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Saibel1969-TireWearModel.pdf
.... AR
. ..
L
Szibe., arid Chen l.:nq Tsai
of
..n. neering
!.=nnica
Carnegie-Ye&lon Li-iversity
PittsbL'rgn, Pa. 15213
X%~vmber 1959
in-erim Report No. 2 for Period Augusz 1.5 to November 15, '969
?t-r--red for
Tire Sytems Section
-= D-f Ve.iie Systems "esearch
NATIONAL BU.,AJ
OF STANDARDS
Washingtc:., D. C. 20234
157-854-5
This report was prepared under National Bureau of Standards
Contract CST-854-5 (funded b- the Air Force 7iit
Dynamics
Laboratory, Delivery Order No. F33615-69-M-50!I/Project
No. 1369/Task No. 136903). The opinion, findings, and conclusions expressed in this publication are those of th2
authors and not necessarily those of the National Bureau of
Standards nor the AF Flight Dynamics Laboratory.
Reproduced hy the
CLEARINGHOUSE
for Federal Scientific & Technical
Information Springfield Va. 22151
Best Ivailable COPY
Table of Contents
Introductior
1i.
Tire Wear Formulas (for automobile)
;.
How the formulas were derived
2.
That can be inferred from the formulas
3.
Factors that affect wear rate
."-. Fatigue Failure of Rubbers
1".
1.
Fatiue curve (.V6hler Curve)
2.
The tearing energy criterion
Activatior Energy and 'Near
1. Thermc-activation equation for abrasion
2.
Temperature effect
Frictional Temperature Rise of Tires
j.
~Iiehmann's derivations
_. Schallamach's derivations
"hermodynamdcs and Rubber reformation
..
2.
Fundamental principles
Experimental investigations
3. The thermo-elastic inversion phenomenon
.
Thermal effects of extension
'.:II.
Energy Balance for Wear Process
"TII.
Influence of the Type of Stress on the Failure of Rubbers and
'eferences
Tables
,:igures
lsston.rs
X.Introduction
Tire- w~ar dapenids on niaxw factors.
ationed In a prev3ious report tliI.
Some of these factors are
A careful study of these factors
reveals that a single tire wear forwa"a can not describe the complicated
9-;2
We will go Into this in more detatil below.
penm~enor. sultablyJ.
In general, different tread materials are differently affected by
Conventional-4y determined road~
those factors which affect wear rate.
do tut
rear r&._TngsBnal~
a
ct-ailed conclusions to be drawn with
diio
respect to the re"Lative Importance of' the factora as thwy refer to tire
wear'under ordinazzy coniditions,
road wear
Anoith-.
problem is that factors Influencing
yW
not be independent of each other.
~4hS
I rpor~~7comparef wear formulas in the literature, weamineS'
the fat igue phenomenon, eard the tempprature effect on wear rate.
T%A
-VfnlooT'
FinaU~y
into rubber deformation as related to bazie thermodynamic
ceicepte and examixe'tbe feasibility of an energy balance approach for
_f
-the wear process.,
-.Tire Wear Formwlao (for autceibile tires)
The following for'iaas have received wide attention
conpt
(E/
K (P
A
g
I(0d) *
a(
Nua.ers in the brackets refer to references.
5
2t5
3)
I-0
2
Lere I = wear rate
IN
0 =slip angle
.:
p
resilience of the wheel
f
wheel stiffness
Yo
abradibility
=
t
temperature coefficient
tire surface temperature
s
reference temperature at which Y
t
C
=o
contat
d = spacing of abrasion pattern
E - modulus of elasticity
Co = ordinary tensile strength
=
6
coefficient of friction
fatigue exponent
a=
rameter specifying track roughness
a = lergth of contact area
K = constant
P = material constant
C
.)
=
material constant
Their origins are as follows
Schallamach [2, 3] starting from the abrasion of an ideally
elastic wheel with an elliptic pressure distribution over the area of
contact, derived a wear formula for slipping wheels.
In this formula
he found that wear rate was not consistent with what was to be expected.
Then he modified it
by considering the effect of an abrasion pattern and
a temperature effect, finally arriving at equation '1).
based on a fLatiguc ho~
[L4-1~vh
Kragho.zky end
n
the geometry' cf the contact sueface derivod the wear forzinula for wheels
r
lith slip as In equation (2).
Sulgin and 'aters
[5] made experimants with large varieties of
different rubber mai-rials.
4
They found that different materials have
different wear mechanisms which introduced deviations when equations (1)
and (2) were used to describe wear rate.
Using a combination of abrasive
and fCatigue wear they set up an empirical relation for tiewear as given
by equation (3).
2.)
'&at car. be inferred from th-2 for-nulias
Aparently equation '1) shows the importance of the slip angle
z
and the tire surface temperature.
included.
a;__
.
- REF
The properties of the wheel! are
The properties of the contact surface ame Impliitiy expressed
by the abradibility term
Rut according to Bulgin and Walter's data [5],.
a highi styrene SBR (Styrene-butadiene rttbber) has a negative temperature
coefficient.
This seems unreasonable.
In equation '2) the tangential1 oa-d (longitudinal and Iateral
force or side slip angle), length of the contact area, roughness, elasticity
a74 strength properties are involved.
The dependence of slip, angle
on wear has a power of (2+06) which is approximately 2.43.
This is the
special characterisic of fatigue wear which distinguishes itself fro
abrasive wear uhich uzually has a power of 2.
