Journal club for December 2023 : Recent trends in modeling of asperity-level wear

J.-F. Molinari 1, S. Z. Wattel 1, L. Ma 2, and R. Aghababaei 2

1 Computational Solid Mechanics Laboratory, Institute of Civil Engineering, Institute of Materials, École Polytechnique Fédérale de Lausanne (EPFL), CH 1015 Lausanne, Switzerland

2 Surface Mechanics and Tribology Group, Department of Mechanical and Production Engineering, Aarhus University, 8000 Aarhus C, Denmark

1 Introduction

Ernest Rabinowicz’s words, spoken two decades ago in his groundbreaking textbook on the friction and wear of materials [1], continue to resonate today: ’Although wear is an important topic, it has never received the attention it deserves.’ Rabinowicz’s work laid the foundation for contemporary tribology research [2]. Wear, characterized as the removal and deformation of material on a surface due to the mechanical action of another surface, carries significant consequences for the economy, sustainability, and poses health hazards through the emission of small particles. According to some estimates [1, 3], the economic impact is substantial, accounting for approximately 5% of the Gross National Product (GNP).

Despite its paramount importance, scientists and engineers often shy away from wear analysis due to the intricate nature of the underlying processes. Wear is often perceived as a ”dirty” topic, and with good reason. It manifests in various forms, each with its own intricacies, arising from complex chemical and physical processes. These processes unfold at different stages, creating a time-dependent phenomenon influenced by key parameters such as sliding velocity, ambient or local temperature, mechanical loads, and chemical reactions in the presence of foreign atoms or humidity.

Figure 1: Publication statistics in the Wear journal, showing a negligible contribution of numerical studies over past decades [4]. Blue bars show the total number of annual publications on the subject. Red bars show the number of publications that studied the subject using a numerical tool, including the finite-element approach, the discrete-particle method, and the molecular-dynamics approach. Statistics are obtained from the Scopus website.

The review paper by Vakis et al. [5] provides a broad perspective on the complexity of tribology problems. This complexity has led to numerous isolated studies focusing on specific wear mechanisms or processes. The proliferation of empirical wear models in engineering has resulted in an abundance of model variables and fit coefficients [6], attempting to capture the intricacies of experimental data.

Tribology faces a fundamental challenge due to the multitude of interconnected scales. Surfaces exhibit roughness with asperities occurring at various wavelengths. Only a small fraction of these asperities come into contact, and an even smaller fraction produces wear debris. The reasons behind why, how, and when this occurs are not fully understood. The debris gradually alter the surface profile and interacts with one another, either being evacuated from the contact interface or gripping it, leading to severe wear. Due to this challenge of scales, contributions of numerical studies in wear research over the past decades sum up to less than 1% (see Fig. 1). Yet, exciting opportunities exist for modeling, which we attempt to discuss here.

While analyzing a single asperity contact may not unveil the entire story, it arguably represents the most fundamental level to comprehend wear processes. This blog entry seeks to encapsulate the authors’ perspective on this rapidly evolving topic. Acknowledging its inherent bias, the aim is to spark controversies and discussions that contribute to a vibrant blogosphere on the mechanics of the process.

The subsequent section delves into the authors’ endeavors in modeling adhesive wear at the asperity level. Section 3 navigates the transition to abrasive wear, while Section 4 explores opportunities for upscaling asperity-level mechanisms to the meso-scale, with the aspiration of constructing predictive models. Lastly, although the primary focus of this blog entry is on modeling efforts, it would be remiss not to mention a few recent advances on the experimental front.

2 Atomistic simulations of adhesive wear

Our discussion focuses specifically on asperity-level wear mechanisms, where a significant overlap occurs between contacting junctions. This results in either severe plastic deformation or the formation of debris. There is of course a large body of work in nanotribology that investigates nanoscale mechanisms, such as atom by atom attrition, that we leave out of the discussion [7–9].

We commence by examining Molecular Dynamics (MD) simulations conducted by Aghababaei, Warner, and Molinari [10]. Their work, for the first time, revealed the adhesive wear process described by Rabinowicz [1] and discussed in a perspective paper [2]. Fig. 2 succinctly summarizes the main finding, a ductile to brittle transition at a critical material length scale. Contact asperities forming junctions below this scale undergo plastic deformation, while larger junctions lead to crack formation, resulting in debris that detaches from the contact interface. This material length scale determines the minimum debris particle size, crucial in understanding the onset of wear and also fine particle air pollution. The minimum debris particle size is determined by Griffith’s argument, addressing whether the stored elastic energy in the sheared contact junction is sufficient to create new crack surfaces. The resulting length scale d* is proportional to the surface energy γ (for very brittle materials), the shear modulus G and inversely proportional to the square of the shear strength τ:

with λ a geometrical factor of the order of unity.

