JKR Theory Research Articles

Enhancement of adhesion strength in viscoelastic unsteady contacts

We present a general energy approach to study the unsteady adhesive contact of viscoelastic materials. Under the assumption of infinitely short-range adhesive interactions, we exploit the principle of virtual work to generalize Griffith’s local energy balance at contact edges to the case of a non-conservative (viscoelastic) material, subjected to a generic contact time–history. We apply the proposed energy balance criterion to study the approach–retraction motion of a rigid sphere in contact with a viscoelastic half-space. A strong interplay between adhesion and viscoelastic hysteretic losses is reported which can lead to strongly increased adhesion strength, depending on the loading history. Specifically, two different mechanisms are found to govern the increase of pull-off force during either approach–retraction cycles and approach – full relaxation – retraction tests. In the former case, hysteretic losses occurring close to the circular perimeter of the contact play a major role, significantly enhancing the energy release rate. In the latter case, instead, the pull-off enhancement mostly depends on the glassy response of the whole (bulk) material which, triggered by the fast retraction after relaxation, leads to a sort of ‘frozen’ state and results in a flat-punch-like detachment mechanism (i.e., constant contact area). In this case, the JKR theory of adhesive contact cannot be invoked to relate the observed pull-off force to the effective adhesion energy, i.e. the energy release rate G, and strongly overestimates it. Therefore, a rigorous mathematical procedure is also proposed to correctly calculate the energy release rate in viscoelastic dissipative contacts.

Configuration space of helical chiral self-assembly of micro/nano-fibers

As undertaking various key functions, helical chiral structures are widely presented in nature. Herein, we develop a general theoretical framework to guide the formation of helical chiral structures through self-assembly of micro/nano-fibers driven by adhesion. By analyzing the spiral contact geometric characteristics of multiple fibers and extending the JKR theory, an analytical model for adhesive contact between helical interfaces is proposed. Further, the complete configuration space of self-assembled fibers depending on the adhesion work, elastic modulus, aspect ratio, and initial helix angle is theoretically obtained. A diversity of helical configurations that far beyond existing experimental findings are predicted, which stems from the existence of multiple energy minimum points on the configuration-energy map. This work reveals the mechanism of adhesion-driven helical chiral structures, and provides theoretical foundation for guiding high-efficient fabrication of helical chiral structures through self-assembly method, which could promote the wide application of chiral structures in fields such as optics, catalysis and drug screening.

Physics of the Sub-Monolayer Lubricant in the Head-Disk Interface

This review presents a series of studies which have demonstrated that the diffusion characteristics of rarefied mobile lubricant films used in modern magnetic disks can be evaluated by a novel diffusion theory based on continuum mechanics, and that the meniscus force of the rarefied film is the major interaction force at the head-disk interface. The limitations of the conventional diffusion and disjoining pressure equations are first shown, and diffusion and disjoining pressure equations for rarefied liquid films are proposed, showing that the diffusion coefficient is in good agreement with the experiment. The experiment, in which glass spheres with radii of 1 and 2 mm collided with magnetic disks of different film thicknesses, showed that attraction similar to the pull-off forces of a static meniscus was measured only at the separation. Furthermore, mathematical analysis of the elastic meniscus contact between a sphere and a plane with a submonolayer liquid film showed that the maximum adhesion force is equal to the meniscus pull-off force and that the contact characteristics become similar to those of the JKR theory as the liquid film thickness decreases. A basic physical model of submonolayer liquid film is also proposed to justify the continuum mathematical equations.

Nonlinear modeling of nanoscale interaction forces between atomic force microscope and carbon nanotubes

This paper introduces a non-linear non-classical model of nanoscale interaction forces between atomic force microscope and carbon nanotubes in a manipulation process. To accurately model the 3D manipulation process, Prandtl-Tomlinson friction model and the Hertz, JKR, and Lundberg contact mechanics theories were employed, taking into account the CNT geometry and manipulation conditions. The applied forces exerted by both the substrate and the probe on the CNT were extracted. The dynamic characteristics of the nanorods and nanotubes were investigated by employing the non-linear modified couple stress Timoshenko beam. Additionally, the cantilever was modeled using the Mindlin plate theory. The findings showed a good agreement between the experimental data of multi-walled carbon nanotube with a diameter of 100 nm, and the theoretical simulations. In addition to critical force and time, CNT bending results were presented using various Timoshenko theories. In order to validate the results of the maximum bending, the outcomes derived from the computational modeling of a polystyrene nanorod using the Euler-Bernoulli and Timoshenko theories were juxtaposed with the existing classical bending outcomes. Finally, the effects of the size parameter were investigated in non-classical polystyrene models. The findings indicated that considering the size effects is important; For instance, the variance between classical and non-classical models can exceed 57% for l/D = 0.5.

