JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 102, NO. B12, PAGES 27,635-27,649,DECEMBER 10, 1997
Distributed damage, faulting, and friction
VladimirLyakhovsky
Instituteof Earth Sciences,The HebrewUniversity,Jerusalem
Yehuda Ben-Zion
Departmentof EarthSciences,Universityof SouthernCalifornia,Los Angeles
Amotz Agnon
Instituteof Earth Sciences,The Hebrew University,Jerusalem
Abstract. We presenta formulationfor mechanical
modelingof geologicalprocesses
in the
seismogenic
crustusingdamagerheology.The seismogenic
layeris treatedasan elasticmediumwheredistributeddamage,modifyingtheelasticstiffness,evolvesasa functionof the
deformation
history.The modeldamagerheologyis basedon thermodynamic
principlesand
fundamentalobservations
of rockdeformation.The theoreticalanalysisleadsto a kinetic
equationfor damageevolutionhavingtwo principalcoefficients.The first is a criterionfor the
transitionbetweenstrengthdegradation
andrecovering(healing),andis relatedto friction.The
secondis a rate coefficientof damageevolutionwhichcanhavedifferentvaluesor functional
formsfor positive(degradation)andnegative(healing)evolution.We constrainthesecoefficientsby fitting modelpredictionsto laboratorydata,includingcoefficientof frictionin sawcut
setting,intactstrengthin fractureexperiments,
firstyieldingin faultingexperiments
under
three-dimensional
strain,onsetandevolutionof acousticemission,anddynamicinstability.
The modeldamagerheologyaccounts
for manyrealisticfeaturesof three-dimensional
deformationfieldsassociated
with an earthquake
cycle. Theseincludeaseismicdeformation,gradual strengthdegradation,
development
of processzonesandbranchingfaultsaroundhighdamageareas,strainlocalization,brittlefailure,andstatedependentfriction.Someproperties
of themodeldamagerheology(e.g.,cyclicstick-slipbehaviorwith possibleaccompanying
creep)areillustratedwith simplifiedanalyticalresults.The developments
of thepaperprovide
an internallyconsistent
frameworkfor simulatinglonghistoriesof crustaldeformation,and
studyingthe coupledevolutionof regionalearthquakes
andfaults.This is donein a follow up
work.
Brittle behavioris often modeledby a rigid elastic-plastic
solidthatis governedby simplestatic-kinetic
frictionor Byer-
1. Introduction
Rocks exhibit a wide variety of theologicalbehaviors lee's law [Brace and Kohlstedt, 1980]. However, such modrangingfrom viscoelastic
deformation
to plasticflow and lo- els do not accountfor detailsof strengthevolutionand they
calizedfaulting.A greatchallengeof theoretical
geodynamic thuscannotbe usedto studyimportantportionsof the deforstudiesis to incorporate
multi theologicalbehavior,including mationfield, suchas nucleationof slip instabilities.The ratefaulting,into modelsthat simulatedeformational
processes
in and state-dependent(RS) friction model [e.g., Dieterich,
the uppercrust.At the presenttime, a generallyaccepted 1979, 1981; Ruina, 1983] providesa frameworkthat can be
methodfor describingtime-dependent
deformationalproc- usedto simulateall importantaspectsof an earthquakecycle,
essesin the brittle-elasticpartsof the lithosphere
is not avail- including stable slip, nucleation of instabilities, rupture
able. The purposeof this paperand a follow-upwork (V. propagation,and healing. However, the RS formulationasLyakhovsky,Y. Ben-Zion, and A. Agnon, manuscript in sumesthat deformationat all stagesoccurson well defined
preparation;hereinafterreferredto aspaper2) is to developa frictionalsurfaces,and it doesnot providea mechanismfor
useful framework
for studies concerned with seismic and
aseismicdeformationsin large domainsof spaceand time.
The overall large-scalestructureof our model is a layered
elastic-viscoelastic
half-spaceincorporating
damagetheology.
In thepresentpaperwe focuson theoryandobservations
relevantto thedamagetheologyandits couplingwith viscousrelaxation.In paper2 we discussthe othercomponents
of the
modelandprovidevarioussimulationexamples.
Copyright1997by theAmericanGeophysical
Union.
Papernumber 97JB01896.
0148-0227/97/97JB-01896509.00
understandingdistributeddeformation.In addition, it is not
clear [e.g.,Andrews, 1989; Ben-Zion and Rice, 1995] to what
extentthe existingRS friction laws are valid for natural conditionsinvolvingcomplexgeometry,largevaluesof slip, slip
rate, and time, etc.
Varioussimpleconceptualschemesbasedon a networkof
blocksand springs[e.g., Burridge and Knopoff, 1967; Carlson and Langer, 1989] have been used to simulate static,
quasi-static,and dynamic sliding processes.
Rice [1993] and
Ben-Zionand Rice [1993, 1995] criticizedthe validity of representinga fault in elastic solid with a block-springarray.
They simulated slip historieson various types of a two-
27,635
27,636
LYAKHOVSKYET AL.:DISTRIBUTEDDAMAGE,FAULTING,AND FRICTION
dimensional(2-D) strike-slipfault in a 3-D elastichalf-space
with modelsincorporating
continuumelasticity.Cowieet al.
