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Edited by: Irina Georgievna Goryacheva, Institute for Problems in Mechanics (RAS), Russia

Reviewed by: Feodor M. Borodich, Cardiff University, United Kingdom; Luciano Afferrante, Politecnico di Bari, Italy

This article was submitted to Tribology, a section of the journal Frontiers in Mechanical Engineering

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The adhesive contact between a parabolic indenter with superimposed roughness and an elastic half space is studied in the JKR-limit (infinitely small range of action of adhesive forces) using the boundary element method with mesh-dependent detachment criterion suggested in 2015. Three types of superimposed roughness are considered: one- and two-dimensional waviness and randomly rough roughness. It is shown that in the case of regular waviness, the character of adhesion is governed by the Johnson adhesion parameter. For our randomly rough surfaces a new adhesion parameter has been identified numerically, which uniquely determines the adhesive strength of the contact.

It is well-known that neutral bodies attract each other by van der Waals forces. However, adhesive forces in macroscopic systems often are negligible. Kendall expressed this with his famous statement “solids are expected to adhere; the question is to explain why they do not, rather than why they do!” (Kendall,

The influence of simplified roughness in form of regular waviness on adhesion in the whole range of roughness amplitudes from “very rough” to “practically smooth” was studied by Johnson in 1995. He introduced a dimensionless parameter governing the adhesion behavior. He found that above a critical value of this “Johnson parameter,” surfaces jump into complete contact even at zero load (Johnson,

In the present paper we numerically study contacts of parabolic bodies with superimposed roughness either in form of a regular waviness or multi-scale roughness. We restrict ourselves to the limiting case of very short ranged adhesive forces which we will call the JKR-limit, because the JKR theory uses exactly the same assumption. Application of the Boundary Element Method for simulation of adhesive contacts in the JKR limit was made possible in 2015 by means of a mesh-size dependent detachment criterion proposed by Pohrt and Popov (Pohrt and Popov,

The structure of this paper is the following. In section Adhesive Contact of One- and Two-Dimensional Wavy Surface With Periodic Boundary Conditions we recapitulate the known analytical results for adhesive contact between a wavy surface (one- and two-dimensional waviness) and an elastic half space and reproduce them by numerical simulation. In section Adhesive Contact of a Sphere With a Two-Dimensional Wavy Roughness the adhesive contact of a curved surface with superimposed waviness is considered. We show that the character of adhesion in this case is essentially governed by the Johnson parameter. In section Adhesive Contact of a Parabolic Indenter With a Random Roughness contacts of curved surfaces with superimposed random roughness are investigated. It is noted that the numerical simulation in sections Adhesive Contact of a Sphere With a Two-Dimensional Wavy Roughness and Adhesive Contact of a Parabolic Indenter With a Random Roughness are carried out only for the pull-off (unloading). Section Conclusion gives conclusions on the above cases.

Basic understanding of the role of roughness on adhesion has been achieved by Johnson (Johnson,

and an elastic half space with effective elastic modulus ^{*} = ^{2}), where

where _{a} = π

where γ is the work of adhesion per unit area. In the following, we will refer to α as the “Johnson parameter.”

The relation is shown in

Dependence of the mean pressure on the contact half-width in “one-dimensional” adhesive contact. Solid lines show analytical solution and symbols present results of numerical simulation with BEM.

In the paper by Johnson (

were also briefly discussed. An analytical solution was given only for the extreme case of large load (approaching the complete contact) based on the asymptotic solution of non-adhesive contact (Johnson et al.,

Here _{b} = π

Dependencies of the average pressure on the normalized contact radius obtained by BEM are shown in

Dependence of the normalized average pressure on the normalized contact radius with two-dimensional waviness

Consider a parabolic shape with a superimposed waviness:

where

Illustration of a regularly

Adhesive contact of a rough sphere.

