Nonreciprocity in acoustic and elastic materials

Bacot, V., Labousse, M., Eddi, A., Fink, M. & Fort, E. Time reversal and holography with spacetime transformations. Nat. Phys. 12, 972–977 (2016).

Fink, M. et al. Time-reversed acoustics. Rep. Prog. Phys. 63, 1933–1995 (2000).

Colton, D. & Haddar, H. An application of the reciprocity gap functional to inverse scattering theory. Inverse Probl. 21, 383–398 (2005).

Bonnet, M. Boundary Integral Equation Methods for Solids and Fluids (Wiley, 1995).

Bonnet, M. & Constantinescu, A. Inverse problems in elasticity. Inverse Probl. 21, R1–R50 (2005).

Andrieux, S., Ben Abda, A. & Bui, H. D. Reciprocity principle and crack identification. Inverse Probl. 15, 59–65 (1999).

MacLean, W. R. Absolute measurement of sound without a primary standard. J. Acoust. Soc. Am. 12, 140–146 (1940).

Ten Wolde, T. Reciprocity measurements in acoustical and mechano-acoustical systems. Review of theory and applications. Acta Acust. United Acust. 96, 1–13 (2010).

Fahy, F. Some applications of the reciprocity principle in experimental vibroacoustics. Acoust. Phys. 49, 217–229 (2003).

de Hoop, A. T. Time-domain reciprocity theorems for acoustic wave fields in fluids with relaxation. J. Acoust. Soc. Am. 84, 1877–1882 (1988).

Howe, M. S. Acoustics of Fluid-Structure Interactions, Ch. 1 (Cambridge Univ. Press, 2010).

Maznev, A. A., Every, A. G. & Wright, O. B. Reciprocity in reflection and transmission: What is a ‘phonon diode’? Wave Motion 50, 776–784 (2013).

Fleury, R., Sounas, D., Haberman, M. R. & Alù, A. Nonreciprocal acoustics. Acoust. Today 11, 14–21 (2015).

Fleury, R., Haberman, M. R., Huang, G. & Norris, A. N. Introduction to the special issue on non-reciprocal and topological wave phenomena in acoustics. J. Acoust. Soc. Am. 146, 719–720 (2019).

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

Kittel, C. Interaction of spin waves and ultrasonic waves in ferromagnetic crystals. Phys. Rev. 110, 836–841 (1958).

Kariyado, T. & Hatsugai, Y. Manipulation of Dirac cones in mechanical graphene. Sci. Rep. 5, 18107 (2015).

Zangeneh-Nejad, F. & Fleury, R. Doppler-based acoustic gyrator. Appl. Sci. 8, 1083 (2018).

Wiederhold, C. P., Sounas, D. L. & Alù, A. Nonreciprocal acoustic propagation and leaky-wave radiation in a waveguide with flow. J. Acoust. Soc. Am. 146, 802–809 (2019).

Aaboud, M. et al. Probing the quantum interference between singly and doubly resonant top-quark production in pp collisions at $$\sqrt{s}=13{\rm{TeV}}$$ with the ATLAS detector. Phys. Rev. Lett. 121, 152002 (2018).

Guo, X., Lissek, H. & Fleury, R. Improving sound absorption through nonlinear active electroacoustic resonators. Phys. Rev. Appl. 13, 014018 (2020).

Popa, B.-I. & Cummer, S. A. Non-reciprocal and highly nonlinear active acoustic metamaterials. Nat. Commun. 5, 3398 (2014).

Rivet, E. et al. Constant-pressure sound waves in non-Hermitian disordered media. Nat. Phys. 14, 942–947 (2018).

Koutserimpas, T. T., Rivet, E., Lissek, H. & Fleury, R. Active acoustic resonators with reconfigurable resonance frequency, absorption, and bandwidth. Phys. Rev. Appl. 12, 054064 (2019).

Slater, J. C. Interaction of waves in crystals. Rev. Mod. Phys. 30, 197–222 (1958).

Wang, D.-W. et al. Optical diode made from a moving photonic crystal. Phys. Rev. Lett. 110, 093901 (2013).

Swinteck, N. et al. Bulk elastic waves with unidirectional backscattering-immune topological states in a time-dependent superlattice. J. Appl. Phys. 118, 063103 (2015).

