Adhesive wear regimes on rough surfaces and interaction of micro-contacts
AAdhesive wear regimes on rough surfaces and interaction of micro-contacts
Son Pham-Ba * , Jean-Franc¸ois Molinari Institute of Civil Engineering, Institute of Materials Science and Engineering,´Ecole polytechnique f´ed´erale de Lausanne (EPFL), CH 1015 Lausanne, Switzerland
We develop an analytical model of adhesive wear between two unlubricated rough surfaces, formingmicro-contacts under normal load. The model is based on an energy balance and a crack initiationcriteria. We apply the model to the problem of self-affine rough surfaces under normal load, which wesolve using the boundary element method. We discuss how self-affinity of the surface roughness, andthe complex morphology of the micro-contacts that emerge for a given contact pressure, challengethe definition of contact junctions. Indeed, in the context of adhesive wear, we show that elasticinteractions between nearby micro-contacts can lead to wear particles whose volumes enclose theconvex hull of these micro-contacts. We thereby obtain a wear map describing the instantaneousproduced wear volume as a function of material properties, roughness parameters and loadingconditions. Three distinct wear regimes can be identified in the wear map. In particular, the modelpredicts the emergence of a severe wear regime above a critical contact pressure, when interactionsbetween micro-contacts are favored.
Keywords: adhesive wear, severe wear, self-affine surface, boundary element method
1. I
NTRODUCTION
Wear is an ubiquitous phenomenon, and yet it is stillhardly predictable. It is usual that factors contributing toor limiting wear are studied via extensive experimentalcampaigns. Different wear regimes are observed in prac-tice, depending on the ambient and loading conditions,sliding velocity and sliding distance.
This paper focuseson the emergence of the severe wear regime in dry con-tact conditions and assuming adhesive wear processes, formaterials of similar hardness.All macroscopically flat-looking surfaces are in realityrough on a range of scales, whether they are man-made or natural. Often, surface roughness is found to be self-similar or self-affine, in which case it can be described by afractal dimension or by a Hurst exponent (which describesthe scaling of the frequency spectrum ) and the frequencyrange of self-affinity. These parameters describe the sur-face geometry and dictate, along with material proper-ties, the contact mechanics. When put into contact witheach other, two rough surfaces create a number of micro-contacts of various sizes at the interface, depending on thenormal load.
The real contact area is, for common load-ing conditions, a small fraction of the apparent area, andin the small load limit, it was shown to be roughly propor-tional to the normal load by a wide variety of analyticaland numerical models.
Upon frictional sliding, wear particles are eventuallycreated at these micro-contacts, when enough elastic en-ergy is accumulated at the contact junctions which resistsliding. The most fundamental approach to understand-ing wear is to first consider a single contact junction. Abreakthrough in the understanding of adhesive wear was * Corresponding author. E-mail address: son.phamba@epfl.ch made thanks to computer simulations that revealed a ma-terial length scale at which a transition between ductileand brittle behavior occurs. This length scale is related tothe smallest particle size that can be detached from a con-tact interface, as first theorized by Rabinowicz in 1958.Recent molecular dynamics (MD) simulations revealedthat contact junctions above this length scale generatewear particles, while surfaces asperities that form smallerjunctions deform plastically. In a system comprising oftwo sliding solids with identical material properties, thecritical junction diameter for the ductile to brittle transi-tion is defined as d ∗ = Λ γ G σ , (1)where Λ ∼ O ( ) is a factor depending on the exact geo-metrical configuration of the system, G is the shear modu-lus, γ is the surface energy, and σ j is the shear strength ofthe junction formed between the two colliding asperities,a parameter ultimately linked to the adhesive strengthor, in case of full adhesion, to the yield strength. Again,we highlight that a wear particle can only be detached atthe junction if d (cid:62) d ∗ . This quantity provides a mechanis-tic rationale to the transition between low wear (plasticdeformation of protruding asperities) and wear particlesgeneration.MD simulations also revealed that two nearby contactjunctions interact elastically when the distance separatingthem is of the order of the junction diameter. Elastic in-teractions result in a crack shielding mechanism, so thatduring wear particle formation, not all cracks can prop-erly develop as they are unloaded by nearby propagatingcracks, leading to the formation of larger wear particles.In this context, instead of forming two wear particles atthe two separate junctions, a single larger wear particle,whose size encloses the two junctions, is created. These1 a r X i v : . [ c ond - m a t . s o f t ] F e b lastic interactions were further explored in more detailsanalytically in a two dimensional setting. Using an en-ergy balance criterion, a wear map was obtained to predicthow a set of multiple tangentially loaded micro-contactstransition from multiple small wear particles to a singlelarger wear particle. Crack shielding mechanisms werealso confirmed in the context of finite-element simulationswith a phase-field approach to fracture. We emphasizethat these studies on contact junctions interactions werecarried in a rather academic context of 2D or quasi-2D se-tups. A recent 3D numerical model explored the effect ofelastic interactions for multiple contact junctions, whoseshapes were optimized for an energetically efficient wearparticle removal process. Elastic interactions between a large number of micro-contacts, with varying sizes and shapes, as occurs duringloading of self-affine surfaces, is largely unexplored andis the focus of the present paper. Fr´erot et al . used theboundary element method (BEM) to numerically simulatethe contact between two rough surfaces and obtain a mapof the micro-contacts. Each micro-contact size is comparedto d ∗ to assess if it can result in the formation of a wearparticle. Ultimately, an instantaneous wear coefficient canbe estimated. Brink et al . followed the same principle butadded the notion of sliding distance to compute a wearvolume over time and obtain a more physically accuratewear coefficient. These models give promising results,but their downside is that they do not take into accountelastic interactions and assume that each micro-contact isisolated from the others, overlooking the potential tran-sition to a severe wear regime. An interesting study byPopov and Pohrt also relies on BEM to compute the con-tact between rough surfaces. They use an energy balancecriterion to determine if a wear particle can be detached,therefore accounting for elastic interactions. However, wewill discuss how the approximation that was made in thereleased energy, while allowing for a high computationalefficiency, does not capture the full effect of the elasticinteractions. As a consequence, they obtain wear particlesthat can enclose several junctions, but do not observe atransition to a severe wear regime.In summary, this paper explores a mesoscale mechanis-tic model for adhesive wear. In the context of self-affinesurfaces in contact, the model aims to explain the emer-gence of different wear regimes as function of normal load.Section 2 extends the two-dimensional model of Pham-Ba et al . to three dimensions. To this end, we describeanalytically the elastic interactions between nearby micro-contacts and discuss how these affect the formation ofwear particles. Then, in Section 3, the model is numeri-cally extended to the contact between self-affine rough sur-faces, resulting in the description of three well-identifiedregimes of wear. A salient feature of the model is that theproduced wear maps are function of well-defined physics- xy z qqE , ν , γ Ω Γ
Figure 1:
Micro-contacts under uniform tangential load on asemi-infinite solid based model parameters, including material properties,surface roughness, and load.
