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.
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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.
The authors declare no competing financial interests.
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
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)
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Coulais, C., Sounas, D. & Alù, A. Static non-reciprocity in mechanical metamaterials. Nature 542, 461–464 (2017). https://doi.org/10.1038/nature21044
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