Creeplike Relaxation at the Interface between Rough Solids under Shear

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Creeplike Relaxation at the Interface between Rough Solids under Shear

T. Baumberger, L. Gauthier

To cite this version:

T. Baumberger, L. Gauthier. Creeplike Relaxation at the Interface between Rough Solids under Shear.

Journal de Physique I, EDP Sciences, 1996, 6 (8), pp.1021-1030. �10.1051/jp1:1996113�. �jpa-00247228�

J.

Phys.

I France 6

(1996)

1021-1030 AUGUST 1996, PAGE 1021

Creeplike Relaxation at the Interface between Rough Solids under Shear

T.

Baumberger (*)

and L. Gauthier

(**)

Laboratoire de

Physique

de la Matière Condensée (***

), École

_Normale

Supérieure,

24 _rue

Lhomond,

75231 Paris Cedex 05, France

(Received

9

January1996,

received in final form 22

April

1996,

accepted

2

May 1996)

PACS.05.45.+b

Theory

and models of chaotic systems

PACS.46.30.Pa Friction, _wear, adherence, hardness, mechanical contacts, ^and

tribology PACS.62.20.Hg Creep

Abstract. We

study

the transient behaviour of _a slider

coupled

to _a

loading spring, creeping

its _way to rest from _a low

velocity sliding

state.

Experimentally,

_a two-stages _process is observed which is

analyzed

with reference to _a

previously

discussed theoretical model for

creeplike

friction

dynarnics

ai trie interface between solids.

Résumé. Nous étudions la relaxation du

glissement

continu _vers le _repos d'un

patin

entraîné

élastiquement.

Un processus en deux

étapes

est observé et

analysé

à l'aide d'un modèle

théorique

de frottement solide _en

régime

de

fluage

interfacial.

Introduction

It bas

long

been realized that _even

slight departures

from trie academic Amontons-Coulomb laws of friction _may have drastic consequences on the

stability

of trie relative

sliding

between solid bodies

[ii. Following

the

pioneer

work of Rabinowicz _on metals

[ii,

the

experimental study

of low

velocity

friction

properties

of various materials has known _a revival for the last

decades,

under the

impulse

of rock mechanicians who studied

laboratory-scaled

crustal faults

consisting

of _e.g.

granite blocks,

in _a

quest

for _a better

understanding

of

earthquakes

mechan- ics [2].

Recently, experiments

on model

systems

^{have been}

reported, featuring

materials such

as

Poly(methylmethacrylate)

_a

transparent, amorphous polymer glass, allowing

^for

optical

observations [3] or Bristol

board,

_a

compound

fibrous matenal

exhibiting unusually

sta- ble and

reproducible dynamic properties,

hence suitable for _a detailed

analysis

of the

sliding

stability

_[4].

All these

experiments

involve the _same basic setup,

consisting

of _a

rough

slider of _mass

M,

loaded

by

_{a spnng} of stiffness

K, remotely

driven at

velocity

V

(Fig. ₁₎

_{on a}

rough

track. Here

by rough,

_{we mean} that the

interface, though nominally flat,

exhibits

asperities

at a

microscopic

(* Author for

correspondence je-mail: tristan@physique.ens.fr)

(** ^{Present address:}

École

_Normale

Supérieure

de Lyon, 46 allée d'Italie, 69364 Lyon Cedex 07, France

*** URA 1437 CNRS

©

Les

Éditions

_de

_Physique

₁₉₉₆

M K V

slider track

Fig.

1. Schematic

spring-block

set-up. Both trie upper block-slider and trie lower track _are coated with nominally fiai "Bristol" board _pieces

(darkened zones).

The slider's lower surface is _a 10 _x 10 cm~

square. The shder of _mass M is

remotely

driven

through

_a

spring

of stiffness K whose

extremity

is translated ai constant

velocity

V.

scale,

_so that the effective contact between _two such surfaces _occurs

through

_numerous

tiny patches

referred _as the micro-contacts

population [si.

