## 2 Materials and Methods
### Constitutive Model
The yield function is defined as follows  
\[F=[\frac{1}{2}(|\widetilde{S_{1}^{\prime}}-\widetilde{S_{2}^{\prime}}|^{a}+| 2\widetilde{S_{2}^{\prime\prime}}-\widetilde{S_{1}^{\prime\prime}}|^{a}+|2 \widetilde{S_{1}^{\prime\prime}}-\widetilde{S_{2}^{\prime\prime}}|^{a})]^{ \frac{1}{2}} \tag{2}\]  
where \(a\) is the exponent describing the lattice structure of the metal where [25] recommends a value of 6 for body-centered lattice and 8 for face-centered lattice. \(\widetilde{S_{k}}(k=1,2)\) is the eigenvalue of the tensor \(\widetilde{S}\). This tensor is the result of linear transformations of the stress tensor \(\sigma\), where  
\[\widetilde{S}^{\prime}=L^{\prime}\sigma\qquad\text{ \ and \ \ \ \ }\widetilde{S^{ \prime\prime}}=L^{\prime\prime}\sigma \tag{3}\]  
Figure 1: Material constitutive modelization by modules. The linear transformation tensors \(L^{\prime}\) and \(L^{\prime\prime}\) are completely determined using eight independent anisotropy coefficients \(\alpha_{1}\) to \(\alpha_{8}\) owing to the orthotropic symmetry of the rolled sheet. \(L^{\prime}\) and \(L^{\prime\prime}\) are the linear transformation tensors. They are constructed from different anisotropy coefficients, \(\alpha_{1}\) to \(\alpha_{8}\), of the material under study. The details of implementation can be found in the studies conducted by [3]. However, there have been developments in advanced yield surface formulations and hardening models aimed at improving the predictive accuracy for complex behaviors of sheet metals such as BBC 2005, Barlat Yld2004-18p, and HAH, kinematic-hardening laws such as the Yoshida-Uemori model, and crystal plasticity approaches as applied to non-stable austenitic steels. The BBC2005 yield function [26] is mathematically equivalent to Yld2000-2d, offering similar accuracy in representing initial anisotropy. For materials exhibiting stronger anisotropy and more complex yield loci, an extended version, BBC2008, was developed. This model provides the flexibility to capture complex yield surfaces, and it has been successfully used to predict caring profiles in aluminum alloys [27]. To address general three-dimensional stress states, Yld2004-18p [28] incorporates up to 18 anisotropy parameters through linear stress tensor transformations. Although its calibration demands additional tests, it predicts the plastic response of highly anisotropic materials with excellent accuracy but a higher computational cost. More recent developments [29] aim to incorporate yield surface evolution. The Homogeneous Anisotropic Hardening (HAH) model blends an anisotropic yield function (Yld2000-2d) with evolution laws for distortional hardening. This allows the yield surface to change shape and position under non-proportional loading, capturing transient anisotropy and evolving r-values. These advanced models, while computationally more intensive, provide significantly improved predictions for strain-path-dependent behavior. With regard to the industrial implementation, many of the discussed models have found their way into commercial FEA software, bridging academic developments with practical use. LS-DYNA has a rich library: Hill48, Barlat Yld91 and Yld2000, and Barlat Yld2004-18p are available, as well as Yoshida-Uemori. Abaqus provides Hill48 and a general anisotropic yield potential that users have used to implement Barlat/Yld models and even HAH formulations. PAM-STAMP and AutoForm, specialized for sheet forming, offer built-in advanced yield criteria. PAM-STAMP includes Hill48, Vegter 2017, BBC 2005/2008, etc., and allows mixed isotropic-kinematic hardening. AutoForm has also incorporated user-defined yield surfaces in recent versions to meet the demands of AHSS forming. Despite these successes, there are trade-offs. More complex models require more material testing for calibration. Industrial users will adopt a new model only if its benefits clearly outweigh the costs. The trend shows an increasing adoption of advanced anisotropy models as the material behavior becomes a limiting factor in simulation accuracy.