Adhesive Contact Problem Research Articles

A Generalized Model for Adhesive Contact Between a Rigid Cylinder and a Transversely Isotropic Substrate

In this paper, a solution to the quasi-static adhesive contact problem between a rigid cylinder and a transversely isotropic substrate is extended to the most general case by taking adhesion hysteresis into account. An analytical solution to the contact stress is obtained by solving the integral equations established on the basis of the Green's function for the two-dimensional transversely isotropic half-space problem. By using equilibrium conditions and Griffith's criterion, the adhesion force and resistant moment to rolling are determined as functions of contact geometries and material properties of the contacting solids. Detailed discussions on the adhesion force and resistant moment are presented for some specific cases, revealing adhesion behaviors that have not been predicted by previous models. As the most generalized solution to the discussed problem, our results would have extensive applications in predicting the adhesion behavior between solids undergoing sophisticated mechanical loadings.

On the two-dimensional solution of both adhesive and non-adhesive contact problems involving functionally graded materials

Abstract This paper presents a semi-analytical algorithm for the determination of the contact half width and surface pressure which results from both adhesive and non-adhesive contact problems involving functionally graded materials (FGM). The inhomogeneously elastic solid comprises a graded elastic coating whose shear modulus depends exponentially on the vertical coordinate and a homogeneously elastic substrate. The solid is assumed to be in a state of plane strain and thus a two-dimensional analysis is performed within this work. Using the work of Chidlow et al. (2011a) as a starting point, we derive a pair of integral equations which may be used to determine approximations to the contact pressure when either the surface deflection or the deflection gradient is known over the contact region. As these integral equations are non-singular, we use Galerkin's method to approximate the contact pressure and it is found that relatively small trial spaces allow accurate computation of the pressure. Information about the prescribed load is then used to formulate an iterative algorithm to determine the contact half width. A selection of numerical results are presented using this method and it is found that the solutions computed here compare favourably with those of other authors. A further investigation is then conducted into the solution of adhesive contact problems using the assumptions of Maugis (1992) and Johnson and Greenwood (2008) to inform the nature of the adhesive stresses outside of the contact. It is found that both JKR-like and DMT-like behaviour can be observed in contact problems involving FGMs.

On the reduction method of dimensionality: The exact mapping of axisymmetric contact problems with and without adhesion

Starting from the classical theory of contact mechanics it is shown that the relationship between load, penetration and contact radius of any axisymmetric contact can be mapped exactly on a one-dimensional system, thus the reduction method of dimensionality is valid for conforming and non-conforming contacts. Furthermore the reduction method has been successfully extended to adhesive contact problems. The mapping of the classical theory of Johnson, Kendall and Roberts derived for spherical contacts as well as its application to axisymmetric contacts of arbitrary shape is possible; all results are reproduced precisely.

Adhesive full stick contact of a rigid cylinder with an elastic half-space

The problem of adhesive contact of a rigid cylinder with an elastic half-space is considered. The proposed adhesive contact model differs from the Johnson–Kendall–Roberts (JKR) model by preserving the influence of the contact shear stresses in the problem formulation and considering the so-called full stick contact due to the large values of the friction coefficient between contacting surfaces, as opposed to the frictionless contact assumed in the JKR model. An analytical treatment of the problem is presented, with the corresponding boundary-value problem formulated in the bipolar coordinates. A general solution in the form of Papkovich–Neuber functions and the Fourier integral transform is used to obtain an exact solution to the formulated boundary-value problem. Comparison of the results with the JKR model shows that accounting for the contact shear stresses leads to smaller contact areas as compared to those predicted by the JKR model.

Influence of complete sticking on the indentation of a rigid cone into an elastic half-space in the presence of molecular adhesion

The problem of the indentation of an absolutely stiff cone into an isotropic elastic half-space when there is molecular adhesion between the contacting bodies is investigated. An exact solution of the problem is given within the Johnson–Kendall–Roberts model under the condition of complete sticking (no-slip condition) between the bodies. The solution is compared with the solution of the same adhesive contact problem in the complete absence of friction.

An easy-to-implement numerical simulation method for adhesive contact problems involving asymmetric adhesive contact

A novel numerical method to solve asymmetric adhesive contact problems in rectangular coordinates has been developed. Surface interaction is modelled using an interface potential, deformation is coupled using Green's functions for a half space, and the resulting system of equations is solved by a relaxation technique. The method can handle arbitrary surface topography and properties. Compared with previous methods, this numerical scheme is much easier to implement and is just as accurate. Here, it is applied to two adhesive contact problems: one between a sphere and a cylinder; and the other between two identical cylinders in oblique contact. The numerical results reveal inaccuracies in elliptical contact theory when the skew angles between the two cylinders are small and the resulting contact is highly eccentric. The pull-off forces show an indiscernible decrease with decreasing value of the skew angle, which is quite different from the elliptical JKR theory. This technique can be used to solve adhesive contact problems that involve partial contact or complex geometry, such as rippled or rough surfaces.