2. A
NALYTICAL MODEL FOR ELASTICINTERACTIONS BETWEENMICRO - CONTACTS
Before diving into the contact between two rough sur-faces, we take a look at a simplified contact between twoflat surfaces, joined at a small number of cold-welded per-fect junctions that we call micro-contacts. To study theelastic interaction between multiple micro-contacts dur-ing adhesive wear, we model them with uniform loads ofmagnitude q acting along the x direction at the surface Γ ofa semi-infinite solid Ω (see Figure 1). Out of the two solidsin contact, only the bottom one is considered because ofsymmetry.In this section, adhesive wear is incorporated in thismodel by introducing two wear criteria, which are thenused to find the definition of a critical size of micro-contact d ∗ for the geometric configuration of Figure 1. We thenstudy analytically the interaction between two circularmicro-contacts and produce a wear mechanism map . Elastic energy -
When loaded, the solid Ω of Figure 1 accu-mulates elastic energy of deformation. If it is made of alinear elastic material, no energy is dissipated in the load-ing process, which implies that the elastic energy is equalto the work of the load on the surface: E el = (cid:90) Γ u · p d Γ , (2)where u and p are respectively the displacement and trac-tion fields on the surface Γ . In our case, the imposedtractions are only in the x direction, so (2) reduces to E el = (cid:90) Γ u x p x d Γ . (3)2ere, the field p x describes the distribution of the tangen-tial tractions in the x direction on Γ , which are equal to q wherever there is a micro-contact, and 0 otherwise (see Fig-ure 1). The surface displacements in the x direction u x aredetermined from the Bousinesq fundamental solution : u ker x → x = π G (cid:20) ( − ν ) r + ν x r (cid:21) , (4)which is the displacement field in the x direction caused bya unit point load at the origin of Ω , also in the x direction. G is the shear modulus of the material, ν the Poisson’sratio, and r is the distance from the origin: r = x + y + z . The full displacement field u x is obtained by linearsuperposition of the contributions of all tractions: u x ( x , y ) = (cid:90)(cid:90) u ker x → x ( x − ξ , y − η ) p x ( ξ , η ) d ξ d η (5) = [ u ker x → x ∗ p x ]( x , y ) , (6)which is a convolution (denoted by the ∗ symbol). Thetractions p x in the x direction also induce displacementsin the y and z direction, but they do not intervene in (3).We can rewrite (3) as E el = (cid:90) Γ [ u ker x → x ∗ p x ] p x d Γ . (7)Note that in order to be calculable analytically for sim-ple cases, the convolution can be turned into a cross-correlation, as shown in Appendix A.1. Adhesive energy -
To detach a wear particle from Ω , newsurfaces have to be created, requiring adhesive energy (orfracture energy). We assume a simple spherical geometry.Therefore, the detachment of a single particle of diame-ter d requires the creation of two hemispherical surfaces,needing an adhesive energy of E ad,1 = πγ d . (8) Criterion -
When a wear particle is detached from the bulk,it can no longer carry a tangential load applied at thesurface. If a particle is detached where micro-contactswere present, those micro-contacts get unloaded and nolonger contribute to the traction field p x , resulting in adecrease ∆ E el of the elastic energy. This energy does notdisappear, and in fact contributes to the formation of thecracks resulting in the particle detachment. Therefore, todetermine if the particle can be fully detached, we considerthe energy ratio R = ∆ E el E ad , (9)and the particle can be detached if R (cid:62) xz a’ b’a b Figure 2:
Cross section view of the required crack nucleationsites for the formation of a spherical wear particle at a micro-contact. Points (a) and (b) are slightly below the junction, andpoints (a’) and (b’) slightly above. The thick red lines show thecracks which must be nucleated in order to detach the particle.Tensile stresses (red arrows) must overcome the tensile strengthof the material. Equal and opposite compressive stresses (bluearrows) appear by symmetry.
