Avalanches in (1 + 1)-dimensional piles: a molecular dynamics study

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Submitted on 1 Jan 1993

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dynamics study

Jysoo Lee

To cite this version:

Jysoo Lee. Avalanches in (1 + 1)-dimensional piles: a molecular dynamics study. Journal de Physique

I, EDP Sciences, 1993, 3 (10), pp.2017-2027. �10.1051/jp1:1993229�. �jpa-00246849�

Classification Physics Abstracts

05.40 46.10 62.20 64.60

Avalanches in (I ₊ I)-dimensional piles:

_a

molecular dynamics

study

Jysoo

Lee

HLRZ-KFA Jiilich, Postfach1913, W-52425 Jiilich, Germany

(Received

I September 1992, revised 13 June 1993

(*),

accepted 16

June1993)

Abstract. We numerically study the piles generated by continuously dropping _a particle

on the top of _a stable pile in

(I

+

I)

dimension. We _{use a} code which has _an implementation

of static friction in the Molecular Dynamics

(MD)

simulations of granular material. We study

several properties of the pile-the time evolution of _mass and slope, and the avalanche _size

(and

duration) distribution. We also study the effect of gravity _on the properties of the pile. We find that the angle of repose @R decreases _as gravity is increased. On the other hand, gravity _seems

to have little effect _on the angle of marginal stability _@Ms. We also find that the angles _(@R and

@Ms) are different between the piles of monodisperse and polydisperse particles. We suggest _a possible explanation for these dependencies of the angles.

1 Introduction.

Systems

of

granular particles (e.g. sand)

exhibit many

interesting phenomena [1-4].

^{One of the} distinct

properties

of

granular

systems ^is that

they

_can behave like both _a solid and _a fluid. One

can pour

(like

_a

fluid)

sand

grains

_{on a}

table,

and

they

form _a stable

pile

with finite

slope (like

a

solid). Using

_a cellular automation model of

sandpile, Bak, Tang

and Wiesenfeld _[5]

(BTW) showed,

in

(2 +1)

and

(3 +1)

dimensions, that when _one

continuously drops grains

_{on a} stable

pile,

the system ^evolves into _a state with _no intrinsic

length

and time scales

(critical state).

Experimental

studies _on real

sandpiles, however,

_are not

fully

consistent with the results of BTW.

Experiments by Jaeger

et al. [6] and

Evesque iii

do not show any

sign

of

criticality.

On the other

hand,

experiments

by

Held et al. [8] showed that the

piles

_are indeed critical for small sizes, ^but cease to be critical for

larger

sizes.

The behavior of

(I

+

I)-d sandpiles

_can be very different from their

(2

+

1)-d

counterparts.

For

example,

the

(I

₊

I)-d

version of the BTW model does not

display

_any critical behavior.

On the other

hand,

there exist

(I

+

I)-d

models which _are critical [9]. ^Whether a

(I

+

I)-d

real

sandpile

shows critical behavior _or not is

certainly

_{an open}

question.

In this paper, we

study

(*) The editor regrets the delay due _to lost mail.

the behavior of realistic

(I

+

I)-d piles using

^Molecular

Dynamics (MD).

The _{very reason}

why

sand _can form _a

pile

is static friction. In other

words,

_one needs _a

finite

threshold to break

a contact between

grains (and

^between a

grain

and _a

wall). Therefore,

_one has to include static friction in the MD simulations of

granular particles

to obtain _a stable

pile. Here,

the

implementation

^{of static friction is done}

by using

^{the scheme of Cundall and Strack}

[10].

^We

study essentially

_a

(I

+

I)-d

version of the

experiments by Jaeger

et al. and Held et al.. In other

words,

we

drop

a

particle

_{on a}

pile,

and wait until the

pile

becomes

stable,

then add another

particle,

and _{so on.} We find that the time evolution of the _mass and

slope

of the

piles

show

complex

structures. We also

study

the avalanche size

(and duration)

distribution _to determine

whether the system ^is critical. The avalanche size distribution is consistent with _a

power-law

form

(critical). However,

_we cannot make _a definite statement, ^since we have studied

only

_very small systems.

