Mean-field solution of a block-spring model of earthquakes

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Mean-field solution of a block-spring model of earthquakes

Didier Sornette

To cite this version:

Didier Sornette. Mean-field solution of a block-spring model of earthquakes. Journal de Physique I,

EDP Sciences, 1992, 2 (11), pp.2089-2096. �10.1051/jp1:1992269�. �jpa-00246688�

Classification Physics Abstracts

05.50 05.70J 62.20M

Mean-field solution of

_a

block-spring model of earthquakes

Didier Somette

Laboratoire de

Physique

de la Matibre Condensde(*), Universitd de

Nice-Sophia Antipolis,

B-P- 71, Parc Valrose 06108 Nice Cedex 2, France

(Received 26 June 1992,

accepted

in final

form

17

July

1992)

Abstract. -A _mean field version of the

Burridge-Knopoff block-spring stick-slip

model of

earthquake

faults is

mapped

onto a

cycled generalization

of the democratic fiber bundle model (DFM). This

provides

an

exactly

soluble model which describes the set of

earthquakes preceding

_a

major earthquake.

We find the coexistence of 1) _a differential

Gutenberg-Richter

distribution d(A A~ ^~'~ of bursts of size A, with _a cut-off A~~

(«,

« )~ as the _{stress «}

- «, and 2) _{a run}

away

occurring

at _a well-defined stress threshold _«,. The total number of bursts of size A up ^to the _run away scales _as D(A A ~'~. The exponent 5/2 reflects the _occurrence of

larger

and

larger

events when

approaching

the _run away

instability

(Omori's law for foreshocks). The

Gutenberg-Richter

and Omori power laws _{are not} associated with _a

stationary

criticality but to fluctuations

accompanying

the nucleation of the _run away. Introducing long _range correlations in the model lead _{to a} continuous

dependence

of the above exponents as a function of the correlation

exponent.

Understanding earthquake dynamics

and

plate

tectonic mechanics is _now the focus of _an

important

research effort in the

physical community [1-9],

since it has been realized that

they provide

_one of the most

interesting physical

realization of

spatio-temporal complex dynamics [10],

_a

subject

of

large

current interest which _covers many different fields

[9,

II

].

Beginning

with the

Burridge-Knopoff

model

[12],

_a

variety

^of

increasingly sophisticated

mechanical models that simulate the _occurrence of

earthquakes

have been studied

[3-8].

These

characteristically produce

_power laws for the _rate of _occurrence of model

earthquakes

and

seem to

display

the

property

^of

self-organized criticality (SOC) [13].

At present, ^there is _no real theoretical

understanding why

such power laws appear in

block-spring stick-slip Burridge- Knopoff

_type models _{or even} in its various cellular automata versions.

Consider the

Burridge-Knopoff block-spring

model of

earthquake

faults

consisting

in _an

array of blocks of _mass m

coupled

to each other

by

harmonic

springs

of

strength

k~ and

pulled

to the

right by

_{an upper}

rigid

surface

acting through pulling springs

of

strength

kp. ^{The blocks} are in contact with _a fixed

rough

substrate

exerting

_a static friction force

F~

on them. The blocks have initial random

positions

and _ate motionless. The upper

pulling

surface

steadily

increases its

displacement

and thus the force exerted _on each block increases.

(*) CNRS URA190.

2090 JOURNAL DE

PHYSIQUE

I N° II

At _some

point,

the total force exerted _{on a}

block,

which is the _sum of the

pulling

force and the action of the block

neighbors,

overcomes

F~

and this block

begins

_to slide. This

sliding

_may

either stop

eventually

_or

trigger

the

sliding

of its

neighbors, possibly cascading

to

produce

a

large

event. The

difficulty

of

solving

this model

theoretically

comes from the

interplay

between

I)

the non-linear threshold behavior of the friction

law, 2)

the fact that the _{event at} time _t

depends

_on all the

previous history

of past events and

3)

the local

coupling

between

neighboring blocks,

which

puts

^{this model in the}

large

^class of

N-body spatio-temporal

chaotic

dynamical

models.

Inspired by

^the standard treatment of

phase

transitions, we now propose a mean field

(MF)

version of this model which

simplifies

^the above

complexity

and allows for _an exact solution.

