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

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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.


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


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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

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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

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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)

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.

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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. 

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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.

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anyway, there is already one code in public domain mathematica

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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.

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