Typical phases of pre-failure damage in granitic rocks under

differential compression

XINGLIN LEI1

Institute of Geology and Geoinformation/GSJ, National Institute of Advanced Industrial

Science and Technology (AIST), Higashi 1-1-1, AIST no. 7, Tsukuba 305-8567, Japan

(e-mail: xinglin-lei@aist.go.jp)1Also at State Key Laboratory of Earthquake Dynamics, Institute of Geology,

Chinese Earthquake Administration, Beijing, China

Abstract: The evolution of pre-failure damage in brittle rock samples subjected to differentialcompression has been investigated by means of acoustic emission (AE) records. The experimentalresults show that the damaging process is characterized by three typical phases of microcrackingactivity: primary, secondary, and nucleation. The primary phase reflects the initial activity ofpre-existing microcracks, and is characterized by an increase, with increasing stress, both inevent rate and b value. The secondary phase involves subcritical growth of the microcrackpopulation, revealed by an event rate increase and a dramatic decrease of the b value. The nuclea-tion phase corresponds to initiation and accelerated growth of the ultimate macroscopic fracturealong one or more incipient fracture planes. During the nucleation phase, the b value decreasesrapidly to the global minimum value around 0.5. The temporal variation of b in every phaseclearly correlates with grain size of the test sample, hence indicating that a comparativelylarger grain size results in a lower b value. In order to investigate the fracture mechanism ofeach phase, a damage model was tested by employing the constitutive laws of subcritical crackgrowth of crack populations with a fractal size distribution.

Abundant experimental evidence shows that macro-scopic shear failure in rock does not occur by thegrowth of a single crack in its own plane. Rather,shear failures are preceded by a complex evolutionof some pre-failure damage (Lockner et al. 1991;Lei et al. 2004). Therefore, studies focusing onboth fracture dynamics and pre-failure damage area subject of widespread interest, having relevancefor several applications, such as safe design ofdeep tunnelling (e.g. Diederichs et al. 2004), andnatural processes such as volcanism and seismology(e.g. Ponomarev et al. 1997; Diodati et al. 2000).The fracturing dynamics and damage evolution instressed materials has been extensively studied inthe laboratory by a number of methods, including(1) direct observation of samples by scanningelectron microscopy (e.g. Zhao 1998) or opticalmicroscopy (Cox & Scholz 1988) operated duringor after a fracture test and (2) monitoring ofthe space distribution of acoustic emission (AE)events caused by microcracking (e.g. Lockneret al. 1991; Lei et al. 1992). The fracturingdynamics and the pre-failure damage can beinferred from AE statistics, as the number of AEevents that is proportional to the number ofgrowing cracks, and the AE amplitudes are bothproportional to the length of crack growth incre-ments in the rock (e.g. Main et al. 1989, 1993;

Sun et al. 1991). Therefore, the AE technique isapplied to the analysis of the microcracking activityinside the sample volume, and it can be performedunder confining pressure, which is very importantfor the simulation of underground conditions. Inaddition, owing to the mechanical and statisticalsimilarities between AE events and earthquakes,AE in rocks is studied as a model of natural earth-quakes (Ponomarev et al. 1997). The disadvantageof the AE technique is that it is insensitive toductile deformation, which does not produceappreciable AE. Therefore, it is applicable only inbrittle regimes.

The recently developed high-speed, multichannelwaveform recording technology permitted monitor-ing with high precision of the AE events associatedwith spontaneously/unstably fracturing processeswithin stressed samples. It was applied to theanalysis of the fracture process of hornblendeschist (Lei et al. 2000b), of two granitic rocks ofextremely different density with a pre-existingmicrocrack (Lei et al. 2000a), and mudstone con-taining quartz veins resembling strong asperities(Lei et al. 2000c). The above studies show thatthe fracturing of several samples containing faultsof widely differing strength exhibit three long-term microcracking phases: primary, secondary,and nucleation (Lei et al. 2004), and that the

From: CELLO, G. & MALAMUD, B. D. (eds) 2006. Fractal Analysis for Natural Hazards.Geological Society, London, Special Publications, 261, 11–29.0305-8719/06/$15.00 # The Geological Society of London 2006.

density of pre-existing cracks is a major controllingfactor in the fracture and pre-failure damage evol-ution of rocks (Lei et al. 2000a). Besides crackdensity, the size distribution of pre-existing cracksis another important factor. Following Tapponier& Brace (1976) we assume that the size distributionof pre-existing cracks is consistent with the grainsize distribution in each test sample; in otherwords, we assume that samples of comparativelylarger grain size have a larger number of cracks oflarger sizes. In order to check this assumption andidentify the typical phases during pre-failuredamage, systematic experiments were carried outat different loading rates on granites of differentgrain size distributions (Lei et al. 2005).One major purpose of a study on pre-failure

damage is to consider the possibility of predictingthe time of the catastrophic failure event fromdamage evolution. The ever-increasing interest insuch predictions is associated with damagemodels based either on statistics of critical phenom-ena or on the experimental knowledge about subcri-tical crack growth. It is well known that subcriticalcrack growth under stress can be considered a resultof stress-aided corrosion at the crack tip. Owing tothis, some fracture laws were proposed based onthe experimental results of a single macroscopicextensional crack (e.g. Das & Scholz 1981;Meredith & Atkinson 1983). They were laterextended to crack populations whose size distri-bution is fractal (e.g. Main et al. 1993). Thesetypes of models predict not only the event releaserate, but also the seismic b value (the exponent ofthe power-law magnitude–frequency relation)(Lei et al. 2005); in addition, they may explainsome precursory anomalies, such as quiescence(Main&Meredith 1991) and b value decrease (Mainet al. 1989) associated with large earthquakes.The present paper aims on the one hand to

confirm the typical phases of pre-failure damage,and on the other hand, to improve availabledamage models. For this purpose, AE data obtained

from a series of experiments are here analysed andsummarized. Experimental data obtained under aconstant stress-rate loading (including some creeptests) on samples of several granitic lithologieswere used for summarizing the typical commonfeatures of pre-failure damage. Such samples wereselected in order to check the roles, for the fractur-ing behaviour, of grain size, of the pre-existingmicrocrack density, and of macroscopic structuressuch as healed joints. The first two sections focuson a review of the experimental and data-processingprocedures. Typical experimental results are sum-marized in the third section; then, an improvedmodel is evaluated. Finally, some plausible physicalmechanisms occurring during every damage phaseare discussed.

