## Abstract

The Weierstrass series was considered in Ciavarella *et al.* (Ciavarella *et al.* 2000 *Proc. R. Soc. Lond. A* **456**, 387–405. (doi:10.1098/rspa.2000.0522)) to describe a linear contact problem between a rigid fractally rough surface and an elastic half-plane. In such cases, no applied mean pressure is sufficiently large to ensure full contact, and specifically there are not even any contact areas of finite dimension. Later, Gao & Bower (Gao & Bower 2006 *Proc. R. Soc. A* **462**, 319–348. (doi:10.1098/rspa.2005.1563)) introduced plasticity in the Weierstrass model, but concluded that the fractal limit continued to lead to what they considered unphysical predictions of the true contact size and number of contact spots, similar to the elastic case. In this paper, we deal with the contact problem between rough surfaces in the presence of adhesion with the assumption of a Johnson, Kendall and Roberts (JKR) regime. We find that, for fractal dimension *D*>1.5, the presence of adhesion does not qualitatively modify the contact behaviour. However, for fractal dimension *D*<1.5, a regularization of the contact area can be observed at a large magnification where the contact area consists of segments of finite size. Moreover, full contact can occur at all scales for *D*<1.5 provided the mean contact pressure is larger than a certain value. We discuss, however, the implication of our assumption of a JKR regime.

## 1. Introduction

Contact between rough surfaces has attracted more and more scientific interest due to the wide implications that it has in many engineering systems. The first attempt to investigate the multiscale nature of elastic contact is due to Archard in the 1950s [1,2]. Afterwards, Greenwood and Williamson (GW model in the following) [3] dealt with this problem and introduced asperity-based models in which the distribution of contacting asperities is replaced by a distribution of Hertzian asperities with equivalent height and curvature. They showed that statistical distribution of height asperities leads approximately to linearity between contact area and normal load. The GW model neglects interactions and coalescing of contact spots. As a result, approaching full contact conditions, the area-load relation progressively deviates from linearity, as shown in [4], where a simplified model is proposed to take into account interaction and coalescing of asperities.

Persson [5] proposed a different approach giving an exact solution in complete contact. Solution in partial contact is obtained by arguing that adding an increment of roughness corresponds to an increment of magnification, so the probability density function of the contact pressure must satisfy a diffusion equation. However, only with successive development of the theory [6], based on the introduction of a universal correcting factor to properly calculate the elastic energy stored at the interface, are more accurate predictions of the contact area obtained. However, the slope of the linear relation between contact area and load continue to differ from the predictions of multi-asperity contact theories and from numerically calculated values (see e.g. [7–10]).

Majumdar & Bhushan [11,12], Borri-Brunetto *et al.* [13] and then Ciavarella *et al.* (CDBJ model in the following) [14] introduced other models. In particular, CDBJ used a Weierstrass profile demonstrating that extended regions of contact are not possible with this model, according to similar conclusions reached by Persson [5]. Gao & Bower [15] introduced plasticity in the Weierstrass model, and concluded that the fractal limit still leads to unphysical predictions of the true contact size and number of contact spots, for both elastic and elastic–plastic solids.

The presence of adhesion may be deleterious in micro- and nano-devices, e.g. in electromechanical switchers [16], but, in some cases, adhesion may be beneficial [17–22]. The area of real contact is largely influenced by adhesion and Gao & Bower [15] argued that adhesion can regularize the ‘ill posed nature of contact even for perfectly fractal surfaces’, mentioning a recent work [23]. The latter shows that adhesion has mainly an effect for surfaces of low fractal dimensions, which in fact are those most commonly found in the real world [24]. Corroboration of these results can be also found in [25], where adhesion of a thin elastic plate to a randomly rough hard substrate is discussed. We follow here Gao & Bower's suggestion to extend CDBJ to include adhesion, and we show indeed that the regularization of the contact area does occur for low fractal dimensions (*D*<1.5, for one-dimensional profiles), as suggested in [23]. The results we obtain are, however, limited by our assumptions, and specifically the way load is redistributed as per the original Archard procedure, and by the fact of using a Johnson, Kendall and Roberts (JKR) calculation for the adhesion mechanics [26]. It is clear that the JKR analysis has a lower limit size for which it can be applied (and this is clear even from a Tabor index which even for a sphere points to JKR giving place to a DMT criterion). This correction is relevant mostly for large fractal dimensions where we find that the contact is a fractal lacunar region similar to the adhesiveless case (see also discussion in [27]). This range of high fractal dimensions is generally considered to be rare [24], but it is clear that some regularization would occur also in this case considering a full potential for the van der Waals forces, instead of the JKR assumption.

