## 2 Materials and Methods  
In this work, an axisymmetric process to stretch a metal sheet is studied through the MSC. Marc finite element software. In particular, as shown in Figure 1, a hemispherical punch characterized by a diameter of 120 mm, during its stroke, stretched a sheet positioned on a die (characterized by an inner diameter of 129.5 mm) and constrained to the edge from the presence of a blank holder (with a diameter of 166 mm) that avoided its slipping inside the die. The problem being axisymmetric, four-noded isoparametric elements with bilinear interpolation were used. A comparison was presented between the use of axisymmetric elements and thick-shell elements to demonstrate that the best choice was the first one. The sheet represented the deformable body and it was discretized using two rows of 166 elements. The mechanical behavior of the material was simulated through the use of a rigid-plastic model; therefore, the effects of the material spring-back phenomenon were neglected. The constitutive equation of the material is expressed by the following power law:  
\[\sigma=K\epsilon^{n}, \tag{1}\]  
where \(\sigma\) and \(\epsilon\) represent the equivalent stress and the equivalent strain respectively, while the strength coefficient K and the work-hardening index n are in constant connection with the material under study. The latter was represented by the aluminum-based alloy AA6060,  
Figure 1: Scheme of the axisymmetric stretching on a sheet with an initial variable thickness (1 is the punch, 2 is the blank and 3 is the die). whose chemical composition by weight was the following: Al-0.6% Si-0.3% Fe-0.1% Mn-0.6% Mg-0.1% Cu-0.15% Zn-0.05% Cr-0.1% Ti. The K and n constants of the material were derived from the results of the tensile tests: their values were equal to 135 MPa and 0.12 respectively. The punch and the die were considered rigid, while the blank holder was replaced by locking the displacement of the node, belonging to the outer edge of the sheet, into contact with the die. The other nodes of the outer edge, as well as the nodes of the sheet on the symmetry axis, were not able to move orthogonally to the axis of symmetry. The simulation of the stretching process involved the contact between a deformable body and two rigid bodies. Furthermore, simulations were carried out in presence of friction among the bodies. Therefore, the appropriate contact option and the modified Coulomb friction model were used. Using these models, a relationship between the tangential f\({}_{\text{t}}\) and the normal f\({}_{\text{n}}\) forces was defined:  
\[\text{f}_{\text{t}}=\text{\text{\text{u}}}\text{f}_{\text{n}}(2/\pi)\text{ arctg}(\text{v}_{\text{r}}/\text{R}_{\text{sv}}), \tag{2}\]  
In this expression, v\({}_{\text{r}}\) represents the relative sliding speed and R\({}_{\text{sv}}\) is the relative sliding speed value below which the friction force was null. The value of the friction coefficient u was set equal to 0 in conditions of perfect lubrication, while the values 0,1 and 0.2 were taken into account in friction conditions, that are commonly used in sheet metal working processes. The analysis of the results also foresees a comparison in conditions of sheet metal instability. This condition was defined by comparing the estimated strains during the process with the formability limit curve obtained from the local Hill theory [27] and the Swift diffusion theory [28]. Therefore, once \(\beta\) is defined as the ratio between the main strains, \(\varepsilon_{max}\) and \(\varepsilon_{min}\), measured in the plane of the sheet, the formability limit parameter (FLP) is defined as:  
\[\text{FLP}=\varepsilon_{max}/\text{FLC}(\varepsilon_{min}) \tag{3}\]  
where FLC (\(\varepsilon_{min}\)) represents an analytical description of the formability limit curve.