Abstract
Adhesive contact of a rigid flat surface with an elastic substrate having Weierstrass surface profile is numerically analyzed using the finite element method. In this work, we investigate the relationship between load and contact area spanning the limits of nonadhesive normal contact to adhesive contact for various substrate material properties, surface energy and roughness parameters. In the limit of nonadhesive normal contact, our results are consistent with published work. For the adhesive contact problem, we employ LennardJones type local contact interaction model with numerical regularization to study the transition from partial to full contact including jumptocontact instabilities as well as loaddepth hysteresis. We have investigated evolution of bonded contact area and pulloff force for various surface roughness parameters, substrate material properties and surface energy. We have identified two nondimensional parameters to adequately explain experimentally observed adhesion weakening and strengthening phenomena. A design chart of the relative pulloff force as function of nondimensional parameters is also presented.
Introduction
Smooth surfaces when brought into close proximity spontaneously jump into contact and require a finite force to pull them apart. This finite pulloff force quantifies adhesion or adhesive strength of the bond between two solids. The weakness or lack of adhesion in most of realworld phenomena is attributed to surface contamination and lack of proximity between surfaces, which is measured on the scale of the range of van der Waals interactions. Surface roughness, which precludes surfaces from coming into close contact, is observed to be the dominant factor in comparison with surface contamination contributing to the loss of adhesion^{1}.
Based on ASME B46.1, even the finest surface finish for industrial applications is about 10–15 times the range of interaction α~1.0 nm. In automotive industry applications, the surface finish ranges from about 75–1250 α. These high ratios of roughness to range of interaction is one of the reasons for adhesion to be a less critical factor in design considerations of industrial machine components. On the other hand, in applications such as sensors and actuators, MEMS devices have high surface to volume ratio with length scales ranging from micron to nanoscale. The reliability of assembly and operation of these devices is observed to primarily depend on adhesion^{2,3,4}. The goal of this work is to contribute to the quantification of adhesion in terms of nondimensional parameters towards developing an engineering design chart.
The loss of adhesion with an increase in roughness is observed in several experiments^{5,6,7,8,9,10,11}. These experiments can be grouped based on a nondimensional parameter \(\beta =(\gamma /{E}^{\ast }\alpha )\), the ratio of surface energy density γ to product of modulus E^{∗} and range of interaction α. Quon et al.^{7} investigated the contact between rough gold on a smooth mica surface which is on the lower end on \(\beta \sim 2\times {10}^{3}\) scale and authors found that there is more than ~80% reduction in adhesion for an increase in roughness by ~1–1.5 α. Gui et al.^{8} investigated the effect of roughness on bondability of Si wafers, which has \(\beta \sim 7\times {10}^{4}\), and reported that spontaneous bonding transitions to loss of bonding for an increase in roughness from 0.1 to 1.0 α Fuller and Tabor^{9} studied the adhesion of soft rubbers on rough perspex surface with β ~ 50–500 and showed a decrease in the rate of loss of adhesion with increase in roughness for more compliant rubber. Fuller and Roberts^{10} performed rolling experiments of soft rubber on a rough surface with β ~ 30–1500 and showed both loss and gain of adhesion with increase in roughness. For β ~> 250, in the low roughness regime, adhesion is observed to increase with roughness until a critical value followed by loss of adhesion. Guduru^{11} performed experiments using even softer rubberlike material with \(\beta \sim {10}^{4}\) on a rigid wavy surface and found similar adhesion strengthening with increase in roughness and attributed it to the contact instabilities and hysteresis in loading and unloading.
The seminal work by Fuller and Tabor^{9} was one of first comprehensive efforts in developing a model to explain the experimental results of adhesion loss due to increase in roughness. In this work, they have used JohnsonKendell_Roberts (JKR)^{12} approximation of adhesion between elastic spheres in combination with GreenwoodWilliamson (GW)^{13} statistical multiasperity contact model to explain loss of adhesion using a dimensionless parameter \(\theta ={E}^{\ast }{\sigma }^{3/2}/{R}^{1/2}\gamma \) where σ, R being root mean square (RMS) roughness and radius of curvature respectively. Maugis^{14} and Morrow^{15} used a similar approach to investigate DMT^{16} approximation and the transition from JKR and DMT respectively using multiasperity contact models for rough surfaces. In all the abovementioned studies, the adhesive contact was at the individual asperity scale and the asperities are noninteracting. These models cannot capture the important observed phenomena such as the jumptocontact instabilities and hysteresis in loading and unloading due to surface roughness. In contrast, the work done in references^{17,18,19,20,21,22,23} have studied contact instabilities during loading and unloading of adhesive microcontact. Komvopoulos^{24} used constitutive relations between interfacial force and separation at the asperity level exhibiting jumpin instability in conjunction with GW multiasperity model and introduced a new adhesion parameter which is the ratio of rms roughness to equilibrium interatomic distance.
