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Ludema1996-FrictionWearMechanismModeling.PDF
WEAR
ELSEVIER
Wear200 (1996) I-7
Mechanism-based modeling of friction and wear
K.C. Ludema
Departmentof MeclumicalEngineering,Universityof Michigan,2250G.G. BrownBuilding,Ann Arbor, M148109.2125, USA
Abstract
Equations for "predicting" friction and wear that product design engineers can readily apply requite new methods to develop. These
include the following.
l. The planned cooperationof several technical disciplines.
2. The selectionof a well studied system.
3. Parametricequations (models) for the observedfriction or wear behavior, including all variablesknown by an interdisciplinary group.
4. Data that cov~ a wide range (e.g. several decades) of most relevant variables.
5. Methods for adjusting parameters in the modelsto closely match the data.
The results of three research efforts are di~ussed in terms of the prospects for developing equations either by curve fiuing the data or by
describing underiying phenomena involved in the sliding process. Finally, possible contributions of ~veral technical disciplines toward
modeling the conditions for scuff prevention ate suggested.
Keywords: Modeling:F d ~ ; Wear
1, Introduction
This paper advocates a new approach to developing quations that designers of mechanical components can confidendy use to predict friction and wear of sliding components.
Product engineers need such equations, or at least some
guidelines in a mathematical format, for optimizing their
overall designs.
The need for a new approach is obvious in that previous
efforts in modeling friction and wear have not kept pace with
most other technical topics. There are many equations available, indeed, more than 300 may be found in the literature of
the last 40 years [ 1], but even the best has very limited use
[2]. The limited utility of the 300 equations is not because
of a shortage of thought on the subject: there at© more than
1000 papers in the literature that contain substantive discus.
sions on mechanisms of friction and wear without proposing
models or equations. (The terms "equations" and' 'models"
will be used interchangeably, though "equations" may b¢
seen as a "utility" item in the design process whereas a
"model" suggests some tentative idea of how systems
b~have.)
The following discussion suggests reasons for the present
problem. The lack of good models is surely the result of the
complicated nature of friction and wear, but little has been
done to plot a rational course through the complexity. Inap0043-1648/961515,00 © 1996Elsevier ScienceS.A. A)i rights reserved
1)11S0043-1648 ( 96 ) 07312-7
propriately simple methods are widely used in research, most
likely derived from a narrow disciplinary focus. Some of this
focus is probably perpetuated by attempts by tribologists to
meet the scholarly standards of the specialty sciences that arc
"adjacent" to their topics rather than by standards that
advance the state of modeling in friction and wear. Thick film
lubrication, incidentally does not come under review here
because it is, of all topics in tribology, the most highly
developed.
The point of this paper is not to focus on deficiencies, but
on future needs. Though existing equations have limited practical value, by studying them we can develop a conviction of
the scope of the work still before us, i.e. we can now sccmor¢
clearly what to do next.
2. The consequences of limited models
The most obvious proof that designers have few useful
equations for friction and wear is the high warranty costs for
problems in friction and wear in all induscies. Warranty costs
are usually highest where technology is the weakest, The
North American auto industry, as one example, incurred war.
rant), costs for brake problems alone, in 1995, of about 1.5
billion dollars ($1.5× 109), Clutch plates in the automatic
transmissions, lock-up clutches in torque converters, wind-
2
K I..udemalWear200f1996) I-7
shield wipers, cam/follower pairs, sliding thrust washers,
belts and ~.nany other sliding parle also cause problems, ultimately because they are not properly designed. By contrast,
wan'anty costs are low in most other design areas for which
there are models and equations, such as for stress analysis,
fatigue prediction, fluid flow, heat balance and many more
topics.
A second consequence of a lack of good models is that
poor models are then nseO. Most often this is seen in the
development of finite element models of sliding systems.
Whereas most finite element modelers me meticulous in
selecting models for stress states, vibration modes and other
phenomena, they cannot pmcee~ as surely in friction and
wear. Thus some questionable models for friction and wear
have become embedded in design methods. One example is
the Coulomb model for friction, by which it is generally
impossible to predict vibrations and unsteady friction over
time.
The above depressing message is probably no different
from that in any other developing field. In due time we would
expect that some equations would emerge within one or
another of the several sub-divisior, s of our technologies,
in the past. We could cite the field of hydrodynamic lubrication as one highly developed field at this time, but 100+
years ago, in the days of 0. Reynolds the subject must have
been as bewildering as friction and wear are today. Perhaps
good models for friction and wear will advance our technology as much as the science of lubrication did.
