Three-dimensional contact problem of the motion of a stamp with friction

There is considered the three-dimensional contact problem of elasticity theory with friction forces collinear to the motion direction. Such a case holds during stamp motion along the boundary of an elastic half-space with anisotropic friction /1/. In the case of an arbitrary friction surface, the mentioned force distribution is satisfied approximately during stamp motion.     

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.     

非均匀法向荷载下半空间的二阶弹性效应问题

本文提供各向同性弹性半空间，在非均匀分布法向荷载下，二阶弹性效应的一个封闭形式解，运用积分变换方法，讨论了按Ｈｅｒｔｚ规律分布的荷载情形；导出了不可压缩各向同性弹性材料的极限解；算出了上述二阶弹性材料问题在ｚ方向的位移和法向应力数值。我们发现，与线弹性情形相比较，在二阶弹性材料中相应位移增大而法向应力减小。     

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.     

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.     

Fourier-transform solution to the problem of steady-state transonic motion of a line load across the surface of an elastic half-space (the corrected Cole-Huth solution)

Considered in the present article is the problem of a line load load moving at a constant transonic speed across the surface of an elastic half-space. Solution of the problem is derived here using the method of Fourier transforms.     

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.     

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.     

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.     

On torsional loading in an axisymmetric micropolar elastic medium

Effects of torsional loading in an axisymmetric micropolar elastic half-space are studied. The components of microrotation displacement, force stress and couple stress are obtained for a half-space subjected to an arbitrary load produced by shearing stress. A special case of a particular type of twist has been discussed in detail for a specific model and the micropolar effects have been shown graphically.     

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.     

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.     

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.     

Strain gradient solutions of half-space and half-plane contact problems

General solutions for the problems of an elastic half-space and an elastic half-plane, respectively, subjected to a symmetrically distributed normal force of arbitrary profile are analytically derived using a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter. Mindlin’s potential function method and Fourier transforms are employed in the formulation, and the half-space and half-plane contact problems are solved in a unified manner. The specific solutions for the problems of a half-space/plane subjected to a concentrated normal force or a uniformly distributed normal force are obtained by directly applying the general solutions, which recover the existing classical elasticity-based solutions of the Flamant and Boussinesq problems as special cases. In addition, the indentation problems of an elastic half-space indented by a flat-ended cylindrical punch, a spherical punch, and a conical punch, respectively, are solved using the general solutions, leading to hardness formulas that are indentation size- and material microstructure-dependent. Numerical results reveal that the displacement and stress fields in a half-space/plane given by the current SSGET-based solutions are smoother than those predicted by the classical elasticity-based solutions and do not exhibit the discontinuity and/or singularity displayed by the latter. Also, the indentation hardness values based on the newly obtained half-space solution are found to increase with decreasing indentation radius and increasing material length scale parameter, thereby explaining the microstructure-dependent indentation size effect.     

The phase velocity of acoustic perturbations near a vibration excitation source

The plane problem of the forced vibrations of an elastic half-space due to the action of a time-periodic force, acting along the tangent to the surface of a medium is considered. It is shown that in the near zone the phase velocity of surface acoustic perturbations varies considerably with distance from the vibration source and considerably exceeds the Rayleigh wave velocity.     

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.     

Rayleigh wave motions in an orthotropic half-space under time-harmonic loadings: A theoretical study

Rayleigh surface wave motion in a homogeneous orthotropic elastic solid half-space under time-harmonic sources is theoretically investigated. In this article, closed-form expressions of the Rayleigh wave field as a result of a time-harmonic concentrated line load are simply determined by using reciprocity theorems. For the verification purpose, analytical predictions are also obtained through Fourier transform technique with the aid of the residue theorem. The solutions found by the two approaches are shown to be mathematically the same. As an example of calculation, surface waves due to a set of buried loadings in the orthotropic half-plane are then examined in detail. A discussion of these results is presented to explore the influence of the loading types on the displacement amplitudes of the generated wave fields.     

Mathematical modeling of seismic and acousto-gravitational waves in a heterogeneous earth-atmosphere model

A numerical-analytical solution to problems of seismic and acoustic-gravitational wave propagation is applied to a heterogeneous Earth-Atmosphere model. The seismic wave propagation in an elastic half-space is described by a system of first order dynamic equations of the elasticity theory. The propagation of acoustic-gravitational waves in the atmosphere is described by the linearized Navier-Stokes equations. The algorithm proposed is based on the integral Laguerre transform with respect to time, the finite integral Bessel transform along the radial coordinate with a finite difference solution of the reduced problem along the vertical coordinate. The algorithm is numerically tested for the heterogeneous Earth-Atmosphere model for different source locations.     

Scattering of elastic waves by nonaxisymmetric three-dimensional dipping layer

Using a boundary method, we investigated the scattering of elastic plane harmonic SH, SV, P, and Rayleigh waves by three-dimensional nonaxisymmetric dipping layers embedded in an elastic half-space. The valley was subjected to incident Rayleigh wave and oblique incident SH, SV, and P waves. The method utilized spherical wave functions to express the unknown scattered field. These functions satisfy the equation of motion and radiation conditions at infinity but they do not satisfy the stress-free boundary conditions at the surface of the half-space. The boundary and continuity conditions are imposed locally in the least-square-sense at several points on the layer interface and on the surface of the half-space. A comparative study was done to examine the validity and limitations of the two-dimensional approximations (antiplane and plane strain models) of three-dimensional models. It is demonstrated that the two-dimensional approximations may be inadequate to represent actual displacement field for three-dimensional irregularities.