There _{are two}

important angles

for _a

pile.

^One is the

angle just

after _an avalanchw-the

angle

^of repose

9R,

and the other is the maximum

angle

^of a stable

pilw-the angle

of

marginal stability 9Ms.

These two

angles

^differ

by

_a few

degrees ill].

In order to understand the mechanisms

responsible

for these

angles,

Evesque et al. studied the

dependencies

of these

angles

on the

density

of

packing

and

gravity

[12].

They

found that the

angles

_are

strongly dependent

on the

density,

but less

sensitively

on

gravity. However, they

did _not

study

the effect of

gravity

for _a fixed

density. Rather, they

studied the time evolution of the densities

during

several

avalanches,

while

gravity

^is

being changed. Here,

we fix the

density,

and

study

the

dependence

of both

angles

_on

gravity.

We _measure 9R _as follows. We put

particles

at

randomly

chosen

positions

in _a box. We wait until the

particles dissipate

their _energy, and fill the box ivithout _any

significant

motion. We then _remove the

right wall,

and wait until the

pile

becomes stable

(motionless).

We define the

angle

of

resulting pile

to be

9R. Now,

_we

slowly

rotate clockwise the box

(with

the

pile inside).

The

pile

is stable until _we rotate more than the

tilting angle

ST The

angle

of

marginal stability

9Ms is estimated from 9R and

ST.

We find that

9R systematically

decreases _as

gravity

is increased. On the other

hand,

9Ms seems to show little _{or no}

dependence

_on

gravity.

We also find that both

angles

_are different between

piles

of

monodisperse

and

polydisperse particles.

This

difference,

_as well _as the _near

independency

of

9Ms,

indicate that the geometry ^{of the}

packing plays

_a dominant role.

2. Interaction of the

particles.

Each

particle

is

represented by spheres

which interact

only

if

they

_are in contact with each other. Consider _two

particles

I and

j

in contact, the force between them is the

following.

Let the coordinate of the center of

particle

I

j)

be B~

(Rj),

and _r

= B~

Rj.

In _two

dimension,

_we

use a new coordinate system ^defined

by

two vectors n

(normal)

and _s

(shear). Here,

_{n =}

r/ (r(,

and _{s are} obtained

by rotating

clockwise _n

by 7r/2.

The normal component

Fp_,

^{of the force}

acting

_on

particle

I

by j

is

lift,

₌

kn(a,

+ aj

(r()~/~ ~fnme(v n), (la)

where _a;

(aj)

is the radius of

particle

I

(j),

and _v

=

dr/dt.

The first _term is the Hertzian elastic

force,

where _km is the elastic constant of the mater1al. The constant ~fn of the second

term is the friction coefficient of _a

velocity dependent damping

term, m~ is the effective _mass, m,m~

/(m;

+

mj).

The shear component

F]_;

^is

^{given by}

Fj_,

₌

-~f~me(v s) sign(ds) min(ka(ds(, p(F),(), (16)

where the first _term is _a

velocity dependent damping

term similar _to that of

equation (la).

The second _term is to simulate static

friction,

which

requires

_a

finite

amount of force

(pF/_;)

to break _a contact

[10]. Here,

p is the friction

coefficient,

ds the total shear

displacement during

a contact, and k~ ^{the elastic} constant of _a virtual

spring.

There _are several studies _on

granular

systems

using

the above interactions

[10, 13-15]. However,

all of

them,

except ^references

[10,

14] ^and

[15],

^do not include static friction. A

particle

_can also interact with _a wall. The force

on

particle

I, in contact with _a

wall,

is

given by equations (I)

with aj = oo and _m~

= m;. A

wall is assumed _to be

rigid, I-e-,

the

position

is fixed.

Also,

the system is under _a

gravitational

field _g. We do not include the rotation of the

particles

in the present simulation. A detailed

explanation

^{of the} interaction is

given

in reference [15].

3.

"Sandpile"

with infinitesimal flux.