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.

Discrete element analysis of diametrical compression of annular agglomerates

The tensile strength is an important material attribute of granular products, and diametrical compression is one of the most common methods to determine the tensile strength of the agglomerate. In this study, three-dimensional diametrical compression tests of annular agglomerates with various bonding strengths, i.e. interfacial energies and internal holes, are for the first time modelled by the Discrete Element Method (DEM) with the JKR theory. The fracture patterns and tensile strengths of annular specimens are analysed and compared with those of disc specimens. It is found that annular specimens show complete tension failures with progressive fractures along the loading diameter towards the central hole, while the disc specimens show wedge-shaped shear fractures near the platens. The results also show that the tensile strengths of annular specimens are linear with the surface energy, whereas increase with the size of internal hole and then decrease with the size of internal hole. A dimensionless parameter is introduced to analyse the failure mechanism of the agglomerate. This parameter quantifies the ratio between the external force exerted by the platens and the cohesive force among bonded particles. The proposed parameter emphasises the crucial influence of the internal hole size of an annular agglomerate and bonding capacity on its tensile strength.

Dry abrasion mechanisms of superhydrophobic nanocomposite coating surfaces

To date, the theoretical studies on dry abrasion on surfaces with microscopic roughness are based on JKR or DMT contact theory. Superhydrophobic nanocomposite coatings with low surface energy and microscopic surface roughness have extremely low real contact area and adhesion to the abrasive when subjected to contact and friction, making the above theories inapplicable. To date, there lack of theoretical studies on the dry abrasion characteristics of superhydrophobic nanocomposite coating surfaces. In response to this problem, a numerical understanding of the frictional force on superhydrophobic nanocomposite coatings is proposed based on Hertz contact theory, and the abrasion mechanisms of superhydrophobic nanocomposite coatings and its influencing factors are analyzed from the perspective of microscopic torque balance. The abrasion process is divided into three stages, and there exists a minimum value of bonding strength between the microstructure on the coating surface and the polymer matrix of the coating that determines whether abrasion will occur. This minimum bonding force is determined by Young's modulus of the polymer matrix, the radius of the microstructure, and the applied load. The conclusions can be a guide for the improvement of the anti-dry-abrasion property of superhydrophobic nanocomposite coatings from the perspective of coating preparation.

The adhesion behavior of the retina

Ocular diseases and treatment related to rhegmatogenous retinal detachment (RRD) are highly correlated with retinal adhesion behavior. Therefore, this paper proposes to study the adhesion behavior of the intact retina. This can provide theoretical guidance for the treatment and research of retinal detachment (RD) related diseases. To systematically analyze this aspect, two experiments were performed on the porcine retina. The pull-off test combined with the modified JKR theory was used to study the adhesion behavior of the vitreoretinal interface, while the peeling test was used to study the adhesion behavior of the chorioretinal interface. In addition, the adhesion phase involved in the pull-off test was simulated and analyzed by building the corresponding finite element method (FEM). The experimental results of adhesion force on the vitreoretinal interface were obtained by pull-off test with five sizes of rigid punch. The experimental value of the pull-off force FPO tends to increase gradually with increasing punch radius in the range of 0.5–4 mm. A comparison of the experimental results with the simulation results shows that they are in a well agreement. And there is no statistical difference between the experimental and theoretical values of the pull-off force FPO. In addition, the values of retinal adhesion work were also obtained by pull-off test. Interestingly, there is a significant scale effect of the retinal work of adhesion. Finally, the peeling test gave a maximum peeling strength TMax of about 13 mN/mm and a stable peeling strength TD of about 11 mN/mm between the retina and the choroid. The pull-off test well shows the process of retinal traction by the diseased vitreous at the beginning of RRD. A comparison of the experimental results with the finite element results verifies the accuracy of the simulation. The peeling test well investigated the adhesion behavior between the retina and the choroid and obtained key biomechanical data (peeling strength, etc.). The combination of the two experiments allows a more systematic study of the whole retina. This research can provide more complete material parameters for finite element modeling of retina-related diseases, and it also can provide the theoretical guidance for individualized design of retinal repair surgery.

