## 2 The e-HAH Model
The two components are expressed as follows:  
\[\mathbf{s}_{\mathrm{c}}=\mathrm{H}\Big{(}\mathbf{\hat{h}}^{\mathbf{s}}: \mathbf{s}\Big{)}\mathbf{\hat{h}}^{\mathbf{s}}, \tag{4}\]  
\[\mathbf{s}_{\mathrm{o}}=\mathbf{s}-\mathbf{s}_{\mathrm{c}}=\mathbf{s}- \mathrm{H}\Big{(}\mathbf{\hat{h}}^{\mathbf{s}}:\mathbf{s}\Big{)}\mathbf{\hat {h}}^{\mathbf{s}}, \tag{5}\]  
To measure the loading path change, the indicator \(\cos\chi\) is introduced according to the following:  
\[\cos\chi=\frac{\mathbf{\hat{h}}^{\mathbf{s}}:\mathbf{\hat{s}}}{\sqrt{\mathbf{ \hat{h}}^{\mathbf{s}}:\mathbf{\hat{h}}^{\mathbf{s}}}\sqrt{\mathbf{\hat{s}}: \mathbf{\hat{s}}}}=\mathrm{H}\cdot\Big{(}\mathbf{\hat{h}}^{\mathbf{s}}: \mathbf{\hat{s}}\Big{)}, \tag{6}\]  
where the normalized deviatoric stress tensor is defined as \(\mathbf{\hat{s}}=\frac{\mathbf{s}}{\sqrt{\mathrm{Hs}\mathbf{s}}}\). The value of \(\cos\chi\) indicates the current loading state against the previous loading path. For example, values of \(1\), \(0\), and \(-1\) denote monotonic loading, cross-loading, and reverse loading, respectively. The evolution laws for the state variables are summarized in three groups. The first group (Equations (7)-(10)) is related to the distortion along \(\mathbf{\hat{h}}^{\mathbf{s}}\):  
\[\frac{\mathrm{d}\mathbf{g}_{1}}{\mathrm{d}\bar{\tau}}=\mathbf{u}_{0}\cdot \mathrm{k}_{2}\bigg{(}\mathrm{k}_{3}\frac{\overline{\sigma}(0)}{\overline{ \sigma}(\bar{\tau})}-\mathbf{g}_{1}\bigg{)}+(1-\mathbf{u}_{0})\cdot\mathrm{k} _{1}\frac{\mathbf{g}_{4}-\mathbf{g}_{1}}{\mathbf{g}_{1}}, \tag{7}\]  
\[\frac{\mathrm{d}\mathbf{g}_{2}}{\mathrm{d}\bar{\tau}}=\mathbf{u}_{0}\cdot \mathrm{k}_{1}\frac{\mathbf{g}_{3}-\mathbf{g}_{2}}{\mathbf{g}_{2}}+(1-\mathbf{ u}_{0})\cdot\mathrm{k}_{2}\bigg{(}\mathrm{k}_{3}\frac{\overline{\sigma}(0)}{ \overline{\sigma}(\bar{\tau})}-\mathbf{g}_{2}\bigg{)}, \tag{8}\]  
\[\frac{\mathrm{d}\mathbf{g}_{3}}{\mathrm{d}\bar{\tau}}=(1-\mathbf{u}_{0})\cdot \mathrm{k}_{5}(\mathrm{k}_{4}-\mathbf{g}_{3}), \tag{9}\]  
\[\frac{\mathrm{d}\mathbf{g}_{4}}{\mathrm{d}\bar{\tau}}=\mathbf{u}_{0}\cdot \mathrm{k}_{5}(\mathrm{k}_{4}-\mathbf{g}_{4}), \tag{10}\]  
Here, \(\mathbf{u}_{0}\) is a unit step function with respect to \(\cos\chi\). In other words, \(\mathbf{u}_{0}=1\) when \(\cos\chi\geq 0\) and \(\mathbf{u}_{0}=0\) when \(\cos\chi<0\). Depending on the sign of \(\cos\chi\), the different anisotropic hardening behaviors are expressed using \(\mathbf{g}_{1}\) and \(\mathbf{g}_{2}\). For example, when \(\cos\chi\geq 0\), \(\mathbf{g}_{1}\) and \(\mathbf{g}_{2}\) determine the magnitude of the Bauschinger effect and transient behavior, respectively, whereas, \(\mathbf{g}_{3}\) and \(\mathbf{g}_{4}\) determine the saturating values of \(\mathbf{g}_{1}\) and \(\mathbf{g}_{2}\), respectively. In the simulation, these four state variables are initialized to take the value of \(1\). In Equations (7)-(10), \(\mathrm{k}_{1-5}\) are material constants related to the evolution rates and amounts of \(\mathbf{g}_{1-4}\). Specifically, \(\mathrm{k}_{1}\) controls the recovery rate of the transient behavior, \(\mathrm{k}_{2}\) and \(\mathrm{k}_{3}\) control the evolution rate and amount of the Bauschinger effect, respectively, and \(\mathrm{k}_{4}\) and \(\mathrm{k}_{5}\) determine the magnitude and rate of permanent softening, respectively. The model parameters, \(\mathrm{k}_{1-5}\), can be identified using the load reversal experiments, i.e., tension-compression test and cyclic shear test. The second group is related to the distortion orthogonal to the \(\mathbf{\hat{h}}^{\mathbf{s}}\) direction.