Trainiti, G. & Ruzzene, M. Non-reciprocal elastic wave propagation in spatiotemporal periodic structures. New J. Phys. 18, 083047 (2016).

Hu, X., Hang, Z., Li, J., Zi, J. & Chan, C. Anomalous Doppler effects in phononic band gaps. Phys. Rev. E 73, 015602 (2006).

Wang, Y. et al. Observation of non-reciprocal wave propagation in a dynamic phononic lattice. Phys. Rev. Lett. 121, 194301 (2018).

Chen, Y. et al. Nonreciprocal wave propagation in a continuum-based metamaterial with space-time modulated resonators. Phys. Rev. Appl. 11, 064052 (2019).

Nassar, H., Chen, H., Norris, A. N., Haberman, M. R. & Huang, G. L. Non-reciprocal wave propagation in modulated elastic metamaterials. Proc. R. Soc. A 473, 20170188 (2017).

Nassar, H., Chen, H., Norris, A. N. & Huang, G. L. Non-reciprocal flexural wave propagation in a modulated metabeam. Extreme Mech. Lett. 15, 97–102 (2017).

Yu, Z. & Fan, S. Complete optical isolation created by indirect interband photonic transitions. Nat. Photonics 3, 91–94 (2009).

Zanjani, M. B., Davoyan, A. R., Mahmoud, A. M., Engheta, N. & Lukes, J. R. One-way phonon isolation in acoustic waveguides. Appl. Phys. Lett. 104, 081905 (2014).

Zanjani, M. B., Davoyan, A. R., Engheta, N. & Lukes, J. R. NEMS with broken T symmetry: Graphene based unidirectional acoustic transmission lines. Sci. Rep. 5, 9926 (2015).

Chaunsali, R., Li, F. & Yang, J. Stress wave isolation by purely mechanical topological phononic crystals. Sci. Rep. 6, 30662 (2016).

Croënne, C., Vasseur, J. O., Matar, O. B., Hladky-Hennion, A.-C. & Dubus, B. Non-reciprocal behavior of one-dimensional piezoelectric structures with space-time modulated electrical boundary conditions. J. Appl. Phys. 126, 145108 (2019).

Marconi, J. et al. Experimental observation of nonreciprocal band gaps in a space-time-modulated beam using a shunted piezoelectric array. Phys. Rev. Appl. 13, 031001 (2020).

Merkel, A., Willatzen, M. & Christensen, J. Dynamic nonreciprocity in loss-compensated piezophononic media. Phys. Rev. Appl. 9, 034033 (2018).

Huang, J. & Zhou, X. A time-varying mass metamaterial for non-reciprocal wave propagation. Int. J. Solids Struct. 164, 25–36 (2019).

Attarzadeh, M. A., Callanan, J. & Nouh, M. Experimental observation of nonreciprocal waves in a resonant metamaterial beam. Phys. Rev. Appl. 13, 021001 (2020).

Torrent, D., Poncelet, O. & Batsale, J.-C. Nonreciprocal thermal material by spatiotemporal modulation. Phys. Rev. Lett. 120, 125501 (2018).

Vila, J., Pal, R. K., Ruzzene, M. & Trainiti, G. A Bloch-based procedure for dispersion analysis of lattices with periodic time-varying properties. J. Sound. Vib. 406, 363–377 (2017).

Wallen, S. P. & Haberman, M. R. Nonreciprocal wave phenomena in spring-mass chains with effective stiffness modulation induced by geometric nonlinearity. Phys. Rev. E 99, 013001 (2019).

Goldsberry, B. M., Wallen, S. P. & Haberman, M. R. Non-reciprocal wave propagation in mechanically-modulated continuous elastic metamaterials. J. Acoust. Soc. Am. 146, 782–788 (2019).

Nassar, H., Xu, X. C., Norris, A. N. & Huang, G. L. Modulated phononic crystals: Non-reciprocal wave propagation and Willis materials. J. Mech. Phys. Solids 101, 10–29 (2017).

Nassar, H., Chen, H., Norris, A. & Huang, G. Quantization of band tilting in modulated phononic crystals. Phys. Rev. B 97, 014305 (2018).

