# 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
Figure 6: Schematic sketch of the used test setup for the bursting and pre-bulging tests. **1**. Upper crosshead with hydraulic adjustment. **2**. Upper clamping device. **3**. Rupture disc. **4**. Lower clamping device. **5**. Lower crosshead. **6**. Pressure gauge. **7**. Shut-off valve. **8**. High-pressure pneumatic energy source. The bulge-forming by elastoplastic deformation is shown on the right-hand side of Figure 5 in terms of the current configuration for an applied working pressure \(p\). Since the material volume can be regarded as constant during plastic deformation, the increase in the sample surface takes place at the expense of its thickness. This results in a variable thickness distribution along the meridian, which decreases steadily towards the apex or pole, respectively. At the apex, even at \(x=0\), the material is stressed by uniform stretching, insofar as it has quasi-isotropic properties. The equi-biaxial stress state is the reason for the global minimum of the specimen thickness to be found there, as the material stress on the bursting membrane reaches its maximum at this position. Please note that the meridional radius of curvature \(\rho_{\varphi}\) depends on the angle \(\varphi\) or the current radial distance \(r\) from the pole, which is characteristic of the bulge problem. The variability of the radius \(\rho_{\varphi}\) along \(r\) decreases with increasing degree of bulging \(\eta=p/p_{\rm B}\). The degree of bulging \(\eta\) expresses the ratio of the instantaneous effective pressure \(p\) to the specific bursting pressure \(p_{\rm B}\) of the diaphragm under consideration. Figure 5 shows the effective disc thickness \(t(r)\) as a function of the radial distance \(r\) with the radius of curvature \(\rho_{\varphi}\) associated with the section shown. It can also be observed that all material points between the pole and the edge experience a vertical displacement \(u_{z}\) as well as a horizontal displacement \(u_{x}=r-r_{0}\) in the radial direction. This is illustrated by the trajectory shown in blue. The first insights into the kinematic relationships described above were already gained in the second half of the 1940s through the experimental work of [PERSON] et al. [14] and by [PERSON] and [PERSON] [15]. They carried out bulge tests on thin circular samples of aluminium alloys and copper. The reason for this is the fixed clamping of the edge of the sample, which, depending on the Coulomb friction force, counteracts the flow of the material and also induces a bending moment that locally disturbs the membrane stress state. In addition to the material properties and the applied degree of bulging \(\eta\), the magnitude of this disturbance also depends on the ratio of the initial thickness \(t_{0}\) to the bulge diameter \(d_{\rm B}\) and the bending radius \(R\). This effect is extremely important for rupture disc production, as the correct choice enables the most homogeneous possible flow of force during forming and thus minimizes the introduction of pre-damage in this transition zone. [PERSON] [16] and [PERSON] [17] demonstrated by means of experimental investigations that the application of membrane theory (neglecting of bending and transverse shear stresses) to the bulge problem is permissible as a good approximation if a minimum threshold of the ratio \(d_{\rm B}/t_{0}\) is not undercut and a sufficiently large distance to the clamped edge is maintained. [PERSON] proposed the ratio \(d_{\rm B}/t_{0}>150\), while [PERSON] later reduced this to \(d_{\rm B}/t_{0}>100\). However, since the bending behavior during elastoplastic deformation also depends on the individual strain hardening of the employed material, it is very questionable whether a criterion based purely on the geometry is sufficient at this point. Nevertheless, all configurations considered here fulfill the more conservative condition \(d_{\rm B}/t_{0}>150\) according to [PERSON] (cf. Table 1) and are therefore regarded to be sufficiently flexible and shear-soft, at least in the area of the pole environment.