Adhesive Contact Problem Research Articles

Weak solvability of elliptic variational inequalities coupled with a nonlinear differential equation

In this paper we establish existence, uniqueness, and boundedness results for an elliptic variational inequality coupled with a nonlinear ordinary differential equation. Under the general framework, we present a new application modeling the antiplane shear deformation of a static frictional adhesive contact problem. The adhesion process has been extensively studied, but it is usual to assume a priori that the intensity of adhesion is bounded by introducing truncation operators. The aim of this article is to remove this restriction.The proof is based on an iterative approximation scheme showing that the problem has a unique solution. A key ingredient is finding uniform a priori bounds for each iterate. These are obtained by adapting versions of the Moser iteration to our system of equations.

Stochastic process model for interfacial gap of purely normal elastic rough surface contact

In purely normal elastic rough surface contact problems, Persson’s theory of contact shows that the evolution of the probability density function (PDF) of contact pressure with the magnification is governed by a diffusion equation. However, there is no partial differential equation describing the evolution of the PDF of the interfacial gap. In this study, we derive a convection–diffusion equation in terms of the PDF of the interfacial gap based on stochastic process theory, as well as the initial and boundary conditions. A finite difference method is developed to numerically solve the partial differential equation. The predicted PDF of the interfacial gap agrees well with that by Green’s Function Molecular Dynamics (GFMD) and other variants of Persson’s theory of contact at high load ranges. At low load ranges, the obvious deviation between the present work and GFMD is attributed to the overestimated mean interfacial gap and oversimplified magnification-dependent diffusion coefficient used in the present model. As one of its direct application, we show that the present work can effectively solve the adhesive contact problem under the DMT limit. The current study provides an alternative methodology for determining the PDF of the interfacial gap and a unified framework for solving the complementary problem of random contact pressure and random interfacial gap based on stochastic process theory.

The adjustable adhesion strength of multiferroic composite materials via electromagnetic loadings and shape effect of punch

Tunable and reversible dry adhesion possess great potential in a wide range of applications including transfer printing, climbing robots, wearable devices/electronics, and gripping in pick-and-place operations. Multiferroic composite materials offer new routines and approaches to achieve tunable adhesion due to their multi-field coupling effects. In this paper, the classical Johnson-Kendall-Roberts (JKR) adhesion model is extended to investigate the adhesive contact problem of a multiferroic composite half-space indented by an axisymmetric power-law shaped punch, whose shape index is denoted by n. The JKR-n adhesion models under the action of the power-law shaped punches with four different electromagnetic properties are set up by means of the total energy method. The explicit analytical expressions relating the indentation load and indentation depth to the contact radius are obtained, which can include the existing results in open literature as special cases. The generalized Tabor parameter and the interfacial adhesion strength applicable to multiferroic composite materials are defined. The effects of the shape index and the electromagnetic loadings on adhesion behaviors are revealed. It is found that both of them have prominent influences on the relationships among the indentation load, indentation depth and contact radius, the contact radius and indentation depth at self-equilibrium state, and the critical contact radius and indentation depth at pull-off moment. The pull-off force under the action of the conducting spherical punch subjected to non-zero electromagnetic loadings is dependent on material properties, which is different from the classical JKR result. More importantly, our analysis indicates that the pull-off force and the interfacial adhesion strength can be adjusted via altering the electromagnetic loadings and the shape index of the punch, which provides new approaches to achieve tunable adhesion.

An Application of the Mossakovskii-Jäger Procedure for Solving Plane Strain Adhesive Contact Problems of Power-Law Graded Elastic Solids

An Application of the Mossakovskii-Jäger Procedure for Solving Plane Strain Adhesive Contact Problems of Power-Law Graded Elastic Solids

Sliding frictional and adhesive coupling contact analysis of FGPM layered half-space under an insulating indenter

This paper proposes an analysis model for the sliding frictional and adhesive coupling contact problem in the plane strain state between the FGPM layered half-space and an insulating indenter. The electro-mechanical properties of FGPM layer vary exponentially along the thickness direction. By applying the Fourier integral transformation and superposition principle to the governing boundary value problem, the general solution for the sliding frictional and adhesive coupling contact problem can be derived by using the Maugis type of adhesion theory and extended Amonton’s law of friction. A Cauchy singular integral equation is further derived for present problem which is then numerically solved. The primary aim of this paper is to provide insight into the likely behavior for the effect of the gradient index, friction coefficient and adhesion parameters on the surface electro-mechanical response of FGPM layered half-space. For given values of parameters R, w, βh and μ, when value of σ0 decreases from infinity to zero there is a continuous transition from the JKR approximation to the DMT approximation. The research not only helps to further understand the frictional and adhesive damage mechanisms of MEMS devices composed of FGPM, but also provides reference basis for FGPM layer experimental analysis and intelligent structure design.

