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On Pastewka and Robbins' Criterion for Macroscopic Adhesion of Rough Surfaces

Mike Ciavarella's picture

On Pastewka and Robbins' Criterion for Macroscopic Adhesion of Rough Surfaces

M. Ciavarella

[+] Author and Article Information

J. Tribol 139(3), 031404 (Nov 30, 2016) (5 pages)

Paper No: TRIB-16-1057; doi: 10.1115/1.4034530

History: Received February 15, 2016; Revised July 18, 2016




Abstract | Introduction | A Simple Asperity Model | Pull-Off | Discussion | Conclusion | References

Pastewka and Robbins (2014, “Contact Between Rough Surfaces and a Criterion for Macroscopic Adhesion,” Proc. Natl. Acad. Sci., 111(9), pp. 3298–3303) recently have proposed a criterion to distinguish when two surfaces will stick together or not and suggested that it shows quantitative and qualitative large conflicts with asperity theories. However, a comparison with asperity theories is not really attempted, except in pull-off data which show finite pull-off values in cases where both their own criterion and an asperity based one seem to suggest nonstickiness, and the results are in these respects inconclusive. Here, we find that their criterion corresponds very closely to an asperity model one (provided we use their very simplified form of the Derjaguin–Muller–Toporov (DMT) adhesion regime which introduces a dependence on the range of attractive forces) when bandwidth α is small, but otherwise involves a root-mean-square (RMS) amplitude of roughness reduced by a factor α−−√α. Therefore, it implies that the stickiness of any rough surface is the same as that of the surface where practically all the wavelength components of roughness are removed except the very fine ones.


Mike Ciavarella's picture

so if anyone is interested, I can discuss privately the developments.........


Mike Ciavarella's picture

Despite these days it is impossible to talk of GW models because the "fractal" community suggests interaction and multiscale effects are completely wrong in GW, here are some GW models with adhesion which seem to work against experiments, showing the main effect of rms roughness, contrary to what the PR model says in the rush to forget all GW, that stickiness depends only on slopes and curvature

Mike Ciavarella's picture

In this new paper accepted in Trib Int we show that the deviation from PR to GW cannot be all attributed to scatter due to imperfect tails of Gaussian surfaces as easily provoqued by low fractal dimension with insufficient size of the domain

Mike Ciavarella's picture

This paper may help shed some ligth why the very numerous crude approximations in PR model may have coincidentally led to reasonable agreement in their set of parameters.   But not for any extrapolation fundamental purpose!   The paper is submitted but probably siib accepted, as reviewers were positive.

1. arXiv:1701.04300 [pdfpsother]

On the use of DMT approximations in adhesive contacts, with remarks on random rough contactsMichele CiavarellaComments: 11 pages, 5 figuresSubjects: Materials Science (cond-mat.mtrl-sci)

Mike Ciavarella's picture

On the use of DMT approximations in adhesive contacts, with remarks on random rough contacts

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The approximation of the DMT theory (to neglect deformations due to attractive forces), is receiving some new interest.

We show it leads to extremely large overestimations of the adhesive forces in the case of spherical contact, even at relatively small Tabor parameters, except at pull-off.

For cylindrical contact, the opposite trend is found: DMT now overestimates pull-off by large factors, but for larger contact areas, the error becomes in the opposite direction.

This may explain why some partial success has been found for rough contacts in some limited range of parameters, considering the geometry of contacts can be very elongated.



The contact between rough surfaces with adhesion is an extremely difficult problem, and the approximation of the DMT theory (to neglect deformations due to attractive forces), originally developed for spherical contact of very small radius, is receiving some new interest. The DMT approximation leads to extremely large overestimations of the adhesive forces in the case of spherical contact, except at pull-off. For cylindrical contact, the opposite trend is found for larger contact areas. These findings suggest some caution in solving rough contacts with DMT models, unless the Tabor parameter is really low. Further approximate models like that of Pastewka & Robbins’ may be explained to work only due to a coincidence of error cancellation in their range of parameters.