Effect of roughness small scales on the adhesion of randomly rough surfaces with high fractal dimension

Surface roughness affects several tribological phenomena like adhesion. In this work, JKR-like experimental tests were performed between Polydimethylsiloxane (PDMS) spherical samples and flat glass substrates with different surface roughness properties. Data were interpreted in the light of classical JKR theory. We find that adhesion is destroyed by roughness in agreement with the most common and well-known results in the literature. However, from the analysis of the Power Spectral Density (PSD) of the surfaces, it emerges that the smaller scales of roughness have a greater influence on the surface adhesion energy in the range of the high fractal dimensions (D > 2.5). In this range, for surfaces with the same root mean square (RMS) roughness amplitude hrms , which is mainly affected by large scale roughness, the pull-off force (i.e., the maximum tensile force reached during the retracting phase) is observed reducing by increasing the RMS slope of the surface roughness , that is by changing the roughness content at the small scales. This result agrees with recent theoretical findings and has never been shown experimentally in the literature to the best of our knowledge.

A coarse grain model with parameter scaling of adhesion forces from liquid bridge forces and JKR theory in the discrete element method

The discrete element method (DEM) is a reliable tool for analyzing powder processes. However, the DEM is problematic for fine particle simulations owing to its huge computational cost. Coarse grain models, where multiple original particles are replaced with one large particle, are a promising solution. In such a model, scaling laws for forces acting on a coarse-grained particle are used to make its behaviors match the behaviors of the original particles. While various scaling laws have been proposed, there is not enough insight into combining several scaling laws and the relationship between scaled parameters. This study newly proposes a coarse grain model with a systematic parameter scaling law for adhesion forces, especially liquid bridge forces. Simulation results are compared with experimental results. Good agreements are achieved for macroscopic particle behaviors such as cascading angles and lifted heights. The adequacy of the proposed coarse grain model is also verified. The behaviors of the coarse-grained particles well match the corresponding behaviors of the original particles. Consequently, this study demonstrates the superiority of the coarse grain model by comparing its results with those obtained without the model.

Effect of Surface Adhesion on the Rough Contact Response Near Complete Contact

The adhesion phenomena between interfaces is widely investigated in engineering and scientific research. Due to the complexity of loading condition and surface topography, the traditional adhesive theory has many limitations. To better understand the adhesive properties of rough surfaces, we release the restrictions of JKR theory and propose a new adhesive model for single asperity. Initiated by this, a discrete rough surface contact model is presented, which extends the application scope of the traditional theory. First, we establish an elastic model by describing the gap between the interfaces accurately, so analytical solutions which are still valid for high contact pressure are proposed. Then, based on this, the exact adhesive solutions for different shapes of indenters are derived, which greatly improves the predicting accuracy of contact relationship and adhesion force. Finally, we use the results in the analysis of rough surface. The effect of both surface roughness and surface energy on the adhesive response of rough contact are studied in detail. The results show that adhesion is more easily to happen for smooth and geometrically continuous surfaces. The overall adhesion effect will be reduced in rough surface analysis with the increasing of surface roughness and the decreasing of surface energy. Our research sheds light on the understanding of the adhesion between solids and provides a theoretical guidance for the design of adhesion biomimetic materials and MEMS systems.

Asymmetric non-slipping adhesion behavior of layered piezoelectric structures

Layered piezoelectric structures have found wide applications in micro-electro-mechanical systems (MEMS) due to their intrinsic electro-mechanical coupling effect. The adhesion effect resulting from various surface forces (e.g., the electrostatic force, capillary force, and Van der Waals force) has become one dominant factor in determining the reliability and service life of the MEMS devices. In this study, the asymmetric non-slipping adhesion behavior of layered piezoelectric structures under the action of different external loadings is investigated. A system of coupled singular integral equations are obtained for the stated problem in the framework of the generalized JKR theory, where the coupling effect between the normal and tangential contact stresses is taken into consideration. The closed-form analytical solutions of the electric displacement and stress fields within the contact zone are derived. The relations between the contact size and applied loadings are set up with the Griffith energy balance criterion. The effect of the mechanical and electric loads and the mismatch strain on the adhesion behavior of layered piezoelectric structures is discussed in detail. It is found that applying the electric load can only strengthen the adhesion effect for different types of layered piezoelectric structures. Both the inclined angle of the mechanical force and the geometric parameter of the contacting objects have significant effects on the adhesion behavior. The results obtained from present paper should be very useful for understanding the adhesion failure mechanism of the MEMS devices composed of layered piezoelectric structures.

