# The Study of Multi-Stage Cold Forming Process for the Manufacture of Relief Valve Regulating Nuts
## 2 Materials and Methods  
The manufacturing process of the relief valve regulating nut was the multi-stage cold forming with five stages. The actual forming machine had five-stage dies. A cold-forging quality AISI 1010 steel wire coil, which was fabricated by China Steel Corporation, Kaohsiung, Taiwan, was used in the cold-forming experiments. The chemical composition of the steel wire is shown in Table 1.  
The cold-forming experiments were conducted using a 20-tonne universal testing machine under a constant ram speed of 5 mm/min at room temperature. A billet of \(\phi\)7.7 mm \(\times\) L24.2 mm was cut by the shear die and moved to the next forming stages. The 3D and cross-section views for the initial billet and forming parts of each stage are shown in Figure 1. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**C** & **Mn** & **P** & **S** & **Si** & **Al** \\ \hline
0.10 & 0.50 & 0.021 & 0.007 & 0.03 & 0.035 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Chemical composition of AISI 1010 steel wires (wt.%). Figure 1: The shapes for initial billet and product parts of five stages. (**a**) 3D views (simulation), (**b**) cross-section views (simulation), (**c**) cross-section views (experiment). There was visible deformation at the ends of the cutoff billet, as shown in Figure 1, due to it being cut to length by shearing. For the first stage, the preparation process includes flattening the billet end and centering for the backward extrusion, which then moves to the second stage through the transfer mechanism, where the extrusion tool mounted in the die (stationary) conducted the backward extrusion. A cavity of \(\phi\)5.2 mm was formed at the bottom end with a depth of 8.0 mm by using a punch tool mounted in the die and with a reduction in area of 43.9% for the backward extrusion. Then, the workpiece was delivered to the third stage, where the upper end of the workpiece was upset and prepared for backward extrusion, and delivered to the fourth stage where the backward extrusion over a moving punch formed a hexagonal cavity at the upper face. Finally, it moved to the fifth stage, where the inner hole was formed by piercing. The deformation energy was determined as follows:  
\[E=\int_{0}^{\Delta L}Fdl, \tag{1}\]  
where \(F\) is the forming force and \(\Delta L\) is the total acted forming stroke. The numerical simulations of the five-stage cold forming were constructed as 3D finite elements using the DEFORM-3D FE code. The workpiece was constructed using tetrahedral elements. Since the tool materials are generally much harder than the workpiece, their deformations were ignored and considered rigid bodies. The material of the workpiece was an AISI 1010 carbon steel billet, which was treated as a rigid-plastic material, following the [PERSON] yield criterion of isotropic hardening. For more accurate simulation results, a cylindrical compression test, with 5 specimens of \(\phi\)7.68 mm \(\times\)\(L\)11 mm (not lubricated at both ends), was performed on a 20-tonne universal testing machine under a constant ram speed of 3 mm/min at room temperature, and the following equations were used to obtain the compression true stress and compression true strain [23]:  
\[\sigma=P/A_{0}, \tag{2}\]  
\[\varepsilon=(h-h_{0})/h_{0}, \tag{3}\]  
\[\sigma_{\mathrm{t}}=\sigma\,(1+\varepsilon)=\sigma\,(h/h_{0}), \tag{4}\]  
\[\varepsilon_{\mathrm{t}}=\ln(1+\varepsilon)=\ln(h/h_{0}), \tag{5}\]  
where \(\sigma\) is the engineering stress, \(\varepsilon\) is the engineering strain, \(P\) is the compression force, \(A_{0}\) is the initial cross-sectional area, \(d_{0}\) is the initial diameter, \(h_{0}\) is the initial height, \(h\) is the instantaneous height, \(\sigma_{\mathrm{t}}\) is the compression true stress, and \(\varepsilon_{\mathrm{t}}\) is the compression true strain. Figure 2 demonstrates the compression true stress-compression true strain diagram of AISI 1010, which was required in the simulation.