Adhesive Contact Problem Research Articles

Friction in an adhesive tangential contact in the Coulomb-Dugdale approximation

ABSTRACTWe study the problem of tangential frictional contact in the presence of adhesion. The model can be considered as a generalization of the theory by Cattaneo and Mindlin to the case where there are “long range adhesive interactions” between the contacting surfaces, which exert an additional pressure on the surfaces even in the absence of an external normal force. The adhesion forces are described by the Dugdale model and the tangential forces in the contact by Coulomb’s law of dry friction. These approximations allow obtaining an analytical solution for the tangential contact problem of a rigid parabolic indenter and a half-space. As in the case of nonadhesive contact, application of an arbitrarily small tangential force leads to slip in the narrow ring-shaped area near the contact boundary. Further increase in tangential load leads to a decrease in the radius of the stick region until sliding expands to the entire contact region. The obtained analytical solution shows that the main governing parameter of the problem is the parameter introduced by Maugis for the normal adhesive contact problem.

Johnson-Kendall-Roberts adhesive contact for a toroidal indenter.

The unilateral axisymmetric frictionless adhesive contact problem for a toroidal indenter and an elastic half-space is considered in the framework of the Johnson-Kendall-Roberts theory. In the case of a semi-fixed annular contact area, when one of the contact radii is fixed, while the other varies during indentation, we obtain the asymptotic solution of the adhesive contact problem based on the solution of the corresponding unilateral non-adhesive contact problem. In particular, the adhesive contact problem for Barber's concave indenter is considered in detail. In the case when both contact radii are variable, we construct the leading-order asymptotic solution for a narrow annular contact area. It is found that for a v-shaped generalized toroidal indenter, the pull-off force is independent of the elastic properties of the indented solid.

Periodic adhesive contact between an elastic anisotropic layer and a wavy rigid halfspace

The paper presents solutions to the periodic adhesive contact problem between an elastic anisotropic layer and a wavy rigid half-space. Different systems including rigid, soft substrate and infinite elastic layer with orientation effect are considered in the paper. The periodic Green function in the form of Fourier series is obtained to formulate an integral equation of the contact pressure. The relation between the contact pressure and the relative approach is derived analytically. The results show that the adhesion effect depends strongly on both material orientation of the elastic layer and the substrate stiffness. These results are in particular useful for the design of new tunable adhesive systems.

A simple method for solving adhesive and non-adhesive axisymmetric contact problems of elastically graded materials

An efficient method is presented for solving axisymmetric, frictionless contact problems between a rigid punch and an elastically non-homogeneous, power-law graded half-space. Provided that the contact area is simply-connected profiles of arbitrary shape can be considered. Moreover, adhesion in the framework of the generalized JKR-theory can be taken into account. All results agree exactly with those given by three-dimensional contact theories. The method uses the fact that three-dimensional contact problems can be mapped to one-dimensional ones with a properly defined Winkler foundation; hence, the method is to be understood as an extension of the method of dimensionality reduction (MDR). A prerequisite of its applicability forms the generality of contact stiffness regardless of the geometry of the axisymmetric profile, which is proved. All the necessary mapping rules are derived and their ease of use explained by a recent example.

Two Kinds of Contact Problems in Dodecagonal Quasicrystals of Point Group 12 mm

In this paper, two kinds of contact problems in 2-D dodecagonal quasicrystals were discussed using the complex variable function method: one is the finite frictional contact problem, the other is the adhesive contact problem. The analytic expressions of contact stresses in the phonon and phason fields were obtained for a flat rigid punch, which showed that: (1) for the finite frictional contact problem, the contact stress exhibited power-type singularities at the edge of the contact zone; (2) for the adhesive contact problem, the contact stress exhibited oscillatory singularities at the edge of the contact zone. The distribution regulation of contact stress under punch was illustrated; and the low friction property of quasicrystals was verified graphically.

Effects of surface tension on the adhesive contact between a hard sphere and a soft substrate

In the present paper, the adhesive contact between a rigid sphere and a compliant elastic substrate with surface tension is analyzed. Using the minimum potential energy principle and considering both the effects of surface tension and adhesion, explicit solutions are obtained for the contact radius and the indent depth for the case without external loading. It is found that surface tension evidently alters the pressure distribution in the contact region and tends to decrease both the contact radius and the indent depth. The present model establishes a bridge between the JKR model for adhesive contact problems and the Young-Laplace's law of surface tension.

