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Surface friction and dynamic mechanical properties of polymers
1’01.. 2 (r()gS/5’,!
10X
SURFACE
FRICTION
AND DYNAMIC
PROPERTIES
MECHANICAL
OF POLYMERS
Results are presented of experiments on the lubricated sliding of metals on polymers over a range of speeds and temperatures. These results indicate a correlation between the frictional behavior of materials and their bulk mechanical properties. Support for the experimental correlations is presented in the form of a theory relating the coefficient of rolling friction to bulk mechanical properties. The general conclusions may bc expected to hold for metals as well as other materials. The theory may also be expected to apply to well lubricated sliding where shearing forces have been minimized. lynder the conditions of lubrication most commonly encountered, the sliding friction is expected to be much more c<,rnplicated; both the shear propertics of tllr bountlary layer ant1 thr hysteresis chamctrristics will be important.
OBERFLtiCHENRElBUNG
TND
MECHANISCHE
DYNAMISCH
EIGENSCHAFTEN
VON
POLYMEREN
Messungen des geschmierten Gleitens van Metallen auf Polymeren dber einen Bereich \-on Geschwindigkeiten und van Temperaturen werden beschrieben. Die Ergebnisse weisen auf cinc Beziehung ewischen dem Reibungsverhalten van Material und den mechanischen Eigenschaftcn der Masse. Der experimentell gefundene Zusammenhang wird durch eine Theorie gestiitzt, die den Reibungskoeffizient der rollenden Reibung auf die mechanischen Eigenschaften der Masse bezieht. Erwartet wird, dass die allgemeinen Schlussfolgerungen sowohl fiir Metalle als such fiir andere Material& gelten. Giiltigkeit der Theorie kann fiir gleitende Reibung bei guter Schmierung, wenn die Schubkrgfte minimal geworden sind, erwartet werden. Dagegen wird untcr den Bedingungen van Schmierung wie diese meistens angetroffen wird, die gleitende Reibung vie1 komplizierter scin; Schubspannungen der Greuzschicht und Hysteresis Erscheinungcn wvc~rtlcntlann wichtl?.
The work of TABOR has provided are important friction1-5.
for determining
His work encouraged
We have found (polymethyl
of rolling
of the friction
and on polyethylene
these materials. In the lubricated
and lubricated
sliding
in more detail.
of steel sliding on Plexiglas
that elastic losses are important
for
sliding of steel on Plexiglas, for example, the curves
and speed at a series of temperatures
relating dissipation I
that elastic hysteresis losses in rubber
us to explore this phenomenon
from measurements
methacrylate)
relating friction
evidence
the coefficients
factor and stress frequency.
parallel quite closely
The dissipation
those
factor, in turn, has
v*L. 2 6958159)
DYNAMIC
MECHANICAL
PROPERTIES
OF POLYMERS
16g
been shown in other work to go hand in hand with mechanical losses in the polymer. That elastic losses in polymers are highly frequency- and temperature-dependent is well known. Furthermore,
in some cases, these losses can be related in a known
manner to molecular properties. It should be possible, then, to develop a relationship between dynamic mechanical, and therefore molecular, properties and rolling friction in those cases where bulk deformation is predominant. This point of view leads one to anticipate a relationship between these bulk properties and lubricated friction in cases where shearing forces have been minimized.
sliding
EXPERIMENTAL
Friction measurements were made by sliding a hemisphe~cally-ended steel rider (0.150 in. dia.), under a load of 108 g, against the polymer, or other sample, under study. The latter consisted of a sleeve, 1/4 in. thick and z in. in outer diameter, slipped over an inner ceramic cylinder enabling thermal insulation during runs at elevated temperatures. The frictional force was measured with the aid of strain gages attached to the rider holder and was continuously recorded. Details of the apparatus have been given previouslye. Preliminary experiments on neoprene lubricated with aqueous sodium stearate indicated the possible importance of bulk properties. The change in friction with speed (Fig. I) exhibited a behavior similar to what one might expect for a loss ‘us. frequency curve. This led us to investigate several other materials and to consider the theoretical aspects. _ Cl2Or
,”
s
t
I
SLlDlNG
Fig. I. Friction
100
10
I
of steel on neoprene
lubricated
SPEED
J
1000
(cm/w)
with sodium stearate.
Normal load = 108 g.
STEEL SLIDING ON PLEXIGLAS
For steel sliding on dry, freshly-machined Plexiglas, inflections in the friction VS. speed curve were obtained (Fig. z). They probably arose from a combination of References p.
182
.\. nr. HI’bxHE,
170
1). (i.
I;LoM
VOL.
2 (1C)jl';/.j(l)
shearing and elastic loss terms. Changing track conditions and past stress history resulted in a lower precision (IO%),
especially at low speeds, than would have been
desired. .4 marked effect of stress history on mechanical losses in polymers has been noted
by
~~~
L
fo-’
L
IO-'
I
SLIDtNG
Fig.
z. Friction
of steel sliding
,
SPEED
on Plexiglas,
,
10 (cmlsecf
101
unlubricated.
Normal
, 103 load
=. 108 g.
The frictional force was found to vary directly with load over the narrow load range of roe-zoo g. Subsequent measurements at much higher loads (::* IOOOg) indicated possible departure from this behavior; consequently, frictional force rather than ,& is plotted in Fig. 2 and in subsequent friction curves. When lubricated with an aqueous solution containing approximately 35% by weight sodium stearate, the friction of steel on Plexiglas as a function of speed was markedly different from that of unlubricated sliding (Fig. 3). The precision was also somewhat greater. The striking result of a change in slope at about 60°C and the greater friction at 78°C as opposed to that at 105°C should be noted. The significance of these features will become more apparent following a discussion of the dynamic bulk properties of Plexiglas. DYNAMIC
MECHANICAL
PROPERTIES
OF PLEXIGLAS
Unfortunately, the measurements of dynamic moduli and elastic losses reported in the literature have been generally made at stress frequencies too low for direct comparison with the results of sliding experiments. A conservative estimate of the frequency of deformation corresponding to a given sliding speed can be made if the Hejerences
p. 182
VOL. 2
(1958/59)
DYNAMIC MECHANICAL PROPERTIES OF POLY’MERS
171
period of cyclic stress is considered as the time required for the rider to move a distance d, where d is the diameter of the apparent area of contact. The value of d is obtainable from the width of the track resulting from initial plastic flow, and in the present experiments this width was about 0.4 mm, or roughly T/IO of the diameter of the rider hemisphere. The resultant conversion factor was therefore I cm/set = 2.5 cycles/set. ~easurement6
of the dynamic mechanical properties of ~lymethyl
methacr~~late
have been reported for frequencies up to xoo cycles/secapsand in one instance up to 2000 cycles/seclO. Measurements
on other polymers at slightly
higher frequencies
have been reported’. Although the frequency ranges for the polymethyl
methacrylate
damping curves reported in the literature and the friction curves from Fig, 3 overlap only slightly,
changes from negative
to positive
slopes appear at 60-70” in both
instances, and the general shapes of the overlapping
I 102
I
I
10
I
SLIDING
SPEED
curves are similar.
