# Impact of Die Radius in a Streamlined Die during Wire Drawing
## 5 Discussion  
To produce high-quality wires, it is necessary to derive a practical strategy based on the results achieved in the above comparative study using FE analysis. To derive practical solutions, we need to consider all design factors, such as strain and stress distribution, temperature rise, drawing force, and damage value. A new non-dimensional indicator for the die radius (\(I_{\text{DR}}\)) was defined as follows to deduce general conclusions by considering several design factors:  
\[I_{\text{DR}}=\frac{d_{\text{m}}}{R_{\text{die}}} \tag{8}\]  
where \(R_{\text{die}}\) is the die radius and \(d_{\text{m}}\) is the mean diameter of the wire, which is represented as follows:  
\[d_{\text{m}}=\frac{d_{\text{i}}+d_{\text{f}}}{2} \tag{9}\]  
The calculated \(I_{\text{DR}}\) values are summarized in Table 2 based on the die radius. Figure 12 shows the variations in the \(SIF\) value, maximum shear stress, maximum effective stress, maximum damage value, temperature rise, and drawing force with the \(I_{\text{DR}}\) value. Roughly, all design factors increased with the \(I_{\text{DR}}\) value. For example, the wire by the die with a radius of 4 had the highest \(SIF\) value, shear stress, effective stress, damage value, and temperature rise. However, in the domain of a small \(I_{\text{DR}}\) value, the design factors did not follow macroscopic trends. For instance, the wire fabricated by the die with a radius of 128 had the smallest \(SIF\) value, shear stress, effective stress, and damage value, but it had a higher temperature rise and drawing force, which means that the optimal die design exists around this \(I_{\text{DR}}\) value. To understand the behavior of design factors with \(I_{\text{DR}}\) value and determine the optimal die design, the drawing behavior of the wire by streamlined die needs to be understood. The effect of the die radius on the design factors is highly related to the die angle (\(\alpha\)) during wire drawing, as shown in Figure 1e. Additionally, the die angle controlled the contact length (\(l_{\text{c}}\)) of the wire and die. According to the geometry of the conventional die (Figure 2), \(l_{\text{c}}\) can be represented as follows:  
\[l_{\text{c}}=\frac{d_{\text{i}}-d_{\text{f}}}{2 sin\alpha} \tag{10}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Die Radius (mm)** & 128 & 64 & 16 & 4 \\ \hline \(I_{\text{DR}}\) **value** & 0.096 & 0.192 & 0.770 & 3.079 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Die radius and related \(I_{\text{DR}}\) value used in this study. Figure 11: Comparison of (**a**) history of drawing force along the drawing direction and (**b**) the steady-state drawing force with the die radius. Figure 13 shows the normalized \(l_{\text{c}}\) with the die angle for the conventional die. As the die angle increases, the \(l_{\text{c}}\) decreases sharply. A high \(l_{\text{c}}\) indicates a larger frictional stress or frictional heating due to the longer contact distance of the wire and die in the deformation zone of the die. In contrast, a low \(l_{\text{c}}\) indicates a higher redundant deformation and surface hardening because of the abrupt and excessive direction change of the metal flow in the deformation zone of the die (Figure 14) [38]. As shown in Figure 6, as the die radius increases, the die angle decreases, leading to an increase in \(l_{\text{c}}\). From the \(l_{\text{c}}\) perspective, the behavior of design factors can be understood. The temperature rise and drawing force were high at a higher die radius (small \(I_{\text{DR}}\) value) because of the great frictional stress and heating effect stemming from the high \(l_{\text{c}}\). In contrast, the higher \(S\!I\!F\) value, effective stress, damage value, and temperature rise were highly related to the unsound metal flow or redundant work at the surface area [38; 42].