The traditional GW multiasperity surface model does not include the longrange elastic interactions of asperities which leads to qualitatively and quantitatively different contact morphologies. Persson^{25} developed a contact mechanics theory for selfaffine fractal surfaces, which approximately includes elastic interactions. Persson’s theory explained the importance of including power spectrum of heights, longrange elastic interactions and showed linearity between loadcontact area and fractal nature of the contact morphology. In the nonadhesive limit, Hyun et al.^{26} performed a comprehensive numerical study of contact between a rigid flat surface and an elastic substrate with selfaffine fractal rough surface and showed that the range of linearity of load and contact area, contact morphology and pressure distribution are quantitatively different from the predictions of GW multiasperity type contact models. These differences in contact morphology due to longrange elastic interactions will have a significant influence on the adhesive contact behavior of rough surfaces. Although the aforementioned references are relevant and seminal works, they are by no means exhaustive and the authors refer to most elegant and comprehensive review on the topic by Ciavarella et al.^{27}.
In the present work, we study the contact between a flat rigid surface and an elastic substrate with rough surface characterized by a Weierstrass function. The interaction is modeled using LennardJones interaction potential with a viscous regularization to handle the jump to contact instabilities. The adhesive contact is solved in ANSYS, a generalpurpose finite element solver, with nonlinear large deformation effects. We have investigated the contact area evolution and pulloff force for various RMS roughness, RMS slope, Hurst exponent, substrate modulus and found two nondimensional parameters to explain both adhesion strengthening and weakening observed in experiments.
Results
Evolution of Contact Area
Real contact area has significant engineering importance in the thermal management of electrical connections in micro and nano devices. The stiffness of joints in aerospace and microelectronic devices to random vibrations depends largely on the real contact area and its morphology^{28}. In this section, we discuss the evolution of contact area in the presence of adhesion and the limiting case of nonadhesive contact.
For different values of β, during approach, the evolution of contact area with load is shown in Fig. 1. Unlike the gradual and linear evolution of real contact area A with load N for nonadhesive contact, the adhesive contact is zero until a net attractive load and suddenly jumps into contact. Moreover, there are several contact instabilities during the development of adhesive contact area. This behavior is similar to the experimental results published in Waters et al.^{29}.
In the nonadhesive limit, it is well established that the contact area is proportional to applied load with range of linearity from about 5–8% relative contact area^{26}. In agreement with published work, the nondimensional quantity \(\kappa =(A{E}^{\ast }{h{\prime} }_{rms}/N)\) is independent of system size and constant within 5–8% relative contact area (see Fig. S4 in supplemental information). For H and v equal to 0.5 and 0.3 respectively, the value of k is about 2.35 which in agreement with Hyun et al.^{26} and lies within that reported by Bush et al.^{30} and Persson^{31}. Here, \({E}^{\ast }=E/(1{\upsilon }^{2})\) is the effective modulus, with E and v being Youngs modulus and poisons ratio. During nonadhesive contact (see Fig. S5 in supplemental information), the evolution of the heights power spectrum clearly shows that the increase in contact area is accompanied by the increase in the length scales over which the contact is accommodated. This contrasts the contact evolution of traditional GW type models which does not include the longrange elastic interactions of the asperities.
In contrast, as shown in Fig. 2, the adhesive contact evolution starts with a jump to contact at a finite net attractive load and increases from state ‘1’ to ‘5’ in a sequence of jump to contact instabilities followed by a decrease in contact area to state ‘8’. To understand the adhesive contact evolution, we have investigated the contact pressure and gap profiles at various states to elucidate the intermittent crack zipping and unzipping as the mechanism of adhesive contact attachment and detachment respectively (see Supplementary Information for further details).
