On the use of jump conditions at a thin inclusion in solving problems of elasticity theory for a half-space with protuberances

We propose a method of approximate solution of problems of elasticity theory for a half-space with protuberances based on the use of jump conditions in the stresses and displacements at a thin elastic element. The problem of determining the stresses reduces to a system of two-dimensional integral equations of Newtonian potential type for determining the contact stresses between the protuberances and the half-space. We consider the case when the elastic characteristics of the material of the protuberances are different from the material of the half-space.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 156–160.     

Torsion of a nonhomogeneous layer

A rotation of a nonhomogeneous, with respect to thickness, layer is considered. Pliability functions are studied that connect the transformations of stresses and displacements. Some estimates, asymptotic formulas, distribution and character of zeros and poles in the complex plane are given. A solution to the problem of rotation of a nonhomogeneous layer with a ring die is constructed. Triple integral equations are reduced to successively solvable Fredholm equations of the second kind and an infinite system of linear algebraic equations that is normal in the sense of Poincaré—Koch.Translated from Dinamicheskie Sistemy No. 8, pp. 22–30, 1989.     

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.     

Determination of contact stresses in problems on bending of a package of transversely isotropic bars

A mechanomathematical model for bending of packages of transversely isotropic bars of rectangular cross section is proposed. Adhesion, slippage, and separation zones between the bars are considered. The resolving equations for deflections and tangential displacements are supplemented with a system of linear differential equations for determining the normal and tangential contact stresses, and boundary conditions are formulated. A scheme for analytical solution of two contact problems—a package under the action of a distributed load and a round stamp—is considered. For these packages, a transition is performed from the initial system of differential equations for determining the contact stresses, where the unknown functions are interrelated by recurrent relationships, to one linear differential equation of fourth order and then to a system of linear algebraic equations. This transition allows us to integrate the initial system and get expressions for the contact stresses.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 761–778, November–December, 2004.     

A Study on the Propagation of Plane Stress Waves across the Thickness of a Plate by the Method of Analytic Continuation in Time

The interaction of plane tension/compression waves propagating within a plate perpendicularly to its surface is considered. The analytic solution is obtained by a modified method of characteristics for the one-dimensional wave equation used in problems on an impact of a rigid body on the surface of a plate. The displacements, velocities, and stresses in the plate are determined by the edge disturbance caused by the initial velocity and the stationary force field of masses of the striker and the plate. The method of analytic continuation in time put forward allows a stress analysis for an arbitrary time interval by using finite expressions. Contrary to a stress analysis in the frequency domain, which is commonly used in harmonic expansion of disturbances, the approach advanced allows one to analyze the solution in the case of discontinuous first derivatives of displacements without calculating jumps in summing series. A generalized closed-form solution is obtained for stresses in an arbitrary cycle n(t), which is determined by the multiplicity of the time of wave travel across the double thickness of the plate. A method of recurrent solution based on calculating the convolution of repeated integrals of the initial form of disturbance at t = 0 is elaborated. The procedure can be used for evaluating the maximum stress and the contact time in a plane impact on the surface of a plate.     

Vertical oscillations of multimass system on elastic half space with a cylindrical cavity

A problem on oscillations of a multimass system (MS) is considered on an elastic half space with a cylindrical cavity. Equations of motions of an MS are given, which are modeled by masses that are connected by springs and dampers. A motion of the half space with a cavity is characterized by a transmitting function,which is known from a solution of a contact problem with vertical oscillations of a die on the half space given. The conditions of interrelation of the MS with the base close the system of algebraic linear equations for determining amplitudes of oscillations of each element of the MS. Translated from Dinamicheskie Sistemy, No. 7, pp. 13–18, 1988.     

The elastoplastic stressed state of a half-plane under local nonstationary heating

We solve the thermoplastic problem for a semi-infinite plate under local nonstationary heating by heat sources. The physical equations are taken to be the relations of the nonisothermic theory of plastic flow associated with the Mises fluidity condition. The solution of the problem is constructed by the method of integral equations and the self-correcting method of sequential loading, where time is taken as the loading parameter. We carry out numerical computations of the stresses in the case of heating a plate with heat output by normal-circular heat sources. We study the problem of optimization of heating regimes in order to introduce favorable residual compressive stresses (from the point of view of hardness) in a given region of a half-plane. Two figures.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 29–34.     

The influence of coupled diffusion of heat and moisture on the state of stress in a plate

The fundamental results of previous reports on the coupling of the processes of diffusion of heat and moisture in solids are reviewed. The influence of the coupling on diffusion into an infinite plate from both its surfaces is discussed with the aim of clarifying what can and cannot be learned from such experiments. The nonuniformity of temperature and moisture content tends to cause nonuniform deformation. The stresses set up are computed for a homogeneous, isotropic plate. These self-equilibrating stresses first increase with time and then die out. For some combinations of material constants, the stresses undergo a reversal. For example, in some cases the surface stresses initially become compressive as moisture begins diffusing into the plate. Subsequently, as the interior of the plate increases in temperature due to the coupling effect, the surface stresses become tensile before eventually subsiding to zeroPublished in Mekhanika Kompozitnykh Materialov, No. 1, pp. 53–61, January–February, 1980.     

