Experimental evidence of self organized criticality in the stick-slip dynamics of two rough elastic surfaces

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stick-slip dynamics of two rough elastic surfaces

S. Ciliberto, C. Laroche

To cite this version:

S. Ciliberto, C. Laroche. Experimental evidence of self organized criticality in the stick-slip dynam- ics of two rough elastic surfaces. Journal de Physique I, EDP Sciences, 1994, 4 (2), pp.223-235.

�10.1051/jp1:1994134�. �jpa-00246899�

Classification Physics Abstracts

05.45 9i.30 05.40

Experimental evidence of self organized criticality in the

stick-slip dynamics of two rough elastic surfaces

S. Ciliberto and C. Laroche

Laboratoire de Physique

(*),

Ecole Normale Supérieure de Lyon, 46 Allée d'Italie, 69364 Lyon, Cedex 07, France

(Received

18 August 1993, ^revised 1 October 1993, accepted 4 November

1993)

Résumé. Nous décrivons _une expérience _sur la dynamique du frottement solide _entre deux surfaces élastiques. La dynamique révèle _un comportement critique auto-organisé _{sur un} grand intervalle des paramètres de contrôle qui sont la vitesse de glissement et la _pression. L'effet de la taille du système _a été étudié. Nous faisons _{aussi une} comparaison avec la dynamique produite

par un système à _une seule dimension et par rapport _{au cas} où les deux surfaces sont rigides.

Abstract. An expenment _on the stick slip dynamics of two elastic

rough

surfaces _is de- scnbed. This dynamics produces _a self organized cntical state _{on a} wide range of control parameters, namely loading speed ^and _pressure. ^The dependence _on the system size has also been studied. A comparison ^with the stick-slip dynamics produced by _{a one} d1nlensional system and by two rigid surfaces is made.

1 Introduction.

Mechanical

models,

based _on the stick

slip dynamics

of blocks connected

by springs,

^and

sliding

on a surface with _a

given

friction law, bave been

recently widely

studied to

give

more

insight

into trie mechanisms of

earthquakes

and friction

iii.

Several of these models [2-6] are

directly

derivable from trie

original

_one of

Burridge

and

Knopoif

_[7], however tue

stick-slip dynamics

bas been also

investigated

_using cellular _automata

[8-12].

^Ail of these models _are characterized

by

_a

loading period

where tue system accumulates _energy and _a

slip period,

^" tue

earthquake",

where part ^{of this} energy is released. An important issue _m this

study

is to understand under what conditions these models present self

organized

critical

(S.O.C.)

states [8,

13].

^In a

geophysical

context S-O-C- could

help

to rationalize severa1observations and _a detailed discussion _on this

point

can be found in references

[16,

13]. ^{For what} concerns tue above mentioned models _one

(*) URA 1325.

wants to know whether it is

possible

to

reproduce

tue

Gutenberg-Richter

Iaw

(G.R.)

_[14] for the number of _occurrence N of

earthquakes

with _a

given

moment

E, specifically NIE)

_cc

^E~~~~

We recall that for real

earthquakes

trie average value of B is around ₁

iii.

Except

for _a few numerical simulations where the G-R law is satisfied tilt tue maximum energy [4, 5,

iii,

in

general NIE)

presents _an anomalous

peak

for

large

E [2], ^{that is to} say trie

scaling

is self-similar

only

for small

earthquakes.

There _{are many reasons} [2, 10]

why

this may

happen

but here _we want to focus

only

_on the

experimental

verification of G-R- and _on trie

comparison

with tue numerical results.

Indeed,

in spite of trie

large

amount of numerical data

on trie above mentioned

models,

to trie best of _our

knowledge,

there _are

only

_a few

laboratory

experiments [12, là,

16],

where trie stick

slip dynamics

bas been tested from trie point of view of its statistical

properties.

Trie great diiliculties of these experiments are that trie surfaces, which _are

shding

_{one over}

another, change

their properties _{as a} function of time because of trie

erosion due to trie continuous friction. Therefore _statistics should be constructed _on very short

time

intervals,

otherwise

they

coula be influenced

by

trie detenoration of _trie surfaces. This effect is

clearly pointed

out in reference [15] ^{where is described} a very

interesting experiment

which allows one tu measure not

only averaged

properties but aise the localisation _{m space} and time of trie

stick-slip dynamics.

This experiment, _{as many} other numerical simulations [2, 9,

loi,

shows _very

dearly

for _a

large

_range of trie control parameters,

namely

_pressure and

sliding speed,

_{a power} law distribution for trie

slip amplitudes

with _an enhanced

probability peak

at

high energies.