In case of adhesive contact, the contact area at net zero load defined as bonded contact area A^{0} is of great engineering importance. As it can be seen from Fig. 1, the bonded contact area A^{0} increases with increasing β. Similar to nonadhesive contact, we have investigated the relation between load and change in contact area from bonded state. In Fig. 3a, we have plotted a nondimensional quantity k^{*}(see Eq. 1), similar to k, as a function of change in contact area about bonded state relative to nominal contact area A_{0}, \((A{A}^{0}/{A}_{0})\).
It turns out that k* is constant and independent of h_{rms} and h′_{rms}. As shown in the Fig. 3b, the value of k* for various values of β approaches the value of k in the nonadhesive limit. The value of k* is found to decay with increase in β as a power law. For an adhesive contact, at constant value of β, this relationship can be used to estimate the value of k* and in turn, the bonded contact area.
Pulloff force and Scaling
Besides jump to contact instabilities, an adhesive contact exhibits hysteresis in an approach and detachment cycle. As shown in loaddisplacement plot (see Fig. 4), our approach can capture both the instabilities and hysteresis during loading and unloading. The maximum load during detachment of contact is defined as the pulloff force. Pulloff force can also be considered as a measure of the strength of adhesion. Similar to the experimental results reported in^{9,10}, as shown in Fig. 5a, the relative pulloff force P_{f}/P_{0} decays with increasing normalized surface roughness h_{rms}/α. Here, \({P}_{f}\) and \({P}_{0}\) are pulloff forces for rough and smooth surfaces respectively. Additionally, at constant \({h{\prime} }_{rms}\), the rate of decay of \({P}_{f}/{P}_{0}\,\)decreases with increase in the value of β. For low values of β, corresponding to stiff solids, the results are in accordance with the experiments on direct wafer bonding^{8}.
To understand the effect of hysteresis on pulloff force, we have investigated the ratio of energy dissipated (D) during a cycle to the amount of work done during approach (W) as a function of \({h}_{rms}/\alpha \) (see Fig. 5b). For each value of β, the onset of dissipation occurs at different values of roughness. Using these critical values of surface roughness, as shown in Fig. 5a, the red dashed line marks the onset of hysteresis dividing the space in regions A and B. In region B, at constant value of \({h}_{rms}/\alpha \), the increase in pulloff force for a given increase in β is larger in comparison to that in region A. This implies that the parameter \({h}_{rms}/\alpha \), acts to weaken and strengthen adhesion in region B and solely has a weakening effect in region A. Additionally, as shown in Fig. 5b, the ratio of bonded to nominal contact area \({A}^{0}/{A}_{0}\) also increases with increase in β. For a given value of β and \({h{\prime} }_{rms}\), higher the value of \({A}^{0}/{A}_{0}\), lower is the rate of decay of relative pulloff force \({P}_{f}/{P}_{0}\) with \({h}_{rms}/\alpha \). Hence, for a given \({h{\prime} }_{rms}\), relative pulloff force is a function of β, \({h}_{rms}/\alpha \) and \({A}^{0}/{A}_{0}\).
In Fig. 6a, at a constant \({h}_{rms}/\alpha \), we have investigated the effect of \({h{\prime} }_{rms}\) on relative pulloff force \({P}_{f}/{P}_{0}\). As it can be seen, we could collapse \(\,{P}_{f}/{P}_{0}\) data by scaling \({h{\prime} }_{rms}\) as \({\gamma }^{\ast }=(\gamma /{E}^{\ast }{g}_{N}{({h{\prime} }_{rms})}^{m})\). Here g_{N} is the smallscale amplitude and the exponent m has an exponential decay with Hurst exponent H (see Fig. 6b). In the fractal limit, the quantity \({g}_{N}{({h{\prime} }_{rms})}^{m}\) scales as \({\lambda }_{N}^{H+(H1)m}\,\) which is convergent for H ≥ 0.5. This convergence of \({\gamma }^{\ast }\) is similar to the conclusions found in references^{32,33,34,35,36,37,38} and in contrast with work done in refs. ^{39,40}. Hence, as shown in Fig. 7, for various values of \(\beta \), \({h}_{rms}\), \({h{\prime} }_{rms}\,\)and H, \(\,{P}_{f}/{P}_{0}\) is a function of two nondimensional parameters \({\gamma }^{\ast }\) and \({h}_{rms}/\alpha \).
