# Numerical Simulation and Analysis of the Manufacturing Process of Pre-Bulged Rupture Discs
## 3 Results and Discussion
In addition to the curves of the absolute stress variables, the dimensionless stress ratio \(R_{\sigma}\) for some degrees of bulging \(\eta\) is also shown in Figure 27 (upper part) for a more detailed characterization of the local distribution of the biaxial stress state. For this purpose, the coordinate of the current radial pole distance \(r\) was normalized using half the bulge diameter \(d_{\rm B}\). Starting from the flat membrane, an approximately constant distribution of the meridional stress \(\sigma_{\varphi}(r)\) is formed during the first stage of deformation. Due to the originally homogeneous thickness allocation \(t(r)=\text{const. }\), this extends almost over the entire bulge diameter \(d_{\rm B}\) until shortly before the bending radius at \(r\approx 45\) mm, where it is locally disturbed by the bending influence prevailing there. For the circumferential stress \(\sigma_{\theta}\), on the other hand, a continuously decreasing curve towards the bending radius can be observed, as the edge restraint counteracts deformation in the hoop direction. This finding can be observed with a degree of bulging of \(\eta=0.2\) in Table 2.8 Since the effective stress \(\overline{\sigma}\) as a measure of the material stress therefore increases in the direction of the pole, a plastic deformation occurs within the bursting membrane that depends on the radial distance \(r\) and has its maximum at the pole itself at \(r=0\) mm. This effect increases as the deformation or degree of bulging \(\eta\) progresses and is the reason for the inhomogeneous thickness profile \(t(r)\
eq\text{const. }\) that develops during the bulge-forming process. As a result, the meridional stress \(\sigma_{\varphi}\) increasingly shows a strictly monotonic decreasing curve in the direction of the edge with \(\mathrm{d}\sigma_{\varphi}/\mathrm{d}r<0\) for \(r>0\). A comparison of \(\sigma_{\varphi}(r)\) and \(\sigma_{\theta}(r)\) with increasing degree of bulging \(\eta\) shows that the stress drop of the circumferential stress with \(\mathrm{d}\sigma_{\theta}/\mathrm{d}r<0\) for \(r>0\) in the direction of the bending radius is not only more extensive than that of the meridional stress \(\mathrm{d}\sigma_{\varphi}/\mathrm{d}r\), but the stress level of \(\sigma_{\theta}\) is moreover approximately constant in the range around \(r\approx 45\) mm. The static frictional capacity induced by the edge restraint counteracts the deformation in the hoop direction, so that the circumferential stress \(\sigma_{\varphi}\) increases only to a limited extent compared to the meridional stress \(\sigma_{\varphi}\) in the area of the bending radius as the degree of bulging \(\eta\) increases. As a result, the stress ratio \(\sigma_{\theta}/\sigma_{\varphi}\) is reduced there by about 20% from an initial \(R_{\sigma}(\eta=0.2)=0.57\) to a final \(R_{\sigma}(\eta=1)=0.46\). Furthermore, the graphs confirm the insight already gained from the analysis of the curvature behavior (cf. Figure 22) of a surface area with spherical bulge geometry that increases with the degree of bulging \(\eta\). This is expressed in the stress behavior by an area with identical membrane stresses increasing from the pole where \(\sigma_{\varphi}=\sigma_{\theta}\) or \(R_{\sigma}=1\) applies (cf. also Figure 27). Only in this region does the effective stress \(\overline{\sigma}\) correspond to the meridional stress \(\sigma_{\varphi}\) or the circumferential stress \(\sigma_{\theta}\) (which also becomes the meridional stress at the pole). However, as soon as \(R_{\sigma}\
eq 1\) applies with a sufficient distance \(r>0\), the effective stress \(\overline{\sigma}\) must be determined from both components according to Equation (7). The numerical analysis also confirms the analytical consideration from Section 2.1 (cf. Equation (9)), where the thickness stress \(\sigma_{t}\) (or minimum principal stress) acting perpendicular to the membrane during bulge forming has no relevance for sufficiently slender bursting membranes and can therefore be neglected.