**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.

Hello, the topic addressed by Ciavarella is very important and interesting but

still mysterious to me. Here are some comments:

1) Contrary to the claim of Ciavarella, experiments have been reported where the contact area

increases with the increase in the frictional shear force, see [1].

This was for a rough rubber surface in contact with a smooth glass plate, but the

rubber asperities can be (approximately) considered as small spherical (or elliptic) bumps.

2) There is no theory so far which can explain the reduction in the contact area with

increasing frictional shear stress as observed in the pioneering study of Vorvolakos and Chaudhury [2].

Applying the theory of Ciavarella et al [3] (a similar theory was already presented and used in Ref. [4]) gives

a transition from JKR-like contact area to Hertz-like contact area at much too small sliding speed

(see Ref. [4]).

3) A decrease in the contact area with increasing shear force is observed in some cases

also when there is no macroscopic adhesion, e.g., for a dry clean finger. I believe this is

due to mechanical non-linearity and could be studied using FEM.

4) It is also possible that the superposition of the normal and parallel deformation fields assumed in most theories up to now

is not accurate enough when the parallel deformations becomes large and coupling effects becomes important.

This problem can probably only be studied using accurate

numerical methods e.g. the FEM. In fact, using FEM, Duong and Sauer have studied the deformations

of an elastic block as a function of increasing tangential force,

and found a strong decrease in the contact area with increasing shear stress, in contrast to “linear” theory which

predict a contact area independent of the applied stress.

They have produced an interesting movie illustrating the

deformation field, see https://doi.org/10.5446/37885

5) Most rubber materials are viscoelastic which may influence the contact mechanics.

Even PDMS, which is usually considered as a nearly perfect elastic material, exhibit viscoelastic

properties which increases the adhesion[5]. However, in the study in [2]

viscoelasticity appears not to be important for the contact area; the contact area

is perfectly constant at the JKR-value up to rather high sliding speeds

(corresponding to high frictional shear stress), which could only

occur if the influence of viscoelasticity on the contact mechanics would be perfectly compensated

by some other mechanism, which appears unlikely to me.

[1] B.A. Krick, J.R. Vail, B.N.J. Persson and W.G. Sawyer,

“Optical in situ micro tribometer for analysis of real

contact area for contact mechanics, adhesion, and sliding experiments”,

Tribology Letters 45, 185 (2012).

[2] K. Vorvolakos and M.K. Chaudhury, Langmuir 19, 6778 (2003).

[3] See the note and references given by M. Ciavarella.

[4] B.N.J. Persson, I.M. Sivebaek, V.N. Samoilov, K. Zhao, A.I. Volokitin, Z. Zhang,

Journal of physics: condensed matter 20, 395006 (2008).

[5] A. Tiwari, L. Dorogin, A.I. Bennett, K.D. Schulze, W.G. Sawyer, M. Tahir, G. Heinrich, B.N.J. Persson,

The effect of surface roughness and viscoelasticity on rubber adhesion”

Soft Matter 13, 3602 (2017).

Thanks to Bo Persson who is one of the most outstanding contact mechanicians alive to joint this “challenge”.

In response to his interesting replies:-

1) only one experiment is very hardly effective in explaining the large amounts of data from other labs. In [1], I suspect there may be viscoelastic effects at large velocities, which of course would completely change the picture, and have nothing to do with the MCD theory, which is simply for elastic materials

2) again, I don’t understand the reference to “sliding speed”. There is no reference to sliding speed in neither MCD theory, nor Savkoor, nor Johnson97. There is a complete misunderstanding here of what we are discussing.

3) of course decrease of contact area can occur due to other effects, as well as an increase of contact area can be due to the classical Bowden-Tabor plasticity junction growth theory which I am sure Bo knows, but so what?

4) it is possible that there are other non-linear effects, but my “challenge” remains: is the MCD theory more simply wrong? Violating the principles of thermodynamics by a mathematical error which creates energy out of nowhere?