Although Rabinowicz initially proposed a similar energetic argument [2] for the release of trapped debris, the simulations of Aghababaei et al. not only confirm the existence of a minimum debris size but also reveal that small contact junctions undergo plastic deformation, continuously smoothing surface roughness during sliding. Ductile processes therefore reduce roughness, eventually forming junctions large enough to result in debris formation at a later stage. This competition between ductile and brittle events is a key observation and leads to roughness reaching a steady state.

While MD simulations are resource-intensive and constrained in time and space, Milanese et al. [11] demonstrated that after debris creation, the surface roughness resembles a self-affine fractal. These 2D simulations, illustrating roughness creation, were recently extended to 3D using a supercomputer [12, 13]. An essential point is that to reveal brittle events and create roughness, the simulation domain must exceed the critical junction size. MD simulations that neglect this length scale only display a flattening of the contact surface, as seen by Spijker [14], where rough aluminum samples were modeled with an EAM potential in a simulation box of 32nm, which is clearly insufficient to represent fracture processes in aluminium. Similarly, the MD simulations in [15, 16] are not conducted at scales that can reveal brittle mechanisms.

Aghababaei et al. simulations have opened up several perspectives, indicating that frictional work is an excellent predictor of debris particle size for contact junctions near the critical size [17], while serving as an upper bound for larger junctions [18]. Brink extended early simulations that used simple Morse potentials to more realistic potentials for ceramics, uncovering a third mechanism—slip for surfaces with reduced adhesion [19].

Naturally, these models can be applied to materials with hardness contrast, exploring the transition to abrasive wear—an area we delve into in the next section.

Figure 2: A ductile to brittle transition controls the debris formation process. a) Aghababaei, Warner and Molinari have shown that adhesive wear is controlled by the geometry of asperities that collide and material properties [10]. b) Contact junctions smaller than a material length scale d* deform plastically during sliding. c) Contact junctions larger than d* result in the formation of a detached debris. These results were initially obtained with simple and quite brittle Morse potentials, that allowed MD simulations to span the parameter space at a low computational cost.      

3 Atomistic simulations of abrasive wear

In the realm of abrasive wear, MD simulations have gained considerable attention. In this section, we mainly review the MD simulations of two-body abrasive wear at the single asperity level.

Starting in the 1980s with the development of Atomic Force Microscopy (AFM) [20], tribology has taken a turn towards bringing about atomic insights into single-asperity abrasive wear [21]. Kato et al. [22] established an abrasive wear mode map, distinguishing the cutting, wedge-forming, and plowing modes. In recent years, many researchers have used MD simulations to study parameters that control abrasive wear and corresponding material removal mechanisms. Fang et al. [23, 24] undertook a systematic exploration of how loading conditions, such as applied load, dwell time, temperature, and geometry influence the wear responses during nanoscratching [25–27].

The interplay of adhesive interactions between the abrasive piece and substrate emerges as an important factor in material removal rate and mechanism [28]. Hu et al. [29] investigated the effects of adhesive strength and load on material transfer during nanoscale sliding, finding that adhesion and abrasion contribute to material transfer, with their relative importance depending on load and work of adhesion. It was also shown [30, 31] that the higher the adhesive strength, the larger the plastic deformation and material transfer. On the other hand, low adhesive strength leads to material detachment without significant subsurface deformation.

Li and Aghababaei [32, 33] studied the effect of abrasion and adhesion on material removal via controlling scratching depth and adhesive strength. As shown in Fig. 3, this study confirmed the existence of a critical adhesive strength, dictating the transition between the atom-by-atom removal and the fragment removal regimes. It can be seen that the larger the scratching depth, the smaller the critical adhesive strength at which the transition in the material removal mechanism occurs. It was also shown that the wear coefficient increases with adhesion strength and scratching depth and eventually saturates to a constant value. The saturation observed is linked to the shift from atomic attrition wear to plasticity-induced wear. This confirms the possibility of obtaining a wear coefficient independent of depth and adhesion when plastic deformation governs abrasive wear.

Figure 3: Image taken from [32] that investigates how abrasion and adhesion influence the material removal mechanisms during abrasive wear. There is a critical adhesive strength, indicating the transition between (i) atom-by-atom removal and (ii) fragment removal regimes. The critical adhesive strength is a function of scratching depth.

Crystalline materials exhibit anisotropy at the micro- and nano-scale due to different crystallographic orientations. This may influence the surface morphology [34, 35] and scratching forces [36]. Extensive MD simulations have been conducted to study the anisotropic scratch response of single crystal Cu [37–39], Ni [40, 41], and Au [42, 43]. Also, researchers investigated the deformation mechanism and tribological properties of nanolayered overcoats through single-asperity abrasive wear models. Price et al. [44] conducted MD simulations to study the impact of coating design parameters, such as layer thickness and composition, on the adhesion of ultra-thin multi-layer DLC coatings to the substrate. Their findings reveal that an intermediate DLC layer with a lower sp3 fraction than the outermost DLC coating layer protects the substrate from plastic deformation under external loading. The layering morphology of various multilayer coatings was studied using MD simulations, strategically aimed at optimizing abrasive wear performance [45–47].