Analysis of adhesive contact of heterogeneous elastic materials

This paper reports a three-dimensional numerical model to solve adhesive elastic contact problem of heterogeneous materials with an engineered rough surface. This model integrates the Maugis-Dugdale (MD) adhesive contact model, the Boussinesq-Cerruti integral equations, the equivalent inclusion method (EIM), and the discrete convolution and fast Fourier transform (DC-FFT) algorithm, aiming at accurate determination of contact results for heterogeneous materials. An adhesive displacement-driven conjugate gradient method (CGM) was also developed to support rapid solutions of contact pressure and contact area. Four cases were analyzed to verify this model with satisfaction. This model was used to analyze the effects of inhomogeneity (location, size, and material properties) and adhesion on micro-adhesive-elastic contact, and the results lead to a curve-fitting equation for quick evaluation of the effects of inhomogeneity properties and adhesion characteristics on normal traction. It was also utilized to study the influence of the roughness level of a type of rough surface on the surface adhesion behavior.

Analysis of a doubly-history dependent variational–hemivariational inequality arising in adhesive contact problem

This paper is devoted to studying a dynamic adhesive contact model where the body is composed of a viscoelastic material with long memory. The adhesion process is modeled by a bonding field on part of the contact surface and on which the contact and the friction are modeled by nonmonotone Clarke subdifferential conditions with adhesion. Meanwhile, on another part of the contact surface, the contact is comprised of a normal compliance contact condition while the memory effects of the obstacle are considered, the friction is modeled by a version of Coulomb's law of dry friction and the friction bound is dependent on the normal stress. The variational form of the model generates a system of a doubly-history dependent variational–hemivariational inequality as well as a differential equation. Then, on the basis of the proof of abstract variational–hemivariational inequality, the existence and uniqueness results of the contact problem are established.

A custom arc‐length Finite Element solver for large deformation adhesive contacts using a k‐d tree accelerated volumetric interaction scheme

AbstractThis paper introduces a Finite Element framework for large deformation adhesive contact between elastic bodies based on volume‐to‐volume interaction. A key component of this framework is a novel, customized arc‐length algorithm designed to follow the solution path through multiple snap‐through and snap‐back instabilities. Another new component is a k‐d tree based algorithm which significantly accelerates the computationally expensive volume‐to‐volume interaction calculations. This optimized Finite Element framework is then used to solve two‐body and three‐body elastic adhesive contact problems with realistic material and interaction parameters, using a 6‐12 Lennard‐Jones interaction potential. The improved efficiency of the k‐d tree based scheme makes it indispensable for interaction calculations. This paper also shows the importance of implementing adhesion‐specific customizations in the arc‐length solver.

Analysis and simulation of an adhesive contact problem governed by fractional differential hemivariational inequalities with history-dependent operator

The main idea of this manuscript is to study an adhesive contact problem with long memory which is governed by a hemivariational inequality and a fractional differential equation. We first prove the existence of a unique solution of the fractional differential hemivariational inequality system. Subsequently, we consider a fully discrete scheme of this system and then focus on deriving error estimates for numerical solutions. To the tail of this manuscript, we present two numerical simulation examples for the adhesive contact problem, which provide numerical evidence to support our theoretical predictions.

The adhesive contact problem for a piecewise-homogeneous orthotropic plate with an elastic patch

A piecewise-homogeneous elastic orthotropic plate, reinforced with a finite patch of the wedge-shaped, which meets the interface at a right angle and is loaded with tangential and normal forces is considered. Using methods of the theory of analytic functions, the problem is reduced to the system of singular integro-differential equations (SIDE) with fixed singularity. Under tension-compression of patch using an integral transformation a Riemann problem is obtained, the solution of which is presented in explicit form. The tangential contact stresses along the contact line are determined and their asymptotic behavior in the neighborhood of singular points is established.

Revisiting the contact splitting hypothesis: An effective route for enhancing adhesion on rough surface

The contact splitting hypothesis (CSH) has been proposed for around 20 years, which suggests that the patterned or fibrillar surfaces enable very efficient biological attachment systems (e.g., Geckos’ seta). However, there is still a debate in academic community on the necessity of the CSH in man-made adhesives, since experiments have indicated that the non-fibrillar surface can also show strong adhesion. This study demonstrates that the surface roughness plays a key role in evaluating the influence of CSH on adhesion by a simple finite element (FE) model. The highly split structures are essentially required for strong adhesion on highly rough surface, but may in turn cause slight adhesion reduction on smooth surface due to the areal loss in the splitting process. With our results, the debate in CSH is explained as an incomplete understanding of an adhesive contact problem, in which the surface roughness is overlooked. Our results further sheds light on the natural selection that climbing animals with heavier body usually evolve finer adhesive structures on their pads, by considering both the surface roughness and its length scale dependence.

