A story of two transitions: From adhesive to abrasive wear and from ductile to brittle regime

Wear is the damaging and gradual removal of material on surfaces. The study of wear is of utmost economical and environmental importance as the replacement of worn-out parts is estimated to represent an annual expenditure of 680 × 10⁹ euros and to require 3% of the global energy expenditure to manufacture.¹ Two types of mechanical wear—wear originating of sliding contact between two surfaces—represent more than 80% of total wear across the world: (a) adhesive wear (when significant adhesive forces exist between surfaces of similar properties) and (b) abrasive wear (when there is a significant contrast of hardness between the surfaces).

Tribological research has been dominated by phenomenological studies looking for general trends or specific empirical laws, which can be useful for engineering design but lacks a foundation in physics. This culminated in the famed review publication of Meng and Ludema⁴ compiling around 182 equations of wear, and more than a 100 parameters, of which none have better generalization capabilities than Archard’s law.

A new generation of tribological research has recently emerged, thanks to the increasing power of computational mechanics. Indeed, during an experiment, observing “live” what happens between two surfaces as they slide against each other can be difficult. Numerical simulations offer a glimpse of the micromechanical world we cannot yet observe.

The family of interatomic pair potentials used to uncover d* had a tunable hardness, which allowed, for the first time, to capture both ductile and brittle adhesive wear behaviors in a single model. By using potentials of the same family, work has been done to explore how applicable are Archard’s assumptions at the asperity level,⁶ the importance of frictional work to predict wear,⁷ and how the critical length scale relates to Reye’s theory of wear rate,⁸ to the adhesive wear behavior of heterogeneous materials,⁹ or to the ductile to brittle transition in cutting.¹⁰ In this work, we propose to study the transition from adhesive to abrasive wear by introducing a contrast of hardness between the two contacting surfaces.

All simulations are run using the MD software Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS)¹¹ and are constrained to be two-dimensional in space. The parameters of the potentials used are detailed in Table I. Unless otherwise mentioned, all units are rendered dimensionless (reduced units) by taking a reference energy ϵ₀, length r₀ (respectively the depth and distance of the potential well), and atomic mass m₀ as well as a unity Boltzmann constant k_B = 1. The resulting reductions are summarized in Table II.

The atomistic simulations use interatomic pair potentials for which short-range and long-range interactions are decoupled and can be tuned individually.¹² This is done by modifying the tail part of the potential without changing the part close to the energy well. Mechanically, this results in the possibility of changing the plastic properties (strength, hardness, etc.) without affecting the elastic ones (stiffness, equilibrium lattice spacing, surface energy, etc.). In doing so, brittle, fracture-dominated behavior can be favored at computationally-affordable scales.^5,6,12

These combination potentials are used as they have an almost linear variation of shear strength, and thus hardness, with the combination parameter.⁹ The characteristics of the three base pair potentials, plotted in Fig. 1, are

• a short-tailed brittle one, P_b (r_cut = 1.42r₀),

• a long-tailed ductile one, P_d (r_cut = 1.71r₀),

• and an intermediate one, P_m (r_cut = 1.56r₀).

The boundaries of r_cut were chosen so that the transition from ductile to brittle behavior happens at a computationally affordable scale: the potential P_b starts to exhibit brittle behaviors at the scale of a few atoms, while P_d remains ductile until the scale of a 100 atoms.

Hardness is often defined as the indentation resistance of a material. However, atomistic simulations of indentation test at the scale of a few tens of atoms is unreliable as many parameters, such as the precise shape of the indenter or its movement relative to the orientation lattice, can result in a factor 10 difference in the measurement. Thus, the hardness is estimated from the shear strength τ with the following relation:¹³  $H=33τ$⁠. While the measurement of shear strength is also dependent on the exact method, we believe that it is more reproducible. The details and justification for the method of shear strength measurement can be found in Wattel et al.,⁹ and relevant results for the current study are reproduced in Table I.

The simulation domain is initially composed of two surfaces and a circular third body. The diameter of the third body is varied between d = 10r₀ and d = 100r₀, and the domain is scaled accordingly as depicted in Fig. 1. The potential used for the top surface and the initial third body is the hardest one, P₀, whereas the potential for the bottom surface is varied between P₁ and P₉, all softer than P₀. Interactions between the soft surface and the hard component are ruled by the softer potential. A normal pressure P is applied on the top and bottom boundary. Four values of the pressure P are used: 0.01, 0.04, 0.07, and 0.1 (rendered dimensionless by the reductions detailed in Table II). The highest pressure is close to the one necessary to crush the third body when the softest potential P₉ is used. Initially, the third body is not in contact with the two surfaces, but the pressure puts them in contact. After a time t = 2000, the bottom boundary is fixed horizontally and the top boundary is submitted to a horizontal velocity v = 0.15. The simulations were run until a sliding distance s = 500d was reached or for 12 h of computation time, whichever came sooner. All simulations reached at least a total sliding distance of s = 250d. The left and right boundary conditions are periodic. Initial velocities are distributed at random and scaled so that the initial temperature is 0.1, which equilibrates quickly to T = 0.05. The atoms are arranged in a hexagonal lattice, with an initial spacing corresponding to the equilibrium one at a temperature of T = 0.05. A six-atom-thick bulk-velocity-corrected Langevin thermostat¹⁴ is present on the top and bottom, set at T = 0.05. The time-integration is done through a Verlet algorithm with a timestep of 0.005.