4 Upscaling

MD simulations implicitly account for many complex mechanisms such as contact, plasticity, damage or fracture, which make them of great use for the numerical investigation of tribology. Nevertheless, a significant drawback is their high computational cost, limiting their applicability to modeling large-sized asperities or surfaces comprising many asperities. In the following, methods to deal with both of these challenges are presented.

4.1 Single asperity beyond the nanometer

4.1.1 Coarse-graining methods

To address the computational cost issue, coarse-graining methods have been recently proposed. Pham-Ba and Molinari [48] propose a Discrete-Element Model (DEM) where each particle represents many atoms. The interaction potential between particles features an elastic branch when two particles are in compressive contact and a cohesive branch when they are pulled away, aiming to emulate the adhesive contact and fracture behavior of atomistic simulations. The potential can be tuned to match desired macroscale properties such as Young’s modulus or surface energy. This model is able to capture the brittle to ductile transition of single asperity while reducing the computation requirements by several orders of magnitude. However, as the particles are non-breaking, a natural size limitation appears: the particle should stay below d* if one wishes to keep the size of debris formed consistent with the critical junction size model. To upscale further, Mollon introduces another DEM framework using both deformable particles and cohesion [49]. Indeed, cohesion has been found to be a necessary ingredient to be able to capture the rolling behavior of third bodies. With these models, [50, 51], a rich set of third-body behavior was uncovered, including a mixed-mode behavior, combining characteristics of the rolling third body behavior and of shear bands. These modes are illustrated in Fig. 4, and were shown to be a critical factor in the origin of friction forces.

It is important to note that both models differ fundamentally from conventional DEM formulations, as their interaction laws are not designed to represent a specific granular material. Rather, parameters are tuned to match homogenized material properties, positioning these models as surrogate representations of damageable continuous media.

Recent efforts [52, 53] developed a concurrent multiscale method, coupling the continuum finite-element domain with discrete coarse-grained atomistic potentials [10]. In this multiscale framework, complexities like large deformation, crack nucleation, and propagation are simulated in the atomistic domain, while far-field boundary conditions are modeled in the continuum domain. It is shown that the method can accurately predict crack formation while reducing the computation cost significantly.

These methods open research opportunities by enabling micromechanical simulations at a larger scale.

Figure 4: (a) Example of a mixed regime, showcasing the generation of wear particles, their aggregation, and subsequent reattachment to the sliding bodies. The gap between the bodies remains consistently sized. The colors denote the initial vertical position of the particles. (b) Example of a shear band scenario. No discernible gap is observed between the two bodies, and the particles exhibit vertical migration. (c) Example of the formation of large wear particles, increasing in size until they match the dimensions of the entire system. Reproduced from [51].  

4.1.2 Continuum methods

While continuum models might struggle to capture the large deformations and debris formation mechanisms seen at a tribological interface, they suffer less from scaling issues. Several recent works [54–56] modeled asperity collision with a phase-field approach for fracture. By adjusting the material ductility and model geometry, they recover the three behaviors discussed above: slipping without significant damage, damage to the asperity’s tip, or crack nucleation at the base of the asperity [56]. Once again, the critical length scale d* proves useful in predicting the predominant behavior. Micromechanics of asperity fracture under static and cyclic loading conditions have also been investigated in recent years [57–60], providing a basis for developing wear models for brittle solids.

A recent investigation of abrasion with size-dependent crystal plasticity by Zhu and Aghababaei [61] indicated that crystallographic anisotropy results in orientation and scratching direction-dependent wear volume and thus coefficient, see Fig. 5. Alternatively, it is shown that a unique wear coefficient can be obtained for crystalline materials when one uses the scratching hardness in Archard’s wear law. Furthermore, it is shown [62] that the degree and nature of size-dependency in the scratch hardness value and pile-up magnitude vary significantly and oppositely with the scratch direction. For instance, while the [001] direction shows the highest degree of size effect in scratch hardness, it presents the lowest pile-up size effect.

Figure 5: Comparison of surface topographies of wear groves on (001)-oriented single crystalline copper (a.) obtained from experiments and simulations after scratching in (b.) 0° and (c.) 22.5°, measured based on [100] crystalline direction. (d.) presents the total wear volume obtained from scratching simulations and experiments for different scratch directions on [001], [101], and [111] crystalline copper surfaces.

On another note, Milanese and Molinari [63] conducted an analytical and numerical study of the mechanics of a rolling particle. The purpose of this investigation was to explain the observation that a particle, while rolling between two surfaces in the presence of substantial adhesion, tends to increase in size, resembling the build-up of a snowman. It was shown that, at the junction of the particle and the bulk, cracks preferably grow into the bulk rather than into the particle, resulting in chips adhering to the particle.

