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.     

Pressure of a plane stamp of nearly circular cross section on an elastic half-space : PMM vol. 40, n 6, 1976, pp. 1143–1145

An approximate method of solving the contact problem of impressing a plane stamp of nearly circular cross section into an elastic half-space is suggested. The friction of the contact surface is neglected. A numerical algorithm for the method is produced. An elliptical and rectangular stamps are considered as examples.There is no general method of solving the problems for stamps of nearly circular cross section. Apart from the classical problem of a plane elliptical stamp, the literature gives solutions for the problems of polygonal stamps, with each problem however requiring a different approach. An approximate solution for the problem of impressing a stamp of nearly circular cross section into an elastic half-space is given in [1]. The method makes it possible to use the same approach to solve the contact problem for an arbitrary region of contact, and to construct an universal numerical algorithm. The program can be adapted to each particular case by making the corresponding changes in the procedure of computing the Fourier coefficients of the equation of the boundary of the area of contact. Below a numerical algorithm for the approximate method in question is given. A more effective formulation of the solution is given for the case of the elliptical stamp.     

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.     

Vibration of a stamp partially adhering to an elastic medium : PMM, vol. 42, no. 6, 1978, pp. 1085–1092

There is examined the problem of vibration of a stamp of arbitrary planform occupying a space Ω and vibrating harmonically in an elastic medium with plane boundaries. It is assumed that the elastic medium is a packet of layers with parallel boundaries, at rest in the stiff or elastic half-space. Contact of three kinds is realized under the stamp: rigid adhesion in the domain Ω₁, friction-free contact in domain Ω₂, there are no tangential contact stresses, and “film” contact without normal force in domain Ω₃ (there are no normal contact stresses, only tangential stresses are present.). It is assumed that the boundaries of all the domains have twice continuously differentiable curvature and Ω = Ω₁ Ω₂ Ω₃.

The problem under consideration assumes the presence of a static load pressing the stamp to the layer and hindering the formation of a separation zone. Moreover, a dynamic load, harmonic in time, acts on the stamp causing dynamical stresses which are of the greatest interest since the solution of the static problem is obtained as a particular case of the dynamic problem for ω = 0 (ω is the frequency of vibration). The general solution is constructed in the form of a sum of static and dynamic solutions.

A uniqueness theorem is established for the integral equation of the problem mentioned and for the case of axisymmetric vibration of a circular stamp partially coupled rigidly to the layer, partially making friction-free contact, the problem is reduced to an effectively solvable system of integral equations of the second kind, which reduce easily to a Fredholm system.

These results are an extension of the method elucidated in [1], where by the approach in [1] must be altered qualitatively to obtain them.     

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.     

Solution of certain boundary value problems of potential theory and elasticity theory for a half-space with a paraboloidal cavity

A method for solving boundary value problems for the Laplace equation in a half space with a paraboloidal cavity or a paraboloidal segment is suggested. Using formulas for the re-expansion of the fundamental solutions of the Laplace equation from a cylindrical to a paraboloidal coordinate system and their inverses, the basic and certain mixed problems are reduced to Fredholm integral equations or systems of equations of the second kind with completely continuous operators in a certain Hilbert space. The problem of torsion of an elastic half-space with a paraboloidal cavity by a stamp linked to part of the surface of the paraboloid and the problem of distribution of electricity on a paraboloidal segment located in the half-space are considered.Translated from Dinamicheskie Sistemy, No. 4, pp. 33–40, 1985.     

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.     

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.     

A dynamic contact problem for an elastic half-plane in the case of high frequency oscillations

Harmonic high frequency oscillations of a rigid stamp coupled without friction tc an elastic half-plane are considered. The main difficulty in constructing the high-frequency asymptotic forms is that of carrying out the effective factorization of the kernel of the basic integral equation. A function is proposed, which takes into account all properties of the kernel, enables it to be uniformly approximated and is easily factorized. Such a solution of the problem of approximate factorization makes it possible to write, in a simple explicit form, the principal term of the asymptotic expression of the solution. The nature of the distribution of contact stresses under the stamp is studied, as well as the compliance of the foundation and phase shift between the applied force and the displacement of the stamp.     

