Abstract
The long-term challenge of formulating an asymptotically motivated wave theory for elastic plates is addressed. Composite two-dimensional models merging the leading or higher-order parabolic equations for plate bending and the hyperbolic equation for the Rayleigh surface wave are constructed. Analysis of numerical examples shows that the proposed approach is robust not only at low- and high-frequency limits but also over the intermediate frequency range.
Highlights
A substantial fresh interest in the mechanics of thin elastic structures, inspired by the demands of modern advanced technologies, is strongly focused on modelling of microscale phenomena, e.g. see recent publications [1,2,3,4,5]
The long-standing problem concerned with the derivation of a two-dimensional (2D) hyperbolic plate theory supporting an asymptotically consistent short-wave behaviour is not yet solved even in the context of linear isotropic elasticity
Along with the simplest composite relation based on the Kirchhoff equation, we suggest a more sophisticated one corresponding to the refined plate equation
Summary
A substantial fresh interest in the mechanics of thin elastic structures, inspired by the demands of modern advanced technologies, is strongly focused on modelling of microscale phenomena, e.g. see recent publications [1,2,3,4,5]. A new prospect for deriving 2D hyperbolic equations has recently arisen due to the development of the asymptotic hyperbolic–elliptic model for the surface Rayleigh wave, see [20,21] and references therein. We attempt to establish 2D composite hyperbolic equations for an elastic plate using the Kirchhoff or refined asymptotic plate equations along with the Rayleigh wave equation; see [29,30] regarding the idea of composite equations incorporating both long- and short-wave limiting forms, for which a typical wavelength is much greater or smaller than the plate thickness. Surface loading in the form of a plane time-harmonic wave is considered
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