A Contemporary Review and Data-Driven Evaluation of Archard-Type Wear Laws Open Access

Perhaps the most intuitive phenomenon within the field of tribology of machine components is sliding contact where at least two surfaces are engaged in contact and relative motion. However, sliding is far from a “simple” concept. Sliding contacts may dissipate energy by frictional interactions and generate wear damage on one or both of the contacting surfaces. Severe sliding wear is therefore a long-recognized damage mode of tribological machine components. To understand, model, and develop material and lubrication solutions for engineering sliding wear, sliding contact experimentation is essential. The controlled environments in these tests allow for isolation of individual tribological parameters to understand their effects on wear of materials.

This work explores 75 different published sliding experimental data sets, acknowledging the variety and commonalities in experimental materials, setups, and operating conditions utilized by the diverse set of researchers. In these reports, 15% of experiments were motivated by the automotive industry, either for applications in internal-combustion engines or electric vehicles; 11% mentioned applications to aerospace components; 27% targeted wear reduction in gears; 55% anticipated applications in sliding bearings; and 9% of experiments were conducted to validate numerical wear models. Other observed applications were for developing wear-resistant coatings, drilling operations, tooling, theoretical work, and artificial human joint replacements.

The three most prominent reported experimental setups observed in the reviewed literature were commercial tribometers in pin-on-disk (36%), ball-on-disk (20%), and reciprocating ball-on-flat (19%) configurations, although some of the research-developed, noncommercial pin-on-disk tribometers would likely fall under the ball-on-disk category as the “pins” involve a semispherical tip. These apparatuses are generally simple, yet versatile, compact, and have certain controllable environments. The surfaces tested with these apparatuses were also readily observable with white light interferometry, laser confocal microscopy, atomic force microscopy, scanning electron microscopy (SEM), and/or energy-dispersive X-ray (EDX) spectroscopy, etc., to characterize worn surfaces and study wear mechanisms. Most commonly, adhesive and abrasive wear were observed, but many experiments also found oxidative wear and some fatigue wear. The vast majority experienced a combination of wear modes as confirmed by the aforementioned techniques. These types of tribometers can easily implement simple-sliding contacts, but some are fully capable of producing rolling and sliding contacts. However, when studying wear resistance, only 9% of experiments reviewed incorporated a slide-to-roll ratio. The other 91% of experiments were performed in simple sliding with one surface stationary and the other moving.

Lubrication science is intimately tied to the studies of wear; however, only 27% of experimental work reviewed was performed in a lubricated setting while the other 73% was conducted under dry sliding—ambient air without added lubrication. Subsequently, 35% of lubricated sliding experiments were conducted specifically for lubrication studies where lubricants were evaluated based on their abilities to reduce wear [1–8]. The majority of the lubricants studied were mineral or synthetic oil based (70%), but water, biodiesels, and even gaseous lubricants were also investigated in some experiments.

Though many metals (e.g., steel, titanium, magnesium, and zirconium alloys) are considered in this review, the focus is on bearing steels (e.g., AISI 52100, 100Cr6, SUJ2, GCR15, or EN31), which are characterized by higher carbon content and martensitic microstructure, and known for their strength, hardness, and fatigue resistance. According to Erişir et al. [9], these qualities make them particularly suitable for applications in aerospace, automotive, railway, energy industries, and more. Although the components of rolling bearings are mainly subjected to rolling contacts between rolling elements and raceways, there may still be sliding interfaces exposed to sliding wear damage, such as the contact between rolling elements and retainers or ring flange faces.

Wear modeling research dates back over 70 years in literature. In 1995, Meng and Ludema [10] performed a comprehensive review of over 5,000 authors from Wear and Wear of Materials conferences from 1957 to 1992. They found 182 unique equations for wear modeling, which contained over 100 different variables and constants. Their findings revealed no common structure in wear-prediction formulae despite many models being derived from similar manipulations of fundamental approaches (empirical, contact mechanics, material failure mechanics). Despite this, one of the oldest yield-pressure-based models of wear, commonly known as Archard's Wear Law, has remained the frontrunner for wear prediction. As a result of Archard's formative paper [11], which followed the work by Holm [12], his wear rate model and many subsequent models take the form of a proportionality between applied mechanical load L, sliding distance s, and material hardness H, as seen in Eq. (1), where V is the expected cumulative wear volume resulting from rubbing. This formulation was original derived for adhesive wear, but it is shown by Halling [13] and Timsit et al. [14] in the Encyclopedia of Tribology [15] that Eq. (1) is also the result of abrasive wear derivation. Furthermore, Liu et al. [16] demonstrated a high degree of accuracy in predicting fretting and oxidative wear by using a slightly modified Archard model, where the hardness value used was for the tribologically transformed material formed on the surface during the fretting process. Given its vast history, the Archard wear equation deserves a revisit with the assistance of modern analytical methods.

Since the Meng–Ludema review [10], wear experimentation has continued with new emerging trends in wear modeling. Goryacheva [18] introduced the idea that the relationship among these variables should not be confined to the linear-like Eqs. (1) or (1a) but have variable exponents that are dependent on material properties, the mechanical interface, and operating conditions. Similarly, Kragelsky et al. [19] present a wear equation, in which the linear wear rate with respect to time is a function of contact pressure and sliding velocity, summarized in their entry in the Encyclopedia of Tribology [15]. In the study of glass polishing, Preston [20] was amongst the earliest researchers to theorize that wear depth was linearly proportional to sliding distance and frictional force at constant velocity. Their works indicated that this relationship could take many forms depending on applications; however, many cases fit the variable exponent relationship had been described by Goryacheva. Furthering Goryacheva's summary, Zhu et al. [21] included a variable exponent on the material-hardness term as well to allow for more flexibility in material representation for their mixed lubrication simulation work.

Zhang et al. [22] performed a comprehensive review exploring classical models and modern wear simulations. Their view of classical models highlighted the Archard model, and they criticized that this famous model simply could not capture the true nature of wear data as other empirical models could. They took issue with the need to experimentally determine the constant of proportionality, k. Additionally, the simple linear relationship in Eq. (1) between wear volume and applied load is insufficient to model experimental data, yet the model remains at the forefront of wear prediction—agreeing with the organizational review work of Goryacheva [18]. Zhang et al. [22] also explored modern simulation techniques, such as the finite element method (FEM) and molecular dynamics simulations. Molecular simulations should be used with caution as Aghababaei et al. [23] found that adhesive wear only presents itself above the tens of microns length scale. The study of wear below this scale should also pay attention to the influences of other forces than the applied load, not apparent in macrowear phenomena. Despite this, wear simulation software shows potential for viable wear predictors; Zhang et al. [22] suggest that improvements in modeling asperity geometry, elastic-plastic behavior, and surface roughness could advance the wear prediction science, which mirrors some factors in wear coefficient k mentioned above.

