Dry Friction as an Elasto-Plastic Response: Effect of Compressive Plasticity

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Dry Friction as an Elasto-Plastic Response: Effect of Compressive Plasticity

C. Caroli, B. Velicky

To cite this version:

C. Caroli, B. Velicky. Dry Friction as an Elasto-Plastic Response: Effect of Compressive Plastic-

ity. Journal de Physique I, EDP Sciences, 1997, 7 (11), pp.1391-1416. �10.1051/jp1:1997137�. �jpa-

00247459�

Dry Friction

_{as an}

Elasto-Plastic Response:

Eflect of Compressive Plasticity

C. Caroli

_(~>*) and B.

Velicky

_(~>~)

(~)

Groupe

de

Physique

des

Solides,

Universit4s Paris VI et Paris

VII,

2

place Jussieu,

75251 Paris, Cedex 05, France

(~) Faculty

of Mathematics &

Physics,

Charles University, Ke Karlovu 5, 12116 Praha 2, Czech

Republic

(Received

22 May

1997,

received in final form and accepted 24

July 1997)

PACS.46.30.Pa

Friction,

_wear,

adherence, hardness,

mechanical contacts, and

tribology PACS.62.20.Hg Creep

PACS.81.40.Pq Friction,

lubrication, and _wear

Abstract. Unlike the well known Amontons-Coulomb laws of dry friction, the recent precise

experimental

studies of low

velocity

friction

dynamics

provide sensitive tests for

a theoretical description. We introduce _a

model,

in which the friction force originates from

multistability

induced

by

interactions of individual pairs of

asperities

whose

properties undergo

_a slow time evolution due to

plastic

_creep induced

by

the

compressive

^forces acting _on the real contacts.

Even _a

crudely

schematic form of the model

reproduces

the time

strengthening

of the static friction force and the

velocity weakening

of the

steady

motion

dynamic

one.

Instability

with respect to the onset of stick

slip

is

found,

but _open questions remain

concerning

the features of the

corresponding bifurcation, pointing

towards the need to extend the model to take into

account shear-induced

plastic

effects.

1. Introduction

Recent

experiniental

advances in

experinients

_on friction have

put

to the fore

again

the idea

that,

_{as was}

already argued long

ago

by

Rabinowicz

iii,

it is

by studying

the

velocity (resp.

tinie)

variations of

dynaniic (resp. static)

friction

coefficients, and,

_niore

generally,

the details of frictional

dynaniics

that _{one can}

get

the best inforniation about the nature of the relevant

physical

_processes and about their

dependence

_on niaterial

properties.

Indeed,

the work of Greenwood et at. [2] ^{has niade clear} that the niost salient feature of

dry

friction

naniely

the fact that the friction force F is

independent

of the

apparent

area of contact and

proportional

to the normal load N

(Aniontons-Coulonib laws)

is

priniarily

_a

geonietrical

effect. That

is, they

have shown

that,

whether the real

(niicro)

contacts between the two surfaces involved _are

conipletely

elastic

(Hertzian)

_or

plastic _[3],

as soon as one takes into account the fact that real surface

profiles present asperities

with randoni sizes and

heights,

the real _area of contact is

(quasi) proportional

to N. The

only

condition for this to hold

(*)

Author for correspondence

(e-niail: [email protected])

©

Les

#ditions

_de

Physique

1997

is that the distributions of

heights

and sizes do not exhibit

slowly decreasing

^{tails. Hence the}

fact that Amontons law is

essentially

material

independent

and does not

yield

_any inforniation

on the details of the

physical

_processes at work.

Absolute values of friction coefficients _~t

=

FIN

_are not very inforniative

either,

_as

they

do not vary very

significantly

_nor

systeniatically

between different classes of niaterials [3].

In order to go further in the

discussion,

it is

iniportant

to recall

experimental

facts which

are still _{now very} often

overlooked, inspite

^{of the}

eniphasis put

_upon theni

by

Rabinowicz and

by

several rock niechanicians

[4].

^The

"high

school niodel" of friction states that:

there is _a static friction coefficient _~ts,

corresponding

to

thi

niininiuni force needed to set a solid into

sliding

niotion _on

top

of another

solid,

the

dynaniic

friction coefficient is

velocity-independent

and _~td < ~ts. ~ts and ~td thus _appear

as two nunibers

characterizing

_a

given couple

^of solids.

This is in fact not true. Well controlled

experinients

_on a wide

variety

of niaterials have

shown that _~ts and _~td exhibit sniall but nieasurable

variations, respectively

with the tinie of contact

prior

to

sliding (waiting

tinie

tw)

^and with the

sliding velocity.

Moreover,

the

dynaniics

of

sliding

is

coniplex. While,

in _sortie ranges of values of the control

paranieters

of the

dynaniical systeni

fornied

by

the slider and the

pulling

niachine

(pulling velocity

V and stiffness of the

pulling systeni K), stationary

niotion at

velocity

V is

observed,

in other

regions

of

parameter

_space, the

stationary reginie

is unstable

against periodic stick-slip.

Intermittencies have also been observed.

That

iniportant physical

inforniation _can be

gained

front the

study

of this

dynaniics

has been

proved by

Dieterich [5] whose

pioneer

work _on transients

following velocity junips

in the

stationary reginie showed,

in

agreenient

with

previous suggestions

of Rabinowicz

ill,

that the

dynaniics

of

dry

friction involves _{a nieniory}

length Do, typically

on the order of niicronieters.

He has

proposed,

and

recently

illustrated

by

direct

optical experinients _[6],

that this

length

is

directly

related with the size of the microcontacts.

