subloading-friction model with a smooth elastic-plastic sliding transition is ... Key Words: Constitutive law, friction, hardening/softening, subloading, time- ...
Journal of Applied Mechanics Vol.9, pp.503-511
Constitutive
equation
(August 2006)
for friction
K.
JSCE
with static-kinetic
friction
transition
Hashiguchi*andS. Ozaki**
* Member Dr. of Eng.andDr. ofAgr.,Prof, Divisionof Bio-production andenvironmental Science, KyushuUniversity(Hakozaki6-10-1,Higashi-ku,Fukuoka812-8581,Japan) **Dr. of Agr.,ResearchAssociate, Department of MechanicalEngineering, TokyoUniversity of Science(Kudankita1-14-6,Chiyoda-ku,Tokyo102-0073, Japan)
A high frictioncoefficientis firstobservedas a slidingbetweenbodiescommences, whichis calledthestatic-friction. Then,thefrictioncoefficientdecreasesapproaching the loweststationaryvalue,whichis calledthekinetic-friction. Thereafter, if the slidingstops fora whileandthenit startsagain,the frictioncoefficientrecoversanda similarbehavior as thatinthe firstslidingis reproduced. Thesearefundamentalcharacteristics in the friction phenomenon, whichhavebeenwidelyrecognizedfor a longtime.In this articlethe subloading-friction modelwitha smoothelastic-plastic slidingtransitionis extendedso as to describethese factsby formulating the rate-dependent hardening/softening rule of sliding-yield surfaceadequately. Key Words:Constitutive law,friction,hardening/softening, subloading,time-dependency
traction
1. Introduction
observed
rating Description equation they
of
has
have
penalty
been
been
i.e.
account
so as to
contact
traction
has
assumed
the
and
loading
models.
model
in
tive
They
by
present
21) within capable inside
the of
the
framework
concept crease
the
of
the
of
rate
the
slid-
the
due
by
the
in increase
ess has
for
by
first
value, not been
coefficient
a while
and
the
the
of
model19) •` , which
rate
of
is
the
transition
incorporating model
the
of
normal
contact
de-
of time
is accompanied velocity
variables
and their
― 503―
appears
has
which
has
which
However,
this
the increase
that
the
is
approaching
described
found
to slide to
first,
it decreases
again,
as that
itself
leads
a proc-
of fric-
as the hardening if the sliding
static-friction
in the initial
ceases recovers
sliding
is repro-
to the loss of objectivity
from
with
the fact
that
of time
as the stop
the arbitrariness
fluctuates.
of material
are
been
it starts
on the judgment
variation
above
law
at rest begin
so far, although
as known
which
netic-friction
then
has been
behavior
equations
The
bodies
the kinetic-friction.
it has
incorpo-
The recovery has been formulated by equations24) the time elapsed after the stop of sliding . However,
depending
the
this
then
varies
sliding
stress
proposed
Further,
friction
coefficient and
formulated
identical
duced24)•`36. 36) including
stitutive
friction
by
the decrease
models.
when
up the peak
and
while
in Coulomb
that
called
is described
surface,
in friction
known
a high
the inclusion
author
widely
static-friction,
friction constitu-
smooth by
the
process3)•`16).
to
described
surface
authors
state Besides,
with
the
plasticity
describing
surface.
plas-
inside
the
other,
called
tion
surface
plastic
subloading
plastic-sliding
coefficient
the
the
conventional
hand,
strain
Further,
subloading
of friction
other
the
plastic
surface.
to the
the
On
model22)
elastic-
called
classification
each
sliding-yield
into account
adopted
stationary
smooth
thus
be
of unconventional the
yield
and
cannot
the
proposed
the
of plastic-sliding
traction be
into
reaching
traction
accumulation
could
describing
of
sur-
taken
It is widely
the
contact
sliding-yield
domain
rate
contact
has
the
which
is taken
exhibiting curve
of
elastic the
Drucker18).
subloading-friction from
to
with
article
interior
in
between
results17)
been
Further,
hardening
displacement
an
the
of
accordance
models
test
the be
due
surface
cyclic
these
the
to
springs
not been
a constitutive
elastoplasticity3)•`16)
isotropic
sliding
velocity
ing-yield
an
the
describe
as
as a rigid-plasticity1),2).
fictitious
and
However,
been
tic-sliding
the
to
the
vs.
phenomenon
first
extended
is incorporated
static-friction.
friction
attained
concept,
faces
the
in experiments23),24)
the nonlinear
Here,
property
has
be
when
the
that
the
noted
to be described
time
of sliding,
especially
it should
in con-
the elapsed
by
internal
rates.