The work reported in this paper evaluates the continued use of the Archard model within the context of sliding wear experiments and focuses on addressing the following questions: is the Archard model still practically useful over 70 years after its ideation and does the Archard theory have inherent value in how it relates wear volume to applied load (or pressure), sliding distance (or sliding velocity), and material hardness. The Archard model is analyzed from four angles: (1) its original ideation by Holm in 1946 [12] and Archard in 1953 [11] and its mechanics nature; (2) extension of its capability by introducing exponents on variables; (3) under what conditions does the Archard model prove insufficient; and (4) a possibility of its nondimensionalization. A data-driven evaluation of the content of an Archard-type model on sets of reviewed experiments is conducted to support this discussion along with reasoning for the model selection, as well as where improvements are necessary.

To extend the generalizability of the Archard model, exponents α, β, and γ were introduced to accommodate the nonlinear impacts of material properties and testing conditions. Recognizing this, the wear coefficient, k, should no longer be dimensionless. Table 1 takes data from the work of Zhu et al. [21] with reference to Goryacheva [18], where presented wear models have been organized according to Eq. (2). A more detailed explanation can be found in the book by Wang and Zhu [24]. These models either directly applied the Archard model for their primary applications or were adjusted by varying the exponents or adding additional terms.

As this is a modification to a wear law that has existed for over 70 years, justification should be provided. One theoretical consideration for the use of $α$ comes from the Hertzian contact analysis, and in point contact, the relationship between applied load and peak hertzian pressure is cubic [25], implying that the pressure range is no longer confined to material plastic flow as stated by the original Archard equation. Kragelsky [26] also discussed a formulation for wear depth per unit sliding distance, proportional to nonlinear contact pressure: $r/s=pα$⁠, where $α=1+bt$⁠, b is a parameter that reflects the microgeometry of the wearing surface, and t is a friction fatigue parameter and a material property. He explained the formulation is experimentally valid for polymers (e.g., plastics, rubbers, plastic-based composites); in dry sliding, b should be small enough so that $α$ approaches unity, but lubricated sliding could provide a setting in which $α$ should be much greater than unity. Alternatively, several of the authors listed in Tables 1 and 2 give justifications for inclusion of $α$ and $β$ from empirical observations. Rhee [27] found that the wear coefficient of Teflon-matrix bearing materials varied by a factor of two if only load and sliding velocity were changed; therefore, Rhee included exponents on the variables as well as a temporal variable to produce better-fit wear models. Similarly, Xue et al. [28] found that in the presence of a fabric liner, the wear coefficient of self-lubricated spherical plain bearings varied with contact pressure and sliding velocity. Moreover, Cayer-Barrioz et al. [29] found the wear coefficient varied with the square of applied load on polymeric fibers. Most instances of the exponential modification to Eqs. (1) or (1a) come from experimentation but lack theoretical backing as Moore et al. [30] explained for their augmentation of diamond tool-bit velocity with exponent $β$⁠.

As for $γ$⁠, it is important to consider that hardness is meant to represent the resistance of a material to plastic deformation (reflected, i.e., by yield strength, $Sy$⁠). Elasto-perfect-plasticity calculations of typical steels in circular contacts suggested $H/Sy$ to be about 2.31 when considering hardness as the surface full-plastic flow pressure, although in many cases, $H/Sy=3.0$ was used [24,31–35]. The relationship among elasticity, plasticity, and work hardening is not simple because hardness-strength correlations often ignore factors like grain size, strain rate, deformation mechanisms, and geometry of indenters [35]. Cheng and Cheng [36] evaluated the relationship between hardness normalized to yield strength and yield strength normalized to Young's modulus for several work hardening exponent values. They found a nonlinear relationship that only showed approximate convergence to $H/Sy$ of 1.7 as $Sy/E$ exceeds 0.1. Their findings explain that hardness depends on $Sy$ and E, and especially the work hardening exponent when $Sy/E$ is low (e.g., for a bearing steel with a yield strength of 2 GPa and Young's modulus of 210 GPa, $Sy/E≈0.01$⁠). It is reasonable then to suggest that exponent $γ$ may be reserved to reflect a nonlinear relationship between hardness and material yield strength responsible for wear.

In an effort to expand upon the collection and organizational work of Goryacheva (1998) where several studies featuring wear models were represented in a general form, 75 more recent wear studies on metals were reviewed beyond those listed in Table 1. Of the reviewed publications, 48 (65%) included a model to predict wear, and 39 (81%) of the wear models reported can be expressed by Eq. (2) for a rate-based change in the depth of the wear scar, as summarized in Table 2. In some cases, as seen in Table 1, the simplicity of the Archard model encourages extension with additional terms listed in both Table 1 and Table 2. For example, Tabrizi et al. [44] extended the proportionality of wear rate to include surface parameters so that the model obtained a better characterization of the worn-surface topography.

Common applications of a wear model include wear prediction or wear characterization. A frequent application of the Archard model has been wear simulation. In many FEM simulations that attempt to model experimental data, the Archard model is used to determine how the wear scar depth evolves over time. The simple linear model is easy to discretize, which can be readily implemented into existing FEM software. Bildik and Yaşar [66] implemented Eq. (2) into ansys to model wear of nonstandardized steels. Their FEM-based model found 85% agreement between experimental wear data and simulation of sliding wear utilizing the Archard model. Xue et al. [28] similarly constructed a FEM wear simulation by making use of the exponents in Eq. (2) to establish a wear depth routine that transitions from an initial wear stage with one set of exponents to a stable wear stage with a different set of exponents. Their experimental data found that $α=1.54$ and $β=−0.5$ best reflect the initial stage of wear; while for the stable stage of wear, they used $α=0.94$ and $β=0.98$⁠. The flexibility of Eq. (2) allows for initial wear depth progression to vary with increased sliding velocity until stability is met and wear depth progression continued more closely in line with the classic Archard model. Their FEM model achieved a relative error of 4.5% in the maximum wear depth, in comparison with experimental data. The difficulty with this approach, however, is to determine theoretically when this switch between exponent sets occurs—once again illustrating a dependence on empirical calibration. In another study, Furustig et al. [58] used the Archard model as a basis to extend a contact-mechanics solver to include an evolution of wear depth. Their implementation of Eq. (2) did not include material hardness explicitly, but hardness (i.e., yield strength) was used to determine the plastic deformation limit before wear initiation. Equation (2) was a good choice since the contact-mechanics solver they used already computed nodal pressures. Their results found good agreement with experimental data for wear depths beyond 100 nm. They suggested that, at smaller scales, wear was dominated by third-body abrasive particles that were unaccounted for by the model. However, their method still used the Archard model to conveniently evaluate wear morphology over the entire contact domain.

