# Numerical Simulation and Analysis of the Manufacturing Process of Pre-Bulged Rupture Discs
## 2 Materials and Methods
### Finite Element Model
The symmetry conditions for the quarter model are ensured by fixed displacements \(u_{x}=0\) for all nodes at the disc edge with \(x=0\) and \(u_{y}=0\) for the nodes at the disc edge with \(y=0\). Therefore, the bursting membrane is supported in its plane at the edges for the entire duration of the simulation. Its vertical node degrees of freedom \(u_{z}\) remain free of displacement boundary conditions so that it can deform unhindered as realistically as possible while the external loads act on it. The lower clamping device is defined as a position-controlled body, whereby its position remains unchanged during the entire simulation period. It thus serves as a support for the bursting membrane. Both bodies are already in contact with each other at the start of the simulation. Unlike the lower clamping tool, the upper tool must necessarily be defined as a load-controlled body, as it is controlled by both displacements and loads during the process under consideration. In order to implement this requirement, a control node is positioned in the center of the model and coupled with the upper contact body. The control node only allows vertical displacement in the \(z\) direction. All other displacement and rotational degrees of freedom are set to zero, i.e., \(u_{x}=u_{y}=\phi_{x}=\phi_{y}=\phi_{z}=0\). An overview of the boundary conditions affecting the quarter model can be found in Figure 15. Some of these process parameters are time-dependent and therefore change over the course of the bulge-forming process. Therefore, these parameters were scaled during the simulation according to their time sequence, as shown in Figure 16. This time sequence also provides information about the modeling implementation of the individual steps \"clamping - pre-bulging - unclamping\" of the real process. In this setting, the pre-defined control node is used to initially apply the hold-down force according to the time function via the upper clamping device (see Figure 16). When the target value of 600 kN is reached, this is kept constant for the duration of the forming process. Analogous to the tests, this hold-down force was selected equally for all simulation variants. Due to the contact pressure acting on the top and bottom of the bursting membrane edge, this is supported in a vertical direction and is therefore now completely statically determined. Furthermore, the working pressure can be applied to the bottom face of the bursting membrane in the form of a homogeneous face load. This is also scaled with a time function according to Figure 16. The bursting pressure \(p_{\text{B}}\) from Table 1, which is determined experimentally and depends on the initial thickness \(t_{0}\), is used as the maximum value. The face load is formulated as a so-called follower force in the current configuration. This ensures that the pressure load vectors are always perpendicular to the membrane surface during deformation. Once the corresponding pressure maximum has been reached, it is completely removed again, resulting in an initial elastic recovery. In the last simulation step, the hold-down force applied to the upper clamping device and kept constant up to this point is removed. At the same time, a time-dependent displacement is applied to this load-controlled contact body via its control node, which gradually moves it in the positive \(z\)-direction and thus opens the existing contact to the edge of the bursting membrane. The reshaped bursting membrane is then free of external constraints, so that the internal stresses introduced into the structure by the elastoplastic deformation can dissipate until a state of equilibrium is reached. The residual internal stresses thus become visible. With this last step, the entire production sequence of the bulge-formingprocess, from a flat sheet metal blank to the finished pre-bulged bursting membrane, is described numerically.