Contact of a rigid cylinder indenting an elastic layer sliding over a rigid substrate

Details of the integral transform solution of the state of stress in a layer resting on but sliding over a rigid substrate, in the presence of interfacial friction, is studied. The free surface of the layer is subject to a localised contact, which is represented as a piecewise linear distribution of tractions, using the Bentall–Johnson procedure. The influence functions needed are derived and their properties discussed and compared with those already available for other interface conditions. Lastly, the procedure is applied to the problem of a shrink fit tyre which, under severe tyre/road tangential loading, can be ‘torn’ around the wheel (here, the substrate).     

The contact problem of a rigid cylinder on an elastic layer

Summary This paper deals with the contact problem of a rigid cylinder pressed on an elastic layer connected rigidly to a rigid base. It is assumed that there is no friction between cylinder and layer and that the cylinder is long enough to ensure a plane deformation. Asymptotic solutions are presented when the ratio of the half width c of the contact area to the thickness b of the layer is small and also when c/b is large. The breakdown of the asymptotic solution for large values of c/b when the material is incompressible, discussed by Koiter [6], is overcome by considering a more general solution of the Wiener-Hopf integral equation encountered. The results of both asymptotic solutions match so well that a satisfactory solution is obtained for all values c/b and for 00.5.     

Collision of an elastic finite-length cylinder with a rigid barrier

The paper proposes an approximate solution describing a collision of an elastic finite-length cylinder with a rigid barrier when the lateral boundary conditions of the first fundamental problem of elasticity are satisfied. A finite-difference approach with respect to time and the integral transform method are used to reduce the original initial-boundary-value problem to a one-dimensional one. It is solved using the matrix Green’s function. The final expressions for displacements are obtained by solving a singular integral equation by the orthogonal-polynomial method. The values of displacements and strains are analyzed for short periods of time __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 9, pp. 74–82, September 2007.     

Indentation of the semi-infinite elastic solid by a concave rigid punch

A solution is obtained for the relationship between load, displacement and inner contact radius for an axisymmetric, spherically concave, rigid punch, indenting an elastic half-space. Analytic approximations are developed for the limiting cases in which the ratio of the inner and outer radii of the annular contact region is respectively small and close to unity. These approximations overlap well at intermediate values. The same method is applied to the conically concave punch and to a punch with a central hole. , , . , . . .     

Some effects of compressibility on the indentation of a thin elastic layer by a smooth rigid cylinder

Summary A truncated cosine series solution to the integral equation for the pressure distribution is evaluated using a digital computer. Tables of results are given which allow the calculation of pressure distribution, tangential surface strain and indentation under load when the contact width is less than four times the layer thickness. Calculations derived from these results are given, which show the effect of layer compressibility.Nomenclature b layer thickness - d indentation at centre of contact width relative to undisturbed position of outer boundary - D Composite diameter term 1/D=1/D ₁+1/D ₂ where D ₁ and D ₂ are roller diameters - E Young's modulus - h ₀ half contact width for infinite cover thickness - h half contact width for finite cover thickness - K h/2b - V(X) displacement normal to layer boundary at X from the centre of the contact width - W load per unit axial width - x co-ordinate in direction tangential to cover surface - X x/h - y co-ordinate normal to cover surface, positive direction into cover itself - C ₁(K) a function of K such that h=h ₀ C ₁(K) - Poisson's ratio     

Deformation of an elastic helix in contact with a rigid cylinder

Summary The deformation of a short helix in contact with a rigid cylinder is investigated. Deformations occur due to bending, torsion and longitudinal elasticity of the helix. Shear deformation is neglected. Some of the equations describing the problem have been given already in Love's Treatise on the Mathematical Theory of Elasticity, in terms of curvature changes. Nevertheless, the equations for small deformations have to be reformulated in terms of displacements and rotations, because contact constraints cannot be expressed in terms of curvature. Friction is neglected, thus the problem is symmetric, and it is sufficient to determine its solution for one half of the helix.Without friction between the cylinder and the helix, the contact problem arises only for a helix longer than one length of twist. For a shorter helix, the global equilibrium conditions cannot be satisfied for nonvanishing contact forces. For the minimum length, there are two noncontact zones, and the helix is in contact with the cylinder only at three points: at the ends and in the middle. For a slightly longer helix, four contact points and three noncontact regions are found. The dependence of the noncontact zones and the contact forces, which are of practical interest, can be calculated as a function of the length of the helix and its geometrical parameters. The case of a very long helix with more than four contact points remains unsolved.     

Contact with friction of a rigid cylinder with an elastic half-space

An exact solution to the problem of indentation with friction of a rigid cylinder into an elastic half-space is presented. The corresponding boundary-value problem is formulated in planar bipolar coordinates, and reduced to a singular integral equation with respect to the unknown normal stress in the slip zones. An exact analytical solution of this equation is constructed using the Wiener-Hopf technique, which allowed for a detailed analysis of the contact stresses, strain, displacement, and relative slip zone sizes. Also, a simple analytical solution is furnished in the limiting case of full stick between the cylinder and half-space.     

