Adhesive contact of an elastic semi-infinite solid with a rigid rough surface: Strength of adhesion and contact instabilities

The effect of adhesion on the contact behavior of elastic rough surfaces is examined within the framework of the multi-asperity contact model of Greenwood and Williamson (1966), known as the GW model. Adhesive surface interaction is modeled by nonlinear springs with a force–displacement relation governed by the Lennard–Jones (LJ) potential. Constitutive models are presented for contact systems characterized by low and high Tabor parameters, exhibiting continuous (stable) and discontinuous (unstable) surface approach, respectively. Constitutive contact relations are obtained by integrating the force–distance relation derived from the LJ potential with a finite element analysis of single-asperity adhesive contact. These constitutive relations are then incorporated into the GW model, and the interfacial force and contact area of rough surfaces are numerically determined. The development of attractive and repulsive forces at the contact interface and the occurrence of instantaneous surface contact (jump-in instability) yield a three-stage evolution of the contact area. It is shown that the adhesion parameter introduced by Fuller and Tabor (1975) governs the strength of adhesion of contact systems with a high Tabor parameter, whereas the strength of adhesion of contact systems with a low Tabor parameter is characterized by a new adhesion parameter, defined as the ratio of the surface roughness to the equilibrium interatomic distance. Applicable ranges of aforementioned adhesion parameters are interpreted in terms of the effective surface separation, obtained as the sum of the effective distance range of the adhesion force and the elastic deformation induced by adhesion. Adhesive strength of rough surfaces in the entire range of the Tabor parameter is discussed in terms of a generalized adhesion parameter, defined as the ratio of the surface roughness to the effective surface separation.     

Numerical study of contact forces for crack closure analysis

Plasticity induced crack closure (PICC) has been widely studied using numerical models. Different numerical parameters can be considered to quantify the opening level, namely one based on the analysis of contact stresses at minimum load. A modified version of this parameter is proposed here, based on nodal contact forces instead of contact stresses. The predictions were found to be similar to those obtained from the contact status of 2nd node behind crack tip. The PICC_contact parameter was also found to be very consistent and adequate for parametric studies of the influence of different physical parameters. The contributions to the opening stress intensity factor of different points along crack flank were found to strongly decrease with distance to crack tip. The cumulative K_open between the crack tip and a distance of 0.1 mm was found to vary from 30% to 100%, increasing with stress ratio, R. Finally, a K solution was developed for punctual forces applied on crack flank and compared with a literature solution for infinite plates. A good agreement was found for plane strain state but significant differences of about 10% were found for plane stress state.     

Hertz contact at finite friction and arbitrary profiles

Axisymmetric contact at finite Coulomb friction and arbitrary profiles is examined analytically and numerically for dissimilar linear elastic solids. Invariance and generality are aimed at and an incremental procedure is developed resulting in a reduced benchmark problem corresponding to a rigid flat indentation of an elastic half-space. The reduced problem, being independent of loading and contact region, was solved by a finite element method based on a stationary contact contour and characterized by high accuracy. Subsequently, a tailored cumulative superposition procedure was developed to resolve the original problem to determine global and local field values. Save for the influence of the coefficients of friction and contraction ratio, it is shown that at partial slip the evolving relative stick-slip contour is independent of any convex and smooth contact profile at monotonic loading. For flat and conical profiles with rounded edges and apices, results are illustrated for relations between force, depth and contact contours together with surface stress distributions. The solution for dissimilar solids in a full space is transformed to a half-space problem and solved for a combination of material parameters in order to first determine interface traction distributions. Subsequently, full field values for the two solids were computed individually. In order to predict initiation of fracture and plastic flow, results are reported for the location and magnitude of maximum tensile stress and effective stress, respectively, for a range of geometrical and material parameters. In two illustrations, predicted results are compared with experimental findings related to initiation of brittle fracture and load-depth relations at nanoindentation.     

