Stability Analysis of an Inflation of Internally-Pressurized Hyperelastic Spherical Membranes Connected to Aneurysm Progression

In this study, we consider a nonlinear second-order ordinary differential equation to describe the inflation of a thin-walled hyperelastic spherical membrane, subjected to an internal distention pressure. We examine the stability of the equilibria of the basic model using three different forms of strain energy functions (SEFs), representing the mechanical properties of elastomers and soft tissues. It is shown that the mechanical stability or instability of the membrane material is associated with the monotonicity or non-monotonicity of the pressure-stretch relation. We define two types of instabilities according to the specific constitutive relation. We prove analytically that a stable inflation of the membrane can retain or can change to an unstable one depending on the specific analytical form of SEF and its material parameters. We derive conditions for the stability/instability of the equilibria of the model with the different SEFs and relate the identified unstable equilibrium states to development and rupture of aneurysms.