Brittle-ductile transition and associated seismicity: Experimental and numerical studies and relationship with the b-value

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Experimental and numerical studies and relationship with the b-value

David Amitrano

To cite this version:

David Amitrano. Brittle-ductile transition and associated seismicity: Experimental and numerical

studies and relationship with the b-value. Journal of Geophysical Research : Solid Earth, American

Geophysical Union, 2003, 108 (B1), pp.2044. �10.1029/2001JB000680�. �hal-00173129�

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Brittle-ductile transition and associated seismicity:

Experimental and numerical studies and relationship

with the b-value

DavidAmitrano

LAEGO,EcoleNationaleSuperieuredesMinesdeNancy,France

Abstract. The acousticemission (AE) and the mechanicalbehavior of granitesam-

ples during triaxial compression tests have been analyzed. The size of AE events displays

power-law distributions,conforming to the Gutenberg-Richter lawobserved for earth-

quakes which is characterizedby the b-value. As the conning pressureincreases, the

macroscopic behaviorbecomesmore ductile. For all dierent stages of the rockmechan-

ical behavior(linear, non-linear pre-peak, non linear post-peak, shearing), there is asys-

tematic decrease of the b-value with increasing conningpressure. A numerical model

based onprogressiveelastic damageand thenite element method allows simulations

of the main experimental observations onAE and of awide range of macroscopic be-

haviorsfrom brittlenessto ductility. The model reproduces adecrease in theb-valuethat

appearsto be related to the typeof macroscopic behavior (brittle-ductile) rather than

to the conning pressure.Both experimental and numerical resultssuggesta relation-

ship betweenthe b-value and the brittle-ductile transition. Moreoverthese resultsare

consistentwith recent earthquakeobservations and give new insightinto the behavior

of the Earth'scrust.

1. Introduction

The mechanicalloading of rocks involveslocal inelastic

processesthatproduceacoustic waveemissions(AE).Non-

linearity of the macroscopic mechanical behavior results

from these microscopic scale processes. For rocks loaded

athighstrainrateandlowtemperature,microfracturingis

considered to be the main inelastic process [Kranz, 1983].

The correlation between AE activity and macroscopic in-

elasticstrainhasbeenestablishedinmanyexperimental[see

Lockner,1993,forareview]andnumerical[e.g.Youngetal.,

2000]studies.

Asmicrofracturingprogresses,cooperativeinteractionsof

cracks take place and lead to the coalescence of a macro-

scopic fracture, i.e. to the macrorupture [Costin, 1983;

Kranz, 1983; Reches and Lockner, 1994; Schulson et al.,

1999]. Thisbehavior has beenexperimentallyobservedby

AEsourcelocation[Lockner etal.,1991].

Themacroscopicbehaviorofrocksrangesfrombrittleness

toductilitydependingonrocktypeandloadingconditions

(i.e. strainrate,conningpressureandtemperature).Many

denitions of brittle-ductilebehavior basedon thetypeof

macroscopicbehaviorhavebeenproposed[JaegerandCook,

1979]. Themostsimpleis basedontheamountofinelastic

deformationbeforethemacrorupture (Figure1). A purely

brittlematerialfailswithoutanyinelasticstrainbeforethe

failure. Bycontrast,apurelyductilematerialstrainswith-

outlossofstrength. Thefailure,ifany,occursafteracon-

siderableamountofinelasticstrain.

Papernumber.

0148-0227/02/$9.00

Fracturing dynamics during mechanical loading, which

canbe studiedthroughAE monitoring, usuallydisplaysa

powerlawdistributionofacousticeventssize.

N(>A)=c:A b

(1)

WhereAisthemaximumamplitudeofAEevents,N(>A)

is the numberof eventswith maximumamplitude greater

thanA,andcandbareconstants. Inalog-logrepresenta-

tion,this distributionappearslinearand bis givenby the

slopeoftheline.

l ogN(>A)=C b:l ogA (2)

This distribution exhibits remarkable similarity to the

Gutenberg-Richter relationship observed for earthquakes

[Gutenberg andRichter,1954].

l ogN(>M)=a bM (3)

WhereN(>M)isthenumberofearthquakeswithamag-

nitudelargerthanM.

Assumingthatthe magnitudeisproportionalto thelog of

the maximal amplitude of the seismic signal, the b-value

obtainedfromthemagnitudeortheamplitudecanbecom-

pared [Weiss, 1997]. Rigorously, the amplitudemeasured

atagivendistancefromthesourceshouldbecorrected for

theattenuation. Nevertheless,theoretical[Weiss,1997]and

experimentalstudies[Lockner,1993]haveshownthatatten-

uationhasnosignicanteectontheb-value.

