Crackling Noise and Universality in Fracture Crackling Noise and Universality in Fracture SystemsSystems

Gianni Niccolini and Gianfranco DurinGianni Niccolini and Gianfranco Durin

Istituto Nazionale di Ricerca Metrologica; Torino, Italy

UnsolvedUnsolved Problems on Noise 2008

Lion, June 3rd

Crackling NoiseCrackling Noise = the system response to continuously changing external = the system response to continuously changing external conditions through discrete, impulsive events. conditions through discrete, impulsive events. Event sizes span a broad range of scales following regular behaviourEvent sizes span a broad range of scales following regular behaviour(i.e., power-law)(i.e., power-law)

Many physical systems crackle (fracture systems); Many physical systems crackle (fracture systems); two examples:two examples:

• The earth responds to the slow strains imposed by continental The earth responds to the slow strains imposed by continental drift with violent and intermittent earthquakes.drift with violent and intermittent earthquakes.

• A concrete specimen emits intermittent and sharp acoustic A concrete specimen emits intermittent and sharp acoustic emissions (AEs) as it is slowly loaded.emissions (AEs) as it is slowly loaded.

The Earth crackles The Earth crackles typical time history of earthquakestypical time history of earthquakes in a spatial region and a time periodin a spatial region and a time period

The Gutenberg-Richter (GR) law:The Gutenberg-Richter (GR) law:

N(N(M≥ Mth)M≥ Mth) 10 10 –bMth

Magnitude with logarithmMagnitude with logarithm of event size:of event size:

M M LogSLogS

→→ N(N(SS≥ ≥ SSthth) ) Sthth–b–b

The range of this law extends from lab scale unnoticeable tremblesThe range of this law extends from lab scale unnoticeable trembles (i.e., the acoustic emissions ) to catastrophic earthquakes: (i.e., the acoustic emissions ) to catastrophic earthquakes:

universalityuniversality ( the same behaviour on a wide range of scales) ( the same behaviour on a wide range of scales)

Crackling noise is defined only by the size response of the systemCrackling noise is defined only by the size response of the systemThere is any reference to its time evolution There is any reference to its time evolution

We analyze time properties of fracture systemsWe analyze time properties of fracture systemsin order to investigate universal features of fracture phenomenain order to investigate universal features of fracture phenomena

Earthquakes and AEs point events in space, time and magnitude

• Hypocentres coordinates

• Initiation time ti

•Size magnitude Mi

Drastic (but useful) simplification

Later on the spatial degrees of freedom will be disregarded

•Consider a fixed region and a fixed time window T (space-time window w)Consider a fixed region and a fixed time window T (space-time window w)

•Consider events with magnitude larger than a threshold Consider events with magnitude larger than a threshold MMthth

•Compute waiting times as the time between consecutive events:Compute waiting times as the time between consecutive events:

ττi i ≡ ≡ ttii –– ttii–1–1 ((Bak et al., 2002; Corral, 2004Bak et al., 2002; Corral, 2004))

Broad scale of timesBroad scale of times(from seconds to years)(from seconds to years)

GR rate (Number of earthquakes perGR rate (Number of earthquakes perUnit time in w) very poor temporalUnit time in w) very poor temporalDescription:Description:

R(MR(Mthth) = N(M) = N(Mthth) / T=< ) / T=< ττ (M (Mthth)>)>–1–1

We consider distributions of waiting times:We consider distributions of waiting times:

D(D(ττ;M;Mthth) = Prob [ ) = Prob [ τ τ ≤ waiting time< τ + d ≤ waiting time< τ + d τ τ ] / d ] / d ττ

Waiting-time probability densities for Italian seismicity Waiting-time probability densities for Italian seismicity

In the period T = 1984-2002 and different MIn the period T = 1984-2002 and different M thth::

Scale transformation of the axes: i.e., measuring waiting times for Scale transformation of the axes: i.e., measuring waiting times for each distribution in units of its meaneach distribution in units of its mean < < τ τ (M(Mthth))> = > = R R –1–1(M(Mthth))

τ →τ → R R (M(Mthth)) τ τ

DD((ττ; ; MMthth)) → D → D((ττ; ; MMthth)) / R / R((MMthth))

All the distributions collapse onto a All the distributions collapse onto a single curve F:single curve F:

D / R = F( R τD / R = F( R τ))

With GR law :With GR law :

DD((ττ; ; MMthth)) 1010bMbMth th = = f f ((1010––bMbMthth ττ ))

Inserting Inserting ((ττ; ; MMthth)) and (and (ττ''; ; MMthth'')) , , ττ'' = 10 = 10bb ττ' ' MMth th '' = = MMthth +1 +1

DD((ττ; ; MMthth)) = = 1010b b DD((1010b b ττ; ; MMth th +1+1))

Scale invariance in the timing of the earthquakesScale invariance in the timing of the earthquakes

E.g., we relate distribution of events with ME.g., we relate distribution of events with M≥ 3 separated by ≥ 3 separated by ττ= 100 = 100 hours with that of events hours with that of events with Mwith M≥ 4 separated by τ≥ 4 separated by τ '' = 10 = 10bb ττ'' =1000 =1000 hours (usually b hours (usually b 1 in seismicity) 1 in seismicity)