Method of homogeneous solutions in the mixed problem for a finite circular cylinder : PMM vol. 43, no. 6, 1979, pp. 1073–1081

The mixed axisymmetric problem of elasticity theory on the torsion of a finite circular cylinder by a stamp is considered. The stamp is fixed rigidly to one plane face of the cylinder, the other plane face is fixed, and conditions for no displacements or stresses are given on the cylinder surface. The problem is investigated by the method of homogeneous solutions [1], which permits obtaining its approximate solution for practically any values of the parameters. Such efficiency of the method is determined by the fact that the solution of the problem reduces to investigating an infinite algebraic system of the Poincaré — Koch normal systems type. When the ratio of the cylinder height to the radius of the stamp is sufficiently large, the system coefficients, the contact stresses, and the other characteristics of the problem are evaluated to any degree of accuracy, and effective asymptotic expressions are obtained for small values of this ratio. Results of numerical computations are presented.

A solution of the problem for the case of a large value of the ratio (R − a) /h and small values of the ratio λ = h / a is obtained in [2].     

The displacements in an elastic half-space when a load moves along a beam lying on its surface

The motion, with constant velocity, of a normal load along an elastic beam lying on an elastic isotropic homogeneous half-space is considered. A method for the approximate calculation of the normal displacements of the surface of the half-space for subsonic velocities of motion is developed. An estimate is given of the expressions obtained and a comparison is made with existing results for the problem of the motion of a point load along a half-space.     

Contact problems for functionally graded materials of complicated structure

An approximate analytical method allowing one to efficiently solve, to a preassigned accuracy, contact problems for materials with properties arbitrarily varying in depth is developed. Its possibilities are illustrated with the example of torsion of an elastic half-space, having a coating inhomogeneous across its thickness, by a circular stamp. All the results obtained are rigorously substantiated. For the approximate solutions constructed, their error is analyzed. The asymptotic properties of the solutions are investigated. The cases of a nonmonotonic change in the elastic properties are considered. In particular, the analytical solutions are examined in the case where the variation gradient of the elastic properties changes its sign many times. The results derived allow one to solve the inverse problems of elasticity theory of inhomogeneous media (e.g., the problem on controlling the variation in the elastic properties of a covering across its thickness).     

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.     

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.     

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.     

Plane contact problem of the indentation of a stamp into an elastic wedge

We consider the contact interaction of a stamp with rectilinear base and an elastic wedge. One of the wedge faces is fixed, and the stamp edge touches the wedge vertex. Using the Wiener–Hopf method, we have obtained an exact solution of this problem. We have also determined the stress distributions in the contact region and on the wedge fixed face as well as the displacements of its free boundary.     

Asymptotic determination of the formation process of nonlinear distortion of one-dimensional pulses in a layered medium : PMM vol.40, n 6, 1976, pp. 1093–1103

Nonlinear effects in the propagation, reflection, and refraction of one-dimensional pulses in a medium consisting of two layers lying on a half-space are considered and analyzed. Properties of layers and of the half-space are different, and stresses are defined by an expansion in powers of strains. The initial pulse of finite duration is specified in the form of boundary condition at the surface of the external layer either for the deformation or for the dislocation rate, and the problem of wave pattern when the initial pulse amplitude tends to zero,i.e. in the case of small nonlinear effects, is solved.Problem is solved by the method of successive integration of nonhomogeneous linear wave equations, in which the solution of the linear problem is taken as the first approximation and the subsequent approximations are derived by approximating the nonlinear terms with the use of the preceding approximation.     

Axisymmetric indentation of a circular stamp with adhesion into an elastic half-space with a spherical cavity

On the basis of the expansion formulas of the vector solutions of the Lame equations in cylindrical and spherical coordinates, the problem of a circular stamp is formulated in the form of an integro-algebraic system of equations. By the method of orthogonal polynomials, it is reduced to a collection of infinite systems of linear algebraic equations, for which the method of reduction is justified. Formulas for the normal and tangential stresses under the stamp are given.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 14–20, 1987.     

An energy approach to the proof of the existence of Rayleigh waves in an anisotropic elastic half-space

An approach based on investigating the energy functional is applied for the first time to the classical problem of Rayleigh waves in an anisotropic half-space with a free boundary. The main object of the investigation is an ordinary differential operator in a variable characterizing the depth. An investigation of the spectrum by variational methods enables a new proof to be given of the existence of a Rayleigh wave in a linear elastic half-space with arbitrary anisotropy, which does not rest on the Stroh formalism.