Quasistatic delamination models for Kirchhoff‐Love plates

AbstractA quasistatic rate‐independent brittle delamination problem and also an adhesive unilateral contact problem is considered on a prescribed normally‐positioned surface in a plate with a finite thickness. By letting the thickness of the plate go to zero, two quasistatic rate‐independent crack models with prescribed path for Kirchhoff‐Love plates are obtained as limit of these quasistatic processes.

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 model of the adhesive component of the sliding friction force

► Energy is dissipated in approach-separation of two asperities. ► Energy loss due to adhesion is calculated. ► Friction is modelled based on energy losses for elementary contacts. ► Friction as a combined effect of roughness and adhesion. The adhesive contact problem is stated and solved for two axisymmetric elastic asperities of different shape. The energy dissipation in an approach-separation cycle is calculated and analyzed. A model is suggested to determine the adhesive component of the sliding friction force based on calculating the energy dissipation in formation and breaking of contacts between asperities in sliding of rough surfaces. The friction force is calculated and analyzed for various roughness parameters and values of the surface energy.

A New Numerical Approach for Adhesive Contacts of Real Engineering Surfaces

Microelectromechanical system (MEMS) and nanotechnology are important directions on the development of the science in twenty-first century. Some of the effects, such as viscous force, surface force, electrostatic force, friction etc., which can be usually ignored on the traditional scale, have become noticeable when the scale has turn to micro or nano scale. Nanotribology is one of the main areas of the indispensable researches on the basic theory and methodology of the effects. The micro/nano adhesive contact which is the foundation of nanotribology is studied in this paper. The earliest study on adhesive contact was done by Bradley. He presented an expression of adhesive force of two contacting rigid spheres. Derjaguin, Muller and Toprov (DMT) gave the relation of the contact area and the applied load of the adhesive contact of two spheres, but they did not consider the elastic deformation due to the adhesive force of the bodies. Johnson, Keudall and Roberts (JKR) provided a theory of the adhesive contact of two elastic spheres. Tabor gave a parameter (Tabor parameter) to interpret the ratio of the elastic deformation with the adhesive force of two contacting bodies. That is to say the DMT model corresponding to small Tabor parameter(<0.1) and the JKR model to large Tabor parameter(>5). Maguis gave a DMT-JKR transition using the Dugdale model in fracture mechanics (M-D model) in the intermediate region between the DMT model and the JKR model. A numerical algorithm of elastic adhesive contact based on the meshless method is presented in this paper. This make it possible to solve the adhesive contact with more complex surface topography and to consider more intricate factors, such as thermal stress, friction, elasto-plastic deformation etc. in the further studies on micro/nano scale adhesive contact problems. The meshless method seems to be a promising approach for contact analyses because of its flexibility in domain descritization and versatility in node arrangements. It can be used to solve a variety of complicated engineering problems. A numerical example of adhesive contact between a micro elastic cylinder and a rigid half-space is carried out to show the feasibility of the algorithm. In the simulation, an effective method of the M-D model is used to save the cost of computation. Compared with the existed solutions, the results solved by the presented algorithm are reasonable.

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.

A note on JKR and DMT theories of contact on a transversely isotropic half-space

In this paper the adhesive contact problem for transversely isotropic materials is discussed. A spherical contact on transversely isotropic, adhesive half-space is studied. Solving an equivalent external circular crack problem in a transversely isotropic body, the modified JKR and DMT theories are developed. The results show that the modified JKR and DMT contact theories for transversely isotropic materials have same form as JKR and DMT expressions for isotropic materials, except for the elastic constant K in the formula, which has a different expression for different materials.

On the applicability of different adhesion models in adhesive particulate flows

An adhesion map provides quantitative criteria for the appropriate selection of adhesion models applicable to a specific adhesive contact problem of fine particles in complex particulate flows. In this paper, three different general adhesion models are used to construct adhesion maps. The applicable regimes on the adhesion map for different approximate adhesion models are determined according to their underlying limitations. It is found that the choice of general model has limited influence on the structure of a constructed adhesion map. On the contrary, the regime of application for each approximate model is sensitive to the approximation level. A three-dimensional, more intuitive adhesion map based on physical parameters of particles is also built. Finally, recent applications of adhesion models in discrete element method (DEM) investigations of fine-particle flow dynamics are briefly discussed.

