Journal of Applied Mechanics Vol.11, pp.271-282
(August 2008)
JSCE
Anisotropic constitutive equation for friction
with transition from static to kinetic friction and vice versa
K. Hashiguchi* and S. Ozaki**
* Member Dr . of Eng. andDr. of Agr., Prof, Dept.Civiland Environ.Eng.,
Collegeof Technology,DaiichiUniversity,
(Kokubu-Chuo1-10-2,Kirishima-shi,Kagoshima-ken899-4395,Japan)
**Dr. of Agr.,AssistantProfessor,Facultyof Engineering,Divisionof SystemsResarch,
YokohamaNationalUniversity(Tokiwadai79-5,Hodogaya-ku,Yokohama240-8501,Japan)
Highfrictioncoefficientis firstobservedwhen a slidingbetweenbodiescommences,which
is calledthe staticfriction. Then,the frictioncoefficientdecreasesapproachingthe lowest
stationaryvalue, which is calledthe kineticfriction. Thereafter,if the slidingstops for a
whileand then it startsagain,the frictioncoefficientrecoversand a similarbehavioras that
in the firstslidingis reproduced.In this articlethe subloading-friction
model1)witha smooth
elastic-plasticslidingtransition(Hashiguchi,2005) is extendedso as to describethe reduction from the staticto kineticfrictionand the recoveryof the staticfriction.The reductionis
formulatedas the plasticsofteningdue to the separationsof the adhesionsof surfaceasperities inducedby the slidingand the recoveryis formulatedas the viscoplastichardeningdue
to the reconstructions
of the adhesionsof surfaceasperitiesduringthe elapseof timeundera
quite high actualcontactpressurebetween edges of asperities.Further,the anisotropyof
frictionis describedby incorporatingthe rotationandthe orthotropyof sliding-yieldsurface.
Key Words: anisotropy; constitutive equation; friction; elastoplasticity;hardening/softening;
subloadingsurfacemodel;viscoplastkity.
1. Introduction
Descriptionof the frictionphenomenonby a constitutiveequation has been attainedfirst as a rigid-plasticity2),3).
Further,it has
beenextendedto an elastoplasticity4)-17)
in whichthe penaltyconcept, i.e. the elastic springsbetweencontact surfacesis incorporatedand the isotropichardeningis taken into accountso as to describethe test results18)
exhibitingthe smooth contacttractionvs.
slidingdisplacementcurve reachingthe static-friction.However,
the interiorof the sliding-yieldsurfacehas been assumedto be an
elasticdomainand thus the plasticslidingvelocitydue to the rate
of traction insidethe sliding-yieldsurfaceis not described.Therefore,the accumulationof plasticslidingdue to the cyclicloadingof
contact traction within the sliding-yieldsurface cannot be describedby these models. They could be calledthe conventional
frictionmodelin accordancewiththe classificationof plasticconstitutivemodelsby Drucker19).
On the other hand, the firstauthor
of the presentarticlehas proposedthe subloadingsurfacemodel20)
21)within
the frameworkof unconventionalplasticity
, which is capable of describingthe plasticstrainrate by the rate of stressinside
•\ 271•\
the yield surface.Based on the conceptof subloadingsurface,the
authors proposed the subloading-frictionmodel1),22)
which describesthe smoothtransitionfrom the elasticto plasticslidingstate
and the accumulationof slidingdisplacementduringa cyclicloading of tangentialcontacttraction.Besides,in this model the reduction of frictioncoefficientwiththe increaseof normalcontacttraction observedin experiments15),23),24)
is formulatedby incoiporating the nonlinearsliding-yieldsurface,whilethe decreasehas not
been taken into accountin Coulombsliding-yieldsurface,which
has beenadoptedwidelyin constitutivemodelsfor frictionso far.
It is widelyknownthat when bodiesat rest beginto slideto
each other,a high frictioncoefficientappearsfirst, whichis called
the staticfriction, and then it decreasesapproachinga stationary
value, called the kineticfriction. However,this process has not
beenformulatedpertinentlyso far, althoughthe increaseof friction
coefficientup to the peak has been described as the isotropic
hardening,i.e. the expansionof sliding-yieldsurfaceas described
above.
