The basic tenet of metamaterials is that the architecture controls the physics1,2,3,4,5,6,7,8,9,10,11,12. So far, most studies have considered defect-free architectures. However, defects, and particularly topological defects, play a crucial role in natural materials13,14,15. Here we provide a systematic strategy for introducing such defects in mechanical metamaterials. We first present metamaterials that are a mechanical analogue of spin systems with tunable ferromagnetic and antiferromagnetic interactions, then design an exponential number of frustration-free metamaterials and finally introduce topological defects by rotating a string of building blocks in these metamaterials. We uncover the distinct mechanical signature of topological defects using experiments and simulations, and leverage this to design complex metamaterials in which external forces steer deformations and stresses towards complementary parts of the system. Our work presents a new avenue to systematically including spatial complexity, frustration and topology in mechanical metamaterials.
Subscribe to Journal
Get full journal access for 1 year
only $15.58 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
All prices are NET prices.
The code that supports the plots within this paper and other findings of this study is available from the corresponding author on reasonable request.
Mullin, T., Deschanel, S., Bertoldi, K. & Boyce, M. C. Pattern transformation triggered by deformation. Phys. Rev. Lett. 99, 084301 (2007).
Coulais, C., Teomy, E., De Reus, K., Shokef, Y. & van Hecke, M. Combinatorial design of textured mechanical metamaterials. Nature 535, 529–532 (2016).
Dudte, L. H., Vouga, E., Tachi, T. & Mahadevan, L. Programming curvature using origami tessellations. Nat. Mater. 15, 583–588 (2016).
Chen, B. G.-g et al. Topological mechanics of origami and kirigami. Phys. Rev. Lett. 116, 135501 (2016).
Paulose, J., Chen, B. G.-g & Vitelli, V. Topological modes bound to dislocations in mechanical metamaterials. Nat. Phys. 11, 153–156 (2015).
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).
Serra-Garcia, M. et al. Observation of a phononic quadrupole topological insulator. Nature 555, 342–345 (2018).
Coulais, C., Sabbadini, A., Vink, F. & van Hecke, M. Multi-step self-guided pathways for shape-changing metamaterials. Nature 561, 512–515 (2018).
Florijn, B., Coulais, C. & van Hecke, M. Programmable mechanical metamaterials. Phys. Rev. Lett. 113, 175503 (2014).
Frenzel, T., Kadic, M. & Wegener, M. Three-dimensional mechanical metamaterials with a twist. Science 358, 1072–1074 (2017).
Bertoldi, K., Vitelli, V., Christensen, J. & van Hecke, M. Flexible mechanical metamaterials. Nat. Rev. Mater. 2, 17066 (2017).
Kang, S. H. et al. Complex ordered patterns in mechanical instability induced geometrically frustrated triangular cellular structures. Phys. Rev. Lett. 112, 098701 (2014).
Nisoli, C., Moessner, R. & Schiffer, P. Colloquium: artificial spin ice: designing and imaging magnetic frustration. Rev. Mod. Phys. 85, 1473–1490 (2013).
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).
McEvoy, M. A. & Correll, N. Materials that couple sensing, actuation, computation, and communication. Science 347, 1261689 (2015).
Reis, P. M., Jaeger, H. M. & van Hecke, M. Designer matter: a perspective. Extreme Mech. Lett. 5, 25–29 (2015).
Wehner, M. et al. An integrated design and fabrication strategy for entirely soft, autonomous robots. Nature 536, 451–455 (2016).
Lao, Y. et al. Classical topological order in the kinetics of artificial spin ice. Nat. Phys. 14, 723–727 (2018).
Grima, J. N., Alderson, A. & Evans, K. E. Auxetic behaviour from rotating rigid units. Phys. Stat. Solidi B 242, 561–575 (2005).
Coulais, C., Kettenis, C. & van Hecke, M. A characteristic length scale causes anomalous size effects and boundary programmability in mechanical metamaterials. Nat. Phys. 14, 40–44 (2018).
Bertoldi, K., Reis, P. M., Willshaw, S. & Mullin, T. Negative poisson’s ratio behavior induced by an elastic instability. Adv. Mater. 22, 361–366 (2010).
Morrison, M. J., Nelson, T. R. & Nisoli, C. Unhappy vertices in artifical spin ice: new degeneracies from vertex frustration. New J. Phys. 15, 045009 (2013).
Syôzi, I. Statistics of kagomé lattice. Prog. Theor. Phys. 6, 306–308 (1951).
Kano, K. & Naya, S. Antiferromagnetism: the kagomé Ising net. Prog. Theor. Phys. 10, 158–172 (1953).
Zandbergen, R. M. A. On the Number of Configurations of Triangular Mechanisms. BSc thesis, Leiden Univ. (2016); https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/zandbergen.pdf
Mermin, N. D. The topological theory of defects in ordered media. Rev. Mod. Phys. 51, 591–648 (1979).
Alexander, G. P., Chen, B. G.-g, Matsumoto, E. A. & Kamien, R. D. Colloquium: disclination loops, point defects, and all that in nematic liquid crystals. Rev. Mod. Phys. 84, 497–514 (2012).
Ning, X. et al. Assembly of advanced materials into 3D functional structures by methods inspired by origami and kirigami: a review. Adv. Mater. Interf. 5, 1–13 (2018).
Kane, C. L. & Lubensky, T. C. Topological boundary modes in isostatic lattices. Nat. Phys. 10, 39–45 (2013).
Sinterit LISA Product Specification (Sinterit, 2014); https://www.sinterit.com/wp-content/uploads/2014/05/LISA_Specification.pdf
Sinterit Flexa Black Specification (Sinterit, 2014); https://www.sinterit.com/wp-content/uploads/2014/05/Flexa-Black-Specification.pdf
Blunt, M. O. et al. Random tiling and topological defects in a two-dimensional molecular network. Science 322, 1077–1081 (2008).
MacMahon, P.A. Combinatory Analysis Vol. 2 (Cambridge Univ. Press, 1916).
Sloane, N. The On-Line Encyclopedia of Integer Sequences (1996); https://oeis.org/A008793
Pellegrino, S. et al. Structural computations with the singular value decomposition of the equilibrium matrix. Int. J. Solids Struct. 30, 3025–3035 (1993).
Audoly, B. & Pomeau, Y. Elasticity and Geometry (Oxford Univ. Press, 2010).
We thank R. Ilan, E. Lerner, B. Mulder, B. Pisanty, E. Teomy and E. Verhagen for fruitful discussions, J. Paulose for co-developing code for the numerical model FH, D. Ursem for technical support and R. Zandbergen for supplying the exact counting data in Fig. 1h. This research was supported in part by the Israel Science Foundation grant no. 968/16.
The authors declare no competing interests.
Peer review information Nature Physics thanks Cristiano Nisoli and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Meeussen, A.S., Oğuz, E.C., Shokef, Y. et al. Topological defects produce exotic mechanics in complex metamaterials. Nat. Phys. 16, 307–311 (2020). https://doi.org/10.1038/s41567-019-0763-6
Physical Review Letters (2020)