Action of an elliptic punch on a transversally isotropic half-space

The 3D contact problem on the action of a punch elliptic in horizontal projection on a transversally isotropic elastic half-space is considered for the case in which the isotropy planes are perpendicular to the boundary of the half-space. The elliptic contact region is assumed to be given (the punch has sharp edges). The integral equation of the contact problem is obtained. The elastic rigidity of the half-space boundary characterized by the normal displacement under the action of a given lumped force significantly depends on the chosen direction on this boundary. In this connection, the following two cases of location of the ellipse of contact are considered: it can be elongated along the first or the second axis of Cartesian coordinate system on the body boundary. Exact solutions are obtained for a punch with base shaped as an elliptic paraboloid, and these solutions are used to carry out the computations for various versions of the five elastic constants. The structure of the exact solution is found for a punch with polynomial base, and a method for determining the solution is proposed.     

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.     

Quasistatic periodic contact problem for a viscoelastic layer,a cylinder,and a space with a cylindrical cavity

This paper considers the contact problem of interaction of a rigid die, a rigid band, and a rigid insert with a viscoelastic layer, a viscoelastic cylinder, and viscoelastic space with a cylindrical cavity, respectively. It is assumed that the die, band, and insert move at a constant velocity along the boundaries of the viscoelastic bodies. In the first stage, the displacement of the boundaries of the above-mentioned bodies is determined as a function of the applied normal loads ignoring friction in the contact area. In the second stage, integral equations are derived to determine contact pressure in the contact problems. In the third stage, approximate solutions of the integral equations are constructed using a modified Multhopp-Kalandia method.     

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.     

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.     

Contact strength of a two-layer covering under friction forces in the contact region

Friction and antifriction composite materials of multilayer structure [1] have recently become very popular in the engineering industry. Antifriction materials are widely used in sliding bearings, and friction materials are widely used in brakes. In the first case, the friction forces between the contacting surfaces are negligible, but in the second case, they are rather large. We use two examples of two plane problems from the theory of elasticity concerning the interaction between a die and a base formed by two elastic layers with different mechanical properties, which are rigidly connected with each other and with an undeformable support, to study how the geometric and mechanical parameters of the problem affect the stress-strain state of such a base, both on its surface and at its internal points, and to find the optimal parameters ensuring the required operation resources of the friction units thus modeled. We assume that the die foot is parabola-shaped or plane, the normal and tangential stresses in the contact region are related to each other by the Coulomb law, and the die is subjected to normal and tangential forces. In this case, the die-two-layer base is in the limit equilibtrium, and the die does not rotate in the process of deformation of the layer. In this setting, the problems were studied in [2] by solving the integral equations (IE) by the asymptotic method of large λ (see [3–7], etc.), which permits finding the effective solution only for relatively large thicknesses of layers compared with the dimensions of the contact region. But in real friction units mentioned above, the layers can have rather small relative thicknesses, and the large λ method cannot be used. We note that the other asymptotic methods (e.g., see [3]) efficient in the case of relatively small thicknesses of layers cannot yet be adapted to the case of friction forces in the contact region. In the present paper, we propose to use the collocation method following the scheme given in [8] to solve the corresponding integral equation of the first kind with logarithmic kernel. This method allows one to obtain sufficiently exact solutions practically for all values of the parameters of the problem with relatively small expenditure of the computer time for modern computers. The contact problem for a two-layer base was used in [9] for a close statement of the problem without friction forces in the contact region.     

Contact problems for a two-layer elastic wedge

We obtain integral equations for plane contact problems for a two-layer wedge (composite) under three types of boundary conditions on one of its sides (absence of stresses, sliding, or rigid fixation). The composite consists of two wedges completely linked with each other, which have different opening angles and elasticity parameters. Using the symbols (Mellin transforms) of the kernels of integral equations for the two-layer wedge, one can derive the symbols of the kernels of integral equations for symmetric problems about a crack in a three-layer wedge or a three-layer strip and for contact problems for a two-layer strip (by passing to the limit in a special way). The complex zeros of the Mellin transform determine the asymptotics of the normal contact pressure at the corner point of the composite as the contact region approaches this point. It is important that this asymptotics is also preserved in three-dimensional contact problems as the die enters the edge of a two-layer wedge (outside the corner points of the die itself). Taking into account this asymptotics, we obtain solutions of the contact problems as the die enters the vertex of the composite. We show that by appropriately choosing the materials and the internal angle of the two-layer wedge one can avoid contact pressure oscillations at the vertex, which occur in the case of a homogeneous wedge and result in loss of contact. The contact pressure at the wedge vertex can be made zero for a composite, while for a homogeneous wedge with the same opening angle it increases unboundedly. We construct asymptotic solutions of the contact problems for a plane die located relatively close or to the vertex of a two-layer wedge or relatively far from the vertex. The asymptotic and other methods were earlier used to solve similar plane contact problems for a homogeneous wedge [1, 2]. In the case of sliding fixation of one of the sides of a plane homogeneous wedge, the closed solution of the contact problem is known for a die entering the corner point [3, p. 131]. Two-dimensional contact problems were studied for a truncated wedge [4] and for a wedge supported by a rod of equal resistance [5]. The out-of-plane shear vibrations were studied for wedge-shaped composites [6, 7]. The spatial contact problems were considered for a homogeneous wedge [8]. The plane contact problem was analyzed for a continuously inhomogeneous wedge one of whose sides was rigidly fixed (the shear modulus continuously depends on the angular coordinate and the Poisson coefficient is constant). For a two-layer composite, which is studied in the present paper, the kernel symbol has different asymptotic properties, which are used in asymptotic methods for solving the problem. A similar distinction of the symbol properties takes place in contact problems for a continuously inhomogeneous layer and a layered packet.     