Further,it has been foundthat if the slidingceasesfor a while
and then it starts again, the frictioncoefficientrecoversand the
similarbehavioras that in the initialslidingis reproduce25)-35).
The
recoveryhas been formulatedby equations includingthe time
elapsedafterthe stop of sliding26),28),30),31),33),35).
However,the inclusionof time itselfleadsto the loss of objectivityin constitutive
equationsas known from the fact that the evaluationof elapsed
time variesdependingon the judgmentof time when the sliding
stops,which is accompaniedwith the arbitrarinessespeciallyfor
the statevaryingslidingvelocityin low
Generallyspeaking,
the variationof materialpropertycannot be describedpertinently
by the elapsefroma particulartimebuthas to be describedby state
of internalvariableswithoutthe inclusionof timeitself
The reductionof frictioncoefficientfrom the staticto kinetic
frictionand the recoveryof frictioncoefficientmentionedabove
are to be the fundamentalcharacteristicsin frictionphenomenon,
which havebeen recognizedwidelyfor a longtime. Differenceof
the staticand kineticfrictionsoften reachesup to severalten percents.Therefore,the formulationof the transitionfromthe staticto
kineticfrictionand vice versa are of importancefor the development of mechanicaldesignin the field of engineering.However,
the rationalformulationhas notbeenattainedso far.
The differenceof frictioncoefficientsis observedin the mutuallyoppositeslidingdirections.It couldbe describedby the rotation of sliding-yieldsurface,whilstthe anisotropyof soilshas been
describedby the rotationof yield surface36)-38).
Further,the difference of the rangeof frictioncoefficientsis observedinthe different
slidingdirections.It could be describedby conceptof orthotropy
of sliding-yieldsurface14).
In this article,the subloading-friction
model1)is extendedso
as to describethe reductionof frictioncoefficientfromthe staticto
kinetic frictionas the plasticsofteningdue to the slidingand the
recoveryof frictioncoefficientas the viscoplastichardeningdue to
the creep phenomenoninduced with the elapse of time under a
high contactpressurebetweenedgesof surfaceasperities.It is further extendedso as todescribethe anisotropyby incorporatingthe
rotationand the orthotropyof sliding-yieldsurface.
(2)
n
is the
and •Z
and
of
unit
outward-normal
denote
I
is the
the
and
second-order
Kronecker's
Vn
vector
scalar
identity
delta ƒÂij
is the normal
the
=1
component
for
at the
tensor
contact
tensor
having
i=j
, ƒÂij
of the sliding
surface,
products,
the
components
=0
velocity,
(•E)
respectively,
for
i •‚
j
.
i.e.
(3)
where the sign of Vn is selected to be plus when the counter body
approaches to the relevant body.
Fig. 1. Contact
traction
f and sliding
velocity
v.
Further, it is assumed that V is additively decomposed
the elastic sliding velocity Ve
into
and the plastic sliding velocity
Vp , i.e.
(4)
with
(5)
(6)
(7)
2. Formulation of the constitutiveequation for friction
The subloading-friction
mode1)is extendedbelow so as to describethe static-kinetic
friction transition,i.e. the transitionfrom
staticand kineticfriction,and vice versa.
where Ves
respectively,
and Vpn
are the
elastic
plastic
part,
of Vn .
The contact traction
f
acting on the body is decomposed
into the normal part, i.e. normal-traction
2.1 Decompositionof slidingvelocity
The slidingvelocity V is definedas the relativevelocityof
the counterbody and is additivelydecomposedinto the normal
and the
part, i.e. tangential traction ft
fn and the tangential
as follows:
(8)
part Vn and the tangentialpart Vt as follows(seeFig. 1):
(9)
(1)
where
whilst n is identical to the normalized
i.e.
•\ 272•\
direction vectors of fn ,
(10)
describing anisotropy due to the rotation around the null traction
point without the normal component of contact surface, while it is
and fn is the normalpart of the contacttraction f , i.e.
assumed
thatitdoesnotevolve
leading
to I = 0 , andthusitholds
(11)
that
(18)
where
the
sign
of
body
is compressed
tions
of
the
velocity
fn
is selected
by the
tangential
are
I-n•Zn•‚t•Zt
counter
contact
not
to
be
body.
traction
necessary
plus
when
Here,
note
and
the
identical
(t•ßft/•aft•a)
in
the
the
that
relevant
the
tangential
in
(19)
direc-
The anisotropy
sliding
general
and
three-dimensional
be-
Now,
let the
hypo-elastic
far
elastic
relation,
small
compared
sliding
whilst
with
velocity
the
the
be given
elastic
plastic
sliding
sliding
by
velocity
by the translation
of
of friction
such
it is
phenomenon
of the sliding-yield
is de-
surface.
Then, it
holds that
is usually
in the
that the anisotropy
scribed by the rotation
the following
velocity
is described
yield surface but the anisotropy of frictional materials
as soils is described
by the rotation36)-38). Therefore,
assumed
havior.
of metals
friction
(20)
phenomenon.
(12)
where
lin and
component,
which
are
it
are
respectively,
related
to the
the
noimal
component
of
t , (•‹)
denoting
material-time
and
derivative
where
s is an arbitrary
rate,
based
on
(•E)
Therefore,
tangential
the corotational
denoted
by
(21)
the
f =0
(13)
with
what
follows,
In
surface
sliding
where
the
skew-symmetric
tensor Ħ
is the
spin
describing
normal
based
rotation
of
the
contact
surface.
an
and
at
are
the
elastic
contact
surface.
not
an absolute
tive
velocity
can
be
in the normal
On
the
velocity
between
adopted
objectivity.
moduli
to the
It follows
and
other
hand,
of a point
two
points
Eq.
the
sliding
on the body
on the
constitutive
from
the tangential
contact
relation
(12)
as
directions
velocity
surface
it is since
and
traction
the
interior
but
of traction
by Eq.
shape
space,
of
that
i.e.
the
slid-
the
inside
(17)
and
that
plastic
surface.
be renamed
as the
swface.
on
the
concept
of subloading
suiface20),21),
we
in-
the
shape
and
spect
to
relathus
it has
sliding-subloading
the
current
surface,
contact traction
which
point
f
always
and
passes
keeps
a similar
orientation
to the
normal
sliding-yield
surface
with
re-
is
the
= const
origin
.
of
Then,
the
contact
traction
space,
sliding-subloading
i.e.
surface
f =0
fulfills
for
the
fol-
it
lowing
geometrical
i)
lines
characteristics.
the
that
All
connecting
ing-subloading
the
(15)
an
arbitrary
surface
normal
called
where the second-order tensor Ce
described
degree-one.
similar
domain
is
to
v
but the
surface,
that
elastic
(21)
the
through
contact
of
the
of contact
assume
Eq.
the
troduce
rigid-body
keeps
by the rate
surface
sliding-yield
Then,
we
a purely
whilst
function
surface
is induced
let the
quantity,
to the origin
is not
velocity
Therefore,
(14)
respect
= const.
scalar
homogeneous
sliding-yield
for ƒÀ
ing-yield
which is derived from
Euler's
the
orientation
as follows:
positive
and
sliding-yield
the
point
its
surface
similarity-center,
inside
conjugate
join
which
at
is
the
point
a unique
the
slidinside
point,
origin
of
the
is the fictitious contact elastic
contact
traction
space
in the
present
model.
modulus tensor given by
ii)
All
ratios
necting
(16)
face
of
two
to
•\ 273•\
an
sizes
Let
responds
ratio,
0 < R < 1
to the
con-
sliding-subloading
conjugate
points
inside
The
coincides
sur-
line-element
identical.
size
of the
sliding-yield
denoted
to the null
the
line-element
the
normal
ratio
with
con-
is
the
slid-
called
ratio
the
of
the
surfaces.
