Constrained inflation of a stretched hyperelastic membrane inside an elastic cone

This paper studies the inflation and interaction mechanics of a flat circular membrane inside an elastic cone under the action of uniform gas pressure. The membrane is assumed to be a homogeneous and isotropic Mooney–Rivlin hyperelastic material, while the conical surface is taken to be a distributed linear stiffness in the direction normal to the undeformed surface. The set of coupled second order nonlinear ordinary differential equations that governs the constrained inflation mechanics is reduced to a set of four first order ordinary differential equations by change of variables. A two dimensional grid search technique using the bisection method is employed to determine the equilibrium configuration of the inflated membrane. The principal stretches and curvatures have been obtained which exhibit some interesting trends. It is observed that the limit point instability can completely disappear (even in the case of neo-Hookean membrane material model) when an inflating membrane interacts with a constraining surface. Most remarkably, pre-stretching the membrane can revive the occurrence of the limit point instability in certain cases leading to a softening behavior. This counterintuitive effect appears to be a shadow of the stretch induced softening behavior observed recently in literature.