fcmfsecl
Fig. 3, Friction of steel sliding on Plexiglas, lubricated with sodium stearate. Normal load = 108 g.
Fortunately,
dielectric losses for polymers can be measured to rather high frequen-
cies. Furthermore, it has been shown that the mechanical and the dielectric losses exhibit similar maxima although the frequencies and temperatures may be differentlltl2. Replotting
the dielectric
loss data of TELFAIR l3 shows that the tan &frequency
curves (Fig. 4) are very similar to the friction-speed (Fig. 3). Superim~sing a log-log
curves obtained in our work
TELFAIR’S data and our friction data over their common ranges on
scale give a better idea of the fit between the two sets of measurements
(Fig. 5). We see from this figure that a sliding speed of I cm/set corresponds to a frequency
of about 18 cycles/set, in fair agreement with the conversion
25 calculated eaxlier on the basis of measured track widths. References p. 182
factor of
.\.
Ijl
.I
31.
ti~‘lr(‘HJi,
I).
G.
1’1.o.v
2
1’01
(lcyjH/jcf
4
.I2
80” 750 700
10-c
Fig.
I
IO
FREQUENCY
( kc /set)
IO-'
4. IXssipation
factor
(tan 6) of polytnethyl
SLlDlW 40 ,-
4 II
5
7
1
IO
20 I
I
102 methncr$ate
IO’ vs. freq~t~n~y.
SPEED (cmhec.) 50
30 III
70
I
100
200
I
I
30
300
I
.I5
789c DS°C _
25
,I0
-8o’C
G
k35T -
-09 to
w v20
.08
8 LL
.07
2
06
tb 2
-I z” a L
45oc
I5
-
* F
46.X .05 $
,/ //
E
25°C 04
IO FRICTION ~ DISSIPATION ----
‘\
25’C
03
, 0.1
I
0.2
Pig. 5. Steel on Plexiglas, Rrfevcnces p. r&k
1
I
1
,
0.4 FREOUENCY
lubricated (-)
I (kc /SW)
I
,
I,
I
2
-3
45
7
and dissipatiun factor of Plexiglas (- - -),
it z 5
J
yo=. 2 (1958~59)
DYNAMIC
MECHANICAL
PROPERTIES
OF POLYMERS
173
STEEL SLIDING ON POLYETHYLENE
In addition to the correlation between the friction and dynamic mechanical properties found for Plexiglas, a brief study. of the effect of branching on the friction of polyethylene indicated a similar correlation for that polymer. The friction experiments were carried out in a manner similar to those for Plexiglas. Steel was made to slide on Alathon IO {branched polyethylene) and on Marlex jo (unbranched polyethylene), under both lubricated and unlubricated conditions. The results are shown in Figs. 6 and 7. The precision was roughly -J-SO/~with the exception of lubricated steel on Alathon IO where reproducibility was unaccouniably lower (1920%).
‘OOg
I
w
G p LL
ALATHON
IO AT 105°C
ALATHON
IO AT 25*C
20 51
‘*
%
MARLEX
50
AT 25*
C I
2
3
45
IO
20 SLIDING
40 SPEED
100
200
I
J
400
(cm/set)
Fig. 6. Friction of steel on Alathon 10 and on Marlex 50. No lubrication. Normal load =
0 A 0 0
5750 w B
ALATHON MARLEX ALATHON MARLEX
IO AT SO AT IO AT 50 AT
108 g.
#‘C IO5*C 25V 25’C
0
L
2
3
4s
IO
SLIDING
20 SPEED
40 km/set
1
100
200
4
0
Fig. 7. Friction of steel on Alathon IO and on Marlex 50. both systems lubricated with aqueous sodium stearate. Normal load = 108 g. References p. 182
.\.
‘74
DYNAMIC KLINE,
SAUER
mechanical above,
a decrease
polymer IOOO
measured
OF POLYETHYLENE
the eftect of branching
was accompanied
by a marked
Translated
that the friction
depended
as well as on their structure
roughly
of steel on unbranched
polyethylene
SHEARING
It should
not be concluded
that shearing stearate.
and possibly
of steel-polymer
other lubricants
There
in addition
values of friction known behavior the features
negligible.
that
properties
between
sacrificed
in an attempt
physical
properties
calculation
et ul.
found for Plexiglas
and poly-
from those systems
lubricated
some shearing
in the lubricant
to sodium stearate,
film
we found that
such as glycerol and cetane, gave low
the results
appeared
to be independent
On the other hand it is difficult in other systems
to believe from the
that it could give rise to all of
theory
the simplest
led us to attempt the observed
between
friction
the formulation
quantities.
to preserve the simplicity
was assumed
was neglected.
represent
correlation
and dynamic
would be found in the case of rolling, where shearing forces are
This consideration
ical relationships
elasticity
KLINE
in Fig. 3 which have just been discussed.
One would expect mechanical
results of
as well. For steel on Plexiglas
but for those lubricants of sodium stearate
tests, this would
from Figs. 0 and 7
comparison.
absent
was undoubtedly
of sliding speed and temperature.
to
was lower than that for steel
mechanical
from the correlations
junctions
200
FORCES
forces were completely
with sodium
of the
varied from
It is apparent
in line with the dynamic
aud
in damping.
on the geometry
into terms of sliding speed in our friction
even though their data do not allow a more detailed
ethylene
decrease
but in general
to a range of H to 40 cmjsec.
polyethylene,
on the dynamic
14. They found that at room temperature
of test used in their experiments
cycles/set.
on branched
I’KOI’EKTIES
of polyethylene
in branching
specimens
correspond
MECHANICAL
\'()I.. 2 (I()SS/.jC))
I). G. I’LOM
HlrE(‘HLS.
