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.     

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.     

Apparatus and applications of a generalized Fourier method for transversally isotropic bodies bounded by a plane and a paraboloid of rotation

We construct new basic solutions of equilibrium equations of a transversally isotropic paraboloid of rotation and obtain summation theorems that express them in terms of cylindrical solutions for a half-space and vice versa. The problem of the action of an axial concentrated force on an elastic transversally isotropic half-space with a fixed inclusion in the form of a paraboloid of rotation is investigated. The problem is solved by a generalized Fourier method and is reduced to a system of integral equations with a Fredholm operator on the condition that intersections of the boundaries of the half-space and inclusion are absent. We consider dependences of stresses on the shape of the paraboloidal inclusion and on the distance between the boundary surfaces and analyze the results of the calculation.     

On the use of jump conditions at a thin inclusion in solving problems of elasticity theory for a half-space with protuberances

We propose a method of approximate solution of problems of elasticity theory for a half-space with protuberances based on the use of jump conditions in the stresses and displacements at a thin elastic element. The problem of determining the stresses reduces to a system of two-dimensional integral equations of Newtonian potential type for determining the contact stresses between the protuberances and the half-space. We consider the case when the elastic characteristics of the material of the protuberances are different from the material of the half-space.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 156–160.     

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.     

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.     

One approach to a magnetostatic problem for bodies with inclusions in an inhomogeneous external field

A direct magnetostatic problem for magnets with a finite-size inclusion is considered in an integrodifferential form. An approach is used that, under certain conditions, reduces the problem to a single integral equation on a two-dimensional manifold-the inclusion surface. As an important illustrative example, finite formulas are derived to compute the resulting field of a magnetic half-space with a spherical cavity in an arbitrary external field.     

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.     

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.     

Propagation of torsional surface waves in a homogeneous layer of finite thickness over an initially stressed heterogeneous half-space

In the present paper, the dispersion equation which determines the velocity of torsional surface waves in a homogeneous layer of finite thickness over an initially stressed heterogeneous half-space has been obtained. The dispersion equation obtained is in agreement with the classical result of Love wave when the initial stresses and inhomogeneity parameters are neglected. Numerical results analyzing the dispersion equation are discussed and presented graphically. The result shows that the initial stresses have a pronounced influence on the propagation of torsional surface waves. It has also been shown that the effect of density, directional rigidities and non-homogeneity parameter on the propagation of torsional surface waves is prominent.     

The vibrations and stability of a prestressed plate on an elastic foundation

The free vibrations of a transversely isotropic prestressed linear elastic half-space, localized close to a free surface, are considered. The free vibrations of a prestressed transversely isotropic infinite plate, lying on an elastic foundation, are also considered. The dispersion equation is analysed as a function of the wave numbers, the elastic properties of the foundation and of the plate and the values of the prestresses. The investigation is confined to cases when the initial stresses are less than the critical values, while the elastic waves do not penetrate into the depth of the foundation but are localized close to the free surface. The stability of the half-space and the plate on an elastic foundation is also considered. When analysing the vibrations and the stability of the plate, the results in the three-dimensional formulation of the problem are compared with the results of the two–dimensional Kirchhoff–Love and Timoshenko–Reissner models.     

Quasi-optimal control of axisymmetric temperature displacements in a given section of a body using internal heat sources

For a thermoelastic half-space we study the problem of constructing in the space of continuous functions a quasi-optimal control of the axisymmetric vertical displacements in a given section parallel to the boundary surface in the form of a sequence that minimizes the original optimality criterion. For a half-space heated over a circular region by a heat flux of constant intensity we carry out a numerical analysis of the behavior of the elements of the minimizing sequence. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 104–109.     

The thermally stressed state of a half-space with a cylindrical foreign inclusion

We propose a method of determining the thermally stressed state of semibounded bodies containing a defect in the form of a cylindrical foreign inclusion. We give the results of numerical study of the thermal stresses as functions of the geometric parameters of the inclusion. Four figures. Bibliography: 4 titles.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 30, 1989, pp. 41–45.     

