Symmetry breaking during inflation of a toroidal membrane

Membrane structures with symmetry often exhibit geometric instability under finite inflation. We observe and study this phenomenon in the case of a flat toroidal membrane with axisymmetry and reflection symmetry. The membrane is modeled as a Mooney-Rivlin hyperelastic material. The stability of the symmetric inflation path is studied perturbatively which reveals a zone of instability. Using higher order perturbation, the asymmetric shape is subsequently determined. As the inflation proceeds, first the axisymmetry is broken spontaneously through a supercritical pitchfork bifurcation, and is later restored at a reverse subcritical pitchfork bifurcation. The extent of the symmetry breaking zone depends on the material and geometric parameters of the toroidal structure. It is found that a stout torus can completely eliminate the occurrence of such symmetry breaking bifurcations on its inflation path.