## 2 Method and Materials
### Process Variables and Finite Element Simulations  
The geometrical attributes that significantly affect cold forging backward extrusion are billet size, reduction ratio, punch angle, and land height. The levels of the identified factors have been listed in Table 2. The mentioned ranges for the geometrical attributes have been selected based on practical intuition and industry practices [29]. \begin{table}
\begin{tabular}{l l l l} \hline \hline \multicolumn{4}{l}{**Chemical Composition-Specifications as per BS 970-1955,**} \\ \multicolumn{4}{l}{**(Inspected \% Weight)**} \\ \hline Elements & AISI 1010 & AISI 1018 & AISI 1045 \\ \hline C & 0.08–0.13 (0.008) & 0.15–0.20 (0.180) & 0.43–0.45 (0.410) \\ Si & 0.15–0.35 (0.195) & \textless{}0.300 (0.190) & \textless{}0.300 (0.300) \\ Mn & 0.30–0.60 (0.545) & 0.60–0.90 (0.690) & 0.60–0.90 (0.710) \\ P & 0.04 Max. (0.013) & 0.04 Max. (0.022) & 0.04 Max. (0.025) \\ S & 0.05 Max. (0.003) & 0.05 Max. (0.09) & 0.05 Max. (0.006) \\ \hline \multicolumn{4}{l}{**Mechanical Properties**} \\ \hline Tensile strength (MPa) & \multirow{2}{*}{180 (325)} & \multirow{2}{*}{220 (410)} & \multirow{2}{*}{310 (565)} \\ Yield (Ultimate) & & & \\ \hline Hardness Vickers (Brinell no.) & 105 (108) & 131 (126) & 170 (163) \\ Reduction area (\%) & 50 & 40 & 40 \\ Elongation (\%) & 28 & 25 & 17 \\ Elastic modulus (GPa) & 200 & 200 & 200 \\ Bulk modulus (GPa) & 140 & 140 & 140 \\ Shear modulus (GPa) & 80 & 80 & 80 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Chemical composition and mechanical properties of steel grades. Figure 1: The research approach for the current investigation. The billet size ratio (z) is the ratio of billet diameter (\(D_{0}\)) to billet length (\(L_{0}\)). The billet diameter is fixed at 30 mm owing to practical constraints, and the length is varied from 9-36 mm to get the ratios of 0.3-1.2, as listed in Table 2. The reduction ratio (\(r\)) is the ratio of the cross-sectional area difference between the billet and the punch to the billet. Since the billet size is fixed to \(\mathcal{O}\)30 mm, the reduction ratio limits are 0.3, 0.4, 0.5, and 0.6; corresponding punch sizes are \(\mathcal{O}\)25, 23, 21, and 19 mm, respectively. The punch angle (\(a\)) is a very important factor that facilitates metal flow during deformation and is decided based on the industry intuition, set to vary from 160-170\({}^{\circ}\). Similarly, land height (\(h\)) is varied between 2-4 mm and is crucial because of the friction during metal flow over the punch about the contact area. The cold forging backward extrusion limitation is set either to the ratio (\(l_{i}\)/\(D_{i}\)) not exceeding 1.6, where \(L_{i}\) is extruded length and \(D_{i}\).extruded diameter; otherwise, the bottom thickness of the extruded cup should not be less than 3-5 mm. This is because of the material deformation constraint, and if the bottom thickness falls less by the value mentioned, undue pressure will mount during deformation on the punch by the counter punch (meant for the ejection of the part from the die container), leading to catastrophic failure. For the experimental design, the full factorial design (FFD) strategy [30] is employed with the four factors; each factor is defined at four levels to plan for FE simulations. The three-dimensional (3D) model assembly combinations, as planned with DoE, comprise punch, billet, and die container, and are modeled using SOLIDWORKS software [31]. As the models are cylindrical, assemblies are segmented to a symmetrical plane of 30\({}^{\circ}\) in order to reduce simulation time and memory space. AFDEX software [32] has been utilized, as illustrated in Figure 2, to simulate the process with 3D assembly model combinations that are saved in a stereo lithographic (.stl) format. The models are discretized with the auto-generated meshing option and about 3000-6000 approximately average number of tetrahydron elements were formed. The inputs for forming simulation, such as material and friction models, are selected from the in-built library.