Adhesive depth-sensing indentation tests: Slopes of the force–displacement curves

Depth-sensing indentation experiments are a very important tool for estimating mechanical properties of modern materials. A review of various aspects of contact problems that used as the theoretical basis for interpretation of depth-sensing nanoindentation experiments is presented. Usually, analytical treatment of the indentation tests is based on analysis of the slopes of the force–displacement curves according to the non-adhesive Hertz contact theory. However, molecular adhesion is crucially important for many physical processes at the micro/nano-scales. Here, depth-sensing indentation techniques are reviewed and analyzed using the recent results obtained for adhesive contact problems. Fundamental relations for adhesive nanoindentation tests are derived for both frictionless and no-slip boundary conditions within the contact region. It is argued that the adhesive effects may be very important for treatment of the nanoindentation tests because the slopes of the force–displacement curves may considerably differ from the slopes derived using the non-adhesive contact theory.

Axisymmetric adhesive contact of multi-layer couple-stress elastic structures involving graded nanostructured materials

Graded nanostructured (GNS) materials has attracted widespread attention in the recent years. The gradually changed microstructures and mechanical properties from the surface to the interior layer in graded nanostructured materials can improve the mechanical properties of the surface. The motivation of this research was to explore the effect of size effect, adhesion and gradient on the mechanical properties in contact between GNS coating and spherical indenter. The multi-layer couple-stress elastic structures is used to simulate the GNS coating with arbitrarily varying material properties. Based on the Maugis-Dugdale theory, the axisymmeric adhesive contact problem for multi-layer couple-stress elastic coated substrate system indented by spherical punch is converted to singular integral equations. Numerically calculation are conducted to obtain the influence of the characteristic material length, adhesion parameter and gradient index on the indentation, surface stress, surface displacement, total attraction force. The present results should be helpful for the understanding of the micro contact mechanics of GNS coating.

Numerical solution of the adhesive rubber-solid contact problem and friction coefficients using a scale-splitting approach

We present a comprehensive investigation of the adhesive rubber-solid contact problem by analytical and numerical methods. Theories of rubber friction are reviewed and a new contact theory is developed. The adhesive contact problem is solved using the boundary element method. The introduction of adhesion leads to full coverage below a certain length scale. Friction coefficients are calculated from the spectral density of the rubber surface by splitting the numerical problem into two length scales. Adhesion is shown to increase friction at low velocities. The influence of filler is modeled by assuming that their size corresponds to a linear cross-over dimension separating the bulk elastomer from the composite. Finally, we discuss open problems and describe a simplified picture of rubber friction.

Solutions of a singular integro-differential equation related to the adhesive contact problems of elasticity theory

Abstract Exact and approximate solutions of a some type singular integro-differential equation related to problems of adhesive interaction between elastic thin half-infinite or finite homogeneous patch and elastic plate are investigated. For the patch loaded with vertical forces, there holds a standard model in which vertical elastic displacements are assumed to be constant. Using the theory of analytic functions, integral transforms and orthogonal polynomials, the singular integro-differential equation is reduced to a different boundary value problem of the theory of analytic functions or to an infinite system of linear algebraic equations. Exact or approximate solutions of such problems and asymptotic estimates of normal contact stresses are obtained.

A bi-potential contact formulation of orthotropic adhesion between soft bodies

An orthotropic adhesion model is proposed based on the bi-potential method to solve adhesive contact problems with orthotropic interface properties between hyperelastic bodies. The model proposes a straightforward description of interface adhesion with orthotropic adhesion stiffness, whose components are conveniently expressed according to the local coordinate system. Based on this description, a set of extended unilateral and tangential contact laws has been formulated. Furthermore, we use an element-wise scalar parameter \(\beta \) to characterize the strength of interface adhesive bonds, and the effects of damage. Therefore, complete cycles of bonding and de-bonding of adhesive links with the account for orthotropic interface effects can be modelled. The proposed model has been tested on cases involving both tangential and unilateral contact kinematics. The test cases allowed emergence of orthotropic interface effects between elastomer bodies involving hyperelasticity. Meanwhile, the model can be implemented with minimum effort, and provides inspiration for the modelling of adhesive interface effects in areas of applications such as biomechanics.

