Evolution of real contact area under shear and the value of static friction of soft materials

doi:10.1073/pnas.1706434115

. 2018 Jan 16;115(3):471-476.

doi: 10.1073/pnas.1706434115. Epub 2018 Jan 2.

Evolution of real contact area under shear and the value of static friction of soft materials

R Sahli¹, G Pallares^{1

2}, C Ducottet³, I E Ben Ali⁴, S Al Akhrass⁴, M Guibert¹, J Scheibert⁵

Affiliations

¹ Laboratoire de Tribologie et Dynamique des Systèmes UMR5513, Université de Lyon, Ecole Centrale de Lyon, Ecole Nationale d'Ingénieurs de Saint-Etienne, Ecole Nationale des Travaux Publics de l'Etat, CNRS, F-69134 Ecully, France.
² Laboratoire d'Innovation Numérique pour les Entreprises et les Apprentissages au service de la Compétitivité des Territoires, Centre des Etudes Supérieures Industrielles, F-34070 Montpellier, France.
³ Laboratoire Hubert Curien UMR5516, Université de Lyon, Université Jean Monnet Saint-Etienne, CNRS, Institut d'Optique Graduate School, F-42023 Saint-Etienne, France.
⁴ Ingénierie des Matériaux Polymères UMR5223, CNRS, Université Claude Bernard Lyon 1, Université Jean Monnet Saint-Etienne, Université de Lyon, Institut National des Sciences Appliquées Lyon, F-69622 Villeurbanne, France.
⁵ Laboratoire de Tribologie et Dynamique des Systèmes UMR5513, Université de Lyon, Ecole Centrale de Lyon, Ecole Nationale d'Ingénieurs de Saint-Etienne, Ecole Nationale des Travaux Publics de l'Etat, CNRS, F-69134 Ecully, France; julien.scheibert@ec-lyon.fr.

PMID: 29295925
PMCID: PMC5776957
DOI: 10.1073/pnas.1706434115

Evolution of real contact area under shear and the value of static friction of soft materials

R Sahli et al. Proc Natl Acad Sci U S A. 2018.

. 2018 Jan 16;115(3):471-476.

doi: 10.1073/pnas.1706434115. Epub 2018 Jan 2.

Authors

R Sahli¹, G Pallares^{1

2}, C Ducottet³, I E Ben Ali⁴, S Al Akhrass⁴, M Guibert¹, J Scheibert⁵

Affiliations

¹ Laboratoire de Tribologie et Dynamique des Systèmes UMR5513, Université de Lyon, Ecole Centrale de Lyon, Ecole Nationale d'Ingénieurs de Saint-Etienne, Ecole Nationale des Travaux Publics de l'Etat, CNRS, F-69134 Ecully, France.
² Laboratoire d'Innovation Numérique pour les Entreprises et les Apprentissages au service de la Compétitivité des Territoires, Centre des Etudes Supérieures Industrielles, F-34070 Montpellier, France.
³ Laboratoire Hubert Curien UMR5516, Université de Lyon, Université Jean Monnet Saint-Etienne, CNRS, Institut d'Optique Graduate School, F-42023 Saint-Etienne, France.
⁴ Ingénierie des Matériaux Polymères UMR5223, CNRS, Université Claude Bernard Lyon 1, Université Jean Monnet Saint-Etienne, Université de Lyon, Institut National des Sciences Appliquées Lyon, F-69622 Villeurbanne, France.
⁵ Laboratoire de Tribologie et Dynamique des Systèmes UMR5513, Université de Lyon, Ecole Centrale de Lyon, Ecole Nationale d'Ingénieurs de Saint-Etienne, Ecole Nationale des Travaux Publics de l'Etat, CNRS, F-69134 Ecully, France; julien.scheibert@ec-lyon.fr.

