Stress Distribution in Granular Media and Nonlinear Wave Equation

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Stress Distribution in Granular Media and Nonlinear Wave Equation

J.-P. Bouchaud, M. Cates, P. Claudin

To cite this version:

J.-P. Bouchaud, M. Cates, P. Claudin. Stress Distribution in Granular Media and Nonlinear Wave Equation. Journal de Physique I, EDP Sciences, 1995, 5 (6), pp.639-656. �10.1051/jp1:1995157�.

�jpa-00247090�

Classification

Physics

Abstracts

05.40+j

46.10+z 83.70Fn

Stress Distribution in Graqular ^Media

and Nonlinear Wave Equation

J.-P. Bouchaud

(~),

^{M. E. Cates} _(~>~) ^and P. Claudin

(~)

(~) ^{Service de}

Physique

de l'Etat

Condensé, CEA,

Orme des

Merisiers,

91191 Gif

sur Yvette

Cedex,

France

(~) ^Cavendish

Laboratory, Madingley Road, Cambridge

CB3

OHE,

UK

(~)

Department

of

Physics

and

A3tronomy, University

of

Edinburgh, King's Buildings, Mayfield Road, Edinburgh

EH9

3JZ,

UK

(Received

3

December1993,

extended version received

l7February1995, accepted 27February 1995)

Abstract. We _propose

phenomenological equations

to describe how forces

"propagate"

within

a

granular

nledium. Trie linear part of these

equations

is _{a wave}

equation,

where the vertical coordinate

plays

trie rote of

time,

and trie horizontal coordinates trie raie of _space. This

means that

(in

two

dimensions)

trie stress propagates

along "light-cones";

trie

angle

of these

canes is related

(but

not

equal to)

trie

angle

of _repose.

Dispersive

corrections to trie picture,

and various types of

nonlinearity

_are discussed. Inclusion of nonlinear terms _may be able to describe trie

"arching" phenomenon,

which has been

proposed

to

explain

trie nonintuitive hor- izontal distribution of vertical pressure

(with

_a local minimum _or

"dip"

under the apex of trie

pile)

observed

experimentally. However,

for

physically

motivated parameter

choices,

a

"hump",

rather than

a

dip,

is

predicted.

This is aise true of _a

perturbative

solution of the continuum

stress equations for

nearly-flot piles.

^The nature of trie force fluctuations _is aise

briefly

discussed.

1. Introduction

Granular media bave been trie focus of intense interest among

physicists

in trie last few years.

Beyond

their obvious

practical

and

engineering importance [1-4], they represent

an

interesting

field of classical

phenomenology,

since their behaviour

presents

bath solid-like and

liquid-like

features

leading

to _a number of

fascinating properties. They

bave also been

proposed

_{as a}

conceptual paradigm

of

"self-organized criticality"

_[5]:

sandpiles

would

spontaneously

evolve towards _a

marginal

state, within which trie

dynamics

is scale and time invariant and where events

(avalanches)

of ail sizes may be

triggered by

_a small

perturbation.

This

proposai

_was

followed by

_a series of

interesting experiments

_on

sandpile dynamics [6-9],

as well _as theoretical and numerical work

[10-13].

Here,

_we want to address trie

comparatively simple problem

of trie statics of

sandpiles,

and

more

precisely

to understand trie

spatial

distribution of _stresses in different

geometries (e.g.,

@

Les Editions de

Physique

1995

heaps

_or

silos).

Dur

major

motivation is the rather unintuitive

experimental

result of Smid

[14]

(see

also the numencal work in reference

[12]),

^which shows that the vertical component ^{of trie} force beneath _a sand

heap

bas _a double-belled

shape,

with _a local minimum at trie center of trie

heapl

This distribution of forces

is, furthermore,

found to be

subject

to wild fluctuations for different realisations of trie _same

macroscopic heap [là,16].

A few theoretical works bave

recently

addressed this

question:

trie force distribution of _a two dimensional model

sandpile,

where disks _are

packed

into a

perfect triangular

lattice and for _zero solid

friction,

is amenable to _an exact

analytical

treatment

il 7-19].

The vertical _stress is found to be _a constant

independent

of the

position

z of the

"pressure

sensor" at trie bottom of trie

heap.

This

result, although

itself not very

intuitive,

faits _to

reproduce

trie anomalous

"dip"

_seen at trie _center of trie

pile.

Trie shear

stress,

on trie other

hand,

is

predicted

to be _a hnear function of trie

position,

which is _zero

(by symmetry)

at trie _center of trie

heap

and of

maximum

magnitude

at trie

edges (dropping abruptly

to zero

there).

Trie

problem

_was discussed in

qualitative

ternis

by

Edwards and Oakeshott

[20]. They argued

that _a characteristic feature of

granular

media under stress is trie spontaneous ^{formation of}

arches,

which "deflect" trie stress

sideways

and thus act _as "stress screens".

They

showed how this

might

lead to a double-bell

shape

for trie vertical force distribution.

