  The injection of a long flexible rod into a two-dimensional domain yields a
complex pattern commonly studied through elasticity theory, packing analysis,
and fractal geometries. "Loop" is a one-vertex entity that is naturally formed
in this system. The role of the elastic features of each loop in 2D packing has
not yet been discussed. In this work, we point out how the shape of a given
loop in the complex structure allows estimating local deformations and forces.
First, we build sets of symmetric free loops and performed compression
experiments. Then, tight packing configurations are analyzed by using image
processing. We found that the dimensions of the loops, confined or not, obey
the same dependence on the deformation. The result is consistent with a simple
model based on 2D elastic theory for filaments, where the rod adopts the shape
of Euler's elasticas between its contact points. The force and the stored
energy are obtained from numerical integration of the analytic expressions. In
an additional experiment, we obtain that the compression force for deformed
loops corroborates the theoretical findings. The importance of the shape of the
loop is discussed and we hope that the theoretical curves may allow statistical
considerations in future investigations.
