## 1 Introduction
In particular, several constitutive equation models for superplasticity have been proposed, such as the uniaxial power-law model [2; 36], uniaxial sinh-law model [2; 37], and uniaxial superplastic-damage constitutive equations [2; 38; 39]. These constitutive equations contain many parameters connected with the material to characterize through experimental tests. Several mathematical functions were reported in [40]; they describe the behavior of superplastic materials for different strain rate values and different phenomena, such as work hardening and grain growth. Figure 1: Scheme of the superplastic-forming process. In the present work, a simple procedure is proposed to characterize 1.0 mm thick sheets of superplastic magnesium alloy AZ31. Through free-forming tests carried out at constant pressure and with the help of the FEM, it was possible to identify the characteristic parameters of the material present in the simplest constitutive function used in the scientific literature relating to the behavior of superplastic materials [1; 2; 41; 42; 24; 43; 25; 44]. This function, which is used in a limited range of strain rates, is represented by a power law:  
\[\sigma=\mathrm{K}\delta^{\mathrm{m}}, \tag{1}\]  
where \(\sigma\) is the flow stress, \(\delta\) is the strain rate, and \(\mathrm{m}\) and \(\mathrm{K}\) are the material parameters. The \(\mathrm{m}\) parameter is called the strain rate sensitivity index. This parameter has a significant role in determining the superplasticity behavior of a material. Its value is higher than 0.3; it was found to vary between 0.4 and 0.8 for superplastic materials [2]. It provides the elongation ability of a material since it gives great resistance to neck propagation during tensile deformation. According to international ISO standards, the sensitivity index to strain rate can be determined by two tensile tests carried out at different strain rates or by a tensile test at jumps of strain rate [45]. In both cases, to obtain the value of the strain rate sensitivity index, once the strain value has been preset, it is necessary to determine the stress corresponding to the impressed strain rate. The constitutive Equation (1) cannot be used to model the grain growth and the phenomenon of cavitation. However, these phenomena depend on the equivalent strain to which the material is subjected. Therefore, Equation (1) is modified in this work by considering the parameters \(\mathrm{m}\) and \(\mathrm{K}\) dependent on the effective deformation, i.e.,  
\[\mathrm{m}=\mathrm{f}(\varepsilon)\;\mathrm{K}=\mathrm{g}(\varepsilon). \tag{2}\]  
The constitutive equation of materials represents a key factor in designing superplastic-forming processes through numerical simulation. The reliability of this equation makes it possible to determine the gas pressure profile that must be imposed on the sheet to form it inside the optimal superplastic range. The constitutive equation of the AZ31 superplastic magnesium alloy is represented by a power law that describes the relationship between stress and strain rate [46]. The parameters of the material in this law (represented by strain hardening index, strain rate sensitivity index, and strength coefficient) were considered constants and were obtained through free-forming tests and a numerical-experimental procedure which requires an inverse analysis. The searched parameters were constant throughout the range of variability of the dimensionless displacement \(\mathrm{H}\). On the other hand, in the present work, the values of the parameters \(\mathrm{m}\) and \(\mathrm{K}\) are determined through free-forming tests, while subdividing the entire dimensionless displacement measured at the apex of the dome, \(\mathrm{H}\), in different ranges of \(\mathrm{H}\), thus indirectly representing strain.