Geometrical Analysis of Avalanches in a 2D Drum

(1)

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Geometrical Analysis of Avalanches in a 2D Drum

Florence Cantelaube, Yann Limon-Duparcmeur, Daniel Bideau, G.H. Ristow

To cite this version:

Florence Cantelaube, Yann Limon-Duparcmeur, Daniel Bideau, G.H. Ristow. Geometrical Analy- sis of Avalanches in a 2D Drum. Journal de Physique I, EDP Sciences, 1995, 5 (5), pp.581-596.

�10.1051/jp1:1995107�. �jpa-00247084�

(2)

Classification Physics Abstracts

46.10 05.40 64.60C

Geometrical Analysis of Avalanches in

_a

2D Drum

Florence Cantelaube

(~),

Yann

Limon-Duparcmeur (1),

Daniel Bideau

(1)

and

G-H Ristow (~)

(~) Groupe Matière Condensée et Matériaux

(*),

Université de Rennes I,

Campus

de Beaulieu,

35042 Rennes Cedex, France

(~) Fachbereich Physik, Philipps Universitàt Marburg, Renthof 6, 35032 Marburg, Germany

(Received

28 July 1994, revised 15 December 1994, accepted 2

February1995)

fibstract.

We present _an experimental and numerical analysis of intermittent flow at the surface of _a 2-dimensional granular ^medium. Two types ^of ^devices are used: _a 2-dimensional drum and _a long rectangular box. Two types ^of grains _are used

(disks

and

pentagons),

and

comparison is made between the dilferent _cases. The experimental data _are compared with the results obtained from molecular dynamics simulations, _usmg roughly trie _same number of

partiales. Dynamical conditions of stability _are determined from the angle of _repose 6r and trie angle ^of maximum stability 6m. Intermittent flow at the "surface" _appears ta be of two types, as

in 3-dimension: small avalanches whose time fluctuations _are characterised by

a power spectrum obeying _a _power law, and large events with well-defined _average _size and duration time.

l. Introduction

Granular media _are not

only

_one of the most often used mater1als in industrial _processes but _are also

massively

present ⁱⁿ our environment. Tueir flow

properties

have been [1] and

are still _now tue

subject

of many studies _m

puysics,

mecuanics and soit

mechanics,

civil and chemical

engineering.

Some

exciting puenomena

_are

investigated,

sucu _as the formation of

convection

cells,

spontaneous

heap

formation, surface

fluidization, avalanches, density

_waves,

or size

segregation [2,3].

^For a review, see for

example

the _paper

by Jaeger

and

Nagel

_[4].

Avalanches have

recently

received _a great attention, as

being

_a

paradigm

for Self

Organized Criticality (SOC

_[SI. A

sandpile

would

by

itself

adjust

its

slope

to _a critical

value, supposed

to be the

"angle

of

repose"

_@r, defined _as the

angle

between the horizontal and the free surface of the

sandpile.

This self

adjustment

is

suggested

to be

analog

to

SOC,

and then characterized

by long-range spatial

and

temporal correlations,

with _a

typical 1If

_power spectrum ^of the

fluctuations. Some cellular automaton models have confirmed this

assumption [6-9]. Except

in small-size systems,

experimental

results

[10-13]

are m contradiction with the

prediction

of SOC model:

peak

distributions of avalanche sizes _on intervals _are

observed,

instead

of1If

(*) URA CNRS N° 804

@

Les Editions de Physique 1995

(3)

582 JOURNAL DE PHYSIQUE I N°5

noise. Tue essent1al _reason of tuis

discrepancy

_comes from tue fact tuat tue

stability

of tue

sandpile

cannot be described

by

_a

unique angle

of _repose _@r, but

by

two

angles (and

_a

hysteresis

is

observed):

the _maximum

angle

^of

^stability

_@m ^measured

^just

^when an avalanche starts and the

angle

^of repose measured

just

after the avalanche. The average dilference between these two

angles

_@m _@r is of the order of 2° for sand in 3D.

Although

SOC is _an

exciting problem,

numerical and

experimentalstudies

of avalanches _are interesting ^because

they

are

essent1ally

linked to two not well-understood

problems concerning granular

materials: friction and

dilatancy.

The

dilatancy

_was first introduced

by Reynolds

_[14] and it is of great

importance

in the deformation of dense

granular

media: due to the

geometricalinterlocking

of

grains,

_a shear-

ing

deformation _cannot be

imposed

to _a non-cohesive

granular

mater1al without

dilating

it.

