# FEM Analysis of Superplastic-Forming Process to Manufacture a Hemispherical Shell
## 2 Materials and Methods
### Forming Process of a Hemispherical Shell: 3D and 2D FEM Analysis  
The superplastic forming process of a hemispherical shell was analysed using numerical simulation based on the finite element method. The analysis was carried out with reference to both shell elements (3D analysis) and axisymmetric elements (2D analysis) available in the commercial software MSC-MARC 2005. Using a 2D axisymmetric model allows for simplifying the finite element approach by reducing the involved elements of the mesh and the required simulation time. They are interchangeable for axisymmetric parts. The parameter used to compare the obtained results was the thickness distribution of the formed hemispherical shell. In performing the comparison, the scheme shown in Figure 1 was used. Specifically, the blank consisted of a circular disc with a radius of 55 mm and an initial thickness of 1.0 mm. Since the forming process is assumed to occur at constant temperature, the problem addressed is simply mechanical in nature. The properties of the material to be deformed are related to the forming temperature. The analysis of superplastic forming problems requires the use of the rigid-plastic flow formulation. In this way, the material assumes a perfectly plastic behaviour. The material's constitutive equation was characterised by a power law of the following type [1]:  
\[\sigma=K\delta^{m}, \tag{1}\]  
In this equation, \(\sigma\) represents the equivalent stress, and \(\delta\) the equivalent strain rate, while the strength coefficient \(K\) and the strain rate sensitivity index \(m\) are material constants. This power law is representative of the behaviour of ideal superplastic materials without strain hardening [1]. As mentioned previously, the material constant that most influences the final thicknesses of the manufactured part is the strain rate sensitivity index, \(m\). It influences both the formability of the superplastic materials and the uniformity of the distribution of the thicknesses in the formed hemispherical shell. Increasing the \(m\) value involves a better formability and a more uniform thickness. A typical superplastic material is characterised by an \(m\) value ranging from 0.4 to 0.8; an intermediate value, i.e., \(m=0.5\)[1], was chosen. For the FEM analysis, no automatic mesh adaptation was used, but the mesh remained static, i.e., the number of finite elements did not change during the simulation. In the 3D simulation, only a quarter of the circular blank was represented using four-node thick shell elements (see Figure 2a). In particular, 1000 elements were used, arranged along 10 radial sectors (see Figure 2b). The element thickness was set equal to 1 mm. The die, which was considered a rigid body, consists of a spherical surface, with a radius of 50 mm, connected to a flat surface via a fillet with a radius of 5 mm. The circular blank is constrained so it does not slide within the die (see Figure 2b). For symmetry reasons, the nodes along the x-z axes were constrained from moving along the x-z directions (see Figure 2c,d). A constant pressure load was applied to the upper face of the elements. The convergence took into account the displacements, and the condition of perfect lubrication was considered. In the 2D simulation, due to the problem's axisymmetry, four-node axisymmetric elements were used to discretise only half of the circular blank's cross-section (see Figure 3). In this case, four-node, isoparametric, quadrilateral elements adapted for axisymmetric applications, present in the calculation code library, were used. A total of 100 elements are used, distributed uniformly along the radius of the blank. The used elements are therefore rectangular with dimensions of 1 mm \(\times\) 0.55 mm. The die consists of a circular curve with  
Figure 1: Scheme of the superplastic forming process to obtain a hemispherical shell. a radius of 50 mm connected to a straight line through a fillet with a radius of 5 mm. In this case, too, the circular blank is constrained to prevent it from sliding within the die.