Non-slipping adhesive contact of a rigid cylinder on an elastic power-law graded half-space

In this paper, adhesive contact of a rigid cylinder on an elastic power-law graded half-space is studied analytically with the theory of weakly singular integral equation and orthogonal polynomial method. Emphasis is placed on the coupling effect between tangential and normal directions which was often neglected in previous works. Our analysis shows that the coupling effect tends to reduce the contact area in the compressive regime. The effect of bending moment on the adhesion behavior is also examined. Like a pull-off force, there also exists a critical bending moment at which the cylinder can be bended apart from the substrate. However, unlike pull-off force, the critical bending moment is insensitive to the gradient exponent of the graded material.     

Two-dimensional contact mechanics of functionally graded materials with arbitrary spatial variations of material properties

A multi-layered model for frictionless contact analysis of functionally graded materials (FGMs) with arbitrarily varying elastic modulus under plane strain-state deformation has been developed. Based on the fact that an arbitrary curve can be approached by a series of continuous but piecewise linear curves, the FGM is divided into several sub-layers and in each sub-layers the shear modulus is assumed to be linear function while the Poisson’s ratio is assumed to be a constant. With the model, the frictionless contact problem of a functionally graded coated half-space is investigated. By using the transfer matrix method and Fourier integral transform technique, the problem is reduced to a Cauchy singular integral equation. The contact pressure, contact region and indentation are calculated for various indenters by solving the equations numerically.     

Rolling contact of a rigid sphere/sliding of a spherical indenter upon a viscoelastic half-space containing an ellipsoidal inhomogeneity

In this paper, the frictionless rolling contact problem between a rigid sphere and a viscoelastic half-space containing one elastic inhomogeneity is solved. The problem is equivalent to the frictionless sliding of a spherical tip over a viscoelastic body. The inhomogeneity may be of spherical or ellipsoidal shape, the later being of any orientation relatively to the contact surface. The model presented here is three dimensional and based on semi-analytical methods. In order to take into account the viscoelastic aspect of the problem, contact equations are discretized in the spatial and temporal dimensions. The frictionless rolling of the sphere, assumed rigid here for the sake of simplicity, is taken into account by translating the subsurface viscoelastic fields related to the contact problem. Eshelby's formalism is applied at each step of the temporal discretization to account for the effect of the inhomogeneity on the contact pressure distribution, subsurface stresses, rolling friction and the resulting torque. A Conjugate Gradient Method and the Fast Fourier Transforms are used to reduce the computation cost. The model is validated by a finite element model of a rigid sphere rolling upon a homogeneous vciscoelastic half-space, as well as through comparison with reference solutions from the literature. A parametric analysis of the effect of elastic properties and geometrical features of the inhomogeneity is performed. Transient and steady-state solutions are obtained. Numerical results about the contact pressure distribution, the deformed surface geometry, the apparent friction coefficient as well as subsurface stresses are presented, with or without heterogeneous inclusion.     

任意轴对称弹性体吸附接触的广义Maugis模型

通过线性叠加Sneddon方法和Lowengrub-Sneddon方法分别给出的解, 得到了一个弹性半空间 轴对称混合边值问题的一般解，进而研究了两个一般轴对称弹性体的正向无摩擦吸附接触问 题. 考虑任意有效的表面形状(要求中心部分首先进入接触)和任意的表面吸附作用，推广 得到了广义Maugis模型. 该模型是一个半解析的模型，它可以分解成表面形状和表面吸附 作用的分别独立影响的两部分，以及一个关联变形和吸附作用的式子. 利用Dugdale模型近 似表面吸附作用，得到了具有任意有效的表面形状的广义M-D模型. 它在强吸附或软材料条 件下的极限形式是广义JKR模型，而在弱吸附或硬材料下的另一个极限形式是广义DMT模型.     

Mechanics of indentation for piezoelectric thin films on elastic substrate

Frictionless normal indentation problem of rigid flat-ended cylindrical, conical and spherical indenters on piezoelectric film, which is either in frictionless contact with or perfectly bonded to an elastic half-space (substrate), is investigated. Both conducting and insulating indenters are considered. With Hankel transform, the general solutions of the homogeneous governing equations for the piezoelectric layer and the elastic half-space are presented. Using the boundary conditions for a vertical point force or a point electric charge, and the boundary conditions on the film/substrate interface, the Green’s functions can be obtained by solving sets of simultaneous linear algebraic equations. The solution of the indentation problem is obtained by integrating these Green’s functions over the contact area with unknown surface tractions or electric charge distribution, which will be determined from the boundary conditions on the contact surface between the indenter and the film. The solution is expressed in terms of dual integral equations that are converted to a Fredholm integral equation of the second kind and solved numerically. Numerical examples are also presented. The comparison between two film/substrate bonding conditions is made. It shows that the indentation rigidity of the film/substrate system is lower when the film is in frictionless contact with the substrate. The effects of the Young’s modulus and Poisson’s ratio of the elastic substrate, indenter electrical condition and indenter prescribed electric potential on the indentation responses are presented.     

