Regular Piling - IRPHE

Phys.

J.

I

IYance

(1997)

7

Aspects Piling

Stochastic Granular

of the

Christophe Eloy (*) Laboratoire

des

Universitd

(Received

and

Milieux

Pierre

PACS.46.10+z

#ric

Curie,

received

Mechanics

PACS.05.40.+j

of

Granular

Network

Cldment

Place

in

final

in

Jussieu,

form

discrete

a

PAGE

1541

Regular

(**)

Hdtdrogbnes (***),

et

4

75252

B.

Paris

86, 05,

Cedex

July 1997, accepted

25

5

France

August 1997)

systems

phenomena,

Fluctuation

PACS.83.70.Fn

Force

Ddsordonnds

Marie

et

May 1997,

14

DECEMBER1997,

1541-1558

random

processes,

and

Brownian

motion

solids

We study the influence of solid friction of the force the network structure on regular bidimensional granular piling. We show how the mechanical equations local equilibrium may couple naturally with a stochastic A Monte Carlo variable. method is designed to the force networks satisfying static equilibrium A conditions. extract statistical ensemble is constructed and we study of its structural and stochastic properties. some We address, in particular, the transfer different load question of the vertical in two cases i,e. the localized force of the of horizontal piling and the distribution to at top response a excess an under a sand pile. Using this very simple forces mechanical model, we test the validity of the various stochastic descriptions as well as the existence of constitutive macroscopic to passage relations. We also address influence of a local friction bias on the macroscopic the question of the equilibrium conditions. static Abstract.

sustaining describing

a

Introduction

1.

The

large nized

Since

fluctuations that

properties of an assembly of non cohesive grains is a long standing diffipioneering experimental work of Dantu [1] (see also [2]), who visualized and long range disordered of the force network, it has been structures recog-

mechanical

static

problem.

cult

the

the

passage

recently, numerical highly disordered proposed, on the

to

a

character base

of

equations describing introduce, in a simple way, media

forces as

an

Les

medium

based

various

of the

on

contact

phenomenology

force and

description would be an algorithmic principles [3-7] distribution.

symmetry

Recent

properties,

arduous

task.

have

evidenced works

theoretical new

sets

of

More the have

continuous

approaches directional relevant granular features such as the propagation of and also the of arches boundary is considered [8-10]; they come existence whenever a alternative the mechanical viewpoint to the standard close equation in the static to way

(*) Now at IRPHE, 12 avenue (**) Author for correspondence (***) URA 800 CNRS

@

continuous

simulations

#ditions

de

Physique

transport

du

Gdndral

of

contact

Leclerc,

forces

13003

(e-mail erctlccr.jussieu.fr)

1997

between

Marseille,

the

France

grains.

These

JOURNAL

1542

DE

PHYSIQUE

I

N°12

quasi static situations, usually based on the assumption of local incipient failure [11]. But far, explicit connection fluctuating properties has been made. On the other hand, to so no original of point view is proposed by Liu et al. [12j (see also Coppersmith et al. [13j) asan from one suming a simple but suggestive stochastic redistribution representation of the force layer of grains to the other. This model was designed to explain experimental measurements or

force

vertical

the

on

grains.

distribution

Following this work,

the

at

bottom

of

stochastic

numerous

three-dimensional

a

models

have

filled

container

with

proposed in order to rengranular assembly [14,15j.

been

disordered of the force character propagation problem in a problem is still the validation of these models on the basic point of view of mechanical Here, we present equations. obtained the force distribution of regular, bidiresults some on A mensional, of fundamental of the between grains. static friction is aspect arrays presence the grains which Coulomb renders the problem multi-valued of in the the representation sense of solid This is a direct Coulomb of the inequalities describing static contacts. consequence equilibrium. We purposely do not refined description of the in a enter contact status more that could be based, for example, on a microscopic modelization of the real surfaces in contact which is a difficult indeed (see for example [16,17j). This is the issue why we stay in reason the frame work of the macroscopic Coulomb modelization. We use a Monte Carlo method to der

the

The

degeneracies

solution

the

remove

and

extract

we

of

sets

exact

solutions

for

the

forces

distri-

Therefore, a statistical study of the force networks is possible and a bridge towards stochastic approach is discussed on rational mechanical grounds [18j. In this context, we a investigate the statistical properties of the force network, the response to a local force excess and the problem of the below a sand pile. We also address the question of distribution stress relations constitutive relating the stress components. tensor bution.

Description

2.

The

piling

ders

with

the

of

Model

bidimensional mono-disperse and array of hard cylin(bidimensional fashion triangular compact a a cannon ball piling). This system would be close to the experimental set-up investigated by Travers The is assumed et al. [2j for regular cylinders. angle 6 60° (see contact constant to be at a Fig. 1a) thus, each bead receives two forces of contact from the layer and distributes upper (See Fig. 1b). In this model, we explicitly neglect forces to the layer downwards contact two the of active between beads at the depth, this would be consituation contacts presence same sistent angle of contact 6 slightly below 60°. Each with locus of arriving and departing an forces is called a vertex. In this case, the of each structure vertex is quite simple but it is easy that this notion might be generalized according to general disordered geometrical to imagine properties of a granular network [19j. The forces are constrained by the solid contact contact friction properties of the material captured in a static coefficient p defined in the Coulomb here is

consider

we

size

and

made

of

weight unity, piled

a

in

=

sense.