Clearly,

_{one may}

distinguish

between _two main classes of

experiments: experiments

of the first kind involve _a

stijfloading

machine

(K

_~

oo)

and focus _on

steady sliding

and the _tran-

sients between two

steady

states

following

_a

driving velocity jump

_[2].

Experiments

of the second kind involve _a

compliant loading

machine of finite

K;

the

dynamical system namely

the material itself

plus

the

loading

machine then

usually

exhibits _a

velocity-

and stiffness-

controlled bifurcation between _a

steady sliding regime

and _a

stick-slip oscillating

_one. The

dynamical study

of the

system

close to trie bifurcation then reveals trie basic features of the

underlying

friction laws

[4].

These

studies, performed

_on materials which

belong

to very different "Masses"

(metals, rocks, polymer glasses, ...)

have

yielded

_a unified

paradigm

of trie low

velocity, dry

friction

dynamics

between two

rougir

sohds which _can be summarized _as follows:

both static and

steady dynamic

friction forces

obey

the Amontons-Coulomb law which states

proportionality

between friction force and normal

load,

with friction coefficients

respectively

_~ls and _~ldi

~ls ^is the friction force at the onset of

sliding

normalized to the load normal to the interface: it increases with the stick time _{Tstj~k Prior} to

macroscopic sliding;

~ld is the normalized friction force

during steady sliding

at

velocity

V: it decreases with V

"velocity weakening" _),

both variations follow the _same functional form

provided

that _an effective _{contact time}

Do/V

_is attached to

steady sliding,

thus

defining

a characteristic

length Do:

~1(~~~~

(V)

_{e ~ls}

(Do IV).

The

slip

distance

Do

is associated with the microscopic process of

renewing

the micro-contacts

population through

shear-induced

breaking

of

asperities;

the mstantaneous friction force for

unsteady shding depends

_on the contact

history

with

Do

_{as a memory}

length.

These

properties

^hold

typically

for

driving

velocities

ranging

_up to _a hundred micrometers Per second.

Sliding

in this range is referred to _as

"creeplike"

motion. On the basis of these

robust

features,

_we have built _a

phenomenological

model of low

velocity

friction

dynamics

which has

proved

to be in excellent

quantitative agreement

with

experimental

data close to the bifurcation

[4].

^{The atm of this} paper is to _use the model free of

fitting

_{parameters to}

analyze

_m details the transient behaviour of the

spring-block system

when

shding

to rest after

a sudden

stop

of the

driving

machine.

N°8 CREEPLIKE RELAXATION AT A SOLID-SOLID INTERFACE 1023

1.

Dynamical Equations

for tl~e

Spring-Block System

The

experimental

features of

dry

friction listed here above have been taken into account

by

Rice and Ruina [6], ⁱⁿ the so-called state- and

rate-dependent

^friction models. A heuristic derivation of similar

dynamical equations

based _{on an} activated

depmning

_process has been

proposed recently

_[4]. Since the details of this derivation _are

extensively

discussed

elsewhere,

_we

just

recall here its main results.

The

spring-block system

^{sketched in}

Figure

1 is described

by

two

dynamical variables, namely

the

time-dependent position ~(t)

of the center of _mass of the shder with

respect

to the track, and the _average

"age" 4l(t)

of the

population

of micro-contacts. Both

dynamical

variables _are

coupled through

_a set of differential

equations.

The first _one expresses the

quasi-static

balance

between the external

loading

force

fext

and _a

time-dependent

friction force:

lext i~, t)

_"

~~ i~di~0/~h)

+ ~

~~i~~/~0)1 i~)

where A _> 0.

The effective _age variable

4l(t)

is defined _{so as} to

interpolate

between the _rest-state limit where

4l(t)

= t and the

steady sliding

_one where

4l(t)

=

Do/V.

