# Numerical Simulation and Analysis of the Manufacturing Process of Pre-Bulged Rupture Discs
## 2 Materials and Methods
### Finite Element Model  
The simulation model presented below is based on the commercial finite element software MSC Marc/Mentat. All simulations were carried out in program version 2023.3. Further details on the modeling technique described below and its numerical implementation in the software can be found in the user manual [26; 27]. The mesh was generated externally using the commercial software MSC Apex 2023.3 and then imported into Marc/Mentat. In view of the quasi-static test procedure presented in Sections 2.1 and 2.2, time-dependent effects were not taken into account in the numerical simulation. Accordingly, a static analysis was chosen and the dependence of the flow stress was considered exclusively on the true plastic strain and its effective value for the multi-dimensional case. The simulations were carried out until the experimentally determined bursting pressures were reached. The very ductile material achieves large plastic strains, which are paired with large displacements. Therefore, the kinematics of deformation and the description of motion are  
Figure 12: True stress–true plastic strain curve (also called flow curve) for annealed AISI 316L/EN 1.4404: Hockett–Sherby approximation using the parameters from Table 7.  
described by an updated Lagrangian formulation. Furthermore, a finite strain formulation with additive decomposition of the strain increments into an elastic and plastic part is used. To reduce the computational effort, the symmetrical geometry of the bursting membrane was utilized and the bulge problem was modelled as a quarter model. The geometry of the bursting membrane was discretized using three-dimensional hexahedral elements in order to adequately address the constraints regarding the bending influence of the edge clamping described in Section 2.1 and to be able to show the distribution of the bending stress over the specimen thickness. Linear elements with eight nodes in an isoparametric formulation and full integration were used for this purpose. To take into account the bending behavior, the bursting membrane was resolved with eight elements over the thickness. The O-grid mesh shown in Figures 13 and 14 was generated in order to discretize the circular disc as structured as possible and in high quality despite the quadrilateral elements (or hexahedral elements). A primary radial alignment with a homogeneous transition to the center of the circle can be seen. The deviation from the ideal element (Jacobi determinant) could thus be significantly minimized compared to a pure radial mesh. The deformable bursting membrane is in contact with the clamping devices during the bulge-forming process. This contact was described using the nonlinear node-to-segment  
Figure 14: Finite element mesh details. (**a**) Top view of the circular disc center. (**b**) Side view of the clamping situation in the area of the bending radius. Figure 13: Finite element model with mesh and definition of the contact bodies. contact algorithm established in Marc/Mentat. Here, the contact status between the contacting bodies is updated at each iteration step. An individual stress-based separation criterion was used as the problem-specific separation criterion. Both clamping devices are modelled as rigid bodies. The geometric body formulation for these two tools reduces the computational effort and avoids numerical difficulties due to very large differences in the modulus of elasticity of the contact bodies. The state of friction between the contact bodies is modeled using the well-known Coulomb friction model. The transition between stick and slip is described in this model with a bilinear step function. A value of 0.1 is used as the coefficient of friction corresponding to a contact of unlubricated metallic surfaces with low roughness. The formulation of the boundary conditions for the bulge-forming process is not trivial. For further consideration, it should be noted that the origin of the coordinate system refers to the center of the disc.