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.     

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.     

Wave fields in a thin layer of an (ideal) liquid covering an elastic half-space (the simplest model of a shelf)

The head interference wave associated with the propagation of the P-wave in an elastic half-space is studied by using as an example the propagation of pressure waves in a liquid layer covering an elastic half-space. The attenuation of such a wave with respect to the distance between a source and a receiver is smaller than that in the classical theory. The wave field is considered both in time and frequency domains. The stationary wave field of the head interference wave is of resonance nature. From the mathematical point of view, the resonance peaks occur when the roots of the dispersion equation pass through a branch point. The minimal attenuation of the stationary wave field is observed in a neighborhood of such resonance peaks. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 225, 1996, pp. 40–61. Translated by N. S. Zabavnikova.     

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.     

Propagation of longitudinal viscoelastic waves in a three-layer medium

The nonstationary problem of propagation of a longitudinal plane one-dimensional stress wave through a plane-parallel viscoelastic layer of finite thickness separating two linear elastic half-spaces with different properties is solved in the linear formulation. A plane wave traveling in one of the half-spaces is normally incident on the boundary of the layer (one-dimensional problem). The field in the other elastic half-space, excited as a result of the multiple reflection of the fronts from the boundaries of the layer, is investigated. Graphs of the small displacements at a given point of the elastic half-space are presented. The solution of the problem is based on the dynamic correspondence principle formulated by Bland [3].Central Scientific-Research Institute of Machine Building, Moscow. Translated from Mekhanika Polimerov, No. 1, pp. 151–156, January–February, 1971.     

On dynamic effects in an elastic half-space under “thermal impact” : PMM vol. 38, n 6, 1974, pp. 1105–1113

The general uncoupled dynamical problem of thermoelasticity for a half-space under the condition of a thermal impact with a finite rate of change in temperature on its boundary is solved by the method of principal (fundamental) functions within the framework of a generalized theory of heat conduction.An elastic steel half-space is analyzed as an illustration. The problem on thermal stresses originating in an elastic half-space due to thermal impact produced by a jump change in temperature on the boundary was first analyzed in [1]. Since the temperature change on the boundary occurs at a finite rate, it is generally impossible to realize the thermal impact considered in [1] physically. The dynamic effects in an elastic half-space under a thermal impact with finite rate of change in the temperature on the boundary have been studied in [2]. For high rates of change of the heat flux we obtain a generalized wave equation of heat conduction [3] taking into account the finite velocity of heat propagation. Hence, the solution of the ordinary parabolic heat conduction equation used in [1, 2] does not correspond to the true temperature field. The problems of [1, 2] have been examined in [4, 5], respectively, within the framework of a generalized theory of heat conduction.     

Instability of thermoelastic interaction between a half-space and a rigid base through a thin liquid layer

We investigate the instability of thermoelastic interaction between elastic and rigid half-spaces through a liquid interlayer under the conditions of heat transfer across the interfaces. Due to the small thickness of the liquid layer, its influence on the temperature field is taken into account by the thermal resistance of the contact between the bodies, which depends on the normal displacement of the boundary of the elastic body. The pressure inside the liquid is equal to the external pressure applied to the bodies. We determined the critical value of the external heat flow for which the instability becomes possible in such a system and studied the dependence of this value on the parameters of the elastic half-space, the thickness of the liquid layer, and its thermal conduction. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 76–82, April–June, 1998.     

Scattering of Plane Elastic Waves on a Small Obstacle Inside a Layer

The problem on the diffraction of a plane elastic wave with horizontal polarization (SH wave) on a small obstacle placed in a layer is investigated. The layer is situated on a half-space with stress-free boundary. The obstacle is assumed to be a circular cylinder with radius small in comparison with the length of the incident wave. The polarization of the incident wave is parallel to the axis of the cylinder. It is proved that the small inhomogeneity radiates as a linear source such that the intensity of the radiation is proportional to the area of the cross-section of the obstacle and the jump of the squared transverse velocities in the layer and in the obstacle. Bibliography: 5 titles.     

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.     

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.     

Vibrations of the surface of an elastic double-layered half-space with a periodic system of cracks

The two-dimensional problem of the normal incidence of a plane transverse wave from the far field on to the free surface of an elastic double-layered half-space, comprising a homogeneous layer attached to a semi-infinite base of a different elastic material, is considered. At the boundary between the two media there is a system of plane cracks, arranged periodically along the separation line, which models the fracture zone at the interface between dense solid rock and soft sedimentary rock. The effect of the fractures on the transmission of a transverse seismic wave generated by a deep-focus earthquake, and of the type of vibrations of the free surface of the ground that result, is studied. It is difficult to predict whether the seismic wave is strengthened or weakened by the fracture zone. The effect of the system of cracks on vibrations of the free surface largely depends on the physical and geometrical parameters and, primarily, on the vibration frequencies.     

The effectiveness of wave barriers on the dynamic stiffness coefficients of foundations using boundary element method

A comprehensive numerical investigation has been carried out to indicate the influence of wave barriers on the complex dynamic stiffness coefficients of the surface supported foundations under dynamic loads. In the considered problems the underground is regarded as an elastic homogeneous half-space and as a layer on the top of a rigid bedrock. The results of the present numerical simulation are obtained by using the substructure approach in the frequency domain which is formulated on basis of the boundary element method (BEM). The validations of the numerical results are shown through comparison with published numerical data. The results point out that using an open trench barrier with appropriate distance surrounding the vibration source, a notable reduction level can be achieved.     

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.     

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.     

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.     

On the problem of determining the parameters of a layered elastic medium and an impulse source

Considering the linear system of elasticity equations describing the wave propagation in the half-space ? ₊ ³ = {x ∈ ?³ | x ₃ > 0} we address the problem of determining the density and elastic parameters which are piecewise constant functions of x ₃. The shape is unknown of a point-like impulse source that excites elastic oscillations in the half-space. We show that under certain assumptions on the source shape and the parameters of the elastic medium the displacements of the boundary points of the half-space for some finite time interval (0, T) uniquely determine the normalized density (with respect to the first layer) and the elastic Lamé parameters for x ₃ ∈ [0, H], where H = H(T). We give an algorithmic procedure for constructing the required parameters.     

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.     

Determination of the boundary of the plastic zone in a mineral vein weakened by a three-dimensional hole

We propose a method of computing the boundary of the plastic zone formed in a neighborhood of the hole during the mining of a mineral. The problem is studied in a three-dimensional formulation. The boundary of the plastic zone is determined from the condition of continuity of the vertical normal stresses acting on the surface of contact of an elastic half-space and an elastoplastic layer. The computation is carried out for a hole having a parallelepipedal shape. One figure. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 6–10, 1991.     

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.     

Scattering of Plane Electromagnetic Waves from a Small Inhomogeneity in a Slightly Curved Layer

The problem on the diffraction of the elastic plane wave of horizontal polarization (SH wave) from a small inhomogeneity lying in a slightly bent layer is investigated. The layer is situated on the boundary of a half-space. The inhomogeneity is assumed to be a cylinder the cross-section diameter of which is small in comparison with the length of the incident wave. The wave is polarized parallel to the axis of the cylinder. It is proved that the small inhomogeneity radiates as a point source the intensity of which is proportional to the area of the cross-section of the inhomogeneity and to the jumps of the squared velocities in the layer and in the inhomogeneity. Bibliography: 10 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 297, 2003, pp. 106–115.