# Numerical Simulation and Analysis of the Manufacturing Process of Pre-Bulged Rupture Discs
## 2 Materials and Methods
### Experimental Analysis and Evaluation of the Bulge-Forming Process
This material fulfills a wide range of requirements and is therefore suitable for the use as a bursting membrane in a variety of process engineering systems. In addition to the very good deformation properties required for the pre-bulging process and the resulting resistance to material fatigue, its chemical composition offers excellent protection against corrosion and also allows it to be used at elevated temperatures of up to around 450 \({}^{\circ}\)C. The chemical composition of the batch from which the samples considered in this work (initial sheet thicknesses 0.102 mm, 0.343 mm and 0.480 mm) were taken is summarized in Table 5.  
These and the mechanical properties listed in Table 6 were taken from the test certificates provided by the steel producer. As the mechanical properties depend not only on the chemical composition but also to a large extent on the forming history and the heat treatment process, these must always be considered on a coil-specific basis. Due to the biaxial stress state, bursting membranes often reach strains during deformation that significantly exceed the uniform elongation \(\delta_{\rm u}\) of the same material in a uniaxial test. In order to reliably describe the elastoplastic deformation of a bursting membrane numerically up to the bursting pressure \(p_{\rm B}\), the relationship between true stress \(\sigma\) and true strain \(\epsilon\) is required for scenarios beyond the limitation given in the uniaxial test setting. The elastic range can simply be represented by the linear relationship between stress and strain according to [PERSON]'s law. Both variables are known to be linked via the modulus of elasticity \(E\). For the plastic range, a material-specific flow curve \(\sigma_{\rm f}(\epsilon_{\rm pl})\) is required, which also accurately describes the material behavior beyond the logarithmic uniform strain \(\epsilon_{\rm u}\). This can be obtained up to the logarithmic uniform strain, e.g., from a flat tensile test, and extrapolated to large plastic strains. A number of empirical approaches exist for this purpose. A simple and practicable option is to use the power law according to [PERSON] [20], which goes back to the preliminary work of [PERSON] [21]. According to this approach, the constitutive law contains two material-specific parameters. These parameters can be interpreted as a scaling factor and strain hardening exponent. The power law is suitable for ductile metals at room temperature. [PERSON]'s approach offers the advantage that the flow curve can be constructed approximately using the procedure according to [PERSON] [22] even if continuous tensile test data are not available. To estimate the two parameters, only the elementary mechanical parameters of a material in the form of the ultimate tensile strength \(R_{\rm m}\) and the uniform elongation \(\delta_{\rm u}\) are required. However, the disadvantage of this power law is that it is less suitable for high-alloyed, stainless steels and can also tend to overestimate strain hardening in the range of large strains above the logarithmic uniform strain \(\epsilon_{\rm u}\), cf. [23]. Exponential approaches often provide a better approximation quality for describing the saturation behavior as realistically as possible. The approach according to [PERSON] [24] offers a wide range of application and high quality for ductile metals. This reads  
\[\sigma_{\rm f}=\sigma_{\rm s}-\exp\Bigl{[}-N\epsilon_{\rm pl}{}^{m}\Bigr{]} \left(\sigma_{\rm s}-\sigma_{0}\right). \tag{17}\]  
In addition to the initial yield stress \(\sigma_{0}\), the Hockett-Sherby approach explicitly contains a saturation flow stress (steady-state flow stress) for the range of large plastic strains in the form of \(\sigma_{\rm s}\). Together with the fitting parameters \(N\) and \(m\), these must be adapted to the measurement data of a tensile test.