# A New Approach to Characterize Superplastic Materials from Free-Forming Test and Inverse Analysis
## 3 Theory  
A superplastic material is generally characterized by the power law below. \[\overline{\sigma}=\text{K}\varepsilon^{\text{n}}\delta^{\text{m}}, \tag{2}\]  
This law puts into relationship the equivalent stress, \(\sigma\), to the equivalent strain, \(\varepsilon\), and to the strain rate, \(\delta\), through the strain rate sensitivity index, m, the strain hardening index, n, and the strength coefficient, K.  
To determine the constants of Equation (2), the conventional procedure [38; 47], which is used to characterize the behaviour of superplastic materials, requires the calculation of the strain rate sensitivity index through two free-forming experimental tests that are carried out at two values of pressure (p\({}_{1}\) and p\({}_{2}\)) that are kept constant. The tests must be carried out until a unitary value of the dimensionless displacement, H, is reached. In this way, it is possible to calculate m using the following expression [4; 37; 43; 48; 49]:  
\[\text{m}=\text{ln}(\text{p}_{1}/\text{p}_{2})/\text{ln}(\text{t}_{2}/\text{t}_ {1}), \tag{3}\]  
Figure 2: FEM scheme of the free-forming test. where \(t_{1}\) and \(t_{2}\) are the times to reach the unit value of the dimensionless displacement, H, under the action of a constant pressure equal to \(p_{1}\) and \(p_{2}\), respectively. To evaluate the n and K constants, the finite element modelling of the free-forming process is commonly used [38, 47]. The normalized time, \(\tau\), is defined as follows:  
\[\tau=t_{H=1}/t_{H=0.5}, \tag{4}\]  
where \(t_{H=1}\) and \(t_{H=0.5}\) are the times needed to reach the dimensionless displacements H = 1 and H = 0.5, respectively. Therefore, once an arbitrary value is assigned to K and the constant m is known, numerical simulations are carried out by varying the n constant. The value of the strain hardening index n is found by minimizing the following function Q(n):  
\[ \begin{split} Q(n)=\left(\frac{(\ln\tau)^{FEM}-(\ln\tau)^{EXP}} {(\ln\tau)^{EXP}}\right)_{P_{1}}^{2}+\left(\frac{(\ln\tau)^{FEM}-(\ln\tau)^{ EXP}}{(\ln\tau)^{EXP}}\right)_{P_{2}}^{2}+\left(\frac{(\varepsilon)^{FEM}-( \varepsilon)^{EXP}}{(\varepsilon)^{EXP}}\right)_{P_{1}}^{2}+\\ \left(\frac{(\varepsilon)^{FEM}-(\varepsilon)^{EXP}}{( \varepsilon)^{EXP}}\right)_{P_{2}}^{2},\end{split}  \tag{5}\]  
which is constituted by quantities, such as normalized time, \(\tau\), and equivalent strain, \(\varepsilon\), of FEM numerical simulations and experimental tests. Similarly, once the value of the hardening index, n, is determined, further numerical simulations of the free-forming test allow for the evaluation of the strength coefficient, K, by minimizing the function F(K) given by the following formula:  
\[F(K)=\left(\frac{(t_{H})^{FEM}-(t_{H})^{EXP}}{(t_{H})^{EXP}}\right)_{P_{1}}^ {2}+\left(\frac{(t_{H})^{FEM}-(t_{H})^{EXP}}{(t_{H})^{EXP}}\right)_{P_{2}}^ {2}, \tag{6}\]  
where t is the time required to reach the unit dimensionless displacement, through FEM numerical simulations and experimental tests, under the action of the constant pressure \(P_{1}\) or \(P_{2}\). In this work, the procedure proposed for the characterization of superplastic materials involves a reduction in the material parameters. In fact, the reference power law is no longer given by Equation (2) but by the following formula:  
\[\overline{\sigma}=C\delta^{m}, \tag{7}\]  
where m and C are parameters that depend on the equivalent strain, \(\varepsilon\). The variability of m with strain is determined experimentally by dividing the dimensionless height, H, into five ranges (0 \(\leq H\leq\) 0.2; 0.2 \(<H\leq\) 0.4; 0.4 \(<H\leq\) 0.6; 0.6 \(<H\leq\) 0.8; 0.8 \(<H\leq\) 1.0). For each range the parameter m can be determined using the following equation:  
\[m=\ln(p_{1}/p_{2})/\ln(\Delta t_{2}/\Delta t_{1}), \tag{8}\]  
in Equation (8), \(\Delta t_{1}\) and \(\Delta t_{2}\) are the times needed to fill the single range of H under the action of the pressure \(p_{1}\) and \(p_{2}\), respectively.