## Theory of equilibria of elastic 2-braids with interstrand
interaction

E.L. Starostin & G.H.M. van der Heijden

Motivated by continuum models for DNA supercoiling we formulate a 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. 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. The other strand is tracked by including in this frame
the normalised closest-approach chord connecting the two strands. The
kinematic constant-distance constraint is formulated at strain level through
the introduction of what we call braid strains. As a result the total
potential energy involves arclength derivatives of these strains, thus giving
rise to a second-order variational problem. The Euler-Lagrange equations for
this problem 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 number of problems
that can be solved analytically and then applied to several new problems that
have not hitherto been treated.
keywords: elastic rods, braids, interstrand interaction, invariant
second-order variational problem, DNA supercoiling

*J. Mech. Phys. Solids* **64**, 83-132 (2014)