# 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  
The bulge-forming process is analyzed within this work up to the onset of bursting of undisturbed sheet metal blanks. Three different sheet thicknesses \(t_{0}\) with the same nominal diameter DN are considered. The material of all samples was taken from the same batch. Bursting tests were first carried out to define the upper limit for the pressure. Five random samples were taken for each thickness \(t_{0}\) and the bursting pressure \(p_{\mathrm{B}}\) was arithmetically averaged. With regard to the bursting pressure \(p_{\mathrm{B}}\) of each thickness \(t_{0}\), a test program with six different degrees of bulging \(\eta\) was defined in order to be able to subsequently measure the change in the geometric configuration during the bulge-forming process. The different boundary conditions for the series of experiments are summarized in Table 1.  
The evolution of the bulge shape as a function of the effective pressure \(p\) was realized successively by measuring the samples of all degrees of bulging \(\eta\) defined in Table 1. To ensure representative results, five samples per bulging level \(\eta\) and initial thickness \(t_{0}\) were also prepared and measured. The procedure can be illustrated using the right-hand side of Figure 8. Starting from the membrane pole, equidistant radial distances \(r_{n}=\{0,10,20,30,40\}\) mm were defined and projected onto the membrane surface with the degree of bulging \(\eta\) after reshaping. These \(n=5\) distances were used uniformly for all samples considered in this work and plotted along an imaginary section of the bursting membrane with a meridional plane \(\theta=\mathrm{const}\). The displacements \(u_{z}(r_{n},\eta)\) in the \(z\)-direction associated with the respective \(x\) positions \(r_{n}\) were then determined using a contacting digital height gauge on a measuring plate. In order to detect possible anisotropy effects affecting the kinematics, all height measurements \(u_{z}(r_{n},\eta)\) were made for four angular positions evenly distributed around the circumference. The measured values show a very low variability with a maximum relative dispersion of about 1% and are therefore only given as an arithmetic mean. This shows that the assumption of axial symmetry is largely valid for the problem at hand. As the displacements were determined starting from the surface of the measuring plate, on which the sample lays flat in addition to the measuring device itself, the measured values must be reduced by \(t_{0}\). The results averaged over all measured samples are shown in Table 2 and represent the fraction of vertical displacement caused by plastic deformation. This procedure allows us to track the kinematics of the membrane surface during deformation with simple means until shortly before bursting. Figure 8: Definition of the measuring points on the rupture diaphragm for measuring the thickness at a degree of bulging \(\eta=0.99\) (left side) and the evolution of the bulge shape during the pressure increase (right side). In addition to the bulge shape, knowledge of the thickness profile along the meridian is of great interest. To limit the measurement effort, this was only determined shortly before bursting at 99% of the bursting pressure \(p_{\rm B}\), which corresponds to a bulging level of \(\eta=0.99\). The procedure for this can be illustrated using the left-hand side of Figure 8. Here, the results for each initial thickness \(t_{0}\) at stage \(\eta=0.99\) correspond to the arithmetic mean of five random samples as well. The entire area between the pole at \(x=0\) and the outer edge at \(x=d_{\rm D}/2=61.5\) mm is considered in order to determine the thickness curve. This should enable a possible reduction in thickness in the edge area due to post-flow processes to be recorded.