## 4 Evaluation of Stress Integration Algorithms for the e-HAH Model
### One Element Analysis
For comparison, the evolutions of the isotropic yield surfaces are also included in the figures. In each figure, three points are marked, namely A, B, and C, where A and C represent the initial and final stress states during the second loading, respectively, whereas B is selected between the two loading points to compare the transient behavior in the second loading path. Figure 4a,b shows that a significant amount of the Bauschinger effect and permanent softening (\(\mathsf{g_{4}}\cdot\overline{\vartheta}=0.8224\cdot\overline{\vartheta}\)) are represented from the e-HAH parameters of MAT 1 under C5T10R. For C5T10 CR, a similar transient behavior and permanent softening were calculated, but the Bauschinger effect (contraction) was found to be much less than for the C5T10R loading path. For MAT 2, the Bauschinger effect is less than in MAT 1 for the C5T10R loading path, though the permanent softening is also pronounced. However, the C5T10 CR loading condition showed a very noticeable stress overshoot upon changing the loading path and subsequent softening. Note that MAT 1 and MAT 2 were virtually designed to represent the characteristics of the e-HAH model. In the following, the accuracy of each investigated stress integration algorithm is assessed with different time increments. The effect of the time-step size on the accuracy of the stress update is evaluated via one-element simulations under the two loading paths. Three different strain increments during tensile loading were considered, i.e., \(5.0\times 10^{-3}\), \(5.0\times 10^{-4}\), and \(5.0\times 10^{-6}\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Material** & \(\boldsymbol{\sigma_{0}/\sigma_{0}}\) & \(\boldsymbol{\sigma_{4/5}/\sigma_{0}}\) & \(\boldsymbol{\sigma_{90}/\sigma_{0}}\) & \(\boldsymbol{\sigma_{b}/\sigma_{0}}\) \\ \hline \multirow{3}{*}{Mat 1,2} & 1 & 1 & 1 & 1 \\ \cline{2-5}  & \(\mathrm{r_{0}}\) & \(\mathrm{r_{45}}\) & \(\mathrm{r_{90}}\) & \(\mathrm{r_{b}}\) \\ \cline{2-5}  & 0.5 & 1.0 & 1.5 & 1.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Anisotropic mechanical properties. The subscripts 0, 45, 90 represent rolling direction (RD), diagonal direction (DD), and transverse direction (TD), respectively. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{10}{c}{Elastic property and isotropic hardening law parameters} \\ \hline Material & [PERSON]’s modulus (GPa) & [PERSON]’s ratio & K (MPa) & \(\mathrm{e_{0}}\) & n \\ \hline MAT 1,2 & 70.0 & 0.33 & & 500.0 & 0.01 & 0.2 \\ \hline \multicolumn{10}{c}{Yld2000-2d coefficients} \\ \hline Material & m & \(\alpha_{1}\) & \(\alpha_{2}\) & \(\alpha_{3}\) & \(\alpha_{4}\) & \(\alpha_{5}\) & \(\alpha_{6}\) & \(\alpha_{7}\) & \(\alpha_{8}\) \\ \hline MAT 1,2 & 8.0 & 0.7826 & 1.1778 & 1.1075 & 0.9753 & 1.0154 & 0.9028 & 0.9989 & 0.9990 \\ \hline \multicolumn{10}{c}{e-HAH model parameters} \\ \hline Material & q & k & \(\mathrm{k_{1}}\) & \(\mathrm{k_{2}}\) & \(\mathrm{k_{3}}\) & \(\mathrm{k_{4}}\) & \(\mathrm{k_{5}}\) & L & \(\mathrm{k_{L}}\) & S & \(\mathrm{k_{S}}\) \\ \hline MAT 1 & 2.0 & 250.0 & 300.0 & 250.0 & 0.25 & 0.80 & 50 & 1.0 & 0.0 From the simulations, the evolution of the flow stress and r-value at the second loading step were evaluated, and the averaged relative errors are reported, using the following equations:  
\[\text{Error}_{\sigma}=\frac{1}{\text{N}}\sum_{\text{m}=1}^{\text{N}}\frac{| \sigma_{\text{m}}^{\text{exact}}-\sigma_{\text{m}}|}{\sigma_{\text{m}}^{\text{ exact}}}\times 100(\%)\text{ and }\text{Error}_{\text{r}}=\frac{1}{\text{N}}\sum_{\text{m}=1}^{\text{N}}\frac{ \left|\frac{\text{r}_{\Theta,\text{m}}^{\text{exact}}-\text{r}_{\Theta,\text{ m}}}{\text{r}_{\Theta,\text{m}}^{\text{exact}}}\right|}{\text{r}_{\Theta, \text{m}}^{\text{exact}}}\times 100(\%), \tag{60}\]  
where N is the total number of data points used for the error estimations.