# Cyclic Elasto-Plastic Behaviour of 51 CrV4 Steel for Railway Parabolic Leaf Spring Design
## 5 Conclusions  
The modelling of the elasto-plastic behaviour in cyclic conditions of the chromium-vanadium alloyed steel (DIN 51 CrV4) standardised for parabolic leaf springs was obtained in this paper. The experimental data concerning the cyclic tests performed previously in [24] were used for determining the parameters of the isotropic and kinematic hardening plasticity model. [PERSON]'s kinematic hardening model along with the isotropic hardening model were considered for modelling the cyclic behaviour. The hardening model parameters were obtained from statistical techniques based on the least squares method. Figure 11: Error 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 for applied strain amplitudes, \(\varepsilon_{a}\), of 1.25%, 1.00%, 0.75%, and 0.50%. Note: The average parameter \(R_{\infty}\) for applied strain amplitudes of 1.25%, 1.00%, 0.75% is \(R_{\infty,max}\) determined according to Equation (12), and parameter \(R_{\infty}\) for the strain amplitudes of 0.50% is \(R_{\infty}\) computed for \(\varepsilon_{a}=0.50\%\). These parameters were considered in the finite element model, whose outcomes were posteriorly compared to the experimental results. Initially, the Chaboche's kinematic hardening properties for stabilised cyclic conditions assuming 90% of the total life criterion were determined. The Ramberg-Osgood elasto-plastic parameters were used for fitting the kinematic coefficients \(C_{i}\) and \(\gamma_{i}\). The best fit was found for three non-linear and one linear back stress, with values of \(C_{1}=81\),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 \(C_{4}=2673.4\), as well as an \(RRMSE\) of 0.2853%. These values were posteriorly compared with the ones obtained from stabilised cyclic conditions for 50% of the total life, since this criterion is usually used for determining the fatigue life in strain-controlled life. Comparing both criteria, we verified that both criteria could be used without existing large differences in the estimators \(C_{i}\) and \(\gamma_{i}\), although the cyclic strength tended to decrease. Additionally, the Chaboche kinematic hardening properties were estimated from estimation models based on monotonic properties, the Modified Universal Slopes method (MUSM), which did not significantly affect the kinematic coefficients when the cyclic yield stress was defined correctly. Concerning the isotropic hardening parameters, the evolution of the yield surface, \(R\), was determined for several strain amplitudes. A relationship for estimation of \(R_{\infty}\) as a function of the applied strain amplitude was assumed. According to the relative root mean square error, a regression model with \(RRMSE\) less than 15% was found. In addition, a limit value for \(R_{\infty}\) was found (\(R_{\infty,max}\)) from the regression model. This value, \(R_{\infty,max}\), was found to be independent of the applied strain amplitude and was equal to \(-232.2\) MPa. Regarding the isotropic saturation rate, \(b_{\infty}\), from the optimisation methods, we found an average value of 0.3705 with an \(RRMSE\) = 42.00%. The high value of \(RRMSE\) was associated with the large scatter verified for different strain amplitudes and the low fitting of \(b_{\infty}\) throughout the test. From the final comparison between the fitting of the constitutive model in transient conditions, we verified that the material's non-Masing material behaviour did not allow the lower branch of the stress of the hysteresis loops to be represented with accuracy. The average value for rate saturation, \(b_{\infty}\), may be used as a conservative value for the maximum and minimum stresses with deviation around 10 and 20%. The area of the hysteresis loops obtained at the end of each test was slightly larger than the experimentally verified and is therefore conservative.