Resolving a controversy about adhesion in sliding contacts

TL;DR

This paper critically examines a recent claim that tangential shear stresses increase contact area in sliding adhesive contacts, providing a thermodynamic analysis that aligns with classical theories and highlights the need for considering dissipative effects.

Contribution

It offers a rigorous thermodynamics calculation that clarifies the relationship between shear stresses and contact area, resolving a controversy and aligning with classical models.

Findings

Thermodynamics analysis shows no increase in contact area due to shear stresses.

The model predicts a stronger reduction in contact area than classical theories.

Dissipative effects are necessary to reconcile theory with experimental observations.

Abstract

An interesting recent paper by Menga, Carbone & Dini (MCD, 2018, [1]), suggests that in sliding adhesive contacts, the contact area should increase due to tangential shear stresses at the interface, assumed to be constant and corresponding to a material constant. This is not observed in the known experiments, and is in sharp contrast with all the classical theories about the transition from stick to sliding, both in the JKR (Griffith like) conditions which involve singular pressure and shear, as well as in full general cohesive models. We offer a rigorous thermodynamics calculation, which suggests in fact there is no qualitative contrast but a very close quantitative agreement, with previous theories. Actually, the model predicts an even stronger reduction of contact area than predicted by Savkoor and Briggs, contrary to experimental observations, so would certainly require some adjustements to consider dissipative effects.

Equations

$$G={\left(\frac{\partial U_E}{\partial A}\right)}_{S,\delta }$$

$${\left(\frac{\partial U}{\partial A}\right)}_{S,\delta }={\left(\frac{\partial U_E}{\partial A}\right)}_{S,\delta }+{\left(\frac{\partial U_S}{\partial A}\right)}_{S,\delta }=G-G_{Ic}=0$$

$$G={\left(\frac{\partial U_E}{\partial A}\right)}_{S,\delta }={\left(\frac{\partial U_E}{\partial A}+\frac{\partial U_P}{\partial A}\right)}_P=G_{Ic}$$

$${\mu }_{sphere}={\left(\frac{R{G}_{Ic}^2}{E^{\ast 2}\Delta r^3}\right)}^{1/3}=\frac{{\left(R{l}_a^2\right)}^{1/3}}{\Delta r}=\frac{{\sigma }_0}{E^{\ast }}{\left(\frac{R}{l_a}\right)}^{1/3}\to \infty$$

$$G=\frac{1}{2E^{\ast }}\left[K_I^2+\frac{2-\nu }{2\left(1-\nu \right)}{K}_{II}^2\right]$$

$$K_I=\frac{P_a}{2a\sqrt{\pi a}}=\frac{p_a}{2}\sqrt{\pi a};K_{II}=\frac{T}{2a\sqrt{\pi a}}=\frac{{\tau }_m}{2}\sqrt{\pi a}$$

$$G_c=G_{Ic}=\frac{1}{2E^{\ast }}\left[{\left(\frac{p_a}{2}\right)}^2+\frac{3}{2}{\left(\frac{{\tau }_m}{2}\right)}^2\right]\pi a$$

$$G_{Ic,eff}=G_{Ic}-\frac{3\pi }{8}\frac{{\tau }_m^{2}a}{E^{\ast }}$$

$$G_{Ic,eff}=G_{Ic}+\frac{4{\tau }^{2}a}{\pi E^{\ast }}$$

$$U_E=U_E^N+U_E^T$$

$$dU={\left(\frac{\partial U_E^N}{\partial \delta }\right)}_{A,\tau }d\delta +{\left(\frac{\partial \left[U_E^N+U_E^T+U_S\right]}{\partial A}\right)}_{\delta ,\tau }dA+{\left(\frac{\partial U^T}{\partial \tau }\right)}_{A,\delta }d\tau$$