Non-slipping adhesive contact of a rigid cylinder on an elastic power-law graded half-space

In this paper, adhesive contact of a rigid cylinder on an elastic power-law graded half-space is studied analytically with the theory of weakly singular integral equation and orthogonal polynomial method. Emphasis is placed on the coupling effect between tangential and normal directions which was often neglected in previous works. Our analysis shows that the coupling effect tends to reduce the contact area in the compressive regime. The effect of bending moment on the adhesion behavior is also examined. Like a pull-off force, there also exists a critical bending moment at which the cylinder can be bended apart from the substrate. However, unlike pull-off force, the critical bending moment is insensitive to the gradient exponent of the graded material.     

Indentation of a spherical cavity in an elastic body by a rigid spherical inclusion: influence of non-classical interface conditions

This paper examines the indentation of an elastic body by a rigid spherical inclusion. In contrast to conventional treatments where the contact between a rigid inclusion and the elastic medium is regarded as being perfectly bonded, we examine the influence of non-classical interface conditions including frictionless bilateral contact, separation and Coulomb friction on the load–displacement behaviour of the spherical rigid inclusion. Both analytical methods and boundary element techniques are used to examine the inclusion/elastic medium interaction problems. This paper also provides a comprehensive review of non-classical interface conditions between inclusions and the surrounding elastic media.     

Vertical impact of the side of a rigid circular cylinder against an elastic half-space

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 25, No. 12, pp. 41–47, December, 1989.     

Contact interaction of a rigid die with an elastic layer during frictional heating

A formulation and solution are presented for the static thermoelastic problem of the sliding of a rigid die on the surface of an elastic layer with a fixed base. Frictional heating which occurs in accordance with Amonton's law is taken in account. With the assumption that the die is insulated, the problem is reduced to a system of two integral equations for contact pressure and a function which is a linear combination of the temperature and heat flux on the contact surface. A numerical algorithm is proposed for solving the system. A study is made of the effect of the reduction brought about in the size of the contact area by the steady generation of heat during the interaction of the layer and die with parabolic and semicylindrical bases. I. Franko University, Lvov, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 36, No. 1, pp. 130–138, January, 2000.     

Off-center impact of an elastic column by a rigid mass

Based on the Timoshenko beam model the equations of motion are obtained for large deflection of off-center impact of a column by a rigid mass via Hamilton's principle. These are a set of coupled nonlinear partial differential equations. The Newmark time integration scheme and differential quadrature method are employed to convert the equations into a set of nonlinear algebraic equations for displacement components. The equations are solved numerically and the effects of weight and velocity of the rigid mass and also off-center distance on deformation of the column are studied.     

Interfacial slip and creep in rolling contact incorporating a cylinder with an elastic layer

A systematic approach for investigating the interfacial behaviour of tyred systems is proposed. A two-dimensional contact model of an elastic strip, shrink-fitted onto a wheel, and subjected to different rolling contact conditions, has been adopted to illustrate the method. The model combines existing techniques to explore individual elastic contact problems and it enables us to characterise the behaviour at the strip/substrate interface caused by loads induced by a quasi-static application of stationary and moving loads on the surface of the layer. The solution is compared to the stationary load case and regimes of local slip, full stick, separation and frictional creep are identified and collated for a variety of loading conditions, materials and geometries. Further, this article presents an investigation of frictional shakedown for layered systems subjected to periodic contact loading. The term shakedown is here referred to as the possibility of developing interfacial residual stresses at the layer/substrate interface such that frictional slip, originally activated by the applied external contact load, ceases after a few loading cycles. The possible applicability of the Melan’s theorem for elastic frictional system is investigated and preliminary results presented.     

Frictional contact problem of a rigid stamp and an elastic layer bonded to a homogeneous substrate

In this study, the frictional contact problem for a layer bonded to a homogeneous substrate is considered according to the theory of elasticity. The layer is indented by a rigid cylindrical stamp which is subjected to concentrated normal and tangential forces. The friction between the layer and the stamp is taken into account. The problem is reduced to a singular integral equation of the second kind in which the contact pressure function and the contact area are the unknown by using integral transform technique and the boundary conditions of the problem. The singular integral equation is solved numerically using both the Jacobi polynomials and the Gauss?CJacobi integration formula, considering equilibrium and consistency conditions. Numerical results for the contact pressures, the contact areas, the normal stresses, and the shear stresses are given, for both the frictional and the frictionless contacts.     

Contact interaction of a rigid heat-conducting punch with an elastic layer involving nonstationary frictional heating

A mathematical formulation is given and a solution is found to the quasistatic contact problem of thermoelasticity for a rigid heat-conducting punch moving over an elastic layer with fixed base. The interaction is accompanied by heating due to frictional forces obeying Amonton’s law. The problem is reduced to a system of integral equations with time-varying limits of integration. The structure of these equations depends on the type of thermal and physical conditions on the contact surface. An algorithm is proposed for the numerical solution of this kind of equations. The variation in the contact pressure and contact area with time is studied __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 12, pp. 35–46, December 2005.