This

work, together

with the recent

systematic exploration by

Heslot et at.

ii,

_8] of the

dynamical phase diagram

of _a paper-on-paper

system,

has

given

renewed

impetus

to the for- mulation of

phenomenological equations describing

the

dynamics

of friction in various

physical

situations. We have

proposed

and used _a

phenomenology inspired

from

previous

work

by

Ruina and Rice [9] ^to

analyze

the

stick-slip

bifurcation of paper, as well _as other

dynamical

features of this

system.

Carlson and Batista

[10]

^{have set} up a

dynamical description

of lubricated friction between

atomically planar surfaces,

while

recently

Persson

ii i]

has formulated _a model for the _same kind of

system

^{in the} case where the lubricant is _a

grafted

film

(which

therefore

does not exhibit _a shear induced

nielting transition).

We will

specialize

front _{now on} to the _case of

dry (~lnt~lbricated) friction.

^The success of the

analysis

of the _paper

systeni

based _on the

phenonienological equations ininiediately

leads to _a further

question:

if the

phenonienological equations

we have used _are, at least to _sortie level of

approxiniation, correct,

it niust be

possible

to

derive,

rather than

nierely

_assunie theni _on the basis of

plausibility argunients,

^front a niore

"niicroscopic"

theoretical

approach.

Also,

the

analysis

of

experiniental

data leads to a

puzzling question.

The forni of the assunied

expression

of the

dynaniic

friction force

iniplies

that:

the

dynamics

_can be reduced _to the translational niotion of the _center of _niass,

this _occurs via noise activated niotion _across

potential

barriers whose

height

increases with

the

"age"

of the microcontacts

(I.e.

the tinie needed to slide _a characteristic distance

Do)-

Such _an

analysis

of

experimental

data

yields

a value of the

strength

of the noise

acting

_on that

single degree

^{of freedom several orders of}

magnitude larger

than the most

"optimistic"

direct estimate. This casts _a doubt at least _on the

physical interpretation

^{of this heuristic}

model,

and

possibly

_on the form of the model itself

(I).

These remarks

point

to the need for

building

_a theoretical frame which would

permit

to

incorporate,

at least in _a

partly phenomenological

_way, the

qualitative knowledge

about the main

physical

mechanisms at work in

dry

friction which _emerges from

experiments.

The recent results of

Baumberger

et al. _on paper and PMMA

[12,13j

confirm

that,

_as first

proposed by

Tabor [3], for most materials under usual

loading

conditions

solid/solid

_contact is associated with _a finite elastic contact

stiffness,

the slow increase with

waiting

time of _/1~ and the

corresponding velocity weakening

of

~ld can be

interpreted

_as due _to the

plastic (or viscoplastic)

evolution of the

microcontacts,

which results in their slow

strengthening.

This is also confirmed

by

the recent observation

by

Dieterich and

Kilgore _[§],

on various

transparent materials,

of the

(quasi logarithmic)

slow

growth

of contact sizes.

That

is, dry

friction _must be modelled _{as an}

elasto-plastic

_response.

2.

Hysteretic

Elastic Model

Al _a first step towards this

goal,

Nozieres and _one of _us formulated _a model

[14j

^{in which} friction results from the

purely

elastic _response of the

asperities

involved in the microcontacts.

We first

briefly

outline here the basic features of this

model,

_as well _as its conclusions and limitations.

Asperities

_are

schematically

modelled _as dilute

pinning

centers,

randomly

distributed _on the two solid

surfaces,

which _are

swept through

_as the slider _moves.

They

are constrained to _move in the

sliding

direction

only (quasi-lD model).

Two

asperities

interact

(I.e.

_a microcontact _or

active

trap

is

created),

when the distance _r between their centers is smaller than their radius a, via _a

pinning potential 4l(r).

This

potential,

which remains at this

stage phenomenological,

can be understood _as

resulting

from the energy of elastic

compression

_necessary for the prongs to retract in order to allow

crossing.

It _may be either

repulsive

or

attractive, depending

_on the local relative

geometry

^{of the} two surface

profiles.

If the

asperity height

is

comparable

to their

width,

its maximum

strength

is of order

Ea~,

with E the elastic modulus of the slider

(~).

Let p~ be the

configuration

coordinate of _a

given trap I,

I.e. the

corresponding interasperity

distance in the

system

ⁱⁿ the absence of shear

elasticity.

In the

real, deformable, system,

it

undergoes,

under the effect of the

pinning force,

_a horizontal

displacement

_t1~, measured from

a reference

point

at

infinity

_on the scale _a, for

example

the

point

^{where the external}

pulling

force is

applied.

The total energy of the

system

^is then the _sum of the

pinning

_energy ^{of the}

contacts between the sheared surfaces and of _an elastic term:

Utot

=

£

_{4~(pz + t~z) +}

L _{~zjiliUj Ii)}

z z,j

The

~q's

define _an elastic stiffness matrix. The elements of its

inverse, (~~l)q,

_measure the horizontal

displacement

_t1~ at the site of contact I

resulting

from _a unit horizontal force

applied

at contact

j. So, (~~~)m

Ge

I/Ea,

while off

diagonal

elements

(~~~)q

_m

I/Edq,

where

dq

is

the distance between contacts I and

j.

^Since contacts _are sparse,

dq

» a, so that ~m Ge

Ea, lzj

G~

Ea~/dq

_<

In.

(~)

Indeed,

the behavior close to a bifurcation

yields only

limited information _{on a}

dynamical

system,

so that the

corresponding

fits should be considered _{as a}

plausibility

test

only.

