Local minima in incompressible elasticity

Gianpietro Del Piero and Raffaella Rizzoni
Universitá di Ferrara
Dipartimento di Ingegneria

In 1979, Gurtin and Spector proved some theorems of stability and uniqueness in finite elasticity. A direct consequence of their results is that, if the natural configuration of a compressible hyperelastic body is "positive" in some precise sense, then there is a neighborhood of the natural configuration in which all solutions of the equilibrium problem are local energy minimizers. At our knowledge, no counterpart of this result is known for incompressible bodies, and this is precisely the problem which forms the object of the present communication.

After establishing some general results, as an example we consider the fundamental solution of the problem of the torsion of a circular cylinder made of an incompressible isotropic material. We show that this solution is a local energy minimizer for sufficiently small values of the angle of twist, and we prove that the amplitude of the stability interval is a decreasing function of the ratio between height and diameter of the cylinder.