Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Static non-reciprocity in mechanical metamaterials


Reciprocity is a general, fundamental principle governing various physical systems, which ensures that the transfer function—the transmission of a physical quantity, say light intensity—between any two points in space is identical, regardless of geometrical or material asymmetries. Breaking this transmission symmetry offers enhanced control over signal transport, isolation and source protection1,2,3,4,5,6. So far, devices that break reciprocity (and therefore show non-reciprocity) have been mostly considered in dynamic systems involving electromagnetic, acoustic and mechanical wave propagation associated with fields varying in space and time. Here we show that it is possible to break reciprocity in static systems, realizing mechanical metamaterials7,8,9,10,11,12,13,14,15,16 that exhibit vastly different output displacements under excitation from different sides, as well as one-way displacement amplification. This is achieved by combining large nonlinearities with suitable geometrical asymmetries and/or topological features. In addition to extending non-reciprocity and isolation to statics, our work sheds light on energy propagation in nonlinear materials with asymmetric crystalline structures and topological properties. We anticipate that breaking reciprocity will open avenues for energy absorption, conversion and harvesting, soft robotics, prosthetics and optomechanics.

Your institute does not have access to this article

Relevant articles

Open Access articles citing this article.

Access options

Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Figure 1: Static non-reciprocity.
Figure 2: Discrete models for non-reciprocal metamaterials.
Figure 3: 2D topological mechanical metamaterial.


  1. Potton, R. J. Reciprocity in optics. Rep. Prog. Phys. 67, 717–754 (2004)

    ADS  Article  Google Scholar 

  2. Lira, H., Yu, Z., Fan, S. & Lipson, M. Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip. Phys. Rev. Lett. 109, 033901 (2012)

    ADS  Article  Google Scholar 

  3. Fan, L. et al. An all-silicon passive optical diode. Science 335, 447–450 (2012)

    ADS  CAS  Article  Google Scholar 

  4. Peng, B. et al. Parity–time-symmetric whispering-gallery microcavities. Nat. Phys. 10, 394–398 (2014)

    CAS  Article  Google Scholar 

  5. Estep, N. A., Sounas, D. L., Soric, J. & Alù, A. Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops. Nat. Phys . 10, 923–927 (2014)

    CAS  Article  Google Scholar 

  6. Fleury, R., Sounas, D. L., Sieck, C. F., Haberman, M. R. & Alu, A. Sound isolation and giant linear nonreciprocity in a compact acoustic circulator. Science 343, 516–519 (2014)

    ADS  CAS  Article  Google Scholar 

  7. Lakes, R. Foam structures with a negative Poisson’s ratio. Science 235, 1038–1040 (1987)

    ADS  CAS  Article  Google Scholar 

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

    ADS  CAS  Article  Google Scholar 

  9. Schaedler, T. A. et al. Ultralight metallic microlattices. Science 334, 962–965 (2011)

    ADS  CAS  Article  Google Scholar 

  10. Silverberg, J. L. et al. Using origami design principles to fold reprogrammable mechanical metamaterials. Science 345, 647–650 (2014)

    ADS  CAS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  12. Nash, L. M. et al. Topological mechanics of gyroscopic metamaterials. Proc. Natl Acad. Sci. USA 112, 14495–14500 (2015)

    ADS  CAS  Article  Google Scholar 

  13. Shan, S. et al. Multistable architected materials for trapping elastic strain energy. Adv. Mater. 27, 4296–4301 (2015)

    CAS  Article  Google Scholar 

  14. Susstrunk, R. & Huber, S. D. Observation of phononic helical edge states in a mechanical topological insulator. Science 349, 47–50 (2015)

    ADS  Article  Google Scholar 

  15. Frenzel, T., Findeisen, C., Kadic, M., Gumbsch, P. & Wegener, M. Tailored buckling microlattices as reusable light-weight shock absorbers. Adv. Mater. 28, 5865–5870 (2016)

    CAS  Article  Google Scholar 

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

    ADS  CAS  Article  Google Scholar 

  17. Maxwell, J. C. L. On the calculation of the equilibrium and stiffness of frames. Phil. Mag. Ser. 4 27, 294–299 (1864);

    Article  Google Scholar 

  18. Betti, E. Teoria della elasticita. Nuovo Cimento 7, 69–97 (1872)

    Article  Google Scholar 

  19. Charlton, T. M. A historical note on the reciprocal theorem and theory of statically indeterminate frameworks. Nature 187, 231–232 (1960)

