We derive analytical solutions based on singular Green’s functions, which
enable efficient computations of scattering simulations or Floquet-Bloch
dispersion relations for waves propagating through an elastic plate, whose
surface is patterned by periodic arrays of elastic beams. Our methodology is
versatile and allows us to solve a range of problems regarding arrangements of
multiple beams per primitive cell, over Bragg to deep-subwavelength scales; we
cross-verify against finite element numerical simulations to gain further
confidence in our approach, which relies upon the hypothesis of Euler-Bernoulli
beam theory considerably simplifying continuity conditions such that each beam
can be replaced by point forces and moments applied to the neutral plane of the
plate. The representations of Green’s functions by Fourier series or Fourier
transforms readily follows, yielding rapid and accurate analytical schemes. The
accuracy and flexibility of our solutions are demonstrated by engineering
topologically non-trivial states, from primitive cells with broken spatial
symmetries, following the phononic analogue of the Quantum Valley Hall Effect
(QVHE). Topologically protected states are produced and coexist along:
interfaces between adjoining chiral-mirrored bulk media and edges between one
such chiral bulk and the surrounding bare elastic plate, allowing topological
circuits to be designed with robust waveguiding; these topologically
non-trivial states exist within near flexural resonances of the constituent
beams of the phononic crystal, and hence can be tuned into a deep-subwavelength
regime.
