Dispersion properties of vortex-type monatomic lattices

Carta, G.; Brun, Michele; Movchan, A. B.; Movchan, N. V.; Jones, I. S.

doi:10.1016/j.ijsolstr.2014.02.026

Cited by 50 publications

(68 citation statements)

References 12 publications

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“…Here, we extend the cartesian formulation given recently in Colquitt et al [19] to the cylindrical cloak configuration. In particular, equations (7) and (10) are normalized in the following way: Cloaking has been affected in these two cases. Then, introducing the following definition for the rigidity and inertial parameters: These quantities are constrained to satisfy the in-plane balance equation…”

Section: Alternative Approach: Plate Subjected To In-plane Forces Andmentioning

confidence: 99%

Transformation cloaking and radial approximations for flexural waves in elastic plates

et al. 2014

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It is known that design of elastic cloaks is much more challenging than that of acoustic cloaks, cloaks of electromagnetic waves or scalar problems of antiplane shear. In this paper, we address fully the fourth-order problem and develop a model of a broadband invisibility cloak for channelling flexural waves in thin plates around finite inclusions. We also discuss an option to employ efficiently an elastic pre-stress and body forces to achieve such a result. An asymptotic derivation provides a rigorous link between the model in question and elastic wave propagation in thin solids. This is discussed in detail to show connection with non-symmetric formulations in vector elasticity studied in earlier work.

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Section: Alternative Approach: Plate Subjected To In-plane Forces Andmentioning

confidence: 99%

Transformation cloaking and radial approximations for flexural waves in elastic plates

et al. 2014

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“…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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“…This phenomenon, known as dynamic anisotropy , has been observed both in lattices with and without spinners 13 , 29–33 .…”

Section: Unidirectional Waveformsmentioning

confidence: 91%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…In the same model, a precise beam deviation is achieved by modifying the arrangement of the gyros within the lattice. The system is modelled as an infinite medium by attaching PML ( Perfectly Matched Layers ) to the boundaries, designed to suppress reflected waves 13,27 .

Figure 1Deviation of a unidirectionally localised wave (“elastic prism”) in an elastic gyro-system. Each macro-cell contains two spinners rotating in opposite directions, as shown in Fig.…”

Section: Introductionmentioning

confidence: 99%

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

Carta

Jones

Movchan

et al. 2017

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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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Dispersion properties of vortex-type monatomic lattices

Cited by 50 publications

References 12 publications

Transformation cloaking and radial approximations for flexural waves in elastic plates

Transformation cloaking and radial approximations for flexural waves in elastic plates

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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