‘Parabolic’ trapped modes and steered Dirac cones in platonic crystals

Abstract

This paper discusses the properties of flexural waves governed by the biharmonic operator, and propagating in a thin plate pinned at doubly periodic sets of points. The emphases are on the design of dispersion surfaces having the Dirac cone topology, and on the related topic of trapped modes in plates for a finite set (cluster) of pinned points. The Dirac cone topologies we exhibit have at least two cones touching at a point in the reciprocal lattice, augmented by another band passing through the point. We show that these Dirac cones can be steered along symmetry lines in the Brillouin zone by varying the aspect ratio of rectangular lattices of pins, and that, as the cones are moved, the involved band surfaces tilt. We link Dirac points with a parabolic profile in their neighbourhood, and the characteristic of this parabolic profile decides the direction of propagation of the trapped mode in finite clusters.

Blockage and guiding of flexural waves in a semi‐infinite double grating

“…These striking features of the scattering pattern around the double grating in an elastic flexural plate are also linked to the earlier paper [6], which addressed dispersion properties and directional localisation of Bloch-Floquet waves in infinite periodic structures.…”

“…It is also relevant to cite [6] where degeneracies were analysed in the structure of the dispersion surfaces in connection to flexural waves in 'platonic crystals' built with a doubly periodic rectangular array of rigid pins. It was also found that the relative spacings in the vertical and horizontal directions are important in order to control the order of degeneracy of multiple roots of the dispersion equation.…”

“…The motivation for the choice of the values of the vertical separation b and the values of the spectral parameterˇcomes from the earlier analysis of dispersion relations in [6] and Figure 7(b) of [6]. Here, we identify an approximation, which gives us the 'effective wavenumber' of the flexural wave trapped between the gratings of pins.…”

“…Analysis of flexural Bloch waves in elastic periodically constrained plates was performed by McPhedran et al [6]. Special attention was given to studies of dispersion relations and, in particular, the vibration modes linked to so-called Dirac cones as well as standing waves and directional anisotropy.…”

“…It is mathematically challenging to identify and explain formally blockages of waves as well as the wave-guiding action possessed by constrained elastic plates. However, the recent paper [6] has provided an insight regarding the right choice of parameters for the model. We note that Bloch-Floquet waves observed in an infinite periodic waveguide differ from a field associated with wave scattering by a semi-infinite scatterer such as a line screen or a structured scatterer, for example, a semi-infinite row of small inclusions.…”