A comment on "A dimensionless measure for adhesion and effects of the range of adhesion in contacts of nominally flat surfaces" by M. H. Muser

TL;DR

This paper critically examines Muser's proposed dimensionless measure for adhesion in flat surface contacts, discussing its implications and limitations within tribological contact theory.

Contribution

It provides a detailed commentary on the validity and applicability of Muser's adhesion measure, offering insights and potential improvements.

Findings

Highlights limitations of the proposed measure

Suggests alternative approaches for adhesion quantification

Clarifies theoretical assumptions in adhesion modeling

Abstract

letter to the Editor of Trib.Int.

Equations

$${\mu }_{sphere}={\left(\frac{R\Delta {\gamma }^2}{E^{\ast 2}\Delta r^3}\right)}^{1/3}$$

$$\Delta {\gamma }_{rss}=\frac{\Delta \gamma }{E^{\ast }}\frac{\tanh \left({\mu }_T\right)}{{\left(h_{rms}^{\prime }\right)}^3}$$

$$\Delta {\gamma }_{rrs}\sim {\zeta }^{\left(2-H\right)\frac{2}{3}-3+3H}\sim {\zeta }^{\left(7H-5\right)/3},\zeta \to \infty$$

$${\zeta }^{(1-5H/3)}<C,$$

$$\frac{h_{rms}}{\Delta r}<{\left(\frac{9}{4}\frac{{\sigma }_0/E^{\ast }}{\Delta r{q}_0}\right)}^{3/5}$$

Full text

A comment on ”A dimensionless measure for adhesion and effects of the range of adhesion in contacts of nominally flat surfaces” by M. H. Muser

M.Ciavarella

Politecnico di BARI. DMMM dept. V Orabona, 4, 70126 Bari.

Hamburg University of Technology, Dep Mech Eng, Am Schwarzenberg-Campus 1, 21073 Hamburg, Germany

email: mciava@poliba.it

Muser attempts in a recent paper (Muser, 2016) to introduce single dimensionless parameters to razionalize the complex problem of the contact of rough multiscale surfaces in the presence of adhesion, and checks his parameters against some numerical simulations with a quite large number of grid points for today’s computational capabilities ($\sim 10^{6}$).

1. The first generalization relates to so called ”Tabor parameter”. The original Tabor parameter (Tabor, 1977) which controls the validity of the DMT (”long-range” adhesion) vs JKR regime (”short-range” adhesion) solution for the sphere, is

[TABLE]

where $R$ is the sphere radius, $\Delta\gamma$ is work of adhesion, $\Delta r$ is the range of attraction of adhesive forces, and $E^{\ast}$ the plane strain elastic modulus. Similarly to Persson and Scaraggi (2014), Muser merely generalizes the Tabor parameter, by considering the relevant ”local” mean radius $R_{c}\simeq 2/h_{rms}^{\prime\prime}$ which largely depends on the truncating wavevector of roughness $q_{1}$, and denominates this Tabor parameter $\mu_{T}$. But while the original Tabor parameter is unambigously defined for the sphere, for the case of multiscale surfaces, we need to return to the ”real” meaning of it, to understand why various possibilities exist for its generalization, and the dimensional analysis Muser suggests is just one of many possible alternatives. In essense, the Tabor parameter defines the limit of validity of the Linear Elastic Fracture Mechanics, like in the ‘small-scale yielding’ criterion. Since the elastic tensile stresses near the boundary of the contact region are given by $\sigma(x)=\frac{K_{\mathrm{I}}}{\sqrt{2\pi x}}\;$, where $x$ is the distance from the boundary, the width of the region where the stress is higher than the theoretical strength $\sigma_{0}$ is s_{0}=\frac{E^{\raisebox{1.99168pt}{}}\Delta\gamma}{\pi\sigma_{0}^{2}}\;, and we prescribe for JKR to be a good approximation that $s_{0}\ll a_{\min}$, where $a_{\min}$ is strictly for a rigorous application of dimensional analysis, the () smallest length scale associated with the geometry of the problem. This could be for example (i) the smallest width of the contact region (and this leads univoquely to the original Tabor parameter in the sphere problem, given the dependence on $R$), or of the separation region, when the contact is almost complete (see Ciavarella, et al. 2019). So, in truth, the ”generalized Tabor parameter” is load-dependent, and this  limits the generalization $\mu_{T}$ of (Muser, 2016) to the small-load limit, as it should be clearly recognized, or anyway when condition (*) is satisfied – which may not be trivial to check.

2) The second attempt of a dimensionless general quantity is the ”dimensionless surface energy”, defined as

[TABLE]

where $h_{rms}^{\prime}$ is the root mean-square gradient of the surface, $\tanh$ is introduced as an empirical fitting between the ”correct” asymptotics in the two limits of small and large Tabor coefficients $\mu_{T}$. Unfortunately, this second attempts sums the uncertainties of the generalization of Tabor parameter, with further ones, like assuming we are in ”unsticky range” and that the area increases due to adhesion by a linear proportion of the load, and probably other, hidden, ones. The positive check with some numerical results suffers from the fact that the numerical results of Muser correspond to limited range in the spectrum of roughness (spanning less than 3 decades, so only reaching from nm to $\mu m$ wavelength size, while clearly most real surfaces will necessarily have roughness on scales much larger than this), for which Muser is not to blame since this is limited by present computational capabilities (similarly narrow spectra are considered by Pastewka & Robbins, 2014).

