Wear 262 (2007) 883–888
A universal wear law for abrasion
Matthew T. Siniawski a,∗ , Stephen J. Harris b , Qian Wang c
a
Loyola 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
Abstract
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
Mechanical and materials engineers have spent more than a
century searching for mathematical models for the wear rates
of materials that are both general and based on fundamental
principles. Finding a wear equation that can be trusted over a
wide range of conditions and materials would have enormous
practical value, allowing laboratory tests to substitute for expensive, potentially dangerous field tests and allowing for smaller
margins of safety to be designed into machinery and structures.
Thus, it is not surprising that Meng and Ludema [1] found more
than 300 proposed wear models and equations in the published
literature. However, such a large number is itself evidence that
this goal has been elusive.
Perhaps the simplest and most widely used model for abrasive wear is that of Archard [2]. Its derivation is straightforward
and intuitive, and it predicts the wear volume of an abraded
material as a function of sliding distance in terms of a wear
coefficient, the applied normal load and the material hardness.
Unfortunately, few systems obey this law over a wide range of
conditions [2]. For many systems the wear coefficient changes
∗
Corresponding author. Tel.: +1 310 338 5849; fax: +1 310 338 2782.
E-mail address: [email protected] (M.T. Siniawski).
0043-1648/$ – see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.wear.2006.08.017
during a test, even when external conditions appear to remain
constant. In 1962, Mulhearn and Samuels [3] presented a simple
model for the abrasion of steel sliding against silicon carbide
paper. Other, more detailed models have also been developed
to predict the abrasive rates of materials under specific conditions. For example, Lawn [4] proposed a model for wear of
brittle solids under fixed abrasive conditions. Sundararajan [5]
presented a model for two-body abrasive wear based on localization of plastic deformation. More recently, Bull and Rickerby
[6] proposed a model of abrasive wear based upon multiple-pass
scratching experiments. Because both the functional forms and
even the parameters considered by these and other models differ,
as a group they offer little guidance as to what are the controlling material and environmental parameters and what are the
fundamental relations among them.
The reason that no abrasive wear equation proposed to date
has general validity may be that the abrasion process is intrinsically extraordinarily complex, with the process varying substantially from one material pair to another and during a single
test. The abrasion rate depends, among other factors, on the
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.
Scanning electron microscope (SEM) images in Fig. 1 show this
evolution for the surface of a boron carbide (B4 C) coating that
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,
and the steel counter-surface shows equally dramatic changes. It
is unlikely that simple analytical models can capture the details
of such processes, while finite element models of the contact
mechanics cannot handle the complexity and three-dimensional
nature of the surfaces and the details of the evolution process.
Furthermore, a fundamental understanding of the abrasion process at the micrometer and nanometer scale is lacking. Therefore,
a completely different approach is required.
2. Model development
Recently, the authors discovered a simple relationship that
describes the kinetics for the abrasion of 52100 steel ball bearings sliding against coupons coated with sputtered B4 C and
diamond-like carbon (DLC) in pin-on-disk (unidirectional dry
sliding) experiments [7–17]. The relationship is
A(n) ≡
V (n)
= A1 nβ ,
d
(1)
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 cycledependence (or time-dependence) of the abrasion rate. It is
an empirical constant that satisfies −1 ≤ β < 0 for the commonly 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.
The simple relationship given by Eq. (1) holds in this system
for n ranging from 1 to at least 104 or 105 cycles and for A(n)
varying over more than 3 orders of magnitude. Fig. 2 gives typ-
ical data illustrating this relationship. By differentiating Eq. (1),
the abrasion rate on any given cycle i is found as
Ai ∼
= (1 + β)A1 iβ .
(2)
Thus, measurements of the abrasion rate during the first few
cycles, which determine A1 and β, allow prediction of the abrasion rate on any subsequent cycle (up to at least n = 20,000 cycles
in Fig. 2). β is a truly remarkable parameter. Experiments with
sputtered B4 C and DLC have shown that it does not change
over at least tens of thousands of cycles, even with the dramatic
changes in surface morphology seen in Fig. 1. It is independent
of the friction coefficient, the surface finish and sliding speeds
between 1 and 20 cm/s, when the frictional heating is small. It
takes the same value for 52100 and SAE 1010 steel ball bearings.