Alternatively, many authors rearrange Eq. (1) and calculate k for their wear results from a tribological test, where material hardness was absorbed into k in this instance, and k was referred to as the “Specific Wear Rate” (SWR) rather than a wear coefficient [5,51–53,57,64,65,68,70,73,74,77,78]. This is reflected in many rows with $α=β=1$ and $γ=0$ in Table 2. One clear observation from Table 2, whether for prediction or characterization, is that there is a consensus that contact load (pressure) and sliding distance (velocity) are critically linked to wear volume (depth). The hardness of materials is only explicitly considered by 13 (33%) author groups. This is a particularly interesting result because hardness testing of specimen materials is a common prerequisite to sliding wear experiments. As mentioned before, hardness measures the material resistance to plastic deformation and is controllable via material selection, surface modification, or heat treatment design. When choosing to lump material hardness into the wear coefficient, researchers were likely to explore differences in material wear performances as independent factors associated with hardness (i.e., SWR in units of mm³/N-m provides a consistent basis for surfaces with various hardness to be compared against). Though most papers did not provide an explicit definition of “wear resistance,” usage across many works suggests that it is a relative term linked with SWR. Materials that produce a low SWR relative to other materials were characterized as having a higher wear resistance. Using SWR to compare wear resistances seems to be more preferred than wear coefficient since quantities used to calculate SWR are known to a high degree of precision.

Another observation of keen interest is that only one of the collected papers, by the previously discussed Xue et al. [28], reported a wear model that made use of noninteger exponents. No other authors reported any models making use of an exponent other than 0 or 1. Compared to the previous work (Table 1), more recent publications do not alter exponents maintained by Archard's law. This is unsurprising for wear characterization, but these exponents are of greater interest for wear theorization. There is a strong justification for maintaining $α=β=γ=1$ as these were derived from simple contact plasticity. However, some researchers questioned the validity of the Archard model. Reichelt and Cappella [69] pointed out that the wear coefficient actually varies with the product of sliding distance and normal load in their dry sliding experiments, making Archard's Law invalid in certain operating regimes. Similarly, results from Xi et al. [59] show a clear reduction in wear rate (mm³/min) from initiation to middle and end stages of the wear tests. The linear relationship between wear volume and load cannot account for an evolution of wear modes as load increased. Taltavull et al. [56] observed the wear behavior of magnesium alloys under varying loads and sliding velocities. They reported that at low loads, primarily oxidative, abrasive, and adhesive wear modes were present. However, increasing the load invited delamination wear as the primary mode. With their pin-on-disk apparatus, cyclic and heavy loads caused subsurface cracks which resulted in layers of material being removed and some became wear debris. This research group also found that, as loading was increased further, there was another transition to plastic-deformation wear. The permanent deformation resulted in further surface damage, leading to hardened and embrittled areas that eventually broke off and joined wear debris. They also observed, from their experiments, that the wear rate did not grow linearly with load as suggested by Archard's Law.

The Archard model, Eq. (1), was also criticized for its simplicity. There are many factors that contribute to wear, which the model could only capture by means of the wear coefficient. A highly prominent factor that is left out of the equation is the sliding velocity of the contact interface. This factor should not be neglected from wear consideration because it plays a critical role in oxidative wear, as well as other tribochemically related surface variations. As sliding velocities increase, flash temperature between sliding surfaces increases, which promotes material oxidation and interfacial tribochemistry. Pasha et al. [79] performed a full factorial set of experiments, which included sliding velocity as a factor for wear of magnesium composites. They observed that the magnesium matrix began to oxidize beyond a certain sliding speed. A combination of SEM and EDX results confirmed the formation of a magnesium oxide layer; a distinct region could be seen clearly with SEM imaging, and the presence of oxide was chemically confirmed by EDX. Wear rates at these higher velocities drop off because the oxide layer provides protection between wear specimens and the metallic counterface. To account for this reduction in wear rate at higher velocities, the Archard model would necessarily require a different wear coefficient. This in turn reduces the generalizability of the model across sliding wear operating conditions unless the wear coefficient is extended to be a function of sliding velocity.

Sliding velocity, however, is not the only consideration for a nonconstant wear coefficient. Choudhry et al. [76] introduced a novel adaptation of the Archard model, in which wear coefficient was a function of deformation energy, which, being a function of normal pressure and displacement, introduced a spatial element to the Archard model. Thus, their model better accounts for the evolution of adhesive-wear particles, in shape and size, throughout the contact zone. Their results showed that wear coefficient clearly varied along the sliding direction, increasing from the leading edge to the trailing edge. While accounting for this variation with deformation energy, they acknowledged that sliding velocity and frictional energy were not accounted for and were thought to have further effects on wear rate. Fouvry et al. [80] accounted for frictional-energy correlated wear volume with the energy dissipated (Ed) in sliding contact where Ed is the product of sliding distance, applied load, and friction coefficient. The calculation of Ed is similar in nature to Eq. (1), and an approximately linear correlation between wear volume and Ed was obtained. An issue pointed out with this model, however, is the reliance on friction coefficient, a property of the interface of contact and rubbing surfaces, and not itself an operating condition.

A creative solution, similar to the work of Choudhry et al. [76], was put forth by Argatov and Chai [81], where they used an artificial neural network to compute the wear coefficient in the Archard-Kragelsky model (⁠$α$ and $β$ from Eq. (2) assumed to be variable and $γ=0$⁠). Their neural network was used for the prediction of wear rates for aluminum matrix composite materials and was trained to map reinforcement percentage and additive particle sizes to the wear coefficient. The architecture of the network integrated into the Archard-Kragelsky model allows the backpropagation training procedure to also learn the optimal $α$ and $β$ values. For their experiments, they achieved high accuracy with $α=0.257$ and $β=1.503$⁠, and most significantly, their model outperformed a pure neural network uninformed by contact mechanics. Their resulting model is one that has a variable wear coefficient attached to an optimized model rooted in contact mechanics, with which the performance prediction outside of the training dataset is more reliable.

The form of extended Archard model in Eq. (3) is preferred for this task over Eq. (2) for two reasons. First, these variables are the most commonly reported in sliding wear experimentation—especially the applied load rather than contact pressure. If a contact pressure is reported, it is usually from a Hertzian-contact calculation stemming from a known applied load. Second, Eq. (2) implies a constant wear rate so long as load and sliding velocity are held constant during sliding wear experiments. The resulting model, therefore, can only yield linear relationships between wear and time/sliding distance (time and sliding distance are perfectly correlated under a constant velocity).

