  A cylindrical rubber fibre subject to twist will also elongate: a
manifestation of Poynting's effect in large strain elasticity. Here, we
construct an analogous treatment for an active rubber fibre actuated via an
axisymmetric pattern of spontaneous distortion. We start by constructing an
exact large-deformation solution to the equations of elasticity for such fibre
subject to imposed twist and stretch, which reveals spontaneous warping and
twisting of the fibre cross-section absent in passive rubbers. We then compute
the corresponding non-linear elastic energy, which encompasses the Poynting
effect, but is minimized by a finite spontaneous twist and stretch. In the
second half of the paper, we apply these results to understand the
twist-contraction actuation of nematic elastomer fibres fabricated with
director-fields that encode helical patterns of contraction on heating. We
first consider patterns making a constant angle with respect to the local
cylindrical coordinate system (conical spiral director curves) and verify the
predicted spontaneous twist, contraction and cross-section deformation via
finite elements. Secondly, we consider realistic director distributions for the
experimentally reported fibres fabricated by cross-linking while simultaneously
applying stretch and twist. Counter-intuitively, we find that maximum actuation
twist is produced by applying a finite optimal twist during fabrication.
Finally, we illustrate that spontaneously twisting fibres will coil into
spring-like shapes on actuation if the ends are prevented from twisting
relative to each other. Such twist-torsion coupling would allow to make a
tendril-like "soft-spring" actuator with low force and high linear stroke
compared to the intrinsic contraction of the elastomer itself.
