Theory of equilibria of elastic braids with applications to DNA supercoiling
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Thu, 07/03 16:00 |
Gert Van Der Heijden (UCL London) |
Industrial and Applied Mathematics Seminar  |
DH 1st floor SR |
| We formulate a new theory for equilibria of 2-braids, i.e., structures formed by two elastic rods winding around each other in continuous contact and subject to a local interstrand interaction. Unlike in previous work no assumption is made on the shape of the contact curve. The theory is developed in terms of a moving frame of directors attached to one of the strands with one of the directors pointing to the position of the other strand. The constant-distance constraint is automatically satisfied by the introduction of what we call braid strains. The price we pay is that the potential energy involves arclength derivatives of these strains, thus giving rise to a second-order variational problem. The Euler-Lagrange equations for this problem (in Euler-Poincare form) give balance equations for the overall braid force and moment referred to the moving frame as well as differential equations that can be interpreted as effective constitutive relations encoding the effect that the second strand has on the first as the braid deforms under the action of end loads. Hard contact models are used to obtain the normal contact pressure between strands that has to be non-negative for a physically realisable solution without the need for external devices such as clamps or glue to keep the strands together. The theory is first illustrated by a few simple examples and then applied to several problems that require the numerical solution of boundary-value problems. Both open braids and closed braids (links and knots) are considered and current applications to DNA supercoiling are discussed. | |||