Inflation of a toroidal membrane within a fluid-filled elastic spherical enclosure

Abstract

The present research investigates the growth based inflation model of an inflated toroidal membrane within a fluid-filled environment enclosed by an elastic spherical cavity. This problem statement resembles the growth of toroidal vesicle membranes within biological cells. The toroidal membrane is described by hyperelastic Mooney–Rivlin model with meridional anisotropy. The rise in internal gauge pressure of the torus causes the surrounding incompressible fluid to exert a distributed radial force on the surface of the elastic sphere, resulting in its deformation. With a subsequent gradual increase in gauge pressure, a contact is initiated as the torus indents onto the inner surface of the elastic sphere. The contact condition is assumed to be frictionless, and a variational formulation is adopted for solving the contact problem. The maximum indentation as well as the generated contact stress are found to be higher with a lesser stiffness of the elastic spherical enclosure. As the contact patch grows, the phenomenon of membrane thinning is predominantly observed at the inner equator of the torus. The growth of the contact boundary varies linearly with increasing torus gauge pressure, but non-linearly with the fluid pressure within the spherical enclosure.