Attarzadeh, M. A. & Nouh, M. Elastic wave propagation in moving phononic crystals and correlations with stationary spatiotemporally modulated systems. AIP Adv. 8, 105302 (2018).

Lurie, K. A. Low frequency longitudinal vibrations of an elastic bar made of a dynamic material and excited at one end. J. Math. Anal. Appl. 251, 364–375 (2000).

Lurie, K. A. An Introduction to the Mathematical Theory of Dynamic Materials (Springer, 2007).

Willis, J. R. Variational principles for dynamic problems for inhomogeneous elastic media. Wave Motion 3, 1–11 (1981).

Willis, J. R. in Continuum Micromechanics (ed. Suquet, P.) 265–290 (Springer, 1997).

Lurie, K. A. Effective properties of smart elastic laminates and the screening phenomenon. Int. J. Solids Struct. 34, 1633–1643 (1997).

Quan, L., Sounas, D. L. & Alù, A. Nonreciprocal Willis coupling in zero-index moving media. Phys. Rev. Lett. 123, 064301 (2019).

Cheng, D. K. & Kong, J.-A. Covariant descriptions of bianisotropic media. Proc. IEEE 56, 248–251 (1968).

Kong, J.-A. Theorems of bianisotropic media. Proc. IEEE 60, 1036–1046 (1972).

Sieck, C. F., Alù, A. & Haberman, M. R. Origins of Willis coupling and acoustic bianisotropy in acoustic metamaterials through source-driven homogenization. Phys. Rev. B 96, 104303 (2017).

Muhlestein, M. B., Sieck, C. F., Wilson, P. S. & Haberman, M. R. Experimental evidence of Willis coupling in a one-dimensional effective material element. Nat. Commun. 8, 15625 (2017).

Cullen, A. A travelling-wave parametric amplifier. Nature 181, 332 (1958).

Hayrapetyan, A., Grigoryan, K., Petrosyan, R. & Fritzsche, S. Propagation of sound waves through a spatially homogeneous but smoothly time-dependent medium. Ann. Phys. 333, 47–65 (2013).

Lurie, K. A. & Weekes, S. L. Wave propagation and energy exchange in a spatio-temporal material composite with rectangular microstructure. J. Math. Anal. Appl. 314, 286–310 (2006).

Milton, G. W. & Mattei, O. Field patterns: a new mathematical object. Proc. R. Soc. A 473, 20160819 (2017).

Torrent, D., Parnell, W. J. & Norris, A. N. Loss compensation in time-dependent elastic metamaterials. Phys. Rev. B 97, 014105 (2018).

Berry, M. V. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. A 392, 45–57 (1984).

Cheng, C. M. & Fung, P. C. W. The evolution operator technique in solving the Schrodinger equation, and its application to disentangling exponential operators and solving the problem of a mass-varying harmonic oscillator. J. Phys. A 21, 4115 (1988).

Chen, H., Yao, L. Y., Nassar, H. & Huang, G. L. Mechanical quantum Hall effect in time-modulated elastic materials. Phys. Rev. Appl. 11, 044029 (2019).

Hatsugai, Y. & Fukui, T. Bulk-edge correspondence in topological pumping. Phys. Rev. B 94, 041102 (2016).

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

Khanikaev, A. B. & Shvets, G. Two-dimensional topological photonics. Nat. Photonics 11, 763–773 (2017).

Lu, L., Joannopoulos, J. & Soljacic, M. Topological photonics. Nat. Photonics 8, 821–829 (2014).

Ma, G., Xiao, M. & Chan, C. T. Topological phases in acoustic and mechanical systems. Nat. Rev. Phys. 1, 281–294 (2019).

Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett. 61, 2015 (1988).

Zhang, Y., Tan, Y., Stormer, H. & Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438, 201–204 (2005).

Kane, C. L. & Mele, E. J. Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005).

Kane, C. L. & Mele, E. J. Z2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005).

Bernevig, B., Hughes, T. & Zhang, S. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006).

Haldane, F. D. M. & Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett. 100, 013904 (2008).

Brendel, C., Peano, V., Painter, O. & Marquardt, F. Snowflake phononic topological insulator at the nanoscale. Phys. Rev. B 97, 020102 (2018).

Fleury, R., Khanikaev, A. B. & Alù, A. Floquet topological insulators for sound. Nat. Commun. 7, 11744 (2016).

