# Numerical Simulation and Analysis of the Manufacturing Process of Pre-Bulged Rupture Discs
## 3 Results and Discussion
As Figure 27 (upper part) shows, during the first phase of bulging, only the pole itself experiences the maximum stress with \(R_{\sigma}=1\), whereby the material particles there plasticize more strongly than their neighbors even at small degrees of bulging \(\eta\). Although the zone within \(R_{\sigma}=1\) increases with the degree of bulging \(\eta\), it can no longer compensate for the plastic deformation protrusion of the pole. Table 6.0 Effective stress \(\overline{\sigma}\) at the bursting membrane pole as a function of the effective pressure \(p\) for bursting membrane with initial thickness \(t_{0}=0.102\) mm up to bursting at \(\eta=1\) (corresponding bursting pressure \(p_{\text{B}}=2.39\) MPa): comparison of simulation-based result with experimentally determined values (**red data points**: using Equation (16) with the simplified method of determining the radius of curvature \(\rho_{\text{s}}\) according to Equation (11) and the measured values of the bulge height \(h\) from Table 2; **black data points**: using Equation (14) and taking into account the local radius of curvature \(\rho_{\text{FEM}}\) obtained from the simulation). Table 6.1 Evolution of the biaxial ratios of principal Cauchy stresses \(R_{\sigma}=\sigma_{\theta}/\sigma_{\varphi}\) (**top plot**) and logarithmic principal plastic strains \(R_{\varepsilon}=\epsilon_{\mathrm{p}\mathrm{I}\theta}/\epsilon_{\mathrm{p} \mathrm{I}\varphi}\) (**bottom plot**) as a function of the current radial distance \(r\) from the pole (normalized with \(d_{\mathrm{B}}/2\)) during bulge-forming process (different degrees of bulging \(\eta\) according to Table 1). Table 6.2 Simulation results of the principal Cauchy stresses \(\sigma_{\varphi}\), \(\sigma_{\theta}\) and \(\sigma_{\mathrm{t}}\) acting in the bursting membrane 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 \(\eta=1\). Once the desired degree of bulging \(\eta\) has been reached during bulge forming, the working pressure \(p\) is reduced and the clamping devices are released so that the reshaped bursting membrane is relieved of external constraints. During this release process, the internal stresses introduced by the elastoplastic deformation are reduced until an equilibrium is reached. These residual stresses remain permanently in the material without further action. Figure 29 gives an impression of the residual stress level to be expected in a bursting membrane after completion of the bulge-forming process. For this purpose, the three principal stresses of the medium-thick membrane with initial thickness \(t_{0}=0.343\) mm are considered, which allow us to distinguish between tensile and compressive stresses. According to the slightly overestimated hardening behavior of the selected material model for this coil (or the specific removal position of the sample from the coil), a final degree of bulging of \(\eta\approx 0.9\) is obtained for the applied simulation boundary conditions. This shows a concentration dominated by tensile stresses around the area of the bending radius (cf. maximum and minimum principal stress), which depends on the chosen degree of bulging \(\eta\) and the membrane thickness \(t_{0}\) as well as the bending radius \(R\). As the pressure increases during bulge forming until the desired degree of bulging \(\eta\) is reached in the interval \(0.1<\xi\leq 0.6\), the superposition of membrane and bending stresses leads to extensive stressing of the material in this region. While tensile stresses dominate on the bottom side, compressive stresses dominate on the upper side. If the working pressure \(p\) is now gradually reduced for \(0.6<\xi\leq 0.7\) after the desired degree of bulging \(\eta\) has been reached, the bulged profile will perform an elastic settlement.