# A New Approach to Characterize Superplastic Materials from Free-Forming Test and Inverse Analysis
## 1 Introduction  
Forming processes are becoming more and more complex to satisfy the client requirements of personalization and accuracy. The development of digital twin tools [1] or neural networks [2] may help to optimize these processes under these requirements. The phenomenon of superplasticity allows some metal alloys to reach high tensile elongations if they are worked in particular forming conditions. Such conditions require a fine and stable-grain structure at the process temperature, a high and constant process temperatures (above half the melting temperature of the material), and very low strain rates that are typically between \(10^{-3}\) and \(10^{-5}\) s\({}^{-1}\). Information on the microstructure of superplastic materials is widely discussed in [3; 4; 5]. The superplastic forming process (SPF) requires the use of a pressurized gas instead of the punch used in the traditional forming process. As shown in Figure 1, the pressurized gas pushes the superplastic sheet to adhere to the surfaces of the die by copying its internal geometry. The superplastic sheet, clamped on the die using a blank holder, is (i) indirectly heated until it reaches the optimum value of the process temperature [6] and is (ii) subjected to the action of a load curve (pressure-time) that guarantees the optimal superplastic conditions. Finite element modelling (FEM) of the forming process is relied upon to accurately design the optimal load curve [7; 8; 9; 10; 11]. The main superplastic materials of industrial interest are titanium-based alloys, which are used in the aerospace industry, biomedical sector, and architectural applications, and aluminum-based alloys, which are mainly used in the aircraft industry [4]. Superplastic materials decrease the weight of the aircraft, thereby consuming less fuel. Thus, the manufacturing processes connected with this class of materials may be considered green technologies to provide an answer to environmental requirements. To take advantage of the high formability and high strength-to-weight ratio as evidence in superplastic aluminum alloy sheets, superplastic forming (SPF) processes have been  
Figure 1: Scheme of the superplastic forming process. employed in the automotive industry. In a single forming step, parts of complex shapes can be manufactured by decreasing the assembly operations and the weight of the parts. Recently, magnesium-based alloys have been studied in superplastic conditions due to the decrease in material cost and an interesting strength/weight ratio [12; 13; 14; 15; 16], even when they present more critical issues [17; 18]. In recent years, quick plastic forming processes have been introduced to speed up hot forming processes, which do not fully exploit the high superplastic properties of aluminum- and magnesium-based alloys [4; 19; 20; 21; 22; 23]. Moreover, since the superplastic forming processes involve sheet metal stretching [24; 25], to better control the thickness distribution of a manufactured part, multiphase forming processes have been used [26; 27; 28], or alternatively, forming processes on a sheet of variable thickness have been proved [29; 30; 31; 6; 22]. To describe the behaviour of superplastic materials, several constitutive models [4; 32; 33; 34; 35; 36] have been proposed; they require the identification of different constants by characterizing the material through several experimental tensile tests. However, it has been demonstrated that the results of the tensile test are not accurate enough to be introduced into the simulation of superplastic forming processes [37; 38; 39; 40]. On the contrary, the free-forming test is rapid and accurate since it involves a deformation condition similar to that of industrial forming processes. The free-forming tests require using a die with a simple geometry that allows for the manufacturing of hemispheres.