Spiral Patterns in the Packing of Flexible Structures

Spiral patterns are found to be a generic feature in close-packed elastic structures. We describe model experiments of compaction of quasi-1D sheets into quasi-2D containers that allow simultaneous quantitative measurements of mechanical forces and observation of folded configurations. Our theoretical approach shows how the interplay between elasticity and geometry leads to a succession of bifurcations responsible for the emergence of such patterns. Both experimental forces and shapes are also reproduced without any adjustable parameters.

Figure 1

Experimental setups and patterns. We used Mylar sheets of thickness $h=0.1\mathrm{mm}$ and bending modulus $B={6.610}^{-4}\mathrm{N}\mathrm{m}$. (a) Setup for conical packing. A sheet of radius 40 cm is pulled through a hole of radius 1 cm. (b) Typical pattern at high confinement ($p=15\%$). Examples of $Y$ points (bifurcations), $C$ curves, and $S$ curves are shown. (c) Typical pattern at lower confinement ($p=3\%$). A double-layered $S$ curve inside a set of $C$ curves forming a smaller effective container. (d) Setup for cylindrical packing. Sheets of height $H=14\mathrm{cm}$ are glued into a cylinder and introduced into a Plexiglass container of inner radius $R=2.6\mathrm{cm}$ and height 38 cm. The sheet is pushed at small velocity ($0.5\mathrm{mm}{\mathrm{s}}^{-1}$) with an aluminum disc of radius 5 cm, and the pushing force $F$ is recorded in a steady state (results in Fig. 3).

Figure 2

Configurations for cylindrical packing. Experiments (left) and numerics (right) are almost indistiguishable. (a), (b) Symmetric configurations with no self-contact ($ϵ=0.016$, short dashed line), one self-contact point ($ϵ=0.31$, continuous line), and two self-contact points ($ϵ=0.66$, long dashed line). (c), (d) First asymmetric configuration for $ϵ=0.85$. The $S$ curve and the surrounding $C$ curves form a yin-yang-like pattern. (e) Yin-yang pattern at high confinement $ϵ=9.4$. The thick $C$ curves are formed of 10 layers. (f) Numerical shape of the $S$ curve in the yin-yang pattern for very large confinements. In this case, the $C$ curves form a rigid circular effective container.

Figure 3

Mean pressure as a function of confinement for cylindrical packing. The pressure is given in units of $B/R^3$. Diamonds correspond to experimental measurements and lines to theoretical results. The dashed line corresponds to metastable asymmetric configurations reflecting the hysteretic character of the transition; the cross signals a termination of the asymmetric branch. Vertical dashed lines correspond to separations between different types of numerical configurations shown as insets numbered from 1 to 9. Single contacts with the container are labeled with force vectors; lineic contacts with the container are delimited by disks.