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.     

Pressure of an elastic half space on a rigid base with rectangular hole in the case of a liquid bridge between them

A model of contact between an elastic half space and a rigid base with a shallow surface rectangular hole is proposed. The hole contains an incompressible liquid and gas. The liquid occupies the middle part of the hole and forms a capillary bridge between the opposite surfaces. The remaining volume of the hole is filled with gas under a constant pressure. The liquid completely wets the surfaces of the bodies. The pressure drop at the liquid–gas interface caused by the surface tension is defined by the Laplace formula. The corresponding plane contact problem for the elastic half space is essentially nonlinear because the pressure of the liquid and the length of the capillary in the contact-boundary conditions are not known in advance and depend on the external load. The problem is reduced to a system of three equations (a singular integral equation for the function of height of the hole and two transcendental equations for the length of the capillary and the height of the meniscus). An analytic-numerical procedure for the solution of these equations is proposed. Dependences of the length of the capillary and the pressure drop at the liquid–gas interface on the external load, volume of liquid, and its surface tension are analyzed. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 150–156, January–March, 2008.     

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.     

The contact problem for an elastic strip and a wavy punch when there is friction and wear

The plane problem of the mutual wear of a wavy punch and an elastic strip, bonded to an undeformable foundation under the condition of complete contact between the punch and the strip is considered. An analytical expression for the contact pressure is constructed using the general Papkovich–Neuber solution, the two harmonic functions in which are represented in the form of Fourier integrals after which the problem reduces to a non-linear system of differential equations. In the case of a small degree of wear of the strip, this system becomes linear and admits of a solution in explicit form. The harmonics, constituting the profile of the punch and the contact pressure, move along the strip with respect to one another and are shifted in time. Conditions are obtained that ensure the hermetic nature of the contact between the wavy punch and the strip when there is friction and wear.     

Optimization of the stress tensor in an elastic anisotropic half-plane

We consider the problem of optimizing the components of the stress tensor and their integral characteristics. The normal and tangential forces prescribed on the boundary of the elastic anisotropic half-plane y≥0 are chosen from certain function classes of curvilinear strip type. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 110–116.     

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.     

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.     

A Contact Problem of the Interaction of a Semi-Finite Inclusion with a Plate

A piece wise-homogeneous plane made up of twodifferent materials and reinforced by an elastic unclusion is considered on a semi-finite section where the different materials join. Vertical and horizontal forces are applied to the inclusion which haz a variable thichness and a variable elasticity modulus.Under certain conditions the problem is reduced to integrodifferential equations of third order. The solution is constructed effectively by applying the methods of theory of analytic functions to a boundary value problem of the Carleman type for a strip. Asymptotic estimates of normal contact stress are obtained.     

Fracture of 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 periodic system of rectilinear cracks. The action of the stringers is re placed by unknown equivalent concentrated forces at the points of their connection with the medium. The boundary-value problem on equilibrium of the periodic system of cracks under the action of external tensile forces is reduced to a singular integral equation, from the solution of which the stress in tensity factors are found. The condition of limiting state of equilibrium of the cracks is formulated based on a criterion of brittle fracture. The stress state in the case where crack faces come into a partial contact is also considered. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 1, pp. 59–72, January–February, 2007.     

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.     

Approximate Description of the Dynamics of Thin Isotropic Elastic Coatings and Interlayers by Using Asymptotics of High Order of Accuracy

Asymptotically accurate low-frequency models for isotropic elastic coatings and interlayers are developed. The main constraint is the requirement on contact conditions for the layer and the base that at least one of the boundary conditions must include the displacement component in an explicit form. The displacement and stress fields in the three-dimensional elastic system are sought in the form of asymptotic expansion into power series of a small parameter — the ratio between the half-thickness of the layer and the minimum length of the wave in the longitudinal direction. The action of the layer is approximated by impedance boundary conditions, which are transferred to the contact surface with the basic, more thick body. These conditions are obtained with an asymptotic error up to and including the sixth order of magnitude. A numerical testing, which is carried out with the example of partial waves, shows a satisfactory accuracy, comparable with that of high-order theories of plates. The results obtained can be utilized in fast algorithms for calculating spectra of natural waves in half-spaces, thick laminated plates, and shallow shells with coatings and interlayers. The physical limit of applicability of the theory developed is the frequency of the first quasi-resonance in the corresponding deformable system. The number of alternating interlayers is unlimited. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 6, pp. 783–794, November–December, 2005.     

Deformation of a composite plate on an elastic foundation by local loads

Bending of an elastic annular composite plate with a light filler lying on an elastic foundation is considered. The plate is subjected to local loads. To describe the kinematics of the package, asymmetric across its thickness, the hypotheses of broken normal is accepted. The reaction of foundation is described based on the Winkler model. A system of equilibrium equations is constructed, and its exact solution in displacements is found. Numerical solutions for a metal-polymer sandwich plate are presented. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 1, pp. 109–120, January–February, 2007.     

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.     

Two problems with mixed boundary conditions for an elastic orthotropic strip

Two problems are considered for an elastic orthotropic strip: the contact problem and the crack problem. Both problems are reduced to integral equations of the first kind with different kernels, containing a singularity: logarithmic for the first problem and singular for the second problem. Regular and singular asymptotic methods are employed to construct approximate solutions of these integral equations. Numerical results are presented.     

Local near-surface buckling of a system consisting of an elastic (viscoelastic) substrate,a viscoelastic (elastic) bond layer,and an elastic (viscoelastic) covering layer

Within the framework of a piecewise homogenous body model and with the use of a three-dimensional linearized theory of stability (TLTS), the local near-surface buckling of a material system consisting of a viscoelastic (elastic) half-plane, an elastic (viscoelastic) bond layer, and a viscoelastic (elastic) covering layer is investigated. A plane-strain state is considered, and it is assumed that the near-surface buckling instability is caused by the evolution of a local initial curving (imperfection) of the elastic layer with time or with an external compressive force at fixed instants of time. The equations of TLTS are obtained from the three-dimensional geometrically nonlinear equations of the theory of viscoelasticity by using the boundary-form perturbation technique. A method for solving the problems considered by employing the Laplace and Fourier transformations is developed. It is supposed that the aforementioned elastic layer has an insignificant initial local imperfection, and the stability is lost if this imperfection starts to grow infinitely. Numerical results on the critical compressive force and the critical time are presented. The influence of rheological parameters of the viscoelastic materials on the critical time is investigated. The viscoelasticity of the materials is described by the Rabotnov fractional-exponential operator. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 6, pp. 771–788, November–December, 2007.     

The contact problem for a piecewise-homogeneous plane with a semi-infinite inclusion

A piecewise-homogenous elastic plate, reinforced with a semi-infinite inclusion, which intersects the interface at a right angle and is loaded with shear forces is considered. The contact stresses along the contact line are determined and the behaviour of the contact stresses in the neighbourhood of singular points is established. Using methods of the theory of analytical functions and integral transformations the problem is reduced to a system of singular integro-differential equations on the semi-axis. The solution is presented in explicit form.     

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.     

Non-linearity of multibody dynamic equations with respect to Lagrange multipliers: application to railway dynamics

In this paper, the dynamics of multibody systems with closed kinematical chains of bodies is considered. The main focus is set on non-linearity of the multibody equations with respect to the Lagrange multipliers. When closed chains are considered, loop cutting procedure is a solution to express the constraint equations associated with the loops. Dynamic equations of the multibody tree-like structure are thus completed with the constraint forces via the Lagrange multipliers. In the considered case of railway vehicles, constraints arise from the contact between the rigid wheels and the rails. Corresponding contact forces applied to the wheels appears via the Lagrange multipliers λ and the tangent creep forces as well. Resulting differential-algebraic equations can be transformed into an ODE system and then time-integrated using the coordinate partitioning method [3], when the system is linear with respect to λ. This paper presents an algorithm allowing us to solve this system in case of nonlinearities with respect to λ, which is typical of wheel/rail contact force models. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

The rolling of a wheel with a reinforced tyre along a plane without slip

A model of a wheel with a reinforced tyre, whose surface is simulated by a flexible strip (tread) attached to parts of two tori (the sidewalls of the tyre) is considered. The disk of the wheel (a rigid body) has six degrees of freedom and is in contact with the plane along part of the tread. Based on several assumptions, the potential energy functional of the deformed wheel is found as a function of the deformations of the centre line of the tread. On the assumption that the wheel is rolling without slip in the region of contact of the tread with the plane along a previously unknown section of the tread, the complete system of equations of motion is obtained. The equilibrium of the wheel and the steady state of rolling in a straight line with given swivel and tilt are investigated, and all characteristics of the motion are found (the contact region, the tyre deformation, and the forces and torques applied to the wheel disk).