Stresses in an elastic plate lying over a base due to strip-loading

The closed-form analytic expressions for the stresses at any point of an elastic plate coupling in different ways to a base as a result of a two-dimensional shear strip-loading are obtained. The contact between the horizontal layer and the base is either smooth-rigid or rough-rigid or welded. The variations of the shear stresses with the horizontal distance have been studied numerically. It is found that the effect of different boundary conditions on the stress field is significant and the stresses for an elastic layer lying over an elastic half-space differ considerably from those of an entire homogeneous elastic half-space.     

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.     

An Erratum in the Paper “A Plane Shear Wave Excited in an Elastic Half-Space by Oscillating Normal Pressure Distributed Nonuniformly”

Excitation of an elastic half-space by an oscillating normal load nonuniformly distributed over its surface is considered. The medium is assumed to be linear, isotropic, and homogeneous. Exterior load distribution is smooth, and the scale of its variation is small compared to wavelengths. A simple expression that describes the shear wave with plane front set, running away from the surface, is found. This wave vanishes if the load is uniform. Bibliography: 1 title.     

A Plane Shear Wave Excited in an Elastic Half-Space by Oscillating Normal Pressure Distributed Nonuniformly

Excitation of an elastic half-space by an oscillating normal load distributed nonuniformly over its surface is considered. The medium is assumed to be linear, isotropic, and homogeneous. The external load is distributed smoothly, and the extent of variation in it is large in comparison with the wavelength. A simple expression describing the shear wave with a plane front set, running away from the surface, is found. This wave disappears if the load is constant. Bibliography: 10 titles.     

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.     

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.     

The plane contact problem of steady thermoelasticity taking heat generation into account

The plane steady contact problem of thermoelasticity when there is heat generation from friction, which arises when an infinite cylindrical punch moves over the surface of an elastic half-space along its generatrix, is considered. It is assumed that heat exchange between the free boundary of the half-space and the surrounding medium obeys Newton's law, while the condition for ideal thermal contact exists in the region in which the solids interact. The problem is reduced to a system of three integral equations in the heat fluxes and temperature. The effect of the thermal and mechanical properties of the cylinder and the half-space on the main contact characteristics is investigated numerically.     

On the impact deformation of an incompressible half-space under the action of a shear load of variable direction

Using the example of a one-dimensional planar problem for a nonlinear elastic incompressible half-space, the loading process is considered in which the shear action on the boundary plane changes with respect to both intensity and direction. It is shown that, in the regions of the space where the nonlinearity of the medium becomes a significant factor, the solution in the nearfront region of the shock wave is determined by a system of nonlinear evolution equations. The general solution of the evolution system is obtained. As an example, a partial solution of the system is considered for one of the simplest boundary conditions. A parametric method is presented for determining the displacements on the basis of solution of the evolution system.     

Sources Acting on the Free Boundary of a Porous Biot Medium and Reflection on This Boundary

A homogeneous isotropic porous Biot half-space z 0 with free boundary z = 0 is considered. This half-space is excited by point sources situated on the boundary z = 0 and applied to the elastic phase. The following four sources are considered: (1) a normal force, (2) a center of tangent forces, (3) a center of rotation, and (4) a tangent force. For all these sources, the wave fields are established. Relationships between the wave fields in a porous medium and the respective wave fields in an elastic medium are investigated. The reflection coefficients on the free boundary of the porous Biot half-space are determined and studied. Bibliography: 11 titles.     

Two-dimensional problem of a two-temperature generalized thermoelastic half-space subjected to ramp-type heating

In this work, we will consider a half-space filled with an elastic material, which has constant elastic parameters. The governing equations are taken in the context of the theory of two-temperature generalized thermoelasticity. A linear temperature ramping function is used to more realistically model thermal loading of the half-space surface. The medium is assumed initially quiescent. Laplace and Fourier transform techniques are used to obtain the general solution for any set of boundary conditions. The general solution obtained is applied to a specific problem of a half-space subjected to ramp-type heating. The inverse Fourier transforms are obtained analytically while the inverse Laplace transforms are computed numerically using a method based on Fourier expansion techniques. Some comparisons have been shown in figures to estimate the effect of the ramping parameter of heating.     

An Axisymmetric Contact Problem with Given Region of Adhesion

The indentation of an elastic half-space by an axisymmetricrigid punch under normal load is considered. The contact areais divided into an inner region with adhesion, the dimensionof which is known beforehand, surrounded by an annular slipzone. Two different cases of the problem, given by the assumptionsof friction-free slip or Coulomb friction, are treated. In thefriction-free case the problem is formulated in terms of a coupledpair of Cauchy type integrals for Abel transforms of the normaland shear stresses at the surface of the half-space. These arecombined to an inhomogeneous Fredholm equation which is solvedby a method of successive approximations, and the stresses areobtained as Abel transform inversions of its solution. In thecase with Coulomb friction the problem is solved by a numericalmethod based on piecewise constant approximations to the surfacestresses.     

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.     

Impedance profile inversion via the first transport equation

The subject of this paper is the inverse reflection problem for a stratified elastic half-space. That is, a linear elastic medium, whose elastic properties depend only on depth from a planar free surface, is stimulated at t = 0 by a plane wave impulsive source. The motion of a typical surface element is recorded for 0 ? t ? 2T. It is shown that this surface trace determines the acoustic impedance of the medium as a function of travel time, to (travel-time) depth T. Moreover, we give a precise characterization of those functions which may appear as surface traces, and show uniqueness, existence, and continuous dependence of the logarithm of the impedance as a function of the surface trace in the Sobolev H¹ topology.     

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.     

Investigation of wave propagation in models of cracked media

An effective two-phase model of wave propagation in a cracked medium in a half-space whose boundary is perpendicular to the cracks is investigated. The boundary conditions for this model are established in the case of the medium being in contact with other fluid and elastic media.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 116–127, 1989.     

Diffraction of a pulsed longitudinal shear wave by a cylindrical cavity in an anisotropic half-space

The problem of the stress concentration on a free tunnel-shaped cavity in an orthotropic half-space with a free flat boundary, on which shear waves in the form of periodic triangular pulses are incident, is solved. The initial problem is reduced to a series of problems of diffraction of harmonic shear waves. The effect of the degree of shear anisotropy on the tangential-stress concentration at the boundary of a cavity with a circular cross section is studied for different positions of the leading edge of the pulse relative to the boundary of the half-space and the cavity.Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 62–66, 1990.     

Transient analysis of a potential pulse in a piezoelectric (622) crystal plate

Transient analysis of the wave propagation due to an electric potential pulse concentrated at a point on the boundary of an infinite piezoelectric plate of the crystal class (622) resting on an infinite elastic medium is considered. The electric potential is studied for an exponential pulse and step pulse. Numerical results are obtained for theβ-quartz plate that rests on aluminium half-space.     

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.     

非均匀切向荷载下弹性半空间二阶效应的一般解答*

继文[1],本文运用积分变换方法,获得了可压缩各向同性弹性半空间在非均匀分布的切向荷载下,二阶弹性效应问题的封闭形式一般解答.