A three‐dimensional surface stress tensor formulation for simulation of adhesive contact in finite deformation

SummaryA three‐dimensional surface adhesive contact formulation is proposed to simulate macroscale adhesive contact interaction characterized by the van der Waals interaction between arbitrarily shaped deformable continua under finite deformation. The proposed adhesive contact formulation uses a double‐layer surface integral to replace the conventional double volume integration to compute the adhesive contact force vector. Considering nonlinear finite deformation, we have derived the surface stress tensor and the corresponding tangent stiffness matrix in a Galerkin weak formulation. With the surface stress formulation, the adhesive contact problems are solved in the framework of nonlinear continuum mechanics by using the standard Lagrange finite element method. Surface stress tensors are formulated for both interacting bodies. Numerical examples show that the proposed surface contact algorithm is accurate, efficient, and reliable for three‐dimensional adhesive contact problems of large deformations for both quasi‐static and dynamic simulations. Copyright © 2015 John Wiley & Sons, Ltd.

Two kinds of contact problems in three-dimensional icosahedral quasicrystals

Two kinds of contact problems, i.e., the frictional contact problem and the adhesive contact problem, in three-dimensional (3D) icosahedral quasicrystals are discussed by a complex variable function method. For the frictional contact problem, the contact stress exhibits power singularities at the edge of the contact zone. For the adhesive contact problem, the contact stress exhibits oscillatory singularities at the edge of the contact zone. The numerical examples show that for the two kinds of contact problems, the contact stress exhibits singularities, and reaches the maximum value at the edge of the contact zone. The phonon-phason coupling constant has a significant effect on the contact stress intensity, while has little impact on the contact stress distribution regulation. The results are consistent with those of the classical elastic materials when the phonon-phason coupling constant is 0. For the adhesive contact problem, the indentation force has positive correlation with the contact displacement, but the phonon-phason coupling constant impact is barely perceptible. The validity of the conclusions is verified.

Fracture mechanics implications for apparent static friction coefficient in contact problems involving slip-weakening laws

We consider the effect of differing coefficients of static and dynamic friction coefficients on the behaviour of contacts involving microslip. The classic solutions of Cattaneo and Mindlin are unchanged if the transition in coefficients is abrupt, but if it occurs over some small slip distance, the solution has some mathematical similarities with those governing the normal tractions in adhesive contact problems. In particular, if the transition to dynamic slip occurs over a sufficiently small area, we can identify a ‘JKR’ approximation, where the transition region is condensed to a line. A local singularity in shear traction is then predicted, with a stress-intensity factor that is proportional to the square root of the local contact pressure and to a certain integral of the friction coefficient–slip distance relation. We can also define an equivalent of the ‘small-scale yielding’ criterion, which enables us to assess when the singular solution provides a good approximation. One consequence of the results is that the static coefficient of friction determined from force measurements in experiments is significantly smaller than the value that holds at the microscale.

Analysis of an adhesive contact problem for viscoelastic materials with long memory

The goal of this paper is to study a mathematical model which describes the adhesive contact between a deformable body and a foundation. The body consists of a viscoelastic material with long memory and the process is assumed to be quasistatic. The adhesive contact condition on the normal plane is modeled by a version of normal compliance condition with unilateral constraint in which adhesion is taken into account, the adhesive contact condition on the tangential plane is described by an adhesive Clarke subdifferential condition, and the evolution of the bonding field is described by an ordinary differential equation. We derive the variational formulation of this problem which is a system of a variational–hemivariational inequality for the displacement field and an ordinary differential equation for the bonding field. Then, we obtain existence and uniqueness results on abstract inclusions and abstract variational–hemivariational inequalities. Finally, we apply the abstract results to prove the existence of a unique weak solution to the contact problem.