Effect of normal load -
The creation of micro-contacts be-tween two surfaces often results from the application of anormal load, which means that all the terms p x , p z , u x and u z contribute to the elastic energy (2). Nevertheless, thechange ∆ E el in elastic energy due to unloading can still besolely attributed to the change of p x and u x , neglecting theeffect of the constant p z . A proof is given in Appendix A.2. Since the formation of wear particles results from theformation of cracks, their creation must start with thenucleation of such cracks, which can only be initiated at apoint on a surface if σ I (cid:62) σ m , (11)where σ I is the first principal stress at this point, which isthe maximum tensile stress if it is positive, and σ m is thetensile strength of the material.Until now, we only considered the bottom solid to sim-plify the problem. Let us also reconsider the top solid fora moment. In order to detach a spherical wear particle,two diametrically opposed cracks have to be initiated, asshown in Figure 2 as thick red lines at locations (a) in thebottom solid and (b’) in the top one. The cracks can benucleated if the tensile stress at those points, shown by redarrows, is sufficiently large. Thanks to the symmetry ofthe loading, we can state that the maximum tensile stressat the point (b’) in the top solid is equal in magnitude tothe maximum compressive stress at the point (b) in thebottom solid, so that the conditions for crack opening canbe defined by looking only at the bottom solid. In sum-mary, when considering only the bottom solid, a particlecan be detached if it has a sufficiently large tensile stressat its trailing edge (a) and a sufficiently large compressivestress at its leading edge (b).In the way our micro-contacts are defined (Figure 1),3he tangential traction field oriented in the x direction hasdiscontinuities from 0 to q at the borders of the micro-contacts, which leads to stress singularities (regions ofinfinite stresses) in the other directions at those places, also leading to infinite principal stresses. Therefore, thecrack initiation criterion (11) is always satisfied on theborders of the micro-contacts. In order to define a critical junction size as inAghababaei et al. , we consider a single circular micro-contact of diameter d . The stored elastic energy stored bytangentially loading the micro-contact can be calculatedanalytically from (7) (see Appendix A.3), leading to E el,1 = ( − ν ) d q G . (12)The corresponding adhesive energy required to detacha hemispherical wear particle under this circular micro-contact is given in (8). From these two expressions, weobtain the energy ratio from (9): R = ( − ν ) dq πγ G . (13)The maximum tangential load q which can be appliedon the micro-contact is equal to the shear strength σ j of thejunction between the two surfaces in contact. After setting q = σ j in the expression of R , we look for the value of d which makes R satisfy the energy balance criterion (10).We find a critical diameter: d ∗ = πγ G ( − ν ) σ (14)which only depends on material parameters. The en-ergy balance criterion is satisfied whenever d (cid:62) d ∗ . Notethat the expression of d ∗ found in this geometrical con-figuration is in accordance with the expression found byAghababaei et al . (1) with a newly defined geometricalfactor.A single sheared micro-contact of diameter d can re-sult in the detachment of a wear particle if d (cid:62) d ∗ andotherwise flows plastically. To study the elastic interactions between multiple micro-contacts in the same manner as the two-dimensionalmodel of Pham-Ba et al. , we start by considering twotangentially loaded circular micro-contacts of diameters d and having a distance l between their centers, as shownin Figure 3. They are tangentially loaded in the x direc-tion with a pressure of magnitude σ j and the line goingthrough both of their centers makes an angle θ with the x axis. xy zq θ dd a qE , ν , γ Ω Γ l Figure 3:
Two circular contacts under uniform tangential load. d a is the ‘apparent’ diameter of a hemispherical particle thatwould encompass both micro-contacts, and θ is the angle be-tween the direction of the load and the line going through thecenters of the micro-contacts. l / d E e l , / E e l , numerical integralanalytic approx. Figure 4:
Comparison of the numerical integration and the an-alytical approximation of the elastic energy stored under twotangentially loaded circular contacts. Here, ν = θ = l is large. Elastic energy -
The calculation of the stored elastic energyin this case requires rewriting (7) to be calculable by hand.A good approximation of the elastic energy is derived inAppendix A.4: E el,2 ≈ ( − ν ) d σ G + π d σ G − ν sin θ l , (15)which is exact in the limit when l (cid:29) d but is still veryaccurate when l reaches l = d , which is when the twomicro-contacts are adjacent. The accuracy is verified bynumerically integrating (7) for fixed values of d and l andcomparing the results with (15), as shown in Figure 4.The expression (15) consists in the sum of two terms.The left term is equal to 2 E el,1 (12) and represents the en-ergetic contributions of each circular micro-contact. Theright terms is proportional to 1/ l and represents the effectof the elastic interactions between the two micro-contacts.When l is large compared to d , the right term vanishes andthe two micro-contacts do not interact: the total energyis just the sum of their energies taken separately. When l decreases, the two micro-contacts get closer and the inter-action term increases (see Figure 4).Note that if the two micro-contacts were to be super-4 (a) One separated d (b) Two separated d a (c) Combined
Figure 5:
Different cases of wear particle formation with twomicro-contacts. The red surfaces show the surfaces created whenthe particle is detached. The case where no wear particle isdetached is not shown. imposed into a single circular micro-contact loaded at 2 q ,the resulting elastic energy would reach 4 E el,1 , as (12) hasa quadratic dependence on q . This would not happen inpractice when q = σ j since the tangential load is limitedto σ j , but it explains why the E el,2 is necessarily boundedbetween 2 E el,1 and 4 E el,1 . Adhesive energy -
With two loaded micro-contacts, severalcases of wear particle formation may arise, as shown inFigure 5. There can be either zero, one or two separatedwear particles at each micro-contact, or a single combinedwear particle encompassing both micro-contacts. We call d a (for apparent diameter) the diameter of the potentiallyformed combined wear particle. We have d a = d + l .In the case of the formation of a single separated parti-cle, † the required adhesive energy to detach the particle isthe same as (8): E ad,1sep = πγ d . (16)In the case of the formation of two wear particles, therequired adhesive energy is twice as big: E ad,2sep = πγ d , (17)and in the case of the formation of a combined wear parti-cle, it is E ad,comb = πγ d . (18) Energy balance criterion -
