## 3 Results
### Refinement of FEM Models: Hardening Rule  
Previous results suggested that the Hollomon extrapolation may be inappropriate to describe the ISF processing loads of the studied aluminum components. Thus, the Voce  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Case ID**} & \multirow{2}{*}{**Material**} & \multirow{2}{*}{**Shape**} & **Wall Angle** & **Experimental** & **FEM** & **Error** \\  & & & **\(\alpha\) (deg)** & \(F_{2}\) (kN) & \(F_{2}\) (kN) & **(\%)** \\ \hline
1 & AA1050-H24 & Pyramid & 45 & 0.309 & 0.341 & 10.4 \\
2 & AA1050-H24 & Pyramid & 60 & 0.282 & 0.314 & 11.3 \\
3 & AA1050-H24 & Pyramid & 70 & 0.277 & 0.265 & \(-4.3\) \\
4 & AISI304 & Pyramid & 45 & 0.801 & 1.331 & 66.2 \\
5 & AISI304 & Pyramid & 60 & 0.857 & 1.531 & 78.6 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Experimental UWA cases used to evaluate the maximum FEM force predictions.  
Figure 5: Experimental and numerical axial force evolution for parts formed with the AA1050-H24 sheet: (**a**) Cases 1, 2; (**b**) Case 3.  
Figure 6: Experimental and numerical axial force evolution for Cases 4 and 5; formed with the AISI-SAE 304 sheet.  
hardening law was implemented as a straightforward alternative to improve the AA1050-H24 force estimations. The Voc's parameters identified from the tensile experiments of this material are incorporated as  
\[\sigma=146.78-24.34\exp\left(-7.67\varepsilon_{p}\right) \tag{8}\]  
Such an enhanced hardening model was applied in Cases 2 and 3. Figure 7 compares the experimental and numerical force evolution for the two cases. The physical (AA1050-H24) \(\alpha=60^{\circ}\) part presented a slight descending phase, starting around 9 mm depth, before ramping up for a second time after 13 mm depth. The force curve of the pyramid with a wall angle of \(\alpha=70^{\circ}\) (Case 3) reveals the onset of fracture, identified by the characteristic drop after the peak, occurring at a depth of 7 mm. Indeed, a slow propagating crack appeared in the part, indicating the forming limit for this material under the employed experimental conditions.  
The numerical force curves also describe the characteristic response for each case. Specifically, for the \(60^{\circ}\) wall angle component, the values stabilize during the falling phase of the experimental curve. Moreover, the \(70^{\circ}\) wall angle curve exhibits a force drop after the peak, at a depth of 8 mm, although with a lower gradient than its physical counterpart. Additionally, the improved FEM peak estimations were less than 4% above the experimental maximum force.  
For the stainless steel, [PERSON]'s hardening law parameters were adjusted by a curve-fitting approach comprising actual SPIF experiments as the data source. Figure 8a depicts the identification process employed. The error minimization problem was solved using the Levenberg-Marquardt method implemented in Matlab(r) release 2010a [40]. Despite the affordable computational time of our models, the large number of iterations required by the optimization algorithm compelled us to fix limits on the procedure. Specifically, the force values of the \(45^{\circ}\) experiment (Case 4 in Table 3), obtained up to 6 mm depth, were assumed as the adjusted work hardening region. After 25 iterations (3.1 h per iteration), important progress in the curve fitting was observed, and the process was stopped. The values adjusted at this point, K = 0.861 GPa and n = 0.287, were considered optimal. The stress-strain curve of both hardening models is shown in Figure 8b, compared to the experimental engineering and true stress-strain curves.  
Figure 7: Comparison of the numerical force evolution of two aluminum parts, based on Vocé’s hardening law, against the experimental results.  
Since the model fitted Case 4, its verification was carried out with Case 5. As shown in Figure 9, the peak force overestimation for the 60\({}^{\circ}\) pyramid was reduced from 79% to 22%.