Topological defects produce exotic mechanics in complex metamaterials

Abstract

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.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Fig. 1: Designing structurally complex, compatible metamaterials.
Fig. 2: Defect design in periodic and complex metamaterials.
Fig. 3: Mechanical detection of defects.
Fig. 4: Forcing pairs of building blocks steers stresses and deformations.

Data availability

The data represented in Fig. 1h and Fig. 3b–e are provided with the paper as source data. All other data that support the plots within this paper and other findings of this study are available from the corresponding author on reasonable request.

Code availability

The code that supports the plots within this paper and other findings of this study is available from the corresponding author on reasonable request.

References

  1. 1.

    Mullin, T., Deschanel, S., Bertoldi, K. & Boyce, M. C. Pattern transformation triggered by deformation. Phys. Rev. Lett. 99, 084301 (2007).

    ADS  Article  Google Scholar 

  2. 2.

    Coulais, C., Teomy, E., De Reus, K., Shokef, Y. & van Hecke, M. Combinatorial design of textured mechanical metamaterials. Nature 535, 529–532 (2016).

    ADS  Article  Google Scholar 

  3. 3.

    Dudte, L. H., Vouga, E., Tachi, T. & Mahadevan, L. Programming curvature using origami tessellations. Nat. Mater. 15, 583–588 (2016).

    ADS  Article  Google Scholar 

  4. 4.

    Chen, B. G.-g et al. Topological mechanics of origami and kirigami. Phys. Rev. Lett. 116, 135501 (2016).

    ADS  Article  Google Scholar 

  5. 5.

    Paulose, J., Chen, B. G.-g & Vitelli, V. Topological modes bound to dislocations in mechanical metamaterials. Nat. Phys. 11, 153–156 (2015).

    Article  Google Scholar 

  6. 6.

    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).

    ADS  Article  Google Scholar 

  7. 7.

    Serra-Garcia, M. et al. Observation of a phononic quadrupole topological insulator. Nature 555, 342–345 (2018).

    ADS  Article  Google Scholar 

  8. 8.

    Coulais, C., Sabbadini, A., Vink, F. & van Hecke, M. Multi-step self-guided pathways for shape-changing metamaterials. Nature 561, 512–515 (2018).

    ADS  Article  Google Scholar 

  9. 9.

    Florijn, B., Coulais, C. & van Hecke, M. Programmable mechanical metamaterials. Phys. Rev. Lett. 113, 175503 (2014).

    ADS  Article  Google Scholar 

  10. 10.

    Frenzel, T., Kadic, M. & Wegener, M. Three-dimensional mechanical metamaterials with a twist. Science 358, 1072–1074 (2017).

    ADS  Article  Google Scholar 

  11. 11.

    Bertoldi, K., Vitelli, V., Christensen, J. & van Hecke, M. Flexible mechanical metamaterials. Nat. Rev. Mater. 2, 17066 (2017).

    ADS  Article  Google Scholar 

  12. 12.

    Kang, S. H. et al. Complex ordered patterns in mechanical instability induced geometrically frustrated triangular cellular structures. Phys. Rev. Lett. 112, 098701 (2014).

    ADS  Article  Google Scholar 

  13. 13.

    Nisoli, C., Moessner, R. & Schiffer, P. Colloquium: artificial spin ice: designing and imaging magnetic frustration. Rev. Mod. Phys. 85, 1473–1490 (2013).

    ADS  Article  Google Scholar 

  14. 14.

    Wang, R. F. et al. Artificial ’spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands. Nature 439, 303–306 (2006).

    ADS  Article  Google Scholar 

  15. 15.

    Castelnovo, C., Moessner, R. & Sondhi, S. L. Magnetic monopoles in spin ice. Nature 451, 42–45 (2008).

    ADS  Article  Google Scholar 

  16. 16.

    McEvoy, M. A. & Correll, N. Materials that couple sensing, actuation, computation, and communication. Science 347, 1261689 (2015).