$${\left(\frac{\partial U_E^N}{\partial \delta }\right)}_{A,\tau }$$

$${\left(\frac{\partial U^T}{\partial \tau }\right)}_{A,\delta }$$

$${\left(\frac{\partial U_S}{\partial A}\right)}_{\delta ,\tau }$$

$${\left(\frac{\partial U_E^T}{\partial A}\right)}_{\delta ,\tau }$$

$$dU=Pd\delta +\left[{\left(\frac{\partial U_E^N}{\partial A}\right)}_{\delta ,\tau }+{\left(\frac{\partial U_E^T}{\partial A}\right)}_{\delta ,\tau }-G_{Ic}\right]dA+Wd\tau$$

$${\left(dU\right)}_{\delta ,\tau }=\left[{\left(\frac{\partial U_E^N}{\partial A}\right)}_{\delta ,\tau }+{\left(\frac{\partial U_E^T}{\partial A}\right)}_{\delta ,\tau }-G_{Ic}\right]dA$$

$$G_I={\left(\frac{\partial U_E^N}{\partial A}\right)}_{\delta ,\tau }=G_{Ic}-{\left(\frac{\partial U_E^T}{\partial A}\right)}_{\delta ,\tau }=G_{Ic}-\frac{3\sqrt{\pi }{k}_w}{4}\frac{{\tau }^{2}a}{E^{\ast }}=G_{Ic,eff}$$

$$a>a_0=\frac{4}{3\sqrt{\pi }{k}_w}\frac{E^{\ast }{G}_{Ic}}{{\tau }^2}=\frac{2}{3\sqrt{\pi }{k}_w}{\left(\frac{K_{Ic}}{\tau }\right)}^2$$

$$\varphi =U=U_E\left(\delta ,w,A\right)+U_S\left(A\right)=U_E^N\left(\delta ,A\right)+U_E^T\left(w,A\right)-G_{Ic}A$$

$$\psi$$

$$=U_E+U_E^T-G_{Ic}A-P^{eq}\delta -W^{eq}\tau$$

$${\frac{\partial \psi }{\partial \delta }|}_{\left(A,\tau \right)}$$

$${\frac{\partial \psi }{\partial \tau }|}_{\left(A,\delta \right)}$$

$${\frac{\partial \psi }{\partial A}|}_{\left(\delta ,\tau \right)}$$

$$={\left(\frac{\partial U_E^N}{\partial A}\right)}_{\delta ,\tau }-\left(G_{Ic}-{\left(\frac{\partial U_E^T}{\partial A}\right)}_{\delta ,\tau }\right)$$

Full text

Resolving a controversy about adhesion in sliding contacts

G.Cricrì1, M.Ciavarella2,3

1Università di Napoli Federico II, DII dept., P.Tecchio, 80. 80125 Napoli (It)

2Politecnico di BARI. DMMM dept. V Orabona, 4, 70126 Bari. (It) mciava@poliba.it

3Hamburg University of Technology, Dep Mech Eng, Am Schwarzenberg-Campus 1, 21073 Hamburg, Germany

Abstract: An interesting recent paper by Menga, Carbone & Dini (MCD, 2018, [1]), suggests that in sliding adhesive contacts, the contact area should increase due to tangential shear stresses at the interface, assumed to be constant and corresponding to a material constant. This is not observed in the known experiments, and is in sharp contrast with all the classical theories about the transition from stick to sliding, both in the JKR (Griffith like) conditions which involve singular pressure and shear, as well as in full general cohesive models. We offer a rigorous thermodynamics calculation, which suggests in fact there is no qualitative contrast but a very close quantitative agreement, with previous theories. Actually, the model predicts an even stronger reduction of contact area than predicted by Savkoor and Briggs, contrary to experimental observations, so would certainly require some adjustements to consider dissipative effects.