(~) ^The track is assumed to be _non deformable.

Finally,

low

velocity sliding

is described _as

quasistatic

motion of the

asperities. Indeed,

the time scale for

sweeping through

a

trap potential

_(Ge

a/V)

is

considerably larger

than the

asperity

acoustic relaxation time

(typically,

^{the time for}

propagating

sound _out of _a

region

of scale

a).

The

analysis

of this schematic model leads _to conclusions which _can be summarized _as follows:

(I)

^As

already pointed

out

long

_ago

by

Tomlinson

[15],

^and even earlier

by

Brillouin

[16], dry

friction

(I.e.

the existence of _a finite

drag

force in the low

velocity

limit and of _a finite static

threshold)

exists

only

if the elastic

trap

_response is multistable. This _{can occur}

only

if the

trap potential

is

strong enough (or, equivalently,

the bulk

rigidity

of the slider weak

enough)

for the condition

lj~

<

14ljj

=

minjd~4ljr)/dr~)j

to be fulfilled. The friction coefficient is then

proportional

to the _area of the

corresponding hysteresis cycle.

(ii)

Elastic interactions between _contacts

play

_a

negligible role,

_as

long

_{as one} focuses _on the average friction force

(they certainly

have _a

major

influence _on the noise

spectrum).

This

behavior,

which contrasts with the _case of vortices _or

charge density

_waves, where _even weak

traps

can

give

rise to

pinning,

results from the

dimensionality

of the

dry

friction

problem.

Here _{we are}

dealing

with _a 2D random set of _contacts

coupled

via _a 3D elastic

med1tlm,

hence the above mentioned decrease of the off

diagonal ~q

with intercontact distance. The _so- called Larkin

length,

which _measures the _space scale

beyond

which elastic interactions become

"relevant"

ii.

_e. induce _non

perturbative cooperative pinning

however small the

trap potentials)

is in the present case

exponentially large _[17],

_so that it _can be considered for all

practical

purposes infinite _as

compared

with real

sample

dimensions in

laboratory

friction

experiments.

(iii)

^The

dynamic

friction coefficient which results is

velocity independent.

The

approxima-

tion of

quasi

static

asperity

motion and _zero

temperature

leads to

describing

contact

breaking

as the instantaneous

jump

of

asperities

out of their

trap potential

when the

spinodal

bifurca- tion

corresponding

to the

metastability

limit is reached. This _can be

improved

_upon

by taking

into account

departures

from

quasistatic equilibrium

due to the finite value of the

asperity

relaxation

time,

which result in _a finite

delay

of the

spinodal jumps.

The

corresponding

correction to

~ld is

velocity strenghtening,

but

negligibly

small in the small-V

regime.

It _can

only

become

important

at

large velocities,

when friction is dominated

by

"viscous"

effects,

depinning

due to

early jumps

above the

metastability barrier,

activated

by

thermal noise.

This

gives

rise to a very small

quasi logarithmic

correction to ~ld, which is

velocity strenght- ening, contrary

to what is observed

experimentally.

So,

it appears that such _a

purely

elastic

model, although

it is

certainly

worth

studying

in _more

depth

in order to

try

and understand the effect of elastic interactions _on the noise

spectrum,

in

particular

via cascade

jumps,

cannot account for the observed time and

velocity

variations of _~ls and _{~ld, nor} for the

instability

of

steady sliding against stick-slip.

Clearly,

what is

missing

in such _a

description

is the above mentioned fact deduced from

experiments

that the deformation _response of

asperities

to the

large

local stresses

they

^bear

is not

purely elastic,

but at least

partly plastic.

^{In this} article _we propose a

first,

schematic version of such _an extended

elasto-plastic

model.

3. Elasto-Plastic Model

This extended model is based _on the idea that the active contacts

undergo

_an irreversible

ageing

_process which is very slow _as

compared

with the

quasi

instantaneous elastic _processes.

The

elasto-plastic

model is then defined _as the elastic _one, in which the characteristics of the elastic

traps undergo

_a slow evolution. Their instantaneous values _are used to describe the

quasi

static

trap equilibrium,

and hence the

trap dynamics.

We therefore first

specify

_more

precisely

the basic elastic

description.

3.I. ELASTIC CONTACT MODEL

(I)

^We assume that

asperities

_can

only

_move

along

the

sliding direction,

thus

making

the

problem quasi

_one dimensional.

They

_are

randomly

distributed _on the surfaces of the slider and of the track.

(ii)

We

neglect

elastic interactions between

traps (see above).

(iii) Traps

are identical.

(iv)

The

trap potential

is taken to be

repulsive

and of the form:

4lix)

= 4~o

ilixla)

_{)x) < a}

j~~

The total _energy of _a

trap

is then:

U =

4l(p

+ t1) ⁺

1tl~ /2 (3)

where p is the

configuration

coordinate of the

trap (see

Sect.

2)

and _t1 its elastic

displacement.

We _assume the

trap

to

displace rigidly

_over its width _{a i,e, t1} to be constant on its width _a

that

is,

we

neglect intra-asperity

elastic strains.

At adiabatic

equilibrium,

_{t1 assumes} the value t1* which minimizes

U,

I,e.:

il'iz*)

₊

fliz* iPla))

= °

14)

where z*

=

z*(pla)

=

(p

+

t1*)la

and the

reciprocal fl~l

ofthe dimensionless

parameter fl

=

~a~ /4lo (5)

measures the

trap pinning strength

in terms of the elastic _energy needed for

displacing

the

asperity by

an amount of the order of its

size,

I.e, for

"breaking"

the contact.