Starting

from this

analysis,

_a model for

sandpile

avalanches has been

proposed,

which takes

dilatancy

and friction

explicitly

into account

[11,15,16].

A macroscopic friction coefficient _was introduced

by

Coulomb [17] as

equalto

_~t

=

tan@r.

This _mean value of friction does trot

explain

the two values of obtained

above,

and their relation to the geometry ^of ^the system, the

shape

of the

grain,

and the

microscopic

^friction is not clear.

Because of its

cyclic

character, a

rotating

drum allows for

experiments

with

long-time

ob- servations in processes characterized

by

short-time scales

(e.g., avalanches),

and it has served

as a

good experimental

tool

recently [18,19].

This geometry ^is also

currently

used in industrial systems, ^such as

cylindrical

_ovens.

In this _paper _we

explore experimentally

and

numerically

the rote of the friction coefficient and the

partiale shape

in

equilibrium

of the

heap,

and tue detail of the statistics of avalanches.

We restrict ourselves _to two

spatial

dimensions in order to be able to observe the geometrical modifications in the system. ^Some

experiments

have

previously

been

performed using

pentagons,

showing clearly

the main importance of face _to face contacts,

compared

to tuat of tue usual

punctual

contacts of

spueres

_or disks [20]. ^In

fact,

in tue _case of pentagons, tue number of

possible

stable

configurations

is

larger,

and _some

large

dilferences _are

expected

in tue definition of tue surface

equilibrium

of a

pile,

from botu the statics and

dynarnics.

Wuile it is

perfect

for

geometrical properties,

tuis

2-dimensionalexperimental

and numerical system poses

problems

_m tue context of SOC: the 2D version of tue model

proposed by Bak, Tang

and Wiesenfeld

(BTW)

_{[Si for} avalanches _is not critical at

ail,

unlike _some otuer cellular

automaton models [21]. ^A ^numencal ^simulation

by

molecular

dynamics

of _a 2D

sandpile

does not

give

clear results

concerning

tuis cntical cuaracter [22].

Tuis _paper _is divided into turee parts: ⁱⁿ tue first part, we present our

experimental

set-up and numerical

techniques.

Tue second part ^is ^devoted to tue

dynamicalequilibrium

conditions of _a

pile,

and tue tuird part _gives _our

experimental

and numerical results _on statistics of

avalanches.

2.

Description

of Our

System

2.1. EXPERIMENTAL SET-UP. Tue watts of the drum _are two

parallel glass

disks with _a

diameter of 60 _cm. The width between these _two disks is

(1+ e)

mm

(e

«

1)

^and _we ^have used

particles (disks

_or

pentagons)

which _are 1 _mm thick. Thus the

overlap

of

partiales

is

impossible,

and _our system is

actually

2-dimensional. The ensemble is driver in rotation

by

_a

motor and _a v-belt. Avalanches _are studied

by filling

half of the drum with 1400 pentagons

with _a side

length

of 6 _mm, _or with 1400 disks with _a diameter of 8 _mm. The

packing

^fraction

(surface

of the

grain

over total

surface)

of the

pile

_is found to be

equal

to o.77 + o.03, _a value

(4)

OE~«-UloewoE~-UloeoE~-UloeoE~

-oeOE«Ulw~moe-oeoEUlwmoeooeoE ---oeoeoe number of pentagons

Fig. 1. Distribution of the _mass of avalanches observed in the drum for pentagons.

comparable

to that obtained _on dense

packings

of pentagons [20]. ^For

disks,

this

packing

fraction _is 0.82 + o.02.

As stated

above,

the surface of the

packing

within the drum is characterized

by

two

angles:

tue

angle

^of _repose _@r and tue _maximum

angle

of

stability

_@m. Tue dilference

A@(=

_@m _@r)

between tue _two is

generally

^of _a few

degrees.

For small systems, tuere is _a critical

lengtu

_1,

defined _as the

length

from which _one

grain

_is _seen

by

_an

angle

A@. If the size of the system is smaller than 1,

hysteresis

cannot be

observed,

and the nature of avalanches is

changed:

^for

example,

_some _power law for the avalanche size distribution _is observed [12]. ^In our case, this

length

1is 3 _cm, which is small

compared

to tue diarneter of tue drum.

Tue rotation speed (uJ) îs âbout ô.25 _rpm, wuicu is

sufficiently

slow to _ensure tuat tue statistics of avalanches is

speed-independent.

