Combinatorial design of textured mechanical metamaterials

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The structural complexity of metamaterials is limitless, but, in practice, most designs comprise periodic architectures that lead to materials with spatially homogeneous features1,2,3,4,5,6,7,8,9,10,11. More advanced applications in soft robotics, prosthetics and wearable technology involve spatially textured mechanical functionality, which requires aperiodic architectures. However, a naive implementation of such structural complexity invariably leads to geometrical frustration (whereby local constraints cannot be satisfied everywhere), which prevents coherent operation and impedes functionality. Here we introduce a combinatorial strategy for the design of aperiodic, yet frustration-free, mechanical metamaterials that exhibit spatially textured functionalities. We implement this strategy using cubic building blocks—voxels—that deform anisotropically, a local stacking rule that allows cooperative shape changes by guaranteeing that deformed building blocks fit together as in a three-dimensional jigsaw puzzle, and three-dimensional printing. These aperiodic metamaterials exhibit long-range holographic order, whereby the two-dimensional pixelated surface texture dictates the three-dimensional interior voxel arrangement. They also act as programmable shape-shifters, morphing into spatially complex, but predictable and designable, shapes when uniaxially compressed. Finally, their mechanical response to compression by a textured surface reveals their ability to perform sensing and pattern analysis. Combinatorial design thus opens up a new avenue towards mechanical metamaterials with unusual order and machine-like functionalities.

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Figure 1: Voxelated mechanical metamaterials.
Figure 2: Combinatorial design.
Figure 3: Pattern recognition and pattern analysis.


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We are grateful to J. Mesman for technical support. We thank R. Golkov, Y. Kamir, G. Kosa, K. Kuipers, F. Leoni, W. Noorduin and V. Vitelli for discussions. We acknowledge funding from the Netherlands Organisation for Scientific Research through grants VICI No. NWO-680-47-609 (M.v.H. and C.C.) and VENI No. NWO-680-47-445 (C.C.) and from the Israel Science Foundation through grant numbers 617/12 and 1730/12 (E.T and Y.S.).

Author information

C.C. and M.v.H. conceived the main concepts. C.C., E.T., Y.S. and M.v.H. formulated the spin problem. E.T. and Y.S. solved the spin problem. C.C and K.d.R. performed the experiments and simulations with inputs from E.T., Y.S. and M.v.H. C.C. and M.v.H wrote the manuscript with contributions from all authors.

Correspondence to Corentin Coulais.

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Competing interests

The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 Motif-based design.

a, 2D representation of the six bricks (x, x+, y, y+, z, z+), and illustration of complex motifs. All complex motifs can be generated by defining two binary vectors with elements ci (column) and rj (row) that govern the placement of z bricks at location (i, j). The remaining sites are then filled with x and y bricks. Respecting parity, this generates all motifs for given c and r. b, The six motifs (A+, …, F+) that are compatible with a 5 × 5 smiley texture. c, A total of 65 smiley metacubes can be designed by varying the stacking order; here A denotes the same motif as A+ but with inversed spins. The x and y spins follow from the choice of motifs.

Extended Data Figure 2 Implementation.

a, b, Computer assisted design of the geometry of the unit cell (a) and a 5 × 5 × 5 cube (b). All our samples were 3D printed with the dimensions a = 11.46 mm, D = 10.92 mm and w = 3.6 mm. To make the wall thickness outside the cube equal to the internal wall thickness, the outer walls are thickened by 0.27 mm.

Extended Data Figure 3 Lock-and-key experiment.

a, Picture of the textured clamp. b, Side view of the experiment.

Extended Data Figure 4 10 × 10 × 10 metacube under uniaxial compression.

a, Motif A+—the cube is designed by stacking motifs A+ and A. b, Opposite face of the one shown in Fig. 1e showing the inverted pattern. c, One of the transverse faces showing a checkerboard pattern.

Extended Data Figure 5 Complex sensory properties of a complex 5 × 5 × 5 metacube with internal smiley texture.

a, Force–compression (Fu) curves for five experiments (thick solid lines), where the colour indicates the external texture shown in Fig. 3c. The black thin lines show fits with a quadratic function F(u) = ku + ηu2 performed in the shaded region 0.8 mm ≤ u ≤ 2.5 mm. b, Corresponding numerical results. c, Scatter plot showing very good correspondence between the stiffness obtained from simulations (ks) and experiments (ke).

Extended Data Table 1 The exact value of Ω for L × L × L metacubes up to L = 14

Supplementary information

Supplementary Information

This file contains Supplementary Text and Data, Supplementary Figures 1-14 and Supplementary Table 1. (PDF 679 kb)

5x5x5 metacube, uniaxially compressed along its minor axis by flat clamps.

As discussed in the main text, it exhibits a pattern transformation, where its building blocks suddenly morph in to alternated bricks of elongated and flattened shape. (MP4 1718 kb)

10x10x10 metacube decorated with square pedestals, which is uniaxially compressed along its minor axis by clamps textured in a checker board pattern (seemethods).

As discussed in the main text, its surface texture morphs into an exactingly designed ”smiley” pattern. (MP4 875 kb)

The opposite face of the same 10x10x10 metacube during a similar experiment.

As discussed in the methods, its surface texture morphs into the inverted ”smiley” pattern. (MP4 822 kb)

A side face of the same 10x10x10 metacube during a similar experiment.

As discussed in the methods, its surface texture morphs into a checkerboard pattern. (MP4 824 kb)

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Coulais, C., Teomy, E., de Reus, K. et al. Combinatorial design of textured mechanical metamaterials. Nature 535, 529–532 (2016) doi:10.1038/nature18960

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