Variational phase-field continuum model uncovers adhesive wear mechanisms in asperity junctions
VVariational phase-field continuum model uncovers adhesive wearmechanisms in asperity junctions
Sylvain Collet , Jean-Fran¸cois Molinari , and Stella Brach ∗ Institute of Civil Engineering, Institute of Materials Science and Engineering, Ecole Polytechnique F´ed´erale deLausanne (EPFL), CH 1015 Lausanne, Switzerland Laboratoire de M´ecanique des Solides, CNRS, ´Ecole Polytechnique, Institute Polytechnique de Paris, Palaiseau,91128, France
Abstract
Wear is well known for causing material loss in a sliding interface. Available macroscopic approachesare bound to empirical fitting parameters, which range several orders of magnitude. Major advances intribology have recently been achieved via Molecular Dynamics, although its use is strongly limited bycomputational cost. Here, we propose a study of the physical processes that lead to wear at the scale ofthe surface roughness, where adhesive junctions are formed between the asperities on the surface of thematerials. Using a brittle formulation of the variational phase-field approach to fracture, we demonstratethat the failure mechanisms of an adhesive junction can be linked to its geometry. By imposing specificcouplings between the damage and the elastic energy, we further investigate the triggering processesunderlying each failure mechanism. We show that a large debris formation is mostly triggered by tensilestresses while shear stresses lead to small or no particle formation. We also study groups of junctions anddiscuss how microcontact interactions can be favored in some geometries to form macro-particles. Thisleads us to propose a classification in terms of macroscopic wear rate. Although based on a continuumapproach, our phase-field calculations are able to effectively capture the failure of adhesive junctions, asobserved through discrete Molecular Dynamics simulations.
Wear is a major aging process of any system experiencing relative motion of its components. The presenceof adhesive forces at the level of the surface roughness, leading to the detachment of wear particles, waslong observed and described as adhesive wear [1–3]. The size of the wear particles (besides having a greatimplication on the global wear rate and thus durability) is strongly related to the resulting pollution. Forinstance, it was measured that 20% of the overall traffic-related fine particle emissions comes from brakewear [4]. These airborne particles are known to cause health hazards, whose intensity is strongly related tothe particle size, as nanoscale fragments can leach and settle into internal organs [5–7]. One of the first butstill most used wear formula, known as the Archard wear law, linearly relates the resulting wear volume tothe normal load [8]. However, this linear relation is only valid for a certain range of applied load [9] andstrongly depends on a fitting parameter, the wear coefficient, that can range over several orders of magnitude[10]. In presence of lower or higher applied load, the relation between wear volume and applied load wasobserved to be sublinear or superlinear respectively [11–15]. As a consequence, a wear process has to beclassified into low, mild or sever wear for an approximate measure of the wear volume to be computed.Major advances in tribology were recently made possible by the use of Molecular Dynamics (MD), wherethe wear process can be investigated directly at the level of the surface roughness [16–20]. In [16], a unifiedapproach was proposed to describe both the plastic flattening of the surface roughness associated to thelow wear regime, and the fracture-induced particle formation associated to the mild wear regime. Authors ∗ Corresponding author: [email protected] a r X i v : . [ c s . C E ] S e p ostulated the existence of a critical length-scale d ∗ , describing the transition from one regime to the otherbased on the size of the adhesive junction: their computations showed that junctions smaller than d ∗ werecharacterized by the gradual smoothing of the asperities, while for sizes larger than d ∗ a wear debris wasdetached from the interface by propagation of two cracks. This model was further developed in [19] toaccount for weaker adhesion forces at the junction. This new formulation allowed to capture a third mode offailure: the relative slip of the asperities, where damage is only located along the junction’s vicinity. Finallyin [18], the transition from mild to severe wear was traced back to the stress field interaction at the contactpoints of an adhesive junction, providing the interesting result in that the size of the wear debris can bededuced from the observed early-stage crack patterns.Although being a powerful tool for modeling wear, Molecular Dynamics carries its own limitations. First,the computational cost of MD simulations limits its application to systems at the nanoscale. Second, as thedomain greatly evolves with time, a thorough study of the effect of the junction’s geometry on the wearprocess is practically out of reach. Hence the need of a more agile numerical formulation.In this paper, we investigate adhesive wear by using a phase-field model of brittle fracture [21, 22],which numerically implements the variational gradient damage formulation [23]. Widely used to predictcrack nucleation and propagation in a variety of materials and geometries [24, 25], this approach doesnot require any a-priori assumption on the crack path and the fracture process is independent from thenumerical discretization, as long as the mesh size is set smaller than a regularization parameter. Withrespect to MD simulations, the variational phase-field model allows us to perform a thorough study ofvarious junctions’ geometries at a relatively low computational cost. It further permits to model much largersystems involving materials that are defined by their macroscopic properties (such as Young’s modulus andfracture toughness), thus not requiring any atomic potential characterization. Notwithstanding those keyadvantages, very few works can however be found in literature on the topic of wear. Particularly noteworthyis the recent contribution [26], where a phase-field fracture model was used to simulate crack patterns inadhesive junctions: results suggested a dependence of the wear process on the geometry of the junction,although the performed simulations did not cover a large enough parameter space to provide exhaustiveconclusions.This is further investigated in the present paper, whose main contribution is twofold. First, we clearlyestablish a relation between the failure patterns of the adhesive junctions and their geometries. Second, wepropose a classification in terms of macroscopic wear rate, qualitatively describing the material loss of thejunction. We are thus able to highlight key differences in the failure of adhesive junctions and provide a newunderstanding on the various failure mechanisms, which can be associated to different wear particle sizes.The outline of the paper is as follows. In Section 2 we first recall the formulation of the variationalphase-field approach to brittle fracture used in this study. The results of the simulations are then presentedin Section 3, by considering perfectly-adhesive