Analysis of the plane contact with discontinuous curvature

The analytical solution for the plane contact between nonconforming C¹ profiles with a curvature discontinuity is obtained and discussed. The initial gap between the undeformed profiles was represented by a double parabolic function having the curvature discontinuity within the contact region. On the basis of the analytical solution, the complete analysis of pressure distribution was obtained. A raw technique to find an approximate solution is presented and comparison to the analytic one provided. For the limit case of strong curvature ratio the asymptotic treatment is discussed in terms of consistency with hypotheses claimed.     

A unified treatment of axisymmetric adhesive contact problems using the harmonic potential function method

A unified treatment of axisymmetric adhesive contact problems is provided using the harmonic potential function method for axisymmetric elasticity problems advanced by Green, Keer, Barber and others. The harmonic function adopted in the current analysis is the one that was introduced by Jin et al. (2008) to solve an external crack problem. It is demonstrated that the harmonic potential function method offers a simpler and more consistent way to treat non-adhesive and adhesive contact problems. By using this method and the principle of superposition, a general solution is derived for the adhesive contact problem involving an axisymmetric rigid punch of arbitrary shape and an adhesive interaction force distribution of any profile. This solution provides analytical expressions for all non-zero displacement and stress components on the contact surface, unlike existing ones. In addition, the newly derived solution is able to link existing solutions/models for axisymmetric non-adhesive and adhesive contact problems and to reveal the connections and differences among these solutions/models individually obtained using different methods at various times. Specifically, it is shown that Sneddon’s solution for the axisymmetric punch problem, Boussinesq’s solution for the flat-ended cylindrical punch problem, the Hertz solution for the spherical punch problem, the JKR model, the DMT model, the M-D model, and the M-D-n model can all be explicitly recovered by the current general solution.     

On oblique contact of creeping solids

Oblique indentation of power-law creeping solids by a rigid die is analysed in three dimensions with perfectly plastic behaviour emerging as an asymptotic case. Indenter profiles are prescribed to be axisymmetric for simplicity but not by necessity. Invariance and generality is aimed at, as the problem is governed by only four essential parameters, i.e. the die profile, p, the indentation angle, γ, the power-law exponent, n, and the coefficient of friction, μ. The solution strategy is based on a self-similar transformation resulting in a reduced problem corresponding to flat die indentation of complete contact. The reduced auxiliary problem, being independent of loading, history and time, was solved by a three-dimensional finite element analysis characterized by high accuracy. Subsequently, cumulative superposition was used to resolve the original problem and global and invariant relations between force, depth and contact area were determined. Detailed results are given for the location and shape of the contact region and stick/slip contours as well as for local states of surface stresses and deformation at flat and spherical indenters. Due to the asymmetry prevailing, it was found that in the spherical case, contact contours proved to be oval and shifted, although with normal and tangential forces only weakly coupled. Finite friction as compared to full adhesion proved to have only a minor effect on global relations. The framework laid down may be applied to the contact of structural assemblies subjected especially to elevated temperatures and also to various issues such as compaction of powder aggregates, flattening of rough surfaces and plastic impact.     

Self-similar problems of elastic contact for non-convex punches

Self-similar problems of contact for non-convex punches are considered. The non-convexity of the punch shapes introduces differences from the traditional self-similar contact problems when punch profiles are convex and their shapes are described by homogeneous functions. First, three-dimensional Hertz type contact problems are considered for non-convex punches whose shapes are described by parametric-homogeneous functions. Examples of such functions are numerous including both fractal Weierstrass type functions and smooth log-periodic sine functions. It is shown that the region of contact in the problems is discrete and the solutions obey a non-classical self-similar law. Then the solution to a particular case of the contact problem for an isotropic linear elastic half-space when the surface roughness is described by a log-periodic function, is studied numerically, i.e. the contact problem for rough punches is studied as a Hertz type contact problem without employing additional assumptions of the multi-asperity approach. To obtain the solution, the method of non-linear boundary integral equations is developed. The problem is solved only on the fundamental domain for the parameter of self-similarity because solutions for other values of the parameter can be obtained by renormalization of this solution. It is shown that the problem has some features of chaotic systems, namely the global character of the solution is independent of fine distinctions between parametric-homogeneous functions describing roughness, while the stress field of the problem is sensitive to small perturbations of the punch shape.     