**Keywords: **Adhesion, JKR model, friction, soft matter, fracture mechanics, mixed mode, cohesive models.

1

Introduction

A recent interesting paper by Menga, Carbone and Dini [1], applied to contact mechanics problem in the presence of adhesion and friction, assumes no dissipation, and yet seems to suggest that the contact area increases in the presence of shear tractions, in particular, assuming them constant as apparently observed in some experiments [2], which corresponds to an effective increase of the mode I surface energy. In fracture mechanics, mixed mode enhances the toughness (the generalization of the concept of surface energy) observed in pure mode I ([3,4,5]), but this is due to dissipative effects, and anyway we shall see it is insufficient to cause area increase (we shall call it ”MCD paradox” in the following), as indeed all the other models of contact, either assuming singular full stick ”Signorini” contact conditions (Savkoor and Briggs, [6]) but without dissipation (”ideally brittle”), or even more general cohesive models (Johnson, [7]) empirically adding the effect of dissipation, invariably find reduction of the contact area. This is experimentally confirmed in both old [6] as well as very recent and careful experiments ([8-13]). Dissipation, at most, *limits *the contact area reduction to a much weaker dependence on tangential force, but doesn’t seem to lead to any increase. Instead, taking into account of dissipation, the MCD paradox would lead to even larger paradoxical increases.

We shall use here a simple energetic treatment, based on two equivalent procedures (a more direct calculation, and another based on Legendre transform), to offer an alternative result, which is completely in line with existing theories and results.

2 Models in adhesive contacts

Let us consider a generic mode I (no shear tractions, see Fig.1) contact problem with contact area $A$, and use the classical thermodynamic treatment as in Maugis ([14], par.3.2) which applies to contact problem classical fracture mechanics. The quantity $G$, which describes the variations of elastic energy with $A$ at constant remote displacement $\delta$, is the (elastic) energy release rate

[TABLE]

where $U_{E}$ is elastic strain energy, and the derivative is calculated at constant entropy $S$ and (remote) displacement$\ \delta$. This can be obtained from an energy balance (Griffith balance) where the energy is the sum of elastic strain energy $U_{E}$ and surface energy $U_{S}=-G_{Ic}A$. The variation of the total energy at equilibrium is zero

[TABLE]

where $G_{Ic}$ is toughness or surface energy111Under force control, instead, we easily get an alternative

$G=\left(\frac{\partial U_{E}}{\partial A}\right)_{S,\delta}=\left(\frac{\partial U_{E}}{\partial A}+\frac{\partial U_{P}}{\partial A}\right)_{P}=G_{Ic}$

(3)

. Classical methods to compute $G$ permit to solve the problem, for example $G$ is estimated from stress intensity factors (in the Irwin equivalent procedure of the energetic one) if we are under Linear Elastic Fracture mechanics framework, or with cohesive models in more general cases (as in Johnson [7] model). The ”JKR-assumptions” assume extremely short range adhesive forces (virtually a delta-function), and hence correspond to the classical Signorini definition of contact as either ”intimate contact” or full separation, for which the contact area is clearly defined. In cohesive models, generally there is a continuous transition between contact and separation, the definition of ”contact area” is blurred, and the simple JKR procedure cannot be applied. Here, JKR stands for Johnson, Kendall and Roberts [15] fundamental contribution, which introduced Griffith-like energy balance in the world of contact mechanics, and was later discussed to be the correct limit for soft and large bodies, or more precisely when Tabor parameter [16]  for the sphere, is large

[TABLE]

where $R$ is the sphere radius, $\Delta r$ is the range of attraction of adhesive forces, and $E^{\ast}$ the plane strain elastic modulus. $E^{\ast}=\left(\frac{1-\nu_{1}^{2}}{E_{1}}+\frac{1-\nu_{2}^{2}}{E_{2}}\right)^{-1}$ and $E_{i},$ $\nu_{i}$ are the Young modulus and Poisson ratio of the material couple. Also, $\sigma_{0}$ is the theoretical strength of the material, and we have introduced the length $l_{a}=G_{Ic}/E^{\ast}$ as an alternative measure of adhesion, usually a very small length scale.

\begin{array}[c]{cc}{\includegraphics[height=213.62303pt,width=376.44006pt]{fig1.eps}}&\\ &\end{array}

Fig.1. Geometry of the problem in pure mode I (without tangential forces)

When extending this calculation to mixed mode (Savkoor and Briggs [6]) in terms of Irwin Stress Intensity Factors we have in principle all 3 modes of fracture, but normally, to retain the axisymmetric simplification, we average the values around the periphery, obtaining