## 2. Adhesion of slightly wavy surfaces

In this section we briefly recall the solution, given in [28], of the contact problem in the presence of adhesion between a two-dimensional elastic half-plane and a rigid body with a sinusoidal profile defined by the function
_{0} are, respectively, the amplitude and wavelength of the sine wave.

### (a) Partial contact

In the absence of adhesion, when contact occurs at the crests of the profile on segments of width 2*a*, i.e. when the mean pressure *E**=*E*/(1−*ν*^{2}), the contact pressure can be written in closed-form as (Westergaard's solution [29])
*a* by

In the presence of adhesion, the solution of the problem can be obtained by superposing on *p*^{′}(*x*) a negative (tensile) pressure *p*^{′′}(*x*), corresponding to the tensile loading of a plane containing an array of equally spaced cracks, each of length 2*b*=λ_{0}−2*a*, whose solution was given by Koiter [30],

The mean tensile pressure *a* by equating the work of adhesion *w* to the elastic strain energy release rate *G*=*K*^{2}_{I}/2*E**, being *K*_{I} the mode I stress intensity factor,
*α*_{0} is a dimensionless parameter depending on the work of adhesion *w*

The net pressure distribution at the contacts is then given by the superposition of *p*^{′}(*x*) and *p*^{′′}(*x*). Consequently, the mean pressure *a*
*x*=0, which is

Figure 1 shows the variation of the mean pressure *πa*/λ_{0}, for different values of the adhesion parameter *α*_{0}.

Since the problem is force controlled, under a compressive load, the decreasing parts of the curves correspond to unstable solutions. Therefore, when the rigid sinusoidal profile is approached to the elastic half-plane with zero mean pressure, contact is immediately established, due to the action of the adhesive forces, and the equilibrium condition at point B is reached. This process is not reversible. In fact, the variation in adhesive surface energy occurring in this case is larger than the elastic strain energy associated with the deformation, the excess being dissipated in the propagation of stress waves. From this point, the pressure follows the equilibrium curve up to the maximum (point C); beyond it complete contact occurs. The maximum mean pressure necessary to establish full contact is a fraction *κ* of *α*_{0}. For *α*_{0}>*α*_{cr}≃0.57, point B disappears and partial contact may not occur with compressive mean pressures. In this case, the solution simply leads to full contact even at zero load. Note, when contact is established, it may be maintained also for negative (tensile) mean pressures, provided *η* is a negative parameter, a function of *α*_{0}. When

Note the parameter *α*_{0} is proportional to the ratio between the adhesive surface energy in one wavelength and the elastic strain energy required to flatten the sinusoid and is related to the modified Tabor parameter *μ* by
*σ*_{0} being the theoretical strength of the interface.

At high values of *α*_{0}, the adhesion energy is sufficiently high to ensure complete contact. In such cases, no finite traction is strictly sufficient to cause pull off according to the JKR theory. Since the JKR is an approximation of the Lennard-Jones force law, pull off initiates as soon as the maximum tensile traction at the troughs of the sinusoid reaches the theoretical strength *σ*_{0}. However, in real cases, separation is obtained for smaller values of the tensile traction, so in [28] it is argued that the presence of interfacial defects, acting like small cracks, must be invoked to initiate pull off.

Alternatively, we can observe that at high values of *α*_{0}, corresponding to *μ*<1, a different approach with respect to the Johnson one should be developed, as discussed in §5.

#### (i) Probability distribution of contact pressure

According to the approach given in [14], the cumulative probability pressure distribution _{0} over which the contact pressure exceeds a given value, *p*, i.e.
*y* is a function of *p*, *α*_{0}, and its closed-form expression is given in appendix A.

The cumulative probability distribution for the contact pressure is then
*Q* and *q* are functions of the mean pressure

### (b) Complete contact

When the mean pressure, *x*=0, is

#### (i) Probability distribution of contact pressure

The cumulative probability pressure distribution _{0} over which the contact pressure is larger than a given value *p*, is obtained from equation (2.17)

Then, the probability density function for contact pressure

## 3. Adhesion of rough surfaces

Consider a rough surface containing a series of superposed sine waves, defined by the Weierstrass function [31,32]
*n*th term has amplitude and wavelength given by, respectively,
*γ*>1 and *D*>1, equation (3.1) defines a plane fractal surface of fractal dimension *D*.

### (a) Complete contact

If the mean pressure

For contact to be continuous the pressure must be positive everywhere so that *γ*>1 and *D*>1, the series (3.5) does not converge, there is no finite value of mean pressure

However, when adhesion is postulated at the interface, depending on the values of *γ*, *D* and *α*_{0}, finite values of mean pressure *n* according to the relation: *α*_{n}=*α*_{0}(*γ*^{(2D−3)n})^{−1/2}. Figure 2 shows the variation of *α*_{n} with *n*, for different values of the fractal dimension *D*.

For *D*>1.5, *α*_{n} reduces with *n*. In this case, complete contact may not occur for finite values of *α*_{n} moves towards zero in the limit of large values of *n*. This entails that, at the short length-scale structures of the rough surface, adhesion is completely destroyed.

For *D*<1.5, the adhesion parameter increases with *n*. In such case, full contact can occur at all scales provided the mean contact pressure

For *D*=1.5, *α*_{n} is constant with *n* and complete contact at all scales occurs only if *α*_{cr} if the mean contact pressure is compressive).

### (b) Partial contact

#### (i) Probability distribution of contact pressure

If we consider a fractal surfaces of the form (3.1) with *γ*≫1, so that there are many waves of scale *n* in one wavelength of scale *n*−1, the mean pressure at the scale *n* can be approximated by the pressure *p*_{n−1} at scale *n*−1, because it changes only slightly over one wavelength λ_{n}. Under these assumptions, in [14] it is shown that the problem can be solved with a recursive application of Westergaard's solution with applied mean pressures which varies during the process. This is essentially the Archard's assumption that the load carried locally at each given location is then redistributed to a sufficiently large number of asperities at the higher level of roughness. This permits an ‘uncoupling’ of scales in the calculation of the redistribution of the pressure from one scale to the next smaller one. Therefore, in our notation, this is applied for Weierstrass profiles with *γ*≫1, solving the contact problem at the scale *n*−1 and then locally redistributing the contact load at the smaller scale *n*. The process shares with the Persson magnification process [5] the idea that roughness appears in the process incrementally with shorter and shorter wavelengths, but the two processes (Archard and Persson's) have some differences and lead to different results [33]. Therefore, equations given in section carry over into this problem, with the substitutions

The function *p*_{n}, given *p*_{n−1}, i.e. *q*_{n−1}(*p*_{n−1}), we can calculate the probability distribution *q*_{n}(*p*_{n}) by summation over all values of *p*_{n−1}. As a result, splitting the integral into two intervals corresponding to partial contact and full contact, we can write

Moreover, the limits of integration of *I*_{1} and *I*_{2} are further restricted by the inequalities given in (2.15), (2.16), (2.23) and (2.24), leading to
*p*_{n}, is the value of the mean pressure for which

Note

At scale *n*=0, we have *q*_{n}(*p*_{n}).