Thermoelastic contact mechanics for a flat punch sliding over a graded coating/substrate system with frictional heat generation

The problem of thermoelastic contact mechanics for the coating/substrate system with functionally graded properties is investigated, where the rigid flat punch is assumed to slide over the surface of the coating involving frictional heat generation. With the coefficient of friction being constant, the inertia effects are neglected and the solution is obtained within the framework of steady-state plane thermoelasticity. The graded material exists as a nonhomogeneous interlayer between dissimilar, homogeneous phases of the coating/substrate system or as a nonhomogeneous coating deposited on the substrate. The material nonhomogeneity is represented by spatially varying thermoelastic moduli expressed in terms of exponential functions. The Fourier integral transform method is employed and the formulation of the current thermoelastic contact problem is reduced to a Cauchy-type singular integral equation of the second kind for the unknown contact pressure. Numerical results include the distributions of the contact pressure and the in-plane component of the surface stress under the prescribed thermoelastic environment for various combinations of geometric, loading, and material parameters of the coated medium. Moreover, in order to quantify and characterize the singular behavior of contact pressure distributions at the edges of the flat punch, the stress intensity factors are defined and evaluated in terms of the solution to the governing integral equation.     

3D contact patch measurement inside rolling tyres

This paper presents a novel method for capturing the 3D profile of the inside of a rolling off-road vehicle tyre at the tyre-road contact region. This method captures the contact region at all times as the vehicle negotiates obstacles. The system uses a pair of inexpensive digital cameras (capable of capturing up to 300 frames per second) and features a purely mechanical stabilisation system to ensure that the cameras capture the contact region at any wheel speed or vehicle acceleration.The captured images are processed using 3D computer vision techniques using an open source computer vision library called OpenCV. Stereo image pairs are used to create clouds of 3D points showing the profile of the inside surface with good accuracy. Various obstacles were traversed with the deformed tyre profile being compared to the undeformed profile. The system improves on current measurement techniques used to measure the contact patch by capturing a large region of the contact patch, providing full 3D surface geometry, as well as remaining centred on the contact patch irrespective of wheel rotation. The system also enables other imaging techniques to be used such as digital image correlation to determine velocity profiles as well as strain measurements.     

Micro-plasticity of surface steps under adhesive contact: Part II—Multiple-dislocation mediated contact hardening

The study of micro-plastic behavior of rough surface contacts is the critical link towards a fundamental understanding of contact, friction, adhesion, and surface failures at small length scales. In the companion paper (Yu, H.H., Shrotriya, P., Gao, Y.F., Kim, K.-S., 2007. Micro-plasticity of surface steps under adhesive contact. Part I. Surface yielding controlled by single-dislocation nucleation. J. Mech. Phys. Solids 55, 489–516), we have studied the onset of surface yielding due to single-dislocation nucleation from a stepped surface under adhesive contact. Here we analyze the contact hardening behavior due to multiple dislocations in a two-dimensional dislocation model. Continuum micro-mechanical analyses are used to derive the configurational force on the dislocation, while a modified Rice–Thomson criterion is used to model dislocation nucleation. Dislocations nucleated from the surface step are stabilized and pile up as a result of the balance between the resolved driving force and the non-zero lattice resistance in the solid. The dislocation pileup will exert a strong back stress to prevent further dislocation nucleation and thus lead to the contact hardening behavior, the degree of which depends on the slip-plane orientation. Particularly, we find that dislocation interactions between two slip planes can make the contact loading order-of-magnitude easy to nucleate multiple dislocations, which is thus named “latent softening”. A mechanistic explanation shows that the latent softening is closely related to the stress-concentration mode mixity at the surface step. Dislocation nucleation will modify the geometric characteristics of the surface step, so that the contact-induced stress state near the step, as described by the mode mixity, changes, which influences the subsequent dislocation nucleation. Our calculations show that the dislocation pileup on one slip plane can even cause the spontaneous dislocation nucleation on the other slip plane without further increase of the contact load. Furthermore, it is found that rough surface contacts at small length scale can lead to the dislocation segregation and the formation of a surface tensile sub-layer. The discrete-dislocation model presented here and in the companion paper provides novel insights in bridging the atomistic simulations and continuum plastic flow analysis of surface asperity contact.     