[TABLE]

where for incompressible materials the factor $\frac{2-\nu}{2\left(1-\nu\right)}$ is equal to $3/2.$ Here,

[TABLE]

where $P=P_{H}-P_{a}$ while $P_{H}=\frac{4E^{\ast}a^{3}}{3R}$ is a compressive Hertzian load and $P_{a}$ is responsible of the contact edge singularity. Here, $a$ is the radius of the circular contact area, $a=\sqrt{A/\pi}$. Hence, this ”ideally brittle fracture” model equilibrium dictates

[TABLE]

and   therefore the contact area will follow the JKR equation, but with a reduced effective surface energy

[TABLE]

which shows also a size effect. Experiments of Savkoor and Briggs clearly evidenced a reduction of the contact area when tangential load was applied, but less than expected from the model, and this clearly indicated dissipative processes.

In the interesting MCD paradox paper ([1]), their calculation leads to their eqt.26 (both under force, or under displacement control in mode I)

[TABLE]

where $\tau$ is here a material constant shear stress in sliding, i.e. the effective surface energy (or toughness of the interface) is increased, rather than decreased in Savkoor’s theory (8), and curiously of a very similar quantity, as $\frac{3\pi}{8}=\allowbreak 1.\,\allowbreak 178\,$ while $\frac{4}{\pi}=\allowbreak 1.\,\allowbreak 273\,$, the difference perhaps being that MCD doesn’t take into account of mode III and of averaging around the periphery. In MCD, it appears also surprising in that there would be an ”effective adhesion”, as $G_{Ic,eff}\rightarrow\frac{4\tau_{0}^{2}a}{\pi E^{\ast}}$, even in the absence of adhesion: who provides this energy? It seems created out of nowhere. Contrast with all experiments we know is also striking [6-13]. We shall therefore proceed to a rigorous thermodynamic treatment from first principles.

3 A first energy calculation

In order to obtain the case of constant shear stress of the MCD paradox, as a first method, we can impose a direct minimization on the total energy $U$ under variations of the relevant extensive parameters. Consider $U\left(S,\delta,\tau,A\right)=U_{E}\left(S,\delta,\tau,A\right)+U_{S}\left(A\right)$ in general as a function of entropy $S$, a contact indentation displacement $\delta$, a shear stress $\tau$, and the contact area $A$. We shall neglect variations of entropy, considering purely reversible transformations, as the MCD paradox does. Further, we shall distinguish the terms due to pressures and shear stresses with superscript $"N"$ and $"T"$, in other words

[TABLE]

Notice that the two energy contributions are completely separated because the contact is assumed to be that of a rigid sphere against an elastomer, which has Poisson’s ratio $v=0.5$. Indeed, more general conditions could be assumed when Dundurs’ second constant is equal to zero (Barber, [17]).

The total differential of energy is then

[TABLE]

where as standard notation, $\left({}\right)_{\delta,\tau}$ indicates that the derivative should be computed with constant $\delta,\tau$. Also, $U^{T}=U_{E}^{T}=\frac{1}{2}\tau W$, as the strain energy is easily computed in this case from the work of the shear tractions in the contact area and the tangential displacements, where $W$ is displaced volume $W=Aw$, and $w$ is the mean horizontal displacement which we can write as $w=k_{w}\tau A^{1/2}/E^{\ast}$, where $k_{w}$ is a dimensionless coefficient of order 1, whose exact value is unimportant here (MCD report $k_{w}=\frac{8}{\left(\pi\right)^{3/2}}\simeq 1.\,\allowbreak 437$). Then, we immediately find

[TABLE]

Hence, substituting back in (11)

