# Numerical Simulation and Analysis of the Manufacturing Process of Pre-Bulged Rupture Discs
## 3 Results and Discussion
In order to the get an impression of the thickness reduction during the entire forming process, Figure 19 illustrates the instantaneous thickness \(t\) at the respective membrane pole over the normalized process time \(\xi\). It can be seen that the pole region undergoes a significant thickness reduction, which finally leads to the rupture of the membrane at the corresponding bursting pressure \(p_{\text{B}}\). Initially, elastic thinning takes place within the initial elastic domain, ranging from \(\xi=0.1\) to about \(\xi=0.13\). This is followed by an extensive thickness reduction during  
Figure 17: Vertical displacement \(u_{z}\) of the bursting membrane pole during the entire simulation or bulge-forming process for the three initial thicknesses \(t_{0}\) considered. the elastoplastic domain until the experimentally determined individual instability limit is reached at \(\xi=0.6\) (degree of bulging \(\eta=1\)). Since the simulation considers pure elastic and plastic deformation of the material without damage, elastic recovery during pressure reduction can now be observed. The relative thickness reduction at the pole to bursting is 29% for the thick membrane, 28% for the medium membrane, and 34% for the thin membrane. Knowing that the chosen constitutive law provides slightly too high strength values for the thick and medium thick sheet, and considering the experimental results from Figure 20, the relative thickness reduction of about 34% in the thin membrane also applies approximately to the two thicker ones. Figure 19: Instantaneous thickness \(t\) of the bursting membrane pole during the entire simulation or bulge-forming process for the three initial thicknesses \(t_{0}\) considered. Figure 18: Bulge height \(h\) at the pole of the bursting membrane as a function of the effective pressure \(p\) for the three different initial thicknesses \(t_{0}\) considered: validation of the simulation results with the experimentally determined values from Table 2.  
Even though the thickness reduction dominates in the region of the membrane pole due to the equibiaxial stress state prevailing there, the surface enlargement of the membrane during bulge forming occurs at the expense of almost the entire membrane cross-section. To get an idea of which area of the bursting membrane must provide which fraction, Figure 20 shows the instantaneous thickness \(t\) of the membrane as a function of current radial distance \(r\) from the pole within a meridional section. The figure illustrates the continuous simulation results for all three initial thicknesses \(t_{0}\) compared to the discrete measured values \(t(r_{m})\) shortly before bursting at the degree of bulging \(\eta=0.99\) (in accordance with the corresponding individual bursting pressures \(p_{\text{B}}\) from Table 1). The thickness curves of the simulation were obtained from the true plastic strain components in a follow-up calculation and the measurement results were taken from Table 3. The comparison of simulation and experiment once again validates the high level of approximation quality of the numerical model for the thin membrane with an initial thickness of \(t_{0}=0.102\) mm. As the membrane thickness \(t_{0}\) increases, the approximation quality decreases. This is due to the fact that the flow curve model shown and implemented in Figure 12 is less effective in describing the plastic properties of the membrane material as the thickness \(t_{0}\) increases (cf. Table 6). The curves demonstrate a notable reduction in membrane thickness across the entire area, extending up to the outer edge of the rupture disc. However, it is notable that the contribution to plastic strain decreases continuously from the pole to the bending radius at \(r=d_{\text{B}}/2\approx 45\) mm. The membrane stress state is disturbed in the area of the bending radius. The additional bending stress component causes a local necking depending on the membrane thickness.