A universal wear law for abrasion

Wear 262 (2007) 883–888

                                             A universal wear law for abrasion
                                 Matthew T. Siniawski a,∗ , Stephen J. Harris b , Qian Wang c
                                      aLoyola Marymount University, 1 LMU Drive, MS 8145, Los Angeles, CA 90045, USA
                                                 b MD 3083, Ford Motor Company, Dearborn, MI 48121, USA
                                  c Northwestern University, Department of Mechanical Engineering, Evanston, IL 60208, USA

                                   Received 14 March 2006; received in revised form 4 August 2006; accepted 31 August 2006
                                                               Available online 2 October 2006

   Finding a wear law that is valid over a wide range of conditions and materials would have enormous practical value. The authors have previously
discovered a simple relationship describing the evolution of the abrasive wear rate of steel sliding against boron carbide-coated coupons, and have
developed a model accounting for its kinetics. The authors show here that this wear equation accurately describes the evolution of abrasive wear
rates for several additional material pairs and contact conditions that were tested, as well as for all of the material pairs for which literature data
could be found. The only material parameters are the initial abrasiveness and the initial rate at which the abrasiveness changes with number of
cycles. No other wear law so simple, accurate and widely applicable is known.
© 2006 Elsevier B.V. All rights reserved.

Keywords: Abrasive wear; Abrasion; Universal model

1. Introduction                                                                  during a test, even when external conditions appear to remain
                                                                                 constant. In 1962, Mulhearn and Samuels [3] presented a simple
    Mechanical and materials engineers have spent more than a                    model for the abrasion of steel sliding against silicon carbide
century searching for mathematical models for the wear rates                     paper. Other, more detailed models have also been developed
of materials that are both general and based on fundamental                      to predict the abrasive rates of materials under specific condi-
principles. Finding a wear equation that can be trusted over a                   tions. For example, Lawn [4] proposed a model for wear of
wide range of conditions and materials would have enormous                       brittle solids under fixed abrasive conditions. Sundararajan [5]
practical value, allowing laboratory tests to substitute for expen-              presented a model for two-body abrasive wear based on local-
sive, potentially dangerous field tests and allowing for smaller                 ization of plastic deformation. More recently, Bull and Rickerby
margins of safety to be designed into machinery and structures.                  [6] proposed a model of abrasive wear based upon multiple-pass
Thus, it is not surprising that Meng and Ludema [1] found more                   scratching experiments. Because both the functional forms and
than 300 proposed wear models and equations in the published                     even the parameters considered by these and other models differ,
literature. However, such a large number is itself evidence that                 as a group they offer little guidance as to what are the control-
this goal has been elusive.                                                      ling material and environmental parameters and what are the
    Perhaps the simplest and most widely used model for abra-                    fundamental relations among them.
sive wear is that of Archard [2]. Its derivation is straightforward                 The reason that no abrasive wear equation proposed to date
and intuitive, and it predicts the wear volume of an abraded                     has general validity may be that the abrasion process is intrin-
material as a function of sliding distance in terms of a wear                    sically extraordinarily complex, with the process varying sub-
coefficient, the applied normal load and the material hardness.                  stantially from one material pair to another and during a single
Unfortunately, few systems obey this law over a wide range of                    test. The abrasion rate depends, among other factors, on the
conditions [2]. For many systems the wear coefficient changes                    evolution of two surfaces, typically at the micro- or nano-scale
                                                                                 [7]; on intimate details of how these two surfaces mate; and on
                                                                                 the changing responses of each of the surfaces to that mating.
 ∗   Corresponding author. Tel.: +1 310 338 5849; fax: +1 310 338 2782.          Scanning electron microscope (SEM) images in Fig. 1 show this
     E-mail address: msiniawski@lmu.edu (M.T. Siniawski).                        evolution for the surface of a boron carbide (B4 C) coating that

0043-1648/$ – see front matter © 2006 Elsevier B.V. All rights reserved.
884                                                    M.T. Siniawski et al. / Wear 262 (2007) 883–888

Fig. 1. SEM images of: (a) unworn and (b) worn B4 C coatings after sliding against 52100 steel for 500 cycles [8]. Evidence indicates that the steel chemically
polishes the boron carbide even as the steel is mechanically polished by the much harder boron carbide [9].

has run against AISI 52100 steel [8]. The changes are profound,                   ical data illustrating this relationship. By differentiating Eq. (1),
and the steel counter-surface shows equally dramatic changes. It                  the abrasion rate on any given cycle i is found as
is unlikely that simple analytical models can capture the details
of such processes, while finite element models of the contact                     Ai ∼
                                                                                     = (1 + β)A1 iβ .                                                       (2)
mechanics cannot handle the complexity and three-dimensional                      Thus, measurements of the abrasion rate during the first few
nature of the surfaces and the details of the evolution process.                  cycles, which determine A1 and β, allow prediction of the abra-
Furthermore, a fundamental understanding of the abrasion pro-                     sion rate on any subsequent cycle (up to at least n = 20,000 cycles
cess at the micrometer and nanometer scale is lacking. Therefore,                 in Fig. 2). β is a truly remarkable parameter. Experiments with
a completely different approach is required.                                      sputtered B4 C and DLC have shown that it does not change
                                                                                  over at least tens of thousands of cycles, even with the dramatic
2. Model development                                                              changes in surface morphology seen in Fig. 1. It is independent
                                                                                  of the friction coefficient, the surface finish and sliding speeds
    Recently, the authors discovered a simple relationship that                   between 1 and 20 cm/s, when the frictional heating is small. It
describes the kinetics for the abrasion of 52100 steel ball bear-                 takes the same value for 52100 and SAE 1010 steel ball bearings.
ings sliding against coupons coated with sputtered B4 C and                       β is even independent of the load, while A1 scales as (load)2/3
diamond-like carbon (DLC) in pin-on-disk (unidirectional dry                      [16].
sliding) experiments [7–17]. The relationship is                                      Borodich, Harris and Keer [18] proposed a mathematical
                                                                                  framework with which to treat the evolution of abrasive wear. It
          V (n)
A(n) ≡          = A1 nβ ,                                                (1)      was suggested that the key to understanding the origin of Eq. (1)
            d                                                                     lies in treating the surface statistically rather than mechanically.
where A(n) is the abrasion rate averaged over the first n cycles,
V(n) the volume of steel removed from the ball during the first
n cycles by the abrasion process, d = 2πrn the distance traveled
by the steel ball over the coated disk at a wear track radius r
and A1 is the abrasion rate (volume of steel removed/meters
traveled) on the first cycle. The value of β controls the cycle-
dependence (or time-dependence) of the abrasion rate. It is
an empirical constant that satisfies −1 ≤ β < 0 for the com-
monly encountered case where the abrasion rate falls with time.
(Since the abrasion rate cannot fall below zero, the average
abrasion rate A(n) cannot fall faster than inversely with n.)
β = −1 corresponds to the case where all of the abrasiveness
is lost after a single cycle, while β = 0 corresponds to the case
where the abrasion rate is constant. β > 0 would apply to systems
where the abrasion rate increases with time. Experiments have
shown that β is approximately −0.8 for the sputtered B4 C–steel
system.                                                                           Fig. 2. Average abrasion rate A(n) as a function of cycles for 52100 steel
                                                                                  sliding against B4 C coatings (black squares, dry sliding with a B4 C surface
    The simple relationship given by Eq. (1) holds in this system                 roughness Ra = 300 nm; white diamonds, dry sliding with a B4 C surface rough-
for n ranging from 1 to at least 104 or 105 cycles and for A(n)                   ness Ra = 10 nm; white squares, lubricated sliding with B4 C surface roughness
varying over more than 3 orders of magnitude. Fig. 2 gives typ-                   Ra = 300 nm).
M.T. Siniawski et al. / Wear 262 (2007) 883–888                                                          885