[1993], Sornetteet al. [1994], and Ward [1996] simulatedthe
development
of fault patternsandregionalearthquakes
in 2-D
elasticsolids;we discussthesemodelsin more detail in paper
2. Locknet and Madden [1991 a, b] developeda numerical
2. Distributed Damage in Rocks
We brieflylist belowsomeindications
of damagein natural rocksandrocksampleswhichformtheobservational
basis
fbr our theoreticaldamagemodel for the crust.Pioneering
studies of fractures and faults treated the crust as an infinite,
perfectlyelasticmedium[e.g.,Anderson,1951]. Subsequent
studiesaccountedfor the finite lengthof faults, and the perturbationto the regionalstressfield due to the proximityof
additionalfaults [Chinnery, 1966 a, b]. Field mappingoften
showsthat the densityof faults dependson the scaleof the
map,sohigherresolution
increases
the numberof faultsin a
given
domain
[Scho&,
1990].
This
complexity
limitsthe use
Reches,1988] or in the field [e.g.,Segalland Pollard, 1983].
that specifythe positionsof isolatedcracksin the
Fracturedistributionsin situ and fragmentationof rocks in of methods
laboratorysamples show fractal-like patterns[e.g., King, deformingregion.
Classical fracture mechanicspostulatesthat in a linear
1983; Turcotte, 1986; Okubo and Aki, 1987; Aviles et al.,
1987]. Thus fYacturenetwork simulationsshouldnot depend elasticsolidan isolatedcrackwill propagateat velocitiesapon specificlengthscales,suchas lengthscalesprescribed
in proachingthe speedof soundin the mediumoncea critical
stressintensityfactorK•.hasbeenreachedor exceeded
at the
regulararrays.
A theologicalmodelof the faultingprocessshouldinclude cracktip [Irwin, 1958]. At lowerstressintensityfactorsthe
subcritical
crackgrowthfromveryearlystagesof the loading, crackremainsstable.A more generalapproachin classical
materialdegradationdue to increasingcrack concentration, fYacturemechanicsis to considerthe strainenergyreleaserate
[e.g.,Freund,1990].Dynamiccrack
macroscopicbrittle failure, post failure deformation,and G duringcrackextension
healing. Suitablevariablesshouldbe definedto characterize extensionoccurs when G reaches a critical value G•.
Thesefracturemechanicsapproaches
havebeenusedsucthe abovedeformationalaspectsquantitativelyin a framework
cessfully
to
predict
catastrophic
crack
propagation
in metals,
compatiblewith continuummechanics
and thermodynamics.
andglasses.In grainymaterials,however,thestress
Amongsuchapproaches
are Robinson's[1952] linearcumu- ceramics,
on the grainscale.Stressintensity
lative creepdamagelaw, Hoff s [1953] ductilecreeprupture field is highlynonuniform
theory, Kachanov's [1958, 1986] brittle rupture theory, factorsK andstrainenergyreleaseratesG arecalculatedmacdue to grain
Rabotnov's[1969, 1988] coupleddamagecreeptheory,and roscopically,neglectingstressconcentrations
sliding.Such
manymodifications
of thesetheories.Severalresearchers
(see contactsandenergyreleasedueto intergranular
the review of Kachanov, [1994]) proposedmodelswith a materialssubjectedto long-termloading show significant
crackextensionat valuesof K and G
scalardamageparameterchangingfrom0 at an undamaged ratesof macroscopic
stateto I at failure. The scalardamagemodelsfit reasonably significantlylower than the critical.This phenomenon
is
well existing experimentalresults,including culminationof knownas subcriticalcrackgrowth [Swanson,1984,Atkinson
damagein concretesubjectedto fatigueloading[Papa, 1993] and Meredith, 1987; Cox and Scholz, 1988].
and damageincreasein 2024-T3 aluminumalloy underdifferThe investigationsof granite fracturingby Yukutake
ent loading and temperature conditions [Hansen and [1989], Locknet et al. [1991], and Reches and Locknet
Schreyer, 1994]. In the studyof Hansen and Schreyer [1994], [1994] showthat fracturingcannotbe describedin termsof
the scalar isotropicdamage model correlateswith all meas- propagation
of a singlecrack.Severalexperimental
studiesreured quantitiesexceptthe changein the apparentPoissonra- vealedthat elasticparametersstronglydependon the defortio. For this reason, Ju [1990] and Hansen and Schreyer mationalhistory(i.e., damageextent),leadingto vanishing
[1994] suggestedupgradingthe damage parameterfrom a elasticmoduli at large stresses
just beforefailure [Locknet
scalarto tensorquantity.Suchan anisotropictensorialdamage and Byeflee, 1980]. While linear elasticfracturemechanics
model contains at least three adjustableparameterswhich assumesthe size of the inelastic zone at the crack tip to be
permitcorrectsimulationof the apparentPoissonratio.
negligiblysmall,the experiments
showthat this zonehas a
multiple-crack
modelfor the failureprocess
of a brittlesolid
whichsimulatesgrowthof microcracks
on a regulararrayof
potentialcracksites.This and similarnumericalmodelsreproducevariouscommonfeaturesof fracturingprocesses,
especiallythoseoccurringin somefabricated
materials,but they
cannotexplainfault patternsobservedin experiments[e.g.,
Variations
of elastic moduli and Poisson's ratio with extent
of damage,underdifferenttypesof load,can alsobe described
usinga nonlinearelasticmodel with scalardamageprovided
that it is scaledproperlywith the ratio of strain invariants.
This has been done in the damagemodel proposedby Lyakroyskyand MYasnikov[ 1984, 1985], Myasnikovet al. [ 1990],
and Lyakhovskyet al. [1993]. Previousapplicationsof this
modelto geodynamicproblemswere given by Ben-Avraham
and Lyakhovsky[1992], Lyakhovskyet al. [1994], andAgnon
and Lyakhovsky[1995]. The scalardamagemodel accurately
reproducesresultsfrom the four point beam test [Lyakhovsky
et al., 1997]. Here we provideadditionaldevelopments
of the
above model, and constrainthe final model parametersby
comparisons
of theoreticalpredictionswith variouslaboratory
results.In paper2 we incorporatethe damagerheologyinto a
model of a 3-D layeredhalf-spaceand provideexamplesof
simulatedpatternsof seismicityandfaulting.
significantsize.