Trie other experimental _paper [12] ^shows a power law _on trie

integrated probability

distribution function for

only

_a value of trie central parameter, thus _we do net know

if this _coeurs in _a wide _range of central parameters. Furthermore tue

frequency

_spectrum of the

friction force _versus time presents ^trie triv1al _power law

f~~

In other words _we do net know if that system ^{is in} a state of self

organized criticality

_[8]. In this _{paper we} describe _an expenment

on

stick-slip dynamics

in which ail the above mentioned

problems

_are solved and the system presents _a ^self

organized

critical state in a wide _range of contrai parameters. Tu obtain this result much _care has been taken in order _tu construct the

sliding

surfaces which _are described

m section 2

together

with tue experimental apparatus. Trie rest of trie paper is

orgamzed

_as

follows. In section 3 _we report trie statistical properties of trie

stick-slip dynamics.

in section 4 _we compare systems ^{with different} properties,

namely elasticity

and

dimensionahty. Finally

trie results _are discussed in section 5.

2. The

experimental

set up.

The very

simple experimental

_{apparatus is} shown in

figure

1. Two expenments ^with different

geometries

bave been

performed,

but _no relevant quantitative difference bas been noticed except

for trie _maximum value of trie

scaling

exportent.

2.1 THE _ANNULAR EXPERIMENT. in trie first _une,

figures

la, 16, le, trie

sliding

surfaces

bave _an annular

shape

of width h

= 1 _cm and _mean diameter D. We used three values of D in order tu

verify

if trie results _are

independent

of trie size of trie surfaces. Trie two

sliding

surfaces

are mounted _on two aluminium disks which _are 2 _cm thick in order to avoid deformations.

One of trie two disks is

kept

fixed and the torque _necessary to compensate trie friction force Ff between trie two

sliding

surfaces is measured

by

_a torque transducer.

Specifica1Iy

_{we measure} Ff _x

D/2.

Trie transducer

signal smtably amplified

and filtered is couverted

by

_a 16 bits

AID

converter. Trie minimum detectable variation of trie friction force Ff is about 3 _x lo~2 _{N. Trie} other aluminium disk _was driven at constant

speed

Q

= 2 _x

V/D by

_a

reduction-gear

and _a de electric motor, here V is trie

loading speed.

A micrometric device a1Iows _us to

change

trie

a)

~~

~~

_{ÎÎfl 5''}

'C°n ~U°b~~

F-D

2 D

,

i'

',

i

j 1

C)

1

steel sJ~eres

~

d

)

§j Wood table

flfl Silicon Rubber

Fn v

steel _s ~ere ^~

Fj

Fig. l. Expenmental apparatus.

a)

Schematic drawmg of the disk expenment, ^trie torque F x D necessary ^to keep the disk fixed is measured by _a torque transducer while the other disk is maintained

at constant angular speed V _x D. b) Disk surface. c) Expanded _cross section of _a disk showing how

the steel spheres _are glued inside the groove with the silicon rubber.

d)

Cross section of the linear geometry expenment; the sbde _is moved _at

a constant speed V and the friction dependent force Ff is nleasured by _a force transducer. The wood table is 180

cm long.

pressure with which the surfaces of trie two aluminium disks _are

pushed against

une another and trie axia1force _is aise measured. Trie two annular

shding surfaces, figures 16,

le, mounted

on the two aluminium disks _were

prepared

in the

following

_way. First of ail _an annular _groove of

depth

1.5 _mm and width 1 _{cm is} made _on trie disk surface

(see Fig. l).

Three grooves with different _mean diameters D bave been

made, specifically

9.3 _cm, là _cm and 29.3 _cm. We put inside trie _grooves steel

spheres

of diameter 2 _{mm m} such _a way that trie distance between

sphere centers _was about two diameters.

Finally

trie grooves were filled with silicon rubber.

In this way an artificia1surface with controlled

roughness

and

elasticity

is constructed. Trie maximum

roughness

is

exactly

o-à _mm

equal

tu trie

sphere

diameter minus trie

depth

of trie groove. Furthermore each

single sphere

is rather free tu _move around its

equilibrium position

because of trie

elasticity

of trie rubber. We will call this surface soft surface _tu be

distinguished

from _an other type ^{of surface which is} very hard and whose

roughness

dues trot

change

toc much because of trie friction. This surface is constructed

by dring

_a mixture of

glass

microspheres

(10

_~m of

diameter)

and

"stycast" glue which,

_once it is dried, ^becomes a very

rigid

surface with _a well defined

roughness.

We will call this surface

rigid

surface.

