# 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
Comparing the hysteresis loops associated with the first cycle, it is visible that the plasticity model could not correctly describe the lower branch of the experimental model. This plasticity model imprecision was verified in the three highest applied strain levels. However, comparing the experimental data associated with the first and steady-stable loops, it is notable that the evolution of the minimum stress was much less than the maximum tensile stress. This behaviour is typical for materials exhibiting non-Masing behaviour [57]. The variation in maximum stress in each cycle for the data under analysis is presented in Figure 10. This figure clearly shows the existence of a non-symmetry associated with the maximum and minimum stress in each cycle of the softening process. In fact, the variation in the minimum stress of the hysteresis loop seems to be only associated with the kinematic hardening component for any applied strain amplitude. This minimum stress tended to remain unchanged throughout the life of the specimen, regardless of the applied strain amplitude. Figure 10: Comparison of the variation in the maximum, minimum, and centre of the elastic surface limit between experimental data for 90% of the total life with the simulation outcomes of the kinematic and isotropic hardening in terms of applied strain amplitude, \(\epsilon_{a}\), of (**a**) 1.25%, (**b**) 1.00%, (**c**) 0.75%, and (**d**) 0.50%. Note: The average parameter \(R_{\infty}\) for (**a–c**) is \(R_{\infty,\text{max}}\) according to Equation (12), and parameter \(R_{\infty}\) for case (**d**) is \(R_{\infty}\) determined for 0.50% (also identified with a star). Concerning the effect of the isotropic saturation rate parameter, \(b_{\infty}\), it is shown that the plasticity model with an average value of \(b_{\infty}\) equal to 0.37 could not correctly describe the first phase of the material softening for the higher strain amplitudes. However, for a strain amplitude of 0.5%, there was a good correspondence between the experimental and numerical behaviour for the maximum stress in each cycle. Notice that \(b_{\infty}\) is an average value assumed constant for several strain amplitudes and throughout the cyclic life. Therefore, despite this value not corresponding to the reality of the material model, \(b_{\infty}=0.37\) appears to be conservative for most of the whole life, as illustrated in Figure 10.  
Now, analysing the hysteresis loops in the steady-state conditions, one verifies that the maximum and minimum stress of the hysteresis loops are very similar when compared with the experimental data. In general, for higher applied strain amplitudes, as presented in Figure 9a,c, the maximum stress tended to be around 6.0%, whereas, for the testing condition in Figure 9b, the maximum stress was slightly inferior, around \(-4.0\%\), in comparison with the experimental data (see Figure 11a). Concerning the minimum stress, the offset of the numerical models was around \(-10.0\%\), as seen in Figure 11b. The maximum deviation occurred between 5 and 30% of the total life, and it reached 16%. For specimens tested with a strain amplitude of 0.5%, a different error analysis needed to be made. In fact, upon analysing Figure 9d, it is verified that from the first cycle until the stabilised cycle, the strain ratio of \(R_{\varepsilon}=0.0\) was never achieved. Over this assumption, the error associated with the minimum value of the stress at the stabilised cycle was normal and above 100%, as seen in Figure 10d. Concerning the hysteresis loop area of \(\sigma-\varepsilon\), one verifies that the area obtained from numerical outcomes tended to be conservative.