Derjaguin and the DMT Theory: A Farewell to DMT?

There is a widespread belief that in the 1970s, two conflicting theories of the adhesion of a spherical particle to a substrate were published: the JKR and DMT theories. And that the dispute was resolved when Tabor introduced a parameter mu, such that for small mu, DMT was correct, while for large mu, JKR was correct. We point out that there never were two theories of contact: the dispute was about the magnitude of the pull-off force (with an implication that since the DMT value was obtained by a thermodynamic method, it must be correct). And what Tabor actually said was simply that for large mu, the neglect of surface forces in the JKR theory was acceptable, and that he distrusted the neglect of deformation by the large surface forces immediately outside the Hertzian contact in the DMT theory. We point out the errors in both the DMT/MDT thermodynamic method and the MDT force method (preferred by MDT) but also argue that once Derjaguin and his collaborators (MYD) established that Hertzian geometry does not occur, no theory based on that geometry should be taken seriously. Derjaguin’s well-merited fame rests on more important contributions.

Impact of particle stiffness on shear-thinning of non-Brownian suspensions

We attempt to identify the impact of particle stiffness on the shear-thinning behavior exhibited by frictional and adhesive non-Brownian suspensions. To this aim, we compare the rheological behavior of three suspensions whose particles have different Young’s moduli. The stiffest particles are soda-lime glass beads with a Young modulus of 70 GPa. The two other kinds of particles are made of polydimethylsiloxane and have been manufactured using a homemade millifluidic device that allowed us to produce batches of sparsely polydisperse particles with tunable Young modulus: 1.8 MPa for the softest ones and 15 MPa for the stiffest. We show that the observed shear-thinning is mainly caused by the presence of adhesive forces between particles. For each of the three suspensions, the shear-thinning behavior is described by the variation of the jamming volume fraction, ϕm, with stress, σ, and the results are interpreted in the light of the model proposed by [Richards et al., J. Rheol. 64(2), 405–412 (2020)]. We show that the magnitude of variation of ϕm with σ is greater the lower the particle stiffness. More precisely, in the adhesive regime, decreasing the particle stiffness leads to a significant decrease in the value of the jamming fraction while, as predicted by the JKR theory, the characteristic adhesion stress, σa, does not vary with stiffness. We, finally, show that, for stresses much higher than σa, the suspensions behave like usual frictional suspensions with a viscosity controlled by both the particle volume fraction and the value of the interparticle friction coefficient.

Thermal-Regulated Adhesion Enhancement and Fast Switching within the Viscoelastic Glass Transition Zone of a Shape Memory Polymer

The effects of temperature and preload on adhesion of an epoxy-based shape memory polymer (ESMP) were investigated experimentally. The ESMP sheet and polydimethylsiloxane (PDMS) control sample were prepared by mold casting. The adhesion measurements were carried out on a home-built adhesion tester using a hemispherical glass indenter as the counter-surface. The reduced modulus was determined by the Oliver-Pharr method and Hertz theory. The viscoelastic energy dissipation was extracted, and the effective work of adhesion was calculated by the fitting method and JKR theory. In the vicinity of glass transition temperature (Tg), the viscoelastic enhancement factor (1 + f(v, T)) is as high as 955 because of the strong viscoelastic effect of the ESMP sheet. The adhesion strength of the ESMP sheet is about 589 kPa under a relatively smaller contact displacement condition (∼44.6 μm). The strong viscoelastic effect induces more viscoelastic energy dissipation that contributes to the effective work of adhesion and leads to strong preload dependence of the adhesion. The pull-off force Fpull-off is demonstrated to linearly depend on Fm1/3Er2/3. In the sharp half glass transition zone (T > Tg), the viscoelasticity and rigidity rapidly decrease with the temperature increasing about 10 °C, leading to a 6-fold reduction in adhesion. The results indicate that the adhesion of the ESMP sheet can be significantly enhanced and meanwhile rapidly switched within the viscoelastic glass transition zone.

A Note by K. L. Johnson on the History of the JKR Theory

The history of the following note is as follows. In 2003, I invited Kenneth Johnson to Berlin to give a talk on adhesion in a seminar at the Institute of Mechanics. His lecture on the topic "Mechanics of adhesion of spherical surfaces" took place on Monday, January 26, 2004. In the run-up to the seminar, Professor Johnson sent me a historical note dated November 18, 2003. In my opinion, this note, which was written in the form of a paper, may be of interest for experts in contact mechanics and tribology. Prof. Johnson did not publish it, so it remained a private communication. For a publication he might have made a revision and would possibly have credited other important contributions. But this we can only guess at, and therefore the note is published below in the form I received it from Kenneth L. Johnson, with only a few misprints corrected. It is interesting as a historical document from Ken Johnson, who played a key role in development of theory of adhesive contacts.