Incidentally

_we remind that

surfaces,

_very similar tu trie soft une, bave been used tu make mathematical models of friction laws between two

rougir

surfaces

il?,

_18].

Unfortunately

in these _two references tue

Ioca1interactions _are different from those of this experiment, thus _a direct

comparison

is net

possible.

However

experiments

of this kind associated with numerical models

il?,

_{18] can} be useful in order tu

give

_more

insight

into trie

study

of friction laws.

2. 2 THE LINEAR EXPERIMENT. Trie second

experimental

set-up, ^shown in

figure id,

used

instead _a linear geometry. A slide is moved at constant

speed

V _{on a}

rougir

surface constructed with steel

spheres emerging

of o-à _mm from _a wood

plate

180 _cm

long

and 20 _cm wide. Trie force _necessary to _move trie slide _at constant

speed

is measured

by

_a force transducer. The normal force

acting

on the surface of the slide _can be

changed by modifying

trie

weight

of trie slide. Trie surface of trie shde _was constructed in trie _same way as trie soft surface described

above. Two slides _are used in order _to

study

trie

dependence

_on trie system size, one 9.5 x18 cm~

and trie other 9.5 _x 10 cm~. Trie

depth

of trie _groove where steel

spheres

_are inserted is 1.5 _mm

as for trie annular surfaces.

We

point

ont that there is _a very

important

^diiference between trie disk experiment and trie linear _one. Indeed in trie disk experiment ^{it is} trie distance between trie two disks that

is

kept

fixed and trie normal force

changes

because of trie

dynamics.

In contrast, ⁱⁿ trie slide experiment, ^{it is trie normal force that remains constant.}

However,

_in

spite

of this

important diiference,

trie results _are similar _m trie two cases, thus _we will net in

general

make _a distinction

between them. Trie _reason for this

similarity

between trie two

geometries

_is

probably

due to trie _very

large compliance

of trie matrix and could be net true _m a very

rigid

_case.

It is aise important to

explain

how trie

spheres actua1Iy

_move.

By Iooking

at

figure

1 _we see that trie

spheres

_are

touching

trie bottom of trie

plate. Thus,

when _a

sphere

_is

pushed against

an

asperity

of trie opposite

surface,

it Carnot pass over this asperity ^{but it} can

only

move

transversely

to trie direction of trie

loading velocity

_m order to go around trie obstacle.

This bas been verified

by making

in trie hnear experiment a transparent ^slide and

looking

from

above,

with _a

videocamera,

trie motion of several

spheres.

This is useful aise _to estimate the maximum

displacement

of _a

sphere

that is about 0.2 _mm.

2,3 PHYSICAL PROPERTIES OF THE SURFACES AND TYPICAL TIME SCALES. Trie stick-

slip dynamics produced by

trie interaction of trie diiferent surfaces put in contact bave been studied in trie _case of trie disks. Three _cases bave been considered: trie

ngid-ngid, rigid-soft

and soft-soft interactions. Trie

rigid-rigid

interaction is net very interesting from _a statistical point view and it wiII be discussed in section 4.2.

Trie

rigid-soft

and soft-soft interactions, which _are trie most important,

give

similar results and

they

^{will be}

analysed

in fuII detail in section 3. Their

dynamics

lias been studied _{as a}

function of trie

applied

_pressure and of tue

loading speed

V. Trie values of trie

apphed

_param-

eters must be

compared

with those obtained

using

trie

physical properties

of trie experiment.

Tue _mass M~ of tue

spheres

is 4 _x 10~~

kg

and trie total number of

spheres

is _m between 200

and 300. The total _mass

MT

involved is about 1

kg.

With trie total _{mass we} indicate trie

sum of

sphere

_masses

plus

either that of tue disk _or that of trie slide

depending

_on ^tue

experi-

mental geometry. ^Tue

Young

modulus of tue E of tue rubber where

spheres

_are embedded is 5 _x 10~

N/m~. Therefore, using

_{as a}

typical Iength

scale tue

sphere

diameter d

= 2 x

10~~

_m,

trie elastic constant k

= E x d is 10~

N/m.

Two

typical

time Saales _can be

defined,

_one based

on trie acceleration of Ma and trie other _on MT: TT ₌

(MT /k)~/~

and Ta =

(Ma /k)~/~

We find that Ta ce 2 _x 10~4

s whereas TT _m 3 _x 10~~s. As it wiII be discussed in section 3, this value of TT is very close _to trie

sliding

time for trie smallest _events measured _on tue friction

force,

_meaning that tue motion is dominated

by

tue total inertia. Tue

darnping

time of tue

oscillation of M~ and MT _are of trie _same order _as Ta and TT

respectively.