Its time evolution _is descnbed

by:

4l = 1

Î4l/Do (2)

The

complete

set of

dynamical equations il

and

(2),

_once linearized about the

steady shding

solution

(~(t)

=

~ld(V) Mg/K

+

Vt; 4l(t)

=

Do/V)

which is stable at

high

values of

K/M, yields

a direct

Hopf

bifurcation between

steady sliding

and

stick-slip

oscillations. The linear

characteristics of the bifurcation _are listed below.

They

will be used _to determine the relevant

parameters

^from

experimental

data.

The critical value of the control

parameter

_{x =}

KDO/Mg

for the

driving velocity

V and _on the bifurcation line is given

by:

~~

~~~ /ÎÎ

~~~

~~~

which has _a

positive

value _smce the

system

^is

velocity weakening

m the low-V

regime

^of interest here.

The critical

pulsation

of the

stick-slip

oscillations reads:

Qcjv)

=

) _{())~~~ (4)}

o

The data of direct measurement of

~ld(V)

are

fitted,

within

experimental

errors

by:

/~d(V)

m mû mi

In(V/Vo) (5)

with Vo an

arbitrary velocity scale,

^{taken in the}

following equal

to 1 _~lm

s~l

such _an

empirical law,

vahd _over several

decades,

has been

widely reported

in metals

Iii,

rocks

[2],

paper

[4,

7]

and

polymer glass

_[8].

However, experimentally,

_a

negative slope

_{for Xc}

ÎIn(V)]

^{is found which}

requires

_some

higher

order corrective terms to be added to

(5):

/Jdiv)

= mû mi

Iniv/Vo

_{+ m2}

iiIniv/Vo

_)i~

16)

This corrective term

generally

remains within the

uncertainty

^of direct friction force _mea- surements. This

empirical expansion

has

proved

to be suflicient _to

quantitatively

account for non-linear charactenstics of the

Hopf

bifurcation [4] and ^it will be assumed in the

following.

2.

Experimental Setup

The

system

is sketched in

Figure

1. Both the slider and the track _are

rigid

metal

plates

coated with _a 2 _mm thick

cardboard, glued

with Araldite. This

board,

of the "Bristol"

visiting

card

type,

is _a manufactured _material which

properties

are very

reproducible

from batch to batch. The surface needs _no

preparation

and is

weakly

sensitive to _wear.

However,

due to the

hydrophillic

character of _paper, the

setup

^{is endosed in} a box and the relative

humidity kept

constant at _a value of

30%.

The _area of the slider surface is 10 _x 10

cm~.

_{The results}

reported

below refer to

sample #1

and

sample #2;

these

correspond

to the _same

pieces

of cardboard but

sarnple #2

has been _worn out

by

hundreds of _runs.

The translation

stage

is motorized

by

_a

geared

down

stepping motor, providing

smooth _mo- tion at velocities

ranging

from ù-1

~lms~l

_to 500

~lms~l,

set

by

_an externat dock

frequency Provided by

_a function

generator.

^The

loading tangential

force _is transmitted

through

a can-

tilever

spring

whose stiffness K is chosen well below the stiffness of the translation

stage (of

order

10~

N

m~~

so that it is the effective softer

part

^of the whole

loading

machine. The _mass of the slider is

adjusted

within the range 0.4 4

kg by adding

10 _x 10

cm~

brass

plates

thus

Providing

_a uniform

loading.

In order _to minimize

torques,

the

tangential loading

force lies _on the

plane

of the interface between slider and track

I?i.

The measurement of the

applied tangential

force is achieved

by measuring

the

spring

de-

flection with _a commercial inductive

position

transducer. The _{rms noise} level of the detection

stage

^is

nm/@, _depending

on the

adjustable

bandwidth of the

amplifier.

A

simple

first order

low-pass

filter _was

used,

whose bandwidth _{was set} to either 10 Hz _or 100

Hz, depending

on trie real time response needed.

Any

bias due to the slow drift of the detector _was avoided

by checking

after each measurement that the datum

corresponding

to _a null deflexion of the

spring

had remained _constant.

3. Results

The _stress relaxation

experiment

consists of the

following procedure:

the slider is driven _{at constant}

velocity

V

by

_a suitable choice of K and M within the

steady sliding

_zone of parameters space thus the

system

is in _a

perfectly

well defined

dynamical state;

at time t =

0,

the remote

driving speed

is the

suddenly

set to _zero and the translation

stage

is hold in _a fixed

position;

the

subsequent displacement ~(t)

of the shder with

respect

to the track is then recorded

through

the deflection ofthe spring. The

origm along

the track is chosen _so that

~(0)

₌ 0.

As shown _m

Figure 2,

the

slider,

whose

velocity

at t

= 0+ is

V,

decelerates and reach _an

asymptotic position

characterized

by

the "cumulative

slip" /l~~(K, M, V)

=

lim[~(t)]t-+m

It is important to note that the slider tends to _an

equilibrium position

under finite shear

stress;

the reduced _stress relaxation

K/l~~ /Mg

remains of the order of _a few