Discussion
The strength of adhesive contact of a rough surface is a strong function of the statistical properties of the surface roughness. It is a ubiquitous natural observation that rougher surfaces adhere weakly or not at all compared to the smoother counterparts. Moreover, softer polymers such as glues, pressure sensitive adhesives tend to be sticky on relatively rougher surfaces. These observations suggest an intimate competition between surfaces roughness and material softness towards both strengthening and weakening adhesion. In this study, we have identified two nondimensional parameters, \({h}_{rms}/\alpha \) and \({\gamma }^{\ast }\), to influence adhesion of rough surface. The two asymptotic limits of this phenomena are the one with weak interactions and stiff solids and the other being strong interactions and soft solids. It is a well established result that surface roughness \({h}_{rms}\) monotonically decreases adhesion close to the limit of weak interactions and stiff solids. But, in the transition towards the softer solids and stronger interactions, \({h}_{rms}\) tends to contribute both to weakening and strengthening of adhesion. This effect is due to increase in energy dissipated due to contact instabilities. Therefore, for a constant \({h{\prime} }_{rms}\), adhesion is solely influenced by \({h}_{rms}/\alpha \). For a given \({h}_{rms}\), especially in the neighborhood of soft solids and strong interactions, strength of adhesion is also influenced by the strain energy stored near the interface at the bonded state. This energy can be interpreted as the available energy to pry the surfaces apart which depends on the contact area and its morphology at the bonded state. As shown in Fig. 8, this energy per unit bonded contact area has a power law scaling with \({h{\prime} }_{rms}\) for cases in the neighborhood of soft solids and strong interactions (higher values of β). In the other limit, there is barely any correlation with \({h{\prime} }_{rms}\) which implies that \({h}_{rms}/\alpha \) is the only relevant parameter for low values of \(\beta \) and \({\gamma }^{\ast }\)influences adhesion for moderate to high values of \(\beta \).
For a physical understanding of the parameter \({\gamma }^{\ast }\), following the energetic arguments of Johnson’s parameter^{41,42}, we have investigated the relation between \({\gamma }^{\ast }\) and the ratio of surface energy \({\Gamma }^{eff}\) to strain energy \({W}^{SE}\) at pulloff. The surface energy \({\Gamma }^{eff}\) denotes the amount of surface energy gained by the system if the surfaces are separated at pulloff. \({W}^{SE}\) is computed by subtracting the strain energy due to average pressure, homogenous part, from the total strain energy stored in the system at pulloff. As shown in the Fig. 6a, with decreasing \({\gamma }^{\ast }\), rapid increase in \({\Gamma }^{eff}\,/{W}^{SE}\) suggests that \({W}^{SE}\) tends to zero faster than \({\Gamma }^{eff}\,\)while approaching the nonadhesive limit. Beyond \({\gamma }^{\ast }\) equal to one, we found a power law scaling between \({\Gamma }^{eff}\,/{W}^{SE}\) and \({\gamma }^{\ast }\). The ratio \({\Gamma }^{eff}\,/{W}^{SE}\) at pulloff when expressed in terms of energy densities as \({\gamma }^{eff}\,/{U}^{SE}{d}^{i}\), length scale \({d}^{i}\,\)emerges as the depth over which the inhomogeneous strain energy decays. It turns out that \({d}^{i} \sim {h}_{rms}.\) Here, \({\gamma }^{eff}\) is computed as \({\Gamma }^{eff}\,/{\lambda }_{0}\). We have also investigated ratio \({\gamma }^{eff}\,/\gamma \) in relation to \({\gamma }^{\ast }\) for various values of \({h{\prime} }_{rms}\) as shown in Fig. 9. Similar to relative pulloff force \({P}_{f}/{P}_{0}\), \({\gamma }^{eff}\,/\gamma \) also has a sharp transition beyond \({\gamma }^{\ast }\)equal to one. Hence, the condition for an appreciable adhesion can be quantified as \({\gamma }^{\ast } > 1\) for \(H\ge 0.5\). To make sense of our criterion, we have looked at Dalhquist’s criterion^{36}, an empirical criterion for pressure sensitive adhesives which bounds the modulus of adhesive to about 0.3 MPa for stickiness. This criterion is an industry standard for the design of pressure sensitive adhesive tapes. For polymers, assuming an average chain length on the order of a micron, computing small scale amplitude \({g}_{n}\) along with taking \(\gamma \sim \,\)50 mJ/m^{2}, \(h{\text{'}}_{rms}=0.01\) and \(H=0.7\), \(E < 0.3\,MPa\) for \({\gamma }^{\ast } > 1\) which agrees very well with Dalhquist’s criterion.
