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SUMMARY:Equilibrium configurations of elastic torus knots (n\,2) - Starost
 in\, E (University College London)
DTSTART:20121206T113000Z
DTEND:20121206T114500Z
UID:TALK41896@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:We study equilibria of braided structures made of two elastic 
 rods with their centrelines remaining at constant distance from each other
 . The model is geometrically exact for large deformations. Each of the rod
 s is modelled as thin\, uniform\, homogeneous\, isotropic\, inextensible\,
  unshearable\, intrinsically twisted\, and to have circular cross-section.
  The governing equations are obtained by applying Hamilton's principle to 
 the action which is a sum of the elastic strain energies and the constrain
 ts related to the inextensibility of the rods. Hamilton's principle is equ
 ivalent to the second-order variational problem for the action expressed i
 n reduced strain-like variables. The Euler-Lagrange equations are derived 
 partly in Euler-Poincare form and are a set of ODEs suitable for numerical
  solution. \n\nWe model torus knots (n\,2) as closed configurations of the
  2-strand braid. We compute numerical solutions of this boundary value pro
 blem using path following. Closed 2-braids buckle under increasing twist. 
 We present a bifurcation diagram in the twist-force plane for torus knots 
 (n\,2). Each knot has a V-shaped non-buckled branch with its vertex on the
  twist axis. There is a series of bifurcation points of buckling modes on 
 both sides of each of the V-branches. The 1st mode bifurcation points for 
 n and n4 are connected by transition curves that go through (unphysical) s
 elf-crossing of the braid. Thus\, all the knots turn out to be divided int
 o two classes: one of them may be produced from the right-handed trefoil a
 nd the other from the left-handed. Higher-mode post-buckled configurations
  lead to cable knots. \n\nIt is instructive to see how close our elastic k
 nots can be tightened to the ideal shape. For the trefoil knot the tightes
 t shape we could get has a ropelength of 32.85560666\, which is remarkably
  close to the best current estimate. Careful examination reveals that the 
 solution is free from self-intersections though the contact set remains a 
 distorted circle.\n\n\n
LOCATION:Seminar Room 1\, Newton Institute
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