Instabilities in Biaxially Loaded Rectangular Membranes and Spherical Balloons Made of Compressible Isotropic Hyperelastic Materials

Abstract

We analyze homogeneous deformations of a rectangular rubberlike membrane loaded by equal normal tensile dead loads on the edges, and of a spherical rubber balloon inflated by a constant pressure. The rubber is modeled as an isotropic compressible hyperelastic material. Three material models, namely the harmonic, the generalized Blatz-Ko, and the St Venant-Kirchhoff models, are employed. It is found that Treloar's instability, i.e. the occurrence of unequal principal stretches in a square membrane under equal normal dead loads, is not admissible in the harmonic and the St Venant-Kirchhoff materials, but is admissible in some Blatz-Ko materials. For each one of the three materials, the pressure-radius relation for the inflation of a spherical balloon does not exhibit the non-monotonicity seen for a Mooney-Rivlin material.