Computer simulations of granular materials: the effects of mesoscopic forces

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Computer simulations of granular materials: the effects of mesoscopic forces

G. Kohring

To cite this version:

G. Kohring. Computer simulations of granular materials: the effects of mesoscopic forces. Journal de

Physique I, EDP Sciences, 1994, 4 (12), pp.1779-1782. �10.1051/jp1:1994115�. �jpa-00247031�

Classification

Physics

Abstracts

02.70 46.10 62.20

Short Communication

Computer simulations of granular materials: the elilects of

mesoscopic forces

G-A-

Kohring

Central Institute for

Applied Mathematics,

Research Center Jülich

(KFA),

D-52425 Jülich,

Germany

(Received

12 October 1994, received in final form 13 October 1994,

accepted

20 October

1994)

Abstract. Trie

problem

of trie

relatively

small

angles

of _repose

reported by

_computer

sim-

ulations of

granular

materials

is discussed. It is shown that this

problem

_can be

partially

understood _as

resulting

from

mesoscopic

forces which _are

commonly neglected

in trie simula- tions. After

including mesoscopic forces,

characterized

by

trie

easily

measurable surface energy, 2D computer simulations indicate that trie

angle

of _repose should _{mcrease as} trie _size of trie

granular

_grains

decreases,

_an elfect not _seen without mesoscopic forces. Trie exact

magnitude

of this elfect

depends

_upon trie value of trie surface energy and trie coordination number of trie

granular pile.

Trie

study

of

granular

materials bas

long

been _an active field of

research, partly

due to the many

interesting physical phenomena

which

granular

materials

give

rise _to and

partly

because of their

importance

for industrial

applications iii.

Due to the advent of _more

powerful computers, especially powerful workstations,

_many scientist and engineers believe that _some of the

phenomena

known _in this field _can be better understood

through

well

planned computer

simulations

[2].

^{This behef rests} on the

premise

that these

phenomena

_are collective _or

emergent

in nature, i e., the constituent grains experience

simple,

well understood interactions with each

other,

but that

unexpected

behavior emerges due to the

large

numbers of

grains

involved.

Hence,

if the

grain-grain

interactions _can be

efliciently prograrnmed

_so that _a

sufliciently large system

can be

simulated,

then it should be

possible

to

study phenomena

which are still

poorly

understood.

The realization of this scheme then rests upon the _use of the "correct" interactions _in the

computer

simulations. Where "correct" _may

depend

_upon the

types

of _{answers in} which

one is interested.

Unfortunately

there is at

present

no

concept

^for

granular

matenals which

corresponds

to the

universality concept

ⁱⁿ statistical

physics

_[3]. For this _reason there is _no

concensus as to how the

grain-grain

interactions should be modeled and how detailed this

modeling

^{should be}

[2].

^{The situation} is

compounded by

the fact that most simulations _are still confined

by

_a lack of

computer

power to two

dimensions,

where

quantitative

_comparison

©

Les Editions de

Physique

1994

1780 JOURNAL DE

PHYSIQUE

I N°12

with

experiment

is not

possible.

Computer

simulations of

granular

materials

using

the molecular

dynamics

_[4]

(or

"distinct element" [2]

approachj generally

_assume that

only

short range, elastic interactions _are

present.

In what _we will call the Hertz contact model for

spherical grains [5],

^the

repulsive

force

acting

normal to the surface of two

colliding grains

is

given by:

~~ ~a2 ÎÎÎ2

_~~

~~ ~~'~~ ~~'

~~~

where Y is the

Young modulus,

_{a is} the Poisson

ratio,

h

= d

Ri R2j

with d

being

the distance between the centers of _mass and R~ the radius of the i-th

particle, n~

is _a unit vector normal to the surface of the

particlesj

_{mr =}

mim2/(mi

₊

m2)j ~~

is the energy

dissipation

rate and

v£,

is the relative

velocity

normal to the contact surface. _o varies between

3/2

for smooth

particles

and 1 for

rough particles.

For the forces

acting tangential

to the contact

surfacej

a standard

approach

is to _assume that the Poisson

hypothesis

is true and to write:

Fll

= min

(mr~ll

v)'~~ ii ~

F~

[) r~l ~~~ (3)

where ~" ^is the shear

dissipation

rate, v)l~, ^{is the} relative

velocity tangential

to the contact

surfacej

and _{~ is} the friction coefficient.

The model based _upon

equations (1-3)

does not show _any

qualitative changes

with variations in

particle

_size.

However,

_as the

particles

become smaller and

smaller~

atomic and molecular

forces will become

important.

But at what

length

scale will such forces

begin

to make _a noticeable contribution to the

physics

of

granular

materials?

When the molecules of two

grains

interact via the _van der Waals

potential~

the

resulting mesoscopic

interaction is called the Hamaker

potential

and has been

experimentally

verified [fil.

However~

for _most

materials,

there _are other

short-range

^{atomic forces which} _are ^far

stronger

than the _van der Waals forces. These forces _can be characterized _on the _mesoscopic scale

by

the surface _energy~

W~

^{of the}

contacting

surfaces and _are attractive in nature. For the _case of

general

surface

forces,

Johnson et ai. derived the

following

expression for the contact of _two

spherical

_{grains [7]:}

~

31 a2 ~~~~

~