3. Reasons for the inadequacy of existing equations
There is some risk in enumerating reasons for inadequacy
of existing equations, though it has been done before [3 ].
The existing equations are, after all, the Woduct of very much
thought and many experiments on the part of very many
competent investigators. However, the form ofequations suggest something of what authors understand about friction and
wear, for example the following.
(a) 'Ihat friction and wear rates vary monotonically over
ranges of most variables. Most investigators appear to select
the narrow ranges of variables in their experiments, just sufficient to indicate trends in the data. Wide ranging tests afterall are costly in time and money, and are difficult to justify
without broader goals.
(b) That th.e combination of mechanisms that control friction and wear are invariant with time, i.e. they are not conditioned by previous events.
(c) That every sliding "situation" can be classified according to an "inherent" wear mechanism, and to prevent wear
it is only necessary to choose a material that resists the identiffed mechanism of wear. Actually, the chosen material takes
its own wear mechanism into the system.
(d) Modeling efforts appear to follow two mutually exclusive philosophies. One is to build equations by measuring
separate dependencies of friction and wear on various mate-
rial properties. The other is to measure friction and/or wear
over some range of chosen variables (speed, load, temperature, etc.), plot the data and then "curve fit" an equation to
the data. This latter tyne of model does not usually incorporate
any information on the mechanisms of friction or wear and
thus has very limited value for predicting friction and wear
for new conditions.
(e) That the chosen variables in the equations can be treated
as independent, leading apparently to Cuee somewhat overlapping expectations:
1. That every variable can be varied independently, with
others held at any convenient value without compromising
the overall conclusion. These expectations are clearly not
valid since few of the material property variables are either
intrinsic or independent of others:
2. That variables not included "'at this time" can be added
at any future date without altering the structure of the
equations; particularly, variables from other fields,
3. That each discipliuecan work effectively withoutconcem
for the variables in other disciplines.
Understanding of friction and wear is a daunting endeavor
also because of the great number of variables that influence
friction and wear rate. In the papers on "wear" the sum of
variables used in the literature exceeds 625, though this number could probably be reduced to about 100 more fundamental
variables. Of these, about 10 would suffice for deep abrasion,
about 20 are requited for dry wear, but more then 50 would
be needed for wear in chemical boundary lubricated contact.
Equations for wear in lubricated sliding would include
(which will again be discussed in a later section):
1. Variables from hydrodynamics, which are mostly those
of fluid mechanics plus methods for describing surface
topography and contact area;
2. Variables from contact mechanics, mostly from linear
elasticity, sometimes from plasticity but seldom from
viscoelasticity;
3. Variables from materials engineering relating to
(quasi-static) substrate material properties and failure
mechanisms, occasionally including fatigue properties;
4. Variables from chemistry relating to the function of active
chemical species in lubricants.
The point that four different disciplines are active in
research is reassuring, but there surely is separation of thought
when such distinct disciplines work in the same field. Unfortunately, there is also some guarding of prerogatives in
solving problems. This is seen in industry and it prevents
shortening of the design cycles for new products. Issues in
friction and wear are often postponed until late in the design
process simply because there is no basis for compelling the
advancement of analysis in these topics.
4. General recommendations
Very obviously the development of equations or models
for friction and wear lags far behind that in other fields. If the
g. lasdemalWear 200(1996)I - 7
reason for this state of affairs is accurately stated above, a
new direction must be taken in research in friction and wear.
The new direction, in both research and in publishing research
pape~ should include the following.
1. Covering a very wide range of several variables Jn all test
programs.
2. Describing the conditions of the tests, materials, mechanical dynamics, environment, test geometries, etc., completely, so that others can cbeck published information
and broaden the range of tentative models.
3. Writing equations in terms of operative mechanisms of
friction and wear in such a way as to reflect actual data.
Three examples will be given in the next section.
4. Engaging in research with investigators of several disciplines. One possible approach will be discussed in a later
section.
5. Using some practical machinery as the "point of departure" for the work and as the target for correlation with
laboratory work. This will prevent the fruitless efforts to
correlate one bench test with another.
S. Functiomd models versus ~.neehanism(s) modek
In the three examples given below, Figs. !-3 are graphs of
data. These data arc particularly useful because of the wide
range of variables used in the experiments. All are for dry
sliding which requires discussion of fewer variables.