  • Adhesion
  • Maugis' theory
  • Rough surfaces
  • DMT theory
  • JKR theory


Article title: On the use of DMT approximations in adhesive contacts, with remarks on random rough contacts

Article reference: JTRI4712

Journal title: Tribology International

Corresponding author: Prof Michele Ciavarella


Sir, when we are generating a self-affine fractal rough surface using the Power Spectral Density, does the Longuet-Higgins theory that "by counting the zeros and the extrema we can estimate the moments m2 and m4" still apply? Thank you. 

Mike Ciavarella's picture

Sir, I don't understand your question.  If you know PSD, you can integrate it to get the moments.  Longuet-Higgins is a theory to study the maxima, minima, zero-crossings, no need to use that to compute m2 and m4.  Please be precise.

Sir, Thank you so much for the prompt reply.

I meant let's say,

a) we generate an isotropic rough surface using PSD and get the moments m2 and m4 by integrating the PSD.

b) Then after generating the rough surface, if we take an arbitrary profile of the surface and count the number of zeros and extrema and using the method mentioned in Longuet-Higgins' paper on isotropic rough surfaces caclulate the moments m2 and m4.  

Will the values of m2 and m4 obtained by methods a) and b) match? Thank you.

Mike Ciavarella's picture

The relation between profile and surface is not that obvious. I suspect many papers around make some mistake (possibly even Pastewka-Robbins one on which this post started!).  In short, Nayak paper of 1971 explains it all, although it may not be the easiest. There is no problem with m0, but there is with m2 --- the slope has a factor 2 difference, because slope in x direction and slòpe in y-direction are independent uncorrelated processes see 

Some observations on Persson's diffusion theory of elastic contact

Thank you for the insight and the link. I was always confused by the interchanging use of Surface PSD and Profile PSD in many of the papers. This mostly clears up that aspect. Nayak's paper is indeed a bit tough to understand.

Why do most of the papers insist on the Gaussianity of height distribution? Like in this paper by Yastrebov et al. they have a detailed discussion on how the lower and upper cut-off wavenumbers affect the Gaussianity of the surface. Is it because all the major asperity contact theories and Persson's theory consider the Gaussian rough surfaces or is their a practical aspect to this as well? Thank you.

Mike Ciavarella's picture

The reason to insist on gaussianity is that the maths is much simpler.  Most people are use to measure roughness assuming it is gaussian, and do not know much statistics.   

In general, gaussian is the very reason of success of Greenwood-Williamson model:  quite a few models before that assumed unrealistic asperity distributions, including a russian one assuming uniform distribution of which I forget the name.  GW paper itself doesn't show a very good evidence of gaussianity of experimental surfaces, except perhaps the top of the distribution.  This is discussed further in a recent paper of mine

On the effect of wear on asperity height distributions, and the corresponding effect in the mechanical responseOn the significance of asperity models predictions of rough contact with respect to recent alternative theories

When the random processes appeared on the scene, the need to assume gaussianity became even more a condition for much easier mathematical development of Nayak and Longuet-Higgins.  Again, this is vaguely based, as any gaussian distribution, on the Central Limit Theorem, which applies with a large number of independent process of about the same variance.

This last condition is not really easy to obtain from a fractal. In a typical fractal, you have essentially a Fourier series whose terms have different "size", and therefore CLT does not apply.   Persson has made his entire career on the gaussian assumption because his model strongly assumes gaussianity.   Notice that the original derivation of Persson's theory is very involute and takes 50 pages of physics journals, whereas I obtain it in two steps from a more mechanical procedure in

Rough contacts near full contact with a very simple asperity model

Yastrebov is right to show that to obtain gaussian fractals, you have to be very careful.  Persson and co. have suggested "roll-off" component of PSD to increase the number of "nearly equal" component and get closer to CLT.  But this is really a distraction and very difficult to understand if real surfaces really have a roll-off or not.   Too much roll-off, and you have no longer a fractal!   So you do need a very large window rather than roll-off, in your random surface, i.e. much larger than lower cutoff in wavenumber, and you should have a good gaussian surface.

But the question remains: are real surfaces gaussian?  Some people now are starting to general Weibull fractals using RMD. I will explain later how to do that.  This is mainly because of my papers above, especially in adhesion, which question this fuss about gaussianity.

Final remark:  the fact that you have a power-law PSD does not imply you have a fractal.  It may simply be a square-form signal whose Fourier decomposition gives a invese cubic power-law!  You would beed to check the phases between different components, or higher-order autocorrelation functions.

Anyway, the sad part of this huge literature, it that it is really academic. The main point of GW is essentially showing the linearity of real contact area and load due to the fact that the number of contact spots increases with load in a way that both area and load grow proportionally.  What Persson's theory found was generally quite academic improvement.

Is it a secret to explain what you are planning to do?

Thank you sir, this addresses many of my doubts and as it happens my doubt on gaussianity, partly, was a result of thoughts on the first two papers you mentioned here.  Correct me if I'm wrong, but from what I've seen from most of the papers that they address only the gaussianity because they are more concerned about the "qualitative" behaviour of the load-separtion & load-area relationships.  

As I am new to this I am currently starting with a 1D roughness which is equivalent to the 2D isotropic rough as in papers by Popov and also in this 2013 paper by Scaraggi et al and analyse it's contact with a plane surface using FEM. So, the questions on finding m2 and gaussianity. Thank you. 

Mike Ciavarella's picture

FEM is not the best way forward for rough contact.  There are full 2D codes available public domain, including surface generators by Lars Pastewka in fact.

the contact mechanics calculator is here:

You can find a rough surface generator here: 

So you need to define a more interesting project!

Wow! I searched a lot about this type of material/codes/notes that could help me kick start but I couldn't find much. These are hugely helpful for me. Are there any other forums/repositories/discussion boards where computational contact is discussed?

     As I am fairly new to not just rough surface contact but to contact itself and already a bit familiar with FE we thought we'll start with something small just to get a feel of rough surface generation and contact. We (i.e me and my guide) haven't yet finalized on the directon. Thank you so much. 

Mike Ciavarella's picture

on numerical methods for rough contacts, almost everything has been said, you may look atThe contact mechanics challenge: Problem definition

That effort is going to appear as a paper soon in Trib. Letters.   We discussed this in a recent Lorentz worshop.  It has taken almost a decade after Persson's paper, to converge on some conclusions.  Mainly because Persson never clearly said if his solution were exact or not:  a very good trick to attract attention and citations!   His solution in the end did result to be approximate, and not much better than asperity one.

His solution for load-separation is even worse, and in fact there is a little work to be done there, which perhaps I will do.

There is a lot more open problems in adhesion of course, as this entire discussion shows.

Mike Ciavarella's picture

anyway, there is already one code in public domain mathematica

Mike Ciavarella's picture

Amodified form of Pastewka-Robbins criterion for adhesionCiavarellaPapangelo - The Journal of Adhesion, 2017 - Taylor & Francis Abstract

Recent numerical investigation on self-affine Gaussian surfaces by Pastewka & Robbins have led to a criterion for “stickiness” based on when the slope of the (repulsive) area-load relationship appears to become vertical in numerical simulations at a ratio of contact area to nominal one (rather arbitrarily) fixed to 1%. Since pull-off and slope of the area-load are two faces of the same medal, a simple check of the results in terms of pull-off shows that Pastewka & Robbins have many more data which fail their criterion than the ones who satisfy it, and this is evident even in their own Figures. As a small improvement, a proposal to modify the criterion to better fit their own data is put forward. However, the pull-off decay seems rather exponential so that it is unclear if their slope criterion really corresponds to a “thermodynamic” limit, and consequently their conclusion that stickiness should depend only on slopes and curvature may be an artefact of their assumption of defining a secant at 1% contact area ratio, rather than a true important property of rough contact. Both the PR criterion and the present modified one imply that for fractal dimension D<2.4, stickiness should increase with resolution, so the problem of truncation of the spectrum seems ill-defined: in fact, PR define rigid self-affine surfaces with rather smooth and well defined slopes, and not a realistic atomic roughness as first studied by Luan and Robbins.

Antonio Papangelo's picture

There is no qualitative contrast between classical asperity models and Persson models or any numerical recent calculation about the slope of area-load curve: the only  geometrical parameter entering the area-load slope is the rms slope of the surface. 