Assuming that both micro-contacts get unloaded when the two separated wear par-ticles or a single combined one are formed, the decreaseof elastic energy ∆ E el in (9) is equal to E el,2 (15). Fromthe expression of ∆ E el and the different expressions of theadhesive energy for each case, we get the energy ratio R = (cid:18) C n + π C inter dl (cid:19) dd ∗ C case (19)where C n =
2, and C inter = − ν sin θ − ν (20)is a constant controlling the amount of interaction, whichonly depends on the Poisson’s ratio and the angle θ andhas always a value between 1/3 and 2/3. Choosing ν = θ = † When there is enough energy to form only a single particle, a slightasymmetry in the system would select one of the two contact junctionsand the other junction would deform plastically. C inter = θ = π /2 (line of the micro-contactsperpendicular to the direction of the load) gives the leastamount of interaction with C inter = C case has a value which depends on the case ofparticle formation. From the different adhesive energies E ad,2sep (17) and E ad,comb (18), we have C case = C case = d / d respectively. Note that the material pa-rameters G , σ j and γ do not appear directly in (19) andwere conveniently replaced by the critical diameter d ∗ (14)which contains all those missing terms.When only one micro-contact out of the two forms awear particle and gets unloaded, the elastic energy goesfrom E el,2 (15) to E el,1 (12) with q = σ j , as the remainingmicro-contact, flowing plastically, still carries a load of σ j . Therefore, the decrease of elastic energy is ∆ E el = E el,2 − E el,1 , which with the expression of the adhesiveenergy E ad,1sep (16) gives the energy ratio (19) with C n = C case = When we derived the energy ratio for a single micro-contact and the formation of a single wear particle (9), wewere able to find a critical diameter d ∗ which easily defineswhich behaviour is expected (plastic flow or formation ofa particle). The expressions of the energy ratio R for twomicro-contacts (19) are more complicated, as they nowdepend on d , l , d ∗ and C inter .Figure 6 shows our best attempt to represent the dif-ferent possible scenarii of wear particle formation as afunction of model parameters. We refer to Figure 6 as a wear map . Each colored region shows where R (cid:62) C case and therefore tells that the indicatedbehaviour is energetically feasible, depending on the con-tact junction size d and the critical junction size d ∗ , bothexpressed relative to l . The colored regions in Figure 6are computed at maximum interaction with C inter = C inter = d = d ∗ in Figure 6 is given forcomparison with the single micro-contact case. Indeed,such system would be in a plastic regime when d < d ∗ ,which is the whole region above the dotted line, and itwould allow the formation of a wear particle when d (cid:62) d ∗ ,which is the region below the line. The wear map predictsthat with two interacting micro-contacts, it is possible toform one and even two wear particles, even if the size ofeach micro-contact is smaller than the minimum required d ∗ . Once more, we emphasize that this is because of elas-tic interactions, which make the available stored elasticenergy larger that trivially expected, as explained by (15).5 l l l l d ll l d ∗ plasticity s e p a r a t e d d = d ∗ Figure 6:
Wear map of the different cases of wear particle forma-tion under tangentially loaded two micro-contacts. The coloredregions are computed at maximum possible elastic interaction( ν = θ = ν = θ = π /2). The wear map hints toward the emergence of multiplewear regimes. The ‘plasticity’ region corresponds to theo-retically no wear volume, as the sliding surfaces only getplastically remodeled. This behavior can be linked to theexperimental observations of low wear. In contrast, thejump in wear volume between the ‘2 separated’ and the‘combined’ regions is significant, and can be related to thetransition between a mild and a severe wear regime.It is possible to derive wear maps for larger numbersof micro-contacts and other arrangements, following thesame derivation for the analytical approximation of theelastic energy (Appendix A.4). However, we will showin the next section that more interesting results can beobtained using numerical simulations with generic roughsurfaces while following the same energetic principles.
3. N
UMERICAL MODEL FOR RANDOM ROUGHSURFACES
In the previous section, we considered systems withone or two circular micro-contacts of a given diameter andwith a simple parameterized arrangement on the surface.In reality, the micro-contacts between two loaded nomi-nally flat surfaces emerge from random surface roughnessat lower scales, and therefore come in various shapes andsizes. We resort to numerical simulations to describe themaccurately.
Distribution of micro-contacts -
We use the open source soft-ware
Tamaas to generate discretized self-affine rough ln ( PSD ( q )) ln ( | q | ) ln ( q l ) ln ( q r ) ln ( q s ) − ( H + ) Figure 7:
Target power spectral density of generated rough sur-faces. Its value is zero at frequencies q < q l and q > q s . H is theHurst exponent, q l , q r , and q s are respectively the frequenciescorresponding to the largest, roll-off and smallest wavelengths. surfaces h ( x , y ) with the power spectral density ‡ shownin Figure 7. The rough surfaces are discretized on a gridof n × n points, with n = L . Theirparameters are the root mean square (RMS) of slopes (cid:112) (cid:104)|∇ h | (cid:105) , the Hurst exponent H and the frequencies q l and q s corresponding respectively to the largest and small-est wavelengths contained in the spectrum, where a fre-quency of the surface is given by q = k π L , also calledwavenumber, and k is a number ranging from 0 to (cid:100) n (cid:101) .We set the roll-off frequency q r at q r = q l . Tamaas , primarily a boundary element method software,is used to efficiently solve the elastic contact between tworough surfaces, equivalently considered as the contact be-tween a rigid rough surface (with equivalent roughness)and a flat deformable elastic solid. Figure 8 shows distri-butions of the contact pressure on a rough surface with H = q l = q s = n /8 for different normal loads.We normalize the normal load :˜ p N = √ π (cid:112) (cid:104)|∇ h | (cid:105) p N E ∗ , (21)where p N is the normal load and E ∗ = E − ν (22)is the effective Young’s modulus. At low normal loads,the normalized normal load ˜ p N is a good approximationof the ratio between the real and the apparent contact area.For given rough surface parameters q l , q s and H , thenormalized load ˜ p N is the only free parameter for thedescription of the rough contact, and it combines the effectof the normal load and the RMS of heights. In the contactsimulations, all the grid points where the local normalpressure is non-zero are in contact. They give the neededlocations of micro-contacts. Detachment of wear particles -
