Axisymmetric Contact Research Articles

Adhesion of a rigid sphere to a freestanding elastic membrane with pre-tension

Abstract Adhesion between a solid sphere and a thin film is a common but crucial issue in the study of biological membranes and two-dimensional materials. To supplement quantitative knowledge of membrane adhesion, this work addresses the axisymmetric adhesive contact between a rigid sphere and a circular freestanding elastic membrane with initial radial tension. For the membranes following linear stretching elasticity, both the JKR- and DMT-type adhesion as well as the transition regime in-between are considered. The dependences of external force with respect to the contact radius and displacement are derived analytically. In essence, the general solution is governed by three dimensionless parameters, reflecting the effects of membrane stretching elasticity, the range of adhesion force and the membrane size, respectively. It is interestingly found that the membrane size does not affect the contact radius and displacement at zero-external force at all and has minor influence on the value of pull-off force. The presented closed form solutions might be useful for the understanding of adhesion behaviors of sphere-membrane systems.

Axisymmetric Hertzian contact problem accounting for surface tension and strain gradient elasticity

In this paper, we investigate the axisymmetric Hertzian contact problem at micro-/nanoscale. The deformation of material bulk is described by a simplified theory of strain gradient elasticity, and the influence of surface tension is integrated based on the surface elasticity theory. Using the Mindlin's potential function method and double Fourier integral transform, the normal surface displacement induced by a concentrated force is derived in a closed form. Following this, the contact between a rigid sphere and an elastic half-space is formulated in terms of singular integral equation, which is numerically solved by applying the Gauss-Chebyshev method. The results indicate that the distribution of contact pressure is distinctly different from that in classical elasticity theory. The indented substrate tends to perform stiffer due to the effects of surface tension and strain gradient elasticity. When the contact radius is comparable with the material length parameter, the indentation force (depth) can be ten (three) times of that given by classical Hertz theory.

Influence of Coating Deformation Law on Characteristics of Axisymmetric Contact Problem at Different Stages of Wear

Influence of Coating Deformation Law on Characteristics of Axisymmetric Contact Problem at Different Stages of Wear

Axisymmetric Contact Problem for a Homogeneous Space with a Circular Disk-Shaped Crack Under Static Friction

Axisymmetric Contact Problem for a Homogeneous Space with a Circular Disk-Shaped Crack Under Static Friction

Initiation of decohesion between a flat punch and a thin bonded incompressible layer

Non-axisymmetric frictionless JKR-type adhesive contact between a rigid body and a thin incompressible elastic layer bonded to a rigid base is considered in the framework of the leading-order asymptotic model, which has the form of an overdetermined boundary value problem. Based on the first-order perturbation of the Neumann operator in the Dirichlet problem for Poisson’s equation, the decohesion initiation problem is formulated in the form of a variational inequality. The asymptotic model assumes that the contact zone and its boundary contour during the detachment process are unknown. The absence of the solvability theorem is illustrated by an example of the instability of an axisymmetric flat circular contact.

On an indentation of a circular cylindrical punch into an elastic half‐space under friction

AbstractThe paper discusses an axisymmetric contact problem of indentation of a circular cylindrical punch with a flat base into a homogeneous elastic half‐space with friction. It is assumed that the shear contact stresses are related to the normal contact pressure by the law of dry friction, in which the coefficient of friction depends on the radial coordinates of the points of the surfaces in contact and is directly proportional to them. With the help of rotation operators, the solution of the stated problem is reduced to the solution of a singular integral equation of the second kind with variable coefficients, and its closed solution in quadratures is constructed. The numerical calculations display the patterns of change in contact stresses and rigid displacement of the punch depending on the maximum value of the friction coefficient.

Smith–Watson–Topper Parameter in Partial Slip Bimodal Oscillations of Axisymmetric Elastic Contacts of Similar Materials: Influence of Load Protocol and Profile Geometry

Based on a very fast numerical procedure for the determination of the subsurface stress field beneath frictional contacts of axisymmetric elastic bodies under arbitrary 2D oblique loading, the contact mechanical influences of loading parameters and contact profile geometry on the Smith–Watson–Topper (SWT) fatigue crack initiation parameter in elastic fretting contacts with superimposed normal and tangential oscillations are studied in detail. The efficiency of the stress calculation allows for a comprehensive physical analysis of the multi-dimensional parameter space of influencing variables. It is found that a superimposed normal oscillation of the contact can significantly increase or decrease the SWT parameter, depending on the initial phase difference and frequency ratio between the normal and tangential oscillation. Written in proper non-dimensional variables, the rounded flat punch always exhibits smaller values of the SWT parameter, compared to a full paraboloid with the same curvature, while the truncated paraboloid exhibits larger values. A small superimposed profile waviness also significantly increased or decreased the SWT parameter, depending on the amplitude and wave length of the waviness. While both the load protocol and the profile geometry can significantly alter the SWT parameter, the coupling between both influencing factors is weak.