Temerature effect is not
included in equation (2).
iFMEq'ization
(3) has the advantage of taking care of the Chang-t of
slip angle dependence, but it can not give azr informtion other than
slip angle.
i
r i
Fig. 7 shows acme re-_
ts from [5] Yhicb f4t equ tion(3
Wall.
Reviewing the deficiencies stated above and realizinz_,116e
1
different properties for df--erent '-nds of
-Mber one cmn easli.y see
the difficulty in setting up a unified equation to describe the tire
wear phenomenon.
3.)
Factors that effect wear rate
Kragh-IsV aend 'Iaepoqshchi
(6] from a fatigue mechanism
demonstrated some of the effects of basic factors on wear.
Fig. 1 shows
how wear depends on pressure, slip, the coeff icie t of friction, the
!'oughress of the surface, the elastic modulus, th3 strength and the
resistance of the rubber to fatigue.
It should be noted, however, that
thene relationships are idealized, since these parameters are interconnected.
For example, pressure influenws both slip (slip decreases W.th increase
in pressure) and coefficient of fricti.n; the stress, strength and
fatigue properties are also interconnected.
Hence, during ex,47imental
determinations of these relationshirs it is essential to select the
samples and test conditions extremsly carefully, so that when one parameter
is changed the others change as little as possible.
Bulgin and Walters (5] studied the wear rate on asperity surfaces
and smooth textured surfaces (whd-ch Schallmach called "sharp abrasives"
yan"blunt abrasives").
In general. on asperities the wear rate is
proportional to applied load, modulus of elasticity, carbon content,
temperature and the sliding distance.
On the smooth surfaces wear rate is
proportional to the power of applied stress,
teerature have an inverse effect.
hile the carbon content and
However, we should notice that
thre~r ii~yexce ii~on as will "eseen from the followina f gures.
teeiFig. 2 gilves the tcmpertr
kinds or~ rubbers.
eet on asperity surfaces for different
Fig. 3 giv.es the case for smooth surfaces. As seen
frm t ie figure, the proport-Ionality between vear and those factors
affet~mg wear rate may onlyv be true in a certain regime.
II.T.
Fat!Li ailr
of "Rubbers
When a titre rolI-s on the road surface it is easy to see the
tVee Tirftzce is --ubject to a cyclic action of extena3ion andi compression.
u~e to this cyclic action on the tire surface, fatigue of rubber will
occur after a certain numiber of cycles is performed, i.e. fracture cracka
and hei
prpagaionwil occur. There are several different approaches
In irnvestigating this phenomeno~n, two of which will be discussed below.
1.)
ratigue curve (Wo-hler curve)
Reankovkil(7] investigated fatigue wear and proposed an
empirical relation to descrlbe the fatigue, life of rtbber
(4)
where npthe number of cycles to failure
a
amplitude of the stress
e onsantwit
=
th pbysical waning of the breaking stress when
n
1; and
6=aconstant expressing the res sance of the material to repeated
loading.
L
Later Yraghelskly et al. [S] chcked the applica~blity of this
relation experimentalAy and established a relationship between normal
(volume) fatigue phenomena and contact (surtace) fatigue.
They found
that there is n3 significant difference betzeen the two for some
-,terials.
It is obvious that in the abrasion of rubber by a fatigue
imehanism, ve are interested in contact fatigue (caused by tangentil force).
R en slippirg occurs, both comression zone and elongation
zone ate formed, whereas the elongation is most dangerous from the
point of view of possible failure.
According to preser.a solutions of
contact problems in elasticity theory [9; 101 the maximuz, stress in
the elongation zone is propy!jrt
al to the product of the maximum contact
pressure ard the ccef-icieit of friction.
Therefore, a first approximation
would be:
K'
where c-d
Pr
Pr
K
the stress, reduced to sl!e
(r
elongation
the mean actual pressure
the maximum contact pressure
=
coefficient of propoziort
lity
f = coefficient of friction
=
the shear stress
Fig. 4 shows fatigue curves, deternined experimentally, for an
unfilled natural ruober vulcanizate.
The results given confirm the
appliCability of equations (10 and (5) and show th-t tho value of 6
(the slope of the fatigue cuires in logarithmice coordinates), which
expresses the ability of a =aterial to resist repeated loading, is
M:
~
alum+, ientical for volume man con'tact fatigue.
F'ig. 5- and 6 give
fatigut curves for t-ead vulcenizates based on. natu-ral rubber and
KB(sodium-celaiJwsed po: ylutadlene rtobtr) and om. Europrene.
Comparison o~f volunia fat1gue aurves wi h frittion-contsett
fatigue curves gives meason to auppose that the breaking streus in the
latter case is the elongation. stress of the surface layver due to the
friction force.
The results obtained -confirm the existence of a cor-
relation between abrasion resistance a-,-A ftigue life of vUlcanizates.
2.)
The tearing energy criterion
Soe
type~s of test pieces (fig. 8) ,were used In determining
the tearig energy and the fatigue life of rubbers [11].
about
A cut, generaflly
ma~i
long, is inserted in one edge end the test piece is repested~r
cycled to a fixed mazznu
extension, the minimum normally being zero.
IMaurernents of the cut length a w. suitable interzals of n cycles
enable the cut growth rate dc/dn to be determined.
For the tensile strip
test piece [13.]