In particular, it was assumed that: (1) asperities become dull and
lose their abrasiveness during the wear process through a pro-
cess that converts the relatively sharp abrasive asperities seen in
Fig. 1a into dull, relatively less-abrasive regions such as the flat
terraces seen in Fig. 1b. Asperities are classified as either sharp
or dull. (2) The abrasion rate is proportional to the area density of
sharp asperities. Thus, if li is the average distance between sharp
asperities on the ith cycle, then the abrasion rate on that cycle
is inversely proportional to li2 . The result is that changes in the
abrasion rate during an experiment are determined by changes
in a single variable, l.
    The particular form of the wear law then depends on the
rule for how l evolves or, equivalently, the rate at which sharp
asperities become dull. Eq. (1) is readily obtained if it is assumed
that li /lj depends only on i/j, because a power-law function is the
only solution to li /lj = f(i/j). This results from an assumption that      Fig. 3. Average abrasion rate A(n) for B4 C coating with a surface roughness
                                                                            Ra = 300 nm against bronze (black squares) and 1100 aluminum (white squares).
the distribution of sharp asperities on the surface remains self-           The white diamonds are for the DLC coating run against 52100 steel. The lines
similar throughout the abrasion process [18]. That is, a 2D map             are linear least square fits to Eq. (1).
showing the location of sharp asperities after n1 cycles would
look, statistically, identical to a map showing the location of             data for every material pair for which data could be found. Thus,
sharp asperities after n2 cycles, except for a scale factor.                it takes just two material constants, A1 and β, to predict the abra-
    An immediate consequence of this power-law form, as is                  sion rate for any load and after any number of cycles for any of
clear from Fig. 2, is that the time (number of cycles) required to          the material pairs for which data was found. Eq. (1) appears to
achieve a given percentage change in l (abrasiveness) increases             be a universal wear law for abrasion.
as i becomes larger. A physical interpretation for this mathemat-
ical result, based on Fig. 1, is that the most prominent asperities         3. Analysis and application
wear down rapidly, while remaining asperities are more and
more difficult to wear down because taller neighboring asper-                   Fig. 2 shows data for 52100 steel sliding against various
ities and terraces shelter them. Thus, this sort of relationship            B4 C coatings under a variety of conditions [11,13,15]. The
is an almost automatic consequence of having a distribution of              white squares represent experiments run under an unformulated
asperity heights, a property of nearly any surface [19]. [Note that         OBOA base Chevron mineral oil, with a viscosity of approxi-
if it were instead assumed that li /lj = f(i − j), then f is an expo-       mately 20 cSt at 40 ◦ C and approximately 4 cSt at 100 ◦ C. The
nential function, and, like radioactive decay, the time required            tests were conducted under boundary lubrication conditions,
to achieve a given percentage change in l is independent of i.              with a calculated minimum film thickness of 0.0138 ␮m. The
In fact, just such an exponential form has been reported for the            presence of a lubricant strongly affects the abrasion rate, reduc-
exceptional case of sandpaper [3], where all grains (asperities)            ing it by around a factor of 3. Significantly, however, the data still
are nominally of the same size. For this case there is no sheltering        fits well to the functional form given in Eq. (1), and the slope β is
effect, since all sharp asperities are equally exposed. As a result,        hardly affected (compare to black squares). This result indicates,
they have a constant probability of becoming dull on any given
cycle. While the results from sandpaper can thus be explained
within the context of the present model, it will be excluded from
further discussion because of its idiosyncratic asperity height
distribution.] The 2/3 power dependence of A1 on load that was
observed comes out of this analysis in a natural way. It is fully
consistent with Hertzian mechanics and with the predictions of
Greenwood and Williamson’s statistical model for elastic con-
tact of rough surfaces [19]. It is interesting to note that the same
2/3 exponent is predicted using an elastic statistical theory, even
though cutting is clearly not elastic.
    Although the authors’ previous work demonstrated the valid-
ity of Eq. (1) for dry and lubricated sliding of sputtered B4 C
against steel and dry sliding of DLC coatings against steel, it
had not been tested for other systems or conditions. This paper
provides new experimental data showing its validity for other
material pairs. These results are then combined with abrasion               Fig. 4. Average abrasion rate A(n) for data obtained from the literature. The
data from the literature that were recast in terms of Eq. (1). Eq.          symbols are the experimental data and the lines are linear least square fits to Eq.
(1) satisfactorily correlates both the authors’ and the literature          (1) which correspond to the listed reference.
886                                                    M.T. Siniawski et al. / Wear 262 (2007) 883–888

Table 1
Summary of the complete model input data and error results
General material   Worn                      Counterpart            Contact                         A1 (mm3 /m)   β         RMS         Model       Reference
category           material                  material               conditions                                              deviation   error (%)