We use the energy balanceand the crack initiation criteria on the distribution ofmicro-contacts to determine the potential wear particleformation sites. Assuming a constant tangential load σ j in the contact areas, the elastic energy can be numerically ‡ It is defined as the squared magnitude of the Fourier transform of h . ∆ E el greater than needed E ad (for a given d ∗ ), save this particle removal; Else, try next largestparticle;4. Repeat until no more particles can be added.After those steps, the remaining elastic energy shouldbe small and insufficient to allow the creation of furtherparticles. The position and size of each created particle isrecorded for analysis.As a result of this procedure, a list of wear particles(position and size) is obtained for any given value of ˜ p N (controlling the micro-contacts) and d ∗ (controlling theductile to brittle transition).Note that this algorithm requires many consecutive ex-plicit calculations of ∆ E el for testing the unloading of eachpossible particle (according to the crack initiation criterion)and is computationally expensive. In comparison, the ap-proach of Popov and Pohrt estimates the energy releaseby integrating the local elastic energy density, without re-computing the contact problem. In essence, this amountsto reusing equation (7) with a local integration, but with-out accounting for the change in p x . We have found thatthis procedure results in an underestimation of elastic in-teractions. The computation of the detachment of wear particles inthe numerical model was tested with a setup consistingof two micro-contacts, for which we previously derivedan analytical theory and a constructed a wear map. Weare choosing the micro-contacts to be aligned with theload ( θ = d for constant values of l and d ∗ . Looking at the wear map(Figure 6), this means moving on an horizontal line fromleft to right. As predicted by the wear map, there is a tran-sition between all the possible behaviors of wear particleformation.The numerically generated wear maps are shown inFigure 10. They are generated by varying the values of d and d ∗ and by computing the possible detachment ofwear particles. Figure 10(a) shows the different behavioursdeduced by the number of formed particles and their vol-ume. It agrees with the analytical wear map (Figure 6)superimposed by dashed lines. Figure 10(b) indicates thecorresponding wear volumes. It shows that in the lower right region of the wear map, a much higher wear volumeis created, which corresponds to the ‘combined’ behav-ior of particle formation. Note that the transition to thisbehavior is quite sharp. The numerical model was run on five randomizedrough surfaces with the roughness parameters H = q l = q s = n /8 (with n = d ∗ ( i.e. for agiven material), the number of particles increases with theload until reaching a maximum value, defining a region‘separated’ where separated particles can be formed, asshown in Figure 11(a) to (g). Then, the number of particlesdecreases with the higher loads, entering the ‘combined’region, where large wear particles can encompass multiplemicro-contacts. The wear volume increases monotonicallywith the load and reaches a plateau (the crossed areas inthe wear maps), which is a non-physical numerical artifactcaused by the fact that wear particles reach the size of thediscretized system.The effects of the material parameters can also be readon the wear maps, since they are contained into d ∗ . Accord-ing to our model, a material with a lower d ∗ , that would beharder or more fragile, should form smaller particles. Themodel also predicts that harder materials are more proneto generate combined particles from neighboring contactjunctions.However, the full volume of debris production is higherfor a harder material, which seems in opposition to Ar-chard’s wear law. This is a limitation of not account-ing for the sliding history, as we discuss further below.It is also a consequence of assuming that our junctionsall carry the material specific shear strength σ j , imply-ing that harder materials are loaded tangentially with alarger force. As more mechanical work is imparted to theinterface for hard materials, this results in larger wear vol-ume production. The exact distributions of shear forces atmicro-contacts should be examined in future work.For an easier interpretation, the wear maps can be rep-resented as curves (Figure 13), where each curve corre-sponds to a constant value of d ∗ and vary with the im-posed load. Every point of each curve is the average be-tween five measurements done with different randomizedrough surfaces, and the standard deviation is indicated.To find the maximum number of particles reached byone curve without being sensitive to the statistical noise,a smoothed version is first computed using a Savitzky-Golay filter of degree 3 on a window of 11 points, and the7 a) ˜ p N = (b) ˜ p N = (c) ˜ p N = (d) ˜ p N = Figure 8:
Boundary element simulations of the micro-contacts and local contact pressures for increasing normal load. Thenormalized normal load is indicated, which also corresponds to the ratio of real contact area to apparent contact area. Brightercolor corresponds to a higher local pressure. For increasing normal load, the number and size of the micro-contacts increase, untilmicro-contacts become large enough to merge together, resulting in a drop in the number of contacts but a sharp rise of theiraverage size. The surface roughness parameters are n = H = q l = q s = n /8. (a) Plasticity (b) (c) (d)
Combined
Figure 9:
Different cases of wear particle formation with two micro-contacts in the numerical model. The blue regions are themicro-contacts. The red circles are the wear particles which can be detached. The small brighter spots are the regions of tensile andcompressive stresses where the crack initiation criterion can be fulfilled. l l l d ll l d ∗ plasticity s e p a r a t e d (a) Wear map l l l d ll l d ∗ − − − − N o r m a li z e d w e a r v o l u m e (b) Wear volume
Figure 10:
Numerical wear maps for two circular micro-contacts. Here, θ = ν = (a) Wear map deduced from the number of formed particles (0, 1 or 2) andtheir volume. The four regions of the analytical wear map (Figure 6) are recovered, and the analytical boundaries are shown withdashed lines. (b)
The wear volume is normalized by L , where L is the side-length of the discretization surface. a) d ∗ L = p N = (b) d ∗ L = p N = (c) d ∗ L = p N = (d) d ∗ L = p N = (e) d ∗ L = p N = (f) d ∗ L = p N = (g) d ∗ L = p N = (h) d ∗ L = p N = Figure 11:
