Stability of twisted rods, helices and buckling solutions in three dimensions
KAUST Grant NumberKUK-C1-013-04
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Abstract© 2014 IOP Publishing Ltd & London Mathematical Society. We study stability problems for equilibria of a naturally straight, inextensible, unshearable Kirchhoff rod allowed to deform in three dimensions (3D), subject to terminal loads. We investigate the stability of the twisted, straight state in 3D for three different boundary-value problems, cast in terms of Dirichlet and Neumann boundary conditions for the Euler angles, with and without isoperimetric constraints. In all cases, we obtain explicit stability estimates in terms of the twist, external load and elastic constants and in the Dirichlet case, we compute bifurcation diagrams for the Euler angles as a function of the external load. In the same vein, we obtain explicit stability estimates for a family of prototypical helical equilibria in 3D and demonstrate that they are stable for a range of tensile and compressive forces. We propose a numerical L2-gradient flow model to study the stability and dynamical evolution (in viscous model situations) of Kirchhoff rod equilibria. In Nizette and Goriely 1999 J. Math. Phys. 40 2830-66, the authors construct a family of localized buckling solutions. We apply our L2-gradient flow model to these localized buckling solutions, demonstrate that they are unstable, study their evolution and the simulations demonstrate rich spatio oral patterns that strongly depend on the boundary conditions and imposed isoperimetric constraints.
CitationMajumdar A, Raisch A (2014) Stability of twisted rods, helices and buckling solutions in three dimensions. Nonlinearity 27: 2841–2867. Available: http://dx.doi.org/10.1088/0951-7715/27/12/2841.
SponsorsThe authors are thankful to referees for useful comments and pointing out interesting works in this field. AM and AR thank Alain Goriely for several helpful discussions and Heiko Gimperlein for discussions about second-order differential operators. The authors also thank John Maddocks, Sebastien Neukirch, Gert van der Heijden, Thomas Lessinnes, Derek Moulton for helpful remarks. AM is supported by an EPSRC Career Acceleration Fellowship EP/J001686/1, an OCCAM Visiting Fellowship and a Keble Research Grant. AR is supported by KAUST, Award No KUK-C1-013-04 and the John Fell OUP fund.