single junctions. The interaction between multiple asperitiesis investigated in Section 4. Conclusions are drawn in Section 5. The following notation is used throughoutthe paper: underlined and double-underlined symbols denote vectors and second-order tensors, blackboardletters indicate fourth-order tensors with the exception of R denoting the real line, the symbol “:” is thedouble-dot product operator. Consider a domain Ω comprised of a linear-elastic isotropic material with stiffness tensor C (with bulkmodulus K and shear modulus µ ) and critical energy-release rate (toughness) G c . We investigate adhesivewear via a variational phase-field model of fracture [21–23]. A regularized energy function E (cid:96) is introduced,expressed in terms of the characteristic length (cid:96) > α ∈ [0 , α = 0corresponds to the intact material, whereas α = 1 to complete fracture. The fracture problem is solved byminimizing E (cid:96) with respect to the displacement field u and the damage variable α ( u ∗ , α ∗ ) = arg min u ∈K u ˙ α ≥ E (cid:96) ( u, α ) (1)where K u is the set of admissible displacement fields with regard to the applied boundary conditions.2n the original developments [21, 22], the regularized energy function E (cid:96) was expressed as E (cid:96) ( u, α ) := (cid:90) Ω (cid:0) W el + W fr (cid:1) d Ω W el = 12 (cid:15) : g ( α ) C : (cid:15)W fr = 3 G c (cid:16) α(cid:96) + (cid:96) |∇ α | (cid:17) (2)where (cid:15) = ( ∇ u + ∇ u t ) / g ( α ) = η + (1 − α ) is a degradation function describing thedecrease in elastic energy as damage progresses and η is a small residual stiffness. The energy E (cid:96) in Eq. (2)is symmetric in tension and in compression, thus material interpenetration can occur under compressiveloadings.To avoid this problem, we adopt the approach proposed by Amor et al. [27] (see also [28] for shearfracture), where the strain energy W el is split in spherical and deviatoric parts and depends on the sign ofthe volume change. Accordingly, we decompose the hydrostatic strain in positive (i.e., tr (cid:15) + = max(0 , tr (cid:15) ))and negative (i.e., tr (cid:15) − = tr (cid:15) − tr (cid:15) + ) parts and introduce the following regularized models with contactconditions. • Positive-hydrostatic (PH). Aiming to reproduce mode-I cracks, fracture is only allowed in materialregions with positive volume change. As such, any increase in the fracture energy W fr is exclusivelydue to a reduction in the spherical part of the tensile strain energy W el = K (cid:0) tr (cid:15) − + g ( α ) tr (cid:15) + (cid:1) + µ (cid:16) (cid:15) d : (cid:15) d (cid:17) (3)where (cid:15) d is the strain deviator. Conversely, compressive volume changes and shear states do notcontribute to the fracture process. • Hydrostatic-deviatoric (HD). With respect to the previous model, mode-II cracks are reproduced byadding the deviatoric part of the strain energy to the coupling with the fracture energy. The resultingregularized expression for the elastic energy is W el = K (cid:15) − + g ( α ) (cid:18) K (cid:15) + + µ (cid:16) (cid:15) d : (cid:15) d (cid:17)(cid:19) (4)where only material regions with positive volume changes or under shear actions are allowed to releasestrain energy through fracture.For our computations, we refer to a non-dimensional form of the previous models. The regularizedfunctional E (cid:96) is divided by E L , where E is a typical value of the Young’s modulus and L is a characteristiclength of the domain. The resulting non-dimensional energy function (cid:101) E (cid:96) reads as in Eq. (2), except that eachquantity is now replaced by its non-dimensional counterpart (cid:101) C = C E , (cid:101) G c = G c E L , (cid:101) u = uL , (cid:101) (cid:96) = (cid:96)L , d (cid:101) Ω = d Ω L , (cid:101) (cid:15) = (cid:15), (cid:101) α = α (5)For easier readability, the tilde above the non-dimensional parameters is dropped in what follows. We numerically reproduce the experiment from Brockley and Fleming [29], addressing wear and debrisformation in sheared junctions. The computational domain Ω ∈ R representing the adhesive junction iscomprised of two asperities in contact to each other, as shown in Figure 1. Both asperities have a triangulargeometry with base D , height H and slenderness ratio H/D . The current and maximal values of the junctionlength are respectively denoted as j and j max . The ratio J = j/j max (with J ∈ [0 , B ≈ D and L ≈ D , while the height ofdomains 2 is set to S ≈ D . Arrows represent the displacement imposed at each time step on both top andbottom horizontal boundaries. The thresholds used for the visualization of the results are displayed on theright. The stress measures are non-dimensionalized with respect to E .as a continuous medium. A quasi-static loading is imposed through the relative horizontal displacement ofthe top and bottom boundaries (sketched by arrows in Figure 1), while the vertical displacement is set tozero. The other boundaries of the domain are free to move in both horizontal and vertical directions. Theincremental value of the applied displacement is set small enough to ensure a damage-free elastic behaviorat early times.The study is performed under plane strain conditions using 2D unstructured meshes with element size δ . The simulations are performed with the open source code mef90 [30]. The constraint minimizationwith respect to the damage α is done using the solvers from PETSc. The minimization with respect to thedisplacement field u is solved using preconditioned conjugated gradients. The numerical fracture toughness[22] is G numc = G c (1 + 3 δ/ (cid:96) ). According to literature [27, 31], a mesh size such as δ/(cid:96) ∈ [1 / , /
5] is able torepresent the crack evolution without any mesh dependency while the ratio (cid:96)/L should tend to zero. In thisstudy, we consider L = D and set the non-dimensional constants as follows: δ = 0 . η = 10 − , ν = 0 . E = 1 and G c = 1, if it is not otherwise specified. The base of the asperities is D = 1. Simulations are firstconducted by assuming the regularization parameter (cid:96) such that (cid:96)/D = 1 /
50 (that is, for (cid:96) = 0 . α ≥ . σ eq = (cid:113) (3 / σ d : σ d , where σ d is the stress deviator, whereas the hydrostatic stress is σ h = tr σ/
2. Equivalent and hydrostatic stresses aredisplayed by applying the thresholds 0 . ≤ σ eq ≤ . − . ≤ σ h ≤ − . . ≤ σ h ≤ . J = 0 . H/D = 0 .
5. The hydrostatic stresses as well as the damage field are displayed according to the thresholds inFigure 1. (a) Time-step = 58: the undamaged state shows both compressive and tensile stress concentrations.The red circles indicate where tensile stresses concentrate, which correspond to potential crack nucleationspots. (b) Time-step = 59: the damaged state computed at the next time-step shows a pair of cracksnucleated inside the junction (at the ’junction’ stress concentration). The nucleation releases the tensilestresses while compressive ones remain and deflect the crack path. Note that the figures show zoom-ins,cropped from a much bigger domain.(a) (b)(c) (d)Figure 3: Visualization of the transition in the crack nucleation pattern in single junctions under PH con-ditions, as a consequence of a change in the ratio
H/D with constant junction parameter J = 0 .