4.2 Going beyond the single asperity

While single asperity contact constitutes the most fundamental mechanism, the majority of sliding contacts of interest involve many micro-contacts. These micro-contacts may or may not interact, affecting the overall behavior. Moreover, the fractal, self-affine, nature of most natural and man-made surfaces means that what is defined as a single asperity can depend on the scale of observation. Thus, it is crucial to extend the understanding gained of the single asperity mechanics - the microscale - to the multi-asperities contact - the mesoscale. Here, we present recent findings on the interaction of asperities and explore how these insights may contribute to advancing the comprehension of macroscale wear laws.

4.2.1 Asperity interactions

Interactions between neighboring asperities play a crucial role in tribological phenomena. When widely spaced, asperities react individually, but as the spacing decreases, their sub-surface stress fields start to overlap, causing them to react together. Aghababei, Brink and Molinari [64] conducted a parametric study using MD simulations of double asperity-to-asperity collision with varying spacing. When the spacing was large compared to the asperity diameter, the two contacts behaved independently. However, as the spacing approached the asperity size, both asperity detached together, as if acting as a single larger asperity, resulting in a proportionally higher wear volume. Son et al. studied the elastic stress field generated by the asperity collision for both the two-dimensional [65] and three-dimensional case [66]. Supplementing the analytical results with MD simulations, maps predicting the expected behavior are derived. Based on spacing and asperity size relative to d*, it could be determined whether plastic, independent or combined behavior would take place in the simulations.

Asperity interaction may explain the transition from mild to severe wear when the normal load is increased [64]. As the load increases, contact patches grow larger and new distinct patches appear. While the increase in real contact area is more or less linear with the load, the patches can start to interact, which results in wear particles larger than the size of individual patches and thus a super linear scaling of the wear rate with load.

4.2.2 Examination of engineering wear laws

Archard’s wear law is widely used in engineering applications. It states that wear is proportional to the sliding distance and normal force, and inversely proportional to the hardness. A wear coefficient, essentially a fitting coefficient, which can take value anywhere between 10-8 to 10-1, accounts for the effect of other parameters and must be, as of now, determined empirically. One of the great challenges of tribology is to determine how this wear coefficient could be derived from first principles.

Systematic investigation of asperity level adhesive wear [18, 67, 68] highlighted that a linear adhesive wear law (i.e. Archard’s relation) can be recovered at the single-asperity level only if the material removal is dominated by plastic deformation, confirming the longstanding Archard’s theoretical hypothesis. Alternatively, the relation breaks down when cleavage fracture or thermally activated atomic detachment governs the loss of material at the asperity level. Additionally, it is shown that [69] in the presence of weak adhesion, sliding is dominated by frictional slipping (i.e., dislocations glide in the contact plane), where occasional atomic cluster detachments and a sublinear wear relation are expected. Alternatively, a linear relationship between the volume of detached material and the frictional work can be obtained when bulk plasticity dominates material removal at the asperity tip. Under this condition, high-shear stresses at the contact trigger the migration of misfit dislocations into the asperity bulk, causing severe plastic deformation via dislocation-mediated interface migration. This result highlights that the state of stresses at contact governs the process of material removal and wear relations at the asperity level. Additionally, it is shown [70] that at the nanoscale, the wear coefficient increases by the adhesion strength and scratching depth and eventually saturates to a constant value. The saturation is associated with the transition from atomic attrition wear mode to plasticity-induced wear.

Figure 6: Schematic of the wear models. (a) Rough-on-rough elastoplastic contact problem, which is solved as a rigid rough on elastoplastic flat contact following Johnson’s assumption [71]. The yellow areas correspond to the contact patches. (b) The initial model proposed by Frérot et al. [72] Contact patches in red form debris. Increasing the hardness (keeping all other model parameters constant) leads to a larger fraction of microcontacts being red, in opposition to Archard’s law. (c) The improved model of Brink et al. [73], which takes the sliding process into account and reconciles model’s prediction with Archard. Reproduced from [73].

Insights from single asperity mechanics and asperities interactions can be applied to the results of BEM (Boundary Element Method) simulations of rough-on-rough contact to bring some elements of answer to the question of the origin of the wear coefficient. BEM simulations allow the efficient modeling of the elastoplastic contact problem between two self-affine rough surfaces [74, 75]. From these simulations, the contact patches can be recovered.

An initial idea [72] was to take the elastic contact map as-is and assume that any contact patch larger than d* would contribute to the wear volume. Popov and Pohrt [76] ran a similar study using elasto-plastic BEM simulations and predicted particle detachment with the ratio of elastic stored energy to adhesion energy, as for the formulation of d*. Brink et al. added an important element, that is the history of the sliding [73]. As the two surfaces slide against each other, new contact patches appear and grow in size before disappearing again. Then, it is assumed that each contact patch will lead to the creation of a wear particle when its size reaches d* and then be ”deactivated” (i.e. cannot form another wear particle), as illustrated in Fig. 6. In this case, increasing the hardness and thus reducing d* leads to the creation of smaller wear particles and thus a smaller wear rate. Thus, by taking into account the sliding movement, the model and Archard’s wear law are reconciled. This model, which contains no fit parameter, yield qualitatively good trends but would need further refinement and experimental data to compare to in order to reach quantitative power.