of the
of the noimal
ing-yield
are
which
ratio
arbitrary
inside
conjugate
of these
the
an
arbitrary
surface
similarity-ratio,
that
(17)
where F is the isotropichardening/softeningfunctiondenoting
the variationof the size of sliding-yieldsurface. ti is the vector
of
two
ing-yield
of
points
that
necting
2.2 Normal sliding-yieldand sliding-subloadingsurfaces
Assumethe followingisotropicsliding-yieldsurfacewith the
isotropichardening/softening,which describes the sliding-yield
condition.
length
surface
by R
traction
subsliding
sliding-subloading
be
(0 •… R •…
state
( f
state
= 0)
(0
< f
called
surface
the
1) , where
as the most
< F
to
normal
slid-
R = 0
cor-
elastic
), and R
state,
=1
to
the normalsliding-yieldstate in whichthe contacttractionlieson
the normal sliding-yieldsurface( f = F ). Therefore,the normal
sliding-yieldratio R playsthe role of three-dimensional
measure
of the degree of approachto the normal sliding-yieldstate.Then,
the sliding-subloading
surfaceis describedby
(22)
The material-timederivativeof Eq. (22) leadsto
(23)
where
(24)
Here,
note
rial-time
by
that
the
derivative
direct
to the
substituting
a•E(ƒ¶a)
= 0
formation
Eq.
for
of
tional
transformation
corotational
an
the
(13)
arbitrary
into
is
for
the
peraturein general.The first and the second terms in Eq. (25)
stand for the deteriorationsand the formations,respectively,of the
adhesionsbetween surfaceasperities.On the other hand, so far
these phenomenahave been describedby separateformulations
for the softeningdue to the slidingdisplacementand the hardening
due to the timeelapsedafterthe stopof sliding.Here,the inclusion
of the time itselfin constitutiveequationS26),28),30),31),33),35)
is not allowed violatingthe objectivitysincethe evaluationof elapsedtime
from the stop of slidingdependson the subjectivityas knownfrom
the statevaryingslidingvelocityin low level.
mate-
is verified
a . The
noting
direct
derivative
verified
the
Eq.(23),
vector
material-time
derivative
of
derivative
the materialconstantsinfluencingthe recoveringrateof F dueto
the elapseof time, whilethey wouldbe functionsof absolutetem-
to
trans-
the
general
corota-
scalar
func-
tion39).
2.3
Evolution
rules
normal
It could
1 ) If the
the
of
the
sliding-yield
be stated
sliding
from
function
experiments
commences,
maximal
minimal
hardening
and
the
ratio
value
of
stationary
the
that
friction
static-fiction
value
of
coefficient
and
reaches
then
it reduces
kinetic-friction.
Fig. 2. Function U(R) for the evolution rule
of the normal sliding-yieldratio R .
first
to the
Physically,
this
It is observed in experiments that the tangential traction inphenomenon
tions
of
bodies
could
the
to
is
ing-yield
and
caused
by
friction
Physically,
the
the
between
elapsed
edges
time
of
could
of
with
initial
after
phenomenon
the recovery
the creep
Taking
is reproduced
the
due
be
the
adhesions
of
under
a quite
high
surface
asperities.
elapse
the
of
time.
to be caused
ing-yield surface and it does not increase any more when it reaches
rule of the normal sliding-yield ratio as follows:
(26)
where U(R) is a monotonicallydecreasingfunctionof R fulfillingthe followingconditions(Fig 2).
(27)
it be
hardening
assumed
due
to
Letthe function U satisfyingEq. (27) be simplygivenby
account
of these
facts,
let the
function
F
evolution
rule
be postulated
(28)
of the isotropic
as follows:
where u
lead
is the
to the analytical
material
constant.
integration
sliding up •ß •aVp•adt
where Fs
creasing
slid-
pressure
(25)
tively. k
the normal
phenomenon.
hardening/softening
minimum
approaching
asperities
contact
let
it increases gradually
time
of sufficient
by the viscoplastic
thereafter
the normal sliding-yield surface. Then, we assume the evolution
with
surface
Then,
slid-
behavior
interpreted
the
is caused
normal
the
coefficient,
sliding
an elapse
that
to the sliding.