This _means that tue system is _over

darnped

and oscillations _{are not} allowed.

Another important time scale is

Tv

"

d/V

which

corresponds

to tue time needed _{to move}

tue

plate

of _a characteristic

Iength;

_we wiII _see that this time

corresponds

to the

Ioading

time between two very

large

events.

Z.

o a

a.z6 _o

@.

~

o

a

o a

@. _o

o on

@.

~.5 1. I.E 2.a 2.5

v (mm/Gec)

Fig. 2. Friction coefficient for soft-soft surfaces _{as a} function of the loading speed. The different symbols correspond to measurements made with different normal forces.

We bave measured _{au average} friction coefficient _~

= <

Ff(t)

_>

/Fn

of _our artificial surfaces

as a function of V

by

_measunng ^tue mean value of tue friction force

Ff(t)

divided

by

tue _mean normal force Fn

(here

< > stands for time

average).

We find that _~ increases

slightly

_{as a}

function of V. Tue _same property bas also been

reported

in reference [15]. ^Tue measurements

of _{~ versus} V _are shown in

figure

2 for tue soft-soft _case. It is important to

point

out that this

is trot a

precise

measurement of ~ because _a measurement of this type ^is

strongly

influenced

by

tue stick

slip dynamics

whose

amplitude

_is

velocity dependent.

A _more accurate evaluation of tue friction coefficient would _require tue suppression of tue

dynarnics;

this _is not easy in tue

case of trie soft surfaces. Thus the behaviour of _{~ as a} function of V shown in

figure

2 is not

quantitative

but it is useful

just

to give an idea of the behaviour of the ratio between the _mean friction force and the

applied

normal force.

15@a

~m

z i Jzz

~

~

~L~

~j 1@Z

ù

_'

/~~

/-.

~

~~

/

>/ ^' ~.

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Î,~~~~~/

/~

~

/?~

/

/~

/

i

~ 7zz i

ij

àze ~----<--,

2C ~'z fié Ba "@3 2~ ù@ 't-C'

i se

c

-~~~~ ~

c

z 3@z

j /

~ /~i

/ /

$ Z@(

~~/ j/

~

~

/l~__ ,/

~

/~ ^{/~ /}

~ ~°~

/ /

^J

£ /

1' ,/

~

~ôzl

_~,

/ /~

~

^/

/

~,'