5. Thehydrostatic stresses and damage are displayed according to the thresholds in Figure 1. (a)
H/D = 0 .
3. (b)
H/D = 0 .
45 and (c)
H/D = 0 .
5: a pair of cracks nucleate inside the junction. The cracks path is deflectedtowards the bulk by the high compressive stresses developed in the junction. (d)
H/D = 0 .
8: a pair of cracksnucleates in the bulk and grows along the asperity basis. Note that the figures show zoom-ins, cropped froma much bigger domain. 5
Fracture analysis of single-asperity junctions
We first perform a set of simulations using the positive-hydrostatic (PH) constraint in Eq. (3), which impliesthat a crack can nucleate only as a consequence of tensile stresses.Figure 2 (a) shows the distribution of compressive and tensile hydrostatic stresses at the time-step whichprecedes the nucleation of cracks. We observe four locations with tensile stress concentrations, which wecollect in two groups. The first one, labeled as ‘junction’, corresponds to the corners formed by the incompleteoverlap of the asperities. The second one, labeled as ‘bulk’, corresponds to the corners where the asperityrises from the bulk. Indeed, for J = 1, the two stress concentrations coincide as the overlap of the asperitiesis complete. Results also reveal the occurrence of compressive hydrostatic stresses, which develop from thejunction -where the two asperities are pressed against each other- and concentrate at the opposite side ofthe ‘bulk’ corners. At the time-step immediately after (Figure 2 (b)), two cracks nucleate at the ‘junction’stress concentrations and the tensile hydrostatic stresses are released through fracture.The influence of the geometry of the asperities on crack nucleation is studied in Figure 3 by varying theslenderness H/D , for a given value of the junction parameter J . Results clearly show a shift in the cracknucleation spot from the ‘junction’ to the ‘bulk’ as the asperities become more slender, thus suggesting alarger loss of material at complete failure. We believe that this can be directly related to the resulting wearparticle size, although the latter cannot be precisely quantified as the final state of the adhesive junctionis not accessible by the model at hand. The reason for this is twofold. First, our approach relies on smallstrains, whereas the failure of a sheared adhesive junction is believed to be also influenced by the largedisplacements that take place [20]. Second, it can be clearly seen in Figures 2 and 3 that the compressivestress field deflects the crack paths towards the bulk. We believe that this is an artifact of the PH model,as in material regions where the volume change is negative the hydrostatic strain energy cannot be releasedthrough fracture. The cracks then deviate towards the bulk, instead of propagating throughout the asperitiesand eventually detaching a debris particle.Nonetheless, MD simulations [16–19] showed that the location of the damage at the early stages of theshearing process provides a good indication on the final state of the junction at failure. In other words,one can assume that a crack nucleating in the bulk will very likely detach a debris particle, whereas failuremechanisms such as slip will most probably not. As such, we associate the formation of two cracks at the‘junction’ as a mechanism that would eventually lead to the formation of a small particle , as the area aroundthe adhesive interface gets separated from the rest of the domain by the cracks (e.g., as in Figure 3 (a),(b) and (c)). This under the assumption that without the cracks being deflected by the compressive stressfield, they would rather propagate in the junction than towards the bottom part of the domain. On theother hand, when a crack nucleates at the ‘bulk’, its path suggests the detachment of the whole junction as asingle debris: Figure 3 (d) shows how the cracks nucleate at the ‘bulk’ stress concentrations and subsequentlypropagate at the asperity base, turning as they are about to detach the full junction. We thus associate thiskind of crack nucleation to the formation of a large particle .The complete study of the geometry’s effect on the failure of the adhesive junction is reported in Figure4, in terms of the slenderness H/D and the junction parameter J . We observe that the transition from smallto large particle detachment follows a clear trend described by a law of the form J ∗ = DH C (6)The latter intersects the dashed line H/D = C when J →
1, which corresponds to the limit case of ajunction formed by completely overlapping asperities. For the material parameters used in this study, thefitting coefficient in Eq. (6) is
C ≈ .
27. Additional simulations performed on domains where all lengths aredoubled while keeping the same δ and (cid:96) further confirmed that the transition from small to large particleformation is governed by the slenderness H/D rather than by the height of the asperity only.The results from Carollo et al. [26] are also reported in Figure 4, blue and red crosses respectivelydenoting small and large particle formation. Authors [26] used a phase-field formulation which differs fromours for the splitting of the elastic strain energy, as well as slightly different boundary conditions. Thenumber of available simulations is not large enough for us to properly compare the outcomes of the two6tudies, however we do notice a certain agreement in the prediction of small particle formation whereasresults tend to disagree when large particles are detached.Figure 4: Evolution of the failure mechanism under PH conditions, as a function of the domain’s geometrywhere each point represents a different simulation. The blue triangles represent junctions that failed throughcracks nucleation inside the junction as shown in Figure 3 (a), (b) and (c). Red dots represent junctionsthat failed through the nucleation of a crack in the bulk as shown in Figure 3 (d). The solid line is a fittingon the points marking the transition from one regime to the other. The crosses represent data points froma similar study [26] which uses a slightly different phase-field formulation. C = 0 . We now consider the hydrostatic-deviatoric (HD) model in Eq. (4), which allows fracture to occur as aconsequence of both shear and positive hydrostatic stresses. This change in the coupling between strainand fracture energy brings failure patterns that cannot be captured using a PH model. In fact, in additionto tension-driven cracks as described in the previous Section, some geometries exhibit failures that involveshear bands (see Figure 5). These regions are characterized by the presence of a diffuse damage and thusare thicker than tensile cracks [27].With respect to what observed under PH conditions, our simulations highlight the occurrence of twoadditional failure modes for the adhesive junction: slip and shear . Slip describes a shear band that formsalong the interface between the two asperities, while shear stands for shear bands that develop horizontally.The major difference is that the second mechanism leads in certain cases to the detachment of a small particle(in the same fashion as described in Section 3.1), whereas the first does not.The transition across the four types of failure as the slenderness
H/D of the asperities increases is shownin Figure 6, for a given value of the junction parameter J . The geometries are the same as those analyzed inFigure 3. For small values of H/D , a shear band forms at the junction interface and the asperities slip pastone another (Figure 6 (a)). An increase in
H/D results in a shear band running throughout the asperityparallel to the macroscopic shear direction (Figure 6 (b)). As the junction becomes even more slender, cracksnucleate hinting to small (Figure 6 (c)) and large (Figure 6 (d)) particle detachment.The complete study of the failure modes of the adhesive junction is reported in Figure 7, as a function ofthe slenderness
H/D and the junction parameter J . Interestingly, the exact same law in Eq. (6) postulatedfor describing the transition between small and large particle detachment still holds under HD constraints.The difference from Figure 4 to Figure 7 thus lies in the area of the chart where there is either small or no7a) (b)Figure 5: Visualization of the junction failure through a slip mechanism under HD conditions, for a junctionwith J = 0 . H/D = 0 .