Metal–coating      52100 steel               B4 C coating 1         Ball-on-disc, dry sliding       2.24E−03      −0.7586    7.0        25          [11]
                   52100 steel               B4 C coating 2         Ball-on-disc, dry sliding       2.83E−03      −0.8501    5.9        22          [13]
                   52100 steel               B4 C coating 2         Ball-on-disc, lubricated        7.81E−04      −0.7678   11.3        29          [15]
                   52100 steel               B4 C coating 3         Ball-on-disc, dry sliding       3.79E−05      −0.5696    5.0        27          [13]
                   52100 steel               DLC coating            Ball-on-disc, dry sliding       2.03E−03      −0.7712    4.0        27          [17]
                   Bronze                    B4 C coating 1         Ball-on-disc, dry sliding       1.14E−02      −0.6922    8.4        28          –
                   1100 aluminum             B4 C coating 1         Ball-on-disc, dry sliding       1.22E−02      −0.5002   10.0        32          –
Coating–metal      DLC coating               Tungsten carbide       Ball-on-disc, dry sliding,      1.27E−04      −0.4179    0.7        11          [20]
                   MoS2 coating              Steel ball bearing     Ball-on-disc, dry sliding       1.33E−06       0.0071    2.1         3          [21]
                   (WTi)C–Ni coating         Tool steel             Block-on-ring, lubricated       9.54E−06      −0.0275    2.0         7          [36]
Metal–metal        CoCrMo                    CoCrMo                 Reciprocating, lubricated       2.97E−03      −0.1667    2.4        15          [24]
                   CoCrMo                    CoCrMo                 Hip wear simulator,             1.14E−03      −0.3503    0.5         2          [25]
                   ZnAlCuSi                  St 37 steel            Rolling, lubricated             1.32E−01      −0.7339    2.7        10          [28]
                   7075-T6 aluminum          Al2 O3                 Ball-on-disc, dry sliding       4.94E−04       0.043     6.3        24          [29]
                   7075-T6 aluminum          Al2 O3                 Ball-on-disc, corrosive         4.01E−05       0.4122    6.3        22          [29]
                   Al–8Fe–4Ce                440C stainless steel   Crossed-cylinder rolling, dry   4.65E−03      −0.2491    3.4        13          [30]
                   Al–13Si                   440C stainless steel   Crossed-cylinder rolling, dry   6.51E−03      −0.2156    3.3        17          [30]
                   Zn–35A1                   440C stainless steel   Crossed-cylinder rolling, dry   1.40E−03      −0.0601    0.9         3          [30]
                   Zn–35Al–Si                440C stainless steel   Crossed-cylinder rolling, dry   2.02E−03      −0.1105    1.6         8          [30]
                   Zn–35Al–3.75Si            440C stainless steel   Crossed-cylinder rolling, dry   3.47E−03      −0.1777    8.8         3          [30]
                   Zn–35Al–5.8Si             440C stainless steel   Crossed-cylinder rolling, dry   2.03E−03      −0.1248    1.1         6          [30]
                   A6061 MMC                 AISI 01 tool steel     Pin-on-disc, dry sliding        1.90E−01      −0.4177    3.6        24          [31]
                   Zn–40Al                   4140 steel             Block-on-ring, dry sliding      8.80E−07       0.0782    3.3        14          [32]
                   Mg–9Al–0.9Zn (AZ91)       52100 steel            Block-on-ring, dry sliding      3.68E−02      −0.0116    1.0         5          [33]
                   Fe–25%TiC                 Steel                  Block-on-ring, dry sliding      1.87E−03       0.0739    0.9         3          [34]
                   Ti–50.3 at%Ni             Cr-steel               Block-on-ring, dry sliding      2.11E−04       0.0361    2.9         9          [35]
                   2Crl3                     Cr-steel               Block-on-ring, dry sliding      2.72E−05       0.0174    2.6        11          [35]
                   Copper                    Carbon steel           Reciprocating, dry sliding      5.51E−02      −0.1629    4.2        16          [26]
                   AISI1045 steel            52100 steel            Pin-on-ring, dry sliding        6.73E−05       0.1495    0.8         4          [27]
Ceramic–metal      Si3 N4                    Steel ball bearing     Dry rolling                     1.24E−04      −0.2156    8.2         7          [22]
Ceramic–ceramic    Si3 N4 (20 wt%HBN)        Si3 N4 (20 wt%HBN)     Pin-on-disc, sliding            7.64E−02      −0.7311    4.5        27          [23]

somewhat surprisingly, that the rate of loss of relative abrasive-                explain why diamond wears quickly when run against steel. The
ness depends only slightly on the presence of a lubricant, at least               smaller value of β for B4 C run against bronze and aluminum
under these boundary lubrication conditions.                                      (compared to steel) may then be due to the fact that there are
    Fig. 3 shows results for bronze and 1100 series aluminum                      no analogous chemical reactions between carbon and bronze
sliding against B4 C, together with sample data for 52100 steel                   or aluminum which can cause the coating asperities to wear
sliding against DLC. Comparing Figs. 2 and 3, it can be seen                      down.
that the initial abrasion rate A1 of the softer materials is roughly                 The published literature is next used to explore the range
one order of magnitude higher than that of steel, as might be                     of validity of Eq. (1). A total of 17 publications [20–36] were
expected. There is also an effect on β, which is in the range −0.5                found in which experiments were described in sufficient detail
to −0.7 for bronze and aluminum compared to −0.8 for B4 C.                        to be analyzed for this study. This literature data is largely,
That is, the abrasiveness of the coating drops more slowly when                   but not exclusively from pin-on-disk, unidirectional systems.
run against the softer materials. It seems intuitively reasonable                 Table 1 briefly describes the contact conditions of each exper-
that softer materials should have a smaller impact on a hard                      iment. Additional details about the specific experimental setup
abrasive coating, but the situation is actually more complex. For                 for each material pair are in the appropriate reference. After
example, β takes the same value when B4 C is run against 52100                    converting to the format presented in Fig. 2, a linear fit was cal-
steel as when it is run against the much softer 1010 steel. It                    culated for each set of published data, with its quality evaluated
would appear that the observed morphological changes (Fig. 1)                     by determining the root-mean-square (RMS) deviation between
that lead to the loss of abrasiveness for B4 C and DLC are not                    the fit and the experimental data. The power-law relationship
caused simply by mechanical processes. Instead, it is likely that                 accurately represents all of the experimental data, as the maxi-
they are due to stress-induced chemical reactions between the                     mum RMS percent deviation in Table 1 is only about 10%, and
carbon in these coatings and steel. Analogous reactions may                       in most cases the deviation is less than half that. Fig. 4 shows
M.T. Siniawski et al. / Wear 262 (2007) 883–888                                                        887