    ADS  Article  Google Scholar 

  20. Love, A. E. H. A Treatise on the Mathematical Theory of Elasticity (Cambridge Univ. Press, 2013)

  21. Timoshenko, S. & Young, D. H. Theory of Structures (McGraw-Hill, 1965)

  22. Casimir, H. B. G. On Onsager’s principle of microscopic reversibility. Rev. Mod. Phys. 17, 343–350 (1945)

    ADS  Article  Google Scholar 

  23. Kane, C. L. & Lubensky, T. C. Topological boundary modes in isostatic lattices. Nat. Phys. 10, 39–45 (2014)

    CAS  Article  Google Scholar 

  24. Chen, B. G., Upadhyaya, N. & Vitelli, V. Nonlinear conduction via solitons in a topological mechanical insulator. Proc. Natl Acad. Sci. USA 111, 13004–13009 (2014)

    ADS  MathSciNet  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  26. Huber, S. D. Topological mechanics. Nat. Phys. 12, 621–623 (2016)

    CAS  Article  Google Scholar 

  27. Kadic, M., Bückmann, T., Schittny, R. & Wegener, M. Metamaterials beyond electromagnetism. Rep. Prog. Phys. 76, 126501 (2013)

    ADS  Article  Google Scholar 

  28. Brûlé, S., Javelaud, E. H., Enoch, S. & Guenneau, S. Experiments on seismic metamaterials: molding surface waves. Phys. Rev. Lett. 112, 133901 (2014)

    ADS  Article  Google Scholar 

  29. Hussein, M. I., Leamy, M. J. & Ruzzene, M. Dynamics of phononic materials and structures: historical origins, recent progress, and future outlook. Appl. Mech. Rev. 66, 040802 (2014)

    ADS  Article  Google Scholar 

  30. Aspelmeyer, M., Kippenberg, T. J. & Marquardt, F. Cavity optomechanics. Rev. Mod. Phys. 86, 1391–1452 (2014)

    ADS  Article  Google Scholar 

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

    CAS  Article  Google Scholar 

  32. Lubensky, T. C., Kane, C. L., Mao, X., Souslov, A. & Sun, K. Phonons and elasticity in critically coordinated lattices. Rep. Prog. Phys. 78, 073901 (2015)

    ADS  CAS  Article  Google Scholar 

  33. 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  CAS  Article  Google Scholar 

  34. Meeussen, A. S., Paulose, J. & Vitelli, V. Geared topological metamaterials with tunable mechanical stability. Phys. Rev. X 6, 041029 (2016)

    Google Scholar 

Download references


We thank D. Ursem for technical assistance. We are grateful to M. van Hecke, V. Vitelli, A. Souslov, Y. Hadad, A. Meeussen and S. Waitukaitis for discussions. C.C. acknowledges funding from the Netherlands Organization for Scientific Research (NWO), VENI grant no. NWO-680-47-445. D.S. and A.A. were supported by the Air Force Office of Scientific Research with grant no. FA9550-13-1-0204, the Office of Naval Research with grant no. N00014-15-1-2685, the National Science Foundation and the Simons Foundation.

Author information

Authors and Affiliations



C.C., D.S. and A.A. developed the concepts. C.C. performed the experiments and the numerical simulations. C.C. and D.S. carried out the theoretical analysis. C.C., D.S. and A.A. wrote the paper.

Corresponding author

Correspondence to Corentin Coulais.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Additional information

Reviewer Information Nature thanks S. Huber and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Extended data figures and tables

Extended Data Figure 1 Pictures of the mechanical metamaterials in their confining frames.

a, b, Fishbone (a) and topological (b) mechanical metamaterials, both with an asymmetry angle θ = π/16.

Related audio

Supplementary information

Supplementary information

This file contains Supplementary Text and Data, Supplementary Figures 1-5 and additional references. (PDF 1129 kb)

Snapshots and image difference of the fishbone metamaterial when actuated at point A (B) from the left (right) hand side with a force F0 (–F0) displayed on the top (bottom).

The video corresponds to the data shown in Figure 1 of the main text. (MP4 25635 kb)

Snapshots and image difference of the topological metamaterial when actuated at point A (B) from the left (right) hand side with a force F0 (–F0) displayed on the top (bottom).

The video corresponds to the data shown in Figure 3 of the main text. (MP4 27446 kb)

PowerPoint slides

Source data

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Coulais, C., Sounas, D. & Alù, A. Static non-reciprocity in mechanical metamaterials. Nature 542, 461–464 (2017).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

Further reading


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.


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing