The interaction of punches on the face of an elastic wedge

The interaction of two punches, which are elliptic in plan, on the face of an elastic wedge is investigated in a three-dimensional formulation for different types of boundary conditions on the other face. The wedge material is assumed to be incompressible. An asymptotic solution is obtained for punches which are relatively distant from one another and from the edge of the wedge. For the case when the punches are arranged relatively close to the edge of the wedge (or reach the edge, the contact area is unknown) the numerical method of boundary integral equations is used. The mutual effect of the punches is estimated by means of calculations. The asymptotic solution of the generalized Galin problem, concerning the effect of a concentrated force applied on the edge of the three-dimensional wedge on the contact pressure distribution under a circular punch relatively far from the edge, is obtained.     

Three-dimensional contact problems with friction for a composite elastic wedge

Contact problems for a composite elastic wedge in the form of two joined wedge-shaped layers with different aperture angles joined by a sliding clamp, where the layer under the punch is incompressible, are studied in a three-dimensional formulation. Conditions for a sliding or rigid clamp or the absence of stresses are set up on one face of the composite wedge. The integral equations of the problems are derived taking account of the friction forces perpendicular to the edge of the wedge. The method of non-linear boundary integral equations of the Hammerstein type is used when the contact area is unknown. A regular asymptotic solution is constructed for an elliptic contact area. By virtue of the incompressibility of the material of the layer in contact with the punch, this solution retains the well known root singularity in the boundary of the contact area when account is taken of friction.     

Three-dimensional contact problems for an elastic wedge with a coating

Three-dimensional contact problems for an elastic wedge, one face of which is reinforced with a Winkler-type coating with different boundary conditions on the other face of the wedge, are investigated. A power-law dependence of the normal displacement of the coating on the pressure is assumed. The contact area, the pressure in this region, and the relation between the force and the indentation of a punch are determined using the method of non-linear boundary integral equations and the method of successive approximations. The results of calculations are analysed for different values of the aperture angle of the wedge, the relative distance of the punch from the edge of the wedge, the ratio of the radii of curvature of the punch (an elliptic paraboloid), and the non-linearity factors of the coating. The results obtained are compared with the solutions of similar problems for a wedge without a coating.     

Boundary-value problems for Helmholtz equations in an angular domain. I

The boundary-value problems are investigated that arise when studying the diffraction of acoustic waves on an infinite cylinder with cross-section of an arbitrary shape situated inside a wedge so that the axis of the cylinder is parallel to the edge of the wedge. The potential theory is worked out which enables one to reduce these boundary-value problems to integral equations on a one-dimensional contour — the boundary of the cross-section of this cylinder. The theorems on existence and uniqueness of solutions to the boundary-value problems and the corresponding integral equations are proved. For this case, a principle of limit absorption is established. Effective algorithms for calculating the kernels of the integral operators are constructed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 403–418, March, 1993.     

Boundary-value problems for the helmholtz equation in an angular domain. II

We investigate boundary-value problems that appear in the study of the diffraction of acoustic waves on an infinite cylinder (with a cross section of an arbitrary shape) placed inside a wedge so that the axis of the cylinder is parallel to the edge of the wedge. The potential theory which enables one to reduce these boundary-value problems to integral equations is elaborated.Translated from Ukrainskii Matematicheskii Zhumal, Vol. 45, No. 4, pp. 500–519, April, 1993.     

Problems of a cut in a three-dimensional elastic wedge

The Ritz variational method is applied to problems of a crack (a cut) in the middle half-plane of a three-dimensional elastic wedge. The faces of the elastic wedge are either stress-free (Problem A) or are under conditions of sliding or rigid clamping (Problems B and C respectively). The crack is open and is under a specified normal load. Each of the problems reduces to an operator integrodifferential equation in relation to the jump in normal displacement in the crack area. The method selected makes it possible to calculate the stress intensity factor at a relatively small distance from the edge of the wedge to the cut area. Numerical and asymptotic solutions [Pozharskii DA. An elliptical crack in an elastic three-dimensional wedge. Izv. Ross Akad. Nauk. MTT 1993;(6):105–12] for an elliptical crack are compared. In the second part of the paper the case of a cut reaching the edge of the wedge at one point is considered. This models a V-shaped crack whose apex has reached the edge of the wedge, giving a new singular point in the solution of boundary-value problems for equations of elastic equilibrium. The asymptotic form of the normal displacements and stress in the vicinity of the crack tip is investigated. Here, the method employed in [Babeshko VA, Glushkov YeV, Zinchenko ZhF. The dynamics of Inhomogeneous Linearly Elastic Media. Moscow: Nauka; 1989] and [Glushkov YeV, Glushkova NV. Singularities of the elastic stress field in the vicinity of the tip of a V-shaped three-dimensional crack. Izv. Ross Akad. Nauk. MTT 1992;(4):82–6] to find the operator spectrum is refined. The new basis function system selected enables the elements of an infinite-dimensional matrix to be expressed as converging series. The asymptotic form of the normal stress outside a V-shaped cut, which is identical with the asymptotic form of the contact pressure in the contact problem for an elastic wedge of half the aperture angle, is determined, when the contact area supplements the cut area up to the face of the wedge.     

