A conic programming approach to the wrinkling of pneumatic membranes using convex potentials

This paper presents a novel conic programming approach for modeling the wrinkling of isotropic hyperelastic membranes subject to nonlinear loading and finite strains, employing convex potentials derived from tension field theory. The incompressible neo-Hookean strain energy is recast as a minimization problem over a set of cones, with a semidefinite constraint on the deformed surface metric. To address pneumatic membranes, a linearized volumetric potential is introduced, reestablishing convexity for follower forces, and Boyle's law is expressed as a minimization problem over the exponential cone. The primal–dual interior point method solves the total potential energy minimization problem, with convergence verified using the analytical solution of a sheared planar membrane. The proposed model, applied to examples of increasing complexity, reveals intriguing membrane behaviors rarely discussed in existing literature. The method is shown to be robust, even when handling highly wrinkled membranes, which are challenging to address using direct nonlinear equilibrium analysis or nonlinear interior point methods. Implementation using high-level automatic code generation tools (FEniCS) results in concise, extensible code. The findings point to several avenues for future research, such as exploring complex material models, mesh refinement, dynamics, and parametric design. Additionally, the linearization process implies potential applicability of the methods to nonlinear problems beyond membrane mechanics.