Different cases of wear particle formation in a rough contact. (a) - (d) When the normal load increases, the number andsize of wear particles increase, following the trend of the micro-contacts. (e) - (h) With a lower d ∗ , particles are generated at lowerloads. Also, at high loads, elastic interactions promote the formation of less numerous and larger particles encompassing multiplemicro-contacts, even if the distributions of micro-contacts are the same as above. ˜ p N d ∗ L plasticity separatedcombined n o n - z e r o m i n . m a x . N u m b e r o f p a r t i c l e s (a) Number of particles ˜ p N d ∗ L − − − − − − − N o r m a li z e d w e a r v o l u m e (b) Wear volume
Figure 12:
Wear maps of the contact between rough surfaces. The map of the number of particles shows clearly distinct regions.Between the ‘separated’ and ‘combined’ regions, the number of particles decreases but the wear volume increases. The wear volumespans multiple orders of magnitude. The crossed area is the region where the numerical simulation validity is not guaranteed,because the size of the wear particles becomes comparable to the size of the simulated system. .00 0.05 0.10 0.15 0.20 ˜ p N N u m b e r o f p a r t i c l e s d ∗ L (a) Number of particles ˜ p N N o r m a li z e d w e a r v o l u m e d ∗ L (b) Wear volume
Figure 13:
Wear curves of the contact between rough surfaces. Each curve follows an horizontal line in the wear maps of Figure 12.The filled areas represent the standard deviation. The first non-zero value of each curve is shown by a star. The maximum numberof particles reached by each curve is shown by a dot. The invalid parts of the curves (shown crossed in the wear maps) are cut. (b)
The evolution of the wear volume with ˜ p N notably shows the transition between a regime with zero wear volume (to the leftof each star) to a regime where the wear volume increases with the load. All the curves follow the same trend, although with anhorizontal shift. For a given curve, between the star and the dot, the wear volume increases steadily. The slope increases drasticallyaround the dot, indicating a transition to a severe wear regime. maximum is determined on the filtered smooth curve. Astudy of the evolution of the wear volumes (Figure 13(b))reveals the emergence of three wear regimes:– There is no wear particle production until the normalload reaches a critical value. This range is the ‘plastic’region: the surfaces are only deformed plastically.This would correspond to the regime of low wear .– Above a critical load, wear increases monotonically.This range goes roughly up to the point of maximumnumber of particles and would correspond the regimeof mild wear .– For loads higher than the point of maximum numberof particles, the slope of the curves increases quickly.This drastic increase of the wear volume would corre-spond to the regime of severe wear .The ability of our model to predict a regime of severeor catastrophic wear is novel among the similar existingmodels, as these models limit the formation of eachwear particle to occur under a single micro-contact. Themaximum instantaneous wear volume is thus limited bythe size and number of micro-contacts, whereas our modeltakes into account elastic interactions and permits the cre-ation of wear particles larger than a single micro-contact.One advantage of the model of Brink et al . is that it simu-lates the sliding process, and reaches a steady state wearrate. This procedure is unfortunately not applicable forour model because of the high computational cost to com-pute even a single pixel of a wear map. In consequence,our model can only predict an instantaneous wear volumeand has no notion of sliding distance. Brink et al. showedthat simulating the sliding history is key to recover Ar-chard’s law, stating that harder materials wear less. The current model gives the opposite trend. A computation-ally efficient procedure to account both for the slidinghistory, as in Brink et al. , as well elastic interactions forwear particle generation, will be the topic of future work. The wear maps and curves of the previous sectionwere computed for a unique set of roughness parame-ters, namely H = q l = q s = n /8. A study oftheir respective effects when being varied is reported inAppendix A.5.Overall, while the number of produced particles varywith the roughness parameters, the wear volumes remainrelatively unaffected. The three previously identified wearregimes also remain untouched. This invariance with theroughness parameters implies that the wear mechanismsare supposedly not affected by the details of the fractaldescription of the rough surfaces in contact, and that theycan be described solely by d ∗ , which includes the mate-rial parameters E , ν , σ j and γ , and by ˜ p N , which is linkedto the imposed normal load p N and the RMS of slopes (cid:112) (cid:104)|∇ h | (cid:105) of the rough surface. However, this only takesinto account an instantaneous measurement of the totalwear volume. Actually, the size of the produced wear par-ticles are dictated by the fractal parameters, and size of thedetached particles may dictate how the surface roughness evolves over time (effectively changing (cid:112) (cid:104)|∇ h | (cid:105) , thus ˜ p N ),and the wear particles themselves may contribute to thetribological properties of the interface, so that their sizewould be a matter of importance.10 . C ONCLUSION
A model of adhesive wear was developed analyticallyand implemented numerically. The model takes into ac-count elastic interactions between several nearby micro-contacts and allows for the formation of combined wearparticles encompassing multiple micro-contacts. A salientresult is that a wear particle is not necessarily formedunder a single junction, which challenges the definitionof what amounts to a contact junction in the context ofadhesive wear. The model is based on two criteria: an en-ergy balance and a crack initiation criteria. It predicts thetransition between a regime of low wear (with zero wearvolume) to a regime of mild wear, and finally to a regimeof severe wear, emerging thanks to the consideration ofthe elastic interactions. The instantaneous wear volume ispredicted from the material parameters, the loading con-ditions and roughness parameters. Hard materials favorelastic interactions and combined wear particles. R EFERENCES
1. Rabinowicz, E. The Least Wear.
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Physical Review Letters (2001).11. Hyun, S., Pei, L., Molinari, J.-F. & Robbins, M. O. Finite-Element Analysis of Contact between Elastic Self-AffineSurfaces. Physical Review E (2004). 12. Yastrebov, V. A., Anciaux, G. & Molinari, J.-F. From Infinites-imal to Full Contact between Rough Surfaces: Evolutionof the Contact Area. International Journal of Solids and Struc-tures
Tribology Letters (2017).14. Aghababaei, R., Warner, D. H. & Molinari, J.-F. CriticalLength Scale Controls Adhesive Wear Mechanisms. NatureCommunications (2016).15. Rabinowicz, E. The Effect of Size on the Looseness of WearFragments. Wear PhysicalReview Letters (2018).17. Pham-Ba, S., Brink, T. & Molinari, J.-F. Adhesive Wear andInteraction of Tangentially Loaded Micro-Contacts.