This work is limited to twodimensions as it is computationally formidable to consider threedimensional case. Although a deterministic Weierstrass profile is considered, it captures selfaffinity of heights observed in real surfaces. We believe that the selfaffinity of any rough surface model is the most important characteristic to adequately capture the contact evolution and scaling.
Conclusions
In this work, we have investigated adhesive contact mechanics of rough surface characterized by a Weierstrass function using a consistent numerical approach in two dimensions. By capturing the contact instabilities and hysteresis, we have elucidated the approach and detachment of an adhesive rough contact to a process akin to interface crack zipping and unzipping. More importantly, we have identified two nondimensional parameters, \({h}_{rms}/\alpha \) and \({\gamma }^{\ast }\), which influence adhesion and shown that \({\gamma }^{\ast } > 1\) can be considered as stickiness criterion.
Methods
Real surfaces such as asphalt road, fractured rock and perspex surfaces are generally rough with a defining property of being selfaffine fractal over several decades of length scales^{43,44}. In this paper, we employ Weierstrass function to model the rough surface profile of twodimensional elastic substrate. The Weierstrass function is given by
The height profile is a spatially periodic function with infinitely many length scales. The amplitude \({g}_{n}\) at each length scale is proportional to \({\lambda }_{n}^{H}\), where H is the Hurst exponent (0 < H < 1). The wavelength \({\lambda }_{n}\) is defined as
The large length scale \({\lambda }_{0}\) which is equal to the system size is subdivided into smaller length scales \({\lambda }_{n}\) by integer powers of \(\xi \) (see Eq. 3).
As real surfaces have a physical lower length scale cutoff such as lattice constant for crystalline materials and fine grain size in disordered materials such as fractured rock surface, we use two times the equilibrium distance of interaction potential as the smallest length scale. In this work, we restrict the length scales to integers by considering \({\lambda }_{0}\) being powers of 2 with \(\xi \) equal to 2 and investigated system sizes from 64 to 16384. We have investigated surfaces with values of H equal to 0.3,0.5 and 0.7.
The general description of the procedure for generating a typical finite element mesh with rough surface profile for various root mean square roughness \({h}_{rms}\) and slope \(h{\text{'}}_{rms}\) is described in Supplementary Information.
To model the adhesive interaction between the flat rigid surface and elastic substrate with rough profile, we have implemented a local contact law using LennardJones (LJ) type force separation law (see Eq. 4).
The length dimension of our simulations is considered in units of equilibrium distance α_{0} of the LJ interaction model. To capture the jump to contact instability during the approach of two surfaces, we have added damping to the LJ force separation law. The added damping to the interaction law simulates the contact behavior as Maxwell type viscoelastic material with nonlinear spring in series with dashpot. The damping parameter and local interaction model was thoroughly validated in the study of adhesion between smooth elastic spheres by comparing against loaddepth curves obtained using RIKS arclength method^{45}. In this work, we have restricted the damping induced energy to a minimum in comparison with the strain energy stored in the body.
The boundary value problem of adhesive contact is simulated in generalpurpose finite element software ANSYS using large strain formulation under quasistatic indentation. The custom interaction model is implemented through a user subroutine USERINTER.
Data availability
The data used in this manuscript can be obtained from the authors upon reasonable request.
References
Kendall, K. Molecular adhesion and its applications: The sticky Universe, (Kluwer Academic, New York 2001).
Maboudain, R. Adhesion and friction issues associated with reliable operations of MEMS. MRS Bulletin 23, 47–51 (1998).
Komvopoulos, K. Adhesion and friction forces in microelectromechanical systems: mechanisms, measurement, surface modification techniques, and adhesion theory. J. Adhesion Sci. Technol. 17, 477–517 (2003).