Mean field model.

The MF treatment of the

paramagnetic-ferromagnetic phase

transition _{amounts to}

replacing

the action of

neighboring spins

on a

given spin by

an effective average

magnetic field,

which is

self-consistently

calculated. The

difficulty

is thus reduced from _a

N-body problem

to _a I

-body problem.

The MF version of the

Burridge-Knopoff

model amounts to

replacing

the forces

exerted

by

the

neighboring

blocks _{on a}

given

block,

acting through

the

springs

_k~,

by

an

average effective force. Consider a

given

initial random static block

configuration

defined

by

the _set of

positive

deviations

(x~,

n ₌ I to N from the

position

at which _zero force is exerted.

This set of deviations defines _{a set} of force thresholds for

sliding given by

(X~

=

F~-k~x~,

n =

I _to

N).

As the upper ^surface

begins

_{to move}

by

a

displacement x(t ),

the incremental force exerted _on each block is _«

= k~

x(t).

When _« becomes

larger

than

Min~ _(X~

=

F~ k~x~,

n = I to N

),

blocks

begin

to slide. In the

spirit

of cellular _automata

modeling,

_{we assume,} in

addition,

that _{once a} block has

stopped

after it has slid _some

distance,

it is unable _to

provide

_any contribution

against

the

pulling

action of the upper surface.

This takes into account the fact that _a block

usually

^{slides ahead} of the

pulling

surface

[3].

We

also _assume that

blocks,

which have slid _{once, are not} allowed _to slide

again

in _a

given cycle

and do not exert any force _on the

pulling

surface. Thus, the total force F

= N« must now be

shared among all

remaining

blocks which have _not slid. If k blocks have slid

(or

_« failed

»),

the

resulting

force _on the

remixing

^{blocks is} now

«N/(N k)

_per block. As the upper surface

moves

further,

_« increases and _an

increasing

number of blocks slide until all the blocks

eventually

^slide in a run away ^event. To _ensure

ongoing

motion after all blocks have slid _once,

we then _assume that the

dynamical sliding

events on each block have

placed

them in _{a new}

random _set of

positive

deviations

(x~,

n = I _to N

).

The

sliding

_process starts

again

and _can then go on

infinitely, cycle

after

cycle.

The

assumption

that blocks which slid _{once never exert} any force is reasonable when

considering

the set of

earthquakes

which

preceed

_a

major

one.

After _a run-away occurs, all blocks _are assumed to exert a force which

depends

_on the

specific

block

positions.

This model. when restricted to a

single cycle,

is identical to a well-known model of rupture

in random

media,

dubbed the _« democratic fiber bundle model

»

(DFB) [14-20],

which has been introduced

initially

to describe the rupture

properties

of

long

flexible cables _or low-twist

yams. It is made of N

independent parallel

vertical fibers with identical

spring

constant k~ and

identically

distributed

independent

random failure thresholds

X~,

n =

I,..,N,

distributed

according

to the cumulative

probability

distribution P (X~

~x)

₌ P

(x).

A total

force F is

applied

to the system ^{and is shared}

democratically

among the N fibers. As

F

increases,

_more and _more fibers break down until the

complete

final rupture. ^The

correspondence

of the _two models is that the rupture ^of a fiber

corresponds

to the

sliding

of _a block. In the

Burridge-Knopoff model,

the disorder _on the failure thresholds

X~

can be viewed

as a

snap-shot

of the

dynamical

evolution of the block

positions

_x~

through

the

relationship

X~ ₌

F~

_{k~ x~.} In _essence, the present ^{MF model} attempts to describe the chaotic

dynamics by

a stochastic

description.

Some

properties

of the DFB model in the continuous limit.

Let _us first recall

briefly

the main known

properties

of the DFB model. Let _us note

Xi, X~,

,

X~,

the ordered

increasing

_sequence of

strengths

of the individual fibers. Under _a total load

F,

_a fraction P

(«

₌ FIN

)

of the threads will be submitted to more than their rated

strength

and will fail

immediately.

