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.
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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.
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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
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.
Notes
This article is a PNAS Direct Submission.
Authors
Affiliations
Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211
Xianchen Xu1
Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211
Honghua Qian1
Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211
Shaoyun Wang
Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211
Department of Mechanics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China
Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211
Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211
Department of Mechanical and Aerospace Engineering, The Hong Kong University of Science and Technology, Hong Kong, China
Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211
Notes
1
Q.W., X.X., and H.Q. contributed equally to this work.
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