## 1 Introduction
### Modeling of Formability Properties  
Formability is an important aspect for the industrial usage of metallic materials [24]. Different models have been used for this task in previous studies, either using a phenomenological approach based on experimental results or using models based on physical principles, usually employing crystal plasticity.  
Some of the parameters most commonly used to evaluate the anisotropic formability properties of sheet metals are the r-values (or Lankford coefficients) along different directions on the sheet plane [25]. The r-value for a specific tensile direction forming an angle \(\alpha\) with the rolling direction is calculated as the ratio between the strain in transversal and normal directions:  
\[r_{\alpha}=\frac{\varepsilon_{y}}{\varepsilon_{z}} \tag{4}\]  
Optimal sheet formability is obtained when the average of the r-values for different \(\alpha\) angles (usually represented as \(\bar{r}\)) tends towards infinity, while their standard deviation (\(\Delta r\)) tends towards zero [26].  
There are numerous examples of studies in which Lankford coefficients are used for the characterization of metal sheets. Any model capable of reproducing the anisotropic behavior of the material in a tensile experiment can be used for this task. Models based on the Taylor-Bishop-Hill theory are usually chosen due to its extreme simplicity (and computational performance) [27; 28]. When further accuracy is necessary, or in order to include further microstructural aspects in the simulation, more advanced crystal plasticity models are employed [29; 30; 31].  
Another important aspect of steel formability is its yielding behavior. Yield surfaces have traditionally been described using analytical expressions. In addition to the classical models of [PERSON] or [PERSON], some of the more commonly used expressions for yield surfaces are the models developed by [PERSON] and [PERSON] [32]. In order to adjust the parameters of these models, a combination of experimental results is usually required, including tensile and biaxial loading experiments, as done for example in the work of [PERSON] [33; 34]. However, it is also possible to calculate yield surfaces using virtual experiments, i.e. simulations. In particular, the combination of crystal plasticity models and analytical yield surfaces have proven to be an useful method for the simulation of complex processes [35; 36; 37].  
The characterization method most directly associated with formability is the realization of forming limit diagrams (FLDs), which represent the maximum deformation that the material can stand under different loading conditions. In fact, different diagrams can be created depending on the criterion used to determine this limit. As happens with yield surfaces, there are several analytical models used to describe forming limits [38; 39]. However, other authors have used crystal plasticity models either to directly find the diagram or to fit the parameters of an analytical model using virtual experiments [40; 41; 42].  
In this work, an advanced model is applied to the calculation of all these properties using virtual experiments. The goal is not to directly evaluate the accuracy of the model, since experimental data are not available for a sensible comparison, but to determine how the microstructural parameters in the model affect the obtained behavior. With this objective, simulations were performed using two different materials, with different microstructures, but the same fitting parameters.