Mathematical modelling of flexural waves in structured elastic plates



Haslinger, Stewart
Mathematical modelling of flexural waves in structured elastic plates. Doctor of Philosophy thesis, University of Liverpool.

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Abstract

This thesis discusses properties of flexural waves in thin elastic plates that incorporate a structured system of gratings of defects. The defects may take the form of inclusions or masses, but we focus on circular inclusions, and in particular, holes with a clamped edge. We place the work in the relatively new field of platonics, which is the study of flexural waves in plates governed by the fourth-order biharmonic plate equation. By analogy with photonic and phononic crystals, the two-dimensional structures in thin elastic plates are known as platonic crystals. We present a novel analysis of trapped modes and transmission resonances in grating stacks, arising from the interaction with plane waves incident on the gratings. We show that the evanescent modes are important in demonstrating interesting and unusual filtering effects. In particular we analyse the previously unstudied effect of elasto-dynamically inhibited transmission (EDIT), where a resonance in transmission is cut in two by a resonant minimum arising from destructive interference. Similar destructive interference-induced phenomena have been observed in other settings, notably classical optical oscillators, metamaterials and plasmonics, but we are the first to do it for flexural plates. The phenomenon of EDIT is a central theme of this thesis, and is linked to the analysis of even and odd Bloch modes in the grating waveguides. We develop a method that identifies the parameters of the model, the relative separations η and lateral shifts ξ of the gratings, and the spectral parameter β and angle of incidence θi of the plane wave, to find EDIT efficiently. The method is powerful and universal, based on a recurrence procedure for the construction of reflection and transmission matrices. A multipole method is employed for circular scatterers and the limiting case of rigid pins, whereby the solutions are determined analytically. Recent developments have also been made with arbitrarily-shaped holes, and other future research is likely to focus on the association with Dirac-like cones that are linked to the standing modes arising in platonic crystals.

Item Type: Thesis (Doctor of Philosophy)
Additional Information: Date: 2014-02 (completed)
Subjects: ?? QA ??
Divisions: Faculty of Science and Engineering > School of Physical Sciences > Mathematical Sciences
Depositing User: Symplectic Admin
Date Deposited: 07 Aug 2014 09:01
Last Modified: 17 Dec 2022 01:41
DOI: 10.17638/00016833
Supervisors:
  • Movchan, Alexander
  • Movchan, Natalia
URI: https://livrepository.liverpool.ac.uk/id/eprint/16833