Keeping It Together: Interleaved Kirigami Extension Assembly

Traditional origami structures can be continuously deformed back to a flat sheet of paper, while traditional kirigami requires glue or seams in order to maintain its rigidity. In the former, nontrivial geometry can be created through overfolding paper, while in the latter, the paper topology is modified. Here we propose a hybrid approach that relies on overlapped flaps that create in-plane compression resulting in the formation of polyhedra composed of freely supported plates. Not only are these structures self-locking, but they have colossal load-to-weight ratios of order $10^4$.

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Paper folding is an old, elegant art, but without glue, tape, or clips, how do you make origami into furniture? After all, the folds and tucks are all made without tearing the paper and so the final construction can be smoothly unfolded to a flat piece of paper. However, the in-plane stresses of the paper can be exploited to frustrate this unfolding without explicit sticking, pasting, or pinning. To achieve this, we change the initial geometry of the paper by making cuts, akin to the methods of kirigami (a variation of origami), but instead of modifying the topology by gluing, we use the cuts to allow for overlapping sections of paper.

In this work, we create structures that can support many times their weight, are light, and can be shipped flat, folding into the desired shape when needed. We demonstrate this with ordinary cardboard and provide an explanation as to how the interleaved assembly of the kirigami edges along with the periodic structure of the folds and cuts creates a mechanism to transfer downward force into shear stress, allowing a piece of cardboard to support 14 000 times its weight without pins or adhesives.

We conclude that the superstrength arises from a change in boundary conditions on the supporting walls. We transfer our construction to plastic sheets and copper foil and demonstrate that the collective effect of the folds, cuts, and stresses can be transferred to other materials. This provides a new set of motifs for lightweight, deployable structures. Someday, you might sit on it.

Figure 1

A diagram of our basic motif. We cut along solid lines and make mountain ($M$) and valley ($V$) folds along the dashed lines. In the final state, the hatched areas overlap. In the inset we show the assembled structure (scale bar 1 cm).

Figure 2

A diagram of the interleaved kirigami extension assembly showing the cooperative frustration between basic units. We cut along solid lines and make mountain ($M$) and valley ($V$) folds along dashed lines and, again, the hatched areas overlap. (a) Template for the threefold structure with inset showing assembled pattern ($\alpha =\pi /3$). (b) Template for the threefold structure with inset showing assembled pattern ($\alpha =2\pi /3$) (scale bar 1 cm).

Figure 3

From left to right: Step-by-step assembly of the periodic plateau array. Note that this assembly requires bending of the sheets and so this structure is not rigid foldable. For clarity, the detailed pattern on the underlying lattice is not shown. Yellow areas are the plateaus or buttes and the gray areas overlap (scale bar 1 cm).

Figure 4

The loading setup. (a) To avoid boundary effects, we only push on fully coordinated plateaus. (b) Upon collapse, some walls buckle in while others buckle out. We denote the outward buckled walls with red lines. Three different samples show that the collapse has a random component, likely due to folding inhomogeneity (scale bar 1 cm).

Figure 5

Experimental curves for the original, pinned, gapped, and disconnected models. Note that the pinned and gapped models are nearly identical: in one case there is no in-plane compression, and in the other, the adjacent edges fail to support each other. We have delineated the end of regime (i) with the vertical dashed line—it is the region where all four models (pinned, gapped, original, and disconnected) have the same response. At larger strain, the “unoriginal” models fail while the original model continues to resist.

Figure 6

Square interleaved kirigami edge assembly. The top panel shows the interlocking pattern. On the bottom, we see that the vertical walls do not convert downward force to in-plane force, preventing the collective response of the design, similar to the disconnected model. We pushed on four square sites with edge length 17.3 mm compared to six equilateral triangle sites with the same edge length so the square model has an even larger contact area by a factor of $8\sqrt{3}/9\approx 1.6$ (scale bar 1 cm).

Figure 7

Numerical buckling images of a sidewall with and without support at two isosceles edges. Solid triangles around the boundary are translational constraints. The sheets are completely restrained along the three longest edges while the shortest edge is allowed to move along the loading direction as depicted by the arrows. The deformation level in images is enlarged by a factor of 4 for clarity.

Figure 8

Numerical images of the first four eigenmodes and the corresponding eigenvalues. The deformation level in images is enlarged by a factor of 4 for clarity.

Figure 9

Pinned, gapped, and disconnected models. (a) Diagrams of cuts and folds. For pinned models, we made cuts along solid lines and added tape on the hatched areas to hold the shape. For the gapped models, we removed the edges of the sidewalls to prevent them from touching adjacent walls upon compression. For disconnected models, we folded six units separately and stuck them at the positions they would sit at were they on the original model. (b) Final states of all three models. Red short lines denote walls buckled outside. Note that outward buckling was the dominant mode in the pinned and disconnected cases (scale bar 1 cm).

Figure 10

The influence of structure density. The models were modified by changing the dimensions of the basic units as depicted and we only pushed on six, fully coordinated units. Note that the yield force was independent of the dimension, implying that the yield pressure scaled as the inverse of the square of the length. (The top plates were all different mass: 69.1, 25.3, and 9 g, for the original, $2/3$, and $1/2$ scale structures, but this difference is well within the noise of the measurement.)

Figure 11

The characterization of materials. (a) Tension tests of 30 mm by 100 mm strips. We cut strips along orthogonal directions. Paper is the only material showing anisotropic behavior. We note that copper, in particular, undergoes plastic deformation under tension as discussed in Ref. [29]. (b) We captured the bending behavior of soft and inhomogeneous materials by using four-legged frames. Inset: Experimental setup. The legs were all 16 mm by 16 mm squares.

Figure 12

The influence of paper thickness. Experimental curves of paper with thickness 0.14, 0.20, 0.28 (original models), and 0.38 mm.

Figure 13

The influence of material difference ($2/3$ scale original structure for copper, photo film, and paper). Top: Copper model (scratched) and polymer film model (unscratched). Bottom: Experimental curves of copper sheet, transparency films, and paper with thickness 0.28 mm (scale bar 1 cm).