  Geometrically frustrated elastic ribbons exhibit, in many cases, significant
changes in configuration depending on the relation between their width and
thickness. We show that the existence of such a transition, and the scaling at
which it occurs, strongly depend on the system considered. Using an asymptotic
approach, we give a uniform framework to study such systems, predict if and
when a phase transition occurs, and what is the elastic energy scaling in each
case. We show that the predictions of our approach match experimental results
in a wide variety of geometric settings.