In most engineeringand rock-like materialsa slowly
propagating
crackis precededby an evolvingdamagezone
distributedaroundits tip [e.g., Bazantand Cedolin, 1991;
Locknetet al., 1991]. The distributeddamagemodifiesthe
elasticcoefficientsin the medium aroundthe tip and hence
controlsthe macrocracktrajectoryand the growthrate [Huang
et al., 1991; Chai, 1993]. The finite size efl•ct of'the fracture
process
zoneis oftentreatedwith modelswhichspecifya cohesivezonenear the cracktip within the planeof the crack
[Dugdale,1960; Barenblatt,1962; Ida, 1972; Palmerand
Rice, 1973, Rubin, 1995 a, b]. This approachis usefulwhen
the crackgeometryis well defined,and in contrastto linear
elastic fracture mechanics,the cohesionzone models do not
containan unphysicalcracktip singularity.
Field observations
suggestthatthe sizeof thedamagezone
(or process
zone)growswith thesizeof thefracture,
in viola-
LYAKHOVSKY ET AL.' DISTRIBUTEDDAMAGE, FAULTING, AND FRICTION
tion of the premisesof the criticalstressintensityfactorapproach [Rubin, 1995 a, b]. This is decisivelydocumented
arounddikes that form by the injectionof magmainto fractures[Delaneyet al., 1986; Baer, 1991; Weinbergeret al.,
1995; Hoek, 1995], andis alsocompatiblewith resultsof Papageorgiouand Aki [1983] who invertedseismicstrongmotion data for earthquakesourceparametersin the contextof
their specificbarriermodel.An early theoreticaldiscussion
of
this phenomenon
is givenby Andrews[1976].
Andrews and Ben-Zion [1997] showedthat earthquake
rupturecan propagatealong an interfaceseparatingdifferent
elastic media in a wrinkle-like
mode associated with little loss
of energyto friction. Thus it is energeticallythvorablefor
rupturesto be locatedalongthe materialinterfacebetweenthe
gougeandthe surrounding
rock,ratherthanwithin the gouge.
In suchcircumstances
the damagezonesof successive
earthquake rupturescontinueto createfresh gougematerial,thus
addingto the overall thicknessof the (damaged)fault zone.
This may help to explain field and laboratorycorrelations
[e.g., Hull, 1988; Robertson,1983] betweengougethickness
and cumulativenumber of earthquakes(or slip) along the
27,637
namics[Onsager,1931' Prigogine,1955; deGrootand Mazur, 1962], which was successfullyapplied to kinetics of
chemicalreactionsand phasetransitions[e.g., Fitts, 1962;
deGroot and Mazur, 1962] and as a basis for variational
methods
of continuous
mediamodels[e.g.,Sedov,1968;Malvern, 1969]. Followingthis framework,Mosolovand Myasnikov [1965] first formulateda variationalapproachto the
model of viscoplasticmedia [see also Ekland and Temam,
1976]. Lyakhovskyand Myasnikov[1985] first usedthe balanceequationsof energyandentropyto establisha thermodynamicalfoundationfor a rheological
modelof damagedmaterial [Myasnikovet al., 1990; Lyakhovskyet al., 1993]. A
similarapproachwas laterusedas the basisof otherdamage
models[e.g.,Valanis,1990;Hansenand Schreyer,1994].
Many workersin continuumthermodynamics
have postulatedthat the free energydensityis a functionof variousstate
variables,including"hiddenvariables"[Colemanand Guttin,
1967; Lubliner, 1972] not availablefor macroscopic
observation. In order to simulatea processof fracturingin terms of
continuummechanics,a nondimensionalintensive damage
variable cz is introduced.
The variable cz can be envisioned
as
fault.
the densityof microcracksin a laboratoryspecimen,or as the
It is importantto consideran additionalpropertyof rocks densityof small faultsin a crustaldomain.The free energyof
when choosinga rheologyfor simulationsof earthquakecy- a solid,F, is assumedto be givenby
cles. Experimentalstudiesof rock deformation[e.g., NishiF = F(T,œij,o
0,
(1)
hara, 1957; Ambartsumyan,1982; Weinbergeret al., 1994]
reveala strongdependence
of elasticcoefficients
on the type whereT and ϥiare the macroscopic
temperature
and Cauchy
of loading, which results in abrupt changesof the elastic tensorof infinitesimalelasticdeformation,respectively,and {x
moduli when the loading reversesfrom tensionto compres- is a nondimensional
damagestatevariable.The elasticstrain
sion. Abrupt changesof elasticpropertiesare commonlyob- tensorϥiis written as the differencebetweena currentmetric
served in grainy materials.For example, the tensile Young tensorg•i and a metrictensordescribingthe irreversibledemodulus of graphite is 20% less than the compressiveone tbrmation,
[Jones,1977]. Jumpsof Young moduli can be 30% for differEij--gij--g•.
(2)
ent typesof iron, and in concretethe compressivemodulus
may be up to 3 times larger than the tensile one It may be represented
throughsmallelasticdisplacements
u•,
[Ambartsumyan,1982]. Resultsof variousexperimentswith
Westerly granite, marble, diabase, and weak granite from
Kola Peninsula,compiledby Lyakhovsky[1990] and Lyakœij+
.