=
Wc
where VI is the strain energy density in the bulk test piece and k ia a
slowly varying function of strain which has been determined Pxperimentally
(12].E
(6)
The cut growth rate de/dn is 'ur i~r-1iI dete=ZdneAd by te
_ea~ringg enen,_y atttiined dwuing a cyde,".
f~i
eq~ttn ()
*rfober
rb.C. ia+tter can ie c .ula~d
t the rzex,1Mium stra!.M 8M]
Uslllg the value of 2k
TMneasUrd cut lemjth c.
Resut s ob-t&reA I-~ thijs wyfor a natura.
gum vulcarnizate arm shaif a given tear4ir.
i_=-nurn
i fIJg. 9.
~un--f
_) cut pr,
-is at- ainea
e-rer
It ha.been ver -fed that
~hic
oc-zurs- 4_5 indepen-dent o-: the test :,ieca u~ed "fig. 10).
Mrprinnts [13] on t1-e prop~gat lon c-f cuts hawe enabled the
fx~igue 1ailure of :mfbbar subjected to- rej.eated simple exzt-ensions to be
quvantitatively underStatA ca a orach-rorsih priocess Vahich counawnces at
aill
natlurally on-- ring f~ta
ri1cum these experimen-Ts two izecbariamls Of2
f low jr aixt growtih have been ide:tlf ed.
One zie flrechanicoMoxidativeV
cu; gwow-~h due -to primarile ne hanica! rupture -&Mich can be facilittated
* bythe t~ ~nce ok
oye;te
athe.r Is ozoe cut gro~th Jbich is a
cowroaive type~ cf process causedb
moI1eculeZ by ozone.
c-hemical breeking of thbe poily-ner
11t is -;.wTnd that no mechanulco-oxidative cut
growtth occurs bc:low a critiesl=2 value of the erie-gy svai.labie for crack
arpgatior.
This eritlcal energy iz! relat-d to primary z~eecular-bond
Atrengths and is apprCxim~ately, coustant Io al
ubes
Fromi its cut-grov-ul characteristias, the fati~gue life of a
rubbal can be accurately
07-iite-wr 3.rnde
for varlous stwmspheric condUJtioi~t.
there ia a mechanical
obsertred '.or waels).
iat-1glue
rsnge of deformatloas and
Coesond-l-r
to -. e critical enry
J3'nit (analogous to the fatigue limit
Below The fatigce limit, flaws at first grow on'-y
slowly in the presence of ozmone until- the critics'! enez-
is reached 3
a[O
%hen failure en~sues fairy rapd~y.
-
Athigher deformations the more
rapid mechanico-oxidative cut growt~h staeus immediately and fatigue lives
are much more shorter (fig. 11).
IV.
Activation Phergy and 'Near
The temperature dependence of the durability (at nominal stress
a~const.) for solids -,nd polymers is expressed by the equation
X.7'
,vhere -r0is a constant numerically similar to the period cf themal
M
oscillation ot the atoms; U is the -activation enerVr for the process of
failure; R is Boltzmann's constant;- T is the abzsolute
teMper~tIA.
L,
The activation eneirgy U debends, for maniy solids, on the- state
decreases accordiing to the fornrla
~of
tres 0,and
where U0 is the activation entnrU of the elementary- failure process in
theabsnceofstrss nd s-sim-1lar in value to the sub ima-tion enerU
for mtals and to the ener&gy of- chemical bonds for polmers; y is a
2F
~coeficient which depends on the- nature -of the structure of the material.
Substituting equation (8)into (70, the temiperatuxe-tme dependence of the
sti:ngth of rubber may be expressed by
To~ exp(U-Q
/?j
This is the equation Zhurkov and fiarzulaev
used in -their.kintic
L14)
zheory of strength.
neo-activation equation for abrasio
'When applying eqjuation (9) to abrasion the rale. of-fatigue
life is then ta]ken by abrasion resistance, I.e. the time nedes~y--t6
Abrade a certain amount of material.
It is more convenient, howeVer,,
to use the rate of' failure, i.e. the rate of wear I whidth is recorded
directly by experiments.
k
t~ere
Thus for abrasion (11
=1 exp-(U0 -A~b/RT)(10)_
is the coef-fidient- of f riction; pp the- force of -trktin and)..
has the sane pkysical -sense as y in equation (9); 17
1k
.
0
is analogous to
The activation barrier U0in abrasion is reduced by V~ pi
But,
since uLcharges only slightly in the region below the flow~ t Mperature
(Tf) with changes in p and T, then u!A'=)- -can be ccnsitlered as constant.
Mhten deterodning The pbysical sense of I mw note that-
is the
Uaniraum fatigue life, dependi~g neither on temperature. nor load.
at suchn a high rate may occur when eit her TI
Y a completely counterbalances U .
0
or
'~is
Failure
so great -that
In such critical coiditions the time
dependence or activation nature of the failure process is lost.
~)Temperature effect
Fig. 12 shows the relationship between ogIund l/T for
several loads.
According to equation (10), these coordinates should
produce a. straighit line, with a slope proportional to activation energy
U, where
pint
th
exerientl
In ~ct
dolie on a straigt line, blat somfftliieS
a "break" U~ observed in the line.
The data were obtained with all test
nmaterlale -abraded- by metal -gauze (16].
