Vortex-type elastic structured media and dynamic shielding

Brun, Michele; Jones, I. S.; Movchan, Alexander

doi:10.1098/rspa.2012.0165

Cited by 51 publications

(79 citation statements)

References 22 publications

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“…Each lattice particle is attached to a gyroscopic spinner (see Fig. 1a), characterised by the spinner constant α, which is a function of the geometry and spin rate of the spinner [30]. The lattice is shown in Fig.…”

Section: Governing Equations and Definitionsmentioning

confidence: 99%

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Wave polarization and dynamic degeneracy in a chiral elastic lattice

Carta

Jones²,

Movchan³

et al. 2019

Proc. R. Soc. A.

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This paper addresses fundamental questions arising in the theory of Bloch–Floquet waves in chiral elastic lattice systems. This area has received a significant attention in the context of ‘topologically protected’ waveforms. Although practical applications of chiral elastic lattices are widely appreciated, especially in problems of controlling low-frequency vibrations, wave polarization and filtering, the fundamental questions of the relationship of these lattices to classical waveforms associated with longitudinal and shear waves retain a substantial scope for further development. The notion of chirality is introduced into the systematic analysis of dispersive elastic waves in a doubly-periodic lattice. Important quantitative characteristics of the dynamic response of the lattice, such as lattice flux and lattice circulation, are used in the analysis along with the novel concept of ‘vortex waveforms’ that characterize the dynamic response of the chiral system. We note that the continuum concepts of pressure and shear waves do not apply for waves in a lattice, especially in the case when the wavelength is comparable with the size of the elementary cell of the periodic structure. Special critical regimes are highlighted when vortex waveforms become dominant. Analytical findings are accompanied by illustrative numerical simulations.

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Section: Governing Equations and Definitionsmentioning

confidence: 99%

“…The matrix T is given by T = t (1) , t (2) . The equations of motion of the chiral lattice can be written in the form [30,31] C…”

Section: Dispersion Properties Of the Chiral Latticementioning

confidence: 99%

Wave polarization and dynamic degeneracy in a chiral elastic lattice

Carta

Jones²,

Movchan³

et al. 2019

Proc. R. Soc. A.

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“…1(d)] and Θ is the amplitude of the harmonic change in θ. Interestingly, the coupling between the mass in the lattice and the gyroscope induces an in-plane gyroscopic inertial force perpendicular to the direction of U tip [24,25,27]:…”

Section: H Y S I C a L R E V I E W L E T T E R S Week Ending 4 Septemmentioning

confidence: 99%

“…However, this gap is topologically trivial, since time-reversal symmetry is not broken and the Chern numbers of the bands are all zero. In order to break time-reversal symmetry, we introduce gyroscopic coupling [25,26,30] and attach each mass in the lattice to the tip of the rotational axis of a gyroscope, as shown in Fig. 1(d).…”

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confidence: 99%

Topological Phononic Crystals with One-Way Elastic Edge Waves

2015

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We report a new type of phononic crystals with topologically nontrivial band gaps for both longitudinal and transverse polarizations, resulting in protected one-way elastic edge waves. In our design, gyroscopic inertial effects are used to break the time-reversal symmetry and realize the phononic analogue of the electronic quantum (anomalous) Hall effect. We investigate the response of both hexagonal and square gyroscopic lattices and observe bulk Chern numbers of 1 and 2, indicating that these structures support single and multimode edge elastic waves immune to backscattering. These robust one-way phononic waveguides could potentially lead to the design of a novel class of surface wave devices that are widely used in electronics, telecommunication, and acoustic imaging. DOI: 10.1103/PhysRevLett.115.104302 PACS numbers: 46.40.Cd, 46.40.Ff, 73.43.-f Topological states in electronic materials, including the quantum Hall effect [1] and topological insulators [2,3], have inspired a number of recent developments in photonics [4,5], phononics [6][7][8][9], and mechanical metamaterials [10][11][12][13]. In particular, in analogy to the quantum anomalous Hall effect [14], one-way electromagnetic waveguides in two-dimensional systems have been realized by breaking time-reversal symmetry [15][16][17][18].Very recently, unidirectional edge channels have been proposed for elastic waves using Coriolis force in a noninertial reference frame [19], but such a rotating frame is very difficult to implement in solid state devices. Moreover, one-way propagation of scalar acoustic waves has also been proposed by introducing rotating fluids [20][21][22]. However, it is important to recognize that elastic waves in solids have both transverse and longitudinal polarizations, while acoustic waves in fluids are purely longitudinal. As a result, it is challenging to achieve topological protection for elastic waves on an integrated platform.Here, we present a robust strategy to create topologically nontrivial edge modes for both longitudinal and transverse polarizations in a solid medium. In particular, we introduce gyroscopic phononic crystals, where each lattice site is coupled with a spinning gyroscope that breaks timereversal symmetry in a well-controlled manner. In both hexagonal and square lattices, gyroscopic coupling opens band gaps that are characterized by Chern numbers of 1 and 2. As a result, at the edge of these lattices both single-mode and multimode one-way elastic waves are observed to propagate around arbitrary defects without backscattering.To start, we consider a hexagonal phononic crystal with equal masses (m 2 ¼ m 1 ) connected by linear springs [red and black rods in Figs. 1(a) and 1(b)]. The resulting unit cell has 4 degrees of freedom specified by the displacements of m 1 and m 2 (U ¼ ½u for wave vectors μ within the first Brillouin zone. Here, ω denotes the angular frequency of the propagating wave and M ¼ diagfm 1 ; m 1 ; m 2 ; m 2 g is the mass matrix. Moreover, K is the 4 × 4 stiffness matrix and is a functi...