of fiction
gradually
as
the
creases almost elastically with the plastic sliding when it is zero but
contact
assumed
of
softening
separa-
between
it be
contraction
recovers
this
let
the reduction
behavior
reconstructions
during
that
after
coefficient
the static
the
by the
asperities
Then,
i.e. the plastic
ceases
identical
to be caused
surface
sliding40).
the
by
of
the
surface,
2 ) If the sliding
friction
interpreted
adhesions
due
reduction
be
and
values
and
rate
Fk
of
m
of
F
(Fs •†
F
are
due
F •†
Fk)
for the static
the
material
to the
plastic
are
and
the
kinetic
constants
sliding,
maximum
frictions,
influencing
and
and
and
respecthe
de-
n
are
of
under
Ro
as follows:
Eq.
R
(26)
with
Eq.
for the accumulated
the
initial
(28)
can
plastic
condition
up - uo0 : R =
(29)
On the other hand, the following function has been used widely so
•\ 274•\
far.
2.5
(30)
However,an analyticalintegrationcannot be obtainedfrom Eq.
formulated
(26) with Eq. (30) and thus Eq. (30) is inconvenientto formulate the return-mappingmethod attractingthe contacttraction to
the subloading suliace41).
2.4 Relationshipsof contact traction rate and slidingvelocity
The substitutionof Eqs. (25) and (26) into Eq. (23) gives
riseto the consistencyconditionfor the sliding-subloading
surface:
Loading
The
in the loading
2. It holds
this
proportionality
factor
, is derived
Eq.
(32)
into
Eq.
(31),
of
(plastic
sliding)
(elastic
v =v
process
process
sliding)
e leading
it holds
it should
The
Vp •‚
process
0 .
Vp
to N•ECe•Ev=
= 0 . Further,
N•ECe•Eve=
be-
N•Ef in
that
in
hand,
the
that
modulus
negative
other
and
be noted
plastic
and
is
the proportionality
facts:
that
unloading
while
0
(Eq.
takes
(36)).
both
signs
hardening/softening
that
the
definite
>>‚•p
in
relaxation
does
lowing
inequality
be
tensor
not
of
positive
materials.
contact
general
plastic
factoră
mc •†
mP
noting
positive
N•ECe•EN
(33)
Substituting
equation
(43)
(32)
is a positive
constitutive
(42)
3.
0)
for the
is given in this section.
(41)
(31)
Assumethatthe directionof plasticslidingvelocityis tangentialto the contactplane and outward-nonnalto the curvegenerated
by the intersectionof sliding-yieldsurfaceand the constantnormal
tractionplane fn = const. , leadingto the tangentialassociated
(>
criterion
in the foregoing
First, note the following
cause
where ă
loading
1. It is required that
in the
flow rule,i.e.
criterion
elastic
and
and
On
thus
it
Ce
holds
postulating
proceed
the
modulus
that
that
infinitely,
let
the
the
fol-
assumed.
as follows:
(44)
(34)
Then, in the unloadingprocess Vp = 0 the followinginequalitieshold dependingon the sign of the plasticmodulusmP ,
i.e. the hardening,perfectly-plasticand softeningstatesfrom Eqs.
and thus
(35)
(34) and (41)-(44).
where
(36)
(37)
SubstitutingEqs. (4) and (35) into Eq. (31),the slidingvelocityis givenby
(45)
Therefore,
the
in which
The
locity,
positive
denoted
proportionality
by the
symbol •È
factor
in terms
, is given
from
of the
Eqs.
sliding
(38)
(38)
sliding
ve-
tive
in
(39)
of ă
is not
the
unloading
and
an
of ă
but
rion
is given
in the
perfectly-plastic
is induced,
loading
as
the
sign
can
unloading
be done
or
negative.
unloading
process
softening
proceeds
On
process.
Thus,
processes
by that
of •È.
the other
the
cannot
from
if the
hand, •È
distinction
be judged
Therefore,
the
the
state,
plastic
is negabetween
by
loading
the
a
sign
crite-
as follows:
The traction rate is derived from Eqs. (4), (15), (32) and
(46)
(39) as follows:
(40)
where < >
an arbitrary
is
the
scalar
McCauley's
variable
bracket,
i.e. <s>
= (s+ •b
s •b)/2
or
(47)
for
s .
in lieu of Eq. (44).