(v)

For the numerical

work,we

take il to be of the form:

wjx)

t 1+

2jxj3 3x2. j6)

The overall

shape

of il is reasonable and its _use is

justified

at this

stage,

^where the model remains schematic. It is

convenient,

as it makes the trap

equilibrium equation (4)

solvable

analytically.

This is shown in the

appendix,

where

analytic expressions

for the

pinning force,

the

spinodal threshold,

etc, are also derived. It is

convenient,

^{for il}

given by (6),

to define

o =

fl/6,

_as the

system

^exhibits

multistability

if

at

fl/6

<1.

17)

lpi

1.

,,

a pc

III. _,

a)

j~i ^(+),,,

,,,,,

bp

b)

Fig.

1.

a) Pinning

force

fp(p)

for _a

single

elastic trap ^with

potential given by equation (2);

cx = 0.13.

The stable branches _are labelled

(+)

and

(-),

the metastable one is dotted. C and C'

are the

spinodal

points. The

hysteretic

_area Ho

(Eq. (12))

is the _sum of _areas marked

(I)

and

(II),

_or,

equivalently,

of

(I)

and

(III), b)

Distribution of

asperities

^with slider at rest. The full lines represent ^the occupied parts of the

(+)

and

(-)

branches under _zero

tangential

load: the discontinuity of the distribution lies at the Maxwell

plateau

MM'; the total force F

= 0, under _a small finite

tangential

load bF _< Fs:

the distribution is

rigidly

shifted

by bp;

bF

(Eq. (13)) corresponds

to the hatched _area.

Moreover,

two _{cases are}

possible, corresponding

to different

physical

situations:

if the slider is very soft

(small a) (or, equivalently,

if trap

pinning

is

strong),

the bifurcation out of the

trapped

state

(point

C in

Fig, la)

takes

place

at y~ =

(pla)~

_> l. In this _case, the

asperities directly jump

out of contact;

moving traps

die when _y

=

pla

reaches _y~. This is the case, for the form of il

given by equation (6),

when

o<cxo=3-2vi ₍₈₎

if _no < a <

I,

the bifurcation _{occurs at y~ <}

I,

where

traps jump

into _a state where 4l

~ 0, they only

die when y reaches I.

We _assume from _{now on} that

pinning

is

"very strong",

I.e. that _{a < ao.} We have checked

that this

assumption,

while

simplifying

to _some extent numerical

calculations,

is

qualitatively unimportant.

The

corresponding

net

pinning

force exerted _on the track

by

the

trap:

fplP)

=

4~'lP

+

11*lP)) 19)

fpi

a)

fPi

-ao no

-ao+Pc(tw)

b)

Fig.

2.

a) Pinning

force _curve

fp(tw)

for the

iystem _aged

_{at rest} under _zero

tangential load,

for tw ₌ 25 to; e = 0.04. Dashed line: elastic _curve of

Figure

I. Full and dotted lines:

occupied

and empty

states for the

aged

system,

b)

Distribution at t

= tw+,

immediately

after the system of

Figure

2a has been

rapidly pulled

_up to its static threshold. Full line:

occupied

states. The upper

edge

of the

distribution reaches

p~(tw).

is

represented

in

Figure la,

where the broken line

corresponds

to unstable

equilibrium

^of the

trap.

It _was shown in reference

[14]

^{that the total friction force} can be written _as:

Fel

~

£ / dpPi~~lp)fplp) (lo)

r

r =

(+, -)

labels the _upper and lower stable branches of the

hysteresis cycle (see Fig, la).

PM) (p)

is the statistical

trap population appropriate

to a

specific experimental

situation. When the

system

^is

sliding

towards

positive p's

at the

imposed velocity V,

the

(+)

branch

only

is

populated,

and this

homogeneously (due

to the randomness of

asperity

distributions _on both the slider and

track)

_up to the

spinodal

limit. Then

[14]:

Pl+)(p)

=

Anp(2a)~~ Pl~)(p)

= 0

Ill)

where A is the

(apparent)

_area of the

slider,

_np the

density

of active

traps (~). Introducing

(~) np =

nno(2a)~,

where

n and _no

are the densities of asperities _on the slider and _on the track.

further the notation

4loHo

for the _area of the

hysteresis cycle

shown in

Figure 2a,

I,_e, the work done when _a trap ^is

being

swept

through,

the total

dynamic

friction force

acting

on the

system

reads:

Fei

"

Anp(2a)~~4loHo. (12)

As mentioned in Section _{2, it is}

velocity independent.

It _was also shown in reference

[14] that,

in such _a

model,

the static threshold force is

equal

to the

dynamic

_one.

The elastic contact stiffness _can also be

evaluated,

with the

help

of

expression (10),

_as follows:

when the horizontal

pulling

force is switched off and the two solids _are left in contact under the normal load

only,

the system

adjusts

from its

previous

state of motion to the condition of

zero horizontal force

by undergoing

a

global

elastic recoil. That

is,

the

trap

distribution

gets

shifted towards

negative p's

until its

discontinuity

reaches the Maxwell

plateau

of the

Fp(p)

curve

(see Fig. 2b).

Imposing

_a small horizontal

displacement

of the

slider, bp

« a, from this

equilibrium state, corresponds

to

shifting rigidly

the trap

population by bp.