We

study

the behavior of _a

pile

with infinitesimal continuous addition of

particles.

The simu- lational setup ^is

essentially

_a

(I

+

I)

dimensional version of the

sandpile

experiments done

by

a)

Fig. I. a) The "sandpiles" _are built _{on a} L shaped wall of width W and height H with _a gravitational

field _g pointing downward. Here, W

= 4, H

= 4 and _g

= 980. We show in

b)

the number of particles,

and in c) ^the angle ^of the piles like the

one shown in a). Both quantities _are measured whenever the

piles become stable.

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120.oo

@100.00

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80.00

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~

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number of _inserted b)

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5.oo

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0.00 (1.40 0.80 1.20 1.60 2.00x10

number of inserted balls _c)

Fig. I.

(continued)

Jaeger

et al. [6] and Held _et al. [8]. ^Consider an L

shaped

wall of width W and

height H,

and

a

gravitational

field of _g

= 980 is

acting

downward

(Fig. la).

We _use CGS units in this paper,

although

we do not

explicitly give

units for each quantities. In the

beginning

of _a

simulation,

we insert _a few

(5

r~

10) particles

_on the bottom

wall,

and calculate the

trajectories

of the

particles,

until the maximum of both

(x

and

y) velocity

_components of the

particles

is less that

a cutoff value _u~ut. If the maximum is smaller than _{u~ut, we} insert _{a new}

particle

_on the left side and two

particle

radius above the

pile.

We

again

calculate the

trajectories

of the

particles,

until the maximum becomes smaller than _u~ut, insert _{a new}

particle,

and iterate this

procedure again. Also,

if _a

particle

falls out of the box, _{we remove} the

particle

from the system.

We want to simulate the situation that the

pile

relaxes to _a static

configuration

after each

addition of _one

particle.

If _a

pile

is

static,

the

velocity

components ^{of all the}

particles

should be zero, which is

only

true if _u~ut

" 0. We

study

the system with _u~ut

"

0.1,

0.01 and

0.001,

and

find _no

systematic

differences. We _{use u~ut}

" 0.I from _{now on.}

Also, changing

the

height

of _a

newly

inserted

particle

to 4 and 12

particle

radii above the

pile produces

_no systematic

change.

The parameters for the interaction between the

particles

_{are km}

= 1.0 _x

10~,

k~ = 1.0 _x

10~,

~fn = 5.0 _x 10~,~fs ₌ ^{5 and} _{p =} 0.2. For the interaction between the

particle

and the

wall,

we use km = 2.0 _x

10~,

and the other parameters are the _{same as} above. We _{use a} fifth order

predictor-corrector

method with time step 5 _x 10~~ In order _to avoid the

hexagonal packing

formed

by particles

of the _same

size,

_we choose the radius from _a Gaussian distribution with average 0.I and width 0.02. The

density

of the

particle

is chosen to be 0.I.

Having

^{defined the} setup for the

simulations,

_{we now}

proceed

to

study

_some

properties

of the

pile.

Whenever _we insert _{a new}

particle,

_we calculate the total number of

particles

m

"mass")

and the

angle

9 of the

pile.

The _mass will later be used _{to extract some} information about the avalanches of the

particles.

The

angle

of the

pile

is calculated _as follows. We divide the box into vertical strips of width

0.2,

which is the _average diameter of the

particles.

In

strip

I, _we define the maximum value of the centers of

particles

to be

h(I). Then,

the

pairs (0.2 I, h(I))

is defined _as the "surface" of the

pile.

We calculate 9 from the least _square fit of

a

straight

line to the surface. All the

angles

are measured in

degrees.

In

figures

16 and lc, _we show _m and 9 measured whenever _{a new}

particle

is inserted. After _a certain transient

period

of

increase,

both

quantities

seem to fluctuate in _a

complicated

_manner.

One of the

important

quantities ^for

characterizing

the behavior of _a

pile

is the

angle

fluctuation d9 and _mass fluctuation dm of the

pile.

