# Numerical Simulation and Analysis of the Manufacturing Process of Pre-Bulged Rupture Discs
## 3 Results and Discussion
Its evolution as a function of the effective pressure \(p\) is plotted as a red curve in Figure 15:7 To determine \(\rho_{\text{FEM}}\), three closely neighboring element nodes located in the area of the pole on a meridian were selected and their \(x\) and \(z\) coordinates were used to describe their relative position to each other during deformation. After the flat sheet metal blank begins to deform, the three element nodes now form a triangle. This is used to determine the local radius of curvature \(\rho=\rho_{\text{FEM}}\) using the three side lengths and the area enclosed by them. Its curve in Figure 23 shows a strongly nonlinear behavior during plastic deformation. For very small degrees of bulging \(\eta\to 0\) or \(p\to 0\), it runs, as expected, to an infinite limit value \(\rho_{\text{FEM}}\to\infty\), since there is no curvature (\(\kappa\to 0\)). Starting from a flat membrane, even a minimal increase in pressure \(\Delta p\approx 0\) leads to a disproportionately large decrease in \(\rho_{\text{FEM}}\). This sensitivity decreases with increasing deformation. For \(p\to p_{\text{B}}\) or \(\eta\to 1\), the bursting membrane made of annealed austenitic stainless steel 316L reaches a polar radius of curvature \(\rho_{\text{FEM}}\) of about 55 mm and is thus still about 20% away from half the bulge diameter \(d_{\text{B}}/2\approx 45\) mm. As Figure 22 already shows, this final radius of curvature is effective over a proportion of about two-thirds of the bulge diameter \(d_{\text{B}}\). Depending on the geometric conditions and degree of deformation, the analytical approach according to Equation (11) is suitable for a simple approximation of the polar radius of curvature \(\rho\). It assumes that the center of curvature is always located on the axis of symmetry (\(z\)-axis) and that the curvature is constant. This approach was used to calculate the polar radius of curvature \(\rho_{\text{s}}\) together with the experimentally determined values of the bulge height \(h\) (denoted as \(h\in\) exp.) and to compare it with the local method based on numerics. The result summarized in Table 4 is plotted as a distribution of discrete data points in Figure 15:8 For a better comparison with the continuous curve of the local radius of curvature \(\rho_{\text{FEM}}\) of the pole determined by the simulation (red curve), a continuous curve was also generated with the help of Equation (11), in which the bulge height \(h\) (denoted as \(h\in\) sim.) determined by simulation was used as a basis (blue dashed curve). This fits with the points based on experimental data, as the evolution of the bulge height \(h\) of the  
Figure 15:9 Simulation result of the meridional radius of curvature \(\rho_{\varphi}\) as a function of the current radial distance \(r\) from the pole: membrane with initial thickness \(t_{0}=0.102\) mm at various degrees of bulging \(\eta\) defined in Table 1 until bursting at \(\eta=1\). simulation corresponds very well with that of the experimental program (cf. Figure 18). A comparison of both curves clearly shows that the deviation decreases with increasing deformation, as the bulge-shape of the bursting membrane successively approaches a spherical contour (cf. Figure 22), and thus, the simplification for large plastic deformations made in Equation (11) is approximately fulfilled. The relative deviation at \(p=p_{\text{B}}\) or \(\eta=1\) is still about 5%, as the ductility of the annealed 316L is not sufficient to achieve a perfectly spherical shape over the entire bulge diameter \(d_{\text{B}}\). The thinning of the pole plays an important role in the failure of an undisturbed bursting membrane. Although the bursting membrane is subjected to a plane stress state over large parts of its geometry, this causes a three-dimensional strain state. Figure 24 shows this in the form of the true or logarithmic principal plastic strains as a function of the effective pressure \(p\) for the thin membrane with initial thickness \(t_{0}=0.102\) mm.