percents.

Two characteristics of the relaxation _{process are}

plotted

in reduced form in

Figure 3, namely

the cumulative

slip /l~~(K/M)

at constant V

(Fig. 3a)

and

/l~~(V)

at constant

K/M

(Fig. 3b),

since it is found

experimentally

that

K/M

is _a relevant

parameter.

It is observed

over a wide range of parameters that

/lz~

decreases with

K/M

and increases with V. Note that it is

expected

that _{at constant}

V, /l~~ (K/M)

_goes to _{zero as}

K/M

_goes to

infinity,

1.e.

when

using

_an

infinitely

stiff

loading

machine hence

working

at

imposed displacement.

On the other

hand,

with _an

infinitely compliant

machine

(K/M

_~

0),

hence

working

at

imposed force,

the slider

displacement ~(t)

is

expected

to grow without limit. These tendencies _are

clearly

N°8 CREEPLIKE RELAXATION AT A SOLID-SOLID INTERFACE 1025

0.3 X ₌

Vt

_

1

0.2

AX

" O.I

~

o run

0 2 3 4 5

t

iS)

1?ig. ^{2. Transient} response

x(t)

of the slider to _a sudden stop of trie

driving

machine. For t < o, trie slider _is

sliding steadily

ai

velocity

V

= 1

pms~~;

_{ai t > o, the}

driving velocity

_is set to _zero and trie slider starts its _way to rest with

an initial

velocity

V _as shown

by

trie tangent fine which is drawn with a

slope

of exactly 1 _pm s~~. _{Trie slider reaches} asymptotically its rest position after

slipping

_over

trie distance Axm. Trie trace

corresponds

to

sample #2

with K ₌ 1-1 _x 10~ Nm~~ and M

= oA

kg.

apparent ⁱⁿ

Figure

3a. The variations of

/l~~

with V

(Fig. 3b)

_{are more} subtle and will be discussed in the next section.

In

qrder

to compare these

experimental

results with the

predictions

of the

model,

it is necessary ^to determine the values of the relevant

parameters (A, Do,

_{mû, mi,}

m2).

These _are determined

experimentally

_as follows: first _mo and _{mi are} deduced from direct ~1(~~~~~

(V)

measurements fitted

by equation (5); then,

the

position

of the bifurcation line in the

plane

(K/M, In(V/Ifi)) Yields Do

and _m2

according

to

equations (3)

and

(6); last,

the

parameter

A is deduced from critical

pulsation

measurements

according

to

equation (4).

Note that the

zeroth order value _mo of the friction coefficient

corresponds

to _an offset in

spring elongation

which is not relevant to the

dynamical analysis proposed

here. Values obtained for

samples

#1

and

#

2 _are listed in Table1.

irable I. Triai and

fitting

values

of

trie

parameters

^inuolued in trie mortel

for

two

samples

<)f Bristol board. Trie triai values _are obtained _as described in the te~t. Trie

jitting

values _are those used in the numerical resolution

of

the mortel

equations

_m the

analysis of

cumulative

slip dependence

with V and

K/M (sample #1,

_see

Fig. 3)

and

of

the whole transient

form (sample

#2,

_see

Fig. $ ).

N R is

for

"non relevant"

jitting parameter.

triai

fitting

trial

mo 0.2 NR 0.3

mi 0.01 0.01 0.013 0.0125

m2 0.0014 0.0033

0.7 0.7

A 0.013 0.012 0.007 0.007

4

3.5 ^° V

= 10

~m.s~~

3 ^°°

2.5

~

~i ~

~

l.5

~'

i

,,

o.5

'''-'

0

lo~~ 10~~ 10

a)

^{x =}

^KDO/Mg

4

~

3.5

,,'

3

~,"

~

2.5 ," °

tJ ,,"

~i 2

~,,,'

~

l.5

-~'"'

i

0.5 x = 7.sxl

i~

o

i io ioo

b)

^V

(~m.s-~)

Fig.

3. Dependence of trie cumulative slip distance Axm _on

(a) K/M

for V ₌ 10

pms~~; (b)

V

for

K/M

= 7A _x 10~ s~~. Trie distance is

expressed

in natural units Do

(here

1

pm).

Trie solid fine _is calculated

by

numencal resolution of

equations iii

and

(2)

with values given m Table I

(sample #1);

trie broken fine

corresponds

to the

analytical approximation

denved in the text.

The

uncertainty

_on these values remains

quite large,

of order

typically 20%, especially

since cumulative _errors

propagate

^from trie estimate of _mi to that of A at the end of the

procedure.

In these

circumstances,

the values determined here above _are taken _as trial values of the

key

parameters. A direct fit of the data

by systematic adjustment

of the

parameters

is _a

heavy task;

we have found it _more eflicient to first make _use of

approximate analytical expressions

to

get

some

insights

_on the way ^to

adjust

the parameters ⁱⁿ the numerical resolution. Derivation of these

analytical approximations

_is

provided

in the

following

section.

Following

this

fitting

Procedure,

we obtained

readily

_{a very}

good agreement

between data and

theory, involving only slight adjustments

of

A,

_mi and _m2, well within

experimental uncertainty. Companson

between

experimental

data and model

predictions

_are shown _m

Figure

3 for cumulative

slip

and

Figure

4 for the whole transient

slip.

N°8 CREEPLIKE RELAXATION AT A SOLID-SOLID INTERFACE 1027

/

/ '

/

CJo

Îl

ii O.I

uP

o

O.Oi

o.oi o-i i io ioo

~ =

Vt/D~

a)

ioo

/ / /

q°

f

_io

if

/

o.oi O.i i io ioo

b ^~