We define d9

=

(92) (9)2

and dm

=

(m2 (m)2,

where the averages are taken _over time. If d9 remains finite in the infinite size

(W

_-

oo) limit,

the

angle

9 does not remain _a constant. One

possible

scenario for this

non-steady

state is that the pile oscillates between the

angle

of repose 9R and the

angle

of

marginal

stability 9Ms, which differ

by

_a few

degrees (See,

_e-g-, Ref. [6]). ^The mass fluctuation dm _can be related to d9

by

dm

=

)

d9 W2. _In

figures

2a and

2b,

_we show

(m)

and dm _{as a} function of W. We average over 10000

samples

for W

= 2,3,4, 5, and 1500

samples

for W

= 6 and 500

samples

for W

= 7. The average mass in

figure

2a increases _as

W2,

_as

expected.

On the other hand, it is not

possible

to find _a

systematic

trend for dm in

figure

2b due _{to too} much noise. Since

the CPU time needed to get one

sample (typically,

10~

r~ 10~

iterations)

of W ₌ 7 is about 4 hours _{on one} CRAY-YMP _processor, it is very hard to get _a

good

estimate of

dm, especially

for

large

systems.

We _now

study

_more details of the avalanche, ^the distribution of their sizes and durations.

Consider two successive steps ^of

inserting

_a

particle,

I and I + I. Let the _mass at step ^{I be} m,.

If m,+i = m, + I, the particle inserted at step ^I just stays in the

pile. Therefore,

_no

particle

flows _out of the system.

Otherwise,

_an avalanche of size _s

= m; + I m,+i has occurred. We define the avalanche size distribution

D(s)

_as the

probability

that _an avalanche of size _{s occurs}

(including

_s

=

0),

when _we add _one

particle

to a

pile.

The distribution

D(s)

should

satisfy

the

following

conditions

f Djs)

= 1,

j2a)

s=o

f

s

Djs)

₌ 1.

(2b)

smo

Equation (2a)

is the normalization of

D(s).

Since the system ^{is in} a

steady

_state, the _average number of

particles dropping

out

by adding

_a

particle

should be unity, which is

equation (2b)

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2.00 3.oo 4.00 5,oo 6.00 7.00

width

b)

Fig. 2. The _average "mass"

a)

and its fluctuation

b)

of the piles for several system sizes.

[16].

^In

figure

3, the distribution

D(s),

obtained for several values of

W,

is shown in _a semi-

log plot. Also,

_we show the

log-log plot

of the _{same curves} in the inset. The

distribution, especially

for

larger W,

_seems to bend

upwards

in the

semi-log plot, suggesting

that it

decays

slower than

exponential.

On the other

hand,

it is also not

possible

to

unambiguously identify

a

power-law regime

in the inset. The _curves shown in

figure

3 _are consistent with

power-law decay,

but

they

_are

simply

insufficient _to

single

out the behavior. One should

study

sufficient

number

(r~ 10000)

of

samples

for

larger

systems, ^{which is not}

possible

_as

explained

before. We also

study

the avalanche duration

distribution,

which is too

noisy

to find _{out any}

systematic

behavior.

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10~

i' ~ ii milj ^.1~ i'r~

~ ~

j~ ^' 10'~ _fi W

" 3

+

10 ^~

i

_w

= 4

j

lo ^~ Q

~ j w

# 5

~"~~

~~

~° ^~ _'

/~

_.5

Cl ..:.-.l',im

~°

i°°

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0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00

s

Fig. 3. The avalanche size distributions

D(s),

measured for several system sizes, _are plotted in _a

semi-log plot. The log-log plot of the _{same curves are} shown in the inset.

4. Effect of

gravity.

We _now

study

the effect of

gravity

_on behaviors of _a

pile

with infinitesimal flux.

Especially,

we are interested in the

gravity dependence

of _two

quantities,

the _{average mass} of the

pile

and its fluctuation. We start from the _same setup ^described

above,

and _we

change gravity. Here,

we define lG

= 980, the value of _g used in the above simulations. In

figure

4, _we show the average mass measured for several different values of _g with W

= 4 and _p

= 0.2. Simulations for

larger

values of

gravity

take

longer

time, since the

particles

have

larger potential

_energy to

dissipate.