AND WOODWAKD
properties
The frequency
>I.
for the material
important
amount
of the calculation. involved;
We feel, however,
the most
of simple mathemat-
A certain
that
fundamental
It is hoped that the results will aid in the selection
of rigor was
An idealized set of
furthermore,
detailed
the model and method aspects
of
of the phenomena.
and interpretation
of new experi-
ments. PRELIMINARY
THEORY
OF ROLLING FRICTION
Consider a sphere with radius a and center base material
whose upper surface
at x = o, y = o, z := - -20 rolling
is bounded
assume that the sphere is very hard compared that the sphere is not deformed
appreciably. 9 +
and it intersects
the base material
y” +
by the x-y
with the other
The equation
(2 + zg)Z =
in the x-y
Heferences
p. 182
material
involved
so
of the sphere is then
a2
plane in a circle of contact
x2 + y" = a2-_20‘J = 12.
on a
plane (Fig. 8). We shall
(I)
described
by (2)
VOL. 2 (I958/59)
DYNAMIC
MECHANICAL
PROPERTIES
OF POLYMERS
I75
Now for each value of x and y within the circle defined by eqn. (2) the surface of the sphere will have a positive value of z, i.e., the base material will be indented. The depth of the indentation,
assuming the latter to be a spherical segment, is given by
# = (a2- xz - y2)1/2 - ,Q = @2_$a
- y2)1/2-(a2 - 12)1/Z
(3)
in the static case; the dynamic case will be discussed later. This differs somewhat from the detailed indentation pressure distribution15.
shape given by HERTZ
who starts with a different
+2 Fig. 8. Idealized deformation by hard sphere.
of base material
Fig, 9. Mechanical model for material having retarded elasticity.
To proceed with our analysis we need to define the physical properties of the base material. For illustrative purposes we will assume that the material properties are represented by the retarded elastic model shown in Fig. 9. In this case the spring, having a modulus constant G, and the dashpot, having a viscosity 7, both resist deformation in the z direction. The pressure, 9, in the z direction is given by
where, it will be noticed, the first term depends on the magnitude of deformation and the second term on the rate of deformation. Since the experiment does not involve a simple shear or tensile experiment, the constant k, having dimensions of a reciprocal length, has been introduced. A value will be obtained for it later. Our assumed material can be considered to have a single characteristic retardation time given by z = q/G
(5)
Few, if any, real materials have such a simple set of physical constants. Most will References p. 182
liavc a series of retardation plastic
deformation.
as found by
times and spring constants
Some phenomena
will require
When making detailed
FITZ~~H~~IP.
in addition
comparisons
proper set of models will riced to be carried through
mass
with experiments
tllc*
the following analysis.
WC will now consider our base material
to be traveling
to the sphere, and the sphere to be rotating
clockwise.
of a point in contact
to a permanent
all of these plus an inertial
in the
.Ydirection rclativc‘
The velocity
in thus z tlirection
with our sphere will be d.r
-“clt
dz
t1t
(a2
__
X2 __
YS I”)‘/”
--
(‘)I
(,2_.;~-:.-,2)1/2
where s is the sliding or rolling speed of the center of the sphere with respect
to the
base material. The pressure is found from eqns. (x), (4) and (6) and is given by p,/& = @a _ ,$ _ y”)l,n ~~.($ _ I”)‘,2 + tsx(az _~ ,r2 .-_ >‘“)-I/“, In the static
case the last term will be zero.
For examples
having no adhesion between the sphere and the base material
will not represent
the pressure
exerted
This fact arises because of the retarded of adhesion material
(7)
a negative
between elasticity
value of $ cannot
will lose contact
the base material of our base material;
be applied
with it at some ncgativc
eqn. (7)
and the sphere. in the absence
to the sphere
and the base
value of X. This is illustrated
in
Fig. 10.
A
B
Pig. IO. Sphere rolling on base material:
Example
A represents
enon at higher
sliding
the case for small valuesof speeds, i.e. larger
for x and y for which this separation zero. Denoting
A plot of the loci of the p := o points References
p. 182
L3
t.s while B represents
ts. Mathematically
occurs by equating
large TS
the phenom-
we can find the values
the pressure
in eqn. (7) to value
becomes
,,,,a2__(,-~)
II.
small TS; Example
the value of x at which this occurs by X and the corresponding
of 22 + j? by 9 the expression
Fig.
Example A -
+ (r_y2
(l_y2
for a few values of zs, a and 1 are shown in
VOL. 2 (x958/59)
DYNAMIC
MECHANICAL
PROPERTIES
OF POLYMERS
I77
Fig. II. Limits of area. of contact for a rolling sphere having a radius of I cm, L = 0.436 cm and TS = o, I/IO, 113 and I.
We will now set out to calculate the total force, W, in the z direction by integrating p over the area of contact. Because of the rather involved limits of integration required by eqn. (8) for the exact case we will investigate only two limiting cases.
I. Small
zs
This case will usually correspond to very low rolling speeds or to materials having low values of 7. Because of the detailed nature of the approximation, however, it will also apply to the unlikely case where the material adheres to the sphere for all positive values of z but suddenly loses adhesion when 2 = o. In these cases the upward value of the viscous force, contributed by the third term of eqn. 7, on the +x side will be just balanced by the downward force on the -x side and the integration over it will be equal to zero. The integrals are y=l WjkG = 4
Jy=0
x
=
(22 -
f x=0
y2) l/2
(PlkG)d&
(9)
since the terms that are left are symmetrical in x2 and y2. Writing this in terms of r2 = x2 + ya and 0, where x = r cos 8, we have References p. 182
A. M. HIFXHE,
n/z W/KG = 4
=f
s 0
i~O1..2 (r958/59)
ROM
G.
1 { (aZ -
i’ 0
2n
1).