On stresses in a strip with a crack

The solution of the problem of a loaded crack in an infinite strip is given using the method of superposition of three problems (a loaded crack in the infinite plane; an infinite homogeneous strip with normal and tangent stresses that are given on nonhomogeneous boundaries; an infinite strip with longitudinal generators which are free from load and an arbitrary load at the end), which makes it possible to satisfy the boundary conditions exactly.Translated from Dinamicheskie Sistemy, No. 9, pp. 65–71, 1990.     

Determination of residual stresses in glass shells by a computational-experimental method

A method is proposed for determining the technological residual stresses in glass shells of revolution based on the solution of inverse problems of shell theory with proper stresses and the use of experimental data obtained by the photoelastic method. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 100–105.     

Dynamics of a Sea Platform Interacting with the Seabed and the Water Medium

Transient vibrations of the system {structure–seabed–water layer} are investigated. The schematic representation of the structure consists of a rigid circular foundation slab and a superstructure in the form of a vertical viscoelastic cylinder with lumped masses. The interaction of the foundation with a viscoelastic half-space (the seabed) is taken into account by using the method of orthogonal polynomials to solve the dynamical contact boundary-value problem. A procedure is developed for including in the analysis the interaction of a cylindrical foundation-supported structure with the water layer (to simulate the dynamics of an oil-drilling platform). The assumed load is seismic action specified by an analog accelerogram of the local area. A computational system is developed for numerical simulation of the vibrations and for determining the forces, displacements, accelerations, and contact stresses.     

Method of nonlinear boundary equations in problems of contact of several elastic bodies

The method of nonlinear boundary equations is applied to develop new formulations of contact problems with unknown contact regions. Our formulation is free from inequality constraints, which enter the method of variational inequalities and the standard formulations of contact problems. Methods of the theory of operator equations are applied to prove that the problems are well-posed.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 62–70, 1987.     

Stress-strain state and stability of composite sandwich shells with a scaling zone between the core and facings

A finite element model is presented for analyzing the strength and stability of sandwich shells of arbitrary configuration with an adhesion failure zone between the core and one of the facings. The model is based on the assumptions that both facings are laminated Timoshenko-type composite shells, only transverse shear stresses in the core and normal stresses in the thickness direction have nonzero values, a free slip in the tangential plane in the adhesion failure zone and unilateral contact along the normal are possible, and the prebuckling state in the stability problem is linear. Biquadratic nine-node approximations for all functions and numerical integration were used. The displacements and rotation angles of the normals toward the facings as well as stresses in the core are taken as global degrees of freedom. The algebraic problem is solved using a special step-by-step procedure of determining the contact area in the scaling zone and employing unilateral constraints for some of the unknowns. Numerical examples are also given.Translated from Mekhanika Kompozitnykh Materialov, Vol. 29, No. 5, pp. 640–652, September–October, 1993.     

Thermoelastic contact interaction of bodies in the presence of surface thermophysical irregularities

We study the thermoelastic contact interaction (in the absence of friction) of half-spaces under conditions of planar deformation in the presence of thin surface thermophysical irregularities that are taken into account by means of generalized conditions of thermal contact with one another. The problem is reduced to solving a system of singular integrodifferential equations with respect to the jumps of temperature and heat flow on the boundary of a section. We analyze the influence of a nonuniform thermal resistance distributed periodically along the surface or localized in one region of it on the distribution of temperature and stresses in the bodies and on their boundary. Four figures.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 23–28.     

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.     

On the solution of mixed problems for a fiber with a vertical circular cylindrical cavity

A method of solving paired integral equations that appear in considerations of mixed problems of elasticity and thermoelasticity theory is given, with the help of generalized integral Weber transforms. The paired equations are reduced to an integral Fredholm equation of the second kind on the semiaxis, which have a discontinuous kernel, or to Fredholm equations of the second kind on a finite interval and infinite systems of linear algebraic equations, which are normal in the sense of Poincare-Koch. As an example, contact problems for an inhomogeneous fiber with a cavity are considered. If the fiber is bonded with the elastic half-space, then a second appproach is realized, which is based on a reduction to an equation with a self-adjoint operator, for which some method of sequential iteractions and the Bubnov-Galerkin method are justified.Translated from Dinamicheskie Sistemy, No. 7, pp. 95–102, 1988.     

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.     

An approach to studying nonsteady-state vibrations of a finite length cylinder

A method is given for analyzing the nonsteady-state vibrations of a finite length cylinder based on using Laplace transforms and approaches developed previously for determining the vibration natural frequencies and forms. A numerical method is demonstrated for determining residues with reference to expressions for displacements and stresses. Quantitative estimates are given for stresses in a cylinder for a specific principle of the change in load with time and features of the time-dependent behavior of displacements in relation to the magnitude of the area of loading at the end are studied.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 129–133, 1987.     

Effect of overload on the integral parameters of a concentric annular die

The non-axisymmetric contact problem in the theory of elasticity is solved for a smooth concentric annular die on a uniform elastic half-space when an overload normal to the boundary of the half-space acts outside the die. A Hankel-Fourier integral transform and triple integral equations are used to reduce the problem to a quasi-regular system of algebraic equations which is solved by a perturbation method and, in general, by a reduction method. The effect of a lumped force on the integral parameters of the die is examined. Graphs are shown which characterize the degree to which the parameters of the problem influence the total force and moment, displacement, and inclination of the die. Kharkov State Economics University. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 111–117, 1999.     

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.