Theory of adhesive contact on multi-ferroic composite materials: Conical indenter

There is still a challenge to realize controllable attachment and detachment in the adhesive devices. New approaches of the controllable or reversible adhesion need to be developed in a wider area. In order to study the influence of the electric and magnetic quantities on the reversible adhesion, the adhesive contact problem between a half-space of the multi-ferroic composite material and a rigid conical indenter is investigated. The classical Johnson–Kendall–Roberts (JKR) and Maugis–Dugdale (MD) models are extended to the framework of magneto-electro-elasticity. The corresponding physical fields are analytically obtained. With the help of the Griffith energy balance relations, the stable equilibrium states of the adhesive contact are established. The corresponding indentation forces and the penetration depths for the JKR and MD models are derived in the closed-form. It is found that the pull-out forces associated with the JKR model depend on the electric potential and magnetic potential in certain cases. The validity is discussed by comparing the obtained solutions with the experimental data in the context of the elasticity. Numerical results show that the adhesive contact behaviors may be adjusted and controlled by changing the electric potential, the magnetic potential and the half apex angle of the rigid conical indenter. The present analytical results not only are the theoretical fundament for the forthcoming indentation experiments at the micro/nanoscopic scale, surface force microscopy (SFM) and atomic force microscopy (AFM), but also serve as guiding bionic design of the robots and grippers with reverse adhesion.

Formulation and existence of weak solutions for a problem of adhesive contact with elastoplasticity and hardening

This paper presents the weak formulation of a quasi-static evolution model for two deformable bodies with uni-directional adhesive unilateral contact on which external loads act. Small deformations and linearized elastoplasticity with hardening are assumed. The adhesion component is rate-dependent or rate-independent according to the choice of the viscosity coefficient of the glue; elastoplasticity is considered rate-independent. The weak formulation is expressed as a doubly non-linear problem with unbounded multivalued operators, as a function of internal and boundary displacements, plastic and symmetric strain tensors, and the bonding field and its gradient. This paper differs from other formulations by coupling the equations defined inside and on the boundary of the solids in functional form. In addition to this novelty, we verify the existence of solutions by a path other than that displayed in similar articles. The existence of solutions is demonstrated after considering a succession of doubly non-linear problems with an unbounded operator, and verifying that the solution of one of the problems is also a solution to the objective problem. The proof is supported by previous results from non-linear Partial differential equations theory with monotone operators.

Explicit transformation between non-adhesive and adhesive contact problems by means of the classical Johnson-Kendall-Roberts formalism.

The classic Johnson-Kendall-Roberts (JKR) contact theory was developed for frictionless adhesive contact between two isotropic elastic spheres. The advantage of the classical JKR formalism is the use of the principle of superposition of solutions to non-adhesive axisymmetric contact problems. In the recent years, the JKR formalism has been extended to other cases, including problems of contact between an arbitrary-shaped blunt axisymmetric indenter and a linear elastic half-space obeying rotational symmetry of its elastic properties. Here the most general form of the JKR formalism using the minimal number of a priori conditions is studied. The corresponding condition of energy balance is developed. For the axisymmetric case and a convex indenter, the condition is reduced to a set of expressions allowing explicit transformation of force-displacement curves from non-adhesive to corresponding adhesive cases. The implementation of the developed theory is demonstrated by presentation of a two-term asymptotic adhesive solution of the contact between a thin elastic layer and a rigid punch of arbitrary axisymmetric shape. Some aspects of numerical implementation of the theory by means of Finite-Element Method are also discussed. This article is part of a discussion meeting issue 'A cracking approach to inventing new tough materials: fracture stranger than friction'.

Contact probing of prestressed adhesive membranes of living cells.

Atomic force microscopy (AFM) studies of living biological cells is one of main experimental tools that enable quantitative measurements of deformation of the cells and extraction of information about their structural and mechanical properties. However, proper modelling of AFM probing and related adhesive contact problems are of crucial importance for interpretation of experimental data. The Johnson-Kendall-Roberts (JKR) theory of adhesive contact has often been used as a basis for modelling of various phenomena including cell-cell interactions. However, strictly speaking the original JKR theory is valid only for contact of isotropic linearly elastic spheres, while the cell membranes are often prestressed. For the first time, effects caused by molecular adhesion for living cells are analytically studied taking into account the mechanical properties of cell membranes whose stiffness depends on the level of the tensile prestress. Another important question is how one can extract the work of adhesion between the probe and the cell. An extended version of the Borodich-Galanov method for non-direct extraction of elastic and adhesive properties of contacted materials is proposed to apply to experiments of cell probing. Evidently, the proposed models of adhesive contact for cells with prestressed membranes do not cover all types of biological cells because the structure and properties of the cells may vary considerably. However, the obtained results can be applied to many types of smooth cells and can be used to describe initial stages of contact and various other processes when effects of adhesion are of crucial importance. This article is part of a discussion meeting issue 'A cracking approach to inventing new tough materials: fracture stranger than friction'.