Finite element modeling of elasto-plastic contact between rough surfaces

This paper presents a finite element calculation of frictionless, non-adhesive, contact between a rigid plane and an elasto-plastic solid with a self-affine fractal surface. The calculations are conducted within an explicit dynamic Lagrangian framework. The elasto-plastic response of the material is described by a J₂ isotropic plasticity law. Parametric studies are used to establish general relations between contact properties and key material parameters. In all cases, the contact area A rises linearly with the applied load. The rate of increase grows as the yield stress σ_y decreases, scaling as a power of σ_y over the range typical of real materials. Results for A from different plasticity laws and surface morphologies can all be described by a simple scaling formula. Plasticity produces qualitative changes in the distributions of local pressures in the contact and of the size of connected contact regions. The probability of large local pressures is decreased, while large clusters become more likely. Loading-unloading cycles are considered and the total plastic work is found to be nearly constant over a wide range of yield stresses.     

An analytically-numerical approach for the analysis of an interface crack with a contact zone in a piezoelectric bimaterial compound

An interface crack with an artificial contact zone at the right-hand side crack tip between two dissimilar finite-sized piezoelectric materials is considered under remote mixed-mode loading. To find the singular electromechanical field at the crack tip, an asymptotic solution is derived in connection with the conventional finite element method. For mechanical loads, the stress intensity factors at the singular points are obtained. As a particular case of this solution, the contact zone model (in Comninou’s sense) is derived. A simple transcendental equation and an asymptotic formula for the determination of the real contact zone length are derived. The dependencies of the contact zone lengths on external load coefficients are illustrated in graphical form. For a particular case of a short crack with respect to the dimensions of the bimaterial compound, the numerical results are compared to the exact analytical solutions, obtained for a piezoelectric bimaterial plane with an interface crack.     

A slightly corrected Greenwood and Williamson model predicts asymptotic linearity between contact area and load

The author presents a very simple and slightly corrected version of the well-known Greenwood and Williamson (GW) model which closely follows the predictions of more complicated and computationally expensive theory such as the Bush, Gibson and Thomas (BGT) theory [Bush, A.W., Gibson, R.D., Thomas, T.R., 1975. Wear 35, 87]. This new model (which I call GW modified in the following) still treats the asperities of the rough surface as spherical cups, but, this time, the curvature of spheres is not constant and instead depends on the asperity height. The GW modified theory is, in particular, able to predict, in the limiting case of large separations, the same asymptotic linearity between contact area and load as in the BGT theory. This, in turn, proves that the BGT asymptotic linear area-load relation is not a consequence of having taken into account that the contact between the asperities is actually elliptic and, therefore, of having included the spread of asperity curvature at a given height (which incidentally makes the treatment very complicated), but a consequence of having included only the influence of asperity heights on the curvature distribution of the summits. I also give a simple explanation for why the GW modified model and the BGT theory follow exactly the same asymptotic behavior. Indeed, I show that the surface summits can be treated as perfectly spherical cups (all those at the same height having the same radius of curvature) as their height is increased to very large values. In fact, in the asymptotic limit of large separation, the mean curvature of summits is shown to increase proportionally to the summit height, whereas the difference of the two principal curvatures approaches a finite constant value. The consequence of this is that the Hertzian contact between the approaching elastic (initially flat) half-space and the summits exactly resemble that between an elastic half-space and a sphere.     

