Static and dynamic stability results for a class of three-dimensional configurations of Kirchhoff elastic rods

• We analyze the stability of a naturally straight elastic rod in 3D.
• We obtain explicit criteria for the static stability of arbitrary extrema.
• We show the equivalence between static stability and dynamic stability.
• We generalize our results to arbitrary quadratic energy functionals.
• We extend our analysis to realistic rods with local damping.

We analyze the dynamical stability of a naturally straight, inextensible and unshearable elastic rod, under tension and controlled end rotation, within the Kirchhoff model in three dimensions. The cases of clamped boundary conditions and isoperimetric constraints are treated separately. We obtain explicit criteria for the static stability of arbitrary extrema of a general quadratic strain energy. We exploit the equivalence between the total energy and a suitably defined norm to prove that local minimizers of the strain energy, under explicit hypotheses, are stable in the dynamic sense due to Liapounov. We also extend our analysis to damped systems to show that static equilibria are dynamically stable in the Liapounov sense, in the presence of a suitably defined local drag force.