5) of course viscoelastic effects are important. So what?

Regards, and thanks for your contribution to the challenge.

MC

Agree, the velocity in itself is irrelevant but in the study of Vorvolakos and Chaudhury they find that the frictional

shear stress increases with increasing sliding speed up to very high sliding speeds

so higher sliding speeds means higher frictional shear stress (which is typical for rubber friction at room temperature). And you know this so why comment on it!

My naive opinion for a rubber sliding against a rough surface is that adhesion hysteresis arising from very short range forces (e.g. hydrogen bonding), pinning defects and viscoelastic drag could increase the area between the two. I’m generally reluctant to express such an unguarded opinion as I’ve not verified its validity with careful experiments. The experiments of Krick et al (Tribology Letters 45, 185, 2012) is quite inspiring in this regard.

Manoj thanks for comment. However to make progress we first need to clarify the easiest clean mathematical problem which seems here in MCD still unclear.

The effect of combined normal and tangential loading on the contact area was studied both theoretically and experimentally in the following references. Hence, the question why this contact area increase due to sliding is never observed is incorrect.

1.V. Brizmer, Y. Kligerman and I. Etsion: A Model for Junction Growth of a Spherical Contact under Full Stick Condition, J. of Tribology, Trans. ASME, Vol. 129 Oct. 2007, pp. 783-790.

2.A. Ovcharenko, G. Halperin, and I. Etsion: In Situ and Real Time Optical Investigation of Junction Growth in Spherical Elastic-Plastic Contact, Wear, Vol. 264 (11-12), May 2008, pp. 1043-1050.

3.D. Cohen, Y. Kligerman, and I. Etsion: The Effect of Surface Roughness on Static Friction and Junction Growth of an Elastic-Plastic Spherical Contact, J. of Tribology, Trans. ASME, Vol. 131 (2), April 2009, p. 021404.

Here are some quotes from Ref. 2 above, regarding the experiment and the mechanism causing the increase of the contact area.

“The contact area evolution during pre-sliding (junction growth) of copper spheres loaded against a hard sapphire flat is investigated experimentally. Tests are performed with a novel test rig for real-time and in situ direct measurements that provide, for the first time, a new insight of the junction growth mechanism. It is found that junction growth at sliding inception can cause up to 45% increase in the initial contact area that is formed under normal preload alone. Good correlation is found between the present experimental results and a theoretical model for medium and high normal preloads.”

“A special marking technique was used to understand the mechanism of junction growth, showing that the contact area is under stick condition. As a result no radial expansion of the initial contact area formed by the normal preload is possible and the contact area increase is due to new points of the sphere, outside the initial contact area, that are coming into contact with the flat.”

Izhak

thanks for your reply. Your increase of contact area is indeed due to plasticity and related to the classical “junction growth” theory of Bowden and Tabor. Here, instead, we are talking of a pure fracture mechanics effects of increase of adhesive energy due to shear streses. The two things are very different!!!!

Regards

Mike

I am a little unsure, what’s actually the topic of this discussion: Is it the general question, whether or under which circumstances tangential loading increases or reduces the contact area, or is it the publication by Menga et al. (2018)?

Menga et al. propose a mechanism, how tangential loading can “enhance adhesion” (regarding the contact area), based on the following assumptions:

1. all deformations are elastic and the normal and tangential contact problem are elastically uncoupled

2. the pressure distribution results from the Hertzian solution superimposed with a rigid body displacement

3. when establishing the equilibrium contact radius, the tangential contact stresses are kept constant

The third assumption has the odd consequence, that for an increasing contact radius both the tangential force and the tangential displacement increase, which results in a term in the thermodynamic potential, that increases the contact radius and thus “enhances adhesion”.

Their thermodynamics is valid, but (alike Ciavarella) I am struggling to think of a mechanism able to ensure the validity of assumption 3 without breaking the first two.

However, Prof. Ciavarella, maybe you can clarify, what precisely you want to discuss here?