[f
,2)1/Z -
Z2)112) r&Cl0 =
ZZ{,‘J - 22)1 /z __~__..__.
._!I”‘_-
-
(a2 -
2
3
Since experimentally we will normally be interested expand eqn. (I r) to find
I
(IQ)
(1 1)
in cases where &a < T we can
We are now in a position to estimate the constant k by comparison with theresults of the detailed elastic calculation of HERTZ. He found for the static case that l-022 I-f71 2 -f-------
3w -4 13
! H
El
E2
>
=-L+l al
~2
(i3)
where the o’s and E’s are the Poisson ratios and Young’s moduli of the two bodies in contact. In our case E, is vory large compared with E, and our base material is a plane. If we set E
G=-
2 + 20
we have from eqn. (13) w=
8
(Ii-4
3 (I - 02)
a3 a
Comparison of eqns. (12) and (14) shows that k
1+u ---WV sv2
3n (l-&f
I
I
4.85 I
(15)
if a 1 = 0.3. Therefore we find that k has the the dimensions of a reciprocal length as required by eqn. (4). The coefficients of friction to be calculated later are insensitive to the value of k; on the other hand, it does enter the expressions relating 1 to IV. Next we would like to calculate the frictional force and the coefficient of friction. This can be done without calculating the energy dissipation as such, by recognizing that the base material exerts a torque on the sphere. The pressure, while symmetrical in y, is not equal in the +x and -z directions. This inequality produces a moment of forces tending to oppose the steady rolling of the sphere. The moment, AI, is given by M/kG = Referettces
p. 182
(p/kG)xclA
(16)
VOL. 2 P958/59)
DYNAMIC
MECHANICAL
PROPERTIES
OF POLYMERS
179
Substituting n/2
Mlkrjs = 4
s 0
1
= (n/3) [2aS +
Expanding
Y3 co@ I3
&de =
(a2-r2)112
I 0
(aaZ2)3/2
-
3a2(a2 -
Z2)1/2]
(17)
for l/a 4 I we have (18)
For steady rolling this torque must be opposed by that due to the frictional acting at a distance a, i.e. F = M/a = 2
kqs f
+ ”
4
The coefficient
of friction,
16 yckrp krp a4 + ___ 64
I2
18 -
force, F,
(19)
aa
1, is then (20)
This surprisingly simple result shows that, to this approximation, the coefficient of rolling friction is a linear function of the sliding speed and depends on the radius of the sphere. It should be possible, then, to estimate the characteristic retardation time of the base material by making friction measurements at low speeds and low loads. If we set s = mav where v is the frequency of rotation of the rolling sphere we find I = 2ntv
This result
2.
Large
(21)
may be more useful in some instances.
values
of zs
In this case we will consider example B in Fig. IO. We will assume that conditions are such that the base material loses contact with the sphere at x = o and that all of the energy used in deforming the sample is lost. Starting with eqn. (7) for the pressure we retain all three terms and integrate as in eqn. (9) over the area of contact for +x values only. In this way we find W/zkG = F
+ References p. 182
-
n(a2 -
n12(a2 -
Z2)3/2 6
-
4
4 [ azsin-1 (+I -l(az-12)1/z]
Z2)lJ2
+
(24
Expanding as before WC:have
The moment, df, is found by the procedure of cqn. (16) subject to the above restrictions on the limits of integration. It is given by ,1/f/&(;
z
lja2 ____.
-
t‘L)3/” _ -1
+
&(&.-.-~K_.-
q
WA +
a* sin --.i_
-l(&z)
_
4
Expanding we have (2s) Using the definitions of cqns. (r(j) and (20) we now have
For zs + 1 this becomes approximately
But l/a is dependent on rs by virtue of eqn. (23).Subject to the same approximations our coefficient of friction becomes
It appears that, in this approximation,
the coefficient of friction is load-dependent
and has a magnitude depending on G as well as z. Our eqns. (21) and (28) are plotted in Fig. 12 using a = I cm, IV = IOO g and G = 106 g/cm2. Only the solid lines are to be taken seriously. The dotted lines are drawn to indicate the trends and are certainly not representative of the shape near the maximum of the curve. It appears to be reasonable to expect, however, that the effect of a single retardation time is felt even at very low frequencies of rolling. Furthermore, the maximum in the friction will probably appear at a value of WC I. To find the shape of the entire curve it appears to be necessary to carry out the calculations using the limits of integration as defined by eqn. (8). 12+3Wacesp.
I82
VOL. 2 (~958~59)
DYNAMIC
MECHANICAL
PROPERTIES
OF POLYMERS
181
From this analysis of the problem we conclude that there exists a relatively simple relation between rolling friction and the bulk physical properties of the materials involved. The coefficient of friction begins to sense the presence of a characteristic frequency of the material long before the rolling frequency reaches the point where the maximum friction is observed.
: :
I :
.Ol t
/’
Fig. 12. Plot of equations g/cma.
I \ \ \
\ \\ \\
\\ \\
(21) and (28) (solid lines) using A =
~cm,
W =
IOO g and G =
106
More detailed treatments of this phenomenon seem to promise a new method for the precise dete~ination of the mecha~cal properties of materials. Apparently, even at this stage, the theory could be used to find approximate values of the characteristic frequencies of metals, ceramics and polymers. DISCUSSION
OF RESULTS
It is apparent from the foregoing analysis that the relatio~~p between rolling friction and speed, for a hard sphere of constant diameter on a softer material of given retardation time r, resembles closely the relationship between mechanical loss and frequency. Extending the analysis for rolling to wall-lub~cated sliding, then, it follows that the slope of the friction vs. speed curve at a given temperature, as in Fig. 3, will depend upon a shift with temperature of the maximum in the friction vs. rsjn relationship. Should these relationships prove generally applicable to #olymers at low and intermediate temperatures, they might also be expected to apply to the friction of metals at elevated temperatures where internal friction is known to be greatly increasedla. Modifications in the theoretical model used to represent the bulk properties (Fig. g) References p.
182
would very likely be necessitated by basic differences in the structures of metals and polymers. Finally, one might expect thv clywnic mechanical properties to impose an ultimate lower limit on the friction in systvtns where shearing forws had been ciiminated by cffrctivc
hountlar\- Inbric;~tion.
