Non-unique response of Mooney–Rivlin model in bi-axial membrane stress

• Instabilities can occur in FE simulations of biaxial membrane stress, when using a Mooney–Rivlin model.
• Numerical examples show when these instabilities occur.
• The boundary conditions, interpolation, and mesh density are important factors.
• The instability can excite dynamic edge instabilities.
• Also other material models must be checked for similar instabilities.

The commonly used two-parameter Mooney–Rivlin incompressible hyper-elastic material model can show non-intuitive responses under certain conditions. This paper shows that critical states with non-unique responses occur at least at very specific bi-axial stress states. This can happen for cases where the constant related to the second invariant of strain is positive, but not for the case with this constant equal to zero (the Neo-Hookean case). The dependence of the instabilities on the ratio between the two constitutive constants is shown by evaluated fold lines. The instability is shown to be related to the imposed boundary conditions. An analytical treatment of the problem shows that dynamic edge effects correspond to the static instability.