A unified treatment of axisymmetric adhesive contact for piezoelectric materials

Nanoindentation technique has been widely applied to characterize the electromechanical properties of piezoelectric materials, where the adhesion effect induced by different surface forces becomes very prominent. The indenter tip should have more general profile rather than just three common simple shapes (flat, cone and sphere) in practical testing. In this study, the classical Johnson-Kendall-Roberts (JKR) and Maugis-Dugdale (M-D) models are extended to investigate the adhesion behaviors of a piezoelectric half-space indented by the power-law shaped punches with different electric properties, whose shape index is denoted as n. The JKR-n and M-D-n adhesion models for both the electrically conducting and electrically insulating punches are set up by means of Griffith energy balance. The Derjaguin-Muller-Toporov-n (DMT-n) adhesion solutions are derived from the corresponding M-D-n adhesion models as the limiting cases. The generalized Tabor parameter and interfacial adhesion strength applicable to piezoelectric materials are defined for the first time, with the former being used to describe the transition behavior from JKR-n model to DMT-n model. The impacts of the electric loading and the shape index on adhesion behaviors are discussed in detail. It is found that the pull-off force increases with the electric potential, which reveals the adhesion strengthening effect caused by electric loading. The shape index has a prominent influence on the pull-off force, interfacial adhesion strength, contact radius at pull-off moment, indentation depth and contact radius at self-equilibrium state, and the transition behavior from JKR-n model to DMT-n model. The results derived from this work not only are helpful to understand the contact behaviors of piezoelectric materials at micro/nano scale, but also provide a theoretical basis for nanoindentation technique in evaluating material properties of piezoelectric mediums.

A mechanically-derived contact model for adhesive elastic-perfectly plastic particles, Part I: Utilizing the method of dimensionality reduction

In this two part series (Zunker and Kamrin, 2024), we present a contact model able to capture the response of interacting adhesive elastic-perfectly plastic particles under a variety of loadings. In Part I, we focus on elastic through fully-plastic contact with and without adhesion. For these contact regimes the model is built upon the method of dimensionality reduction which allows the problem of a 3D axisymmetric contact to be mapped to a corresponding problem of a 1D rigid indenter penetrating a bed of independent Hookean springs. Plasticity is accounted for by continuously varying the 1D indenter profile subject to a constraint on the contact pressure. Unloading falls out naturally, and simply requires lifting the 1D indenter out of the springs and tracking the force. By accounting for the incompressible nature of this plastic deformation, the contact model is able to capture multi-neighbor dependent effects such as increased force and formation of new contacts. JKR type adhesion is recovered seamlessly within the method of dimensionality reduction by simply allowing the springs to ‘stick’ to the 1D indenter’s surface. Because of the mechanics-focused formulation of the contact model, only a few physical inputs describing the interacting particles are needed: particle radius, Young’s modulus, Poisson ratio, yield stress, and effective surface energy. The contact model is validated against finite element simulations and analytic theory—including Hertz’s contact law and the JKR theory of adhesion. These comparisons show that the proposed contact model is able to accurately capture plastic displacement, average contact pressure, contact area, and force as a function of displacement for contacts as well as particle volume within the elastic to fully-plastic regimes.

A General Approximate Solution for the Slightly Non-Axisymmetric Normal Contact Problem of Layered and Graded Elastic Materials

We present a general approximate analytical solution for the normal contact of layered and functionally graded elastic materials for almost axisymmetric contact profiles. The solution only requires knowledge of the corresponding contact solution for indentation using a rigid cylindrical flat punch. It is based on the generalizations of Barber’s maximum normal force principle and Fabrikant’s approximation for the pressure distribution under an arbitrary flat punch in an inhomogeneous case. Executing an asymptotic procedure suggested recently for almost axisymmetric contacts of homogeneous elastic media results in a simple approximate solution to the inhomogeneous problem. The contact of elliptical paraboloids and indentation using a rigid pyramid with a square planform are considered in detail. For these problems, we compare our results to rigorous numerical solutions for a general (bonded or unbonded) single elastic layer based on the boundary element method. All comparisons show the quality and applicability of the suggested approximate solution. Based on our results, any compact axisymmetric or almost axisymmetric contact problem of layered or functionally graded elastic materials can be reduced asymptotically to the problem of indenting the material using a rigid cylindrical flat punch. The procedure can be used for different problems in tribology, e.g., within the framework of indentation testing or as a tool for the analysis of local features on a rough surface.

