Reflection from a semi-infinite stack of layers using homogenization

A canonical scattering problem is that of a plane wave incident upon a periodic layered medium. Our aim here is to replace the periodic medium by a homogenized counterpart and then to investigate whether this captures the reflection and transmission behaviour accurately at potentially high frequencies.We develop a model based upon high frequency homogenization and compare the reflection coefficients and full fields with the exact solution. For some material properties it is shown that the asymptotic behaviour of the dispersion curves are locally linear near critical frequencies and that low frequency behaviour is replicated at these critical, high, frequencies. The homogenization approach accurately replaces the periodic medium and the precise manner in which this is achieved then opens the way to future numerical implementation of this technique to scattering problems.     

Universality of the frequency spectrum of laminates

We show that the frequency spectrum of two-component elastic laminates admits a universal structure, independent of the geometry of the periodic-cell and the specific physical properties. The compactness of the structure enables us to rigorously derive the maximal width, the expected width, and the density of the band-gaps – ranges of frequencies at which waves cannot propagate. In particular, we find that the density of these band-gaps is a universal property of classes of laminates. Rules for tailoring laminates according to desired spectrum properties thereby follow. We show that the frequency spectrum of various finitely deformed laminates are also endowed with the same compact structure. Finally, we explain how our results generalize for laminates with an arbitrary number of components, based on the form of their dispersion relation.     

Wavelength selection of ripples in a vertically vibrating dynamically thick granular layer due to density-wave refraction

A numerical study was made on the wavelength selection mechanism of the ripples observed on the surface of a granular layer that is oscillated vertically. Multiple collisions of the one-dimensional array of beads show a time-dependent particle distribution, which induces a density wave. The magnitude of the wave velocity is estimated by the theory of elasticity, which reveals the refraction of the density wave in a quasi-two-dimensional granular layer. Our theory explains how the vertical excitation of particles determines the horizontal characteristic scale on the surface, or the wavelength of the ripples, which is applicable even to the saturation regime of a dynamically thick granular layer, e.g., the one that is thick enough to allow an immobilized region in the lower part.     

Reflection and refraction of rays in pre-stressed solids

Rays in viscoelastic, pre-stressed solids are considered. The eikonal equation and the transport equation are derived for both transverse and longitudinal polarizations. The analogue of the standard Snell's law is established. Then the amplitudes of reflected and transmitted rays are determined. The pre-stress is shown to affect the geometry of reflection-refraction processes and the amplitudes of the rays emanating from the interface. Pre-stress effects are also found on traction-free surfaces.

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Sommario Si considerano raggi in solidi viscoelastici soggetti ad uno stress iniziale. L'equazione dell'iconale e l'equazione del trasporto sono dedotte sia per polarizzazione trasversale, sia per polarizzazione longitudinale. Stabilita la forma appropriata della legge di Snell, si determinano le ampiezze dei raggi riflessi e trasmessi. Si mostra che lo stress iniziale interviene sia negli angoli sia nelle ampiezze dei raggi reflessi e trasmessi. Effetti dello stress iniziale si presentano persino nel caso di riflessione sulla superficie libera di un semispazio.

     

Edge waves in asymmetrically laminated plates

We study edge waves propagating along the edge of an asymmetrically laminated elastic plate for which the out-of-plane component of displacement is coupled with the in-plane components. A Stroh-like formulation is used to show that such a plate can support at most two edge waves. An efficient method for computing the edge-wave speeds is proposed and explained through examples.     

Elasto-optic technique for measurement of elastic wave propagation in solids

The modulation of the optical path of the beam of a laser vibrometer in a specimen under acoustic excitation is measured at two planes, separated by a precisely known distance. The phase shift and the decrease in magnitude are used to calculate the phase velocity and attenuation, respectively. The method is demonstrated for a homogeneous specimen, and the results compare favorably with those obtained by a conventional ultrasonic technique. The method is then applied to measure specular and first diffraction-order reflection from a coplanar periodic array of particles in an elastic matrix and phase velocity spectra in a tetragonal periodic particulate composite. As expected, in a periodic composite the establishment of dispersive Floquet-type waves is observed throughout the entire periodic particulate composite.     

