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
~~.