Local energy minimizers in incompressible elasticity

The natural (stress-free) state of an elastic body is usually assumed to be a global energy minimizer. However, it is not known in general whether the property of being even a local minimizer is preserved by equilibrium configurations close to the natural state. In this paper we first outline a sufficient condition for a local energy minimum, obtained in [4], which applies to incompressible anisotropic hyperelastic bodies of arbitrary shape. This condition is then applied to the torsion problem for an isotropic circular cylinder. For it, we show that Rivlin’s fundamental solution is a local energy minimizer over a small, but finite, range of angles of twist, whose size depends on the slenderness ratio of the cylinder