As -we can see from the figures, If temperature changes from
rate-1-
0,tb
toi.6orC
G
change b y a factor up to 50 vey signfict
F-g.13 shows -the -effedt of teirperature -on the abras1non
gaze- (faig
-f:il the--f
ewear -a
on abrasive paper (abrasive wear).
dw-ine ssated---a
ate
based onthe so called
-AlOecu1ar-kinettC -theorY of failure,. -according-to, Which abrasion Is the
failure of cheijMcal bonds asa a eult -of cer'tain fluctuations -in the
thermal movements of -moecules. Vhen the tangerti
epugh to- eliminAte, the--barrier
foce is great
then each therIal movemet results
in aebrasion.,
'If -abrasion is- stujdied fwda the point of view. of mcro-failure
-
E
is, a fatiguxe process: [15, 17, 18] failure of
-neeeAbrasion
then,
thd surface takes P1UPceafter repeated defoiaation by projekdons of
abradant.
Notice that In fig., 12 the slope of the lines decreases with
Increases in load.
12
r.
ti~nal Temperature Rise of Tires
Viehm,?zn's derivtis~
Viebmann [19] at temed to detei-adne the Crla.onal temperatures
-7-4!h may occur in tires under various conditions.
hienomenollogical theories of heat conduction.
Rt was based on
The temperature rise
Was round to be localized in boundary layers of t.he order of 1From-Lie estimated taluas of the temner-atures (see table 1), he
concluded that the abrasion *.fich occurs under normal driving conditions
is 1mraly mechanical. On!iy under extreme conditions, sach as in very
rapi5d acceleration, or in case of locked whel
art±ua, a
htw
-
call- 11'contact bridgestt or surface elevations
-may
vie expect high frictional
tempieratur'es, *hic-h lie far' above the decomposition t--e-mperature
of rubber.
2.)
Scbaillamach's derivations
Schalliarach (20]
A
sumd tire ar.- road are taken t
ei
Ideal contact-with each other, and the temperature rise of' a surface
folemer, is calculated as if it were in the surface of a
body.
e
infint
Further assumptimio s that heat flows onlY at righ angles to the
contact, surface because the low t.herma condtivit-y Of rubber anlw
lateral heat flow to be neglected.
The problem to be solved Is thus
reduced to an-one-dimensiornal one.
Starting from a theoirj of inear heat conduction, in a semiInfinite body [21] and through some lengt
derivations he obtainled a
maxiam frictional temperature rise 0 Which occurs at the rear of the
M
contact area of a slippIng wheel-
VAL
___
where
K r = thermal diffusivity of road
= thermal conductivity of road
r
K
=
thermal adiffusivity of tire
=thermai conductivity of- tire
V
--
=
circumferential velocity which for mbderata -slip, can
the tratvelix
Sequated-with
i
:velocity
-L=the load of the tire
-~a =length of contact area
:
b
-width of contact area
t =the .total sliding time
sa= slip
!
i
In the derivations Schallamnach did not give the numerical values for
i--
equation -(3.2), however he gave an expression for-e'z)/e m (where Z is the
co-ordintea t rigt angles to the contact surface)
t
(12a
)
14
where erf is-the erxor function, c stands for
The value of c is small enough at mioderate slip for the term in square
brackets to be replaced by unity [2].
The ratio C-(,)/O eped
th
onl-y on Z/2(Kt)' and is shown as a function of this quAintiLty in Fig. 14.
21
A typical figure for-1 fdt tpead rubber is 1.7x10 -3 am /see; the sliding
time t Is of the ordt-r of 10- sece. at a travelling speed of 30 mphx.
With these numerical values. the depthn 2' at which the temperature has
dropped to Iper cent of the surface temperature Is about 0.15 mm.
The
heat -into which the frictional energy has been converted is therefore
confi'd to a very' thin surface layer.
This fact is consistent with
that-pointed out by ihmann[9]
Brunner [22] has determined the temperature profile-s-across
the thickness of the tread of a travelling tire by means of thermocouples
embedded in and on the tire.
The drawn-out curve in Fig. 15,
A~n
from- Brunners paper, sho-s the temperature distribution i R a natural
rubber tire travelling at 80 krmi'h and exhibits a -well defincml maximum.
Brunner's iwiasurements do not cover the critical region. near the =uside
surface where, in an~y case, no constant ter-Trrature distribution can be
mnintained because of the periodically occuring frictional temDerature rise.
The dotted part of the curve in f ig. 15. indicates schAema-wceL3,v the
Instantaneous fricticnal temperature acecording to Schal1amadh [20] -ihen
superimposed on Brunnerfe curve.
For clarity, Th1ontlsa~
of this transient has been expanded about ten-fold.
VI.
Thermodynamics and Rubber Deformation
i.)
Fmidamental principles
The f irst law of thermodynamics providea us writh a definition
of Internal energy.. namely
SE-SQ ±4W
(6
This-equition states that the increase in internal en~ergy dE in any
change taking place in a system is equal to the sum of the heat added
t -o-t
AtC, and- -the mork.-peiformed on it,
TV.
7n4 second la. defines
ent rbly change-dS. in any reversible process by the relation
De*fine Helxioltz f ree- energ
A by- the relation
Treloar (23] used the fact that the c2henge in He~mholtz free energy in
8fl isothernmi process is-equal to the work done on the system by the
external forces and that-the tension is equal to the change in Helrholtz
free energy per unit extension.
