Contact interaction of cylindrical shells of different lengths

We solve a problem of determination of the contact rigidity of a structural joint of a steel thin-walled pipe under the action of internal pressure with a shroud made of a composite material using a special technology. A mathematical model of the contact interaction of the pipe and shroud modeled by cylindrical shells of different length was constructed using the classical Kirchhoff–Love theory of shells. We obtain an analytic solution of the contact problem under conditions of ideal contact of the elements of the structural joint by the conjugation method. A numerical analysis of the influence of geometric and physicomechanical characteristics of the shroud on the contact pressure and rigidity of the shrouded pipe is presented.     

Mathematical modeling of wear-resistance processes of bodies with coatings

We study a mathematical formulation of problems of contact interaction of coated bodies when the material is worn. We state the physical relations that connect the intensity of wear with the characteristic physical quantities. For the case of a half-plane with a thin elastic coating the problem is reduced to singular integro-differential equations whose solution is obtained in the class of orthogonal polynomials. We give estimates of the influence of the characteristics of the coating and the base on the wear process. One figure, two tables.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 51–55.     

Analysis of contact stresses in structural joints of composite shell with steel banding

Based on the generalized Timoshenko-type shell theory, a numerical-analytical procedure for determining contact stresses from the interaction between a cylindrical composite shell and rigid bandings is proposed. Specific cases of loading and contact interaction (ideal contact through an adhesive interlayer) are considered. The contact problems are reduced to the solution of a Fredholm integral equation of the second-kind. A calculation analysis is performed. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 1, pp. 109–120, January–February, 2000.     

Stability of a thin liquid layer,spreading on a quasi–rotating disk

In many industrial processes solids are coated to obtain specific surface properties, as e.g. corrosion resistance, mechanical (wear) resistance, optical, or electrical properties. Even today many coating processes are not fully understood and the choice of parameters is largely based on experience. Hence, a prediction of the complete hydrodynamic process and the appearance of instabilities in its dependency on the parameters appears highly desirable. This would serve to optimize the quality of the coating. A common coating technique is the so-called spin coating. The coating agent is dissolved or suspended in a liquid, brought onto the solid, spread by rotation, and the carrier liquid is finally removed by evaporation or by chemical reactions. In this article an evolution equation is derived from lubrication theory, valid for thin liquid layers. The model involves a dynamic contact angle, centrifugal, capillary, and gravitational forces. The evolution equation can be solved analytically, provided the capillary number is small. Then a coupled linear stability analysis of the contact line and the free interface is performed. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On wheel–rail contact problem with material properties varying with depth

Numerous laboratory experiments indicate that the use of a layer or a coating material attached to the conventional steel body reduce the magnitude of contact stress. Therefore in this paper we solve numerically the wheel–rail contact problem with friction and wear assuming the existence of a small elastic layer on the rail surface. Material properties of this layer are changing with its depth. The friction between the bodies is governed by Coulomb law. In contact zone Archard's law of wear is assumed. We take special features of this rolling contact problem and use so-called quasistatic approach to solve this contact problem. Finite element method is used as a discretization method. The numerical results including the distribution of normal stress along the contact boundary are provided and discussed. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Impression of a semiinfinite stamp in an elastic strip with regard for friction and adhesion

We consider the problem of contact interaction between a semiinfinite stamp with rectilinear base and an elastic strip with one rigid side. Friction forces in the contact region are taken into account. These forces lead to the division of the contact region into slipping and adhesion zones. With the use of the Wiener–Hopf method, a system of integral equations is reduced to an infinite system of algebraic equations. The computational results of stresses and strains at the boundary and at inner points of the elastic strip are presented. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 138–149, January–March, 2008.     

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     

Numerical Simulation of Free Oscillations of Elastic Bodies with a Thin Coating

We propose an approach to the investigation of problems on free oscillations of elastic bodies with a thin coating. The method consists of applying a combined mathematical model which is based on the three-dimensional equations of elasticity theory in the domain of a body and on the two-dimensional equations of the theory of shells of the Timoshenko type in the domain of a thin coating. The systems of these equations are related by the conditions of conjugation on the surface of contact. For the numerical analysis of the eigenvalue problem, we used a scheme of the finite-element method constructed by using approximations of different dimensionality.     