It is clear from the

figure

that the average mass of the

pile

decreases for

larger

_g.

In order to understand the

dependence,

_we first consider the

following possibility.

Since _we fix the

dropping height

^of _a

particle,

the momentum of the

particle

when it collides with the

pile

is greater ^for

larger gravity.

The

larger

momentum could make the

pile

less

stable,

and results in smaller _{average mass.} We _use different

dropping heights

^{for different values of} _g to

make the _momentum of the

particle

about the _same. We indeed find that

9R systematically

decreases _as the

dropping height

is increased.

However,

the effect due to the momentum of

dropping

_one

particle

is too small

(typically,

less than _one

degree)

to account for the observed

gravity dependence.

We _now consider another

possibility.

The

dynamics

of the

pile

_can be divided into _two stages. One is to build _a

pile

until it becomes unstable

(angle

of

marginal stability, 9Ms),

and the other is _to make the

pile

stable

again by

an avalanche

(angle

of _repose,

9R).

In order to

study

the

gravity dependence

of each

angle,

_we measured 9R and ST _as described in section I.

We

study

the

gravity dependence

of both

angles,

and combine the results to find the behavior of

9Ms.

The interaction parameters we use in this simulation _are

kept

to be the _{same as} above.

In

figure

5a, _we show 9R for different values of g. The

angle

9R is measured

by

the method

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° _{loo oo}

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80 on

n.0n 0.50 n0 50 2 nn 2~l) ~ 0n 3.5n

gravity [G]

Fig. 4. The average mass of the piles for different values of gravity. Here, W

= 4 and _p

= 0.2.

described before. Each

point

is obtained

by averaging

_over 10

samples.

^One can see that 9R

clearly

decreases for

larger gravity.

We interpret the result _as follows. When _an avalanche is

started,

the

particles,

which _are

falling down, probably

have _a

larger

momentum for

larger

gravity.

Since the

pile

is less stable in _a collision with

particles

with

bigger

momentumj 9R becomes smaller for

larger

_g [17]. ^This argument ^is also consistent with the above

study

of the

momentum of the

dropping

ball.

Next,

_we consider the

tilting angle,

ST We show ST _{vs. g in}

figure 5b,

where the averages

are taken _over 10

samples.

The

angle

seems to get

larger

for

bigger

^{values of} g,

although

the _curve is too

noisy.

We _now estimate 9Ms from the _two

angles,

9R and ST- We propose

the

following approximation.

Consider _a

pile

of

angle 9R,

rotated

by

ST

(Fig. 6). Along

the

bottom

@,

we define the component ^{of the force normal}

(tangential)

to the surface to be Fn

(F~).

Since the

pile

is

marginally

stable _at the

tilting angle,

F~ ₌

pfn,

where _p is the friction

coefficient between _a

particle

and _a wall. We _now fill the

triangle

OAB with the _same material

as the

pile,

and _remove the bottom

@.

_{The forces}

(F[, F() along

^the new bottom OB _can be

obtained,

in the first

approximation, by rotating

the

(Fn, F~) by

9T. That is

F[

_{= cos} ST Fn + sin ST F~

F(

₌ sin ST Fn + cos ST F~.

(3)

The

stability

of the _new

pile

_can be checked

by calculating

the ratio

F] IFS. Using

equation

(3),

the ratio becomes

~

,

~

/~ ~~~~T

F[

I + p ^tan ST j~~

The ratio becomes less than p for small

positive

ST, therefore _a pile with

angle

9R + 9T is stable. When the

angle

is increased

additionally by 9',

the

pile

becomes

marginally

^{stable. If} ST « I, it is

plausible

^{that 9'} = A9T _up to

O(9[),

where A is _a

proportionality

constant.

Furthermore, _we can show [18], ^from an

approximate

stress distribution inside _a

pile,

that 9' is indeed

proportional

to

9i.

_Therefore, 9Ms

'- 9R +

(1+ A)9T.

