Granta Gravel model of sandpile avalanches: towards critical fluctuations?

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Granta Gravel model of sandpile avalanches: towards

critical fluctuations?

Pierre Evesque

To cite this version:

Short Communication

Granta

Gravel

model of

_sandpile

avalanches: towards critical

fluctuations?

Pierre

_Evesque(*)

Laboratoire de Matériaux et Structures de Génie Civil

_(**),

BP

_Orly

Sud n°

155,

F-94396

_Orly

Aero-gare

Cedex,

France

(Received 6 July

1990,

accepted

6

_September

₁₉₉₀₎

Résumé. 2014 En nous

_inspirant

du modèle de "Granta

Gravel",

nous _proposons une

_approche

théo-rique

des avalanches de billes ou de sable

_{qui postule}

des

_{grains rigides,}

des

_pertes

_par frottement

solide,

des déformations

_plastiques

et des effets de dilatance. On montre en

_particulier,

_que la taille des avalanches est _{controlée par} la différence entre la densité d de

_{l’empilement}

et une densité

cri-tique

dc.

Lorsque d

=

_dc,

_les

pertes peuvent

être

_dissipées

localement et le _processus d’avalanche _peut

présenter

des fluctuations

_{critiques. Lorsque}

d >

_dc,

la théorie

_prédit

des avalanches

_{macroscopiques}

et un

_phénomène

du

_premier

ordre. Notre

_{description jette}

un

_pont

entre la théorie d’état

_critique

auto-organisé

de R Bak et al. et les résultats

_{expérimentaux.}

Abstract. 2014 A

theory

which

_predicts

the size of

_sandpile

avalanches is

_given.

It is based on the so-called Granta Gravel model which assumes

_{rigid grains, plastic yielding}

and friction

_losses;

it also takes into account

_dilatancy

effects.

_According

to our

model,

the avalanche size is controlled

_by

the difference between the real

_{pile density d}

and a critical

_density

dc:

macroscopic

avalanches

_(i.e.

first order

_process)

are obtained when d >

_dc,

since the

_slope

of the

_pile

becomes unstable for an

_angle

larger

than the friction

_angle

but critical fluctuations of avalanche sizes

_(i.e.

second order

_process)

are

_expected

when d = dc. This

theory

makes _a link between the

_theory

of

_{Self-Organized}

_Criticality

of sand avalanches and

_experimental

results.

Classification

Physics

Abstracts 46.10 - 47.20 - 05.60

Recently,

Bak,

Thng

and Wiesenfeld

_{[1] (BTW)}

have introduced a

_simple

model to understand the

_1/f

noise

_problem.

It is based on a

_process

which creates

_{self-organized}

critical states so that the

_{physical properties}

of the

_system

are

_governed

_{by scaling}

laws and critical

_exponents

[1-5],

(as

in a

_second

order

_phase

_{transition). Interestingly,}

these authors have used their model to describe

_grain

avalanches at a

_pile

surface and have

_{predicted large}

fluctuations of the event sizes. On the

_contrary,

recent

_experimental

works

_[6-9]

on such bead avalanches have not found such

large

fluctuations.

2516

Besides we have established

_[10]

a

_parallel

between bead avalanches and

_experimental

results

obtained with a classical triaxial cell

_[11-14]

on sands and soils.

Futhermore,

it

_appears

that the

main behaviours obtained within such triaxial cell tests _may be

_{explained by}

a

_theory

of Schofield and Wroth

_(SW)

_{[11] ,}

called the Granta Gravel model and based on a few

_simple

_assumptions.

In this

_paper,

we want to use the Granta Gravel

_{approximations}

and a Schofield-and-Wroth-like

approach

to

_investigate

the effect of

_dilatancy

_[15]

on the avalanche

_problem.

We demonstrate

that

_dilantancy

_[15]

controls the avalanche

_size,

_through

the difference d -

_de

of the

_{pile density}

d and the so-called "critical

_density"

_{de :}

the

_slope

_tg(9)

of the free surface _may

_overpass

the friction

coefficients M =

_tg

(03A6e)

for

_piles

denser than

_dé..

According

to the Granta Gravel

_model,

the

_grains

are

_rigid,

the

_sample

deformation occurs

by

plastic yielding

and the _energy losses are

_governed

_by

a

_unique

solid friction coefficient M =

tg(O,)

which is

_independent

of the material

_{compactivity.}

The

_sample

_might

exhibit

_dilatancy

[15],

but it reâches

_{asymptotically}

the so-called "critical state of soil" at

_large

_plastic

_yielding.