l/

>,/

~~~à

l~ le m 4Q

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se c

iz@@ -j

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ii ji

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o Il

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' ~

( ' ' '

Ezz

,

^{i ' [} _iii

^~

j

?~j, lj~l ii

^'

~m ~ Ii ,j< jj

jj

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7 1'

'

~@@ ^Î

~ _' Î

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@@~»>

»>,» »»<>~"'>'>

5 je i~ 2@ 25 2@ 25 ~C

1Sec)

Fig. 3. Friction force _versus t1nle for diierent loading speeds. a) ^V = 0.04

mal/s;

b) V

= o-1

nlm/sj

c) V =1.25

mal/s.

3. The statistical

properties

of stick

slip.

We _now

analyse

tue statistical

properties

of tue stick

slip dynamics

in _our system. To do this

we first record

Fit)

and _an

example

for tue soft-soft _case is shown in

figure

3. We

dearly distinguish loading cycles

^{where the} two surfaces _are stuck and

Fit)

increases _up to time t;

where it reaches _a relative maximum. At that

point

tue

slip period begins

and Iasts until time te where

Fit)

reaches _a relative minimum and then it

begins

to _mcrease

again.

We define the

amplitude

of tue

slip

event _as

M(ti)

=

Fit; F(te).

We _can estimate tue Iateral

displacements

of

spheres corresponding

to tue minimum Mm;n

and maximum

Mmax

value of

Mit).

We find

Mm;n

= 3 x10~~ N and

Mmax

= 3 N

respectively.

Assuming

that tue minimum events are

produced by only

_one

sphere

_we find that _~lmjn

=

Mm;n /k

₌ 3 x10~~ _{mm which} is _a very reasonable value because it is about

1/7

of the maximum

displacement

_we have measured

(see

Sect.

2.2).

In contrast ~lmax =

Mm~x/k

= 3 _mm which is not reasonable because

spheres

cannot be moved _on such _a

long

^distance. Therefore _one deduces that trie

largest

events are

produced by

_an ensemble of _at least 100

spheres

which in average are

displaced

of about _~min. Notice that _~lmin

/TT

identifies _a characteristic

sliding velocity

( _m

mm/s.

Indeed _we will _see in trie _next section that ail

interesting

statistical

features

disappear

when V >

(,

that

is,

when

loading speed

becomes

comparable

to

sliding speed.

3. 1 PROBABILITY DISTRIBUTION. We _measure tlie

probability

distribution function

P(M)

of M

finding

_a transition to a power Iaw distribution for sma1I velocities V. At Ieast 10~ _events bave been recorded _to construct

P(M).

This is shown _m

figure

4a for tue soft-soft surface.

We _see that at

hîgh

velocities

P(M)

presents

just

_a

large plateau

with _an

exponential

cut-off whereas _at small velocities _{a power} Iaw _appears with _an exponent _i ^which is

increasing

for

decreasing

V and reaches trie value of about 2 for V _- 0. This _{is seen} in

figure

4b where trie exponents _i of tue _power Iaws _are

reported

_{as a} function of V for _two different types ^{of surface}

interactions

(rigid-soft

and

soft-soft).

We found that tue cut-off of

P(M),

1-e-tue

amplitude

of tue

largest

events,

depends

_on the norma1force. This is shown in

figure

5a where trie

P(M)

obtained

by using

^diiferent normal forces _are

reported.

We

clearly

_see that trie exponent ^of _power law does not

depend

_on trie

normal force but trie

amplitude

of trie

largest

events increases with

increasing Fn.

Furthermore

we bave checked that trie cut-off of

P(M) depends

_on the system size. This is done

by using

either different D in trie disk

experiment

_or trie small and

big

slides in trie Iinear

experiment.

In

figure

5b trie

P(M)

obtained from trie

large

slide _is

compared

with that obtained from the small _one. The

Ioading speed

_was V

= o-à

mm/s

and the normal pressure p =

Fn/S

was trie _same in trie two cases, here S is trie _area of trie

sliding

surface. It is important to

stress that tue normal pressure, and not tue normal

force,

remains constant

by changing

tue

system ^{size. Indeed tue relevant} parameter _is tue normal force

acting

_on each

sphere

which is about

Fn /S,

because tue number of

spheres

_per unit _area is

always

tue _saine. This is also tue

case in numerical simulations [2-7] ^{where trie normal force}

acting

on each block _is tue control parameter. ^In

figure

5b _{we see} that tue maximum

amplitude

of M increases with tue system size.

3. 2 POWER SPECTRA. TO stress trie fact that trie system ^is

really

in _a state of S.O.C. when trie

Ioading speeds

_are slow _we show _m

figure

6 trie _power spectra ^of

Fit)

for _two different V.

We

clearly

_{see a}

large

^diiference m tue _two spectra. At

high

V trie spectrum is broadened with

a

Sharp

cut off. In contrast at slow V trie spectrum

decays

in

f~~

for _at least _une order of

magnitude.

This _means that trie derivative of

Fit) decays

in

f~~

Indeed

by looking

at tue

