# Cyclic Elasto-Plastic Behaviour of 51 CrV4 Steel for Railway Parabolic Leaf Spring Design
## 4 Results and Discussion
### Determination of the Isotropic Hardening Properties  
The isotropic hardening law properties, \(R_{\infty}\) and \(b_{\infty}\), in Equation (6) were determined separately. Initially, the parameter \(R_{\infty}\) was determined, and \(b_{\infty}\) was subsequently determined, both using the unconstrained optimisation scheme. Considering the determination of the isotropic hardening parameters following the unconstrained optimisation scheme for the variation in \(R_{\infty}\) (Equation (12)), then, in accordance with Equation (15), it is assumed that  
\[\hat{\textbf{y}}_{i}=R_{\infty,max}(1-\exp[k_{0}(\varepsilon_{a,i}-\varepsilon_ {0})])\]  
and  
\[\textbf{y}_{i}=R_{\infty,j}.\]  
In the numerical procedure, the lower and upper bounds of regressors, \(R_{\infty,max}\), \(k_{0}\), and \(\varepsilon_{0}\) were found. After finding the side bounds, a pool of permissible solution of regressors was created in order to be considered as the initial design vector, \(\textbf{x}=\{R_{\infty,max},k_{0},\varepsilon_{0}\}\). From a pool of optimal solutions, the best optimal solution was considered. It was assumed that the best optimal solution has the lowest value of the objective function. According the algorithm, the optimal solution of **x** is \(R_{\infty,max}=-232.2\) MPa, \(k_{0}=-380.55\), and \(\varepsilon_{0}=0.32\%\), with a relative root mean square error of \(RRMSE=14.83\%\). Figure 7 depicts the evolution of the amount of cyclic softening as a function of the applied strain amplitude. According to this variation, the existence of a cyclic softening was considered after a threshold strain value, \(\varepsilon_{0}\). The threshold strain \(\varepsilon_{0}\) corresponds to the strain for a stress of 648 MPa, which is approximately equal to a cyclic yield strength of \(\sigma_{01}=\) 676.7 MPa, but this is 4.24% inferior. From Figure 7, it is verified that \(R_{\infty,max}\) was reached after 2.0% of strain amplitude, which is in accordance with [75; 76; 77]. Regarding the isotropic saturation rate parameter, \(b_{\infty}\), its value was determined according to the method in [41; 89] such that the response curves are assumed to be defined as follows:  
\[\hat{\textbf{y}}_{i}=1+\exp(-\hat{b}_{\infty,i}\textbf{p}_{i})\]  
and  
\[\textbf{y}_{i}=R_{i}/R_{\infty}.\]  
Figure 8 depicts the normalised variation of the yield surface size evolution, \(R/R_{\infty}\), throughout the cyclic testing until 90% of the total life to failure. Both the mean curve and  
Figure 7: Evolution of the amount of cyclic softening as function of the applied strain amplitude. standard deviation bands concerning the mean value of \(b_{\infty}\) are also presented. From the comparison between the experimental data and numerical outcome, it is notable that cyclic tests performed at \(\varepsilon_{a}=0.375\%\) exhibited a non-coherent evolution, due to the large differences in rate size evolution. In contrast, cyclic tests carried out at higher strain amplitudes than 0.375% exhibited a better agreement with respect to the numerical approach. Considering the 90% of total-life criterion, most cyclic tests appeared not to have an actual stabilisation plateau, as suggested by the numerical approach [41]. However, one of the cyclic tests, at \(\varepsilon_{a}=0.375\%\), appeared to have a stabilised state. With respect to the quality of the fitting curve, it is noted that in the initial cyclic softening phase, from \(10^{-2}\) until 1, the regression curve tended to exhibit a lower saturation rate, whereas in the middle phase, the softening evolution followed the same trend as the experimental results. This behaviour has been also found in some materials [89,101]. The average value found for \(b_{\infty}\) was \(0.3705\pm 0.1628\), with an \(RRMSE\) of \(42.00\pm 13.22\%\). According to [89], \(b_{\infty}\) is usually assumed to be between 50 and 0.5 considering the half-life criterion. In spite of \(b_{\infty}\) being slightly lower than 0.5, \(b_{\infty}\) represents the average value obtained from different tests for a full cyclic test.