Study of the Response to Pulses and Possible Prediction of Catastrophes

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Study of the Response to Pulses and Possible Prediction of Catastrophes

Muktish Acharyya, Bikas Chakrabarti

To cite this version:

Muktish Acharyya, Bikas Chakrabarti. Study of the Response to Pulses and Possible Prediction of Catastrophes. Journal de Physique I, EDP Sciences, 1995, 5 (2), pp.153-158. �10.1051/jp1:1995119�.

�jpa-00247047�

Classification

Physics Abstracts 91.30 -41.70

Short Communication

Study of the Response to Puises and Possible Prediction of Catastrophes

Muktisll

Acharyya(*)

and Bikas K.

Chakrabarti(**)

Salla Institute of Nuclear Physics,

1/AF,

Bidhannagar, Calcutta-700064, India

(Received

19 November 1994, revised 6 December 1994, accepted 14 December

1994)

Abstract. We show in _a numerical study of the Burridge-Knopoff spring-block model of

earthquake, that if _one applies _very weak puises

(fixed

amount of

momentum)

to any particular

block at regular intervals and studies the _{response on} the _same block

(the

stress developed _on

it), then accurate predictions of the timings of the imminent earthquakes _are possible. This _is

shown by studying the

(local)

puise susceptibility _xp, which _grows exponentially with the inverse of the time interval from the next

(global)

earthquake and saturates to _a

(nonuniversal)

value

dependent _on the model parameters

(fixed

for _any particular _serres of

earthquakes).

Considerable _progress bas

recently

been made in the statistical

study

of trie

(fracture,

break- down _or

earthquake) strength

_or

magnitude

distribution of disordered solids

(porous media,

random composites

etc.),

of

granular packings,

_or of trie

dynamical

mortels of

earthquake,

etc.

iii.

Several

simple

mortels

(at

_a semi-microscopic

level)

bave been introduced for mechan- ical [2] and electrical _[3] failure of such

randomly

disordered

media,

modelled

by randomly

disordered lattices with individual bonds

breaking irreversibly.

These theoretical results about the fracture _or breakdown

strength

distribution _etc. have also been checked in several _exper- iments [4]. ^Failure processes

playmg

intrinsic roles in many systems ^of industrial importance and in many natural

catastrophic (disaster) phenomena,

the above mentioned statistical stud-

ies

establishing

the nature of

(non-self averaging) fracture-strength distribution,

its

(critical) fluctuations,

consequent _power laws for trie _average

strength

of such solids, etc.

iii,

_are of extreme

importance. Although

_a

significant

_amount of literature has been

developed

for these studies of failure

strength distribution,

not much studies have been made

regarding

the

dynam-

ics of

microscopic

failures. Since the

microscopic

failures _are irreversible and therefore

require

intermediate redistributions of the forces

(potentials),

the

equations

^{for the}

dynamics

of failure

are

intrinsically

nonlinear and

dissipative.

The formulation and studies of such

equations

_are

necessary for the search of any precursor effect of trie macroscopic failure

(if

trie macroscopic iailure _is at most _a critical and not a

fully

chactic

phenomenon).

In _some recent

experiments

_[5]

(*) E-mail: [email protected].

(**) E-mail: [email protected].

© Les Editions de Physique 1995

154 JOURNAL DE PHYSIQUE I N°2

on the

dynamics

of the cracks in thin

glass plates,

with thermal stresses, the

dynamics

_seems

to under go a sequence of

(numerous

but

reproducible

instabilities

(not

sensitive to every de- tait of the fluctuations in initial

conditions).

The

dynamics

of fracture is thus observed _to be

mostly

critical, on the verge of chaos

(depending

_on the

velocity

of the crack

tip),

but not

quite ^chaotic [6]. ^{Similar is the} case for the

dynamical (Burridge-Knopoff

_I?i

type)

mortels of

earthquake

_[8], where also _one gets the

Guttenberg-Richter

type _power ^{law for the}

magnitude (strength)

variation of the

density

of

quakes (failure

distribution

being critical).

There has been indications

recently

_[9] that _some linear

properties je-g-,

the ratio of shear modulus and bulk

modulus)

show abnormal

(but universal) properties

_near the breakdown

(macroscopic failure)

point of disordered solids. The detailed

study

of such

(linear)

_response

(before breakdown)

would thus

provide prior

indication of the

catastrophe. However,

such

indications _are found to be _very selective and it bas not been

possible

to

study

in _cases of important

catastrophic

failures like trie

earthquake

etc. The

study

of trie response of such

dynan1icaI

systems to short-duration

pulses

indicates that another kind of linear response,

namely

the

pulse-susceptibility [10],

^{has indeed universal behaviour} near the fail,Jre

point.

It locates very

accurately

the

magnetic

transition

point

[10], ^{and the}

self-organized

critical

point~ il Ii

in various models [12]. ^We present here the results of _our numerical

study

of

pulse

response in the uniform

Burridge-Knopoff

model

(Carlson-Langer

_[8]

), indicating

the

possibility

of accurate

prediction

of

"earthquakes"

in the model.

