Instabilities in a sandpile under vibration

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Instabilities in a sandpile under vibration

B. Chakrabarti, M. Acharyya

To cite this version:

B. Chakrabarti, M. Acharyya. Instabilities in a sandpile under vibration. Journal de Physique I, EDP

Sciences, 1992, 2 (4), pp.389-392. �10.1051/jp1:1992104�. �jpa-00246493�

Classification

Physics Abstracts

05.40 05.60 05.70 46.10

Short Communication

Instabilities in

_a

sandpile under vibration

B-Ii- Chakrabarti and AI.

Acharj'ya

Saha Institute of Nuclear PIIj<sics,

1/AF

Bidhannagar, Calcutta-700064, India

(Received

25 November1991, accepted in final form 15 January

1992)

Abstract. ~Ve consider the mechanical stability of _a

(static)

pile _or heap of grains, modglled

as hard spheres. l&ie consider the limiting stability of

a random

(stochastic)

heap of hard

spheres with perfectly smooth surfaces. Roughness of the surfaces of the spheres gives additional contribution to the angle of repose. These considerations for the stability of the heap _are then extended for vibrations and _a scenario is developed to explain the convcctive _current iii the vibrating heap.

A

heap

of sand

ill

maintains under

gravity,

_a

ineclianically

determined

angle

called the

angle

of _repose,

beyond

which the _excess material is lost

through

avalanches. This

angle

is not

unique

and has _a small _range due to _a

hysteresis

via friction _or

(realistic)

disorder [2]. A

sandpile

_may

be viewed _{as an}

interesting

classical system

having

_some mechanical and

dynamical

features both from solid and

liquid

_[3]: the

heap

is solid-like belo~v t-he

angle

of repose

(ir)

and loses its

rigidity beyond

that. liiith the

application

of

vibrat.ion,

_say transverse to the horizontal base of the

heap,

the

heap

loses the mechanical

stability (of

the static

heap

belo~v the static

angle

of _repose fir)

and,

^with _proper

boundary condition,

_a convective current

(with

dour>nward flo~v

along

the free surface and

upward

flow

along

the vertical

centre)

sets in and the

heap

maintains

dynamic angle

of _repose

(9[)

less than its static value (9~)

[4-6].

A A,lonte Carlo simulation of such

granular

materials under vibration has also

reproduced

the

dynamical

relaxation of fir [7]. ^Two

plausible

scenarios have been forwarded [4,5] to

explain

the

dynamical

behaviour

of the

granular

material under vibration. In the scenario put forward

by

de Genncs [8], the

heap undergoes

^successive "active" and

"passive" periods

wider,,ibration.

During

the

up~vard

movement of the

base,

the

heap

remains solid

(passive period). During

the don,nward _movement of the base

(active period),

the

grains

^fall

freely

if the accelerat.ioii

(ow~

^m-here a and _{w are}

the

amplitude

and

frequency respectively

of the

vibrating base)

of the base movement exceeds that

(g)

due _to

gravity.

The surface tension of the

dry granular

material

being

_zero, the

bottom surface of the

pile

will become

extremely rough

and it is

speculated

that

fingers

~vill be formed whidi in effect induce the convective current. The external parameter,

driving

the

dynamics

of the

sandpile

under vibration is the acceleration

(aw~)

^{of the}

vibrating platform

390 JOURNAL DE PHYSIQUE I N°4

and if it exceeds that

(g)

of the

gravity,

convective current sets

in, according

to this scenario.

In order _to

explain

the

slope

relaxation

(from

_fir to

9[

^{with the} onset of

vibration)

behaviour of the

sandpile

under

vibration, Jaeger

et al. IS] used the well established concept of

"granular

temperature".

It views the increase of vibration _energy

(a~w~)

as

effectively increasing

the

"temperature"

_or random kinetic energy of the

grains

in the system ^due to the metastable nature of the

randomly packed grain configurations:

ordered vibrational _energy

giving

rise to random kinetic _energy of the

grains.