Deng, Y., Lu, M. & Jing, Y. A comparison study between acoustic topological states based on valley Hall and quantum spin Hall effects. J. Acoust. Soc. Am. 146, 721–728 (2019).

Khanikaev, A. B. et al. Photonic topological insulators. Nat. Mater. 12, 233–239 (2013).

Mousavi, S., Khanikaev, A. & Wang, Z. Topologically protected elastic waves in phononic metamaterials. Nat. Commun. 6, 8682 (2015).

Pal, R., Schaeffer, M. & Ruzzene, M. Helical edge states and topological phase transitions in phononic systems using bi-layered lattices. J. Appl. Phys. 119, 084305 (2016).

Huo, S.-y, Chen, J.-j, Feng, L.-y & Huang, H.-b Pseudospins and topological edge states for fundamental antisymmetric Lamb modes in snowflakelike phononic crystal slabs. J. Acoust. Soc. Am. 146, 729–735 (2019).

Darabi, A. & Leamy, M. J. Reconfigurable topological insulator for elastic waves. J. Acoust. Soc. Am. 146, 773–781 (2019).

Cha, J., Kim, K. W. & Daraio, C. Experimental realization of on-chip topological nanoelectromechanical metamaterials. Nature 564, 229–233 (2018).

Peano, V., Brendel, C., Schmidt, M. & Marquardt, F. Topological phases of sound and light. Phys. Rev. X 5, 031011 (2015).

Prodan, E. & Prodan, C. Topological phonon modes and their role in dynamic instability of microtubules. Phys. Rev. Lett. 103, 248101 (2009).

Nash, L. M., Kleckner, D., Vitelli, V., Turner, A. M. & Irvine, W. T. Topological mechanics of gyroscopic metamaterials. Proc. Natl Acad. Sci. USA 112, 14495–14500 (2015).

Zhou, X. M. & Zhao, Y. C. Unusual one-way edge state in acoustic gyroscopic continuum. Sci. China Phys. Mech. Astron. 62, 14612 (2019).

Khanikaev, A. B., Fleury, R., Mousavi, S. H. & Alù, A. Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice. Nat. Commun. 6, 8260 (2015).

Souslov, A., van Zuiden, B. C., Bartolo, D. & Vitelli, V. Topological sound in active-liquid metamaterials. Nat. Phys. 13, 1091–1094 (2017).

Souslov, A., Dasbiswas, K., Fruchart, M., Vaikuntanathan, S. & Vitelli, V. Topological waves in fluids with odd viscosity. Phys. Rev. Lett. 122, 128001 (2019).

Ding, Y. et al. Experimental demonstration of acoustic Chern insulators. Phys. Rev. Lett. 122, 014302 (2019).

Yang, Z. et al. Topological acoustics. Phys. Rev. Lett. 114, 114301 (2015).

Yves, S., Fleury, R., Lemoult, F., Fink, M. & Lerosey, G. Topological acoustic polaritons: robust sound manipulation at the subwavelength scale. New J. Phys. 19, 075003 (2017).

Zhang, L., Ren, J., Wang, J. & Li, B. Topological nature of the phonon Hall effect. Phys. Rev. Lett. 105, 225901 (2010).

Wang, P., Lu, L. & Bertoldi, K. Topological phononic crystals with one-way elastic edge waves. Phys. Rev. Lett. 115, 104302 (2015).

Salerno, G., Ozawa, T., Price, H. & Carusotto, I. Floquet topological system based on frequency-modulated classical coupled harmonic oscillators. Phys. Rev. B 93, 085105 (2016).

He, C. et al. Topological phononic states of underwater sound based on coupled ring resonators. Appl. Phys. Lett. 108, 031904 (2016).

Chen, H., Nassar, H., Norris, A. N., Hu, G. K. & Huang, G. L. Elastic quantum spin Hall effect in kagome lattices. Phys. Rev. B 98, 094302 (2018).

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

Ningyuan, J., Owens, C., Sommer, A., Schuster, D. & Simon, J. Time-and site-resolved dynamics in a topological circuit. Phys. Rev. X 5, 021031 (2015).