10X
SURFACE
FRICTION
AND DYNAMIC
PROPERTIES
MECHANICAL
OF POLYMERS
Results are presented of experiments on the lubricated sliding of metals on polymers over a range of speeds and temperatures. These results indicate a correlation between the frictional behavior of materials and their bulk mechanical properties. Support for the experimental correlations is presented in the form of a theory relating the coefficient of rolling friction to bulk mechanical properties. The general conclusions may bc expected to hold for metals as well as other materials. The theory may also be expected to apply to well lubricated sliding where shearing forces have been minimized. lynder the conditions of lubrication most commonly encountered, the sliding friction is expected to be much more c<,rnplicated; both the shear propertics of tllr bountlary layer ant1 thr hysteresis chamctrristics will be important.
OBERFLtiCHENRElBUNG
TND
MECHANISCHE
DYNAMISCH
EIGENSCHAFTEN
VON
POLYMEREN
Messungen des geschmierten Gleitens van Metallen auf Polymeren dber einen Bereich \-on Geschwindigkeiten und van Temperaturen werden beschrieben. Die Ergebnisse weisen auf cinc Beziehung ewischen dem Reibungsverhalten van Material und den mechanischen Eigenschaftcn der Masse. Der experimentell gefundene Zusammenhang wird durch eine Theorie gestiitzt, die den Reibungskoeffizient der rollenden Reibung auf die mechanischen Eigenschaften der Masse bezieht. Erwartet wird, dass die allgemeinen Schlussfolgerungen sowohl fiir Metalle als such fiir andere Material& gelten. Giiltigkeit der Theorie kann fiir gleitende Reibung bei guter Schmierung, wenn die Schubkrgfte minimal geworden sind, erwartet werden. Dagegen wird untcr den Bedingungen van Schmierung wie diese meistens angetroffen wird, die gleitende Reibung vie1 komplizierter scin; Schubspannungen der Greuzschicht und Hysteresis Erscheinungcn wvc~rtlcntlann wichtl?.
The work of TABOR has provided are important friction1-5.
for determining
His work encouraged
We have found (polymethyl
of rolling
of the friction
and on polyethylene
these materials. In the lubricated
and lubricated
sliding
in more detail.
of steel sliding on Plexiglas
that elastic losses are important
for
sliding of steel on Plexiglas, for example, the curves
and speed at a series of temperatures
relating dissipation I
that elastic hysteresis losses in rubber
us to explore this phenomenon
from measurements
methacrylate)
relating friction
evidence
the coefficients
factor and stress frequency.
parallel quite closely
The dissipation
those
factor, in turn, has
v*L. 2 6958159)
DYNAMIC
MECHANICAL
PROPERTIES
OF POLYMERS
16g
been shown in other work to go hand in hand with mechanical losses in the polymer. That elastic losses in polymers are highly frequency- and temperature-dependent is well known. Furthermore,
in some cases, these losses can be related in a known
manner to molecular properties. It should be possible, then, to develop a relationship between dynamic mechanical, and therefore molecular, properties and rolling friction in those cases where bulk deformation is predominant. This point of view leads one to anticipate a relationship between these bulk properties and lubricated friction in cases where shearing forces have been minimized.
sliding
EXPERIMENTAL
Friction measurements were made by sliding a hemisphe~cally-ended steel rider (0.150 in. dia.), under a load of 108 g, against the polymer, or other sample, under study. The latter consisted of a sleeve, 1/4 in. thick and z in. in outer diameter, slipped over an inner ceramic cylinder enabling thermal insulation during runs at elevated temperatures. The frictional force was measured with the aid of strain gages attached to the rider holder and was continuously recorded. Details of the apparatus have been given previouslye. Preliminary experiments on neoprene lubricated with aqueous sodium stearate indicated the possible importance of bulk properties. The change in friction with speed (Fig. I) exhibited a behavior similar to what one might expect for a loss ‘us. frequency curve. This led us to investigate several other materials and to consider the theoretical aspects. _ Cl2Or
,”
s
t
I
SLlDlNG
Fig. I. Friction
100
10
I
of steel on neoprene
lubricated
SPEED
J
1000
(cm/w)
with sodium stearate.
Normal load = 108 g.
STEEL SLIDING ON PLEXIGLAS
For steel sliding on dry, freshly-machined Plexiglas, inflections in the friction VS. speed curve were obtained (Fig. z). They probably arose from a combination of References p.
182
.\. nr. HI’bxHE,
170
1). (i.
I;LoM
VOL.
2 (1C)jl';/.j(l)
shearing and elastic loss terms. Changing track conditions and past stress history resulted in a lower precision (IO%),
especially at low speeds, than would have been
desired. .4 marked effect of stress history on mechanical losses in polymers has been noted
by
~~~
L
fo-’
L
IO-'
I
SLIDtNG
Fig.
z. Friction
of steel sliding
,
SPEED
on Plexiglas,
,
10 (cmlsecf
101
unlubricated.
Normal
, 103 load
=. 108 g.
The frictional force was found to vary directly with load over the narrow load range of roe-zoo g. Subsequent measurements at much higher loads (::* IOOOg) indicated possible departure from this behavior; consequently, frictional force rather than ,& is plotted in Fig. 2 and in subsequent friction curves. When lubricated with an aqueous solution containing approximately 35% by weight sodium stearate, the friction of steel on Plexiglas as a function of speed was markedly different from that of unlubricated sliding (Fig. 3). The precision was also somewhat greater. The striking result of a change in slope at about 60°C and the greater friction at 78°C as opposed to that at 105°C should be noted. The significance of these features will become more apparent following a discussion of the dynamic bulk properties of Plexiglas. DYNAMIC
MECHANICAL
PROPERTIES
OF PLEXIGLAS
Unfortunately, the measurements of dynamic moduli and elastic losses reported in the literature have been generally made at stress frequencies too low for direct comparison with the results of sliding experiments. A conservative estimate of the frequency of deformation corresponding to a given sliding speed can be made if the Hejerences
p. 182
VOL. 2
(1958/59)
DYNAMIC MECHANICAL PROPERTIES OF POLY’MERS
171
period of cyclic stress is considered as the time required for the rider to move a distance d, where d is the diameter of the apparent area of contact. The value of d is obtainable from the width of the track resulting from initial plastic flow, and in the present experiments this width was about 0.4 mm, or roughly T/IO of the diameter of the rider hemisphere. The resultant conversion factor was therefore I cm/set = 2.5 cycles/set. ~easurement6
of the dynamic mechanical properties of ~lymethyl
methacr~~late
have been reported for frequencies up to xoo cycles/secapsand in one instance up to 2000 cycles/seclO. Measurements
on other polymers at slightly
higher frequencies
have been reported’. Although the frequency ranges for the polymethyl
methacrylate
damping curves reported in the literature and the friction curves from Fig, 3 overlap only slightly,
changes from negative
to positive
slopes appear at 60-70” in both
instances, and the general shapes of the overlapping
I 102
I
I
10
I
SLIDING
SPEED
curves are similar.