The force bF _necessary to

perform

this

displacement corresponds

to the hatched _area shown in

Figure

2b. So:

bF _m

~~ (A fp)bp (13)

where

Alp

is the

discontinuity

of

fp

across the Maxwell

plateau. Therefore,

the shear _contact stiffness:

l~ m

~)~ (Alp ). l14)

3.2. COMPRESSIVE PLASTIC EVOLUTION _OF THE CONTACTS. We _now want to

modifj

the

above elastic model in order to take into account the fact

that,

since the real _area of contact

(the

total

trap area)

is much smaller than the

apparent

_one, when the

traps

start their active life

(either

because the two solids have

just

^been put ^into non

moving

contact or because motion is

proceeding), they

_are submitted _to normal stresses much above the

yield

stress Y of the

material. This

gives

rise to slow irreversible

plastic

deformation of the

asperities:

_as observed

by

Dieterich and

Kilgore _[6],

contact sizes grow

slowly

_{on average,}

quasi logarithmically.

In this

perspective,

_we base _our model _{on a}

simplified description

of the evolution of contacts in _{a non}

moving system recently proposed by

^{Brechet and} Estrin

[18j. They

model the

asperities

as

cylinders

of

equal

initial radius and

height

with _a flat horizontal base. Their scenario of

plastic

evolution then involves two

phases.

In _{a very} short

"pre-initial" phase,

fast

plastic

flow results in _an increase of the

asperity

radius _up to _a value _ao such that the normal stress borne

by

each

asperity

is reduced _to about the

compressive yield

stress Y.

Plastic evolution then

proceeds through

_a

second,

slow

phase

of

thermally

activated creep

across energy barriers biased

by

the

applied

stress a, the

dynamics

of which

obeys

an

Eyring-

Nabarro law

(~):

2a~~ (da /dt)

₌

do exp(a IS) (15)

