J. Kaplunov

High-frequency homogenization for periodic media

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.

On Timoshenko-Reissner type theories of plates and shells

Abstract Linear TR theories (Timoshenko-Reissner theories) of isotropic plates and shells are discussed. These theories take into account the transversal shear deformation and rotation inertia. The main subject under consideration is the construction of these theories by the asymptotic method and the related error estimates for static and dynamic problems. In the dynamic case a method is suggested for the extension of the range of applicability of the TR theory.

A low-frequency model for dynamic motion in pre-stressed incompressible elastic structures

An asymptotic one–dimensional theory, with minimal essential parameters, is constructed to help elucidate (two–dimensional) low–frequency dynamic motion in a pre–stressed incompressible elastic plate. In contrast with the classical theory, the long–wave limit of the fundamental mode of antisymmetric motion is non–zero. The occurrence of an associated quasi–front therefore offers considerable deviation from the classical case. Moreover, the presence of pre–stress makes the plate stiffer and thus may preclude bending, in the classical sense. Discontinuities on the associated leading–order wavefronts are smoothed by deriving higher–order theories. Both quasi–fronts are shown to be either receding or advancing, but of differing type. The problems of surface and edge loading are considered and in the latter case a specific problem is formulated and solved to illustrate the theory. In the case of antisymmetric motion, and an appropriate form of pre–stress, it is shown that the leading–order governing equation for the mid–surface deflection is essentially that of waves propagating along an infinite string, a higher–order equation for which is derived.

Eigenvalue of a semi-infinite elastic strip

A semi-infinite elastic strip, subjected to traction free boundary conditions, is studied in the context of in-plane stationary vibrations. By using normal (Rayleigh–Lamb) mode expansion the problem of existence of the strip eigenmode is reformulated in terms of the linear dependence within infinite system of normal modes. The concept of Grams determinant is used to introduce a generalized criterion of linear dependence, which is valid for infinite systems of modes and complex frequencies. Using this criterion, it is demonstrated numerically that in addition to the edge resonance for the Poisson ratio ν=0, there exists another value of ν≈0.22475 associated with an undamped resonance. This resonance is best explained physically by the orthogonality between the edge mode and the first Lamé mode. A semi-analytical proof for the existence of the edge resonance is then presented for both described cases with the help of the augmented scattering matrix formalism.

Free localized vibrations of a semi-infinite cylindrical shell

Free vibrations of a semi-infinite cylindrical shell, localized near the edge of the shell are investigated. The dynamic equations in the Kirchhoff-Love theory of shells are subjected to asymptotic analysis. Three types of localized vibrations, associated with bending, extensional, and super-low-frequency semi-membrane motions, are determined. A link between localized vibrations and Rayleigh-type bending and extensional waves, propagating along the edge, is established. Different boundary conditions on the edge are considered. It is shown that for bending and super-low-frequency vibrations the natural frequencies are real while for extensional vibrations they have asymptotically small imaginary parts. The latter corresponds to the radiation to infinity caused by coupling between extensional and bending modes.

Edge waves and resonance on elastic structures: An overview:

Over 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.

An Asymptotically Consistent Model for Long-Wave High-Frequency Motion in a Pre-Stressed Elastic Plate

A one-dimensional asymptotic model is derived to elucidate the effect of pre-stress on long-wave high-frequency two-dimensional motion in an incompressible elastic plate. Solutions for the leading-order displacements and pressure increment are derived in terms of the long-wave amplitude; a governing equation for which is derived from the second-order problem. This equation is shown to become elliptic for certain states of pre-stress. Loss of hyperbolicity is shown to be synonymous with the existence of negative group velocity at low wavenumber. A higher-order theory is constructed, with solutions obtained in terms of both the long-wave amplitude and its second-order correction. An equation relating these is obtained from the third-order problem. The dispersion relations derived from the one-dimensional governing equations are also obtained by expansion of the corresponding exact two-dimensional relations, indicating asymptotic consistency. The model is highly relevant for stationary thickness vibration of, or transient response to high-frequency shock loading in, thin-walled bodies and also fluid-structure interaction. These are areas for which the effects of pre-stress have previously largely been ignored.

Three-dimensional edge waves in plates

This paper describes the propagation of three-dimensional symmetric waves localized 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 for any non-zero Poisson ratio in a plate with traction-free faces, the eigenfrequencies related to the first-order wave are complex valued.

A long-wave model for the surface elastic wave in a coated half-space

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.

High-frequency homogenization for checkerboard structures: defect modes, ultrarefraction, and all-angle negative refraction

The counterintuitive properties of photonic crystals, such as all-angle negative refraction (AANR) [J. Mod. Opt.34, 1589 (1987)] and high-directivity via ultrarefraction [Phys. Rev. Lett.89, 213902 (2002)], as well as localized defect modes, are known to be associated with anomalous dispersion near the edge of stop bands. We explore the implications of an asymptotic approach to uncover the underlying structure behind these phenomena. Conventional homogenization is widely assumed to be ineffective for modeling photonic crystals as it is limited to low frequencies when the wavelength is long relative to the microstructural length scales. Here a recently developed high-frequency homogenization (HFH) theory [Proc. R. Soc. Lond. A466, 2341 (2010)] is used to generate effective partial differential equations on a macroscale, which have the microscale embedded within them through averaged quantities, for checkerboard media. For physical applications, ultrarefraction is well described by an equivalent homogeneous medium with an effective refractive index given by the HFH procedure, the decay behavior of localized defect modes is characterized completely, and frequencies at which AANR occurs are all determined analytically. We illustrate our findings numerically with a finite-size checkerboard using finite elements, and we emphasize that conventional effective medium theory cannot handle such high frequencies. Finally, we look at light confinement effects in finite-size checkerboards behaving as open resonators when the condition for AANR is met [J. Phys. Condens. Matter 15, 6345 (2003)].