Abstract
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.
This is a preview of subscription content, access via your institution
Access options
Subscribe to this journal
Receive 51 print issues and online access
$199.00 per year
only $3.90 per issue
Rent or buy this article
Prices vary by article type
from$1.95
to$39.95
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
References
Lakes, R. Foam structures with a negative Poisson’s ratio. Science 235, 1038–1040 (1987)
Mullin, T., Deschanel, S., Bertoldi, K. & Boyce, M. C. Pattern transformation triggered by deformation. Phys. Rev. Lett. 99, 084301 (2007)
Grima, J. N. & Evans, K. E. Auxetic behavior from rotating squares. J. Mater. Sci. Lett. 19, 1563–1565 (2000)
Schaedler, T. A. et al. Ultralight metallic microlattices. Science 334, 962–965 (2011)
Nicolaou, Z. G. & Motter, A. E. Mechanical metamaterials with negative compressibility transitions. Nat. Mater. 11, 608–613 (2012)
Babaee, S., Shim, J., Weaver, J. C., Patel, N. & Bertoldi, K. 3D soft metamaterials with negative Poisson’s ratio. Adv. Mater. 25, 5044–5049 (2013)
Kadic, M., Bückmann, T., Schittny, R. & Wegener, M. Metamaterials beyond electromagnetism. Rep. Prog. Phys. 76, 126501 (2013)
Florijn, B., Coulais, C. & van Hecke, M. Programmable mechanical metamaterials. Phys. Rev. Lett. 113, 175503 (2014)
Coulais, C., Overvelde, J. T. B., Lubbers, L. A., Bertoldi, K. & van Hecke, M. Discontinuous buckling of wide beams and metabeams. Phys. Rev. Lett. 115, 044301 (2015)
Schenk, M. & Guest, S. D. Geometry of Miura-folded metamaterials. Proc. Natl Acad. Sci. USA 110, 3276–3281 (2013)
Waitukaitis, S., Menaut, R., Chen, B. G. & van Hecke, M. Origami multistability: from single vertices to metasheets. Phys. Rev. Lett. 114, 055503 (2015)
Silverberg, J. L. et al. Using origami design principles to fold reprogrammable mechanical metamaterials. Science 345, 647–650 (2014)
Bückmann, T., Thiel, M., Kadic, M., Schittny, R. & Wegener, M. An elasto-mechanical unfeelability cloak made of pentamode metamaterials. Nat. Commun. 5, 4130 (2014)
Paulose, J., Meeussen, A. S. & Vitelli, V. Selective buckling via states of self-stress in topological metamaterials. Proc. Natl Acad. Sci. USA 112, 7639–7644 (2015)
Liu, Z. et al. Locally resonant sonic materials. Science 289, 1734–1736 (2000)
Wannier, G. H. Antiferromagnetism: the triangular Ising net. Phys. Rev. 79, 357–364 (1950)
Sadoc, J. F. & Mosseri, R. Geometrical Frustration (Cambridge Univ. Press, 1999)
Zykov, V., Mytilinaios, E., Adams, B. & Lipson, H. Robotics: Self-reproducing machines. Nature 435, 163–164 (2005)
Ware, T. H., McConney, M. E., Wie, J. J., Tondiglia, V. P. & White, T. J. Voxelated liquid crystal elastomers. Science 347, 982–984 (2015)
Harris, M. J., Bramwell, S. T., McMorrow, D. F., Zeiske, T. & Godfrey, K. W. Geometrical frustration in the ferromagnetic pyrochlore Ho2Ti2O7 . Phys. Rev. Lett. 79, 2554–2557 (1997)
Wang, R. F. et al. Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands. Nature 439, 303–306 (2006)
Castelnovo, C., Moessner, R. & Sondhi, S. L. Magnetic monopoles in spin ice. Nature 451, 42–45 (2008)
Nisoli, C., Moessner, R. & Schiffer, P. Colloquium: Artificial spin ice: designing and imaging magnetic frustration. Rev. Mod. Phys. 85, 1473–1490 (2013)
Grünbaum, B. & Shephard, G. C. Tilings and Patterns (Freeman, 1987)
Kirkpatrick, S., Gelatt, C. D. Jr & Vecchi, M. P. Optimization by simulated annealing. Science 220, 671–680 (1983)
Mezard, M., Parisi, G. & Virasoro, M. A. Spin Glass Theory and Beyond (World Scientific, 1987)
Cho, Y. et al. Engineering the shape and structure of materials by fractal cut. Proc. Natl Acad. Sci. USA 111, 17390–17395 (2014)
Leong, T. G. et al. Tetherless thermobiochemically actuated microgrippers. Proc. Natl Acad. Sci. USA 106, 703–708 (2009)
Shepherd, R. F. et al. Multigait soft robot. Proc. Natl Acad. Sci. USA 108, 20400–20403 (2011)
Overvelde, J. T., Kloek, T., Dhaen, J. J. & Bertoldi, K. Amplifying the response of soft actuators by harnessing snap-through instabilities. Proc. Natl Acad. Sci. USA 112, 10863–10868 (2015)
Acknowledgements
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
Authors and Affiliations
Contributions
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.
Corresponding author
Ethics declarations
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 (F–u) 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).
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)
Source data
Rights and permissions
About this article
Cite this article
Coulais, C., Teomy, E., de Reus, K. et al. Combinatorial design of textured mechanical metamaterials. Nature 535, 529–532 (2016). https://doi.org/10.1038/nature18960
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/nature18960
This article is cited by
-
Curvature tuning through defect-based 4D printing
Communications Materials (2024)
-
Self-bridging metamaterials surpassing the theoretical limit of Poisson’s ratios
Nature Communications (2023)
-
Bounded Wang tilings with integer programming and graph-based heuristics
Scientific Reports (2023)
-
Surface morphing control of mechanical metamaterials using geometrical imperfections
Journal of Materials Science (2023)
-
A three step recipe for designing auxetic materials on demand
Communications Physics (2022)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.