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.     

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.     

A receding contact problem for an anisotropic elastic medium consisting of a layer and a half plane

This paper considers a frictionless receding contact problem between an anisotropic elastic layer and an anisotropic elastic half plane, when the two bodies are pressed together by means of a rigid circular stamp. The problem is reduced to a system of singular integral equations in which the contact stresses and lengths are the unknown functions. Numerical results for the contact stresses and the contact lengths are given by depending on various fibre orientations.     

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.     

Axisymmetric contact problem for a biphasic cartilage layer with allowance for tangential displacements on the contact surface

A unilateral axisymmetric contact problem for a biphasic cartilage layer indented by a rigid punch is considered. The refined linearized kinetic relationship which takes into account both the radial and tangential displacements of the boundary points of the biphasis cartilage layer is imposed. The obtained analytical solution is valid over long-time periods and can be used for increasing loading conditions. It can be used as it is and as a benchmark for verification of FEM model accuracy.     

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.     

Anisotropic disc loaded by a layer of forces in an unbounded elastic medium

Novosibirsk State Technical University, Novosibirsk 630092. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 36, No. 3, pp. 146–150, May–June, 1995.     

Rayleigh waves in an orthotropic elastic half-space overlaid by an elastic layer with spring contact

Meccanica - In this paper, the propagation of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer of arbitrary uniform thickness is investigated. The layer...     

Probability density functions of contact forces for cohesionless frictional granular materials

A theory is developed for the probability density functions of contact forces for cohesionless, frictional granular materials in quasi-static equilibrium. This theory is based on a maximum information entropy principle, with an expression for information entropy that is appropriate for granular materials. Entropy is maximized under the constraints of a prescribed stress and that the normal component of the contact force is compressive and that the tangential component of the contact force is limited by Coulomb friction. The theory results in a dependence of the probability density function for the tangential contact forces on the friction coefficient. The theoretical predictions are compared with results from discrete element simulations on isotropic, two-dimensional assemblies under hydrostatic stress. Good qualitative agreement is found for means and standard deviations of contact forces and the shape of the probability density functions, while the quantitative agreement is fairly good. Discrepancies between theory and simulations, such as the difference in shape of the probability density function for the normal force and the observed dependence on elastic properties of the exponential decay rate of tangential forces, are attributed to the fact that the method does not take into account any kinematics, which are essential in relation to elastic effects.     

Winkler地基模型上功能梯度材料涂层的摩擦接触问题

研究Winker地基模型上功能梯度材料涂层在一刚性圆柱形冲头作用下的摩擦接触问题。功能梯度材料涂层表面作用有法线向和切线向集中作用力。假设材料非均匀参数呈指数形式变化,泊松比为常量,利用Fourier积分变换技术将求解模型的接触问题转化为奇异积分方程组,再利用切比雪夫多项式对所得奇异积分方程组进行数值求解。最后,给出了功能梯度材料非均匀参数、摩擦系数、Winker地基模型刚度系数及冲头曲率半径对接触应力分布和接触区宽度的影响情况。     

Contact problem for an orthotropic half-space

Numerical and analytical solutions of the 3D contact problem of elasticity on the penetration of a rigid punch into an orthotropic half-space are obtained disregarding the friction forces.A numericalmethod ofHammerstein-type nonlinear boundary integral equations was used in the case of unknown contact region, which permits determining the contact region and the pressure in this region. The exact solution of the contact problem for a punch shaped as an elliptic paraboloid was used to debug the program of the numerical method. The structure of the exact solution of the problem of indentation of an elliptic punch with polynomial base was determined. The computations were performed for various materials in the case of the penetration of an elliptic or conical punch.     

Plane contact problem for periodic laminated composite involving frictional heating

Summary The stationary problem of a rigid thermally insulated punch sliding over the boundary surface of a periodic two-layered thermoelastic half-space is considered. The heat generated in the contact area is assumed to be caused by frictional forces. The problem is formulated within the framework of thermoelasticity with microlocal parameters, and it is reduced to a system of two integral equations, which is solved numerically. The effects connected with the composite structure are analyzed.     

The contact problem for a class of orthotropic elastic solids

The contact problem between two orthotropic solids is examined. The problem is solved by using Lodge's method, which permits the transformation of the boundary-value problem of an anisotropic solid to a form identical with the corresponding problem of an isotropic medium. The proposed solution is then compared with known results of certain cases and it is observed that it producesHertz's solution when used for an isotropic case,Lodge's solution when applied to contact between an orthotropic solid and a rigid plane and, finally,Love's solution if the solid is transversely isotropic with the axis of material symmetry perpendicular to the rigid plane of contact.     

Dynamic analysis of two-dimensional frictional contact by linear complementarity problem formulation

Two-dimensional dynamic contact is formulated as a linear complementarity problem using contact compatibility conditions and the Coulomb friction law. For the time integration of dynamic response, displacements are approximated as second-order polynomials and a new scheme of considering velocity discontinuities is presented. The efficiency of the method is shown with three numerical examples.