Of the 75 reviewed wear studies, only the data of 39 (52%) were suitable for model optimization. Note that these 39 papers create a subset of reviewed research that shares some overlap with Table 2, but a paper being featured in Table 2 has no bearing on its selection for this task. The papers selected [1,4,5,8,9,16,44,51–53,57,59,62–66,68,71, 73,75,77,78,82–97] were judged by whether their experimental results can support the generation of a set of three-dimensional data sufficient for the optimization of Eq. (4) and containing applied load, sliding distance, and material hardness. Surprisingly, this criterion eliminated 48% of the reviewed papers for five primary reasons:

1. Wear rates are reported on a length scale with no means to calculate or estimate wear volume.

2. Material hardness not reported.

3. Wear volumes are reported in a format that can hardly be associated with the accompanying load and/or sliding distance.

4. In instances of metals sliding against composites/ceramics, no metal wear volume are reported.

5. Fewer than four datapoints in a dataset, insufficient for four-variable optimization.

For completeness, there were several reviewed papers [98–111] that were eliminated from the calibration study that have not been previously mentioned nor displayed in Table 2.

The selected papers were processed to create individual datasets. Data presented in a tabular format were collected without difficulty. However, of all reviewed papers, 60 (82%) presented wear volume data in a graphic format. The wear volume data presentation, however, has no consistent format; therefore, a tool called WebPlotDigitizer [112] was used to meticulously scale and extract individual datapoints from each selected paper that presented wear volume in a graphic format.

In many cases, wear volumes were not explicitly reported, instead, they were presented in different forms. It was straightforward to convert those in SWR or with respect to sliding distance to volume. However, some reported wear volume rates, which were handled by using the trapezoidal integration to obtain cumulative datapoints to be fit into Eq. (3). Trapezoidal integration was also used in some instances of reported material hardness because many results present hardness as a function of depth from the surface. In these cases, the trapezoidal integration was used to obtain an average value from the surface to a meaningful depth. Additionally, the reviewed papers have reported material hardness in units of Rockwell C Hardness, Vickers Microhardness, gigapascals, Knoop Hardness, Shore Hardness, and Brinell Hardness. As these units are associated with different testing apparatuses, there is no robust method of converting from one to another precisely. For consistency across data sets, hardness measurements were directly converted or approximated in units of gigapascals. Per their definition, Vickers, Knoop, and Binell measurements could be directly converted to units of pressure. Rockwell C and Shore hardness, however, were linearly interpolated from conversion tables found in ASTM International [113].

The Archard model is primarily used for steady-state wear modeling after the running-in period of wear [114]. Other approaches or extensions of the Archard model have been developed to accommodate transient wear at the onset of sliding contact [115–117]. The curation of the 39 datasets does not explicitly exclude wear data from the running-in period; however, 90% of experimental datapoints were recorded after 100 m (or 96% greater than 20 m) of sliding wear. None of the eligible experiments reported signs of or intentionally induced severe wear.

Because the datasets are curated from published papers not created with the intent of being used to optimize Eq. (3); rather, they were from studies on heat treatments, coatings, lubricants, etc., and many do not contain any variation in some variables. Thus, the optimization of an exponent associated with a variable that remains constant across all datapoints can always be converted back to an exponent of 1, and the conversion factor be absorbed by the wear coefficient. Therefore, the optimization routine only minimized Eq. (4) with respect to variables that have variations in the dataset, and exponents attached to constant variables were set to 1.

A graphical summary of optimized exponents from models trained on each of the 39 datasets is presented in Fig. 1. To draw comparisons about the optimized models that more closely reflect the Archard model against those that do not, a Gaussian mixture model (GMM) is used to separate each set of the optimized exponents into two clusters, hereby named the Archard cluster and the Alternate cluster. The Archard cluster is aptly named as the cluster of exponents reside closely to where the classic Archard wear law would at $α=β=γ=1$⁠. The GMM is an ideal unsurprivised machine-learning method for segmenting the exponent data since it not only develops cluster means but also cluster covariance matrices. This allows the GMM to consider the shape of clusters and recognize if there are densely clustered points within a more variable cluster. This ability is most clearly presented in Fig. 1(c), where the Archard cluster is surrounded, on all sides, by Alternate models. It is important to note that while naming the Archard cluster, no prior information about the Archard Model having $α=β=γ=1$ was given to the GMM. This cluster is formed purely from the variation present in the exponent data; no information about wear coefficients was provided to the GMM either.

The GMM classified 31 exponent triples near the Archard model while eight were found to be dissimilar, belonging to the Alternate cluster. The means, $μ$⁠, and univariate standard deviations, $σ$⁠, of each cluster can be found in Table 3. Representing the coordinates in the above system as $(α,β, γ)$⁠, the GMM found that the means of the Archard Cluster are at (0.958, 0.958, 0.879) and those for the Alternate cluster are at (1.50, 1.01, 6.82). This indicates that the 31 optimized models fell within the neighborhood of the classic Archard model. Interestingly, the Alternate cluster has $α$ and $β$ mean values also near the classic Archard model, but its $γ$ mean resides at 6.82, which presents outside the established typical range. A closer look at the standard deviations shows that the Alternate cluster has much larger variations in $α$ and $γ$ but little in $β$⁠, as compared to those for the Archard cluster. This does warrant further exploration, however. The Alternate cluster is smaller with only eight points, and of those points, only three were optimized for $α$ and three for $β$ (i.e., only three datasets contained any variation in load and sliding distance each). This can explain the low variation in $β$ given that most of the points were fixed at 1.0, but it also implies how drastic the variation is in the three points where $α$ is not equal to 1.0 to create such a high standard deviation. This can be most easily seen in Fig. 1(d). Additionally, the small variation in $β$ and its mean proximity to 1.0 indicates that any reported wear data from transient, running-in modes of wear have little effect compared to what would be expected if large amounts of transient wear were present [115–117]. Contrary to the other exponents, seven of the eight points in the Alternate cluster were optimized for $γ$⁠, but it generated the highest standard deviation across both clusters. This indicates that many of the optimizations rely heavily or even solely on variation in material hardness, resulting in models that veered far from the classic Archard model.