Experimental procedure

Rock samples

Table 1 lists the analysed lithologies and theirmechanical properties, including grain size,density of pre-existing crack, and the minimum/maximum P-velocities (Vmin, Vmax) along the axialdirection. Because most pre-existing cracks arelikely to be closed following stress increase, thedifference Vmax2 Vmin can be considered as aqualitative measurement of pre-existing microcrackdensity. Some samples contain one or more veins(healed joints), which evolved during compressioninto the ultimate fracture plane. The size andspatial distribution of pre-existing microcracks arecontrolled by the grain size distribution (Tapponier& Brace 1976); hence, on the scale range frommicrocrack size to sample dimension, rocks of largergrain size generally respond more heterogeneouslyduring the fracturing process. The Westerly granite(WG) is fine-grained and has the smallest grainsize. The Inada granite (IG), the Tsukuba granite(TG), the Mayet granite (MG) and the granitic

Table 1. Some mechanical properties of the test samples

Rock type Grain size (mm)range/major

Density ofpre-existing

cracks

Axial Vp (km s21)min/max

Westerly granite (WG) ,2/,1 High 4.8/5.6Oshima granite (OG) 1–5/�2 High 4.4/5.8Inada granite (IG) 1–10/�7 High 4.2/5.8Granitic porphyry (GP) 1–10/�6 Very low 5.8/6.0Tsukuba granite (TG) 1–30/�10 High 4.2/5.8Mayet granite (MG) 1–30/�15 Low 5.2/6.1S-C cataclasite (SC) 1–10/�5 Very high 4.2/5.6Nojima granite (NG) 1–10/�5 High 4.7/5.7

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porphyry (GP) are typical coarse-grained litholo-gies. The Oshima granite (OG) is classified intointermediate-grained rock, and has a mean grainsize greater than WG but smaller than IG. Consist-ent with the value of Vmax2 Vmin, the density ofpre-existing cracks is high in WG, OG, IG, andTG, whereas it is relatively low in MG, and GP isalmost crack-free (Lei et al. 2000a).The WG, OG, and IG are suited for investigating

the role of grain size on fracture behaviour,because they have similar mineralogical contentand density of pre-existing microcracks, contrastingwith significantly different grain size distribution.TheMG and TG samples were selected for checkingthe role of heterogeneity at large scales eitherbecause they contained some large grains orbecause they contained healed joints. A foliatedgranitic cataclasite (CG) and its host Nojimagranite (NG) were also used, in order to investigatethe fracture behaviour of such weak rock in or near afault zone. The SC and NG samples were collectedfrom drill cores on the Nojima active fault zone ata depth of �200 m. The Nojima fault, located insouthwest Japan, ruptured in the 1995 Kobe earth-quake (Lin 2001). The experimental results obtainedfrom the MG samples were preliminarily presentedby Jouniaux et al. (2001). In the present study, thehypocentre of every AE event was redeterminedafter checking every arrival time. Such hard workgreatly improved the precision of the hypocentres.The test samples were shaped into cylinders of

125 mm or 100 mm length and 50 mm diameter.All samples were dried under normal room con-ditions for more than one month and then theywere compressed at a constant stress rate or at aconstant stress (creep test) at a stress level of�95% the nominal fracture strength at room temp-erature (air-conditioned at �258C). During thedeformation, the confining pressure was maintainedat a constant level of 40 or 60 MPa within the brittleregime. Under these conditions, the rock samplesnormally fractured along a compression shearfault oriented at �308 with respect to the axis ofmaximum compressive stress.

Experimental apparatus

Figure 1 shows the functional block diagram of theexperimental set-up along with important details ofthe loading apparatus, AE recorder, and other dataacquisition systems. The assembled pressure vesselwas placed in a loading frame and high-pressure fluidlines attached for external confining pressure.Hydraulic oil was used for the confining pressuremedium. As many as 32 PZTs (piezoelectric trans-ducers, compressional mode, 2 MHz resonant fre-quency, 5 mm in diameter) were mounted on thesample surface for detecting the AE signals

produced by microcracking events. The signal ispre-amplified by 40 dB before feeding into thehigh-speed waveform recording system, which hasa maximum sampling rate of 40 ns and a dynamicrange of 12 bits. Two peak detectors were used tocapture the values of the maximum amplitudes,from two artificially selected sensors, after 20 or40 dB pre-amplifiers. An automatic switching sub-system was designed for sequentially connectingevery selected sensor, in total as many as 18, tothe pulse generator for velocity measurement.Such a high-speed AE monitoring system canrecord the maximum amplitude and waveform ofthe AE signals with no major loss of events, evenfor AE event rates of the order of several thousandevents per second such as are normally observedbefore a catastrophic failure.

The AE hypocentres were determined by using thearrival times of the P wave and the measured P-velocities during every test. Location errors are gen-erally less than 1–2 mm for fine-grained rocks andslightly greater for coarse-grained rocks. Duringevery such test, the trigger threshold for waveformrecording is about 10 times larger than the thresholdfor peak detection (i.e. for the detection of themaximum amplitude of the AE signal). In addition,at least four precise P arrival times are required forhypocentre determination. As a result, the hypocen-tre data are a subset of magnitude data. Besides AEmeasurement, eight 16-channel, cross-type straingauges were mounted on the surface of the testsamples for measuring the local strains along theaxial and circumferential directions. Stress, strain,and confining pressure were digitized at a resolutionof 16 bits and sampling interval of millisecond order.The local volumetric strain (en) was calculated fromthe axial (ea) and circumferential (ec) strains accord-ing to the equation en ¼ eaþ 2ec. The mean strainsof the test sample were estimated by averaging theselocal measurements.

Data processing

The statistical parameters which better characterizethe pre-failure damaging behaviours are

(1) energy release rate and cumulative energyrelease including event rate and cumulativenumber;

(2) b value in the frequency–magnitude distri-bution; and

(3) fractal dimension of the AE hypocentre.

In the following sections it will be briefly illustratedhow the above parameters were estimated.

Normalized time-to-failure

Rock fracturing is generally characterized as a fasttransient phenomenon showing a high degree of

PRE-FAILURE DAMAGE IN GRANITIC ROCKS 13

nonlinearity before the final failure. Plots of AEdata v. time-to-failure in logarithmic scale aretherefore helpful for presenting the details of thedamaging process. The normalized time-to-failureis defined by:

~t ¼tf � t

tf � t0ð1Þ

where tf is the failure time, t0 is an artificial startingtime that can be set, for example, as the onset time ofAE activity or any other chosen time. In the presentstudy, for modelling convenience, t0 was set as thetime of the maximum b value. These time instantscorrespond to the phase change from an initialrupture to subcritical crack growth of the crack popu-lation. This is discussed in detail in later sections.

AE magnitude and energy release rate

Following the definition of the body-wave earth-quake magnitude, the magnitude (M) of an AEevent is determined according to the log of the

maximum amplitude (Vmax) of the AE signal, as itrepresents the vibration velocity of the elastic wave(Lei et al. 2003), M / logVmax ¼ AdB/20; here AdB

is the maximum amplitude in dB. Such definitionresults in magnitudes consistent with earthquakemagnitudes. However, because the calibration par-ameter of the PZT sensor is unknown, themagnitudesthat were obtained ought to be considered only asrelative values. The amplitude threshold for the AEsignals was set at 45 dB. Hence the AE magnitudewas actually calculated using M ¼ (AdB2 45)/20.We note that the dynamic range of the system is100 dB. As a result, every event with maximumamplitude equal to, or larger than, 100 dB would berecorded as M ¼ 2.75. However, such saturationcould be easily corrected by using the continuationtime of the AE signal (Lei 2003). It should be notedthat the maximum amplitude was digitized with 11digits. The distinguishable magnitude interval istherefore 0.25.