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Journal of the Mechanics and Physics ofSolids, arXiv:2011.12281 [cond-mat] (2020).20. Fr´erot, L., Aghababaei, R. & Molinari, J.-F. A MechanisticUnderstanding of the Wear Coefficient: From Single to Mul-tiple Asperities Contact.
Journal of the Mechanics and Physicsof Solids arXiv:2004.00559 [physics] (2020).22. Popov, V. L. & Pohrt, R. Adhesive Wear and Particle Emis-sion: Numerical Approach Based on Asperity-Free Formu-lation of Rabinowicz Criterion.
Friction Contact Mechanics (Cambridge UniversityPress, Cambridge, 1985).24. Fr´erot, L., Anciaux, G., Rey, V., Pham-Ba, S. & Molinari,J.-F. Tamaas: A Library for Elastic-Plastic Contact of Peri-odic Rough Surfaces.
Journal of Open Source Software Journalof Applied Physics
PeerJ Computer Science e103 (2017). . A PPENDIX
A.1. Turning convolutions into cross-correlations
Following the definition of u x caused by p x in the text, we can find any component i of the displacement as u i ( x , y ) = (cid:90)(cid:90) u ker j → i ( x − ξ , y − η ) p j ( ξ , η ) d ξ d η (23) = [ u ker j → i ∗ p j ]( x , y ) , (24)which is a convolution, where i and j can be either of the three coordinates x , y or z and the Einstein summationconvention is used. The full expression of the elastic energy can be written as E el = (cid:90) Γ [ u ker j → i ∗ p j ] p i d Γ , (25)now taking into account the components of u and p in all directions.Alternatively, the integrand [ u ker j → i ∗ p j ] p i of the elastic energy can be written as [ u ker j → i ∗ p j ] p i = (cid:90)(cid:90) u ker j → i ( x − ξ , y − η ) p j ( ξ , η ) p i ( x , y ) d ξ d η (26) = (cid:90)(cid:90) u ker j → i ( ξ (cid:48) , η (cid:48) ) p j ( x − ξ (cid:48) , y − η (cid:48) ) p i ( x , y ) d ξ (cid:48) d η (cid:48) , (27)which, injected into the E el expression, gives E el = (cid:90)(cid:90) [ u ker j → i ∗ p j ] p i dx dy (28) = (cid:90)(cid:90)(cid:90)(cid:90) u ker j → i ( ξ (cid:48) , η (cid:48) ) p j ( x − ξ (cid:48) , y − η (cid:48) ) p i ( x , y ) dx dy d ξ (cid:48) d η (cid:48) (29) = (cid:90)(cid:90) u ker j → i ( ξ (cid:48) , η (cid:48) )[ p j (cid:63) p i ]( ξ (cid:48) , η (cid:48) ) d ξ (cid:48) d η (cid:48) (30)which now contains a cross-correlation, denoted by the (cid:63) symbol. Using a lighter notation: E el = (cid:90) Γ u ker j → i [ p j (cid:63) p i ] d Γ . (31) A.2. Effect of the normal load on the calculation of the gain of elastic energy
Let us consider a surface with micro-contacts, loaded tangentially and vertically. The traction field is p = p x e x + p z e z , (32)and the elastic energy, obtained with (31), is therefore E el = (cid:90) Γ (cid:16) u ker x → x [ p x (cid:63) p x ] + u ker x → z [ p x (cid:63) p z ] + u ker z → x [ p z (cid:63) p x ] + u ker z → z [ p z (cid:63) p z ] (cid:17) d Γ . (33)When the micro-contacts are unloaded, they can no longer carry the tangential load, so p x goes to 0. However, thenormal load remains, so that the unloaded elastic energy is ∆ E el = (cid:90) Γ (cid:16) u ker x → x [ p x (cid:63) p x ] + u ker x → z [ p x (cid:63) p z ] + u ker z → x [ p z (cid:63) p x ] (cid:17) d Γ , (34)where u ker x → z = π G (cid:104) ( − ν ) xr (cid:105) = − u ker z → x (35)and u ker x → x is given by (4). In the particular case where p x ( x , y ) = p x ( − x , − y ) and p z ( x , y ) = p z ( − x , − y ) , we have p x (cid:63) p z = p z (cid:63) p x , so that the unloaded elastic energy becomes ∆ E el = (cid:90) Γ u ker x → x d Γ , (36)which is independent of p z . Therefore, in this particular case, the unloaded elastic energy does not depend on thenormal load, if it is conserved during the unload of the tangential load. The symmetry conditions on p x and p z arefulfilled in the simple analytical cases derived in this paper, and they are also satisfied (approximately) in the case of acontact between rough surfaces, which should be statistically similar upon axial symmetry.12 .3. Calculation of the elastic energy for a single circular micro-contact Only the component p x of p is non-zero, and we can write p x as p x ( x , y ) = c q ( x , y ) q (37)where q is the value of the uniform tangential load and c q ( x , y ) is a function describing the shape of the micro-contact,in this case equal to 1 when r = (cid:112) x + y < d /2 and 0 otherwise. p x (cid:63) p x is easier to calculate than u ker x → x ∗ p x , whichmeans that we can use (31) to calculate the elastic energy. We have p x (cid:63) p x = ( c q (cid:63) c q ) q (38)which is an autocorrelation, calculable geometrically. As c q is a circle of diameter d /2, [ c q (cid:63) c q ]( x , y ) is equal to the areaof the intersection between two circles of diameter d /2 with a distance r = (cid:112) x + y between their centers: C ( x , y ) = [ c q (cid:63) c q ]( x , y ) = d (cid:16) rd (cid:17) − r (cid:112) d − r , (39)where we called C the autocorrelation of c q . In (31), this autocorrelation multiplies u ker x → x (4) u ker x → x = π G (cid:20) ( − ν ) r + ν x r (cid:21) ,which has a 1/ r component and a x / r component. Using polar coordinates and with the help of the Python symboliclibrary