~'~~~~~

a =

Y(alar)~.

~~~~

(~)

Note that the

thermally

activated _processes

underlying

expression

(15)

_are concerned with struc- tural rearrangements on the atomic scale

le,

_{g. vacancy or} interstitial diffusive

jumps,

local changes of

conformation of long

molecules,

_or of small

glassy clusters...).

That is, thermal noise at _room temper-

ature is

usually

^sufficient to drive them quite

efficiently

while, _as mentioned in Section I, it is much

too small to activate

jumps

of the _center of _mass

"macroscopic" degree

of freedom.

Equations (15, 16) immediately yield:

a~

=

ajjl

_{+ e}

_Injl

₊

t/t~)j j17)

e and to are related to the material parameters

do

and S

(the

strain rate

sensitivity) by:

e =

S/Y; tj~

=

e~Y/SexpjY/S). j18)

These considerations will _now be extended

phenomenologically

in order _to define _a

model, which, though incomplete

and

crude, yet incorporates

_a number of

important physical ingre-

dients in _a consistent _manner.

Before

proceeding

to describe this in _more

detail,

let _us

briefly explain

the

general point

of view which _we

adopt

here. The authors of reference

[18j

^{had in mind} creep due to the

compressive plasticity

of metals _as the

underlying microscopic mechanism,

but the derivation

given is,

in essence,

phenomenological,

and

applies equally

well to a

variety

^of

materials,

_e.g.

polymeric and/or glassy,

where

quasi logarithmic

slow

ageing

a is also known to take

place, although

via different

microscopic

_processes such _as local conformational

changes

_or cluster

rearrangement. Equations (15-18) represent

the

phenomenological

basis of _our

mesoscopic model, which, though crude,

has the virtue of

being

consistent _as far _{as we are} concerned here with the frictional

dynamics

and statics of _a solid-solid multicontact interface under the

combined effect of elastic forces and of _creep caused

by compressive

^{forces alone. We}

expect

this model to be

relevant,

_as far _as such effects _are

concerned,

to a wide

variety

of

materials,

which

only

have in _common the

phenomenological logarithmic ageing

law.

As

pointed

out in Section

I,

_we exclude from consideration

plastic

evolution of the contacts due to shear stress. We will comment _on this

strong assumption

in the last section.

Our model of

elasto-plastic

contact is then

specified by

the

following assumptions:

ii)

The

ageing

law

(12)

is assumed to hold both in the static _case and for motion at low

sliding

velocities. We define the

plastic

_age of each active

trap

^I _as #~ = t

t~,

^where t~ is the time _at which the

corresponding pair

of

asperities

^first came into contact. The characteristic radius of _an active trap ^{then increases with time} as:

a~jt)

=

a~jl

_{+ e}

Injl

+

#zit)/t~)j~/~

=

aoar14zit)) i19)

where _ao is the characteristic initial size of the trap, ^while ar is _a dimensionless

ageing

function

depending

_on the two

phenomenological parameters

^defined in

equation (18).

_{e measures} the creep ^{rate. Its}

typical

order of

magnitude

is 0.01

[18, 12j.

^The characteristic time to defines _an extrinsic time scale for the friction process, which should be

compared

with the dwell time for static

friction,

_or with the trap

sweeping

time in the

dynamic

_case. ^In view of its

exponential dependence

_on the ratio

Y/S,

and of the

material,

_pressure and

temperature dependence

of

do,

no

typical

order of

magnitude

_can be

safely assigned

to it _a

priori.

(ii)

^We assume the elastic

compliance

~ to scale _as

a(t),

that is

~zit)

=

~oiazit)lao), 12°)

while the

potential

of each active

trap

I at time t retains the form

(2),

with the

scaling

laws

a ~

art)>

_4~o ~

4~olt)

= 4~oo

tart)lao)~. ₁₂₁₎

These

assumptions

are made in

correspondence

with the

scaling

estimates

given

in Section 2.

For the elastic

compliance,

the

scaling

is _a

purely geometrical effect,

while for the

potential

strength,

the

scaling

is chosen _to reflect the creep induced _age

strengthening

of _an

elastically compressed

contact which has been

initially

crushed to size _ao and is therefore not

purely

hertzian. With these

assumptions,

the dimensionless

pinning strength

is not affected

by

the creep process:

cx =

const, (22)

remains time

independent

and

provides

_a

signature

^of a

given

trap.

The above

scaling assumption

is at least

qualitatively

consistent with the above estimates

(~

_{Ge a,} 4lo Ge

a~)

for the elastic _response of

asperities

which have been

initially

crushed

plastically

to size _ao, and _are therefore _not

purely

hertzian.

4. Static

Ageing

and Static Friction

Let _{us now} consider what appears

(at

least at first

sight)

the most

simple ageing situation, namely

^the case where the slider and track _are

put

^into non

moving

contact at time 0. At time t > 0 all the active

traps

^{have the} same age:

#I(t)

₌

t,

I,e. the same size

a(t). Since,

under the

assumption

of constant _a, the

equilibrium

elastic

displacements _t1]

scale _as

a(t),

the net

pinning

force at time t on a

trap

of

configuration

coordinate _{p~, age}

#~(t),

is:

/~~~~~ '~~~~~

Ill

4~°°~P'

lz. lAll _i~~~

that is, the

hysteretic

_curve of

Figure

I grows in size while

retaining

its

shape (~).

Let _us denote

Pi (p)

the normalized statistical distribution of

trap configuration

coordinates at time t. The total horizontal force _on the slider reads:

Flt)

=

Np / _{dPz L}

Pl~~lP~)fji~lP~, 4~lt)) 124)

where

Np

₌

Anp

is the total number of active

traps.

Assume from _{now on} that the static friction coefficient is measured

according

to the

following (realistic) protocol:

the

system

is put ^into contact and

kept

under _zero horizontal load

during

a

waiting

^time tw. At this

instant,

a

rapidly increasing

^horizontal force is switched on, and slider motion is recorded. Let _us call

F~(tw)

the

corresponding

static threshold.

At the time of first contact, a = ao, F

= 0. The

system adjusts instantaneously by

elastic recoil _so as to

satisfy

the _zero horizontal

loading

condition. We _assume

asperities

to be dis-

tributed

homogeneously

in the absence of contact. The distribution

P(~~ (p)

^is then

necessarily

of the form

Pi+~ lP)

₌ ^CB

[(PM P)lP

+

ao)1 125a)

Pi~~ iP)

₌

CBIIP PM) iao

_P)1

125b)

with fl the Heaviside function and _pM the

position

of the Maxwell

plateau.

_pM

# 0 for _our

x-symmetric potential

il. The

normalizing

constant C

=

[2ao]~~

As time

elapses, asperities

which _{are not}

engaged

into contact

()p)

>

ao)

^do not

evolve,

while the

potentials

of the active traps ^evolve

according

to

(21). Meanwhile, P(p)

remains

(~) ^{Note that} expression

(15) implies

that _we consider that the duration of the

"preinitial"

stage of Brechet and Estrin is

negligible

_on the scale of the times of interest

(typically, waiting

times in static friction

experiments

_are at least

seconds,

while trap crossing times in the low velocity

dynamic regime

of interest here _are

larger

than

10~~s).

fs I

1.25 $

1.2

1 15 3

1.1 2

1 05

1

o oi o-i i-o io, loo.

7 -

Fig.

3. Dimensionless static friction force

fls

_versus reduced

waiting time

_{T =}

t/t~,

for _e

= 0.01

(1),

0.02

(2),...,

0.05

(5).

unchanged,

_so

that,

for t <

tw,

the distribution of net

pinning

forces has the structure shown in

Figure

2a.

When the

pulling

force is switched _on, the slider _moves

globally (say,

to the

right) by

_some finite

bp,

which results in a

rigid

translation of P

by

the _{same amount.} Irreversible motion starts

ii,

_e, the static threshold is

reached)

when the

discontinuity

of P reaches the

spinodal

bifurcation

point,

I,e. when

bp

=

p~(tw)

^determined

^by: p~(tw)

₌

a(tw)y~.

As the

pulling

force is increased

further,

the

(+)

contacts with _{p > p~} start

snapping

off down to the

(-)

branch of the

hysteresis

_curve. Note that the

asperities

which have been thus

pulled

into contact

(-ao

_{< p < -ao} +

pc(tw))

feel the "fresh"

potential 4l(p,

a =

ao).

The static friction force is

given by equation (24),

with:

P)~~ (P)

~

P(~~(P

+

Pc(tw)). (26)

This distribution is shown in

Figure

2b. Then:

~~~~~ il~ _~~~~~~ ill(twi~~'l~~~~ Iii lal~~~~~~~~~~~'l~~~~ Ill

)1

(27)

Given

expression (17)

for

a(t), Fs only depends

_on the reduced time variable _T

=

tw/to.

The reduced static force

fls(T),

measured in units of

Fej (Eq. (12)), depends

_on the _creep

parameter

e and _on the reduced

pinning strength

_cx. In all the numerical illustrations to

follow,

_we have chosen _a

= 0,13.

Figure

3 shows _a

plot

of

fls(T),

for several values of _e. It does not

simply

scale _as

fp ii,

_{e. as}

a) it)).

This stems from two facts:

ii)

at the instant when

sliding starts,

the set of active

traps

is

composed

for

part

^of contacts which have

experienced plastic ageing,

for

part

of newborn _ones which

got engaged

when the

pulling

force _was

applied,

(it)

^{the relevant}

^spinodal

^{limit is that for the}

aged

traps, increased

by ageing

from the elastic value _y~ao to

y~aoar(tw).

