# Cyclic Elasto-Plastic Behaviour of 51 CrV4 Steel for Railway Parabolic Leaf Spring Design
## 3 Material and Analysis Procedure
### Experimental Procedure and Statistical Techniques  
The kinematic and isotropic mechanical properties were determined for chromium-vanadium alloyed steel. The cycle tests carried out in [24] were considered to determine the cyclic hardening properties and subsequent verification in the computational model by analysing the hysteresis loops. The sample set consisted of seven flat-shaped specimens (nominal total length, \(L\), of 135 mm, width on grips, \(W=20\) mm, and a transition radius, \(R\), of 12 mm) with a net cross-section, \(A_{0}\), of 38.98 \(\pm\) 0.28 mm\({}^{2}\) (gauge width, \(w_{o}\), of 6.04 \(\pm\) 0.04 mm and thickness, \(h_{o}\), of 6.46 \(\pm\) 0.01 mm), and a nominal gauge length, \(L_{0}\), of 15 mm (see Table 3 and Figure 3), which resulted in a gauge geometry with an enough strength to avoid buckling effects. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Material** & **C** & **Fe** & **Si** & **Mn** & **Cr** & **V** & **S** & **Pb** \\ \hline DIN 51 CrV4 (1.815) & 0.47–0.55 & 96.45–97.38 & \(\leq\)0.40 & 0.70–1.10 & 0.90–1.20 & \(\leq\)0.10–0.25 & \(\leq\)0.025 & \(\leq\)0.025 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Standard chemical composition of 51 CrV4 steel grade in %wt. Figure 2: Microstructure of the chromium–vanadium alloyed steel following the thermal treatments according ISO 683-14 standard [24]. The cyclic specimens were tested under constant strain-controlled loading conditions according to the ASTM E606 standard [91] for a strain ratio, \(R_{c}\), of 0.0 and an average strain rate \(\mathrm{d}\varepsilon/\mathrm{d}t\) of 0.8%/s. The applied strain amplitudes were 0.375% (two samples), 0.50% (two samples), 0.75%, 1.00%, and 1.25%. Cyclic tests were conducted in an INSTRON 8801 servo-hydraulic machine (INSTRON, Norwood, MA, USA) with a load cell of 100 kN with a dynamic INSTRON 2620-202 clip-on gauge with a range of \(\pm\)2.5 mm, as illustrated in Figure 4.  
The kinematic and isotropic hardening properties were estimated using statistical techniques based on the least squares method. In the case that the problem could be described by a linear function, the ordinary least squares (OLS) was considered, whereas if the problem was non-linear, an unconstrained numerical optimisation scheme was assumed. Figure 4: The servo-hydraulic machine used for the constant strain-controlled amplitude testing. Figure 3: Geometry of the specimens used in cyclic tests, featuring geometric representation with identification of the geometrical parameters and the detail of the surface finishing (Dt. A1). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \(w_{o}\) & \(h_{o}\) & \(A_{o}\) & \(L_{o}\) & \(L\) & \(W\) & \(R\) & \(R_{\epsilon}\) & \(\mathbf{d}\varepsilon/\mathbf{d}t\) \\
**[mm]** & **[mm]** & **[mm\({}^{2}\)]** & **[mm]** & **[mm]** & **[mm]** & **[mm]** & **[mm]** & **[\%/s]** \\ \hline
6.04 & 6.46 & 38.98 & 15 & 135 & 20 & 12 & 0.0 & 0.8 \\ \(\pm\)0.04 & \(\pm\)0.01 & \(\pm\)0.28 & & & & & & & \\ \hline \end{tabular}
\end{table} 
Table 3: Summary of the specimen geometry considered in cyclic testing and general testing parameters. For a linear response function, the minimisation problem results in the equation system in (14)  
\[\hat{\mathbf{\beta}}=\arg\min_{\hat{\mathbf{\beta}}\in\Omega}||\mathbf{y}-\hat{\mathbf{y}}_{i}||^ {2}=\big{(}\mathbf{X}^{\prime}\mathbf{X}\big{)}^{-1}\mathbf{X}^{\prime}\mathbf{y}, \tag{14}\]  
where \(\mathbf{X}\) is the design matrix, and \(\mathbf{X}^{\prime}\) is its transposition. \(\hat{\mathbf{\beta}}\) is the estimator vector containing estimators, which in this case is \(\beta_{0}\) and \(\beta_{1}\)[92].