The existence of this "critical

state",

which is

_only

a _{characteristic state, is} a

_major

result of

_many

exper-imental studies

_using

the triaxial cell method on soil and sand

_[11-14J;

it is also a

_{major assumption}

in this

_theory.

This "critical state" is characterized

_by

its

_density

_de,

which is

_independent

of the

material

_history

but which

_depends

on the

_"pressure"

_p

_(i.e.

mean

_stress),

_(de(p));

this state is assumed to be

_isotropic

for the sake of

_simplicity.

The

_{packing density}

d _may be different from

de

in an other state than the "critical state".

Consider now a

_homogeneous

_pile

with a constant

_density

d and a flat free surface inclined at

an

_angle

0 with the horizontal

_plane,

as sketched in

_figure

la and

_(P)

a

_{plane parallel}

to and at

a distance h from this free

_surface;

this

defines

a slice of material which lies on

_top

of the

_plane;

be _p and _q the forces

_respectively

normal and

_{perpendicular}

to the

_plane,

_applied

_by

thé slice

to the

_plane.

The

_positive

direction for _p

_{and q}

will be

_downwards;

_p is

_always

> 0. Consider

now a

_slight

variation

_bp

and

_bq

of _p and _q, which

_might

induce a small

_yielding

which will be

characterized

_by

a volume

_change

bêv,

_equivalent

to a

_{displacement perpendicular}

to the

_plane,

and a

_displacement

_ôeq

_parallel

to the

_plane;

both are localized at the interface. take

_positive

ôev

as volume contractions and

_{positive bêq}

as downwards

_{displacements; negative}

bc,

will then

indicate

_dilatancy.

So,

according

to the

_{plastic yielding theory}

in an

_isotropic

medium

_{[11] ,}

local

(p,

q ; êv,

-q)

->

_(p

_{+ ap,}

_q +

_âq;

E, +

_bêv,

_Eq

+

_ôeq)

_{yielding obeys}

a

_stability

_criterium,

which

states that no

_yielding

occurs if

Futhermore,

as

_{during yielding}

the released _energy is

_dissipated

_through

solid

_friction,

character-ized

_by

M,

one has:

Thus,

cansidering

a

_sliding

down case

_(i.e.

_bEq

>

_0,

_bq

>

₀₎

at the limit

_{of stability}

and

_combining

equation (la)

and

_(1b)

lead to:

Integrating equation (lc)

leads to the

_{family of yield}

curves

_{(py, qY):}

where In

_(pu)

+ 1 is the

_constancy

of

_integration.

We shall see that the

_{point (pu,}

qu = M _x

_pu)

is the so-called "critical state". A

_{typical yield}

curve is sketched in

_figure

_lb;

the maximum of

py, e x _pu, is reached for _qy =

0;

the maximum of _qy, M x _pu, is reached for _py =

pu. One may

define

_{tg( 1»}

=

qy/py

as the

_{pseudo-friction}

coefficient of the

_yielding

_point

_(py,

_qY);

as near a

Fig.

1. -

a)

Sketch of a

_pile

with a free surface inclined at an

_angle

0 with the horizontal

_plane.

The forces

p and q are

_applied

to the bottom

_part

of the

_{pile by}

the _upper one.

_b)

The

_yield

curve, i.e. the set of

_{(p, q)}

point

of the _pq

_plane

for which

_instability

is

reached,

when

_dilatancy

effects are taken into account.

_c)

The three

_typical

behaviours of "Granta Gravel" when submitted to a triaxial test:

_{({ )}

dense

_packing,

( - - - -)

loose

_{packing, (-.}

-.

-)

Granta Gravel critical state

_density);

_p and _q are defined in

_figure

la,

Eq is the strain in direction 1 and ev the volume

decrease,

(cl >

0 for volume

decreases).

Now,

if the

_system

at

_{(py, qy) is}

_sliding

down when

_applying

_{(bp >}

_0,

_{bq >}

_0),

_{equation (1b)}

predicts

the behaviors of local

_density:

Using

now the

_experimental

evidence of the existence of an

_asymptotic

_behavior,

the so-called

"critical

_state",

characterized

_by

its

_density

_de

and its friction coefficient

_tg

_{«)e) = M}

_gives:

** _The

system

obeiying

équation

(2a)

is in its critical state, so that d =

dc,

M =

tg

Ce = M x _{p at}

_sliding;

_sliding

near a free surface occurs when

em q>,

thus

d, Om

and 4> are

time

_independent.