4. The equivalent stress as well as the damage field are displayed according tothe thresholds in Figure 1. (a) Time-step= 38: the domain is still intact, the highest values of equivalentstress are located along the interface between the two asperities. (b) Time-step= 39: a shear band developsalong the interface and the stresses are completely released. Note that the figures show zoom-ins, croppedfrom a much bigger domain.(a) (b)(c) (d)Figure 6: Visualization of the transition in the crack nucleation pattern in single junctions under HDconditions, as a consequence of a change in the ratio
H/D with constant junction parameter J = 0 .
5. Thehydrostatic stresses and damage fields are displayed according to the thresholds in Figure 1. The geometriesare the exact same as in Figure 3 (PH model). (a)
H/D = 0 .
3: a single shear band nucleates along theinterface between the two asperities. (b)
H/D = 0 .
45: a single shear band nucleates horizontally from one ofthe stress concentration in the junction. (c)
H/D = 0 .
5: a pair of cracks nucleates inside the junction. Thecracks path is deflected towards the bulk by the high compressive stresses developed in the junction. (d)
H/D = 0 .
8: a pair of cracks nucleates in the bulk along the asperities’ basis. A shear band also develops fromthe opposite corner, suggesting a large particle detachment. Note that the figures show zoom-ins, croppedfrom a much bigger domain.particle detachment. In fact the left side of the chart, where junctions are relatively small and asperities8igure 7: Evolution of the failure mechanisms under HD conditions, as a function of the domain’s geometry,where each point represents a different simulation. The green stars represent the junctions that failed througha slip mechanism, as shown in Figure 6 (a). The black downward triangles represent the junctions that failedby developing one (hollow markers) or two (dense markers) straight shear bands, the first case being shownin Figure 6 (b). The blue upward triangles represent junctions that failed through crack nucleation insidethe junction, as shown in Figure 6 (c). The red dots represent junctions that failed through the nucleationof cracks in the bulk, as shown in Figure 6 (d). The solid line is the exact same function as show in Figure 4and it now marks the transition from small or no-particle formation to large particle detachment. C = 0 . H/D increases, the slope of the interfacerises and the shear bands grow horizontally through the junction rather than along the interface. For themost slender asperities, we even observe two parallel shear bands that detach a small particle from thejunction.More generally, we observe that almost all simulations that lie under the transition curve are now showingfailure mechanism which involve shear bands rather than cracks triggered by opening stresses. The onlysimulations that still present tension-driven cracks inside the junction (blue upside triangles) are located inthe proximity of the transition curve. Hence, our results indicate that the transition from small or no-particle(slip) to large-particle formation is associated with a change from shear bands to tension-driven cracks.
The transition from shearing an asperity with negligible mass loss to the detachment of a large debrisparticle has already been observed in MD simulations [16]. The first mechanism led to the ductile failure ofthe junction, via the progressive smoothing of the asperities involved in the sliding contact. On the otherhand, brittle failure occurred in asperities with larger junction size, eventually producing a wear debris.Authors explained the transition between these two regimes by the introduction of a critical length scale d ∗ (see Eq. (3) in [16]), below (resp., above) which ductile (resp., brittle) failure occurred.Our phase-field simulations further demonstrate the existence of such a critical length-scale. In fact, wefind that the transition from relatively low mass loss to large particle formation can be captured by the law9n Eq. (6). For a given slenderness H/D of the asperity and material properties, Eq. (6) uniquely identifiesa critical junction length J ∗ above which the junction detaches as a large wear debris. On the other hand,junction’s lengths smaller than the critical value J ∗ lead to small or no particle formation. Importantly, thetransition curve (6) is exactly the same for both PH and HD models, as the value obtained for the constantparameter C is the same in both formulations.The transition law (6) can be analytically derived based on the arguments presented in [16].We start by noticing that the regularization length (cid:96) of the phase-field model defines the nucleation stress σ = 3 G c E (cid:96) (1 − ν ) (7)which corresponds to the tensile strength of the material under uniaxial loading and in plane strain conditions[27]. Comparisons with experiments [25] demonstrated that, by choosing the regularization length (cid:96) such that σ c matches the tensile strength, the phase-field formulation is able to capture crack nucleation in a varietyof materials and geometries. Moreover, by recasting the variational HD formulation in a thermodynamicframework, crack nucleation can be shown to be governed by the stress criterion F ( σ ) = σ + 3 µK σ − µE σ (8)where σ eq (resp., σ m+ ) is the equivalent deviatoric (resp., positive hydrostatic) stress invariant and σ c is thetensile strength in Eq. (7). The PH formulation leads to a criterion similar to that in Eq. (8), expressed onlyin terms of the positive hydrostatic stress σ m+ .Our numerical computations showed that, for a given (cid:96) and depending on the geometry of the junction,the formation of a wear debris can be triggered by tensile stresses σ m+ or shear stresses σ eq (see Figure 7).Based on the fracture criterion (8), in both cases the elastic energy E el released by the detachment of a weardebris can be approximately computed as E el = 2 A deb σ c E (9)where A deb is the surface area of each of the particles forming the wear debris. On the other hand, thefracture energy related to the formation of two cracks of length L crack reads as E fr = 2 L crack G c (10)Assume that an asperity junction with given length J ∗ and slenderness H/D fails by the detachment oftwo triangular debris, as a consequence of the nucleation of two horizontal cracks at the ‘junction’ stressconcentrations in Figure 2 (a). The surface area of each debris particle is then A deb = α ( J ∗ ) HD/
2, whilethe crack length is L crack = βJ ∗ D . Coefficients α and β account for the fact that the actual shape ofthe detached debris might be other than triangular, and that the crack path might be not parallel to theasperity’s basis. Following [16] we note that, in order for a wear debris to be detached, the elastic energystored in the junction must overcome the fracture energy, that is E el ≥ E fr . This energy condition yields J ∗ ≥ βα EG c σ H (11)which, in view of Eq. (7), results in J ∗ ≥ β α (cid:0) − ν (cid:1) (cid:96)H (12)Finally, by recalling that the regularization length (cid:96) has been defined as a small fraction of the asperity’ssize D (i.e., (cid:96) = D/
