Abstract
We study the scattering of a long longitudinal radiating bulk strain solitary wave in the delaminated area of a two-layered elastic structure with soft ("imperfect") bonding between the layers within the scope of the coupled Boussinesq equations. The direct numerical modelling of this and similar problems is challenging and has natural limitations. We develop a semi-analytical approach, based on the use of several matched asymptotic multiple-scale expansions and averaging with respect to the fast space variable, leading to the coupled Ostrovsky equations in bonded regions and uncoupled Korteweg-de Vries equations in the delaminated region. We show that the semi-analytical approach agrees well with direct numerical simulations and use it to study the nonlinear dynamics and scattering of the radiating solitary wave in a wide range of bi-layers with delamination. The results indicate that radiating solitary waves could help us to control the integrity of layered structures with imperfect interfaces.
Highlights
The discovery of solitons as extremely stable localised coherent structures1 is intrinsically linked with the discovery of the Inverse Scattering Transform (IST) for the Kortewegde Vries (KdV) equation—the method for the solution of a large class of initial-value problems on the infinite line
Developed as a purely analytical technique,7,8 in recent years the IST formed the basis for the development of efficient numerical approaches to the analysis of nonlinear problems, most notably within the framework of another famous integrable model, the Nonlinear Schr€odinger (NLS)
We studied the scattering of a long radiating bulk strain solitary wave in a delaminated bi-layer with a soft bonding between the layers
Summary
The discovery of solitons as extremely stable localised coherent structures is intrinsically linked with the discovery of the Inverse Scattering Transform (IST) for the Kortewegde Vries (KdV) equation—the method for the solution of a large class of initial-value problems on the infinite line. The latter has shown that solitons, when present, constitute the main part of the long-time asymptotics of initial-value problems for localised initial data, and this is the reason why solitons proved to be a very important part of the physical world we live in, across all scales.. The method has found a new application in our studies of the scattering of long longitudinal bulk strain solitons in a symmetric perfectly bonded layered bar with delamination.. Experimental studies of the excitation of the resonant radiation by localised waves have been a prominent theme in nonlinear optics and a number of other physical settings, see, for example, the reviews and the references therein It was shown, within the framework of a complex lattice model, that long nonlinear longitudinal bulk strain waves in a bi-layer with a sufficiently soft bonding can. The direct numerical simulations are expensive; we use our semi-analytical method to study the scattering of radiating solitary waves in a wide range of complex imperfectly bonded bi-layers with delamination, giving an elaborate description of the possible dynamical effects.
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