Phys.

J.

£Yance

I

5

(1995)

631-638

1995,

JUNE

631

PAGE

Classification

Physics

Abstracts

62.20Mk

64.60Ak

Communication

Short

Universal

Scaling

for

J.-C.

Log-Periodic Rupture Stress

Anifrani(~),

Renormalization Correction Group to Prediction Acoustic Emissions From

Floc'h(~),

C. Le

D.

Sornette(~>~)

Souillard(~)

and B.

(~) Aérospatiale, B-P. Il, 33165 St Médart en Jalles, France Condensée, CNRS URA 190 Université de Physique de la Matière (~) Laboratoire Antipolis, B.P. 71, 06108 Nice Cedex 2, Fiance (~) X,RS, Parc-Club, 28 rue Jean Rostand, 91893 Orsay Cedex, Fiance

(Received

Based

Abstract.

point,

critical

we

fro~n

acoustic

15-20i~

below

the

spherical

pressure constitute

scaling

the

similar

tested

of

observation

vanous

in

Acoustic

(AE)

e~nissions

occurring (seismic events) to

m

a

failure

stress

January 1995)

31

heterogeneous systems with good reliability

constant

stress

rate

up

to

similar

is

maximum

a

context

framework. raturai

Our

waves

of

the

universal

method

could

(earthquakes,

sciences

range of materials,

beell

AE

most

dilfers

Les

most

material detect

prediction

final

©

from

ongm in methods

(among the rupture...)

load

produced by sudden and

structures

periodic

corrections

to

applied usefully to volcanic eruptions, etc.) be

processes,

~novement

from

one

other

stressed

in

largest scale dehcate technique and con easily be the

testing have

trot

out.

the

its

a

borne

smallest

a

our

(motion of dislocations). It is a rather and interpret the whole since each loading is unique, tests structure to use by noise. Notwithstanding trie development of AE contaminated structural numerous procedures [1, 2], hopes to use AE as a reliable practical method for failure prediction the

to

precision

and

approach is to fit the experimental signais to scaling theory for rupture in terms of complex a new successfully on an application, namely high mdustrial liber-matrix results composites. As a by,product, our

~nechanical

are

wide

of

at

of

natural

a

the

systems,

rupture

the

basis from

is

renormalization-group predicting problems m the

in

trie

measurements

deduced

made

tanks

first

that

Trie

stress.

method

Trie

January 1995, accepted

26

predict

to

emission

failure

exponents.

idea

how

expression

mathematical

fractal

revised

trie

on

show

(m 51~) a

1994,

December

20

Nice-Sophia

de

various

and

itself,

not

causes

then

de

an

determine

such

develop

as an

rupture.

Editions

in

existing geometrical is to first

scheme

Physique

testing (NDT) methods

nondestructive

other

1995

crack

externat

source, discontinuities.

which

nucleation

in

that detects

the AE

signal

movement,

has

while

Thus, the general basis for developing relevant phenomena are precursory and growth, liber-matrix delammation, liber

detected

algorith~n which

and in that it

allows

AE

one

to

relate

these

precursors

to

the

JOURNAL

632

PHYSIQUE

DE

I

N°6

the general case where global rupture is not controlled by a single event growth of a single crack), as occurs in a ho~nogeneous system with very few defects, but rather by a succession of events, in general corresponding to diffuse damage, possibly culminating m trie catastrophic growth of a dominating crack. In this case, we daim that failure only be predicted by viewing rupture as a cooperative phenomenon: it is can through the separated analysis of the of the recorded hits that not rupture succession can be predicted but by somehow synthetizing all trie information embedded in these in a events global framework. From a formal point of view, our approach to rupture prediction consists in viewing the final stage of rupture as a kind of critical point, which be described by a fixed point in can below). renormalization scheme This description (see is inspired from a wealth of group a work in trie statistical physics community, mainly based on exact solutions of solvable recent