While clustering was only performed on optimized exponents (i.e., wear coefficient data was not shown to the GMM), a comparison of the wear coefficients belonging to each cluster reveals again that the Alternate cluster demonstrates a clear deviation from the Archard cluster. Its wear coefficient is a point of contention with the Archard model; its high variability can make it difficult to transfer insight from tribometer testing to real application. Table 4 gives five distribution metrics of wear coefficient for each cluster. The maximum, minimum, and median are standard distribution metrics, and the interquartile range (IQR) and geometric mean are included to account for the wear coefficients spanning many orders of magnitude. The IQR is defined as the difference between values representing the 75th and 25th percentile of the data range and can give similar insight into the spread of the wear coefficients as standard deviation would. The geometric mean is defined as the $nth$ root of the product of $n$ values. With values spanning many orders of magnitude, the geometric mean is less dominated by extremely large values than a typical arithmetic mean.

It becomes immediately clear from Table 4 that the wear coefficients associated with the Archard cluster have a smaller range, spread, and center (median and geometric mean) than the Alternate cluster. The Archard cluster reflects a balanced distribution that demonstrates approximate lognormality. Evidence of this is the agreement between the orders of magnitude of the median and geometric mean and the shape of its distribution in Fig. 2. The center of the wear coefficients from the Archard cluster also reflects similar values to those found in the reviewed literature. Recall the established a typical range of wear coefficient between $10−7$ and $10−2$⁠, which seem to align with the optimized wear coefficients found here for the Archard cluster. Once again, it is worth mentioning that by performing a fractional dimensional analysis, a possible cause of the spread of wear coefficient distribution for the Alternate cluster is the dimensional inconsistency within the cluster. Exponents in the Archard cluster are much closer together, which makes the units of the wear coefficient more consistent and closer to unitless. After performing the review and optimization, another clear reason for lumping material hardness into the wear coefficient becomes apparent. As previously mentioned, the reviewed papers have reported material hardness in a variety of units with some only having approximate conversions. Thus, a wear coefficient could change in the order of magnitude just by the choice of Rockwell C hardness versus Vickers Microhardness, for example. Using SWR as a function of load and sliding distance makes it invariant to the choice of material hardness measurement and useful to compare wear-resistance performance across different testing apparatuses.

The eight sets of optimized exponents grouped into the Alternate cluster use a variety of hardness measurements including Vickers, Rockwell C, Shore, and gigapascals. This, along with the many units represented by the Archard cluster, suggests that a particular choice of material hardness measurement is not the root cause of the high variation in exponents and wear coefficients characteristic of the Alternate cluster. It should be evaluated whether the presence of various wear modes is responsible for deviation from the Archard cluster. A summary of the reported wear modes for each study grouped by cluster is presented in Table 5. In many experiments, multiple wear modes were identified (e.g., abrasive and adhesive wear) through such methods as optical or Sem imaging, as well as XDS analysis. A $χ2$ analysis is conducted to determine if there is any statistical difference between the distribution of identified wear modes between each cluster. For an alpha level (statistical significance and not related to Eq. (3)) of 0.05 and 4 degrees-of-freedom (df), the critical $χ2$ value is 9.488 (i.e., an observed $χ2$ value should exceed this value to be statistically significant). The df depends on the number of rows $nrows$ and columns $ncolumns$ in Table 5 through the relationship $df=(nrows−1)×(ncolumns−1)$⁠. The data presented in Table 5 yields a value of 5.94. This does not surpass the critical value; therefore, this analysis shows no statistical difference between present wear modes in each cluster.

Table 6 may offer an explanation, detailing metrics used to track the optimization performance of Eq. (3) on each dataset. The R² metric, often referred to as the coefficient of determination, measures how much variance in the data is captured by model predictions. The root-mean-squared error (RMSE) is the square root of the squared deviation between model predictions and experimental data (seen as the error bars of Figs. 3–8). The mean absolute error (MAE) is a similar metric but considered only the absolute value of the difference between predictions and data, so that it is less sensitive to outlier predictions than RMSE. R² is a dimensionless quantity residing between 0.0 (capturing no variance) and 1.0 (capturing all variance) that comments on how much the model can explain the variance present in the data; whereas both RMSE and MAE are dimensional quantities that share the prediction units (wear volume). Though these metrics are observed closely for an individual optimization routine, they cannot be compared across datasets since the target wear volume spans orders of magnitude from $10−7$ to $102 mm3$ across all datasets used in the optimization routine. Thus, mean absolute percentage error (MAPE) was also calculated which normalizes MAE to the target wear volume. These numbers are reported in percentages and can be compared across datasets. Therefore, Table 6 includes the statistical results for R² and MAPE in the metrics.

In Table 6, the mean and median of the R² distributions indicate that the optimized models clustered into the Alternate cluster tend to render more variance in the data than do those in the Archard cluster. Additionally, both metrics of spread show that there is little variation among those in the Alternate cluster. For MAPE, the Archard cluster has a higher mean but a similar median, compared to the Alternate cluster; the Archard cluster also demonstrates higher measures of spread for MAPE. This indicates that there are some outlier optimization fits in the Archard cluster that have significantly higher errors. Reasons for this will be explored in the commentary on Fig. 6 in Sec. 4. According to the more outlier robust measurements (median and IQR), most of the errors of the data in the Archard cluster are centered slightly lower than those in the Alternate cluster, but they experience greater variation from that center.

The observation over the behavior of these exponents in each cluster and the difference in metric distributions suggests that the Archard model is consistently average, and models that deviate to the Alternate cluster are highly specialized and overfit to individual experiments. Using the exponents near $α=β=γ=1$ typically results in a model that can capture a slight majority of the variance presented in the data with a wear coefficient that is a part of a seemingly lognormal distribution. This supports the continued sentiment that the Archard model seems to capture some underlying mechanisms responsible for sliding wear, but there are clearly other factors that the wear coefficient cannot make up for. Choosing exponents that stray from the Archard model can generate highly accurate models, but these models may lose generality. From a theoretical standpoint, more can be learned about the mechanisms of wear from a group of similar models with error than from another group of dissimilar models that overfit to specific wear tests.

