Calculation of contact temperatures generated by the rotation of a shaft in a bearing

The problem of the distribution of contact stresses resulting from the interaction between a journal and its bearing was considered in [1]. This paper deals with the problem of temperature distribution in the area of contact of a rotating cylindrical shaft and a bearing. The process is assumed to be stabilized.The problem reduces to an integral equation with respect to the contact temperature at the shaft surface.An approximate method is proposed for solving the integral equation which had permitted the derivation of a simple approximate formula for the contact temperature within any range of variation of the parameters of this problem.     

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.     

Interaction of a die with a layered elastic foundation

Plane and axisymmetric contact problems for a three-layer elastic half-space are considered. The plane problem is reduced to a singular integral equation of the first kind whose approximate solution is obtained by a modified Multhopp-Kalandiya method of collocation. The axisymmetric problem is reduced to an integral Fredholm equation of the second kind whose approximate solution is obtained by a specially developed method of collocation over the nodes of the Legendre polynomial. An axisymmetric contact problem for an transversely isotropic layer completely adherent to an elastic isotropic half-space is also considered. Examples of calculating the characteristic integral quantities are given. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 165–175, May–June, 2006.     

Constant velocity motion of a strip die on the boundary of a viscoelastic base

We consider a contact problem on the interaction of a rigid strip die with the boundary of a viscoelastic base. We assume that the die moves at a constant velocity on this boundary and is indented into it by a constant normal force. Friction in the die—surface contact region is neglected. The die base is corrugated in the direction perpendicular to the direction of motion. At the first stage, we determine the displacement of the base boundary due to the normal load applied to it. Then, at the second stage, we derive the integral equation of the contact problem for determining the contact pressure. At the third stage, we construct an approximate solution of this integral equation by using the modified Multhopp—Kalandiya method.     

A receding contact plane problem between a functionally graded layer and a homogeneous substrate

In this paper, we consider the plane problem of a frictionless receding contact between an elastic functionally graded layer and a homogeneous half-space, when the two bodies are pressed together. The graded layer is modeled as a nonhomogeneous medium with an isotropic stress–strain law and over a certain segment of its top surface is subjected to normal tractions while the rest of this surface is free of tractions. Since the contact between the two bodies is assumed to be frictionless, then only compressive normal tractions can be transmitted in the contact area. Using integral transforms, the plane elasticity equations are converted analytically into a singular integral equation in which the unknowns are the contact pressure and the receding contact half-length. The global equilibrium condition of the layer is supplemented to solve the problem. The singular integral equation is solved numerically using Chebychev polynomials and an iterative scheme is employed to obtain the correct receding contact half-length that satisfies the global equilibrium condition. The main objective of the paper is to study the effect of the material nonhomogeneity parameter and the thickness of the graded layer on the contact pressure and on the length of the receding contact.     

A receding contact axisymmetric problem between a functionally graded layer and a homogeneous substrate

In this paper, we consider the axisymmetric problem of a frictionless receding contact between an elastic functionally graded layer and a homogeneous half-space, when the two bodies are pressed together. The graded layer is modeled as a nonhomogeneous medium with an isotropic stress–strain law and is subjected over a part of its top surface to normal tractions while the rest of it is free of tractions. Since the contact between the two bodies is assumed to be frictionless, then only compressive normal tractions can be transmitted in the contact area. Using Hankel transform, the axisymmetric elasticity equations are converted analytically into a singular integral equation in which the unknowns are the contact pressure and the receding contact radius. The global equilibrium condition of the layer is supplemented to solve the problem. The singular integral equation is solved numerically using orthogonal Chebychev polynomials and an iterative scheme is employed to obtain the correct receding contact length that satisfies the global equilibrium condition. The main objective of the paper is to study the effect of the material nonhomogeneity parameter and the thickness of the graded layer on the contact pressure and on the length of the receding contact.     

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

The dynamic contact problem of a plane punch motion on the boundary of an elastic half-plane is considered. The punch velocity is constant and does not exceed the Rayleigh wave velocity. The moving punch deforms the elastic half-plane penetrating into it so that the punch base remains parallel to itself at all times. The contact problem is reduced to solving a two-dimensional integral equation for the contact stresses whose two-dimensional kernel depends on the difference of arguments in each variable. A special approximation to the kernel is used to obtain effective solutions of the integral equation. All basic characteristics of the problem including the force of the punch elastic action on the elastic half-plane and the moment stabilizing the punch in the horizontal position in the process of penetration are obtained. A similar problem was considered in [1] and earlier in the “mode of steady-state motions” in [2, 3] and in other publications.     

Motion of a rigid punch at a low velocity along the boundary of a viscoelastic half-plane

The contact problem of the interaction of a rigid punch with a viscoelastic half-plane is considered. The dependence of the displacement of the boundary of half-plane on the normal load applied to it is determined, and the integral equation for determining the contact pressure is derived and solved by the method of “small λ”. Distributions of contact pressures under the punch are graphically represented.     

Impact of a plane die on an elastic orthotropic half-plane

To study the process of impact of a rigid body on the surface of an elastic body made of a composite material, we consider a nonstationary dynamic contact problem about the impact of a plane rigid die on an elastic orthotropic half-plane. The problem is reduced to solving an integral equation of the first kind for the Laplace transform of the contact stresses under the die base. An approximate solution of the integral equation is constructed with the use of a special approximation to the symbol of the kernel of the integral equation in the complex plane. The inverse Laplace transform of the solution results in determining the scalar contact stress field on the die base, the force exerted by the die on the elastic medium, and the vertical displacement field of the free surface of the orthotropic medium out side the die. The solutions thus obtained permit studying specific features of the process of die penetration into an orthotropic medium and the strain properties of the medium.     

A frictional receding contact plane problem between a functionally graded layer and a homogeneous substrate

This paper investigates the plane problem of a frictional receding contact formed between an elastic functionally graded layer and a homogeneous half space, when they are pressed against each other. The graded layer is assumed to be an isotropic nonhomogeneous medium with an exponentially varying shear modulus and a constant Poisson’s ratio. A segment of the top surface of the graded layer is subject to both normal and tangential traction while rest of the surface is devoid of traction. The entire contact zone thus formed between the layer and the homogeneous medium can transmit both normal and tangential traction. It is assumed that the contact region is under sliding contact conditions with the Coulomb’s law used to relate the tangential traction to the normal component. Employing Fourier integral transforms and applying the necessary boundary conditions, the plane elasticity equations are reduced to a singular integral equation in which the unknowns are the contact pressure and the receding contact lengths. Ensuring mechanical equilibrium is an indispensable requirement warranted by the physics of the problem and therefore the global force and moment equilibrium conditions for the layer are supplemented to solve the problem. The Gauss–Chebyshev quadrature-collocation method is adopted to convert the singular integral equation to a set of overdetermined algebraic equations. This system is solved using a least squares method coupled with a novel iterative procedure to ensure that the force and moment equilibrium conditions are satisfied simultaneously. The main objective of this paper is to study the effect of friction coefficient and nonhomogeneity factor on the contact pressure distribution and the size of the contact region.     

Analytical solution of the contact problem for a system of bodies under collective wear

The contact problem is considered for a system of bodies subject to wear on a common base. The deformation properties of the bodies and the base are described by the Winkler model. The problem is reduced to a system of ordinary differential equations and an integral equation of hereditary type with difference kernel. The solution of the problem is constructed with the use of the Laplace transform. The asymptotics of the solution at large times is studied. The continuity conditions for the contact of each of the bodies with the base are derived.     

Thermoelastic contact mechanics for a flat punch sliding over a graded coating/substrate system with frictional heat generation

The problem of thermoelastic contact mechanics for the coating/substrate system with functionally graded properties is investigated, where the rigid flat punch is assumed to slide over the surface of the coating involving frictional heat generation. With the coefficient of friction being constant, the inertia effects are neglected and the solution is obtained within the framework of steady-state plane thermoelasticity. The graded material exists as a nonhomogeneous interlayer between dissimilar, homogeneous phases of the coating/substrate system or as a nonhomogeneous coating deposited on the substrate. The material nonhomogeneity is represented by spatially varying thermoelastic moduli expressed in terms of exponential functions. The Fourier integral transform method is employed and the formulation of the current thermoelastic contact problem is reduced to a Cauchy-type singular integral equation of the second kind for the unknown contact pressure. Numerical results include the distributions of the contact pressure and the in-plane component of the surface stress under the prescribed thermoelastic environment for various combinations of geometric, loading, and material parameters of the coated medium. Moreover, in order to quantify and characterize the singular behavior of contact pressure distributions at the edges of the flat punch, the stress intensity factors are defined and evaluated in terms of the solution to the governing integral equation.     

Axisymmetric contact problem for an elastic half-space and a circular cover plate

The contact interaction problem for a thin circular rigid cover plate and an elastic half-space loaded at infinity by a tensile force directed in parallel to the boundary of the half-space is considered. It is assumed that the cover plate is not resistant to bending deformations. The problem can be reduced to an integral equation of the first kind whose kernel has a logarithmic singularity. The equation is solved approximately by the Multhopp-Kalandia method. The resulting approximate solution is compared with the previously obtained asymptotic solution.     

