## 1 Introduction  
Superplastic alloy sheet forming (SPF) is used to manufacture parts of complex shapes in one step. It is based on the phenomenon of superplasticity, i.e., the possibility of some materials reaching very high deformations if submitted to particular forming conditions. Materials with superplastic behavior have a fine-grained structure (generally 10 \(\upmu\)m) and are stable during the forming process [1]. The grain refinement can be carried out by hot working at temperatures close to those connected with the superplastic deformation of the material (in this regard, the classes of materials subjected to this grain refinement are the following: duplex alloys such as \(\upalpha/\upbeta\) titanium alloys, duplex stainless steels, \(\upalpha/\upbeta\) copper alloys, and eutectic alloys such as Pb-Sn alloy) [2]. Some materials (such as duplex materials) can undergo grain refinement by phase separation (e.g., \(\upalpha/\upbeta\) Ti alloys and Zn-Al alloy). Other materials (such as low-alloy steels, ultrahigh-carbon steels, and Cu-Al alloys) can be subjected to grain refinement through phase transformation [2]. Techniques based on the use of severe plastic deformation are used to produce very fine grain sizes. The techniques based on this principle include equi-channel angular extrusion, high-pressure torsion, accumulative roll bonding, and friction stir processing [2]. The cyclic extrusion compression process technique was used for an aluminum-based alloy to obtain an ultrafine grain structure in [3, 4]. During a superplastic-forming process, peculiar forming conditions are a process temperature higher than half of the absolute melting temperature of the material and a strain rate between 10\({}^{-3}\) and 10\({}^{-5}\) s\({}^{-1}\) depending on the material [1, 2, 5]. The high formability of superplastic sheets allows reducing the assemblies and the weight of the components that are used in the transport sector (particularly in the aerospace and automotive industries) [2]. The main classes of superplastic materials used for industrial applications are aluminum and titanium-based alloys [1]. Titanium-based alloys have excellent mechanical properties (high strength/weight ratio), excellent corrosion resistance, and exceptional biocompatibility [2]. In the past, superplastic forming and diffusion bonding for titanium-based alloy systems have been used in the aerospace industry. More recently, titanium alloys have been successfully used in the biomedical sector and architectural applications [2]. Aluminum-based alloys are lightweight, have a good strength-to-weight ratio, and have outstanding corrosion resistance [2]. The primary user of aluminum-based alloy SPF has been the aircraft industry, which requires higher cost and produces fewer parts that justify the use of slower machining techniques. Some manufacturers have successfully applied superplastic forming to produce automobile parts in aluminum alloy with 50% weight reduction. Furthermore, 8090 Al superplastic aluminum alloys have been employed to produce military aircraft panels, reducing weight by approximately 25% and production cost by 70% [2]. Magnesium alloy sheets cannot be formed successfully at room temperature due to a tightly packed hexagonal crystal structure. However, it has been observed that magnesium alloys, once heated, can reach high elongations even if the main deformation mechanism is not grain boundary sliding [6; 7; 8; 9; 10]. In recent decades, magnesium alloys have been used due to the continuous decrease in material manufacturing cost and the high strength/weight ratio, which enables structural applications. However, it has been highlighted that, at high temperatures, magnesium alloys are subjected to the phenomena of cavitation, grain growth, and interaction, which limit their deformability [11]. These phenomena are taken into account in predictive models, e.g., to predict grain growth during the forming process [12].