Trie aim of this _paper is to

provide

_{a more}

general theory

within which _to discuss these

effects,

in trie form of _{a set} of

phenomenological equations

for trie

propagation

of forces

(Section 2.1),

or more

correctly

stresses

(Section 2.2)

within _a

granular

medium. We restrict _our attention to a two-dimensional model in which trie

heap

is

triangular

rather than

conical;

this allows

us to take trie discussion further than would be

possible

in _a fuit three-dimensional

geometry.

Results for trie fuit three- dimensional _case will be

presented

elsewhere

[21].

^A

key

idea is that trie forces

obey

trie _wave

equation _[22],

where trie vertical coordinate _z

plays

trie rote of

time,

and _z trie rote of _space. Trie

analogue

of trie

"velocity

of

light"

is viewed _{as a}

phenomenological parameter,

^which we relate _to trie

angle

of _repose. When solved in _a

heap _geometry (Section 2.3),

this

equation essentially reproduces

trie results of Liffman et ai.

[17-19].

One _can aise salve this

equation

_{m a} 2-D "Silo"

geometry (Section 2.4)

_we find that trie load

beanng

_on trie bottom

plate

first rises

hnearly

with trie

height

and then saturates when this

height

^{reaches trie}

diameter of trie bottom

plate,

in

agreement

^{with standard}

observations,

and

classicaltheory

_[2].

We then argue, in Section

3,

that trie _wave

equation

is in fact _a

simplification,

and discuss

some of the most relevant additional ternis which should be

included,

in trie

spirit

of _a

phe- nomenological approach.

One of these

(Section 3.1)

is _a

dispersive

_or diffusion-like _term which

smears out trie

light

_cone and allows _one to recover, in

2D,

smooth force

profiles

in response

to a

point perturbation.

Trie "diffusion

constant",

_so

defined,

bas trie dimension of _a

length

1 which

separates

a

regime

of "small"

sandpiles

for which trie

"light

cone" is

totally

blurred out, from

"large" sandpiles

^{where trie}

light-cone picture

becomes vahd. Another _set of _ternis

(Sec-

tion

3.2)

describe non-linear

effects;

_we consider what _we

expect

to be trie most

important

^of these. We then discuss whether this modified

equation might (for

certain parameter

choices) reproduce

the central

"dip"

in trie vertical force

(Section 3.3). Unfortunately

this does _net

appear

likely

for

physically

motivated

parameter

choices.

In Section 4 _we return to trie classical continuum

equations

for stresses _m

granular

media.

As it is

well-known,

these _are

fundamentally incomplete; however,

it is

commonly

assumed that trie _two

principle

stresses in trie material _are

proportional everywhere

_[3,

_4].

This _can be

"justified" by asserting

that trie materai is

everywhere

at trie threshold of

stability (described by

_a Coulomb

criterion)

_or else _can be viewed _{as a}

global "pre-averaging"

of _{a more}

complicated

equation.

In either _case, trie

resulting equations permit

a

perturbative

solution for trie _case of _a

"nearly-flat" pile.

Trie nonlinear

equations,

to trie

leading order,

_are

closely

related to trie

phenomenological

_ones discussed in Section 2 and

3;

in this limit trie

parameters

are agam

F

F

R

j

Fig.

l. Basic

configuration

of three

grains.

The 6

angle

defines the

packing

geometry. ^The forces F~, Fz

imposed by

the upper

grains (net shown) plus

the

weight

of the

grain

_are transmitted and

shared

by

the bottom

grains

in order to _ensure mechanical

equilibrium.

such _as to

predict

a

"hump"

rather than _a

dip. Although

_{we are} net able _to

explain

trie

dip

in this paper, we think _our

approach

_may form trie basis of future

developments.

Further work bath _on trie

microscopic

derivation of _our

equations

and _on trie

experimental proof (or dismissal)

of _our

suggestions

is

obviously

needed _to transform _our ideas into _a

proper

theory.

Trie

implementation

of _our

equations

in three dimensions

is,

_as mentioned

above,

trie

subject

of

ongoing

^work

[21].

^There are, even in two

dimensions,

_some

interesting

extensions of _our

approach.

Due _can, for

example,

consider trie effect of random variations of trie local

properties,

such _as trie

density

_or trie

"velocity

of

light"

which in trie

strongly

_non-

linear

regime

should lead to self similar _pressure fluctuations

analoguous

to those observed in surface

growth

models

[23-26].

A brief discussion of these ideas is

given, along

with _our

conclusions and _some

suggestions

^{for future}

work,

in Section 5.

2. The

"Wave-Equation"

The

Heap

and the Silo

2.1. A NAIVE APPROACH. Let _us first establish _Dur

phenomenological

_wave

equation by

considering

three

grains configured

_as in

Figure

1. Let

F~(z, z), Fz(z, z)

denote

respectively

trie _x and _z

components

^{of trie total force}

acting

_on trie _upper

hemisphere

of trie

grain

^centered

at

point

_{z, z.}

(Note

that _z is measured downward from trie _apex of trie

pile; gravity

acts in trie +z

direction.)