Depending on the initial size of the wear particle and the hardness of the surfaces, two main behaviors could be identified. For larger size and harder materials, the wear particle rolls smoothly between the two surfaces and gradually detaches chips from both the soft and hard surfaces. The chips tend to be larger for the softer material. Even though material is also transferred back from the wear particle to the surface, the wear particle size still increases with time. The gradual removal of chips from both surfaces results in an onion-like structure of the wear particle, such as is shown in the middle snapshot of Fig. 2 and further discussed later.

The second behavior is observed for the wear particle of smaller initial size and for softer surface. The wear particle tends to get stuck in the softer surface, removes large pieces of the softer material that may be larger than the particle, and/or breaks in several pieces and re-agglomerates. The snapshot (b) of Fig. 2 illustrates this behavior. The wear rate appears to be greater than in the first mode. However, as the deformations involved are large, they tend to reach the boundary conditions early in the simulation, after which the results are meaningless. This renders a quantitative analysis of the wear rate difficult. A surprising observation is that a softer surface actually leads to greater deformation in the hard initial third body, which goes against the initial intuition. One possible explanation is that the hard particle gets embedded more in the soft material, which prevents it from rolling and thus leads to plastic shearing-based behavior.

In Fig. 2, a map of the two behaviors is plotted. The axes are the initial particle diameter and the hardness of the soft surface. Each point represents four simulations at varying pressure. The gradient from yellow to black indicate whether most of the simulations showcased the first, gradual wear behavior or the second one, respectively. The results were discriminated by judging simulations snapshots by eye. Also plotted is the critical length scale d* of the soft surface, which appears to be a good delimiter of both domains.

Thereafter, unless otherwise mentioned, the results’ analysis will be restrained to simulations showcasing the gradual wear behavior (more specifically, the 48 simulations with the softer surface potential P1 to P7, initial particle size 80–100 atoms). In this wear regime, the wear particle detaches chips from both surfaces and the resulting structure of the wear particle reflects this process. In Fig. 3, snapshots of the wear particle for surfaces with low contrast of hardness (a) and greater contrast of hardness (b) are shown. As chips are detached, they agglomerate concentrically around the particle. In the case of the higher contrast of hardness, as chips from the hard surface are significantly smaller than the one from the soft surface, they are easy to distinguish individually. For low contrast of hardness, the structure is more muddled. From these snapshots, it is possible to extract only the atoms that originally belonged to the hard surface [top right of Figs. 3(a) and 3(b)]. The chips of hard matter can be isolated by the following algorithm: first, a graph is created in which two atoms are connected if the distance (accounting for periodic boundary condition) separating them is less than a pre-defined cut-off, for us, 1.5r₀. Then, atoms are considered to be part of the same cluster if there is a connected path between them. The resulting clustering is highlighted by the colors. The chips from the soft surface only form one or a few big clusters as they are mostly all connected; thus, no clustering analysis could be applied to them. These clusters are related to the original chips of matter removed from the surfaces by the accretion process but do not correspond to them one to one: a cluster might be composed of the amalgamation of several chips or an original chip could be sheared apart in several pieces. Indeed, the initial circular core of the wear particle loses its shape, and this is evidence that the particle undergoes significant deformation as it rolls.

Also present in Fig. 3 is a histogram of cluster sizes and a cumulative curve of cluster weight, both plotted on a double logarithmic scale. The second graph represents how much of the atoms belong to a cluster below a certain size. For simulation with a lower contrast (P₁ to P₅), the cumulative distribution appears to follow a power law. For the one with the highest contrast, the power law has two branches with different exponents: the large clusters represent a higher proportion of the total volume. One explanation could be that the higher quantity of soft matter aggregated quickly covers and protects the hard cluster. They get embedded in a soft matrix that prevents them being broken by or redeposited by the rolling motion. Thus, the higher proportion of large clusters might not be due to larger initial chips being removed but by preventing them being broken up.

To estimate the wear volume at the end of the simulations, the wear particle was manually outlined. Then, atoms from the initial wear particle were removed, and the remaining atoms were counted. This count is considered to be the wear volume. To compare to Archard’s wear law, the average wear rate measured at the end of the simulation is plotted against the constant normal force applied on the top and bottom boundaries. For all but the simulation with the softest potential P₉, the normalized worn volume is proportional to the pressure, in accordance with Archard’s law, and the slopes of the linear regression give the wear coefficients. Different slopes are obtained for simulations with different soft surfaces. Even though the soft materials are very similar (same lattice, same elastic stiffness, for example), the only difference being the tail of the potential, Archard’s law does not catch all the dependence of hardness on wear rate. In Fig. 4(b), the measured wear coefficients are plotted against the hardness, and a power law fit of exponent −1.9 is found. This is close to experimental results of wear rate with two surfaces of contrasting but relatively close hardness^3,15 where such power law scaling was also found with exponents between 2 and 6.

