Action of an elliptic stamp moving at a constant speed on an elastic half-space : PMM vol. 42, no. 6, 1978, pp 1074–1079

The action of a rigid stamp moving at a constant speed, on the boundary of an elastic half-space, is investigated. It is assumed that the frictional forces between the stamp and the surface of the half-space are absent. The integral equation obtained in [1] yields formulas for the pressure, for the case when the area of contact between the stamp and the half-space has an elliptic form.     

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.     

Boussinesq indentation of a transversely isotropic half-space reinforced by a buried inextensible membrane

The normal indentation of a rigid circular disk into the surface of a transversely isotropic half-space reinforced by a buried inextensible thin film is addressed. By virtue of a displacement potential function and the Hankel transform, the governing equations of this axisymmetric mixed boundary value problem are represented as a dual integral equation, which is subsequently reduced to a Fredholm integral equation of the second kind. Two important results of the contact stress distribution beneath the disk region as well as the equivalent stiffness of the system are expressed in terms of the solution of the Fredholm integral equation. When the membrane is located on the surface or at the remote boundary, exact closed-form solutions are presented. For the limiting case of an isotropic half-space the results are verified with those available in the literature. As a special case, the elastic fields of a reinforced transversely isotropic half-space under the action of surface axisymmetric patch loads are also given. The effects of anisotropy, embedment depth of the membrane, and material incompressibility on both the contact stress and the normal stiffness factor are depicted in some plots.     

Three-dimensional contact problem for a prestressed elastic body : PMM vol. 42, no. 6, 1978, pp 1080–1084

A half-space of an incompressible neo — Hookean [1,2] material subjected to a homogeneous bi-axial tension or compression along its boundary, is considered. A small deformation caused by the action of a smooth rigid stamp on the boundary of the half-space is superimposed on the initial finite deformation. An integral equation is obtained for the contact pressure. A solution of this equation is obtained for an inclined elliptic stamp with a flat base, and for an elliptic stamp with a curved base, for the cases when the extension coefficients in two directions are either identical, or differ little from each other. The influence of the inital loading on the distribution of the contact pressure, the displacement of the stamp and the form of the contact zone, is analysed.     

Stress field around an arbitrary thin inclusion in a transversely isotropic elastic half-space

We consider a thin flat inclusion of arbitrary shape located inside a transversely isotropic elastic half-space in the plane parallel to its boundary z = 0. An arbitrary tangential displacement is prescribed on the inclusion. The boundary of the half-space is stress-free. We need to find the complete field of stresses and displacements in this half-space. A governing integral equation is derived by the generalized method of images, introduced by the author. The case of circular inclusion is considered as an example. Two methods of solution of the governing integral equation are derived. A detailed solution is presented for the particular cases of radial expansion, torsion and lateral displacement of the inclusion. The solution is also valid for the case of isotropy. The governing integral equation for the case of isotropy is derived.     

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.     

The Kramers problem: Velocity slip and defect for a hard sphere gas with arbitrary accommodation

The half-space problem of rarefied gas flow (the Kramers problem) is considered for the linearized Boltzmann equation and arbitrary gas-surface interaction. Accurate numerical results for the velocity slip coefficient and velocity defect are obtained for the rigid sphere interaction and Maxwellian boundary condition.     

Interacting indentors on a poroelastic half-space

This paper examines the interaction between two rigid circular indentors on a poroelastic half-space. The resulting mixed boundary value problem, when formulated in the Laplace transform domain, yields an infinite set of Fredholm integral equations. These integral equations are then solved for some special cases. Numerical results for the case of a single indentor show a good agreement with those obtained by using Heinrich and Desoyer's assumption. For the case in which the radius of one indentor reduces to zero (interaction between a rigid indentor and an externally placed load), the resulting equations are solved by a semi-inverse method to give analytical solutions for the resultant force and moment required to maintain the indentor with no normal displacement. When the indentor is subjected to an axial load but allowed to undergo an additional settlement and tilt, numerical results are presented to demonstrate the manner in which Poisson's ratio and the drainage boundary conditions influence the consolidation of the half-space. Numerical results are also given to illustrate the interaction between two identical indentors when ratio of the radius to the spatial distance between them is small.     

基于表面阻抗张量的界面滑移波动态失稳分析

基于Stroh公式和表面阻抗张量理论,提出了研究界面滑移波动态失稳问题的一种新的方法．该方法将表面阻抗张量概念推广到复波速域,并将摩擦接触界面上的边界条件以表面阻抗张量表示．最终将边值问题化归为求解一个复多项式在单位圆内的根．以弹性半空间与刚体平面相对稳态摩擦滑移为例进行了详细的分析,导出了一个4次复特征方程并讨论了方程在单位圆内解的特性,给出了滑移界面波失稳条件的显式解析表达式．     

Numerical Study of Properties of Quasilocal Plane Waves of the Modal Type in the Case of a Thin Low-Velocity Layer that is in Contact with an Elastic Half-Space

Numerical study of properties of quasilocal plane waves of the modal type propagating deep into a medium is carried out with the example of the model of a low-velocity elastic layer in the case of rigid contact with the underlying half-space. It is established that the genesis of these waves is closely related to singular complex roots of the dispersion equation of the problem. Eighteen variants of the model differing by relative parameters of the problem, which have a physical meaning, are considered. For every variant, seismograms of modal and body waves are computed and a comparison of them in intensity is carried out. Bibliography: 4 titles. Dedicated to P. V. Krauklis on the occasion of his seventieth birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 182–196.     

The indentation of a flat punch into a rigid-plastic half-space

The indentation of a flat punch into a rigid-plastic half-space is modelled by a centred field of slip lines with rotation of the rectilinear free boundary about the corner point of the punch. Adjacent to the rectilinear boundary, there is a rigid, stress-free region which is calculated using a velocity hodograph and determines the curvature of the initial horizontal boundary of the half-space during indentation up to the steady-state stage of the motion of the punch in the unbounded rigid-plastic medium.     

A note on the dispersion of Love waves in layered monoclinic elastic media

The dispersion equation for Love waves in a monoclinic elastic layer of uniform thickness overlying a monoclinic elastic half-space is derived by applying the traction-free boundary condition at the surface and continuity conditions at the interface. The dispersion curves showing the effect of anisotropy on the calculated phase velocity are presented. The special cases of orthotropic and transversely isotropic media are also considered. It is shown that the well-known dispersion equation for Love waves in an isotropic layer overlying an isotropic half-space follows as a particular case.     

The contact problem of electroelasticity for a plane elliptic die

We obtain an exact solution of the problem of the stress-strain state of an elastic piezoelectric half-space acted on by a rigid elliptic die with a flat base. The axis of symmetry of the body coincides with the direction of the field of preliminary polarization of the body. The solution is confined to the case of translational displacement of the die. We determine the quantities that characterize the mechanical and electric fields that arise in the region of contact of the die with the half-space. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 40–52.     

Comparison of methods for computing interference elastic waves in thin-layered media. 2

The paper is an immediate continuation of the paper where the solution of the problem on the propagation of low-frequency waves in thin-layered media by the dispersion equation method was considered in detail. In the present article, the solution of a similar problem is given for an elastic layer and a half-space, which are in rigid contact, by the method of superposition of complex plane waves. Bibliography: 17 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 217–232.     

Liouville Property for Solutions of the Linearized Degenerate Thin Film Equation of Fourth Order in a Halfspace

We consider a boundary value problem in a half-space for a linear parabolic equation of fourth order with a degeneration on the boundary of the half-space. The equation under consideration is substantially a linearized thin film equation. We prove that, if the right hand side of the equation and the boundary condition are polynomials in the tangential variables and time, the same property has any solution of a power growth. It is shown also that the specified property does not apply to the normal variable. As an application, we present a theorem of uniqueness for the problem in the class of functions of power growth.     

Possibility of Love wave propagation in a porous layer under the effect of linearly varying directional rigidities

The present paper investigates the Love wave propagation in an anisotropic porous layer under the effect of rigid boundary. Effect of initial stresses on the propagation of Love waves in a fluid saturated, anisotropic, porous layer having linear variation in directional rigidities lying in contact over a pre-stressed, inhomogeneous elastic half-space has also been considered. The dispersion equation of phase velocity has been derived and the influence of medium characteristic such as porosity, rigid boundary, initial stress, anisotropy and inhomogeneity over it has been discussed. The velocities of Love waves have been calculated numerically as a function of KH (where K is the wave number and H is the thickness of the layer) and are presented in a number of graphs.     

Un modèle paraxial de propagation de la lumière : problème aux limites pour l'équation d'advection Schrödinger en coordonnées obliques

We study the Schrödinger equation which comes from the paraxial approximation of the Helmholtz equation in the case where the direction of propagation is tilted with respect to the boundary of the domain. Our primary interest is in the boundary conditions successively in a half-space, then in a quadrant of $\text{R}{}^2$. We also sketch a numerical method for this problem. To cite this article: M. Doumic et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Solution of problems of the theory of elasticity for a half-space with planar boundary cracks

We propose a method of solving three-dimensional problems of the theory of elasticity for a half-space containing planar boundary cracks. The problem is reduced to a system of integro-differential equations for determining the functions that characterize the opening of the crack during deformation of the halfspace. The kernels of the equations, besides having poles, also have a fixed singularity at the points of intersection of the surface of the crack with the boundary of the half-space. The equations obtained are solved numerically for the case of cracks that are part of a circular region. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 58–63.     

The interaction of punches on a transversely isotropic half-space

Three-dimensional contact problems on the interaction of two similar punches on an elastic transversely isotropic half-space (five elastic constants) are investigated, when the isotropy planes are perpendicular to the boundary of the half-space. In this connection the stiffness of the half-space boundary depends on the direction. The kernel of the integral equation of the contact problems is represented in a quadrature-free form using the theory of generalized functions. This form of the kernel enables it to be regularized at singular points and enables Galanov's method to be used to solve the contact problem with an unknown contact area.     

Temperature jump in degenerate quantum gases with the Bogoliubov excitation energy and in the presence of the Bose-Einstein condensate

We construct a kinetic equation modeling the behavior of degenerate quantum Bose gases whose collision rate depends on the momentum of elementary excitations. We consider the case where the phonon component is the decisive factor in the elementary excitations. We analytically solve the half-space boundary value problem of the temperature jump at the boundary of the degenerate Bose gas in the presence of a Bose-Einstein condensate.