~~~0

1?ig. ^4.

a)

Transient relaxation

slip

_m reduced units

((r)

for

K/M

= 2.8 x 10~

s~~,

V

= 1

pms~~

(open circles)

and V

= 10

pms~~ (fuit circles).

The sohd fines

correspond

to numencal resolution of the model

equations

with the parameter ^{values of}

sample

#2

(see

Tab.

I).

The _same set

of

values _is

used

for

bath _curves. The dashed fines

correspond

to the initial stage ^of

creeplike

motion at constant

<:ontact age described in the text.

b)

Numerical calculation of the contact age m reduced units

çi(r)

<.orresponding

to the data at V

= 1 ~Lms~~ of

Figures

2 and 4a.

Asymptotics

_are shown _m dashed hnes.

4.

Analytical Analysis

of

Dynamical Equations

It is convenient to work with

reduced,

dimensionless variables and

parameters, setting: (

=

z/Doi

_{T =}

Vt/Doi 4

_"

V4l/Doi

_{X "}

KDO/Mg.

For _T >

0, ((t)

and

#(t)

_are the solution of ihe set of differential

equations:

-XÎ

_"

lLd(V/#) lLd(V)

+ Ain

14)) (7)

~~

= l

#~~

with

((0)

=

0; #(0)

= 1

(8)

This has been solved

numerically using

_a standard fourth order

Runge-Kutta procedure (Figs.

