Stability Estimates for a Twisted Rod Under Terminal Loads: A Three-dimensional Study

Majumdar, Apala; Prior, Christopher; Goriely, Alain

doi:10.1007/s10659-012-9371-8

Cited by 11 publications

(37 citation statements)

References 26 publications

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“…Similarly, we can show that the second variation of the rod-energy in (19), about Θ 0 , is negative for in (??). The negativity of the second variation for a particular choice of (α, β) suffices to demonstrate the instability of Θ 0 for forces F L 2 > Aπ 2 1 − M 2 C 2 A 2 [3,10]. This completes the proof of Proposition 3.…”

Section: Propositionsupporting

confidence: 56%

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

Majumdar

Raisch²

2014

Nonlinearity

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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.

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Section: Propositionsupporting

confidence: 56%

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

Majumdar

Raisch²

2014

Nonlinearity

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“…The initial condition can be any arbitrary function 'sufficiently close' to the isotropic equilibrium. We study the nonlinear 'static' stability of the isotropic equilibrium by computing the second variation of the LdG energy in (2), the positivity of which is a sufficient criterion for nonlinear 'static' stability in one spatial dimension [5,13]. The second variation of the associated dimensionless LdG energy about S = 0 is given by…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Transitions through critical temperatures in nematic liquid crystals

et al. 2013

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We obtain 'dynamic' estimates for critical nematic liquid crystal (LC) temperatures with a slowly varying temperature-dependent control variable. We focus on two critical temperatures : the supercooling temperature below which the isotropic phase loses stability and the superheating temperature above which the ordered nematic states do not exist. In contrast to the static problem, the isotropic phase exhibits a memory effect below the supercooling temperature. This delayed loss of stability is independent of the rate of change of temperature and depends purely on the initial value of the temperature.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

On global and local minimizers of prestrained thin elastic rods

2017

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We study the stable configurations of a thin three-dimensional weakly prestrained rod subject to a terminal load as the thickness of the section vanishes. By Γ -convergence we derive a one-dimensional limit theory and show that isolated local minimizers of the limit model can be approached by local minimizers of the three-dimensional model. In the case of isotropic materials and for two-layers prestrained three-dimensional models the limit energy further simplifies to that of a Kirchhoff rod-model of an intrinsically curved beam. In this case we study the limit theory and investigate global and/or local stability of straight and helical configurations. Through some simple simulations we finally compare our results with real experiments. arXiv:1606.04524v1 [math.AP] 14 Jun 2016 2 MARCO CICALESE, MATTHIAS RUF, AND FRANCESCO SOLOMBRINO been derived since the pioneering papers by Le Dret and Raoult [16] and by Friesecke, James and Müller [11,12] by many authors [1,27,28,29,33,34] under different modelling assumptions.More recently the problem above has gained increasing attention in the case of prestrained bodies. A number of results have appeared in the case of 3-d to 2-d dimension reduction in [4,10,17,18,35] and many interesting questions have been raised. On one hand the above problem has been left undiscussed in the case of 3-d to 1-d dimension reduction (see [2,3] for a similar problem in the theory of nematic elastomers where the dimension reduction is performed in two subsequent steps 3-d to 2-d and 2-d to 1-d), on the other hand recent experiments in [19] suggest to consider it from a rigorous mathematical point of view. In few words in [19] the authors take two long strips of elastomer of the same initial width, but unequal length. The short strip is stretched uniaxially to be equal in length to the longer one. The initial heights are chosen so that after stretching the bi-strip system has a rectangular cross section. The two strips are then glued together side-by-side along their length. The bi-strips appear flat and the initially shorter strip is under a uniaxial prestrain. As a last step of the experiment, the external force needed to stretch the ends of the bi-strip is gradually released so that the initially flat bistrip starts to bend and twist out of plane. It may evolve towards either a helical or hemihelical shape (more complex structure in which helices with different chiralities seem to periodically alternate), depending on the cross-sectional aspect ratio. In particular, a big enough aspect ratio favors the formation of a helix, whereas a small aspect ratio favors that of a hemihelix.The analysis in [19] is simplified first assuming that the system is one-dimensional, so that a Kirchhoff-rod approximation is used, and then analyzing stability of configurations close to the straight rod by matching asymptotics in a restricted class of competitors. On one hand the results appear to be mathematically unsatisfying, on the other hand a rigorous derivation of the complete observed behavior seems to be...

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Stability Estimates for a Twisted Rod Under Terminal Loads: A Three-dimensional Study

Cited by 11 publications

References 26 publications

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

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

Transitions through critical temperatures in nematic liquid crystals

On global and local minimizers of prestrained thin elastic rods

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