Bhushan, B. Adhesion and stiction: Mechanisms, measurement techniques, and methods of reduction. J. Vac. Sci. Technol. B. 21, 2262–2296 (2003).
Rabinovich, Y. I., Adler, J. J., Ata, A., Singh, R. K. & Moudgil, B. M. Adhesion between nanoscale rough surfaces. J. Colloid. and Inter. Sci. 232, 17–24 (2000).
Beach, E. R., Tormoen, G. W., Drelich, J. & Han, R. Pulloff force measurements between rough surfaces by atomic force microscopy. J. Colloid. and Inter. Sci. 247, 84–99 (2002).
Quon, R. A., Knarr, R. F. & Vanderlick, T. K. Measurement of the deformation and adhesion of rough solids in contact. J. Phys. Chem. B. 103, 5320–5327 (1999).
Gui, C., Elwenspoek, M., Tas, N. & Gardeniers, G. E. The effect of surface roughness on direct wafer bonding. J. Appl. Phys. 85, 7448–7454 (1999).
Fuller, K. N. G. & Tabor, D. The effect of surface roughness on the adhesion of elastic solids. Proc. R. Soc. Lond. A. 345, 327–342 (1975).
Fuller, K. N. G. & Roberts, A. D. Rubber rolling on rough surfaces. J. Phys. D: Appl. Phys. 14, 221–239 (1981).
Guduru, P. R. & Bull, C. Detachment of a rigid solid from an elastic wavy surface: Experiments. J. Mech. and Phys. of solids 55, 473–488 (2007).
Johnson, K. L., Kendall, K. & Roberts, A. D. Surface energy and the contact of elastic soilds. Proc. R. Soc. Lond. A. 324, 301–313 (1971).
Greenwood, J. A. & Williamson, J. B. P. Contact of nominally flat surfaces. Proc. R. Soc. Lond. A. 453, 1277–1297 (1997).
Maugis, D. On the contact and adhesion of rough surfaces. J. Adhesion Sci. and Tech. 10, 161–175 (1996).
Morrow, C., Lovell, M., Ning, X. & JKRDMT, A. transition solution for adhesive rough surface contact. J. Phys. D: Appl. Phys. 36, 534–540 (2003).
Derjaguin, B. V., Muller, V. M. & Toporov, Y. P. Effect of contact deformation on the adhesion of particles. J. Colloid and Inter. Sci. 53, 314–326 (1975).
Kadin, Y., Kligerman, Y. & Etsion, I. Unloading an elasticplastic contact of rough surfaces. J. of the Mechanics and Physics of Solids 54, 2652–2674 (2006).
Kadin, Y., Kligerman, Y. & Etsion, I. Jumpin induced plastification of approaching adhesive microcontacts. J. Applied Physics 103, 013513 (2008).
Kadin, Y., Kligerman, Y. & Etsion, I. LoadingUnloading of an elasticplastic adhesive spherical microcontact. Journal of Colloid and Interface Science 321, 242–250 (2008).
Kadin, Y., Kligerman, Y. & Etsion, I. Cyclic loading of an elasticplastic adhesive spherical microcontact. J. Applied Physics 104, 073522 (2008).
Sahoo, P. & Banerjee, A. Asperity interaction in elasticplastic contact of rough surfaces in presence of adhesion. J. Phys. D: Appl. Phys. 38, 2841 (2005).
Sahoo, P. & Banerjee, A. Asperity interaction in adhesive contact of metallic rough surfaces. J. Phys. D: Appl. Phys. 38, 4096–4103 (2005).
Sahoo, P. Adhesion friction for elasticplastic contacting rough surfaces considering asperity interaction. J. Phys. D: Appl. Phys. 39, 2809 (2006).
Song, Z. & Komvopoulos, K. Adhesive contact of an elastic semiinfinite solid with a rigid rough surface: Strength of adhesion and contact instabilities. Int. J. Solids and Structures 51, 1197–1207 (2014).
Persson, B. N. J. On the elastic energy and stress correlation in the contact between elastic solids with randomly rough surfaces. J. Phys. Condens. Matter. 20, 312001–312003 (2008).
