Investigation of low-frequency normal waves in a Biot layer surrounded by an elastic medium

The porous Biot layer surrounded by two elastic half-plates is considered. For this medium, two dispersion equations are established. These dispersion equations correspond to symmetric and antisymmetric parts of the medium. The investigation of these equations is conducted in the region of low frequencies. The imaginary roots of these equations in this region determine low-frequency normal waves. Bibliography: 6 titles.     

Shock waves in an elastic medium with cubic anisotropy

Low-intensity shock waves, propagating along the principal diagonal of a cube in an incompressible elastic medium possessing cubic symmetry, are considered. The form of the shock adiabatic in the phase plane of shears is obtained. Sections corresponding to non-decreasing entropy at the discontinuity and the conditions of evolutionarity of the discontinuity on it are indicated. The structure of the shock waves is investigated.     

Scattering of cylindrical waves by an elastic inclusion in an electrically conducting compressible medium

We study the problems of diffraction of cylindrical waves at an elastic homogeneous weakly conducting cylinder in a liquid weakly conducting medium and at a magnetohydrodynamic liquid cylinder in a magnetoelastic medium.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 127–131.     

Scattering of elastic waves through a heterogeneous medium

In this paper we study the elastic wave diffraction in R ³ through a heterogeneous medium, with periodic structure, which occupies a bounded domain. We show that, as the period tends to zero, the solution tends, in some sense, to the solution corresponding to the diffraction by an obstacle made of the classical “homogenized medium”. An analogous result is also proved for the scattering frequencies and the associated scattering functions.     

The propagation of elastic waves in a viscous-fluid-saturated porous medium

Elastic shock waves in a viscous-fluid-saturated porous medium are investigated. The porosity is only taken into account with respect to pores communicating with one another, and isolated pores are considered as elements of the elastic part of the porous skeleton. It is shown, using the theory of discontinuity, that in such a medium there are two types of vortex-free waves and one equivoluminal wave. Differential equations and their solution for determining the change in the wave-front intensity are obtained. The effect of the fluid viscosity and porosity on the propagation of spherical waves is demonstrated using an example.     

Love waves of SH type in an inhomogeneous transversely isotropic elastic medium

The asymptotics of high-frequency Love waves, which are analogous to transverse surface SH waves, is considered for a special type of anisotropy (transverse isotropy) of elastic media. The wave field is represented as a sum of the space-time (ST) caustic expansion and two additional ST ray series for faster (relative to the transverse surface wave) body waves, decaying exponentially with depth. Near the surface, the coefficients of the ST caustic and ray series, as well as the eikonals of waves, are determined in the form of expansions in a small parameter, which characterizes the proximity of the caustic of the ray field to the surface. With regard for the specific structure of the elasticity tensor of a transversely isotropic medium, the surface is treated as a plane. Interrelations between the parameters of elasticity, which are consistent with the conditions of the positivity of the elastic deformation energy and provide for the origination of the surface waves considered, are obtained.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 243–262.This work was supported by the Russian Foundation for Basic Research under grant No. 96-01-00666.     

Diffraction of elastic waves by an inclined crack in a layer

For the problem of the diffraction of normal modes by an inclined crack in an elastic layer, an integral equation with explicit representation of the Fourier symbol kernel is derived in the form of the product of matrices. The algorithm for calculating the wave fields, based on the analytical representations obtained, enables a rapid parametric analysis to be carried out of the influence of the size and orientation of the crack on the transmission of travelling waves. The influence of the inclination of the crack on the effects of resonance trapping and localization of wave energy, which were established previously for the case of a horizontal crack, is analysed.     

Propagation of quasielastic Rayleigh waves in an inhomogeneous,anisotropic, elastic medium

The propagation of modulated oscillations along the smooth surface free from normal stresses of an inhomogeneous, anisotropic, elastic body with parameters which vary little over a characteristic wavelength is considered.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 234–245, 1979.The author thanks the participants of the seminar on the propagation and diffraction of waves of the Leningrad Branch of the Mathematics Institute (LOMI), in particular, V. S. Buldyrev and V. M. Babich, for fruitful discussions and their attention to the work.     

