# A New Approach to Characterize Superplastic Materials from Free-Forming Test and Inverse Analysis
## 4 Results and Discussion  
The experimental activity consisted of submitting a disc in AZ31 superplastic magnesium alloy to the free-forming test carried out at a temperature of 753 K and a constant pressure of 0.2 and 0.4 MPa, respectively. Each test was repeated three times for a total of six tests. Table 1 shows the value of the m parameter that was measured through the conventional procedure (Equation (3)) proposed in this work for which the m parameter varies inside each range of H (Equation (8)). Table 2 shows the values of n and K obtained using the conventional procedure and the values of parameter C for each range of H obtained using the proposed procedure. The reliability of the parameters characterizing the magnesium-based alloy AZ31 (identified through the conventional procedure and the procedure proposed in this work) was estimated by comparing the results of the FEM simulation of free-forming processes with those coming from the experimental activity. In particular, the comparison results are represented by the forming times achieved at each configuration of the deformed metal sheet. The metal sheet configuration is associated with the dimensionless displacement achieved at the apex of the same metal sheet, H. Figure 3a,b show the forming times associated with each dimensionless displacement achieved in two free-forming processes conducted at the constant pressures (p\({}_{1}\) = 0.2 MPa and p\({}_{2}\) = 0.4 MPa) by a sequence of small circles. At the same time, Figure 3a,b show the results of the simulations conducted using the following as parameters characterizing the material:  
* The parameters coming from the conventional procedure (C.P.) adopted to identify them (dotted line--FEM C.P. );
* The parameters coming from the proposed procedure (P.P.) adopted to identify them (solid line--FEM P.P. ). \begin{table}
\begin{tabular}{c c c} \hline \hline \begin{tabular}{c} **Dimensionless Displacement H** \\ \end{tabular} & \
\begin{tabular}{c} **Strain Rate Sensitivity Index,** \\ **m, of Conventional Procedure** \\ \end{tabular} &
\begin{tabular}{c} **Strain Rate Sensitivity Index, m,** \\ **of Proposed Procedure** \\ \end{tabular} \\ \hline \(0\leq\text{H}\leq 1\) & 0.381 & / \\ \hline \(0\leq\text{H}\leq 0.2\) & / & 0.703 \\ \hline \(0.2\leq\text{H}\leq 0.4\) & / & 0.465 \\ \hline \(0.4\leq\text{H}\leq 0.6\) & / & 0.495 \\ \hline \(0.6\leq\text{H}\leq 0.8\) & / & 0.406 \\ \hline \(0.8\leq\text{H}\leq 1\) & / & 0.284 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Evaluation of the strain rate sensitivity index, m, of the conventional and proposed procedures. \begin{table}
\begin{tabular}{c c c c} \hline \hline \begin{tabular}{c} **Dimensionless Displacement H** \\ \end{tabular} & \begin{tabular}{c} **Strain Hardening Index,** \\ **n, of Conventional Procedure** \\ \end{tabular} & \
\begin{tabular}{c} **Strength Coefficient, K, of** \\ **Conventional Procedure** \\ \end{tabular} &
\begin{tabular}{c} **Strength Coefficient, C, of** \\ **Proposed Procedure** \\ \end{tabular} \\ \hline \(0\leq\text{H}\leq 1\) & 0.1 & 163.24 & / \\ \hline \(0\leq\text{H}\leq 0.2\) & / & / & 862.875 \\ \hline \(0.2\leq\text{H}\leq 0.4\) & / & / & 241.357 \\ \hline \(0.4\leq\text{H}\leq 0.6\) & / & / & 346.950 \\ \hline \(0.6\leq\text{H}\leq 0.8\) & / & / & 198.300 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Evaluation of the hardening index, n, and the strength coefficients, K and C.  
Figure 3 shows a qualitative comparison between the obtained results. It can be seen that the use of the m and C parameters, which vary with H, allows for the obtaining of t-H curves via FEM nearer to the experimental ones than the constants m, n, and K do. It was quantified that the comparison between the numerical and experimental results through the parameter \(\Delta\).