It _was seen

[loi,

both in _a Monte Carlo

study

and in the numerical solutions of the _mean field equation of motion of _an

Ising

magnet, that if _one

applies

_an externat

magnetic

field

pulse

of

height hp

and width ôt _on such _a system m

equilibrium,

and if the _response

magnetization pulse

is characterized

by

its

height

_mp and

(half)

width At

(> ôt),

then both the

pulse susceptibility

xp =

mp/hp

and the width ratio

Rp

₌

At/ôt incorporates (senses)

the

equilibrium (critical)

fluctuations of the

unperturbed

_system. Both xp and

Rp

have

sharp divergence (peak)

at the order-disorder transition

point

of the

unperturbed (hp

₌

0)

system. ^The

precision

in the location of the transition point can be made accurate with somewhat

langer pulse amplitude hp

^and

^extremely

^{small width ôt}

(it

_may be noted that the

pulse susceptibility

_xp reduces to

the

ordinary susceptibility

_as

hp

- o and ôt _-

cc). Similary,

_we found [12] ⁱⁿ a computer simulation

study

that _a

pulsed

addition of

"particles"

at _a central site for _a short time ôt in the BTW mortel

Ill],

after its avalanche

dynamics

has

stopped,

_{causes a} successive series of local avalanches

persisting

for _a response time At

(after

which the

dynanlics

stops

again).

The width ratio

Rp

₌

At/ôt

_was found to

diverge

with _{a power} law _as the _average

"slope"

approaches

its

self-organized

critical value. In

fact,

_{one can}

predict

quite

accurately

this critical.

value of the

slope by determming

the

extrapolated

_average

slope

value where the inverse width ratio vanishes

(with

some

appropriate

_power it

gives

linear

fit)

_[12]. These _successes of such

dynamic (pulse response)

measurements to locate the

(self-organized)

critical

points prompted

us to

investigate

similar _response behaviour in the

earthquake

mortels [7, 8].

We have studied the

stick-slip

mortel of

earthquake

_I?i. In this

mortel,

_a linear array of N blocks, ^each

coupled

to its nearest

neighbours by

elastic

springs

and each connected _{to a}

rigid

support

(at

the

top) by

elastic

springs,

_is put _{on a}

uniformly

_moving

rough platform.

The equations of motion

(after

linear scale

transformation)

for _any of the

blocks,

in the uniform

spring

constant

Burndge-Knopoff

mortel

(Carlson-Langer [8]),

can be written _as:

d~Uj /dT~

₌ ^l~

(Uj+1 2Uj

+

Uj-i Uj #[20u

+

20(dUj /dT)], (1)

where, Uj

^{denotes the}

displacement

of the

j-th

block from its

equilibrium position,

l~ is the ratio of the

spring

constants, a is a constant

(linear

transformation

factor),

and _{u is} the uniform

pulling velocity

of the

rough (frictional) platform.

The nonlinear friction force _is

given by

the

function

#(y)

=

sign(y) /(1

+

(y().

43 (Q)

io

3

0

o ioo00 20000 30000 40000 50000 60000 70000 60000 ?0000 >00000

o

~ (b)

4

z

g _~

~

-2

-4

-6

-o

o 10000 20000 30000 40000 so000 60000 70000 00000 yoo00 >ooooo

o.42

o4 (C)

om

o.30

1 ~

'~ '

~u ^.

~ l

i t

~' °'~ .j

Î ^{~ ÎÎr}

h.

Î _Î.

'~ ~, ^«.

.

~-

^~

" *4~ ^°

o ioo00 20000 30000 40000 so000 G0000 >0000 H0000 90000 io0000

à

Fig. 1.

(a)

The time variation of the total elastic energy ET of the entire system.

(b)

shows the time variation of the elastic stress

(force)

/hUk developed _on the block _on which puise has been applied

(k

= 50

here).

(c) Time variation of inverse of the logarithm of the puise susceptibility (Xp). The linearly extrapolated saturation _(+w 0.25) point

(time)

ofthis quantity _gives the possible location

(time)

of the

imminent earthquake (compare with the major _energy release points _in

la)).

156 JOURNAL DE PHYSIQUE I N°2

In such

dynamical models,

the elastic stresses

(energy) developed

compete with the

(velocity dependent)

frictional force _as

long

_as the blocks stick to the

moving platform.

As it

fails,

there

occurs local failures

(shppage

for _a finite number of

blocks)

_or

global

failures

(simultaneus

slippage

of almost all the

blocks).