With some effort, equations for the lines shown in Figs. 13 could he generated. Should one encounter these same identical materials and surrounding conditions again, these equations would help to "predict" friction and wear rate for the
new case. But, more useful, and perhaps simpler equations
could possibly be developed by recognizing the major materials processes at work in these three sliding pairs. It is the
modeling of data curves through the actual friction and wear
wocesses that is being urged in this paper.
3
between !0 and 20 cut s -t would suggest that wear rate
increases as temperature increases, but suddenly decreases
when the temperature exceeds 300 °C. Surely, equations
based on isolated tests run over narrow ranges of sliding
speed, even for the same material and under identical conditions would be very confusing.
However, with some explanation, i.e. with a "word
model" of controlling events in Lancaster's research, the
results would become much more comweheasible. The transition between mild and severe wear was Woposed by Lancaster to be controlled by the thickness of oxide. The oxide
thickness is a function of two factors, namely the time available to reoxidi~ a denuded region on the sliding surfaces,
and the rate of formation of the oxide: these are sketched in
Fig. 2. The time available to oxidize is inversely related to
the sliding speed in repeat pass sliding as with a pin on ring.
The rate of formation of oxide is influenced by temperature,
'
~
In~
in oxygen
- -
3O0*0
10
j lO'
|
~o~
i
I
o.1
I
1
$~
i
1o
I
'too
I
todd
sped (¢nvsec)
Fig. I. Wearratevs. ~';.:lingspeed,with3 ks load.
I
av.al.~._ for
r
ox"
5.1. Model I
The first data, in Fig. 1, are due to Lancaster [4] who
measured the wear rate of a 60Cu--40Zn brass pin on a high
speed steel (HSS) ring over a very wide range of sliding
speed end temperature. He classified wear in relative terms,
mild and severe, severe in the region of the high regions of
the curves and mild elsewhere. The differences in wear rate
are expressed in orders often. The transition between severe
wear and mild wear is influenced by atmosphere, as well as
sliding speed and ambient temperature.
Wear modeling would consist of writing equations for the
curves in Fig. 1 in such a way as to reflect the influence of
temperature and oxygen concentration. As to the influence of
temperature, by simple curve fitting, for example between
0.1 and 0.2cm s-i, the slope for thedataat 100°C would be
negative whereas the slopes for the data from tests at higher
temperature would be positive. A test run in the range
Rg. 2. Wearratevs. loadfor 1050sleelsof thn=~ , ; .
¢mnUgl~
t
~ lO
~..4, 1
io "z
o,I
lO
O.Ol
20~
,o.,
o.
400
600
t ~
Fig. 3, The friction of a solid pin on a ~
tim plate,
4
g. Ludema/Wear200(1996)1-7
which is the ambient temperature plus the teml~crature rise
due to sliding.
Fig. 3 shows the relation between wear rate, the coefficient
of friction and electrical cou-trot resistance over a range of
temperature in Lancastor's work. Apparently at the higher
temperatures there is sufficient oxide to electrically separate
the metals, and to increase friction. These data call to question
as well. the simple assumptions that wear rate may be predicted from a friction nw,asurement, and that oxides arc
always' 'abrasive' '.
5.2. Model 2
The second example comes from the work of Welsh [5]
who measured wear rates vs. load for 0.5%C steel on steel,
using a pin-ou-ring configuration. He found transitions
between severe (high) wear and mild wear in the form shown
in Fig. 4. Three curves were selected for illustrative purposes.
The large transitions ( = 2.5 orders of ten) in the data for the
softest steel seem impossible and yet they are real: these data
have been vcrified by research students many times.
These data can again be explained in terms of the influence
of oxides as before. In the experiments of Welsh the average
thickness of the oxide film diminishes with increasing load
but becomes thicker at higher loads due to high contact temperature. Increasing the hardness diminish~ 'the extent of
transition between the two modes of wear. Very likely, the
l
lO
1o 1o3
t0
k~cl.gmm
Fig. 4. Influenceofcompetingfaclmsthat controloxidelilrathickness.
O
A
S
~
I
I~eh
Inx q crata taken at meCkml tempermre
~
~," ~omdmumna
log ( s k ~ ~
x Iranslcxma~ fz:lor, s vl
Fig. 6. Tnmfoming of data to a single curve.
critical oxide thickness is less for hard substrates than for the
soft substrate.