•\ 275•\
(56)
3. Specific sliding-yield surfaces
It can be stated from experiments that the friction coefficient
decreases with the increase of contact pressure15),23),24),42).There-
(57)
fore, the normal sliding-yield surface cannot be described appropriately by the Coulomb sliding-yield surface in which the tangential contact traction and the normal contact traction are linearly re-
where
lated to each other using the angle of external friction and the ad-
(58)
hesion. In what follows, the sliding-yield surface with thenonlinear relation of tangential contact traction and normal contact trac-
Further,it holdsfromEqs. (16) and (55)-(57)that
tion is assumed below, by which the reduction of friction coefficient with the increase of normal contact traction is described.
The closed
normal
sliding-yield
and the sliding-subloading
surfaces
can be described
(59)
by putting
(48)
(60)
as follows:
(49)
where
(50)
M is the material constant denoting the traction ratio
(=ft/fn) at the maximum point of ft . The simple examples of the function g(X) in the sliding-yield function
in Eq. (48) are as follows:
(61)
(62)
(51)
(52)
(63)
(53)
(54)
(64)
All the sets of Eqs. (17) and (48) with Eqs. (51)-(54)
exhibit the closed surfaces passing through the points
fn = 0 and fn = F at ft = 0 . Eq. (51) and (52) are
based on the original Cam-clay yield surface43)and the
modified Cam-clay yield surfaces44),respectively, for soils.
Eq. (53) exhibits the tear-shaped surface1)45),46)which is
reversed from the surface of Eq. (51) on the axis of normal contact traction. Eq. (54) exhibits the parabola1).
It holds for Eq. (48) that
The substitutionof Eqs. (16) and (59)-(64) into Eqs. (38)
and (40) leadsto the slidingvelocityvs. contacttractionrate and
its inverserelationare givenas follows:
(55)
(65)
•\ 276•\
(76)
The substitutionof Eqs.(16) and (72)-(76) into Eqs.(38)
and (40) leads to the slidingvelocityvs. contactfractionrate and
its inverserelationare givenas follows:
(66)
(77)
On the other hand, the normal sliding-yield and the sliding-subloading
surfaces for the circular cone of the Coulomb fric-
tion condition is given by putting
(67)
as follows:
(68)
where ƒÊ
is the
in the identical
friction
form
coefficient
with
and
Eq.(25)
the evolution
rule
is given
as follows:
(69)
ƒÊs
and ƒÊk
and
the
cients,
are
minimum
frictions,
respectively.
function
of
f
of ƒÊ
and
i.e.
f(f,ƒÀ)
surfaces
shape
constants
and
thus
static
in
in degree-zero,
sliding-subloading
conical
material
the
friction
Eq.(67)
and
in Eq.(68)
expand/contact
designating
and
the
kinetic
is
normal
the
maximum,
increase/decrease
anisotropy
site sliding directions can be described by the aforementioned
and
surfaces
to orthotropic
The difference of fiction coefficients in the mutually oppo-
homogeneous
sliding-yield
with
the
4. Extension
coeffi-
are open
(78)
having
rota-
tional anisotropy. However, the difference of the range of friction
a
coefficients in the different sliding directions cannot be described
by the rotational anisotropy. In order to extend so as to describe it,
R.
let the concept of orthotropy be further incorporated below.
It holds for Eq.(68) that
(70)
(71)
Further,it holds fromEqs.(16) and (57) that
(72)
(73)
(74)
(75)
Fig. 3. Surface asperity model suggesting
and the orthotropic anisotropy.
the rotational
The simple surface asperity model is illustrated in order to
obtain an insight into the anisotropy in Fig.3. Here, the directions
•\ 277•\
M1
in the inclinationof surfaceasperitieswould lead to the rotational
anisotropy,and the anisotropicshapes and intervalsof surfaceasperitiesto the orthotropicanisotropy.Now, choosingthe bases el
and e*2in the directionsof the maximumand the minimumprincipal directionsof anisotropy,respectively,and letting e*3coincide with n so as to make the right-handcoordinatesystem
(e*1,e*2,),
it canbe writtenas
(79)
while
the spin Ħ
of the base
(e*1,
e*2,
e*3)
is described
as
(80)
Eq.(79)
is rewritten by
(85)
where
as follows:
(86)
(81)
In Fig. 4 the section of the sliding-yield surface with the
rotational and the orthotropic anisotropy is depicted in the
coordinate system with the bases (e*1,e*2).