Figures 3 and 4 give some insight into why highly specialized models can achieve strong regression metrics at the cost of generalization. Liu et al. Conducted a surface modification study to determine how the addition of ammonia during the heat treatment process affects material hardness and wear resistance [84]. They recorded the rate of mass loss at specific intervals over the full sliding distance of 3200 m at the constant load of 200 N for surfaces treated with several rates of ammonia addition. Therefore, variation in the data was present over the material hardness and sliding distance, with which $k$⁠, $β$⁠, and $γ$ were all optimized, and the results are $2.8×109$⁠, 1.32, and 24.63, respectively. These optimized parameters yield 0.97 for R², $4.65×10−7 mm3$ RMSE, and 15.2% MAPE. As seen in Fig. 3, each material hardness series in a progression over time/sliding distance displays a slightly stronger than linear progression in wear volume. Subsequently, a $β$ value slightly higher than 1.0 was found by the optimization routine. To make up for the variation between each material hardness data series, the optimization was forced to rely on a choice of $γ$ and k in an inverse relationship that would result in a sufficient separation between predictions at any sliding distance; this suggests that Eq. (3) may be unable to capture the mechanisms responsible for differentiating thermo-chemically modified surfaces. Comparing the variation present in input hardness data to the expected variation in the output wear volume data, the six different surfaces have a 0.0789 GPa sampling standard deviation in hardness, and wear volumes span an order of magnitude. Equation (3) then requires extreme choices of $γ$ and k to map the small variation in hardness to the comparatively large variation seen in wear volume. In simpler terms, a more realistic choice of $γ$ and k would collapse all prediction lines down into one line that would be incapable of accurately regressing through the data. This is more clearly seen by the Archard-model regression series shown in Fig. 3. These curves are generated by fixing $γ=1$ such that only k is optimized. In this case, it is seen that the Archard model has poor a performance on the data since all predictions have essentially collapsed to the same curve.

This then calls into question whether the variation in wear volume is purely a result of increased hardness. Liu et al. concluded, from SEM imaging, that adhesive and oxidative wear were the primary wear mechanisms for the 52100 steel samples while smoother wear scars with no oxides were present on samples treated with ammonia [84]. The worn surface might have been affected by tribochemistry related to sliding, resulting in less oxidation and adhesion between surfaces, which were not directly related to surface hardness. This suggests Eq. (3) be too simple to model the underlying mechanisms involved in this type of wear of the thermo-chemically modified surfaces. Chemomechanics may be needed to better understand and quantify microstructural and tribochemical differences between the wear behaviors of the surfaces fabricated by Liu et al. [84] to better inform a wear model.

The optimization using Eq. (3) can also produce exponents dissimilar to those in the Archard model when material hardness is the only source of variation in a dataset. Türedi et al. [85] performed a study on the effects of tempering temperatures on 52100 steel hardness via sliding the 52100 steel specimens against alumina and 52100 steel counterfaces. They recorded worn volumes of both specimens and counterfaces at the end of testing under a constant applied load and equal sliding distances. Subsequently, the only source of variation in the data was material hardness, so that $k$ and $γ$ should be the only optimized quantities in Eq. (3). Their values are 0.00787 and 3.45, respectively, and generate an R² value of 0.94, RMSE of $0.143 mm3$⁠, and MAPE of 42.8%. Figure 4 shows the testing data and modeling results. The points plotted at 2.18 GPa correspond to the highest tempering temperature of the materials. There are four additional points corresponding to lower temperatures and six points for the wear of the counterfaces. Other than the two points for high-temperature tempering, there is little variation in the wear volume for the high variation in material hardness. This, combined with the jump in hardness from the points at 2.18 GPa, presents a difficult fit for the inverse relationship between volume and hardness in Eq. (3). This difficulty is confirmed by the triangle series that shows how Eq. (1) would perform on this dataset. It is seen that the Archard model underpredicts the results associated with low hardness and overpredicts those with high hardness since $γ=1$ is not a sufficiently strong exponent. To adjust for the jump followed by the flat region, a high value of $γ$ is necessary to reduce the prediction error. Note that this is now the opposite challenge, a large variation in material hardness, for the model to what previously seen in Fig. 3, but still presents a scenario that the classic Archard model would not be well followed. This dataset also illustrates another difficulty in wear modeling—materials may be worn away from both contacting surfaces. It is common to make a counterface much harder than the specimen so that wear is predominantly found on the specimen. However, the data from Türedi et al. [85] includes wear of the specimen and the counterface. A single optimized Eq. (3) produces an Alternate model that handles this particular dataset with six 52100 steel specimens, three 52100 steel counterfaces, and three alumina counterfaces; however, it can hardly be expected that a general wear model could characterize a complex wear interface between differing materials by accepting a single value for hardness. It remains unclear how or whether hardness data of both contacting surfaces should be incorporated.

The previous two cases have been examples of how overspecializing the Archard model can produce overfit models that drastically differ from one another. Optimized models that clustered near the classic Archard model, however, still presented clear situations where the Archard model was insufficient. One of the clear distinctions between Eqs. (2) and (3) is the presence of the time rate of change term. Equation (2) can account for variable sliding velocity but cannot inherently express any wear-rate transitions (e.g., the only way to model wear rate changes from running-in to mild and to severe wear is through manipulation of $β$⁠). Equation (3) allows for nonlinear wear in time/sliding distance through $β$ (e.g., $β>1$ suggests that wear rate becomes more severe as wearing continues) but cannot distinguish between experiments of varying sliding velocity. This can result in predictions observed in Fig. 5. These predictions were made with $k=0.000147$⁠, $α=0.52$⁠, $β=0.92$⁠, and $γ=2.14,$ resulting in 0.88 for R², $0.000318 mm3$ for RMSE, and 125% for MAPE, and the exponent triples put this optimization fit in the Archard cluster.

A close observation of the data series in Fig. 5 reveals that for the eight independent data series [86] (i.e., constant load, hardness, and sliding velocity with varied sliding distance), there are only four visible prediction series. This is a product of the fact that Eq. (3) cannot distinguish between sliding velocities. Thus, the optimization routine is forced to find regression lines that best split the difference between sliding distance series with the same loading and hardness but differ in sliding velocity. These results support the common criticism that the Archard model is too simplistic and the motivation of authors like the aforementioned Choudhry et al. [76] and Argatov and Chai [82] who developed Archard models with variable wear coefficients. However, any modification of the Archard model has only been done in a local space. Sliding wear is a highly multidimensional problem, and the experiments that Eq. (3) was fit contain issues of lubrication, surface roughness/textures, heat treatment, coating, surface modification, temperature, and others. For instance, operating conditions like load and sliding speed can be held constant, but lubrication, surface roughness, and flash temperature can hardly be. The Archard model has no theoretical support to model these aspects outside of a variable wear coefficient. The principal issue then becomes determining how to develop a wear coefficient that can account for the variations in lubrication, surface roughness, temperature, humidity, deformation, and any other number of the hundred parameters that Meng and Ludema [10] identified.

