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# Mathematical modelling of nonlinear waves in layered waveguides with delamination

thesis
posted on 2018-06-12, 14:59 authored by Matthew R. Tranter
The propagation of nonlinear bulk strain waves in layered elastic waveguides has many applications, particularly its potential use for non-destructive testing, where a small defect in the bonding between the layers of a waveguide can lead to a catastrophic failure of the structure. Experiments have shown that strain solitons can propagate for significantly longer distances than the waves used in current methods, and therefore they are of great interest. This thesis considers two problems. Firstly, we consider the scattering of nonlinear bulk strain waves in two types of waveguides: a perfectly bonded layered waveguide, and a layered waveguide with a soft bond between the layers, when the materials in the layers have similar properties. In each case we assume that there is a region where the bond is absent - a delamination. This behaviour is described by a system of uncoupled or coupled Boussinesq equations, with conditions on the interface between the sections of the bar. This is a complicated system of equations, and we develop a direct numerical method to solve these equations numerically. A weakly nonlinear solution is then constructed for the system of equations, describing the leading order reflected and transmitted strain waves. In the case of a layered elastic bar with a perfect bond we obtain Korteweg-de Vries equations, and in the case of a soft bond between the layers, where the properties of the layers are close, we obtain coupled Ostrovsky equations describing the propagation of the reflected and transmitted waves in each layer of the waveguide. In the delaminated regions of the bar, Korteweg-de Vries equations are derived in every case and therefore we make use of the Inverse Scattering Transform to provide theoretical predictions in this region. The modelling in each case is extended to the case of a finite delamination in the waveguide, and we study the effect of re-entering a bonded region on a strain wave. In each case considered we develop a measure of the delamination length in terms of the change in amplitude of the incident wave, and furthermore the structure of the wave provides further insight about the structure of the waveguide. Numerical simulations are developed using finite-difference techniques and pseudospectral methods, and these are detailed in the appendices. Finally, we consider the initial value problem for the Boussinesq equation with an Ostrovsky term, on a periodic domain. The initial condition for this equation does not necessary have zero mean on the interval. The mean value is subtracted from the function so that a weakly nonlinear solution to the problem can be constructed where all functions in this expansion have zero mean. This is necessary as the derived Ostrovsky equations have zero mean. The expansion is constructed in increasing powers of $\sqrt{\epsilon}$ up to and including $\O{\epsilon}$, where $\epsilon$ is a small amplitude parameter in the equation. We compare the results for a wide range of values of $\gamma$ (the coefficient of the Ostrovsky term) and varying mean values for the initial condition, to confirm that the expansion is valid. A comparison of the errors shows that the constructed expansion is correct and the errors behave as predicted by the expansion. This was further confirmed for non-unity coefficients in the equation.

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## Department

• Mathematical Sciences

## Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

2018

## Notes

A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.

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