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.