A rigorous physicist would

immediately try a possible extrapolation to $\Delta\gamma_{rss}$ when the spectrum becomes infinitely large: defining ”magnification” $\zeta=q_{1}/q_{0}$, where $q_{0}$ is some reference short wavevector truncation, and $q_{1}$ is the large wavevector truncation, and assuming a power law PSD spectrum for simplicity as it is common, $h_{rms}^{\prime}\sim\zeta^{1-H}$, while $h_{rms}^{\prime\prime}\sim\zeta^{2-H},$ where $H$ is Hurst exponent which is $H=3-D$ where $D$ is fractal dimension of the surface) and hence $\tanh\left(\mu_{T}\right)\sim$ $\left(h_{rms}^{\prime\prime}\right)^{-1/3}$ so that

[TABLE]

which goes to zero if $\left(7H-5\right)/3<0$ or $H<5/7=\allowbreak 0.714$. However, we know from alternative semi-analytical investigations which permit to explore wide range of PSD spectra (Joe et al., 2018), that surfaces may be sticky or non-sticky even in this limit but independently on $H$, as the contact solution converges and doesn’t depend on the ”local” quantities $h_{rms}^{\prime\prime},h_{rms}^{\prime}$, that the reference to ”DMT” or ”JKR” regimes is misleading since the ”generalized Tabor parameter” goes to zero but the problem may remain not defined by a DMT analysis, and the main parameters ruling the problems are the macroscopic quantities rms heights $h_{rms}$ and the short wavevector truncation $q_{0}:$ hence, this  ”dimensionless surface energy” quantity is meaningless.

Notice incidentally that similar apparent contradictions occur in the Pastewka-Robbins’s suggested criterion for stickiness based on numerical observations (see Ciavarella, et al. 2019), which can be cast in the form

[TABLE]

where $C$ is a positive constant, which in the limit $\zeta\rightarrow\infty$, would seems to suggest similar results than Muser (2016), except for $H<0.6$ (and the difference in the limit $H$ may be due to different ways of extrapolating numerical results or to different precisions). Viceversa, Violano et al. (2019) suggest that for low fractal dimension $(D\simeq 2.2)$ rough hard surfaces stick for

[TABLE]

which naturally and simply corresponds to well known empirical Dalhquist criterion (Dalhquist, 1969a, 1969b), which for 50 and more years has simply postulated that effective adhesives should have elastic modulus lower than about $1$ MPa, regardless of the local $h_{rms}^{\prime\prime},h_{rms}^{\prime}$ values of roughness, which would anyway be extremely difficult to measure in real life. Similar results were obtained in a much simpler theory (Ciavarella, 2018, Ciavarella & Papangelo, 2019) which incidentally fit Pastewka-Robbins’ (2014) results regardless of the local $h_{rms}^{\prime\prime},h_{rms}^{\prime}$ which seem so crucial in the Muser (2016) (and also Pastewka-Robbins’ 2014) numerical interpolations.

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References

Barber, J. R. (2018). Contact mechanics (Vol. 250). Springer. Berlin.

Ciavarella, M. (2018). A very simple estimate of adhesion of hard solids with rough surfaces based on a bearing area model. Meccanica, 53(1-2), 241-250.

Ciavarella, M., Joe, J., Papangelo, A., & Barber, J. R. (2019). The role of adhesion in contact mechanics. Journal of the Royal Society Interface, 16(151), 20180738.

Ciavarella, M., & Papangelo, A. (2019). Extensions and comparisons of BAM (Bearing Area Model) for stickiness of hard multiscale randomly rough surfaces. Tribology International, 133, 263-270.

Ciavarella, M., Xu, Y., & Jackson, R. L. (2019). The generalized Tabor parameter for adhesive rough contacts near complete contact. Journal of the Mechanics and Physics of Solids, 122, 126-140.

Dahlquist, C. A. in Treatise on Adhesion and Adhesives, R. L. Patrick (ed.), Dekker, New York, 1969,2, 219.

Dahlquist, C., Tack, in Adhesion Fundamentals and Practice. 1969, Gordon and Breach: New York. p. 143-151.

Joe, J., Thouless, M. D., & Barber, J. R. (2018). Effect of roughness on the adhesive tractions between contacting bodies. Journal of the Mechanics and Physics of Solids, 118, 365-373.

Müser, M. H. (2016). A dimensionless measure for adhesion and effects of the range of adhesion in contacts of nominally flat surfaces. Tribology International, 100, 41-47.

Pastewka, L., & Robbins, M. O. (2014). Contact between rough surfaces and a criterion for macroscopic adhesion. Proceedings of the National Academy of Sciences, 111(9), 3298-3303.

Persson,

B. N. J. , Scaraggi, M. (2014) Theory of adhesion: Role of surface roughness, J. Chem. Phys. 141, 124701.

Tabor, D. (1977) Surface forces and surface interactions. J. Colloid Interface Sci. 58, 2.

Violano, G., Afferrante, G., & Ciavarella, M. (2018), On stickiness of multiscale randomly rough surfaces. arXiv preprint arXiv:1810.10960.