50 in the performed calculations), Eq. (12) can be recast as J ∗ ≥ DH C (13) Note that here we make the assumption that prior to failure the stress distribution in the junction is relatively uniform andthat failure is triggered by either one of the two stress invariants. In this sense, the resulting expression for E el is an estimateof the elastic energy released upon the detachment of a wear debris. C collects all the non-geometrical parameters and accounts for some of the introduced as-sumptions (shape of debris particle and crack paths, uniform stored energy).Hence, for an asperity junction with given slenderness H/D and ratio (cid:96)/D , Eq. (13) identifies the criticaljunction length J ∗ above which a wear debris is formed. This holds for both PH and HD formulations.Although based on simple observations, these calculations allow us to retrieve the transition curve in Eq. (6).They further permit to highlight the dependency of J ∗ on the ratio (cid:96)/D . Calculations in Section 3.3 lead us to extend our analysis to the three-dimensional space identified by theparameters J , H/D and (cid:96)/D . The aim is to understand whether the observed wear mechanisms change asa function of (cid:96)/D . Computations are conducted by using a HD variational formulation, for junctions withgiven asperity basis D , elastic moduli and fracture toughness.Due to the extensive number of simulations needed to reproduce an entire chart ( J, H/D ) for differentratios (cid:96)/D , we restrict our attention to a representative selection of junction’s geometries. With referenceto Figure 7, the following two groups are considered. • Geometries G , with junction length J = 0 . H/D ∈ [0 . , . , . , . , . (cid:96)/D =0 .
02, those junctions failed under the action of shear stresses , through either a slip mechanism or thedetachment of a small wear particle. • Geometries G , with junction length J = 0 . H/D ∈ [0 . , . , . , . (cid:96)/D = 0 . tensile stresses , resulting in the formation of a small or large wear debris.The ratio (cid:96)/D has to be small enough for the regularized phase-field formulation to Γ-converge to thesharp Griffith’s model. We thus choose (cid:96)/D ∈ [0 . , . , . , . , . , . D of the computational domain is at least 30 times bigger than the regularization parameter (cid:96) . For eachvalue of (cid:96)/D , the mesh size δ is chosen such that the numerical toughness G numc = G c (1 = 3 δ/ (cid:96) ) is the samein all the performed simulations.Obtained results are reported in Figure 8. A grey shaded area indicates the plane ( J, H/D ) correspondingto the ratio (cid:96)/D = 0 .
02, for which simulations in Figure 7 were conducted. With respect to those results,the following is observed. • Geometries G in Figure 8 (a). For large values of (cid:96)/D , failure is mainly triggered by shear stresses.Depending on the slenderness H/D of the asperity junction, two wear mechanisms are observed: smallasperities (i.e., with relatively small values of
H/D ) slip one past the other, whereas more slenderjunctions (i.e., with larger slenderness
H/D ) fail through nucleation of shear bands and detachmentof a small wear particle. As the ratio (cid:96)/D decreases, both those shear mechanisms are progressivelyreplaced by tension-driven fracture. For the smallest regularization length, small particle formation isobserved for all the considered values of
H/D . • Geometries G in Figure 8 (b). Large ratios (cid:96)/D result in junctions failing by either slip, small or largeparticle formation, depending on the slenderness H/D of the asperity. In particular, small junctionsexperience shear-driven failure, whereas more slender ones fail through the nucleation of tensile cracks.As the ratio (cid:96)/D decreases, the slip mechanism is replaced by small particle formation triggered bytensile stresses. On the other hand, no influence of (cid:96)/D is observed on junctions detaching a largeparticle debris.Therefore, reducing the regularization length (cid:96) promotes tension-driven failure to the detriment of shear-driven mechanisms. Interestingly, such a transition from slip to particle formation could only be observedvia Molecular Dynamics [19] by varying the strength (cid:101) τ adh of the adhesive interface with respect to thematerial bulk strength (cid:101) τ b (kept constant). According to this study, a junction that detaches a wear particlein presence of a fully adhesive interface ( (cid:101) τ adh / (cid:101) τ b = 1) can on the other hand fail through a slip mechanismwhen weakly-bonded ( (cid:101) τ adh / (cid:101) τ b < σ c varies with (cid:96) .We further observe that, according to Eq. (7), the material strength σ c increases as the regularizationlength (cid:96) reduces. The maximum strain energy E el in Eq. (9) (that is, the energy stored in the junction11rior to failure) increases correspondingly, while the fracture energy E fr remains the same. This does notparticularly affect junctions that fail through the detachment of a large particle, as they already meet theenergy condition E el ≥ E fr . Conversely, relatively small junctions (as those shown in Figure 8) are particularlysensitive to changes in the material strength, as an increase in σ c would allow them to store more energyin a rather small debris particle A deb . Hence, as (cid:96) decreases, E el might in fact overcome E fr . Indeed, this iswhat we observe in Figure 8 as the ratio (cid:96)/D reduces: for the smallest value, all the considered geometriesfail by detaching a wear debris. As briefly recalled in the Introduction, the most extreme wear rates observed might be associated with thedetachment of macro-particles whose relevant length-scale exceeds a single junction [18]. Thus, any realisticapproximation of the wear rate must account for the presence of clusters of junctions rather than consideringeach of them as an isolated system. We therefore study how two junctions interact with each other whengetting closer together. The simulations follow the same procedure as described in Section 2.2, although thecomputational domain now consists of two junctions as represented in Figure 9. The regularization lengthis such that (cid:96)/D = 0 . real ( j r ) and an apparent ( j a ) contact length.The former is the actual contact length developed within the junction, while the latter is the total lengthof the cluster which also includes the space ∆ between the asperities. We use the horizontal projection j p = JD/ j r and j a with the projected critical junction length j ∗ p = j p J ∗ = D H C (14)which represents the junction length at which a single junction with a slenderness H/D shows a transitionfrom small to large particle formation. This normalization parameter is computed from the transition curveidentified in the study of single asperities. The real junction length is then defined as the normalized sum ofthe projected junction lengths j p (see Figure 9) j r = j p j ∗ p (15)which corresponds