[
16
He then obtained the following expressions:
(dflTa
OI
~
(~/
-EA
(1re
9)__
~
(20)
f is the tensile force, and 1.is the length, measured in the
awhere
direction of the force.
Equation (19) gives the entropy change per
unit extension in terms of a measurable quantity (Zf/BT)Z, the temperature
coefficient of tension at constant length.
MEL
relationship for the correspondir
Equation (20) gives a
internal energy change.
These two
equations are of fundamental importance in rubber elasticity, since they
-provide a direct means of determining experimentally both the internal
energy and entronr changes accompalying a deformation.
The only experimental
data hich are necessary to provide for this purpose are a set of
equilibrium values of the tension at constant lene;h over a range of
temperatures.
If,
for example, the curve cc' in fig. 16 represents the
variation with temperature of the force at constant length, its slope
Mat
the point P, which is (f/BT)I,
is by equation (19), equal to the
entropy change per unit extension (MS/)T when the rubber is extended
isothermally at the temperature T.
in a correspondintgn
way, the intercept
of the tangent to the curve at P on the vertical axis T=O (absolute zero),
is f-T(bf/;T)A, which by equation (20) is equal to the internal energy
change per unit extension 'fE/A)T.
The course of the internal energy and entrogr changes accompanying
_
-_
--
-- -- -
------
the deformation may thus be obtained by direct inspection of the
*
stress-temperature curves.
In particular, if these curves are linear
(as In Meyer and Ferri's data, Fig. 17) both internal energy and
entropy terms are independent of temperature.
If, in addition, the
stress-temperature relation is represented by a straight line passing
through the origin, the internal energy term is zero.
It follows
that in this case the elastic force arises solely from the change in
entropy.
2
•
2.)
Experinental investigations
.-
A typical curve, representing the behavior of a rubber
vulcanized with 8% of sulphur, is illustrated in fig. 18. The stress
is seen to be very nearl3y proportional to the absolute temperature
W-er a range of about 120 degrees.
Tt i' therefore concluded that
the- elastic tension i.s due almost entirely to the entropy term, in agree-merit with the prediction of the kintic theory of elasticity.
213 0°
At about
the temperature coefficient of tension exhibitd an abrupt reversal
of sign. This corresponds to the transition to the glass-hard state,
where the rubber- loses its characteristic extensibility.
The behavior
then corresponds to that of an ordinary hard solid, with the internal
energy term dominant.
Fig. 19 shows som other stress-temperature curves for the same
imterial. The results confirm iWyer and Ferri's original conclusions in
general way. Calculation of the terms (B/ B)T and (IF/a.)Tfrom the
curves in fig. 19, by neans of equations (19) and (20), yielded the result
shown in Fig. 20.
It is evident from this figure that the form of the
force-elongation curve is effectively due to the entrolF term (aS/BAL)T.
-t
Execept at small extensions (<200 per cent) the internal energy term
accounts for not more than one-sixth of -the observed tension fror freeene.r&r change.
At low extensions, however, the internal energy changes
aiofthe same ordier of magnitude as the entroW term,. while the lat.ter,,
invwtead of remaining negative (as required by the kinetic theory),. charges
sign at an extension corresponding to the thermo-elastic inversion point,
3)The thermo-elastic inversion phenomenonin fig. 19 one can observe that whilst above 10 per cent elongatiOU
the tension at constant length increases vith rising temperatre
at loer
elongations the variation with temperature is in the oppoite diretion.,
IN
M~yer and Ferri Interpreted this tbenno-elastic inversion phenomenon in
terms of the expansion of volume of the rubber on heating.
Thi expansion
will clearly have the eff ect of tending- to Increase the length at constant
__
stress, ,Rhich is equivalent to reducing the tension at constant length.
At very low stresses the reduction in tension by therma-l" expansion exceedsthe inerease- of tension to be expected from the kinetic theoy of elasticity;
the thermo-elastic inversion point is the elongation at wich these two
effecits exactly balance.
The fact that volume changes enter into the thermo-elastic
phenome~non of rubber in no way limits the applicability of the fuldemental equations (19) and (20) for the derivation of the infternal energ
and entropy changes on extension.
These relations are purely phenomenological;
theyj do not imp].y aror particular mechanism.
It is in the interpretation-
not in the derivation-of the internal energy andA ent rowr changes that
physical or molecular concepts may usefully be introduced.
In this
-~Connexio,
the explanation of the thermo-elestic inversi on put forifardI
byl Myer a-rd Ferri is of the utmost significance since it points to the
msportance of hitherto neglected voluma: changes in -the discusslon of the
SL~a
RM
Mernmcdynamics of rubber elasticity.
4.)Theral. effects oil extension
It has been found that when rubber is stretched adiabatically
its temperature will rise.
Also -It was -neticed that a piece of rubber,
extended by a constant. load, contracted when the te.-mperatura w~as raised.
Which means that at constant length the tension increased with increasing
ME
temnarature.
By the second law of thennodynamics, as re-rsne
(17), the evolution of heet
yeuto
-dCO) in a reversible change gives a direct
vla
measure of the change of entropy in the process. ?L' heti
eaive.
on the extension of avy material, t--he entropy ch ei
Conversely, if heat is absorbed, the entropy c&nnge Is positive,
I
On
account of the smallness of the heat effects in rubber. it is usual to
measure the change of temperature in an adiabatic deformation rather than
the heat evolved in an isothermal deformation,
__
*
11nder adlabatic conditions
by definition, the entrop7 change Is zero, and the cangeite
erur
is given by the relation (sitnce Wt"=6/j
where C. is the specific heat at cstant length, and ~C/Z3A) T is the
isothermal heat of extension.
