Nonlinear analysis of soft elastic membranes using the position-based finite element formulation

The structural analysis of membranes made of soft elastic materials is challenging due to the presence of many geometric and material nonlinearities. In the finite element method, a total Lagrangian formulation can be adopted, where large displacements and strains are measured from a fixed reference configuration. Moreover, the nonlinear elastic response can be accounted for by a suitable hyperelastic material model. With this approach, however, the secant stiffness matrix turns out to be unsymmetric - which is undesirable from the computational point of view - and not univocally determined as a function of the displacement vector. To overcome this issue, a position-based finite element formulation (PFEF) can be adopted, where the nodal position in the current configuration is used in place of the displacement as the main unknown of the problem. As a result, a symmetric secant stiffness matrix is obtained. A further advantage of the PFEF is that any hyperelastic material model can be easily implemented. The proposed approach is illustrated through the analysis of some example problems.