Geometrical feature of the scaling behavior of the limit-point pressure of inflated hyperelastic membranes

Abstract

The occurrence of the limit-point instability is an intriguing phenomenon observed during stretching of hyperelastic membranes. In toy rubber balloons, this phenomenon may be experienced in the sudden reduction in the level of difficulty of blowing the balloon accompanied by its rapid inflation. The present paper brings out a link between the geometry and strain-hardening parameter of the membrane, and the occurrence of the limit-point instability. Inflation of membranes with different geometries and boundary conditions is considered, and the corresponding limit-point pressures are obtained for different strain-hardening parameter values. Interestingly, it is observed that the limit-point pressure for the different geometries is inversely proportional to a geometric parameter of the uninflated membrane. This dependence is shown analytically, which can be extended to a general membrane geometry. More surprisingly, the proportionality constant has a power-law dependence on the nondimensional material strain-hardening parameter. The constants involved in the power-law relation are universal constants for a particular membrane geometry.