5 Outlook on experiments

To complement the understanding gathered from numerical simulations, systematic wear experiments at small scales are crucial. They enable validation and refinement of numerical simulations, offering a holistic perspective on wear modeling from nano to engineering scales. Such experiments offer invaluable insights into the fundamental mechanisms governing adhesive and abrasive wear at the nanoscale level. Consistent with the prediction of the critical junction size model, proposed by Aghababaei et. al., asperity level wear experiments pictured two distinct wear mechanisms: plastic smoothening [77–81] and asperity fracture [82, 83].

By employing precise control and measurement capabilities, researchers have explored the interactions between individual asperities [77] and surfaces [84, 85]. Leriche et. al [84], proposed an novel experimental technique enabling the detection of wear volumes as small as 0.016μm3 over contact areas large enough to offer insights on interactions between asperities. Leriche et. al [85] showed that Si3N4 wears through either atomic attrition or ductile removal enhanced by subsurface damage, depending on the magnitude of the local Si3N4-on-Si contact pressure. Surprisingly, they observed that Si3N4 wear in the pre-sliding wear experiments did not scale with the energy dissipated through friction, suggesting that contact formation and adhesive wear played an important role in the wear process. Recent advances in X-ray and computed tomography also enable studying the contact mechanics between non-transparent surfaces [86]. It is obvious that in-situ wear experiments across varying scales become essential, as these experiments provide real-time data that validate numerical simulations and advance our fundamental comprehension of wear mechanisms.

6 Concluding remarks

We conclude this blog entry by reemphasizing the critical importance of tribology in industry and for a sustainable economy. The view of the authors’ is that there is a substantial place for growth in Science in tribology. Computer simulations and nanoscale experimental measurements of wear processes have opened new opportunities for predictive wear models. Certainly, we have forgotten many relevant references in the summary above. We are hoping this entry will generate a lively discussion to complement our biased and incomplete list of references.

References

1

E. Rabinowicz, Friction and wear of materials (Wiley, New York, 1995).

2

E. Popova, V. L. Popov, and D.-E. Kim, “60 years of rabinowicz’ criterion for adhesive wear”, Friction 6, 341–348 (2018).

3

H. Jost, “Tribology — origin and future”, Wear 136, 1–17 (1990).

4

R. Aghababaei, “On the origins of third-body particle formation during adhesive wear”, Wear 426-427, 22nd International Conference on Wear of Materials, 1076–1081 (2019).

5

A. Vakis, V. Yastrebov, J. Scheibert, L. Nicola, D. Dini, C. Minfray, A. Almqvist, M. Paggi, S. Lee, G. Limbert, J. Molinari, G. Anciaux, R. Aghababaei, S. Echeverri Restrepo, A. Papangelo, A. Cammarata, P. Nicolini, C. Putignano, G. Carbone, S. Stupkiewicz, J. Lengiewicz, G. Costagliola, F. Bosia, R. Guarino, N. Pugno, M. Müser, and M. Ciavarella, “Modeling and simulation in tribology across scales: an overview”, Tribology International 125, 169–199 (2018).

6

H. Meng and K. Ludema, “Wear models and predictive equations: their form and content”, Wear 181–183, 443–457 (1995).

7

B. Gotsmann and M. A. Lantz, “Atomistic wear in a single asperity sliding contact”, Physical Review Letters 101, 10.1103/physrevlett.101.125501 (2008).

8

T. D. B. Jacobs and R. W. Carpick, “Nanoscale wear as a stress-assisted chemical reaction”, Nature Nanotechnology 8, 108–112 (2013).

9

N. N. Gosvami, J. A. Bares, F. Mangolini, A. R. Konicek, D. G. Yablon, and R. W. Carpick, “Mechanisms of antiwear tribofilm growth revealed in situ by single-asperity sliding contacts”, Science 348, 102–106 (2015).

10

R. Aghababaei, D. H. Warner, and J.-F. Molinari, “Critical length scale controls adhesive wear mechanisms”, Nature Communications 7, 10.1038/ncomms11816 (2016).

11

E. Milanese, T. Brink, R. Aghababaei, and J.-F. Molinari, “Emergence of self-affine surfaces during adhesive wear”, Nature Communications 10, 10.1038/s41467-019-09127-8 (2019).

12

T. Brink, E. Milanese, and J.-F. Molinari, “Effect of wear particles and roughness on nanoscale friction”, Physical Review Materials 6, 10.1103/physrevmaterials.6.013606 (2022).

13

J. Garcia-Suarez, T. Brink, and J.-F. Molinari, Roughness evolution induced by third-body wear, 2023.

14

P. Spijker, G. Anciaux, and J.-F. Molinari, “Dry sliding contact between rough surfaces at the atomistic scale”, Tribology Letters 44, 279–285 (2011).