Adhesive contact of elastic spheres revisited: numerical models and scaling

For the purpose of determining the relevant scaling, the fundamental and widely applicable problem of adhesive contact between elastic solids is revisited. A comprehensive and accurate finite-element modelling is undertaken. A local contact law, consistent with the current level of modelling, is used. The analysis of the results yields the following conclusions. For a broad range of physically reasonable contact laws, and for low values of the Tabor parameter, a simple modification of adhesive range entering in the Tabor parameter allows for one-parameter scaling of the problem. For high values of the modified Tabor parameter, the problem requires description in terms of two non-dimensional parameters, one of which represents the magnitude of the contact surface stretch. The contact surface stretch correction is significant for a wide range of problems with spheres smaller than the threshold size, which, for a broad range of materials, is 300 nm to 100 μm, depending on the adhesive energy and elastic compressibility.

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.

The adhesive contact problem for a piezoelectric half-space

The problem of adhesive contact for a piezoelectric half-space is considered. The problem is reduced to the solution of two coupled integral equations and one given separately. These equations are solved exactly. Explicit expressions are found for the contact stresses, the displacement outside the contact region and for electric physical quantities. The numerical results presented for some piezoelectric materials show that the influence of anisotropy on the analysed physical quantities is significant.

Hybrid FEM/HTA analysis of friction contact in elastic and elastoplastic plane stress problems

The Hopfield–Tank neural network evolutionary equations (called for short HTA – HT Analog) were modified for the analysis of friction contact interface. The modified HTA was used for modeling a function in the hybrid program FEM/HTA and applied to the analysis of two numerical examples. The first example, taken from [Mistakidis ES, Panagiotopoulos PG. Numerical treatment of the nonmonotone (zig-zag) friction and adhesive contact problems with debounding: approximation by monotone subproblems. Comput Struct 1993;(47):33–46], is related to the analysis of nonmonotone (zig-zag) contact of elastic strips. The second example concerns the tension of elastoplastic strip with a rigid bolt inserted into a circular hole of the strip. The example generalizes the analysis performed in [Pabisek E. Neural analysis of elastoplastic plane stress problem with unilateral constraints. Comput Assis Mech Eng Sci, in press], considering Coulomb’s friction contact in the bolt–hole inference.

Adhesive contact problem of axisymmetric miniature circular plates with central rigid bump

Considering the adhesive effect and geometric nonlinearity, the adhesive contact between an elastic substrate and a clamped miniature circular plate with two different central rigid bumps under the action of uniform transverse pressure and in-plane tensile force in the radial direction was analyzed. And an analytical solution is presented by using the perturbation method. The relation of surface adhesive energies with critical load to detach the contacted surfaces is obtained. In the numerical results, the effects of adhesive energy, in-plane tensile force, rigid bump size and contact radius on the critical load are discussed, and the relation of critical contact radius with the gap between the central rigid bump and the substrate for different adhesive energies is investigated.

Effect of large deformation and material nonlinearity on the JKR (Johnson–Kendall–Roberts) test of soft elastic materials

AbstractWe study in detail the effect of large deformation and material nonlinearity on the JKR (Johnson–Kendall–Roberts) theory of adhesive contact for two systems. The first is a Neo‐Hookean hemisphere in adhesive contact with a smooth rigid substrate. The second is a smooth rigid spherical indenter in adhesive contact with a Neo‐Hookean half space. We show that our results are special cases of a general theory that models large deformation adhesive contact of spherical lenses. This theory shows that the solution of any large deformation JKR (LDJKR) problem can be obtained from the solution of a corresponding large deformation Hertz (LDH) problem. Using this theory, we extend the small strain JKR theory to the large deformation regime, the only restriction being that the materials are nonlinear elastic or hyperelastic. The adhesive contact problem for the two systems is solved using two methods. In method one, the LDJKR theory is obtained using finite element simulation results for a corresponding LDH problem; in method two, we solve the adhesive contact problems directly using a cohesive zone model to quantify adhesive interaction. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2912–2922, 2006

Evaluation of elastic modulus of materials by adhesive (no–slip) nano–indentation

Results of Mossakovskii's analysis for the elastic, adhesive (no–slip) contact between a rigid, axisymmetric punch and an isotropic, elastic half–space are developed in order to show that a simple relation exists between the contact stiffness, the contact area and the elastic modulus. The relation is similar to the Bulychev–Alekhin–Shorshorov equation, which is commonly used for evaluation of elastic modulus of materials by nano–indentation. The final formula differs from the frictionless case by a factor that depends on the Poisson ratio of the material. The bounds for the values of the factor are estimated. Similar to the frictionless analysis performed by Pharr, Oliver and Brotzen, the relation is not dependent on the geometry of the punch.