A video film of tue drum in rotation is

analyzed by

_an

image processing

_program

(visilog 4.1.3)

_on _a work station. From

tuat,

_we obtain the statistics of tue

weigut,

^duration and

separation

time of

avalanches,

and _we _are able to _measure the

angle

at tue

beginning

and at tue end of _an avalanche. Tue results obtained _on pentagons

by

tuis metuod _are

compared

_to tue results tuat _we have

previously

obtained _on

disks,

and

by

numerical simulations.

For eacu avalanche event, we record _grey scale

images rigut

before and

rigut

after tue avalanche and

digitize

tuem _using tue

image processing

_prograrn. Tuese

pictures

_are bina- rized witu _a thresuold in order to represent pentagons

by

values of1 and the

background by

o.

On the

picture

taken after _an

avalanche,

_we

impose

a

negative

rotation

by #

₌ uJtd, where id is the duration of tuis

avalanche,

to correct tue drum rotation in _our calculation of the _mass of the avalanche. If _one tilts tue voles _m the pictures

by

_a

geometrical

operation

(dilation)

and

tuen subtracts

tuem,

the

resulting

picture shows tue surface of tue

avalanche,

wuicu _we will colt

weigut.

In order to separate

large

events from small events, we have carried out two types of

analysis

of these statistics. A

simple

observation of _our

experiments,

or of avalanches _at tue surface of _an mclined box filled

by sand, clearly

shows that these two dilferent types of avalanches

coexist:

they

will be described later.

Figure gives

the _mass distribution of avalanches. It is important to note tuat in tuis

figure,

tue

probability

to observe _very

big

avalanches is cut off due to finite _size elfects in the drum which prevent the system ^from

reaching

_as

higu

_a value

as m the static _case. For the

analysis

of

large

events, we suppress ^events with _a surface smaller

(5)

o

@ C

©

2ce

OE

, o

~ o

O

~ ~ ~

@ _~

£l

E _io0

o ~ o

° cxpcnmcnul

~ C

~ .

~cantcd

o

0

0 5 10 15 20 25 30

deltatheta

Fig. 2. Experimental

(0)

and tueoretical

(-)

relationships between the angle ^dilference ^and ^the avalanche _mass, expressed in number of grains.

tuan 25 pentagons.

In order to

study

tue

spatial

distribution of avalanches _more

precisely,

_we bave built _an

experimental

device _very similar to tuat used in Drake's

experiment

_[23]. It is _a 2-dimensional

box, consisting

of two

rectangular glass planes,

200 _cm

long

and 30 _cm

uigh.

Tue widtu

between tue two

planes

is about 1.2 _mm. Tue inclination

angle

of tuis box witu respect to tue horizontal _cari be controlled

by

_a system ^of

pulleys.

As _in the

drum,

_we tilt tuis cell witu styrene disks witu _a diameter of 8 _mm, _or witu pentagons ^whose

sidelengtu

is 6 _mm. It is necessary to _use 2400

particles

to obtain

size-independent results,

witu _a

pile

about ter

particles uigu.

Actually

tuere _are _never _more tuan rive

layers

of

grains

whicu fait when _an avalanche settles.

Thus tue surface _on whicu the

partiales

flow is

really

constituted of otuer mobile

particles

like

m the drum.

2.2. EXPERIMENTAL VALIDATION.

Using

pentagons allows _us to have _an

amorphous

_system

without

crystalline

_zones, which _con induce _some _non trivial correlations in collective rotations.

Crystalline

structures

clearly modify

the conditions of the static and

dynamic equilibrium.

We have also studied

amorphous

systems ^of

equal

disks in the

drum,

and in the 2D

box, by gluing

a

rough layer

of disks of dilferent sizes at the externat

watt,

_or

by mixing particles

of dilferent sizes.

Figure

2 shows the avalanche _mass _versus A@, the dilference of the

angle

before and alter

an avalanche. The

points

have been obtained

by

the image

processing

_program where _we took tue rotation of the drum

dunng

tue avalanche _into account. The

straight

fine is the theoretical

curve calculated via the relation

~

Î

_~~

where L is the

length

of the

pile,

C the

packing fraction,

and

Sp

the surface of _one pentagon.

It is obtained

directly by writing

the surface

occupied by

pentagons in _an arch of radius

L/2

and of

angle

@.

