The axisymmetric torsional contact problem of a functionally graded piezoelectric coated half-space

In this article, we study the axisymmetric tor-sional contact problem of a half-space coated with func-tionally graded piezoelectric material (FGPM) and subjected to a rigid circular punch. It is found that, along the thick-ness direction, the electromechanical properties of FGPMs change exponentially. We apply the Hankel integral trans-form technique and reduce the problem to a singular integral equation, and then numerically determine the unknown con-tact stress and electric displacement at the contact surface. The results show that the surface contact stress, surface azimuthal displacement, surface electric displacement, and inner electromechanical field are obviously dependent on the gradient index of the FGPM coating. It is found that we can adjust the gradient index of the FGPM coating to modify the distributions of the electric displacement and contact stress.     

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.     

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.     

The axisymmetric viscoelastic contact problem with rotational friction

The boundary value problem that arises when a mechanically rough rigid punch of arbitrary axisymmetric profile is pressed against the surface of a linear aging viscoelastic half space and is also made to rotate about its axis, so that there is total slip between the contacting surfaces is analysed and solved. The moment required to make the punch rotate, on the assumption that the coefficient of friction obeys a power law or is a constant, and the total normal pressure acting on the punch may each be evaluated in terms of the history of the radius of the contact area. Application is made to the special cases where the punch has the form of (i) a cone, (ii) a paraboloid of revolution and (iii) a flat ended cylinder. Apart from case (iii) where the contract area is constant we can only find an explicit expression for the moment in terms of total pressure so long as the contact area is increasing. The case of constant total pressure and Maxwell viscoelastic material is examined in more detail.     

Analytical solution of axisymmetric contact problem about indentation of a circular indenter into a soft functionally graded elastic layer

The paper addresses a contact problem of the theory of elasticity,i.e.,the penetration of a circular indenter with a flat base into a soft functionally graded elastic layer.The elastic properties of a functionally graded layer arbitrarily vary with depth,and the foundation is assumed to be elastic,yet much harder than a layer.Approximated analytical solution is constructed,and it is shown that the solutions are asymptotically exact both for large and small values of characteristic dimensionless geometrical parameter of the problem.Numerical examples are analyzed for the cases of monotonic and nonmonotonic variations of elastic properties.Numerical results for the case of homogeneous layer are compared with the results for nondeformable foundation.     

Reissner-Sagoci problem for a homogeneous coating on a functionally graded half-space

This paper is concerned with the axisymmetric elastostatic problem related to the rotation of a rigid punch which is bonded to the surface of a nonhomogeneous half-space. The half-space is composed of an isotropic homogeneous coating in the form of layer, which is attached to the functionally graded half-space. The shear modulus of the FGM is assumed to vary in the direction of axis Oz normal to the boundary as μ₁(z) = μ₀(1 + αz)^β, where μ₀, α, β are positive constants. The punch undergoes rotation due to the action of the internal loads. By using Hankel's integral transforms, the mixed boundary value problem is reduced to dual integral equations, and next, to a Fredholm's integral equation of the second kind, which is solved numerically for the case of β = 2. The final results show the effect of non-homogeneity on the shear stresses and an unknown moment of punch rotation.     

压电涂层-功能梯度压电界面层-基底结构的二维接触问题研究

在多层压电元件中,由于界面处材料成分和性质的突变,常常导致界面处应力集中,使得界面处出现开裂或蠕变现象,从而大大缩短了压电元件的使用寿命。功能梯度压电材料作为界面层,可有效的缓解界面材料不匹配导致的破坏。本文主要研究利用功能梯度压电材料界面层连接压电涂层和基底,分析三层结构在圆柱型压头作用下的力电响应。利用傅里叶积分变换技术,本文将压电涂层-功能梯度压电层-基底结构在刚性圆柱压头作用下的二维平面应变接触问题转化为带有柯西核的奇异积分方程。运用高斯-切比雪夫积分公式,将奇异积分方程转化为线性方程组并对其进行数值求解,得到压电涂层-功能梯度压电层-基底结构在圆柱形压头作用下的应力分布和电位移分布。数值结果表明,梯度压电材料参数的变化对结构中的力电响应具有重要的影响。本文研究结果对于利用功能梯度压电界面层消除界面处的应力不连续导致的界面破坏具有重要的理论指导意义,研究结果可为功能梯度压电材料界面层的设计提供帮助。     