The AE event rate N ¼ dn/dt gives the simplestmeasurement of the frequency of microcrackingactivity. The event rate can be calculated for

Fig. 1. Block diagram of the experimental apparatus used for rock fracture tests. Up to 32 PZT sensors weremounted onthe surface of the test sample (a rock cylinder of 125 mm length and 50 mm diameter). The signal is pre-amplified by40 dB before feeding into the high-speedwaveform recording system,which has amaximumsampling rate of 40 ns and adynamic range of 12 bits. Two peak detectors were used to capture the values of the maximum amplitudes, from twoartificially selected sensors, after 20 or 40 dB pre-amplifiers. An automatic switching subsystem was designed forsequentially connecting every selected sensor, in total 18, to the pulse generator for velocity measurement. Stress, strain,and confining pressure were digitized at a resolution of 16 bits and sampling interval of millisecond order.

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either a fixed number of events (dn) or a fixed timeinterval (dt). The two methods give similar resultsbut may include different details. The generalizedenergy release correlates with magnitude:

Ei / 10CMi (2)

where C is a constant. The most important cases areC ¼ 0.75 and 1.5, which correspond to the Benioffstrain and the classic energy release, respectively.The energy release rate can be estimated bysumming Eq. (2) within a given unit time interval:

E ¼Xi

10CMi : (3)

The cumulative energy release is simply defined by

XE ¼

ðE dt: (4)

The cumulative event numberP

N can be con-sidered as a special case of Eq. (4) with C ¼ 0.

The b value and recurrence time

The Gutenberg–Richter relationship for magnitude–frequency distributions holds both for earthquakesand for AE events in rock samples (Scholz 1968;Liakopovlou-Morris et al. 1994). The cumulativenumber (Nm) of events with magnitude M orgreater is a function of the magnitude (Gutenberg& Richter 1944):

log10 Nm ¼ a� bM: (5)

In Eq. (5), a and b are constant; an estimate of the bvalue can be obtained by using either the least-squares method or the maximum likelihoodmethod (Aki 1965; Utsu 1965; Bender 1983). Forgiven n events, the latter method gives

b ¼n log10 ePn

i¼1 (Mi �Mc þ DM=2)

¼log10 e

M �Mc þ DM=2(6)

where Mc is the cutoff magnitude and DM is thedifference between successive magnitude units,which is 0.25 for the AE data. The approximatestandard error of the b value estimate is s ¼

b=ffiffiffin

p(Aki 1965), and the confidence limit of the

estimation is given by

s(b) ¼ 2:30b2, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

i¼1

(Mi �M)2=(n(n� 1))

s(7)

where M denotes the mean magnitude estimation(Shi & Bolt 1982). In the present paper, the bvalues were sequentially calculated for consecutivegroups of 500 to 8000 events (n ¼ 500–8000) withan increment of n/4 events. A larger number ofevents leads to comparatively smoother curves,and it can be used for investigating the long-termtrend. In contrast, a smaller number of eventsleads to a larger scatter, which reflects the complex-ity of the damage evolution. In the present paper,bLS and bMLH (or b) are used to denote the least-square b value and the maximum likelihood bvalue, respectively. In most experimental studies,the b value was calculated from the signals recordedby one AE sensor. Therefore, the focal distancefrom the source should be corrected for the attenu-ation. Nevertheless, the theoretical study by Weiss(1997) has shown that attenuation has no significanteffect on the b value. In the present study, thesignals from two sensors, mounted at differentpositions on the sample surface, were fed to peakdetectors for recording the maximum amplitude. Itwas thus found that two sets of amplitude dataresult in consistent b values, although there arecertain differences in the short-term fluctuations(Fig. 2).

On the basis of the a and b values, the prob-abilistic recurrence time Tr for an event withmagnitude equal to, or greater than, a chosen M isestimated by

Tr ¼ DT=10a�bM: (8)

Here DT denotes the time length for which the a andb values are derived. In the present paper Tr was

Fig. 2. Maximum likelihood b values calculated fromAE signals obtained from two sensors (#1 – solid blackline, #2 – solid grey line) are given as a function of thelog of the normalized time-to-failure. The b values weresequentially calculated for consecutive data sets of 1000events with an increment of 250 events. Also shown(grey rectangle in upper right) are the positions of AEsensors #1 and #2 with respect to the WG sample.

PRE-FAILURE DAMAGE IN GRANITIC ROCKS 15

calculated for the maximum distinguishable magni-tude of 2.75.

Fractal dimension of hypocentre

distribution

A multifractal analysis was applied to the hypo-centre distribution of AE events in order toexamine quantitatively the spatial clustering ofpre-failure damage. The multifractal concept is anatural extension of the (mono) fractal concept tothe case of heterogeneous fractals. It was observedthat the AE hypocentres, particularly in coarse-grained rocks, show heterogeneous fractal charac-teristics (Lei et al. 1993). In the present study,the generalized correlation-integral (Kurths &Herzel 1987), defined by the following equation,was used:

Cq(r) ¼1

n

Xnj¼1

nj(R � r)

n� 1

� �q�1" #1=(q�1)

(q ¼ �1, . . . , �2, �1, 0, 1, 2, . . . ,1)

(9)

where nj(R , r) is the number of hypocentre pairsseparated by a distance equal to, or less than, r, qis an integer, and n is the total number of AEevents analysed. If the hypocentre distributionexhibits a power law for any q, Cq(r) /Dq, thehypocentre population can be considered as a mul-tifractal, and Dq defines the fractal dimension thatcan be determined by the least-squares fit on alog–log plot. In the case of a homogeneousfractal, Dq does not vary with q. However, in thecase of a heterogeneous fractal, Dq decreaseswith increasing q and it is called the spectrum ofthe fractal dimension. It can be easily provedthat D0, D1, and D2 coincide with the informationdimension, with the capacity dimension, and withthe correlation dimension, respectively. The differ-ence between D2 and D1 is an index representingthe degree of heterogeneity of the fractal set.Normally, D20 is an efficiently precise estimationof D1.Fractal analysis was applied routinely either for

every group of 100–200 hypocentres, or for everygroup of 2000–8000 events, in order to inspectthe temporal evolution of the fractal structureand the correlation between fractal dimensionand b value. For the later case, that is, with afixed number of events, as mentioned in the pre-vious section, the hypocentre data are a subset ofthe magnitude data. The average actual numberof hypocentres, which were precisely determinedand were thus available for fractal analysis inevery group, is normally in the order of severalhundreds. The minimum number of hypocentres

required for applying fractal analysis was artifi-cially set as 100 for determining reliable D values.The fractal dimension of every group with anumber of hypocentres less than 100 was notdetermined.

