# Numerical Simulation and Analysis of the Manufacturing Process of Pre-Bulged Rupture Discs
## 3 Results and Discussion
As the bending stress component increases with increasing distance from the neutral axis, this effect becomes more influential as the membrane thickness increases. In addition, the structure at this location is subjected to a variable contact pressure distribution that is superimposed on the internal stress state. This relationship can be seen by comparing the three curves for different membrane thicknesses. While the thin membrane with \(t_{0}=0.102\) mm does not show any significant local necking due to its very low thickness, the local disturbance of the membrane stress state becomes clear for the two thicker ones. In addition, a reduction in thickness is visible along the clamped edge of the bursting membrane. This is strongly dependent on the ratio of hold-down force \(F_{\text{cyl}}\) to effective pressure \(p\) as well as the coefficient of friction between the contact partners. The results show a thickness curve along the edge of the rupture disc that depends on the initial thickness \(t_{0}\) and thus on the effective pressure \(p\). For the same nominal diameter (DN 80 mm in this case), the required effective bulge-forming pressure \(p\) increases with the initial thickness \(t_{0}\) of the disc and thus the resulting tangential force in the contact area. If this force exceeds the static friction force applied, the clamped particles will slide locally until a new equilibrium is reached. During the sliding process, the released part of the membrane can contract more or less freely and thus locally exceeds the yield stress of the membrane material. This process continues successively from the inside to the outside as the pressure increases further and can be observed with the two thicker membranes. Depending on the effective pressure \(p\), an individual bulge shape of the bursting membrane is generated during the bulge-forming process. Figure 21 shows this profile along a meridional section for the three membranes investigated with initial thicknesses \(t_{0}\) of 0.102 mm, 0.343 mm and 0.480 mm at various degrees of bulging \(\eta\) defined in Table 1. The permanent vertical displacement \(u_{z}\) due to plastic deformation was determined. Since each material particle also undergoes a horizontal displacement \(u_{x}\) during the bulge-forming process (with the exception of the pole in the axially symmetric approach), the radial distance \(r\) from the pole must be taken into account for the representation of the bulge shape in the current configuration with \(r=r_{0}+u_{x}\). The continuous curves represent this relationship for the numerical simulation, while the discrete points show the measurement results. These are taken from Figure 15:0 The numerical results of the thin membrane again show excellent agreement with the experimental data. This is true for the entire manufacturing process for all degrees of bulging \(\eta\) and thus for all configuration modes up to the edge zone of the membrane influenced by the contact with the bending radius. In the case of the two thicker membranes, however, it is again evident that the material model used describes their flow behavior only in a rough approximation. Although the error is still relatively small in the range of small plastic strains, it increases with increasing deformation due to an over-estimated strain hardening behavior. Looking at the plots of the bulge shapes in Figure 21, one can initially assume a spherical geometry of the domed-bursting membranes, except for the area just before the bending radius. For clear identification, Figure 22 shows the curvature behavior of the thin membrane with an initial thickness of \(t_{0}=0.102\) mm during the bulge-forming process along a meridional section. For this purpose, the simulation results were processed accordingly and the principal radius of curvature \(\rho_{\varphi}\) of the meridian was plotted for some degrees of bulging \(\eta\) defined in Table 1 as a function of the current radial distance \(r\) from the membrane pole.