Three-dimensional contact problems with friction for a composite elastic wedge

Contact problems for a composite elastic wedge in the form of two joined wedge-shaped layers with different aperture angles joined by a sliding clamp, where the layer under the punch is incompressible, are studied in a three-dimensional formulation. Conditions for a sliding or rigid clamp or the absence of stresses are set up on one face of the composite wedge. The integral equations of the problems are derived taking account of the friction forces perpendicular to the edge of the wedge. The method of non-linear boundary integral equations of the Hammerstein type is used when the contact area is unknown. A regular asymptotic solution is constructed for an elliptic contact area. By virtue of the incompressibility of the material of the layer in contact with the punch, this solution retains the well known root singularity in the boundary of the contact area when account is taken of friction.     

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.     

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.     

The impact of a plane punch on an elastic half-plane

The transient dynamic contact problem of the impact of a plane absolutely rigid punch on an elastic half-plane is considered. The solution of the integral equation of this problem in terms of the unknown Laplace transform of the contact stresses at the punch base is constructed by a special method of successive approximations. The solution of the transient dynamic contact problem is obtained after applying an inverse Laplace transformation to the solution of the integral equation over the whole time range of the impact process, and the law of the penetration of the punch into the elastic medium is determined from a Volterra-type integrodifferential equation. The conditions for the punch to begin to separate from the elastic half-plane are formulated from the solution obtained, and all the stages of the separation process are investigated in detail. The law of the punch motion on the elastic half-plane and the width of the contact area, which varies during the separation, are then determined from the solution of the Volterra-type integrodifferential equation when an additional condition is satisfied.     

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.     

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.     

Interaction of a rigid punch with a base-secured elastic rectangle with stress-free sides

The plane contact problem of the indentation of a rigid punch into a base-sucured elastic rectangle with stress-free sides is considered. The problem is solved by a method tested earlier and reduces to a system of two integral equations in functions describing the displacement of the surface of the rectangle outside the punch and the normal or shear stress on its base. These functions are sought in the form of the sum of trigonometric series and an exponential function with a root singularity. The ill-posed infinite systems of algebraic equations obtained as a result of this are regularized by introducing small positive parameters. Because the matrix elements of the systems, and also the contact stresses, are defined by poorly converging numerical and functional series, the previously developed method of summation of these series is used. The contact pressure distribution and the dimensionless indenting force are found. Examples of a plane punch calculation are given.     

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 dynamics of the motion of a flat punch on the boundary of an elastic half-plane

The dynamic contact problem of the motion of a flat punch on the boundary of an elastic half-plane is considered. During motion, the punch deforms the elastic half-plane, penetrating it in such a manner that its base remains parallel to the boundary of the half-plane at each instant of time. In movable coordinates connected to the moving punch, the contact problem reduces to solving a two-dimensional integral equation, whose two-dimensional kernel depends on the difference between the arguments for each of the variables. An approximate solution of the integral equation of the problem is constructed in the form of a Neumann series, whose zeroth term is represented in the form of the superposition of the solutions of two-dimensional integral equations on the coordinate semiaxis minus the solution of the integral equation on the entire axis. This approach provides a way to construct the solution of the two-dimensional integral equation of the problem in four velocity ranges of motion of the punch, which cover the entire spectrum of its velocities, as well as to perform a detailed analysis of the special features of the contact stresses and vertical displacements of the free surface on the boundary of the contract area. An approximate method for solving the integral equation, which is based on a special approximation of the integrand of the kernel of the integral equation in the complex plane, is proposed for obtaining effective solutions of the problem that do not contain singular quadratures.     

3‐D Contact problems for elastic wedges with Coulomb friction

3‐D quasi‐static contact problems for elastic wedges with Coulomb friction are reduced to integral equations and integral inequalities with unknown contact normal pressures. To obtain these equations and inequalities, Green's functions for the wedges, where one face of the wedges is either stress‐free or fixed, are needed. Using Fourier and Kontorovich–Lebedev integral transformations, all the stresses and displacements in the wedges can be constructed in terms of solutions of Fredholm integral equations of the second kind on the semiaxis. The Green's functions can be calculated as uniformly convergent power series in (1‐2ν), where νis Poisson's ratio. An exponential decay of the kernels and right‐hand sides of the Fredholm integral equations provides the applicability of the collocation method for simple and fast calculation of the Green's functions. For a half‐space, which is a special case of an elastic wedge, the kernels degenerate and the functions reduce to the well‐known Boussinesq and Cerruti solutions. Analysing the contact problems reveals that the Green's functions govern the kernels of the above mentioned integral equations and inequalities. Under the assumption that the punch has a smooth shape, the contact pressure is zero on the boundary of the unknown contact zone. Solving the contact problems with the help of the Galanov–Newton method, the normal contact pressure, the contact zone and the normal displacement around the contact zone can be determined simultaneously. In view of the numerical results, the influence of the friction forces on the punch force and the punch settlement is discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

The contact problem for a two-layer spherical base

Analytical methods for solving problems of the interaction of punches with two-layer bases are described using in the example of the axisymmetric contact problem of the theory of elasticity of the interaction of an absolutely rigid sphere (a punch) with the inner surface of a two-layer spherical base. It is assumed that the outer surface of the spherical base is fixed, that the layers have different elastic constants and are rigidly joined to one anther, and that there are no friction forces in the contact area. Several properties of the integral equation of this problem are investigated, and schemes for solving them using the asymptotic method and the direct collocation method are devised. The asymptotic method can be used to investigate the problem for relatively small layer thicknesses, and the proposed algorithm for solving the problem by the collocation method is applicable for practically any values of the initial parameters. A calculation of the contact stress distribution, the parameters of the contact area, and the relation between the displacement of the punch and the force acting on it is given. The results obtained by these methods are compared, and a comparison with results obtained using Hertz, method is made for the case in which the relative thickness of the layers is large.     