JOURNAL DE PHYS>QUE T 4, N' 2 FEBRUAR~ >994 9

~

~.~

/

~

~.e

~

~

-i 1-Il ^~ _O

r5 ~

~ i ~ Q

' ~

0-

zÎ

*

©

~

~ r

co

j

o U b

~ _#

'

o z-

>,m~-,<-,> ~~m~~Tr ,~, i

Z

d-B 1-Z >.5 2.~ P-b 3.~

~ _~'m/~éCJ

~ i i

Lo _g

ie(

^h)

Fig. 4.

a)

Event size distribution

P(M)

for different loading speeds. Starting from the bottonl _curve the corresponding V are, Ù-1 n1n1/s, 0.2

nlm/s,

Ù-S n1n1/s, 1-S

nlm/s

and 2.2

nlm/s.

The vertical scale

is arbitrary because the _curves have been shifted in order to clearly show their shapes _on the _saule

graph, b) Exponent of the _power laws of

P(M)

_{as a} function of the loading speed for different normal forces and different experimental set-ups.

lu)

and

IA)

correspond to the soft-soft interaction _in the disks and linear expennlent respectively. (*) correspond to trie soft-ngid _{case on} the disk expennlent.

i

~

b

~ W5

~

i 1

~ o

j c5

o 1 o

~-3.

o

i

1 ~

8

~, i ~

La g~~ ~i) La

g~~ ~i)

Fig.

5.

a)

Companson of

P(M)

obtained at the _same loading speed

IV

₌ Ù-Sn1n1/s) but for different nornlal forces using the large slide; Fn

= SA N

(continuous fine);

Fn

" 10.8 N

(long

dashed fine;

Fn = 20.4 N

(short

dashed fine). The straight fine _{is a} fit M~~'~. b) Conlpanson of P(M) obtained

for the _saule loading speed IV ₌ Ù-S

nlm/s)

but for two different system sizes m the linear geonletry;

(continuous fine)

small shde,

(circles)

large shde. The normal _pressure Fn/S _was the _saule

in the two

nleasurements.

f-3

h

-9

m O.l la

Fig. 6. Power spectra of

F(t)

for two different loading speeds V,

a)

V ₌ Ù-1

nlm/s

and

b)

V

=

1.4

mal/s.

suape

of trie

signa1in figure

3a _we understand tuat tue

Iargest

contribution to trie derivative of

F(t)

is

coming

from trie fast

decreasing

parts

(trie slip phase)

whose

amplitude

M bas _{a power} law distribution _{as we} bave _seen in

figure

4. Notice however that _a series of discontinuities in trie

signal

it is not

enough

to get an exponent -3 in trie _power spectra but it is important

that also trie time of _occurrence of _events bas _{a non}

trivia1distribution,

otherwise tue triv1al

exponent -2 is obtained. Tuerefore _we bave also measured tue distibution

P(r)

of tue _time

r between two successive events of any

amplitude.

Trie results _are

reported

in

figure

7 for _a sma1I and _a

high velocity.

We _see

tuat,

at V ₌ 0.1

mm/s, FIT)

_{presents a} clear _power Iaw

r~~

with

fl

m 1. In contrast at V ₌ 1A

mm/s FIT)

is

simply

broadened with _no

specific

ieature.

Thus _one concludes that at sma1I V not

only P(M)

bas _a power Iaw

dependence

but also tue power spectra ^and

P(r).

3.3 COMPARISON oF POWER LAW EXPONENTS. Trie

relationship

between M and trie

actualenergy

released

during

tue

slip period

is not clear _not

only

in _our

experiment

but also in numerical models. M could be

proportional

eituer to tue number _n of

spueres

involved in

tue _{event or to} E. Tue

simplest equation

_is M

= k _~ln witu ~1tue

slip

distance and k tue elastic constant. However tue _same M could

correspond

eituer _{to a} small number of

spueres

wuicu

undergo

_a

large displacement

_or to many

spueres

and _a small

displacement.

Tuus tuis

relationship

_is not obvious because it may

depend

_on tue

specific

system and _on tue most

probable configuration

alter _an

eartuquake.

Indeed in reference [2] tue

jump

in friction force is related witu tue

"eartuquake"

_moment, wuereas tuis _seems not to be tue _case in tue

experiment

of reference [15] ^{wuere tue} measure of tue total force is Iess sensitive tuan tue local

analysis.

Furthermore in the numerical model of reference [4] ^it has been found [19] ^{tuat M} cc

nf

_witu

(

_+~

O(i).

In tuis model tue

eartuquake

moment E _cc n° with _{I.à < a < 2}

depending

_on

control parameters.

In

general

_{we con} write tuat E _cc M~ witu 1 < c < 2 tuus B ₌

(i- ii /c. As1

_- 2 for V _- 0

we see tuat o-s < B < wuicu _{is in} tue _range of wuat is observed _in

realeartuquake

statistics.

A _{more precise} evaluation of B could be done

only by measuring

tue local

displacement

of tue steel

spueres during

tue events to

exactly

^{correlate tue friction force} to tue moment.

~

~

~

z

, cj

z

n~~

n

m ~

_ m

~ mm

- g

~ ""

~m

iÎ ?

qjfl~Zm~

~ J

,~q%

, m

i ,-

~'~

o 1 2 3

lcgl~)

Fig. 7. Probability distribution function of the time ₇ between two _successive events of any ampli- tude. V =11.1

mal/s (black

squares) and V ₌ IA

mm/s (white

squares).

4. Other systems.

4.1 THE _ONE DIMENSIONAL CHAIN. We bave checked under what conditions _Dur system

could

reproduce

^trie

peak

of

P(M)

at

high

M observed in many numerical simulations [2].

There _are indeed two ways ^to enhance trie

P(M)

for

large

M. One is to increase trie normal force and trie other to force trie system to be _as close _as

possible

