Inflation and instabilities of a spherical magnetoelastic balloon

TL;DR

This paper investigates the complex stability and bifurcation behaviors of a spherical magnetoelastic balloon under inflation, revealing how magnetic fields and material properties influence symmetry-breaking and shape transitions.

Contribution

It introduces a detailed stability analysis of a magnetoelastic balloon, identifying bifurcation points and the effects of magnetic fields and material parameters on shape stability.

Findings

Symmetry-breaking bifurcation leads to pear-shaped equilibrium.

Magnetic field intensity influences bifurcation suppression.

Stable and unstable pear-shaped solutions form a loop with shape transitions.

Abstract

This study explores the instabilities during the axisymmetric inflation of an initially spherical magnetoelastic balloon, modeled as a magnetizable Ogden material, under combined internal pressure and a non-uniform magnetic field generated by current-carrying coils. The nonlinear interplay of geometric and material effects leads to governing equations sensitive to bifurcations and instabilities. A coordinate singularity at the poles of the balloon is identified within the system of governing differential equations, which is resolved through an appropriate choice of field variables and L'H\^opital's rule. Stability analysis reveals that as inflation progresses, axisymmetry is broken through a supercritical pitchfork bifurcation, resulting in a pear-shaped equilibrium. This symmetry is later restored through a reverse subcritical pitchfork bifurcation, forming an isolated loop of pear-shaped solutions containing stable and unstable branches in the case of a six-parameter Ogden material model (SPOM). The onset of symmetry-breaking bifurcations is influenced by material parameters and magnetic field intensity, with critical values beyond which such bifurcations are suppressed. Both symmetry-preserving and pear-shaped configurations are stable under small asymmetric perturbations in both magnetic and non-magnetic cases. Snap-through transitions between pear-shaped and axisymmetric configurations are also observed.