On the contact interaction of dissimilar anisotropic half-planes with thermally insulated gaps at the interface

We obtain a system of singular integral equations for the problem of thermoelastic contact of two dissimilar anisotropic half-planes with a system of thermally isolated gaps on the line of contact. Analysis of the contact interaction parameters was carried out for half-planes made of identical materials using the condition that on the surface of one of them there is a symmetric hole of prescribed shape. We give a scheme for numerical solution of the problem for holes of different profiles. Translated fromMatematichni Metodi i Fiziko-mekhanichni Polya, Vol. 40, No. 1, 1997, pp. 117–124.     

Wave-field asymptotics in the problem of diffraction by a planar junction of thin layers and boundary-contact conditions

The problem of diffraction by a planar junction of thin layers covering a perfectly conducting substratum is considered, and its asymptotic solution is constructed. The wave field in the vicinity of the junction of the layers is described by a function of the boundary layer. Based on the asymptotics obtained, the generalized impedance boundary condition, which simulates thin layers, and the contact conditions are derived. The uniqueness of the solution of a model problem is discussed. Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 157–171. Original Translated by M. A. Lyalinov.     

Comparison of methods for computing interference elastic waves in thin-layered media. 2

The paper is an immediate continuation of the paper where the solution of the problem on the propagation of low-frequency waves in thin-layered media by the dispersion equation method was considered in detail. In the present article, the solution of a similar problem is given for an elastic layer and a half-space, which are in rigid contact, by the method of superposition of complex plane waves. Bibliography: 17 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 217–232.     

Regularity of the free boundary in a neighborhood of a contact point in the obstacle problem

The obstacle problem for an arbitrary linear elliptic equation is considered. The regularity of the boundary of the noncoincident set is studied in a neighborhood of a contact points of the boundary of a domain. The C1-regularity of the boundary and the continuity up to contact points of the second-order derivatives of the solution are proved. Bibliography: 3 titles. Translated fromProblemy Matematicheskogo Analiza, No. 19, 1999, pp. 101–104.     

The method of volterra integral equations in contact problems for thin-walled structural elements

We reduce the solution of contact problems in the interaction of rigid bodies (dies) with thin-walled elements (one-dimensional problems) to Volterra integral equations. We study the effect of the model describing the stress-strain state of plates on the type of integral equations and the structure of their solutions. It is shown that taking account of reducing turns the problem into a Volterra integral equation of second kind, which has a unique solution that is continuous and agrees quite well with the results obtained from the three-dimensional theory. In the case of a theory of Timoshenko type the problem is reduced to a Volterra three-dimensional theory. In the case of a theory of Timoshenko type the problem is reduced to a Volterra integral equation of first kind that has a unique continuous solution; but for dies without corners the Herz condition does not hold (p(a) ≠ 0), and the contact pressure assumes its maximal value at the end of the zone of contact. For thin-walled elements, whose state can be described by the classical Kirchhoff-Love theory, the integral equation of the problem (a Volterra equation of first kind) has a solution in the class of distributions. The contact pressure is reduced to concentrated reactions at the extreme points of the contact zone. We give a comparative analysis of the solutions in all the cases just listed (forces, normal displacements, contact pressures). Three figures, 1 table. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 96–103. Original article submitted March 15, 1997.     

Boundary integral equations for a body with inhomogeneous inclusions

On the basis of the partially singular differential equations of the stationary problem of heat conduction and the quasi-static problem of thermoelasticity, written taking account of conditions of nonideal thermomechanical contact, we derive boundary integral equations for a body with inhomogeneous inclusions. We propose a method of solving these equations taking account of the order of the principal term of the asymptotics of the solution in neighborhoods of the corners of the contact surfaces. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 37–41.     

On the behavior of free boundaries near the boundary of the domain

Let u be a solution to the obstacle problem in a domain Ω⊂ℝ ⁿ . In this paper, the behavior of the free boundary in a neighborhood of ϖΩ is studied. It is proved that under some conditions the free boundary touches ϖΩ at contact points. Bibliography:4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1995, pp. 5–19. Translated by T. N. Rozhkovskaya.     

An asymptotic solution of the Signorini problem for a beam lying on two rigid supports

An asymptotic solution is constructed to the Signorini problem for a two-dimensional thin beam that is in possible contact with two rigid supports. For the position of points where the beam leaves the base, an asymptotic formula is derived by analysis of the boundary-layer phenomenon near these points. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 324, 2005, pp. 43–60.     

The quasisteady contact problem of thermoelasticity for a two-layered cylinder under frictional heating and nonideal thermal contact

In this article we give an analytic solution of the polar-symmetric quasisteady thermoelastic contact problem for a two-layer hollow circular cylinder. The problem is solved taking account of frictional heat production and thermal resistance on the mutually tangent surfaces of the components of the cylinder. On the exterior boundary of the two-layer system we study the condition of Winkler elastic fixing. In the solution we apply the Laplace transform with respect to time. We carry out a numerical analysis whose results are shown as graphs. Translated fromMatematichni Metodi i Fiziko-mekhanichni Polya, Vol. 40, No. 1, 1997, pp. 104–110.     

