Maugis Dugdale Model Research Articles

A reinvestigation on combined dry and wet adhesive contact considering surface tension

The present study theoretically explores combined dry and wet adhesive contact between a rigid sphere and elastic semi-half substrate, in which dry contact is encircled by liquid bridge. We consider threefold effects of liquid bridge on contact behavior, namely Laplace pressure induced by the curved surface of liquid meniscus, surface tension at the triple-phase junction and alternation of adhesion energy between solid surfaces ascribed to liquid immersion. A clear novelty in this study is the investigation on the effect of surface tension at the vapor-liquid-solid junction on the adhesive contact response, in contrast to previous studies. The model solution predicts that the contact behavior and adhesive strength are strongly dependent on surface wettability (manifested by contact angle), liquid volume and the contact system's rapidity in achieving thermodynamic equilibrium. It is found that the transition of the pull-off force is evidently different from Maugis-Dugdale model in terms of a couple of interesting characteristics. Moreover, it is unveiled that the jump instabilities and hysteresis of force-separation curves are highly affected by surface wettability and liquid volume. These theoretical results can not only shed lights on the mechanism of liquid-mediated adhesion employed by animals and plants, but also provide us inspiration for development of biomimetic adhesive devices.

An Analytical Adhesion Model for Elastic Contact Electrification

Abstract Contact electrification is a universal phenomenon that commonly occurs in almost every solid–solid contact pair. The tribo-charges deposited on two surfaces by contact electrification can significantly affect adhesion; however, contact electrification is often overlooked in the study of adhesive contact. Here, we develop an analytical model to investigate electroadhesion during the contact phase between two initially uncharged dielectric surfaces, namely, an elastic parabolic surface and a rigid flat. A system of nonlinear equations is derived to describe the relationship between the indentation, normal load, radius of contact area, and radius of the charged zone using the Barthel–Maugis–Dugdale model (Barthel, 1999, “Modelling the Adhesion of Spheres: When the Form of the Interaction Is Complex, Colloids. Surf., A., 149, pp. 99105.). The analytical results show good agreement with the numerical results of the full self-consistent contact model. When contact electrification leads to a higher tribo-charge density and a larger charged zone, it has a greater impact on the normal traction, interfacial gap, force-approach curves, jump-out, and dissipated energy. The analytical model developed in this study serves as the foundation for advances in rough surface electroadhesive contact and electroadhesion testing, and it sheds light on the usage of adhesive joints in ultra-high vacuum environments and outer space, where contact electrification has a significant impact.

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.

An upper bound for viscoelastic pull-off of a sphere with a Maugis-Dugdale model

ABSTRACT We develop an extension of the Maugis-Dugdale solution for viscoelastic spheres. We show that we can define two characteristic Tabor parameters, a larger one corresponding to relaxed modulus and a smaller one for instantaneous modulus of the material. Only if both are very large (corresponding to the JKR regime, ) the pull-off load increase due to viscoelastic effect is possibly very large at large pulling speeds, as given by existing solutions and approximately equal to the ratio , and otherwise the amplification at very high speeds is much reduced and we give a very simple upper bound of the pull-off load as a function of the relaxed Tabor parameter, independently on the exact form of the viscoelastic linear modulus. An example detailed calculation is given for standard material and constant velocity of load reduction. A dependence on preload is found.

Indentation mapping of stretched adhesive membranes

Unilateral adhesive contact between a rigid indenter and a uniformly stretched membrane of arbitrary shape is considered. The generalized Johnson–Kendall–Roberts (JKR)-type and Derjaguin– Muller–Toporov (DMT)-type models of non-axisymmetric adhesive contact are presented for short- and long-range adhesion, respectively, and the JKR–DMT transition is established in the framework of the generalized Maugis–Dugdale model. A refined method of matched asymptotic expansions is applied to construct the leading-order asymptotic model for indentation mapping of freestanding two-dimensional materials with an axisymmetric probe, using the approximate analytical solution obtained in explicit form for an infinite membrane in the limit of short-range adhesive contact with an indenter in the form of an elliptic paraboloid. The cases of a spherical indenter and a rectangular membrane are studied in detail.

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.