15

P. Stoyanov, P. A. Romero, R. Merz, M. Kopnarski, M. Stricker, P. Stemmer, M. Dienwiebel, and M. Moseler, “Nanoscale sliding friction phenomena at the interface of diamond-like carbon and tungsten”, Acta Materialia 67, 395–408 (2014).

16

M. R. Sorensen, K. W. Jacobsen, and P. Stoltze, “Simulations of atomic-scale sliding friction”, Physical Review B 53, 2101–2113 (1996).

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 114, 7935–7940 (2017).

18

J. Garcia-Suarez, T. Brink, and J.-F. Molinari, “Breakdown of reye’s theory in nanoscale wear”, Journal of the Mechanics and Physics of Solids 173, 105236 (2023).

19

T. Brink and J.-F. Molinari, “Adhesive wear mechanisms in the presence of weak interfaces: insights from an amorphous model system”, Physical Review Materials 3, 10.1103/physrevmaterials.3.053604 (2019).

20

G. Binnig, C. Gerber, E. Stoll, T. R. Albrecht, and C. F. Quate, “Atomic resolution with atomic force microscope”, Europhysics Letters 3, 1281 (1987).

21

E. Gnecco, R. Bennewitz, and E. Meyer, “Abrasive wear on the atomic scale”, Phys. Rev. Lett. 88, 215501 (2002).

22

K. Kato, “Nanoscale analyses of wear mechanisms”, in Nanotribology: critical assessment and research needs, edited by S. M. Hsu and Z. C. Ying (Springer US, Boston, MA, 2003), pp. 45–54.

23

T.-H. Fang, W.-J. Chang, and C.-I. Weng, “Nanoindentation and nanomachining characteristics of gold and platinum thin films”, Materials Science and Engineering: A 430, 332–340 (2006).

24

Y.-C. Liang, J.-X. Chen, M.-J. Chen, Y.-l. Tang, and Q.-S. Bai, “Integrated md simulation of scratching and shearing of 3d nanostructure”, Computational Materials Science 43, 1130–1140 (2008).

25

Y. Gao, C. Lu, N. Huynh, G. Michal, H. Zhu, and A. Tieu, “Molecular dynamics simulation of effect of indenter shape on nanoscratch of ni”, Wear 267, ICAP 2008, 1998–2002 (2009).

26

P.-z. Zhu, Y.-z. Hu, H. Wang, and T.-b. Ma, “Study of effect of indenter shape in nanometric scratching process using molecular dynamics”, Materials Science and Engineering: A 528, 4522–4527 (2011).

27

I. Alabd Alhafez, A. Brodyanski, M. Kopnarski, and H. M. Urbassek, “Influence of tip geometry on nanoscratching”, Tribology Letters 65, 1–13 (2017).

28

N. Myshkin, M. Petrokovets, and A. Kovalev, “Tribology of polymers: adhesion, friction, wear, and mass-transfer”, Tribology International 38, Tribology International A Celebration Issue, 910–921 (2005).

29

X. Hu, S. Sundararajan, and A. Martini, “The effects of adhesive strength and load on material transfer in nanoscale wear”, Computational Materials Science 95, 464–469 (2014).

30

J. Song and D. Srolovitz, “Adhesion effects in material transfer in mechanical contacts”, Acta Materialia 54, 5305–5312 (2006).

31

J. Song and D. J. Srolovitz, “Mechanism for material transfer in asperity contact”, Journal of Applied Physics 104, 124312 (2008).

32

L. Ma and R. Aghababaei, “On the effect of adhesive strength and scratching depth on material transfer during nanoscale scratching”, Tribology Letters 70, 26 (2022).

33

L. Ma and R. Aghababaei, “On the Uniqueness of Wear Coefficient for Abrasive Wear at Nanoscale”, Journal of Tribology 145, 062101 (2023).

34

J. Zhu, Q. Zhou, Y. Huang, B. Zhou, and J. Wang, “Surface deformation of single crystalline copper on different nano-scratching paths”, Journal of Materials Science 56, 10640–10652 (2021).

35

H. Liu, P. Zhao, Y. Guo, D. Li, Y. Wang, S. Sun, and J. Wu, “Material removal behaviors of fcc metals in nanoscale and microscale scratching: theoretical model and experiments”, Journal of Materials Processing Technology 312, 117855 (2023).

36

J. Lin, F. Jiang, Q. Wen, Y. Wu, J. Lu, Z. Tian, and N. Wang, “Deformation anisotropy of nano-scratching on c-plane of sapphire: a molecular dynamics study and experiment”, Applied Surface Science 546, 149091 (2021).

37

P. Zhu and F. Fang, “Study of the minimum depth of material removal in nanoscale mechanical machining of single crystalline copper”, Computational Materials Science 118, 192–202 (2016).