Again, equations can probably be written to describe the
curves in Fig. 4, though with sou~ difficulty and they would
have little predictive value for suggesting new materials. A
cautionary note may be derived from Fig. 4 concerning running accelerated tests. Iftbe sliding pair of Welsh is to operate
with a load of 50 g, an accelerated test run with 500 g load
would suggest that these materials are notgood choices. Conversely, if the sliding pair is to operate with a load of 500 g,
an accelerated test run with 5000 g load would erroneously
indicate that these materials are very good choices.
5.3. Model3
A third example is the friction of a steel pin sliding on a
flat plate of P T ~ [6] shown in Fig. 5. Dam over a wide
range of sliding speed again appear chaotic or incomprehensible. However, these data can be understood if"visco-elastic" transforms are applied to the data. ~
is a visco--clastic
material in which the mechanical behavior at high temperalure and high strain rate is very similar to that at tow temperature and low strain rote. Essentially, strain rate and
temperature, in a proper relationship, have the same effect.
Individual data curves taken at high temperature can each be
shifted a different amount to the left and individual data
curves taken at low temperature can each be shifted adifferent
amount to the right to produce the smooth curve shown in
Fig. 6. This curve has the shape of a damping loss curve for
P T ~ , which would suggest that sliding produces some cyclic
straining in the FIFE. Though friction is due not only to
damping loss, the point is clear that some visco.elastic phenomena ate involved which serves to develop a single curve
to characterize ~ friction of this polymer. An equation or
model can now be found to describe this single curve, and it
will include some of the properties of the polymer.
6. Developing a scuff prevention-boundary lubrication
model through interdisciplinary work
0.05
I
-
.~ .
•4
÷''".~ .........
-3
-2
-1
0
• Omums ~ spw (cnvsec)
Fig.5. Frictionandwarm'.
It is clear to most investigators that interdisciplinary work
requires much effort to coordinate. The impediments had so
far been more obvious than the benefits, However, action will
probably not arise from a recognition of need alone: some
framework for interdisciplinary work must be proposed, and
one is suggested below. It is on the subject of hounctmy
K. LudemaI Wear200 (1996} 1-7
lubrication and scuff prevention. Boundary lubrication here
refers to the formation of protective surface substances by
chemical reaction between a lubricant and a solid surface.
Scuffing is defined as a surface damage that results from
inadequate fluid film lubrication. (Not~ that this definition of
scuffing refers mostly to the initiating events that l e ~ to the
several forms of surface change, various stages of which have
names and most of which are assumed to involve adhesion.)
When a shaft and bearing slide together at sufficiently high
speed while immersed in fluid, the shaft can be supported
upon a film of fluid of such thickness that no contact occurs
between the passing solid bodies. However, during starting
and stopping, or during periods of overload or at high temperature when fluid viscosity is reduced, the supporting fluid
film becomes too thin to maintain separation between the
sliding members.
The simplest expression for the point at which a fluid film
becomes inadequate is the point at which "asperities" in the
opposing surfaces "touch" each other. It is further supposed
in t~le simplest view, that when these asperities "touch" each
other they will bond together, a condition known as "adhesion". This concept is so compelling that authors repeat it
with little critical comment. The evidence for this concept
seems simple enough: inadequately lubricated_ parts "weld
together". In ~eality, adhesion in the sense of primary bonding cannot be implicated as the initiator of scuffing: there are
o±er and parallel causes.
Scuff resistance of sliding pairs requires, some if not all of
the following.
1. The existence of a soft solid coating on hard bodies to
l~roduce low friction (which will accommodate a high
normal stress with low heating). A viscous substance
might also function well under some conditions except
that it would be squeezed out of the contact region upon
standing still.
2. The rate of loss of film (from at least one of the surfaces)
should not exceed the rate of gain over a long period of
time.
3. Given that there is no known solid that will indefinitely
resist removal by sliding, a method is required to renew
the film.
4. Film renewal should preferentially occur in the microscopic regions in which loss is taking place.
It appears that the only reasonable method of film renewal is
by passing a chemically active fluid over a "receptive" solid
surface. Film renewal by this method has been well studied,
but macroseopically and phenomenologicaUy,without much
attention to micro-scale events. The desired theological properties of films is not known, nor are the mechanisms of incipient scuff healing and other vital details.