** _The

system

which

_obeys

_{equation (2b)}

is characterized

_by

d

_de,

_qy/py

=

tg(&)

M;

sliding

near the free surface occurs at

_{E), = e}

_03A6e.

When

_sliding

occurs, d

_{(then 03A6)}

increases

to tend to

_de

_{(and 4>,),}

so that

_yielding

_stops

_{spontaneously}

if 0 is

_{stopped increasing.}

**

The

_system

which

_{obeys equation (2c)}

has d >

dc,

qy/py

=

tg(03A6)

_>

_M;

_sliding

near a free surface occurs at

_6m

= & >

&c.

When

_sliding

occurs, d

_{(and 03A6)}

decreases to tend to

_dc

_(and

03A6c),

so that

_yielding

cannot

_stop

if 0 is

_kept

_constant;

it can

_only

be

_{stopped by tilting quickly}

0 to a value smaller than the new 4. When this is not

_performed,

one observes a

_macroscopic

2518

at a value smaller than

03A6c.

The size of this event scales as

L2

x

(03A6- 03A6c)

x

e/4, (where

L and e

are

_respectively

the

_length

of the

_slope

and the transverse size of the free

_surface).

We have established

_equation

_{(2) by taking}

into account friction

_losses,

_dilatancy

effects and the existence of

_the

so-called "critical state" defined

_by

its

_density

de

and towards which a

_system

evolves

_{asymptotically;}

in turn,

_{equation (2) predicts}

the existence

_{of macroscopic}

avalanches,

the size of which scales as the

_pile

volume,

when the

_pile

is inclined

_continously

and when its

_density

is

_larger

than

_de.

Another

_point

which is worth

_noting,

since it is the usual _{way soil mechanists} sum

_up

this

_theory,

is the

_{following :}

assume that

_somebody

is able to control e in such a

_way

that the

_pile

is

_kept

at

its limit of

_yielding;

then

_somebody

_may

draw the evolution

_qY/pY

=

tg(4)

_versus

eq

and that one

of êv vs.

_eq

for a

_{given density}

d

_(such

an

_expèriment

is not difficult when d

de,

but needs to tilt back

_quickly

0 when d

de,

before the avalanche has

_started).

We have

_reproted

in

_figure

1c the three

_typical

behaviours of these curves and have assumed 03A6 -

03A6c

to be

_proportional

to

_d-de,

so that

_dilatancy

is a

_pertinent

_parameter:

** _{when d}

de,

-v and e decrease

_continuously

as

_yielding

(Eq)

_increases,

which means

_pile

expension

and

_{pile hardening;}

* * _{when d}

>

de,

ev and

0 are

_decreasing,

which means

_{pile expension}

and

_{pile softening;}

* * _{when d}

=

de, c v

and 0 are

_kept

constant to their

critical

values all

_along

_sliding.

Thèse three series of curves are a classical soil-mechanics _summary of this

_theory,

since

_they

_may be

_compared

to

_{typical experimental}

results obtained with a triaxial cell

_{[10- 14].}

Consider now a

_pile

denser than

_{de ;}

it exhibits

_strongly

nonlinear and unstable

_behaviors;

so, the existence of

_slight

d fluctuations in the

_pile

_implies

that small strains occur

_firstly

in few loca-tions and induce there local

_dilatations,

which means also

_softening

of the

material,

so that new

strains are

_expected

to occur there

_{preferentially;}

we

_expect

then

_large

strain localizations. This

is

_commonly

observed indeed.

Another

_point

which is worth

_commenting

is the

_following:

it is

_{experimentally}

observed,

at

least at

_large

_p, that the "critical"

_density

de

increases with _p, so that one assumes in

_general

the "critical state" law

_[11-14]:

However,

_equation

_(3a)

cannot hold near a free

surface,

since one

_expects

_de

to tend to a finite value

_dco,

as found

_by

_{[16] ,}

but it is still

_expected

that

_de

increases

_slightly

with p near _p = 0.

So,

consider a

_homogeneous

_pile,

with a free

_surface,

with a constant

_density

_d,

_everywhere

denser than the "critical state". The bottom

_part

of this

_pile

is then nearer from "critical state" and from

sliding

than its

_upper

_part.

A

_question

arises

_[17]

then:

_why

do we observe surface avalanches

rather than

_deep

earth

_slidings?

A

_plausible

_explanation

is that this avalanche event is made more

unstable due to noise: if one

_grain

inside the

_pile

is unstable it slides down

_{spontaneously}

and loses

a

_potential

_energy

_Ep

which has to be

_dissipated

inside the

_pile.