PMID: 29295925
PMCID: PMC5776957
DOI: 10.1073/pnas.1706434115

Abstract

The frictional properties of a rough contact interface are controlled by its area of real contact, the dynamical variations of which underlie our modern understanding of the ubiquitous rate-and-state friction law. In particular, the real contact area is proportional to the normal load, slowly increases at rest through aging, and drops at slip inception. Here, through direct measurements on various contacts involving elastomers or human fingertips, we show that the real contact area also decreases under shear, with reductions as large as 30[Formula: see text], starting well before macroscopic sliding. All data are captured by a single reduction law enabling excellent predictions of the static friction force. In elastomers, the area-reduction rate of individual contacts obeys a scaling law valid from micrometer-sized junctions in rough contacts to millimeter-sized smooth sphere/plane contacts. For the class of soft materials used here, our results should motivate first-order improvements of current contact mechanics models and prompt reinterpretation of the rate-and-state parameters.

Keywords: area of real contact; elastomer; rate-and-state friction; rough contact; static friction.

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Conflict of interest statement

The authors declare no conflict of interest.

Figures

**Fig. 1.**
Monitoring the area of real contact of elastomeric multicontact interfaces. (A) Sketch of the experiment. (B) Typical image of a PDMS/cross-linked PDMS multicontact. $A^{R} / A^{A} ≃ 2.05 %$ . $R_{q} = 20 μ$ m. $P = 2.08$ N. (Scale bar: 1.87 mm.) (B, *Inset*) Zoom-in on a microjunction. Red (resp. blue): contour for $Q = 0$ (resp. under shear, at the onset of sliding). (Scale bar: 100 $μ$ m.) (C) Typical concurrent evolution of the area of real contact (blue) and the tangential force (red), of the multicontact interface shown in B, as a function of time (1 point of 10 shown). $A_{0}^{R}$ ( $A_{s}^{R}$ ): initial area (at static friction). $Q_{s}$ : static friction force. $P$ = 2.08 N. $V$ = 0.1 mm/s.

**Fig. 2.**
Area reduction and static friction. (A) $A^{R}$ vs. $Q$ , for a PDMS/glass multicontact submitted to various normal loads $P$ (1 point of 130 shown). $R_{q}$ = 26 $μ$ m. $V$ = 0.1 mm/s. Solid curves: quadratic fits of the form of Eq. 1. Solid straight line: linear fit through data points corresponding to the onset of macroscopic sliding. See *Materials and Methods* for the value of $σ$ . (B, *Inset*) Static friction force estimated using Eq. 3, $Q_{s}^{e s t i m a t e d}$ , vs. its measured value, $Q_{s}^{m e a s u r e d}$ , for all experiments, including different velocities. (B, main plot) $(σ A_{0}^{R} - Q_{s}^{e s t i m a t e d}) / σ A_{0}^{R}$ , as a function of $(A_{0}^{R} - A_{s}^{R}) / A_{0}^{R}$ . In both plots, the solid straight line has slope 1 and goes through the origin. Purple, PDMS/glass interfaces; yellow, PDMS/grafted PDMS; orange, PDMS/cross-linked PDMS; stars, multicontacts; circles, smooth sphere/plane contacts; blue diamonds, fingertip/glass contacts. $V$ = 0.1 mm/s except light purple circles ( $V$ =0.05 mm/s, 0.1 mm/s, 0.5 mm/s, 1 mm/s for PDMS/glass at $P$ = 1.1 N). (C) $A^{A}$ vs. $Q$ , for a smooth PDMS/glass sphere/plane contact, presented as in A. One point of 70 is shown. $R$ = 9.42 mm. $V$ = 0.1 mm/s. See *Materials and Methods* for the value of $σ$ . (D) Images of the sphere/plane contact in C for $P$ = 0.55 N. (D, *Left*) $Q$ = 0. (D, *Right*) $Q$ = $Q_{s}$ . (Scale bars: 1 mm.)

