# FEM Analysis of Superplastic-Forming Process to Manufacture a Hemispherical Shell
###### Contents  
* 1 Introduction
\t* 2.1 Motivation of the material's strain rate
\t* 2.2 The material's strain rate
\t* 2.3 The material's strain rate
\t* 2.4 The material's strain rate
\t* 2.5 The material's strain rate
\t* 2.6 The material's strain rate
\t* 2.7 The material's strain rate
\t* 2.8 The material  
of a punch, a gas is used to push the sheet into the die cavity, and it is precisely regulated to control the material's strain rate. The optimal strain rate for superplastic materials is generally very low, ranging between \(10^{-3}\) and \(10^{-5}\) s\({}^{-1}\)[1]. At the same time, the sheet's temperature must be high, reaching values above 0.5 \(T_{m}\) (where \(T_{m}\) is the material's melting temperature in Kelvin). This temperature must be kept constant throughout the forming process [1]. In [2], concerning producing a component for the aerospace industry, two hemispherical shells were welded together from circular blanks of constant thickness, made of superplastic titanium alloy, to obtain a spherical product. The two half-shells, produced by superplastic forming, showed a non-uniform final thickness distribution. Specifically, the thickness distribution decreased from the edge to the pole of the two produced hemispherical shells. This distribution strongly depends on the material's properties (particularly, on the material's strain rate sensitivity index (\(m\))). Therefore, to reduce thickness non-uniformity, operating with the highest values of \(m\) is necessary [1]. Forming a superplastic material while maintaining a high strain rate sensitivity index requires the forming gas to follow a precise pressure-time loading curve. This can be excellently predicted with the aid of the finite element method (FEM). Therefore, for the superplastic material to exhibit its best formability throughout the entire process, it is necessary to control the strain rate step by step to keep the maximum strain rate within the material's optimal superplastic range. Over time, several authors have highlighted the need to propose predictive or a posteriori techniques to determine the pressure-time curve. A method for calculating the pressure cycle that can be associated with a finite element code to simulate superplastic forming processes was considered in [3]. A method using a predictor-corrector algorithm that predicts/corrects a new pressure value based on a previously known value and on the material constitutive equation was presented in [4]. The optimal pressure-time curve was calculated by a simple linear interpolation of the pressure value obtained in the previous iteration steps [5]. To optimise the pressure-time curve, a microstructure-based constitutive model was used in [6]. The aim of the work was to proceed at an optimal strain rate to ensure uniform deformation and reduce the forming time. In [7], the finite element model developed to predict the optimal pressure-time curve was used to compare the performances of the 2D and 3D approaches. In [8], an original predictor-corrector algorithm interfaced to the finite element calculation code was presented to determine the optimal pressure-time curve. The employed algorithm allows for keeping the maximum strain rate as close as possible to the optimal strain rate value while reducing the forming times. In [9], an automatic pressure calculation algorithm was presented both to avoid sheet metal necking and to reduce the forming times. It was shown in the literature that it is still possible to control the thickness of the formed product in an SPF process using the multi-phase forming technique; it requires that the process occurs in multiple phases: 1--pre-forming phase (using a pressurised gas or the action of a punch) of the blank at constant thickness with the task of preparing the sheet metal; 2--subsequent forming phases (the last phase occurs through the action of a pressurised gas). In [10], a two-stage approach was adopted that made the thickness distribution of cylindrical parts more uniform.