# Numerical Simulation and Analysis of the Manufacturing Process of Pre-Bulged Rupture Discs
## 3 Results and Discussion  
The extensive results of the simulations are presented in the following and complemented by the experimental results from Section 2.2. The bulge-forming process was considered in the simulation for all three initial membrane thicknesses \(t_{0}=\{0.102,\,0.343,\,0.480\}\) mm of the annealed 316L stainless steel until the individually experimentally determined bursting pressure \(p_{\text{B}}\) from Table 1 was reached. The numerical and experimental results are compared at the discrete degrees of bulging \(\eta\), also specified in Table 1, where \(\eta\) is the ratio of the effective pressure \(p\) currently acting on the bursting membrane and its individual bursting pressure \(p_{\text{B}}\). Figure 17 illustrates the vertical displacement \(u_{z}\) of the bursting membrane pole during the bulge-forming process for all three considered thickness levels \(t_{0}\). According to the set up of the numerical model, the results are shown until the removal of the upper clamping device at normalized time \(\xi=0.8\). Pressurization begins at the end of clamping at \(\xi=0.1\) and continues until individual bursting pressure \(p_{\text{B}}\) is reached at \(\xi=0.6\) (corresponding to the degree of bulging \(\eta=1\)). All displacement curves start with an abrupt increase in deformation due to the lack of bending stiffness of flat membranes. Then, a phase of monotonous elastoplastic deformation follows. All three curves show an increase in slope  
Figure 16: Scaling functions of the corresponding boundary conditions during the bulge-forming simulation. Figure 15: Boundary conditions of the finite element model. that occurs just before the bursting pressure \(p_{\text{B}}\) is reached. This is an indication that the strain hardening of the material is less and less able to compensate for the reduction in membrane cross-section with increasing effective pressure \(p\). Then, an elastic recovery takes place when the pressure drop begins at \(\xi=0.6\). This shows that the elastic deformation component of the vertical pole displacement \(u_{z}\) is only about 1% for all three thickness levels. It can also be seen that the removal of the hold-down force at the final stage from \(\xi=0.7\) has no significant effect on the polar displacement \(u_{z}\). Figure 18 shows the experimental and numerical results for the bulge height \(h\) at the membrane pole as a function of the effective pressure \(p\) for all three thickness levels \(t_{0}\), whereby the measurement results were taken from Table 2. The bulge height \(h\) represents the final polar height of the rupture disc. That is, only the permanent portion caused by plastic deformation. The graph therefore gives an impression of the expected polar bulge height \(h\) at a given effective pressure \(p\). For \(t_{0}=0.102\,\text{mm}\), the simulation accurately reproduces the experimental results since the constitutive law nicely fits the material properties of the respective membrane material. For the other two thickness levels, the simulation systematically underestimates the polar bulge height \(h\) due to the softer material properties of the primary material batches, cf. Table 6. Even if the absolute values of the tests do not exactly match the simulation results, at least the deformation behavior is qualitatively correct. At this point, a softer material model that better accounts for the specific properties of the two coils with thicknesses \(t_{0}=0.343\) mm and \(t_{0}=0.480\) mm would lead to more accurate simulation results. However, it should be noted that there may also be slight variations in material properties within a coil, which should be taken into account if necessary. During the bulge-forming process, the thickness \(t\) of the bursting membrane locally decreases due to elastic and plastic deformation, whereby the plastic part clearly dominates. The minimum thickness is found at the pole of the membrane.