Modelling adhesive contact problems involving a layered elastic solid and cylindrical indenter using Lennard Jones potential

This paper presents an iterative algorithm that solves for the displacement and sub-surface stresses induced within a layered elastic solid adhering to a rigid cylindrical indenter under lightly loaded conditions. The solid is assumed to comprise a functionally graded coating of finite thickness bonded to a homogeneous substrate of infinite extent and is assumed to be in a state of plane strain which allows a two-dimensional analysis to be performed. The Lennard–Jones potential is used to model the adhesive force acting between the indenter and solid whilst the effects of surface adhesion are characterised using Tabor’s parameter.A selection of numerical results for the adhesive contact problem are presented which indicate that the maximum pressure and induced sub-surface stresses increase dramatically as Tabor’s parameter increases. It is also found that the shear modulus and thickness of the coating have a significant effect on material behaviour with harder coatings experiencing significantly larger tensile stresses but smaller surface displacement than softer coatings. The present investigation allows us to deduce that at smaller scales, surface adhesion can be instrumental in causing wear or potential material failure if coatings are improperly designed.

Stress-driven local-solution approach to quasistatic brittle delamination

A unilateral contact problem between elastic bodies at small strains glued by a brittle adhesive is addressed in the quasistatic rate-independent setting. The delamination process is modeled as governed by stresses rather than by energies. This leads to a specific scaling of an approximating elastic adhesive contact problem, discretized by a semi-implicit scheme and regularized by a BV-type gradient term. An analytical zero-dimensional example motivates the model and a specific local-solution concept. Two-dimensional numerical simulations performed on an engineering benchmark problem of debonding a fiber in an elastic matrix further illustrate the validity of the model, convergence, and algorithmical efficiency even for very rigid adhesives with high elastic moduli.

Adhesive contact analysis for anisotropic materials considering surface stress and surface elasticity

In this study, an adhesive contact problem for anisotropic materials is analyzed by considering surface stress and surface elasticity. The displacement field on the surface is obtained from the surface Green’s function considering the surface stress and surface elasticity. The displacement due to the adhesive force, i.e., the van der Waals force, calculated from the Lennard–Jones potential is used in the analysis. The adhesive force is calculated from the distance between two surfaces. First, an adhesive contact problem of a rigid spherical indenter and an isotropic substrate with various material properties is analyzed under a condition in which no surface mechanical property is considered, and the results are compared with the Johnson–Kendall–Roberts theory in order to validate the calculation algorithm of the analysis. Next, a substrate with orthotropic properties is subjected to adhesive contact analysis. When the elastic modulus in the normal direction to the substrate surface increases, the maximum adhesion force increases, similar to the case of the isotropic substrate. However, when the elastic modulus in the tangential direction of the substrate surface is varied, the maximum adhesion force does not vary much. Finally, an anisotropic half-substrate is subjected to adhesive contact analysis considering the surface mechanical property. This analysis reveals that when the values of the surface mechanical property are varied, the maximum adhesion force changes, similar to the case of varying the elastic modulus in the tangential direction of the surface.

An adhesive contact problem for an incompressible non-homogeneous elastic halfspace

In this paper, we examine the axisymmetric adhesive contact problem for a rigid circular plate and an incompressible elastic halfspace where the linear elastic shear modulus varies exponentially with depth. The analytical solution of the mixed boundary value problem entails a set of coupled integral equations that cannot be solved easily by conventional integral transform techniques proposed in the literature. In this paper, we adopt a computational scheme where the contact normal and contact shear stress distributions are approximated by their discretized equivalents. The consideration of compatibility of deformations due to the indentation by a rigid indenter in adhesive contact gives a set of algebraic equations that yield the discretized equivalents of the contacts stresses and the axial stiffness of the medium.

The JKR-type adhesive contact problems for power-law shaped axisymmetric punches

The JKR (Johnson, Kendall, and Roberts) and Boussinesq–Kendall models describe adhesive frictionless contact between two isotropic elastic spheres, and between a flat-ended axisymmetric punch and an elastic half-space respectively. However, the shapes of contacting solids may be more general than spherical or flat ones. In addition, the derivation of the main formulae of these models is based on the assumption that the material points within the contact region can move along the punch surface without any friction. However, it is more natural to assume that a material point that came to contact with the punch sticks to its surface, i.e. to assume that the non-slipping boundary conditions are valid. It is shown that the frictionless JKR model may be generalized to arbitrary convex, blunt axisymmetric body, in particular to the case of the punch shape being described by monomial (power-law) punches of an arbitrary degree d≥1. The JKR and Boussinesq–Kendall models are particular cases of the problems for monomial punches, when the degree of the punch d is equal to two or it goes to infinity respectively. The generalized problems for monomial punches are studied under both frictionless and non-slipping (or no-slip) boundary conditions. It is shown that regardless of the boundary conditions, the solution to the problems is reduced to the same dimensionless relations among the actual force, displacements and contact radius. The explicit expressions are derived for the values of the pull-off force and for the corresponding critical contact radius. Connections of the results obtained for problems of nanoindentation in the case of the indenter shape near the tip has some deviation from its nominal shape and the shape function can be approximated by a monomial function of radius, are discussed.