Initially the

(elastic)

_{energy mcreases} and then

suddenly

falls

(causing

_an avalanche _or

quake) releasing

the _excess elastic energy. As the released energy

is often _more than the

"excess",

the system

again

becomes "sub-critical". The _energy starts

0.38

0.35

1 _0.34

'~

d

_~° 0.32

~W

0~3 4

, ,,

o,za # _,

~

, ^,

,,

,

°"~~

-~L,~ _-' Î-~-

~

Àooo0 42000 44000 w6000 wsooo soooo s2000 s4000

45

40

33

30

y~

₂₅

20

là

io

~

40000 42000 44000 4600n 46000 50000 52000 54000

Îô

Fig. 2. A close _view of the puise susceptibility variation _{near some major} events _or earthquakes

(enlarged

from _a portion ^of Fig. 1). The time variation of the total elastic _{energy is} also compared.

to build _up

again

and after _some time it

falls;

and the _process continues. The strain energy released in any such failure is identified _as the

magnitude (strength)

of the

earthquake

in this mortel system. We have solved the above set of

equations using

fourth-order

Runge-Kutta

method

taking

100 blocks

(N

=

100).

We have checked the distributions _etc. of

earthquakes (the Guttenberge-Richter law)

for the mortel and

reproduced

the previous results _[8] for various initial randomness of the block velocities and positions. ^{We then}

apply

_very weak

pulses

_{to an}

arbitrarily

chosen

(central; k-th)

block

(giving

fixed

arbitrary

ôÙk _'~ 0.01for _a duration ôt

+w

0.01;

k ₌ 50

here)

_{at a}

regular

time interval of T

= 1000 We then calculate

again

the total elastic _energy

ET(" (£j l~(Uj+j 2Uj

+

Uj-j)~

₊

U))/2)

at each time step and

plot

them

against

time

(Fig. la).

In all the

figures

_we show the

typical

results for N

=

100,

l~ ₌ 10, a

=

2.5,

_u

= 0.01,

although

the

general

features _we discuss _are observed for any combinations of these parameter values. We also show in

Figure

1, the time variation of the elastic stress

AUk

(" l~(Uk+1

2Uk ₊

Uk-i) Uk) developed

_on the k-th block, on which the

pulses

_are

being applied (Fig. lb).

We also show there

lin Fig. lc),

the time variation of the

pulse

susceptibility

_Xp, defined _as xp =

AUjf /Uf

with

AUjf denoting

the maximum of AUk within the

pulse period

and

Uf

=

f/~ ^ôÙkdt.

We find xp +w

exp[A(t~~ /(t~~ t)]

for t < t~~, ^where

A(t~~

^is a constant and t~~ ^{denote the onset time of} the n-th

earthquake

_or

catastrophe.

It

may be noted

(from Figs.

lc and

2)

^that

log(xp)~~ finally

saturates to a value around 0.25 before the

earthquakes.

Indeed the linear

extrapolation (with time)

to this saturation value of

(log xp)~~ gives quite

accurate prior indication about the location

(time)

of the imminent

earthquake (compare

with the

large

elastic energy releases for the entire system, ^{shown in}

s

+

«

4 -~

~

fi

_j_

x 3 , +

ce o

~ 4

-1'

2

+ +

4 +

4 ~~

0

-2 -1 s -1 -o,s 0 O,s 1,s

Loq [El

Fig. 3. Earthquake _energy distribution, givmg the variation of the number N

(El

of quakes releasing

elastic energy E:

(Q)

For unperturbed system.

(+)

For the

same system perturbed with the application

of puises.

158 JOURNAL DE PHYSIQUE I N°2

Figs.

l and

2).

This is

quite

similar to the observation of Sahimi and Arbabi _[9]

regarding

the convergence of the elastic constant ratio to a universal value before fracture.

However,

the saturation value

(for (log xp)~~)

in this _case is found _to be nonuniversal and

depends

_on the parameters of the

dynamics

_{(1, a} and _u; the saturation value decreases _{as u}

increases). Still,

the saturation value

being

fixed for _a

particular

series of

earthquakes,

the above

procedure

for the

prediction

^of the

earthquake

^{should be useful.} In

Figure

3, we

just

show the distribution

NIE)

of

earthquakes releasing

_energy E. The distribution indeed follows the

Guttenberg-Richter

type _power ^law

decay (N(E)

+~

E~P, _fl

+~ 2.0 1?i

),

^both in the

unperturbed

system

(without puise)

and the system

perturbed by

weak

pulses (on

^{the k-th}

block)

at

regular

intervals. It may be noted

however, although

the _nature of the distribution remains

unchanged,

because of the sensitivity ^{of the initial}

conditions,

the exact

magnitude

and the

timings

of the

quakes

_are

somewhat different in the two _cases.

In _{summary, we} find that

by applying

_very weak

(mechanical) pulses

_{on any}

particular block,

and

by studying

the local response

(studying

the

pulse susceptibility

_Xp at the _sanie

block),

_one

can locate very

accurately

almost all

(more

than 75 per

cent)

of the

major (global) earthquakes (for

the entire

system)

in the

Burndge-Knopoff

type model.

Acknowledgments

We _are

extremely

thankful to D. Stauffer for _a careful

reading

^{of the} manuscript ^{and for critical} remarks.

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