Active metamaterials for realizing odd mass density

Edited by John Rogers, Northwestern University, Evanston, IL; received June 9, 2022; accepted March 29, 2023

May 18, 2023

120 (21) e2209829120

Significance

The conventional mechanical metamaterials with inner resonators are characterized as homogenized solids with symmetric effective mass density tensors to interpret subwavelength wave attenuation mechanism. In this work, we present a class of active metamaterials described by an odd mass density tensor which is no longer symmetric and whose nonzero asymmetric part arises from active and nonconservative forces. The unconventional wave phenomena caused by the odd mass density are demonstrated experimentally and numerically. The directional wave amplification is also illustrated by controllable feed-forward electric circuits. This finding may contribute to a wave manipulation strategy in nondestructive structural health monitoring, sensing, and vibration suppression and control.

Abstract

Solids built out of active components have exhibited odd elastic stiffness tensors whose active moduli appear in the antisymmetric part and which give rise to non-Hermitian static and dynamic phenomena. Here, we present a class of active metamaterial featured with an odd mass density tensor whose asymmetric part arises from active and nonconservative forces. The odd mass density is realized using metamaterials with inner resonators connected by asymmetric and programmable feed-forward control on acceleration and active forces along the two perpendicular directions. The active forces produce unbalanced off-diagonal mass density coupling terms, leading to non-Hermiticity. The odd mass is then experimentally validated through a one-dimensional nonsymmetric wave coupling where propagating transverse waves are coupled with longitudinal ones whereas the reverse is forbidden. We reveal that the two-dimensional active metamaterials with the odd mass can perform in either energy-unbroken or energy-broken phases separated by exceptional points along principal directions of the mass density. The odd mass density contributes to the wave anisotropy in the energy-unbroken phase and directional wave energy gain in the energy-broken phase. We also numerically illustrate and experimentally demonstrate the two-dimensional wave propagation phenomena that arise from the odd mass in active solids. Finally, the existence of non-Hermitian skin effect is discussed in which boundaries host an extensive number of localized modes. It is our hope that the emergent concept of the odd mass can open up a new research platform for mechanical non-Hermitian system and pave the ways for developing next-generation wave steering devices.

Get full access to this article

Purchase, subscribe or recommend this article to your librarian.

Data, Materials, and Software Availability

Acknowledgments

This work is supported by the Air Force Office of Scientific Research under Grant No. AF 9550-18-1-0342 and AF 9550-20-0279 with Program Manager Dr. Byung-Lip (Les) Lee and the Army Research Office under Grant No. W911NF-18-1-0031 with Program Manager Dr. Daniel P. Cole.

Author contributions

Y.C. and G.H. designed research; Q.W., X.X., H.Q., S.W., Y.C., and G.H. performed research; Q.W., Y.C., and G.H. analyzed data; G.H. supervised the research; and Q.W., R.Z., Z.Y., H.M., Y.C., and G.H. wrote the paper.

Competing interests

The authors declare no competing interest.

Supporting Information

Movie S1.

Animation with control for transverse incidence.

Movie S2.

Animation without control for transverse incidence.

Movie S3.

Time-dependent 2D simulations.

Movie S4.

Time-dependent 2D measurement.

References

Z. Liu et al., Locally resonant sonic materials. Science 289, 1734–1736 (2000).

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

K. Bertoldi, V. Vitelli, J. Christensen, M. Van Hecke, Flexible mechanical metamaterials. Nat. Rev. Mater. 2, 1–11 (2017).

H. Huang, C. Sun, G. Huang, On the negative effective mass density in acoustic metamaterials. Int. J. Eng. Sci. 47, 610–617 (2009).

H. Yasuda, J. Yang, Reentrant origami-based metamaterials with negative Poisson’s ratio and bistability. Phys. Rev. Lett. 114, 185502 (2015).

K. T. Tan, H. Huang, C. Sun, Optimizing the band gap of effective mass negativity in acoustic metamaterials. Appl. Phys. Lett. 101, 241902 (2012).

