## 4 Evaluation of Stress Integration Algorithms for the e-HAH Model
### One Element Analysis
#### 4.1.2 C5T10 CR Loading Condition  
Figures 9 and 10 show the flow stress and r-value evolutions under the C5T10 CR cross-loading path for MAT 1 and MAT 2, respectively. The accuracy and stability of the investigated stress-integration algorithms for MAT 1 and MAT 2 are summarized in Tables 5 and 6, respectively. Similar to the C5T10R reverse loading path, the overall accuracy for the flow stress and r-values increases upon decreasing the strain increment as expected. However, due to the anisotropic hardening and its orthogonal distortion in the e-HAH model, distinct features can be also noticed, which mostly occur under the large strain increment \(\Delta\varepsilon_{\text{xx}}=5\times 10^{-3}\). The evolution of the r-value predicted by the CPM scheme shows an abnormal behavior as can be observed in Figure 9a. This is because the semi-explicit algorithm does not account for the residual minimization for a flow rule. Moreover, as shown in Figure 9a,b, the flow stress and r-value predicted by the CPM and EBM-4R schemes show a different trend with respect to the algorithms using 14 residual--based EBMs. The evolution of the two state variables, g\({}_{1}\) and g\({}_{2}\), under a strain increment \(\Delta\varepsilon_{\text{xx}}=5\times 10^{-3}\) are presented in Figure 11a,b. Under this loading path, the transient behavior of CPM and EBM-4R is controlled by g\({}_{2}\), as the loading path indicator \(\cos\chi\) is negative for both algorithms (Figure 11c). However, for the other algorithms that include the exact solution, \(\cos\chi\) is positive, which leads g\({}_{1}\) to have a major effect on the transient behavior. This circumstance represents the dominant cause behind the abnormally predicted flow stress curve and r-value evolutions for the two algorithms. In other words, the undesirable evolution of the state variables results in significantly deviated stress updates in the e-HAH model, as shown in Figure 11d. Similar to MAT 1, oscillating flow curves are also predicted by the CPM and EBM-4R approaches for MAT 2 when the time increment is not small enough. In this loading path, the fluctuating flow behavior is attributed to the state variable g\({}_{\text{L}}\) associated with latent hardening (Figure 12). Note that the MAT 2 flow curve is influenced by g\({}_{\text{L}}\) more significantly than for MAT 1, due to the slower evolution of \(\mathbf{\hat{h}^{s}}\). Regarding the computational time, all investigated numerical algorithms present rather similar values, especially when compared with the much larger differences encountered in the accuracy values, particularly under large strain increments. However, it is important to note that under the C5T10 CR loading path, the finite difference gradient--based algorithm (EBM-14R/ND) is characterized by a \(\sim\)10-15% longer CPU time, compared to the analytical gradient algorithm (EBM-14R/AD). Figure 8: Evolutions of the state variables g\({}_{1}\) for different stress integration algorithms with strain increment \(\Delta\varepsilon_{\text{xx}}=5\times 10^{-3}\): (a) MAT 1 and (b) MAT 2. Figure 9: The evolution of the flow stresses and r-values of MAT 1 under C5T10 CR path for different stress integration algorithms with strain increments: (**a**,**b**) \(\Delta\varepsilon_{\text{xx}}=5\times 10^{-3}\), (**c**,**d**) \(\Delta\varepsilon_{\text{xx}}=5\times 10^{-4}\) and (**e**,**f**) \(\Delta\varepsilon_{\text{xx}}=5\times 10^{-6}\). Figure 10: The evolution of the flow stresses and r-values of MAT 2 under C5T10 CR path for different stress integration algorithms with strain increments: (**a**,**b**) \(\Delta\varepsilon_{\text{xx}}=5\times 10^{-3}\), (**c**,**d**) \(\Delta\varepsilon_{\text{xx}}=5\times 10^{-4}\) and (**e**,**f**) \(\Delta\varepsilon_{\text{xx}}=5\times 10^{-6}\). Figure 11: The evolutions of (**a**) \(g_{1}\), (**b**) \(g_{2}\), (**c**) \(\cos\chi\) for different numerical algorithms under C5T10 CR loading path. (**d**) Yield loci for EBM-4R and EBM-14R/AD with \(\mathbf{\hat{h}^{s}}\).