fcmfsecl
Fig. 3, Friction of steel sliding on Plexiglas, lubricated with sodium stearate. Normal load = 108 g.
Fortunately,
dielectric losses for polymers can be measured to rather high frequen-
cies. Furthermore, it has been shown that the mechanical and the dielectric losses exhibit similar maxima although the frequencies and temperatures may be differentlltl2. Replotting
the dielectric
loss data of TELFAIR l3 shows that the tan &frequency
curves (Fig. 4) are very similar to the friction-speed (Fig. 3). Superim~sing a log-log
curves obtained in our work
TELFAIR’S data and our friction data over their common ranges on
scale give a better idea of the fit between the two sets of measurements
(Fig. 5). We see from this figure that a sliding speed of I cm/set corresponds to a frequency
of about 18 cycles/set, in fair agreement with the conversion
25 calculated eaxlier on the basis of measured track widths. References p. 182
factor of
.\.
Ijl
.I
31.
ti~‘lr(‘HJi,
I).
G.
1’1.o.v
2
1’01
(lcyjH/jcf
4
.I2
80” 750 700
10-c
Fig.
I
IO
FREQUENCY
( kc /set)
IO-'
4. IXssipation
factor
(tan 6) of polytnethyl
SLlDlW 40 ,-
4 II
5
7
1
IO
20 I
I
102 methncr$ate
IO’ vs. freq~t~n~y.
SPEED (cmhec.) 50
30 III
70
I
100
200
I
I
30
300
I
.I5
789c DS°C _
25
,I0
-8o’C
G
k35T -
-09 to
w v20
.08
8 LL
.07
2
06
tb 2
-I z” a L
45oc
I5
-
* F
46.X .05 $
,/ //
E
25°C 04
IO FRICTION ~ DISSIPATION ----
‘\
25’C
03
, 0.1
I
0.2
Pig. 5. Steel on Plexiglas, Rrfevcnces p. r&k
1
I
1
,
0.4 FREOUENCY
lubricated (-)
I (kc /SW)
I
,
I,
I
2
-3
45
7
and dissipatiun factor of Plexiglas (- - -),
it z 5
J
yo=. 2 (1958~59)
DYNAMIC
MECHANICAL
PROPERTIES
OF POLYMERS
173
STEEL SLIDING ON POLYETHYLENE
In addition to the correlation between the friction and dynamic mechanical properties found for Plexiglas, a brief study. of the effect of branching on the friction of polyethylene indicated a similar correlation for that polymer. The friction experiments were carried out in a manner similar to those for Plexiglas. Steel was made to slide on Alathon IO {branched polyethylene) and on Marlex jo (unbranched polyethylene), under both lubricated and unlubricated conditions. The results are shown in Figs. 6 and 7. The precision was roughly -J-SO/~with the exception of lubricated steel on Alathon IO where reproducibility was unaccouniably lower (1920%).
‘OOg
I
w
G p LL
ALATHON
IO AT 105°C
ALATHON
IO AT 25*C
20 51
‘*
%
MARLEX
50
AT 25*
C I
2
3
45
IO
20 SLIDING
40 SPEED
100
200
I
J
400
(cm/set)
Fig. 6. Friction of steel on Alathon 10 and on Marlex 50. No lubrication. Normal load =
0 A 0 0
5750 w B
ALATHON MARLEX ALATHON MARLEX
IO AT SO AT IO AT 50 AT
108 g.
#‘C IO5*C 25V 25’C
0
L
2
3
4s
IO
SLIDING
20 SPEED
40 km/set
1
100
200
4
0
Fig. 7. Friction of steel on Alathon IO and on Marlex 50. both systems lubricated with aqueous sodium stearate. Normal load = 108 g. References p. 182
.\.
‘74
DYNAMIC KLINE,
SAUER
mechanical above,
a decrease
polymer IOOO
measured
OF POLYETHYLENE
the eftect of branching
was accompanied
by a marked
Translated
that the friction
depended
as well as on their structure
roughly
of steel on unbranched
polyethylene
SHEARING
It should
not be concluded
that shearing stearate.
and possibly
of steel-polymer
other lubricants
There
in addition
values of friction known behavior the features
negligible.
that
properties
between
sacrificed
in an attempt
physical
properties
calculation
et ul.
found for Plexiglas
and poly-
from those systems
lubricated
some shearing
in the lubricant
to sodium stearate,
film
we found that
such as glycerol and cetane, gave low
the results
appeared
to be independent
On the other hand it is difficult in other systems
to believe from the
that it could give rise to all of
theory
the simplest
led us to attempt the observed
between
friction
the formulation
quantities.
to preserve the simplicity
was assumed
was neglected.
represent
correlation
and dynamic
would be found in the case of rolling, where shearing forces are
This consideration
ical relationships
elasticity
KLINE
in Fig. 3 which have just been discussed.
One would expect mechanical
results of
as well. For steel on Plexiglas
but for those lubricants of sodium stearate
tests, this would
from Figs. 0 and 7
comparison.
absent
was undoubtedly
of sliding speed and temperature.
to
was lower than that for steel
mechanical
from the correlations
junctions
200
FORCES
forces were completely
with sodium
of the
varied from
It is apparent
in line with the dynamic
aud
in damping.
on the geometry
into terms of sliding speed in our friction
even though their data do not allow a more detailed
ethylene
decrease
but in general
to a range of H to 40 cmjsec.
polyethylene,
on the dynamic
14. They found that at room temperature
of test used in their experiments
cycles/set.
on branched
I’KOI’EKTIES
of polyethylene
in branching
specimens
correspond
MECHANICAL
\'()I.. 2 (I()SS/.jC))
I). G. I’LOM
HlrE(‘HLS.