Effects of the moving speed of a rigid stamp on contact behaviors of anisotropic materials based on real fundamental solutions

The paper studies contact problem of a rigid stamp moving at a constant speed over the surface of anisotropic materials. The solution method is based on Galilean transformation, Fourier transform and singular integral equation. The stated mixed boundary value problem is reduced to a Cauchy type singular integral equation based on real fundamental solutions, which is solved exactly in the case of a rigid flat or cylindrical stamp. Explicit expressions for various stresses are obtained in terms of elementary functions. In particular, explicit formula is derived to determine the unknown contact region for the cylindrical stamp. For a flat stamp, detailed calculations are provided to show the influences of dimensionless moving speed on the normal and in-plane stress. For a cylindrical stamp, the effects of dimensionless moving speed, the mechanical loading and the radius on the contact region, the normal and in-plane stress are analyzed in detail.     

Indentation of a Rigid Body into an Elastic Plate

The problem of a punch shaped like an elliptic paraboloid pressed into an elastic plate is studied under the assumption that the contact region is small. The action of the punch on the plate is modeled by point forces and moments. The method of joined asymptotic expansions is used to formulate the problem of one–sided contact for the internal asymptotic representation; the problem is solved with the use of the results obtained by L. A. Galin. The coordinates of the center of the elliptic contact region, its dimensions, and the angle of rotation are determined. The moments which ensure translational indentation of the punch are calculated and an equation that relates displacements of the punch to the force acting on it is given.     

Application of generalized images method to contact problems for a transversely isotropic elastic layer on a smooth half-space

The idea, first used by the author for the case of crack problems, is applied here to solve a contact problem for a transversely isotropic elastic layer resting on a smooth elastic half-space, made of a different transversely isotropic material. A rigid punch of arbitrary shape is pressed against the layer’s free surface. The governing integral equation is derived; it is mathematically equivalent to that of an electrostatic problem of an infinite row of coaxial charged disks in the shape of the domain of contact. The case of circular domain of contact is considered in detail. As a comparison, the method of integral transforms is also used to solve the problem. The main difference of our integral transform approach with the existing ones is in separating of our half-space solution from the integral transform terms. It is shown that both methods lead to the same results, thus giving a new interpretation to the integral transform as a sum of an infinite series of generalized images.     

Uniform indentation of the elastic half-space by a rigid rectangular punch

The problem considered is that of a rigid flat-ended punch with rectangular contact area pressed into a linear elastic half-space to a uniform depth. Both the lubricated and adhesive cases are treated. The problem reduces to solving an integral equation (or equations) for the contact stresses. These stresses have a singular nature which is dealt with explicitly by a singularity-incorporating finite-element method. Values for the stiffness of the lubricated punch and the adhesive punch are determined: the effect of adhesion on the stiffness is found to be small, producing an increase of the order of 3%.     

Bending of an Elastic Anisotropic Plate with a Circular Opening Stiffened Over Finite Areas

A contact problem is solved for an infinite anisotropic plate weakened by a circular opening, stiffened by inclusions of variable stiffness, and subjected to bending. For the unknown contact force of interaction between the plate and an inclusion, an integro-differential equation is derived and then reduced to an infinite system of linear algebraic equations. The system is analyzed for regularity.     

Sliding frictional contact between a rigid punch and a laterally graded elastic medium

Analytical and computational methods are developed for contact mechanics analysis of functionally graded materials (FGMs) that possess elastic gradation in the lateral direction. In the analytical formulation, the problem of a laterally graded half-plane in sliding frictional contact with a rigid punch of an arbitrary profile is considered. The governing partial differential equations and the boundary conditions of the problem are satisfied through the use of Fourier transformation. The problem is then reduced to a singular integral equation of the second kind which is solved numerically by using an expansion–collocation technique. Computational studies of the sliding contact problems of laterally graded materials are conducted by means of the finite element method. In the finite element analyses, the laterally graded half-plane is discretized by quadratic finite elements for which the material parameters are specified at the centroids. Flat and triangular punch profiles are considered in the parametric analyses. The comparisons of the results generated by the analytical technique to those computed by the finite element method demonstrate the high level of accuracy attained by both methods. The presented numerical results illustrate the influences of the lateral nonhomogeneity and the coefficient of friction on the contact stresses.     

一类Signorini问题的边界变分不等式

本文讨论了一类简化的Signorini问题。首先将原问题和一个边值问题建立联系，其次将原问题的解分解为不带不等边界条件的变分方程的解和一个变分不等式的解。然后利用边值问题的边界积分方程将变分不等式等价地化解为边界变分不等式。这样原求区域上的第一类椭圆变分不等式问题化解为求一个区域上的变分方程和一个边界变分不等式。最后说明了边界变分不等式解的存在唯一性。文末计算了柱面和半无限刚性基础的摩擦接触问题。结论表明文中方法具有较好的精度。