β is even independent of the load, while A1 scales as (load)2/3
[16].
Borodich, Harris and Keer [18] proposed a mathematical
framework with which to treat the evolution of abrasive wear. It
was suggested that the key to understanding the origin of Eq. (1)
lies in treating the surface statistically rather than mechanically.
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
roughness Ra = 300 nm; white diamonds, dry sliding with a B4 C surface roughness Ra = 10 nm; white squares, lubricated sliding with B4 C surface roughness
Ra = 300 nm).
M.T. Siniawski et al. / Wear 262 (2007) 883–888
In particular, it was assumed that: (1) asperities become dull and
lose their abrasiveness during the wear process through a process 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
the distribution of sharp asperities on the surface remains selfsimilar throughout the abrasion process [18]. That is, a 2D map
showing the location of sharp asperities after n1 cycles would
look, statistically, identical to a map showing the location of
sharp asperities after n2 cycles, except for a scale factor.
An immediate consequence of this power-law form, as is
clear from Fig. 2, is that the time (number of cycles) required to
achieve a given percentage change in l (abrasiveness) increases
as i becomes larger. A physical interpretation for this mathematical result, based on Fig. 1, is that the most prominent asperities
wear down rapidly, while remaining asperities are more and
more difficult to wear down because taller neighboring asperities and terraces shelter them. Thus, this sort of relationship
is an almost automatic consequence of having a distribution of
asperity heights, a property of nearly any surface [19]. [Note that
if it were instead assumed that li /lj = f(i − j), then f is an exponential function, and, like radioactive decay, the time required
to achieve a given percentage change in l is independent of i.
In fact, just such an exponential form has been reported for the
exceptional case of sandpaper [3], where all grains (asperities)
are nominally of the same size. For this case there is no sheltering
effect, since all sharp asperities are equally exposed. As a result,
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 contact 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 validity 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
data from the literature that were recast in terms of Eq. (1). Eq.
(1) satisfactorily correlates both the authors’ and the literature
885
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 white diamonds are for the DLC coating run against 52100 steel. The lines
are linear least square fits to Eq. (1).
data for every material pair for which data could be found. Thus,
it takes just two material constants, A1 and β, to predict the abrasion rate for any load and after any number of cycles for any of
the material pairs for which data was found. Eq. (1) appears to
be a universal wear law for abrasion.
3. Analysis and application
Fig. 2 shows data for 52100 steel sliding against various
B4 C coatings under a variety of conditions [11,13,15]. The
white squares represent experiments run under an unformulated
OBOA base Chevron mineral oil, with a viscosity of approximately 20 cSt at 40 ◦ C and approximately 4 cSt at 100 ◦ C. The
tests were conducted under boundary lubrication conditions,
with a calculated minimum film thickness of 0.0138 ␮m. The
presence of a lubricant strongly affects the abrasion rate, reducing it by around a factor of 3. Significantly, however, the data still
fits well to the functional form given in Eq. (1), and the slope β is
hardly affected (compare to black squares). This result indicates,
Fig. 4. Average abrasion rate A(n) for data obtained from the literature. The
symbols are the experimental data and the lines are linear least square fits to Eq.
(1) which correspond to the listed reference.
