Characterisation and stability of localised bulging/necking in inflated membrane tubes

TL;DR

This paper analyzes the initiation, growth, and stability of localized bulging in inflated hyperelastic membrane tubes, revealing that kink-wave solutions are likely stable while solitary-wave solutions are unstable.

Contribution

It provides an analytical characterization of bulging behavior, including the initiation pressure, growth dynamics, and stability analysis of different bulging solutions in membrane tubes.

Findings

Initiation pressure equals the limiting pressure in uniform inflation.

Bulges grow continually until reaching a critical size and then propagate.

Kink-wave solutions are probably stable, solitary-wave solutions are unstable.

Abstract

We consider localised bulging/necking in an inflated hyperelastic membrane tube with closed ends. We first show that the initiation pressure for the onset of localised bulging is simply the limiting pressure in uniform inflation when the axial force is held fixed. We then demonstrate analytically how, as inflation continues, the initial bulge grows continually in diameter until it reaches a critical size and then propagates in both directions. The bulging solution before propagation starts is of the solitary-wave type, whereas the propagating bulging solution is of the kink-wave type. The stability, with respect to axially symmetric perturbations, of both the solitary-wave type and the kink-wave type solutions is studied by computing the Evans function using the compound matrix method. It is found that when the inflation is pressure-controlled, the Evans function has a single non-negative real root and this root tends to zero only when the initiation pressure or the propagation pressure is approached. Thus, the kink-wave type solution is probably stable but the solitary-wave type solution is definitely unstable.