N. V. Glushkova

3-D elastic stress singularity at polyhedral corner points

Abstract The three-dimensional stress singularity at the top of an arbitrary polyhedral corner is considered. Based on the boundary integral equations, the problem is reduced by the Mellin transform to a system of certain one-dimensional integral equations. The orders of stress singularity are spectral points of the integral operators while angular distribution and intensity factors are found as residues at those points. Numerical results are obtained by means of the Galerkin discretization scheme using expansions in terms of orthogonal polynomials with the proper weights. Some of the results illustrating the orders dependence on the elastic properties and corner geometry for a wedge-shaped punch and a crack, for an elastic trihedron and for a surface-breaking crack are given.

Blocking property of energy vortices in elastic waveguides

Energy vortices in time-averaged energy flows of time-harmonic fields are considered. The purpose of the paper is to verify the supposition that the energy flux along an elastic waveguide with an obstacle is blocked completely at the stop frequencies by the vortices. It is hoped that this work will draw attention to the analogy between energy and fluid flows which is potentially fruitful for understanding various wave phenomena. The normal-mode diffraction by a surface punch on an elastic layer is taken as an example. For tracing energy streamlines in the near field up to the obstacle, a special efficient semianalytical approach has been developed. To illustrate the stated supposition, plots of transmission coefficients versus frequency and figures of the near-field streamline structures are given.

Energy vortices and backward fluxes in elastic waveguides

Abstract The behavior of energy fluxes in a multilayer halfspace is investigated under loading by harmonic stresses. The multilayering produces new phenomena compared to a homogeneous halfspace. In such system energy vortices and backward energy propagation are possible, which satisfy mathematical rigor and physical conditions of energy radiation. Various examples of these phenomena are given and the conditions under which they arise are formulated.

Natural resonance frequencies, wave blocking, and energy localization in an elastic half-space and waveguide with a crack

Sharp stopping of time-harmonic wave transmission in elastic structures with defects is considered as a manifestation of the well-known trapped mode effect. It is associated with natural resonance poles lying close to the real axis in the complex frequency plane. Nonresonant wave blocking may also occur due to antiphase combination of the incident and scattered waves. The present paper is aimed to give an insight into such phenomena using an analytically based computer model which strictly takes into account all wave interactions in a cracked structure. Numerical examples are restricted to the case of a line horizontal crack in a half-plane or in a layer (2D in-plane motion), that is, nevertheless, quite enough to demonstrate two kinds of the Rayleigh wave stopping mechanisms (resonant and nonresonant) as well as a possibility of pure real natural resonance frequencies and of a full blocking effect with energy localization.

Influence of porosity on characteristics of rayleigh-type waves in multilayered half-space

Poroelastic fluid-saturated multilayered half-space is considered; particle motion in this half-space is described by Biot-Frenkel equations for two-phase media. A brief description of the derivation of integral and asymptotic representations for wave fields excited by a given surface load is presented; the influence of the porous microstructure on the form of dispersion curves and amplitude characteristics of excited traveling waves is analyzed. It is demonstrated using numerical examples that the amplitude of additional modes occurring due to the presence of a microstructure can be essentially larger than the amplitudes of principal modes present in the waveguide with purely elastic layers.

Blocking of traveling waves and energy localization due to the elastodynamic diffraction by a crack

The semianalytic methods developed for solving the problems of elastodynamic diffraction by a horizontal strip-like crack are used to analyze the transmission and reflection of Rayleigh waves in a half-plane and normal modes in a layer with the aim to determine the parameters of blocking and study the blocking mechanism. The resonance blocking is shown to be accompanied by energy localization near the crack. For the case of a crack in a half-plane, a possibility of nonresonance blocking is revealed. The transmission and reflection coefficients are analyzed as functions of frequency, crack size, and crack depth. Numerical examples of energy streamline and power distribution structures are given for the resonance and nonresonance blocking, as well as examples of the behavior of stress intensity factors at the crack tips.

Surface waves in materials with functionally gradient coatings

Surface wave excitation and propagation in a half-space with a continuous dependence of elastic properties on depth has been considered. The total wave field generated by a given surface load can be represented as a convolution of the Green’s matrix of the medium with the vector of surface stresses, while the traveling surface waves are described by the residues from the poles of the Green’s matrix Fourier symbol. Comparison of the gradient and multilayer models shows that with a high enough number of partitions (layers), the dispersion properties and amplitude-frequency characteristics of surface waves in FGMs are described by the curves obtained upon a steplike approximation of gradient properties; however, with high contrast properties, the multilayer model can be more time-consuming. The effect of the vertical inhomogeneity of the medium on the surface wave characteristics has been analyzed for a series of typical dependences occurring in micro- and nanocoatings due to diffusions or technological features of sputtering and gluing of protective films.

The local Green's function method in singularly perturbed convection-diffusion problems

Previous theoretical and computational investigations have shown high efficiency of the local Green’s function method for the numerical solution of singularly perturbed problems with sharp boundary layers. However, in several space variables those functions, used as projectors in the Petrov-Galerkin scheme, cannot be derived in a closed analytical form. This is an obstacle for the application of the method when applied to multi-dimensional problems. The present work proposes a semi-analytical approach to calculate the local Green’s function, which opens a way to effective practical application of the method. Besides very accurate approximation, the matrix stencils obtained with these functions allow the use of fast and stable iterative solutions of the large sparse algebraic systems that arise from the grid-discretization. The advantages of the method are illustrated by numerical examples.

The effect of anomalous transparency of the water-air interface for a volumetric sound source

Anomalous transparency consists in the passage at certain frequencies of the majority of a source’s radiated energy through an interface, which usually gives strong reflection. Earlier, this effect was established for low-frequency point sources located in a fluid bounded by an air medium. In the case of volumetric sources, additional scattering of waves occurs between the interface of the media and the emitter surface; and the character of the manifestation of this effect is unclear. This work, using the solution to the integral equation corresponding to a boundary value problem, examines the emission of wave energy by spherical sources of different radius and its distribution between the energy flow passing through the water-air interface into the upper half-space and the energy flow going to infinity in the lower half-space. It has been established that the size of the source has virtually no effect on the energy distribution in the low-frequency range, i.e., on the anomalous transparency effect. We also analyze how the relative dimensions of spherical sources affect the energy characteristics in the mid- and high-frequency range.

Resonance blocking of traveling waves by a system of cracks in an elastic layer

Wave processes that occur in an elastic layer when waves traveling in it are diffracted by a system of horizontal cracks are investigated. Integral representations of wave fields are constructed in terms of the convolution of Green’s matrices and unknown jumps of displacements at the cracks. The displacement jumps are determined from the boundary integral equations, which are obtained from the initial boundary-value problem with the boundary conditions at crack faces being satisfied. The spectrum of the integral operator is studied for different variants of mutual crack arrangement and is compared with the spectrum of the corresponding operators for individual cracks; the relationship between the spectrum and the blocking effects is analyzed. The possibility of obtaining an extended frequency band of waveguide blocking in the case of groups of cracks is demonstrated.