After this first step, ^{the total load will be} redistributed

by

the transfer of _stress from the broken links _to the other unbroken links. This transfer will in

general

^induce

secondary

failures which in _tum induce

tertiary

_ruptures and _{so on.} The

properties

of this

rupture problem

is obtained

by noting

that the total bundle will _not break

under _a load F if there _{are n} links in the bundle each of which _can withstand

F/n. In other

words,

if the first k I weakest links _are

removed,

the bundle will resist under _a force smaller than _or

equal

to

F~= (N-k+ I)X~ (I)

since there remain

(N-k+1)

links of

breaking strength larger

^than or

equal

to

X~. ^The

strength F~

of the bundle is then

given by F~

= max

((N

k ₊

I) X,

: I _w k _w N

)

which goes to N

max~[x(I -P(x))]

in the limit N

- + oJ. It _can be shown

[15]

that

F~ obeys

a central limit theorem

according

to which the

probability

that the

global

failure threshold

F~

be

equal

to F is P

(F~

=

F

(2

grN )~ ^~'~

exp(- (F

No

)~/2 Nx(),

_{and thus}

F~

_{converges to} No where 0

=

x~(I

P

(x~))

is the

unique

maximum of

x(I

P

(x))

at

x = x~. The number

k(«)

of links which have failed under the force F

=

N« is

[19]

k(«)=

NP

(x(«)), (2)

where

x(«)

is defined

by

« =

x(«) Ii

P

(x(«))j (3)

The number of

remaining

links is therefore N

[I

P

(x(« ))]

=

F/x(«

and

x(«

is thus the force exerted _on each

surviving

fibers.

Consequently,

the _stress

(« )-strain (e

characteristic of the system ^is

x(~r)

=

kp E(~r ). (4)

Just before

complete

^failure of the

bundle,

the total number of broken links is

k~

₌ NP

(xo).

The

remaining

N

[I

P

(xo)]

fibers break down

suddently

ⁱⁿ one

sweeping

_run

away event when

« - 0 I.e.

x(«

- xo. For F _w

F~,

I-e- _{« w} 0,

x(«

is in the

neighborhood

of xo and can be

expressed

under the form

x(«)

_xo

=

A

(0 «)~'~

for _«

- 0

,

(5)

where A is _a coefficient which

depends

_upon the cumulative distribution P

(x).

The _stress- strain characteristics

«(e)

is thus

quadratic

near

global

rupture ^with a

vanishing slope

at

« =

0

[19].

The number of links which have failed under the _stress F is

given by k(« )/N

=

P

(xo)

B

(0

_«

)

^~'~ for _«

- 0

,

(6)

2092 JOURNAL DE

PHYSIQUE

I N° II

where B

depends

_upon the cumulative distribution P

(x). Expression (6) implies

a very

rapid

increase of the number of broken links _{as «}

- 0 for which

k(« )

tends _to NP

(xo)

with _a

slope exhibiting

_{a square} root

singularity.

These laws

imply

_an

interesting dependence

of the _rate of elastic energy release _as the _run away event is

approached.

For _a

given applied

stress «, the elastic energy stored in the bundle of

surviving

fibers is

E(«)= e(«)x(«)k(«),

where

e(«)

is

given by equation (4),

x(«) by equations (3)

and

(5)

and

k(«) by equations (2)

and

(6). Using equations (3-4),

_we obtain

E(«)=Neik~e-«1. (7)

If the system ^{is driven at} a constant strain _rate

de/dt,

dE/dt goes ^to a constant as

« - 0. If, on the other hand, the system ^{is driven at} a constant stress rate, the rate

dE/dt of elastic energy release

diverges

when

approaching global

failure

according

_to

dE/dt

(0

_«

)~

^~'~ This

corresponds

to a marked average increase of rupture

activity prior

to the _run away

analogous

to omori's law for foreshocks

[21].

These results also underline the

sensitivity

of the behavior with respect to the

loading path.

In the models studied in reference

[3]

for

instance,

_a constant strain rate was chosen. It is not clear whether this is the

case in _nature since _a

given

fault is surrounded

by

_an elastic medium deteriorated

by

_many

other faults which

interact, leading

_to ill-defined

boundary

conditions.