The

experimental points

_are the result of

averaging

the _masses obtained in each interval of _one

degree. Except

for small events

(which

_are,

by

_nature, dilferent irom the

large

events descnbed

by

this

formula),

_a ^reasonable agreement is obtained

(the

fluctuations observed _are

(6)

essent1ally experimental),

and _we conclude that the

experimental weight

is

roughly proportional

to the

angle

dilference of the

large

avalanches.

2.3. NUMERICAL MODEL. In

1979,

Cundall and Strack

proposed

_a model to describe the forces

acting

at the

microscopic

level _on

grains

in _an

assembly

of

spheres

which

they

called the distinct element method [24]. ^Since ^then ^this ^method ^has ^been ^extended ^and

slightly

modified to

describe,

_among other

things,

the mechanical

sorting

^of

grains

_[25], the shear-induced

phase

boundaries

[26],

the

origin

^of convection cells

[27-29],

and the outflow from _a

hopper [30-32].

For the sake of

simplicity,

_we will consider

spherical particles,

but if _one includes _a static friction mechanism

(thereby neglecting

the rotation of the individual

sphere),

_we _can well

reproduce

the

experimental

results obtained with pentagons, ^which we will discuss below.

Whenever two

partiales

_are doser thon the _sum of their

radii,

_1-e-, when

they

have been

deformed, they

interact via normal and shear forces. The force _on

partiale1,

caused

by partiale j

is

given by:

F~j

=

Fnii

₊

Fsé (1)

Fn

=

-kn(r~

+ rj

(ri rj)ii)" ~inme~(f~ fj)11 (2)

Fs

₌

-sign(ôs) min(ks jôsj,/J jfnj) (3)