Thius the temperature change in a quick
20
extension from the initial length I
to the final legth A is obtained
b-y integration of (21) with respect to A.
A-
Thus
Tf(S/laeLde
Fig. 21 is a curve given by Joule 0.859). to
(22)
used a thermo-
coupie 4 - cont wdth the rabber for this purpose. The curie shows an.
initial cooling folihwed by a rapidy rising heating effect as the
extension was increased.
Later on a slight difference between the
effects on atenion and on retraction re-spectively -gas fon,
due to
the i rreversibe uhenomonon of re anticr
The data given in fig. 21
are mean f igures for extensIon -md mrac4 iOn.
At higher elongations the variatios' in the heaet effect.
tMO
variations in internal energy chan~ie. ar-e more complex
iceabV on the type of rubber.
rubber are sno.
cuve is
in fig. 22.
reversible.
Typical data for a vnlcanized latex
Beyond this point the heatig Increases rather
the coolirg on retraction being
greater than the heating on extens-on.
Hwever, some other .materials
m
ien renon,
have the opposite
Energ
anI depend
-n to about 233 per cent extension the
Sharply and is no longer reversible
Ii
lNi
Balance for Wear Process
Chnan
oahf241
Chenea and Roa.h
rt
2~4], based on the remval of wear marticles
from material as a wear process, derived soe equations for mass balance
and energy balance.
7
After assuming an isothermal and steady atate, the
mechanical wear be-tween identicell mterlials can be expresse-d by
ct S
(23)
(24
-nz
where
h
=total
change in surface -1vel
ttime
f frietidir-coefficient
V -slidingE veloct
S =surfaceenergyr- per -unit -area-of- er
-
articles
-P- -density
E=interbal-energy per unit -mast of' wear- particles
:a-
shap
factor- for wear particles
m-= mass*of the particlep
=frequ4ency-function
-From equation 123) Wbdt can -be interpreted as- wear xate.
On. the r-ight
-hannd Gie riormal load, sliding velocity, surface energyand internal
lenergy -are -all,-involved'. However, the *Wdarof tires is mos lyineptd
as the fatigue: fakilure-of rubber.
werperticles" w
o
esial-
Thus-the application of "removal of
for the tire wear process.Afuhe
indquacy is -the isothermnal assumpition, -since-we Xnwtire- wear will
never be -an-isotheitul process as we saw in the -section on frictional
2
1emperature rise.
Although the approach does not fit the tire -wesr
process too well the i.7ea of using the energy- balance for a -Wear pr~ocessmay be a promising way of solving the problems.
__
Rabino micz [28] has attempted to account for wear in- metals asa fracture process by equating the energy~ necessary to form -fresh -surfacesfound in the wear particles to the enerr -necessaryr to produce -a fracture.
Suchi calculations have not been particularly successful up toVow and--it
seepis that energy methods are better applied through chemical and
mechanical activation processes.
'ITl.
influence of the TYpea of Stress on the -Failure of Rubbers and El-astomiersThe most critical type of stressed state for elastomers in tension
under vhich various -types of failure may be- observed; rupture, :shear, obr-
a combination-of rupture and shear (fig-, :23).
A oplcte
6)served in the specimen when-in som eighborhood sdmQUIt~r
sili_
gro
Is
t_ ad
each ot~her end then join by local shear.
E
Types of f ailure, are f requenly- met with. -in elastoms-ric s9tripe
_[25]
-which are used to make sealing rings *here under-gisat -And -prqlftnge
compression there appear, due to faiAule, -slight tears in, the pakig AWh
1re
eihra
nagle of 450 to the direction -of the conipres6 or. -(fig.24
Or pamrallel to the,-contacting surfaces (fig. 25) whiere the side of thedeformed packing bulges -most. The recaglrcosscinoLh
Packing takes- On the shape-of a "barrel#, (fig. 26), whereby in -the pinrts3
A and B under-great compression a tension appears In-the lateral direction.
The greater the-compression of the packing, themr the fr ee side-surfaces
bulge.
Along Ve- line AB of that nbarrel"' a growth of WmAl
23
tears takes place from the surface into the depth of the zaterial (27].
it
should be noted in conclusion that very little
wFork has
been done on the failure of polymers and glasses under commlicated
stress systems (26].
,
j\abe COPJ
_24-
LS
Saibel, E. A., Interim eport for period
)ay 15 to August 15, 1969 to
.!BS.
12"
Waahington, D. C.
Sc1aiMMU
ach, A.
and Turner, D.
,
ear, 3 (1960) p. 1-25
r-:oach, K. A. and Scha1laieh, A., Wear,
4 (1961) p. 356-37
'Kraghelsk,
. V. and %epoM ashhi, 7. F., Wear. 8 (1965) 33-319
- 'i gi, D. and IWValters, '...,
Fifth Intern. Rubr-Tech. Cbnf., Brighton,
1967
_- -K'aghels_-y,
I..
PLblishing Co.,
-
-a±X
ITc., New York, p. 3-13
[7]
Reznikovsld.i, U. I.,
[8]
Kra
-
[9]
[0]
-T
Nepomnashchi, E. v., Abrasion of
Rubber, Palmerton
Soviet Rubber Tech.
