TY - JOUR

T1 - Infinitely long isotropic Kirchhoff rods with helical centerlines cannot be stable

AU - Borum, Andy

AU - Bretl, Timothy

N1 - Funding Information:
We are thankful to Federico Fuentes and Alex Townsend for helpful conversations. This work was supported by the NSF under Grant No. IIS-1320519. The work of A.B. was supported by the NSF-GRFP under Grant No. DGE-1144245 and by the NSF under Grant No. DMS-1645643.

PY - 2020/8

Y1 - 2020/8

N2 - It has long been known that every configuration of a planar elastic rod with clamped ends satisfies the property that if its centerline has constant nonzero curvature, then it is in stable equilibrium regardless of its length. In this paper, we show that for a certain class of nonplanar elastic rods, no configuration satisfies this property. In particular, using results from optimal control theory, we show that every configuration of an inextensible, unshearable, isotropic, and uniform Kirchhoff rod with clamped ends that has a helical centerline with constant nonzero curvature becomes unstable at a finite length. We also derive coordinates for computing this critical length that are independent of the rod's bending and torsional stiffness. Finally, we derive a scaling relationship between the length at which a helical rod becomes unstable and the rod's curvature, torsion, and twist. In a companion paper, these results are used to compute the set of all stable rods with helical centerlines.

AB - It has long been known that every configuration of a planar elastic rod with clamped ends satisfies the property that if its centerline has constant nonzero curvature, then it is in stable equilibrium regardless of its length. In this paper, we show that for a certain class of nonplanar elastic rods, no configuration satisfies this property. In particular, using results from optimal control theory, we show that every configuration of an inextensible, unshearable, isotropic, and uniform Kirchhoff rod with clamped ends that has a helical centerline with constant nonzero curvature becomes unstable at a finite length. We also derive coordinates for computing this critical length that are independent of the rod's bending and torsional stiffness. Finally, we derive a scaling relationship between the length at which a helical rod becomes unstable and the rod's curvature, torsion, and twist. In a companion paper, these results are used to compute the set of all stable rods with helical centerlines.

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U2 - 10.1103/PhysRevE.102.023004

DO - 10.1103/PhysRevE.102.023004

M3 - Article

C2 - 32942476

VL - 102

SP - 023004

JO - Physical Review E

JF - Physical Review E

SN - 2470-0045

IS - 2

M1 - 023004

ER -