where

m~mj

~~~

mi + mj

and

ôs ₌

Î

(é~

éj )é

dl

Here _ri denotes tue

position

vector of tue i~~

partiale,

ù is tue unit _vector

pointing

from the

partiale

to

j

and § is _a unit vector

perpendicular

to ii

rotating

clockwise. Tue constant kn stands for tue

"Young modulus",

ks for tue

spring

constant in the suear direction to mimic static friction

[32],

^and ôs for tue total shear

displacement during

the contact of the two

partiales.

me~ is the effective _mass. Tue shear force Fs bas two

regimes

where tue transition is

given by

the friction coefficient _~t due _to the Coulomb

Fs

critenon. For _a

= 1, we are left with Hooke's law and for _a ₌

3/2,

_we

investigate

the Hertzian contact force for

slightly

^deformable disks.

Our units _are chosen in such _a _way that the average

partiale

diameter is unity ^with a mass

of

unity.

The

pseudo "Young

modulus"

kn

is

equal

to 10~

N/m~/~,

and _we used _a time step ^of Ai ₌ 2 _x 10~~ _s which assured numerical

stability.

The

relationship

between _our

damping

parameter _~in and the

experimentally

_more

commonly

used restitution coefficient _e _con be derived

exactly

for _a ₌ 1 from equations

(1)

and

(2)

and is of tue form

In(e)

'/n C~

~

For _a ₌ 1.5, _one

gets

a very weak

velocity dependence

but the above

equation

_can still be used

over a wide range. Tuis has been verified for _our model

by

_one of _us [30].

To avoid tue ordered

triangular lattice,

_we _use _a distribution of

partiale

sizes wuere tue size of tue

biggest particle

is four times tue _size of tue smallest

partiale.

^The drum _is ualf filled with

partiales

^and contains

rouguly

1450 of tuem. We bave tested _our simulation for dilferent rotation

speeds

and have found the three

experimentally

well-known

regimes

of distinct

avalanches,

continuous flow and

centrifugated regime

_[18]. The rotation

speed

^has ^been ^chosen

as uJ = 0.25 _rpm in order _to be in the distinct avalanche

regime

^which is that

experimentally

used,

_as discussed in the

preceding

section.

(7)

0.2 0.2

P(8M)

(a)

P(8R)

(b)

0.ifi 0.ifi

0.12 0.12

0.08 0.08

0.04 0.04

0 0

30 40 60 60 30 40 60 60

BM

8R

Fig. 3. Probability distribution of the maximum angle of stability 6m

(a)

and of the angle ^of _repose 6r

(b)

of _an avalanche. These distributions _are obtained experimentally with pentagons.

As it has been

argued

in Section 2.2, the _mass of _an avalanche _is

proportional

to the

angle

dilference A@

= @m-@r. Since the latter is measured _more

easily

in _our

simulations,

_we will _now describe the

procedure.

At each iteration, ^the

rotating

drum is divided into 100 vertical

strips

and the position of the

highest partiale

in each strip is searched. From these 100

partiales,

_we obtain the

angle

of the surface

by

_a

least-squares

fit. The

angle

is

averaged

_over 1000 iterations and _we check _our automated

algorithm by comparing

the results with _a visual manual fit for

dilferent _times.

3.

Dynamic Equilibrium

of trie Pile and

Geometry

We have _seen that in the intermittent

regime (avalanches),

_a

heap

of sand is cuaracterized

by

two

angles:

tue _maximum

angle

of

stability

_@m and tue

angle

^of _repose _@r. Tuese

angles

_are

non-trivial functions of tue coefficient of friction between

grains,

tue geometry of tue system

at tue individual scale

(shape

of tue _grains, etc. and _at tue

global

scale.

3.1. EXPERIMENTAL ANALYSIS. _Each

experimental angle

is determined _as trie result of

an

averaging

_process

(least-squares metuod)

_over tue drum

diameter,

in order _to smooth the

roughness

of the surface.

Experimentally,

it is clear that there _is _a _more _or less broad distribution of the measured

angle

of repose @r and of the maximum

angle

of

stability

@m. The distributions for pentagons ^obtained

by measuring

260 avalanches _are

given

m

Figure

3: the

mean value of

@r is 37° and that of _@m is 45°.

Experimentally

too, ^for

amorphous (1.e.,

^with disk _size

distribution) packings

of

disks,

_we have obtained _@r

= 24° and _@m

= 33°. Two _reasons

con

explam

this dilference:

first,

because of face-to-face contacts, each pentagon ^is more stable thon the

corresponding

disk. But there _are also strong correlations in the relative rotations of pentagons [33], ^and ^tue

beginning

of _an avalanche and its motion _are _more difficult in this

case: a

pile

of pentagons ^is more stable tuan _a

pile

of disks.

As suown in

Figure

4, tue

experimental angle

dilference A@, wuicu _measures tue _range of avalanche

sizes,

_is _very wide: _it _ranges from

nearly

_zero to

nearly

20

degrees,

witu _a _mean value of 9°. One _notes tuat tue

angles

between 2 and 14

degrees

are almost

equally probable.

(8)

o.io

P(A8)

0.08

o-où

0.04

0.02

0

0 10 20

A8

Fig. 4. Probability distribution

P(A6)

of the dilference A6 between the starting and stopping angle