Invokingthe orthotropicanisotropyproposedby Mroz and
Stupkiewicz(1994),let Eq. (48) with Eq. (49) taken accountof
the rotationalanisotropybe extendedas follows:
The subscript i takes 1 or 2 and is not summedeven when it is
repeated
It holds fromEqs. (16) and (85) that
(87)
(82)
(83)
where
(84)
, and M2 are the material constants standing for the values of
M
in the maximum and the minimum principal directions of ani-
sotropy, respectively.
(88)
(89)
(90)
Fig. 4. Sliding-yield
surface with the rotational
and the orthotropic
anisotropy.
Thepartialderivativesfor Eq. (82) are givenas
―278―
(91)
(94)
Eq. (67) with Eq. (68) for Coulombfrictionconditionwith
the rotationalanisotropyis extendedto the orthotropicanisotropy
as follows:
(95)
(96)
(92)
where
The substitutionof Eqs. (16) and (88)-(92) into Eqs. (38)
and (40) leads to the slidingvelocityvs. contacttractionrate and
its inverserelationare givenas follows:
(97)
C1 and C2 are the materialconstants.The partialderivativesfor
Eq. (95) are givenas follows:
(93)
(98)
Further,it holds fromEqs. (16) and (98) that
(99)
(100)
―279―
(101)
(102)
(105)
(103)
The substitutionof Eqs. (16) and (99)-(103)into Eqs. (38)
and (40) leads to the slidingvelocityvs. contacttractionrate and
its inverserelationare givenas follows:
The calculationfor slidingwiththe orthotropicanisotropyhas
to be performedinthe coordinatesystemwiththe principalaxesof
orthotropy,i.e. (e*1,e*2,n).
5. Linear slidingphenomenon
We examinebelow the basic responseof the presentfriction
modelby the numericalexperimentsand the comparisonwithtest
data for the linearslidingphenomenon(Fig.1) without a normal
slidingvelocityleadingto
(106)
The traction rate vs. sliding velocityrelation for Eq. (82)
with Eq. (83) under the condition(106) is given from Eqs.
(40), (90)-(92) by
(104)
(107)
while it holds that fn=const.
― 280―
The tractionrate for Eq. (95) with Eq. (96) is given from
Eqs. (40) and(101)-(107) by
locity. This advantage is of importance especially for the analysis of cyclic friction phenomena
in which a loading and an
unloading are repeated
6. The difference of friction coefficients in the mutually opposite
sliding directions and the difference of the range of frictioncoefficients in the different sliding directions are described by the
rotational and the orthotropic anisotropy, i.e. the rotation and the
orthotropy of sliding-yield surface.
The constitutive equation of friction formulated in this article
would be applicable widely to friction phenomena between solids.
It will be extended so as to be applicable to rubber-like material exhibiting a large nonlinear elastic behavior in thefuture.
References
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(108)
6. Concluding remarks
The constitutive model for friction is formulated by extending
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locity due to therate
ing-yield
surface
surface in which the plastic sliding ve-
of contact traction inside the normal slid-
isdescribed
exhibiting
the smooth
tic-plastic transition. It is inevitable for the prediction
elasof the
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in wheel rotation on a solid surface for instance.
4. The reductionof frictioncoefficientwith the increaseof normal
contacttractionis describedby incorporatingthe nonlinearsliding-yieldcondition.
5.A judgmentwhetheror notthe slidingyield conditionis fulfilled
is not requiredin the loadingcriterionfor the plasticslidingve-
•\ 281•\
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(Received:April 14,2008)
•\ 282•\