# 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
If the instantaneous diameter of the measurement circle is denoted by \(d\) and a spherical pole geometry can be assumed, the logarithmic principal plastic strain is as follows:  
\[F_{\text{cyl}}=\frac{\pi}{4}\Big{[}{d_{\text{D}}}^{2}-(d_{\text{B}}+2R)^{2} \Big{]}p_{\text{c}}, \tag{1}\]3The local radius of curvature \(\rho\) of a spherical pole geometry (or \(\rho_{\varphi}\) and \(\rho_{\theta}\) for the general case) can be determined very precisely by calculation if the coordinates \(x_{i}\) and \(z_{i}\) (or additionally \(y_{i}\) for the general case and the condition that the coordinate axes coincide with the main axes) of three neighboring points \(P_{i}\) on the circle of curvature are known and their positions are as close together as possible. Then, the geometric relationships resulting from the stretched triangle can be used to determine the local radius of curvature. A more straightforward approach can be taken if it is assumed that the entire bulge shape of the bursting membrane in the meridional section can be approximated by a circular arc. The quality of this approximation increases with increasing degree of bulging \(\eta\). However, it is assumed that the center of curvature is located on the axis of symmetry. In our notation, this is the \(z\)-axis. Then, only the bulge diameter \(d_{\rm B}\) and the current polar bulge height \(h\) must be known. This results in the simple relationship for the radius of curvature  
\[F_{\text{cyl}}=\frac{\pi}{4}\Big{[}{d_{\text{D}}}^{2}-(d_{\text{B}}+2R)^{2} \Big{]}p_{\text{c}}, \tag{1}\]4  
This expression can be easily derived using Pythagoras' theorem and the geometric relationships shown in \[F_{\text{cyl}}=\frac{\pi}{4}\Big{[}{d_{\text{D}}}^{2}-(d_{\text{B}}+2R)^{2} \Big{]}p_{\text{c}}, \tag{1}\]5 At this point, the equality of the two principal radii of curvature according to \(\rho_{\varphi}=\rho_{\theta}=\rho\) is assumed. This applies to the area of the pole for the axially symmetric case. Using the law of logarithmic or true strain and the material incompressibility during plastic deformation, the current sheet thickness \(t\) in the pole can finally be determined:  
\[F_{\text{cyl}}=\frac{\pi}{4}\Big{[}{d_{\text{D}}}^{2}-(d_{\text{B}}+2R)^{2} \Big{]}p_{\text{c}}, \tag{1}\]6  
Since the polar sheet thickness \(t\) and the polar radius of curvature \(\rho\) can be determined by measuring simple geometric quantities, and the effective fluid pressure \(p\) acting on the bursting membrane is already known from a pressure gauge, the effective pole stress \(\overline{\sigma}\) can be determined. This gives the functional relationship  
\[F_{\text{cyl}}=\frac{\pi}{4}\Big{[}{d_{\text{D}}}^{2}-(d_{\text{B}}+2R)^{2} \Big{]}p_{\text{c}}, \tag{1}\]7  
Taking into account Equations (8), (10) and (12), the failure criterion of an undisturbed forward acting rupture disc made of a ductile and quasi-isotropic metal follows from  
\[F_{\text{cyl}}=\frac{\pi}{4}\Big{[}{d_{\text{D}}}^{2}-(d_{\text{B}}+2R)^{2} \Big{]}p_{\text{c}}, \tag{1}\]8  
If the radius of curvature \(\rho\) is determined using the simplified approach according to Equation (11), it can be expressed in terms of the easily measured polar bulge height \(h\). The functional relationship is then  
\[F_{\text{cyl}}=\frac{\pi}{4}\Big{[}{d_{\text{D}}}^{2}-(d_{\text{B}}+2R)^{2} \Big{]}p_{\text{c}}, \tag{1}\]9  
Substituting Equation (11) into Equation (14) finally yields the practical expression  
Figure 5:0  
Equations (14) or (16) can be used to experimentally determine the effective stress \(\overline{\sigma}\) at the membrane pole. If the influence of the rolling texture cannot be neglected, the direction-dependent principal stresses \(\sigma_{\varphi}\) and \(\sigma_{\theta}\) according to Equations (4) and (5) can be used to determine the effective stress \(\overline{\sigma}\) according to [PERSON]. It should be noted that the two in-plane principal plastic strains \(\epsilon_{\rm pl,\varphi}\) and \(\epsilon_{\rm pl,\theta}\) can then no longer be assumed to be equivalent, but must be determined individually.