To first order in e,

expression (27)

reduces to:

bfls(T)

₌

Fs(7)

'~

~~~

~~~~ ~ ~~ ~~~~~

where

Ho

is the total _area of the elastic

hysteresis cycle (Eq. (12))

and:

y~

Hi

₌

/ _dy il'(zl+)(y)). _(28b)

-1+y~

The

experimental

results _on

bits

obtained

by

Heslot et al. [7] for paper and

by Baumberger

et al.

[12]

^{for PMMA} are well fitted

by

linear lnT laws.

This, compared

with

equation (28a),

entails that the

plastic

time to for these materials is smaller than the shortest

tw,

of order I s, in the

experiments.

The measured dln

bits Id

lnT _{are on} the order of _a few

percent,

which

confirms the fact that _E indeed lies in the

10~~

_range.

It should _now be clear that the

expression

for the static force derived above is valid

only

for the assumed

experimental protocole (waiting

under _zero horizontal

load).

Had _we for

example

assumed

that,

_{as soon as} contact between the slider and track is

established,

_a horizontal force is

applied,

with _a value such that the

system

remains in the reversible elastic

regime,

the

initial distribution

Po(p)

would have had its

discontinuity

at a non zero value of p defined

by

the contact elastic stiffness. The translated distribution

corresponding

to

incipient sliding

would therefore contain _more

ripe

contacts and less newborn _ones, both ~ls and

d~ls Id Intw

would therefore be

larger

than those measured for

ripening

under _zero horizontal

loading.

Such _a trend is indeed observed

experimentally _[19]. However,

this

qualitative agreement

^has to be considered with extreme _care.

Indeed,

_our crude model of

plastic

evolution does not take into account the

possible

contribution of shear induced

plasticity.

The above remark leads _us to

emphasize

that the static friction coefficient is not a

uniquely

defined characteristic of _a

couple

of materials. It

depends

_on the

experimental protocole

used

to _measure it. This fact is often overlooked because the relative variations of _{~ls are} small. It is

nevertheless of

conceptual importance since,

as

already

_appears, it is

precisely

these variations which contain most of the information about the

physical

_processes at

play

in friction. We will

come back to this

point

later.

5.

Stationary Dynamic

Friction

Consider _now the _case where the slider is

moving

to the

right

at the constant

velocity

V. In this

stationary regime, asperities get engaged

at _a constant rate into active

traps

^and

dragged

across the

trap potentials

_up till when

they jump

out of contact

(bifurcate down)

at the _same

rate. In the absence of _any

ageing

_process, this _means that

they populate uniformly

all the

states of branch

(+)

of the elastic

hysteresis

_curve.

In the presence of

plastic ageing,

^the contacts evolve _as the

system

is

moving. Trap

I becomes active

(contact

I is

created)

at time t~ when:

p~(t~)

₌ -ao. At time

t,

_{p~ = -ao} +

V(t _t~).

The age of _a

trap #~(t)

= t t~ is therefore _a linear function of its

configuration

coordinate:

4zlPz)

=

lpi

₊

ao)/V ₁₂₉₎

corresponding

to a trap ^radius

ajpj

=

a~jl

_{+ e}

Injl

+

jp

₊

a~j /Vt~jj~/~ j30j

fP

,,"]

," '

," ^'

."

." '

," '

."

," j

,-"

," '

,"

;"

P ~

-ao ao P~(U)

Fig.

4.

Pinning

force _curve for

steady

motion at reduced

velocity

_u

=

V/K

₌ 0.I. Full _curve:

dynamically

aged occupied states. Dashed _curve: reference elastic _curve.

The set of active

traps

now

populates uniformly

the

(+

states of instantaneous

equilibrium

of the

continuously deforming potential U(p, tlja(p)). So,

_{a trap} with

configuration

coordinate _p

experiences

a net

pinning

force

fj~~~lP)

=

lalP)lao)~4~ooai~lY'lz~+~lPlalP)). 131)

The

corresponding hysteresis

curve is shown in

Figure

4.

Contrary

to what _was the case for static

ageing,

it cannot be deduced from the

purely

elastic _curve

by

a

simple scaling,

due to the fact

that,

in

steady motion,

contact age

continuously

increases with _p.

The bifurcation takes

place

for _p

=

p~(V),with

_p~ defined

by Pcla(p~)

= y~

(32)

where,

due to _our

scaling assumption (20, 21),

_{y~ is} a fixed number.

So,

the normalized distribution

Pl+)(p)

=

(2ao)~~; Pl~)(p)

= 0.

(33)

The reduced

steady dynamic

friction force then

finally

reads:

flj~~~(V)

= Fj~~~

/Fei

"

H(V) /Ho (34a)

~~~~

Hjv)

=

/~' ^~P~2j

~~~y,

~j+)

P

)j ^j3~~)

_~~ ao ^~

aoar(p)

which _can be

rewritten,

when

setting plan(p)

= y, ^as:

HIV)