Archard’s law breaks down for the simulations with the softest material. These simulations are at the edge of the ductile domain and starts displaying significant plastic shearing. As shearing is regulated by the tangential force, the dependence on the normal pressure reduces. This suggests that a law based on the tangential work instead of the normal pressure may be more relevant in this case or in general, as have been proposed before.⁶ 

Next, the wear of the hard and soft surface is studied independently. Even though one surface is harder, both get worn away during the sliding simulations. Knowing from which surface the wear is originating from can help predict the expected useful service life of an abrading tool for example. Experimentally,³ the ratio between the worn volume from the soft surface and from the hard surface appears to follow a power law of the ratio of the hardness of both surfaces with an exponent between 2 and 3. The same ratios are plotted in logarithmic scales in plot (c) of Fig. 4, and a scaling of exponent of about 2.5 is also apparent. However, we do not wish to draw significant conclusion on the existence of a power law because of the large scatter in the data and the small range of hardness contrast, less than one order of magnitude, in the simulations. As of now, to the best of our knowledge, no fist-principle explanation has been provided for this scaling, and data, both experimental and numerical, are scarce. Finally, an interesting result is that the wear of the hard surface is not only weakly dependent on the hardness of the soft surface but is also counterintuitively slightly higher against softer surfaces. This is evidenced by plot (d) of Fig. 4, which shows the pressure-corrected volume worn from the hard surface for the different hardness contrasts. These results suggest that an abrading tool would be worn away at a similar rate irrespective of the tooled material hardness but would cut softer material faster, meaning that wear of the cutting tool would be better predicted by operating time rather than the amount of material cut away.

The tunable model potentials used in this work are not meant to represent any specific material but rather to allow the exploration of different mechanisms by varying a single parameter regulating ductility properties. However, they have been useful in further understanding the underlying mechanisms of wear and related process.^5,8,9 The model used in this work has been able to reproduce power scaling laws observed in experiments about wear rates and ratio of wear volume between hard and soft surfaces. This gives confidence that the model may be able to capture the mechanisms at the origin of these scaling laws.

For the wear of surfaces by gradual removal of matter by the third body, the surface is worn away chip by chip. Thus, the total worn volume is equal to the product of the average chip volume $V̄$⁠, the frequency of chip removal f, and total sliding distance s. It could be qualitatively observed in the simulations that the chips tend to be thicker for softer surfaces. While the initial fracture angle of the chip depends on the size of the third body,¹⁹ if we postulate, following these observations, that its thickness is proportional to d*, then $V̄∝d*∝1/H2$⁠, with H being the hardness. Assuming two surfaces of different hardness, the ratio of worn volume is $Vsoft/Vhard∝fsoft/fhardHhard/Hsoft2$⁠. This provides a partial explanation of the power scaling between worn surface ratio and hard ratio. This hypothesis could be tested with further experiments and numerical simulations.

Finally, Archard’s wear law, while formulated initially for adhesive wear, is often used for other types of wear, including abrasive wear. In recent years, a few studies have explored the application of Archard’s wear law at the asperity-level. It was found that the frictional work was a better predictor of wear volume⁶ than Archard’s wear law. However, Archard’s law could be recovered when the wear is plasticity-driven.⁷ For fracture-driven wear as well as when the asperity size is much larger than the critical length scale d*, the scaling of frictional work with wear was found to be sublinear.^7,8 As a direction of further study, it would be interesting to see how the results of these studies are applicable to abrasive wear at the asperity-level.

Simulations of three-body wear were run with surfaces of contrasting hardness. Two main behaviors could be isolated: one of gradual accretion of the wear particle by removal of chips from both surfaces and one displaying a more erratic behavior with a significant amount of plastic deformation and removal of pieces larger than the third body size. It was found that the critical length scale d* was a good predictor of this ductile to brittle transition. For simulations displaying the behavior of gradual accretion, the wear coefficients as well as the ratio of matter worn from each surface follow power scaling laws found in experiments. Finally, these simulations allow us to look into the structure of the wear particle: the accumulation of chips create a concentric, onion-like structure. The chips originating from the hard surface could be isolated and their size is analyzed: the cumulative distribution of chip sizes also follows a power law, whose exponent depends on the hardness of the soft surface. However, the study was limited to a narrow range of hardness contrast due to the computational restriction, and we hope that this paper will spark interest in experimentalists to further investigate this relatively unexplored area of tribology.

This work was supported by the Swiss National Science Foundation under the grant “Wear across scales” (Grant No. 197152).

The authors have no conflicts to disclose.

S. Z. Wattel: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). J.-F. Molinari: Conceptualization (equal); Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal).