Self-Ordering of Buckling, Bending, and Bumping Beams

A collection of thin structures buckle, bend, and bump into each other when confined. This contact can lead to the formation of patterns: hair will self-organize in curls; DNA strands will layer into cell nuclei; paper, when crumpled, will fold in on itself, forming a maze of interleaved sheets. This pattern formation changes how densely the structures can pack, as well as the mechanical properties of the system. How and when these patterns form, as well as the force required to pack these structures is not currently understood. Here we study the emergence of order in a canonical example of packing in slender structures, i.e., a system of parallel confined elastic beams. Using tabletop experiments, simulations, and standard theory from statistical mechanics, we predict the amount of confinement (growth or compression) of the beams that will guarantee a global system order, which depends only on the initial geometry of the system. Furthermore, we find that the compressive stiffness and stored bending energy of this metamaterial are directly proportional to the number of beams that are geometrically frustrated at any given point. We expect these results to elucidate the mechanisms leading to pattern formation in these kinds of systems and to provide a new mechanical metamaterial, with a tunable resistance to compressive force.

Figure 1

(a) An oyster mushroom that has been left in the open air to dry (approximately one day between consecutive pictures). During this drying process, the gills become compressed along their length, causing them to buckle (middle) and then align (right). (b) Experimental observations of a similar phenomenon in a system of slender parallel plates compressed in an Instron. (c) A numerical experiment of the same system. Videos of a typical experiment (Supplemental Material, Video 1 [30]) and simulation (Supplemental Material, Video 2) are available.

Figure 2

The emergence of order for (a) two beams and (b),(c) many beams. (a),(i) Illustrated examples of $T_i$ ($+1$, blue; $-1$, orange) (a),(ii) Experiments (green) overlaid with simulations (blue and orange) of two beams that buckle toward each other. There are two metastable “clumped” $\overline{T}=0$ states (vertically and rotationally symmetric), however, when $\mathrm{\Gamma }>{\mathrm{\Gamma }}_c$, $\overline{T}=1$. (a),(iii) ${\mathrm{\Gamma }}_c$ increases with the normalized distance between two beams $d/{\ell }_s$ (experiments, blue diamond; simulations, red square; model, black line; Supplemental Material, Sec. II [30]) (b) $\overline{T}$ for experiments and simulations of many beams ($N=26$, $W=140\mathrm{mm}$, ${\ell }_s=54\mathrm{mm}$, $h=1.54\mathrm{mm}$, average $d=5.4\mathrm{mm}$, number of ensemble measurements: experiment, 25; simulation, 100) (c) $\overline{T}$ increases with $\mathrm{\Gamma }$ with a rate highly dependent on $N$, $d$, $W$, $h$, and ${\ell }_s$.

Figure 3

Understanding beam alignment. (a) A step-by-step illustration of clump-hole annihilation, with simulations at increasing $\mathrm{\Gamma }$ from top to bottom. Clumped beams are colored purple, and all others are colored by their tropism ($T_i=1\to \mathrm{blue}$, $T_i=-1\to \mathrm{orange}$). (b) Illustration of the parameters in our mathematical model. (c) The ensemble value of $\overline{T}$ as a function of $\alpha$ [prediction from Eq. (2), red]. Error bars are the same as in Fig. 2, left off for clarity.

Figure 4

Ordering affects compressive stiffness. (a) Snapshots of a simulation at $\mathrm{\Gamma }=1.18$ (i) and $\mathrm{\Gamma }=1.38$ (ii), superimposed with the vertical compressive force ($F_i$, blue circle) and bending energy ($U_{bi}$, red square) of each beam, normalized by the force and bending energy of a beam that is not part of a clump ($F_1$, $U_{b1}$). (b) The total compressive force (blue circle) and bending energy (red square) of the same system of beams as a function of $\mathrm{\Gamma }$, normalized by the compressive force and bending energy of the system if all the beams were buckled in the same direction. Predictions from our approximated model [Eq. (3)] are blue filled circle ($F$) and red filled square ($U_b$). Vertical lines show places where a clump and hole annihilate (Supplemental Material, Video 4 [30]). (c) Normalized force and bending energy against $1+N_t/N$ together with our predictions from Eq. (3) (simulation $U_b$, squares; predicted $U_b$, red line; simulation $F$, circles; experiment $F$, diamonds; predicted $F$, blue line). Gray in the legend indicates coloration by $\mathrm{\Gamma }$.