If trie

angle

^between contacts is

(see Fig. 1),

and if trie

tangential

force T is

everywhere equal

to zero, mechanical

equilibrium requires

that:

S"-S~°~

~~~

and

~~

i

~~

tan~ ôÎ

_~~~

where _p is trie

density

of

grains.

We bave chosen units where _g

= 1 and trie

partiale

radius is also

unity;

_a is _a dimensionless

geometrical

constant

(which

could be absorbed into

p).

Furthermore,

_we have taken the continuum

limit,

which _assumes that trie forces _vary

slowly

on trie scale of _a

partiale

radius. Note that

equations (1)

and

(2)

_are

precisely

trie continuum

version of trie

equations

considered in references

[17-19].

Now,

in _a real

heap,

trie

packing

is somewhat disordered

(local

variations of Ù, in trie

shape

and size of trie

grains, density

of

vacancies, etc.) _[19],

and _{as a} result trie central

difliculty

lies

m trie correct

description

of solid

friction,

which may or may ^not be "mobilized"

(that is,

it

is easy ^to

put

^bounds on frictional

forces,

but hard to calculate their actual

values).

A way to

by-pass

this

difficulty

is to construct

phenomenologicalequations

at _a

coarse-grained leuel,

based _on

symmetry,

conservation and "relevance"

arguments;

this

procedure

bas proven useful

m vanous contexts

(see,

_e-g.,

[23-26]).

In trie

present

case, trie "force

propagation" equations

should

satisfy

trie

following

requirements:

1) The

mechanicalequilibrium

of _a

layer imposes

that trie

equations

should bave trie form:

ôzfz

=

-ô~ Jz

+

weight

of trie

grains ÔzF~

₌

-Ô~J~

where

Jz, J~

_can be viewed _as "currents"

(treating

_{z as a} timelike

variable),

_so that

by integration

_{over z} these will reduce to

boundary contributions,

_as

they

should.

ii)

^The

^equations

^{should be left invariant when} z - -z and

F~

_-

-F~ simultaneously.

iii)

^The

^equations

^{should be invariant when} z - -z,

Fz

_-

Fz, F~

_-

-F~

and _{g - -g}

simultaneously (g

is trie

gravity).

Note indeed that

"upper hemispheres"

become "lower

hemispheres"

under this transformation.

On this

basis,

trie

simplest phenomenological equations

should

read,

_m two dimensions:

([~

=

-ki([~ +ap(x,z) (3)

and

Î _~~Î

~~~

where _p is trie local

density, ki

"

(this

is shown

below),

while

k2

is a

phenomenological

parameter describing (in

_{an average}

way)

trie

geometry

^of contacts, trie effect of _{a non zero} solid

friction,

etc. Note that _we bave two

equations

m two unknowns.

2.2. CONTINUUM MECHANICB APPROACH.

Equations (3)

and

(4)

con be considered _a

poor man's view of

equilibrium

continuum mechanics. For _an

isotropic

medium there _are three

independent

stress tensor

components

_{azz, a~z =} az~, and _a~~

(We

_use here trie conventional

physics notation,

rather than that often used in

powder

mechanics _m which _a~z

= -az~

[3]).

In terms of

these,

trie condition for static

equilibrium

_is

simply

_.a

= bulk

forces,

_or

explicitly:

~)j~

⁺

~)(~

⁼

^{P(x,z) (5)}

and

~ÎÎ~

^~

~ÎÎ

~ ~~~

The

requirement

of static force balance thus

provides

_a

priori only

two

equations relating

trie three

mdependent components

of trie _{stress tensor.}

Identifying Fz

with _azz

(up

to a trivial factor of

a), F~

with _a~z and

ki

+

1,

one discovers that

equations (3)

and

(4)

are

equivalent

to

equations (5)

and

(6)

taken

together

with _a certain constitutive relation between _a~~ and azz. This

simply

reads:

a~~ "

k2azz (7)

Such _a relation _was first

proposed

in 1895

by

Janssen _m bis

study

of silos

[27] (Janssen

also made further

simplifying assumptions

which _we do not

adopt here). Although quite

reasonable

physically,

this is

certainly only

_an

approximation

_{[3]. We} shall below

(Section 4)

^{relate this} to other

possible

constitutive relations and show that

equation (7)

_can for _one of these be viewed

as trie first term of _a

perturbation expansion,

in which _a~z also _{appears at}

higher

order.

Within trie continuum mechanics

approach,

_one

conventionally

introduces _a

limiting

^coeffi-

cient of internai

friction,

sin

#,

such that

0<r<dsin# (8)

"~~~~~

r =

j(a~~ azz)~ /4

₊

a]zj~/~ (~)

d =

(a~~

+

azz)/2 (10)

The

parameters

d and _{r are}

respectively

trie _mean and trie difference of trie _two

principal

stresses;

#

is assumed

independent

of

position (these equations

_are written with _no

assumption

about trie orientation of trie

principal

_axes in trie

(z, z) plane).

It is

elementary

_[3] then to show that

#

is

equal

to trie

angle

of _repose. We _use this fact below.