Hyun, S., Pei, L., Molinari, J.F. & Robbins, M. O. Finite element analysis of contact between elastic selfaffine surfaces. Phys. Rev. E. Stat. Nonlin. Soft Matter. Phys. 70, 026117–026130 (2004).
Ciavarella, M., Joe, J., Papangelo, A. & Barber, J. R. The role of adhesion in contact mechanics. J. R. Soc. Interface 16, 1–22 (2019).
Akarapu, S., Sharp, T. & Robbins, M. O. Stiffness of contact between rough surfaces. Phys. Rev. Lett. 106, 204301–204304 (2010).
Waters, J. F., Lee, S. & Guduru, P. R. Mechanics of axisymmetric wavy surface adhesion: JKRDMT transition solution. Int. J. Solids and Struc. 46, 1033–1042 (2009).
Bush, A. W., Gibson, R. D. & Thomas, T. R. The elastic contact of a rough surface. Wear 35, 87–111 (1975).
Persson, B. N. J. Elastoplastic contact between randomly rough surfaces. Phys. Rev. Lett. 87, 116101–116104 (2001).
Ciavarella, M. & Papangelo, A. Extensions and comparisons of BAM (Bearing Area Model) for stickiness of hard multiscale randomly rough surfaces. Tribology International 133, 263–270 (2019).
Violano, G., Afferrante, L., Papangelo, A. & Ciavarella, M. On stickiness of multiscale randomly rough surfaces, arXiv preprint arXiv 18101, 0960 (2018).
Ciavarella, M. Universal features in “stickiness” criteria for soft adhesion with rough surfaces, https://doi.org/10.1016/j.triboint.2019.106031 (2019).
Persson, B. N. J. & Tosatti, E. The effect of surface roughness on the adhesion of elastic solids. The Journal of Chem. Phys. 115, 5597–5610 (2001).
Dahlquist C Tack., Adhesion fundamentals and practice, New York: Gordon and Breach, 143–51 (1969).
Joe, J., Scaraggi, M. & Barber, J. R. Effect of finescale roughness on the tractions between contacting bodies. Tribol. Int. 111, 52–6 (2017).
Joe, J., Thouless, M. D. & Barber, J. R. Effect of roughness on the adhesive tractions between contacting bodies. J. Mech. Phys. Solids 118, 365–373 (2018).
Pastewka, L. & Robbins, M. O. Contact between rough surfaces and a critesion for macroscopic adhesion. Proceedings of the National Academy of Sciences 111, 3298–3303 (2014).
Muser, M. H. A dimensionless measure for adhesion and effects of the range of adhesion in contacts of nominally flat surfaces. Tribology International 100, 41–47 (2016).
Johnson, K. L. The adhesion of two elastic bodies with slightly wavy surfaces. Int. J. Solids and Struc. 32, 423–430 (1995).
Ciavarella, M. & Papangelo, A. A generalized Johnson parameter for pulloff decay in the adhesion of rough surfaces. Phys. Mesomechanics 21, 67–75 (2018).
Persson, B. N. J. Contact mechanics of randomly rough surfaces. Surface Sci. Reports 61, 201–227 (2006).
Persson, B. N. J., Albohr, O., Tartaglino, U., Volokitin & Tosatti, E. On the nature of surface roughness with application to contact mechanics, sealing, rubber friction and adhesion. J. Phys.: Condens. Matter. 17, R1–R62 (2004).
Radhakrishnan, H. & Mesarovic, S. D. Adhesive contact of elastic spheres revisited: numerical models and scaling. Proc. R. Soc. A. 465, 2231–2249 (2009).
Acknowledgements
H.R. and S.A. wish to appreciate the support and encouragement of ANSYS Inc. S.A. wish to express thanks to Prof. Mark Robbins for introducing to the important field of contact mechanics of rough surfaces.
Author information
Authors and Affiliations
Contributions
S.A. has designed the research; H.R. and S.A. performed the research; H.R. and S.A. analyzed the data and made the plots; S.A. wrote the paper; H.R. ran the simulations.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Radhakrishnan, H., Akarapu, S. Twodimensional finite element analysis of elastic adhesive contact of a rough surface. Sci Rep 10, 5402 (2020). https://doi.org/10.1038/s41598020611879
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41598020611879
This article is cited by

Is there more than one stickiness criterion?
Friction (2022)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.