One _can then

effectively

_use the idea of thermal relaxation induced

by

the

"temperature"

T _or the

intensity

of the vibration. The relaxation _{occurs over}

"typical"

barrier

height

^h determined

by

the

geometry

^{of the}

grains:

The relaxation

probability

is

~- exp

(-h/kBT) kBT

₌

a~w~,

where kB is the Boltzmann _constant. The

driving

parameter

(a~w~)

^{in this scenario is thus different from that}

(aw~)

^{of the}

previous

_one. The realistic

experiments

_{[4, 5] are} not yet accurate

enough

to resolve between the scenarios.

We consider here

(theoretically)

the

problem

of the

dynamics

of sand

grain heap

under vibration. In order to

investigate

the

problem,

_we first look into the

origin

of the mechanical

(meta) stability

of _a static

(conical) heap

under

gravity.

Let _us model the

grains

_as

(tiny)

hard

spheres

with smooth surfaces

(having

_no

significant

surface

friction). Also,

let _{us assume} that the horizontal

platform

_or

surface,

_on which the

heap

is _grown, is _very

sticky,

_so that the first

layer

of balls gets

quenched

in random

positions.

This

rigidity

of the first

layer, together

with the hard-core

repulsion

between the

spherical grains

_or

beads,

_can stabiIize _a

heap.

It may be noted that the

"boundary condition",

of _a

sticky platform surface,

is

tricky

and is not

directly applicable

to normal

sandpiles (for

which the horizontal

platform

_are not

sticky). However, given this,

the

angle

of _repose fir ₌

9)

is

purely geometry dependent

and stochastic _or statistical in

origin.

In

fact,

this

angle

of _repose for _a

heap

of

randomly packed

smooth

spheres [9-11],

comes from _a dhcontinuous

change

of the

crystallographic angle

of _a

regular

close

pack:

In the

hexagonal

close

pack (HOP),

when the arrangement ^is

regular

and

packing

fraction is maximum

(volume packing

fraction pHcP ^£t 0.74

[9,11])

_one has fir ₌ 9HCP _"

cos~~(1/3)

Gt 70.5° _for

stability

under

gravity;

while for random

packing, stabilizing

^under

gravity,

the

packing

makes _a discontinuous

change

[9],

to the random close

packing (RCP)

value

(pRCP

<

0.64)

and the

angle

of _repose reduces

(from

the

crystallographic value)

to about

ill]

_firGt

18°.

Thb

angle

^of _repose, for _a

heap

of random hard _core

spheres

with smooth

surfaces,

is thus

purely

statistical in

origin

and arises out of _a

discontinuous transition from HOP

"solid")

to RCP

"liquid"

structures _as in Bamal's model of

solid-liquid

transition [9], ^driven

by

thermal noise _or

by

random

shaking:

_pHCP +

0.74,

9HCP _E~~

70.5°,

to pRCP ^££ 0.64 and fir £t 9RCP ££ 18° [9,

II].

It would be

quite interesting

to check this stochastic value fir in a Monte Carlo simulation of random hard

spheres forming

_a

ballistic

deposition

under

gravity (cf.

Refs,

ii,10]). Anyway,

the

numbers, given above,

of

course,

depend

_on the

geometries

of the

grains

and the above

comparisons

_are

possible only

for

spherical grains

for which results of extensive studies _are available

[9,11].

Usually, however,

there _are other factors present and

they

contribute to or over this bare minimum

of (coming

from stochastic

geometric considerations).

This additional contribution

A9r

_comes

essentially

from surface friction of the

grains

which _are not

usually

_very smooth and

the mechanical

stability

of the free surface of the

heap,

at this

angle

fir

(= of

+

A9r)

,

arises

mainly

from _a

competition

between static frictional force ₇

along

the

sandpile

surface and the

tangential

_component of

weight WT

of the beads in the

heap

_or

pile.

The

weight

of each bead

on the free surface at any horizontal

layer

is

equalIy

shared

(on average)

^and

supported by

_a

number ZeR of similar beads within the solid

angle

x

29r

in the next

layer.

The total number of beads for _a RCP structure is

ZRCP

and effective number ZeR of beads within the solid

angle (x 29r)

, can be estimated _as

Z«p (x 29r) /2x.