**Fig. 3.**
Area reduction across scales: $α_{A}$ vs. $A_{0}^{A}$ (PDMS/glass interface). Circles: sphere/plane contacts for all $R$ . $V$ = 0.1 mm/s. $+$ : raw data for microjunctions within multicontacts ( $R_{q}$ = 26 $μ$ m). Squares: average of the positive raw data divided into 40 classes. Bars show SD within each class. Line: guide for eyes with slope −3/2. *Inset*: $α_{R}$ vs. $A_{0}^{R}$ for the same multicontacts. *Inset* line: guide for eyes with slope −1.

**Fig. 4.**
Area reduction in human fingertip contacts. (A) $A^{R}$ vs. $Q$ , for various normal loads $P$ (1 point of 190 shown). $V$ = 0.1 mm/s. Curves: quadratic fits of the form of Eq. 1. Line: linear fit through the data corresponding to the onset of macroscopic sliding, passing through origin. See *Materials and Methods* for the value of $σ$ . (B) Relative area difference between initial and final contact. B, left: area of real contact, $A^{R}$ (error bar: segmentation threshold modified by $\pm$ 3 gray levels). B, center: area of apparent contact, $A^{A}$ (error bar: same as B, left). B, right: individual area of 10 selected microjunctions (colored in C) that remain in contact all along the experiment (error bar: $\pm$ SD). (C) Binarized image of a typical contact ( $P$ = 1.57 N). Red line: contour of the apparent area of contact. (Scale bar: 3 mm.) C, *Left*: $Q$ = 0. C, *Right*: steady sliding.

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References

1. Bowden FP, Tabor D. The Friction and Lubrication of Solids. Oxford Univ Press; Oxford: 1964.
1. Ovcharenko A, Halperin G, Etsion I. In situ and real-time optical investigation of junction growth in spherical elastic–plastic contact. Wear. 2008;264:1043–1050.
1. Krick BA, Vail JR, Persson BNJ, Sawyer WG. Optical in situ micro tribometer for analysis of real contact area for contact mechanics, adhesion, and sliding experiments. Tribol Lett. 2012;45:185–194.
1. Archard JF. Elastic deformation and the laws of friction. Proc R Soc Lond A. 1957;243:190–205.
1. Dieterich JH, Kilgore BD. Direct observation of frictional contacts: New insights for state-dependent properties. Pure Appl Geophys. 1994;143:283–302.

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[1] Bowden FP, Tabor D. The Friction and Lubrication of Solids. Oxford Univ Press; Oxford: 1964.

[2] Bowden FP, Tabor D. The Friction and Lubrication of Solids. Oxford Univ Press; Oxford: 1964.

[3] Ovcharenko A, Halperin G, Etsion I. In situ and real-time optical investigation of junction growth in spherical elastic–plastic contact. Wear. 2008;264:1043–1050.

[4] Ovcharenko A, Halperin G, Etsion I. In situ and real-time optical investigation of junction growth in spherical elastic–plastic contact. Wear. 2008;264:1043–1050.

[5] Krick BA, Vail JR, Persson BNJ, Sawyer WG. Optical in situ micro tribometer for analysis of real contact area for contact mechanics, adhesion, and sliding experiments. Tribol Lett. 2012;45:185–194.

[6] Krick BA, Vail JR, Persson BNJ, Sawyer WG. Optical in situ micro tribometer for analysis of real contact area for contact mechanics, adhesion, and sliding experiments. Tribol Lett. 2012;45:185–194.

[7] Archard JF. Elastic deformation and the laws of friction. Proc R Soc Lond A. 1957;243:190–205.

[8] Archard JF. Elastic deformation and the laws of friction. Proc R Soc Lond A. 1957;243:190–205.

[9] Dieterich JH, Kilgore BD. Direct observation of frictional contacts: New insights for state-dependent properties. Pure Appl Geophys. 1994;143:283–302.

[10] Dieterich JH, Kilgore BD. Direct observation of frictional contacts: New insights for state-dependent properties. Pure Appl Geophys. 1994;143:283–302.

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Evolution of real contact area under shear and the value of static friction of soft materials

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Evolution of real contact area under shear and the value of static friction of soft materials

Authors

Affiliations

Abstract

Conflict of interest statement

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