Sympy , we get the integrals of the products with the components: (cid:90) Γ r C d Γ = π d (cid:90) Γ x r C d Γ = π d E el,1 = (cid:90) Γ π G (cid:20) ( − ν ) r + ν x r (cid:21) C q d Γ (42) = π G (cid:20) ( − ν ) π d + ν π d (cid:21) q (43) = ( − ν ) d q G . (44) A.4. Calculation of the approximate elastic energy for two circular micro-contacts
In this case, following the notation of Appendix A.3, c q is made of two circular regions of diameter d with aspace l between their centers and having the line connecting their centers making an angle θ with the x axis. Theautocorellation C of c q in this case can be written as a function of C (39) for a single micro-contact: C ( x , y ) = C ( x , y ) + C ( x − l cos θ , y − l sin θ ) + C ( x + l cos θ , y + l sin θ ) , (45)which has a centered component 2 C ( x , y ) and two side components. The centered component simply gives a 2 E el,1 contribution to the total elastic energy. The integrals of the products of the side components with the terms 1/ r and x / r of u ker x → x have to be approximated by assuming that x and r do not vary much in the region where the sidecomponents of C are non-zero. We have : (cid:90) Γ r C ( x ± l cos θ , y ± l sin θ ) d Γ ≈ l (cid:90) Γ C ( x ± l cos θ , y ± l sin θ ) d Γ (46) = π d l , (47) (cid:90) Γ x r C ( x ± l cos θ , y ± l sin θ ) d Γ ≈ cos θ l (cid:90) Γ C ( x ± l cos θ , y ± l sin θ ) d Γ (48) = π d cos θ l . (49)13sing (31), we finally get: E el,2 = (cid:90) Γ π G (cid:20) ( − ν ) r + ν x r (cid:21) C q d Γ (50) ≈ E el,1 + π G (cid:20) ( − ν ) π d l + ν π d cos θ l (cid:21) q (51) = E el,1 + π G (cid:20) π d l + ν π d ( cos θ − ) l (cid:21) q (52) = ( − ν ) d q G + π d q G − ν sin θ l . (53) A.5. Effect of the roughness parameters on the wear maps
The effects of the parameters H , q l and q s is assessed by running the simulations listed in Table A.1 and comparingthe curves of number of particles and wear volume. The Figure A.1 shows such comparison for a single value of d ∗ / L = (cid:112) (cid:104)|∇ h | (cid:105) is also a roughness parameter,but its effect is already taken into account in the normalized imposed load ˜ p N . Table A.1:
List of roughness parameters for the production of wear maps and curves
H q l q s repetitions n /8 50.5 8 n /8 10.3 8 n /8 50.8 4 n /8 10.8 16 n /8 10.8 8 n /4 10.8 8 n /16 1 The Figures A.1(a) and (d) show the effect of the Hurst exponent H . One physical interpretation of H in the contextof self-affine rough surfaces is that a surface of size L with a roughness of characteristic height R can be viewed ona window of size α L , and the new roughness viewed on this window would have a roughness with characteristicheight α H R . It means that for H =
1, the surface roughness always look the same in the range of self-affinity ( i.e. with q l < q < q s ) regardless of the scale of observation. A surface with a smaller H will look flatter if zoomed-out androugher if looked at from a smaller scale. In Figure A.1(a), we see that the rough surfaces with a lower H can producemore wear particles, but smaller, as the overall wear volume (Figure A.1(d)) is surprisingly independent of H .The frequency parameters q l and q s control the region (scaling) of fractal self-affinity of a rough surface. q l controlsthe lower frequencies, so a lower value means higher large scale features. q s controls the smaller length scales, so ahigher value means smaller rough features. The trends shown by the Figures A.1(b) and (c) are in accordance withthis description: at higher values of q l , a rough surface look flatter because the lower frequency shapes are absent,which promotes more contact on the smaller ‘bumps’ on the surface and thus the creation of more wear particles. Thetrend is the same when q s decreases, as more smaller bumps appear and contribute in the rise of the number of wearparticles. Still, the total wear volume remains only weakly affected by the change of these roughness parameters.14 .00 0.05 0.10 0.15 0.20 ˜ p N N u m b e r o f p a r t i c l e s H = H = H = (a) ˜ p N N u m b e r o f p a r t i c l e s q l = q l = q l = (b) ˜ p N N u m b e r o f p a r t i c l e s q s = n /4 q s = n /8 q s = n /16 (c) ˜ p N N o r m a li z e d w e a r v o l u m e H = H = H = (d) ˜ p N N o r m a li z e d w e a r v o l u m e q l = q l = q l = (e) ˜ p N N o r m a li z e d w e a r v o l u m e q s = n /4 q s = n /8 q s = n /16 (f) Figure A.1:
Effect of the roughness parameters on the wear curves. Here, d ∗ / L = n = H = q l = q s = n /8 and H = q l = q s = n /8 were computed with five repetitions, so their wear curve isaveraged and a standard deviation is shown. The curves for the other sets of parameters are computed from only one rough surfaceand thus are more subject to statistical noise./8 were computed with five repetitions, so their wear curve isaveraged and a standard deviation is shown. The curves for the other sets of parameters are computed from only one rough surfaceand thus are more subject to statistical noise.