# Numerical Simulation and Analysis of the Manufacturing Process of Pre-Bulged Rupture Discs
## 3 Results and Discussion
The calculation of the local curvature involves the first- and second-order derivatives of the displacement values \(u_{z}\) with respect to the associated radial coordinate \(r\). The numerical calculation employs a forward difference quotient for these derivatives. All curves show a non-negligible dependence of the curvature behavior on the degree of bulging \(\eta\). While it is clear that the area of the pole has a spherical contour during almost the entire bulge-forming process, i.e., is constantly curved, this is not the case for the remaining area of the bulge diameter \(d_{\text{B}}\) of the bursting membrane. At the beginning of bulging, the curves have a negative slope from the pole to just before the bending radius. As a result, the meridional radius of curvature \(\rho_{\varphi}\) decreases continuously towards the edge, causing the curvature \(\kappa_{\varphi}\) to increase. The curvature behavior changes with increasing degree of bulging \(\eta\) and thus increasing plastic deformation, as the influence of the bending moment induced by the edge restraint is reduced. As a result, the slope of the curves starting from the pole tends to zero over an increasingly large range, so that the shape of the membrane approaches a spherical contour until it bursts at \(\eta=1\). It then has this spherical shape at least over a range of about two-thirds of its bulge diameter \(d_{\text{B}}\). Shortly before the bending radius at \(r\approx 44\) mm, a peak of the meridional radius of curvature \(\rho_{\varphi}\) can be seen. The second-order derivative \(\text{d}^{2}u_{z}/\text{d}r^{2}\) has a zero crossing at this position, because the curvature \(\kappa_{\varphi}\) of the bursting membrane has an inflection point before the bending  
Figure 15:1 Thickness \(t\) of the bursting membrane meridian due to plastic deformation as a function of the current radial distance \(r\) from the pole for the degree of bulging \(\eta=0.99\) defined in Figure 15:2 validation of the simulation results for the three initial thicknesses \(t_{0}\) considered with the experimentally determined values from Figure 15:3  
radius. At the inflection point itself, the membrane is not curved (\(\kappa_{\varphi}\to 0\) or \(\rho_{\varphi}\to\infty\)). The center of curvature, which lies below the rupture diaphragm starting from the pole until this inflection point is reached just before the bending radius, now moves upwards and is positioned above the diaphragm. In the region of the bending radius at \(r\approx 45\) mm, the meridional radius of curvature \(\rho_{\varphi}\) assumes the value of the bending radius of \(R=1.5\) mm. It can be seen that this area increases as the degree of bulging \(\eta\) increases, since the bursting membrane fits more and more closely to the bending radius of the upper clamping device as the plastic deformation progresses. As expected, there is no curvature after the bending radius in the area of the clamped edge for \(r>46\) mm (\(\kappa_{\varphi}\to 0\)). Therefore, the meridional radius of curvature \(\rho_{\varphi}\) of the bursting membrane in this area is \(\rho_{\varphi}\to\infty\). Figure 15:4 Permanent displacement \(u_{z}\) due to plastic deformation as a function of the current radial distance \(r\) from the pole (bulge shape) at various degrees of bulging \(\eta\) defined in Figure 15:5 validation of the simulation results with the experimentally determined values from Figure 15:6  
The local polar radius of curvature \(\rho\) associated with a characteristic pressure value \(p\) is required in the case of analytical calculations to determine the material stress at the bursting membrane pole. Due to the assumption of axial symmetry, \(\rho\) has no directional dependence in the area of the pole. For the thin diaphragm with an initial thickness of \(t_{0}=0.102\) mm, the required radius \(\rho\) was determined numerically from the finite element results and denoted as \(\rho_{\text{FEM}}\).