## 2 The e-HAH Model  
The enhanced homogeneous yield function based anisotropic hardening (e-HAH) model was proposed to simulate the path-dependent anisotropic hardening responses of metallic materials. This model is based on the distortional hardening concept, without depending on kinematic hardening, and is capable of reproducing complex flow behaviors, such as cross-hardening or cross-softening [37; 38; 39]. The distortion of the initially isotropic yield function is controlled by a second-order tensor and eight scalar variables. The second-order tensor, named the microstructure deviator, \(\hat{\mathbf{h}}^{\mathbf{s}}\) monitors the loading path changes and determines the direction of the distortion. The scalar variables control the magnitude of the distortions in terms of the loading path changes. In this section, the main equations of the constitutive model, along with the related evolution of the state variables, are briefly summarized. The complete derivations of the constitutive model can be found in [37; 38; 39]. The yield surface (\(\Phi\)) of the e-HAH model is described as the following:  
\[\Phi=\left\{\left(\sqrt{\psi(\mathbf{s})^{2}+\psi(\mathbf{s}_{\mathrm{P}})^{2 }}\right)^{\mathrm{q}}+\mathrm{f}_{1}^{\mathrm{q}}\Big{|}\hat{\mathbf{h}}^{ \mathbf{s}}:\mathbf{s}-\Big{|}\hat{\mathbf{h}}^{\mathbf{s}}:\mathbf{s}\Big{|} \Big{|}^{\mathrm{q}}+\mathrm{f}_{2}^{\mathrm{q}}\Big{|}\hat{\mathbf{h}}^{ \mathbf{s}}:\mathbf{s}+\Big{|}\hat{\mathbf{h}}^{\mathbf{s}}:\mathbf{s}\Big{|} \Big{|}^{\mathrm{q}}\right\}^{\mathrm{1/q}}=\overline{\sigma} \tag{1}\]  
where \(\overline{\sigma}\) is the equivalent stress, often identified from the uniaxial stress-plastic-strain curve as a reference state, and q is a constant. The two variables \(\mathrm{f}_{i}=\left(\mathrm{g}_{i}^{-\mathrm{q}}-1\right)^{\mathrm{q}}\) with \(\mathrm{i}=1\), \(2\) control the distortion of the yield surface along the direction of \(\hat{\mathbf{h}}^{\mathbf{s}}\). At the start of the simulation, \(\hat{\mathbf{h}}^{\mathbf{s}}\) is initialized as \(\hat{\mathbf{h}}_{ini}^{\mathbf{s}}=\frac{\mathbf{s}_{ini}}{\sqrt{\mathrm{H}_{ \mathrm{init}}\mathbf{s}_{ini}}}\). Here, \(\mathbf{s}_{ini}\) is the deviatoric stress corresponding to the initiation of the plastic deformation, and H is a constant, which is set to \(8/3\) in this work. In Equation (1), \(\sqrt{\psi(\mathbf{s})^{2}+\psi(\mathbf{s}_{\mathrm{P}})^{2}}\) represents the distorted yield function, which is used to reproduce the cross-hardening or softening; in the original HAH model, it is replaced by any first-order homogeneous yield function \(\phi\)[37]. \(\psi(\mathbf{s})\) and \(\psi(\mathbf{s}_{\mathrm{P}})\) are defined in Equations (2) and (3), respectively, using the collinear (\(\mathbf{s}_{\mathrm{c}}\)) and orthogonal (\(\mathbf{s}_{\mathrm{o}}\)) components of the deviatoric stress tensor (\(\mathbf{s}\)) with respect to \(\hat{\mathbf{h}}^{\mathbf{s}}\). \[\psi(\mathbf{s})=\phi(\mathbf{s}^{\prime\prime})=\phi\bigg{(}\mathbf{s}_{ \mathrm{c}}+\frac{1}{\mathrm{g}_{\mathrm{L}}}\mathbf{s}_{\mathrm{o}}\bigg{)}, \tag{2}\]  
\[\psi(\mathbf{s}_{\mathrm{p}})=\phi(\mathbf{s}_{\mathrm{p}^{\prime\prime}})= \phi\bigg{(}\frac{4(1-\mathrm{g}_{\mathrm{s}})}{\mathrm{g}_{\mathrm{L}}} \mathbf{s}_{\mathrm{o}}\bigg{)}, \tag{3}\]where \(\mathbf{s}^{\prime\prime}=\mathbf{s}_{\mathrm{c}}+\frac{1}{\mathrm{g}_{\mathrm{L}}} \mathbf{s}_{\mathrm{o}}\) and \(\mathbf{s}_{\mathrm{p}}=4(1-\mathbf{g}_{\mathrm{s}})\mathbf{s}_{\mathrm{o}}\) are the two tensors, and \(\mathbf{g}_{\mathrm{L}}\) and \(\mathbf{g}_{\mathrm{S}}\) are the two corresponding state variables. The tensors \(\mathbf{s}^{\prime\prime}\) and \(\mathbf{s}_{\mathrm{p}}\) are used for controlling the cross-hardening and softening, respectively.