Statistical model of nearly complete elastic rough surface contact

In the area of homogeneous, isotropic, linear elastic rough surface normal contact, many classic statistical models have been developed which are only valid in the early contact when real area of contact is infinitesimally small, e.g., the Greenwood–Williamson (GW) model. In this article, newly developed statistical models, built under the framework of the (i) GW, (ii) Nayak–Bush and (iii) Greenwood’s simplified elliptic models, extend the range of application of the classic statistical models to the case of nearly complete contact. Nearly complete contact is the stage when the ratio of the real area of contact to the nominal contact area approaches unity. At nearly complete contact, the non-contact area consists of a finite number of the non-contact regions (over a finite nominal contact area). Each non-contact region is treated as a mode-I “crack”. The area of each non-contact region and the corresponding trapped volume within each non-contact region are determined by the analytical solutions in the linear elastic fracture mechanics, respectively. For a certain average contact pressure, not only can the real area of contact be determined by the newly developed statistical models, but also the average interfacial gap. Rough surface is restricted to the geometrically-isotropic surface, i.e., the corresponding statistical parameters are independent of the direction of measurement. Relations between the average contact pressure, non-contact area and average interfacial gap for different combinations of statistical parameters are compared between newly developed statistical models. The relations between non-contact area and average contact pressure predicted by the current models are also compared with that by Persson’s theory of contact. The analogies between the classic statistical models and the newly developed models are also explored.     

Modeling slow deformation of polygonal particles using DEM

We introduce two improvements in the numerical scheme to simulate collision and slow shearing of irregular particles. First, we propose an alternative approach based on simple relations to compute the frictional contact forces. The approach improves efficiency and accuracy of the Discrete Element Method （DEM） when modeling the dynamics of the granular packing. We determine the proper upper limit for the integration step in the standard numerical scheme using a wide range of material parameters. To this end, we study the kinetic energy decay in a stress controlled test between two particles. Second, we show that the usual way of defining the contact plane between two polygonal particles is, in general, not unique which leads to discontinuities in the direction of the contact plane while particles move. To solve this drawback, we introduce an accurate definition for the contact plane based on the shape of the overlap area between touching particles, which evolves continuously in time.     

An electrically conducting interface crack with a contact zone in a piezoelectric bimaterial

A plane problem for an electrically conducting interface crack in a piezoelectric bimaterial is studied. The bimaterial is polarized in the direction orthogonal to the crack faces and loaded by remote tension and shear forces and an electrical field parallel to the crack faces. All fields are assumed to be independent of the coordinate co-directed with the crack front. Using special presentations of electromechanical quantities via sectionally-analytic functions, a combined Dirichlet–Riemann and Hilbert boundary value problem is formulated and solved analytically. Explicit analytical expressions for the characteristic mechanical and electrical parameters are derived. Also, a contact zone solution is obtained as a particular case. For the determination of the contact zone length, a simple transcendental equation is derived. Stress and electric field intensity factors and, also, the contact zone length are found for various material combinations and different loadings. A significant influence of the electric field on the contact zone length, stress and electric field intensity factors is observed. Electrically permeable conditions in the crack region are considered as well and matching of different crack models has been performed.     

Modeling slow deformation of polygonal particles using DEM

We introduce two improvements in the numerical scheme to simulate collision and slow shearing of irregular particles. First, we propose an alternative approach based on simple relations to compute the frictional contact forces. The approach improves efficiency and accuracy of the Discrete Element Method (DEM) when modeling the dynamics of the granular packing. We determine the proper upper limit for the integration step in the standard numerical scheme using a wide range of material parameters. To this end, we study the kinetic energy decay in a stress controlled test between two particles. Second, we show that the usual way of defining the contact plane between two polygonal particles is, in general, not unique which leads to discontinuities in the direction of the contact plane while particles move. To solve this drawback, we introduce an accurate definition for the contact plane based on the shape of the overlap area between touching particles, which evolves continuously in time.     

The axisymmetric partial slip contact problem of a graded coating

In this paper, the axisymmetric contact problem with the partial slip condition for a functionally graded coated half-space which is indented by a rigid spherical indenter is considered. The material properties are assumed to vary along the thickness of the coating. A series of linear functions of the thickness are used to model the functionally graded coating with the arbitrarily varying shear modulus. The contact problem is formulated in terms of a set of Cauchy singular integral equations by employment of Hankel integral transforms technique and transfer matrix method. By using the uncoupled solution and the coupled solution,the coupled equations are solved. The effect of the coating’s gradient on the normal and radial tractions in the whole contact region is presented. The results show that the contact tractions and the size of the stick zone can be altered by adjusting the gradient of the coating. This may have potential applications in the resistance of contact deformation and damage.