# Numerical Simulation and Analysis of the Manufacturing Process of Pre-Bulged Rupture Discs
## 3 Results and Discussion
This explains the degressive nature of the \(\overline{\sigma}\)-\(p\) behavior within the initial elastic range. The inflection point defines the transition to the elastoplastic domain, where plastic deformation dominates. There is an approximately proportional relationship between \(\overline{\sigma}\) and \(p\) over large ranges. Since a significant part of the mechanical energy introduced into the structure is dissipated during plastic deformation, the increase in stress \(\mathrm{d}\overline{\sigma}/\mathrm{d}p\) in the plastically dominated domain is visibly smaller than before during fully elastic deformation. From a pressure level of about 85 to 90% of the individual bursting pressure \(p_{\text{B}}\), an increasingly progressive stress behavior within the pole structure up to bursting can be observed, similar to the strain behavior in Table 2.4 The effective stress \(\overline{\sigma}\) applied to the pole under the bursting pressure of \(p_{\text{B}}=2.39\) MPa has a final value of approximately 975 MPa. Using the methodology presented in Section 2.1, the effective polar stress \(\overline{\sigma}\) can also be approximated on the basis of experimental data. This was initially carried out with the help of Equation (16), in which the polar radius of curvature \(\rho_{\text{s}}\) is taken into account in the form of the simplified method according to Equation (11). In order to calculate the stress \(\overline{\sigma}\) associated with a pressure level \(p\), the two variable parameters \(h\) and \(d\) are needed as well. These were taken from the measurement results of Tables 1, 2 and 4, where the bulge height \(h\) corresponds to the displacement \(u_{z}\) contained in Table 2 at \(r_{n}=0\) mm. The resulting stress response is shown in the form of red data points and labeled with \(\approx\overline{\sigma}\) (\(\exp.\,\text{with}\,\rho_{\text{s}}\))\(\approx\) in Table 2.5 Here, a relative deviation from the numerical simulation of about 5 to 10% becomes visible, whereby the error behavior corresponds to that of the radii of curvature plotted in Table 2.6 This discrepancy is therefore due to the simplifications made to determine the radius of curvature \(\rho_{\text{s}}\) using Equation (11). If instead the numerically determined local radius of curvature of the simulation \(\rho_{\text{FEM}}\) is used and the stress approximation is performed using Equation (14) (\(p\) and \(d\) are used analogously from Tables 1 and 4), the discrete curve is obtained using the black data points. This is marked with \(\approx\overline{\sigma}\) (\(\exp.\,\text{with}\,\rho_{\text{FEM}}\))\(\approx\) and agrees extremely well with the numerical result of the simulation. Thus, it can be seen that both methods lead to the same result with corresponding parameter quality, which again proves the high quality of the numerical model and the validity of the methodology for analytical stress calculation based on experimental data presented in Section 2.1. While an equibiaxial stress state with \(\sigma_{\theta}/\sigma_{\varphi}=1\) is established at the pole of an axially symmetric bursting membrane, a stress ratio of \(R_{\sigma}=\sigma_{\theta}/\sigma_{\varphi}<1\) results in increasing radial distance \(r\), in which the meridional stress \(\sigma_{\varphi}\) dominates. This situation is examined below, and the evolution of the stress distribution during the bulge-forming process is shown in Figure 26 for various degrees of bulging \(\eta\) defined in Table 2.7 The plot shows the relevant membrane stress components \(\sigma_{\varphi}\) and \(\sigma_{\theta}\) for the thin bursting membrane with an initial thickness of \(t_{0}=0.102\) mm as a function of the current radial distance \(r\) from the pole. These stresses are defined as Cauchy stresses and can also be interpreted as principal stresses.