to a system with two junctions. The apparent contact length is defined as the normalizedsum of the asperity length and the distance ∆ j a = 1 j ∗ p (cid:18) D + ∆2 (cid:19) (16)When considering the formation of a macro-particle that encapsulates several asperities, it is natural toexpect the cracks to form in the bulk rather than in the junction. It is indeed very unlikely that junctionsthat independently fail with a mechanism where the crack grows only within the junction would interactto form a macro-particle. We thus expect that the macro-particle formation would only be located abovethe transition curve shown on Figures 4 and 7 (red dots). The comparative study of PH and HD modelsshows that this region is independent of the chosen formulation. We thus perform the interaction study byreferring to the PH formulation. As the failure mechanism strongly depends on the geometry, we consider junctions that correspond tovarious modes of failure to conduct a thorough study of the interaction. The rectangular markers in Figure10 show the { H/D, J } couples used to study the failure of a domain with two junctions. For each ofthese geometries, we perform several simulations by varying the distance ∆ between the asperities, which Here we make the assumption that a macro-particle can only be formed from two junctions which individually fail undertension (that is, above the transition curve) while the geometries that fail under shear (below the transition curve) are assumedto never interact. Indeed, a comparative study under HD conditions is needed to assess the role played by shear stresses andwill be the object of future investigation. / D ‘ / D . . . . . . J Slip (shear)Small particle (shear)Small particle (tension) (a) H / D ‘ / D . . . . . . J Slip (shear)Small particle (tension)Large particle (tension) (b)
Figure 8: Evolution of the failure mechanisms in the three-dimensional space defined by the junction’s length J , the asperity’s slenderness H/D and the ratio (cid:96)/D . HD conditions are assumed. (a) Geometries G with J = 0 . H/D ∈ [0 . , . , . , . , . G with J = 0 . H/D ∈ [0 . , . , . , . (cid:96) of the phase-field model is varied such that (cid:96)/D ∈ [0 . , . , . , . , . , . D = 1. A grey shaded area indicates the plane( J, H/D ) corresponding to the ratio (cid:96)/D = 0 .
02, for which simulations in Figure 7 were conducted. As inFigure 7, the hollow markers represent the junctions that failed by developing one straight shear band, asshown in Figure 6 (b). 13
H/D and junction parameter J . The two junctions are are separated by an horizontaldistance ∆. The length j p is the horizontal projection of the junction.Figure 10: Representation of the geometries used to study the interaction between two junctions. The chartshows the failure mechanisms for single asperities under PH conditions (i.e., Figure 4), where the rectangularmarkers highlight the geometries used for the interaction study. The black rectangles represent geometriesfar from the transition curve, which lead to the results represented as dense points in the interaction studyin Figure 12. The grey rectangles represent geometries sitting on each side of the transition curve, whichlead to the results represented as transparent points in Figure 12. C = 0 . j r ) andapparent ( j a ) contact lengths. The obtained results are shown in Figure 12.We observe that geometries that lie far from the transition curve in Figure 10 (black rectangular markers),give rise to very well defined regions in Figure 12 (dense markers). On the other hand, the geometries locatedclose to the transition curve in Figure 10 (grey rectangular markers) tend to produce results that slightlyoverlap in Figure 12 (transparent markers). Regardless of the distance ∆, every junction that independentlyfails forming a small particle shows the same behavior when considered as a group. In Figure 12, thiscorresponds to the area identified by small real contact lengths j r (blue triangles). Conversely, two distinct14a) (b)Figure 11: Interaction study under PH conditions, for a cluster of junctions with J = 0 . H/D = 0 . . D . A pair of cracks independently nucleates in the bulk around eachjunction. The resulting wear product is identified as large debris formation. (b) The same junctions as in(a), where the horizontal distance is now reduced to ∆ = D . Only one pair of cracks nucleates and thetwo junctions will eventually detach as a single wear debris. This phenomenon is identified as interactingjunctions or macro-particle formation. Note that the figures show zoom-ins, cropped from a much biggerdomain.Figure 12: Interaction study, evolution of the failure mechanisms as a function of the real j r (Eq. 15) andapparent j a (Eq. 16) contact lengths. The blue triangles represent independent failures of the two junctionsthrough the nucleation of cracks inside the junction, resulting in small particle formation. The red dotsrepresent independent failures of the two junctions through the nucleation of cracks inside the bulk (seeFigure 11 (a)), resulting in large particle formation. The large green dots represent interacting failures ofthe junctions through the nucleation of a single pair of cracks that embodies the two junctions (see Figure 11(b)), resulting in the formation of a macro-particle debris. The dense markers are the results of interactionstudies performed on geometries represented by dense markers in Figure 10, which lie far from the transitioncurve. Transparent markers correspond to junction geometries represented by transparent points in Figure10, which lie on either side of the transition curve. 15egions are observed for larger values of j r . Geometries with a high j a exhibit a crack pattern similar tothat shown in Figure 11 (a) (red dots). These points correspond to junction couples where the distance ∆ islarge enough to prevent interaction, resulting in the detachment of a large particle debris. Geometries with alower j a show a crack pattern similar to that in Figure 11 (b) (large green dots). In these cases, the distance∆ is small enough for the interaction to occur and the two junctions detach together as a macro-particle. This paper investigates adhesive wear in asperity junctions by using a variational phase-field approach tobrittle fracture [21–23]. We consider two regularized formulations with contact conditions, as proposed in[27]. In the positive-hydrostatic model (PH), fracture is only allowed in material regions with positive volumechange, while the hydrostatic-deviatoric fomulation (HD) also includes shear-driven fracture.We first consider the case of a single junction comprised of two triangular asperities in contact to eachother. The crack paths observed in the comparative study of various geometries lead to the identificationof various failure patterns. This gives rise to a classification in distinct failure mechanisms (schematicallysummarized in Figure 13 (a)–(d)), each of them is characterized by a triggering stress (shear or tension) anda debris formation pattern (no particle, small particle and large particle detachment).The study of single junctions under PH conditions shows that tension-driven cracks results in the forma-tion of either a small (Figure 13 (c)) or large particle (Figure 13 (d)), depending on the slenderness
