Edge bending wave on a thin elastic plate resting on a Winkler foundation

J Kaplunov,G A Rogerson,D A Prikazchikov

doi:10.1098/rspa.2016.0178

J Kaplunov, G A Rogerson + Show 1 more

Open Access

https://doi.org/10.1098/rspa.2016.0178

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Abstract

This paper is concerned with elucidation of the general properties of the bending edge wave in a thin linearly elastic plate that is supported by a Winkler foundation. A homogeneous wave of arbitrary profile is considered, and represented in terms of a single harmonic function. This serves as the basis for derivation of an explicit asymptotic model, containing an elliptic equation governing the decay away from the edge, together with a parabolic equation at the edge, corresponding to beam-like behaviour. The model extracts the contribution of the edge wave from the overall dynamic response of the plate, providing significant simplification for analysis of the localized near-edge wave field.

Highlights

It is well known that the bending edge wave localized near the edge of a thin elastic plate is dispersive in contrast to the classical Rayleigh wave on an elastic half-space
We extend the previous results to the bending edge wave on a plate supported by a Winkler foundation, modelling a deformable half-space as a set of elastic springs
The proposed approach leads to insightful general observations, for example it underlines the dual parabolic–elliptic nature of the dispersive bending edge wave on an elastically supported plate contrasting with the hyperbolic–elliptic nature of the Rayleigh wave

Summary

Introduction

It is well known that the bending edge wave localized near the edge of a thin elastic plate is dispersive in contrast to the classical Rayleigh wave on an elastic half-space. Construction of an appropriate mathematical theory for the Rayleigh wave resulted in the general representation of the wave field in terms of harmonic functions, originated by Friedlander [11] and followed by Chadwick [12]. It is shown that the promoted assumption of the beam-like behaviour allows construction of the wave field in terms of an arbitrary single harmonic function, generalizing the known bending edge wave of a sinusoidal profile. The proposed approach leads to insightful general observations, for example it underlines the dual parabolic–elliptic nature of the dispersive bending edge wave on an elastically supported plate contrasting with the hyperbolic–elliptic nature of the Rayleigh wave.

Statement of the problem

Homogeneous wave of arbitrary profile

Perturbation scheme

Parabolic equation on the edge

Near-resonant harmonic excitation

Concluding remarks