Steady-State Frictional Heat Generation on a Periodic Sliding Contact

We investigate the motion of a periodic system of rigid, isolated, parabolic dies along the surface of a half-space. The action of friction forces results in heat generation in the contact region. We assume that the surfaces of the dies are thermally insulated and the half-space is a heat conductor. We reduce the problem under consideration to a set of two integral equations for the contact temperature and pressure. We solve these equations numerically and investigate the influence of thermal deformation on the distributions of temperature and pressure and on the dimensions of the contact region.     

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.     

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.     

Elastic stresses in a half-space with a spherical foreign inclusion

We solve an axisymmetric problem of the theory of elasticity for a half-space with a spherical foreign inclusion given a preliminary heating or subject to an autostrain. It is shown that the stresses on the surface of the half-space may exceed the stresses on the boundary of contact when the distance from the inclusion to the surface is sufficiently small. Three figures. Bibliography: 6 titles.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 30, 1989, pp. 29–36.     

Contact problems of thermoelasticity for a half-space of a functionally gradient material

We propose a method of solution of problems of thermoelasticity for an inhomogeneous half-space. We assume that Poisson's ratio of a material in the half-space is constant, and Young's modulus and the coefficients of linear thermal expansion and thermal conduction vary exponentially with distance from the surface of the half-space. As an example, we consider the contact problem on sliding an inhomogeneous body along the surface of a rigid base with regard for frictional heating. I. Franko L'viv University, L'viv; Warsaw University, Warsaw. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 45–56, April–June, 1998.     

The contact problem with the bulk application of intermolecular interaction forces (a refined formulation)

A refined formulation of the contact problem when there are intermolecular interaction forces between the contacting bodies is considered. Unlike the traditional formulation, it is assumed that these forces are applied to points within the body, rather than to the surface of the deformable body as a contact pressure, and that the body surface is load-free. Solutions of the contact problems for a thin elastic layer attached to an absolutely rigid substrate and for an elastic half-space are analysed. The refined and traditional formulations of the problem when there is intermolecular interaction are compared. ©2013     

Hertz problem for a rigid punch moving across the surface of a semi-infinite elastic solid

The elastodynamic problem of a rigid punch moving at a constant sub-Rayleigh speed across the surface of an elastic half-space is investigated in the present paper. The unknown contact region is determined as part of solution from the unilateral or Signorini conditions. Numerical results are plotted showing how the eccentricity of the contact ellipse changes with the punch speed. Some asymptotic properties of the solution for the case where the punch speed is comparable with the Rayleigh wave speed are explored in details.     

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.     

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.     

Axially symmetric problem of local separation of an elastic half-space from a rigid base due to a point source of cooling

An axially symmetric problem on the unilateral frictionless contact of an elastic half-space and a rigid thermally insulated base under the action of a subsurface source of cooling, which leads to local loss of contact over a circular region, is considered. The intensity of the source at which the local loss of contacts begins is determined. The dependence of the geometrical parameters of the gap between the surfaces of the bodies on the intensity of the source of cooling and its location relative to the surface is analyzed in detail.     

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.     

Solutions of half-space and half-plane contact problems based on surface elasticity

Analytical solutions for the problems of an elastic half-space and an elastic half-plane subjected to a distributed normal force are derived in a unified manner using the general form of the linearized surface elasticity theory of Gurtin and Murdoch. The Papkovitch–Neuber potential functions, Fourier transforms and Bessel functions are utilized in the formulation. The newly obtained solutions are general and reduce to the solutions for the half-space and half-plane contact problems based on classical linear elasticity when the surface effects are not considered. Also, existing solutions for the half-space and half-plane contact problems based on simplified versions of Gurtin and Murdoch’s surface elasticity theory are recovered as special cases of the current solutions. By applying the new solutions directly, Boussinesq’s flat-ended punch problem, Hertz’s spherical punch problem and a conical punch problem are solved, which lead to depth-dependent hardness formulas different from those based on classical elasticity. The numerical results reveal that smoother elastic fields and smaller displacements are predicted by the current solutions than those given by the classical elasticity-based solutions. Also, it is shown that the out-of-plane displacement and stress components strongly depend on the residual surface stress. In addition, it is found that the new solutions based on the surface elasticity theory predict larger values of the indentation hardness than the solutions based on classical elasticity.     

A dynamic contact problem which reduces to a singular integral equation with two fixed singularities

The problem of the harmonic sheat oscillations of an elastic strip, coupled to an elastic half-space is considered. Using the method of integral transformations, the problem is reduced to a singular integral equation in the contact stresses in the region where the strip and the half-space are coupled when there are two fixed singularities at points bounding the integration intervals. One of the main results of this paper is the method of solving this equation numerically, taking into account the true singularity of the solution and based on the use of special quadrature formulae for singular integrals. The approximate solution obtained provides the possibility of numerically investigating the effect of the oscillation frequency and the ratio of the elastic constants of the strip and the half-space on the stress distribution in the contact area.     

The sliding of viscoelastic bodies whenthere is adhesion

The plane contact problem of the sliding without friction of a rigid cylinder over a viscoelastic half-space when there is adhesion is solved, neglecting the inertial properties of the half-space. The distribution of the contact pressure, the size and position of the contact area, and the deformation force of resistance to motion of the cylinder are investigated as a function of the adhesion properties of the surfaces, the mechanical characteristics of the half-space and the sliding velocity of the cylinder.     

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.     

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.     

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.     

Axisymmetric contact of a punch of polynomial profile with an elastic half-space when there is friction and adhesion

The axisymmetric problem of the contact interaction of a punch of polynomial profile and an elastic half-space when there is friction and partial adhesion in the contact area is considered. Using the Wiener–Hopf method the problem is reduced to an infinite system of algebraic Poincare–Koch equations, the solution of which is obtained in series. The radii of the contact area and of the adhesion zone, the distribution of the contact pressures and the indentation of the punch are obtained.     

The solution of the contact problem for a circular plate

An approach, based on Rektorys’ theorem on the minimum of a quadratic functional which, without any fundamental difficulties, can be used for diverse contact problems, is used to solve the problem of the contact interaction of a circular flexible plate with an elastic half-space.     

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.