# Impact of Die Radius in a Streamlined Die during Wire Drawing
## 2 Experimental and Numerical Procedures
### Numerical Procedure  
Depending on the process parameters, a wire undergoes inhomogeneous deformation during wire drawing [12; 28]. In addition, owing to the heat generated by the plastic deformation of the wire and the frictional heating between the wire and die, the temperature rise varies with the wire area during the wire drawing process [29]. In other words, a complex distribution of stain, stress, and temperature occurs with area of the wire during wire drawing. Therefore, finite element (FE) analysis was utilized to evaluate the temperature, stress, and strain distributions of the drawn wire because it provides reliable results for temperature and deformation distribution in a deformed specimen. DEFORM FE commercial software with version 11.0 (Scientific Forming Technologies Corporation, Columbus, OH, USA) was used to simulate the wire drawing process. Taking advantage of the axisymmetric nature of the wire drawing process, a two-dimensional axisymmetric module was selected to reduce the computational cost. In addition, a rather short workpiece length of 100 mm was chosen based on several length tests to keep the calculation time low. The initial 13 mm-diameter wires were drawn to 11.63 mm using the conventional die and the four streamlined dies with die radii of 4, 16, 64, and 128 mm as shown in Figure 3. The conventional die can be considered a streamlined die with an infinite die radius, which is referred to as \"inf.\" hereafter. The total number of elements in the wire was approximately 12,000. The die and wire were assumed to be a rigid body and isotropic material, respectively. The elastic behavior of the die and wire was not considered in this simulation since wire drawing is a bulk-forming process with a large plastic deformation. Accordingly, the constitutive behavior of the wire can typically be described using [PERSON]'s law as follows:  
\[\sigma=K\varepsilon^{n} \tag{2}\]  
where \(K\) is the strain-hardening coefficient and \(n\) refers to the strain-hardening exponent. The \(K\) and \(n\) values were derived by curve fitting of the RT tensile curve, as shown in Figure 4: \(\sigma=1934\varepsilon^{0.46}\) MPa. The following shear friction model was used in this study because the wire drawing process results in the formation of relatively high pressure,  
\[\tau=mk \tag{3}\]  
where \(\tau\) and \(k\) are the shear stress on the contact surface and the shear yield stress of the material, respectively. The shear friction factor (\(m\)) in the wire-die interface was chosen to be 0.1 [30; 31], and other process conditions and physical properties were kept similar to the real experiments. The ambient and initial wire temperatures were approximately 21 \({}^{\circ}\)C. The temperature rise (\(\Delta T\)) during wire drawing was numerically calculated using the following equation [32; 33]:  
\[\Delta T=\frac{\Delta Q}{\rho C_{p}}=\frac{\beta}{\rho C_{p}}\int_{\epsilon_{ 1}}^{\epsilon_{2}}\sigma d\varepsilon \tag{4}\]  
where \(Q\), \(\rho\), \(C_{p}\), and \(\beta\) refer to the heat energy, density, specific heat capacity, and fraction factor from mechanical work to heat energy, respectively. These values are listed in Table 1 based on references [32; 34; 35]. \(\beta\) is assumed to be 0.9 because the mechanical energy stored within the deformed wire as elastic energy is small [32; 33]. In this study, the thermal properties of the wire and die were assumed to be constant because of the relatively small temperature change of the wire and die during the process: from 20 to 150 \({}^{\circ}\)C.