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M.T. Siniawski et al. / Wear 262 (2007) 883–888
Table 1
Summary of the complete model input data and error results
General material
category
Worn
material
Counterpart
material
Contact
conditions
A1 (mm3 /m)
β
RMS
deviation
(%)
Model
error (%)
Reference
Metal–coating
52100 steel
52100 steel
52100 steel
52100 steel
52100 steel
Bronze
1100 aluminum
B4 C coating 1
B4 C coating 2
B4 C coating 2
B4 C coating 3
DLC coating
B4 C coating 1
B4 C coating 1
Ball-on-disc, dry sliding
Ball-on-disc, dry sliding
Ball-on-disc, lubricated
Ball-on-disc, dry sliding
Ball-on-disc, dry sliding
Ball-on-disc, dry sliding
Ball-on-disc, dry sliding
2.24E−03
2.83E−03
7.81E−04
3.79E−05
2.03E−03
1.14E−02
1.22E−02
−0.7586
−0.8501
−0.7678
−0.5696
−0.7712
−0.6922
−0.5002
7.0
5.9
11.3
5.0
4.0
8.4
10.0
25
22
29
27
27
28
32
[11]
[13]
[15]
[13]
[17]
–
–
Coating–metal
DLC coating
Tungsten carbide
1.27E−04
−0.4179
0.7
11
[20]
MoS2 coating
(WTi)C–Ni coating
Steel ball bearing
Tool steel
Ball-on-disc, dry sliding,
3.1%RH
Ball-on-disc, dry sliding
Block-on-ring, lubricated
1.33E−06
9.54E−06
0.0071
−0.0275
2.1
2.0
3
7
[21]
[36]
CoCrMo
CoCrMo
CoCrMo
CoCrMo
2.97E−03
1.14E−03
−0.1667
−0.3503
2.4
0.5
15
2
[24]
[25]
ZnAlCuSi
7075-T6 aluminum
7075-T6 aluminum
Al–8Fe–4Ce
Al–13Si
Zn–35A1
Zn–35Al–Si
Zn–35Al–3.75Si
Zn–35Al–5.8Si
A6061 MMC
Zn–40Al
Mg–9Al–0.9Zn (AZ91)
Fe–25%TiC
Ti–50.3 at%Ni
2Crl3
Copper
AISI1045 steel
St 37 steel
Al2 O3
Al2 O3
440C stainless steel
440C stainless steel
440C stainless steel
440C stainless steel
440C stainless steel
440C stainless steel
AISI 01 tool steel
4140 steel
52100 steel
Steel
Cr-steel
Cr-steel
Carbon steel
52100 steel
Reciprocating, lubricated
Hip wear simulator,
lubricated
Rolling, lubricated
Ball-on-disc, dry sliding
Ball-on-disc, corrosive
Crossed-cylinder rolling, dry
Crossed-cylinder rolling, dry
Crossed-cylinder rolling, dry
Crossed-cylinder rolling, dry
Crossed-cylinder rolling, dry
Crossed-cylinder rolling, dry
Pin-on-disc, dry sliding
Block-on-ring, dry sliding
Block-on-ring, dry sliding
Block-on-ring, dry sliding
Block-on-ring, dry sliding
Block-on-ring, dry sliding
Reciprocating, dry sliding
Pin-on-ring, dry sliding
1.32E−01
4.94E−04
4.01E−05
4.65E−03
6.51E−03
1.40E−03
2.02E−03
3.47E−03
2.03E−03
1.90E−01
8.80E−07
3.68E−02
1.87E−03
2.11E−04
2.72E−05
5.51E−02
6.73E−05
−0.7339
0.043
0.4122
−0.2491
−0.2156
−0.0601
−0.1105
−0.1777
−0.1248
−0.4177
0.0782
−0.0116
0.0739
0.0361
0.0174
−0.1629
0.1495
2.7
6.3
6.3
3.4
3.3
0.9
1.6
8.8
1.1
3.6
3.3
1.0
0.9
2.9
2.6
4.2
0.8
10
24
22
13
17
3
8
3
6
24
14
5
3
9
11
16
4
[28]
[29]
[29]
[30]
[30]
[30]
[30]
[30]
[30]
[31]
[32]
[33]
[34]
[35]
[35]
[26]
[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]
Metal–metal
somewhat surprisingly, that the rate of loss of relative abrasiveness depends only slightly on the presence of a lubricant, at least
under these boundary lubrication conditions.
Fig. 3 shows results for bronze and 1100 series aluminum
sliding against B4 C, together with sample data for 52100 steel
sliding against DLC. Comparing Figs. 2 and 3, it can be seen
that the initial abrasion rate A1 of the softer materials is roughly
one order of magnitude higher than that of steel, as might be
expected. There is also an effect on β, which is in the range −0.5
to −0.7 for bronze and aluminum compared to −0.8 for B4 C.
That is, the abrasiveness of the coating drops more slowly when
run against the softer materials. It seems intuitively reasonable
that softer materials should have a smaller impact on a hard
abrasive coating, but the situation is actually more complex. For
example, β takes the same value when B4 C is run against 52100
steel as when it is run against the much softer 1010 steel. It
would appear that the observed morphological changes (Fig. 1)
that lead to the loss of abrasiveness for B4 C and DLC are not
caused simply by mechanical processes. Instead, it is likely that
they are due to stress-induced chemical reactions between the
carbon in these coatings and steel. Analogous reactions may
explain why diamond wears quickly when run against steel. The
smaller value of β for B4 C run against bronze and aluminum
(compared to steel) may then be due to the fact that there are
no analogous chemical reactions between carbon and bronze
or aluminum which can cause the coating asperities to wear
down.