Note that

equations (3)

and

(4)

cari be

generalized

to three dimensional

assemblies,

if _one restricts to _cases where _a~~

= a~~ and a~~ ^{e 0}

(y

is trie third

dimension).

Then

equations (3,4)

read:

)

"

~Q 'É

₊

_p(£>

Z)

(~~)

and

~~~

=

-k2ÙFz ₍₁₂₎

where

É

_{is trie force field in trie}

plane perpendicular

to z, 1-e-,

(a~z,a~z).

This notation

obviously

includes trie 2D

formulation,

and _we sometimes _{use it}

below, though

3D calculations

are deferred _{to a} future _paper

[21].

2.3. THE WAVE

EQUATION

AND THE HEAP. We shall first _assume that _p is

uniform, p(z, z)

= po.

Eliminating

either

É

or

Fz

from

equations (11,12),

_we find that

É

_and

_Fz,

_each

separately, obey

trie _waue

equation:

~Î~

=

cÎv~F ₍₁₃₎

with _a

"velocity

^of

light"

_co +

$.

This bas

immediately

_an

interesting

consequence: in two dimensions _a

perturbation

localized in space ^at

(z', z') only

affects

points along

trie

"light-cane"

ix x'j

₌

co(z z'),

with _{z >}

z') emanating

from

x',

z'.

Thus,

_a localized force _on trie surface of _a

powder layer

should _{create a}

sharp

"echo" of stress at two points _on the bottom of the

layer,

which should in fact allow _one to _measure co

(Fig. 2).

Trie existence of trie

"light-canes"

is

strongly suggested by birefringence pictures

of stressed

plexiglass

2D bead

packings [28],

and aise

perhaps

in 3D

[16]. Note, however,

that in

3D,

a localized

perturbation

will not remain localized _on trie

light

_cone but rather

spreads

out

nonuniformly

within trie _cone

(with

a

singularity

at its

surface) _[21, 29].

A second direct consequence of

equations (5,6)

is that

only heaps

with

opening half-angle greater

^than

tan~~

_{co can} be

stable;

since trie

boundary

conditions at trie surface of trie

pile

are

jÉj

=

Fz

=

0,

the

"light

cones"

originating

^front _any

point

within the

pile

cannot intersect its surface

(these boundary

conditions arise since there _can be _no normal stress at trie

surface;

.i

h

~

c h

'KÎI

Fig.

2. "Echo"

experiment:

_a locahzed force _is

imposed

at the top ^of a sand

loyer, leading (in 2D)

to _an extra contribution to the vertical pressure ^at the bottom which is localized around _a circle of

radius coh and of width

/À.

In 3D however, the extra contribution extends within the whole _cane, with _a

singularity

at the

edges.

2

,

' '

15

,' ',

>'

')

^'

S K>Ù

-1

,z 5 5 -o 5 o5 5 2 5

x

Fig.

3. Force

profiles

_{as a} function of the radial coordinate _~,

as

computed through

the _wave

equation (plain fines),

and _as obtained from numerical simulations

il?]

_or

taking

into account

dispersion

elfects

(dotted fines).

combined with

equation (7),

this _means that ail stresses vamsh

there). Below, however,

_we find _a

stronger

condition _on trie

opening angle,

dictated

by equation (8),

and then _use this to find trie relation between _co and trie internai friction

angle #.

One _can

proceed

to solve

completely

for the forces _{m two} dimensions for different _geome-

tries, by looking

for solutions of the form

Fz(x, z)

=

f+(x

₊

coz)

₊

f-(x coz)

_{+ poz} and

F~(z, z)

₌

g+(x

₊

coz)

+ g-

(x coz).

Let _us describe the force distribution for _a

pile

of _open- mg

half-angle #

of the

heap

chosen

(as

shown _necessary

above)

such that _{c e}

tan(~)

> co.

Taking

_{x =}

0,

_{z =} 0 _as trie _apex of trie

heap,

and

imposmg

that

F~(cz, z)

=

Fz(cz, z)

_e o, _we find

(see Fig. 3):

~°~~ for 0 < _z < coz

F

(~ ~j

^{(C + Co)}

(i~)

~

C~~~~C()

_{~~~ ~~} ~~~ _~ ~ ~~~

~~~

F«(x, z)

=

~ÎÎ~

_{~~~ ~ ~}

~ ~ ~°~

(là)

(C~

C()