The solution of the Hertz axisymmetric contact problem

The main terms of the asymptotic form of the solution of the contact problem of the compression without friction of an elastic body and a punch initially in point contact are constructed by the method of matched asymptotic expansions using an improved matching procedure. The condition of unilateral contact is formulated taking account of tangential displacements on the contact surface. An asymptotic solution of the problem for the boundary layer is constructed by the complex potential method. An asymptotic model is constructed, extending the Hertz theory to the case where the surfaces of the punch and elastic body in the vicinity of the contact area are approximated by paraboloids of revolution. The problem of determining the convergence of the contacting bodies from the magnitude of the compressive force is reduced to the problem of calculating the so-called coefficient of local compliance, which is an integral characteristic of the geometry of the elastic body and its fixing conditions.     

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 punch in the form of an elliptic paraboloid into the plane boundary of an elastic body

The three-dimensional contact problem for an elastic body of arbitrary geometry with a single plane face, into which a punch in the shape of an elliptic paraboloid is indented, is considered. The curvilinear boundary of the body is partially clamped, and the remaining boundary (outside the contact region) is stress-free. It is assumed that the dimensions of the contact area are small compared with the characteristic dimension of the body. Using the method of matched asymptotic expansions a model problem of unilateral contact without friction is derived for the boundary layer, which is solved using the apparatus of Hertz's theory. Asymptotic models of the contact interaction of different degrees of accuracy are constructed, including corrections to the geometry and clamping conditions of the elastic body. The sensitivity of the parameters of the elliptic region of the contact to these factors is investigated.     

Three-dimensional contact problems for an elastic wedge with a coating

Three-dimensional contact problems for an elastic wedge, one face of which is reinforced with a Winkler-type coating with different boundary conditions on the other face of the wedge, are investigated. A power-law dependence of the normal displacement of the coating on the pressure is assumed. The contact area, the pressure in this region, and the relation between the force and the indentation of a punch are determined using the method of non-linear boundary integral equations and the method of successive approximations. The results of calculations are analysed for different values of the aperture angle of the wedge, the relative distance of the punch from the edge of the wedge, the ratio of the radii of curvature of the punch (an elliptic paraboloid), and the non-linearity factors of the coating. The results obtained are compared with the solutions of similar problems for a wedge without a coating.     

An Axisymmetric Contact Problem with Given Region of Adhesion

The indentation of an elastic half-space by an axisymmetricrigid punch under normal load is considered. The contact areais divided into an inner region with adhesion, the dimensionof which is known beforehand, surrounded by an annular slipzone. Two different cases of the problem, given by the assumptionsof friction-free slip or Coulomb friction, are treated. In thefriction-free case the problem is formulated in terms of a coupledpair of Cauchy type integrals for Abel transforms of the normaland shear stresses at the surface of the half-space. These arecombined to an inhomogeneous Fredholm equation which is solvedby a method of successive approximations, and the stresses areobtained as Abel transform inversions of its solution. In thecase with Coulomb friction the problem is solved by a numericalmethod based on piecewise constant approximations to the surfacestresses.     

Singular integral equation method for contact problem for rigidly connected punches on elastic half-plane

In the contact problem of a rigid flat-ended punch on an elastic half-plane, the contact stress under punch is studied. The angle distribution for the stress components in the elastic medium under punch is achieved in an explicit form. From obtained singular stress distribution, the punch singular stress factor (abbreviated as PSSF) is defined. A fundamental solution for the multiple flat punch problems on the elastic half-plane is investigated where the punches are disconnected and the forces applied on the punches are arbitrary. The singular integral equation method is suggested to obtain the fundamental solution. Further, the contact problem for rigidly connected punches on an elastic half-plane is considered. The solution for this problem can be considered as a superposition of many particular fundamental solutions. The resultant forces on punches are the undetermined unknowns in the problem, which can be evaluated by the condition of relative descent between punches. Finally, the resultant forces on punches can be determined, and the PSSFs at the corner points can be evaluated. Numerical examples are given.     

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.     

A Two-dimensional Contact Problem with Given Region of Adhesion

The two-dimensional indentation of an elastic half-plane bya rigid punch under normal and tangential load is considered.The contact area is divided into an inner region with adhesion,the dimension of which is known beforehand, surrounded by tworegions in which inward slip takes place. The problem is formulatedin terms of a coupled pair of Cauchy type integrals for thenormal and shear stresses at the surface of the half-plane.In the case of friction-free slip these integrals are combinedto an inhomogeneous Fredholm equation which is solved by a methodof successive approximations. In the case when the inward slipis governed by Coulomb friction, the problem is solved by anumerical method.