[TABLE]

but as we assumed $\tau=const$, $\delta=const$ then

[TABLE]

the energy is minimum for $dU=0$ or

[TABLE]

and therefore we have the problem is equivalent to a mode I problem with reduced effective surface energy. Notice incidentally that this generalizes trivially the definition of energy release rate. The result exactly corresponds to Savkoor and Briggs [6] when $\tau=\tau_{m}$ (i.e. in the limit of full sliding), and for $k_{w}=\frac{3\pi}{8}\frac{4}{3\sqrt{\pi}}=\allowbreak 0.89,$ whereas we have reported $k_{w}=\frac{8}{\left(\pi\right)^{3/2}}\simeq 1.\,\allowbreak 437$, which means the models are even quantitatively very close, but actually the present model would lead to an even stronger reduction of contact area than predicted by Savkoor and Briggs [6], contrary to experimental observations. The result of this more direct energy calculation are also qualitatively close to a full cohesive (more general) model of Johnson [7] in the limit of full sliding and when dissipation is neglected222Johnson suggests Savkoor and Briggs case is when his function $f_{1}=1$ or $\alpha=1+1/2g$ (where $g=\frac{\tau_{0}\overline{s}_{0}}{\sigma_{0}h_{0}}$, being $\tau_{0},\sigma_{0}$ cohesive stresses, and $\overline{s}_{0},h_{0}$ are the limit displacements of the Maugis-Dugdale equivalent model in shear and opening, respectively). See the original paper for details., as Johnson’s results clearly show not much difference from the Savkoor Briggs results even when we are quite far from the assumptions of the Griffith-LEFM-JKR case both for mode II and/or for mode I.

The present model holds up to $G_{Ic,eff}>0$. Hence, when

[TABLE]

one should rather expect that adhesion is completely destroyed, leading to the standard Hertzian solution. This is quite easy for stiff materials. For example, in the experiments in UHV in the AFM by Carpick et al. [18], it was measured $G_{Ic}=0.19J/m^{2}$ and a uniform frictional stress $\tau=0.84GPa$. Also, $E^{\ast}=44GPa$ for a platinum tip contact with mica, and hence $a_{0}\sim 10nm$. In very soft materials, we can reach much higher values of contact area with the presence of friction and adhesion.

4

Legendre transform alternative procedure

We can build a thermodynamic potential formulation (see for example Maugis [14], par.3.2) where explicit variables of the problem are $\left(\delta,w,A\right)$ where $\delta$ is the vertical indentation, $w$ is the tangential imposed displacement, and $A$ is contact area, by considering first the internal energy $\phi$

[TABLE]

where notice that we have continued to explicitly split the normal and tangential components of energy $U_{E}^{N}\left(\delta,A\right),U_{E}^{T}\left(w,A\right)$ since the elastic contact problem is uncoupled. However, since we want to switch the control on $\left(\delta,\tau,A\right)$ rather than $\left(\delta,w,A\right)$, we minimize a new thermodynamic potential obtained by considering the Legendre transform

[TABLE]

where we wrote the superscript ”eq” to remind that the partial derivatives are computed at equilibrium. This thermodynamic potential represents the energy which can effectively when we impose the contraints $d\delta=d\tau=0$. Hence, writing the minimum conditions on the thermodynamic potential represent the equilibrium conditions

[TABLE]

and finally

[TABLE]

which corresponds to the same result as the previous direct calculation, obviously. The difference with respect to the other procedure is that with the Legendre transform, we have defined in a single thermodynamic potential a single function ”free energy” to minimize, which contains all the equilibrium conditions of the system, given the other constraints .

5 Conclusion

The thermodynamics simple treatment for JKR type of adhesion, with a constant value of shear stress in the contact area, has shown results that are qualitatively and quantitatively similar to existing to existing Savkoor and Briggs or Johnson cohesive models, namely that there is a large reduction of the contact area under reversible ”ideally brittle” conditions with no dissipative effects, as in the context of the MCD paradox theory. Even more than this, the model shows an even larger reduction of contact area than predicted by Savkoor and Briggs, and to fit experimental observations, some adjustements are needed to consider dissipative effects. This resolves a controversy generated in the interesting MCD paper, who had suggested instead that the contact area should increase, a rather surprising result considering it was in contrast with all previous theories, and all previous experiments.

6

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