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“…Chiral properties can be conferred on a discrete elastic medium by installing a system of gyros (or spinners), as proposed by Brun et al 12 for the first time. Wave polarisation, dynamic anisotropy and forced response in the frequency domain for an elastic lattice with gyros were discussed extensively by Carta et al 13 .…”

Section: Introductionmentioning

confidence: 99%

“Deflecting elastic prism” and unidirectional localisation for waves in chiral elastic systems

Carta

Jones

Movchan

et al. 2017

Sci Rep

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1For the first time, a design of a "deflecting elastic prism" is proposed and implemented for waves in a chiral medium. A novel model of an elastic lattice connected to a non-uniform system of gyroscopic spinners is designed to create a unidirectional wave pattern, which can be diverted by modifying the arrangement of the spinners within the medium. This important feature of the gyro-system is exploited to send a wave from a point of the lattice to any other point in the lattice plane, in such a way that the wave amplitude is not significantly reduced along the path. We envisage that the proposed model could be very useful in physical and engineering applications related to directional control of elastic waves.An object is defined as "chiral" if it cannot be superimposed onto its mirror image 1 . In electromagnetism, chiral metamaterials have been designed to exhibit negative refraction [2][3][4] and to achieve dynamic tunability 5 . In elasticity, chiral assemblies of structural elements have been proposed to realise an effective medium with negative Poisson's ratio 6,7 and to modify the dispersive properties of a lattice system through generating band-gaps and directional preference of waves 8,9 . An elastic metamaterial with a chiral microstructure was devised by Zhu et al. 10 to yield negative refraction effects at the sub-wavelength scale. More recently, Tallarico et al. 11 have designed a chiral interface by introducing tilted resonators into a triangular elastic lattice, which behaves as a flat elastic lens.Chiral properties can be conferred on a discrete elastic medium by installing a system of gyros (or spinners), as proposed by Brun et al. 12 for the first time. Wave polarisation, dynamic anisotropy and forced response in the frequency domain for an elastic lattice with gyros were discussed extensively by Carta et al.13 . The same model was used by Wang et al.14 to induce propagation of edge modes around defects in the transient regime. A gyroscopic metamaterial breaking time-reversal symmetry was presented by Nash et al. 15 . A model with micro-inertia having effective frequency-dependent moment of inertia was developed by Milton and Willis 16 by introducing a spinning top into a seemingly rigid body.Dynamic properties of elastic discrete media have attracted much interest in the last two decades, since discrete systems allow for an analytical treatment. In particular, elastic lattices can be designed analytically to exhibit band-gaps at predefined frequencies 17,18 . Disorder and imperfections can also be included in the mathematical formulation of the model by using the notion of localisation factor [19][20][21] . In this paper, we establish unidirectional localisation and directional control of waves in an elastic triangular lattice, where each mass is coupled with a gyroscopic spinner. For convenience, this medium will henceforth be referred to as a gyro-system. In a different way to previous works 12,13 , here we assume that the system of spinners consists of two types of gyros, havin...

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Vortex-type elastic structured media and dynamic shielding

Cited by 51 publications

References 22 publications

Wave polarization and dynamic degeneracy in a chiral elastic lattice

Wave polarization and dynamic degeneracy in a chiral elastic lattice

Topological Phononic Crystals with One-Way Elastic Edge Waves

“Deflecting elastic prism” and unidirectional localisation for waves in chiral elastic systems

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