Therefore,

the

loading path

of _a real fault is

probably

intermediate between the pure constant strain _{or stress} rate,

which may

explain why

increased foreshock

activity

is not

always

observed before _a main

large earthquake.

Burst

properties

of the DFB model due to the discrete nature of the elements

(fibers

or

blocks).

When

looking

at the fiber

(block) scale,

simultaneous failure of many fibers _{can occur} due to

«small scale» fluctuations of the

strength

of bundle subsets.

Indeed,

the sequence

(F~)

of external loads

given by equation (I)

_at which the fibers would fail do not form _a

monotonically increasing

sequence

[20] (see Fig. I).

The random variables X~ are indeed put ⁱⁿ

increasing

order but

they

are

multiplied by

a

monotically decreasing

factor

(N

₊ I

k)

as

k increases.

on the average, we have P

(X~)

₌ k/N which

implies

that

(X~

~ i

X~)

₌

[dP/dx

_(x~] N and

(Fk+

i

F~)

=

[I

P

(X~)] [dP/dx(x~]~ ^X,

=

[dP/dx(x ]~

^d

(x[I

P

(x)])/dx(x

(8)

Note that

(F~

~ i

F~)

is

positive

as

long

as

X~

_{~ xo} ^{and vanishes} on

approaching

the _run

away event as

~~k

+

~

k)

(X0 X

) (9)

The fluctuations of F

~ ate

given by [F

~_~ i F~]~ [X~

~ i

X~

^~

(X~

~ i

X~)

^~

which is finite and

weakly dependent

^of x.

Starting

from _a stable

configuration corresponding

to some value

F~,

_a simultaneous rupture of A

fibers,

which _can be called _{an event}

or burst of

size A, _occurs if

F~

_~

F~

for k ₊ I _w n Sk ₊ A and

F~

~~ ~ i m

F~.

Since the _set of failure

N

°

N

P(x ~) k

~f

~~o

o

o °

o

o o

~

~°~o o~

~

o o

o

Fig.

I.

Strength Ft

of _a bundle of N fibers _{as a} function of the number k of broken fibers (below) _:

magnified

view of the

dependence

of

F~ showing

the random walk in the space of forces Fk, the role of time being

played

by k.

thresholds

X~

are

independent

random

variables,

the function

F~ undergoes

a random walk where the

displacement

is

F~

and the time is k

(see Fig. I).

This random walk is biased

by

_a

drift

equal

to

(F~~

i

F~)

which vanishes at the

instability

threshold

(Eq. (9)).

The bias is

large

far away from the

instability

threshold and

only

small bursts _occur. As x-x~,

(F~~

i

F~)

- 0 and the fluctuations

completely

dominate the burst occurrence.

In the random walk

picture,

it is easy ^to obtain the

probability

that _a burst of size

A _occurs after k fibers have been

broken,

I-e- at _some value

F~.

This

corresponds

to the

probability

_pi

(A )

of first _retum to the

origin

of _a random walker

(see Fig.

I

).

In the absence of any

bias,

the

probability

to be found at the

origin

after A

steps decays

_as

^A~~'~

and the

probability

to retum

for

the

first

time at the

origin

is pi

(A) A~~'~ _{[22]. Thus,}

the local differential distribution

d(A)

of bursts of size A is

given by

d(A)~pi(A)~ ^A-3'2 (io)

This law

(10)

holds for Am

A~~~(x),

where

[A~~~(x)]~'~ _(x~ x) A~~~(x),

is such that the

drift

(x~ x) brings

_a

displacement

smaller than the

typical

random walk excursion in the time A~~~

(x)

_:

A~~~

(x) (xo x)~

^~ for _x

- xo

(I1)

2094 JOURNAL DE PHYSIQUE I N~ ii

Equation (II) implies

that _{an event} of size A may occur

only

when

A~~~(x)

WA, I-e- xo -x w A~ ^~'~ The

powerlaw (Il)

is reminiscent of the Omori's law for foreshocks often

observed for real

earthquakes [21]

_:

A~~~(x) (0 «)~

_~,

using equation (5).

Let _us now calculate the total number D

(A)

of bursts of size A.

