"Deflecting elastic prism" and unidirectional localisation for waves in chiral elastic systems

Abstract

For 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.

Figure 1

Deviation 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. 2b. The unidirectionally localised wave is generated by a harmonic displacement of amplitude 0.01, indicated by the arrow. The amplitude of the total normalised displacement, defined in the next sections, is plotted in the figure.

Figure 2

(a) Three-dimensional representation of the gyro-system, consisting of a triangular lattice connected to a system of gyros; (b) periodic unit cell of the system; (c) schematic representation of a gyroscopic spinner, where ψ, θ and ϕ are the angles of spin, nutation and precession, respectively.

Figure 3

Dispersion surfaces (left) and relative cross-sections for ${\tilde{k}}_y=0$ (center) and ${\tilde{k}}_x=0$ (right) with the following values of the spinner constants: (a) ${\tilde{\alpha }}_1=0.4,{\tilde{\alpha }}_2=0.8$, (b) ${\tilde{\alpha }}_1=0.4,{\tilde{\alpha }}_2=1.6$, (c) ${\tilde{\alpha }}_1=1.2,{\tilde{\alpha }}_2=1.6$, (d) ${\tilde{\alpha }}_1=-1,{\tilde{\alpha }}_2=1$.

Figure 4

(a) Slowness contours and (b) amplitude field of the total normalised displacement, calculated at $\tilde{f}={\tilde{f}}^{\mathrm{GB}}=0.94$ in a gyro-system with equal and opposite spinner constants (${\tilde{\alpha }}_1=0.9,{\tilde{\alpha }}_2=-0.9$), under a vertical harmonic displacement of amplitude 0.01. The lattice in (b) is approximately a square of side length equal to 60 and it is surrounded by PML, the parameters of which have been tuned to maximise the reduction of reflected waves.

Figure 5

(a) Slowness contours and (b) amplitude field of the total normalised displacement, determined at $\tilde{f}={\tilde{f}}^{\mathrm{GB}}=0.86$ in a gyro-system with different spinners $\left({\tilde{\alpha }}_1=0.8,{\tilde{\alpha }}_2=-0.9\right)$; (c) dependence of the Gaussian beam frequency on the spinner constant ${\tilde{\alpha }}_1$ when ${\tilde{\alpha }}_2=-0.9$. The geometry of the lattice in (b) is the same as in Fig. 4b; the harmonic displacement has amplitude 0.01 and it is imposed on a node attached to a spinner with $\tilde{\alpha }={\tilde{\alpha }}_2$.

Figure 6

Amplitude fields of the total normalised displacement obtained for different spinner constants, resulting from a vertical harmonic displacement of amplitude 0.01 applied to a node associated with the smaller absolute value of the spinner constant.

Figure 7

Bending of a Gaussian beam in a lattice with spinner constants ${\tilde{\alpha }}_1=0.8$ and ${\tilde{\alpha }}_2=-0.9$ at ${\tilde{f}}^{\mathrm{GB}}=0.858,$ when the medium is surrounded by (a) PML or (b) a lattice without gyros. The insets on the left show how the spinners are arranged in proximity of the interfaces, indicated by dashed lines.

Figure 8

(a) Closed waveforms produced by a harmonic displacement of amplitude 0.01 and frequency ${\tilde{f}}^{\mathrm{GB}}=0.858$, indicated by the arrow, with two different configurations of the gyro-system, detailed in the figures; (b) resonant modes of the same lattices with Dirichlet boundary conditions in correspondence with the frequency ${\tilde{f}}^{\mathrm{GB}}$; (c) wave patterns obtained by placing the source at a different position.