**Abstract**

A paper by Menga, Carbone & Dini (MCD) recently published in prestigeous journal, suggests that in the contact with adhesion, the effect of tangential forces, and in particular of shear stresses assumed constant at the interface and equal to a material constant, an increase of contact area should occur which instead is never observed. We also notice that this is in contrast with all the previous theories about the transition from stick to slip, not only in the JKR-Griffith like framework, but even in the full cohesive models. Several paradoxical implications are also found, namely that even in the absence of surface adhesion, this increase would persist like if surface energy were to emerge from the tangential problem. It is very interesting therefore to resolve this paradox, which concerns the very fundamentals of adhesion and friction, ie. of tribology. What is wrong in the thermodynamics treatment, or in the experiments? Perhaps there is an error? It is always useful to make progress from these apparently striking results.

## Introduction

In fracture mechanics, it is well known that mixed mode enhances the toughness observed in pure mode I, due to dissipative effects. However, the Menga, Carbone and Dini (MCD, 2018) paper finds this increase of toughness (surface energy, in particular), due to a new effect, purely from energy balance at the interface, without the need to assume any irreversible process (dissipation). The case of area enhancement is rather counterintuitive, as experimentally it is confirmed (Ciavarella, 2018, Papangelo & Ciavarella, 2019, Sahli et al. 2019, Papangelo et al. 2019) that it does not occur, so MCD argue that pressure fluctuations destroy this effect in reality.

Fracture mechanics concepts were firstly applied by Johnson, Kendall and Roberts (JKR-theory, 1971) to adhesion between elastic bodies, are applicable to contact even in the presence of friction, as mixed-mode fracture mechanics problem, as done first by Savkoor and Briggs (1977) who also conducted experiments between glass and rubber. They found an effective surface energy which is reduced due to tangential force. Experiments clearly evidenced a reduction of the contact area when tangential load was applied, but less than expected from the prediction. Experimental findings also found development of Schallamach wave which tend to permit slip without affecting the contact.

More recent experiments continue to confirm contact area reduction at both macroscopic and even smaller scales (Sahli et al. 2018, Sahli et al. 2019, Waters and Guduru, 2009) and various recent other papers (Ciavarella, 2018, Papangelo & Ciavarella, 2019, Sahli et al. 2019, Papangelo et al. 2019) have generalized the LEFM Savkoor and Briggs-like models to elliptical shapes of contact, and to include dissipation. But the contact area still decreases with tangential load.

MCD obtain, instead, from a thermodynamic treatment, an effective surface energy given by their eq. 26:

(1)

where is the material constant “shear strength” at the interface, is plane strain elastic modulus, and is the radius of the contact circle. However, this result appears paradoxical on various grounds.

- How can there be effectively adhesion even in the limit , there would be an “effective adhesion”, as ? Where does this energy come from? Under large compressive normal forces, the contact area would be large, which implies an additional energy in any contact which would violate Hertz theory. Yet, Hertz theory has been largely validated in many machines even in sliding contacts, without the need to consider this possibly unbounded increase of the effective energy;
- why the present experiments do not show this area enhancement?

## Conclusion

The tribology community should gain some insigth if this paradoxical result is better understood.

## Author

M.Ciavarella, Politecnico di BARI. DMMM dept. V Orabona, 4, 70126 Bari. email: [email protected]

## References

Ciavarella, M. (2018). Fracture mechanics simple calculations to explain small reduction of the real contact area under shear. Facta universitatis, series: mechanical engineering, 16(1), 87-91.

Johnson, K. L., 1997, Adhesion and friction between a smooth elastic spherical asperity and a plane surface. In Proceedings of the Royal Society of London A453, No. 1956, pp. 163-179).

Johnson, K. L., Kendall, K. & Roberts, A. D. 1971 Surface energy and the contact of elastic solids. Proc. R. Soc. Lond. A 324, 301–313.

Menga, N., Carbone, G., & Dini, D. (2018). Do uniform tangential interfacial stresses enhance adhesion?. Journal of the Mechanics and Physics of Solids, 112, 145-156.

Papangelo, A., Scheibert, J., Sahli, R., Pallares, G., & Ciavarella, M. (2019). Shear-induced contact area anisotropy explained by a fracture mechanics model. Physical Review E, 99(5), 053005.

Papangelo, A., & Ciavarella, M. (2019). On mixed-mode fracture mechanics models for contact area reduction under shear load in soft materials. Journal of the Mechanics and Physics of Solids, 124, 159-171.

Sahli, R., Pallares, G., Papangelo, A., Ciavarella, M., Ducottet, C., Ponthus, N., & Scheibert, J. (2019). Shear-induced anisotropy in rough elastomer contact. Physical Review Letters, 122(21), 214301.

Savkoor, A. R. & Briggs, G. A. D. 1977 The effect of a tangential force on the contact of elastic solids in adhesion. Proc. R. Soc. Lond. A 356, 103–114.

Sahli, R. Pallares, G. , Ducottet, C., Ben Ali, I. E. , Akhrass, S. Al , Guibert, M. , Scheibert J. , Evolution of real contact area under shear, Proceedings of the National Academy of Sciences, 2018, 115 (3) 471-476; DOI: 10.1073/pnas.1706434115

Waters JF, Guduru PR, 2009, Mode-mixity-dependent adhesive contact of a sphere on a plane surface. Proc R Soc A 466:1303–1325.

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