Here,

address

we

(nucleation

and

models

and

scaling

laws

often

view

to

up

a

in

trie

unable

are

to

effective

which

approaches

specific N-body

shown

method

by

trie

thus

(evolving) including

an

properties,

only

far

from

point,

cntical

rupture

certain

of trie

opposite

modelling

in

of

existence

trie

at

average suitable

are

of the

nature

is

consists

characterized

These

parameter. the

capture

Dur

literature, medium

bave

which

[3-6].

rupture

mechanical

heterogeneous system as an increasing global damage an and

total

of

time

advocated

simulations,

numerical

extensive

on

rupture and

are

predictors. industrial application on which our method bas In order to fix ideas, we shall consider an been tested extensively. Trie complexity of such systems can be taken as a test of the robustness of our technique. made of carbon libers The systems are spherical tanks pre-impregnated in a wrapped up around a metallic liner. We have tested dilferent (carbon resin matrix materials (titanium steel) libers, dilferent for liner), kevlar metallic compounds the well as or or as (radius of 0.2 to 0.42 m). AE signais are obtained from three to six dilferent tank dimensions (resonant frequency of150 kHz) placed at equal distances on the equator transducers acoustic of a given sphencal tank by mcreasing at a the internal constant rate (3 to 6 bar/s) water tank. Figure and thus the exerted the typical of 1 data the stress represents set pressure on a AE energy internal the mstantaneous rate dE/dt as a function of the applied to pressure p up threshold pr, obtained by simple addition at each time step of the intensities measured rupture complex all transducers. Note the intermittent of the operating data, with structure quiet on periods separated by bursts of widely dilferent amplitudes, and trie large increase of trie AE 713 bars). rate on the approach to failure (occurring in the present at Pr case energy In order to test the cntical point concept on this particular set of systems, we checked for therefore

useless

as

=

the

of

existence

a

power

la~A~,

dE/dt where

close

is

a

to

Pr

approach

to

sc-called

a

to

critical

within

failure

is

about mdeed

exponent. less fitted

than very

=

Eo(Pr

In all

cases

51~, the well by

P)~" explored,

dramatic a

(1)

power

found

we

of

increase

law

(1),

with

that

AE an

for p

energy

exponent

sufficiently rate

a

=

on

1.5

the +0.2

specific sample or previous history as long as the critical zone (approximately Trie power law (1) was trie interval [o.95pr; pr] bas not been reached in preceding loads. m p), using trie measured either by representing Log(dE/dt) as a function of Log(pr checked dilferent dilferent tanks (and two This is illustrated partitions for one in Figure 2 for three Pr. tank) which exhibits clearly trie asymptotic linear dependence when suiliciently close to Pr. behavior whose slope determines We also studied We find that ail data tend to a straight o. (dE/dt)~l/° (with o 1.5) as a function of p. Agam, an asymptotic straight line is obtained for Pr better than ll~. estimate These two whose with the abscisse intersect gave a very good mdependent of

the

=

fitting

procedures

are

not

independent

but

put

dilferent

weights

on

trie

data

point.

Therefore,

RENORMALIZATION

N°6

GROUP

ACOUSTIC

FOR

EMISSIONS

633

SYSTEM 6000

sooo

o

o

4000

°

°

~

~~~~

1o o

u~

°

°

~

2000 ~

é

~

~~~~

~

°

o

~ °

o

o °

o o °

o

o

°

]

~

o

o

o

0 550

500

Fig.

AE

Instantaneous

1. p at

pressure

constant

energy

dE/dt in bar/sec up ta

rate

of 6

rate

pressure

600 PRESSURE

system

1

the

as

650

700

function

a

threshold

rupture

of pr

applied

the =

713

internai

bars.