Problems of cuts in a composite elastic wedge

Problems of strip and elliptical cuts (tensile cracks) in the middle of a three-layer elastic wedge are investigated in a three-dimensional formulation. Free or rigid clamping conditions or the stress-free condition are stipulated on the outer surfaces of the composite wedge. The problems are assumed to be symmetrical about the plane of the cut. The wedge-shaped layer containing the cut is incompressible and hinged along both faces with two other layers. The integral equations of the problems with respect to the opening of the cut are derived. Inverse operators are obtained for the operators occurring in the kernels of these equations. The relation between problems on cuts and the corresponding contact problems for a composite wedge of half the aperture angle is used. The method of paired integral equations is used for the case of a strip cut emerging from the edge of the wedge. The problems are reduced to Fredholm integral equations of the second kind in certain auxiliary functions, in terms of the values of which the normal stress intensity factors are expressed. A regular asymptotic solution is constructed for the case of an elliptic cut.     

A new version of boundary integral equations and their application to dynamic three-dimensional problems of the theory of elasticity

A version of boundary integral equations of the first kind in dynamic problems of the theory of elasticity is proposed, based on an investigation of the analytic properties of the Fourier transformant of the displacement vector, rather than on fundamental solutions. A system of three boundary integral equations of the first kind with Fredholm kernels is constructed, and the equivalence of the initial boundary-value problem on the vibrations of a bounded region and the system of boundary integral equations obtained is investigated. A version of the numerical realization, which combines the ideas of the classical method of boundary elements and the Tikhonov regularization method, is proposed. The results of numerical experiments are given.     

The Contact Problems of the Mathematical Theory of Elasticity for Plates with an Elastic Inclusion

The contacts problem of the theory of elasticity and bending theory of plates for finite or infinite plates with an elastic inclusion of variable rigidity are considered. The problems are reduced to integral differential equation or to the system of integral differential equations with variable coefficient of singular operator. If such coefficient varies with power law we can manage to investigate the obtained equations, to get exact or approximate solutions and to establish behavior of unknown contact stresses at the ends of elastic inclusion.      

The three-dimensional contact problem for a wedge-shaped valve

The three-dimensional contact problem for an elastic wedge-shaped valve, situated in a wedge-shaped cavity in an elastic space, is investigated. A regular asymptotic method is used to solve the integral equation of this problem. The method is effective for a contact area relatively far from the edge of the wedge-shaped cavity. Calculations are carried out. The solutions of the three-dimensional auxiliary problems on the equilibrium of an elastic wedge-shaped cavity and an elastic wedge are based on well-known Green's functions, constructed using Fourier and Kontorovich–Lebedev integral transformations.     

An asymptotic method of solving transient dynamic contact problems of the theory of elasticity for a strip

An asymptotic method is proposed for solving transient dynamic contact problems of the theory of elasticity for a thin strip. The solution of problems by means of the integral Laplace transformation (with respect to time) and the Fourier transformation (with respect to the longitudinal coordinate) reduces to an integral equation in the form of a convolution of the first kind in the unknown Laplace transform of contact stresses under the punch. The zeroth term of the asymptotic form of the solution of the integral equation for large values of the Laplace parameter is constructed in the form of the superposition of solutions of the corresponding Wiener-Hopf integral equations minus the solution of the corresponding integral equation on the entire axis. In solving the Wiener-Hopf integral equations, the symbols of the kernel of the integral equation in the complex plane is presented in special form — in the form of uniform expansion in terms of exponential functions. The latter enables integral equations of the second kind to be obtained for determining the Laplace-Fourier transform of the required contact stresses, which, in turn, is effectively solved by the method of successive approximations. After Laplace inversion of the zeroth term of the asymptotic form of the solution of the integral equations, the asymptotic solution of the transient dynamic contact problem is determined. By way of example, the asymptotic solution of the problem of the penetration of a plane punch into an elastic strip lying without friction on a rigid base is given. Formulae are derived for the active elastic resistance force on the punch of a medium preventing the penetration of the punch, and the law of penetration of the punch into the elastic strip is obtained, taking into account the elastic stress wave reflected from the strip face opposite the punch and passing underneath it.     

Integral equations in the linear theory of elasticity in semiinfinite domains

We investigate linear integral equations on a semiaxis that appear in the course of construction of solutions of boundary-value problems in the theory of elasticity in such domains as a semiinfinite strip or a cylinder. By using the Mellin transformation and the theory of perturbations of linear operators, we establish general results concerning the solvability and asymptotic properties of solutions of the equations considered. We give examples of application of the general statements obtained to specific integral equations in the theory of elasticity. Translated from Ukrainskii Matematicheskii Zhumal, Vol. 50, No. 5, pp. 613–622, May, 1998.     

