We consider the motion of an infinitely long, naturally curved, planar Elastica. The Elastica is flattened onto a rigid impenetrable substrate and held by its endpoints. When one of its endpoints is released, it is set off in a curling motion, which we seek to describe mathematically based on the non-linear equations of motions for planar elastic rods undergoing finite rotations. This problem is used to introduce the technique of matched asymptotic expansions. We derive a non-linear solution capturing the late dynamics, when a roll comprising many turns has formed: in this regime, the roll advances at an asymptotically constant velocity, whose selection we explain. This contribution presents an expanded version of the results published in Callan-Jones et al. (Phys. Rev. Lett. 2012).