# Numerical Simulation and Analysis of the Manufacturing Process of Pre-Bulged Rupture Discs
## 3 Results and Discussion
With a further increase in pressure, the plastic deformation of both components proceeds with progressive characteristics according to their ratio, since the material hardening is less and less counteracting the structural stress. From a level of about 85 to 90% of the bursting pressure \(p_{\text{B}}\), a further visible increase in the gradient of the polar plastic strain can be seen until bursting. The strain hardening of the material, which is gradually approaching saturation, is less and less able to compensate for the reduced load-carrying behavior of the structure due to the already reduced cross-section. A small increase in pressure \(\text{d}p\approx 0\) therefore causes a potentially sensitive strain response \(\overline{\epsilon}_{\text{pl}}\) or \(|\text{d}\epsilon_{\text{pl},t}|\), which can lead to the collapse of the polar structure. Therefore, the permissible operating pressure of tension-loaded rupture discs should be below this threshold due to the unpredictable failure caused by material fatigue or creep. The strain components of the membrane pole shown in Figure 24 are the result of the mechanical stress on the structure due to the differential or effective pressure \(p\) acting on it. The variables relevant for the stress on the pole are mainly characterized by the meridional  
Table 2.3 Logarithmic principal plastic strains \(\epsilon_{\text{pl},\varphi}\), \(\epsilon_{\text{pl},\theta}\) and \(|\epsilon_{\text{pl},t}|\) as well as effective elastic strain \(\overline{\epsilon}_{\text{el}}\) acting on 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 results with experimentally determined values using Equation (10). stress \(\sigma_{\varphi}\) and the circumferential stress \(\sigma_{\theta}\). Both components act as tensile stresses and are also principal stresses. Due to the underlying axial symmetry, both stresses are identical at the pole of the membrane. As already shown in Section 2.1, these can be summarized similarly to the effective strain measure \(\overline{\epsilon}_{\text{pl}}\) to the effective stress \(\overline{\sigma}=\sigma_{\varphi}=\sigma_{\theta}\), which will be used to describe the polar stress state in the following. For the thin membrane with initial thickness \(t_{0}=0.102\) mm, the relationship between the effective stress \(\overline{\sigma}\) of the pole and the effective pressure \(p\) is plotted in Figure 25 for the entire bulge-forming process until the individual bursting pressure \(p_{\text{B}}\) (resp. \(\eta=1\)) is reached. The membrane stresses \(\sigma_{\varphi}\) and \(\sigma_{\theta}\) included in the calculation of the effective stress \(\overline{\sigma}\) are exclusively Cauchy stress components. The red curve shows the continuous curve determined by the simulation, with the graph splitting into two characteristic domains. A stress behavior \(\overline{\sigma}(p)\) proportional to the course of the elastic strain \(\overline{\epsilon}_{\text{el}}(p)\) can be seen over the entire bulge-forming process (cf. Figure 24). Beginning with the initial pressurization of the initially flat bursting membrane, a significant increase in stress is initially observed within the initial elastic range. Please note that the nonlinear curve has a degressive characteristic, which is caused by the nonlinear kinematics similar to that of the strain \(\overline{\epsilon}_{\text{el}}\) (cf. Figure 24). Due to its extremely low bending stiffness, the initially flat membrane has no significant resistance to the pressurization, so that especially the material particles of the pole experience very large displacements in the vertical \(z\)-direction. As the deformation increases, the gradient of the vertical pole displacement \(\mathrm{d}u_{z}/\mathrm{d}p\) gradually decreases as the bulging membrane gains extensional stiffness.