The constant A is,

however,

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7 oo

6 50

6.00

~

0J _5.50

~

~ 500

£

~ ₄₅₀

~ 400

3.50

3,no

0 00 1.25 2 50 375 5.00 6.25 7.50 8.75 10 00

gravity [G]

b)

Fig. 5. a) The angle of _repose @R and b) the tilting angle @T for different values of gravity, obtained

by removing _a wall and rotating the box (see text).

too much

dependent

_on ^the details of the

approximation.

If A

r~ 2, ^there is _no

systematic dependence

of

9Ms

_{on g.} We

conclude,

from the

tilting simulations,

that 9R

clearly

decreases

as g is increased. On the other

hand,

9Ms _seems to show very weak _{or no}

dependence

_{on g.}

How _{can we} understand the _near constancy of 9Ms _over

gravity?

Consider contacts between the

particles land

between _a

particle

^and a

wall).

The

stability

of _a contact is determiRed

by

the ratio

f~ / fn,

where

fn( f~)

is the normal

(shear)

force between the _contact. If the network

of contacts remains

unchanged,

both forces

fn's

and

f~'s depend linearly

_{on g.} Since all the

ratios

fn/f~

_are constant for different g, the ratio

Fn/F~

is also

independent

of _{g, so} is the

4

F~ °R

F~

D

F( F]

H,r _o

Fig. 6. A pile of angle _{@R is} tilted by @T. The forces along M _{is related to the force} along $ by

rotation.

stability

of the

pile.

This could

explain

the _near constancy of _{HMS over}

gravity. However,

_one should note that if

gravity

is

large enough

to

change

the contacts of the

network,

then the above argument ^does not hold.

Finally,

_we repeat the

tilting experiment

with the

particles

of the _same radius

(0.I).

The

angle

_HR (HT) ^is measured to be 24.97 +1.14

(6.90

+

0.83), compared

to 18.0 + 0.6

(4.8 +1-1)

obtained before. These differences _can be

explained

from the "network

picture"

_as follows. In

a

pile,

_we draw _a line between

particles

which _are in contact with each other. The _stress dis- tribution inside the

pile depends

_on ^the structure of the network. Since _we expect ^{the network}

to be different between the

monodisperse (triangular lattice)

and

polydisperse (disordered

lat-

tice) particles,

it is

quite possible

that the _stress distribution

(and

HR, HMS) also

depends

_on the

"dispersity".

Furthermore, the network picture can also

explain

the

dependence

_{of HMS on} the

density

of

packing,

which is found

experimentally by Evesque

et al. [12]. ^If one fixes the

density

of the

packing,

_one ^also

specifies

_a certain structure of the network. In other

words,

networks of different densities should have different structures, ^which can result to different HMS. These

arguments, of _course, should be checked

by

direct calculations of the stress distribution.

5 Conclusion.

We

study

properties ^of

(I

+

I)-d sandpiles by

molecular

dynamics

simulations. The avalanche size distribution of

piles,

built

by continuously adding particles,

is consistent with _{a power} law

behavior.

However,

since the size of the systems we

study

is too

small,

_we cannot make _a definite statement about the

asymptotic

behavior. The CPU time needed for

obtaining

the distribution is

significant,

_even for _a small system size

(150 particles).

The _reason is that

we have to wait _a

long

time

(typically,

^10~ _r~ ^10~

iterations)

for the

pile

to

relax,

for each addition of _a

particle

_on the

pile.

We also

study

the

dependences

of HR and HMS on

gravity.

we find that HR ^is decreased, as

gravity

is increased. On the other hand, HMS shows _{no or} little

dependence

_on

gravity.

We try to

explain qualitatively

the dependence

using

the "network"

~f f~rces It would be interesting if _{one can}

quantitatively

relate the stress distribution

(and

o~s)

to

properties

of the network.

Acknowledgements.

thank H. J. Herrmann for _many useful discussions.

References

ill

Savage S. B., Adv. Appl. Mech. 24

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[18] This calculation is done by using the approximate stress distribution obtained in reference [15].