Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates

An elastic analog of graphene is introduced and analyzed. The system consists of a honeycomb arrangement of spring-mass resonators attached to a thin elastic layer, and the propagation properties of flexural waves along it is studied. The band-structure calculation shows the presence of Dirac points near the $K$ point of the Brillouin zone. Analytical expressions are found for both Dirac frequency and velocity as a function of the resonator's parameters. Finally, the bounded modes of infinitely long ribbons of this honeycomb arrangement are analyzed. The presence of edge states, which are studied using multiple scattering theory, is shown. It is concluded that these structures can be used to control the propagation of flexural waves in thin plates.

Figure 1

Schematic view of the system studied in this work. An array of resonators is attached to a thin elastic plate. Each resonator is modeled as a point mass attached to the plate by a spring, $m_R$ being its mass, and the resonant frequency ${\omega }_R$ of the mass-spring system the relevant parameters.

Figure 2

Geometry of the honeycomb arrangement of resonators. It is seen that there are two resonators per unit cell. In this analogy, every resonator is equivalent to carbon atoms in graphene.

Figure 3

Band structure for the flexural waves propagating the system of attached resonators with the honeycomb arrangement. In this example, ${\gamma }_R=10.0$ and ${\Omega }_{R}a=4\pi$. At the $K$ point the doubly degenerated conical-like dispersion relation called a Dirac cone can be seen.

Figure 4

Normalized Dirac frequency (frequency of the vertex of the cone in the band structure) as a function of the normalized resonant frequency of the spring-mass system for different ${\gamma }_R$ values.

Figure 5

Normalized Dirac velocity as a function of the normalized resonant frequency of the spring-mass system for different ${\gamma }_R$ values.

Figure 6

Schematic view of a ribbon with zigzag edges. The left and right arrows indicate that the system is infinite along the horizontal direction ($x$ axis). The vertical rectangle defines the unit cell that is repeated along the $x$ axis and is employed for the calculation of the dispersion relation. The horizontal dashed rectangle defines a “line.” The depicted structure has five lines.

Figure 7

The red lines represent the dispersion relations computed for ribbons with zigzag edges with one, two, three, and four lines. The dispersion relations are limited by the free-wave dispersion relation, since outside this region the modes are leaky modes. It is clear how bulk modes are formed as we increase the number of lines but edge states appear between $\Omega a\approx 2$ and $\Omega a\approx 3$.

Figure 8

Dispersion relation of a ribbon made of ten lines with zigzag edges. The gray area is the band structure projected on the edge direction, showing clearly the difference between edge states and bulk states. The white spots in the gray regions correspond to roots not encountered by the searching algorithm due to its lack of precision.

Figure 9

Multiple scattering simulation of an edge state excited in a single-line zigzag ribbon. The source is oscillating at a normalized frequency $\Omega a=2.6$. Note that although the state is guided along the ribbon, there are also leaky waves.

Figure 10

Multiple scattering simulation of a finite structure with only zigzag edges. The excitation is a point source located in the upper-right corner and oscillating at a normalized frequency $\Omega a=2.566$. The edge state is clearly observed, although a weak bulk state is also excited.

Figure 11

Multiple scattering simulation of a structure larger than that studied in Fig. 10 with zigzag edges. In comparison with Fig. 10, now more bulk states are excited.