Meylan, MMcPhedran, RSmith, Michael James A.2014-01-092013http://hdl.handle.net/2292/21370This thesis considers the propagation of flexural waves through structures embedded in thin, elastic plates. In a similar manner to photonic and phononic crystals, two-dimensional structures in thin elastic plates are known as platonic crystals (PlaCs) and can be formed using scatterers of any shape arranged in any regular periodic geometry. The time-harmonic waves that propagate through these structures are governed by the fourth-order biharmonic plate equation, in contrast to the second-order Helmholtz equation which governs wave propagation through other crystals and media, such as deep water. Here we consider two-dimensional structures which are comprised of circular scatterers, pins, and arbitrary shapes arranged in a square array. In addition to these two-dimensional PlaCs, we consider wave scattering by other platonic structures, such as one-dimensional arrays, finite clusters and single bodies. For circular and pinned geometries the solution can be obtained analytically using multipole methods, but for arbitrary shapes this is not possible. Here we outline a solution method for an arbitrarily shaped smooth scatterer using boundary integral equations, which arise from an appropriate decomposition of the biharmonic operator. The resulting system of equations can then solved by implementing boundary conditions at the body edge and using boundary element methods (BEMs). To verify the BEM solutions we use the multipole solutions for circular geometries, which are outlined here. For a two-dimensional array of arbitrarily shaped bodies one can construct the scattering and transfer matrices corresponding to a one-dimensional array, and then search for admissible Bloch factors. This permits the construction of band surfaces which reveal when Bloch-Floquet waves can propagate through the array to infinity. We also demonstrate that PlaCs can steer and disperse flexural waves analogously to light in photonic crystals and elastic waves in phononic crystals, and we validate this directly using localised Gaussian beams for pinned clusters. For such pinned structures the solution can be obtained directly using the fundamental solution to the biharmonic plate equation, and using these compact incident waves we are able to confirm the existence of a number of interesting diffraction effects, including negative refraction in thin plates. We also demonstrate that strong energy localisation is possible within the defects and waveguides of pinned PlaCs, and show a number of analogues to well known optical phenomena such as platonic polarisers in two-dimensional arrays of arbitrarily shaped scatterers. A solution is also presented for a single arbitrary body and for one-dimensional arrays of arbitrary geometry.Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated. Previously published items are made available in accordance with the copyright policy of the publisher.https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmhttp://creativecommons.org/licenses/by-nc-sa/3.0/nz/ThesisCopyright: The Authorhttp://purl.org/eprint/accessRights/OpenAccessQ112904030