Exact analysis of the orientation-adjusted adhesive full stick contact of layered structures with the asymmetric bipolar coordinates

The adhesion failure has become one dominant factor in determining the reliability and service life of miniaturized devices subject to loadings with arbitrary orientations. This article establishes an adhesive full stick contact model between an elastic half-space and a rigid cylinder loaded in any direction. Using the Papkovich-Neuber functions, the Fourier integral transform, and the asymmetric bipolar coordinates, the exact solution is obtained. Unlike the Johnson-Kendall-Roberts (JKR) model, the present adhesive contact model takes into account the effects of the load direction as well as the coupling of the normal and tangential contact stresses. Besides, it considers the full stick contact which has large values of the friction coefficient between contacting surfaces, contrary to the frictionless contact supposed in the JKR model. The result shows that suitable angles can be found, which makes the contact surfaces difficult to be peeled off or easy to be pressed into.

     

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 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.     

Sliding Contact with Friction at Arbitrary Constant Speeds on a Pre-Stressed Highly Elastic Half-Space

Resisted by Coulomb friction, a rigid indentor slides at a constant arbitrary speed on a generalized neo-Hookean half-space under pre-stress. A dynamic steady-state situation in plane strain is assumed, and is treated as the superposition of contact-triggered infinitesimal deformations upon finite deformations due to pre-stress. Exact solutions are presented for both deformations, and the infinitesimal component exhibits the anisotropy typically induced by pre-stress, and wave speeds that are sensitive to pre-stress. In view of the unilateral constraints of contact, these and other critical speeds define the sliding speed ranges for physically-acceptable solutions. In particular, a Rayleigh speed is the upper bound for subsonic sliding. Solutions are further constrained by the unilateral requirement that contact zone shear must oppose indentor/half-space slip. The generic parabolic indentor is used for illustration, and it is found that traction continuity at the contact zone leading edge is lost for supersonic sliding and at the single sliding speed allowed in the frictionless limit in the trans-sonic range. A range of acceptable pre-stresses is also identified; for pre-stresses that lie out of range, either a negative Poisson effect occurs, or the Rayleigh wave disappears, thereby precluding sliding in the subsonic range. This revised version was published online in August 2006 with corrections to the Cover Date.     

Steady Sliding Frictional Contact Problems in Linear Elasticity

The existence and uniqueness of an equilibrium solution to frictional contact problems involving a class of moving rigid obstacles is studied. At low friction coefficient values, the steady sliding frictional contact problem is uniquely solvable, thanks to the Lions-Stampacchia theorem on variational inequalities associated with a nonsymmetric coercive bilinear form. It is proved that the coerciveness of the bilinear form can be lost at some positive critical value of the friction coefficient, depending only on the geometry and the elastic properties of the body. An example presented here, shows that infinitely many solutions can be obtained when the friction coefficient is larger than the critical value. This result is paving the road towards a theory of jamming in terms of bifurcation in variational inequality. The particular case where the elastic body is an isotropic half-space is studied. The corresponding value of the critical friction coefficient is proved to be infinite in this case. In the particular case of the frictionless situation, our analysis incidentally unifies the approaches developed by Lions-Stampacchia (variational inequalities) and Hertz (harmonic analysis on the half-space) to contact problems in linear elasticity.     

Axisymmetric frictionless contact of functionally graded materials

The main interest of this study is a new method to solve the axisymmetric frictionless contact problem of functionally graded materials (FGMs). Based on the fact that an arbitrary curve can be approached by a series of continuous but piecewise linear curves, the FGM is divided into a series of sub-layers with shear modulus varying linearly in each sub-layer and continuous at the sub-interfaces. With this model, the axisymmetric frictionless contact problem of a functionally graded coated half-space is investigated. By using the transfer matrix method and Hankel integral transform technique, the problem is reduced to a Cauchy singular integral equation. The contact pressure, contact region and indentation are calculated for various indenters by solving the equations numerically. An erratum to this article can be found at     

Two-dimensional sliding frictional contact of functionally graded materials

A multi-layered model for sliding frictional contact analysis of functionally graded materials (FGMs) with arbitrarily varying shear modulus under plane strain-state deformation has been developed. Based on the fact that an arbitrary curve can be approached by a series of continuous but piecewise linear curves, the FGM is divided into several sub-layers and in each sub-layers the shear modulus is assumed to be a linear function while the Poisson's ratio is assumed to be a constant. In the contact area, it is assumed that the friction is one of Coulomb type. With this model the fundamental solutions for concentrated forces acting perpendicular and parallel to the FGMs layer surface are obtained. Then the sliding frictional contact problem of a functionally graded coated half-space is investigated. The transfer matrix method and Fourier integral transform technique are employed to cast the problem to a Cauchy singular integral equation. The contact stresses and contact area are calculated for various moving stamps by solving the equations numerically. The results show that appropriate gradual variation of the shear modulus can significantly alter the stresses in the contact zone.     