Pr-P i

~~

~~~ u ~

'"'"'~,

2G

g

~

loo

'"

ioooo

~

F-

G

x

j~

'~,~,~

C4

k,

ÎiÎ

~

io

o

o

w

10

looo

o

o

100

1000

10000

ENERGY Fig. for

being straight log-log 2

Upper nght:

2.

each

system,

m

represented hne scale.

has

The

a

dE/dt

three twice

slope

straight

in

dilferent for cY

=

fine

log-scale systems:

two

dilferent

I.à.

Lower

has

a

slope

function

a

as

system

1

crame-graimng

dilferential

left: of

a

=

-2.

of pr

p

m

log-scale,

(0)

usmg

the

(+); system (0): 11 intervals; (+): 27 N(Eb) of burst distribution

(O); system

2

and

measured 3

pr

(x), system

intervals. energy

The

Eb

in

JOURNAL

634

verifying expected,

exportent

This

with

the

detail

N°6

and

I.à

which

central

energy

'Eb'

components of burst

consistently equal

B

in [8] in

ta

1+

Gutenberg-Richter

of trie trie

AE

(1).

the power law insensitive ta the

of

existence ta

tank

reminiscent

in

I

distribution

exponent

an

studied

empirical

upon

differential

trie

law, which is

been

bas

checking the equal

for

"universal"and

depend

however

Ej~~~~~,

+~

systems investigated. earthquake sizes [7],

found

shows

2 aise

N(Eb)

law

a

important

is is

could

It

Figure

damage. power

a

the

realization.

tank

of

consistency

their

PHYSIQUE

DE

As

specific

trie

nature

which

is

aise

0.2 for ail

the

distribution

of

context.

prediction should in possible by extrapolation of AE data using expression (1). Note that this scheme is similar ta that proposed by Voight a few years ago ta describe and predict rate-dependent material failure, based on trie use of an empirical power law relation obeyed by a measurable of quantity [9,10]. However, this procedure is unpractical due ta trie trie demain narrowness of validity of the law (1) (critical region [0.95pr; pr] in most explored cases), preventing a prediction at pressures less than, say, 0.97 pr. should allow one ta make predictions at much lower scheme A useful Consider for pressures. instance attempt ta predict pr by monitoring trie AE up ta, say, 0.85 pr, 1-e- up ta about 610 an shown in Figure 1. Inspection of Figure 1 shows that, apparently, very little bars in trie case of trie whole AE data is contained in trie AE set up ta 610 bars. Furthermore, trie dependence release as a function of trie of trie rate of AE energy applied pressure up ta 610 instantaneous bars bas apparently nothing ta do with a power law such as (1) and trie prediction thus seems hopeless. fundarnental idea is that trie concept of rupture criticality embodies Dur usable inmore than just trie power law (1) valid in trie critical demain. formation We argue that specific outside trie critical region can lead ta "universal" recognizable signatures in trie AE precursors of trie final data rupture. In order ta identify these signatures, we first note that an expression Armed

with

principle

these

of trie

tests

concept

of

critical

rupture,

be

(1) can be obtained from trie solution of a suitable renormalization group (RG) [11]. formalism, introduced in field theory and in cntical phase transitions, arnounts ta view N-body problem as a succession of I.body problems with effective properties varying with

like

Trie

RG

trie

of

scale

observation.

based

It is

trie

on

existence

of

a

scale

invanance

of trie

trie

underlying physics

physical scale and distance from trie critical axis: in Dur problem, trie damage rate and therefore AE rate point in trie central parameter p' related those through another suitable non-linear ta at at a given pressure p are pressure a #(x) with x p' transformation formalism then provides the general pr pr p. The RG of the functional equation that trie physical observable obey (for trie simplest structure must which

allows

define

ta

one

a

mapping

between

=

=

of

case

Due.parameter

a

RG): iLF(x)

For

trie

sake of

trie

AE

energy We

number.

connection

cntical

simplicity, that

assume

between

point is

on

we

which

rate,

this trie

bave

introduced

goes to zero trie function

formalism attractor

=

at

and trie

of trie

trie

trie

F(x)

F(4(z))

is

(2)

rupture

continuous

cntical

F(z)

notation

critical

and

=

point that