to _{a one} dimensional _one. In the first _case the

explanatîon

is

quite simple.

Indeed _a

large

normal force makes the system

rigid

because trie

spheres

_are

pushed against

the disk surface and _{are net} free _{to move.} Therefore the system is very similar to the interactions of

only

two

rigid

blocks

(see

Sect.

4b).

In the

second _{case, m} order to reduce the system one dimensional _we put

just

one row of

spheres

embedded with silicon _{m a very} thin _groove with _a width of 3 _mm. In such _{a way} the

spheres

cari move

mainly

m the direction of the

loading speed

V. We studied the stick

slip

of this

single

_row of

spheres sliding

on either the hard surface described above _or another

single

_row.

We found _a

peak

^for

large

M in trie

P(M)

and trie result is

reported

in

figure

8. The existence

of _{a power} law for small M

depends

_on V _as in the system ^described in section 3. This _means

that trie

peak

in trie

P(M)

at

large

M is

related,

at least in _our

experiment,

to trie dimension of the system ^and to the fact that the

spheres

have not much freedom to _move

transversely

to the direction of the

Ioading speed.

^Indeed

by making

_a

Iarger

_groove the

peak

at

large

M

disappears.

A similar

explanation

^is also discussed in in reference _[2] for the block mortels and

m reference

iii]

for automata. However there _{are some} other _one dimensional chains [4] wuose

-1

~ i

~

~~

*

Ô~*,

r5

,

fi _-2 (~L

~ _l/~ § h,

~

' 'l'~ ;,/"~' ^ù- '~

~/~

~_ ,P ,/" _& § ( _{é à}

q~ j

R C0 * à

o Q ' ~

~

o ~ $

[

Î

~ _~

~ ~ ~

j ~ ô

~ ^{* ~}

~

~ ~ n

i

, ^à

©~ ô ~

~5.

i 5

LUg10( ti)

~~'

i i

Fig.

9

Lo g~~ ~

Fig.

8

Fig. 8.

P(M)

for _a one-d1nlensional chain of elastically coupled steel spheres. The continuous fine corresponds to nleasurements obtained wuen the one-d1nlensional chain is sliding _on the hard surface

described in the text at V

= 0.04

mm/s.

The straight fine has _a slope 1.64. lnstead the dashed fine corresponds to measurements obtained when the _one dimensional chain _is sliding _on another _one-

dimensional chain at V

= 0.7

mm/s.

Fig. 9.

P(M)

m the _case of the rigid-rigid surfaces. (*) and (o) correspond to nleasurements made

with the _saule normal force Fn =10 N. The loading speeds _were V

= 0.05

mm/s

and V

= 0A

mm/s,

respectively. Instead

IA)

correspond to measurements with V

= 0A

mm/s

but at _a normal force

Fn ₌ 20 N.

dynarnics

does not

produce

the

peak

in tue

P(M).

These cuains _are Ioaded _{in a} diiferent _way,

suggesting

^{tuat also tue}

loading

metuod _may

play

_{a very}

important

role.

4.2 THE RIGID-RIGID SURFACES.

Finally

_we bave studied tue

dynamics produced by

tue

interaction of two

rigid surfaces,

described in section 2. In

doing

this

experiment

_we took

care to bave tue _{same masses} involved and

rouguly

tue _same

elasticity

of tue

experimental

apparatus except

tuat,

tue surface

being rigid,

trie

elasticity

_was

just

on tue disk axis mstead of

being

distributed ail _over tue

interacting

surfaces. Time TT defined in section 2.2 is about

tue _same. Tuis kind of

experimental

set-up presents _no interest from tue point ^of view of G-R

because tue system ^{is reduce} to tue interaction of two blocks like in tue Turcotte model

il,

3]. ^Thus

only

_a cuaotic behaviour bas been obtained At small

V,

these chaotic states _are

perturbed by

trie surface randomness which is trot

enough

to construct _a selfsimilar

scaling

law and

only

_a broad distribution of

P(M)

is observed. Tuis is suown in

figure

9 wuere tue

P(M)

obtained for diiferent normal _pressures and

loading speeds

_are

compared.

We _see tuat at

uigu

V _a

periodic

state witu _a well defined

amplitude

_is recovered. Wuereas _at small V tue

P(M)

_are ^broadened because of tue existence of cuaotic _states

perturbed by

_a random noise.

We mention uere tuis expenment on tue

rigid-rigid

interaction

just

to state tuat randomness

atone,

eituer

produced by

tue

experimental

_{noise or}

by

tue surface

rouguness,

cannot construct

power Iaws in

P(M).

Thus _one condudes that trie results descnbed in trie previous sections _are not

produced by

tue fact that trie steel

spuere

_array is not

perfectly regular.

Trie

important

fact is that each

protuberance

of trie

surface, although interacting

with the other _Que, is free to _move in two directions.

5. Discussion and conclusions.

In this _{paper we} bave

analysed

trie

dynamics

of

stick~slip produced by

trie friction of two elastic surfaces which bave _an artificial and well defined

roughness

made of steel

spueres.

When _one of these surfaces

(called

soft-surface _m trie

text)

is moved _{on a} similar _one, trie

spheres produce

Iocal interactions which _are

elastically coupled

_among them. This kind of interaction _{is very}

