# Robust Design of Deep Drawing Process through In-Line Feedback Control of the Draw-In
## 2 Materials and Methods
### FE Modeling  
The FE model of the deep drawing process of the part in Figure 1 was developed by defining a blank in DC05 steel with a thickness of 0.75 mm and initial dimensions of 450 mm \(\times\) 680 mm. DC05 steel was chosen since it is a cold-rolled low-carbon steel with excellent drawing and welding performance, widely adopted to manufacture various automotive parts [31]. The mechanical properties of this steel in terms of flow curve, yield surface, and forming limit curve (FLC), taken from the AutoformR10(r) library material database, are shown in Figure 3.  
The flow curve in Figure 3a is defined using a combination of the Swift and Hockett-Sherby approximation [32]. Equation (1) shows the expression for the calculation of flow stress. Table 1 collects all parameter values of the expression in Equation (1). \[\sigma=(1-\alpha)\Big{\{}C\times\Big{(}\varepsilon_{pl}-\varepsilon_{0}\Big{)} ^{m}\Big{\}}+\alpha\bigg{(}\sigma_{sat}-(\sigma_{sat}-\sigma_{i})e^{-a_{pl}^ {p}}\bigg{)} \tag{1}\]  
Yield surface in Figure 3b was defined according to the Hill48 model [33]. The yield surface according to the Hill model is assumed to be a quadratic function in stress space. The yield surface is defined using the three r values \(r_{0}\), \(r_{45}\), and \(r_{90}\) (plastic strain ratio with \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline \(\alpha\) & 0.25 \\ C & 515.5 MPa \\ \(\varepsilon_{0}\) & 0.00607 \\ \(\sigma_{sat}\) & 413.1 MPa \\ \(\sigma_{i}\) & 146.5 MPa \\ \(a\) & 4.47 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of Swift and Hockett-Sherby model. Figure 3: (**a**) Flow curve, (**b**) yield surface, and (**c**) FLC of DC05 steel. respect to 0\({}^{\circ}\), 45\({}^{\circ}\), and 90\({}^{\circ}\) of rolling direction) and the initial yield stress \(\sigma_{0}\) in the rolling direction. Table 2 shows the values of Hill48 model parameters adopted in this study. The parameter \(r_{m}\) represents the average value of the anisotropy and it is calculated as \(r_{m}=\left(r_{0}+2r_{45}+r_{90}\right)/4\). The Biax parameter is the biaxial stress factor, and it allows the yield surface to be expanded or contracted at the equi-biaxial stress points. This material-dependent value lies in the range from 0.8 to 1.2. If the biaxial stress factor is set equal to 1.0, then the classical Hill48 model is used. The Arcelor V9 model was adopted to generate the FLC in Figure 3c. The uniform elongation (A\({}_{\text{g}}\),90) and the tensile strength along the 90\({}^{\circ}\) rolling direction were imposed, equal to 24.4% and 298.5 MPa, respectively. The tools modeled in FE model are shown in Figure 4, from top to bottom: the punch, the blank holder and the die. The die and the punch were taken as rigid bodies, and the blank holder was taken as a force-controlled tool. During the drawing process, the punch velocity was set equal to 8 mm/s. Lubrication conditions were considered for the friction in the simulation. The Coulomb model with stick-slip modeling was used in the FE model. The friction coefficient applied was assumed between all tools and the blank. The friction coefficient value was calibrated based on the experimental data. Once the FE model was developed, two sets of numerical simulations were performed. The first one was designed to identify the influence of the drawing depth and the blank holder force on the quality of the drawn component. The parameters p and BHF were evaluated in the range of 25 mm-35 mm and 450 kN-950 kN, respectively. The quality of the drawn component was evaluated in terms of thinning and surface deflections. The first defect appeared prevalently at the angled parts of the component highlighted with T\({}_{\text{A}}\) and T\({}_{\text{B}}\) in Figure 5; the latter occurred in the central area of the component (region S\({}_{\text{D}}\) in Figure 5). Thinning is a defect indicative of the material formability, and it indicates the change in thickness of the blank (relative to the original thickness) during the process.