# Numerical Simulation and Analysis of the Manufacturing Process of Pre-Bulged Rupture Discs
## 3 Results and Discussion
Since the stress behavior of the thick membrane with \(t_{0}=0.480\) mm is very similar to that of the thin membrane illustrated in Figure 26, the slenderness ratio \(d_{\rm B}/t_{0}\leq 185\) is considered at least sufficient for AISI 316L in annealed condition. While Figure 26 illustrates the principal stresses \(\sigma_{\varphi}\) and \(\sigma_{\theta}\), Figure 28 shows the distribution of the three associated principal plastic strains \(\epsilon_{\rm pl,\varphi}\), \(\epsilon_{\rm pl,\theta}\) and \(\epsilon_{\rm pl,\theta}\) as a function of the current radial distance \(r\) from the membrane pole. These are defined as logarithmic, i.e., true strain values, and also refer to the thin bursting membrane with an initial thickness of \(t_{0}=0.102\) mm as well as to the degrees of bulging \(\eta\) defined in Table 2.9 The plots show a characteristic similar to the stress behavior, although it differs in some details. In addition to the meridional \(\epsilon_{\rm pl,\varphi}\) and circumferential \(\epsilon_{\rm pl,\varphi}\) strain components acting within the membrane surface, there is now also the component \(\epsilon_{\rm pl,t}\) acting in the normal or thickness direction, which is caused by the plane stress state due to the surface enlargement of the membrane and thus describes the thinning of its meridian. The curves along the radial coordinate \(r\) also confirm the dominance of the plastic strain component \(\epsilon_{\text{pl},t}\) of the thickness or normal direction. Here, \(|\epsilon_{\text{pl},t}|\) up to the edge is the effective plastic strain \(\overline{\epsilon}_{\text{pl}}\). Due to plastic incompressibility, this is the sum of the principal plastic strains in the membrane plane according to \(\overline{\epsilon}_{\text{pl}}=|\epsilon_{\text{pl},t}|=\epsilon_{\text{pl}, \varphi}+\epsilon_{\text{pl},\theta}\). As a result of the radial distribution of the two principal stresses \(\sigma_{\varphi}\) and \(\sigma_{\theta}\) within the membrane plane during the bulge-forming process (cf. Figure 26), the principal plastic strains \(\epsilon_{\text{pl},\varphi}\) and \(\epsilon_{\text{pl},\theta}\) are maximum at the pole and even identical for the axially symmetric problem (but only there). Consequently, the thinning dominates at the pole and initiates the plastic instability of the system from there. The circumferential strain \(\epsilon_{\text{pl},\theta}\) decreases more towards the edge than the meridional strain \(\epsilon_{\text{pl},\varphi}\) and has an approximate value of zero there depending on the contact boundary condition, which counteracts the deformation in the circumferential direction. Consequently, \(|\epsilon_{\text{pl},t}|\approx\epsilon_{\text{pl},\varphi}\) is valid in the region of the bending radius. For further identification of the strain behavior within the membrane plane, a strain ratio according to \(R_{\epsilon}=\epsilon_{\text{pl},\theta}/\epsilon_{\text{pl},\varphi}\) was introduced analogous to the biaxial stress ratio \(R_{\sigma}\) and plotted in Figure 27 (lower part) over the normalized radial coordinate of the current configuration \(2r/d_{\text{B}}\) for some degrees of bulging \(\eta\). As the degree of bulging \(\eta\) increases, \(R_{\epsilon}\) grows in the interval \(0<r<d_{\text{B}}/2\), whose characteristic is similar to that of \(R_{\sigma}\). However, due to the disappearance of the circumferential strain \(\epsilon_{\theta}\) in the boundary region, a delayed increase compared to the stress ratio \(R_{\sigma}\) can be observed here. In contrast to \(R_{\sigma}\), the ratio of the principal plastic strains during the whole bulge-forming process only applies to the pole itself \(R_{\epsilon}=1\). Since the entire deformation history must be considered for the plastic strain, a strictly monotonically decreasing curve for \(R_{\epsilon}\) results for all degrees of bulging \(\eta\) (also for \(\eta=1\)) starting from the pole.