6.1. The wGrdmodel
Assume a single asperity within a very large "conjunction" region as shown in Fig. 7(a). It is immersed in chemically active fluid when the surfaces are wide apart. When
a.
S
~
fluid |ira1
,,,,,--,- conlu~
,
roolon,-,,-,~,
,,.,-- ~ ] u ~ t o n region--.--~,
Fig. '7. Anasperityin a conjunctionreOon, near an opposing,~urf~.
newly manufactured, the asperity of interest has a shape and
has near neighbors whose overall profiles are described in
terms of surface roughness. Its surface is of different composition than the substrate, perhaps it is covered with an
oxide, or sulfide, or perhaps its surface is dcpletc~l of some
original constituents.
As the conjunction region around the targeted asperity
comes near to the opposing surface, load is transferred from
one to the other through a thinning fluid film. Heat is generated in the lubricant, in the surface coating and in the substrate. The asperity is likely also su'ained, perhaps plastically,
changing its shape. A soft coating forms from the chemical
species in the fluid film (shown in Fig.7(b)), only to be
partially removed by sliding contact. The asperity, however
is intermittently "out of contact", then supported by a very
thin film of fluid which contains a different concentration of
active chemical species than in the general conjunction
region. The rate of growth of soft film on the asperity will
depend on the surface temperature of the asperity, thcamount
of plastic flow of the asperity, the concentration of active
species in the fluid, and the rate of diffusion of these active
species toward the asperity substrate. The modeling exercise
should be directed toward such specific end goals as specifying the optimum film thickness and properties, the optimum
surface topography, the optimum break-in process (artificial
or otherwise), optimum start-up and stopping routines, optimum control on temporary overload, etc.
It may be seen that the building of a functional model
requires many steps, involving all disciplines that had previously made contributions to the prevention ofscufling. The
disciplines and their potential contributions is suggested.
6,1.1. Hydrodynamics
The calculation of fluid film thickness is a major contribution to designing of successful high speed machinery, but
there is nothing in the art of hydrodynamics whereby the
effective lower limit of fluid film thickness may be estimated,
particularly in the presence of chemically active lubricants.
However the calculation of the flow rate and temperature of
fluid passing through the microscopically thin conjunction
(carrying active chemical substances) woold be a majorcon-
6
IC.1.~ltmaI Wear200 (1996)!-7
tribution toward development of a functional scuffprevention
model.
6.1.2. Solid mechanics
T~.e ~:.~lidmechanics community subscribes to the hypothesis that scuffing will occur when two asperities contact each
o,her, Some also suggest that adhesion is not operative until
a particular amount of plastic deformation has occurred in
contacting asperities. This concept is embedded in the "plasticity index", which is an expression for the relative ease with
which asperities plastically deform when contacted by
another asperity. Asperity slope as well as height are seen as
important. There are no definite sta:ements to explain why a
particular amount c~"plastic flow is required to cause (or
allow) adhesion, though some suggest that surface coatings
(oxides perhaps ) are broken up or flaked off by plastic flow,
leaving a nascent surface (exposed atoms) that will metallurgicaliy oond together. Inherent in this concept is an
assumptionthat oxides are (absolutely) brittle and that when
"flaked off" the materials disappear from the system. Some
aspects of this idea may indeed be valid, but this simple
expression is not realistic: the asperities with and without
oxide or other coatings operate in a chemical environment
that makes the simple distinction~oxide-coated versus oxide.
free surface, meaningless.
The potential contribution from solid mechanics to scuff
modeling would be a calculation of the distribution of contact
points within a conjunction, the distribution of strain and
temperature in asperities and in coatings, the change in shape
(topography) of surfaces, and the load carrying capacity of
the composite interface region.
6.1.3. Lubricant chemistry
Lubricant formulation has been a third major contributor
to the successful design of sliding elements in machinery.
Over a century ago engineers knew that lubricants varied
considerably in their effectiveness according to their geographic origin and chemical treatment, in the earliest view,
polar molecules were thought to adsorb to two metal surfaces,
thereby decre_a~ingthe ~c~l~aceenergy o~ each and reducing
their tendency for adhesion. More recent research showed
th~.t the differences between lubricants lay in their different
,~bemical make-up, and this finding led to the formulation of
a vast array of very sophisticated lubricant products.