Effect of surface tension on the behavior of adhesive contact based on Maugis‒Dugdale model

This present study reconsiders the effect of surface tension on the behavior of adhesive contact between a rigid sphere and an elastic half-plane, in which the adhesive interactions are supposed to follow the Dugdale laws. The adhesive contact issue is transformed into two inter coupled non-linear integral equations which are governed by two parameters: λ (Maugis adhesion parameter) and S (elastocapillary number). By means of iteration method, numeric results are obtained. Analogous to the traditional Maugis-Dugdale (M-D) model, the results provide transition of the pull-off force between JKR and DMT type contact models through the Maugis adhesion parameter λ with a fixed parameter S. On the other hand, with a fixed adhesion parameter λ, we also present the transitions of the pull-off force between four extreme models, named M-D, JKR, Bradley models and Young–Dupre equation, through adjustment of the parameter S. Finally, we find the uniformity and discontinuity of the pressure distribution are affected by the combination of three factors: applied load P, adhesion parameter λ and elastocapillary number S.

An investigation on the behavior of anisothermal adhesive contact based on Maugis–Dugdale model

The present study considers the effect of heat conduction on the behavior of adhesive contact between two elastic spheres of different temperatures, where the adhesion forces are supposed to follow the Dugdale model. The thermal adhesive contact issue is transformed into two intercoupled nonlinear integral equations which are governed by two parameters: λ (adhesion parameter) and M. By means of iteration method, numeric results are obtained. Analogous to the traditional Maugis-Dugdale (M-D) model, the results provide transition of the pull-off force between JKR and DMT type contact models through the adjustment of the adhesion parameter λ with fixed parameter M. On the other hand, with a fixed adhesion parameter λ, it is indicated that the shape of the force-contact radius curve and the pull-off force are dependent on the parameter M. Finally, we find the uniformity of the pressure distribution is affected by the applied load F and adhesion parameter λ.

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

This article is devoted to studying the adhesive contact problem for a spherical indenter on a half-space of the transversely isotropic multi-ferroic composite materials in a systematic manner. In view of magneto-electric properties of the spherical punch, four sets of mixed boundary value problems are taken into account. By virtue of the generalized potential theory method, the physical quantities on the contact surface, corresponding to the Johnson-Kendall-Roberts (JKR) and Maugis-Dugdale (MD) models, are explicitly obtained in the closed form. The expressions of indentation force and penetration depth associated with the Derjaguin-Muller-Toporov (DMT) model, which is the limiting case of the MD model, are derived through a limiting procedure. In particular, the analytical solutions to an external circular crack problem, which is new to literature, are presented as an auxiliary. Energy release rates pertinent to the JKR and MD models are derived and the corresponding equilibrium relations are established according to the Griffith energy balance. The adhesive contact behaviors including the pull-out force especially for three classical models are elaborated. The adjustment and control mechanism by the magnetic and electric means on adhesive contact behaviors is revealed. Numerical calculations are performed to present the obtained analytical results, compare the adhesive models and analyze the effect of magneto-electric properties on adhesive models. The obtained solutions are expected to be crucial to the forthcoming experiments on characterization of properties of multi-ferroic composite media, especially on the micro/nonoscopic scale.

The effect of edge compliance on the adhesive contact between a spherical indenter and a quarter-space

In this work, the effect of the edge compliance on the adhesive, frictionless contacts near the edge of a quarter-space is analyzed in terms of stress, deformation, and contact stiffness. The study relies on the numerical implementation of the conjugate gradient method (CGM) to adhesive contacts near the edge of a quarter-space. It extends the previous CGM developed for contact analysis of non-adhesive and adhesive contacts on half-space to the matrix formulation of the quarter-space problem. The considered adhesive contact interaction is in terms of the Maugis–Dugdale model to provide insight into the transitional behavior from non-adhesive to various type of contact adhesive regimes. It is found that the edge compliance affects the contact mechanics of adhesive contacts as a function of position, applied force, and adhesive parameter. The analysis provides calculations for quantities like depth indentation and contact stiffness that can be found directly in indentation-type experiments performed on samples with edge geometries. Such understanding and characterization of edge contacts can be used to unambiguously account for the mechanical response of nanoscale structures near their edges and detect possible mechanical variability introduced by fabrication and processing.