^{~~~ ~~}

~~~ _~ ~ ~~~

In trie

(hypothetical)

limit where _{c -}

c)

trie normal stress is thus _a constant while trie shear

stress is _a linear function of _x

(note

that this must vanish

by symmetry

for _x

=

0)

^with

step

discontinuities _at the boundanes. Trie ratio

) _{(1.e., j~)}

_bas

a maximum value of

#,

_which

is achieved

throughout

the

"wings"

^{of trie}

pile)

_(x( >

coi _(1.e.,

_{exterior ta}

a

light

_cone

starting

at trie

tip).

Note that trie solid friction coefficient between trie

grains

and trie bottom

"plate"

must be at least

equal

to

c( /c

for trie

heap

to be stable.

Using

this

solution,

_{we can now}

find,

_as

promised,

_a relation between _co and trie

angle

of repose

#. Substituting

_our constitutive

equation (7)

^and trie solutions

(14,15)

into

equations (8,9,10),

_we find trie

mequality

~~4

~~ ~

(1+ c()2 ^sin~ (1- c()~

^~~~~

The minimum

possible

value of _c

corresponds

to _a

pile

at the

limiting (repose) angle

of static

stability,

that

is,

C > Cr OE

(17)

tan

#

Companng

this with

equation (16) gives,

after _a httle

rearrangement

~~ _~~

1 +

1an~

#

^~~~~

This shows that _co is

strictly

less thon the

angle

^of repose in ail _cases.

Correspondingly,

trie

"wings"

of trie stress

pattem

are

always

of finite _extent: there _are

always regions

in trie outer

parts

^{of trie}

pile

where trie vertical stress rises

linearly

with distance.

Dur

preliminary

numerical results for the 3D _case show

broadly

similar features _to those

presented

above for 2D

[21]. Specifically,

there is _a

broad,

almost _{constant zz stress}

plateau

below the apex of the

pile.

This _contrasts with _a calculation based _toc

hterally

_on the

light-

cone idea: if in three dimensions the stress _were to carry

only along light

_canes, a semicircular

profile

for _azz would be

expected _[30]. However,

_as noted above

[29],

^the wave

equation

with two

(transverse) spacelike

and _one timelike

(z)

variable does net have

Sharp light-canes

m

contrast to either 3+1 or1+1 dimensions. Fora further

discussion,

_see reference

[21].

Note

that, despite adoption

of the Janssen-like constitutive

equation (Eq. (7)),

_we have net assumed

las

he

did)

that the

principle

stress _{axes are} horizontal and

vertical; indeed,

_our

prediction

of _a finite

F~

shows that

they

_are net _so oriented.

Any

such

assumption

would thus be

self-contradictory

if

equation (7)

_were to be retained. The

assumption

that the

principle

axes

are so

oriented,

combined with _a second

assumption (very

often made

[3])

^{that the materai is}

everywhere

at the

point

of failure

(more precisely, incipient

active

failure,

for which _azz >

a~~)

would lead to the choice

c(

= k2 =

hÎfl,

which dilfers from _{our own. We} discuss this distinction further below

(Section 4).

Equations (14,15)

_are rather similar to the results

recently

obtained in the discrete models of references

[17-19].

A numerical simulation of _an ordered

triangular packing

was also

performed

[17],

with results _m

good agreement

with the theoretical

predictions (but

_see

below).

The main dilference between _our work and these discrete models is that _we consider the

velocity

_co to

-w w

~

"',

0,,"'

x

",

1

1

2n-1

,, ',

,, ',

,, 1

2n ,,

', ,,

3 ', 2 2n+1

',

,, ',

,, ',

2n+2

c~z

$

Fig.

4. Definition of

regions

1, 2 and 3 within _a

silo,

and

integer

_n

appearing

in

equations (19-22).

be _a

phenomenological parameter,

and bave shown that _c > cr

(trie

_repose

angle condition),

where _cr

strictly

exceeds _co

(see Eq. (18)).

In trie discrete

models,

_co is fixed

by

trie chosen

geometry

^{of the}

packing,

and _{c = co} is often assumed.

2.4. THE SILO. We _now consider _a certain amount of

powder filling

_a box of

height

h and width 2W. At the

retaining watts,

_x

=

+W,

_one should

impose

_a Coulomb friction condition which reads _a~z

= /Ja~~, where _/J is trie

particle/watt

friction coefficient. This _assumes trie watt friction is

fully

mobilized

(we

retum to this

later). Using equation (7),

_one thus find that trie

boundary

condition to be used _on

Fz, F~

reads

cofz(+W, z)

₌

uF~(+W, z),

where _{u e}

flf

In _a 2D

situation,

we find that trie functional form of trie solution is dilferent in each of

ÎÉe regions

shown in

Figure

^4. For

regions

of

type 1,

trie forces _are

given by:

Fz(x, z)

=

~°~"

+ ~°

~" ~)

~

[coz W(2n

₊

_u)] (19)

co co u +

F~(x, z)

= pox

Ù (20)

~"

_{+ l}

where _n is _an index which increases with

depth

_as shown in the

figure.

In regions of type

2,

one has instead

Fz(x, z)

=

~°~"

+

~/) _{~" )~} ^[ucoz

(x(

W(u~

₊ 2nu + u

1)] (21)

co co u u +

F~(x, z)

_{= pox} + ~° ^"

[coz

vx

W(2n

₊