Axisymmetric contact problem with regard for friction and wear

Axisymmetric contact problem with regard for friction and wear

Tangential contacts of three-dimensional power-law graded elastic solids: A general theory and application to partial slip

New technologies in additive manufacturing have enabled the cost-effective production of gradient materials with desired multifunctional properties. In mechanical engineering, such functionally graded materials (FGMs) are often used within mechanical joints, as they ensure higher strength as well as wear resistance and prevent fretting fatigue. However, due to the variations in their composition and structure, theoretical predictions of their contact-mechanical behavior are significantly complicated. In this work, a general theory for solving tangential contacts between 3D power-law graded elastic solids of arbitrary geometry is presented. For multiple contacts, such as those occurring between two nominally flat but rough half-spaces, the well-known Ciavarella–Jäger theorem is established accompanied by a discussion of tangential coupling. Nevertheless, the focus of the work is on axisymmetric single contacts under arbitrary unidirectional tangential loading, for which closed-form analytical solutions are derived based on the Mossakovskii–Jäger procedure. In comparison to the results of common approximate methods, the solutions include the non-axisymmetric components of tangential displacements, which are indispensable for the accurate determination of the relative slip components and thus the surface density of frictional energy dissipation in the partial slip regime. Although a simplified approach is used for the calculation of the dissipated energy density, the results in the limiting case of homogeneous material are in excellent agreement with those from a full numerical computation. As an application example, the complete solutions for the tangential contact of parabolically shaped power-law graded elastic solids in the partial slip regime are derived and the influence of the material gradient as well as Poisson’s ratio on the surface density of dissipated energy is investigated.

Simulation based analysis to study the effect of compression ring crown height in ring liner assembly on the performance of four stroke petrol engine

Among different piston rings of an Internal Combustion Engine (IC), the top compression ring-liner assembly shared a momentous amount of the total frictional losses, particularly at the Top Dead Center (TDC) and Bottom Dead Center (BDC) where boundary lubrication subsists. In the present study, the lubrication mechanism in piston ring cylinder liner conjunction for top compression ring using one dimensional Reynolds equation assuming axisymmetric contact was studied. The hydrodynamic and mixed lubrication regimes were taken during the study. Gumbel boundary conditions were used for Reynolds equation solution. For boundary pressure calculation, first, combustion chamber pressure variation was determined, then, interring pressure was calculated assuming orifice volume method by solving mass flow rate through crevice volumes using Runge-Kutta 4th order method. The Reynolds equation was solved using finite difference method. Whole solution process was repeated for different engine speed and crown height of top compression ring to study the effect of them in wear and power loss. First, the solution was found for 3000, 5000 and 7500 RPM keeping constant crown height of 10 µm. Average power loss was found to be increased with the increasing engine speed. There was increased in oil film thickness around mid-stroke with the increased engine speed, but there was no significant change in film thickness with the engine speed at the TDC and BDC positions. Then, the solution was again repeated for different RPM varying the crown height. The compression ring with lower crown height showed the better wear performance by increasing the minimum film thickness in critical zones of engine cycle for all RPM. The crown height of 5,7 and 9 µm gave the minimum power loss (77.89, 163.47 and 297.33 W) for 3000, 5000 and 7500 rpm. Moreover, the power loss in the range of 5-9 µm had no significant differences.

Axisymmetric contact problem of piezoelectric coating-substrate system with functionally graded piezoelectric interfacial layer

In this paper, the electromechanical response of piezoelectric coating-functionally graded piezoelectric (FGP) interfacial layer–piezoelectric substrate system under the rigid spherical punch is considered. It is assumed that the piezoelectric coating and piezoelectric substrate are connected by the FGP interfacial layer which is simulated by a multi-layer model. The axisymmetric frictionless contact problem of piezoelectric coating–substrate system with FGP interfacial layer is formulated in terms of singular integral equation with the method of transfer matrix and Hankel integral transform technique. The effect of the gradient of material properties on the stress components and electric displacement components in piezoelectric coating–substrate system is gained by solving the equation numerically. The present results could serve as a useful reference for the design of piezoelectric coating–substrate system.

A Simple Semi-Analytical Method for Solving Axisymmetric Contact Problems Involving Bonded and Unbonded Layers of Arbitrary Thickness

In the present work, a recently extended version of the method of dimensionality reduction (MDR) for layered elastic media is applied for the first time using a semi-analytical approach. It is based on a priori knowledge of the cylindrical flat punch solution which is determined numerically using the boundary element method (BEM). We consider arbitrary indenters of revolution producing a circular area of contact with bonded and unbonded layers of arbitrary thickness. The proposed method reduces the contact solution to the numerically efficient evaluation of simple one-dimensional integrals. We further show that the solution of JKR-adhesive contacts with layers and contacts with linear-viscoelastic layers is straightforward using the well-known MDR formalisms. A specific focus has been devoted to study the thickness effect in different application examples. Comparisons with the literature and finite element simulations show very good agreement with the proposed method.