A study of a unilateral and adhesive contact problem with normal compliance

The aim of this paper is to study a quasistatic unilateral contact problem between an elastic body and a foundation. The constitutive law is nonlinear and the contact is modelled with a normal compliance condition associated to a unilateral constraint and

A Unified Treatment of Axisymmetric Adhesive Contact on a Power-Law Graded Elastic Half-Space

In the present paper, axisymmetric frictionless adhesive contact between a rigid punch and a power-law graded elastic half-space is analytically investigated with use of Betti's reciprocity theorem and the generalized Abel transformation, a set of general closed-form solutions are derived to the Hertzian contact and Johnson–Kendall–Roberts (JKR)-type adhesive contact problems for an arbitrary punch profile within a circular contact region. These solutions provide analytical expressions of the surface stress, deformation fields, and equilibrium relations among the applied load, indentation depth, and contact radius. Based on these results, we then examine the combined effects of material inhomogeneities and punch surface morphologies on the adhesion behaviors of the considered contact system. The analytical results obtained in this paper include the corresponding solutions for homogeneous isotropic materials and the Gibson soil as special cases and, therefore, can also serve as the benchmarks for checking the validity of the numerical solution methods.

Similarity considerations in adhesive contact problems

The classical ‘JKR’ solution for the force required to separate two elastic spheres is independent of the elastic moduli. We demonstrate that this remarkable simplification is a rigorous consequence of the quadratic character of the spherical profile and hence that it must also apply exactly to more general quadratic profiles such as the contact of two ellipsoids. The same feature is also responsible for the fact that Bradley's ‘rigid sphere’ solution depends only on the interface energy and not on the detailed character of the adhesive force law.

Adhesive contact delaminating at mixed mode, its thermodynamics and analysis

An adhesive unilateral contact problem between visco-elastic heat-conductive 1 bodies in linear Kelvin-Voigt rheology is scrutinized. The flow-rule for debonding the adhesive is considered rate-independent, unidirectional, and non-associative due to dependence on the mixity of modes of delamination, namely of Mode I (opening) and of Mode II (shearing). Such mode-mixity dependence of delamination is a very pronounced (and experimentally confirmed) phenomenon typically considered in engineering models. An anisothermal, thermodynamically consistent model is derived, considering a heat-conductive viscoelastic material and the coupling via thermal expansion and adhesion-depending heat transition through the contact surface. We prove the existence of weak solutions by passing to the limit in a carefully designed semi-implicit time-discretization scheme.

An energy‐momentum‐conserving temporal discretization scheme for adhesive contact problems

SUMMARYNumerical solution of dynamic problems requires accurate temporal discretization schemes. So far, to the best of the authors’ knowledge, none have been proposed for adhesive contact problems. In this work, an energy‐momentum‐conserving temporal discretization scheme for adhesive contact problems is proposed. A contact criterion is also proposed to distinguish between adhesion‐dominated and impact‐dominated contact behaviors. An adhesion formulation is considered, which is suitable to describe a large class of interaction mechanisms including van der Waals adhesion and cohesive zone modeling. The current formulation is frictionless, and no dissipation is considered. Performance of the proposed scheme is compared with other schemes. The proposed scheme involves very little extra computational overhead. It is shown that the proposed new temporal discretization scheme leads to major accuracy gains both for single‐degree‐of‐freedom and multi‐degree‐of‐freedom systems. The single‐degree‐of‐freedom system is critically analyzed for various parameters affecting the response. For the multi‐degree‐of‐freedom system, the effect of the time step and mesh discretization on the solution is also studied using the proposed scheme. It is further shown that a temporal discretization scheme based on the principle of energy conservation is not sufficient to obtain a convergent solution. Results with higher order contact finite elements for discretizing the contact area are also discussed. Copyright © 2012 John Wiley & Sons, Ltd.