Three-dimensional Contact of a Rigid Roller Traversing a Viscoelastic Half Space

In this paper, the authors consider a nontrivial three-dimensionalviscoelastic contact problem which has some physical significance.(Although the subject of the analysis is an elliptical roller,it is only a small step onward to the consideration of a crownedcylindrical roller.) Generally, it is the intractability of the mathematics whichhinders analytic solution of true three-dimensional problemsin (visco)elasticity. The traditional method of surmountingthis difficulty is to reduce the problem to two dimensions,either by choosing a suitable geometry, or by using an appropriateco-ordinate system. The elastic line integral theory representsanother approach; certain approximations are used to simplifythe governing equations, thus allowing the solution of the problem. After the development of a viscoelastic analogue of the Boussinesqequation valid for the solution of quasi-steady state viscoelasticcontact problems, analysis proceeds making use of near fieldand extended line integral approximations. Results are generatedshowing the velocity dependence of several physical parameters,including the size and shape of the contact zone. One additionalpoint of interest is uncovered, namely the presence of a pressurepeak near the leading edge of the contact zone.     

Solution of three-dimensional problems of the theory of elasticity for a picewise homogeneous medium with constant poisson''s ratio : PMM vol. 43, no. 6, 1979, pp. 1122–1125

Singular integral equations of the theory of elasticity are studied for a piecewise homogeneous medium with the same Poisson's ratio. It is shown that a solution can be obtained using the method of successive approximations. Use of the potential method for the fundamental problems of the theory of elasticity leads to singular integral equations of second kind [1], In the case of the second internal and external problems, and of the first internal problem, the spectral properties of the integral operators allow the use of the method of successive approximations to obtain a solution.     

Concentration of Mechanical Stresses near a Hole in a Piezoceramic Layer

A new procedure for solving three-dimensional mixed antisymmetric problems of elasticity theory for a layer weakened by through tunnel holes is proposed. The boundary-value problem is reduced to a system of one-dimensional singular integral equations consisting of 3k (k = 1, 2, ...) equations. The calculation results for characteristic stresses are presented.     

Reduction of three-dimensional dynamical elasticity theory problems with arbitrarily located plane slits to integral equations

By generalizing a method described earlier /1/ for reducing three-dimensional dynamical problems of elasticity theory for a body with a slit to integral equations, integral equations are obtained for an infinite body with arbitrarily located plane slits. The interaction of disc-shaped slits located in one plane is investigated when normal external forces that vary sinusoidally with time (steady vibrations) are given on their surfaces.

Problems of the reduction of dynamical three-dimensional elasticity theory problems to integral equations for an infinite body weakened by a plane slit were examined in /1, 2/. The solution of the initial problem is obtained in /1/ by applying a Laplace integral transform in time to the appropriate equations and constructing the solution in the form of Helmholtz potentials with densities characterizing the opening of the slit during deformation of the body. The problem under consideration is solved in /2/ by using the fundamental Stokes solution /3/ with subsequent construction of the solution in the form of an analogue of the elastic potential of a double layer.     

The problem of the equilibrium of a helical spring in the non-linear three-dimensional theory of elasticity

The problem of the loading of a helical spring by an axial force and a torque is considered using the three-dimensional equations of the non-linear theory of elasticity. The problem is reduced to a two-dimensional boundary-value problem for a plane region in the form of the transverse cross section of the coil of the spring. The solution of the two-dimensional problem obtained enables the equations of equilibrium in the volume of the body and the boundary conditions on the side surface to be satisfied exactly. The boundary conditions at the ends of the spring are satisfied in the integral Saint-Venant sense. The problem of the equivalent prismatic beam in the theory of springs is discussed from the position of the solution of the non-linear Saint-Venant problem obtained. The results can be used for accurate calculations of springs in the non-linear strain region, and also when developing applied non-linear theories of elastic rods with curvature and twisting.     

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.     

Potential theory for problems of diffraction on a layer between two parallel planes

We investigate the boundary-value problems that appear when studying the diffraction of acoustic waves on obstacles in a layer between two parallel planes. By using potential theory, these boundary-value problems are reduced to the Fredholm integral equations given on the boundary of the obstacles. The theorems on existence and uniqueness are proved for the Fredholm equations obtained and, hence, for the boundary-value problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 5, pp. 647–662, May, 1993.     

Mixed problems of axisymmetric elasticity theory for some bodies of revolution

We consider the problem of axisymmetric elasticity theory for a space with an elongated ellipsoidal cavity with mixed boundary conditions of smooth contact on the cavity surface and the main mixed problem of axisymmetric elasticity theory for a hyperboloidal layer formed by the two surfaces of a two-cavity hyperboloid of revolution symmetrical about the plane z = O. The problems are solved by the method of p-analytical functions. The solution of the first problem is reduced to solving a Fredholm integral equation of the second kind. We investigate the behavior of the normal stress near the boundary lines. The solution of the second problem is reduced to solving a system of two Fredholm integral equations of the second kind. Existence and uniqueness of the solution is proved for this system.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 88–101, 1989.