Furthermore, in surface modification, coating, and heat treatment studies, the Archard model is forced to rely on material hardness to develop a wear relationship. In many instances, the data fails to present an inverse relationship that the Archard model can do. Figure 6 shows four sets of wear data from such experiments, which were for heat treatment studies. Each of the four data sets was clearly outlined with the heat treatment process that the metals had undergone, and hardness measured prior to tribotesting. Surface treatment was variable, while all other operating conditions were held constant. The optimization of Eq. (3) results in a failure to capture variance in the wear volume data. None of these fits on the four data sets shown below achieved R² scores above 0.5, which indicates that most of the variance in the data is unexplained by Eq. (3). The error bars present in the figures represent the RMSE of the predictions, which are significantly larger than the fits previously seen (Figs. 3 and 4), although all four of the fits seen in Fig. 6 were in the Archard cluster. To avoid falling into local minima, multiple initial starting points were attempted for each optimization run, but they were all realistic choices in terms of similarity to the classic Archard model and previously discussed typical ranges of values for wear coefficient and exponents; thus, in several cases as seen below, an optimization run may converge quickly to an optimal point that does not minimize Eq. (4) well because Eq. (3) is incapable of modeling the presented data. This case results in a poor fit but still belongs the Archard Cluster since there is no path away from the initial exponents that reduces Eq. (4). In machine-learning terms, Eq. (3) has high bias, or its optimization underfits the data (i.e., Eq. (3) is too simple to capture the variance present in the data). Moreover, all the microstructure changes and property variations involved in steel heat treatments were distilled down to a single feature: material hardness. It is clear from the optimization performance of cases like these that variation in hardness alone presents difficulty for wear modeling. Recalling that hardness is meant to inform a wear model of material plasticity, it may be necessary to strength Eq. (3) with a better model of plasticity than solely relying on hardness testing. Additionally, in a study of wear-protective tribofilms on steels, Khan et al. [119] found that rather than a correlation between wear and hardness, alloy composition was more closely tied to wear severity. Their results indicated that steels with a higher chromium content were more able to produce tribofilms that act as protection layers against wear. Yet again, this emphasizes a greater reliance on chemo-mechanics than what Eq. (3) can currently support. Thus, the MAPE results for those optimization fits relying purely on hardness are extremely large and contribute to the previously discussed outliers in the distribution of the MAPE metrics of each cluster.

Finally, Figs. 7 and 8 can help demonstrate that despite its significant drawbacks, the extended Archard model featured in Eq. (3) may still serve as a strong theoretical wear predictor in some instances. Tan et al. [87] conducted 18 different sliding wear tests under varying sliding distances, velocities, and loads. The first nine were performed to calibrate a wear coefficient k for a fatigue-based analytical wear model. Then, their calibrated model was used to make predictions on the latter nine experiments. In the model optimization results plotted in Fig. 7, Eq. (3) was optimized for $k$⁠, $α$⁠, and $β$ over all 18 experiments resulting in values of 0.00529, 0.58, and 0.81, respectively. These parameters produced values of 0.99 for R², 0.227 for RMSE, and 10.3% for MAPE. A noticeable difference from Fig. 3 and 5 is the lack of lines connecting data predictions over a range of sliding distances. This is because Tan et al. [87] performed a partial factorial experiment so that no two datapoints share the same load and sliding velocity. Thus, there cannot be any connecting lines showing a progression through sliding distance since each point is generated with a different set of operating conditions. A legend would include 36 labels for data and predictions; therefore, it was not included as to not obstruct the data. As indicated by the high regression accuracy, the Eq. (3) white circle predictions correspond to the nearest black square data points. The variation in sliding distances can be seen in Fig. 7; experimental loads varied from 20 to 150 N and sliding velocities from 0.016 to 0.345 m/s. The optimization metrics and the graphical representation suggest that this model clustered in the Archard cluster performs exceedingly well with R² being 0.99, indicating that nearly all the variance in the data could be accounted for by Eq. (3). It is also interesting to note that this exceptional performance is seen on a dataset that does not contain any variation in material hardness.

Zhang et al. [82] conducted two experiments with 52100 steel: one varying sliding distance with fixed load and the other varying load with fixed sliding distance. A single optimized Eq. (3) was accordingly optimized for $k$⁠, $α$⁠, and $β,$ which resulted in values of $6.92×10−6$⁠, 1.10, and 1.00, respectively. These parameters produced an R² value of 0.98, RMSE of 0.00130, and MAPE of 45.2%. Figure 8 illustrates the regression lines through experimental data. The optimized model exponents are remarkably close to the classic Archard model in Eq. (1) with a wear coefficient within typically observed values. This result and the fit seen in Fig. 7 give clear indication that there is still merit to the conclusions Archard drew about wear in 1953. These two data sets demonstrate that Eq. (3) is fully capable of capturing nearly all variance present in wear data in isolated conditions. Neither Tan et al. [87] nor Zhang et al. [82] used lubricants, varied material construction, ambient conditions, and did the best they could to isolate surface roughness. Consequently, models similar to the Archard model perform very well. However, it is clear from the discussions about Figs. 3–6 and that in Sec. 3 that this model is insufficient for practical application where the effects from variations in surface material (e.g., microstructure, roughness, coatings, textural modifications, etc.), lubrication, sliding velocity, and more are largely unknown.

Likewise, Eq. (2) can be modified as well to develop a rate-based formulation.

For linear work hardening, H can be normalized by plastic tangential modulus $ET$⁠. Moreover, it is debatable, for mild wear, that should $Ep$ be replaced by equivalent modulus E* defined as $1E*=1−v12E1+1−v22E2$⁠, where E_i and v_i are Young's modulus and Poisson's ratio of contact materials 1 and 2. This replacement emphasizes the contact nature of the interfacial pressure. The choice of s_r to be a lifetime wear-length estimate would result in a set of nondimensional inputs that fall into a typical range of values between 0 and 1. This is favorable from an analytical and data-processing standpoint. Rather than dimensional values, these normalization constants support a conceptualization of how each input contributes to wear. For example, if the interfacial pressure approaches one-half of the yield strength, this scenario could be conceptualized as loading at A% to yield initiation, where A depends on contact type and material Poisson's ratio [24]. Furthermore, values normalized to this smaller range are more preferrable for an optimization routine to determine variable exponents α, β, and γ. This scales all inputs to largely similar ranges, reduces the size of gradient and Hessian computations, and allows for more uniform optimization steps—all of which can improve optimization efficiency and convergence.

Model nondimensionalization does not affect the results presented in the previous two sections for α, β, and γ. However, the amplitude of wear coefficient k may be different from those reported before. Nevertheless, the conclusions about these variables are still the same because nondimensionalization only introduces several more constants. Moreover, the augmented wear coefficient remains dimensionless in Eq. (7); however, it is now related to more material and surface parameters. It should be pointed out that while roughness $Rq0$ is included in Eqs. (7) and (7a), the role of surface roughness and its variation in wear are complicated, other parameters like skewness and kurtosis are also varying as wear progresses, as explained in the artificial neural network wear simulation work by Ao et al. [120] and the experimental results by Pickens et al. [121].