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.     

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.     

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.     

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.     

Indentation of a smooth axisymmetric punch into a transversely isotropic layer

We consider an axisymmetric contact problem for a transversely isotropic layer on a rigid base. The problem is reduced to a Fredholm integro-differential equation of the first kind for the contact pressure under the punch. The integral equation is solved by the collocation method [1]. Some numerical results are presented.     

Electro-mechanical frictionless contact behavior of a functionally graded piezoelectric layered half-plane under a rigid punch

The frictionless contact problem of a functionally graded piezoelectric layered half-plane in-plane strain state under the action of a rigid flat or cylindrical punch is investigated in this paper. It is assumed that the punch is a perfect electrical conductor with a constant potential. The electro-elastic properties of the functionally graded piezoelectric materials (FGPMs) vary exponentially along the thickness direction. The problem is reduced to a pair of coupled Cauchy singular integral equations by using the Fourier integral transform technique and then is numerically solved to determine the contact pressure, surface electric charge distribution, normal stress and electric displacement fields. For a flat punch, the normal stress intensity factor and electric displacement intensity factor are also given to quantitatively characterize the singularity behavior at the punch ends. Numerical results show that both material property gradient of the FGPM layer and punch geometry have a significant influence on the contact performance of the FGPM layered half-plane.     

On the mathematical problems of composite materials with a doubly periodic set of cracks

In this paper,the mathematical problem of the second fundamental problem of composite materials with a doubly periodic set of arbitrary shape cracks are investigated,and the interface are arbitrary smooth closed contours.At first,we establish mathematical models by using Muskhelisvili complex variable methods,change the primitive problems into searching complex stress functions which satisfy four boundary value problems and construct forms of the solution,then,under some general restrictions it is reduced to normal type singular integral equation,the unique solvability is proved mathematically     

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。     

Adhesive Frictionless Contact Between an Elastic Isotropic Half-Space and a Rigid Axi-Symmetric Punch

In this paper we consider the problem of adhesive frictionless contact of an elastic half-space by an axi-symmetric punch. We obtain integral equations that define the tractions and displacements normal to the surface of the half-space, as well as the size of the contact regions, for the cases of circular and annular contact regions. The novelty of our approach resides in the use of Betti’s reciprocity theorem to impose equilibrium, and of Abel transforms to either solve or substantially simplify the resulting integral equations. Additionally, the radii that define the annular or circular contact region are defined as local minimizers of the function obtained by evaluating the potential energy at the equilibrium solutions for each pair of radii. With this approach, we rather easily recover Sneddon’s formulas (Sneddon, Int. J. Eng. Sci., 3(1):47–57, 1965) for circular contact regions. For the annular contact region, we obtain a new integral equation that defines the inverse Abel transform of the surface normal displacement. We solve this equation numerically for two particular punches: a flat annular punch, and a concave punch.     

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.     

Boundary Integral Analysis of the Symmetric Dynamic Problem for an Infinite Bimaterial Solid with an Embedded Crack

The symmetric frequency domain problem for two ideally bonded elastic half-spaces with a perpendicular plane crack is considered. It is reduced to the boundary integral equation (BIE) with integration over the limited crack region. The contact conditions on the bimaterial interface are satisfied identically in the initial stage of obtaining the equation. After boundary element solution of the equation, the stress concentration in the vicinity of a penny-shaped crack under time-harmonic loading of constant amplitude is studied. The mode I stress intensity factors as functions of angular coordinate of a crack front point and wave number for various relations between the material parameters are computed. The crack depth relative to the bimaterial interface is determined, when the effect of the material dissimilarity on the crack can be neglected.     

The symmetrical adhesive contact problem for circular elastic cylinders

The rolling contact problem involving circular cylinders is at the heart of numerous industrial processes, and critical to any elastohydrodynamic lubrication analysis is an accurate knowledge of the associated contact pressure for the static dry problem. In a recent article [1] the authors have obtained new horizontal pressure distributions, both exact and approximate for various problems involving the symmetrical contact of circular elastic cylinders. In [1] it is assumed that only the circumferential horizontal displacement is prescribed in the contact region while the vertical circumferential displacement is left arbitrary and is assumed to take on whatever value is predicted by the deformation. The advantage of this assumption is that the problem reduces to a single singular integral equation which by transformations can be simplified to an integral equation involving the standard finite Hilbert transform. Here we consider the more general displacement boundary value problem within the contact region, and to be specific we examine the problem with zero vertical circumferential displacement and prescribed horizontal circumferential displacement. The solution of this problem involves two coupled singular integral equations for the horizontal and vertical pressure distributions. Basic equations and some approximate analytical solutions are obtained for symmetrical contact of circular elastic cylinders by both parallel plates and circular cylinders which are either rigid or elastic. Numerical results for the approximate analytical solutions are given for contact by rigid parallel plates and rigid circular cylinders.     

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.