Foehr, A., Bilal, O. R., Huber, S. D. & Daraio, C. Spiral-based phononic plates: From wave beaming to topological insulators. Phys. Rev. Lett. 120, 205501 (2018).

Miniaci, M., Pal, R., Morvan, B. & Ruzzene, M. Experimental observation of topologically protected helical edge modes in patterned elastic plates. Phys. Rev. X 8, 031074 (2018).

He, C. et al. Acoustic topological insulator and robust one-way sound transport. Nat. Phys. 12, 1124–1129 (2016).

Ni, X., Weiner, M., Alù, A. & Khanikaev, A. B. Observation of higher-order topological acoustic states protected by generalized chiral symmetry. Nat. Mater. 18, 113–120 (2019).

Rycerz, A., Tworzydło, J. & Beenakker, C. Valley filter and valley valve in graphene. Nat. Phys. 3, 172–175 (2007).

Xiao, D., Yao, W. & Niu, Q. Valley-contrasting physics in graphene: magnetic moment and topological transport. Phys. Rev. Lett. 99, 236809 (2007).

Ma, T. & Shvets, G. All-Si valley-Hall photonic topological insulator. New J. Phys. 18, 025012 (2016).

Chen, X.-D., Zhao, F.-L., Chen, M. & Dong, J.-W. Valley-contrasting physics in all-dielectric photonic crystals: Orbital angular momentum and topological propagation. Phys. Rev. B 96, 020202 (2017).

Xiao, M. et al. Geometric phase and band inversion in periodic acoustic systems. Nat. Phys. 11, 240–244 (2015).

Lu, J. et al. Observation of topological valley transport of sound in sonic crystals. Nat. Phys. 13, 369–374 (2017).

Ye, L. et al. Observation of acoustic valley vortex states and valley-chirality locked beam splitting. Phys. Rev. B 95, 174106 (2017).

Lu, J., Qiu, C., Ke, M. & Liu, Z. Valley vortex states in sonic crystals. Phys. Rev. Lett. 116, 093901 (2016).

Pal, R. K. & Ruzzene, M. Edge waves in plates with resonators: an elastic analogue of the quantum valley Hall effect. New J. Phys. 19, 025001 (2017).

Vila, J., Pal, R. K. & Ruzzene, M. Observation of topological valley modes in an elastic hexagonal lattice. Phys. Rev. B 96, 134307 (2017).

Liu, T.-W. & Semperlotti, F. Experimental evidence of robust acoustic valley Hall edge states in a nonresonant topological elastic waveguide. Phys. Rev. Appl. 11, 014040 (2019).

Chaunsali, R., Thakkar, A., Kim, E., Kevrekidis, P. & Yang, J. Demonstrating an in-situ topological band transition in cylindrical granular chains. Phys. Rev. Lett. 119, 024301 (2017).

Chen, H., Nassar, H. & Huang, G. L. A study of topological effects in 1D and 2D mechanical lattices. J. Mech. Phys. Solids 117, 22–36 (2018).

Makwana, M. P. & Craster, R. V. Geometrically navigating topological plate modes around gentle and sharp bends. Phys. Rev. B 98, 22184105 (2018).

Lepri, S. & Casati, G. Asymmetric wave propagation in nonlinear systems. Phys. Rev. Lett. 106, 164101 (2011).

Lepri, S. & Pikovsky, A. Nonreciprocal wave scattering on nonlinear string-coupled oscillators. Chaos 24, 043119 (2014).

Cui, J.-G., Yang, T. & Chen, L.-Q. Frequency-preserved non-reciprocal acoustic propagation in a granular chain. Appl. Phys. Lett. 112, 181904 (2018).

Yousefzadeh, B., Ramirez, B. & Daraio, C. Non-reciprocal dynamic response of a bilinear lattice in APS Meeting Abstracts (APS, 2019).

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

Wallen, S. P. et al. Static and dynamic non-reciprocity in bi-linear structures in Proc. Meetings on Acoustics 21ISNA Vol. 34 065002 (ASA, 2018).

Deng, B., Wang, P., He, Q., Tournat, V. & Bertoldi, K. Metamaterials with amplitude gaps for elastic solitons. Nat. Commun. 9, 3410 (2018).

