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The Generalised Ply : two twisted filaments with a line of contact
Application : DNA supercoiling (the Strick-Croquette-Bensimon experiment).
- Take two parallel strands of elastic rod and twist them a certain number of turns.
- Then glue to ends together, and allow each side to rotate freely.
- You have just made a ply :
In fact, considering the ply loaded by a wrench (tension G together with twisting moment N), we can write a more general potential energy :
Minimizing this energy with the condition that the rod stay on a cylinder, we get the equilibrium equation for the generalised ply :
We can use this equation to derivate the asymptotic behaviour of a highly twisted rod :
- Take a straight rod and approach the two ends in order half their distance. Then start to twist the rod :
- Increase twist is even further : the rod will start to contact itself. At first with separated point of conctact, but then the contact become continuous :
- At a certain point along the continuation, a line of contact emerges among the discrete contact points :
- Eventually the contact line will developp along more and more length. The rod shape will look something like :
- The equilibrium equation enables us to draw the reponse curve for a generalised ply together with its asymptote :