Experimental results

Systematic experiments on granitic rock sampleswere performed at different loading rates, in orderto investigate the statistical behaviour of pre-failure damage and to set an appropriate damagemodel. A large number of AE events, from tens tohundreds of thousand, were observed during everytest. As an example, Figure 3 shows the AErecord of a typical test of IG sample subjected toa constant loading rate at 27.5 MPa min21. Asmay be seen, the AE data, particularly the energyrelease rate and the b value, exhibit some typicalphases of pre-failure damage, (see also Lei et al.2003, 2004) and microcracking activity initiatedat a stress level of �35% of the sample’s fracturestrength. The event rate generally increases, withincreasing stress, up to the failure point. FollowingLei et al. (2003, 2004), in the following sections, athree-phase model including a primary, a second-ary, and a nucleation phase, will be used fordescribing such behaviour.

The AE data obtained from the IG, OG, andWG samples are partly summarized in Figure 4,where it is clearly shown that the three distinctphases are a universal feature during rock fracturein the brittle domain. The common features ofeach phase are summarized in the followingsections.

Primary phase

During this phase, microcracking activity wasinitiated at stress levels of 30–60% of fracturestrength of every sample. Such stress valuesstrongly depend on the density and size distributionof pre-existing cracks, and such distributions arecontrolled by the grain-governed heterogeneitywithin the sample. In general, a lithology withcomparatively larger mean grain sizes or withhigher pre-existing microcrack density exhibits alower initiation stress and a higher AE activity.During the primary phase, the event rate is lowand it increases slightly with the increase ofstress or time. The b value increases, with increas-ing stress, from an initial value of 0.5–1.2 to 1.0–1.4. The initial and final values also depend on thedensity and size distribution of the pre-existingmicrocracks. In the fine-grained WG, the primaryphase is not clearly distinguishable, involvingonly a small number of events, with a high

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initial b value and a small increase (from �1.2to �1.4, Fig. 4e, f). In contrast, the primaryphase appears obvious in the coarse-grained IG,being characterized by a large number of AEevents with a low initial b value and large increase(from �0.5 to �1.2, Fig. 4a, b). The intermediate-grained OG of mean grain size between WG andIG exhibits a somewhat intermediate behaviourbetween WG and IG, with an increasing b valuefrom �0.7 to �1.3 (Fig. 4c, d). Such resultsclearly show that the temporal variation of theb value during the primary phase stronglydepends on the grain size distribution of everytest sample.

Secondary phase

The typical feature of the secondary phase is amicrocracking activity in which the event rateincreases, with increasing stress or time, and the bvalue decreases from its maximum at the end ofthe primary phase. Average values for themaximum b value in WG, OG, and IG are 1.4,1.3, and 1.2, respectively (Fig. 4). Such maximumresults clearly correlated with the major grain sizeof the test sample; a comparatively larger grainsize results in a lower b value. In the next section,it will be shown that the energy release duringthe secondary phase appears to be consistent with

Fig. 3. An example of AE data obtained from a typical experiment on an IG sample. (a) Differential stress,average axial/circumferential/volumetric strains v. time. (b) AE energy release rate and event rate calculatedconsecutively for every 10 s. (c) b Value, recurrence time (Tr) value and fractal dimension (D2) v. time. P, S, and Nin (b) denote the primary, secondary and nucleation phases, respectively. The time interval and standard error for all band D2 data are shown by the horizontal and vertical bars, respectively.

PRE-FAILURE DAMAGE IN GRANITIC ROCKS 17

a damage model based on the constitutive laws forstress corrosion applied to subcritical crack growthof crack populations.

Nucleation phase

The nucleation phase includes the nucleation andaccelerated growth of failure-related featuresassociated with the macroscopic fracture of thetest sample. It is characterized by a rapidly increas-ing event rate, and by a rapidly decreasing b valuetending to a global minimum of about 0.5.The AE data were collected using a high-speed

acoustic emission waveform recording system andapplied to granitic rock samples subjected to con-fined compression tests; these results are summar-ized in Figure 5. As may be observed, the pre-failure damage evolution in these samples essen-tially consists of a typical three-phase pattern. Inthe fault zone, high AE activity was observedfrom the start of the test. In contrast, the AE activitybegan at �95% of the strength of those samplescharacterized by a very low density of the

pre-existing microcrack. The event rate in the GPsample of Figure 5f was very low before the finalfracture, had a very short secondary phase, and anucleation phase containing only a few foreshocks.However, more than 1000 aftershocks wererecorded after the main fracture. Generally, thewaveform recorder would be saturated before thefinal dynamic faulting. The b value in the MG,NG, and TG samples of Figure 5a–c, e increasedduring the primary phase from 1.0–1.2 to �1.4.

In all cases, the transition from the primary phaseto the secondary phase corresponds to themaximum b value and it can be easily determined.However, the change from the secondary phase tothe nucleation phase appears to depend on thesample’s grain size and on the pre-existing micro-scopic and macroscopic structures. When dealingwith intact and homogeneous samples, the nuclea-tion phase appears to be a natural extension of thesecondary phase. Conversely, in a sample contain-ing a macroscopic structure (Fig. 5a–c) or a verylow density of pre-existing cracks (Fig. 5f), thetransition occurs abruptly and coincides with the

Fig. 4. The b value, fractal dimension D2 and event rate N, obtained from tests using the IG, OG, and WG samples under constant stress-rate loading at either 2 MPa min–1 or 27.5 MPa min–1. ‘PjS’ and ‘SjN’ denote thetransitions from the primary to the secondary and from the secondary to the nucleation, respectively. Themaximum-likelihood b values were sequentially calculated for consecutive groups of 1000 events with arunning step of 250 events. The fractal dimensions (D2) were calculated for groups of 8000 events, in which thenumber of precisely determined hypocentres is more than 100.

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minimum fractal dimension. The damaged fault zonerock, in sample SC, shows an abrupt onset of thenucleation phase (Fig. 5d,e). In such cases, a diffu-sion process of AE hypocentres, with an increasingfractal dimension, was observed during the nuclea-tion phase (Fig. 5b–d). The time duration of thenucleation phase varies in a wide range, from afew seconds to �100 s, and it depends first onsample lithology and pre-existing structures, andsecondly on loading conditions. In general, a com-paratively lower loading rate results in a longernucleation phase, and a homogeneous sample has acomparatively shorter nucleation phase. In samplescontaining large-scale heterogeneous structures,such as joints or large (cm scale) grains, someshort-term, large-amplitude fluctuations can beobserved in the b value and in the AE rate.

Damage model

Subcritical crack growth may occur under stress asa result of stress-aided corrosion at the crack tip.