3, 4),

however it is worth

making

some remarks

conceming

the

analytical

content of _equa- tions

(7, 8).

Where _a "first order" friction law

assumed, namely ~ld(V)

= mo mi

In(V), /l~~

would be

independent

of V. This is

readily

_{seen on}

equation (7)

which then reads:

-XÎ

"

AIn(#Pd(/dT)

with

fl

= 1+ _mi

IA. Any

reference to the

previous driving velocity

vanishes and the cumulative

slip /l~~

=

Do lim[((T)]~-+~

is

given formally by

the

velocity- independent expression:

/l~~

=

Do

^~ ln

Il

⁺

) /~ _{#~~dT (9)}

X _o

A crude overestimation of the

integral

_can be

performed by using

_an

approximation

of the _age

#(T), namely #(T)

= 1for 0 _<

T < and

#(T)

= T for _T >

(Fig. 4b).

The

integration

is then

straightforward

and

yields /l~~

_m

DCA lx In[1+ x/A(1+ A/mi))

This estimate _can be much

improved by replacing

_mi

by

the real

slope

at the

starting velocity ~li(V)

=

-d~ld/d(In V), owing

to the fact that most of the relaxation

slip

is

performed

when trie slider has _a

velocity

close _to the initial V

(Fig. 2). Using equation (3)

_one writes:

~~coix v)

_~

Dol

ⁱⁿ

Ii

+

i _Ii

₊

^~lv~ _{)1 ii°)}

with x > Xc

(V), ensuring steady sliding prior

to

holding.

This

analytical expression provides

a

good

first

approximation

to

experimental

results

(Fig. 3).

Just above the bifurcation _curve, 1e. for _{x m}

x~(V),

the normalized stress relaxation

/l~1~

=

K/l~~ /Mg

reads:

/l~1~

m

AIn(2

_{+ xc}

IA)

_m A

(11)

where _use has been made of _{x~ m mi *}

A,

_{a very} crude

approximation

the effect of which

is softened

by

the slow

logarithmic

variation. This relation

provides

_an

mdependent

_way of

estimating

parameter A. As _a final

remark,

it is

interesting

to note that A _appears here

as a characteristic stress

drop

after

infinitely long

transients with _a

compliant

machine while Dieterich has noted that A is also related _to the short time transient

following

_a

velocity jump

with _a stiff machine [2].

We _now address the time variation

x(t)

of the relaxation

slip,

_as illustrated

by Figure

4a.

It consists

schematically

of _two

stages:

the first _one

(t

<

Do/V)

is _a

creephke

motion at

quasi-constant

_age 4l

=

Do IV;

resolution of

(7)

_is then

straightforward

and

yields:

((T)

=

~

ln

(1

^{+ ~}

T) (12)

X A

during

which most of the relaxation

slip

is achieved

(Fig. 4a).

The second

stage (t

»

Do/V)

is _a self-decelerated motion due to the fast

ageing

of contacts

according

to

4l(t)

r~ t

(Fig. 4b)

which leads _to the

asymptotic /l~~

with _{a power} law

decay:

1(T)

_m

ii ( _exp1-j (j _{TmilA (13)}

where agam the

logarithmic

friction law

(5)

has been assumed for the resolution of

(9).

This behaviour _{is a} dear illustration of the

strengthening

of micro-contacts with ageing.

N°8 CREEPLIKE RELAXATION AT A SOLID-SOLID INTERFACE 1029

5.

Creep

_versus Inertial Relaxation

lLet _us

emphasize

that the set of

dynamical equations

used _so far has been established under the

hypothesis

of

quasi-static

motion. Dtherwise

stated,

the kinetic _energy of the slider

plays

_no

i.ole in the

stopping

_process which is

entirely

controlled

by dissipation during

the

creeplike slip.

