# 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
For the sake of completeness, it should be mentioned that in an experimental-iterative process for the rupture disc parameters investigated in this work, a contact pressure \(p_{\text{c}}\) of around 60 MPa on average was identified as the lower limit value. This corresponds to a hold-down force of around 300 kN, which the actuating cylinder must provide as a minimum in order to be able to produce all the parameter sets listed in Table 1 with acceptable forming results. The average contact pressure \(p_{\text{c}}\) and the resulting hold-down force or cylinder force \(F_{\text{cyl}}\) of the hydraulic adjustment are related to the test setup shown in Figures 5 and 6 via the relationship  
\[F_{\text{cyl}}=\frac{\pi}{4}\Big{[}{d_{\text{D}}}^{2}-(d_{\text{B}}+2R)^{2} \Big{]}p_{\text{c}}, \tag{1}\]  
where \(d_{\text{D}}\) denotes the outer diameter of the clamped metallic disc, \(d_{\text{B}}\) is the bulge diameter and \(R\) is the bending radius of the upper clamping device. To ensure reproducible pre-bulging and bursting results, which arise not only from the high safety requirements for bursting discs, it is essential that the sheet metal blank to be formed is aligned symmetrically with its forming tool. A special clamping tool was therefore developed for positioning the blank. The upper and lower units of this clamping tool interlock via a fitting seat. The diameter of the fitting bore of the lower clamping device was adapted to the outer disc diameter \(d_{\text{D}}\) of the nominal diameter DN 80 mm under investigation. The clamping  
Figure 5: The rupture disc kinematic and geometry in the meridional section before and during the bulge-forming process. unit is also arranged centrally to the hydraulic adjustment device via a shim disc in order to guarantee a uniform flow of force. If the material properties of the rupture disc to be manufactured can be considered approximately isotropic (quasi-isotropic) and also homogeneously distributed within the structure, the uniform pressurization during the bulge-forming process will result in an approximately axially symmetrical profile. The mid-surface of this profile can be described mathematically by the rotation of a plane curve around a straight line lying in its plane. This straight line can be regarded as the axis of symmetry \(z\) in relation to the present application of the pre-bulging of rupture discs, as it intersects the meridian vertically at its center of gravity at \(x=0\). The geometric situation is illustrated in Figure 5 by a section along the meridian. The left-hand side shows the reference configuration of the bulge-forming process, in which the constant ambient pressure \(p_{1}=p_{2}=p_{0}\) acts on both sides. If the pressure in the working space between the bursting membrane and the lower clamping device is increased via the supply of a working medium, a differential pressure \(\Delta p=p_{1}-p_{2}\) acting on the membrane is created, which is referred to as \(p\) in the following and is to be interpreted as the effective pressure given in MPa. Compressed air, which was available in sufficient quantities using a high-pressure compressor and accumulator, was used as the working medium in the experimental work carried out in this contribution. The effective pressure \(p\) acting on the bursting membrane initially produces an approximate elastic indentation of the sample at the beginning of the deformation process. The configuration reacts very sensitively to an increase in pressure in the bulge-forming process depending on the material properties and sample geometry, so that the rate of pressure increase should be selected according to the specific problem. In order to ensure the reproducibility and comparability of the results within the test program and also to minimize time-dependent effects in the flow behavior by means of a quasi-static test procedure, a very low rate of \(0.1\,\mathrm{MPa}\cdot\mathrm{s}^{-1}\) was selected for all samples and this was kept approximately constant via a shut-off valve over the entire duration of the test.