# Cyclic Elasto-Plastic Behaviour of 51 CrV4 Steel for Railway Parabolic Leaf Spring Design
## 2 Cyclic Elasto-Plasticity for Steels
### Empirical Estimation of Hardening Variables  
The estimation of kinematic and isotropic hardening variables is normally carried out considering test data under strain-controlled cyclic loading conditions. Under these conditions, the kinematic hardening evolution is represented by Equation (8) as  
\[\sigma_{a}=\sigma_{y}^{\prime}+\frac{\Delta\alpha}{2}=\left(\sigma_{y}+R_{y}^{ \prime}\right)+\sum_{i=1}^{n_{a}}\frac{C_{i}}{\gamma_{i}}\tanh\left(\gamma_{i} \varepsilon_{a}^{p}\right), \tag{8}\]  
with \(R_{y}^{\prime}\) denoting the variation in the elasticity limit surface for the stabilised-state hysteresis loop and \(C_{i}\) and \(\gamma_{i}\) denoting fitting parameters fitted from Equation (8). Equation (8) allows for taking into account both the ratcheting effect and the stress saturation effect [41; 74]. According to some research works, the limit of the stress-stabilised cyclic curve is reached for strain amplitudes of 2% and 3% in high-strength steels, which means that after 2% and 3%, the increase in the stress amplitude with the strain amplitude is quite low [75; 76; 77]. This slight increase in hardening stress may be considered by summing to the Equation (8) a linear component of stress, given by [PERSON]'s model, which results in  
\[\sigma_{a}=\left(\sigma_{y}+R_{y}^{{}^{\prime}}\right)+\sum_{i=1}^{n_{a}-1} \frac{C_{i}}{\gamma_{i}}\tanh\left(\gamma_{i}\varepsilon_{a}^{p}\right)+\frac {2}{3}C_{i}\varepsilon_{a}^{p}. \tag{9}\]  
The fitting of the parameters \(C_{i}\) and \(\gamma_{i}\) can be carried out considering the hysteresis loop in stabilised conditions for the largest strain amplitude analysed, or by the elastoplastic cyclic curve. According to the literature in [41; 73; 78; 79], the Ramberg-Osgood, RO, model is one of the most suitable models for fitting the [PERSON]'s parameters in steel materials. The RO model represents a stable elastic-plastic hardening that relates the total strain amplitude, \(\varepsilon_{a}^{t}\), to the stress amplitude, \(\sigma_{a}\), through a power law, of the form [80], such that  
\[\varepsilon_{a}^{t}=\varepsilon_{a}^{e}+\varepsilon_{a}^{p}=\frac{\sigma_{a}}{ E}+\left(\frac{\sigma_{a}}{K^{\prime}}\right)^{\frac{1}{p^{\prime}}}, \tag{10}\]  
with \(\varepsilon_{a}^{e}\) and \(\varepsilon_{a}^{p}\) representing the elastic and plastic parts of the total strain amplitude. \(E\) denotes the Young's modulus of the material, \(n^{\prime}\) denotes the cyclic strain hardening exponent, and \(K^{\prime}\) denotes the cyclic hardening strength coefficient. Despite of being an ancient model, several researchers still use this equation model (10) to characterise the elasto-plastic behaviour of a material under cyclic loadings [24; 73; 81; 82; 83; 84; 85; 86; 87]. Additionally, the parameters of the evolution of the elasticity limit surface (Equation (7)) can be obtained from strain-controlled cyclic testing data. Then, the elastic limit surface (Equation (7)) can be represented in terms of the normalised maximum stress, \(R^{*}\), as a function of the accumulated plastic strain such that  
\[R^{*}=\frac{R}{R_{\infty}}=\frac{\sigma_{max}(N_{i})-\sigma_{max}(N_{i=1})}{ \sigma_{max}(N_{i=s})-\sigma_{max}(N_{i=1})}, \tag{11}\]  
with \(\sigma_{max}(N_{1})\), \(\sigma_{max}(N_{i=s})\), and \(\sigma_{max}(N_{i})\) denoting the observed maximum stress associated with the first cycle, the stabilised cycle, and the \(i\)-th cycle, respectively. \(R_{\infty}\) relies on the imposed strain amplitude and represents the difference between the yield stress for the stabilised cycle and for the first cycle. With the amount of softening/hardening, \(R_{\infty}\) increases with the increase in the applied strain amplitude, up to its maximum value, \(R_{\infty,max}\). In several investigations [75; 76; 77], it was found that for medium- and high-strength steels, the elasticity limit surface variation remains unchanged for an applied strain higher than around 2% up to 3%.