~~~

3 1 a2

~~~)

Ri

₊

R2

^~~~

Obviously, equation (4) represents

a force which is attractive for small values of h and

repulsive

for

larger

values and indeed this _was verified in

subsequent

experiments [7].

The _maximum value of the attractive force _in

equation (4)

is

given by:

Fmax

₌

(~rW /~~( (5)

1 +

2

For a

typical

value for W of W

r- 11~l~

erg/cm~,

_{we can} ask when this attractive force _is

equal

to the

weight

of _a

particle.

This _occurs when

4/3~rR(g

=

3/2~rWRiR2/(Ri

₊

R2).

For

particles

with _a

density

of 2

g/cm~

_we find that the attractive force is

equal

to the

particle weight

when

Ri

_is

approximately: Ri

* 2.5 _mm. For

grain

sizes smaller than this value the mesoscopic

forces will make

significant

contributions _to

emergent

^behavior of the

granular

material. This

value of

Ri

_{may seem}

qmte large~

but is related to the initial

assumptions: smooth,

clean

particles.

If the

particles

_are not clean W is reduced and if the

particles

_are not smooth then

o is closer to _one than

3/2. Hence~

for most real

systems

this critical radius would be reduced.

Now,

_one of the most

popular experimental systems

consist of

glass

beads

having

_a radius of l~.15 _{mm [8~}

9].

^These

glass

beads _{are an} order of

magnitude

smaller than the critical radius calculated above.

Hence,

_computer simulations which

rely

_on

equation il)

alone _may be

inadequate

for

modehng

these

experiments.

One

possibility

for

detecting

the _presence of these

mesoscopic

forces _is

through

the so-called

angle-of-repose- Experimentalist

and

engineers

have

long

used this

quantity

to characterize

granular materials, yet

simulations have

repeatedly

found

angles

which _are much smaller than the

experimental

values

[ll~].

We have carried out

computer

simulations

using equation

_(4)~ to determine the

angle

^of

repose for _a

system

of

spherical particles.

The simulations consist of

dropping

the

particles

_one

at a time onto _a flat

plate

and _measuring the _maximum

angle

obtained

by

the

resulting pile.

The

particles

_are

log-normally

distributed with _{a mean} diameter of l~.15 _mm and _a standard deviation of l~.1~5 mm

(The

distribution _is cut off at l~.21~ _mm and l~.ll~

mm.)

The

Young

modulus _was taken to be Y

= ll~~ Pa and _a time

step

^of1.l~ x 11~~6 seconds was used.

Figure

la shows the results for _zero surface energy~ W

= l~ and those in 16 for W

= 31~

erg

/cm~. IA relatively

small value of W is used to _mimic

experiments

^{where the}

particles

_may

°

@

_fi

~ l dl ,,~

~ ^'' <( ^'» '~

t ~ .) l ^~

~i< t Î ', .t '~'

o' f 4 _< '~ à.

'&' ~~ '~ '~' ~.' 'ç ~~,

~>

t" S' " ', _,'<

= 'w'

<., ,j ~ _' ,i'.

~' ~,

MM, ~'< '.

~ÎÎÎ~

~Ù

a)

b)

C)

Fig.

l

a)

Mean

particle

radius 015 _mm. Plate _is 15 _mm wide Surface _energy, W

= 0. Trie

angle

of _repose is

approximately

14°.

b)

Same _as la but W

= 30

erg/cm~.

Trie

angle

of respose is

approxirnately

^20°.

c)

^Sarne as

lb~

but _mean

particle

radius _is 15 _mm and trie

plate

_is 1500 _mm wide.

The

angle

of _{respose is}

approximately

^14°.

1782 JOURNAL DE

PHYSIQUE

I N°12

not be _very

clean.)

Both

systems

were simulated for 121~ simulation seconds _i-e-, 1.2 _x 11~~ time steps. ^Each

figure

shows the

largest slope

obtained

during

these 121~ seconds. The

slope

in

figure

16 is

approximately

21~° while that of

figure

la

approximately

14°.

Hence,

the _presense of

mesoscopic

forces leads to

larger~

_more realistic

angles-of-repose-

On the other

hand, figure

lc shows _a

system

^{with the} same surface _{energy as}

16,

but where all

particles

_are ll~l~ times

larger,

_1-e-, the _mean diameter _is là _mm. Here _again the

angle

^of _repose is smaller than in 16 and

nearly

the _{same as} in

la,

1-e-, the

mesoscopic

forces _are

unimportant

for

larger particles.

Experiments

indicated

angles

of _{repose on} the order of 31~° for

systems

^of

glass

beads with

the _sizes

given

above [8].

However,

_m 3

dimensions~

the coordination number for each

particle

Ii.e.

the number of

neighbors)

is about SI~%

larger

than _in 2

dimensions,

_so that

particles

_on

the surface would feel _a

larger

attractive force due _to the mesoscopic

part

^of

equation (4)~

thus

leading

to _an increase in the measured

angle

of _repose.

In

conclusion,

simulations show that the

angle

of repose should _mcrease when the

grain

size

decreases,

for

clean~

smooth

particles.

As for as we

know,

this _is the first time that such _a

dependence

has been

explicitly postulated.

All

previous suggestions

aimed at

increasing

the

angle

of repose found in computer simulations _were made to work

independently

of trie

grain

size and

independently

of mesoscopic forces _[11~]. It should be

relatively simple

for

experimentalists

to test the

validity

of the

present

conclusions

by measuring

^the

angle

of repose as a function of the

grain

sizes for materials with

diffenng

surface

energies.

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Schwedes J

j

"Fheflverhalten _von

Schüttgütern

_m Bunkern"

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GmbHj Weinheimj 1968)j

Rietema K-j "The Dynamics of Fine Powders"

(Elsevier,

London York,

1991);

Mehta

A.j

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