Nonlinear analysis of hyperelastic membranes

Thin membranes made of soft materials are encountered in many natural systems and technological artefacts. The mechanical modelling of such objects is challenging due to the many sources of material and geometric nonlinearities. Leaving apart inelastic effects (which indeed may be relevant in real-world problems), the nonlinear material behaviour can be described by adopting a suitable hyperelastic model. Geometric nonlinearities arise due to the large in-plane displacements and strains, as well as the large out-plane displacements (e.g., in the case of inflated membranes). Moreover, because of their limited bending stiffness, soft thin membranes are almost unable to sustain compressive stresses. As a result, loss of equilibrium stability may occur in the forms of both global buckling and local wrinkling. As a matter of fact, the nonlinear analysis of hyperelastic membranes requires the use of effective and efficient computational methods. Within the finite element method (FEM), many different approaches have been proposed. These, however, generally lead to cumbersome formulations with nonsymmetric expressions of the secant stiffness matrix and complicated expressions of the tangent stiffness matrix. Moreover, such formulations are usually restricted to a single specific constitutive model. In this work, the position-based finite element formulation (PFEF) is adopted, which yields simple and symmetric expressions of the elastic stiffness matrices. Furthermore, the PFEF enables a straightforward implementation of any hyperelastic constitutive model, including those proposed to account for wrinkling and anisotropy. The effectiveness of the proposed approach is demonstrated through the analysis of some illustrative examples. The structural responses are obtained by using an improved arc-length method, capable of tracing the equilibrium paths with high accuracy.