An order of

_magnitude

of

_Ep

is

m _{x g} x _a,

_(where

m, a

and g

are the

_grain

mass, the

grain

diameter and the

_{gravity respectively);}

dissipation

occurs

_{partly through}

dilatation

6,-,,

6év = -6d,

so that

_Ep

= m x a _{x g} = bd _x

p. This

means that bd is _p

_dependent

and the nearer from the free surface the

_grain

_is,

the

_larger

bd it

creates and the more efhcient it is. Such an

_explanation

could

_explain

the

_{experiniental}

results on sand

_{sliding reported}

in

_[17].

However,

another

_explanation

of the

_unstability

of the free surface

could be based on the fact discovered

_by

_Reynolds

_[15]

that

_dilatancy

is less

_important

near a flat surface than in the

_pile.

Let us now come back to the discussion of the avalanche-size

_problem

in a finite

_pile.

It is

first order transition

_problem,

due to its size

_scaling.

This has been

_{already suggested by Jaeger et}

aL

_[8]

on the basis of the

_{existing discontinuity}

between the initial and final

_slopes.

Futhermore,

from a soil-mechanist’s

_point

of

_view,

it is wellknown that the

_dilatancy

effect

exhibited

_by

Granta Gravel could have been taken into account

_by

_introducing

a cohesion force

u between.

_{grains: (J’}

is

_positive

when the

_pile

is at rest and d >

de

and 0, = 0 when the critical

state is

reached,

i.e. when avalanche occurs. This leads also

_{unambiguously}

to a first order

_phase

transition

_analogy,

since

_{releasing o,}

from u to 0 releases

_abruptly

an amount of

_energy

E and

releases a "latent heat" of transition.

This

_analogy

_may be sketched in another _way: the

_pile,

which is

_flowing

_down,

is in its critical

state, so that

_{yielding (or flow)}

occurs at a constant volume. This

_implies

a Poisson coefficient

equal

to

_0.5,

which is the Poisson coefficient of a

_{perfect liquid.}

Thus the _energy E of

grain-decohesion _may be viewed as a true latent heat of

_melting.

Let us now consider a

_pile

with a

_density

d

_equal

to

_{de ;}

it slides

_exactly

at

_Om

=

_Ob,,,

and the _energy

E needed for

_{grain-decohesion}

is 0.

So,

the transition between

_sliding

and not

_sliding

will become second order. This

_pile

_may

_{perhaps obey}

the

_scaling

laws of the

Bak,

_Thng

and Wiesenfeld model. It seems then to us that the

_{sandpile-avalanche}

_process

is similar to a

_liquid-gas

transition which occurs at a

_given

_pressure

P for a

_given

_temperature

T and which is in

_general

first order with a latent heat of

_liquefaction

_EL.

_However,

it exists a critical

_{temperature Te}

for which

_EL

= 0

and where the transition is second order. So it is

_tempting

to draw the

_analogy

between

_sandpile

yielding

and

_liquid-gas

transition: d ->

T, & - &c

->

EL

and

de

->

Te,

and it is worth

_trying

to

consider the critical state of Granta Gravel as a critical

_point

in a

_phase

transition model.

Thus,

we

_attempt

_thinking

that a

_{self-organized criticality}

_may be found in

_sandpile

avalanches under some restrictive conditions: for

_instance,

a

_precise

control of the surface -and of the volume-densities is needed in order to

_keep

the surface of the

_pile

in its critical state and the

_pile

volume

at a

_{density larger}

than

de.

In such a _{case, the surface will look like} a fluid when 0 =

Om

=

I>e, so

that the continuous model of Hwa and Kardar

_[5]

_may be valid and describe the surface flow.

However,

there is still an unclear

_problem:

is our

_{macroscopic approach,}

based on a 3-D continuous medium

_(which

assumes the

_topology

of

_{neighborhoods}

to be

_independent

of time due to the

_{plastic yielding hypothesis) compatible}

with a

_{one-grain-by-one-grain sliding}

_process

as it is described in the BTW model

_[1-5]

and with a surface

_problem?

Acknowledgements.

This work has benefitted from

_interesting

and fruitful discussions with Dr R

_Bak,

Dr S.

_Roux,

Prof. R

Habib,

Prof.

J.

Biarez,

Prof. J.

_Salencon,

Prof. A.

Zaoui,

Dr M.P.

_Luong,

Dr P.-Y. Hicher and Dr J.

Desrues,

who are then

_{gratefully acknowledged.}

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