# 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
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**C** & **Si** & **Mn** & **S** & **P** & **Cr** & **Ni** & **Mo** & **Cu** & **N** \\ \hline
0.016 & 0.244 & 1.44 & 0.0016 & 0.035 & 16.55 & 10.06 & 2.1 & 0.495 & 0.037 \\ \hline \end{tabular}
\end{table} 
Table 5: Chemical composition of the considered batch of AISI 316L/EN 1.4404 austenitic stainless steel as mass fractions in %. In the literature, there are currently very few convincing studies in which the flow behavior of a cold-rolled and annealed AISI 316L or EN 1.4404 with low thickness values has been analyzed. One promising experimental work was published by [PERSON] et al. [25]. The authors investigated the anisotropic flow behavior of 1.2 mm thick sheets of the required material with a comparable manufacturing process. In addition to uniaxial flat tensile tests, they also carried out hydraulic bulge tests at room temperature and under quasi-static conditions in order to be able to characterize the hardening behavior for strains above the logarithmic uniform strain. To mathematically describe the flow behavior and to extrapolate the flow curve, the measured data from the tensile test were adapted to the Hocket and Sherby approach using a parameter fit and the approximation quality was verified by comparing it with the results of the bulge test. The model parameters presented by [PERSON] et al. are listed in Table 7.  
The yield strength \(R_{\text{p0.2}}\) and the ultimate tensile strength \(R_{\text{m}}\) were extracted from the reference by [PERSON] et al. in order to classify and check the transferability to the primary membrane materials considered in this work. The corresponding values of \(R_{\text{p0.2}}\) and \(R_{\text{m}}\) can also be found alongside the elastic constants \(E\) and \(\
u\) in Table 6. A comparison of the strength characteristics indicates that the plastic behavior of the reference material with a thickness of 1.2 mm corresponds most closely to that of the thinnest bursting membrane with a thickness of \(t_{0}=0.102\) mm. Since both the experienced degree of deformation of the flat products in the rolling process and the boundary conditions of the annealing process are unknown, this can be regarded as quite a realistic starting point for the bulge-forming process. Due to the scientific lack in the current literature and the fact that the flow curve model developed by [PERSON] et al. is an experimentally validated reference, it is used as the basis for the simulations to be carried out in this contribution. The employed flow curve is shown in Figure 12. This model was implemented for the numerical simulation of all three bursting membrane thicknesses \(t_{0}\) as part of the present work. Together with the elastic constants \(E\) and \(\
u\) from Table 6 and the assumption of isotropic material behavior, the constitutive law is thus fully defined. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Initial** & **Young’s** & **Poisson’s** & **Yield Strength** & **Ultimate** & **Total** \\
**Thickness \(t_{0}\)** & **Modulus \(E\)** & **Ratio \(\
u\)** & **R\({}_{\text{p0.2}}\)** & **Tensile** & **Elongation at** \\  & & & & **Strength \(R_{\text{m}}\)** & **Fracture \(\delta_{\text{t}}\)** \\ \hline
0.102 mm & & & 322 MPa & 678 MPa & 0.63 \\
0.343 mm & - & - & 302 MPa & 623 MPa & 0.56 \\
0.480 mm & & & 270 MPa & 612 MPa & 0.57 \\ \hline
1.2 mm & 194 GPa & 0.33 & \(\sim\)339 MPa & \(\sim\)650 MPa & - \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Mechanical properties of the considered batch of annealed AISI 316L/EN 1.4404 austenitic stainless steel as a function of the initial sheet thickness \(t_{0}\) according to the supplier test certificates (upper three rows). The bottom row shows the properties of the reference material that serves as the basis for the employed numerical model according to [25]. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\sigma_{\text{s}}\) & \(\sigma_{0}\) & \(N\) & \(m\) \\ \hline
1445.44 MPa & 339.48 MPa & 1.81 & 0.89 \\ \hline \hline \end{tabular}
\end{table}