Stability and bifurcation of helical equilibrium of a thin elastic rod

The helical equilibrium and its stability of a thin elastic rod with noncircular cross section and intrinsic curvature and twist under application of force and torque are discussed in this paper. A stability condition of the helical equilibrium in the general case is derived in first approximation. The helical equilibrium of a rod without intrinsic curvature and twist is stable when the bending stiffness of the rod about the binormal axis is larger than the bending stiffness about the normal axis, and is always stable for a rod with circular section. The same condition is valid for a rod with intrinsic twist under pure torsion, or a planar ring with intrinsic twist, too. Greenhill's formula as stability condition of a stretched and twisted straight rod with circular cross section applies also when the rod has intrinsic twist, and the influence of the asymmetry of the cross section on the stability diagram of a straight rod with noncircular cross section is demonstrated. In case of a rod with noncircular cross section under pure torsion each helical equilibrium corresponds to a singular point of Euler angles, and a static bifurcation of the number and stability of possible helical equilibriums with the change of intrinsic twist is presented.