(3)
3Xj 3Xi
hovskyet al. [1993], showhigh sensitivityof rock elasticityto
the type of loading.
The strainrate tensoris given as a temporalderivativeof the
It is reasonable to assume that the extent of the latter noncurrent metric tensor
linearity in the elasticresponseof rocks dependsstronglyon
dgij
the state of damage. Perfectly intact and undamagedrock
(4)
eij= dt .
shouldnot display nonlinearelasticityfor small strains.On
the other hand, a rock that is highly damagedalong a plane The balanceequationsof the internalenergyU and entropyS
can respondto uniaxial extensionnormalto the damageplane accountingtbr irreversiblechangesof viscous deformation
with small elastic stress,whereasit will respondto uniaxial andmaterialdamage[e.g.,Malvern, 1969] havethe form
compressionin a mannersimilarto intactrock.
dU
d
1
In the followingsectionswe firstdiscussthe thermodynam= m(F + TS)- --Gijeij
- giJi ,
(5)
dt
dt
p
ics of damagegrowth in elastic solid. Then we constructa
phenomenological
modelthat relatesdamageto the elasticresponsein an internally consistentmanner. Finally, we constrainthe obtainedmodelparametersby comparisons
of theoreticalpredictionswith experimentalresults.
wherep is massdensity.Here Ji is heatflux and F is local entropy production.Both Ji and F result from dissipativeirre3. Model of Medium With Distributed Damage
versibleprocesses
suchas internalfrictionandcreationof new
surfaces.Substituting(5) into (6) and using an equationfor
3.1. General Thermodynamic Formulation
productionof free energy[e.g.,Gibbs, 1961]
Here we present the constructionof a new rheological
3Fdœ..3F
model accountingfor elasticdeformation,viscousrelaxation,
and evolution of damage (material degradationas well as
healing). We follow the approachof irreversiblethermody- andthe definitionof the Cauchystresstensor
dF
- -SdT
+3eiJu+-•-•
da,
(7)
27,638
LYAKHOVSKY ET AL.: DISTRIBUTED DAMAGE, FAULTING, AND FRICTION
modulusof the samesolidwithoutcracks;conversely,
crack
3F
(Jij -- D--
c•œi
j'
(8) closure
under
increasing
compressive
stress
causes
a gradual
increasein the modulus.Openingand closureof microcracks
leadto abruptchangesof elasticpropertiesuponstressrever-
the localentropyproduction
maybe represented
as
Ji
pT2
1 o dgi• 1 OFdot
• ij dt T 3otdt
sal from tensionto compression
[e.g., Weinbergeret al.,
1994]. Various formulationsattemptto model suchphenomena. The modelsof Ambartsumyan-Khachatryan
[AmbartsuThefirsttermof equation
(9a)describes
entropy
production
by royart,1982] and Jones[1977] assumethat the compliance
heat conduction,the secondterm is due to dissipationfor a (Poisson'sratio divided by Young'smodulus)changeswhen
viscousflow, andthe thirdtermis relatedto the damageproc- the associated stress component reverses. Hansen and
ess. We neglect heat productionby radioactivedecay and Schreyer [1995] consideropeningand closingof microcracks
chemicalprocesses.
Theseprocesses
are independent
to first to simulateactivationand deactivationof damagein termsof
order;hence,asis commonlyassumed,eachtermin (9a) must continuum mechanics. For materials with a weak nonlinear
be positive.The part of the entropyproductionrelatedto the response,Lornakin and Rabotnov [1978] assumedthat the
elasticmoduli dependonly on the typeof loading.To evaluate
damageprocess,F•, is
the damageeffects,Lyakhovskyet al. [1997] derivethe mac1 •F dot
Fa =
.
(9b) roscopicstress-strainrelations tbr a 3-D elastic solid with
T 3ot dt
noninteractingcracks embeddedinside a homogeneous
maWe expandF• as a Taylor serieswith respectto dot/draround trix, andtestthe solutionagainstrock-mechanics
experiments.
an equilibriumstateFo(ot)where dot/dr=0:
The cracksconsideredare orientedperpendiculareitherto the
F --•V•T+
(9a)
(ot
dot)
*F,(ot)dot
i dø•)
2>0.(9c)contract.
maximum
tension
axis
ormaximum
compression
axis.
In
the
Fak.
'dt
) _F0(ot)
-•-+F2
(ot'lk-•)
first
case
they
dilate
during
loading,
inofthe
second
they
The
solution
for the
elasticwhile
energy
such
a solid
Here F• and 1-'2are expansioncorfficients.In the caseof constantdamagethe deformationalprocessis reversibleand entropy productionis zero (F,=0). This conditionimplies that
Fo=0. The entropyproductionF, shouldbe nonnegativefor
any level of damageand directionof its evolutionincluding
healing(damagedecrease)and destruction(damageincrease).
That is possibleonly if the secondterm of the Taylor series,
F•, is identicallyzero and F2>0. Thus the quadraticterm in
(9c) is the dominantterm of the Taylor series.Back substitution into (9b) gives
(dot)
2 13Fdot
F2[,
dt) - T3ot
dt'
was derived following the sell-consistentschemeof Budiansky and O'Connell [1976]. Following the formulation discussedby Lyakhovskyet al. [1997], the elastic potentialis
written
as
U-
-•- laI2-TIl
,
whereX,andILtareLameconstants,
II:œkk
andI2=œiiœij
aretwo
independent
invariants
of thestraintensor
œii,
and¾is anadditionalelasticmodulus(summation
notationis assumed).