Scattering of plane elastic waves by a rigid circular cylindrical inclusion imbedded in a random medium

The problem of scattering of a plane elastic wave by a rigid circular cylindrical-inclusion imbedded in a random medium is analyzed on the assumptions that the medium differs slightly from a homogeneous medium, is independent of axial space variable and is statistically homogeneous and isotropic in other space variables. By perturbation method, equations satisfied by the mean wave potentials in the exterior of a thin boundary layer wrapping around the inclusion are derived which are correct through terms of order ϵ², where ϵ measures the randomness of medium. These equations are the same equations satisfied by the wave potentials when ϵ = 0, except the original propagation constants being replaced by their corresponding effective propagation constants. This enables one to treat random problems as deterministic ones of which solutions are usually known. Once the mean wave potentials are found, the mean displacement vector can be calculated in the usual manner.     

Recovery of residual stress in a vertically heterogeneous elastic medium

We study the problem of identifying residual stress within athin subsurface layer in an elastic medium occupying a region = {(x₁, x₂, x₃) , R³: 0 < x₃ < L, where L } in space,where all parameters depend only on the depth x₃. Under thetheoretical framework of linear elasticity with initial stress,the incremental elasticity tensor of each material point iswritten as a sum of two terms, namely the elasticity tensorand the acoustoelastic tensor, both of which are taken hereas isotropic functions of their arguments. By imposing impulsiveloads and measuring the displacements at the boundary x₃ = 0,we recover the residual stress and its gradient there. If theresidual stress has a diagonal form, we can recover the residualstress inside the subsurface layer.     

The motion of a rigid spherical inclusion in an elastic medium under the action of plane waves

Two problems on the motion of a rigid spherical inclusion in an elastic medium under the action of a nonstationary, longitudinal, plane, compressional wave and a harmonic shear wave are considered. Using the method of vector eigenfunctions and the integral Laplace transform with respect to time their precise solutions are constructed. Numerical results are given which illustrate the dependence of characteristics of motion of the inclusion on parameters of the falling wave.Translated from Dinamicheskie Sistemy, No. 9, pp. 37–47, 1990.     

Phase transition for models of an elastic medium with residual stress

A variational problem on martensite-austenite phase transitions in a continuous medium is considered. The energy functional of this problem depends on two parameters: the temperature, which runs all real values, and the positive surface tension coefficient. A half-plane is divided into three open zones. In the first zone, only the martensite one-phase equilibrium state is realized. In the second zone, only the austenite one-phase equilibrium state is realized, whereas, in the third zone, any equilibrium state is a two-phase one. On the interface surfaces separating the zones, only those equilibrium states are realized that are typical for adjoining zones. In the homogeneous and isotropic case, the explicit solution to the problem is given provided that the surface tension coefficient is zero. Bibliography: 12 titles. Illustrations: 4 figures. Translated fromProblemy Matematicheskogo Analiza, No. 17, 1997, pp. 153–191.     

Scattering of emerging normal elastic waves for a crystalline plate in a liquid medium

Frequency dependences are given for the imaginary components of the three lowest energy scattering branches of the dispersion spectra of symmetric and antisymmetric normal emerging waves propagating along the elastically equivalent directions in a plate of single-crystal Rochelle salt immersed in water. Donetsk State University. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 156–161, 1999.     

Propagation and interaction of short waves in a homogeneous transversally isotropic elastic medium

The Cauchy problem for the equations of motion of a homogeneous transversally isotropic elastic medium is considered. For its solution, a short-wavelength asymptotic expansion is constructed, which is also applicable near specific directions. The resonance set, i.e., the set of points at which the ray expansion cannot be used is described.