×

Interfacial waveforms in chiral lattices with gyroscopic spinners. (English) Zbl 1404.74010

Summary: We demonstrate a new method of achieving topologically protected states in an elastic hexagonal system of trusses by attaching gyroscopic spinners, which bring chirality to the system. Dispersive features of this medium are investigated in detail, and it is shown that one can manipulate the locations of stop-bands and Dirac points by tuning the parameters of the spinners. We show that, in the proximity of such points, uni-directional interfacial waveforms can be created in an inhomogeneous lattice and the direction of such waveforms can be controlled. The effect of inserting additional soft internal links into the system, which is thus transformed into a heterogeneous triangular lattice, is also investigated, as the hexagonal lattice represents the limit case of the heterogeneous triangular lattice with soft links. This work introduces a new perspective in the design of periodic media possessing non-trivial topological features.

MSC:

74A50 Structured surfaces and interfaces, coexistent phases
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Raghu S, Haldane FDM. (2008) Analogs of quantum-Hall-effect edge states in photonic crystals. Phys. Rev. A 78, 033834. (doi:10.1103/PhysRevA.78.033834) · doi:10.1103/PhysRevA.78.033834
[2] Wang Z, Chong YD, Joannopoulos JD, Soljačić M. (2008) Reflection-free one-way edge modes in a gyromagnetic photonic crystal. Phys. Rev. Lett. 100, 013905. (doi:10.1103/PhysRevLett.100.013905) · doi:10.1103/PhysRevLett.100.013905
[3] Wang Z, Chong YD, Joannopoulos JD, Soljačić M. (2009) Observation of unidirectional backscattering-immune topological electromagnetic states. Nature 461, 772-775. (doi:10.1038/nature08293) · doi:10.1038/nature08293
[4] He C, Chen XL, Lu MH, Li XF, Wan WW, Qian XS, Yin RC, Chen YF. (2010) Left-handed and right-handed one-way edge modes in a gyromagnetic photonic crystal. J. Appl. Phys. 107, 123117. (doi:10.1063/1.3374470) · doi:10.1063/1.3374470
[5] Khanikaev AB, Mousavi SH, Tse WK, Kargarian M, MacDonald AH, Shvets G. (2013) Photonic topological insulators. Nat. Mater. 12, 233-239. (doi:10.1038/nmat3520) · doi:10.1038/nmat3520
[6] Lu L, Joannopoulos JD, Soljačić M. (2014) Topological photonics. Nat. Photon. 8, 821-829. (doi:10.1038/nphoton.2014.248) · doi:10.1038/nphoton.2014.248
[7] Gao W, Lawrence M, Yang B, Liu F, Fang F, Béri B, Li J, Zhang S. (2015) Topological photonic phase in chiral hyperbolic metamaterials. Phys. Rev. Lett. 114, 037402. (doi:10.1103/PhysRevLett.114.037402) · doi:10.1103/PhysRevLett.114.037402
[8] Skirlo SA, Lu L, Igarashi Y, Yan Q, Joannopoulos JD, Soljačić M. (2015) Experimental observation of large Chern numbers in photonic crystals. Phys. Rev. Lett. 115, 253901. (doi:10.1103/PhysRevLett.115.253901) · doi:10.1103/PhysRevLett.115.253901