point problem

(dE/dt)~~, xr

=

#(z) stems

0; /J is is

from

trie a

real

inverse

dilferentiable. trie fact

of

positive that

Trie trie

point of trie RG flow. Then, trie function #(z), usually used to extract trie qualitative behavior as

fixed

so-called RG flow, is trie stability of fixed points and to deduce trie corresponding critical exponents. Let us of a fixed point but is for simplicity that trie critical point is net only on trie attractor assume indistinguishable from it, as can be done by a suitable change of variable. Then, if z 0 denotes linearized transformation, fixed point (#(0) AT +.. is trie corresponding 0) and #(x) a solution of (2) close to x 0 is given by equation (1) with a =Log/J/LogÀ. then trie critical interested in trie general To go beyond this local analysis m trie region, we are which

well

generates

as

trie

=

=

=

=

RENORMALIZATION

N°6

solutions then

trie

of

equation (2). To get them, let

general

period Log/J,

GROUP

F(x)

solution

related

is

us

to

ACOUSTIC

FOR

Fo(x)(= x")

that

assume

Fo(x)

in

terms

EMISSIONS

of

a

periodic

is

635

a

special solution,

function

p(x),

with

a

as:

F(x)

Fo(x)p(logfo(x))

=

(3)

Since logfo(x) aLogx, this leads to a periodic (in Logx) correction to trie dominating scaling (1). Equivalently, this log-periodicity cari be represented mathematically by a complex x"' cos(a"Logz) gives trie first term in critical since Re(z"'+~"") Fourier exponent, =

a

=

(3). This expression thus introduces universal oscillations series expansion decorating trie mathematical of such bas been identified asymptotic powerlaw (1). Trie existence corrections quite early [12] in RG solutions for trie statistical mechanics of critical phase transitions, but bas been rejected for translationally invariant systems, since a period (even in a logarithmic scale) implies the existence of one or several characteristic scale which is forbidden in these the of heterogeneous For quenched translational invariance does systems. rupture systems, trie hold due to the of static inhomogeneities [13] and trie fact that new damage not presence which are not averaged out by thermal fluctuations. Hence, such at specific positions occurs allowed and should for order log-periodic be looked embody trie physics corrections in to are of damage in trie non-critical region. of

trie few known It is interesting to mention where such a behavior bas been observed. cases Probably the first theoretical suggestion of the relevance to physics of log-penodic corrections forward by Novikov to describe trie influence of intermittency in turbuto scaling bas been put flow typically breaks up into lent flows [14]. Loosely speaking, if an unstable eddy in turbulent smaller eddies, three eddies, but then into 10 existence 20 two or not suspect trie or one can of a preferable scale factor, hence trie log-periodic oscillations. A clear-cut experimental verification of trie generation of log-penodic oscillations by discrete fractals bave scale invariant carried on on Sierpmsky networks of normal-metal links, in which trie normalbeen man-made superconductive oscillations function of trie transition temperature presents log-periodic as a applied magnetic field [15]. Complex tractai exponents bave also been argued to describe trie of trie mammalian lung [16]. To our knowledge, there is no experibranching architecture mental

verification

of this

fact

but

renormalization

dipolar Ising systems il?] and

group

glasses [18]

calculations

of

critical

of

behavior

complex spm results cntical These taken very cautiously by their authors could be the signature exponents. of a spontaneous generation of discrete scale invariance due to the interplay of the physics (interaction) and the quenched heterogeneity. Boolean delay equations involving two time lags with an irrational ratio, used recently to model the climate vanability, aise exhibit superdiffuse behavior oscillations [19]. Here again, we have an example of a discrete with log-periodic scale which is spontaneously generated, in trie present case by the threshold invariance dynamics feedback and nonlinear involved in trie Boolean delay equations. Finally, it bas been pointed random

Dut

that

vibration