_1)] (22)

(1

+

u)

u + 1

~

The behaviour in

regions

of

type

^{3 is} _as for

type 2,

except tbat

F~ changes sign

_on reflection in the

midplane.

Let _us

analyse

the limit where the

height

of the silo is much

larger

than the width

W,

_so that

(1+ u)"

_» 1 in the above formulae. One finds that

Fz(x,

h _ci

f~

₊

_{Mi" fz(x, h),}

_and

F~(x, h)

_{ci pox} +

())" f~(x, h),

where

f~,z

_are linear functions of

x and h with coefficients

~

Pressure on the wajj

~

our wlut<on

( 2

e

é - Janssen's solu>ion

)

~~ ~

g

fi ^{0 3}

g

§m 0 6

( Î~

~ 0 4 °

fl

~

E 0 2 j =30 deg

~ Qq20deg

0

0 2 4 6 8 10 12 14

d<mens<onless depth

Fig.

5. Outward normal stress

(a~~)

at trie watts and _at the bottom of the silo _{as a} function of

height

of the silo. This is

compared

with the classical Janssen calculation.

of order _one. We find

that,

for small watt frictions _u, the total

weight

T _on the bottom

plate saturates,

_as the

height increases,

to _a value of

%~:

T _~

2W~°~"

il e~~~°/"~"j (23)

Co

(Note

that trie limit of _zero wall

friction,

_{u ~ co, recovers an}

appropriate

result

).

^This is similar to Janssen's basic

finding,

and

corresponds

to

qualitative experimental

observations

[2].

^Trie

presence of

partly "absorbing" boundary

conditions "screens out" trie

weight

^{of far} _away

grains,

which _are

supported by

trie walls and not trie base of trie container. One

major

difference with Janssen's

original

conclusions is

that,

due to

multiple

"reflections" _on trie

walls,

trie force field is

modulated,

both in _z and in the horizontal

direction,

since the forces

depend

_on trie

integer

n as shown in

Figure

4.

(Janssen

assumed _an

X.independent pressure.)

Trie

decay length (

=

W/k21~

for trie

exponential asymptote

is similar to that

given

in trie classical

analyses, although,

_as noted above in connection with

equation (16),

these involve _a different relation

between

k2

=

c(

^{and trie} _repose

angle #

from trie _{one we} propose. These alternatives could be

distinguished,

in

principle, by measuring (all

three

of)

trie

wall-friction, angle

of _repose,

and

decay length;

_we bave _not found _accurate

enough

data in trie literature to make this

comparison.

Trie

equations presented

above show _some

unphysical

features for _u < 1. One _can confirm

that,

before this regime is

entered,

trie wall friction coefficient _~t

= tan

#w

^exceeds that of trie

powder itself,

tan

#. (This

_occurs at u = u~ =

(2

+

tan~~ #)~/~

>

v5.) _Clearly

_{in this}

_regime

trie wall friction will not be

fully mobilized, contradicting

_one of _our

assumptions.

Trie

simplest

resolution is to think of _a ~'virtual

wall", just

inside trie real _one, whose friction is trie _{same as} trie bulk

powder;

m that _{case, one may set u}

= u~. Dur

assumption

that trie wall friction _is

fully

mobilized

will,

in any case, be most rehable for weak wall friction

(large u)

which leads to trie

simple exponential asymptote

discussed above.

In

Figure 5,

we show the outward normal stress

(a~

at the walls _{as a} function of

height

of trie silo. This _is

compared

with trie classical Janssen calculation with

k2

₌

hÎ~.

In

Figure 6,

_we

report

trie outward normal stress at trie walls and at trie bottom of _a wide silo

(W

_»

z)

as a function of trie frictional

angle,

_as obtained within _our

method,

and

compared

with _an

approximate

solution

(Coulomb's

method of

wedges [3])

^{of trie} stress

continuity equation

based

horizontal normal stress a~~= Paz f(#,#~)

l 2

tan §~/ tan § 0 63

u

0.8

our solubon

c

0.6

a

04 ^°

u

o

0 2 Coulomb's method

.

~

of wedges _u

o c

0

0 20 40 60 80 1OÙ

4 (fnctional angle)

Fig.

6. Outward normal stress

(a~~)

at the watts and at the bottom of _a wide silo

(W

>

z)

_{as a}

function of the frictional

angle,

_as obtained within _Dur framework, and compared with _an approximate solution

(Coulomb's

method of

wedges

[3]) of trie stress

continuity equation

based _on the constitutive

assumption

of

incipient

failure

everywhere.

on trie constitutive

assumption

of

incipient

failure

everywhere (see

Section

4).

Trie last t~vo methods

give

_very similar

results;

_{our own} is distinct but not

extremely

different. A

larger

dilference would be _seen in _a

plot

of trie vertical normal stress

(azz);

this arises

mainly

from trie

larger

constant of

proportionality k2

used in _our model

(for

_a

_given angle

of

repose).

Preliminary

numerical work _on trie three dimensional _case shows

broadly

similar behaviour to that in two dimensions

[21].

^{Trie situation with} a

"surcharge"

at trie top ^{of trie silo reveals}

interesting "slip" phenomena

which will be

reported

in reference

[21].

3.

