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# One and two-dimensional propagation of waves in periodic heterogeneous media: transient effects and band-gap tuning

Ellis Barnwell

[Thesis].University of Manchester;2015.

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## Abstract

In this thesis, the propagation of transient waves in heterogeneous media and the tuning of periodic elastic materials are studied. The behaviour of time harmonic waves in complex media is a well understood phenomenon. The primary aim of this text is to gain a deeper understanding into the propagation of transient waves in periodic media. The secondary aim is to explore the time harmonic behaviour of two dimensional pre-stressed elastic media and investigate the plausibility of band gap tuning.We begin this text by investigating the reflection of pulses from a semi-infinite set of point masses (we call ``beads'') on a string. The reflected pulse is formulated using Fourier transforms which involve the harmonic reflection coefficient. We find that the reflected amplitude of a harmonic wave depends on its frequency. We then ask whether it is possible to find an effective reflection coefficient by assuming the beaded portion of the string is given by some effective homogeneous medium. An effective reflection coefficient is found by assuming the homogeneous medium has the wavenumber given by the infinite beaded string. This effective reflection coefficient is compared to the exact reflection coefficient found using the Wiener-Hopf technique.The results from studying the reflection problem gave inspiration to chapter 4, which focuses on the time dependent forcing of an infinite beaded string that is initially at rest. We again use the Fourier transform to find a time dependent solution. The z transform is then used, after sampling the solution at the bead positions. We impose a sinusoidal loading which is switched on at a specified time. In doing this we are able to explore how the system behaves differently when excited in a stop band, a pass band and at a frequency on the edge between the two.An exact solution for the infinite beaded string is found at any point in time by expanding the branch points of the solution as a series of poles. We compare this exact solution to the long time asymptotics. The energy input into the system is studied with the results from the exact solution and long time approximation showing agreement. Interesting behaviour is discovered on the two edges between stop and pass bands.In chapter 5 the effect of a nonlinear elastic pre-stress on the wave band structure of a two dimensional phononic crystal is investigated. In this chapter we restrict ourselves to incompressible materials with the strain energy functions used being the neo-Hookean, Mooney-Rivlin and Fung. The method of small-on-large is used to derive the equation for incremental elastic waves and then the plane wave expansion method is used to find the band structure.Finally, chapter 6 focuses on the same geometry with a compressible elastic material. The strain energy function used is the one suggested by Levinson and Burgess. We use the theory of small-on-large to derive the incremental equations for coupled small amplitude pressure and shear waves in this material. In both compressible and incompressible materials we show how it is possible to control the stop bands in a material by applying a large elastic pre-stress.

## Keyword(s)

Band gap; Bloch wave; Heterogeneous media; Metamaterial; Nonlinear elasticity; Phononic crystal; Small-on-large; Wave propagation