Best regards

Dear Dr willert

Are you sure MCD thermodynamics is correct? If their potential is correct, we have a constant tangential force, but being the shear stress also constant then contact area is prescribed and the calculation with the energy terms fails. There is a serious problem.

[email protected]

Why does the potential imply that the tangential force is constant? In fact, if they formulated the potential in terms of the tangential force and kept that constant, the contact area would actually decrease due to the tangential load.

The tangential contributions to the potential are the respective elastic energy and the work done by the tangential force. Calling that “Legendre transform” may be fancy, but why not?

Although, if I understand it correctly, there might be a different conceptual problem: On the one hand the indenter is assumed to be sliding with prescribed velocity, on the other hand the thermodynamics of the tangential contact is treated, as if it where under no-slip conditions. I don’t see, how this paradox is resolved. Or am I getting it wrong?

Dear friends

There is a corrigendum although I am not sure I agree either

https://www.sciencedirect.com/science/article/pii/S0022509619308609

The controversy discussed above may finally have been solved!

In an interesting study Lengiewicz et al [1] have found that

the contact area reduction observed in earlier

papers (see references above) can be explained by a theory based

on non-linear elasticity without invoking adhesion!

This is also what we have suggested in some earlier papers in particular the

Tribology Letters[2]. In another paper [3] we observed

that for a dry clean human finger there is no macroscopic adhesion to a flat glass

plate (due to the large surface roughness of the skin)

so the fact that for this case too the contact area is reduced upon application

of a tangential force must clearly be a non-linear elastic effect.

That adhesion itself may not result in a reduction in the contact area upon

application of a tangential force was also shown theoretically to be the case

in the paper by Menga, Carbone and Dini[4]. We observed something similar in

MD simulations[3], and new MD simulations (unpublished)[5] for a linear elastic

cylinder sliding on a rigid flat surface shows the same: no reduction in

the contact area upon application of a tangential force (the earlier paper

[3] was for rigid cylinder on elastic rectangular block which is

more complex (and interesting) due to the importance of opening and

closing crack tips at the edge of the contact region[6]).

If the reduction in the contact area upon application of a tangential force

can be interpreted as an effective stiffening of the rubber elastic properties

by the tangential deformations, then we expect also that the penetration is decreasing

by increasing tangential force. This could explain the experimental observation

in Ref. [2] that a sharp tip indented in a rubber surface with a given normal force moves

upwards when a parallel force is applied in addition to the normal force.

Within (small deformation) linear elasticity theory this result is unexpected as there

is no coupling between the parallel and the perpendicular deformations when the Poisson ratio

is equal to 0.5 (incompressible solid), as is the case in the rubbery region

for rubber-like materials.

[1] J. Lengiewicz, M. de Souza, M. Lahmar, C. Courbon, D. Dalmas, S. Stupkiewicz, J. Scheibert,

Finite deformations govern the anisotropic shear-induced area reduction of soft elastic contacts.

https://arxiv.org/abs/2005.02107

[2] J.S. Persson, A. Tiwari, E. Valbahs, T.V. Tolpekina, B.N.J. Persson,

On the use of silicon rubber replica for surface topography studies,

Tribology Letters 66, 140 (2018).

[3] Wang, J., Tiwari, A., Sivebaek, I., and Persson, B. N. J. (2020). Sphere and cylinder

contact mechanics during slip. arXiv preprint arXiv:2002.02226.

[4] N. Menga, G. Carbone, D. Dini,

Do uniform tangential interfacial stresses enhance adhesion?(vol 112, pg 145, 2018)

JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS 133 (2019).

[5] Wang, J., Tiwari, A., Sivebaek, I., and Persson, B. N. J (in preparation).

[6] Wang, J., Tiwari, A., Sivebaek, I. M., and Persson, B. N. J (2020). On the

origin of sliding friction: Role of lattice trapping. arXiv preprint arXiv:2002.09300.