Devaux, T., Cebrecos, A., Richoux, O., Pagneux, V. & Tournat, V. Acoustic radiation pressure for nonreciprocal transmission and switch effects. Nat. Commun. 10, 3292 (2019).

Shi, Y., Yu, Z. & Fan, S. Limitations of nonlinear optical isolators due to dynamic reciprocity. Nat. Photonics 9, 388–392 (2015).

Sounas, D. L., Soric, J. & Alù, A. Broadband passive isolators based on coupled nonlinear resonances. Nat. Electron. 1, 113–119 (2018).

Sounas, D. L. & Alù, A. Nonreciprocity based on nonlinear resonances. IEEE Antennas Wirel. Propag. Lett. 17, 1958–1962 (2018).

Nesterenko, V. F. Propagation of nonlinear compression pulses in granular media. J. Appl. Mech. Tech. Phys. 24, 136–148 (1983).

Cebrecos, A. et al. Asymmetric propagation using enhanced self-demodulation in a chirped phononic crystal. AIP Adv. 6, 121601 (2016).

Harbola, U., Rosas, A., Esposito, M. & Lindenberg, K. Pulse propagation in tapered granular chains: An analytic study. Phys. Rev. E 80, 031303 (2009).

Mojahed, A., Bunyan, J., Tawfick, S. & Vakakis, A. F. Tunable acoustic nonreciprocity in strongly nonlinear waveguides with asymmetry. Phys. Rev. Appl. 12, 034033 (2019).

Moore, K. J. et al. Nonreciprocity in the dynamics of coupled oscillators with nonlinearity, asymmetry, and scale hierarchy. Phys. Rev. E 97, 012219 (2018).

Kosevich, Y. A. Fluctuation subharmonic and multiharmonic phonon transmission and kapitza conductance between crystals with very different vibrational spectra. Phys. Rev. B 52, 1017 (1995).

Nesterenko, V., Daraio, C., Herbold, E. & Jin, S. Anomalous wave reflection at the interface of two strongly nonlinear granular media. Phys. Rev. Lett. 95, 158702 (2005).

Liang, B., Guo, X. S., Tu, J., Zhang, D. & Cheng, J. C. An acoustic rectifier. Nat. Mater. 9, 989–992 (2010).

D’Ambroise, J., Kevrekidis, P. G. & Lepri, S. Asymmetric wave propagation through nonlinear PT-symmetric oligomers. J. Phys. A 45, 444012 (2012).

Merkel, A., Tournat, V. & Gusev, V. Directional asymmetry of the nonlinear wave phenomena in a three-dimensional granular phononic crystal under gravity. Phys. Rev. E 90, 023206 (2014).

Devaux, T., Tournat, V., Richoux, O. & Pagneux, V. Asymmetric acoustic propagation of wave packets via the self-demodulation effect. Phys. Rev. Lett. 115, 234301 (2015).

Li, K. & Rizzo, P. Nonreciprocal propagation of solitary waves in granular chains with asymmetric potential barriers. J. Sound. Vib. 365, 15–21 (2016).

Darabi, A. et al. Broadband passive nonlinear acoustic diode. Phys. Rev. B 99, 214305 (2019).

Mojahed, A., Gendelman, O. V. & Vakakis, A. F. Breather arrest, localization, and acoustic non-reciprocity in dissipative nonlinear lattices. J. Acoust. Soc. Am. 146, 826–842 (2019).

Gliozzi, A. S. et al. Proof of concept of a frequency-preserving and time-invariant metamaterial-based nonlinear acoustic diode. Sci. Rep. 9, 9560 (2019).

Grinberg, I., Vakakis, A. F. & Gendelman, O. V. Acoustic diode: Wave non-reciprocity in nonlinearly coupled waveguides. Wave Motion 83, 49–66 (2018).

Bunyan, J. & Tawfick, S. Exploiting structural instability to design architected materials having essentially nonlinear stiffness. Adv. Eng. Mater. 21, 1800791 (2019).

Fu, C., Wang, B., Zhao, T. & Chen, C. Q. High efficiency and broadband acoustic diodes. Appl. Phys. Lett. 112, 051902 (2018).

Blanchard, A., Sapsis, T. P. & Vakakis, A. F. Non-reciprocity in nonlinear elastodynamics. J. Sound. Vib. 412, 326–335 (2018).