The quasi-static rupture velocity V has been foundexperimentally to be related to the stress intensityfactor K by Charles’s power law (Charles 1958):

V ¼ dc=dt ¼ V0(K=K0)l (10)

where V0 is an initial/detectable velocity and K0 isthe corresponding stress intensity factor, c is thecrack length (or half-crack length), and l is referredto as the stress corrosion index and has a typicalvalue of 20–60 for polycrystalline rocks undertensile condition (Das & Scholz 1981). Thispower law is fundamentally based on experimentalobservations of mode-I crack growth of a singletensile macrocrack. It can be straightforwardlymodified for a crack population having a fractalsize distribution by introducing an effective meanvalue for each parameter (Main et al. 1993).However, the physical bases for such an extensionhave not been discussed in detail. Hereafter, c,K and V indicate the mean values of crack

Fig. 5. The b value, fractal dimension D2 and event rate N, obtained from rock samples under constant stress-rateloading. ‘PjS’ and ‘SjN’ denote the transitions from the primary to the secondary and from the secondary to thenucleation phases, respectively. The maximum-likelihood b values were sequentially calculated for consecutive groupsof 500 events with a running step of 125 events. The fractal dimensions (D2) were calculated for groups of 200hypocentres.

PRE-FAILURE DAMAGE IN GRANITIC ROCKS 19

length, stress intensity factor, and growth velocity,respectively.In a generalized form, the stress intensity factor

can be expressed as KI,II,III ¼ Ysffiffiffic

p, where Y is a

dimensionless geometric constant and s is theremote applied stress. Under loading conditions ofconstant stress rate w, s ¼ s0þ wt, s0 is thestress at t ¼ 0, and the crack growth velocity canbe expressed as

dc

dt¼ V0

c

c0

� �l=2s

s0

� �l

¼ V0

c

c0

� �l=2

(1þ wt)l, w ¼ v=s0 (11)

where c0 is the mean crack length at t ¼ 0. Thesolution of the above differential equation underthe initial condition c ¼ c0 at t ¼ 0, is

c¼ c0 1�(l�2)v0((1þwt)lþ1�1)

2w(lþ1)c0

� �2=(2�l)

, w= 0

c¼ c0 1�(l�2)v0

2c0t

� �2=(2�l)

, w¼ 0:

(12)

The mean crack length c increases nonlinearly withtime. A failure time can be defined as tf so that capproaches infinity:

tf (w)¼1

w

2w(lþ1)c0

(l�2)v0þ1

� �1=(lþ1)

�1

!,w=0

tf¼2c0

(l�2)v0, w¼0:

(13)

Under constant stress (w ¼ 0), Eq. (13) reduces tothe same equation given by Das & Scholz (1981).Equation (12) can be simplified as

c�c0 1�t=tf (w)ð Þ2=2�l, w=0

c¼c0 1�t=tfð Þ2=2�l, w¼0:

(14)

Under constant stress-rate loading Eq. (14) is a goodapproximation of the exact solution (12). Equation(14) can be further simplified by using the normal-ized time-to-failure defined in Eq. (1):

c¼c0~t2=2�l

, ~t.0: (15)

This is again a power law. Meredith & Atkinson(1983) showed experimentally that the event rateN of the microcrack activity during the subcritical

crack growth of a single tensile crack also exhi-bits a nonlinear dependence on the stressintensity, showing a form similar to Charles’spower law:

N¼N0(K=K0)l0 : (16)

Within this context, l0 is referred to as the ‘effec-tive’ stress corrosion index and is found to beequal to n within a few percent in brittle rocks(Main & Meredith 1991). From Eq. (16), theevent rate can be represented by

N¼ 1�t=tfð Þ2l0=2�l(1þwt)l

0

: (17)

This model is roughly representative of the AEdata in granitic samples under differential com-pression (Lei et al. 2005). However, the fit ofthe results typically shows l0 ¼ 12 and l ¼ 25–45. We note that l0 is only a half to a third of l.This is most likely because l is excessively highfor compression tests due to the occurrence ofmixed fracture models. Generally, a shear crackshould be more stable than a tensile crack. Formodelling purposes, it is better to refer toenergy release rather than event rate.

Under differential compression, the observabledata are the magnitude and the hypocentre of theAE events. Thus, a model ought to be derived forthe release rate of energy, instead of the incrementof crack length. It was found both observationallyand theoretically (Kanamori & Anderson 1975)that there is a power-law scaling relation betweenthe release of elastic energy and the rupturearea (S ):

E ¼ 101:5Mi / S3=2; (18)

hence, the increase rates of fracture area can beestimated from AE data by

_SAE(t)=_SAE(0) ¼X

101:5Mi� �2=3

¼X

101:0Mi (19)

where the sums are over all events in an individualperiod of unit time at the given time t. It should bementioned that Eq. (19) provides an underestimate,because of increasing energy loss with the increaseof the mean crack length from the AE sourcethrough the receiver. The major factors associatedwith such loss include (1) frictional heat, (2) scatter-ing of elastic waves through a damaged volume,and (3) progressive replacement of tensile crackingby shear cracking. The total effect of all thesefactors is assumed to be a function of the mean

X. LEI20

crack length c in terms of a power law:

_SAE ¼ _Sc�m ¼ 2_cc1�m, m � 0: (20)

Therefore, the basic equation for modelling the AEdata can be expressed by

E1(t)=E1(0) ¼ (1� t=tf )lþ2�2m=2�l(1þ wt)l: (21)

Here, El ¼P

101:0Mi indicates the energy releaserate evaluated by the measured AE magnitude.The fit of Eq. (21) to AE data obtained by typical

experiments is shown in Figure 6. As a general rule,except for the primary phase (~t . 1), AE energyrelease data can be well represented by the afore-mentionedmodel. The best parameters, whichmatchthe AE data, are m ¼ 2.4–3.6 and l ¼ 8–16,depending on the grain size and lithological type.

Such values of l are significantly lower than thetypical value of 25–45 obtained by using Eq. (16)for the event rate, and appear more reasonable.The fine-grained samples have smaller m andlarger l values (m ¼ 3.0, l ¼ 16) than coarse-grained samples (m ¼ 3.6, l ¼ 12), whereaswesterly granite displays smaller m and l(m ¼ 2.8, l ¼ 8–10). It is obvious that both m andl are independent of the loading rates.

The first term in Eq. (21), which governs the non-linear acceleration of damage, can be ignored whent � tf. However, it dominates when t approaches tf.It is clear that if a rock sample contains some pre-existing macroscopic cracks or joints, which canpotentially serve as the ultimate fracture plane, theoverall deformation behaviour of the materialcould be governed by such cracks, following theirinitiation, as a result of high nonlinearity in themodel. In such cases, the damage model is basically

Fig. 6. Key AE data of three granites (IG, OG, WG) having different grain size distribution, under two loading rates of2.0 and 27.5 MPa min–1. The time axis corresponds to the normalized time-to-failure on a log scale, for presentingthe acceleration of the evolution of the pre-failure damage. The dashed lines indicate the fit of the results of thesubcritical crack growth model with the energy release rate (E1). ‘PjS’ and ‘SjN’ denote the transitions from theprimary to the secondary and from the secondary to the nucleation, respectively.