[9] Klitzing KV, Dorda G, Pepper M. (1980) New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Phys. Rev. Lett. 45, 494-497. (doi:10.1103/PhysRevLett.45.494) · doi:10.1103/PhysRevLett.45.494
[10] Siroki G, Huidobro PA, Giannini V. (2017) Topological photonics: from crystals to particles. Phys. Rev. B 96, 041408(R). (doi:10.1103/PhysRevB.96.041408) · doi:10.1103/PhysRevB.96.041408
[11] Luo Y, Lei DY, Maier SA, Pendry JB. (2012) Broadband light harvesting nanostructures robust to edge bluntness. Phys. Rev. Lett. 108, 023901. (doi:10.1103/PhysRevLett.108.023901) · doi:10.1103/PhysRevLett.108.023901
[12] Goldman N, Juzeliūnas G, &quote;Ohberg P, Spielman IB. (2014) Light-induced gauge fields for ultracold atoms. Rep. Prog. Phys. 77, 126401, 1-60. (doi:10.1088/0034-4885/77/12/126401) · doi:10.1088/0034-4885/77/12/126401
[13] Pendry JB, Martin-Moreno L, Garcia-Vidal FJ. (2004) Mimicking surface plasmons with structured surfaces. Science 305, 847-848. (doi:10.1126/science.1098999) · doi:10.1126/science.1098999
[14] Jin D, Christensen T, Sojačić M, Fang NX, Lu L, Zhang X. (2017) Infrared topological plasmons in graphene. Phys. Rev. Lett. 118, 245301,1-6 . (doi:10.1103/PhysRevLett.118.245301) · doi:10.1103/PhysRevLett.118.245301
[15] Nalitov AV, Solnyshkov DD, Malpuech G. (2015) Polariton Z topological insulator. Phys. Rev. Lett. 114, 116401,1-5. (doi:10.1103/PhysRevLett.114.116401) · doi:10.1103/PhysRevLett.114.116401
[16] Ni X, He C, Sun XC, Liu XP, Lu MH, Feng L, Chen YF. (2015) Topologically protected one-way edge mode in networks of acoustic resonators with circulating air flow. New J. Phys. 17, 053016. (doi:10.1088/1367-2630/17/5/053016) · doi:10.1088/1367-2630/17/5/053016
[17] Yang Z, Gao F, Shi X, Lin X, Gao Z, Chong Y, Zhang B. (2015) Topological acoustics. Phys. Rev. Lett. 114, 114301. (doi:10.1103/PhysRevLett.114.114301) · doi:10.1103/PhysRevLett.114.114301
[18] Chen ZG, Wu Y. (2016) Tunable topological phononic crystals. Phys. Rev. Appl. 5, 054021. (doi:10.1103/PhysRevApplied.5.054021) · doi:10.1103/PhysRevApplied.5.054021
[19] Souslov A, van Zuiden BC, Bartolo D, Vitelli V. (2017) Topological sound in active-liquid metamaterials. Nat. Phys. 13, 1091-1094. published online on 2017-07-17. (doi:10.1038/nphys4193) · doi:10.1038/nphys4193
[20] He C, Ni X, Ge H, Sun XC, Chen YB, Lu MH, Liu XP, Chen YF. (2016) Acoustic topological insulator and robust one-way sound transport. Nat. Phys. 12, 1124-1129. (doi:10.1038/nphys3867) · doi:10.1038/nphys3867
[21] Khanikaev AB, Fluery R, Hossein Mousavi S, Alú A. (2015) Topologically robust sound propagation in an angular-momentum-biased graphene like resonator lattice. Nat. Commun. 6, 8260. (doi:10.1038/ncomms9260) · doi:10.1038/ncomms9260
