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.     

Axisymmetric contact of a punch of polynomial profile with an elastic half-space when there is friction and adhesion

The axisymmetric problem of the contact interaction of a punch of polynomial profile and an elastic half-space when there is friction and partial adhesion in the contact area is considered. Using the Wiener–Hopf method the problem is reduced to an infinite system of algebraic Poincare–Koch equations, the solution of which is obtained in series. The radii of the contact area and of the adhesion zone, the distribution of the contact pressures and the indentation of the punch are obtained.     

The indentation of a spherical punch into an ideally plastic half-space when there is contact friction

Calculations are presented of the indentation of a spherical punch into an ideally plastic half-space under condition of complete plasticity and taking account of contact friction, which is modelled according to Prandtl and Coulomb. Friction leads to the formation of a rigid zone at the centre of the punch when there is slipping of the material on the remaining part of the contact boundary. Limit values of the friction coefficients are obtained for which the rigid zone extends over the whole of the contact boundary. The dependence of the indentation force on the radius of the plastic area is in good agreement with experimental data.     

The three-dimensional contact problem with friction for an elastic wedge

Solutions of three-dimensional boundary-value problems of the theory of elasticity are given for a wedge, on one face of which a concentrated shearing force is applied, parallel to its edge, while the other face is stress-free or is in a state of rigid or sliding clamping. The solutions are obtained using the method of integral transformations and the technique of reducing the boundary-value problem of the theory of elasticity to a Hilbert problem, as generalized by Vekua (functional equations with a shift of the argument when there are integral terms). Using these and previously obtained equations, quasi-static contact problems of the motion of a punch with friction at an arbitrary angle to the edge of the wedge are considered. In a similar way the contact area can move to the edge of a tooth in Novikov toothed gears. The method of non-linear boundary integral equations is used to investigate contact problems with an unknown contact area.     

The wear of a punch when it slides randomly on a thin elastic layer

The three-dimensional problem of the wear of a punch, which slides randomly on a thin elastic layer is considered. Using the deformation model of an asymptotically thin layer and the procedure for averaging the wear law in random directions of the sliding of the punch, a differential equation is obtained for the kinetics of the punch wear, an analytical solution of which is constructed by the method of characteristics. It is established that a characteristic feature of the evolution of the shape of the worn surface of the punch is its equidistant displacement in the contact plane. An expression for the rate of this displacement is obtained.     

Asymptotic solution of contact problemsfor a relatively thick elastic layer when there are friction forces in the contact area

The plane contact problem of the theory of elasticity of the interaction between a punch, having a base in the form of a paraboloid,and a layer, taking Coulomb friction in the contact region into account, is considered. It is assumed that either the lower boundary of the layer is fixed or there are no normal displacements and shear stresses on it, and that normal and shear forces are acting on the punch. Here, the punch-layer system is in a condition of limit equilibrium, and the punch does not turn during the deformation of the layer. The case of quasi-statistics, when the punch moves evenly over the layer surface, can be considered similarly in a moving system of coordinates. The problem is investigated by the large-λ method (see [1–3], etc.), which is further developed here, namely, simple recurrence relations are derived for constructing any number of terms of the series expansion of the solution of the corresponding integral equation in negative powers of the dimensionless parameter λ related to the thickness of the layer.     

The contact problem when there are friction forces in the contact area for a three-component cylindrical base

The plane contact problem of elasticity theory on the interaction when there are friction forces in the contact area of an absolutely rigid cylinder (punch) with an internal surface of a cylindrical base, consisting of two circular cylindrical layers rigidly connected to one another and with an elastic space, is considered. The layers and space have different elastic constants. A vertical force and a counterclockwise torque, act on the punch, and the punch – base system is in a state of limiting equilibrium,. An exact integral equation of the first kind with a kernel represented in an explicit analytical form, is obtained for the first time for this problem using analytical calculation programs. The main properties of the kernel of the integral equation are investigated, and it is shown that the numerator and denominator of the kernel symbols can be represented in the form of polynomials in products of the powers of the moduli of the displacement of the layers and the half-space. A solution of the integral equation is constructed by the direct collocation method, which enables the solution of the problem to be obtained for practically any values of the initial parameters. The contact stress distributions, the dimensions of the contact area, the interconnection between the punch displacement and the forces and torques acting on it are calculated as a function of the geometrical and mechanical parameters of the layers and the space. The results of the calculations in special cases are compared with previously known results.     

Solving the problem of contact equilibrium of a punch and an elastic body

The variational problem of contact equilibrium of a punch and an elastic body is considered. An equivalent formulation of the problem is given in variational inequality form. Existence and uniqueness of the solution is investigated in a particular case. A penalty method is proposed for approximate solution of the problem.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 97–103, 1985.     

The axisymmetric contact problem for an elastic layer loaded with a deformable cover plate

The axisymmetric problem of the contact interaction of an elastic cover plate with an elastic layer, loaded at infinity with a uniform stretching force, directed parallel to the boundaries of the layer, is considered. The cover plate resists stretching but does not resist bending. The contact shearing stress under the cover plate, the displacement of the points of the cover plate and the deformation distortion coefficient of the elastic layer are determined.     