Lubricant chemists and chemical engineers developed
"additives" largely by two methods: by tracing the changes
in chemical composition during chemical reactions, in the
presence of oxygen and metals; and by testing candidate
commercial lubricants in test machinery. The test protocols
include the use of practical machinery (engines forexample)
and such bench tests as the four-ball tester, the block-oncylinder and the pin-on-disk. Though Lhe results of bench
tests never assuredly correlated with the performance of
engines (and other machinery), the lubricant specialists
developed sufficient skill to interpret the results properly.
Tf~epotential contribution of chemists to functional modeling of scuffing would be in characterizing the (solid) compounds that form on sliding surfaces from the constituents in
the lubricant, the rote of formation of films, their thicknesses,
their strength of bonding to a solid surface, and the mechanical properties (resistance to being wiped off).
6.1.4. The materials community
Physicists and melallurgi~ts showed many decades ago that
clean metals will adhere together by contact alone, without
heating. They also showed that a monolayer of adsorbed
substances significantly reduces this adhesion.
The role of the materials community in the development
of long lived sliding surfaces has generally been to connect
the properties and (micro-) structure of sliding solids with
their resistance to the various forms of wear. They have developed heat treatments, surface treatments and surface coatings
for very many specific tribological needs. They have also
mapped surface damage, substmte deformation and the composition of transfer particles/layers on sliding surfaces with
a view to recommending better materials for various applications. They have a valuable array of tools available to
observe and characterize material to a scale of a few atomic
dimensions.
However, though they have the tools for very detailed
analysis of surfaces, they had not until recently studied the
chemical conversion coatings (boundary lubricants) on surfaces. In fact, when specimens are submitted for analysis it
is their practice to rigorously clean offthe surfaces m prevent
contaminating their instruments, thereby removing the scuff
resisting coatings that a~sori~ chemically from the lubricant! However, recently, the art of "nano-tribology" has
developed, arising from studies of the frictional behavior of
computer hard disks. "He,no-instruments" ar~ now used to
characterize the adsorbed substances on surfaces.
The potential contributions of the materials community to
functional modeling of scuffing could be to:
1. Develop ways to characterize the composition and
mechanical properties of chemical conversion coatings
(boundary lubricants) on sliding surfaces, preferably in
real time;
2. Assist in formulating solids (monolithic or coated) that
interact with lubricants to form effective surface coatings.
6.1.5. The system dynamics community
The task is far from complete when all assumed variables
and relationships between them have been hypothesized. The
next step is to estimate the validity of the assumptions. This
i~ done by obtaining data from the system under study. These
data will inevitably include some distortion of the actual data
by the instrumentation. In the methods of system identification, used in system dynamic analysis, both the model and
the data are digitized and compared. Disturting characteristics
of the data gathering instrumentation can then be separated
out, leaving more "real" datato be compared with themodel.
Where there is a mis-match, methods are available for adjust-
K. l~dema I Wear200 {i996) 1-7
ing the parameters in the model until an improved fit is
achieved with data. If the fit remains poor in some regions,
or if adjusting some parameters seems only to shift the location of mismatch between the model and data, new variables
can be incorporated into the model, perhaps variables that
had previously been set aside as having little relevance. Several iterations of this kind can produce acceptable and applicable models. A good model cannot be achieved by leaving
out any important variables in an initial system model, nor
e~tn it be achieved without comparison of the model with
data.
7. Conclusion
It clearly is easy to recommend changes in methods of
modeling of friction and wear but difficult to implement. The
7
benefits of interdisciplinary modeling in friction and wear are
as obvious as the benefits ofmedeUng the simplerphenomena
in our technology. Doubtless we will see such models appearing in our technical journals in the near future.
References
[ I ] H.C.Mengand K.C.Ludema,Wearmodelsand predictiveeclualions:
their form and content, Wear, I81-183 (1995) 443-457.
[2] J.F. Arehard,J. AppL Phys., 24 (1953) 981.
[3] Amer.See.foFTeeingand Mat~als,Affi'MbTP 1105,199I,preseated
at Tdboiogi=fl Modeling for Meelumical Designers Syrup., San
Francisco,CA,October1990.
[4] J.K.Lmxeastet,Proc. Roy. Soc, (London),273A (1963) 466.
[5l N.C.Welsh,Philo~. Trans. Roy. Soc. (London), 2, 2~7/t (1965) 51.
[6] K.C. Luckmaand D. Tabor, Ffi~on and viscoelasticpropmlie~of
polymericsolids,RubberChem.Technol., 41 (1965) 462--476.
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