Analysis of a unilateral contact problem taking into account adhesion and friction

In this paper, we investigate a contact problem between a viscoelastic body and a rigid foundation, when both the effects of the (irreversible) adhesion and of the friction are taken into account. We describe the adhesion phenomenon in terms of a damage surface parameter according to Frémond?s theory, and we model unilateral contact by Signorini conditions, and friction by a nonlocal Coulomb law. All the constraints on the internal variables as well as the contact and the friction conditions are rendered by means of subdifferential operators, whence the highly nonlinear character of the resulting PDE system. Our main result states the existence of a global-in-time solution (to a suitable variational formulation) of the related Cauchy problem. It is proved by an approximation procedure combined with time discretization.     

The plane contact problem of steady thermoelasticity taking heat generation into account

The plane steady contact problem of thermoelasticity when there is heat generation from friction, which arises when an infinite cylindrical punch moves over the surface of an elastic half-space along its generatrix, is considered. It is assumed that heat exchange between the free boundary of the half-space and the surrounding medium obeys Newton's law, while the condition for ideal thermal contact exists in the region in which the solids interact. The problem is reduced to a system of three integral equations in the heat fluxes and temperature. The effect of the thermal and mechanical properties of the cylinder and the half-space on the main contact characteristics is investigated numerically.     

The calculation of the energy losses in a sliding elastic contact with friction and wear (the plane problem)

The plane problem of the sliding contact of a punch with an elastic foundation when there is friction and wear is considered. Assuming the existence of a steady solution in a moving system of coordinates, relations are derived between the sliding velocity, the wear, the contact stresses and the displacements for an arbitrary dependence of the wear rate on the contact pressure. Taking into account the presence of a deformation component of the friction force, an equation is written for the balance of the mechanical energy for the punch - elastic base system considered. It is shown that the equality of the work of the external force in displacing the punch to the losses due to friction and the change in the shape of the foundation due to wear is satisfied when the work done by the contact stresses on the increments of the boundary displacements is equal to zero, and the frictional losses must be determined taking into account the non-uniformity of the distributions of the shear contact stresses and the sliding velocity in the contact area. Two special cases of the foundation in the form of a wide and narrow strip are considered, for which the total coefficient of friction is calculated, taking into account the deformation component of the friction force.     

Contact problems of the theory of elasticity with friction and adhesion

An exact closed solution of the plane contact problem for a semi-infinite stamp is constructed for the case when the free boundary of the half-plane is under a load (problem 1), or for an analytic solution, to any prescribed accuracy, of the problem of a finite stamp impressed into an elastic half-plane under the action of a central vertical forceP (problem 2), or under the action of the above force P, a horizontal force T and a pair of forces with moment M (problem 3). In all three cases the region of contact consists of a zone of adhesion and fraction, and the stamp has a plane profile.     

An inverse contact problem in the theory of elasticity

In this paper, we discuss an inverse problem in elasticity for determining a contact domain and stress on this domain. We show that this problem is an ill‐posed problem, and we establish the uniqueness and L²‐conditional stability estimation for the stress. Copyright © 1999 John Wiley & Sons, Ltd.     

The contact problem of elasticity theory for bodies with cracks

The plane contact problem of a stamp impressed into an elastic half-plane containing arbitrarily arranged rectilinear subsurface cracks is formulated and investigated by asymptotic methods. Partial or total overlapping of the crack edges is assumed. The problem reduces to a system of linear singular integrodifferential equations with side conditions in the form of equalities and inequalities. An analytic solution of the problem is obtained in the form of asymptotic power series /1/ in the relative dimension of the greatest crack. Dependences of the first terms of the asymptotic expansions of the desired functions on the mutual location of the cracks and the contact domains, the pressure and friction stress distributions, and the crack size and orientation are determined. Numerical results are presented.

Analysis of the influence of the stress-free boundary of the half-plane on the state of stress and strain of the elastic material near the cracks is presented in /2, 3/. The problem of a crack in an elastic plane whose edges overlap partially is also examined in /3/ by numerical methods.     

A dynamic unilateral contact problem with adhesion and friction in viscoelasticity

The aim of this paper is to study an interaction law coupling recoverable adhesion, friction and unilateral contact between two viscoelastic bodies of Kelvin–Voigt type. A dynamic contact problem with adhesion and nonlocal friction is considered and its variational formulation is written as the coupling between an implicit variational inequality and a parabolic variational inequality describing the evolution of the intensity of adhesion. The existence and approximation of variational solutions are analysed, based on a penalty method, some abstract results and compactness properties. Finally, some numerical examples are presented.     

A dynamic unilateral contact problem with adhesion and friction in viscoelasticity

The aim of this paper is to study an interaction law coupling recoverable adhesion, friction and unilateral contact between two viscoelastic bodies of Kelvin–Voigt type. A dynamic contact problem with adhesion and nonlocal friction is considered and its variational formulation is written as the coupling between an implicit variational inequality and a parabolic variational inequality describing the evolution of the intensity of adhesion. The existence and approximation of variational solutions are analysed, based on a penalty method, some abstract results and compactness properties. Finally, some numerical examples are presented.     

On Kolosov’s relations in the plane problem of elasticity theory

Three different Kolosov-type expressions for stresses in elastic bodies are given.     

A mixed dynamic problem of the theory of elasticity for a piecewise homogeneous plane

We study the process of splitting of a compressed piecewise homogeneous medium under high-speed motion of a wedge with the formation of a crack of unknown extent ahead of the wedge. The motion of the wedge occurs along the interface between physico-mechanical properties of the piecewise homogeneous plane. We carry out numerical studies. We obtain asymptotic representations of the elastic potentials at the limiting velocities of motion of the wedge. Two figures. One table. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 48–58.     

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.     

The spatial contact problem for an elastic layer and wavy punch when there is friction and wear

The spatial (three-dimensional) problem of the wear of a wavy punch sliding over an elastic layer bonded to a rigid base, assuming there is complete contact between the punch and the layer, is considered. It is assumed that there is Coulomb friction and wear of the punch. An analytical expression for the contact pressure is constructed using the general Papkovich–Neuber solution, the harmonic functions in which are represented in the form of double Fourier integrals, after which the problem reduces to a linear system of differential equations. It is established that the harmonics constituting the shape of the punch and the contact pressure are shifted with respect to one another in time along the sliding line of the punch. The velocity of this shift depends on the longitudinal and transverse frequencies of the harmonic, that is, dispersion of the waves is observed.     

Taking the particle interaction into account in a new concept of nucleation theory

Taking the interaction of particles into account in a new theory of nucleation leads to the occurrence of a phase transition in the Van der Waals theory. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 303–304, August, 2008.     

Fundamental solution of a plane problem of elasticity theory for a composite anisotropic medium

The plane problem on the action of an arbitrarily oriented concentrated force, applied at some point of an elastic plane, composed of two different anisotropic half-planes, is considered. By a special choice of a particular solution the problem reduces to a well-known differential equation of the anisotropic theory of elasticity with discontinuous coefficients. The latter reduces, by the method of the integral Fourier transform, to the Riemann boundary value problem. Expressions for the stresses and displacement derivatives at an arbitrary point of the plane are obtained. The application of the obtained results is illustrated on the example of a problem on an elastic linear inclusion (strap).Translated from Dinamicheskie Sistemy, No. 4, pp. 40–45, 1985.     

Complex potentials for the plane problem of elasticity theory for a multiconnected anisotropic body with cracks

Results are given for a study and modernization of basic relationships obtained previously by the author for generalized complex potentials of crack theory generated by solving contact problems in the case of multiconnected anisotropic plates. In contrast to the author's previous work devoted to this question, general presentations of complex potentials, and boundary and additional conditions for finding them, are simplified. Use of them is shown in the case of a crack and concentrated forces, and when a plate has a finite number of cracks along a single line. Independence of stress intensity factor on anisotropy parameters is demonstrated in the last case.Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 24–34, 1990.     

A plane problem in elasticity theory for a half-strip with given normal displacement at the base

This paper presents a solution for the elasticity problem for a half-strip with load-free lateral edges and given normal displacement with zero tangential stress at the base. The functions describing the load at the base are not assumed smooth. Translated fromDinamicheskie Sistemy. Vol. 12. pp. 8–14. 1993.     

Neumann problem for the operator of elasticity theory

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 9, pp. 1279–1281, September, 1989.