Aspowerlawsindicatescaleinvarianceandbecauseofthe

similaritiesinthephysicsofthephenomena(wavepropaga-

tioninducedbyfastsourcemotion),AEofrockobservedin

(3)

thelaboratoryhasbeenconsideredasasmall-scalemodelfor

theseismicityinrockmasses(rockbursts)orintheEarth's

crust (earthquakes) [Scholz, 1968]. Observations of both

earthquakesandAEshowvariationsoftheb-valueintime

and spacedomainswhichareusuallyexplained usingfrac-

turemechanicstheory and/or the self-organizedcriticality

(SOC)concept. Mogi[1962]suggestedthattheb-valuede-

pends onmaterialheterogeneity,alowheterogeneity lead-

ing to a low b-value. Scholz [1968] observed that the b-

valuedecreasesbeforethemaximumpeakstressisachieved

and arguedfor anegativecorrelation between b-valueand

stress. Mainet al.[1989] observed the samevariationbut

invokedanegativecorrelation betweentheb-valueand the

stress intensity factor K. Following this idea, Main et al.

[1989] proposed dierent patterns of b-value variation be-

fore macrorupture, driven by the fracture mechanics and

the type of rupture(brittle-ductile). Therelationship be-

tweentheb-valueandthefractaldimensionDofAEsource

locations was alsoinvestigated[Lockner andByerlee,1991]

andshowedadecreaseofb-valuecontemporarytothestrain

localization,i.e. toadecreaseofD-value.

Mori and Abercombie [1997] observeda decrease of the

b-value with increasing depth for earthquakes in Califor-

nia. They suggestedthat the b-decrease was related to a

diminutionoftheheterogeneityasdepthincreases. System-

atic tests of the dependenceof the b-value ondepthhave

beenrecentlyperformedbyGerstenberg etal.[2001]which

conrmtheseresults. Thedepthdependenceoftheb-value

havealsobeenobservedforthewesternAlpsseismicity[Sue

etal.,2002] andforearthquakessequencealongtheAswan

LakeinEgypt[Mekkawietal.,2002].

Other authorshaveusedcellularautomata[Chen etal.,

1991; Olami et al., 1992] or lattice solid models [Zapperi

etal.,1997]tosimulatepower-lawdistributionofavalanches

which appear to be associated with a ductile macroscopic

behavior. Numericalmodelsbasedonelasticdamage[Tang,

1997; Tangand Kaiser, 1998] succeedinsimulatingbrittle

behavior. Discreteelementmodels simulating macroscopic

behavior rangingfrombrittletoductileandpower-lawdis-

tributions of earthquakes have also been proposed [Wang

et al.,2000; Liet al.,2000; Place andMora,2000]. Wang

et al.[2000] argue that the b-value depends onthe cracks

density distribution but donot report a relation between

theb-valueandthetypeofmechanicalbehavior. Amitrano

etal.[1999]proposedamodelwhichsimulatesbothductile

andbrittlebehaviorand showthattheb-valuedependson

themacroscopicbehavior.

These resultssuggestthat arelationshipbetweentheb-

valueandthemacroscopicbehaviormayexist. Thepresent

paperreportsresults onAEmonitoringofgranite samples

duringtriaxialcompressiontestsandnumericalsimulations.

We studythe eectof the conning pressure on both the

macroscopicbehaviorandtheb-value.

2. Experimental procedure

2.1. TestedRock

Asetof34triaxial compressiontestswereperformedon

Sidobre granite. This rock contains 71% feldspar, 24.5%

quartz,4% mica and 0.5%chlorite. Thegrainsizesare in

therange1-2mmfor thefeldspar,0.5-1mmforthequartz

and0.5-2mmforthemica[Isnard,1982]. Thedensityis2.65

and the continuity index obtained by sound velocity mea-

surement(sonicvelocitymeasuredonthesampledividedby

soundvelocityisabout4800m/s.Themeanuniaxialcom-

pressive strength is 160MPa,Young's modulusis 60 GPa

andthePoisson'sratiois0.24. Thesampleswere40mmin

diameterand80mminlength.

2.2. Experimentaldevice

A hydraulic press of 3000 kN capacity was used. The

conningpressure was appliedby meansof atriaxial cell.

