# A New Approach to Characterize Superplastic Materials from Free-Forming Test and Inverse Analysis
## 2 Materials and Methods  
In this work, a circular sheet with a thickness of 1 mm and a radius of 40 mm in AZ31 superplastic magnesium alloy AZ31 (average grain size 10 \(\upmu\)m) was used to evaluate the parameters characterizing the material using a new characterization procedure. This alloy, whose chemical composition by weight is Mg-3%Al1%Zn, is widely used for industrial applications [4]. It was submitted to the free-forming test at constant pressure with two pressure values (p\({}_{1}\) = 0.2 MPa and p\({}_{2}\) = 0.4 MPa) at a temperature of 753 K.  
The equipment used to perform the free-forming test is composed of two semi-dies with a cylindrical shape (one inside the other) and wrapped with an electrical resistance [46]. Once the sheet was placed between the two half-dies, the electrical resistance heated it up to the process temperature that is controlled by an automatic system to supply electricity. At this point, gas was introduced into the die at constant pressure, p\({}_{1}\) or p\({}_{2}\). The height h reached at the apex of the dome was measured independent of the forming time, t, through a measuring laser put above the die. This height was usually dimensionless with respect to the radius of the cylindrical die, a. In this way, the dimensionless height H was evaluated using the formula below.  
\[\mathrm{H}=\mathrm{h}/\mathrm{a}, \tag{1}\]  
Finite element modelling used the commercial software MSC.Marc(r) (version 2024.1) and was based on the rigid-plastic flow formulation. The numerical simulation used an axisymmetric model characterized by 128 elements with four nodes [47]. A FEM scheme of the free-forming test is shown in Figure 2 where a datum reference frame was introduced with x-axis oriented as the symmetry axis. The boundary conditions are as follows:  
* For the node in contact with the die and placed on the external edge of the metal sheet, \(\text{ux}=0\) and \(\text{uy}=0\) (where u is the displacement of the node);
* For the nodes on the external edge (not in contact with the die) \(\text{uy}=0\);
* For all the nodes on the axis of symmetry (condition of axisymmetry) \(\text{uy}=0\);
* The upper edges of the metal sheet are subjected to a uniform pressure (in one case \(p=0.2\) MPa and in the other case \(p=0.4\) MPa).  
It was possible to identify two bodies: the metal sheet (i.e., a deformable body that is discretized in finite elements) and the die (i.e., a rigid body). The presence of a blank holder was simulated by introducing appropriate boundary conditions at the nodes located on the outer edge of the sheet. In particular, the node in contact with the die had zero displacement along the x and y directions. The nodes on the edge that were not in contact with the die and were constrained not to move along the y-direction. For the conditions of axisymmetry, the nodes on the symmetry axis had a zero displacement along the y direction (that is, along the direction orthogonal to the symmetry axis). The pressure of the forming gas was applied uniformly to the top face of the sheet. The value of the forming pressure as well as the constitutive equation of the material was introduced through the adoption of a subroutine.