On the edge-wave of a thin elastic plate supported by an elastic half-space (original) (raw)
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The paper is concerned with a bending edge wave on a thin orthotropic elastic plate resting on the Winkler-Fuss foundation. The main focus of the contribution is on derivation of a specialised reduced model accounting for the contribution of the bending edge wave to the overall dynamic response, allowing simplified analysis for a number of dynamic problems. The developed formulation includes an elliptic equation associated with decay over the interior, and a beam-like equation on the edge governing wave propagation accounting for both bending moment and modified shear force excitation, thus highlighting a dual parabolic-elliptic nature of the bending edge wave. A model example illustrates the benefits of the approach.
Edge bending wave on a thin elastic plate resting on a Winkler foundation
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2016
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.
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This Letter deals with an analysis of bending edge waves propagating along the free edge of a Kirchhoff plate supported by a Winkler foundation. The presence of a foundation leads to a nonzero cutoff frequency for this wave, along with a local minimum of the associated phase velocity. This minimum phase velocity corresponds to a critical speed of an edge moving load and is analogous to that in the classical 1D moving load problem for an elastically supported beam.
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The propagation of waves along an elastic layer of uniform thickness has been an area of active research for many years. Many contributions have been made to the study of small amplitude wave propagation in a linear isotropic elastic layer, almost all in respect of traction-free boundary conditions. In this paper the associated dispersion relation is briefly reviewed. We also consider two other non-classical types of the boundary conditions, the so-called fixed and free-fixed problems. The associated dispersion relations are first investigated numerically, from which it is shown that no analogues of classical bending or extension exists. The fixed-free case is of particular interest as, unlike the other two cases, it does not decompose into symmetric and anti-symmetric parts. A representation of the associated dispersion relation is established in terms of the fixed and free dispersion relations. Long wave approximations of the phase are obtained; it is envisaged that these approximations will allow future establishment of appropriate long wave models.
Three-dimensional edge waves in plates
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2008
The paper describes the propagation of three-dimensional symmetric waves localised near the traction free edge of a semi-infinite elastic plate with either traction free or fixed faces. For both types of boundary conditions, we present a variational proof of the existence of the low order edge waves. In addition, for a plate with traction free faces and zero Poisson ratio, the fundamental edge wave is described by a simple explicit formula, and the first order edge wave is proved to exist. Qualitative variational predictions are compared with numerical results, which are obtained using expansions in three-dimensional Rayleigh-Lamb and shear modes. It is also demonstrated numerically that whatever non-zero Poisson ratio in a plate with traction free faces, the eigenfrequencies related to the first order wave are complex valued.