Thestinessofthecompleteloadingsystem(press,piston,

samplesupport)isabout10 9

N/m. Theaxialdisplacement

oftheplatenswasmeasuredbyanLVDTsensor. Thesam-

plestrainwasestimatedfromthedisplacement,takinginto

account the stiness of the loading system (shortening of

the pistonand the samplesupport) and the lengthof the

sample. Theaxialdisplacementratewaskeptconstantnear

1m/sexceptduringthemacrorupturewhendynamicfail-

ureoccurs. A resonant transducer(PhysicalAcousticCor-

poration,peakfrequency:135kHzeectiverangefrequency

: 100 kHz - 1 MHz) was applied on the outside part of

thecellpistonwhichwas usedas awaveguide. Thetrans-

ducerwasconnectedtoa40dBpreamplier(PAC1220A)

withadapted lters(20 kHz-1.2MHz)and thento anAE

analyzer(Dunegan-Endevo3000 Series)with40dB ampli-

cation whichperformed the AEcounting. Inparallel the

signals weredigitalized after preamplicationby meansof

afastacquisition board(ImtecT2M50, 8bits). Thesam-

plingfrequencywas 5MHz andthelengthofthe recorded

signalswas2048 samples,whichcorrespondsto aduration

of 410s. Thesignal recordingtrigger was set to 15mV

andthemaximalamplitudeto1V.Theboardmemoryseg-

mentationallowedusto recordseveralhundred signalsper

secondwithoutdeadtime.

2.3. Deformationmode

TheSidobregranitesamplesweredeformedundervarying

conditionsofconningpressure,rangingfrom0to80MPa.

Theaxialdisplacementwasappliedataconstantstrainrate

exceptduringthemacrofailure whichisunstable. Loading

wascontinuedafterfailureuntilthedisplacementalongthe

macrorupturesurfacereachedseveralmillimeters.

2.4. Dataprocessing

TheAEcountingwasdirectlyobtainedfromtheanalyzer.

ThisparameterappearedtobewellcorrelatedwiththeAE

energycalculatedfromthedigitalizedsignals. Theslopeof

thecumulativeAEcountingcurverepresentstheAEactiv-

ity. The digitalized signals were processed to extract the

maximalamplitudeandtheenergyforeachsignal.

Theb-valuewasobtainedfromtheinversecumulativedis-

tributionof theeventsmaximal amplitude. This distribu-

tionwasttedinaleastsquaresensebyalinearfunctionin

alog-logdiagram. Theslopeofthiscurvegavetheb-value.

Theerrorofestimationoftheb-valuehasbeencalculatedfor

acondencelevelof95%. Theb-value was rstcalculated

forallrecordedeventsduringeachtest. Inordertoobserve

variationsoftheb-valueduringthedierentstagesofeach

test,theb-valuewasalsocalculatedseparatelyforeventsoc-

curringduringeachstageof themechanicalbehavior. The

minimumnumberofeventsthatwasusedforcalculatingthe

b-valuewasxedat200;accordingtoPickeringetal.[1995]

this population size is acceptable to calculate the b-value

withagoodaccuracy,i.e. withstandarddeviationlessthan

(4)

3. Experimental results

3.1. Mechanicalbehavior

Asetof34testshavebeenperformedwithconningpres-

sure ranging from 0 to 80 MPa. Figure 2 shows typical

results obtained for a conning pressure of 60 MPa. We

identifyfourstagesinthemechanicalbehavior,asobserved

onthe()curve,andtheacousticemissionactivity.

Stage 1 is dened by the rst linearpart of the ()

curve. The initial part of this stage is inuenced by the

closing of microcracks as indicated by the increase in the

slope ofthestress-strain curve. Afterthat,themechanical

behavior is linearand is notaected bymicrocracks. The

AEactivityisverylowandcanbeattributedtotheclosure

orshearingofprexistingcracks[LocknerandByerlee,1991].

Theb-valueismaximum.

Stage2beginswiththeappearanceofanon-linearbe-

havior. It corresponds to a strain hardening stage as the

strengthincreaseswithrock deformation. AEiscausedby

cracks propagation that aects the macroscopic behavior.

TheAEactivityincreasesdrastically bytheendofstage2

andtheb-valuedecreases(gure6).

Stage3correspondstothepost-peakbehaviorpreced-

ing themacro-rupture. Therocksdisplay strainsoftening,

asthestrengthdecreaseswithincreasingstrain. AEispro-

duced by the propagation and coalescence of cracks. AE

activityreachesitsmaximumvalueandtheb-valueismin-

imum. Thisstageendswiththemacro-failure whichisun-

stable. As addressed earlier by Wawersick and Fairhurst

[1970],thisinstabilityoccurswhenthesamplestrengthde-

creaseswithstrainfasterthantheapparatusunloads. The

startingandendingpointoftheunstablefailurearestrongly

machine-dependent andare notrelevant tothe description

of the rock samplebehavior. It is generally assumedthat

thenucleationofamacroscopicdiscontinuityoccurssimul-

taneouslywiththeunstablefailure.