    Article  Google Scholar 

  17. 17.

    Reis, P. M., Jaeger, H. M. & van Hecke, M. Designer matter: a perspective. Extreme Mech. Lett. 5, 25–29 (2015).

    Article  Google Scholar 

  18. 18.

    Wehner, M. et al. An integrated design and fabrication strategy for entirely soft, autonomous robots. Nature 536, 451–455 (2016).

    ADS  Article  Google Scholar 

  19. 19.

    Lao, Y. et al. Classical topological order in the kinetics of artificial spin ice. Nat. Phys. 14, 723–727 (2018).

    Article  Google Scholar 

  20. 20.

    Grima, J. N., Alderson, A. & Evans, K. E. Auxetic behaviour from rotating rigid units. Phys. Stat. Solidi B 242, 561–575 (2005).

    ADS  Article  Google Scholar 

  21. 21.

    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).

    Article  Google Scholar 

  22. 22.

    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).

    Article  Google Scholar 

  23. 23.

    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).

    ADS  Article  Google Scholar 

  24. 24.

    Syôzi, I. Statistics of kagomé lattice. Prog. Theor. Phys. 6, 306–308 (1951).

    ADS  MathSciNet  Article  Google Scholar 

  25. 25.

    Kano, K. & Naya, S. Antiferromagnetism: the kagomé Ising net. Prog. Theor. Phys. 10, 158–172 (1953).

    ADS  Article  Google Scholar 

  26. 26.

    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

  27. 27.

    Mermin, N. D. The topological theory of defects in ordered media. Rev. Mod. Phys. 51, 591–648 (1979).

    ADS  MathSciNet  Article  Google Scholar 

  28. 28.

    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).

    ADS  Article  Google Scholar 

  29. 29.

    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).

    Google Scholar 

  30. 30.

    Kane, C. L. & Lubensky, T. C. Topological boundary modes in isostatic lattices. Nat. Phys. 10, 39–45 (2013).

    Article  Google Scholar 

  31. 31.

    Sinterit LISA Product Specification (Sinterit, 2014); https://www.sinterit.com/wp-content/uploads/2014/05/LISA_Specification.pdf

  32. 32.

    Sinterit Flexa Black Specification (Sinterit, 2014); https://www.sinterit.com/wp-content/uploads/2014/05/Flexa-Black-Specification.pdf

  33. 33.

    Blunt, M. O. et al. Random tiling and topological defects in a two-dimensional molecular network. Science 322, 1077–1081 (2008).

    ADS  Article  Google Scholar 

  34. 34.

    MacMahon, P.A. Combinatory Analysis Vol. 2 (Cambridge Univ. Press, 1916).

  35. 35.

    Sloane, N. The On-Line Encyclopedia of Integer Sequences (1996); https://oeis.org/A008793

  36. 36.

    Pellegrino, S. et al. Structural computations with the singular value decomposition of the equilibrium matrix. Int. J. Solids Struct. 30, 3025–3035 (1993).

    Article  Google Scholar 

  37. 37.

    Audoly, B. & Pomeau, Y. Elasticity and Geometry (Oxford Univ. Press, 2010).

Download references

Acknowledgements

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.

Author information

Affiliations

Authors

Contributions

M.v.H and Y.S. conceived the project. E.C.O. conceived the structural characterization of defects. A.S.M. conceived the mechanical detection of defects. A.S.M. and E.C.O performed simulations. A.S.M. performed experiments. All authors contributed to the analysis of the results and to the writing of the manuscript.

Corresponding author

Correspondence to Anne S. Meeussen.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

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.

Supplementary information

Supplementary Information

Supplementary Figs. 1–6 and captions.

Source data

Source Data Fig. 1

Numerical data used to generate Fig.1h.

Source Data Fig. 3

Numerical data used to generate Fig.3b–e.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

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

Download citation

Further reading