#### (ii) Contact area

The conditional probability of contact area, given *p*_{n−1}, is unity for complete contact and 2*ψ*_{a}/*π* for partial contact, being *ψ*_{a} solution of the following equation

Therefore, according to [14], the total contact area *A*_{n} at scale *n*, can be calculated by

Figure 3 shows the contact area *A*_{n}/λ_{0} as a function of the scale *n* in a semi-logarithmic plot, for *γ*=5 and a fractal dimension *D*=1.75(>1.5). Plots are given for different values of the adhesion parameter at scale *n*=0, and *a*) and *b*).

Similar plots are shown in figure 4 for *γ*=5 and a fractal dimension *D*=1.35(<1.5) and two different mean contact pressures: *a*) and *b*).

In the absence of adhesion (*α*_{0}=0), the same results shown in [14] are recovered. In this case, the contact area continuously decreases with *n*, showing there is no contact segment of finite size in the fractal limit. The contact area will be constituted by segments of infinitesimal size. This result is independent of *γ*, fractal dimension *D* and mean pressure *n*, the relation between the total contact area and the scale number tends to a straight line with a negative slope, which does not depend on *n*, the curves for low pressures are steeper and the total contact area deviates from the power-law form. At high pressures, full contact occurs at low scales, but increasing *n* partial contact dominates the contact behaviour.

In presence of adhesion (*α*_{0}>0), we observe a different behaviour depending on the value of the fractal dimension *D*. When *D*>1.5 (figure 3), no qualitatively differences in results are observed with respect to the case without adhesion. In fact, in the limit of relatively large scales, the total contact area tends to a decreasing linear curve, so showing a limiting power-law fractal behaviour. This result occurs also at high mean pressure and *α*_{0}.

When *D*<1.5 (figure 4), the contact area reaches a constant value as the magnification is increased and full contact occurs at the short length-scale structures of the rough surface. In this case, therefore, fractality is destroyed by adhesion and the total contact area is constituted by segments of finite size. The present results are in agreement with [25], where numerical calculations are performed about the problem of adhesion of plates on rough surfaces. In this case, the threshold fractal dimension is obviously 2.5. It is shown that, for *D*<2.5, the area of contact reaches a constant value as the magnification is increased and the solid may rest in full contact with the short length-scale structure of the rough surface. On the contrary, when *D*>2.5, a continuous decrease of the contact area with magnification is observed and only partial contact may occur at the interface between the solid and the fine scale surface roughness.

## 4. Detachment from complete contact

There is no doubt that in the case of high fractal dimensions *D*>1.5, since we predict a sparse fractal contact area, with nowhere areas of full contact, we should also have a small pull-off adhesive load. It remains more interesting to see the case of low fractal dimensions *D*<1.5, for which we expect a finite total contact area, which is characterized by areas of separations, interspersed with areas of ‘full’ contact, where successive small wavelengths do not change the apparent converged solution. We can therefore define a general procedure to compute the pull off from a generic ‘defect’, where a defect could be an existing area of separation, or, in the limit case of starting from full contact, a defect postulated at the interface due, for example, to the presence of small dust particles or trapped air, in analogy to what Johnson [28] suggests for the single sinusoid when full contact is achieved during the loading stage. In particular, Johnson [28] proposed a very simple formula to calculate the tensile mean pressure *b*_{f} and located at a trough
*b*_{f}≪λ. The original Johnson's formula is slightly different presumably due to typesetting errors.

For a rough contact, we can calculate the strength of adhesion as the conditions under which the ‘defect’ (or an area of separation) grows. We assume the defect with size 2*b*_{f} located at a trough of the fundamental sine wave with amplitude _{0}. Moreover, we consider the case in which complete contact at all scales can occur (*D*<1.5). In such cases, the mean contact pressure required to have full contact is given by equation (3.6). The condition of zero surface forces within the defect determines a negative tensile pressure acting on the surfaces of the flaw (−*b*_{f}≤*ξ*≤*b*_{f})
*b*_{f} is of the order of magnitude of the wavelength λ_{m}, the above expression can be simplified
*m* terms of the series are requested to close the defect.