Stage4correspondstothemacrorupturesurfaceshear-

ing. AE iscaused bythe ruptureofsurfaceasperitiesand

by gougefracturing. Theshearstrengthisnearlyconstant

orslowlydecreasesandtheAErateslowlydecreases.

3.2. Brittle-ductiletransition

Foreachtestwecalculated 1 3 and1 attheendof

eachstage. Figures 3and 4display theseresults for allof

the tests. Stress and strainare plottedas functionsof the

conningpressure.

In orderto estimate the brittle-ductile characterof the

mechanicalbehavior,wequantiedtherangeoftheinelastic

behaviorbeforemacroruptureusingtwoparameters. Oneis

representing the inelastic strain,

in:

, and the second the

stress range of stage2, in:. in: is the dierence be-

tweenthe stress at theendof stage1 and thepeakstress

(endofstage2). Thisparameterquantiesthestressampli-

tudeofthe strainhardeningstage. Inelastic strain, in:,is

obtainedbysubtractingthe elasticstrain,

el:

,tothetotal

strain,

tot:

,whichiscurrentlymeasured:

in:

=

tot

el:

(4)

Theelasticstrainiscalculatedusingtheelasticmodulus

estimated in the linear stage(dotted line in gure2) and

thecurrentvalueof1 3.

el:

=E

initial

:(1 3) (5)

Figure 5 presents the mean values of the dierential

stress,

1

3

,andtheaxialstrain,

1

,asafunctionofthe

conningpressure. in: and in: are alsoplotted. Since

the starting and ending points of the unstable failure are

strongly machine-dependent, we restrict the discussion to

thevaluesmeasuredattheendofthelinearstage1andat

thepeak(endofthestage2).

For

3

=0,whichcorrespondstoanuniaxialcompression

test,thesamplefailedimmediatelyafterstage1. Theinelas-

ticbehavior range(i.e. in: and in:) isnearlyzero; that

correspondsto a purelybrittlebehavior. Asthe conning

pressureincreases,thestresslevelsforeachstageincreaseat

dierentrates. Inparticularthepeakstressincreasesfaster

thanthe stress at the end ofthe linear stage, which indi-

catesanincreaseinstrainhardening. Inthesamemanner,

theamountofinelasticstrainbeforethepeakincreaseswith

theconningpressure. Hence,therangeofinelasticbehav-

ior(i.e. in: andin:) increaseswithincreasingconning

pressure. Thisindicatesthatthepre-peakbehaviorbecomes

progressivelymoreductile. Therangeofconningpressure

thatwetesteddoesnotcovertheentirebrittle-ductiletran-

sition. Nevertheless,theanalysisofthebrittle-ductilechar-

acteristicwhichwasperformedongranite,arockcommonly

consideredasbrittle,showsthateventhismaterialbecomes

increasinglyductileatrelativelylowconningpressure(the

maximum conning pressure of 80 MPa corresponds to a

3.2 km depthfor a natural geostatic stress eld). Similar

results were obtainedby Brace et al. [1966] ongranite for

largerconningpressurerange. They observedanincrease

intherangeofinelasticbehavior,beforethepeak,forsam-

pletriaxiallyloadedwithconningpressurerangingfrom0

to800MPa. However,asoftenobservedforgranite,theef-

fectoftheconningpressureonthebehaviorremainsminor

astheloadingeverleadstoanunstablefailure.

3.3. b-valuepressure dependence

Inordertoexaminetherelationshipbetweentheb-value

andtheconningpressure,wecalculatedtheb-valueforall

AEeventsdetectedduringeachstageofeachoneofthetests.

Figure6adisplaysthe cumulativedistributions ofAEam-

plitudeforarepresentativetestperformedat3=60MPa,

showingseparatelytheeventsrecordedduringeachmechan-

icalstage. As classically observed,the b-value ismaximal

duringstage1,thendecreasesduringstage2andreachesits

minimalvalueduringstage3.

Figure 6b displays the AE amplitude distributions for

a set of 4 tests at conning pressures ranging from 0

to 80 MPa. Each distribution includes all of the events

recorded during the test. The b-value decreases with in-

creasingconningpressure.

Figure 7 displays the b-value corresponding to the dif-

ferentstages ofmechanicalbehaviorfor all ofthe tests,as

afunctionof 3. Figure 8presentsthe mean b-values for

each stage as a functionof the conningpressure (a) and

ofthedierentialstress(b). Theb-valueisnegativelycor-

related withboth the conningpressure and withthe dif-

ferential stress. This behavior is observed for each of the

stages as wellas for the b-values calculated for the entire

test(i.e. withoutgroupingtherecordedeventsaccordingto

thestages).