AND WOODWAKD
properties
The frequency
>I.
for the material
important
amount
of the calculation. involved;
We feel, however,
the most
of simple mathemat-
A certain
that
fundamental
It is hoped that the results will aid in the selection
of rigor was
An idealized set of
furthermore,
detailed
the model and method aspects
of
of the phenomena.
and interpretation
of new experi-
ments. PRELIMINARY
THEORY
OF ROLLING FRICTION
Consider a sphere with radius a and center base material
whose upper surface
at x = o, y = o, z := - -20 rolling
is bounded
assume that the sphere is very hard compared that the sphere is not deformed
appreciably. 9 +
and it intersects
the base material
y” +
by the x-y
with the other
The equation
(2 + zg)Z =
in the x-y
Heferences
p. 182
material
involved
so
of the sphere is then
a2
plane in a circle of contact
x2 + y" = a2-_20‘J = 12.
on a
plane (Fig. 8). We shall
(I)
described
by (2)
VOL. 2 (I958/59)
DYNAMIC
MECHANICAL
PROPERTIES
OF POLYMERS
I75
Now for each value of x and y within the circle defined by eqn. (2) the surface of the sphere will have a positive value of z, i.e., the base material will be indented. The depth of the indentation,
assuming the latter to be a spherical segment, is given by
# = (a2- xz - y2)1/2 - ,Q = @2_$a
- y2)1/2-(a2 - 12)1/Z
(3)
in the static case; the dynamic case will be discussed later. This differs somewhat from the detailed indentation pressure distribution15.
shape given by HERTZ
who starts with a different
+2 Fig. 8. Idealized deformation by hard sphere.
of base material
Fig, 9. Mechanical model for material having retarded elasticity.
To proceed with our analysis we need to define the physical properties of the base material. For illustrative purposes we will assume that the material properties are represented by the retarded elastic model shown in Fig. 9. In this case the spring, having a modulus constant G, and the dashpot, having a viscosity 7, both resist deformation in the z direction. The pressure, 9, in the z direction is given by
where, it will be noticed, the first term depends on the magnitude of deformation and the second term on the rate of deformation. Since the experiment does not involve a simple shear or tensile experiment, the constant k, having dimensions of a reciprocal length, has been introduced. A value will be obtained for it later. Our assumed material can be considered to have a single characteristic retardation time given by z = q/G
(5)
Few, if any, real materials have such a simple set of physical constants. Most will References p. 182
liavc a series of retardation plastic
deformation.
as found by
times and spring constants
Some phenomena
will require
When making detailed
FITZ~~H~~IP.
in addition
comparisons
proper set of models will riced to be carried through
mass
with experiments
tllc*
the following analysis.
WC will now consider our base material
to be traveling
to the sphere, and the sphere to be rotating
clockwise.
of a point in contact
to a permanent
all of these plus an inertial
in the
.Ydirection rclativc‘
The velocity
in thus z tlirection
with our sphere will be d.r
-“clt
dz
t1t
(a2
__
X2 __
YS I”)‘/”
--
(‘)I
(,2_.;~-:.-,2)1/2
where s is the sliding or rolling speed of the center of the sphere with respect
to the
base material. The pressure is found from eqns. (x), (4) and (6) and is given by p,/& = @a _ ,$ _ y”)l,n ~~.($ _ I”)‘,2 + tsx(az _~ ,r2 .-_ >‘“)-I/“, In the static
case the last term will be zero.
For examples
having no adhesion between the sphere and the base material
will not represent
the pressure
exerted
This fact arises because of the retarded of adhesion material
(7)
a negative
between elasticity
value of $ cannot
will lose contact
the base material of our base material;
be applied
with it at some ncgativc
eqn. (7)
and the sphere. in the absence
to the sphere
and the base
value of X. This is illustrated
in
Fig. 10.
A
B
Pig. IO. Sphere rolling on base material:
Example
A represents
enon at higher
sliding
the case for small valuesof speeds, i.e. larger
for x and y for which this separation zero. Denoting
A plot of the loci of the p := o points References
p. 182
L3
t.s while B represents
ts. Mathematically
occurs by equating
large TS
the phenom-
we can find the values
the pressure
in eqn. (7) to value
becomes
,,,,a2__(,-~)
II.
small TS; Example
the value of x at which this occurs by X and the corresponding
of 22 + j? by 9 the expression
Fig.
Example A -
+ (r_y2
(l_y2
for a few values of zs, a and 1 are shown in
VOL. 2 (x958/59)
DYNAMIC
MECHANICAL
PROPERTIES
OF POLYMERS
I77
Fig. II. Limits of area. of contact for a rolling sphere having a radius of I cm, L = 0.436 cm and TS = o, I/IO, 113 and I.
We will now set out to calculate the total force, W, in the z direction by integrating p over the area of contact. Because of the rather involved limits of integration required by eqn. (8) for the exact case we will investigate only two limiting cases.
I. Small
zs
This case will usually correspond to very low rolling speeds or to materials having low values of 7. Because of the detailed nature of the approximation, however, it will also apply to the unlikely case where the material adheres to the sphere for all positive values of z but suddenly loses adhesion when 2 = o. In these cases the upward value of the viscous force, contributed by the third term of eqn. 7, on the +x side will be just balanced by the downward force on the -x side and the integration over it will be equal to zero. The integrals are y=l WjkG = 4
Jy=0
x
=
(22 -
f x=0
y2) l/2
(PlkG)d&
(9)
since the terms that are left are symmetrical in x2 and y2. Writing this in terms of r2 = x2 + ya and 0, where x = r cos 8, we have References p. 182
A. M. HIFXHE,
n/z W/KG = 4
=f
s 0
i~O1..2 (r958/59)
ROM
G.
1 { (aZ -
i’ 0
2n
1).
