# Numerical Simulation and Analysis of the Manufacturing Process of Pre-Bulged Rupture Discs
## 3 Results and Discussion
The black curve illustrates the strain components \(\epsilon_{\text{pl},\varphi}\) and \(\epsilon_{\text{pl},\theta}\) of the pole lying in the membrane plane. Due to the underlying axial symmetry, these can be equated and are therefore both referred to as \(\epsilon_{\text{pl}}\). They agree very well with the experimental results on the true principal plastic strain \(\epsilon_{\text{pl},\mu}\) determined in Section 2.2 and summarized in Table 2.0 These are shown in Figure 24 as black data points under \({}^{\approx}\epsilon_{\text{pl}}\) (exp.)\({}^{\sim}\). Assuming that the material is incompressible during plastic deformation, the surface strain \(\epsilon_{\text{pl}}\) can easily be converted into the portion \(\epsilon_{\text{pl},\mu}\) acting normal to the surface. This corresponds to the strain component in the thickness direction and describes the thinning of the membrane pole during the bulge-forming process. The magnitude of the simulation result is denoted by \(|\epsilon_{\text{pl},\mu}|\) and illustrated by the red curve. It can be shown that \(|\epsilon_{\text{pl},\mu}|\) corresponds to the effective plastic strain \(\overline{\epsilon}_{\text{pl}}\) according to von Mises (also called equivalent plastic strain) under the assumption of equibiaxial stress, and therefore, \(\overline{\epsilon}_{\text{pl}}=|\epsilon_{\text{pl},\mu}|\) results. This applies to the area of the pole, where \(\overline{\epsilon}_{\text{pl}}\) represents the strain measure relevant for thinning and thus for failure. For comparison with the experimental data, the measurement results of the surface strain \(\epsilon_{\text{pl},\mu}\) were converted to those in the thickness direction. These are plotted as red data points with \(\approx\)\(|\epsilon_{\text{pl},\mu}|\) (exp.)\({}^{\sim}\). At the beginning of the deformation, the plastic strains initially show a constant curve on the abscissa, as the material deforms purely  
Table 2.1 Polar radius of curvature 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 the simulation-based result \(\rho_{\text{FEM}}\) (local determination at the pole) and the simplified approximation \(\rho_{\text{s}}\) using Equation (11), taking into account the simulated and measured polar bulge height \(h\). elastically up to a degree of bulging of \(\eta\approx 0.05\). In order to be able to classify the elastic portion of the total deformation, the effective elastic strain \(\overline{\epsilon}_{\text{el}}\) (equivalent elastic strain) was also used as the true strain variable in Table 2.2 Compared to the total deformation, the elastic component is therefore relatively small and amounts to only about 1% when the bursting pressure \(p=p_{\text{B}}\) or the bulging level \(\eta=1\) is applied. After passing the initial elastic domain, the pole starts to deform plastically for \(p\geq 0.12\) MPa or \(\eta\geq 0.05\). In the elastoplastic range, both plastic strain components \(\epsilon_{\text{pl}}\) and \(|\epsilon_{\text{pl},t}|\) are characterized by a continuous increase in their slope. During this second phase of deformation, the elastic strain shows an approximately proportional character over large parts with a significantly reduced slope to the initial elastic domain. According to the initially very strong hardening behavior of the material under consideration (cf. the slope of the flow curve in Figure 12), the pole initially reacts to an increase in pressure \(\text{d}p>0\) starting from the initial yield point \(\sigma_{0}\) at \(p\approx 0.12\) MPa with a moderate increase in plastic strain \(\epsilon_{\text{pl}}\) within the plane. This can initially be characterized as approximately proportional to the pressure. Due to the ratio of \(|\epsilon_{\text{pl},t}|/\epsilon_{\text{pl}}=2\), the part acting in the thickness direction \(\overline{\epsilon}_{\text{pl}}\) or \(|\epsilon_{\text{pl},t}|\) is weighted correspondingly more heavily.