## 1 Introduction
The disadvantage associated with the low strain rates that characterize the superplastic phenomenon (i.e., long forming times) has been overcome in recent years with the introduction of quick plastic-forming (QPF) processes. These processes (currently used for aluminum and magnesium alloys) are significantly shorter than the conventional superplastic-forming process: by increasing the strain rate and renouncing the high superplastic ductility, moderate elongations may be obtained [2; 13; 14; 15; 16; 17]. The technological scheme of the QPF process is the same as that used for conventional superplastic forming; it requires the use of a pressurized gas, which, in place of the traditional punch used in a cold forming process, pushes the sheet inside of the die to copy its internal geometry. The sheet metal is previously fixed on the die through a blank holder and is heated until the optimal process temperature is reached and maintained [16]. Figure 1 shows a scheme of the SPF process. The figure shows a channel to put gas (generally argon) inside the forming chamber. The optimum process temperature guarantees the best superplastic properties of the material. Similarly, the pressure-time load curve must also be carefully designed to ensure optimal superplastic conditions. This operation is obtained using finite element modeling (FEM) which can provide the most accurate detailed information on the deformation process [18; 19; 20; 21; 22]. The optimal pressure-time loading curve was predicted, through an original algorithm, for the free-bulging process on an AZ31 magnesium alloy sheet [23]. However, in some cases, such as to determine the characteristic parameters of the material, the adoption of a constant-gas pressure cycle is not excluded during a free-forming test. Such a test requires formation through an elongated cylindrical die. In this way, the sheet assumes the shape of a bubble. It has been demonstrated that the free-forming test leads to the characteristic parameters of the material more simply and accurately than the tensile test does [24; 25]. The free-forming test, compared to the latter, stresses the material in the same conditions as an industrial process does (i.e., a biaxial stress state), and their specimens are easier to produce. In contrast, it is not standardized and requires designed in-house equipment and a process control system. In [26], it was shown that a product with a hemispherical shape made of superplastic material, which was formed by a sheet of constant thickness, had a nonuniform final thickness distribution because of the geometric constraints. The obtained thickness distribution varied, decreasing from the edge to the pole of the sheet. In [27], it was shown that this nonuniform thickness distribution decreased as the strain rate sensitivity index m increased. Making the final thickness distribution uniform requires either the adoption of a multiphase forming process [28; 29; 30; 31] or the use of a sheet with a variable initial thickness [16; 32; 33; 34; 35]. Multistage forming processes require designing, through finite element modeling, a preforming stage for thinning the sheet in those areas (usually the edge of the sheet) that are most stretched in the actual forming phase in correspondence with the last area of the metal sheet that comes into contact with the walls of the dies [28; 29; 30; 31]. These preforming operations are essential even if they increase the final time of the overall formation. The use of sheets with a variable initial thickness [16; 32; 33; 34; 35] is more effective in terms of the overall forming time and weight of the component (which is partially reduced compared to the use of a sheet with a constant initial thickness) but requires a more accurate study (generally carried out using finite element analysis) to find the optimal value of the thickness distribution of the initial sheet. It has been observed that the behavior of superplastic materials is highly sensitive to strain rate [2].