Plane waves in a hypoplastic half-space under periodic boundary conditions

The paper presents a numerical study of the propagation of plane waves in a half-space occupied by a granular material, with periodic boundary conditions for velocity or stresses prescribed at the boundary of the half-space. The constitutive behaviour of the material is described by a simplified hypoplastic equation which takes into account different values of the stiffness for different directions of deformation, and the coupling between shear and volumetric strains owing to dilatancy. These two features are responsible for a nonlinear character of longitudinal waves and for the generation of longitudinal motion by transverse disturbances. It is shown that longitudinal and transverse boundary disturbances produce qualitatively the same longitudinal waves at large distances from the boundary. As a longitudinal wave propagates, the amplitude of oscillations decreases and eventually vanishes, resulting in a single non-oscillating wave.Received: 10 September 2002, Accepted: 31 March 2003 Correspondence to: Y. A. Berezin     

复合材料弹性常数测定的超声波技术

本文简要地回顾了利用超声波脉冲技术测定复合材料弹性常数的历史，概述了现状、基本原理与基本方法，并对今后的研究方向提出了看法．     

Numerical interaction of boundary waves with perfectly matched layers in two space dimensional elastic waveguides

Perfectly matched layers (PMLs) are now a standard approach to simulate the absorption of waves in open domains. Wave propagation in elastic waveguides has the possibility to support back-propagating modes (propagating modes with oppositely directed group and phase velocities) with long wavelengths. Back-propagating modes can lead to temporally growing solutions in the PML. In this paper, we demonstrate that back-propagating modes in a two space dimensional isotropic elastic waveguide are not harmful to a discrete and finite width PML. Analysis and numerical experiments confirm the accuracy and stability of the PML.     

复合材料弹性常数测定的超声波技术

本文简要地回顾了利用超声波脉冲技术测定复合材料弹性常数的历史,概述了现状、基本原理与基本方法,并对今后的研究方向提出了看法．      

Impact response of an interface crack in a hybrid piezoelectric laminate

Summary  In a hybrid laminate containing an interfacial crack between piezoelectric and orthotropic layers, the dynamic field intensity factors and energy release rates are obtained for electro-mechanical impact loading. The analysis is performed within the framework of linear piezoelectricity. By using integral transform techniques, the problem is reduced to the solution of a Fredholm integral equation of the second kind, which is obtained from one pair of dual integral equations. Numerical results for the dynamic stress intensity factor show the influence of the geometry and electric field. Received 29 June 2001; accepted for publication 3 December 2001     

Reflection and Transmission of Elastic Waves at the Interface Between Water and a Double Porosity Solid

The reflection and transmission of an incident plane wave at an interface between water and a fluid-saturated double porosity solid are investigated. The properties of the three compressional waves and one shear wave in double porosity media are discussed in detail. The effect of the viscosity, permeability, and porosity on the phase velocity and attenuation of four bulk waves is presented. Comparison between the double porosity theory and the Biot theory reveals that there exists obvious difference in the phase velocity, attenuation and the reflection and transmission coefficients. Two cases of boundary conditions, i.e., the sealed-pore and the open-pore boundaries, are introduced in the numerical calculation. At last, the effect of the fracture permeability and porosity on the reflection and transmission coefficients considered. It is revealed that the amplitudes of the reflected and transmitted waves strongly depend the boundary condition, fracture permeability, and porosity.     

Effects of the local resonance in bending on the longitudinal vibrations of reticulated beams

This work investigates the dynamic behaviour of reticulated beams obtained by repeating a unit cell made up of interconnected beams or plates forming an unbraced frame. As beams are much stiffer in tension–compression than in bending, the longitudinal modes of such structures (governed by tension–compression at the macroscopic scale) can appear in the same frequency range as the bending modes of the elements. The condition of scale separation being respected for compression, the homogenization method of periodic discrete media is used to rigorously derive the macroscopic behaviour at the leading order. In the absence of bending resonance, the longitudinal vibrations of the structure are described at the macroscopic scale by the usual equation for beams in tension–compression. When there is resonance, the form of the equation is unchanged but the real mass of the structure is replaced by an effective mass which depends on the frequency. This induces an abnormal response in the neighbourhood of the natural frequencies of the resonating elements. This paper focuses on the consequences on the modal properties and the transfer function of the reticulated structure. The same macroscopic mode shape can be associated with several natural frequencies of the structure (but the deformation of the elements at the local scale is different). Moreover the vibrations are not transmitted when the effective mass is negative. These phenomena are first evidenced theoretically and then illustrated with numerical simulations.     

