# Cyclic Elasto-Plastic Behaviour of 51 CrV4 Steel for Railway Parabolic Leaf Spring Design
## 4 Results and Discussion
### Determination of the Kinematic Hardening Properties  
When determining the kinematic hardening parameters, hysteresis loops for a life of 90% were considered. The hysteresis loops for a 50% of the total life were also used considered, but only for comparison. Thus, considering the RO cyclic curve (Equation (10)) for 90% of the total life, cyclic RO parameters of \(K^{\prime}=1576.2\) MPa, and \(n^{\prime}=0.0918\) with a coefficient of determination, an \(R^{2}=0.8113\) was obtained. The kinematic hardening properties were determined by fitting the curve of Equation (9) to the cyclic ROcurve (Equation (10)) using the unconstrained numerical optimisation scheme previously described, where Equation (15), was calculated such that  
\[\mathbf{\hat{y}}=\sum_{i=1}^{n_{\alpha}-1}\frac{C_{i}}{\gamma_{i}}\tanh\Bigl{(} \gamma_{i}\varepsilon_{a}^{p}\Bigr{)}+\frac{2}{3}C_{i}\varepsilon_{a}^{p}\]  
and  
\[\mathbf{y}=K^{\prime}(\varepsilon_{a}^{p})^{n^{\prime}}.\]  
with the number of back stress being \(n_{\alpha}=1,2,3,4\). The selected model was the model that provided the highest minimisation of the objective function, hence the smallest \(RRMSE\) and the smallest number of back stress, and that ensured the condition \(C_{i}\), and \(\gamma_{i}>C_{i+1}\) and \(\gamma_{i+1}\), with \(i=1,\ldots,n_{\alpha}\). As a result of this condition, the selected Chaboche's kinematic model was constituted by four back stresses (three non-linear and one linear). Since the optimal solution is dependent on the input parameters, the process to obtain the best initial design vector, \(\mathbf{x}=\{C_{i},\gamma_{i}\}\), with \(i=1,\ldots,n_{\alpha}\), was made from the generation of a random sample. The generation of the random values was made using a pseudo-generator of random values designed as Mersenne Twister that was a twisted GFSR random number generator of 32 bits MT19937 [98; 99]. Side bounds were introduced in the model to constrain the regression parameters \(C_{i}\) and \(\gamma_{i}\). The lower and upper side bounds were assumed to be, respectively, \(10^{-7}\) and \(10^{7}\) for both parameters. Parameters were fitted until the value of fracture strain came out to \(\varepsilon_{f}=7.53\%\). Notice that the fourth back stress was linear in order to take into account the effect of ratcheting and mean stress relaxation effect [44]. Considering the Chaboche parameter fitting method for four-back stress kinematic hardening models (three non-linear and one linear), the optimal parameters were \(C_{1}=81\),701.2 MPa, \(\gamma_{1}=610.2\), \(C_{2}=22\),035.4 MPa, \(\gamma_{2}=159.14\), \(C_{3}=6332.2\) MPa, \(\gamma_{3}=39.93\), and \(C_{4}=2673.4\) MPa, with a relative root mean square, \(RRMSE\), of 0.2853%. The value of each fitting parameter, \(C_{i}\), and \(\gamma_{i}\) is presented in Table 4.  
Figure 5: Cyclic hysteresis curves for the first cycle and cycles corresponding to the 50% and 90% of total life. Figure 6 depicts the decomposition of the total back stress tensor, \(\alpha_{t}\), in the model with three non-linear, \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\) components and one linear \(\alpha_{4}\) component. From Figure 6--on the right side--it is notable that there is a good agreement between the variation in back stress components with the the assumptions of [PERSON]'s model. The \(\alpha_{1}\) and \(\alpha_{2}\) values had higher impacts in the initial hardening until 2% of the plastic strain amplitude. On the other hand, the \(\alpha_{3}\) value contributed gradually to increase the kinematic hardening from the beginning of the plastic deformation process. Lastly, the \(\alpha_{4}\) value linearly increased the total back stress tensor. In addition, Figure 6--on the left side--compares the decomposition of the total stress, \(\sigma\), with the total back stress, \(\alpha_{t}\), and the size of the elasticity limit surface.