# 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
For this purpose, \(m=12\) equidistant measuring points (except in the area of the bending radius) \(r_{m}=\{0,5,10,15,20,25,30,35,40,50,55,60\}\) mm were defined and also projected onto the membrane surface after forming. All measuring points are again located along an imaginary section of the bursting membrane with a meridional plane \(\theta=\) const. A digital micrometer with two measuring tips and a resolution of 1 um was used to measure the wall thickness \(t\) at the positions \(r_{m}\). For this purpose, the samples were previously divided into two halves along the meridional section of the measuring points using a TRUMPF TruLaser Cell 3000 laser cutting machine (cf. Figure 9). The results of the discrete thickness distribution along the meridian are shown in Table 3.  
\begin{table} \begin{tabular}{c c c c c c c c} \hline
**Initial** & **Radial** & \multicolumn{6}{c}{**Displacement \(\mathbf{u_{z}}\) in mm at Specific Degree of Bulging \(\mathbf{\eta}\)**} \\
**Thickness** & **Distance \(\mathbf{r_{n}}\)** & **\(\mathbf{\eta=0.2}\)** & **\(\mathbf{\eta=0.4}\)** & **\(\mathbf{\eta=0.6}\)** & **\(\mathbf{\eta=0.8}\)** & **\(\mathbf{\eta=0.95}\)** & **\(\mathbf{\eta=0.99}\)** \\ \hline \multirow{6}{*}{0.102}  
Figure 10 gives an impression of the test program carried out. It shows an image that represents all test points in the form of a random sample. Starting from the flat sheet metal blank on the left side of the image, the samples were arranged after the bulge-forming process as a function of the degree of bulging \(\eta\) and the initial thickness \(t_{0}\). The bulging level \(\eta\) increases from left to right and the initial sheet thickness \(t_{0}\) increases from top to bottom. The uncontrolled bursting which is characteristic for undisturbed/unnotched membranes can be seen at the right-hand side.  
In addition, the logarithmic principal plastic strain \(\epsilon_{\text{pl}}\) acting within the sheet plane of the membrane pole was determined in the test program and the previously described method for determining the strain was applied. The corresponding results are summarized in Table 4.  
Here, only the thin bursting membrane with an initial thickness of \(t_{0}=0.102\) mm was considered. As a reference, two measuring circles with diameters of \(d_{0}=5\) mm and \(D_{0}=10\) mm were initially applied to the center of the flat sheet metal blank in order to be able to experimentally determine the polar strain at two different support points. A TRUMPF TruMark Station 5000 labeling laser was used for this purpose. Afterwards, the increase in diameter of the two measuring circles was determined at the different degrees  
\begin{table}
\begin{tabular}{c} **Logarithmic** \\ **Measuring Circle \(D_{0}\)** \\ **in mm** \\ \end{tabular} \\ \hline
0.2 & 162.94 & 5.07 & 0.0139 & 10.10 & 0.0101 \\
0.4 & 96.63 & 5.20 & 0.0393 & 10.35 & 0.0349 \\
0.6 & 75.17 & 5.40 & 0.0772 & 10.70 & 0.0685 \\
0.8 & 61.83 & 5.65 & 0.1226 & 11.25 & 0.1192 \\
0.95 & 54.32 & 5.95 & 0.1745 & 11.85 & 0.1717 \\
0.99 & 53.22 & 6.10 & 0.1994 & 12.15 & 0.1969 \\ \hline \hline \end{tabular} 
\end{table}
Table 4: Measurement and calculation results of the logarithmic principal plastic strain \(\epsilon_{\text{pl}}\) of the pole during the bulge-forming process for bursting membrane with initial thickness \(t_{0}=0.102\) mm. Figure 10: Overview of a set of specimens from the test series with a nominal diameter DN of 80 mm after bulge forming. Upper row: initial thickness \(t_{0}=0.102\) mm. Middle row: initial thickness \(t_{0}=0.343\) mm. Lower row: initial thickness \(t_{0}=0.480\) mm.