Axisymmetric thermoelastic contact vibration between a viscoelastic half-space and a rotating spherical punch

Axisymmetric thermoelastic contact vibration between a viscoelastic half-space and a rotating spherical punch

Mutual remodeling of interacting nanodroplets and vesicles

Liquid-liquid phase separation within the cytosol leads to the formation of protein-enriched droplets inside cells. These droplets known as biomolecular condensates have ultra-low interfacial tensions and fulfill a vast range of functions inside cells. Biomolecular condensation can take place at the plasma membrane and generate mechanical forces on membranes as a result of membrane wetting. But little is known about the wetting of membranes by biomolecular condensates. In this study, we utilize energy minimization to explore a wide range of parameters and determine the dependence of membrane wetting phenomena on interfacial tension, bending rigidity, line tension, and spontaneous curvature. We observe that interacting nanodroplets and vesicles mutually remodel one another. In addition, we determine the parameter regimes for which the droplet-membrane systems exhibit axisymmetric and non-axisymmetric contact lines. Our results provide insights into understanding intracellular processes and physical mechanisms based on the mutual remodeling of droplets and membranes.

Axisymmetric frictionless contact between an elastic layer thickness and a circular base under a rigid punch

The study presented in this work deals with analytical methods for an axisymmetric problem of an elastic layer partially reposing on a rigid circular base, and is indented along the upper surface with a rigid punch. The contact between the medium and the base is smooth. This boundary value problem is transformed into a system of dual integral equations. In contrast to the classical approach consisting in resolving the corresponding Fredholm equation of the second kind, the latter equations are obtained from an infinite algebraic system of simultaneous equations, where the particular case [Formula: see text] is verified. The results of this system are also obtained numerically. The normal displacement, normal stress, and the stress singularity factor are given analytically and shown graphically with discussion. By comparison with those predicted by the finite element method, the accuracy of the numerical method is approved.

Analytic contact solutions of the Boussinesq and Cattaneo problems for an ellipsoidal power-law indenter

Based on Galin’s theorem for the indentation of an elastic half-space by a polynomial punch and Barber’s theorem for the determination of the contact area in elastic normal contact problems, an exact contact solution is developed for the frictionless indentation of an elastic half-space by an ellipsoidal power-law punch. Within the Cattaneo–Mindlin-approximation of tangential contacts, the tangential contact problem with friction can be reduced to the frictionless normal contact via the Ciavarella–Jäger principle, based on which also the tangential contact solution for the ellipsoidal power-law indenter is given. All known solutions for special cases (axisymmetric contact, elliptical Hertzian contact) are exactly recovered. A comparison of the pressure distribution obtained analytically for the 4-th power ellipsoidal indenter with a corresponding numerical solution based on the boundary element method shows no noticeable differences. The proposed solution procedure can also be used exactly for an arbitrary finite superposition of ellipsoidal power-law profiles, and, in an approximate sense, is applicable for arbitrary monotonous ellipsoidal profiles.

The Effect of Adhesion on Indentation Behavior of Various Smart Materials

The nanoindentation technique plays a significant role in characterizing the mechanical properties of materials at nanoscale, where the adhesion effect becomes very prominent due to the high surface-to-volume ratio. For this paper, the classical adhesion theories were generalized to study the contact behaviors of various piezoelectric materials indented by conical punches with different electric properties. With the use of the Hankel integral transform, dual integral equations, and superposing principle, the closed-form solutions of the physical fields for the Johnson-Kendall-Roberts (JKR) and Maugis-Dugdale (M-D) models were obtained, respectively. The contribution of the electrical energy to the energy release rate under the conducting punch was taken into consideration. The relationships between the contact radius, the indentation load, and the indentation depth were set up using the total energy method for the JKR model and the Griffith energy balance for the M-D model, respectively. Numerical results indicate that increasing the half cone angle of the conical punch enhances the adhesion effect, which can significantly affect the accuracy of the results of characterization in nanoindentation tests. It was found that the effect of electric potential on adhesion behaviors is sensitive to different material properties, which are not revealed in the existing studies of axisymmetric adhesive contact of piezoelectric materials and multiferroic composite materials. The load-displacement curves under conical punches with different half cone angles have very different slopes. These results indicate that the half cone angle has a prominent effect on the characterization of mechanical properties of piezoelectric solids in nanoindentation tests.