The secondordertermwith the new modulus¾accounts
for
microcrack
openingandclosurein a damagedmaterial.The
incorporates
nonlinearelasticityevenfor an infinitesimal
(9d)term
strain,andit simulates
abruptchangein theelasticproperties
From (9b) - (9d) the equationof damageevolutionhas the
form
whentheloading
reverses
fromcompression
to tension.
Using
(8), the stresstensoris derivedfrom (11) as
dot
3F
•
= -C•
dt
Oot'
(•0)
where C=I/ F2T is a positivefunctionof the statevariables
relation(12) can be rewrittento mimic the
describing
the temporalrateof the damageprocess.
We note The stress-strain
usual
tbrm
of
Hook's
law by introducingeffectiveelastic
that(10) describes
notonlydamageincrease,
butalsoa procmoduli
ess of material recoveryassociatedwith healing of microcracks,
whichis favoredby high confiningpressure,
low
¾. ge I
(13)
shearstress,
andespecially
hightemperature.
3.2. ElasticModuli of a DamagedMaterial
where
thestrain
invariant
ratio•=Ii/x/I2
characterizes
thetype
of deformation as discussed below.
Theelasticproperties
of a damaged
solidshould
depend
on
Lyakhovskyand Myasnikov[1987, 1988] discussedthe
thedamagelevel,andquantification
of thishasbeenthesub- relationbetweenseismicwavevelocityandstateof stressfor
jectof muchresearch
[e.g.Kachanov,
1993].An undamagedthe damagetheologymodelwe use.They foundthat small
solidwith ot=0is modeledby an ideallinearelasticmaterial amplitudeharmonicwavespropagate
in this modelas in a
governedby Hook's law. At the otherextreme,a materialwith
linearanisotropicelasticsolidwith elasticstiffnesstensorde-
ot=l is denselycrackedandlosesits stability.Belowwe de- pendingon the initial stateof strain.Three differentmodesof
scribea nonlinear
elasticbehavior
of damaged
materialforall wavesexist,oneP waveandtwo S waves.Thusin spiteof its
valuesof thedamage
parameter
(0<ot<l),including
strainlo- initial isotropicformulation
anda scalardamageparameter,
calization and brittle failure.
the present nonlinear elastic model accountsfor a stressMany experimentalstudiesmeasurenonlinearstress-strain inducedanisotropy.
relationsfor rocksand rock-likematerials.For example, The energyin theform of (11) is usedbelowto describethe
Walsh[1965] showedthat Young's modulusof a cracked elasticbehaviorof a damagedmaterialwith intermediate
valelasticsolidunderuniaxialcompression
is smallerthanthe uesof the parameterot(0<ot<l).
LYAKHOVSKY ET AL.: DISTRIBUTED DAMAGE, FAULTING, AND FRICTION
27,639
Table1.Matrix
Ell
Ell
E22
E•]
œ33
• + 2g- )•
• - T(el+ e2) • - T(el+ e3)
+)•el2- 2ye
I
+)•ele2
X,-T(e2 +e3)
+)•ele2
+)•e}- 2Te
2
+)•e2e3
X,-T(e2 +e3)
X,+2g-T•
+)•e2e3
+•e• - 2ye
3
+Y•ele3
œ13
œ23
0
0
0
0
0
0
0
0
0
0
0
+)•.;ele3
)C-T(e! +e2) X+2B-)•
œ33 X,-T(eI +e3)
œ12
ϥ2
0
0
0
2g-•
œ13
0
0
0
0
2B- T•
0
œ23
0
0
0
0
0
2B-T•
,
Hereei - Ei / •2 isa normalized
valueofthedeformation
alongprincipal
axes"i".
(15) coincidewith conditionsof positivityof the eigenvalues
of the acousticmatrices which correspondto two different
Two differentmathematicalconditionsare appropriatefor
analyzingmaterial stability.The first is convexityof the elas- polarizationsof shear waves. Thus loss of convexityof the
tic energy which provides a unique solution of the static elasticenergyin the presentmodelalso providesthe criterion
problem [Ekland and Ternare, 1976]. This criterion was for strainlocalizationusedby Rudnickiand Rice [1975].
adoptedandexpandedby R. Hill, T.Y. Thomas,J. Mandel, C.
3.4. Kinetics of the Damage Process
Trusdell, and others [e.g. Bazant and Cedolin, 1991]. The
Equation(10) providesa generalform of damageevolution
secondis ellipticity of the elasto dynamic equation [e.g.,
Rudnicki and Rice, 1975]. These two conditions are not al- compatiblewith thermodynamicprinciples.Practicaluse of
ways identical, especially for nonlinear elasticity [e.g., theequationrequiresan additionalfunctionalrelationbetween
Otandthethreeelasticmoduli•, g, and
Schreyer and Neilsen, 1996 a, b]. The first condition is a thedamageparameter
strongerone,andconvexitymaybe lostpriorto the ellipticity. ¾. With the current level of experimentalconstraints,some
3.3. Loss of Convexity and Strain Localization
For that reason we start with the first condition
for material
simpleassumptions
shouldbe made.Hencewe assumelinear
dependencies
of theelasticmoduli•, g, andT on damage:
stability.
The maximum possiblevalue of the damageparameterOt
X,: 3•o+ OtX,
r,
for a givenstraintensore!iis definedby the requirement
of
g = go + Otgr,
convexityof the elasticenergyU of (11). This conditionim-
(16)
pliespositivity
of all eigenvalues
of thematrix32U/OEijOEkl
whose dimensionis 6x6 for six independentcomponentsof whereX=•, g=go,and7=0 correspond
to initialelasticmoduli
the straintensor(e•, E22,E33,El2,El3,E23).The matrix compo- of the uncrackedmaterial. Combiningequations(10), (11),
nentsin the coordinatesystemof the principalaxesare given and (16) yieldsan equationof damageevolution
in Table 1. The first eigenvalueis equalto
x1= 2g- ht•= 2ge > 0.