PRE-FAILURE DAMAGE IN GRANITIC ROCKS 21

applicable. However, the observed data cannot berepresented by a single fit. It should be mentionedthat for a small value of loading rate w includingcreep test (w ¼ 0), neither m nor l can be determinedbecause, for any given constant F, relation (21) givessimilar results for every set of (m, l) satisfyingm ¼ [(1þ F)lþ 2(12 F)]/2. However, m and lcan be determined from AE data obtained fromtests with a loading rate efficiently larger than zero.In addition, the relative stress intensity factor

can be estimated from the AE release rate as

K

K0

¼ E1(t)=E1(0)ð Þ1=lþ2(1�m): (22)

Meredith & Atkinson (1983) showed that theseismic b value is in fact negatively correlatedwith the stress intensity K normalized by the frac-ture toughness Kc. Because K/K0 can be estimatedfrom Eq. (22), the correlation of b and K can beexpressed alternatively as

b ¼ b0 � a K=K0ð Þ: (23)

Figure 7 shows the maximum-likelihood b valuesagainst the normalized stress intensity factor

estimated by Eq. (22) in fine-grained and incoarse-grained samples both under fast(27.5 MPa min–1) and low (2 MPa min–1) stressrates. The experimental data basically fit the linearcorrelation defined by Eq. (23); however, datawith b values less than 0.8 are systematically notrepresented by a linear trend. Such scattered datacoincide with the last stage, immediately precedingthe catastrophic failure. During this stage, largerand larger events frequently occur. A greatnumber of events are embedded within the tail ofthe previous event. Moreover, events of amplitudelarger than 100 dB were clipped and recordedas an event of a relative magnitude of 2.75. Asa result, both the b value and K/K0 are likelyto be underestimated during the last stage.

Discussion

Fracture mechanism of the primary

damage phase

It is clearly recognized that every damage phase isgoverned by somewhat different fracture mechan-isms and that the statistical behaviour ofevery typical phase strongly depends on thedensity and size distribution of pre-existing micro-cracks, which are controlled by grain sizedistribution within the test sample. The damagemodel and its fit to experimental data providedsome strong insights into the pre-failure damageprocess in the brittle domain.

Based on experimental results, it appears reason-able to assume that the major mechanism of micro-cracking in the primary phase is associated withsome kind of initial rupture of pre-existing flaws.The primary phase shows behaviours that cannotbe represented by the damage model based onsubcritical crack growth of crack populations.This supports the hypothesis that microcrackingduring the primary phase is related to the initialrupture of the pre-existing flaws, rather than tosubcritical crack growth. The pre-existing micro-cracks are probably healed prior to loading, andconsequently the initial rupture proceeds moreeasily by separating the crack walls rather than bycrack growth. Because a number of mechanismscan originate local tensile stress, and because theextension strength of every crack is much lowerthan shear strength, it follows that microcrackopening is expected to be the predominant crackingmode during the primary phase. It can be wellunderstood that the primary phase shows anincreasing b value with increasing stress, becauselarger pre-existing microcracks have a higherprobability of rupturing at comparatively lowerstress.

Fig. 7. For two granitic samples WG and IG, themaximum likelihood b values during the secondary andnucleation phases plotted v. K/K0 estimated from theAE energy release rate. Note that K0 at time t0corresponds to the start of the secondary phase, or inother words, the start of the subcritical crack growth.

X. LEI22

As shown in Figures 4 and 5, the event numberduring the primary phase decreases significantly.In addition, the fractal dimension of the hypocentresshowed no significant stress dependence. Comparedto smaller grain sizes, the comparatively largergrain size samples show a lower initial b, a largermaximum b, and a lower D2. As the distributionof pre-existing microcracks is controlled by grainsize distribution, it is observed that (1) the largermean grain sizes correspond to a larger number oflarge cracks and (2) the event rates, the b values,and the fractal dimensions all appear to be consist-ent with the hypothesis that the primary phase isassociated with the initial activity of pre-existingcracks. Therefore, during the primary phase, the bvalue and fractal dimension reflect the size dis-tribution and spatial distribution of pre-existingmicrocracks, respectively. Although both par-ameters appear to depend on grain size distributionof the test sample, no intrinsic correlation wasobserved between them.

Fracture mechanism of the secondary

and nucleation damage phases

The damage model illustrated above representsappropriately the AE energy release rate duringthe secondary and nucleation phases. This impliesthat the fracture laws governing the mode-I micro-cracking during a single extensional growth of amacrocrack, also control the mixed mode of micro-cracking activity under differential compression.Although the brittle behaviour of rocks undercompression is complex, it is clear that on a grainor microcrack scale, a variety of mechanisms giverise to localized tensile stress. The formation andgrowth of shear faults in a very fine-grained rockunder triaxial compression is therefore guided bythe development of a process zone encompassingtensile microcracks at the fault tip (Lei et al.2000b). Under differential compression, many AEevents also show a nonquadrantal distribution ofthe P first motions, hence suggesting that theymay record the growth of typical wing cracks, thatis, a shear cracks with tensile tails (Lei et al.2000b). Other possible mechanisms causing localtensile stress are:

(1) contacts between grains;(2) bending of elongated grains; and(3) indentation of sharp-cornered grains into

neighbouring ones.

As a result, independently of the remote appliedstress, mode-I tensile cracking is likely to be themajor mechanism for creating a new crack surfaceand for contributing to subcritical crack growth.Furthermore this common micro-mechanismprovides a possible physical base for justifying

why fracture laws derived from extension tests ofsingle macroscopic cracks can also be used forrepresenting the statistical behaviour of mixed-mode crack populations. The fact that the proposedmodel represents very well the AE energy releaserate during the secondary and nucleation phasesalso supports the assumption that the damagingprocess involves mainly the subcritical growth ofpre-existing microcracks. Following increase ofthe mean crack length, the energy release rateincreases nonlinearly, while the b value decreasesalmost linearly with the normalized mean stressintensity factor, which was estimated by the AErelease data.

The nucleation phase is characterized by a pro-gressively increasing release rate and by a rapidlydecreasing b value, which ultimately reaches aglobal minimum of 0.5. This is the most interestingand important phase during the catastrophic fractureof rock samples that contain heterogeneous faults.The term ‘nucleation’ is used here rather than‘tertiary phase’ because it better describes thenucleation process of the ultimate, unstable frac-ture. Once the final fault is initiated at one or atseveral key locations, which could be the edges ofasperities or of previously fractured areas, the fault-ing process will be governed by the acceleratinggrowth of a few very large cracks, or by the pro-gressive fracture of the major asperities on thefault surface.

The proposed damage model, when applied to thenucleation phase, emphasizes that the data fit is notsatisfactory, due to the governing behaviour of afew large cracks. Following the increase of themean crack length of the crack population, inter-action between cracks becomes in fact more andmore important. This interaction results in fluctu-ations of both the energy release and b value;consequently, macroscopic heterogeneity, such aspre-existing joints or asperities on the final fracturesurface, have a strong influence on the damageevolution of the nucleation phase. As alreadymentioned, the duration time of the nucleationphase varies in a wide range, from a few secondsto �100 s, depending on the density of the pre-existing microcracks as well as the loading rate ofstress. For any given lithology containing a faultwith unbroken asperities, a creep test at constantstress would show a comparatively longer nuclea-tion phase than that recorded at constant stress-rate load (Lei 2003).