Numerical simulation of the pull-off was again carried out using the BEM under the displacement-controlled condition. In this series of simulations we did not use periodic boundary conditions but free boundaries like in Hertz-theory. Due to the macroscopic curvature, a new dimensionless parameter λ/_{ad} defined as the greatest tensile force corresponding to the minimum of the force-displacement relation. The absolute values of this force for different amplitudes of waviness are shown in

To prove this hypothesis, we replotted

Dependence of adhesive force on the Johnson parameter

The dependence of adhesive force on the Johnson parameter can be approximated as

The maximum tensile force during adhesive detachment of a sphere with superposed regular waviness can thus be expressed as

In this section we consider the adhesion of a rough sphere. On the one hand, it is widely accepted that roughness can significantly reduce adhesion. In Fuller and Tabor (

Consider a parabolic profile with curvature radius _{0}, _{1} and the rms roughness _{0}, _{1}. The exponent of the power-law dependency, _{min} = 2π/_{max} = π/Δ_{0} = _{min} and _{1} = _{max}. Note that the roughness produced in the described way does not necessarily represent real surfaces, which normally show strong phase correlation (Borodich et al.,

The pull-off was simulated for a number of increasing amplitudes of roughness and for different Hurst exponents. A few examples of force-distance curve are shown in

Force-distance curves for different roughnesses and Hurst exponents

Dependence of adhesive force on ^{*}.

As we have seen in sections Adhesive Contact of One- and Two-Dimensional Wavy Surface With Periodic Boundary Conditions and Adhesive Contact of a Sphere With a Two-Dimensional Wavy Roughness, the main governing parameter for adhesion of wavy surfaces is the Johnson parameter. In the case of randomly rough surfaces, this parameter cannot be determined from the waviness because the roughness is a superposition of a large number of waves with various amplitudes. If any, a series of Johnson parameters for different scales would have to be defined. If a single governing parameter similar to the Johnson parameter does exist in this case, it may depend on all material and geometric parameters of the contact problem: elastic modulus ^{*}, surface energy γ, surface roughness _{0} and _{1}. Based on the results presented in

Note that if there is only one wave vector _{0} = _{1}, equation (11) reduces to the original Johnson parameter equation (3). When the force of adhesion is plotted as function of α^{*}, all results collapse to a single dependency shown in

Different from the case of regular waviness in section Adhesive Contact of a Sphere With a Two-Dimensional Wavy Roughness, the pull-off force here doesn't drop dramatically at some critical value apparently, so we give an analysis of the relation of pull-off force and the adhesion parameter in this transition region. It is found that the following Gaussian-type expression gives a good approximation:

It covers the range from very low force |_{ad}/_{JKR}| ≈ 0 transiting to the value |_{ad}/_{JKR}| ≈ 1 for α^{*} = 0.344. In the range for larger α^{*}, enhanced adhesion is not always observed for large Hurst exponents, so an analytical approximation is not given in this study. Furthermore, there is not a clear critical value of α^{*} as in the case of regular waviness, therefore we give here only a value of adhesion parameter α^{*} = 0.26 for |_{ad}/_{JKR}| ≈ 0.5.

_{ad}/_{JKR}| ≈ 0.8. In the third column, a large roughness value is chosen with |_{ad}/_{JKR}| < 0.1. If the roughness is small enough, the contact area is almost compact. This is particularly pronounced for large

Series of contact configurations during the pull-off process.

The adhesive detachment of curved surfaces with two-dimensional waviness from an elastic half space was numerically simulated using the boundary element method. We found that waviness can both increase and decrease the adhesive force compared to the smooth JKR solution. The deviation from JKR behavior depends mainly on the unmodified Johnson adhesion parameter α. At some critical value

In the case of added random roughness, the simulation was carried out for different Hurst exponents and amplitudes of roughness. In all cases we observed that the adhesive force at small roughness amplitudes is increased. For roughness amplitudes above some critical value, the adhesion is decreased significantly. This observation is in agreement with experiments, such as those conducted by Briggs and Briscoe (

VP and RP conceived the study. VP carried out analytical analysis. QL carried out numerical simulations. All authors drafted and reviewed the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.