(14)
dot=
_Cp(•i•2
+gri2¾ri,•2/,(17)
dt
This conditionimplies stability against simple-sheardefor- whichmay be rewrittenin the form
mation.The secondandthird eigenvaluessatisfythe quadratic
equation
-I-
dt _Cdi2(
[,2¾,. •
dot
x2- (4g- 3'•3+3•)x+(2g- •) 2+
+(2it- •)(3•. - •) + (•.• - •,2)(3- • 2) = 0.
The rootsof this equationare nonnegativeif
(2it- •) 2+ (2it- •)(3•. - •) +
+(•3;•-72)(3-•2) > 0.
'
(18)
The positivecoefficientCd,givenby Cp¾,.,
describes
therateof
damageevolutionfor a given deformation.
To use equation(18) in a 3-D damageevolutionmodel, we
employtwo additionalconstraints.The first is that thereexists
(15) acritical
strain
invariant
ratio
• which
corresponds
toaneu-
If either(14) or (15) is not satisfied,the elasticenergyis not a
convexfunctionof the straintensorand the staticproblemhas
multiple solutions.It can be shown that conditions(14) and
tral statebetweenhealingand degradationof the material.As
will be shownin a later section,this is a generalizationof
t?iction,which is a widely observedconstitutivebehaviorin
rocksand otherbrittle materials.High shearstrainrelativeto
27,640
LYAKHOVSKY
ETAL.:DISTRIBUTED
DAMAGE,
FAULTING,
ANDFRICTION
.
4. General Properties of Damaged Material
_
i
•'0.8
:STABLE
0.6
,
',
i
,,
.
i
i
',
(eq.15)
convexity'
:
i
' (eq.14) '
!
!
i
!
!
i
!
i
!
i
'
convexity " Loss "
!
i
0.4
Loss
',
i
4.1. One-Dimensional
.
o
-2
I
,
-1.5
-1
Youngmoduluson damage,E=E0[1-ot],reducesequation
:
!
I
!
I
I
I
-0.5
0
0.5
•
Deformation
For a uniaxial strain, assuminga linear dependenceof
!
g=g0
i
ß
In this section,three generalpropertiesof the model are
analyzedusinganalyticalsolutions.
The first solutionfor a 1D extension
problemillustrates
strainanddamagelocalization
in a previouslyweakenedzone.The second,2-D case,suggeststhat there is no stresssingularityarounda fully destroyedzone.The last exampleshowsthatthe interaction
betweenthe damageevolutionandviscousrelaxationresultsin
stick-slipshearmotion.
(20) to
.
•
1
dot
=A(•U•
2
1.5
Typeof deformation
(•,)
(21)
whereu is displacement
depending
onlyon the x coordinate,
andA is a coefficientwhichdepends
on the typeof loading.
Sincein the 1-D casethe straininvariantratiois •=+1, A is
Figure 1. Thick line givesthe maximumvalue(otc,-)
of the
damage
parameter
otasa functionof straininvariant
ratio
Therange-x/3<•<•corresponds
tostable
behavior
with
healing.For •>•,0thereis materialdegradation
leadingto loss
of stabilityaccording
to equation(14) or (15).
oneof two constants:
Cd(1-•>)for tensionor Cd(-1-•-oo
) for
compression.
Positiveandnegativevaluesof A correspond
to
fracturingandhealing,respectively.
For the 1-D casewe consideronly a fracturingprocess.
We investigate
the damageevolutionof a bodywith unit
compaction
(0>•,>•,) or extension
(•,>0>•) leadsto degrada- lengthandfixeddisplacements
at theboundaries,
u(0)=0 and
tion, while high compactionwith absenceof or low shear u(1)=uo,in a one-dimensional
deformation.Theseboundary
component
(•<•) leadsto healingof thematerial.Thecoeffi- conditionsgive the body stress
cient• maybe estimated
t¾omtheonsetof damage-induced
instabilityor first yieldingin rock mechanicsexperiments.
An
intuitive possibility tbr the secondconstraint,discussedby
Lyakhovsky[1988] for a similar model, is that of a constant
dx
{5- u0
.
E0(1-ot)
(22)
'
bulkmodulus
(Xø+2/3g
ø)under
isotropic
compaction
(•=-•/3). andthe strainis given by
However,back substitutionto (18) giveswith this assumption
a zerorateof healing
for•=-x/3.Thisis a significant
short-
3u
dx
a•=uoEo-(1-ot)
Eo'(1-ot)
'
(23)
coming for a model expectedto describeearthquakecycles
0
containingboth degradationand healing. Agnon and Lyakhovsky [1995] slightly changedthis assumptionand chose Substitutionof (23) into (21) resultsin an equationof damage
only the modulusX to be constant.Under their condition(16) evolutionfor the investigatedbody,
hasthe fbllowing form
Z = Z0 = const; g = go+ot•0¾r; 5'= ot¾,,
(19)
where 5',-is calculatedfrom the conditions(14) and (15) of
convexitylossfor the maximumvalue of the damageparame-
1
dot
=Au•.(1
- ot)1dx
dt
-ot
ß
(24)
0
For a uniforminitial condition(otlt=o=const),
equation(24) has
of the
showsthe dependence
of the criticaldamageon the strainin- the solutionot(t)=At,leadingto a completedestruction
variantratio for )•=[h•. In this case,condition(15) is realized bodywithfinitetime.For a nonuniform
initialcondition
(otlt=0
•:const),the solutionof (24) maybe writtenin thetbrm
firstfor •>• andprescribes
thescale5',-.