Summarizing the main findings of the nucleationphase, it may therefore be emphasized that (1) thisphase records the quasi-static nucleation of thefinal fault; (2) the damaging process is associatedwith the progressive fracture of several unbrokenasperities on the fault surface; and (3) the AEevents caused by the fracture of individual

PRE-FAILURE DAMAGE IN GRANITIC ROCKS 23

asperities exhibit similar characteristics to those ofnatural earthquakes, including foreshock, main-shock, and aftershock events. The foreshocks,which initiated at the edge of the asperity, occurwith an event rate that increases according to apower law of the temporal distance to the main-shock, and with a decreasing b value (from �1.1to �0.5). One or a few mainshocks are theninitiated at the edge of the asperity or at the frontof the foreshocks. The aftershock period is charac-terized by a remarkable increase and subsequentgradual decrease in the b value, and by a decreasingevent rate, thus obeying the modified Omori law,which has been well established for earthquakes.The fracture of neighbouring asperities is initiatedafter a mainshock associated to a specific asperity,presumably due to redistribution of the strainenergy. The entire process results therefore in theenhancement of stress concentration around thenearest neighbouring intact asperities. The pro-gressive fracturing of multiple, coupled asperitiesresults in some short-term precursory fluctuationsboth in the b value and in the event rate.However, it is believed that at constant stress-rateloading conditions, this kind of process could beaccelerated dramatically.

The significance of the b value

Since the 1960s, b-value variations have beendirectly related to the local stress conditions(Scholz 1968). In the present study, great effortshave been devoted towards investigating the phys-ical significance of the b value in the magnitude–frequency relation, and it as been shown that, infact, the state of stress plays the most importantrole in determining the value of b, which is well rep-resented by a linear relation with the stress intensityfactor (Fig. 7). However, the present study alsoshows that, at a given stress condition, the b valuesstrongly depend also on rock heterogeneity, as thisis a major factor governing pre-failure damage andfracture dynamics. However, the concept of hetero-geneity is somewhat scale-dependent. With regardto sample size, for example, the greatest grain sizecan be considered as an index of rock heterogeneity.However, for every microcrack, because its size isgenerally less than the greatest grain, a samplewith a comparatively smaller grain may showgreater heterogeneity.In general, available experimental data clearly

show that the b value, in every phase, obeysthe general relationship: bIG , bOG , bWG, hencesuggesting that a comparatively larger grain sizeresults in a lower b value. Furthermore, experimen-tal data show that b values may vary, depending onstress, between 0.5 and 1.5, and that fault zonerocks have a comparatively smaller b value than

its host rock of the same lithology (Fig. 5d, e). Onthe other hand, samples with low pre-existingcrack density also have significantly lower bvalues than those with high pre-existing crackdensity (Fig. 5f). Finally, both experiments andnumerical simulations show that the b value ofAE events is controlled by variations of the internalfriction angle, which are induced by variations inconfining pressure (Amitrano 2003).

In conclusion, the b value can be considered agood parameter for measuring the capability ofrocks to release accumulated strain energy. Withthis aim, possible correlations between the bvalue and crack interaction, as well as assessmentof the dependence of the b value on fracture mech-anisms, may represent major items for futurestudies.

Damage localization and failure

nucleation

It has been shown that the AE hypocentre distri-bution shows complicated clustering behavioursduring the pre-damage evolution. Nevertheless, asshown in Figure 8, there are several commonfeatures that may be of interest:

(1) In coarse-grained samples, grain size has therole of characteristic scale, leading to aband-limited fractal or bifractal structure.Large bands (L) and small bands (S) may

Fig. 8. Generalized correlation integral of the AEhypocentres in coarse-grained and jointed samples. Notethat the AE hypocentre exhibits some heterogeneousfractal and bifractal structures; the fractal dimensionsestimated for large bands (L) and small bands (S), atq ¼ 2 and 22 are also shown. The numbers inparenthesis are estimation errors. P and S indicate theprimary and secondary phases, respectively. N1indicates the earlier stage of the nucleation phase. Thenumber n indicates the number of hypocentres used forthe calculation of the correlation integral.

X. LEI24

likely show different fractal dimensions.However, in fine-grained samples, this kindof characteristic scale could not be found,possibly because it is smaller than the avail-able precision of hypocentre determination.

(2) AE hypocentre distribution exhibits multifrac-tal features, with changing values of D2–D22,indicating heterogeneity changes duringdamage evolution (Fig. 8).

(3) Damage localization associatedwith the growthof a fault is characterized byminimumD valuesmeasured at the onset of the nucleation phase(Fig. 4c, Fig. 5a–e).

The final phase of damage evolution is ofoutstanding interest for short-term prediction ofcatastrophic fracture development. However, thetransition from the secondary to the nucleationphase is strongly affected by the homogeneity ofthe test sample. In samples containing one ormore pre-existing joints, a strong localization wasobserved along some discontinuities that were ulti-mately ruptured (Fig. 9; see also Satoh et al. 1996).Clear damage localization was also observed insamples with a few large (cm scale) grains, wherethe final rupture was mostly controlled by thegrain boundaries of some large grains. Sampleswith high pre-existing crack density, such as thefoliated granitic cataclasite from the Nojima faultzone, show less brittle behaviour and a gradualdamage localization followed by clear diffusion.

This results in a significant decrease and subsequentincrease in fractal dimension (Fig. 5d).

In the case of intact rocks, loaded at a constantstress rate, during the final stage the microcrackingoccurred so frequently that it was impossible todistinguish every event from the AE waveforms.Hence, it was impossible to clarify the damagelocalization on the basis of the AE hypocentredistribution. By using the event rate as a feedbacksignal to control the loading system, the nucleationphase, which would otherwise have taken only afew seconds, could be extended to several hoursduration, and the quasi-static nucleation could bemapped using the AE hypocentres (see alsoLockner et al. 1991). In addition, the faultingnucleation could be controlled by using an asymme-trical loading cell (Zang et al. 2000) and the resultwas a final shear fracture, initiated from some arti-ficially determined point, at a relatively lower stresslevel and lower AE background. In such cases,the damage localization was clearly established.

Relationship between the b value and

the fractal dimension

In a fractal fault system, the fractal dimension (DS)of the distribution of fault length is correlatedwith the b value of earthquakes:

DS ¼ 2b: (24)

Fig. 9. AE hypocentres in three coarse-grained granites. AE hypocentres during the pre-nucleation (grey circles)and nucleation (black circles) phases. A clear localization of the damage was observed in two samples only. Duringthe nucleation phase, the AE rate was so high that most events occurred on a very noisy background, and it wasdifficult to determine a sufficient number of P arrival times for hypocentre determination; as a result, the damagelocalization could not be identified.

