## 4 Evaluation of Stress Integration Algorithms for the e-HAH Model
### Forming Problem: S-Rail Forming and Springback  
In this section, the proposed stress integration algorithms are applied to the simulation of large-scale industrial part forming with purely implicit FE software. Though the implicit FE solver was previously employed for large-scale models, this was often limited to simple material constitutive laws, such as isotropic and kinematic hardening. Therefore, this is the first time that the distortional hardening model with cross-hardening or softening (that is, the e-HAH model) is applied to the simulation of industrial forming processes with a static implicit FE solver. By contrast, numerous previous studies have employed the combined dynamic explicit and implicit solvers for complex forming and elastic-driven springback simulations, respectively. In particular, this happens for the case in which the elastic-plasticity constitutive laws become more complex, such as in the present anisotropic hardening model (e-HAH). In this study, the S-rail part forming and springback simulations were performed as a benchmark for the industrial forming process [45]. The simulations were carried out using the static implicit solver, i.e., the ABAQUS/Standard. From the simulation results, the springback, equilibrium iterations, time increment, and computational time during the forming step were comparatively analyzed using the different CPM, EBM-14R/AD, \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithm** & \(\Delta\epsilon_{\text{xx}}\) & \(\text{Error}_{\sigma}\) (\%) & \(\text{Error}_{\tau}\) (\%) &
\begin{tabular}{c} **Total CPU** \\ **Time (Sec)** \\ \end{tabular} \\ \hline \multirow{3}{*}{CPM} & \(5\times 10^{-3}\) & 22.2 & 155 & 1.6 \\  & \(5\times 10^{-4}\) & 0.56 & 19.4 & 5.9 \\  & \(5\times 10^{-6}\) & 0.65 & 0.57 & 196 \\ \hline \multirow{3}{*}{EBM-4R} & \(5\times 10^{-3}\) & 22.4 & 43.8 & 0.5 \\  & \(5\times 10^{-4}\) & 0.61 & 2.86 & 6.2 \\  & \(5\times 10^{-6}\) & 0.65 & 1.60 & 210 \\ \hline \multirow{3}{*}{EBM-14R/AD} & \(5\times 10^{-3}\) & 0.38 & 2.78 & 0.8 \\  & \(5\times 10^{-4}\) & 0.88 & 1.89 & 6.0 \\ \cline{1-1}  & \(5\times 10^{-6}\) & 0.66 & 1.57 & 222 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Average relative errors in the flow curves and r-values of MAT 2 under C5T10 CR. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithm** & \(\Delta\epsilon_{\text{xx}}\) & \(\text{Error}_{\sigma}\) (\%) & \(\text{Error}_{\tau}\) (\%) &
\begin{tabular}{c} **Total CPU** \\ **Time (Sec)** \\ \end{tabular} \\ \hline \multirow{3}{*}{CPM} & \(5\times 10^{-3}\) & 22.2 & 155 & 1.6 \\  & \(5\times 10^{-4}\) & 0.56 & 19.4 & 5.9 \\  & \(5\times 10^{-6}\) & 0.65 & 0.57 & 196 \\ \hline \multirow{3}{*}{EBM-4R} & \(5\times 10^{-3}\) & 22.4 & 43.8 & 0.5 \\  & \(5\times 10^{-4}\) & 0.61 & 2.86 & 6.2 \\  & \(5\times 10^{-6}\) & 0.65 & 1.60 & 210 \\ \hline \multirow{3}{*}{EBM-14R/AD} & \(5\times 10^{-3}\) & 0.38 & 2.78 & 0.8 \\  & \(5\times 10^{-4}\) & 0.88 & 1.89 & 6.0 \\ \cline{1-1}  & \(5\times 10^{-4}\) & 0.88 & 1.84 & 7.6 \\ \cline{1-1}  & \(5\times 10^{-6}\) & 0.66 & 1.58 & 260 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Average relative errors in the flow curves and r-values of MAT 1 under C5T10 CR. and EBM-14R/ND algorithms. Real automotive sheet metals, composed of dual-phase steel (DP780) and stainless steel (STS), were employed in the simulations. Note that DP780 exhibits a significant cross-contraction (or softening) behavior under loading path changes, due to the large Baushinger effect induced by its martensitic islands embedded in the ferrite matrix. In contrast, the STS sheet exhibits stress over-shooting or cross-hardening behavior under loading path changes. The material properties and their related model parameters are listed in Table 7. The FE model set-up, including tool and sheet dimensions, is presented in Figure 13.  
The thickness of the blank sheet was set to 1.2 mm, and a blank holding force of 200 kN was applied during the forming simulation. The friction coefficient between the blank and tools was set to 0.05.