Adhesive elastic contact between a symmetric indenter and an elastic film

This paper examines the frictionless adhesive elastic contact problem of a rigid sphere indenting a thin film deposited on a substrate. The result is then used to model the elastic phase of micro-nanoscale indentation tests performed to determine the mechanical properties of coatings and films. We investigate the elastic response including the effects of adhesion, which, as the scale decreases to the nano level, become an important issue. In this paper, we extend the Johnson–Kendall–Roberts, Derjaguin–Muller–Toporov, and Maugis–Dugdale half-space adhesion models to the case of a finite thickness elastic film coated on an elastic substrate. We propose a simplified model based on the assumption that the pressure distribution is that of the corresponding half-space models; in doing so, we investigate the contact radius/film thickness ratio in a range where it is usually assumed the half-space model. We obtain an analytical solution for the elastic response that is useful for evaluating the effects of the film-thickness, the interface film–substrate conditions, and the adhesion forces. This study provides a guideline for selecting the appropriate film thickness and substrate to determine the elastic constants of film in the indentation tests.     

Effects of interfacial friction on flaw tolerant adhesion between two dissimilar elastic solids

Flaw tolerance refers to a state in which a pre-existing crack-like flaw does not propagate even as the material is stretched to failure near its theoretical strength. Such an optimal scenario can be achieved when the characteristic length scale is reduced to below a critical value. So far, the critical conditions to achieve flaw tolerance have been discussed mostly for homogeneous materials or for two dissimilar materials in frictionless or perfectly bonded adhesion. In this paper, we consider the role of friction in flaw tolerant adhesion between two dissimilar elastic solids. We adopt a frictional contact model in which slip is allowed wherever the shear stress along the interface reaches a threshold value defined as the friction strength. The critical length scale for flaw tolerance is derived analytically for a penny-shaped crack and for an external circular crack. Compared to the cases of frictionless contact, we find that interfacial friction can reduce the critical length scales for flaw tolerance by up to 12.5%.     

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.     

Contact vibration analysis of the functionally graded material coated half-space under a rigid spherical punch

This paper studies the contact vibration problem of an elastic half-space coated with functionally graded materials (FGMs) subject to a rigid spherical punch. A static force superimposing a dynamic time-harmonic force acts on the rigid spherical punch. Firstly, we give the static contact problem of FGMs by a least-square fitting approach. Next, the dynamic contact pressure is solved by employing the perturbation method. Lastly, the dynamic contact stiffness with different dynamic contact displacement conditions is derived for the FGM coated half-space. The effects of the gradient index, coating thickness, internal friction, and punch radius on the dynamic contact stiffness factor are discussed in detail.     

Spatial contact problems for a prestressed incompressible elastic layer

Two problems are considered on frictionless indentation of a stamp into the upper face of a layer with a homogeneous field of initial stresses present in the layer. The model of an isotropic incompressible nonlinearly-elastic material determined by the Mooney potential is used. The following two cases are studied: the lower face of the prestressed layer is rigidly fixed, and the lower face of a prestressed layer is supported by a rigid foundation without friction. It is assumed that the additional stresses due to the action of the stamp on the layer are small as compared with the initial stresses. This assumption makes it possible to linearize the problems of determining the additional stresses. In what follows, the problems are reduced to solving two-dimensional integral equations (IE) of the first kind with symmetric irregular kernels with respect to the pressure in the contact region. As an example, the case of an elliptic (in plan) stamp acting on a layer is considered. The spatial contact problem for a prestressed elastic half-space was first considered in [1].     

The Stress Response of Functionally Graded Isotropic Linearly Elastic Rotating Disks

The purpose of this research is to investigate the effects of material inhomogeneity on the response of linearly elastic isotropic solid circular disks or cylinders, rotating at constant angular velocity about a central axis. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e., materials with spatially varying properties tailored to satisfy particular engineering applications. The analog of the classic problem for a homogeneous isotropic rotating solid disk or cylinder is considered. The special case of a body with Young"s modulus depending on the radial coordinate only, and with constant Poisson"s ratio, is examined. For the case when the Young"s modulus has a power-law dependence on the radial coordinate, explicit exact solutions are obtained. It is shown that the stress response of the inhomogeneous disk (or cylinder) is significantly different from that of the homogeneous body. For example, the maximum radial and hoop stresses do not, in general, occur at the center as in the case for the homogeneous material. Furthermore, for the case where the Young"s modulus increases with radial distance from the center, it is shown that radially symmetric solutions exist provided the rate of growth of the Young"s modulus is, at most, cubic in the radial variable. It is also shown for the general inhomogeneous isotropic case how the material inhomogeneity may be tailored so that the radial and hoop stress are identical throughout the disk. This revised version was published online in August 2006 with corrections to the Cover Date.     

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.