This

assumption

of

"equal sharing"

of

weight

is

certainly quite simplistic

and not actual

(see

_e.g. Ref.

ill]).

In any case, the average

weight

(tangential-tc-the-surface component)

_per bead

WT/Zefr

should not exceed the bead surface friction coefficient _7, for minimal

stability.

This

gives

then _{7 =}

2xWT/((x- 29r) .Zrcp],

or, fir ₌

(x/2) (1- 2WT/7Zrcp). Actually,

this is _an

implicit equation

in fir and fir ^is to be determined

self-consistently

_as

WT

" W sin fir, ^{where W} ₌ mg is the

weight

of the

grains

of

mass m. This has the desired feature

ill]

that the

angle

of _repose fir increases and

approches x/2

_as the surface friction ₇ increases and becomes _very

large. Also,

that fir ^{decreases with}

increasing weight

of the

grains;

_{or, more}

specifically,

fir decreases with

increasing

effective

acceleration due _to

gravity. Although

this estimate

gives

the

right

trend for fir variation with

large

₇ etc., the minimum value of fir "

fif

for _{7 =} 0 _can not be obtained from this kind of calculation _as

of

_comes ^from

purely

statistical considerations

(of

_course with the

boundary

condition that the

platform

is

sticky,

_as discussed

earlier),

and not from such force balance considerations.

The effect of

(transverse

_or

vertical)

vibration

(of

the horizontal

platform)

_on the conical

heap

will thus

depend

_on the type of

heap

_or

pile:

If the

pile

is

statistically

stabilised at

of

then

obviously

the

heap

behaves like

a solid and there is _no effect of vibration until the acceleration

(a~w~)

^{of the}

platform

exceeds that

(g)

^of

gravity,

when the free fall of the

grains during

the

"active" half

period might give

rise to _a fluidization

following

the scenario

suggested by

de Gennes _as discussed earlier.

However,

_more

commonly,

_fir

(> _of)

is determined

by

^the force balance condition due to surface

friction,

_as discussed

earlier,

and the effect of _any vibration is then nontrivial. The

dependence

of Rr on the acceleration

(g)

due to

gravity (through WT

₌ W sin

fir,

W

= mg, the

tangential weight

component of the

grain)

_can

clearly explain

the desired decrease in fir from its static value _to the vibration

dependent (or

^the acceleration a~w~

dependent) _9[.

It also _can

explain

the

origin (giving

the

right direction)

of the convective current.

During

the

upward

movement of the

platform,

the eTective _g

(in W)

increases

by aw~,

the acceleration of the

platform.

This will reduce fir

(not necessarily linearly

with increase in effective

weight W,

as the

tangential

component WT _{" mg} sin fir decreases with

decreasing fir) during

the

upward

motion of the

platform.

There will thus be avalanches

along

the free surface.

Note

that,

this is

exactly opposite

to de

Gennes'scenario,

where the

heap

would behave like

a solid

during upward

motion

(passive period). During

downward motion of the

platform,

eTective W decreases and fir will tend to increase and the material lost

during

the

upward

motion will then be taken in to increase fir

during

downfall. This will

give

the convective current and also the correct direction of the current

(in

de Gennes scenario current direction would be

opposite). Note,

that this kind of

explanation

cannot support ^{the existence} of _a

threshold acceleration [4] of the

platform _(aw~

>

g)

^{for convective current: While the} concept of a threshold acceleration is

appropriate

for the

dynamics

of _a

single particle (in

the

pile),

it is not valid for the

pile.

We believe that _more controlled

experiments

_{are necessary} to

clarify

these issues.

Acknowledgements.

Part of this work _was done when _one of _us

(BKC)

was

visiting

the Tata Institute of Fundamental

Research, Bombay.

He is also

grateful

to S.S. Manna and D. Dhar for _some useful discussions.

We thank the referee for _some useful comments and for

bringing

the _papers

by

Liu et al. [2]

and Visscher et al. [10] to _our notice.

392 JOURNAL DE PHYSIQUE I N°4

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