Nonlinear dynamical theory of the elastica

Abstract

SynopsisThe dynamical behaviour of a slender rod is analyzed here in terms of a generalization of Euler's elastica theory. The model includes a linear stress-strain relation but nonlinear geometric terms. Properties of the rod may vary along its length and various boundary conditions are considered. A rotational inertia term that is neglected in many theories is retained, and is essential to the analysis. By use of the equivalence of an energy and a Sobolev norm, and by reformulation of the equations as a semilinear system, global existence of solutions is proved for any smooth initial data. Equilibrium solutions that are stable in the static sense of minimizing the potential energy are then proved to be stable in the dynamic sense due to Liapounov.

On the Cauchy Problem for a Dynamical Euler's Elastica

“…Our theorem should be compared with results of Caflisch and Maddocks [1], who have given a proof of global existence for a planar dynamical elastica. Their equations of motion include an additional rotational inertia term proportional to 1 2 |∂ t u| 2 ds in the kinetic energy T (see their Eq.…”

Transitions through critical temperatures in nematic liquid crystals

“…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…”

“…Thus, the isotropic equilibrium is stable for A > A T and A T is strictly less than the homogeneous supercooling value, A = 0. Having demonstrated the nonlinear 'static stability' of the isotropic phase for A > A T , we can use Liapounov's direct method (with the LdG energy as a Liapounov function) to prove the dynamic Liapounov stability of the isotropic phase for A > A T [5]. This delayed loss of stability is purely a consequence of the boundary effects.…”

“…the solution branch starts bounded away from the critical value, A = 0. The two competing equilibria near A = 0 are the lower parabolic equilibrium, S − (A) in (5), and the isotropic branch, S = 0. For small A, S − (A) can be approximated by…”

Dynamic curling of an Elastica: a nonlinear problem in elastodynamics solved by matched asymptotic expansions

“…A linear version of these equations is classically used to analyze dynamic buckling (Golubovic et al, 1998;Gladden et al, 2005;Schindler and Kolsky, 1983;Audoly and Neukirch, 2005) as well as the stability of equilibrium solutions (Caflisch and Maddocks, 1984). The dynamics of an Elastica has also been characterized by means of amplitude equations Tabor, 1996, 2000), which is a type of a weakly nonlinear expansion.…”