With the assumptions(19), equation(18) is rewrittenin a
simpleform containingonly two unknownmodelparameters
ter (ot=l), whenthe straininvariantratio is •=•. Figure1
dot
--dt = CdI2(•-•0) .
(20)
The two modelparameters
• andCd are assumed
in our
ot(x,t)
=1 El7
•[E0
(1
-ot(x,0))]'
-t'(t), (25)
wheref(t) dependson ot(x,0).Equation(25) impliesthat the
deformationlocalizesat a point xo which is the maximumof
the initial damagedistribution.To see that, assumethat at
modelto be materialproperties.As will be discussed
in a subsome time t' the elastic modulus in the interval [x0-c, x0+c]
sequentsection,the parametersmay be constrainedby results
maybe approximatedby a parabola
of rock mechanicsexperiments.Comparingour modelpredictions with laboratorydata of rate and state-dependent
friction,
E(x,t,
) = E0(1
- ot(x,t'
))= a2(x-x0)2
+b2.
(26)
representingaveragepropertiesof sliding surfaces,we find
that in sucha contextthe coefficientCd doesdependon dam- Substitutionof (25) and the parabolicapproximation(26) of
approximationof
age. Accordingly,we adoptin that sectiondifferentrate coef- the modulusinto (23) yieldsa corresponding
the straindistributionin the vicinity of the pointx0,
ficients(equation42) for materialdegradationandhealing.
LYAKHOVSKY
ETAL.'DISTRIBUTED
DAMAGE,
FAULTING,
ANDFRICTION
3u
abuo
3x 2arctan(ac/b)[a2
(x_Xo)2
+b2
]
-
abuo
(27)
27,641
rateof thedeformational
process
will increase
andpartof Go
will becomekineticand radiateacousticwaves.In this case,
only a portionof the initial elasticenergyis convertedto a
surfaceenergy.
-• 8(X-Xo).
)2+b2]
•z[a2(X-Xo
4.2. Two-Dimensional
Equation(27) showsthat the deformationlocalizesin the vicinity of the point xo, where the initial damagedistribution
was maximum,duringa processwherethe damageat x0 approachesunity or elasticmodulusgoesto zero(b/auo-->0).
Now considerthe elasticenergytransferfrom the relaxing
part of the body to surfaceenergyof the localizeddamaged
zone. We assume that (x(x,0)=(x2 over a small interval of
lengthL, and O((X,0)=O•
I (<0(2)elsewhere.In this case,the solution of equation(21) may be representedas ot(x,t)=o•2(t)at
points initially belongingto the interval L, and (x(x,t)=tx•(t)
elsewhere.On the basisof (24), the two time-dependent
functions(x• and (x2satisfy
døtl
=Au(•
.(l-L)(1-ct2)+L(1-oti)
1- ct2
dt
Stress Concentration
Extrapolating
the resultsof the previousexampleto a 2-D
case, one may expect strain and damage localization in a
small region which leadsto stressconcentrationsimilar to lin-
ear elasticity.However,continuousdamageevolutionuntil
totaldestruction
eliminates
theclassical
stresssingularity.
To
illustratethat,we analyzestressamplification
arounda circularholein a 2-D platesubjected
to remoteisotropic
extension.
In a cylindrical
coordinate
system,
onlytheradialcomponent
to•=oXr)of the elasticdisplacements
is nonzero,and the stress
tensoris givenby
'
(31)
(28)
I
dot2
=Au•)
dt
(1-
1- o•
L)(1 - or2)+ L(1 - c•)
j2.
Using(31) in theequationof equilibriumfor the linearelastic
materialgivesa generalsolutionin theform
B
to(r) = Ar +--.
(32)
For small o•1 and 52 and with D->O, equations(28) dictateat
the initial stageof the damageevolutionan increaseof (x2with
a rate greater than the rate of increaseof (Xl. Thus the ratio
(x2/(x•increaseswith time. This is in line with the previousre-
radiusR in the linearelasticplatesubjected
to remoteexten-
sult on strain localization
sionp is
for a continuous
distribution.
The
damage processlocalizesin the interval where the initial
damage is high. At the final stage of the evolution, with
ot2•>l, equations(28) havethe solutions
o•1 = const;
do•
2=
•
Auo
2
St L2(1_0•1)
2'
The damageprocesscontinuesonly in the small L interval,
where (x(t) achieves a unit value at finite time and macroscopicfailure of the body occurs.The energytransferredinto
r
From (32) the stressdistribution around a circular hole with
On.= p 1-
,
000=p 14-
.
Thesolution
hasa stress
amplification
attheboundary
- 2p)anda 1/r2decay.
Asa result
ofthisamplification,
the
damageprocessstartsat the edgeof the holeandit is local-
thehighdamage
region,G=Jovdt,
canbe writtenfromthe izedin a thinboundary
layerhavinghighgradients
ofdamage
andelasticmodulivariations.
The elasticdisplacement
to(r)
hasa corresponding
highgradient
nearr=R,andin thatregion
previousresultsas
G= L(1-L)u02E0(1-Oil
x
x
I [(1
(1_
0•2)&
2 _L)]3
_0•1
)L
+(1_
0t2)(1
dt.
thetermdogdris dominantin relations(31) for thestressten-
(29) sor.Neglecting
thetermogrfora finite
radius
R ofthehole,
Integrating(29) from a time when ix2= (x2to the time of destructionwhen ot2=l, the energyflux G remainsfinite and is
givenby
2(1 - L)
E0u02(1L)[(1-(Zl)L+
2(1-•2)]
2[(1-o•!
