  The importance of nonlinearities in material constitutive relations has long
been appreciated in the continuum mechanics of macroscopic rods. Although the
moment (torque) response to bending is almost universally linear for small
deflection angles, many rod systems exhibit a high-curvature softening. The
signature behavior of these rod systems is a kinking transition in which the
bending is localized. Recent DNA cyclization experiments by Cloutier and Widom
have offered evidence that the linear-elastic bending theory fails to describe
the high-curvature mechanics of DNA. Motivated by this recent experimental
work, we develop a simple and exact theory of the statistical mechanics of
linear-elastic polymer chains that can undergo a kinking transition. We
characterize the kinking behavior with a single parameter and show that the
resulting theory reproduces both the low-curvature linear-elastic behavior
which is already well described by the Wormlike Chain model, as well as the
high-curvature softening observed in recent cyclization experiments.