The question arises with adhesion. Pastewka-Robbins suggested that the slope in this case becomes dependent additionally on rms curvature, and not on rms amplitude, whereas asperity models (Fuller-Tabor is the only one, the proper rough random surface one is not in the Literature but we are about to publish it) involve also rms amplitude. 

However, also Pastewka-Robbins do find that pull-off depends on rms amplitude. So in their case there is a curious threshold: for non-sticky surfaces, they say there is no dependence on rms amplitude, whereas for sticky one there is, as they also find.


Can you beleive this?


Mike Ciavarella's picture

First of all, PR use a "truncated potentials", a convenient numerical representation of the Lennard-Jones potential, but certainly a numeric artefact.  Their potential is truncated at a short distance, a0+delta_r where delta_r is of the order of a0 itself (the atomic distance), and therefore, there is an artificial effect there: after 2 atomic steps, there is no longer adhesive force.  This perhaps hides some effects of the rms amplitude?

In a true situation, Lennard-Jones extends to infinity, and it is clear that this fact already lowers the significance of their finding. Even at very long distances, in theory one should always have some attractive force, and this cannot be zero, and this means the area-load bends in the tensile quadrant always. Their criterion therefore has a lot to do with their arbitrary definition of the truncated potential.


Mike Ciavarella's picture

An interesting problem is the following:  what happens for a truly flat surface?  Even with a truncated potential, it is easy to show that the decaying adhesive force induces an instability --- even if the surface were flat. I have done the calculation for this instability, and it turns out, for the Lennard-Jones potential, that this instability occurs if we have a periodic wavelenght of the order of 50 a0.  Therefore, not very large at all.  Two flat surfaces would be in equilibrium at distance a0 and show the theoretical Lennard Jones pull-off force (theoretical strength) only if they were constrained not to assume wavy configurations.

This in fact could be an interesting problem to study, which however requires a numerical solution. Anyone interested?

Mike Ciavarella's picture

Obviously despite there is an instability, the surfaces will jump into contact uniformly much before this instability appears.  Sorry.

Mike Ciavarella's picture

I see this just accepted paper on

Tribology International

Available online 3 March 2017

In Press, Accepted Manuscript — Note to users


 Effect of fine-scale roughness on the tractions between contacting bodies


Persson’s theory for the elastic contact of rough surfaces is modified to include the compliance associated with an interface force law such as the Lennard-Jones law. We determine the effect of adding a small packet of waves on the probability distribution function [PDF] of the local interfacial gap (including the effect of elastic deformation). This procedure is then used iteratively to develop an algorithm for determining the PDF for a rough surface with a prescribed power spectral density. The results show that for untruncated quasi-fractal surfaces, the PDF then converges at large wavenumber, in contrast to the result when only elastic deformation is taken into account. If the roughness is restricted to wavenumbers greater than a certain critical value, the algorithm predicts a converged relation between nominal traction and mean gap that can be regarded as a modified interfacial force law describing the influence of just the fine-scale roughness on the contact. In particular, the area under the resulting curve can be interpreted as a measure of interface energy as reduced by this roughness. The remaining macroscopic features of the surface can then be described using the JKR methodology in combination with this modified interface energy.



This also seems in contrast to PR --- since the procedure converges when fine details are added, and hence the stickiness and any other property of the solution CANNOT depend on slopes and curvatures!   Rather, this study also shows the main dependence on the rms amplitude.

Antonio Papangelo's picture

ArticleinJournal of Adhesion Science and Technology · March 2017 DOI: 10.1080/01694243.2017.1304856

 A simple asperity model using random process theory is developed in the presence of adhesion, using the Derjaguin, Muller and Toporov model for each individual asperity. A new adhesion parameter is found, which perhaps improves the previous parameter proposed by Fuller and Tabor which assumed identical asperities – the model in all his variants for the radius always gives a finite pull-off force, as in Fuller and Tabor, and contrary to the exponential asperity height distribution, where the force is either always compressive, or always tensile. It is shown that a model with spheres having a radius only dependent on height is a reasonable approximation with respect to models having also a distribution of radius curvatures – the three models differ considerably, as opposed to the adhesionless case where these details did not matter. The important surface parameters in the theory determining the pull-off force are the three moments m0, m2, m4. The asymptotic form of the model at large separation is solved in closed form. As the theoretical pull-off of aligned asperities having the same radius (the average value) increases with the square root of the Nayak bandwidth of the roughness, and as asperity models are known to describe less well the surface at large bandwidth parameters, the limit behavior at large bandwidths remains uncertain.
Link to the paper:  