[f
,2)1/Z -
Z2)112) r&Cl0 =
ZZ{,‘J - 22)1 /z __~__..__.
._!I”‘_-
-
(a2 -
2
3
Since experimentally we will normally be interested expand eqn. (I r) to find
I
(IQ)
(1 1)
in cases where &a < T we can
We are now in a position to estimate the constant k by comparison with theresults of the detailed elastic calculation of HERTZ. He found for the static case that l-022 I-f71 2 -f-------
3w -4 13
! H
El
E2
>
=-L+l al
~2
(i3)
where the o’s and E’s are the Poisson ratios and Young’s moduli of the two bodies in contact. In our case E, is vory large compared with E, and our base material is a plane. If we set E
G=-
2 + 20
we have from eqn. (13) w=
8
(Ii-4
3 (I - 02)
a3 a
Comparison of eqns. (12) and (14) shows that k
1+u ---WV sv2
3n (l-&f
I
I
4.85 I
(15)
if a 1 = 0.3. Therefore we find that k has the the dimensions of a reciprocal length as required by eqn. (4). The coefficients of friction to be calculated later are insensitive to the value of k; on the other hand, it does enter the expressions relating 1 to IV. Next we would like to calculate the frictional force and the coefficient of friction. This can be done without calculating the energy dissipation as such, by recognizing that the base material exerts a torque on the sphere. The pressure, while symmetrical in y, is not equal in the +x and -z directions. This inequality produces a moment of forces tending to oppose the steady rolling of the sphere. The moment, AI, is given by M/kG = Referettces
p. 182
(p/kG)xclA
(16)
VOL. 2 P958/59)
DYNAMIC
MECHANICAL
PROPERTIES
OF POLYMERS
179
Substituting n/2
Mlkrjs = 4
s 0
1
= (n/3) [2aS +
Expanding
Y3 co@ I3
&de =
(a2-r2)112
I 0
(aaZ2)3/2
-
3a2(a2 -
Z2)1/2]
(17)
for l/a 4 I we have (18)
For steady rolling this torque must be opposed by that due to the frictional acting at a distance a, i.e. F = M/a = 2
kqs f
+ ”
4
The coefficient
of friction,
16 yckrp krp a4 + ___ 64
I2
18 -
force, F,
(19)
aa
1, is then (20)
This surprisingly simple result shows that, to this approximation, the coefficient of rolling friction is a linear function of the sliding speed and depends on the radius of the sphere. It should be possible, then, to estimate the characteristic retardation time of the base material by making friction measurements at low speeds and low loads. If we set s = mav where v is the frequency of rotation of the rolling sphere we find I = 2ntv
This result
2.
Large
(21)
may be more useful in some instances.
values
of zs
In this case we will consider example B in Fig. IO. We will assume that conditions are such that the base material loses contact with the sphere at x = o and that all of the energy used in deforming the sample is lost. Starting with eqn. (7) for the pressure we retain all three terms and integrate as in eqn. (9) over the area of contact for +x values only. In this way we find W/zkG = F
+ References p. 182
-
n(a2 -
n12(a2 -
Z2)3/2 6
-
4
4 [ azsin-1 (+I -l(az-12)1/z]
Z2)lJ2
+
(24
Expanding as before WC:have
The moment, df, is found by the procedure of cqn. (16) subject to the above restrictions on the limits of integration. It is given by ,1/f/&(;
z
lja2 ____.
-
t‘L)3/” _ -1
+
&(&.-.-~K_.-
q
WA +
a* sin --.i_
-l(&z)
_
4
Expanding we have (2s) Using the definitions of cqns. (r(j) and (20) we now have
For zs + 1 this becomes approximately
But l/a is dependent on rs by virtue of eqn. (23).Subject to the same approximations our coefficient of friction becomes
It appears that, in this approximation,
the coefficient of friction is load-dependent
and has a magnitude depending on G as well as z. Our eqns. (21) and (28) are plotted in Fig. 12 using a = I cm, IV = IOO g and G = 106 g/cm2. Only the solid lines are to be taken seriously. The dotted lines are drawn to indicate the trends and are certainly not representative of the shape near the maximum of the curve. It appears to be reasonable to expect, however, that the effect of a single retardation time is felt even at very low frequencies of rolling. Furthermore, the maximum in the friction will probably appear at a value of WC I. To find the shape of the entire curve it appears to be necessary to carry out the calculations using the limits of integration as defined by eqn. (8). 12+3Wacesp.
I82
VOL. 2 (~958~59)
DYNAMIC
MECHANICAL
PROPERTIES
OF POLYMERS
181
From this analysis of the problem we conclude that there exists a relatively simple relation between rolling friction and the bulk physical properties of the materials involved. The coefficient of friction begins to sense the presence of a characteristic frequency of the material long before the rolling frequency reaches the point where the maximum friction is observed.
: :
I :
.Ol t
/’
Fig. 12. Plot of equations g/cma.
I \ \ \
\ \\ \\
\\ \\
(21) and (28) (solid lines) using A =
~cm,
W =
IOO g and G =
106
More detailed treatments of this phenomenon seem to promise a new method for the precise dete~ination of the mecha~cal properties of materials. Apparently, even at this stage, the theory could be used to find approximate values of the characteristic frequencies of metals, ceramics and polymers. DISCUSSION
OF RESULTS
It is apparent from the foregoing analysis that the relatio~~p between rolling friction and speed, for a hard sphere of constant diameter on a softer material of given retardation time r, resembles closely the relationship between mechanical loss and frequency. Extending the analysis for rolling to wall-lub~cated sliding, then, it follows that the slope of the friction vs. speed curve at a given temperature, as in Fig. 3, will depend upon a shift with temperature of the maximum in the friction vs. rsjn relationship. Should these relationships prove generally applicable to #olymers at low and intermediate temperatures, they might also be expected to apply to the friction of metals at elevated temperatures where internal friction is known to be greatly increasedla. Modifications in the theoretical model used to represent the bulk properties (Fig. g) References p.
182
would very likely be necessitated by basic differences in the structures of metals and polymers. Finally, one might expect thv clywnic mechanical properties to impose an ultimate lower limit on the friction in systvtns where shearing forws had been ciiminated by cffrctivc
hountlar\- Inbric;~tion.