## 3 Results and Discussion
### Discussion  
Numerical simulations of stamping are sensitive to the exact description of boundary conditions and constitutive equations (yield curve and hardening law) describing material behavior during plastic deformation. In the numerical simulation of the cup test the Hill 48 yield law and Hollomon hardening law were used because of their simple constitution from tensile test data (Table 2). Are they adequate enough to describe material behavior and the tendency to increase the maximal drawing force when increasing the blankholder force? [PERSON] [27] investigated the effect of the yield criterion (von Mises, Hill 48, and Barlat Yield 91) combined with the Swift hardening law on the numerical results of redrawing the mild steel. Predicted punch force evolution was close to the experimental one, whatever the yield criterion adopted. Nevertheless, the cup wall thickness distribution was strongly influenced by the yield criteria, being clearly overestimated by the von Mises yield criterion. The same result was reached by [PERSON] [28] when studying the effect of anisotropic yield criteria (Hill 48, Barlat 91, and CPB06) during deep drawing of a cylindrical cup from advanced high-strength steels. In the presented study the predicted maximum forces are lower than the measured one as shown in Table 5. It agrees with [27], where the numerically-simulated force evolution for the 1 st stage (cup drawing) was slightly lower than the experimental one for each yield law studied. Thus, from the view of yield laws, the numerically-predicted punch force in the presented study is not influenced by the yield law. When concerning the hardening law, [PERSON] [29] evaluated different models, such as [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], and Ramberg-Osgood, to determine which best represented the plastic behavior of austenitic (304) and ferritic (439A) stainless steels. They evaluated stress-strain curves obtained from tensile tests and the Ludwik law fit best for the austenitic stainless steel, while [PERSON] fit best for ferritic stainless steel. [PERSON] [30] also determined that the [PERSON] hardening model was the best fit to the experimental data when they studied the influence of hardening model to the thickness distribution of a rectangular cup from AISI 304 stainless steel. Although the trend of thickness distribution predicted by the simulation matched with the experiment, more thinning was observed in the simulation than in the experiments. The comparison of stress-strain curves of [PERSON]'s hardening law for materials used in this study is presented in Figure 9. It is clear that higher strains are predicted for stainless steel AISI 304 due to higher value of the strain hardening exponent (Table 2). Considering it is the austenitic stainless steel and the nature of its plastic deformation, other phenomena should be included in explaning the deformation process of this steel, such as adiabatic deformation. This is because the dominant material variable that determines whether or not adiabatic shearing deformation will occur is undoubtedly the rate of work hardening of the material [31]. The simulation software and hardening law, and others presented in previous sections as well, do not include this phenomena, so that might explain the higher difference in drawing forces measured experimentally to those numerically calculated for austenitic stainless steel. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Material**} & **Strip Drawing Test** & \multirow{2}{*}{**Cup Test**} & \multirow{2}{*}{**\(\mathbf{\Delta f=f_{m}-f_{c}}\)**} \\  & **f\({}_{1,2}\) (\(\boldsymbol{\cdot}\)) Equation (2)** & & **f\({}_{\mathbf{m}}\) (\(\boldsymbol{\cdot}\))** & **f\({}_{\mathbf{c}}\) (\(\boldsymbol{\cdot}\))** & **\(\mathbf{\Delta f=f_{m}-f_{c}}\)** \\ \hline DC 05 & 0.11 \(\pm\) 0.012 & 0.102 & 0.086 & 0.016 \\ AISI 304 & 0.23 \(\pm\) 0.049 & 0.199 & 0.168 & 0.031 \\ \hline \hline \end{tabular}
\end{table}