PRE-FAILURE DAMAGE IN GRANITIC ROCKS 25

This result derives from the interrelationshipsbetween the frequency–magnitude (Gutenberg &Richter 1944), moment–magnitude (Aki 1967),and moment–source area (Kanamori & Anderson1975). It would be interesting also to establishwhether a similar correlation exists between (1)the b value and the fractal dimension of thespatial distribution of earthquake hypocentres, or(2) the fractal dimension and the spatial distributionof active faults (see also Cello et al. 2006, thisvolume). Regarding point (1), the AE data byLockner et al. (1991) show that a decrease of theb value appears to be correlated, with no timeshift, with strain localization, that is, with adecrease of the D value. Regarding point (2), bothpositive and negative correlations are reported(e.g. Oncel et al. 2001). The present study furnishessome additional information on this criticalproblem, as it has been observed that, during theprimary phase, the b value increased with increas-ing stress, but the fractal AE hypocentre dimensiondid not show any systematic variation. The b valueand fractal dimension reflect, in fact, the size distri-bution and spatial distribution of pre-existingmicrocracks, respectively. In other words, there isno intrinsic correlation between the b value andfractal dimension.However, as already mentioned, in some cases,

well-defined damage localization and subsequentdamage diffusion were observed while evolvingfrom the secondary to the nucleation phases. Thechange from decreasing to increasing fractal dimen-sion corresponds to the onset of the nucleationphase. However, during the localization–diffusionprocess, the b value as a long-term average exhibitsa monotonous decrease until the final dynamic frac-turing. As a result, a positive correlation between band D2 was observed during the localization stage,while a negative correlation was observed duringthe diffusion stage. Figure 10 shows the fractaldimension D2 against the b values obtained fromthe tests shown in Figures 4 and 5.

Predictability of rock failure

The experimental results indicate that, either theaccelerated energy release rate or the decreasing bvalue can be used to predict the final catastrophicevent successfully. However, for the natural cases,particularly for the prediction of large earthquakes,the situation appears to be much more difficult.First, because large earthquakes are normallynucleated at depths of between a few km and tensof km, it is impossible to obtain sufficient infor-mation from the surface-based seismic monitoringnetworks. Secondly, most large earthquakes occuralong well-developed active faults, and aregoverned by some kind of mixed mechanism

including frictional sliding on the fault surfaceand rock fracture. In spite of this, some precursoryphenomena are known to appear months and evenyears before large earthquakes and volcaniceruptions occur (e.g. Imoto 1991; Hurukawa 1998;Vinciguerra 1999; Zuniga & Wyss 2001; Nuanninet al. 2005).

The proposed three-phase damage model maytherefore be profitably used for interpreting thesequence of phases preceding catastrophicfailure events, including natural processes such asvolcanic eruptions, mining-induced rock bursts,and earthquakes. The decrease of the b value andthe onset of the nucleation phase, which generallycorresponds to the minimum value of the fractaldimension of hypocentres, are key indicators ofthe preparatory stage of a catastrophic event.

More generally, short-term prediction dependson reliable precursory phenomena. A reliable pre-cursor should follow well-defined empirical laws,and should be related to some plausible physicalmechanism. Experimental approaches under well-controlled conditions are considered useful forfinding possible precursors and their related phys-ical mechanism. To this aim, the present studyclearly shows that a comparatively greater hetero-geneity results in a longer and more complicatednucleation phase, with a larger number and moretypes of precursory anomalies. Differently stated,the catastrophic failure event can be better predictedin a comparatively more heterogeneous rock mass.It is expected that the interaction between neigh-bouring cracks becomes increasingly significant asa result of the progressive increase of both densityand length of every crack. In heterogeneous rocks,such as in coarse-grained granites, the interactionsare markedly affected by the heterogeneous struc-ture, thus causing fluctuations with large amplitude

Fig. 10. The fractal dimension D2 of AE hypocentres v.the b value obtained from the tests shown in Figures 4and 5. In some cases, D2 correlates linearly with b asD2 ¼ 2b þ constant (see text for details).

X. LEI26

in the damage statistics. Such a strong dependenceof pre-failure damage on local structure indicatesthat the predictability of catastrophic events isalso site-dependent. Pre-failure damage evolutionlikely varies greatly in each case; hence, it is par-ticularly important to resolve, for a given targetsite, the local geological structure. Systematicstudies on the simulation of various geologicalstructures are in fact meaningful for predictionpurposes. Future studies need therefore to befocused on establishing quantitative correlationsbetween fluctuations of energy release and the het-erogeneous structure of the crust.

Conclusions

AE data from granitic rock samples indicate thatthe pre-failure damage process is characterized bythree typical phases of microcracking activity: theprimary, secondary, and nucleation phases. It wasassessed that the evolution of the three-phasepre-failure damage is a common feature withingranitic lithologies having different grain sizedistributions, as well as macrostructures such asjoints. The primary phase reflects the initialopening or ruptures of pre-existing microcracks,and it is characterized by an increase, with increas-ing stress, both of the event rate and the b value.The secondary phase involves the subcriticalgrowth of the microcrack population, revealed byan increase, with increasing stress, in the energyrelease rate and a decrease in the b value. Thenucleation phase corresponds to the initiation andaccelerated growth of the ultimate fracture alongeither one or several incipient fracture planes.During the nucleation phase, the b value decreasesrapidly down to the global minimum value of 0.5.Beside the experimental study, a theoretical

analysis was performed, in order to improve thedamage model based on the constitutive laws ofsubcritical crack growth for crack populationswhose size distribution is fractal. Instead of model-ling the event rate as in some earlier studies, theenergy release rate was fitted by using thisimproved model. The model can represent verywell the AE energy release rate in three granitesof different grain size distribution, implying thatthe fracture laws (the same laws as those inmode-I microcracking during a single extensionalgrowth of a macrocrack) actually governed themixed mode of the microcracking activity underdifferential compression.The fractal dimension of the AE hypocentres

appears also to be consistent with the fracture mech-anisms peculiar of every phase. During the primaryphase, the fractal dimension depends on grain sizeand it shows no significant temporary variation.

Hence it reflects the spatial distribution of pre-existing cracks. Some more or less pronounceddecrease of the fractal dimension was observed,being associated with the rapid decrease of the bvalue. The fractal dimension decreased down to aminimum value around the onset of the nucleationphase. A diffusion process of AE hypocentres wasalso observed, in some cases, following the onsetof the nucleation phase. The progressive develop-ment of the fracturing through heterogeneousrocks of large grain size, and through samples thatinclude macroscopically heterogeneous structuressuch as joints, results in some short-term precursoryfluctuations, both in the b value and in the energyrelease rate.

The results of this study indicate that the precursor-based predictability of catastrophic failure is highlydependent on pre-existing heterogeneity andloading conditions. The most important factors arethe density and size distributions of pre-existingcracks. The three-phase model proposed in thisstudy appears to be meaningful for transferringthe experimental results to real situations associatedwith both artificial applications and natural pro-cesses affecting crustal rocks.

This research was supported by the Japan Society forthe Promotion of Science under grant 14340128, theMinistry of Science and Technology of China undergrant 2004BA601B01 and the National Natural ScienceFoundation of China under grant 40127002. Helpfulreviews by G. P. Gregori and F. Storti are gratefullyacknowledged. I would also like to express my gratitudeto G. Cello for his helpful comments and suggestions.

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