D(A)

is the

product

^of the number _n

(A )

of bursts which _can be

larger

than A

by

the

probability

_pi

(A )

that _a burst is of size A. To determine

n(A),

we note that on average, dk

m

(A) (x)

dn, where dk is the number of fibers broken in dn bursts. We do not have _a strict

equality

because _a finite fraction of the fibers

breaks down in

elementary

bursts of size

unity.

The average size

IA ) (x)

of _a burst is

given by j aj ~~)

=

~~~~~~

_{d pi}

_(d)

da

~'

[Amaxo)1~~~ (~° ~~ ~~

_~ ^~~~

~

using equation (5). Using equations (5)

and

(6)

which show that dk N dx for

(xo x)

«

I,

we have dn

-~

N

(xo x) dx,

which allows to obtain

D

(A )

_pi

(A ) ~°

A~~~

(x) (dn/dx)

dx pi

(A )/A

A ^~'~

(12)

xo A-1'2

The

weighting

factor

A~~

_in

D(A)

results from _two

ingredients

_:

I)

due _to the cut-off

A~~~(x)

introduced

by

the

bias,

bursts of size

larger

_or

equal

to A _occur

only

when

x is

sufficiently

close _to xo

2)

_{as x}

- xo, there _are fewer and fewer bursts since

they

_are

larger

and

larger.

This result

(12)

has been

previously

derived

[20] using

_a detailed

probability analysis

and has been checked

numerically.

The present

simple

derivation in _terms of biased random walks

clarifies its

physical origin

and the fundamental _reason for the

universality

of the MF

exponents. Furthermore, it shows the coexistence of

I)

_a differential local distribution of bursts

given by equation (10)

^with an exponent

3/2

and

2)

the total number of bursts of size A

given by equation (12)

with a

larger

_exponent

5/2. Applied

to

earthquakes,

this result suggests ^{that there is} no contradiction in

observing

_a small _« b-value _»

(= 1/2)

in _a restricted time interval and _a

larger

« b-value _»

(= 3/2)

when the time interval is extended _up to the

occurrence of _a great

earthquake.

Our result may

provide

_a clue for the observed drift of b-

values often observed before an

impending earthquake.

Note that the

large

_« ^b-value _»

(= 3/2)

which is

predicted

here for the cumulative distribution of

earthquakes

_{up to} the

largest

run-away is in

good quantitative

agreement with the recent

study [22, 23],

which indicates that 0.8 _w b _w 1.2 for small

earthquakes

and b

m 1.5 for

larger earthquakes

for which the rupture dimension is

larger

^than the

downdip

width of the

seismogenic layer.

Effect of

long~range

correlations.

The main difference between _our MF DFB model and the

original Burridge-Knopoff stick-slip

model is the existence of additional correlations in the

latter,

which _are

dynamically

constructed from the action of the local

couplings

between the blocks. It is

possible

to obtain _a

feeling

for the effect of such correlations

by introducing

in the MF model _a correlation in the set of failure thresholds

X,

and therefore in the

(F~)

_sequence. It is assumed that this correlation reflects the _one which is built up

dynamically

in the

complete Burridge-Knopoff

block model. This

procedure

is thus _an

attempt

to go

beyond

_mean field treatment.

However,

the conclusions _must be limited since no

procedure

is

given

to fix the value of the correlation

exponent,

^{which is}

kept

_{as an}

arbitrary input _parameter.

We thus _assume that the

F~'s

_are correlated such that the correlation function

C

(n)

=

(F~ F~~~) (F~) (F~ ~~)

n~Y is

long ranged. Consequently,

the standard devi-

ation

=

(F()

^~'~ scales _as A ^~ with _a

superdiffusive

exponent ^1/2 w v =

I

y/2

w I for

0 _w y w I and _v

=

1/2 for y ~ l

[22].

Correlations such that y ~ l do not lead _to any modifications _to the

previous analysis.

We thus restrict _our attention _to the _case 0 my w I.

The

probability

to retum for the first time _at the

origin

after A steps ^{is modified and reads}

[24]

_pi

(A)

A~ ^{~~ ~~}

-~

A~ ~~ +Y'~~

Using

this _new

powerlaw dependence,

_we then follow step

by

step ^the same

reasoning

_as described above. A~~~^{is such that} [A~~~