H/D and on the relative overlap J of the asperities forming the junction. We find that the transition from smallto large particle formation can be described by the criterion J ∗ = D/H · C (transition curve), where C is aconstant parameter depending on the material properties. With respect to the failure mechanisms observedunder PH conditions, the HD formulation allows us to capture the occurrence of two additional failure modes, slip and shear . When slip occurs, a shear band forms along the interface between the asperities and no debrisis produced (Figure 13 (a)). On the other hand, when a shear mode is observed, a horizontal shear banddevelops within one or both asperities and the junction fails with either no debris formation (Figure 13 (b))or by releasing a small particle (Figure 13 (c)). The criterion J ∗ = D/H · C , expressed in terms of the samematerial constant C , now describes the change in the stresses triggering failure from tension ( J > J ∗ ) to shear( J < J ∗ ). Interestingly, the condition for large particle formation ( J > J ∗ ) is independent from the chosen(a) (b) (c)(d) (e)Figure 13: Schematic representation of the observed failure mechanisms. Red curves and shaded areasrespectively represent cracks and the produced wear debris. (a) Slip: a shear band propagates along theinterface between the asperities; no debris formation occurs. (b) Single crack in the junction: a single shearband propagates straight from one of the stress concentrations in the junction; no debris formation occurs.(c) Small particle formation: a pair of cracks propagates in the junction; small debris formation occurs.(d) Large particle formation: a pair of cracks propagates in the bulk; large debris formation occurs. (e)Macro-particle formation: two junctions interact and a pair of cracks propagates in the bulk.16oupling between elastic energy and damage field. These results are in agreement with observations reportedin Molecular Dynamics studies [16] highlighting the existence of a critical length scale, which uniquely iden-tifies the transition from relatively low mass loss to large particle detachment. Such a condition for particleformation is analytically derived based on the arguments presented in [16]. This leads us to investigate theinfluence of the phase-field regularization parameter on the observed failure mechanisms. Results show thatreducing the regularization length (that is, increasing the material strength) promotes tension-driven failureto the detriment of shear-driven mechanisms, as a transition from slip to particle formation is observed.In the second part of the paper, we study the interaction between two asperity junctions and show howmicrocontact interactions can be favoured in some geometries to form macro-particles (Figure 13 (e)). Weperform the interaction study by using a PH formulation, as we expect macro-particle detachment fromgeometries which independently produce a large debris ( J > J ∗ ). We find that the two junctions startinteracting as the distance between them decreases, eventually detaching as a single wear debris or macro-particle. The transition from a cluster of non-interacting junctions releasing a large debris to a cluster ofinteracting junctions producing a macro-particle is clearly identified in terms of real and apparent contactlengths.Finally, a central point in our analysis consists in the relation between the debris formation patternsand the macroscopic wear rate induced by the failure of the adhesive junctions. Since mild and severe wearregimes are usually associated to particle detachment through crack propagation, we expect our phase-fieldapproach to be effective in the prediction of those wear processes. The same logic would however put thelow wear regime out of our reach, since this wear form is commonly associated to the complete alterationof the domain through large deformations. Although serving as a first step and approximation for morerefined calculations, the proposed results do deliver some useful insights on the low wear regime, as thefailure mechanisms (a) and (b) in Figure 13 are driven by shear and do not lead to any particle formation,which indeed are two features usually associated to this wear form.Based on the discussion above, we propose a classification in terms of wear rate of the five failuremechanisms shown in Figure 13. • The mechanisms that do not form any wear debris (Figure 13 (a) and (b)), which are driven by shear,can be associated to a low wear regime. • The mechanisms that form small or large wear debris (Figure 13 (c) and (d)), which are driven bytension, can be associated to a mild wear regime. • The mechanisms that form macro-particles (Figure 13 (e)), which are driven by tension, can be asso-ciated to a severe wear regime.This change in the wear process at the microscopic level could explain the change in the type of relationbetween wear rate and applied load at the macroscopic one. It is however to be noted that, in a large-scalefriction process, several failure mechanisms can simultaneously contribute to the overall wear, in which caseprobabilistic approaches are more suitable to estimate the macroscopic wear rate [33].To summarize, we were able to recover clear transitions in the crack nucleation patterns using a phase-field variational approach to fracture. Based on the results from Molecular Dynamics simulations found inliterature, we proposed to associate each crack nucleation pattern to a failure mechanism. The geometry ofthe adhesive junctions is understood to be responsible for the change in failure mechanism, which can beexpressed as a function of the junction length and the slenderness of the asperities. This transition curvecould be recovered regardless of the coupling between the damage and the elastic energy. For geometrieslocated above the transition curve, the spacing between the junctions determines if they interact to detachas a single macro-debris. Prescribing different coupling between the damage field and the elastic energyallowed us to assess the triggering mechanisms underlying the various failure processes: large and macro-particle formation is associated with tension-driven cracks while small or no particle formation is typicallyassociated with the formation of shear bands. With this distinction, we could recover all failure mechanismsassociated to adhesive junctions found in the literature: slip, gradual smoothing, small, large and macro-debris formation.Finally, this study showed the potential of variational phase-field formulations in studying adhesive wearin brittle materials. Moving forward, future work shall investigate the tribological behavior of elastic-plastic17unctions, which on the other hand is hardly accessible by discrete numerical models such as those based onMolecular Dynamics.