Effect of wear on the contact stress distribution in sliding bearings with plastic bushes

The contact problem is considered for a thin elastic annular layer compressed by a rigid cylinder with initial contact over a considerable area. Relations for calculating the maximum contact stresses in sliding bearings with plastic bushes are obtained with allowance for the wear of the bush. From an analysis of these relations it follows that, as the bush wear increases, the maximum contact stresses fall appreciably. This offers a partial explanation of the fall in wear rate as the bearing is run in.Bryansk Machine-Building Plant. Translated from Mekhanika Polimerov, No. 6, pp. 1046–1051, November-December, 1969.     

A frictionless contact problem for elastic–viscoplastic materials with normal compliance: Numerical analysis and computational experiments

Summary. In this paper we consider a frictionless contact problem between an elastic–viscoplastic body and an obstacle. The process is assumed to be quasistatic and the contact is modeled with normal compliance. We present a variational formulation of the problem and prove the existence and uniqueness of the weak solution, using strongly monotone operators arguments and Banach's fixed point theorem. We also study the numerical approach to the problem using spatially semi-discrete and fully discrete finite elements schemes with implicit and explicit discretization in time. We show the existence of the unique solution for each of the schemes and derive error estimates on the approximate solutions. Finally, we present some numerical results involving examples in one, two and three dimensions. Received May 20, 2000 / Revised version received January 8, 2001 / Published online June 7, 2001     

Existence of stationary solutions of the Navier–Stokes equations in two dimensions in the presence of a wall

We consider the problem of a body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modeled by the incompressible Navier–Stokes equations in an exterior domain in a half space, with appropriate boundary conditions on the wall, the body, and at infinity. Here we prove existence of stationary solutions for this problem for the simplified situation where the body is replaced by a source term of compact support.     

An unflexural refined model of deformation of multilayered plates on an undeformable foundation

The stress-strain state of multilayered plates on an undeformable foundation is investigated. The design diagram of a transversely loaded plate is formed by supplementing it with a symmetric one about the contact surface of the foundation. The double-thickness plate obtained becomes loaded bilaterally and symmetrically about its midsurface. In such a way, only unflexural deformations can be modeled, which reduces the number of unknowns and the general order of differentiation of the resolving system of equations. The refined continual model developed takes into account the deformations of transverse shear and transverse compression in high iterative approximations. The cases of a rigid contact between the foundation and the plate and frictionless slip of the plate over the foundation are considered. Calculations confirm the efficiency of such an approach, which allows one to obtain solutions qualitatively and quantitatively close to three-dimensional ones. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 5, pp. 643–654, September–October, 2006.     

Stability of Composite Cylindrical Shells with Noncoincident Directions of Layer Reinforcement and Coordinate Lines

The stability problem is solved for cylindrical shells made of a laminated composite whose directions of layer reinforcement are not aligned with coordinate axes of the shell midsurface. Each layer of the composite is modeled by an anisotropic material with one plane of symmetry. The resolving functions of the mixed variant of shell theory are approximated by trigonometric series satisfying boundary conditions. The stability of the shells under axial compression, external pressure, and torsion is investigated. A comparison with calculation data obtained within the framework of an orthotropic body model is carried out. It is shown that this model leads to considerably erroneous critical loads for some structures of the composites. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 5, pp. 651–662, September–October, 2005.     

Contact with break-off under bending of an elastic strip with a rigid disk

We consider the problem of the theory of elasticity of the contact interaction of a rigid circular disk and an elastic strip, which rests upon two supports with disturbance of contact in the middle part of the contact region. On the basis of the Wiener–Hopf method, an integral equation of the problem is reduced to an infinite system of algebraic equations. The size of the zone of break-off of the boundary of the strip from the disk and the distribution of contact stresses are determined.     

Gravity-driven thin-film flow using a new contact line model

In this paper, we consider how a new model for the motion ofa contact line, proposed by Shikhmurzaev (1993, Int. J. MultiphaseFlow, 19, 589-–610), affects predictions for the gravity-drivenflow of a thin film down an inclined plane. We find that forsufficiently thin films, the model reduces to Navier slip withthe contact angle equal to its static value, while for thickerfilms the model has a character of its own, with a slip regionthat becomes larger, the thicker the film and a contact anglethat increases as the thickness of the film increases.     

