Deformation of loops in 2D packing of flexible rods

The injection of a long flexible rod into a two-dimensional domain yields a complex pattern commonly studied through the elasticity theory, packing analysis, and fractal geometries. 'Loop' is a one-vertex entity that 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 perform compression experiments. Then, tight packing configurations are analyzed using image processing. We find that the dimensions of the loops, confined or not, obey the same dependence on the deformation. The results are 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.

Automated summary

Injecting a long flexible rod into a two-dimensional domain leads to a complex pattern that can be analyzed through elasticity theory, packing analysis, and fractal geometries. In this study, the relationship between the shape of a loop, local deformations, and forces in 2D packing is explored. Experimental results suggest that the dimensions of loops, whether confined or not, follow the same dependence on deformation, supporting a simple model based on 2D elastic theory.