of _an avalanche, obtained experimentally with pentagons.

Tuis

large

_range is due to tue fact

tuat,

for pentagons, tue

possibility

to find _an

equilibrium (individual

_or

collective)

is greater

compared

to tuat for disks

3.2. NUMERICAL APPROACH. It is difficult to make _a

quantitative expenmental analysis

of tue influence of tue friction coefficient between

grains

on the

angles

_@r and _@m. In return,

using

our numerical

simulations,

since the introduced

spring

constant ks is ratuer

artific1al,

we may obtain _a reasonable value of tue

angles defining

tue

equilibrium

of the

heap

from

some test runs. We fill the drum

halfway

with

partiales

of dilferent diameters

(which

is the

way of

introducing

disorder in the numencal

simulations)

and start to rotate it. After _some thousands of

iterations,

_we stop the drum rotation and watt for the system to relax: _we control the relaxation

by monitoring

^the kinetic _energy. We _measure the obtained

angle

of _repose _@r for dilferent values of _k~ and of tue Coulomb turesuold _~t. Tue results _are

presented

in

Figure

5, in whicu the continuous litre

corresponds

to the Coulomb relation _@r

=

tan~~

_~t _[32].

A linear

relationship

_is obtained for _@r _as _a function of ks for small values of _p. For

larger

values of _~t, _a saturation elfect is observed. In order to obtain

equilibrium

conditions in _nu- merica1 simulations and in

experiments

_on pentagons, we have

adjusted

_our simulations

by choosing

_~t

= o-G for _a value of ks

= 1000 N.m~~

Doing this,

_we obtain à numerica1value of

@m near the

expenmental

_one.

Figure

6 shows the distribution of

angles

obtained

numerically

for

non-rotating

disks

averaged

_over 112 avalanches: _we obtain _@m

= 45° and _@r

= 40°.

Therefore _one has to

keep

in mind that, in spite ^of an apparent

quantitative

agreement, there _are strong

geometrical

dilferences between tue two systems

(disks

and

pentagons),

_as

discussed above: tue numerical value of _@r is dilferent from tue

experimental

_one. Here too tue

experimental packings

of pentagons

comparatively

^olfer _a

larger

number of

equilibrium

states than in

numericalsystems

of

disks,

_even with _a strong ^friction: ^tuis

explains

tuat tue distributions of

angles

_are

larger

for

experimental

pentagons

(see Fig. 3)

than for numerical disks. These distributions _are trot

symmetrical,

and _we believe that this is due for _a

large

part

to limite size elfects

(see below).

4. Avalanches

4.1. DEFINITIONS. An avalanche is here defined _as the motion of at least _one

particle

down tue inclined surface.

Starting

^from ^tuis

definition,

_we _must

distinguisu

between _two

(9)

(10)

(11)

(a) (b)

20000 ioooo

p J~

ioooo ioooo

0 0

0 20 40 fi0 80 100 0 20 40 60

t Îs) ^t (s)

a) b)

Fig. 8.

(a)

Fluctuations _m pixel of two successive pictures obtained experimentally in the 2D drum filled with pentagons. Measurements _are taken _every 1° of rotation;

(b)

_same result obtained

numerically. Measurements _are taken _every o.15° of rotation.

Tuis initiates _a collective

back-avalanche,

wuicu is _a

global

loss of

equilibrium involving

ail

partiales

of tue

pile leading

to a total reconstruction of tue

surface,

wuicu is

finally

inclined from tue horizontal

by

_@r.

Small events

only

involve _one _or few

grains

_m metastable

equilibrium

_on the surface of tue

pile.

Tuis loss of

equilibrium

_is

individual,

and is of the type ^observed ^if one

places

_one

grain

on a

rougu

inclined surface built

by gluing glass spueres

_on it [36,

37].

Wuen it loses its

equilibrium,

tue

grain

falls

down,

and

during

its

motion,

it bas _some collisions witu otuer

grains,

and stops _or _may create _a collective motion.

Orly

_a weak and

generally

local modification in tue surface

morpuology

is observed

along

_a

lengtu

_1,

being

tue free

patu

of tuis event,

generally

smaller tuan tue

lengtu

of tue free "surface". Tue _occurrence of tuese events

depends

_on tue

morpuology

of the

pile surface,

_on the inclination

angle

and _on its fluctuations.

In the

following

part, _we will descnbe tue statistics of tuese two dilferent _events.

4.2. SMALL EVENTS

4.2.1.

Experimental A~ialysis.

Bretz et ai. [34] ^bave ^suown ^tuat ^tue ^surface ^of a

sandpile

_is

subject

to small fluctuations wuen _one

slowly

increases the

angle

of _repose. Tuese small _events

occur between

large

avalanches: this is what _we observe in both

experiments

_in the 2D drum

and in _a 2D inclined chute.

They

evolve less than 30

particules (disks

_or

pentagons).

Figure

8a illustrates the relative importance ^{of each} type of events, ^and ^how

they

_are time- distributed: it

gives

the _mass fluctuation _at tue surface of tue 2D

pile

^of

pentagons

m tue drum.

To obtain this

(small)

_mass

variation,

we

analyse

the pictures, ^but

only

between identified