The generalized Tabor parameter for adhesive rough contacts near complete contact

Recently, the first author has obtained a model for adhesive contact near full contact under the JKR assumptions. The model shows, in the common case of low fractal dimensions, an ‘unbounded’ adhesion enhancement when larger and larger upper “truncation wavenumber” is considered in the spectrum of roughness, i.e. when we increase “magnification”. Here, using a more general Maugis–Dugdale model, we show that a generalized multiscale Tabor parameter can be defined which shows a transition to a non-hysteretic regime, dependent on the root-mean-square (rms) slope of the surface. The contact area returns in the “fractal limit” to the adhesionless one. Two examples of rough surfaces from the literature are considered to show the full dependence on magnification of the adhesive solution. The choice of the truncation of the spectrum remains fundamentally arbitrary.

Plane Contact and Adhesion of Two Elastic Solids With an Interface Groove

A systematic study is performed on the plane contact and adhesion of two elastic solids with an interface groove. The nonadhesion and Johnson–Kendall–Roberts (JKR) adhesion solutions for a typical groove shape are obtained in closed form by solving singular integral equations and using energy release rate approaches. It is found that the JKR adhesion solution depends solely on a dimensionless parameter α and the groove is predicted to be unstably flattened with no applied load when α≥0.535. Furthermore, the corresponding Maugis–Dugdale adhesion model has been revisited with three possible equilibrium states. By introducing the classical Tabor parameter μ, a complete transition between the nonadhesion and the JKR adhesion contact models is captured, which can be recovered as two limiting cases of the Maugis–Dugdale model. Depending on two nondimensional parameters α and μ, where α2 represents the ratio of the surface energy in the groove to the elastic strain energy when the grooved surface is flattened, different transition processes among three equilibrium states are characterized by one or more jumps between partial and full contact. Larger values of α and μ tend to induce more energy loss due to adhesion hysteresis. Combination values of α and μ are also suggested to design self-healing interface grooves due to adhesion.

Influence of the Tabor parameter on the adhesive normal impact of spheres in Maugis–Dugdale approximation

We study the influence of the adhesive interaction range on the coefficient of restitution in the normal impact of spheres by analytical and semi-analytical means based on a Maugis–Dugdale model for the adhesive stress. It is found, that—depending on the value of the Tabor parameter—the impact may start and end with and without direct contact between the surfaces. The hysteretic loss of energy takes its maximum value in the JKR-limit, which for even quite large values of the Tabor parameter noticeably overestimates this loss. Our findings can be applied for the rigorous study of adhesive collisions of elastic particles with and without friction.

Numerical method for the adhesive normal contact analysis based on a Dugdale approximation

Modeling adhesion between two contacting surfaces plays a vital role in nano-tribology. However, providing analytical models, although desirable, is mostly impossible, in particular for complex geometries. Therefore, much attention has to be paid to numerical modeling of this phenomenon. Based on the adhesive stress description of the Maugis-Dugdale model of adhesion, which is credible over a broad range of engineering applications, an extended Conjugate Gradient Method (CGM) has been developed for adhesive contact problems. To examine the accuracy of the proposed method, the common case of the adhesive contact of a rigid sphere on an elastic half-space is investigated. To further evaluate the accuracy of this method, the adhesive contact of a rigid sphere over a wavy elastic half-space is also studied for different combinations of the amplitude and wavelength. There is good agreement between the analytical solution and the values predicted by the proposed method in the force-approach curves. Moreover, the calculation of pull-off force at a bisinusoidal interface between two surfaces is carried out for various cases to study the effects of different influential parameters including work of adhesion, elastic modulus, radius curvature at a crest, and the wavelength ratio. A curve is fitted on the calculated pull-off force in order to express it as an analytical relation. Similar to the JKR and DMT expressions for the pull-off force of a rigid ball on an elastic half-plane, the fitted curve is not affected by the elastic modulus and is linearly dependent on the radius of curvature and the work of adhesion. In addition, a power law governs the relation between pull-off force and the wavelength ratio. In the end, it is shown that roughness can either increase or decrease the adhesive force at a rough interface depending on the degree of the roughness.

Revisiting the Maugis–Dugdale Adhesion Model of Elastic Periodic Wavy Surfaces

The plane strain adhesive contact between a periodic wavy surface and a flat surface has been revisited based on the classical Maugis–Dugdale model. Closed-form analytical solutions derived by Hui et al. [1], which were limited to the case that the interaction zone cannot saturate at a period, have been extended to two additional cases with adhesion force acting throughout the whole period. Based on these results, a complete transition between the Westergaard and the Johnson, Kendall, and Roberts (JKR)-type contact models is captured through a dimensionless transition parameter, which is consistent with that for a single cylindrical contact. Depending on two dimensionless parameters, different transition processes between partial and full contact during loading/unloading stages are characterized by one or more jump instabilities. Rougher surfaces are found to enhance adhesion both by increasing the magnitude of the pull-off force and by inducing more energy loss due to adhesion hysteresis.