and

properties

wave

by log-periodic corrections band edges [20]. However,

on

discrete

bave

fractal

shown

structures

trie

existence

should

be

of

characterized

leading singular behavior for trie density of states close to the presented below, results these periodic corrections, that our on are trie first obtained uncontrolled heterogeneity and, to our containing in structures ones an are knowledge, bave not been observed previously m trie context of rupture. Figure 3 shows a fit (continous fines) obtained using trie solution of equation (3) applied to (represented by trie symbols) obtained on two differents (systems 1 tanks AE data sets two intervals of varying width in order and 2). Here, as in Figure 2, we group trie AE bits in time trie solution of equation (3), p(z) bas been expanded in Fourier to reduce trie noise. To express dominant series and we bave kept only trie term:

dE/dt

=

to

the

Eo(pr

p)~~ il

+ C

cos

(#

+

2xLog(pr

p) /LogÀ)].

(4)

JOURNAL

636

PHYSIQUE

DE

I

N°6

iooooo

sysTEM

ioooo

~~

i

Î

j Q~~ f

©~

~~

iÎà

~j

~

1ÙÙ

~

~ x

X

~

10 SYSTEM

550

500

600

650 PRESSURE

Theoretical fit using equation (3) shown in Fig. 3. continous dilferents tanks (systems 1 by the symbols) obtained two on symbols corresponds to dilferent coarse-graining.

2

750

700

fines

AE

of two

2).

and

data

each

For

(represented

sets

dilferent

the

system,

resulting

mathematical expression bas a priori six unknown normalizing parameters: a Eo, the cntical exponent a, trie pressure pr at rupture, the amplitude C of trie oscillatory conditioning, we impose a correction, its penod LogÀ and phase #. To bave better I.à from our previous fit shown in Figure 2, since we expect it to be obtained "universal" within a

The

factor

=

materials.

Mass of

used

We with

trie

to

Log(pr

AE

This

trie

another,

as

seen

in

in

subtle

a

m

Figure

3

can

observe

behavior

for

Figure

borne

time

We

theory

Using this provide which we

a we

first ail

erasmg

results

prediction

for

successive

the

remark

to

bits

AE

presented

at

Figure

is

less

are

pure

a

be

can

mathematical

whose

also

obvious

power

exploited crucial

a

law to

structure

but

(C

are

one

further

to

to

bursts

m

of

correlated

are

universal

property. to

necessary

dramatically

improve

test

a

also

system

account

Eq. (4)).

0 in

=

is

mdeed

constant

Note

3.

from

different

very

penodicity

trie

measured in

burst

intermittent

that

by companng distributions system to system 2: trie universal; however, when they exist, on average they

trie data as coarse-grain above an data upper

conditions

trie

same

trie

pressure

deduce

these

between

by trie data, as amplitude can be

and

descent method steepest importance of the oscillatory

establish

prediction. validity, smce

the

its

that a good fit with five adjustable freedom to bas too much parameters argue Trie basic goal of any theory is m its predicting safe proof of any theory. power, follows. Using a given AE data set such as those presented previously, test as now

could

one

that

show

now

important

fit.

non-hnear

a

the

Figure 3 trie m correctly accounting for

It is

1.

from

determine

to

combines

out

amplitude are not according to trie loganthmic oscillations, oscillations For system 2, trie log-penodic distorsion from for trie still significative and

time

variables

[21], which

correlation be

to

seems

distribution

burst

One

shown

unknown

method

law

power

data

p) implies rate.

pressure

that

with five

us

method.

leading

the

of the

structure

leaves

Hessian

inverse

corrections

m

This

Levenberg-Marquardt

the

for

pf~~~~~~~~

but

global as

one

which

rupture. of trie

in

Figures

pressure

would We

five

2 and

3.

This

pmax. bave stopped

then

apply the

variables

of

this

We

then

mimics

an

pressure

non-linear

fit

fit.

Figure

pmax to

4

this

shows

data

set

by

performed under without reaching

expenment

trie

at

another

construct

truncated

for

system

file

and with

N°6

GROUP

RENORMALIZATION

FOR

ACOUSTIC

EMISSIONS

637

16

fi

ii

14

j/

12

.,/"'1

(

10

/: Ii

( Î O

~ i

~ j

4

à

~~.