Two instructive instabilities in non-linear elasticity: Biaxially loaded membrane, and rubber baboons

Elastic bodies buckle under compressive loads, i.e. solutions become non-unique, they bifurcate and the body becomes unstable. Similar phenomena occur in tension as is evidenced here by the symmetric biaxial loading of a square membrane. Symmetry breaking removes the non-uniqueness. Under non-symmetric loading the load-deformation curves become non-monotone, consequently a hysteresis occurs which is the reflection of a fold-type catastrophy. This instructive instability was discovered by Kearsley [1]. Here we investigate it more fully and present some additional aspects. Balloons have non-monotone pressure-radius relations which suggestnon-trivial stability properties. A stability analysis is presented for two interconnected balloons. In this we follow - and expand on - the analyses presented by Dreyer et al. [2] and Kitsche et al. [3].