Modeling contact of indenter with elastic half-space with adhesive attraction assigned in arbitrary form

A model of the contact of an elastic indenter with an elastic half-space with adhesive attraction has been proposed. The adhesive attraction has been specified as an arbitrary dependence of the adhesive pressure that acts outside the contact zone on the gap between the indenter and the half-space. The shape of the indenter has been described by a power function with an arbitrary exponent. The constructed model has been used to study the applicability of a model with the single-step adhesive potential (the Maugis–Dugdale model). The results of calculations carried out using the Lennard-Jones potential have been compared to the results obtained using linear and exponential potentials.

Influence of the Tabor Parameter on the Roughness-Induced Adhesion Hysteresis

Influence of the Tabor parameter on the roughness-induced adhesion hysteresis was investigated. To achieve this, the adhesive contact model of single asperities was considered by incorporating the Maugis-dugdale model and its corresponding extension firstly. Further more, the load-approach relationship of adhesive contact between a rough surface and a flat was analyzed. The dissipation energy during a load and unload cycle is derived for general values of the Tabor parameter. It was found that the adhesion hysteresis becomes weaker gradually with the increase of the adhesion parameter, and it becomes stronger with the decrease of the Tabor parameter at the same adhesion parameter. The adhesion hysteresis for a special case that rough surfaces with DMT(Deryagin-Muller-Toporov)-type asperities is also discussed.

Transition of contact behavior induced by interfacial adhesion

Large-scale molecular dynamics simulations are performed to study the adhesive contact between a rigid spherical tip and an elastic flat substrate. We focus on he relations between the real contact area and the external load and between the repulsive force and attractive force on contact interface. The simulated results are consistent with the corresponding continuum contact theories, which are Hertz model, Ggeenwood-Williamson (or Persson) model and Maugis-Dugdale model according to surface roughness and interfacial adhesion. We show that there are same relations between the real contact area and the repulsive force for both non-adhesive and adhesive contact, which means that the effect of adhesion on contact behavior can be equivalent to that of a virtual load. We demonstrate that the attractive force on contact interface increases with the real contact area in a power-law function, with a power exponent larger than 1 for the atomic-scale smooth tip and with a power exponent smaller than 1 for the atomic-scale rough tip.

An Adhesion Model for Elastic-Contacting Fractal Surfaces in the Presence of Meniscus

The strong stiction of adjacent surfaces with meniscus is a major design concern in the devices with a microsized interface. The present research concerns the elastic adhesion of rough fractal surfaces in the presence of a thin liquid film. A rough fractal surface is characterized with a two-variable Weierstrass–Mandelbrot fractal function. The microcontact model of the single asperity is established in terms of the fractal parameters. The adhesion model from meniscus is developed with the Dugdale approximation of the Laplace pressure to consider the adhesive interaction within/outside the contact area. Then the Maugis–Dugdale model and its extension are used to solve the elastic adhesive interaction for the two approaching fractal surfaces by incorporating the fractal surface model. Simulations of the external force versus the interface stiffness, surface roughness, and relative humidity are performed, respectively. The simulation results show that the interface stiffness, surface topography, and relative humidity can heavily influence the interface adhesion of rough surfaces with meniscus.

Averaging of a finely laminated elastic medium with roughness or adhesion on the contact surfaces of the layers

The governing relations of a laminated elastic medium with non-ideal contact conditions in the interlayer boundaries are obtained by an asymptotic averaging method. The interaction of rough surfaces is described by a non-linear contact condition which simulates the local deformation of the microroughnesses using a certain penetration of the nominal surfaces of the elastic layers. The cohesive forces, caused by the thin adhesive layer, are described within the limits of the Frémond model which includes a differential equation characterizing the change in the cohesion function. A piecewise-linear approximation of the initial positive segment of the Lennard–Jones potential curve is proposed to describe of the adhesive forces between smooth dry surfaces. A comparison is made with the solution obtained within the limits of the Maugis–Dugdale model based on a piecewise-constant approximation. Solutions of the above problems are constructed taking account of the possible opening of interlayer boundaries.