!se1
y, I. " ., 1Reznikcrskii.
,.14, Brodgkii, G. I. and NepQm=ashei,
B. F,
Scet Rubber Tech., 1965, 24,
No. 9, 30
S.erin,
,Cotact strength off aterial, 1946,
2, see riference
Kovaitskls
, B.
-,11V, Akad. Nauk SSSR, .tdel. Tekhm.
Nauk 1942, No. 9
t
n R.S. andA.- G., J-O -P-oqiyer Sd., 0, (1953),
291-318-
ensmth.
. -,., .T.of AP.
Polymer Sci., 7-(1963), 933-1002
Lk, C. J. and Lxley. p. B.,
t'-.
t_[16]
[171
[18]
(19]
[20]
21]A
1960, 19, Fo. 9, 32
P-cc..
rof. Physical Basis of Yield and
7racture, -ofor, 176-186, 1966
qhrkov, S. 1$. and Narsuiaev, B. N., Zh.
Tekhn, Fiz., 1953, 23, No. 10,
1 1677
atrer, S. Bo, Thesis to the Karpov Pysico-technlical
Institute, 1964
Ratner, S. B. toki. Akad. NAuk S&SR,
1960,. 135, No. 2, 294
atrer, S. B., Dokl. Akad. Nauk SSSR, 1963,
150, No. 4, 848
Kr8ge3-.k,
I. 1., Friction and wear, 1962
Viehmann, W., Rubber Chemistry andTech.,
Vol. 31 (1958), p. 925-40
Schallamach, A., J. of the IRI, vol.
, o. 1, 1967
Carslaw, H. S. and Jaeger, J. C., 1958,
Conduction of Heat in Solids,
lliit
zford
[22]
Brumner, W., 1938, Deutsch Kraftf ahrtforschung, Heft 2
(23]
Treloar, L. R. G.. The Physics of Rubber MIasticity, 1958, Oxford. Second
Edition
[24]
Chenea, P. F. and Roach, A. E.,
Lubrication Engineering, Marcb/April,
1956
p. 123
V25
Bartenev.
(26]
Haward, R. N., The strenh cf Plastics and Glass, !:ew York.
.27]
V. . , Thim. prom. No. 8, 15 (1955)
1949
Bartenev, G. M. and ZWer. Yu. S., Strength and Failure of Visco-Elestic
3terials, 1969, Pergamn Press
j16
Rabi
ic,
E. and Fos*ter, FL. G.,
1964: P. 306-12
Trans. ASIAE Series D. Vc.. 86, No. 2;
TABLE I
FRICTION TEMPERATURES FOR TARI0T-1 DRIVNG
CONDITIO)B AND W-46D MATERILS
Friction temperature,
Fri
lon~eB
-i
P:~1*g
ezditims
~
vahicl u tre
siiage
Asnl
Basalt
Concrete
0C
Iron Asphbalt
35
10
2
60
35
25
:3.5 -140
Basalt
Concrete
40
lbca~izatlion- inIron- -the bouzdary layer,
c
-9
=3%
Gzved road.
A-2
50 !qVhri
slippage
3 Spfli
4.
-
6.5 X 107
100-
36
9 X.10 -3
-(650)
100
-7Xlb3
-!0%
t-Ire,
10~ce/e.240
bIoked skiddimg
tire, max. 120 kriVbr-
160
-(1100)
(700)
24
100
-(1000)
(400)-
Pesure
Slip
Coefficient
Roughness
of
friction
I
I
.
..
Assistance
to fatigue
Suength
lasuic mndulus
The dependence of wear on the main
parameters
Fig. 1
Ouvodic":
Sts'rcne
50
4N0-
40-
-0
F
fo;o
Ason
NR
100.0
X
6.3G
V
.4'V
C.
'
4 ji.
I,
I
(I
i~
S
4:j
Te'rn..ro're
!
. t
t
7G 24
20
BR
p
.
COCUbuX):
(,C)
0
z0
40
W,
TemPC,o1ure (*.I
(10
KoO
Fig. 3 Wear rate on smooth Surfaces
Fig. 2
Wear rate on asperities surfaces
45
W0
0$4
5
4
log n. number of cycles
FIA. 4-Fatigue curves obtained with the -PUPS- app~ralus for an unfllcd
matural rubber vuicanizate: 0. *-radius of curvature of indentor 20 mm:
A. A-radius of curvature of indenior 1IS mm; 0. U-radius of curvature or
indentor I mm I-volume fatigue; 2-contact fatigue; 3--contact fatigue
JI
1
AL
V__
IV
-I
1
7
0 nV1
tii.
Number of cycles
Faa .5-Fatigue curves, obtained on the 'Tsiklomctr**appjraitz,. for ruroprcnc
tread vulcanizaies: *-rpe~ated clongjtlion;Q--coniadc fatiguez (log r - log
an): a-E-22 kgfcm:- b-E 28 k;fcm ; c-E - 32S5 kg! cnm. (The
ordnatc axes should apparently be a. or i. E being the cliiic modulust of
the rubber; Fd.)
log
it.
number
of cycles
0
log at. numbr of cycles
F1o 6Tiu cu"vs obtained on the "PUPS- apparatus. for natural rubber
(8)%a:d SK ( trea vulanizates: . 0-radius of curvature of izndentor
OS mm; A. A-radius of curvature or Mnentor 1-0 mm: 12.5-radius of
curvture of indentor 1-5 nun; f. O-radius of curvature of indentor 20 mmi
1-volume fatigue; 2--contact fatigue (log a
-
log n); 3--contact fatigue
(lot-log 4
Natural rubber/20 IHAF
-P
Q~1-
R
Naua
Natural rubber/30 HAF
-be/0H
T
-
S Intell500/50HAF
TAdiprene C/50 HAF
U Polybutadiene/50 HAP
.