Liang, B., Yuan, B. & Cheng, J.-C. Acoustic diode: Rectification of acoustic energy flux in one-dimensional systems. Phys. Rev. Lett. 103, 104301 (2009).

Kuznetsov, Y. A. Elements of Applied Bifurcation Theory Vol. 112 (Springer, 2013).

Di Bernardo, M. et al. Bifurcations in nonsmooth dynamical systems. SIAM Rev. 50, 629–701 (2008).

Nadkarni, N., Arrieta, A. F., Chong, C., Kochmann, D. M. & Daraio, C. Unidirectional transition waves in bistable lattices. Phys. Rev. Lett. 116, 244501 (2016).

Raney, J. R. et al. Stable propagation of mechanical signals in soft media using stored elastic energy. Proc. Natl Acad. Sci. USA 113, 9722–9727 (2016).

Geniet, F. & Leon, J. Energy transmission in the forbidden band gap of a nonlinear chain. Phys. Rev. Lett. 89, 134102 (2002).

Maniadis, P., Kopidakis, G. & Aubry, S. Energy dissipation threshold and self-induced transparency in systems with discrete breathers. Physica D 216, 121–135 (2006).

Yousefzadeh, B. & Phani, A. S. Energy transmission in finite dissipative nonlinear periodic structures from excitation within a stop band. J. Sound. Vib. 354, 180–195 (2015).

Boechler, N., Theocharis, G. & Daraio, C. Bifurcation-based acoustic switching and rectification. Nat. Mater. 10, 665–668 (2011).

Yousefzadeh, B. & Phani, A. S. Supratransmission in a disordered nonlinear periodic structure. J. Sound. Vib. 380, 242–266 (2016).

Lu, Z. & Norris, A. N. Non-reciprocal wave transmission in a bilinear spring-mass system. J. Vib. Acoust. 142, 021006 (2020).

Yousefzadeh, B., Wang, Y. & Daraio, C. Asymmetric wave propagation in a modulated magnetic lattice in APS Meeting Abstracts (APS, 2018).

Dobrykh, D., Yulin, A., Slobozhanyuk, A., Poddubny, A. & Kivshar, Y. S. Nonlinear control of electromagnetic topological edge states. Phys. Rev. Lett. 121, 163901 (2018).

Pal, R. K., Vila, J., Leamy, M. & Ruzzene, M. Amplitude-dependent topological edge states in nonlinear phononic lattices. Phys. Rev. E 97, 032209 (2018).

Konarski, S. G., Haberman, M. R. & Hamilton, M. F. Frequency-dependent behavior of media containing pre-strained nonlinear inclusions: Application to nonlinear acoustic metamaterials. J. Acoust. Soc. Am. 144, 3022–3035 (2018).

Trainiti, G. et al. Time-periodic stiffness modulation in elastic metamaterials for selective wave filtering: Theory and experiment. Phys. Rev. Lett. 122, 124301 (2019).

Shen, C., Zhu, X., Li, J. & Cummer, S. A. Nonreciprocal acoustic transmission in space-time modulated coupled resonators. Phys. Rev. B 100, 054302 (2019).

Bartolo, D. & Carpentier, D. Topological elasticity of nonorientable ribbons. Phys. Rev. X 9, 041058 (2019).

Sounas, D. L. & Alù, A. Fundamental bounds on the operation of Fano nonlinear isolators. Phys. Rev. B 97, 115431 (2018).

Li, Z.-N., Wang, Y.-Z. & Wang, Y.-S. Tunable nonreciprocal transmission in nonlinear elastic wave metamaterial by initial stresses. Int. J. Solids Struct. 182–183, 218–235 (2020).

Kac, M. Can one hear the shape of a drum? Am. Math. Mon. 73, 1–23 (1966).

Kochmann, D. M. & Bertoldi, K. Exploiting microstructural instabilities in solids and structures: From metamaterials to structural transitions. Appl. Mech. Rev. 69, 0505801 (2017).

Lagrange, J.-L. Du Mouvement des Fluides Compressibles et Élastiques Vol. 2 of Cambridge Library Collection - Mathematics, 337–354 (Cambridge Univ. Press, 2009).