                                                                                    This experimental result makes plausible the assumption that the
                                                                                    surfaces remain self-similar.
                                                                                        For B4 C–steel and DLC–steel systems, over the range of con-
                                                                                    ditions considered, β is independent of load, number of cycles,
                                                                                    friction coefficient, sliding speed, surface finish and the presence
                                                                                    or absence of a lubricant. These remarkable results suggest that
                                                                                    β is a fundamental property of each abrasion pair. The authors
                                                                                    plan to explore the significance of β in future work.

                                                                                    4. Conclusions

                                                                                       It was shown that a wear equation derived to account for abra-
                                                                                    sion between boron carbide coatings and 52100 steel applies
                                                                                    to all abrasive material pairs for which appropriate data are
Fig. 5. Range of the wear model parameters. The black squares are for the           available, including all available data from the literature, except
metal–coating category, the grey squares are for the coating–metal category, the    for sandpaper. This exception is believed due to the use of
white squares are for the metal–metal category, the black diamond is for the        monodisperse silicon carbide grains that are glued to the paper.
ceramic–metal data set and the grey diamond is for the ceramic–ceramic data
                                                                                    Otherwise, the only parameters required for any material pair
set. The ellipses illustrate the approximate range of each category.
                                                                                    are the initial abrasiveness and the initial rate at which the abra-
                                                                                    siveness decreases with cycles, both of which can be readily
the quality of the fits, where the numbers that correspond to the                   obtained from rapid, simple laboratory experiments. With these
references identify the data.                                                       parameters, the abrasion rate at any point in the future can be
    β is negative for the majority of the cases, indicating that                    predicted with considerable accuracy. No other relationship so
the abrasion rate usually decreases with time. This is interpreted                  simple, accurate and widely applicable as Eq. (1) is known to
to indicate that the aerial density of sharp asperities decreases                   exist for the analysis of wear.
with time. For two of the metal–metal cases, β is modestly pos-                        A key enabler for understanding abrasion is the recogni-
itive, indicating that the abrasion rate increases with time. This                  tion that wear is too complex to treat from a purely mechan-
increase in abrasion may be due to sharp wear debris particles                      ical or deterministic perspective. The combination of detailed
created during the abrasion process.                                                morphological information from SEM images together with a
    A wide range of values for A1 and β is evident from Table 1.                    statistical perspective has improved the understanding of the
However, some relationship between them appears to exist                            kinetics of abrasion. It is expected that this sort of combined
within each general material category, as illustrated in Fig. 5,                    mechanical–statistical approach will be a fruitful one for study-
which plots β against A1 . The ellipses illustrate approximate                      ing other forms of wear. Further work is underway to understand
ranges of various material combinations. According to Fig. 5,                       the material and environmental properties that determine β.
β is generally small when A1 is large. In other words, highly
abrasive material pairs tend to lose abrasiveness faster than less-                 Acknowledgements
abrasive material pairs. The relative importance of chemical and
physical mechanisms and any relationship to hardness have yet                         The authors would like to acknowledge the National Science
to be elucidated.                                                                   Foundation for funding for this research, as well as Mr. Wes
    These results show that the abrasion rate can be predicted                      Bredin for assistance with the literature review and Dr. Gordon
with only two parameters, A1 and β. Some of the factors                             Krauss for experimental assistance.
that control A1 are known. For example, the extreme sensitiv-
ity of abrasion rate to relative hardness has been well docu-
mented [2] for abrasive metal and ceramic powders. The authors
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