## 2 Materials and Methods  
### Constitutive Model  
Constitutive models embody a mathematical framework for describing the mechanical behavior of the material. From the simplest to the most advanced model, all of them rely on a diverse number of parameters to represent various mechanical characteristics of the material. It is imperative to adequately quantify these parameters that feed the equations which govern the adopted constitutive model in order to run reliable numerical simulations. This calibration of the model represents the mechanical characterization of the actual material. Mechanical characterization is made through experimental tests. Normally, it is not possible to experimentally measure each of the parameters necessary to construct a constitutive model representing a material, but can be accomplished in an indirect method from other measurements. In this study, a comprehensive experimental framework was employed. Initially, tensile tests were performed to establish the baseline stress-strain response, which allowed for the calibration of the yield stress and hardening behavior by delineating the elastic-to-plastic transition and the post-yield strain hardening. Complementing the tensile tests, hydraulic bulge tests were conducted, wherein the material was subjected to biaxial tension. This testing method minimizes stress concentrations typically encountered in uniaxial tests, thereby generating a more uniform deformation field. The uniform, biaxial stress distribution inherent in the bulge test delays the onset of localized necking, which is a common failure mode in tensile tests, allowing the material to sustain larger deformations before failure. Additionally, Miyauchi tests were utilized to assess directional properties and the evolution of anisotropy under deformation. The Miyauchi test data help calibrate parameters that influence the shape and orientation of the yield surface. Finally, tension-compression tests reveal the material's asymmetry in tension versus compression. Incorporating these data into the calibration process adjusts the yield function to account for differences in yielding under compressive versus tensile loading, which is important for accurately predicting behavior in forming simulations where compressive stresses are common. A material model can be constructed as an assembly of different modules, in which each one of these modules is the numerical description of a characteristic of the mechanical behavior of the material. Figure 1 represents the material constitutive modelization separated by modules. Yield Criteria  
Exceeding the elastic limit of a material implies exiting the elastic regime to enter the plastic state. When considering a multiaxial stress case, such as those occurring in drawing processes, in order to find that point of regime shifting, a dependence between the principal stresses is required. Mathematically, this is expressed by an implicit function, which is referred to as the yield function:  
\[f(\sigma_{1},\sigma_{2},\sigma_{3},Y)=0 \tag{1}\]  
where \(\sigma_{1}\), \(\sigma_{2}\), \(\sigma_{3}\) are the principal stresses and \(Y\) is the yield stress obtained from a simple uniaxial test (tension, compression or shearing). In a three-dimensional space where the three spatial axes are the principal stresses, Equation (1) depicts a surface commonly known as the _yield surface_. In the case of plane stress (\(\sigma_{3}=0\)), the yield surface degenerates into a curve in the principal stresses plane. There is a significant contribution of the plastic anisotropy of sheet metal on the final stress distribution produced during a forming process [24]. In this study, we will apply the non-quadratic anisotropic yield function developed to describe plane strain states, which are commonly referred to as YId2000 [3].