Dynamical problems on longitudinal shear of a half-space with nonhomogeneities

Problems on diffraction of shear waves at cavities and rigid inclusions in a half-space with a fixed boundary and in one with a boundary which is free from forces are considered. The problem is reduced to a singular integral equation in the case of rigid inclusion in a half-space and to a Fredholm equation of the second kind in the case of a cavity.Translated from Dinamicheskie Sistemy, No. 9, pp. 47–54, 1990.     

Diffraction of pulse shear waves in a tunnel cavity of curvilinear section in an anisotropic massif

The problem of diffraction of longitudinal shear waves in the form of periodic triangular pulses by a cylindrical tunnel cavity in a rectilinearly orthotropic unbounded massif is reduced to a number of problems on the diffraction of harmonic shear waves of different relative length. Each of the problems on stationary diffraction is solved in series in the basis solutions of shear vibrations equations by using an affine coordinate transformation of the original domain. It is proposed to use a point method of least squares to seek the coefficients of the series. The influence of the degree of anisotropy of the elastic properties of the massif on the shear stress concentration at the boundary of a circular section is analyzed for a different position of the incident triangular pulse front relative to center of the cavity.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 70–76, 1989.     

Elastic stress concentration near boundary defects in the case of a longitudinal shear deformation

We study the concentration of stresses due to two boundary defects located in an elastic half-space with stress-free boundary loaded at infinity with a constant shear load. The problem is reduced to solving singular integral equations for the cases in which the half-space contains defects of different types: two cracks, two inclusions, and a crack and an inclusion, whose solutions are sought by the method of mechanical quadratures. The interaction of the defects as they approach each other and the influence of their relative sizes are studied numerically. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

Diffraction of a monochromatic plane shear wave on a reinforced cylindrical cavity in an elastic half-space

This article examines a boundary-value problem concerning the diffraction of a monochromatic plane shear wave on a reinforced cylindrical cavity in an elastic half-space. It is assumed that longitudinal shear stresses are absent and that the normal displacements over the entire boundary are specified. Through the use of a special form of the Lamé representation in cylindrical coordinates, the problem is reduced to the determination of scalar functions which satisfy the Helmholtz equation. The coefficients of the Fourier expansions of these functions in the angular coordinate are written as the sum of Fourier and Weber integrals. The densities of these integrals are determined exactly. A specific example is examined.Translated from Dinamicheskie Sistemy, No. 5 pp. 42–49, 1986.     

A solution for transient surface waves of the B-G type in a dissipative piezoelectric crystal

We derive a solution to the problem of shear horizontal electroacoustic surface waves in a piezoelectric half-space. We formulate a time-domain dynamic problem accounting for time dispersion for both electric and elastic fields and use a separation of variables to express the solution in terms of a wave propagator. Transient surface waves of the B-G type are found to propagate with a constant speed and exponentially decay in space. Their amplitude vanishes at large distances from the boundary as the reciprocal of the depth. Dispersive and non dispersive solutions are compared.     

Reflection of plane waves from the boundary of a pre-stressed compressible elastic half-space

The effect of pre-stress on the propagation and reflection ofplane waves in an incompressible isotropic elastic half-spacehas been examined recently by the authors (Ogden & Sotiropoulos,1997). In the present paper the corresponding analysis for compressiblematerials is detailed. In the two-dimensional context consideredfor incompressible materials the (homogeneous) plane waves werenecessarily shear waves. By contrast, in the compressible contextpure shear waves can propagate only in specific directions inthe considered principal plane and, in a general direction,a quasi-shear wave may be accompanied by a quasi-longitudinalwave, as is the case in the anisotropic linear theory. The dependenceof the (in-plane) slowness section on the pre-stress (and finitedeformation) and on the choice of constitutive law is elucidated.This information is used to determine the reflection coefficientsfor reflection of either a (quasi-) shear wave or a (quasi-)longitudinal wave from the boundary of the half-space and tocharacterize the different cases which arise depending on thegeometry of the slowness section. The theoretical results are illustrated by numerical calculationsfor the range of possible types of behaviour with referenceforms of strain-energy function and different states of finitedeformation and to the question of stability of the half-space.     

An energy approach to the proof of the existence of Rayleigh waves in an anisotropic elastic half-space

An approach based on investigating the energy functional is applied for the first time to the classical problem of Rayleigh waves in an anisotropic half-space with a free boundary. The main object of the investigation is an ordinary differential operator in a variable characterizing the depth. An investigation of the spectrum by variational methods enables a new proof to be given of the existence of a Rayleigh wave in a linear elastic half-space with arbitrary anisotropy, which does not rest on the Stroh formalism.     

