An asymptotic procedure based upon a two-scale approach is developed for wave propagation in a doubly periodic inhomogeneous medium with a characteristic length scale of microstructure far less than that of the macrostructure. In periodic media, there are frequencies for which standing waves, periodic with the period or double period of the cell, on the microscale emerge. These frequencies do not belong to the low-frequency range of validity covered by the classical homogenization theory, which motivates our use of the term 'high-frequency homogenization' when perturbing about these standing waves. The resulting long-wave equations are deduced only explicitly dependent upon the macroscale, with the microscale represented by integral quantities. These equations accurately reproduce the behaviour of the Bloch mode spectrum near the edges of the Brillouin zone, hence yielding an explicit way for homogenizing periodic media in the vicinity of 'cell resonances'. The similarity of such model equations to high-frequency long wavelength asymptotics, for homogeneous acoustic and elastic waveguides, valid in the vicinities of thickness resonances is emphasized. Several illustrative examples are considered and show the efficacy of the developed techniques.

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The paper deals with the three-dimensional problem in linear isotropic elasticity for a coated half-space. The coating is modelled via the effective boundary conditions on the surface of the substrate initially established on the basis of an ad hoc approach and justified in the paper at a long-wave limit. An explicit model is derived for the surface wave using the perturbation technique, along with the theory of harmonic functions and Radon transform. The model consists of three-dimensional 'quasi-static' elliptic equations over the interior subject to the boundary conditions on the surface which involve relations expressing wave potentials through each other as well as a two-dimensional hyperbolic equation singularly perturbed by a pseudo-differential (or integro-differential) operator. The latter equation governs dispersive surface wave propagation, whereas the elliptic equations describe spatial decay of displacements and stresses. As an illustration, the dynamic response is calculated for impulse and moving surface loads. The explicit analytical solutions obtained for these cases may be used for the non-destructive testing of the thickness of the coating and the elastic moduli of the substrate.

SummaryA two-scales approach, for discrete lattice structures, is developed that uses microscale information to find asymptotic homogenized continuum equations valid on the macroscale. The development recognises the importance of standing waves across an elementary cell of the lattice, on the microscale, and perturbs around the, potentially high frequency, standing wave solutions. For examples of infinite perfect periodic and doubly-periodic lattices the resulting asymptotic equations accurately reproduce the behaviour of all branches of the Bloch spectrum near each of the edges of the Brillouin zone. Lattices in which properties vary slowly upon the macroscale are also considered and the asymptotic technique identifies localised modes that are then compared with numerical simulations. IntroductionDiscrete mass-spring lattice systems form classical models of crystal vibrations in solid state physics (1, 2) and were used by Newton, Kelvin, Born, and many others, to model and interpret wave phenomena and these models were instrumental in the development of wave mechanics; a review of the historical literature is contained in Brillouin's monograph (2). Lattice models remain a valuable and instructive way of modelling and understanding fundamental wave phenomena in crystal lattices and cellular structures (3). A common characteristic behaviour of such models is that they exhibit band-gaps, (4), namely bands of frequencies for which waves do not propagate through the atomic lattice. More recently such discrete models have been used for engineering structures (5) with a view to designing smart structures capable of filtering out various frequencies. These discrete systems, exhibiting band gap behaviour, are closely related to periodic continuous media, for instance photonic crystal fibres (6,7,8), for which band-gaps occur and that have numerous and varied industrial applications; in some limits there is a direct analogy between the discrete and continuum models (9). In the mechanics of cellular structures and lattices, considerable knowledge has been gained about the static and low-frequency behaviour in terms of homogenized models, but comparatively less is known of their high frequency behaviour.A common feature of both discrete and continuum models, that are defect-free and infinite in extent, is that the periodicity of the structure leads to dispersion relations between

This study revisits Timoshenko beam theory (TBT). It discusses at depth a more consistent and simpler governing differential equation. The so-called second spectrum is also addressed. Then, we provide the asymptotic justification of the aforementioned differential equation along with detailed discussion of the boundary and initial conditions. The paper also presents remarks of historical character, in the context of other pertinent studies.

Abstract:Explicit asymptotic formulations are derived for Rayleigh and Rayleigh-type interfacial and edge waves. The hyperbolic-elliptic duality of surface and interfacial waves is established, along with the parabolic-elliptic duality of the dispersive edge wave on a Kirchhoff plate. The effects of anisotropy, piezoelectricity, thin elastic coatings, and mixed boundary conditions are taken into consideration. The advantages of the developed approach are illustrated by steady-state and transient problems for a moving load on an elastic half-space.

Copyright @ 2011 SAGE PublicationsOver 50 years have elapsed since the first experimental observations of dynamic edge phenomena on elastic structures, yet the topic remains a diverse and vibrant source of research activity. This article provides a focused history and overview of such phenomena with a particular emphasis on structures such as strips, rods, plates and shells. Within this context, some of the recent research highlights are discussed and the contents of this special issue of Mathematics and Mechanics of Solids on dynamical edge phenomena are introduced

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