large

events. We

plot

the dilference in

pixels

^between two successive

pictures

taken with _an interval of time

equal

to 0.67 _s

imposed by

the resolution of the

image processing

_program and

exactly

the _sonne

procedure

_as descnbed in Section 2.1 is

applied

to measure the fluctuation of

weight.

The _area obtained

by

tue subtraction of the two pictures results from the

reorganization

of the

(12)

Î _t~

= z

~ n

m

~ n

fl o

u

20 40 60 80 100 120 140 160 180 200 20 40 60 80 100 120 140 160 180 200

a)

_startIng position (cm)

b)

_~~~~~'~~ _P°~'~'°~ _l~~l

Fig. 9. Probability for

a disk to be the seed of _a small event at distance L from the top of the pile,

in the inchned chute:

(a)

pentagons;

(b)

disks.

heap

surface

during

tue rotation of1°

(rotàtion _angle

of tue drum

during

tue interval of time of 0.67

s).

We

apply

tue _same

procedure

at eacu rotation

by

_one

degree

until the next

large

event. The statistics obtained _are the _sum of these

analyses.

Even if the

experimental

spectrum obtained from the _square of the

amplitude

of these _mass fluctuations could be fitted

by

_a

straight

litre with _a

slope

of the order

of1.5,

the statistics

are not sufficient _to _be affirmative

jbout

_any _power law behaviour. The numerica1simulations

can

explore

the fluctuations of the

pile

surface

during

_an interval of time reduced _to 0.1 _s.

Tuus _we suall _use tuese numerical simulations to describe _a better

analysis

of tuese small _mass fluctuations in tue

following

section.

According

to results obtained _on tue inclined

plane [36, 37],

^it ^is ^reasonable to expect a

uomogeneous

distribution of tue

starting positions along

tue line for tuese small events: _one

con believe tuat _at tue surface of _a

packing,

tue

probability

to find

partiales

in metastable

equilibrium

is tue _same

everywuere

_on tue surface.

We

plotted

_on

uistograms presented

in

Figures

9a and 9b tue

probability

for _a small event

to _occur at _a distance L from the top ^for pentagons

(a)

and disks

(b),

obtained in the indined

chute. This

probability

is not constant

along

the

surface,

neither for

disks,

_nor for

pentagons.

On the contrary, we observe that the

probability

of the

starting position

decreases

nearly linearly

with L. This evolution _is smoother for disks thon for pentagons. Nevertheless small events are intercalated between

large

events, wuich lead to _a

complete reorganization

of the tue surface. After the reconstruction of the

pile,

the successive small _events will

destroy

the metastable states below its

starting

position _up to its capture

by

the bulk: the surface is

progressively

washed. Tuis

explains

trie results suown

Figure

9.

4.2.2. Numencal

Analysis.

In tue _case of numerical studies of small events, we map every o.1 _s _our

grain configuration

to _a soc _x 500

binary grid

wuere _a 1 in tue cell indicated that

more tuan 501~ of the cell _area _is covered

by

a

partiale.

Two successive grids _are ^subtracted from another and the dilference is measured in cells. This dilference is _our measurement for the small surface fluctuations.

A

typical

time trace of _our numerical system ^is ^shown ⁱⁿ

Figure

10. In part

(a)

_we show the

angle

of the surface

averaged

over looo iterations

(which

has been obtained

by dividing

^the

drum in loo vertical

regions, looking

for the coordinate of the

highest partiale

and

making

_a

least-squares fit).

In part

(b)

_we show the total kinetic _energy of the

partides.

One _sees how eacu

peak

in the energy

corresponds

to a drastic

change

in tue surface

angle corresponding

to _a

iouxNAL DB pHYsioçB i T.s,w s,MAy 199s n

(13)

2000

e 1',1

Il

,ji

'

jçoo

~~

.,~

~(

~Î. Ji'

~

~~ _1,~

,

' J 1200

v

j

Si t 'j

) ' soc

~~ 400

, ,

~

6o go 100 120

j~ ,~ fi0 80 100 120 2° 4°

1trie (si ^tinte (si

a) b)

Fig. 10.

(a)

Surface angle fluctuation obtained numerically;

(b)

corresponding total kinetic energy fluctuations.

cuaracteristic avalanche. Also notable _are tue small fluctuations

(small events)

_in botu

grapus (anotuer

_source for fluctuations _are tue statistical _errors of _our measurement

technique).

From these numerical results, _we con draw

Figure

8b. The noise

(at

_a

frequency roughly equal

to là

Hz)

observed in this

figure,

then

concerning

^numerical

simulations,

is

essentially

due to the numerical method. Between

large

events, we observe _a succession of _very small events

(one

has to

keep

in mind that the

large

event distribution is cut off

by

^the ^finite size of

our

system).

As before _we

plotted

the _power spectrum ^{of the} square of the

amplitude

of the _mass fluctuations. The data

only ranged

_over _one order of

magnitude

with _an _error of

roughly

o-à- The numerical

results,

shown in

Figure

ii _are fitted

by

_a

straight

line with _a

slope of1.8,

not