Action of a shear stress pulse on the surface of a viscoelastic anisotropic half-space

The boundary value problem of the behavior of a viscoelastic half-space subjected to a surface shear stress pulse is solved in the linear formulation on the basis of dynamical correspondence principle. The medium occupying the half-space possesses the property of transverse isotropy. The exact solution of the problem is obtained using integral transforms. The solution is analyzed with reference to a Maxwell-Voigt model.Moscow Region. Translated from Mekhanika Polimerov, No. 5, pp. 933–937, September–October, 1969.     

Lamb Problem for a Half-Space Covered with a Two-Axially Prestretched Layer

Within the framework of a piecewise homogeneous body model, with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies, an approach for investigating the Lamb problem for a half-space covered with a layer is developed. It is assumed that the half-space and the cover layer have two-axial initial stresses operating parallel to the layer plane. To the free face of the layer, a normal point force changing harmonically with time is applied. For solving the corresponding boundary value problems, the double-exponential Fourier transformation is employed. An algorithm for obtaining numerical results is proposed, which is examined in particular cases. Numerical results for the influence of prestretching the cover layer on the interfacial stresses are presented.     

A self-similar wave problem in a Prandtl-Reuss elastoplastic medium

We consider a self-similar piston problem in which stresses on the boundary of a half-space are changed instantaneously. The half-space is filled with a Prandtl–Reuss medium in a uniform stressed state. It is assumed that the formation of shock waves is possible in the medium. We prove the existence of a solution to the problem in the cases when two or all three stress components are changed at the initial moment.     

Classical Solvability of a Model Problem in a Half-Space, Related to the Motion of an Isolated Mass of a Compressible Fluid

For an arbitrary finite time interval, the unique solvability of a linear half-space problem is obtained in Hölder classes of functions. The problem arises as the result of the linearization of a free boundary problem for the Navier--Stokes system governing the unsteady motion of a finite mass of a compressible fluid. The boundary conditions in the linear problem are noncoercive because of the surface tension acting on the free boundary. This fact presents the main difficulty in the problem, while the differential system in itself is parabolic in the sense of Petrovskii. The principal idea of the investigation is to reduce the noncoercive problem to a coercive one with zero coefficient of the surface tension. Bibliography: 6 titles.     

Reflection of plane waves from the stress-free isothermal and insulated boundaries of a nonlocal thermoelastic solid

The generalized thermoelasticity theory based upon the Green and Naghdi model II of thermoelasticity as well as the Eringen’s nonlocal elasticity model is used to study the propagation of harmonic plane waves in a nonlocal thermoelastic medium. We found two sets of coupled longitudinal waves which are dispersive in nature and associated with attenuation. In addition to the coupled waves, there also exists one independent vertically shear type wave which is dispersive but without any attenuation. All these waves are found to be influenced by the elastic nonlocality parameter. Furthermore, the shear type wave is found to to be associated with a critical frequency, while the coupled longitudinal waves may have critical frequencies under constraints. The problem of reflection of the thermoelastic waves at the stress-free insulated and isothermal boundary of a homogeneous, isotropic nonlocal thermoelastic half-space has also been investigated. The formulae for various reflection coefficients and their respective energy ratios are determined in various cases. For a particular material, the effects of the angular frequency and the elastic nonlocal parameter have been shown on the phase speeds and the attenuation coefficients of the propagating waves. The effect of the elastic nonlocality on the reflection coefficients as well as the energy ratios has been observed and depicted graphically. Finally, analysis of the various results has been interpreted.     

Rayleigh waves from a point source on a boundary free of tensions

The solutions of equations of elasticity theory that have a discontinuity only on a boundary free of tensions (Rayleigh waves) are considered. Initial data for the complex intensity of the surface Rayleigh waves are found in two simple media. The first elastic medium fills a half-space with Lamé parameters and density dependent on depth. The second medium is bounded by a curve determined by a natural equation. The parameters of the second medium depend on the arc length along the curve. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 132–149.     

On Love-type waves in a finitely deformed magnetoelastic layered half-space

In this paper, the propagation of Love-type waves in a homogeneously and finitely deformed layered half-space of an incompressible non-conducting magnetoelastic material in the presence of an initial uniform magnetic field is analyzed. The equations and boundary conditions governing linearized incremental motions superimposed on an underlying deformation and magnetic field for a magnetoelastic material are summarized and then specialized to a form appropriate for the study of Love-type waves in a layered half-space. The wave propagation problem is then analyzed for different directions of the initial magnetic field for two different magnetoelastic energy functions, which are generalizations of the standard neo-Hookean and Mooney?CRivlin elasticity models. The resulting wave speed characteristics in general depend significantly on the initial magnetic field as well as on the initial finite deformation, and the results are illustrated graphically for different combinations of these parameters. In the absence of a layer, shear horizontal surface waves do not exist in a purely elastic material, but the presence of a magnetic field normal to the sagittal plane makes such waves possible, these being analogous to Bleustein?CGulyaev waves in piezoelectric materials. Such waves are discussed briefly at the end of the paper.     