~n _a

posteriori justification

of this

assumption

_is

provided by

the calculation of the relaxation

in the

"pure"

inertial _case, 1-e- when friction _can be described with _an instantaneous friction coefficient

~ld(Î). Then,

the

equation

of motion

during

relaxation

slip

reduces to:

MÎ ₊ K~

=

Mg[~ld(V)

_~ld

là)] (14)

with

~(0)

= 0 and

k(0)

= V. For _{our purpose}

here,

it is

enough

to consider _a zeroth order

<ipproximation

with

~ld(V)

a constant. The resolution of

(14)

_is then

straightforward: ~(t)

=

V/uJo sin(uJot),

with _mû

"

(K/M)1/~

the

eigenpulsation

of the harmonic oscillator. This solution

is valid for t <

ir/(2uJo)

where the slider

stops

and sticks. The cumulative

slip

in the inertial

case is then:

/l~$

=

V(&I/K)1/~ (là)

For data

plotted

in

Figure 4a, K/M

m 3

x105 s~~,

V

= 10

~lms~l

and

/l~in

m 2 _x

10~~

~lm. Within the range of

K/M

and V

reported here, /l~$

is

always

much smaller than lhe observed

/l~~

which is of order _a few

Do

Note that in this _case, the criterion

/l~t

_<

/l~~

i eads

Do IV

»

(M/K)~/~

and involves _a characteristic _creep time

Do IV

and _an inertial time

(M/K)~/2:

the _same criterion has been

previously

associated to

qualitative changes

_m the lJehaviour of the

system along

the bifurcation line

[4-7].

6. Conclusion

We have addressed the

problem

"how does _a

remotely

driven slider

stops?" experimentally,

and

we have

interpreted consistently

_our results with the

help

of _a

dynamical

model for low

velocity sliding

friction. As _a main

physical result,

it is found that this transient process involves _two

stages, namely

a

creeplike

stress relaxation at

quasi-constant

contact

strength

followed

by

_a

short

slowing-down

due to

age-strengthenmg

of trie micro-contacts

population. Moreover,

fine details of the relaxation bave been

analyzed

^and found in full

quantitative agreement

with the numerical characteristics of the

system

close to the

stick-slip~steady sliding

bifurcation.

Finally,

the whole behaviour of the

system

_over decades of

sliding velocity

and

driving

machine stiffness is

accurately

described

by

_a

four-parameters equation

set which shall be considered

,is a touchstone for further

"microscopic" developments

_[9].

fLeferences

Iii

Bowden F. P, and Tabor

D.,

The friction and lubrication of sohds

(Clarendon Press, Oxford, 1950);

Rabinowicz

E.,

Friction and _wear of materials

(John Wiley

and

Sons, 1965).

[2] Dieterich J.

H.,

J.

Geophys.

Reu. B 84

(1979) 2161;

Scholtz C.

H.,

The mechanics of earth-

quakes

and

faulting (Cambridge University Press, 1990) chap.

2 and references therein.

[3] Dieterich J. H. and

Kilgore

B.

D., Pageoph.

143

(1994)

283-302.

[4] Heslot

F., Baumberger T.,

Perrin

B.,

Caroli B. and Caroh

C., Phys.

Reu. E 49

(1994) 4973-4988; Baumberger T., Caroli,

C. Perrin B. and Ronsin

D., Phys.

Reu. E 51

(1995)

4005-4010.

[si

^{Greenwood J. A. and Williamson J. B.}

P.,

Proc.

Roy.

Soc.

(London)

A 295

(1978

300-319.

[6] Rice J. R. and Ruina A.

L.,

J.

Appl.

Mech. 50

(1983)

343-349.

[7]

Baumberger T.,

in

Physics

of

sliding friction,

B.N.J. Persson and E. Tosatti Eds.

(Kluwer, 1996).

[8]

Baumberger T., unpublished.

[9] Caroli C. and Nozieres

P.,

_m

Physics

of

shding friction,

B-N-J- Persson and E. Tosatti Eds.

(Kluwer, 1996).