model of chromium-vanadium steel 51 CrV4. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \(\mathbf{E}\) & \(\mathbf{v}\) & \(\mathbf{\sigma_{y}}\) & \(\mathbf{R_{\infty}}\) & \(\mathbf{b_{\infty}}\) & \(\mathbf{\sigma_{y}^{\prime}}\) & \(\mathbf{C_{1}}\) & \(\mathbf{\gamma_{1}}\) & \(\mathbf{C_{2}}\) & \(\mathbf{C_{3}}\) & \(\mathbf{C_{4}}\) & \(\mathbf{\gamma_{3}}\) & \(\mathbf{\left[MPa\right]}\) \\
**[GPa]** & & [MPa] & [MPa] & & [MPa] & & [MPa] & & [MPa] & & [MPa] & \\ \hline
202.5 & 0.29 & 1089.2 & \(-\)232.2 & 0.3705 & 676.7 & 80,097.1 & 648.4 & 20,486.2 & 164.9 & 5650.1 & 40.64 & 2302.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Elasto-plastic parameters that define the Hooke, Voee, and Chaboche hardening models for steady-state cyclic conditions at 90% of the total life to failure. Figure 8: Evolution of the normalised variation of the yield surface size evolution throughout the cyclic testing until 90% of the total life reached. (\(\mu_{R}\)—average curve for the normalised variation of the yield surface size evolution, \(\sigma_{R}\)—standard deviation range (2 \(\times\sigma_{R}\) and 3 \(\times\sigma_{R}\)) for the normalised variation of the yield surface size evolution). For each strain amplitude condition, enough number of cycles were applied such that the stabilised behaviour would be possible to observe. The adopted strain amplitudes were only 0.5, 0.75, 1.00, and 1.25%. For strain amplitudes lower than 0.50%, elastic hysteresis loops were obsrved after the first cycles, and therefore, the strain amplitude level of 0.375% was not considered in the analysis. In a first phase, the hysteresis loops of the material numerical model were evaluated by comparing the first cycle and the cycle associated with the loop corresponding to 90% of the total life. Higher strain amplitudes were considered for the first analysis. Figure 9d depicts the comparison between the experimental data in the first and stable hysteresis loops with the numerical outcomes in terms of the stress and total applied strain for the presented strain amplitudes. From the analysis of the monotonic part, it is noted the existence of a lag in the initial hardening between experimental and numerical outcomes. This lag is associated with the definition of the isotropic hardening law such that the linear hardening coefficient  
Figure 9: Comparison between experimental data for the first cycle and the loop corresponding to 90% of the total life with the simulation outcomes of the kinematic and isotropic hardening in terms of applied strain amplitude, \(\varepsilon_{a}\), of (**a**) 1.25% (90% \(N_{f}\) = 243 cycles), (**b**) 1.00% (90% \(N_{f}\) = 540 cycles), (**c**) 0.75% (90% \(N_{f}\) = 1080 cycles), and (**d**) 0.50% (90% \(N_{f}\) = 2610 cycles). Note: The average parameter \(R_{\infty}\) for (**a-c**) is \(R_{\infty,max}\) according to Equation (12), and parameter \(R_{\infty}\) for case (**d**) is \(R_{\infty}\) determined for \(\varepsilon_{a}\) = 0.50% (also identified with a star). (as described in Voce's law) was assumed to be null and also due to the material plastic properties variability. On the other hand, the maximum stress for each applied loading level tended to be very close between the models as illustrated in Figure 9a-c. In the case of Figure 9d, it is verified that the yield strength was much higher than the others, generating a higher difference associated with the hardening material. However, this difference tended to remain constant, since the experimental and numerical curves tended to be collinear.