Under the above assumptions, the stress intensity factor to the ends of the crack can be calculated as
*b*_{f} the critical stress

## 5. Discussion about the fractal limit

The JKR solution may become inappropriate in the limit for a sinusoid of high amplitude/wavelength ratios, as unfortunately is the case of the sinusoids in the Weierstrass or, very similarly, in a power law continuum fractal spectrum. Using the latter case for the notation, the amplitude over wavelength ratio follows a law *H*<1 (but particularly worrying for low *H*, i.e. large fractal dimensions). Starting, therefore, with the case of large fractal dimensions, where Johnson's parameter *α* tends to decrease and therefore at sufficiently low wavelength tends to zero as *w*=*p*_{0}*ε*, where *p*_{0} is the theoretical strength of adhesion, and *ε* is the characteristic length scale in the Lennard-Jones potential. This means that, contrary to the intuitive expectation, for small wavelengths (5.1) seems to diverge (and in particular at one point would even go beyond the theoretical strength itself), although this tension should then be multiplied by the number of contacts and it is unclear *a priori* if the actual pull-off load is finite or not. However, the JKR solution clearly becomes inappropriate at a sufficiently small wavelength, since the amplitude *ε* of the Lennard-Jones force law, and attractive tractions in the separation regions will then have a significant effect. Wu [34] used a numerical method to solve the elastic contact problem with the exact force law for a single sinusoid and showed that the JKR solution is in error when the modified Tabor parameter *μ*<1. In the fractal limit *μ*≪1, a Bradley solution may be preferred to a JKR solution, and this has already been commented on by Scaraggi & Persson [27]. It can be easily shown that a Bradley solution in the limit of relatively large values of

Note that if we were to use Lennard-Jones or Maugis kind of approximation, the curvature of the asperities, which is crucial in the JKR model, has in reality no effect and it is the amplitude of the roughness that is critical, and we might get full contact also for *D*>1.5.

## 6. Conclusion

The present results show that for a rough profile defined by the Weierstrass function, the contact behaviour can significantly change in the presence of adhesion. In particular, for fractal dimension *D*>1.5, no qualitatively differences are observed with respect to the case without adhesion. The contact area is defined by a fractal set, in the sense that contact is constituted by segments of infinitesimal size, so the total contact area tends regularly to zero. For fractal dimension *D*<1.5, complete contact occurs at the short length-scale structures of the rough surface. In this case, the total contact area is restricted to segments of finite dimension in the limit of sufficiently large scales. Moreover, for *D*<1.5, full contact on all scales can occur provided the mean contact pressure exceeds

## Data accessibility

Data plotted in Figs 3 and 4 are deposited in Dryad : (http://dx.doi.org/10.5061/dryad.ms046).

## Author' contributions

L.A. and M.C. conceived of the study; L.A. designed the study, developed the model, carried out the calculations and drafted the manuscript; M.C. and G.D. helped draft the manuscript. All authors gave final approval for the publication.

## Competing interests

We have no competing interests.

## Funding

The Italian Ministry of Education, University and Research supported this research activity within the project PON02-00576-3333604.

## Appendix A. Calculation of the function *y*

After some algebraic manipulations, inequality (2.11) can be rearranged as
*y* is, in turn, the solution of
*y* only depends on *p*, *α*_{0}.

Being (6.3) a third degree polynomial equation, solutions can be found with the method due to Scipione del Ferro and Tartaglia, published by Gerolamo Cardano in 1545 [35],

- Received April 14, 2015.
- Accepted August 25, 2015.

- © 2015 The Author(s)

Published by the Royal Society. All rights reserved.