# FEM Analysis of Superplastic-Forming Process to Manufacture a Hemispherical Shell
## 3 Experimental Tests  
The laboratory tests were performed using the superplastic magnesium-based alloy AZ31 at 440 \({}^{\circ}\)C, whose chemical composition by weight is Mg-3%Al-1%Zn. In general, magnesium-based alloys are not very ductile at room temperature, but it has been demonstrated that, at high temperatures, some of them can achieve superplastic conditions, thus increasing their ductility [1]. Some magnesium alloys applied industrially are presented in [1]. Figure 3: (**a**) Hemispherical shell schematised using 2D four-node axisymmetric elements; (**b**) boundary conditions on the outer node in direct contact with the die; (**c**) boundary conditions on the nodes located along the axis of axisymmetry; (**d**) application of a constant pressure at the top edge of the elements. The constitutive equation of the magnesium-based alloy AZ31 is reported in [31]. The optimised superplastic behaviour was found at a temperature of 440 \({}^{\circ}\)C and a strain rate of \(3.5\times 10^{-4}\) s\({}^{-1}\). The constitutive equation used is of the following type:  
\[\overline{\sigma}=K\varepsilon^{n}\delta^{m}, \tag{4}\]  
In this equation, \(\overline{\sigma}\) is the stress, \(\varepsilon\) the strain, and \(\delta\) the strain rate, while the constants \(K\), \(n\) and \(m\) assume the following values: \(K=289.9\), \(m=0.407\) and \(n=0.12\). The material constants were determined by free forming tests. The tests were conducted at constant pressure (using the pressure \(p=0.275\) MPa once and \(p=0.4\) MPa once). The output of the tests is represented by a curve of displacement at the apex of the dome versus the forming time. Therefore, the material constants were obtained through the free forming test, i.e., a test conducted at constant pressure that does not involve contact between the sheet and the forming die. To determine the material constants, an inverse analysis technique associates the results of an experimental free forming activity conducted at constant pressure with those from a numerical simulation of the forming process itself [29; 31]. Unlike a monoaxial tensile test, the free forming test submits the material to be tested to a state of biaxial stress that is closer to what happens during a real superplastic-forming process. The experimental free forming activity was conducted at two different pressure levels (\(p_{1}=0.275\) MPa and \(p_{2}=0.4\) MPa) with the aim of acquiring the thicknesses and forming times necessary to reach well-defined geometric configurations. The constant-pressure superplastic-forming tests consisted of the creation of a hemispherical shell (characterised by a radius of 30 mm) starting from:  
* A constant-thickness blank with \(s_{0}=0.94\) mm;
* A conical section blank with \(s_{max}=1\) mm and \(s_{min}=0.9\) mm. The experimental tests involved the use of a prototype of equipment located at the University of Cassino Laboratory. The scheme of the equipment is shown in Figure 4, while all the details of the equipment can be found in [27; 28; 29]. In brief, the sheet to be formed is placed between a circular lower die (or blank holder), which contains the die cavity, and an outer upper die wrapped by an electric heating band. This band's purpose is to maintain the sheet at a constant and uniform temperature. Furthermore, to prevent the sheet from sliding within the die cavity, the two dies are  
Figure 4: Scheme of the equipment used for the experimental tests. clamped. The sheet's temperature (approximately 440 \({}^{\circ}\)C) is achieved using the electric heating band and maintained at the required value by a temperature control system. A hole in the outer die allows pressurised air from an external compressor to be conveyed, which is controlled by a proportional valve. During the forming test, a measuring laser monitors the displacement at the apex of the forming dome, creating a graph as a function of forming time.