Acknowledgment
The authors thank the Zeno Karl Schindler foundation for its financial support andProf. Kaushik Bhattacharya for providing a working space as well as computational resources at the Cali-fornia Institute of Technology. 18 eferences [1] R. Holm, in Electrical Contacts . Springer-Verlag, Berlin, 1946.[2] E. Rabinowicz, “The effect of size on the looseness of wear fragments,”
Wear , vol. 2, no. 1, pp. 4–8,1958.[3] E. Rabinowicz and R. Tanner, “Friction and wear of materials,”
Journal of Applied Mechanics , vol. 33,p. 479, 1966.[4] T. Grigoratos and G. Martini, “Brake wear particle emissions: a review,”
Environmental Science andPollution Research , vol. 22, no. 4, pp. 2491–2504, 2015.[5] J. M. Samet, F. Dominici, F. C. Curriero, I. Coursac, and S. L. Zeger, “Fine particulate air pollution andmortality in 20 us cities, 1987–1994,”
New England journal of medicine , vol. 343, no. 24, pp. 1742–1749,2000.[6] C. A. Pope Iii, R. T. Burnett, M. J. Thun, E. E. Calle, D. Krewski, K. Ito, and G. D. Thurston, “Lungcancer, cardiopulmonary mortality, and long-term exposure to fine particulate air pollution,”
Jama ,vol. 287, no. 9, pp. 1132–1141, 2002.[7] U. Olofsson, “A study of airborne wear particles generated from the train traffic—block braking simu-lation in a pin-on-disc machine,”
Wear , vol. 271, no. 1-2, pp. 86–91, 2011.[8] J. Archard, “Contact and rubbing of flat surfaces,”
Journal of applied physics , vol. 24, no. 8, pp. 981–988,1953.[9] J. Archard and W. Hirst, “The wear of metals under unlubricated conditions,”
Proceedings of the RoyalSociety of London. Series A. Mathematical and Physical Sciences , vol. 236, no. 1206, pp. 397–410, 1956.[10] H. Meng and K. Ludema, “Wear models and predictive equations: their form and content,”
Wear ,vol. 181, pp. 443–457, 1995.[11] H. Kitsunai, K. Kato, K. Hokkirigawa, and H. Inoue, “The transitions between microscopic wear modesduring repeated sliding friction observed by a scanning electron microscope tribosystem,”
Wear , vol. 135,no. 2, pp. 237–249, 1990.[12] K. Hokkirigawa, “Wear mode map of ceramics,”
Wear , vol. 151, no. 2, pp. 219–228, 1991.[13] Y. Wang and S. M. Hsu, “Wear and wear transition modeling of ceramics,”
Wear , vol. 195, no. 1-2,pp. 35–46, 1996.[14] S. M. Hsu and M. Shen, “Wear prediction of ceramics,”
Wear , vol. 256, no. 9-10, pp. 867–878, 2004.[15] K. Kato and K. Adachi, “Wear of advanced ceramics,”
Wear , vol. 253, no. 11-12, pp. 1097–1104, 2002.[16] R. Aghababaei, D. H. Warner, and J.-F. Molinari, “Critical length scale controls adhesive wear mecha-nisms,”
Nature communications , vol. 7, p. 11816, 2016.[17] R. Aghababaei, D. H. Warner, and J.-F. Molinari, “On the debris-level origins of adhesive wear,”
Proceedings of the National Academy of Sciences , vol. 114, no. 30, pp. 7935–7940, 2017.[18] R. Aghababaei, T. Brink, and J.-F. Molinari, “Asperity-level origins of transition from mild to severewear,”
Physical review letters , vol. 120, no. 18, p. 186105, 2018.[19] T. Brink and J.-F. Molinari, “Adhesive wear mechanisms in the presence of weak interfaces: Insightsfrom an amorphous model system,”
Physical Review Materials , vol. 3, no. 5, p. 053604, 2019.[20] E. Milanese, T. Brink, R. Aghababaei, and J. F. Molinari, “Emergence of self-affine surfaces duringadhesive wear,”
Nature Communications , vol. 10, no. 1, pp. 1–9, 2019.1921] B. Bourdin, G. A. Francfort, and J.-J. Marigo, “Numerical experiments in revisited brittle fracture,”
Journal of the Mechanics and Physics of Solids , vol. 48, no. 4, pp. 797–826, 2000.[22] B. Bourdin, G. A. Francfort, and J.-J. Marigo, “The variational approach to fracture,”
Journal ofelasticity , vol. 91, no. 1-3, pp. 5–148, 2008.[23] G. A. Francfort and J.-J. Marigo, “Revisiting brittle fracture as an energy minimization problem,”
Journal of the Mechanics and Physics of Solids , vol. 46, no. 8, pp. 1319–1342, 1998.[24] B. Bourdin, J.-J. Marigo, C. Maurini, and P. Sicsic, “Morphogenesis and propagation of complex cracksinduced by thermal shocks,”
Physical review letters , vol. 112, no. 1, p. 014301, 2014.[25] E. Tann´e, T. Li, B. Bourdin, J.-J. Marigo, and C. Maurini, “Crack nucleation in variational phase-fieldmodels of brittle fracture,”
Journal of the Mechanics and Physics of Solids , vol. 110, pp. 80–99, 2018.[26] V. Carollo, M. Paggi, and J. Reinoso, “The steady-state archard adhesive wear problem revisited basedon the phase field approach to fracture,”
International Journal of Fracture , vol. 215, no. 1-2, pp. 39–48,2019.[27] H. Amor, J.-J. Marigo, and C. Maurini, “Regularized formulation of the variational brittle fracture withunilateral contact: Numerical experiments,”
Journal of the Mechanics and Physics of Solids , vol. 57,no. 8, pp. 1209–1229, 2009.[28] G. Lancioni and G. Royer-Carfagni, “The variational approach to fracture mechanics. a practical appli-cation to the french panth´eon in paris,”
Journal of elasticity , vol. 95, no. 1-2, pp. 1–30, 2009.[29] C. Brockley and G. Fleming, “A model junction study of severe metallic wear,”
Wear , vol. 8, no. 5,pp. 374–380, 1965.[30] B. Bourdin, “mef90/vdef: variational models of defect mechanics,” 2019.[31] M. Ambati, T. Gerasimov, and L. De Lorenzis, “A review on phase-field models of brittle fracture anda new fast hybrid formulation,”
Computational Mechanics , vol. 55, no. 2, pp. 383–405, 2015.[32] S. Pham-Ba, T. Brink, and J.-F. Molinari, “Adhesive wear and interaction of tangentially loaded micro-contacts,”
International Journal of Solids and Structures , 2019.[33] L. Fr´erot, R. Aghababaei, and J.-F. Molinari, “A mechanistic understanding of the wear coefficient:From single to multiple asperities contact,”