alfected

by

the "numerica1noise" which _appears at

frequency

15 Hz.

4.2.3. Conclusion. These numerical and

experimental results,

which

give

tue power laws for tuese small events, _are

qualitatively

similar _to tuose obtained in 3 dimensions

[12, 34]. However,

because of the small _size of _our systems, _no definitive conclusion _cari be drawn.

4.3. LARGE EVENTS.

Large

events involve trie whole

pile.

As it is _a

global

_process, which is trot modified

by

small events, the

probability P(L)

to observe _an avalanche

starting

at _a

distance L from the top of the

pile

is

expected

to be constant.

P(L),

determmed in _our inclined

chute,

is

given

in

Figure

12a for pentagons ^and ⁱⁿ

Figure

12b for disks.

Except

for small _or

large

values of

L, P(L)

is

roughly

constant in both _cases. Near the

bottom,

the slide of _a

row of

partiales

_is blocked

by

the wall and _near the top, the pressure on a

grain

is screened

near the wall: this

explains why P(L)

is smaller at the _two ends of the distribution. These two screening

lengths

_are _a little

larger

for pentagons than for

disks,

as the

stability

of these

grains

is

larger. They

are of about 40

particle

diameters.

Accordingly,

finite _size effects _are

expected

to be

significant

in _our experiments in a drum: for disks in _a

drum,

the

histogram

is

comparable

to that of

Figure

12a.

Figure

13 shows that

P(Ù)

is not a constant of the starting position for pentagons in the drum: tue first

partiale

wuicu flows down tue

pile

is located

mostly

at tue top ^{of tue} ^surface.

Thjs dilference in

P(L')

between disks and pentagons con be correlated to tue

geometrical

(14)

le-05

~p

ie-06

ie-llî

le-os

Ie-09

le-10

o-1 10 10']

f[H2]

Fig. Il. Power spectrum relative to surface fluctuations obtained numerically.

£~ _~

é ~

W £

n @

o n

ÉO

20 40 60 80 100 120 140160 180 200 20 40 60 80 100120 140 160180 200

startIng position (cm) startIng position (cm)

a) b)

Fig. 12.

P(L),

probability for _an avalanche to start at _a distance L from the bottom of the pile _m

tue box, _uprsus L:

(a)

pentagons;

(b)

disks.

configuration

of tue top ^of tue

pile just

^before tue avalanche

(see Fig. 14):

^for

disks,

tue surface of tue

pile

in tue drum _is

regular (Fig. 14a),

^wuereas for pentagons, tue

suape

is very dilferent _near trie top

(Fig. 14b). Figure

14a shows tue

configuration

of tue top of ^tue surface for _a mixture of small and

large disks;

but tuere _is _no

change

in tuis

pile geometry

wuen built

only

witu

large partiales.

Because tue

slope

is steeper ⁱⁿ ^tuis part, ^tue top of tue pentagon

pile

becomes _very instable wuen becomes of tue order of _@m. Therefore tue avalanches start

preferentially

at tue top.

Figure

15a shows the distribution of tue duration time of _an avalanche and

Figure 15b,

tue

separation

time between two running avalanches. The

uistograrn

of the duration looks

quite

suarp, and tue _mean duration time is 2.7 seconds. Tue

separation

distribution _is broader thon tue duration _one and gives a mean separation time of 8-9 seconds.

(15)

ce + W W O ce + W w O ce + W w

ce ce ce ce ce

Starting length (cm)

Fig. 13. Probability for _an avalanche to start at _a distance L' from the top ^of ^trie pile in the drum, for pentagons.

i

b) a)

Fig. 14. Photographs

showmg

the geometry of the surface of the pile _m trie drum:

(a)

disks;

(b)

pentagons.

5. Conclusion

While made _m

relatively

small systems, our

study

in 2 dimensions has confirmed the bimodal character of avalanches _at the surface of _a

pile:

small events, ^wuose size ditribution is described

by

_a _power

law,

and cuaracteristic

(large)

events wuose _size and time duration _are

well-averaged.

Tuese results _are

qualitatively

tue _same wuetuer _one

uses

disks,

_or

pentagons

in expenments.

Tue numerical simulations _are

essentially

used _as _a

complementary

tool to

study

in _more detail tue influence of _some

puysical

parameters wuich cannot be reached

experimentally (friction,

energy, ^etc. ).

(16)

0.4

~~~~ l~) Plt) jb)

0.3 0.12

0.2 0.08

o-1 0.04

0 0

0 2 4 fi 0 4 8 12

t [s] t [s]

Fig. 15.

(a)

Distribution of avalanche duration;

(b)

distribution of time separation between two successive avalanches, obtained experimentally with pentagons.

Beyond

tuese results, we bave outlined tue strong ^influence ^of ^tue

partiale shape

and friction coefficient _on tue

angle

^of _repose _or of maximum

stability. Finally,

because of tue _extreme

sensitivity

of tue avalanche to tue local and

global equilibrium,

_we bave suown that

geometrical

limite _size elfects _are important ⁱⁿ ^these systems.

Acknowledgments

This work has benefited from useful discussions with J-P-

Troadec,

A. Hansen and S. Roux. It has been

partly supported by

the GdR CNRS

"Physique

des Milieux

Hétérogènes Complexes".

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