H. Huang, C. Sun, Wave attenuation mechanism in an acoustic metamaterial with negative effective mass density. New J. Phys. 11, 013003 (2009).

M. Kadic, T. Bückmann, R. Schittny, P. Gumbsch, M. Wegener, Pentamode metamaterials with independently tailored bulk modulus and mass density. Phys. Rev. Appl. 2, 054007 (2014).

H. Zhu, S. Patnaik, T. F. Walsh, B. H. Jared, F. Semperlotti, Nonlocal elastic metasurfaces: Enabling broadband wave control via intentional nonlocality. Proc. Natl. Acad. Sci. U.S.A. 117, 26099–26108 (2020).

Z. Wang, Q. Zhang, K. Zhang, G. Hu, Tunable digital metamaterial for broadband vibration isolation at low frequency. Adv. Mater. 28, 9857–9861 (2016).

H. Nassar, Y. Chen, G. Huang, Polar metamaterials: A new outlook on resonance for cloaking applications. Phys. Rev. Lett. 124, 084301 (2020).

X. Xu et al., Physical realization of elastic cloaking with a polar material. Phys. Rev. Lett. 124, 114301 (2020).

T. Bückmann, M. Thiel, M. Kadic, R. Schittny, M. Wegener, An elasto-mechanical unfeelability cloak made of pentamode metamaterials. Nat. Commun. 5, 1–6 (2014).

T. Bückmann, M. Kadic, R. Schittny, M. Wegener, Mechanical cloak design by direct lattice transformation. Proc. Natl. Acad. Sci. U.S.A. 112, 4930–4934 (2015).

J. M. Kweun, H. J. Lee, J. H. Oh, H. M. Seung, Y. Y. Kim, Transmodal Fabry–Pérot resonance: Theory and realization with elastic metamaterials. Phys. Rev. Lett. 118, 205901 (2017).

R. Zhu, X. Liu, G. Hu, C. Sun, G. Huang, Negative refraction of elastic waves at the deep-subwavelength scale in a single-phase metamaterial. Nat. Commun. 5, 1–8 (2014).

Y. Lai, Y. Wu, P. Sheng, Z. Q. Zhang, Hybrid elastic solids. Nat. Mater. 10, 620–624 (2011).

Y. Wu, Y. Lai, Z. Q. Zhang, Elastic metamaterials with simultaneously negative effective shear modulus and mass density. Phys. Rev. Lett. 107, 105506 (2011).

H. Yasuda et al., Origami-based impact mitigation via rarefaction solitary wave creation. Sci. Adv. 5, eaau2835 (2019).

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

H. Nassar et al., Nonreciprocity in acoustic and elastic materials. Nat. Rev. Mater. 5, 667–685 (2020).

X. Xu et al., Physical observation of a robust acoustic pumping in waveguides with dynamic boundary. Phys. Rev. Lett. 125, 253901 (2020).

Q. Wu, H. Chen, H. Nassar, G. Huang, Non-reciprocal Rayleigh wave propagation in space-time modulated surface. J. Mech. Phys. Solids 146, 104196 (2021).

M. I. Rosa, M. Ruzzene, Dynamics and topology of non-Hermitian elastic lattices with non-local feedback control interactions. New J. Phys. 22, 053004 (2020).

Y. Xia et al., Experimental observation of temporal pumping in electromechanical waveguides. Phys. Rev. Lett. 126, 095501 (2021).

L. Sirota, R. Ilan, Y. Shokef, Y. Lahini, Non-Newtonian topological mechanical metamaterials using feedback control. Phys. Rev. Lett. 125, 256802 (2020).

C. Scheibner et al., Odd elasticity. Nat. Phys. 16, 475–480 (2020).

C. Scheibner, W. T. Irvine, V. Vitelli, Non-Hermitian band topology and skin modes in active elastic media. Phys. Rev. Lett. 125, 118001 (2020).

