### 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.