decrease
of friction
and the recovery to be the
fundamental
coefficient
from
of friction
coefficient
behavior
the static-
of friction
to ki-
mentioned between
bodies, which have been recognized widely. Difference of the static- and the kinetic-frictionoften reaches up to several tens percents. Therefore, the formulation of the transition from the static-to the kinetic-frictionand vice versa is of importancefor the development of mechanical design in the field of engineering. However, the rational formulationhas not been attained so far. In this article the subloading-frictionmodel22)is extended so as to describe the decrease of friction coefficient from the static- to kinetic-frictionby the softening due to the plastic-sliding and the recovery of friction coefficient by the hardening due to the creep deformationunder the contactpressure. The extended model would be called the time-dependent
(6) and,
(•‹)
normal the
denoting component
corotational
the
corotational
and
tangential
rate f
of the
rate, fn
and
component, traction
ft
are
the
respectively,
vector
f,
of
i. e.,
(7) which
are
related
to
the
material-time
derivative
denoted
by
(•E) as follows:
(8)
subloading-frictionmodel. 2. Formulation of the constitutive equation for friction where
The subloading-frictionmodel proposed recently by the authors(22)is extended below to the time-dependent friction model describing the static-kineticfriction transition, i.e., the transitionfrom static-to kinetic-frictions,and vice versa.
the
rigid-body
skew-symmetric
rotation
contact
penalty
elastic
the
tensor Ħ contact
parameters
moduli
contact
of
in the
surface.
an
representing
normal
Thus,
designates
surface.
and
it follows
and
the
the
Eq.
are
fictitious
tangential
from
the
at
contact
directions
(5)
the
to
the
that
(9) 2.1 Decomposition of sliding velocity The sliding(relative)velocity r between contact surfaces is additively decomposed into the normal component rn and the tangentialcomponent rt as follows: (1)
where the second-order
tensor
tic modulus tensor between
Ce
is the fictitious contact elas-
contact surfaces
into the normal and tangential
and is decomposed
components,
i. e.,
(10) with
which are given as
(11) (2) where
n
(•E)
and •¬
tively,
is the
and
sumed
that
alty)-sliding
unit
outward-normal
denote I
the
is the r
is
velocity
scalar
identity
and
the
tensor.
additively re
vector
On
decomposed and
the
at the tensor the
contact
products, other
into
hand,
the
plastic-sliding
surface, respecit is as-
elastic
(pen-
velocity
rP,
i. e.,
2.2 Normal-sliding and sliding-subloading surfaces Assume the following sliding-yieldsurface with isotropic hardening/softening,which describes the sliding-yield condition. (12) where •a•a
(3) with
designates
terior but
surface.
the
that
Eq.
that
(12)
be in
accordance
face19)•`21),
we
introduce
let
normal-sliding
with
the
the
size
the
elastic
by the
the
that
in-
domain
variation
surface
of
described
surface.
concept
of
subloading
sur-
velocity be given by always
(5)
passes
shape
and
respect
to
through
a same the
ing-subloading
where
the
assume
is induced
isotropic
of
a purely
Therefore,
renamed
Next,
is the
variation
we
is not
velocity
surface.
F
the
follows,
surface
plastic-sliding
inside
and
denoting
In what
sliding-yield
the
traction by
First, let the elastic-sliding
of
magnitude,
function
of sliding-yield
(4)
the
hardening/softening
fn and ft are the normal
component
the
sliding-subloading
the
current
orientation
to
zero surface
traction fulfills
traction
surface, f and
the
normal-sliding
point
f=0.
the
following
keeps
which a similar
surface Then,
the
geometrical
with slidchar-
and tangentia acteristics.
component,
respectively,
of the traction
vector
f
applied
to a i)
All
lines
connecting
an
arbitrary
point
inside
the
slid-
unit area of contact surface, i. e. ing-subloading
•\ 504•\
surface
and
its
conjugate
point
inside
the
normal-sliding
called the ii)
All
the
contact
traction
ratios
of
necting face
two to
surface
that
length points of
an
two
mal-sliding
surface
the
similarity-ratio,
the
sizes
denoted
by
of
arbitrary
state
subsliding
mal-sliding
state
(f=F).
an
the
arbitrary
are
a
unique
is
the
present
of
model.
line-element
conjugate
identical.
pressure. Then, let it be assumedthat the recovery is caused by the expansion of the normal-slidingsurface,i.e. the isotopic hardening due to the creep deformationunder the contact pressure.
consur-
line-element inside
The
coincides
time. Physically, this phenomenon could be interpreted to be caused by the reconstructionof the adhesion of surface asperities between contact bodies subjectedto the contact
point, origin
sliding-subloading
points
which
the
ratio with
is the
nor-
Taking account of these facts, let the evolutionnile of the isotropichardening function F be postulatedas follows:
called ratio
of
surface
to
surfaces. of the sliding-subloading
surface
R (0•¬R•¬1),
null traction
surface
in the
the similarity-ratio
the normal-sliding
to the
space
conjugate
of these
at
which
inside
connecting
Let
join
similarity-center,
called
where
(f=0) state
be
the
as the most (0