[22] Evans DV, Linton CM. (1993) Edge waves along periodic coastlines. Q. J. Mech. Appl. Math. 46, 643-656. (doi:10.1093/qjmam/46.4.643) · Zbl 0817.76005 · doi:10.1093/qjmam/46.4.643
[23] Adamou A, Craster RV, Llewellyn Smith SG. (2007) Trapped edge waves in stratified rotating fluids: numerical and asymptotic results. J. Fluid Mech. 592, 195-220. (doi:10.1017/S0022112007008361) · Zbl 1128.76009 · doi:10.1017/S0022112007008361
[24] Mousavi SH, Khanikaev AB, Wang Z. (2015) Topologically protected elastic waves in phononic metamaterials. Nat. Commun. 6, 8682. (doi:10.1038/ncomms9682) · doi:10.1038/ncomms9682
[25] Pal RK, Ruzzene M. (2017) Edge waves in plates with resonators: an elastic analogue of the quantum valley Hall effect. New J. Phys. 19, 025001. (doi:10.1088/1367-2630/aa56a2) · doi:10.1088/1367-2630/aa56a2
[26] Kariyado T, Hatsugai Y. (2015) Manipulation of Dirac cones in mechanical graphene. Sci. Rep. 5, 18107. (doi:10.1038/srep18107) · doi:10.1038/srep18107
[27] Vila J, Pal RK, Ruzzene M. (2017) Observation of topological valley modes in an elastic hexagonal lattice. Phys. Rev. B 96, 134307. (doi:10.1103/PhysRevB.96.134307) · doi:10.1103/PhysRevB.96.134307
[28] Maling B, Craster RV. (2017) Whispering Bloch modes. Proc. R. Soc. A 472, 20160103. (doi:10.1098/rspa.2016.0103) · Zbl 1371.78018 · doi:10.1098/rspa.2016.0103
[29] Süsstrunk R, Huber SD. (2015) Observation of phononic helical edge states in a mechanical topological insulator. Science 349, 47-50. (doi:10.1126/science.aab0239) · doi:10.1126/science.aab0239
[30] Huber SD. (2016) Topological mechanics. Nat. Phys. 12, 621-623. (doi:10.1038/nphys3801) · doi:10.1038/nphys3801
[31] Brun M, Jones IS, Movchan AB. (2012) Vortex-type elastic structured media and dynamic shielding. Proc. R. Soc. A 468, 3027-3046. (doi:10.1098/rspa.2012.0165) · Zbl 1371.74154 · doi:10.1098/rspa.2012.0165
[32] Carta G, Brun M, Movchan AB, Movchan NV, Jones IS. (2014) Dispersion properties of vortex-type monatomic lattices. Int. J. Solids Struct. 51, 2213-2225. (doi:10.1016/j.ijsolstr.2014.02.026) · doi:10.1016/j.ijsolstr.2014.02.026
[33] Carta G, Jones IS, Movchan NV, Movchan AB, Nieves MJ. (2017) ‘Deflecting elastic prism’ and unidirectional localisation for waves in chiral elastic systems. Sci. Rep. 7, 26. (doi:10.1038/s41598-017-00054-6) · doi:10.1038/s41598-017-00054-6
[34] Wang P, Lu L, Bertoldi K. (2015) Topological phononic crystals with one-way elastic edge waves. Phys. Rev. Lett. 115, 104302. (doi:10.1103/PhysRevLett.115.104302) · doi:10.1103/PhysRevLett.115.104302
[35] Nash LM, Kleckner D, Read A, Vitelli V, Turner AM, Irvine WTM. (2015) Topological mechanics of gyroscopic metamaterials. Proc. Natl Acad. Sci. USA 112, 14 495-14 500. (doi:10.1073/pnas.1507413112) · doi:10.1073/pnas.1507413112
[36] Thomson W. (1894) The molecular tactics of a crystal. Oxford, UK: Clarendon Press.