1-0
-
2-0 30 40
Sep Ongta (0)
Figure 7 Logarithmic graphs of'
tyre abrasion against
slip angle, showing
that the wyear rate
increases as a power
of' the slip angle,
varYing fromn 3.8
40 3*2
t
,Strain energy
density W
unstrained
height
t
TI
Strain energy
density W
F
T=2F t
T=Wh
T=2kWc
The tearing energy criterion. Various types of test
Figure 8.
piece which can be used for cut-growth measurements in rubber: left
to right, tensile strip, pure shear and 'trousers' test pieces; the
equations define the tearing energy for each test piece.
10---o2
I0-4
I08
-
010A
102
104
106
.
T (er e-2)
Figure 9. The relationship between cut growth and fatigue. Cut-growth
results for natural rubber vulcanizate A obtained in the laboratory
atmosphere at a frequency of 100 cycles/min (test pieces allowed to
relax to zero strain on each cycle).
.
10-3
r-4
"
10x
00
-5K
106
105 5106
3.065
i56
*
17
10'
Tearing energy T (erg cm- 2)
Figure 10. The tearing energy criterion.
Typical
cut-growth results plotted against energy for
the
various types of test piece: * tensile strip,
X
Itrousersr, o pure shear.
1
Z10 5
04
+
I.
I
S103*A
Itoo
Iv
)Ifilmulm strain (%)
Figure 11. Fatigue behavior of "nrious rubbers
in the
laboratory atmosphere
4G,
.9
(minim ' s.rain zero): A, natural
rubber (vulcanizate A); B, SBR (vulcanizate
B);
butyl rubber; D, polychloroprene; E, butadiene- C,
acrylonitrile copolymer; F, synthetic cispo~Jsoprene;
polybutadine. Rubbers A, D, F are strain crystallizing,
C is weakly crystallizing, and B, E, C are noncrystall-zing.
N
JI,,
'1we
a •'
' '
1
Jo
Vt
AS
A&
IVV
Tremperature dependence of abrasion on
a gauze: l-h.p, polyethylene, II-PVC
+ 30% plasticizer; III-PVC +80%
IV-P.9A + 40% plasticizer.
plasticizer;
--!
40/c-0.65; d45 a-2
kfg/cm64; b-1-3;,
Load:
0.3; e-0-15; f-0-06.
Fig. 12
J17
50AV
4
ow7.Vjr
" I I/,
Fig. 13
Effect of temperature on the
relative abrasion or, a gauze
(curves I and 2) and on
abrasive paper (ciirves 2 and
4); 1 and 3-Pro; 2 and 4h.p. polyethylene.
I
'°
03-
2
E
0
04-
04..
02 04 06 O6
I*0 12
14 1&
o '~o , t
Pig. 14
The distribution
of the frictional
temperature rise
below the surface
of a slipping
tyre
670
W
~THICKESS
OF TYIIE
0
c,&
Fig. 15
F
L
I
2
Fll
curve: temperature
distribution in a tyre
after Brunner; interrupted
curve: frictional temperature
rise superimposed on BrLner's
cur, for an arbitrary slip
/C
*C
0
Fig. 16
Absolute temperature
Slope and intercept of
stress-temerature curve
12
tC'
• --g o '-
260
_-,-o-33%.
300
320
Temperature,
340
360
*K
Fig. 17 Force at constant length as
function of absolute temperature. Elongations as indicated.
(Meyer and Ferri, 1935)
17'5-S-
-
-
-
-
-
E
10
Temperature OK
Fig. 18 Force at constant length as
function of temperature.
Elongations as indicated.
(Anthony, Caston, and
Guth, 1942)
E
I.4
of temperature. Elongations as
indicated. (Anthory, Caston, and
Guth, 1942)
E/i
Fig. 20
Changes in internal ener
entroa (S) accompmin
of rubber.
'¢th,
(E) and
extension
(Anthary, Caston and
1942)
0
/
U
2
:
AsA
EC
-
-02-
a-02
SEkgaft
Fig. 21
per cent.
Temperature rise in adiabatic
extension. + Joule's experiments.
o James and Guth's experiments.
-- James and Guth, Theoretical
K>r
I"I
/
10 2DO
Pig. 22
300
400
EWNojoan-Percu*.
500
Temperature change in adiabatic
extension (or retraction).
_Extension. -- Retraction.
(Dart, Anthomy, and Guth, 1942)
Fig. 23
A
Types of failure of technical
rubber under tension. 1. Shear
2. Shear and rupture. 3. Rupture.
*
R,.a-.
--
Fig- 24 Failure of' rubber sample under
compression (chief tear under
an angle Of' 450)
Fig- 25 Failure of' rubber sample
under compression (tear
onI the lateral. surface
* Of' the rubber packing)
rA
Fig. 26 Schematic def'ormnation of'
rubber specimen under corn[rsinIihu
[ln
sipn
tespotn
sufcs(omto
Fly UP