GR law only says that for each event with M GR law only says that for each event with M ≥ 4 there are 10 ≥ 4 there are 10 with M with M ≥ ≥ 3 3

The case of Italian seismicityThe case of Italian seismicity T =1984-2002T =1984-2002 Mth ranging from 2.5 to 5Mth ranging from 2.5 to 5 9096 events with M9096 events with M≥2.5≥2.5 For each threshold we perform a scale transformation of the axes For each threshold we perform a scale transformation of the axes ττ and D : and D :

ττ → 10 → 10 –aMth–aMth ττ , , D → 10 D → 10 cMth cMth DD

→→

f is well fitted by a generalised f is well fitted by a generalised gamma distributiongamma distribution::

f(f() ) –(1––(1–))exp[(exp[(––/x)n]/x)n] 10 10–aM–aMthth ττ

We have 5 fitting parameters (also the rescaling is part of the fitting We have 5 fitting parameters (also the rescaling is part of the fitting procedure):procedure):

aa 0.95, 0.95, cc 0.96, 0.96, 0.36, 0.36, xx 1.27, and 1.27, and nn 1.15 1.15

aa and and cc are compatible with with the GR are compatible with with the GR bb-value-value ( b ( b 0.99) 0.99)

confirming the validity of the scaling law confirming the validity of the scaling law

of the form:of the form:

DD((ττ; ; MMthth)) 1010bMth bMth = f (10= f (10––bMthbMth ττ ))

Accumulated number of earthquakes Accumulated number of earthquakes

as a function of timeas a function of time We consider only the periods of We consider only the periods of

Stationary seismicity (in red) Stationary seismicity (in red)

Assisi earthquake, (M = 5.8, Sept 26,

1997)

aa 0.95, 0.95, cc 0.96, 0.96, 0.47, 0.47,

xx 1.22, and 1.22, and nn 1.13 1.13

The power law is flatter (i.e., smaller The power law is flatter (i.e., smaller clustering degree)clustering degree)

Comparison with universality exhibited for periods of stationary activity in fracture systems (spanning a huge Comparison with universality exhibited for periods of stationary activity in fracture systems (spanning a huge range of scales):range of scales):

0.7, x 0.7, x 1.53, and n 1.53, and n 1 1 (Corral)(Corral)(( 0.8, 0.8, xx 1.4, and 1.4, and nn 1 1 (Davidsen et al.)(Davidsen et al.)

In particular In particular is lower; may it depend on difficult identification of real stationary periods (small magnitudes, i.e., is lower; may it depend on difficult identification of real stationary periods (small magnitudes, i.e., < 2.5, are disregarded being ill-defined)?< 2.5, are disregarded being ill-defined)?

Waiting-Time Distributions for AE in Concrete Waiting-Time Distributions for AE in Concrete FractureFracture 100 × 15 × 15 cm100 × 15 × 15 cm33 concrete beam loaded up to failure acording to concrete beam loaded up to failure acording to

three-point bending test geometrythree-point bending test geometry Test performed at constant displacement rate 10Test performed at constant displacement rate 10–3–3 mm/s mm/s 5 PZT 5 PZT transducers appliedtransducers applied

Identification of the fracture process zone:Identification of the fracture process zone:red line is the actual fracturered line is the actual fracture

black points represent the fracture as a series of localised AE black points represent the fracture as a series of localised AE sourcessources

We take only AE data associated with effective source locations (=event detection by two or more transducers(=event detection by two or more transducers ) )

accumulated number of AEs filtered in this way accumulated number of AEs filtered in this way grows roughly linearly in timegrows roughly linearly in time

where where VthVth = 200, 400 and 800 = 200, 400 and 800 VV are three threshold voltagesare three threshold voltages

and related Waiting time distributions

After rescaling : After rescaling : ττ→ 10 → 10 –aMth–aMth ττ ,, D → 10 D → 10 cMthcMth D D

Collapse onto a curve:Collapse onto a curve:

f(f() ) –(1––(1–) ) exp[(–exp[(–/x)n]/x)n]

aa 0.60, 0.60, cc 0.61, 0.61, 0.73, 0.73, xx 1.53, and 1.53, and nn 1.241.24

a and c close to GR b-valuea and c close to GR b-value (b=0.57) (b=0.57)confirming the scaling lawconfirming the scaling law

Exponents in good agreement with the proposed Exponents in good agreement with the proposed universal exponents characterising fracture universal exponents characterising fracture processes under stationary conditions:processes under stationary conditions:

0.7, x 0.7, x 1.53, and n 1.53, and n 1 1 (Corral) (Corral) 0.8, 0.8, xx 1.4, and 1.4, and nn 1 (Davidsen) 1 (Davidsen)

the GR law, LogN(Mth) – bMth with b 0.57 and and MMthth = Log = Log V Vthth

ConclusionsConclusions

We have confirmed the existence of scaling collapse for various waiting-time distributions,We have confirmed the existence of scaling collapse for various waiting-time distributions, implying the implying the existence of scale invariance for waiting times over a broad range of scalesexistence of scale invariance for waiting times over a broad range of scales

We have verified that the GR law is included in this general lawWe have verified that the GR law is included in this general law

We have not completely verified fulfilment of a universal scaling law for fracture processes under stationary We have not completely verified fulfilment of a universal scaling law for fracture processes under stationary conditions:conditions:

Thank you for your attentionThank you for your attention