Effect of pressure on pull-off of flat cylindrical punch adhered to circular membrane

ABSTRACT Pressure can sometimes be utilized to increase or decrease the strength of adhesion. This phenomenon is analyzed for pull-off of a flat, circular cylindrical punch from a circular membrane. The pressure is applied, and then the punch is pulled upward. Both force control and displacement control are considered, and the pressure may be positive (acting upward) or negative (acting downward) on the membrane. The membrane is assumed to be linearly elastic, and Föppl-von Kármán theory is used to allow moderate displacements. The JKR theory of adhesion is adopted. A transversality (debonding) condition is derived, and then the effects of the work of adhesion, Poisson’s ratio, residual stress, and the relative radius of the punch to that of the membrane are investigated. With an increase in pressure, the pull-off force decreases, the pull-off deflection of the punch relative to the initial flat membrane level increases, and the total pull-off punch deflection from the pressurized membrane may increase or decrease.

A JKR-Like Solution for Viscoelastic Adhesive Contacts

A closed-form solution for the adhesive contact of soft spheres of linear elastic material is available since 1971 thanks to the work of Johnson, Kendall, and Roberts (JKR). A similar solution for viscoelastic spheres is still missing, though semi-analytical and numerical models are available today. In this note, we propose a closed-form analytical solution, based on JKR theory, for the detachment of a rigid sphere from a viscoelastic substrate. The solution returns the applied load and contact penetration as functions of the contact radius and correctly captures the velocity-dependent nature of the viscoelastic pull-off. Moreover, a simple approach is provided to estimate the stick time, i.e., the delay between the time the sphere starts raising from the substrate and the time the contact radius starts reducing. A simple formula is also suggested for the viscoelastic pull-off force. Finally, a comparison with experimental and numerical data is shown.

Parameter calibration of discrete element method modelling for cohesive and non-spherical particles of powder

Numerical analysis via the DEM has been applied to various powder processes. To obtain simulation results of high reliability, a calibration method for the input DEM parameters must be established. This paper therefore proposes a calibration procedure for non-spherical cohesive particles. The rolling friction coefficient and surface energy for cohesive force based on the JKR theory were calibrated. Initially, the dynamic angles of repose were calculated using various parameter combinations to obtain a two-dimensional contour map. Subsequently, the DEM parameters were determined by comparing them with the experimental data. Finally, the theoretical static angle of repose was calculated using the selected parameters to verify its agreement with the experimental data, which confirmed its validity. Furthermore, the validity of the calibrated parameters was evaluated by mixing simulations in a V-shaped mixer. Through the proposed calibration procedure, the DEM parameters of several materials were obtained successfully, and their wide applicability was confirmed.

Adhesion of elastic wavy surfaces: Interface strengthening/weakening and mode transition mechanisms

Surface roughness involved in elastic adhesion of solids may lead to an unstable loading process, which makes experimental characterization and theoretical modeling full of challenge. The full self-consistent model (FSCM) is the most accurate adhesion contact model in the continuum framework, but the strongly nonlinear coupling relationship between surface gap and interaction is very difficult to solve even with high-precision numerical calculation. In this paper, several techniques including the arc-length control method, the Riemann–Stieltjes integral, the adaptive mesh and the Newton–Raphson iterative method were employed in a new solving program to overcome the difficulty of numerical calculation. This calculation method has high efficiency and precision, and it gives a full force–displacement curve containing all possible stable and unstable equilibrium states. The full force–displacement curves for the adhesive contact of wavy surfaces are zigzag and some additional oscillations or minor roundabouts are observed, which have not been detected in the extended JKR and Maugis–Dugdale models. With the increase of surface roughness, the pull-off force first increases and then decreases, because the adhesion mode transforms from the simple contact mode to the multiple one. It is found that the pressure oscillation induced by waviness leads to interface strengthening for small roughness, and the limitation of the theoretical strength and the development of interface cavitation result in interface weakening for large roughness. Compared to the FSCM, the JKR theory significantly overestimates the pull-off forces for large surface roughness. The extended Maugis–Dugdale model is also invalid for large roughness, and if the step cohesive stress is determined by applying a condition of the identical pull-off force at the rigid limit, it can work well for small roughness.