S. Shankar, A. Souslov, M. J. Bowick, M. C. Marchetti, V. Vitelli, Topological active matter. Nat. Rev. Phys. 4, 380–398 (2022).

Y. Chen, X. Li, C. Scheibner, V. Vitelli, G. Huang, Realization of active metamaterials with odd micropolar elasticity. Nat. Commun. 12, 1–12 (2021).

M. Brandenbourger, X. Locsin, E. Lerner, C. Coulais, Non-reciprocal robotic metamaterials. Nat. Commun. 10, 1–8 (2019).

A. Ghatak, M. Brandenbourger, J. Van Wezel, C. Coulais, Observation of non-Hermitian topology and its bulk-edge correspondence in an active mechanical metamaterial. Proc. Natl. Acad. Sci. U.S.A. 117, 29561–29568 (2020).

M. Brandenbourger, C. Scheibner, J. Veenstra, V. Vitelli, C. Coulais, Limit cycles turn active matter into robots. arXiv [Preprint] (2022). https://doi.org/10.48550/arXiv.2108.08837 (Accessed 6 November 2022).

M. I. Rosa, M. Mazzotti, M. Ruzzene, Exceptional points and enhanced sensitivity in PT-symmetric continuous elastic media. J. Mech. Phys. Solids 149, 104325 (2021).

C. Shi et al., Accessing the exceptional points of parity-time symmetric acoustics. Nat. Commun. 7, 1–5 (2016).

Q. Wu, Y. Chen, G. Huang, Asymmetric scattering of flexural waves in a parity-time symmetric metamaterial beam. J. Acous. Soc. Am. 146, 850–862 (2019).

X. Zhou, G. Hu, Analytic model of elastic metamaterials with local resonances. Phys. Rev. B 79, 195109 (2009).

X. Zhou, X. Liu, G. Hu, Elastic metamaterials with local resonances: An overview. Theor. Appl. Mech. Lett. 2, 041001 (2012).

Q. Wu, G. Huang, Omnidirectional wave polarization manipulation in isotropic polar solids. Int. J. Solids Struct. 241, 111481 (2022).

X. Liu, G. Hu, C. Sun, G. Huang, Wave propagation characterization and design of two-dimensional elastic chiral metacomposite. J. Sound Vib. 330, 2536–2553 (2011).

Y. Wu, Y. Lai, Z. Q. Zhang, Effective medium theory for elastic metamaterials in two dimensions. Phys. Rev. B 76, 205313 (2007).

K. Yokomizo, S. Murakami, Non-Bloch band theory of non-Hermitian systems. Phys. Rev. Lett. 123, 066404 (2019).

F. K. Kunst, V. Dwivedi, Non-Hermitian systems and topology: A transfer-matrix perspective. Phys. Rev. B 99, 245116 (2019).

E. J. Bergholtz, J. C. Budich, F. K. Kunst, Exceptional topology of non-Hermitian systems. Rev. Mod. Phys. 93, 015005 (2021).

S. Yao, Z. Wang, Edge states and topological invariants of non-Hermitian systems. Phys. Rev. Lett. 121, 086803 (2018).

Y. Ashida, Z. Gong, M. Ueda, Non-Hermitian physics. Adv. Phys. 69, 249–435 (2020).

Q. Wu, X. Zhang, P. Shivashankar, Y. Chen, G. Huang, Independent flexural wave frequency conversion by a linear active metalayer. Phys. Rev. Lett. 128, 244301 (2022).

Information & Authors

Information

Published in

Proceedings of the National Academy of Sciences

Vol. 120 | No. 21
May 23, 2023

Classifications

Copyright

Data, Materials, and Software Availability

Submission history

Received: June 9, 2022

Accepted: March 29, 2023

Published online: May 18, 2023

Published in issue: May 23, 2023

Keywords

odd mass density
elastic metamaterial
non-Hermitian mechanical system
energy phase transition
non-Hermitian skin effect

Acknowledgments

Author Contributions

Competing Interests

The authors declare no competing interest.