  In this paper, we speculate on a possible application of Liquid Crystal
Elastomers to the field of soft robotics. In particular, we study a concept for
limbless locomotion that is amenable to miniaturisation. For this purpose, we
formulate and solve the evolution equations for a strip of nematic elastomer,
subject to directional frictional interactions with a flat solid substrate, and
cyclically actuated by a spatially uniform, time-periodic stimulus (e.g.,
temperature change). The presence of frictional forces that are sensitive to
the direction of sliding transforms reciprocal, 'breathing-like' deformations
into directed forward motion. We derive formulas quantifying this motion in the
case of distributed friction, by solving a differential inclusion for the
displacement field. The simpler case of concentrated frictional interactions at
the two ends of the strip is also solved, in order to provide a benchmark to
compare the continuously distributed case with a finite-dimensional benchmark.
We also provide explicit formulas for the axial force along the crawler body.
