弹性层间的单退让接触问题

本文基于所有接触面间光滑的假设,研究同时受压的两弹性层间的单退让平面接触问题. 利用Fourier变换把平面弹性方程转化为奇异积分方程. 然后利用Gauss-Chebyshev求积公式和迭代法求其数值解.最后给出了数值算例,分析了剪切模量与上层接触半径对退让半径和接触应力的影响.     

Frictional contact problem for a rigid cylindrical stamp and an elastic layer resting on a half plane

The frictional contact problem for a layer resting on a homogeneous half plane is handled using linear elasticity theory in this study. The layer is in contact with a rigid cylindrical stamp that is on the layer and applies a concentrated force in the normal and tangential directions. Friction between the component couples of layer–stamp and layer–half plane is taken into account. The problem is reduced to a system of singular integral equations, in which the contact pressures and the contact areas are the unknowns, and it is treated using Fourier transforms and the boundary conditions for the problem. The system of singular integral equations is solved numerically using the Gauss–Jacobi integration formula with equilibrium and consistency conditions. Numerical results for the contact pressures and the contact areas are given as a solution for both the frictional and the frictionless cases. This work is the first study that investigates the effect of friction on the receding contact problem of a layer and a half plane with two contact areas.     

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 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 of a rigid indenter and an anisotropic linear elastic layer

We use the Stroh formalism to study analytically generalized plane strain deformations of a linear elastic anisotropic layer bonded to a rigid substrate, and indented by a rigid cylindrical indenter. The mixed boundary-value problem is challenging since the a priori unknown deformed indented surface of the layer contacting the rigid cylinder is to be determined as a part of the solution of the problem. For a rigid parabolic prismatic indenter contacting either an isotropic layer or an orthotropic layer and a flat rigid punch indenting a half space, the computed solutions are found to agree well with those available in the literature. Parametric studies have been conducted to delimit the length and the thickness of the layer for which the derived relation between the axial load and the indentation depth caused by the rigid cylinder is valid. The indentation of a face centered cubic crystal with the plane of indentation oriented differently from the principal planes of symmetry has also been studied to illustrate the applicability of the technique to general layers made of anisotropic materials. Results presented herein can serve as benchmarks with which to compare solutions obtained by other methods.     

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.     

A two dimensional crack problem for an elastic layer bonded to an infinite elastic medium

In this paper we consider the problem of determining the distribution of stress in the neighbourhood of a crack in an infinitely long strip bonded to semi-infinite elastic planes on either side. By the use of Fourier transforms we reduce the problem to solving a single Fredholm integral equation of the second kind. Analytical expressions up to the order of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiabes7aKnaaCaaaleqabaGaeyOeI0IaaGym% aiaaicdaaaaaaa!41AF!\[\delta ^{ - 10} \], where 2 is the thickness of the strip for 1 are derived for the shape of the deformed crack and for the crack energy. Some numerical results have been displayed graphically.

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Zusammenfassung In dieser Arbeit betrachten wir das Problem der Spannungsverteilung in der Nachbarschaft eines Sprunges auf ethem unendlich langen Band welches an beiden Seiten an halbseitig-unendliche elastische Platten aufgeheftet ist. Mit Hilfe von Fourier-Transformationen reduzieren wir das Problem zu einer einzelnen Fredholm Integralgleichung der zweiten Art. Für die Sprung-Energie und die Gestalt des deformierten Sprunges leiten wir analytische Ausdrücke bis zur Ordnung % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiabes7aKnaaCaaaleqabaGaeyOeI0IaaGym% aiaaicdaaaaaaa!41AF!\[\delta ^{ - 10} \] her, wobei 2 für 1 die Dicke des Bandes ist. Einige numerische Resultate haben wir graphisch veranschaulicht.

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This work was supported by National Research Council of Canada through NRC-Grant No. A4177. This work was completed while the author was visiting the University of Glasgow.     

弹性半平面中边界裂纹研究的新方法

讨论了载荷作用在裂纹面上的弹性半平面边界裂纹问题.研究以线弹性断裂力学为基础,采用复变函数方法以及Riemann-Hilbert(R-H)边值问题的一般理论,将问题分拆为含有限裂纹的全平面问题与无裂纹的半平面问题的叠加,计算得到裂纹尖端的应力强度因子.与文献结果比较,该方法具有精度高的优点.     

Double frictional receding contact problem for an orthotropic layer loaded by normal and tangential forces

With the increasing research in the field of contact mechanics, different types of contact models have been investigated by many researchers by employing various complex material models. To ascertain the orthotropy effect and modeling parameters on a receding contact model, the double frictional receding contact problem for an orthotropic bilayer loaded by a cylindrical punch is taken into account in this study. Assuming plane strain sliding conditions, the governing equations are found analytically using Fourier integral transformation technique. Then, the resulting singular integral equations are solved numerically using an iterative method. The weight function describing the asymptotic behavior of the stresses are investigated in detail and powers of the stress singularities are provided. To control the trustworthiness and correctness of the analytical formulation and to compare the resulting stress distributions and contact boundaries, a numerically efficient finite element method was employed using augmented Lagrange contact algorithm. The aim of this paper is to investigate the orthotropy effect, modeling parameters and coefficients of friction on the surface and interface stresses, surface and interface contact boundaries, powers of stress singularities, weight function and to provide highly parametric benchmark results for tribological community in designing wear resistant systems.

     

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.     

Axisymmetric contact problem for a layer on an elastic half-space with initial stresses

International Applied Mechanics -     

The axisymmetric double contact problem for a frictionless elastic layer

The general axisymmetric double contact problem for an elastic layer pressed against a half space by an elastic stamp is considered. The problem is solved under the assumptions that the three materials have different elastic properties, the contact along the interfaces is frictionless and only compressive normal tractions can be transmitted across the interfaces, and, in the case of the elastic stamp, the local radius of curvature of the stamp is large compared to the stamp-layer contact radius. The problem is reduced to a system of singular integral equations in which the contact pressures are the unknown functions. The solution is obtained and extensive numerical results are given for three stamp geometries, namely, rigid and elastic spherical stamps, and a flat-ended rigid cylindrical stamp. The results show that in the case of a flat-ended rigid cylindrical stamp the radius b of the contact area between the layer and the subspace is independent of the magnitude P of the total transmitted load and in all other cases b will depend on P.     

Nonstationary indentation of a rigid blunt indenter into an elastic layer: A plane problem

The nonstationary indentation of a rigid blunt indenter into an elastic layer is studied. An approach to solving a mixed initial-boundary-value problem with an unknown moving boundary is developed. The problem is reduced to an infinite system of integral equations and the equation of motion of the indenter. The system is solved numerically. The analytical solution of the nonmixed problem is found for the initial stage of the indentation process __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 3, pp. 55–65, March 2008.     

Thermal stress intensity factors for a crack in an anisotropic half plane

On the basis of the two-dimensional theory of anisotropic thermoelasticity, a solution is given for the thermal stress intensity factors due to the obstruction of a uniform heat flux by an insulated line crack in a generally anisotropic half plane. The crack is replaced by continuous distributions of sources of temperature discontinuity and dislocations. First, the particular thermoelastic dislocation solutions for the half plane are obtained; then the corresponding isothermal solutions are superposed to satisfy the traction-free conditions on the crack surfaces. The dislocation solutions are applied to calculate the thermal stress intensity factors, which are validated by the exact solutions. The effects of the uniform heat flux, the ply angle and the crack length are investigated.     

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.     

The problem of the simple smooth crack in an infinite anisotropic elastic medium

The problem of the simple smooth curvilinear crack in an infinite anisotropic elastic medium under conditions of generalized plane stress or plane strain and under the supposition that the plane of the problem is a plane of elastic symmetry of the anisotropic medium is reduced to a complex Cauchy-type singular integral equation along the crack together with a condition of single-valuedness of displacements around the crack by using the complex potentials technique. Application to the case of a straight crack is also given.     

Quasiperiodic contact problem for an elastic half-plane

Chuvash State University, Cheboksary. Translated from Prikladnaya Mekhanika, Vol. 28, No. 6, pp. 22–28, June, 1992.