38

G. Xiao, Y. He, Y. Geng, Y. Yan, and M. Ren, “Molecular dynamics and experimental study on comparison between static and dynamic ploughing lithography of single crystal copper”, Applied Surface Science 463, 96–104 (2019).

39

J. Hu, X. Liu, and Y. Wei, “Effect of plasticity and adhesion on the stick-slip transition at nanoscale friction”, Tribology International 164, 107230 (2021).

40

D. V. Lychagin, A. I. Dmitriev, A. Y. Nikonov, and E. A. Alfyorova, “Crystallographic and geometric factors in the shear development in (001) fcc single crystals: molecular dynamics simulation and experimental study”, Crystals 10, 10.3390/cryst10080666 (2020).

41

Z. Song, X. Tang, X. Chen, T. Fu, H. Zheng, and S. Lu, “Nano-indentation and nano-scratching of pure nickel and niti shape memory alloy thin films: an atomic-scale simulation”, Thin Solid Films 736, 138906 (2021).

42

S. Z. Chavoshi and S. Xu, “Nanoindentation/scratching at finite temperatures: insights from atomistic-based modeling”, Progress in Materials Science 100, 1–20 (2019).

43

AFM-Based Nanoscratching: A 3D Molecular Dynamics Simulation With Experimental Verification, International Manufacturing Science and Engineering Conference (June 2014), V001T03A011.

44

M. R. Price and B. Raeymaekers, “Quantifying adhesion of ultra-thin multi-layer dlc coatings to ni and si substrates using shear, tension, and nanoscratch molecular dynamics simulations”, Acta Materialia 141, 317–326 (2017).

45

T.-H. Fang, C.-H. Liu, S.-T. Shen, S. D. Prior, L.-W. Ji, and J.-H. Wu, “Nanoscratch behavior of multi-layered films using molecular dynamics”, Applied Physics A 90, 753–758 (2008).

46

Y. Cao, J. Zhang, Y. Liang, F. Yu, and T. Sun, “Mechanical and tribological properties of ni/al multilayers—a molecular dynamics study”, Applied Surface Science 257, 847–851 (2010).

47

K.-J. Seo and D.-E. Kim, “Deformation behavior of crystalline cr-ni multilayer coatings by using molecular dynamics simulation”, Lubricants 10, 10.3390/lubricants10120357 (2022).

48

S. Pham-Ba and J.-F. Molinari, “Adhesive wear with a coarse-grained discrete element model”, Computer Methods in Applied Mechanics and Engineering 397, 115124 (2022).

49

G. Mollon, “Solid flow regimes within dry sliding contacts”, Tribology Letters 67, 10.1007/s11249-019-1233-0 (2019).

50

G. Mollon, “The soft discrete element method”, Granular Matter 24, 10.1007/s10035-021-01172-9 (2021).

51

S. Pham-Ba and J.-F. Molinari, “Role of minimum adhesive wear particle size in third-body layer properties”, Tribol. Lett. 71 (2023).

52

S. Niknafs, M. Silani, F. Concli, and R. Aghababaei, “A coarse-grained concurrent multiscale method for simulating brittle fracture”, International Journal of Solids and Structures 254-255, 111898 (2022).

53

M. Voisin-Leprince, J. Garcia-Suarez, G. Anciaux, and J.-F. Molinari, “Finite element method–discrete element method bridging coupling for the modeling of gouge”, International Journal for Numerical Methods in Engineering 124, 1474–1488 (2022).

54

V. Carollo, M. Paggi, and J. Reinoso, “The steady-state archard adhesive wear problem revisited based on the phase field approach to fracture”, International Journal of Fracture 215, 39–48 (2018).

55

S. Collet, J.-F. Molinari, and S. Brach, “Variational phase-field continuum model uncovers adhesive wear mechanisms in asperity junctions”, Journal of the Mechanics and Physics of Solids 145, 104130 (2020).

56

S. Brach and S. Collet, “Criterion for critical junctions in elastic-plastic adhesive wear”, Physical Review Letters 127, 10.1103/physrevlett.127.185501 (2021).

57

H. Xu and K. Komvopoulos, “Fracture mechanics analysis of asperity cracking due to adhesive normal contact”, Int J Fract 281 (2013).

58

H. Xu and K. Komvopoulos, “A fracture mechanics analysis of asperity cracking due to sliding contact”, International Journal of Solids and Structures 171, 1–9 (2019).

59

M. Malekan, M. K. Budzik, H. M. Jensen, and R. Aghababaei, “Fracture analyses of surface asperities during sliding contact”, Tribology International 159, 106939 (2021).

60

M. Malekan, “Fracture loads of sharp and blunt asperities under normal and tangential loading”, Tribology International 177, 107949 (2023).

61

J. Zhu, X. Li, Q. Zhou, and R. Aghababaei, “On the anisotropic scratching behavior of single crystalline copper at nanoscale”, Tribology International 175, 107794 (2022).

62

J. Zhu and R. Aghababaei, “On the size effect in scratch and wear response of single crystalline copper”, Tribology International 186, 108573 (2023).

