# Characterization and Finite Element Modeling of Microperforated Titanium Grade 2
## 3 Results and Discussion
### Finite Element Results  
The mechanical properties of both base and perforated materials were calculated from tensile tests. Since the rolling direction does not play a significant role in the elastic regime, the elastic modulus was calculated as the average of nine tensile test results. In that way, the base material has an elastic modulus of 115,000 MPa and microperforated material of  
Figure 11: Comparison of Nakajima results.  
Figure 12: Punch load–stroke curves for all geometries in longitudinal orientation.  
82,000 MPa. The plastic behavior (true stress-strain curves) in the longitudinal orientation for both materials is shown in Table 5, discretized every 0.01 units.  
FLC strains were obtained as the average of three repetitions for each geometry, which are shown in Table 6 in tabular form.  
The anisotropy is introduced in the FE model by means of the previously introduced Rij stress ratios. These ratios were unknown, and a DoE was implemented to find them. The full three-level factor design is summarized in Table 7.  
The holder force and punch stroke were gathered for each FEM case. After, the outputs were calculated for each case as the difference between the FEM and experimental data (Equation (11)). Then, nineteen quadratic models were fitted to predict the differences between the FEM results, and the experimental data were fitted. The following equation is an example of these quadratic models:  
\[Y_{7}=339.5+129.14\cdot R_{22}-50.4\cdot R_{22}^{2}-591.9\cdot R_{33}-22.5\cdot R _{22}\cdot R_{33}+250.8\cdot R_{33}^{2}-27.4\cdot R_{12}-20.9\cdot R_{33}\cdot R _{12}+18.7\cdot R_{12}^{2} \tag{13}\]  
Finally, an optimization by gradient descent algorithm was launched. The objective of the optimization was to minimize all the distances between the Nakajima test results and the FE model results, ergo, to minimize all quadratic models at the same time. Usually, it is impossible to minimize all objectives together due to the conflicts between objectives, and the solution is generally a compromise. The optimum values that achieve this compromise solution are the anisotropy ratios summarized in the Table 8.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Geometry**} & \multicolumn{3}{c}{**Strains**} \\ \cline{2-3}  & **Minor** & **Major** \\ \hline G1 & \(-\)0.187 & 0.299 \\ G2 & \(-\)0.148 & 0.267 \\ G3 & \(-\)0.099 & 0.217 \\ G4 & \(-\)0.061 & 0.178 \\ G5 & \(-\)0.045 & 0.156 \\ G6 & \(-\)0.008 & 0.122 \\ G7 & 0.087 & 0.106 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Forming Limit Curves for microperforated material.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Strain**} & \multicolumn{3}{c}{**Stress**} \\ \cline{2-3}  & **BM** & **PM** \\ \hline  
\hline \hline \end{tabular}
\end{table}
Table 5: True stress–strain curves in longitudinal orientation.  
Finally, the experimental and finite element crack results can be compared visually. Figure 13 compares the results of geometry G4 in the longitudinal orientation after the Nakajima test is completed. Both results have similarities in terms of crack location and strains distribution. However, there are some differences due to the nature of the simulation. The FEM results present double symmetry because the model was a quarter of the whole test, while the experimental results present approximately symmetrical results.