An axisymmetric problem of an embedded crack in a graded layer bonded to a homogeneous half-space

In an attempt to simulate non-uniform coating delamination, the elasto-static problem of a penny shaped axisymmetric crack embedded in a functionally graded coating bonded to a homogeneous substrate subjected to crack surface tractions is considered. The coating’s material gradient is parallel to the axisymmetric direction and is orthogonal to the crack plane. The graded coating is modeled as a non-homogeneous medium with an isotropic constitutive law. Using Hankel transform, the governing equations are converted into coupled singular integral equations, which are solved numerically to yield the crack tip stress intensity factors. The Finite Element Method was additionally used to model the crack problem. The main objective of this paper is to study the influence of the material non-homogeneity and the crack position on the stress intensity factors for the purpose of gaining better understanding on the behavior of graded coatings.     

Anti-plane problem of periodic cracks in a functionally graded coating–substrate structure

The static and dynamic anti-plane problem for a functionally graded coating–substrate structure containing a periodic array of parallel cracks, which are perpendicular to the boundary, is considered. Integral-transform techniques are employed to reduce the problem to the solution of an integral equation with hypersingular kernels. Numerical results are presented to show the influence of geometry, material properties and material gradient parameter on the fracture behavior.     

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。     

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.     

The axisymmetric contact problem of the theory of elasticity for a half-space in the presence of body forces

We consider the axisymmetric problem of determination of the stress-strain state in an elastic half-space in the case of a circular line of separation of the boundary conditions on the boundary plane z = 0. We assume that on the entire boundary z = 0 the tangential stress _rz = 0, while inside the circle r a (z = 0) the normal displacement u_z is known and in its exterior the normal stress _z is given. In addition, we assume that body forces are acting in the half-space. The investigation of problems of similar kind presents interest in connection with the application of A. A. Il'yushin's method of elastic solutions to the problem of the indentation of punches into a nonlinear-elastic, in particular, into an elastoplastic half-space.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 110–115, July–August, 1971.The author wishes to thank M. Ya. Leonov for his valuable suggestions during the preparation of this paper.     

A general solution of the axisymmetric contact problem for biphasic cartilage layers

A unilateral axisymmetric contact problem for articular cartilage layers is considered. The articular cartilages bonded to subchondral bones are modeled as biphasic materials consisting of a solid phase and a fluid phase. It is assumed that the subchondral bones are rigid and shaped like bodies of revolution with arbitrary convex profiles. The obtained closed-form analytical solution is valid over time periods compared with the typical diffusion time and can be used for increasing loading.     

Winkler partial slip solution for harmonic oscillations in steady rolling contact problems

A Winkler model (Kalker’s simplified theory) is adopted for solving analytically partial slip rolling contact problem in the first order perturbation form of small periodic oscillations of generally both normal and tangential load about a steady state. At present, only numerical investigations exist for this problem, with various approximations to deal with the transient effects (often, simply neglected), and particularly the effect of varying normal load and hence contact area, has not been investigated in detail, despite the problem of corrugation is essentially driven by the change of normal load.The linear perturbation analysis is used to obtain closed form expressions for the receptances of the tangential load. Also, similar expressions are obtained for the energy dissipation, which is correlated with the local wear.     

Two-dimensional thermoelastic contact problem of functionally graded materials involving frictional heating

The two-dimensional thermoelastic sliding frictional contact of functionally graded material (FGM) coated half-plane under the plane strain deformation is investigated in this paper. A rigid punch is sliding over the surface of the FGM coating with a constant velocity. Frictional heating, with its value proportional to contact pressure, friction coefficient and sliding velocity, is generated at the interface between the punch and the FGM coating. The material properties of the coating vary exponentially along the thickness direction. In order to solve the heat conduction equation analytically, the homogeneous multi-layered model is adopted for treating the graded thermal diffusivity coefficient with other thermomechanical properties being kept as the given exponential forms. The transfer matrix method and Fourier integral transform technique are employed to convert the problem into a Cauchy singular integral equation which is then solved numerically to obtain the unknown contact pressure and the in-plane component of the surface stresses. The effects of the gradient index, Peclet number and friction coefficient on the thermoelastic contact characteristics are discussed in detail. Numerical results show that the distribution of the contact stress can be altered and therefore the thermoelastic contact damage can be modified by adjusting the gradient index, Peclet number and friction coefficient.