=

/) _dv

_lY'

z~+~lv)lallPlv)1

~~~()°~)j ^134C)

Expanding

this

expression

to lst order _{in e, one}

finds,

in the low

velocity

limit V <

i

= ao

/to,

that:

flj~~~ ^{I Ge}

~eIn(V/Il). ₍₃₅₎

2

/jlst)i

d 1.25

~

l. 2

1 15 2

1.i

I

1.05

1

o.oi o-i i-o lo. loo.

u -

Fig.

5. Dimensionless

steady dynamic

friction force _{1l$~~~ versus} reduced

velocity

_u

=

V/V~,

for

e = 0.01

(1);

0.02

(2);...

0.05

(5).

flj~~~

(V)

_as

computed

from

expressions (34a, b)

is

plotted

in

Figure

5 for several values of the creep

parameter

_e. As is

intuitively expected,

it exhibits _a

quasi logarithmic velocity weakening behavior,

which saturates to I for V _»

(,

the characteristic

velocity

at which the time for

sweeping

across a

trap

becomes smaller than the

plastic

time to. Such _a

saturating

behavior has been observed _{on paper [7]} for

sliding

velocities _on the order of _a few _{10 ~lm}

s~~.

If the

characteristic

trap

size _ao

is,

_{as seems}

natural,

identified with the

"memory length" Do (see below),

of the order of1 _~lm for this

material,

_a

rough

fit of

experimental

data _on the

steady

dynamic

friction coefficient

yields

_a value of to on the order

of10~~

s. No saturation

regime

has been observed _on

glassy

PMMA _{at room}

temperature

^{for V} < 100 ~lm

s~~ [20j,

^which

points

to the fact

that,

in this case, to <

10~~

_s

(~).

6.

Comparison

between Static and

Steady Dynamic

Friction

Following Dieterich,

several authors have

proposed

that the

dry

friction

dynamics

is character- ized

by

_{a memory}

length Do,

which _can be

interpreted

_as the

sliding length

_necessary to

destroy

a set of contacts and

replace

them

by

fresh _ones. This has led Dieterich [4j ^and Scholtz _{[5] to}

propose that the static and

stationary dynamic

friction coefficients _are related

by

the

"scaling

law"

t1diV)

₌

t1siDo/V) ₁₃₆₎

which has been

reasonably

well confirmed

by

low

velocity experimental

data _{on very} different classes of

materials, ranging

from rocks to

glassy polymers. "Reasonably"

here _means

that,

within

experimental

_accuracy, the

slopes d~l~/dIntw

and

d~ld/dInv

_are

roughly equal.

Fits

performed according

to

(36)

then

yield

values of

Do.

These _{range, as}

already mentioned,

(~ Preliminary results [19] on PMMA at temperatures of order 90 ^°

C,

close below the

glass

transition,

seem to show _a trend towards saturation of ~Jd in the range V _m 100 ~Jm

s~~,

which would suggest an

increase of t~ in these regime. However, this must be considered only _{as an} indication which should be confirmed

by

further

study.

, ', ~

fli

', 'h,

",~ ',j

~ ~~

"', _~', /~~

', ',~ _'h,~

l I ',

"',,

~

",~

_",,~

"',

l 05

,~~~~~

l

"""'~~~

0.01 0.I 1.0 10. 100.

u -

Fig.

6. Plot of flj~~~

iv (full line)

and of

fls(aoc/twK)

for various values of the

fitting

parameter

C:

(- -.)

C

= I; (. ^C = 2, 3, 4;

(- --)

C

= 2.45.

in the micrometer range I.e. _are

roughly

of the _same order of

magnitude

_as the _contact sizes observed

optically by

Dieterich

[6j.

The

qualitative physical

basis of statement

(36) is,

we

think, unquestionable: dry

friction

dynamics

is

essentially

controlled

by

the

interrupted

slow irreversible evolution of _a sparse random set of contacts. This

dynamical

birth and death _process is characterized

by

_a

length

on the order of the average contact size. Since these

physical ingredients

are

precisely

the basis of _our

model,

we are now in a

position

to evaluate its

quantitative validity

in _more detail.

A

first, important point,

is the

following.

We have _seen in Section ₄ that the rate of

strength- ening

of static friction

depends

_on the

ageing

conditions

during

the

waiting

time

(I,e,

_on the

experimental protocole),

while ~ld is a

uniquely defined, intrinsic, quantity.

This remark suffices to prove that

(36)

cannot be _a

quantitatively

exact relation.

In order to test its heuristic value for

analysis

of

experimental data,

_we have

compared

_Fj~~~

(Eq. (34))

and F~

(Eq. (27)) corresponding

to

waiting

under _zero horizontal

load, following

the

fitting procedure

used for

experimental

data. That

is,

_we

plot

in

Figure

6 Fj~~~ and F~

for _e

=

0.04,

_versus,

respectively, log(V/Il)

and

log(arc/twi).

C is _a

floating

constant with

respect

to which _one

performs

_{a one}

parameter

fit. The Dieterich

length

is then

Do

=

Cao.

It is _seen in

Figure

6 that _a

global

fit is not

possible. However,

the two _curves can be

brought

into

asymptotic

coincidence in the

high V/small

tw

limit, by

the choice

Do

Ge

2.45ao,

which is the traversal

length (I

+

yc)ao

for the

pinning strength (o

=

0.13)

which _we have used.

In

practice,

as

already mentioned, high

V saturation has been

clearly observed,

_up to now,

only

_on the paper-on-paper

system.

The other

existing data,

which _are

reasonably log-linear,

seem to be ascribable to the low

V/large

tw limit. In this range we

expect,

on the basis of

equations (28a)

and

(35),

the

slopes

of the Fj~~~ ^and

Fs

_curves to be

different, although

of

comparable magnitudes. This, together

with the fact that the

corresponding

variations of _~ls and _{~ld are}

numerically small,

_means that very

precise experiments

are needed for this difference to be evidenced. It has indeed been observed

recently

in the recent

experiments

of

Baumerger

and Berthoud

[20j

on the

temperature dependence

of friction of PMMA-on-PMMA.