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 problem of the bending of a beam lying on an elastic base

The plane contact problem of the transmission of a normal force of specified strength onto an elastic anisotropic, wedge-shaped plate by an elastic beam of variable flexural stiffness is considered. The beam is coupled to one of the edges of the plate and its other edge is stress-free. The solution of the problem is obtained in closed form by reducing it to a Karleman boundary-value problem with shear for a strip. A conclusion is reached concerning the nature of the discontinuity of the normal contact stress at the vertex of the wedge.     

Closed-form solution considering the tangential effect under harmonic line load for an infinite Euler–Bernoulli beam on elastic foundation

The dynamic response of an infinite Euler–Bernoulli beam resting on an elastic foundation, which considers the tangential interaction between the beam and foundation under harmonic line loads, is developed in this study in the form of a closed-form solution. Previous studies have focused on elastic Winkler foundations, wherein the tangential interaction between the bottom of the beam and the foundation is not considered. In this study, a series of separate horizontal springs is diverted to the contact surface between the foundation and beam to simulate the horizontal tangential effect. The horizontal spring reaction is assumed proportional to the relative tangential displacement. As the geometric equation and linear-elastic constitutive equation of beam under the condition of small deformation have been presented based on the basic principle of elasticity mechanics, the analysis model is built and the governing differential equations about normal and tangential deflections of beam are deduced. Double Fourier transformation and the residue theorem are used to derive the closed-form solution to this problem. The proposed solution is then validated by comparing the degraded solution with the known results and comparing the numerical solution with the analytical solution. We also discuss the case in which the load direction is not vertical to the beam. Results can be used as a reference for engineering design.     

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 contact problem for a piecewise-homogeneous plane with a finite inclusion

A piecewise-homogeneous elastic orthotropic plate, reinforced with a finite inclusion, which meets 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. By using the methods of the theory of analytic functions, the problem is reduced to a singular integro-differential equation in a finite section. Using an integral transformation, a Riemann problem is obtained, the solution of which is presented in explicit form.     

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.     

Elementary solution of contact problems for a transversely isotropic elastic layer bonded to a rigid foundation

The contact problem for an arbitrary punch acting on a transversely isotropic elastic layer bonded to a rigid foundation is solved by the generalized images method developed by the author earlier. The problem is reduced to that of an electrostatic problem of infinite row of coaxial charged disks in the shape of the domain of contact. The solution can be obtained by the method of iteration, collocations or any other standard procedure for solving integral equations. Exact inversion can be obtained in the case of a circular domain of contact. The mean value theorem can be used for estimation of the resultant force and tilting moment acting on a punch of arbitrary shape and circular domain of contact. A limiting case of general solution gives the solution for an isotropic layer.     

The antiplane problem of the motion of a point shear load over a two-layer elastic base at a constant velocity

The problem of modelling the motion of a force disturbance in an elastic medium that is heterogeneous over its depth is investigated. It is in an antiplane formulation in a moving system of coordinates that all possible versions of the ratio of the velocity of motion of the surface point shear load to the velocities of the shear waves in the layers of the two-layer elastic base are examined. Cases of a subsonic regime (SBR) in the upper and lower layers, of a supersonic regime (SPR) in the upper layer and an SBR in the lower layer, and of an SBR in the upper layer and an SPR in the lower layer are studied using the Fourier transform and the theory of residues. The last two cases are extremely interesting from the mathematical point of view, as here, on the boundary between the layers, the solutions of elliptic and hyperbolic equations meet, and previously unknown features arise in the displacements that,it seems, should also occur in the solution of the corresponding plane problem. The case of an SPR in the upper and lower layers is investigated using a special method for successive allowance for the incident, reflected and refracted shock wave fronts. In all cases, expressions are obtained for the displacements in the layers, and their characteristic features are investigated.     

Boussinesq indentation of a transversely isotropic half-space reinforced by a buried inextensible membrane

The normal indentation of a rigid circular disk into the surface of a transversely isotropic half-space reinforced by a buried inextensible thin film is addressed. By virtue of a displacement potential function and the Hankel transform, the governing equations of this axisymmetric mixed boundary value problem are represented as a dual integral equation, which is subsequently reduced to a Fredholm integral equation of the second kind. Two important results of the contact stress distribution beneath the disk region as well as the equivalent stiffness of the system are expressed in terms of the solution of the Fredholm integral equation. When the membrane is located on the surface or at the remote boundary, exact closed-form solutions are presented. For the limiting case of an isotropic half-space the results are verified with those available in the literature. As a special case, the elastic fields of a reinforced transversely isotropic half-space under the action of surface axisymmetric patch loads are also given. The effects of anisotropy, embedment depth of the membrane, and material incompressibility on both the contact stress and the normal stiffness factor are depicted in some plots.     

Elementary solution of contact problems for a transversely isotropic elastic layer bonded to a rigid foundation

The contact problem for an arbitrary punch acting on a transversely isotropic elastic layer bonded to a rigid foundation is solved by the generalized images method developed by the author earlier. The problem is reduced to that of an electrostatic problem of infinite row of coaxial charged disks in the shape of the domain of contact. The solution can be obtained by the method of iteration, collocations or any other standard procedure for solving integral equations. Exact inversion can be obtained in the case of a circular domain of contact. The mean value theorem can be used for estimation of the resultant force and tilting moment acting on a punch of arbitrary shape and circular domain of contact. A limiting case of general solution gives the solution for an isotropic layer. (Received: August 11, 2003)     

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 contact problem with the bulk application of intermolecular interaction forces (a refined formulation)

A refined formulation of the contact problem when there are intermolecular interaction forces between the contacting bodies is considered. Unlike the traditional formulation, it is assumed that these forces are applied to points within the body, rather than to the surface of the deformable body as a contact pressure, and that the body surface is load-free. Solutions of the contact problems for a thin elastic layer attached to an absolutely rigid substrate and for an elastic half-space are analysed. The refined and traditional formulations of the problem when there is intermolecular interaction are compared. ©2013     

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.     

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.     

The periodic contact problem of the plane theory of elasticity. Taking friction,wear and adhesion into account

A solution of the plane problem of the contact interaction of a periodic system of convex punches with an elastic half-plane is given for two forms of boundary conditions: 1) sliding of the punches when there is friction and wear, and 2) the indentation of the punches when there is adhesion. The problem is reduced to a canonical singular integral equation on the arc of a circle in the complex plane. The solution of this equation is expressed in terms of simple algebraic functions of a complex variable, which considerably simplifies its analysis. Asymptotic expressions are obtained for the solution of the problem in the case when the size of the contact area is small compared with the distance between the punches.     

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.     

Impression of a semiinfinite stamp in an elastic strip with regard for friction and adhesion

We consider the problem of contact interaction between a semiinfinite stamp with rectilinear base and an elastic strip with one rigid side. Friction forces in the contact region are taken into account. These forces lead to the division of the contact region into slipping and adhesion zones. With the use of the Wiener–Hopf method, a system of integral equations is reduced to an infinite system of algebraic equations. The computational results of stresses and strains at the boundary and at inner points of the elastic strip are presented. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 138–149, January–March, 2008.     

Averaging of a finely laminated elastic medium with roughness or adhesion on the contact surfaces of the layers

The governing relations of a laminated elastic medium with non-ideal contact conditions in the interlayer boundaries are obtained by an asymptotic averaging method. The interaction of rough surfaces is described by a non-linear contact condition which simulates the local deformation of the microroughnesses using a certain penetration of the nominal surfaces of the elastic layers. The cohesive forces, caused by the thin adhesive layer, are described within the limits of the Frémond model which includes a differential equation characterizing the change in the cohesion function. A piecewise-linear approximation of the initial positive segment of the Lennard–Jones potential curve is proposed to describe of the adhesive forces between smooth dry surfaces. A comparison is made with the solution obtained within the limits of the Maugis–Dugdale model based on a piecewise-constant approximation. Solutions of the above problems are constructed taking account of the possible opening of interlayer boundaries.     

Hertzian contact problem: Numerical reduction and volumetric modification

A technique for the analytical formulation and numerical implementation of an elastic contact model for rigid bodies in the framework of the Hertzian contact problem is described. The normal elastic force and the semiaxes of the contact area are computed so that the problem is sequentially reduced to a scalar transcendental equation depending on complete elliptic integrals of the first and second kinds. Based on the classical solution to the Hertzian contact problem, an invariant volumetric force function is proposed that depends on the geometric characteristics of interpenetration of two undeformed bodies. The normal forces computed using the force function agree with results obtained previously for non-Hertzian contact of elastic bodies. As an example, a ball bearing is used to compare the contact dynamics of elastic bodies simulated in the classical Hertzian model and its volumetric modification.     

Vibrations of a beam between two rigid stops: strong solutions in the framework of vector-valued measures

Abstract

We study a problem that models the dynamics of an elastic beam vibrating between two rigid stops, so we use the Signorini non-penetration condition to describe the contact process. This allows for impacts and velocity jumps. Motivated by the need to better understand this kind of dynamics, we introduce a new formulation of the problem in the framework of vector-valued measures, somewhat similar to the case of a discrete mechanical system. We prove the existence of a strong solution and establish the main properties of the reaction shear stress that acts on the system at impacts, which is a measure with a singular part.