Interfacial effects in electromagnetic coupling within piezoelectric phononic crystals

In this paper, we discuss waves in piezoelectric periodic composite, with the emphasis on the connection between the electromechanical coupling and the effects of dispersion of Bloch-Floquet waves. A particular attention is given to structures containing interfaces between dissimilar media and localization of the electrical fields near such interfaces.     

Particle and drop dynamics in the flow behind a shock wave

The results of an investigation of the dynamics of hard particles and liquid drops in the flow behind a transmitted shock wave are presented. From the equation of motion of a particle in the shock wave, relations for the displacement, velocity and acceleration as functions of time and certain velocity-relaxation parameters taking into account the properties of the gas and the aerodynamic drag of the particles are obtained for unsteady flow around the particles at an acceleration of 10³–10⁴ m/s². It is shown that the velocity-relaxation parameters are universal. Approaches to finding the aerodynamic drag of freely-accelerating bodies from the dynamics of their acceleration after being suddenly exposed to the flow are considered. It is established that under these conditions the drop dynamics observed can be well described in terms of the same velocity-relaxation parameters with account for linear growth of the transverse drop size. All the kinematic functions obtained are confirmed experimentally.     

To the boundary and back—a numerical study

This study identifies the key parameters upon which energy absorption at artificial boundaries depends. A thorough numerical study is presented, of typical reflections from open computational boundaries, for problems governed by hyperbolic systems of equations. The emphasis is on systems, where it is often the combination of all boundary procedures that determine the quality of boundary treatment. We study dissipative numerical models which have so far not been analysed to the same extent as non-dissipative models and employ a Law-Wendroff-type scheme as a prototype. While it is widely accepted that dissipative models tend to give fewer problems than non-dissipative ones, we show a variety of cases where substantial reflections do occur even in ID and quasi-ID set-ups, where theory predicts best results. This can partly be explained by the vanishing of dissipation in the far field. Group velocity analysis, justifiable on the grounds of weak dissipation, predicts a pathological behaviour which is confirmed by numerical experiments. We demonstrate strong focusing of asymptotic errors generated at the artificial boundary. Internal reflections due to slowly expanding grids are shown for non-linear systems. The need for high-frequency boundary conditions naturally arises and combined low-high-frequency boundary recipes following Higdon, Vichnevetsky and Pariser are adapted to systems and tested. Partial cures are also discussed, mainly in terms of pointing out their theoretically limited potential.     

薄膜涂层材料中的广义瑞利波

基于弹性动力学的线性理论，建立了涂层材料中广义瑞利波传播的理论分析模型，并 且由波动方程和边界条件推导了波的频散方程．分析了慢层和快层对相速度频散的影响，给 出了不同层厚-波长比和不同涂层-基体密度比情况下广义瑞利波相速度的理论解．算例分 析分别比较了慢层和快层结构中波的相速度、群速度，以及随深度衰减的位移与应力振 幅．另外，相速度曲线和位移振幅曲线与文献中给出的结果吻合，验证了理论模型和分析过 程的正确性．     

Collective Bow Shock Ahead of a Transverse System of Spheres in a Supersonic Flow Behind a Moving Shock Wave

The results of investigating the dynamics and physical conditions of formation of a collective bow shock ahead of a system of spheres with the line of centers normal to the supersonic flow behind a traveling shock wave are presented. Two types of shock-wave patterns that necessarily precede the formation of the collective shock wave and correspond to regular and Mach interaction of the bow shocks were detected experimentally. On the basis of a local gasdynamic-discontinuity interference theory, quantitative criteria of the existence of these regimes and of the formation of a common shock wave are determined. These criteria are confirmed in a series of experiments for the transitional regimes.     

Evolution of plane disturbances in hypoplastic granular materials

Received July 5, 2000 / Published online February 14, 2001