(x )]

^~

(xo x)

_A~~~

(x),

which

yields A~~~(x)~ (xo -x)~

^~'~~~~~ The average burst size

IA) (x)

is

given by

(A ) (x)

=

[A~~~ (x )]

^~

(xo

x

)~

^{~'~~ ~~} We thus

finally

get A~~(X)

D

(d )

_pi

(d) (dn/dX)

dX

,

I-e- l

D(A)

_pi

(A)/(A)

A~ ^{~~ ~~} A~ ^~~+Y'~~ for 0 _w y w

(13)

Expression (13)

exhibits a power law behavior with _an exponent

showing

_a continuous

dependence

on the

mpture

^threshold correlation

exponent

^for 0 w y w I. It is

interesting

to note that the

limiting

_{case y -} 0

yields

_{a «} b-value _» b

=

I

(defined

_as the

exponent

of the

integral

of

D(A)

from I to

A),

which is the value determined

numerically

in the

Burridge-Knopoff

block-spring

models for the distribution of

earthquake

^sizes

[3]. According

to the

present theory,

it is

tempting

to

interpret

this value b

=

I _as the

signature

of the

longest possible

correlation range of the block

positions.

Our

analysis provides

another way to look _at the _non- universal behavior of the

Gutenberg-Richter exponent,

as

being dependent

on the level

of

correlation. This

concept

^{should be}

compared

with _a recent numerical

study

of _a

quasi-static

two-dimensional cellular automaton version of the

Burridge-Knopoff spring-block

^model of

earthquakes [8], according

to which the

exponent

was found _to

depend

_on the level

of

conservation in the cellular automaton. The relation between the level of conservation

[8]

and the

degree

^of correlation

suggested

here remains _to be clarified.

Concluding

remarks.

The _« democratic fiber bundle _» MF

approximation

^of the

Burridge-Knopoff block-spring

model of

earthquakes

has been shown _to exhibit various power law

analogous

to the famous

Gutenberg-Richter [10, 25]

and Omori

[2 Ii

laws for

earthquake

occurrence. In this MF model, these power laws _{are not} associated with

criticality

in _a pure sense but describe the fluctuations

prior

to the nucleation of the _run away event.

Indeed,

there is _a close

analogy

between the burst fluctuations observed before the _occurrence of the _run away event and the

droplet

fluctuations before the nucleation of the critical

droplet

which destabilizes into the _new

phase

in _a standard

first order

phase

transition

[26].

The

exponent

5/2 found here for the distribution

D

(A)

is also characteristic of _mean field

percolation

and

spinodals.

We _note that the

powerlaw

distribution of _events

(see Eqs. (10, 12, 13))

reflects the discrete nature of the

model,

I-e- is associated with the existence of _a

microscopic

^{block scale.} This small scale cut-off

is, however,

of _no

importance

for the continuous

properties

of the deformation since _a central limit theorem holds.

The exponents ^{of the burst size} distributions _are universal which

implies

that the results _are insensitive to the details of the distribution of block

positions

^after each _run away ^event. This result

justifies

_our

assumption

which _ensures

continuing motion, corresponding

to the _so- called

« seismic

cycle

_» model for the _recurrence of great

earthquakes [25],

that each _run away event amounts to a reshuffle of the

breaking

thresholds for the _next

dynamical

_process. Our

analysis

shows that there is _no contradiction in

observing

_power laws in the distribution of

earthquakes

between _{run away} events and the idea of _a seismic

cycle.

Of _course, it is

quite

JOURNAL DE PHYSIQUE i T 2, N' i >, NOVEMBER 1992 75

2096 JOURNAL DE

PHYSIQUE

I N° I I

possible that,

in

reality,

the power law extends down to the

largest earthquakes [1, 2, 4c-d, 5]

and that _our results _are

specific

to the MF model.

Acknowledgments.

I _am

grateful

to J.-P. Bouchaud and L.

Knopoff

for

extremely stimulating discussions,

to P. Cowie for useful remarks and _a critical

reading

of the

manuscript

and to A. Hansen for

providing

ref. 20

prior

to

publication.

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