[37] Prall D, Lakes RS. (1997) Properties of a chiral honeycomb with a Poisson’s ratio of −1. Int. J. Mech. Sci. 39, 305-314. (doi:10.1016/S0020-7403(96)00025-2) · Zbl 0894.73018 · doi:10.1016/S0020-7403(96)00025-2
[38] Spadoni A, Ruzzene M. (2012) Elasto-static micropolar behavior of a chiral auxetic lattice. J. Mech. Phys. Solids 60, 156-171. (doi:10.1016/j.jmps.2011.09.012) · doi:10.1016/j.jmps.2011.09.012
[39] Spadoni A, Ruzzene M, Gonella S, Scarpa F. (2009) Phononic properties of hexagonal chiral lattices. Wave Motion 46, 435-450. (doi:10.1016/j.wavemoti.2009.04.002) · Zbl 1231.82083 · doi:10.1016/j.wavemoti.2009.04.002
[40] Bacigalupo A, Gambarotta L. (2016) Simplified modelling of chiral lattice materials with local resonators. Int. J. Solids Struct. 83, 126-141. (doi:10.1016/j.ijsolstr.2016.01.005) · doi:10.1016/j.ijsolstr.2016.01.005
[41] Zhu R, Liu XN, Hu GK, Sun CT, Huang GL. (2014) Negative refraction of elastic waves at the deep-subwavelength scale in a single-phase metamaterial. Nat. Commun. 5, 5510. (doi:10.1038/ncomms6510) · doi:10.1038/ncomms6510
[42] Tallarico D, Movchan NV, Movchan AB, Colquitt DJ. (2016) Tilted resonators in a triangular elastic lattice: chirality, Bloch waves and negative refraction. J. Mech. Phys. Solids 103, 236-256. (doi:10.1016/j.jmps.2017.03.007) · doi:10.1016/j.jmps.2017.03.007
[43] D’Eleuterio GMT, Hughes PC. (1984) Dynamics of gyroelastic continua. J. Appl. Mech. 51, 415-422. (doi:10.1115/1.3167634) · doi:10.1115/1.3167634
[44] Hughes PC, D’Eleuterio GMT. (1986) Modal parameter analysis of gyroelastic continua. J. Appl. Mech. 53, 918-924. (doi:10.1115/1.3171881) · Zbl 0608.73064 · doi:10.1115/1.3171881
[45] D’Eleuterio GMT. (1988) On the theory of gyroelasticity. J. Appl. Mech. 55, 488-489. (doi:10.1115/1.3173705) · doi:10.1115/1.3173705
[46] Yamanaka K, Heppler GR, Huseyin K. (1996) Stability of gyroelastic beams. AIAA J. 34, 1270-1278. (doi:10.2514/3.13223) · Zbl 0894.73043 · doi:10.2514/3.13223
[47] Hassanpour S, Heppler GR. (2016) Theory of micropolar gyroelastic continua. Acta Mech. 227, 1469-1491. (doi:10.1007/s00707-016-1573-x) · Zbl 1382.74010 · doi:10.1007/s00707-016-1573-x
[48] Hassanpour S, Heppler GR. (2016) Dynamics of 3D Timoshenko gyroelastic beams with large attitude changes for the gyros. Acta Astron. 118, 33-48. (doi:10.1016/j.actaastro.2015.09.012) · doi:10.1016/j.actaastro.2015.09.012
[49] Carta G, Jones IS, Movchan NV, Movchan AB, Nieves MJ. (2017) Gyro-elastic beams for the vibration reduction of long flexural systems. Proc. R. Soc. A 473, 20170136. (doi:10.1098/rspa.2017.0136) · Zbl 1404.74078 · doi:10.1098/rspa.2017.0136
[50] Zhang J, Qiu X. (2014) Non-symmetric deformation of lattices under quasi-static uniaxial compression. Int. J. Mech. Sci. 78, 72-80. (doi:10.1016/j.ijmecsci.2013.11.004) · doi:10.1016/j.ijmecsci.2013.11.004
[51] Gibson LJ, Ashby MF. (1997) Cellular solids: structure and properties, 2nd edn. Cambridge, UK: Cambridge University Press.
[52] Cserti J, Tichy G. (2004) A simple model for the vibrational modes in honeycomb lattices. Eur. J. Phys. 25, 723-736. (doi:10.1088/0143-0807/25/6/004) · Zbl 1161.82325 · doi:10.1088/0143-0807/25/6/004
[53] Hou JM, Chen W. (2015) Hidden symmetry and protection of Dirac points on the honeycomb. Sci. Rep. 5, 17571. (doi:10.1038/srep17571) · doi:10.1038/srep17571
[54] He WY, Chan CT. (2015) The emergence of Dirac points in photonic crystals with mirror symmetry. Sci. Rep. 5, 8186. (doi:10.1038/srep08186) · doi:10.1038/srep08186
[55] Carta G, Jones IS, Brun M, Movchan NV, Movchan AB. (2013) Crack propagation induced by thermal shocks in structured media. Int. J. Solids Struct. 50, 2725-2736. (doi:10.1016/j.ijsolstr.2013.05.001) · doi:10.1016/j.ijsolstr.2013.05.001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.