63

E. Milanese and J.-F. Molinari, “A mechanistic model for the growth of cylindrical debris particles in the presence of adhesion”, International Journal of Solids and Structures 203, 1–16 (2020).

64

R. Aghababaei, T. Brink, and J.-F. Molinari, “Asperity-level origins of transition from mild to severe wear”, Physical Review Letters 120, 10.1103/physrevlett.120.186105 (2018).

65

S. Pham-Ba, T. Brink, and J.-F. Molinari, “Adhesive wear and interaction of tangentially loaded micro-contacts”, International Journal of Solids and Structures 188–189, 261–268 (2020).

66

S. Pham-Ba and J.-F. Molinari, “Adhesive wear regimes on rough surfaces and interaction of micro-contacts”, Tribology Letters 69, 10.1007/s11249-021-01471-5 (2021).

67

K. Zhao and R. Aghababaei, “Adhesive wear law at the single asperity level”, Journal of the Mechanics and Physics of Solids 143, 104069 (2020).

68

R. Aghababaei and K. Zhao, “Micromechanics of material detachment during adhesive wear: a numerical assessment of archard’s wear model”, Wear 476, 23rd International Conference on Wear of Materials, 203739 (2021).

69

K. Zhao and R. Aghababaei, “Interfacial plasticity controls material removal rate during adhesive sliding contact”, Phys. Rev. Mater. 4, 103605 (2020).

70

L. Ma and R. Aghababaei, “On the Uniqueness of Wear Coefficient for Abrasive Wear at Nanoscale”, Journal of Tribology 145 (2023).

71

K. Johnson, J. Greenwood, and J. Higginson, “The contact of elastic regular wavy surfaces”, International Journal of Mechanical Sciences 27, 383–396 (1985).

72

L. Frérot, R. Aghababaei, and J.-F. Molinari, “A mechanistic understanding of the wear coefficient: from single to multiple asperities contact”, Journal of the Mechanics and Physics of Solids 114, 172–184 (2018).

73

T. Brink, L. Frérot, and J.-F. Molinari, “A parameter-free mechanistic model of the adhesive wear process of rough surfaces in sliding contact”, Journal of the Mechanics and Physics of Solids 147, 104238 (2021).

74

L. Frérot, M. Bonnet, J.-F. Molinari, and G. Anciaux, “A fourier-accelerated volume integral method for elastoplastic contact”, Computer Methods in Applied Mechanics and Engineering 351, 951–976 (2019).

75

L. Frérot, G. Anciaux, V. Rey, S. Pham-Ba, and J.-F. Molinari, “Tamaas: a library for elastic-plastic contact of periodic rough surfaces”, Journal of Open Source Software 5, 2121 (2020).

76

V. L. Popov and R. Pohrt, “Adhesive wear and particle emission: numerical approach based on asperity-free formulation of rabinowicz criterion”, Friction 6, 260–273 (2018).

77

T. Sato, Z. Milne, M. Nomura, N. Sasaki, R. W. Carpick, and H. Fujita, “Ultrahigh strength and shear-assisted separation of sliding nanocontacts studied in situ”, Nature Communication 13 (2022).

78

T. Sato, T. Ishida, L. Jalabert, and H. Fujita, “Real-time transmission electron microscope observation of nanofriction at a single Ag asperity”, Nanotechnology 23, 505701 (2012).

79

V. Vahdat, D. S. Grierson, K. T. Turner, and R. W. Carpick, “Mechanics of Interaction and Atomic-Scale Wear of Amplitude Modulation Atomic Force Microscopy Probes”, ACS Nano 7, 3221–3235 (2013).

80

B. Gotsmann and M. A. Lantz, Atomistic Wear in a Single Asperity Sliding Contact, https://doi.org/10.1103/physrevlett.101.125501, Sept. 2008.

81

T. D. B. Jacobs and R. W. Carpick, Nanoscale wear as a stress-assisted chemical reaction, https://doi.org/10.1038/nnano.2012.255, Jan. 2013.

82

C. Brockley and G. Fleming, “A model junction study of severe metallic wear”, Wear 8, 374–380 (1965).

83

J. Liu, J. K. Notbohm, R. W. Carpick, and K. T. Turner, “Method for Characterizing Nanoscale Wear of Atomic Force Microscope Tips”, ACS Nano 4, 3763–3772 (2010).

84

C. Leriche, S. Franklin, and B. Weber, “Measuring multi-asperity wear with nanoscale precision”, Wear 498-499, 204284 (2022).

85

C. Leriche, C. Xiao, S. Franklin, and B. Weber, “From atomic attrition to mild wear at multi-asperity interfaces: the wear of hard si3n4 repeatedly contacted against soft si”, Wear 528-529, 204975 (2023).

86

F. Zhang, J. Liu, X. Ding, and Z. Yang, “A discussion on the capability of x-ray computed tomography for contact mechanics investigations”, Tribology International 145, 106167 (2020).