Mike Ciavarella's picture

While the asperity model shows qualitatively similar results to Pastewka and Robbins for the pull-off value (and therefore main dependence on m0, as suggested also by the recent very elegant Persson's extended solution by Junki Joe Scaraggi and Barber), the area-load slope (not necessarily a very interesting quantity) predicted by PR is in contrast to asperity models.  Anyone joining the effort to solve the puzzle?

Mike Ciavarella's picture

 PR argue that their findings suggests stickiness in strong contrast to the classical findings of FT, whose emphasis in on rms amplitude of roughness, whereas their findings are independent on it.

I wonder how this can be reconciled with the pull-off experiments in Fig.11 from the literature on our recent paper 

Mike Ciavarella's picture

Mulakaluri, N., Persson, B.: Adhesion between elastic solids with randomly rough surfaces: comparison of analytical theory with molecular-dynamics simulations. Europhys. Lett. 96, 66003 (2011)   [PDF]  

In all the figures, from fig.2 to fig.5, the MAIN parameter ruling pull-off is RMS roughness.... against PR criterion.   The puzzle continues...

Mike Ciavarella's picture

I found Caltech people suggested recently an interesting way to model JKR adhesion, which had been suggested already by Popov people in Berlin --- in fact I told the Caltech people in fact that Popov found it before and they cited in the end.

 Any interest to experiment it? It is based on springs that have a highest traction, which depends on mesh size with a power law that turn out reproducing JKR singularity Adhesive contact simulation of elastic solids using local mesh-dependent detachment criterion in boundary elements methodR PohrtVL Popov - Facta Universitatis, Series: Mechanical …, 2015 - Using the concept of stress intensity factors, we suggest a way to include adhesion 
into boundary elements simulation of contacts. A local criterion concerning the maximum 
admissible surface stresses decides whether the adhesive bonds in particular grid points fail 
or not. By taking into account the grid spacing, a robust methodology is found. Validation is 
done using the theoretically derived cases of JKR adhesion.Citato da 3 Articoli correlati Tutte e 5 le versioni Cita Salva

Mike Ciavarella's picture

Surface generator: artificial randomly rough surfaces

In general this may be useful, with related routines

Mike Ciavarella's picture

This interesting new paper shows for the first time some simulations of the Guduru increase of adhesion pull-off due to roughness, decaying then after reaching a maximum, also for random roughness (although the effect is small).

Needless to say, there is no reference to the strange result of Pastewka-Robbins, but rather pull-off is always shown to depend on rms roughness as most commonly expected.

Mike Ciavarella's picture


This recent paper shows a yet more radical view on PR results, where the conclusion is just the opposite: slopes and curvatures have no role at all on stickiness!

A very simple estimate of adhesion of hard solids with rough surfaces based on a bearing area model

Mike Ciavarella's picture

We recently found that Pastewka and Robbins criterion, on which we were always suspicious about the "fractal limit", is indeed wrong, as the numerical observations are limited to very narrow surface roughness spectra, and PR criterion is in contrast with results of BAM of Ciavarella, Persson-Scaraggi, and Persson Tosatti.

See here:


Universal features in “stickiness” criteria for soft adhesion with rough surfacesAuthor links open overlay panelM.CiavarellaShow moreAdd to MendeleyShareCite rights and content Highlights


We derive a stickiness criterion from the simple Persson and Tosatti theory of adhesion of rough solids.

We derive another stickiness criterion from the BAM (Bearing Area Model) theory of Ciavarella.

We compare the two derived new criteria with that Violano et al., and Pastewka and Robbins and Muser.

We find Persson–Tosatti, BAM, and Violano criteria give very close results, and are mainly dependent on macroscopic quantities, while Pastewka and Robbins and Muser criteria differ from the previous three in that they depend on the truncation of the spectrum of roughness.


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