Lamb, H. On reciprocal theorems in dynamics. Proc. Lond. Math. Soc. 1, 144–151 (1887).

von Helmholtz, H. L. F. Theorie der Luftschwingungen in Röhren mit offenen Enden. Crelle’s J. 57, 1–72 (1860).

Clebsch, A. Théorie de l’Élasticité des Corps Solides (Dunod, 1883).

Maxwell, J. C. L. On the calculation of the equilibrium and stiffness of frames. Philos. Mag. 27, 294–299 (1864).

Betti, E. Teoria della elasticita. Il Nuovo Cimento (1869-1876) 7, 158–180 (1872).

Rayleigh, L. Some general theorems relating to vibrations. Proc. Lond. Math. Soc. s1-4, 357–368 (1873).

de Hoop, A. Handbook of Radiation and Scattering of Waves: Acoustic Waves in Fluids, Elastic Waves in Solids, Electromagnetic Waves Ch. 7 (Academic, 1995).

Achenbach, J. D. Reciprocity in Elastodynamics (Cambridge Univ. Press, 2003).

Payton, R. G. An application of the dynamic Betti-Rayleigh reciprocal theorem to moving-point loads in elastic media. Q. Appl. Math. 21, 299–313 (1964).

Muhlestein, M. B., Sieck, C. F., Alù, A. & Haberman, M. R. Reciprocity, passivity and causality in Willis materials. Proc. R. Soc. A 472, 20160604 (2016).

Godin, O. A. Reciprocity and energy theorems for waves in a compressible inhomogeneous moving fluid. Wave Motion 25, 143–167 (1997).

Onsager, L. Reciprocal relations in irreversible processes. I. Phys. Rev. 37, 405–426 (1931).

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

Banerjee, D., Souslov, A., Abanov, A. G. & Vitelli, V. Odd viscosity in chiral active fluids. Nat. Commun. 8, 1573 (2017).

Faust, D. & Lakes, R. S. Reciprocity failure in piezoelectric polymer composite. Phys. Scr. 90, 085807 (2015).

Zhai, Y., Kwon, H.-S. & Popa, B.-I. Active Willis metamaterials for ultracompact nonreciprocal linear acoustic devices. Phys. Rev. B 99, 220301 (2019).

Su, X. & Norris, A. N. Focusing, refraction, and asymmetric transmission of elastic waves in solid metamaterials with aligned parallel gaps. J. Acoust. Soc. Am. 139, 3386–3394 (2016).

Hwan Oh,J., Woong Kim, H., Sik Ma,P., Min Seung, H. & Young Kim, Y. Inverted bi-prism phononic crystals for one-sided elastic wave transmission applications. Appl. Phys. Lett. 100, 213503 (2012).

Colombi, A., Colquitt, D., Roux, P., Guenneau, S. & Craster, R. V. A seismic metamaterial: The resonant metawedge. Sci. Rep. 6, 27717 (2016).

Li, X.-F. et al. Tunable unidirectional sound propagation through a sonic-crystal-based acoustic diode. Phys. Rev. Lett. 106, 084301 (2011).

Shen, C., Xie, Y., Li, J., Cummer, S. A. & Jing, Y. Asymmetric acoustic transmission through near-zero-index and gradient-index metasurfaces. Appl. Phys. Lett. 108, 223502 (2016).

Cremer, L., Heckl, M. & Petersson, B. A. T. Structure-Borne Sound: Structural Vibrations and Sound Radiation at Audio Frequencies Ch. 2 (Springer, 2005).

Shorter, P. J. & Langley, R. S. On the reciprocity relationship between direct field radiation and diffuse reverberant loading. J. Acoust. Soc. Am. 117, 85–95 (2005).

Rayleigh, L. On the application of the principle of reciprocity to acoustics. Proc. R. Soc. Lond. 25, 118–122 (1876).

Morse, P. M. & Ingard, K. U. Theoretical Acoustics Ch. 11 (McGraw-Hill, 1968).

Newland, D. E. Mechanical Vibration Analysis and Computation Ch. 11 (Courier Corporation, 2013).

Yousefzadeh, B. & Daraio, C. in Proc. 26th Int. Congress Sound and Vibration (IIAV, 2019).

Tellegen, B. D. The gyrator, a new electric network element. Philips Res. Rep. 3, 81–101 (1948).