# Numerical Simulation and Analysis of the Manufacturing Process of Pre-Bulged Rupture Discs
## 2 Materials and Methods
### Experimental Analysis and Evaluation of the Bulge-Forming Process
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Initial** & & & & & & & & & & & \\
**Thickness** & & & & & & & & & & & & \\ \(t_{0}\) **in mm** & \(\mathbf{r_{1}=0}\) & \(\mathbf{r_{2}=5}\) & \(\mathbf{r_{3}=10}\) & \(\mathbf{r_{4}=15}\) & \(\mathbf{r_{5}=20}\) & \(\mathbf{r_{6}=25}\) & \(\mathbf{r_{7}=30}\) & \(\mathbf{r_{8}=35}\) & \(\mathbf{r_{9}=40}\) & \(\mathbf{r_{10}=50}\) & \(\mathbf{r_{11}=55}\) & \(\mathbf{r_{22}=60}\) \\ \hline
0.102 & 0.067 of bulging \(\eta\) specified in Table 1 during the bulge-forming process. The current diameters \(d\) and \(D\) of the measuring circles were determined by means of an optical microscope using a digital caliper gauge and arithmetically averaged over five random samples per bulging level \(\eta\). The two measuring circles take on a slightly elliptical shape with increasing working pressure \(p\). The reason for this kind of anisotropy can be found in the rolling texture. The main axis of the ellipse coincides with the rolling direction. The secondary axis is perpendicular to it. This effect becomes macroscopically visible from a degree of bulging of \(\eta\approx 0.8\). However, with a maximum relative deviation between the two axes of around 3% at \(\eta=0.99\), the influence is rather small, confirming the previously made assumption of quasi-isotropy. The diameters \(d\) and \(D\) are therefore given as the arithmetic mean of both axes. The measurement method is illustrated in Figure 11. The left-hand side shows the reference configuration (flat sheet metal blank), while the right-hand side shows the current configuration when the bulging level \(\eta=0.99\) is reached shortly before bursting. To calculate the radius of curvature \(\rho\), the measured values of the polar bulge height \(h\) from Table 2 (corresponds to \(u_{z}\) for \(r_{n}=0\)) were used and the simplified method with \(\rho_{\mathrm{s}}\) according to Equation (11) was applied. The logarithmic principal plastic strain \(\epsilon_{\mathrm{pl}}\) in the area of the membrane pole can then be determined using Equation (10). Properties of the Rupture Disc Membrane Material and Modeling of Its Elastic-Plastic Material Behavior  
The metallic membrane used for rupture discs is usually referred to as a bursting membrane. The primary material for bursting membranes is typically produced in a cold rolling process in order to guarantee the high tolerance requirements. In this process, the strip is rolled down to the required final thickness in several rolling passes at room temperature. The grains are aligned in the rolling direction, which, depending on the degree of deformation and chemical composition of the rolled material, can lead to a directional dependence of the deformation properties on the angle relative to the rolling direction (plane anisotropy). At the same time, the dislocation density increases as a result of cold forming, which increases the strength of the material on the one hand, but reduces its ductility and formability on the other. As a consequence, the material becomes brittle. For these reasons, the primary material of a bursting membrane must be subjected to heat treatment in the form of recrystallization annealing after cold rolling, at least if it is a pre-bulged membrane. This production step eliminates to a great degree the undesirable  
Figure 11: Method for measuring polar principal plastic strain. (**a**) Flat sheet metal blank (initial thickness \(t_{0}=0.102\) mm) as reference configuration with two measuring circles \(d_{0}\) and \(D_{0}\) at the pole. (**b**) Current configuration at degree of bulging \(\eta=0.99\) with slightly elliptical shape of the original measuring circles (dependence of diameter \(d\) and \(D\) on rolling direction). properties caused during cold rolling by forming new grains. Therefore, the material considered in this work can be regarded as quasi-isotropic. As already mentioned, this is the stainless, austenitic chromium-nickel-molybdenum steel AISI 316L or EN 1.4404 in annealed condition.