Solitary wave propagation in elastic bars with multiple sections and layers

Abstract

In this paper we present a numerical scheme for solving a system of Boussinesq-type equations. This can correspond to longitudinal displacements in a multi-layered elastic bar with delamination, with conditions on the interface between the sections of the bar. The method is initially presented for two coupled equations in each section and multiple sections in the bar, and later extended to any number of layers. Previous works have presented a similar method constructed using finite-difference methods, however these only solved for two sections of the bar at a time which limited the scope of studies using these methods.The new method presented here solves for all sections at a given time step and therefore the transmitted and reflected waves in each section of the bar can be studied. The new results are shown to be in excellent agreement with previously obtained results and a further study is performed showing that the delamination width can be inferred from the changes to the incident soliton. The generalised form of this method, for any number of sections and layers with coupling terms independent of time derivatives, can be used to study the behaviour of longitudinal waves in more complicated waveguides in future studies.

Scattering of an Ostrovsky wave packet in a delaminated waveguide

“…We will use a semi-analytical method, as the KdV and Ostrovsky equations were derived analytically but must be solved numerically. To compute the direct numerical simulations of the system (1)-( 4) we make use of the method presented in [28], with ∆x = 0.01 and ∆t = 0.01.…”

Nonlinear Longitudinal Bulk Strain Waves in Layered Elastic Waveguides

“…This direct method was limited to only two sections in the structure at a time, so it could not be used for a short delamination region. The method was extended in [22] so that it can solve for multiple sections in the bar and for multiple layers.…”

“…In physical applications it is often of interest to consider the case when the delamination is finite. In this paper we will present a single case for a bi-layer with three sections, similarly to [22]. We take the same parameters as for Figure 5 and present the results in Figure 6, where we denote the strains in section 1 as e (1) , and so on for sections 2 and 3.…”

“…However, as the length of the delamination region is reduced, the amplitude of the soliton in the second bonded region will tend towards that of the incident soliton. This gives a quantitative measure of the delamination length [22].…”

Special issue of Wave Motion - “Nonlinear waves in solids: In memory of Alexander M. Samsonov”

“…M.R. Tranter considers the propagation of a solitary wave in non-homogeneous layered bars described by a system of coupled Boussinesq-type equations with piecewise-constant coefficients subject to continuity of displacement and stress on the boundaries with the help of numerical simulations [8]. The numerical scheme is applied to describe the transmission and reflection of longitudinal waves in layered waveguides with delamination, extending previous studies.…”