# Numerical Simulation and Analysis of the Manufacturing Process of Pre-Bulged Rupture Discs
## 2 Materials and Methods
### Experimental Implementation of the Bulge-Forming Process and Theoretical Methods for Evaluation
Membrane theory simplifies the general rotational shell problem in such a way that important structural-mechanical relationships can be formulated using simple analytical methods. This approach is therefore used below to determine the structural stresses in the pole region of interest for pre-bulged bursting discs. Within the membrane theory, the external surface load acting vertically on the shell center surface is transferred purely via stresses parallel to the shell center surface. The arising stress state is comparable to that of the plane stress state. The normal stress of the bursting membrane acting perpendicular to the center surface of the shell or in the thickness direction is determined by the mean value of the external and internal pressure. For bulge-forming, this is \(\sigma_{t}=-p/2\) and is to be interpreted as the minimum principal stress of the bursting membrane. Since the condition \(t/\rho<<1\) is fulfilled for most pre-bulged rupture discs, their magnitude is considerably smaller than that of the principal stresses parallel to the center surface of the shell. If a sufficiently large distance to the edge is maintained, the relationships known from the membrane theory of thin shells of revolution can be transferred to the bulge problem of many common bursting membrane dimensions. According to [PERSON] [18] or [PERSON] [19], the static equilibrium of forces in the direction of the shell normal leads to the well-known membrane equation  
\[\frac{\sigma_{\varphi}}{\rho_{\varphi}}+\frac{\sigma_{\theta}}{\rho_{\theta}}= \frac{p}{t}\,, \tag{2}\]  
which can be used to describe the stress situation of the membrane pole of an axially symmetrically loaded bursting disc. The stresses \(\sigma_{\varphi}\) and \(\sigma_{\theta}\) correspond to the principal stresses in the membrane plane, while \(\rho_{\varphi}\) and \(\rho_{\theta}\) represent the associated principal radii of curvature on the differential element with instantaneous thickness \(t\). Different principal curvatures can also occur at the pole if, for example, direction-dependent deformation properties caused by a rolling texture influence the deformation. The stress resultants of a differential element of a bursting membrane under uniform internal pressure \(p\) are shown in the principal axis system in Figure 7a. To determine the two principal stresses \(\sigma_{\varphi}\) and \(\sigma_{\theta}\) from Equation (2) under the general consideration of two independent principal curvature semi-diameters \(\rho_{\varphi}\) and \(\rho_{\theta}\), a further equation is required. This can be found most easily via the static equilibrium of the forces in the vertical \(z\)-direction according to Figure 7b [19]. This is  
\[\pi r^{2}p=2\pi rt\sigma_{\varphi}\text{sin}\varphi. \tag{3}\]  
By means of the relationship \(r=\rho_{\theta}\text{sin}\varphi\), the principal stress along the meridional or tangential direction can be determined directly from Equation (3) as  
\[\sigma_{\varphi}=\frac{p\rho_{\theta}}{2t}. \tag{4}\]  
By combining Equations (2) and (4), the missing principal stress in the circumferential or hoop direction can be determined according to  
\[\sigma_{\theta}=\frac{p\rho_{\theta}}{2t}\bigg{[}2-\frac{\rho_{\theta}}{\rho_ {\varphi}}\bigg{]}. \tag{5}\]  
Figure 7: Principal membrane stresses in a thin shell of revolution under axisymmetric pressure loading. (**a**) Central surface of an infinitesimal element cut out of the shell, which is bounded by the two meridional planes \(\theta\) and \(\theta+d\theta\) and by the two parallel circles with the corresponding normal angles \(\varphi\) and \(\varphi+d\varphi\). (**b**) Meridional section through the crown of the shell above the circle of latitude \(\varphi\). If the material is considered quasi-isotropic and the high symmetry requirements for the experimental setup are met, as in this work, the following identity results exclusively for the two principal radii of curvature for the pole: \(\rho_{\varphi}=\rho_{\theta}=\rho\).