Beyond

the Wave

Equation

In this section _we consider various extensions to the

simplified description

olfered _so far. The first of these

(dispersion)

_can be mcluded while

retaimng

_a hnear

equation.

The remammg

ones involve the

study

of

nonlineanty.

For

generality (since

we do not solve

them!)

_1v.e

give

our

equations

for the full 3D case,

using

the notation of

equations (11,12).

3.1. DISPERSION. The first obvious omission from _our

phenomenological equations II,12) (or (3,4))

is _a diffusion-like term, to be added _to the

nght

hand side of

equation (12)

)

=

-k2ÙFz

₊

~(g)V~É

This term _can be _{seen as} a modification of trie constitutive

relation, equation (7),

made

by adding

an extra term m

ô~a~z

to trie

right

hand side. Note however that

~(g)

must vanish in trie absence of

gravity (g

=

0)

since trie diffusive _term violates trie _z

- -z symmetry

[31].

Trie elfect of this diffusion term is to smooth out trie

sharp

features of _our solutions to trie

wave

equations

_given ^{above. In the} case of the sand

heap,

the result is sketched

quahtatively

Fig.

7. Chain-like

configuration

of grains which deflects the forces

sideways,

transporting them with _an effective

"velocity"

17.

in

Figure

3. This behaviour is indeed confirmed

by

the numerical simulations of reference

[17].

The

dispersion

constant _~ could be estimated

by measuring

how much the

"light

cone" is smeared in _a 2D "echo"

experiment:

_see

Figure

2.

(As

noted

above,

in the 3D _case the response is

spread throughout

the _cone, which nonetheless has

sharp "edges"

when _~

= 0

[21] ).

D _cari be used to define _a charactenstic

length

1, so that trie Péclet number for _a

sandpile

of

height h, Pe(h)

=

@,

_{is of order 1 when h} ci1

(trie

Péclet number is trie usual dimensionless

measure of the

importance

^of

propagation

relative to

dispersion).

Smaller

sandpiles correspond

to a

regime

^where

dispersion

_is dominant and the

"light-cone"

loses its

physical significance.

3.2. ARCHING _AND COMPRESSION EFFECTS. A _more

interesting

_way of

going beyond

the

wave

equation

is to

generalize

the constitutive relation

(7)

_{so as} to allow for trie influence of

a~z. For _reasons of

symmetry,

one should take in any local

neighbourhood (x,z)

trie first

non-linear corrections _as

a~ "

k2azz

+

À°Îz

~

~'°Îz

_{~ ~~~~}

where and À' _{are new}

phenomenological coefficients,

which must have the dimension _[À]

=

[a]~~.

The

simplest possible assumption

would be to take À, À' _{as constants.}

However,

this is not

really

consistent _on dimensional

grounds. Indeed,

in trie

simplest

_case of _a

packing

of hard

grains, there _cannot be _any intrinsic scale with which _{to compare} the local stresses

apart

^from

azz itself

(for

softer _{grains, a} natural stress scale would be the

Young

^modulus

E,

to which could be

related).

We _are thus led to the

proposal that,

for hard

grains (E

_~

co),

À, À' are

locally proportional

to

llazz(x, z).

^This

argument

can be rendered _more

convincing by finding

the

leading

nonlin-

eanty by

_a

pertubation

scheme around _a small

intergrain

friction limit

(for

shallow

sandpiles),

as is done in Section 4 below. If _we

accept it,

then the À' _term

merely

shifts the value of

k2.

Turning

to the

terni,

then

setting

=

Îazz,

_we find that _a term

-ÀÙ (~

must be added to the

right

hand side of

equation (12). Physically,

this _can be

thought

_as

desÉnbing

the presence of "arches" in the

sandpile,

whose role _was

argued

to be crucial

by

Edwards and Oakeshott

[20].

Arches _are "chain-like"

configurations

of

grains (such

_as the _one sketched in

Figure 7),

which act to

transport É along

the chains. In the

equation (12)

for the shear force

É,

_{this effect}

cari

be

mathematically

described _{as an} additional

"velocity-like"

terni

-Ù[Î7É]

to appear, with

a local

velocity

17 ₊

À( proportional

to the shear force itself. This

argument suggests

^that

> 0

(as

found

perturbatively

in Section

4).

JOLRNAL DE PHYSIQmL -T. 5,N°6,JUNE 1995 25

A second set of nonlinear terms _are those

describing

the

dependence

of the local

density

_p

m

equation (3)

on the local state of stress. The

leading

order terms allowed

by

_{symmetry are,}

naively:

p = po ⁺

aifz

+

a2F)

₊

a3É~

_+...

₍₂₅₎

Again,

_{we are} faced with trie

problem

of

making

these

higher

terms bave trie correct

dimensions;

this

might require

introduction of _new

physical

scale to characterize

compression,

_or

(in

trie _case of

a3) negative

_powers of

Fz

_as described above.

Assuming

for

simplicity

that there _{is no new}

stress

scale,

trie

leading

behaviour _{is p}

= po

+13É~ /F)

which describes _a

dilatancy effect,

