Inflated circular membrane in contact with finite indentors of different geometries

This paper investigates the contact problem of an air-inflated circular membrane with a finite rigid indentor having three different geometric profiles, namely flat-face, conical, and spherical. Initially, the axisymmetric inflation problem of a thin circular membrane is studied under uniform pressurization. The material is assumed to be homogeneous, isotropic, and incompressible, which is described by the two-parameter Mooney-Rivlin hyperelastic model. An indentor with finite radius is pressed quasi-statically against the inflated membrane, preserving the axisymmetric nature of deformation. The contact problem is formulated for both frictionless and no-slip contact conditions. A set of coupled nonlinear second order partial differential equations for both contact and non-contact regions are solved using a shooting method coupled with an optimization algorithm. The inflated membrane profiles in contact with different indentor geometries, principal stretch ratios, and Cauchy stress resultants are obtained. The possibility of having multiple contact zones and their interaction on different faces of the indentor is also explored. The force-displacement (stiffness) curves for this finite indentor contact problem show the existence of a critical contact force, which limits the force bearing capacity of the inflated structure. This critical force is found to be higher for larger strain-hardening of the material and higher indentor radius. The junction of contact and non-contact regions for flat-faced and conical indentors is found to be the critical section due to slope discontinuity. However, for the spherical indentor, the pole of the membrane is most prone to rupture due to membrane thinning effect.