E.V. Nolde

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.

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.

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)].

Long wave asymptotic integration of the governing equations for a pre-stressed incompressible elastic layer with fixed faces

Abstract A model for long wave motion in the vicinity of the cut-off frequencies of a pre-stressed incompressible elastic layer is constructed. In contrast with most earlier studies, the faces of the layer are assumed fixed, rather than free, and in consequence there is no fundamental mode. Appropriate asymptotically approximate equations are derived and integrated through a systematic perturbation process. Models are presented for both anti-symmetric and symmetric motion. In the former case, leading-order solutions for displacement components and pressure increment are found in terms of the long wave amplitude, with an equation for this essential parameter obtained from the second-order problem. In the symmetric case, the essential parameter is the incremental pressure and the cut-off frequencies are defined through a transcendental equation of the form tan Λ=Λ . In both cases the governing equation for the appropriate essential parameter may become elliptic for certain states of pre-stress. The paper also details the derivation of higher-order theories.

A Refined Asymptotic Model of Fluid-Structure Interaction in Scattering by Elastic Shells

A refined asymptotic model of fluid-structure interaction in scattering by elastic shells is proposed. The model takes into consideration transverse compression of a shell by a fluid and some other phenomena. As an illustration, scattering of a plane acoustic wave by a circular cylindrical shell is considered. Comparison of numerical data corresponding various approximate approaches is provided.

An asymptotic analysis of initial-value problems for thin elastic plates

An asymptotic technique is developed to analyse initial-value problems in application to the three-dimensional theory of thin elastic plates. Various sets of long-wave initial data are considered, with arbitrary distribution along the plate thickness. To account for the initial data, composite asymptotic expansions are employed, utilizing both two-dimensional low-frequency and high-frequency models. The former correspond to the classical theories of plate bending and extension, and their refinements, whereas the latter are associated with the long-wave motions occurring in the vicinities of thickness stretch and shear resonance frequencies. Six cases of iteration process are revealed depending on the symmetry of the initial data and their thickness variation. For each case, approximate two-dimensional initial conditions are derived, including higher-order corrections for the two-dimensional low-frequency refined plate theories. The validity of the proposed approach is justified by comparison with the exact solution of the model plane problem for initial data with uniform distribution along the thickness and sinusoidal distribution along the mid-plane. The methodology of the paper has potential for more general initial-value problems specified over a narrow domain, including many of those commonly met in physical applied mathematics.

Analysis of transient waves in thin structures utilizing matched asymptotic expansions

SummaryTransient extensional waves in thin structures are analyzed. The structure motion is governed by the Love theory in the case of rods and the theories with modified inertia corresponding to higher order asymptotic approximations of the 3-D dynamic equations of elasticity in the case of plates and shells. The effect of a small viscosity is involved on the basis of the Voigt model. The asymptotic technique utilizing matched expansions is developed. The inner (boundary layer) expansion is applicable in the narrow vicinity of the quasi-front (the extensional wave front in the classical structural theories), while the outer expansion is applicable near the loaded edge of the structure. Three types of the quasi-front [the Poisson (elastic) quasi-front, the viscous quasi-front and the mixed quasi-front] are revealed.

On a class of three-phase checkerboards with unusual effective properties

Abstract We examine the band spectrum, and associated Floquet–Bloch eigensolutions, arising in a class of three-phase periodic checkerboards. On a periodic cell [ − 1 , 1 [ 2 , the refractive index, n , is defined by n 2 = 1 + g 1 ( x 1 ) + g 2 ( x 2 ) with g i ( x i ) = r 2 for 0 ⩽ x i 1 , and g i ( x i ) = 0 for − 1 ⩽ x i 0 where r 2 is constant. We find that for r 2 > − 1 the lowest frequency branch goes through origin with linear behaviour, which leads to effective properties encountered in most periodic structures. However, the case whereby r 2 = − 1 is very unusual, as the frequency λ behaves like k near the origin, where k is the wavenumber. Finally, when r 2 − 1 , the lowest branch does not pass through the origin and a zero-frequency band gap opens up. In the last two cases, effective medium theory breaks down even in the quasi-static limit, while the high-frequency homogenization [R.V. Craster, J. Kaplunov, A.V. Pichugin, High-frequency homogenization for periodic media, Proc. R. Soc. Lond. Ser. A 466 (2010) 2341–2362] neatly captures the detailed features of band diagrams.

Composite wave models for elastic plates

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.