that is, the

density

is affected

by

the shear stress. The effect _can have either

sign [32], depending

_on

the initial

density

of trie

powder.

In

practice,

it _seems to _us

unlikely

that

large density changes (more

than _a few

percent, anyway)

_are induced

by

the

gravitational

stresses and _we therefore do _net introduce nonlinear

density

terms into

equation (3).

In

contrast, k2

in

equation (4)

could be _a

strong

^{function of} trie

density (and

thus of trie

stresses)

and _so vary

markedly

even

though

trie

density change

is itself small. This effect

is, however, already

included in the

general expansion

^of the constitutive

equation, equation (24),

and

gives

a contribution to À, of order

136k2/ôp.

This contribution could have either

sign

but is

likely

to be

positive

at

high densities,

since

although

the

density

decreases _on

shearing (a3

_<

0, dilatancy), k2

in

equation (7)

is

likely

to be _a

decreasing

function of _p

(see

Section

4,

and

Eq. (2),

with

presumably

increasing

_{as p}

decreases). Neglecting

the aforementioned

density ternis,

we obtain

finally:

)

=

-Ù É

₊

po

(26)

)

=

-k2ÙFz

+

~V~É fÙ ~

(27)

These _{are, we}

believe,

_a

physically

motivated

generalization

of _our

phenomenological equations,

to

leading

nonlinear order.

3.3. DTP OR HUMA ? We now look at how these nonlinear terms influence trie 2D force

profiles

obtained

earlier;

_we treat the nonlinear terms _{as a}

perturbation.

We _are mterested in

piles large compared

to1and therefore

neglect

^the

dispersion

term. To decide whether there

is a

dip

_{or a}

hump

_we need consider

only

trie

leading

order

x-dependent

term in trie stress mcrement _near trie

sandpile

centre.

Using

trie _same notations _as

above,

_we find this _as:

AÎ(~, z)

₌ ^-AÀ-_~2

(28)

where A is a

positive

constant which

depends

_{on cu} and _c

(in

Section

4, algebraic

details _are

given

^{for trie} _case ^of c ~

co).

In

Figure 8,

we show stress

profiles

obtained

by numerically iterating equations (26,27)

to first order in small À. As

previously mentioned,

_{we can}

only explain

trie observed

dip

in trie vertical _stress

if,

for _{some reason,} is

negative, which, however,

contradicts trie

physical _arguments

for its

sign given

m trie previous section

(and

in trie next

one).

It is

interesting

to

speculate

on trie

possible

role of trie "effective

velocity"

¹⁷ induced

by

arches _on

É,

_if

one were to carry this idea

beyond

trie

perturbation expansion

into _a

"strongly

nonlinear"

region.

A

possible

form for this function _is sketched in

Figure 9;

trie

interpretation

of trie maximal

velocity

V* would be _as follows. For _a

sufficiently high sandpile (and

hence for

large F~),

_{one can} create a void

by

_removing trie grains from _{a cone} of opemng half-

angle tan~~(V* _co)

below trie

pile

without

causing

trie whole _{structure to}

collapse.

Note

08 03

4=30deg deg

w 07 ~"~r

w 07

( 8 à

a a

~ j g a a

( 06

, , ( 06 # §

~ ~ j ~ ^à a

(

₀₅ t ~

(

₀₅

/ (

c ~ s c : a

~ ^s s ~ a a

04 ° 04 $

~ s m o . .

> » . > o »

# 03 ~ # 03 ~ $

Î ~ ~ ~

°

02 1 ~ ^° ₀₂ ~ $

~ . . o . .

°l

/ (ÎÎÎÎÎ=10.2

₀₁

/ (ÎÎ~ÎÎÎÎO.2

. . m

2 ₁₅

ai ~z b) ~z

Fig.

8. Perturbative solution of the non-lmear _wave

equation

with _~ % 0 and

(a)1

= +0.2

("hump"

or

(b) ¹⁼

-0.2

("dip").

For _a discussion of trie discontinuities at _{~ =}

+coz,

_see alter

equation (36)

below.

v

v*

F

Fig.

9. Possible

shape

of the "effective

velocity"

V induced

by arching

_{as a} function of the shear stress F~.

that trie model considered

by

Edwards and Oakeshott

[20] simply corresponds

_{to co}

= 0 and

V =

V*sign(F~),

_1-e-, trie _curve of

Figure

9 is

approximated by

_{a step} function. This

suggests

that trie

"arching effect",

which _was

argued

_on intuitive

grounds by

Edwards and Oakeshott to be

responsible

^for trie "double-belled"

shape

of _trie

stress, might only

be

properly

modelled

by including higher

order non-linearities

(designed

to reflect trie form of

Fig. 9)

into trie framework

considered

jbove.

4. Perturbation

Theory

in trie Limit of "Flat" Piles

In this section _we return ta a discussion based _on continuum mechanics

equations (5,6), along

with trie

inequality

^for trie static friction embodied in

equations (8-10).

Recall that trie _pa- rarneters d and _{r are} trie _mean and trie difference of trie two

principal stresses, respectively,

which bave

arbitrary

orientation in trie

(x, z) plane.

As

previously mentioned,

for trie

problem

ta be

mathematically well-posed,

trie

inequality

0 < r <

dsin#

must be

replaced by

_a consti- tutive relation that allows _one of trie three

mdependent components

^of trie stress tensor ta be