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 geometrically nonlinear elongation of a plane with an elliptic inclusion

We deduce relations in geometrically nonlinear formulation for the stresses and displacements on the boundary of an elliptic cut containing an inclusion with elastic characteristics different from the elastic characteristics of the base material. We find the dependence of the stress concentration on the size of the tensile force. We show the necessity of taking account of the geometric nonlinearity for low-modulus materials with nonuniformities. We give the dependence of the stress concentration on the size of the tensile forces for various ratios of elastic constants for the inclusion and the base material.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 31, 1990, pp. 89–95.     

One-dimensional wave processes in a magneto-elastic medium

We construct different solutions of one-dimensional magnetoelastic problems. We analyze the process of propagation of disturbances in a magnetoelastic half-space with finite conductivity when a uniformly distributed force load and magnetic field intensity are prescribed on the surface. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 65–69.     

An axially symmetric contact problem for a half-space with an elastically reinforced cylindrical cavity

We consider the pressure of a plate on a half-space with a round cylindrical cavity. The surface of the cavity is reinforced by elastic elements that are modeled by very general operators. The problem is reduced to a Fredholm integral equation of second kind. A detailed study is made of the case of reinforcement described by the Winkler law. An approximate solution is obtained in the form of the asymptotics with respect to the radii of the plate and the cavity.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 80–85.     

Radiation of short waves by a pair of plates backed by a semi-infinite set of ribbed stiffeners

The following acoustic diffraction problem is considered. The upper half-space is filled by an acoustic medium. Two semi-infinite thin plates are situated on the boundary of the upper half-space. One of the plates is backed by a semi-infinite periodical set of ribbed stiffeners. The source of an acoustic field is positioned on the other plate. The problem is reduced to an infinite system of linear algebraic equations. Such a system can be solved in a shortwave approximation. An expression for the acoustic field at a large distance from the junction of the plates is obtained. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 38–46. Translated by A. V. Badanin.     

Determination of the stress state in a half-space in the vicinity of cylindrical defects appearing on the surface under torsional vibrations

We solve the problem of the determination of the stress state under torsional vibrations of a half-space with a cylindrical defect (a crack or a thin rigid inclusion) that crosses its surface. The method of solution is based on the use of discontinuous solutions of the equations of torsional vibrations and consists in the reduction of the initial boundary-value problems to integral equations for the unknown jumps of an angular displacement or a tangential stress.     

On some contact problems for composite half-spaces PMM vol. 31, no. 6, 1967, pp. 1001–1008

Solutions are presented herein of some contact problems connected with the torsion of a composite half-space. In the general case the problem of the torsion of a composite elastic half-space is examined by means of the rotation of a stiff finite cylinder welded into a vertical recess of this half-space. Moreover, the following particular problems on the torsion of such a half-space are considered.

1. 1) A composite half-space with a vertical elastic infinite core, twisted by means of the rotation of a stiff stamp affixed to the upper endplate of the elastic core.

2. 2) A half-space with a vertical cylindrical infinite hole, twisted by means of the rotation of a stiff finite cylinder welded into the upper part of this hole.

In the general case the solution of the problem reduces to the solution of an integral equation of the second kind on a half-line. The question of the solvability of this fundamental integral equation is investigated, and it is shown that its solution may be constructed by successive approximations.

Let us note that the problem of the torsion of a homogeneous half space and of an elastic layer by means of rotation of a stiff stamp has been considered by Rostovtsev [1], Reissner and Sagoci [2], Ufliand [3], Florence [4], Grilitskii [5] and others.

The problem of the torsion of a circular cylindrical rod and the half-space welded to it which are subject to a torque applied to the free endface of the rod has been considered by Grilitskii and Kizyma[6].

The torsion of an elastic half-space with a vertical cylindrical inclusion of some other material by the rotation of a stiff stamp on the surface of this half-space has been considered in [7], wherein it has been assumed that the stamp is symmetrically disposed relative to the axis of the inclusion and lies simultaneously on both materials.