Multiple possibilities of nondimensionalization imply that Archard's law is still primitive to unfold the full nature of adhesive wear.

The review and data-driven analyses on the Archard-type models and related issues suggest a strong need for further mechanics-based enhancement should it be enriched into a general wear law. Several difficulties have to be considered, which are that (i) surface evolution during wear is the result of complicated and combined impacts of multiple factors like material-property modification, deformations, surface asperity fatigue accelerated by sliding [122], interfacial condition variations (pressure, flash temperature, contact stresses, etc.), tribochemical activities, and behaviors and influences of wear debris in terms of their generation, removal, fracture, embedment, or agglomeration; (ii) although the Archard wear model may be used to explain adhesive and abrasive mechanisms to some extent, there are often other mechanisms intertwining during wear, which can be more complicated if electromagnetic fields are involved; (iii) wear characterization and measurement usually only reflect the end states of worn surfaces without revealing the wear process itself, and (iv) a small difference in the macroscopic wear-test configuration and conditions may result in a significant difference in wear data, which is one of the major reasons for wear data discrepancy. Moreover, friction is only explicitly considered by Fouvry et al. [80] to correlate wear to energy dissipation; although wear is not directly proportional to friction, the latter does affect flash temperature and surface-subsurface stresses [24]. Clearly, multifield mechanics is needed and a mechanics-supported machine-learning method is anticipated. Ultimately, wear modeling is a highly interdisciplinary problem that should be enforced by including knowledge distilled from multiple mechanics theories.

Sliding wear experiments are effective means to evaluate wear-resistance of materials and develop data for wear modeling. This paper reviewed 75 papers featuring sliding wear experiments, and 39 of which were selected for an optimization routine adapting a general Archard model to current wear data. The following can be concluded:

1. The Archard model is still widely prevalent in recent sliding-wear experimental works. The Archard model was featured in 39 (81%) of the 48 (65%) papers that made mention of a wear model. To simulate wear evolution, the Archard model is still a good choice for its simplicity and inclusion of variables already computed from a contact-mechanics solvers. Care must be taken to clearly define the quantities to best represent the contact interface of interest (i.e., hardness of which surface, rate of cumulative wear volume, and sliding distance, etc.). This model is also widely adapted to quantify materials' wear resistances by calculating the wear coefficient (named specific wear rate in this context) subjected to a given applied load and measured total sliding distance.

2. A calibration study was done to tune the exponents on applied load, sliding distance, and material hardness in the Archard wear model, and a Gaussian mixture model was used to determine which models clustered near the classic Archard model where $α=β=γ=1$ and which did not. There were eight datasets that belonged to the latter, the Alternate cluster, and they were found to have highly variable hardness exponents and wear coefficients that did not fall within typical ranges informed by literature and physical law. Diverging from the classic Archard model allowed these models to overfit to specific experimental data in which the classic Archard model would not be able to explain the imbalance in variation between wear volume and material hardness. Conversely, 31 models clustered near the classic Archard model and obtained wear coefficients with a lognormal distribution centered near typical values found in literature. On average, these models were able to capture a slight majority of variance in the data with a more consistent grouping of exponents. This supports the current understanding that the Archard model captures some underlying relationships involved in material wear but still lacks completeness.

3. The Archard model's scalar wear coefficient and reliance on material hardness make it an incomplete description of the physical process governing sliding wear. In capturing only three variables involved in sliding wear, its simplicity prevents the Archard model from distinguishing data obtained from various parameters beyond material hardness. Experiments observing changes in material wear as a result of varying sliding velocity, surface texture, temperature, etc. cannot be modeled by Archard's law without fitting a new wear coefficient each time one of these factors is altered. Furthermore, data from multiple authors did not support the perfectly inverse relationship between material hardness and wear volume put forward by Eq. (1). Lumping all the material phenomena changes that occur during heat treatment, surface coating, or surface modification studies into a material hardness parameter is inadequate for general wear prediction.

4. Nondimensionalized forms of the Archard model are proposed to preserve the unitless meaning of wear coefficient. For simple wear modeling, Eq. (6) may be adopted, and to consider the effects of work hardening and plasticity, Eq. (7) may be adopted. Variables in both Eqs. (6) and (7) may be exponentiated while retaining the meaning of the wear coefficient. Interfacial pressure may be appropriately scaled by the wearing material's yield strength, hardness by plastic modulus or Young's modulus to incorporate elastic-plastic tendencies, and sliding distance by a parameter defined as a reference length. It is suggested that a nondimensional Archard model may prove more generalizable and more convenient for wear-data processing.

The authors would like to acknowledge the supports by the National Science Foundation Graduate Research Fellowship, Army Research Laboratory, and Center for Surface Engineering and Tribology at Northwestern University. The views and conclusions in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. The authors would like to thank Professor Y. W. Chung, Northwestern University for his helpful suggestions.

• National Science Foundation Graduate Research Fellowship (Grant No. DGE-2234667; Funder ID: 10.13039/100000001).

• Army Research Laboratory under Cooperative Agreement Numbers (Grant Nos. W911NF-20-2-0230 and W911NF-20-2-0292; Funder ID: 10.13039/100006754).

$Ar$ =

reference contact area, m²

$E$ =

Young's modulus, Pa

$Ep$ =

plastic modulus, Pa

$ET$ =

plastic modulus for linear hardening, Pa

$E*$ =

equivalent modulus, Pa

$f$ =

minimization function to calibrate Eq. (3) parameters

$H$ =

material hardness, Pa

$H¯$ =

nondimensional hardness

$k$ =

wear coefficient

$k¯$ =

nondimensional wear coefficient

$L$ =

applied load, N

$L¯$ =

nondimensional applied load

$N$ =

number of data points in a data set

$p$ =

interfacial pressure, Pa

$r$ =

wear scar depth, m

$Rq0$ =

root-mean-squared roughness of a wearing surface, m

$s$ =

sliding distance, m

$s¯$ =

nondimensional sliding distance

$sr$ =

reference sliding length, m

$Sy$ =

yield strength of a material, Pa

$V$ =

wear volume, m³

$V¯$ =

nondimensional wear volume

$u$ =

sliding velocity, m/s

$α$ =

loading exponent

$β$ =

sliding distance/velocity exponent

$γ$ =

hardness exponent

$μ$ =

cluster mean

$σ$ =

cluster univariate standard deviation