The periodic contact problem for an elastic half-space

A method of solving the periodic contact problem for a system of indentors of arbitrary shape and an elastic half-space is proposed. Different versions of the arrangement of the indentors, at one and at several levels, are considered. The results are used to analyse the effect of the parameters of the microgeometry of the characteristics of a discrete contact and the stressed state of solids possessing regular microrelief.     

On the formulation and investigation of a spatial contact problem for elastic bodies under mixed friction conditions

A spatial contact problem is formulated and investgated for rough elastic bodies which touch each other under mixed friction conditions: the elastic bodies are separated in one part of the contact domain by a layer of viscous incompressible liquid (lubricant), while in the other they are in direct contact (such conditions are characteristic for roller bearings, gear transmissions, etc.). The problem is reduced to a system of nonlinear integro-differential and integral equations and inequalities in the contact domain, part of the external boundary, and a number of inner boundaries that are unknown in advance, but separate the lubricated and unlubricated zones. Special cases are problems of dry and completely lubricated contact. A formulation is given for the problem for the case when the materials of the bodies are identical. The problem of mixed friction is considered in strongly drawn out contact. Sections of the contact domain in which the interaction between the bodies is direct or by means of the lubrication layer are investigated using asymptotic methods.     

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.     

Asymptotic solution of the axisymmetric contact problem for an elastic layer of incompressible material

The solution of the axisymmetric contact problem for an elastic layer made of incompressible material and clamped along the base is constructed by regular and singular asymptotic methods.     

Optimization of the contact pressure in the problem of the interaction of a punch and an elastic medium

The problem of optimizing the pressure distribution under a rigid punch, which interacts without friction with an elastic medium filling a half-space, is investigated. The shape of the punch is taken as the initial variable of the design, while the root mean square deviation of the pressure distribution, which occurs under the punch, from a certain specified distribution, plays the role of the minimized functional. The values of the total forces and moments, applied to the punch, are assumed to be given, which leads to limitations imposed on the pressure distribution by the equilibrium conditions. It is shown that the optimization problem allows of decomposition into two successively solvable problems. The first problem consists of finding the pressure distribution which makes the optimized quality functional a minimum. The second problem is reduced to the problem of obtaining directly the optimum shape of the punch that yields the pressure distribution found. The optimization problem is investigated analytically for punches of different shape in plan. The optimum shapes are given in explicit form for punches with rectangular bases.     

The diffraction of plane elastic waves by a delaminated rigid inclusion when there is smooth contact in the delamination region

A solution of the problem of the diffraction of harmonic elastic waves by a thin rigid strip-like delaminated inclusion in an unbounded elastic medium, in which the conditions for plane deformation are satisfied, is proposed. We mean by a delaminated inclusion an inclusion, one side of which is completely bonded to the elastic medium, while the second does not interact in any way with it, or this interaction is partial. It is assumed that the conditions for smooth contact are satisfied in the delamination region. The method of solution is based on the use of previously constructed discontinuous solutions of the equations describing the vibrations of an elastic medium under plane deformation conditions. The problem therefore reduces to solving a system of three singular integral equations in the unknown stress and strain jumps at the inclusion. An approximate solution of the latter enabled formulae to be obtained that are convenient for numerical realization when investigating the stressed state in the region of the inclusion and its displacements when acted upon by incident waves.     

An effective asymptotic method in the axisymmetric frictionless contact problem for an elastic layer of finite thickness

A brief review of asymptotic methods to deal with frictionless unilateral contact problems for an elastic layer of finite thickness is presented. Under the assumption that the contact radius is small with respect to the layer thickness, an effective asymptotic method is suggested for solving the unilateral contact problem with a priori unknown contact radius. A specific feature of the method is that the construction of an asymptotic approximation is reduced to a linear algebraic system with respect to integral characteristics (polymoments) of the contact pressure. As an example, the sixth‐order asymptotic model has been written out. Copyright © 2015 John Wiley & Sons, Ltd.     

A contact problem for a viscoelastic plate and an elastic beam

Under consideration is the problem of contact of a viscoelastic plate with an elastic beam. To characterize the viscoelastic deformation of the plate, the hereditary integrals are used. The differential formulation of the problem with the conditions in the form of a system of equalities and inequalities in the domain of possible contact is presented, and its equivalence to a variational inequality is proved. The unique solvability of the problem is proved as well as the existence of the time derivative of the solution. A limit problem is also considered as the bending rigidity of the plate tends to infinity.     

The wear contact problem with partial slippage

A method of analysing the variation in the stress state and shape of wearable bodies, subjected to the action of an oscillating shear load with partial slippage of the surface in the contact area is proposed. The method is based on the introduction of two time scales: the time of a single cycle of variation in the shear load and the time corresponding to a specified number of cycles. To estimate the shape variation of the surface due to wear a linear relation between the wear rate and the contact pressure and velocity of relative slippage is used. Cases of complete or partial removal of the wear particles from the friction zone are considered. As an example, the kinetics of the variation in the stress state in the contact of an elastic indentor, having a flat base and rounded edges, with an elastic half-space of the same material, is investigated. Analytical expressions for calculating the asymptotic values of the stresses and shape of the worn surface are obtained.