The published literature is next used to explore the range
of validity of Eq. (1). A total of 17 publications [20–36] were
found in which experiments were described in sufficient detail
to be analyzed for this study. This literature data is largely,
but not exclusively from pin-on-disk, unidirectional systems.
Table 1 briefly describes the contact conditions of each experiment. Additional details about the specific experimental setup
for each material pair are in the appropriate reference. After
converting to the format presented in Fig. 2, a linear fit was calculated for each set of published data, with its quality evaluated
by determining the root-mean-square (RMS) deviation between
the fit and the experimental data. The power-law relationship
accurately represents all of the experimental data, as the maximum RMS percent deviation in Table 1 is only about 10%, and
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 conditions 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
Fig. 5. Range of the wear model parameters. The black squares are for the
metal–coating category, the grey squares are for the coating–metal category, the
white squares are for the metal–metal category, the black diamond is for the
ceramic–metal data set and the grey diamond is for the ceramic–ceramic data
set. The ellipses illustrate the approximate range of each category.
the quality of the fits, where the numbers that correspond to the
references identify the data.
β is negative for the majority of the cases, indicating that
the abrasion rate usually decreases with time. This is interpreted
to indicate that the aerial density of sharp asperities decreases
with time. For two of the metal–metal cases, β is modestly positive, indicating that the abrasion rate increases with time. This
increase in abrasion may be due to sharp wear debris particles
created during the abrasion process.
A wide range of values for A1 and β is evident from Table 1.
However, some relationship between them appears to exist
within each general material category, as illustrated in Fig. 5,
which plots β against A1 . The ellipses illustrate approximate
ranges of various material combinations. According to Fig. 5,
β is generally small when A1 is large. In other words, highly
abrasive material pairs tend to lose abrasiveness faster than lessabrasive material pairs. The relative importance of chemical and
physical mechanisms and any relationship to hardness have yet
to be elucidated.
These results show that the abrasion rate can be predicted
with only two parameters, A1 and β. Some of the factors
that control A1 are known. For example, the extreme sensitivity of abrasion rate to relative hardness has been well documented [2] for abrasive metal and ceramic powders. The authors
found the same relationship for steel run against DLC and
B4 C coatings [37], where the value of A1 is the same as that
for steel run against ceramic powder abrasives of comparable hardness. Fig. 3 is consistent with this result. The wellunderstood dependence of A1 on load, as discussed above, is
much weaker than its dependence on hardness. The sharpness
of the coating asperities also influences the abrasion behavior
[13].
In contrast to A1 , the factors that control β are not understood. The validity of Eq. (1) means that β stays constant during
these abrasion tests. It is worth noting that the authors have
previously shown [12] that the B4 C surfaces are fractal, and the
fractal dimension also stays constant during these abrasion tests.
It was shown that a wear equation derived to account for abrasion between boron carbide coatings and 52100 steel applies
to all abrasive material pairs for which appropriate data are
available, including all available data from the literature, except
for sandpaper. This exception is believed due to the use of
monodisperse silicon carbide grains that are glued to the paper.
Otherwise, the only parameters required for any material pair
are the initial abrasiveness and the initial rate at which the abrasiveness decreases with cycles, both of which can be readily
obtained from rapid, simple laboratory experiments. With these
parameters, the abrasion rate at any point in the future can be
predicted with considerable accuracy. No other relationship so
simple, accurate and widely applicable as Eq. (1) is known to
exist for the analysis of wear.
A key enabler for understanding abrasion is the recognition that wear is too complex to treat from a purely mechanical or deterministic perspective. The combination of detailed
morphological information from SEM images together with a
statistical perspective has improved the understanding of the
kinetics of abrasion. It is expected that this sort of combined
mechanical–statistical approach will be a fruitful one for studying other forms of wear. Further work is underway to understand
the material and environmental properties that determine β.
Acknowledgements
The authors would like to acknowledge the National Science
Foundation for funding for this research, as well as Mr. Wes
Bredin for assistance with the literature review and Dr. Gordon
Krauss for experimental assistance.
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