Solution of certain boundary value problems of potential theory and elasticity theory for a half-space with a paraboloidal cavity

A method for solving boundary value problems for the Laplace equation in a half space with a paraboloidal cavity or a paraboloidal segment is suggested. Using formulas for the re-expansion of the fundamental solutions of the Laplace equation from a cylindrical to a paraboloidal coordinate system and their inverses, the basic and certain mixed problems are reduced to Fredholm integral equations or systems of equations of the second kind with completely continuous operators in a certain Hilbert space. The problem of torsion of an elastic half-space with a paraboloidal cavity by a stamp linked to part of the surface of the paraboloid and the problem of distribution of electricity on a paraboloidal segment located in the half-space are considered.Translated from Dinamicheskie Sistemy, No. 4, pp. 33–40, 1985.     

Reflection of plasma waves from a plane boundary

We state and analytically solve the linearized problem of the reflection of plasma waves from the half-space boundary in gas plasma for the first time. We consider mirror and diffusion boundary conditions and find the reflection coefficient as a function of the original problem parameters. We analyze the long-wave limit. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 3, pp. 498–510, March, 2007.     

Transient response of SH waves in a layered half-space with sub-surface and interface cracks

The transient scattering of SH waves by sub-surface and interface cracks parallel to the free surface in a layered elastic solid is investigated. The problem in frequency domain is solved by using a hybrid method which combines the finite element method of the near field with the boundary integral representation of the far field. The transient responses are then obtained by inverting the spectra via fast Fourier transform with the incident pulse Ricker of wavelet. Numerical results are presented for the surface displacements, dynamic stress intensity factors and wave motion in the layered half-space. Furthermore, the propagations of reflected, diffracted, and direct impact waves at any instant are clearly identified by the present method. To understand the mechanism of elastic wave interaction is very important in the field of ultrasonic non-destructive evaluation (NDE) and fracture mechanics studies.     

Love waves in a fluid-saturated porous layer under a rigid boundary and lying over an elastic half-space under gravity

In this paper, mathematical modeling of the propagation of Love waves in a fluid-saturated porous layer under a rigid boundary and lying over an elastic half-space under gravity has been considered. The equations of motion have been formulated separately for different media under suitable boundary conditions at the interface of porous layer, elastic half-space under gravity and rigid layer. Following Biot, the frequency equation has been derived which contain Whittaker’s function and its derivative that have been expanded asymptotically up to second term (for approximate result) for large argument due to small values of Biot’s gravity parameter (varying from 0 to 1). The effect of porosity and gravity of the layers in the propagation of Love waves has been studied. The effect of hydrostatic initial stress generated due to gravity in the half-space has also been shown in the phase velocity of Love waves. The phase velocity of Love waves for first two modes has been presented graphically. Frequency equations have also been derived for some particular cases, which are in perfect agreement with standard results. Subsequently the lower and upper bounds of Love wave speed have also been discussed.     

Some quasi-static problems for a viscoelastic half-space

Some problems for a viscoelastic half-space are solved in the case of noncommutative operators. A solution of the equilibrium equation analogous to the Boussinesq-Papkovich solution is constructed. The problem of a normal pressure acting on the boundary of a viscoelastic half-space is solved. Two forms of this solution are obtained and both are used in the following problems, the problem of a concentrated load moving over the boundary of a half-space and the problem of a circular rigid stamp. The case of periodic motion of a periodic load is investigated with reference to the example of motion in a circle. At constant Poisson's ratio the solution of the problem of a stamp can be used for determining the creep or relaxation function.Mekhanika Polimerov, Vol. 2, No. 3, pp. 392–402, 1966Presented 12 November 1965 at the Riga Conference on Polymer Mechanics.     

The Love's problem for hyperbolic thermoelasticity

In our paper we investigated the initial-boundary value problem for elastic layer situated on half space of another elastic medium. In this medium the thermomechanical interactions were taken into consideration. The system of equations with initial-boundary conditions describes the phenomenon of wave propagation with finite speed. In our problem there are two surfaces ie. free surface and contact surface between layer and half space. On the free surface are setting boundary conditions for normal and tangent surface force. We consider two types of contact between layer and half-space: rigid contact and slip contact. The initial-boundary value problem was solved by using integral transformations and Cagniard-de Hoope methods. From the solution of this problem follows that in layer and half space exist some kind of thermoelastic waves. We investigated moreover the conditions which should be fullfiled for propagation of Rayleigh and Love's type waves on the contact surface between layers and half space. The results obtained in our investigation were used in technical applications especially engineering design and diagnostics of roads and airfields. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)