Stability of twisted rods, helices and buckling solutions in three dimensions

Abstract

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 L 2 -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 L 2 -gradient flow model to these localized buckling solutions, demonstrate that they are unstable, study their evolution and the simulations demonstrate rich spatio-temporal patterns that strongly depend on the boundary conditions and imposed isoperimetric constraints.

On global and local minimizers of prestrained thin elastic rods

“…In the last section of the paper we investigate the local stability of straight and helical configurations under terminal loads. A similar problem, with the already discussed restrictions and in absence of prestrain, appeared in [22,24,25]. In Theorem 5.2 we compute a critical value (explicitly depending on the boundary conditions) f crit of the terminal load f such that, for f > f crit the straight configuration is a L 2 local minimizer, while for f < f crit it is not.…”

“…To this end in Proposition 4.3 we compute first and second variations of the functional E 0 and eventually prove in Theorem 4.7 that if the second variation of E 0 at a critical point R is a positive-definite quadratic form, then R is a strict local minimizer in L 2 . We remark that we do not use Euler angles to rewrite E 0 in contrast to [22,24,25]. On one hand this makes our problem mathematically more complicated since the domain of our energy functional is not a linear space, on the other we gain in generality since we have to pose no a-priori restrictions to the configurations in order to avoid polar singularities.…”

“…In our opinion the second aspect constitutes the real novelty of this paper together with some second order criteria for strict local minimality of 1-d configurations. Second order criteria for Kirchhoff-rod energies already appeared in the literature [22,24,25] but lacking any rigorous relation to the 3-d problem (see below for other differences between the two approaches). Furthermore we make use of our criteria to test the stability of the 1-d straight and helical configurations in terms of critical forces.…”

Necessary and sufficient conditions for minimizers in one-dimensional problems with natural boundary conditions

“…The author of [14] considered the conjugate points of the derivative h , but then the desired conditions require the computation of an index at each conjugate point and just the sum of those indices can determine the fact whether u 0 is a weak minimizer. The reference [14] has been cited by many papers: Some of the citing papers use the complicated theory in [14] for scalar problems with special Lagrangians (see [12], for example), some use various ad-hoc estimates to obtain at least partial results in the vector-valued case (when the theory in [14] does not seem to apply, see [13], for example) and some refrain from considering the natural boundary conditions because of the complexity of the theory in [14], see [5], for example, where the authors write: ". .…”

Stability of a strut with one end fixed and the other pinned only at the first bifurcation point

“…Koiter [1] proved stability for a strut with hinged ends under infinitesimal disturbance. In recent years, Majumdar et al [2,3] studied stability for equilibria of a naturally straight rod subject to terminal loads in three dimensions and obtained explicit stability estimates. Whether the potential energy of a system holds the minimum is a central problem in the stability estimates.…”