The cauchy problem for a nonlinear integro-differential transport equation

In the present paper, we study the Cauchy problem for a nonlinear time-dependent kinetic neutrino transport equation. We prove the existence and uniqueness theorem for the solution of the Cauchy problem, establish uniform bounds int for the solution of this problem, and prove the existence and uniqueness of a stationary trajectory and the stabilization ast→∞ of the solution of the time-dependent problem for arbitrary initial data. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 677–686, May, 1997. Translated by A. M. Chebotarev     

Influence of viscosity on the structure of shock waves in the MKdV model

A “non-self-adjoint” integrable MKdV model with boundary conditions of “step-like” type is considered. The time-asymptotic behavior of the solution for the Cauchy problem is obtained. A similar problem for the dissipative perturbation of this problem is discussed. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 199, 1992, pp. 37–42. Translated by R. F. Bikbaev.     

Axisymmetric contact of anisotropic shells of rotation with rigid and elastic foundations

A class of problems are investigated on determining the stressed-strained state of anisotropic shells of rotation that are in axisymmetric one-sided contact with rigid and elastic surfaces. The shells are under the action of surface and contour loads. For some combinations of these quantities the shell may break away from the surface. To determine the contact zone, the method of successive approximations is utilized. In contrast to most investigations in which the contact zone is first determined, the method proposed makes use of a special quantity characterizing the size of the contact zone. The load on contours is determined from the solution to the problem on the stressed state of the shell and the condition specified on the boundary of the contact zone. Some examples of solving concrete problems are given. Bibliography: 5 titles. Translated fromObchyslyuval’ na ta Prykladna Matematyka, No. 76, 1992, pp 70–74.     

Bending of a semi-infinite cantilevered plate weakened by a cut with contacting edges

We solve the problem of the bending of a semi-infinite cantilevered plate containing a cut perpendicular to a clamped edge. Contact of the edges of the cut is taken into account in the two-dimensional formulation on the basis of the model of contacting edges on the face of the plate. We study the effect of the boundary on the distribution of the contact reaction and compute the coefficients of force and moment intensity and determine the breakinge load. We compare the results obtained with the solution of the problem not taking account of the contact of the edges of the cut. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 83–86.     

The dynamics of a two-component fluid in the presence of capillary forces

In the present paper we study the qualitative behavior ast→∞ of the solution of the Cauchy problem for a system of equations describing a dynamics of a two-component viscous fluid. The model under consideration takes into account the mutual diffusion of the fluid components as well as their capillary interaction. We describe the ω-limit set of trajectories of the dynamical system generated by the problem. It is proved that the stationary solution of the problem, is a homogeneous stationary distribution of one of the components, is asymptotically stable. Any other stationary solution is not asymptotically stable and is even unstable if there are no close stationary solutions corresponding to a smaller energy level. Translated fromMatematicheskie Zametki, Vol. 62, No. 2, pp. 293–305, August, 1997. Translated by A. M. Chebotarev     

Lamb's inverse axially symmetric problem

The problem of restoring the velocity of transversal waves in an elastic half-space by a known wave field on the boundary of a medium is studied. The problem is reduced to a system of “integral” equations similar to the Volterra one. Theorems on existence in the small and on uniqueness of the solution are proved. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 51–67. Translated by T. N. Surkova.     

The contact problem of electroelasticity for a plane elliptic die

We obtain an exact solution of the problem of the stress-strain state of an elastic piezoelectric half-space acted on by a rigid elliptic die with a flat base. The axis of symmetry of the body coincides with the direction of the field of preliminary polarization of the body. The solution is confined to the case of translational displacement of the die. We determine the quantities that characterize the mechanical and electric fields that arise in the region of contact of the die with the half-space. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 40–52.     

Applied aspects of studying the roots of dispersion equations on nonprincipal sheets of a complex plane

In quantitatively estimating interference wave fields by methods of function theory, the initial integration path is subjected to deformation in the complex (ζ) plane. As a result of this deformation, sometimes the contour λ may go into other sheets of the Riemann (ζ) surface. This forces us to study the integrands and, in particular, the dispersion equation of the problem on the other sheets of the (ζ) plane. Similar questions are discussed for the case where a layer SH is in contact with two elastic half-spaces. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 225, 1996, pp. 62–90. Translated by T. N. Surkova.     

Scattering of a plane wave on a periodic curve with trapping domains

The problem of scattering on a periodic curve is considered. The asymptotic solution of the problem is constructed, and its principal terms are presented. The justification of the asymptotic solution found is provided. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 75–85. Translated by I. V. Kamotskii     

Determination of the temperatures and heat flows for a conical slide support taking account of frictional heating

We consider an axisymmetric problem of heat conduction taking account of frictional heating in a conetorus pair that models the functioning of a conical support. The bodies are pressed together and are rotating about a common axis. Heat is generated in the region of contact of the bodies due to frictional forces. Outside the region of contact there is heat exchange with the surrounding medium. The thermal contact between the two bodies is nonideal. The problem is reduced to a system of integral equations whose solution is constructed by the method of successive approximations. We give the results of numerical studies of the temperature distribution and heat flows from the geometric and thermophysical parameters of the body. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 19–27.     

Unique solvability of the Cauchy problem for a system of Maxwell equations with initial data on a timelike surface

In this paper, a uniqueness theorem on the solution of the Cauchy problem for a system of Maxwell equations is proved in the case where the coefficients ε and μ are analytic functions of coordinates and the initial data are given on an “immovable” surface Σ=Γ×[0≤t≤2T], where Γ is an analytic surface in R³. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 30–37. Translated by N.S. Zabavnikova.