The stand

system

system acting

force for of

the

forces

equations

bead is represented in Figures 1b, c. The on one upwards and the lower case letters stand for the forces describing static equilibrium of the central bead is:

(-Ni

N2

+ ni

+ n2

sine

+

(Ti

T2

ti +

t2)

cos

6

=

upper

letters

cases

downwards.

The

1

(-Ni+N2+ni-n2)cos6+(-Ti-T2+ti+t2)sins=0

(1)

Ti+T2+ti+t2~0. Note

that

explicitly

we

the

consider

local

here rotational

only equilibrium degree of freedom.

not

for This

the is

translation degrees of usually expressed in the

two

but

also

standard

N°12

FORCE

THE

NETWORK

IN A

GRANULAR

REGULAR

PILING

1543

~~ (a)

S~ Q~

Qi

~~

' s

s

~

tl')

Fig.

Description

1.

the

indicates

arrows

of

the

a)

model.

force

positive

local

(c>

c)

axes.

Sketch

of

Horizontal

piling

the

and

vertical

b)

structure.

Contact

forces,

the

charges.

media, at a coarsed grained level, via the symmetry of condition, though necessary, might not be sufficient to provide a relevant description of static equilibrium for a granular assembly which is notoriously discontinuous at the granular level. Now, we express the vertical and horizontal charges as a function of normal and tangential equations

mechanical

the

stress

forces.

contact

We

qi

~

q2

for

the

~

si

=

s2

=

jaz~j).

=

ni

sine

n2

sine cos

-n2

ti + t2

6 + ti cos6

the

We

show

this

A

fundamental

(see Fig. 1c). the charges

extend

The

to

charges

transmitted

vertical

force

transmitted

cos

vertical

force

transmitted

sine

we

from

quantity is the the top beads on

transmitted

on

the

1

down,

on

2

down,

force

transmitted

on

horizontal

force

transmitted

on

hold

relations

with

upper

1

down,

bead 2

down,

bead

letters.

case

of Liu et al. [12j is that each grain redistributes between the total of charge it has received from the layer just above. amount

on

redistribution

can

be

described

in

a

simple

stochastic

fashion.

by the central bead from the top beads 1 and 2 are respectively Qi and Q2 and received the beads 1 and 2 downwards We have: are qi and q2 respectively. monitor

is

the

total

vertical

charge C

received

(2)

gravity with a value of single bead weight taken as unity. charge Z, i.e. the projection of the contact forces received the horizontal direction, respectively Si and 52. These forces are beads downwards, respectively si and s2 and we have:

the

assume

Another

bead bead

horizontal

C=Qi+Q2+1=qi+q2. Here,

on

model

neighbors downwards, quantity

6

6

identical

statistical

which

cos

+ t2 sine

two

to

This

have:

ni

of the

core

continuum

upwards,

beads

The

of

(ja~zj

tensor

presence

of

horizontal

Z=Si+52=si+s2.

(3)

JOURNAL

1544

Therefore, in following

horizontal

the

the

set

of

this

p

s2,

and

equation

set:

si

=

the

it

provided this

four

and

~

~l + ~2

~

tan 6

q2

~

~

Qi

+

~

the

known

values

the

of

parameter

physical >

0.

angle

in

n2

p

for

Second order

the

case

each

The

constraints. is the

to

where

contact

first

is

for

+

+

52)

for

choice

and s2

q2, si

qi,

(4)

(Sl

~~~ ~

~~~ ~ ~~~

tan 6

(Si

tan 6.

perfectly

are

would

parameter

free

a

determined

that

52)

+

arbitrary

made

be

cannot

the

of qi and q2 depend explicitly choice of p). Moreover, the

values

the

forces

contact

problem,

it

i.e.

pni

I.e.

and jt2)


ni

on

the

choice

by

two

0

and

Coulomb

the

within

contained


j

fi

_I

i~

I.(

..'

l~

l

~'

;,

if-'

~'j

t

l''

~','

~~,

~)

)

~i

,'I1 ~".) ii '~

1$

'

1547

c)

jj~~' (

PILING

observe

the

presence

=

=

in the

=

=

~

=

=

=

P~~lf) =

However,

we

the

of the

limits

field and

distribution we

observed

coefficients. is

always of

Figure 4 that capacities, P~~(f). Also, we

observe

But the

in

numerical that an

order

either measurements, conditions. geometry

the

saturation

important

point

4f expl-2f).

distribution

the

could

never

monitored

the

we

of this

first

to

consider

P( f) first that

modified

is

obtain

moment

is

191

a

moment

takes the

by

the

a

af

of the

longer depth

value

of the

value of p and, in to the mean-

similar

distribution

force for

average consistent

distribution lower

friction

fluctuation

magnitude of the average force. This result is with many experimental [12j or numerical [6, 7j obtained for different friction and Nevertheless, contrarily to those previous claims we never really got

of

JOURNAL

1548

PHYSIQUE

DE

N°12

I

u i

°°~

-MF. "'

I