Application of the method of direct cutting-out to the solution of the problem of longitudinal shear of a wedge with thin heterogeneities of arbitrary orientation

The method of direct cutting-out consists of modeling of a finite body, in particular, with thin heterogeneities, using a much simpler problem for a bounded or a partially bounded body with thin heterogeneities located in the same manner and the presence of additional cracks or absolutely rigid inclusions of fairy large length, which are modeled by the boundary conditions of a bounded body. The method is tested on the problems of antiplane deformation of a symmetrically loaded crack in a wedge with free faces and an absolutely rigid inclusion placed with some tension in a wedge with restrained faces. For an elastic inclusion, we construct generalized conditions of interaction, which enable us to unify the procedure of giving different boundary conditions in the case of using the method of direct cutting-out.     

The classical and generalized love problems in a domain of low-frequency interference waves of signal type. I

Low-frequency interference waves propagating through a thin elastic layer, which is in rigid contact with an elastic half-space, are considered. A new method for computing the fields of low-frequency waves suggested by G. I. Petrashen allows us to compute the fields of these waves and to draw some conclusions concerning their behavior. In particular, we may conclude that the thin upper layer significantly distorts a signal sent deep into a medium. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 225, 1996, pp. 5–39. Translated by N. S. Zabavnikova.     

An analysis of contact deformation of auxetic composites

The adaptive mode of frictional interaction has been studied as a self-locking effect upon contact deformation of isotropic and anisotropic auxetic materials with a negative Poisson ratio. This effect manifests itself in the fact that the bearing capacity of the joint rises with increasing shear load. In particular, the parameters of stress state (contact load, tangential stresses, slippage, etc.) were determined for a double-lap joint under conditions of compression with or with out shear. The contact interaction was analyzed by the finite-element method for three profiles of symmetrically located contact elements (plane, cylindrical, and wedge-shaped). The Poisson ratio was varied within the range theoretically admissible for isotropic elastic media. Analogous calculations were also performed for a joint with a deformed element made of an anisotropic auxetic composite, whose reinforcement angle was varied. The maximum loads, tangential stresses, and slippage are obtained as nonlinear functions of Poisson ratio (in the isotropic case) and reinforcement angle of the composite material. The stress concentration and the increased ultimate shear forces are also estimated. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 5, pp. 681–692, September–October, 2006.     

Statistical hydrodynamics of magnetic fluids. I. the nonequilibrium statistical operator method

The Zubarev nonequilibrium statistical operator is used to describe the generalized hydrodynamic state of a magnetic fluid in an external magnetic field. The magnetic fluid is modeled with “liquid-state” and “magnetic” subsystems described using the classical and quantum statistics methods respectively. Equations of the generalized statistical hydrodynamics for a magnetic fluid in a nonhomogeneous external magnetic field with the Heisenberg spin interaction are derived for “liquid-state” and “magnetic” subsystems characterized by different nonequilibrium temperatures. These equations can be used to describe both the weakly and strongly nonequilibrium states. Some limiting cases are analyzed in which the variables of one of the subsystems can be formally neglected. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115, No. 1, pp. 132–153, April, 1998.     

A Crack with Interfacial Bonds in an Isotropic Medium Strengthened with a Regular System of Stringers

An isotropic medium containing a system of foreign transverse rectilinear inclusions is considered. Such a medium can be interpreted as an infinite plate strengthened with a regular system of ribs (stringers) whose cross section is a very narrow rectangle. The medium is weakened by a rectilinear crack. The action of the stringers is replaced with unknown equivalent concentrated forces at the points of their connection with the medium. A model of a crack with areas where its faces interact with each other is investigated. This interaction is modeled by introducing bonds (adhesion forces) between faces in the crack tip zone. The boundary-value problem on equilibrium of the crack under the action of external tensile forces is reduced to a nonlinear singular integral equation, from the solution of which the tractions in the bonds are found. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 6, pp. 773–782, November–December, 2005.