interesting

because it _can be considered _as trie

experimental

realisation of theoretical models used to

study

friction forces

il?,

_18]. A direct

comparison

^of our results with tuose of reference

[17,

18] is not

actua1Iy possible

because the local interactions in _our

experiments

and in the

models _are not the _saine. However this expenment can be useful _to check the

validity

^{of models}

of friction where the local interactions _are made _more similar _to the _one described in this paper.

The stick-slip

dynamics

of the soft surfaces described in section 2 has been studied _as

a function of the control parameters,

namely loading speed,

normal

force,

_system size and geometry. ^{We have found that the}

dyuamics

is _a

complex

_one.

Specifically,

wheu the

loading speed

tends to _zero tue

dynamics

presents, ^for a wide _range of trie control parameters, a self

organized

critical state. Indeed tue event size distributions has _{a power} law

dependence

witu

a cut-oi which is _a function of the system size. The

frequency

_power spectra ^of the friction force

decay algebraically

with _a non-trivial exportent

showing

that the time evolution of trie force is

just

^trie

integral

of _a

il f

noise.

We bave also Iooked under what conditions trie

P(M)

_{presents a}

peak

for

large

M. We

bave shown in section 4 that this

peak

_can be

easily

observed if _our system is reduced _{to a} one-dimensional chain in which trie steel

spheres

_are forced _{to move}

mainly

_{m one} direction.

This is in agreement ^{with several numerical} simulations [2, 9,

loi.

Finally

_we have also shown that randomness atone _is not

enough

to construct power laws but it is

really

_necessary that the system is

composed by

_many diiferent parts which may interact

by

_means of _an elastic medium.

The relevance of this kind of

experiments

in the

study

of

earthquake

statistics is

doubtful,

because friction is just an aspect of

earthquakes il,

_13]. In contrast these experiments ^and the associated mortels _can be _more useful for

understanding

friction laws. In any case we have noticed in section 3 that the value of B obtained

by

the exponent i of the _power Iaw _m

P(M)

is o-à _< B _{< as} for

eartuquakes.

A _more precise

relationsuip

between B and _{i can} be found

by

_measunng the Iocal

displacement

of the steel

spheres.

This will also be _very

important

to

study

the mecuanisms of friction Iaws.

As _a

conclusion,

tue most

important

result of tuis _paper is tuat _{a very}

complex dynamics

with many features of S-O-C- _can be

produced by

tue

stick-slip dynamics provided

that _a two dimensional system ^is

composed by

_many diiferent

interacting

parts and that the

loading penod

is mucu smaller than tue cuaractenstic times of tue system defined in terms of elastic

constants. There _{were a} few

numencalexamples

[4, 5,

iii

wuere tuis bas been shown but _no

dear

experimental

evidence tuat tuis could be obtaiued _{ou a}

laboratory

scale.

Acknowledgements.

We

acknowledge

S.

Fauve,

F. Lund, G.

Paladin,

M. Susa de Vieira and J. P. Vilotte for useful discussions. Tuis work bas been

part1ally supported by

tue

"Region Rhône-Alpes".

References

[ii

For _a review _see for example:

a)

Scholz G. H., The Mechanics of earthquakes and faulting

(Cambridge

University Press,

1990);

b)

Turcotte D. L., Fractal and Chaos in Geology and Geophysics

(Cambridge

_{University Press,}

1992).

(2] Carlson J-M-, Langer J. S., Phys. Rev. A 40 6470

(1989);

Phys. Rev. A 44 884

(1991).

[3] Huang J., Turcotte D. Geophys. Res. Lent. 17 223

(1990);

Nature 348

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234.

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(1992)

6288

(Si Knopofl L., Landoni J. A., Abinate M. S., Phys. Rev A 46

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7445.

[6] Schmitbulll J., Vilotte J. P., Roux S., Europhysics Lett. 21

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375.

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OLami Z., Feder H-J-S-, Christensen K., Phys. Rev. Let. 68

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[12] Feder H.J.S., Feder J,, Phys. Rev. Lent. 66

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(13] ^Sornette A., Sornette D., Europhys. Letters 9

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Sornette D., Virieux J., Nature 357

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403; Sornette D., J. Phys. Hance 2

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Valette D. P., Gollub J. P., Phys. Rev. A 47

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Ellegaard

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[18] Pisarenko D., ^Mora P., to be published in Pure and Applied Geophysics, Special issue _on local friction.

[19] ^{de Sousa Vieira} M., pnvate communication.