## 4 Evaluation of Stress Integration Algorithms for the e-HAH Model
### One Element Analysis  
In the one-element analysis, the loading condition consists of a compression-tension (C-T) test. The accuracies of the predicted flow stress and r-value were compared for the investigated numerical algorithms. Two loading paths were simulated for the C-T test. The first path consists of a 5% compression along the rolling direction (RD) followed by a 10% tension along the same direction. This case is denoted as \"CST10R,\" where \"R\" represents the fact that the loading path is a \"Reversed\" loading condition. During the compression along the RD, the rate of the yield function distortion represented by the transient behavior and Bauschinger effect is maximal in the opposite side of the compressive loading, or tension, along the RD. The second loading path consists of a 5% compression along the RD followed by a 10% tension at an angle of 54\({}^{\circ}\) to the RD. Note that the 54\({}^{\circ}\) angle corresponds to the cross-loading state, where the condition \(\hat{\mathbf{h}}^{\mathbf{s}}:\hat{\mathbf{s}}=0\) is satisfied (the exact value of the angle for the cross-loading condition is 54.74\({}^{\circ}\)). This case is denoted as \"CST10 CR,\" where \"CR\" stands for \"cross\" loading. This loading path was selected because the cross-loading effect with rotation of the microstructure deviator becomes maximized at the loading direction with the condition. A four-node shell element with reduced integration in the ABAQUS/Standard, also denoted S4R, was used. The boundary conditions for the above cases are schematically shown in Figure 3.  
The r-value along the \(\theta\) angle with respect to the RD is defined in Equation (59) and calculated via the nodal displacements of an element, according to the following:  
\[\tau_{0}=\frac{\epsilon_{\text{width}}^{\text{P}}}{\epsilon_{\text{thickness}}^{ \text{P}}}=-\frac{\epsilon_{2}^{\text{P}}}{\epsilon_{1}^{\text{P}}+\epsilon_ {2}^{\text{P}}},\text{ where }\left\{\epsilon_{1}^{\text{P}}=\ln\Bigl{(}1+\Delta u_{1}^{ \text{P}}\Bigr{)}\text{ and }\epsilon_{2}^{\text{P}}=\ln\Bigl{(}1+\Delta u_{2}^{ \text{P}}\Bigr{)}\right\}, \tag{59}\]  
where \(\Delta u_{1}^{\text{P}}\) and \(\Delta u_{2}^{\text{P}}\) are the element nodal displacements along the RD and transverse direction, respectively, during tension, and the superscript \"p\" represents the plastic part of the strain or displacement. The mechanical properties of the two model materials and the constitutive parameters for the initial yield function and e-HAH model are listed in Tables 1 and 2, respectively. The model materials \"MAT 1\" and \"MAT 2\" have distinctive e-HAH parameters, but other properties, such as elasticity, initial yield function, and isotropic hardening, are identical. For isotropic hardening, the Swift hardening power law, \(\varpi(\overline{z})=\text{K}(\text{e}_{0}+\overline{z})^{\text{n}}\), is applied for both materials. Figure 3: Boundary conditions for the one element compression-tension analysis: (**a**) 5% compression along RD followed by (**b**) 10% tension along RD, (**c**) 5% compression along RD followed by (**d**) 10% tension along 54\({}^{\circ}\) to RD. Dashed line and the solid line represent before deformation and after deformation, respectively. Figures 4 and 5 show the reference flow stress curves and the corresponding evolution of the e-HAH surfaces for MAT 1 and MAT 2, respectively. Since the exact solutions are not available, a reference curve with a sufficiently small time-step is assumed to be exact. The reference flow curves were obtained using the CPM scheme implemented into the dynamic explicit FE ABAQUS/Explicit software with the VUMAT user material subroutine. The average strain increment was set to \(2.4\times 10^{-8}\). Figures 4a and 5a illustrate the flow stress curves under the C5T10R loading path, whereas Figures 4c and 5c display the same curves under the C5T10 CR loading path. The evolutions of the yield surfaces are provided in Figures 4b and 5b for C5T10R, and in Figures 4d and 5d for C510 CR.