)L+(1-•2)(1-L)J
2
stresscomponentthere becomeszero in addition to the radial
component.Stresscomponentsaroundthe circular hole in a
damaged
materialremainfiniteevenfor infinitelysmallradius
of curvature.Insteadof singular-likestressdistribution,the
G=Eøu(•(1-øtl)
-L
andusingtheboundary
condition
o,T= 0, giveinstead
of (32)
thecondition
to=const
(dogdr=O)
at theboundary.
If thedamageat the boundary
reaches
its criticalvalue,the tangential
(30) modelpredicts
an evolving
highgradient
damage
or process
zone.Thegeometry
oftheprocess
zoneandtherateof damage
evolution
arecontrolling
factors
forboththecracktrajectory
andthe rateof crackgrowthin mostengineering
materials
For the limiting situation L-->O the energy transfer is
G0=l/2E0u02(1-Otl).
In thiscaseit is seenthatall theelastic
[Chudnovsky
et al., 1990;Huanget al., 1991].
energyof the relaxedpart of the bodyis transferredto surface 4.3. Stick-Slip Motion
energyof thedamagedzone.If the rateof the damageprocess
The previouscasesneglectviscousstressrelaxationand
(given by the constantA) is sufficientlylarge, and/or the dealonlywithelasticbehavior
of damaged
material.Herewe
lengthof the initially damagedpart is sufficientlysmall, the analyzein one dimensionthe behaviorof a viscoelasticdam-
27,642
LYAKHOVSKY ET AL.: DISTRIBUTED DAMAGE, FAULTING, AND FRICTION
age material subjectedto a constantshearstrain rate. For
The materialevolutiondisplaysanotherstyleif the ratio
simplicity,theviscosityof the materialrl is assumed
constant. tv/tdis of the order of 1 (Figure 2). Relativelylow applied
The constitutive
relationfor a Maxwell viscoelastic
bodyis
strainrates(e-!-•cJq) correspond
to a setof trajectories
tend-
+•,
d/•_•/
x
2Xl
e---
dt
ing
to
stable
creep.
The
shear
modulus
isincreased
up
to
its
x=2•!e.
Higher
strain
rates
(e>>[mcJ•!)
also
produce
material
(34) maximum
valueandtheshearstress
approaches
thevalue
wherex is shearstressand e is total strainrate (equation4).
As discussedabove,the shearmodulusg (equation19) is assumedto be a linear functionof the damageparameter.Thus
the equationof damageevolution(20) maybe represented
by
healingat the initial stageof evolution,but the criticalelastic
strain ec,.is achieved before the shear modulus obtains the
value gcr=•!e/ecr
(horizontaldashedline in Fig. 2). At this
pointthe damageevolutionreversesits direction,anda degradation stagebegins, leading to a stressdrop. The dynamic
stressdrop, which is not analyzedhere, quickly reducesthe
elasticstrainto somelow value, keepingzeroshearmodulus.
There is a locusof trajectoriesthat startsfrom the line g=0
whereec,.is a critical straincorresponding
to the onsetof ma- and showssignificantmaterial recoveringtogetherwith interial degradation.For a givendilationI1, ec,-corresponds
to a creaseof elasticstrain.When the shearstressis largeenough,
certain•. Whenthedeformation
is largerthanecrthedamage thedamageevolutionreversesdirectionagain,anda new degincreases,and when it is lower the damagedecreases.Thus radationstagebegins,leadingto the next stressdrop.If the
the elastic modulus changestogetherwith the damage be- strain rate is so high that the value g•r is larger than the
tween zero (loss of convexityand stressdrop) and its maxi- maximum shear modulus of the material with zero damage,
mum value. Within theselimits the systemof equations(34) thesteadycreepcannot be realized,andonlystick-slipbehavand (35) has a singularsaddlepoint corresponding
to the un- ior occurs.This processforms a repeatinglimit cycle which
stableequilibriumsolution
physicallycorresponds
to stick-slipshearmotionof the viscoeBe
lastic
damage
material.
'c= 2qe; g = •.
(36)
dg=
at Cd•0Tr
• -%2r
'
(35)
œcr
The two couplednonlinearequations(34) and (35) describe
thetemporalvariationsof shearstressandshearmodulusin a
damagematerialsubjected
to a constant
rateof shearstrain.
The Maxwell relaxationtime is tv=Xl/g,while a characteristic
time scale of the damage processis of the order of
5. Estimation
of Model
Parameters
Savageet al. [1996] draw a connectionbetweenmacroscopic
frictionmeasuredon saw-cutspecimensand internalfriction
that characterizes
shearfractureof intact rock. They write the
strength
of
an
intact
rock as the sumoverthe planeof the intd=[14•/Cd•.(œ2--œ2cr),
or td=[14)/CdTr•)I2(•--•})
fora 3-D problem.
cipient
fault
of
both
friction on closed microcracksand
The ratiotv/tdcontrolsthe styleof evolutionof the mechanical
system.A small tv/tdimpliesthat the shearstress'c can in- strengthof the remaininggrains.The approachtakenhereexcreasefbr a givenstrainratewithoutsignificantchangein the tends that connection.We focus our attentionon confining
pressuressufficientfor closureof microcracks,so stressconelastic modulus of the material. If this stress causes the elastic
centration
may ariseonly oncethe shearstressmeetsthe fricstrainto be lessthan critical, then stablecreepis realized.A
higherlevel of the appliedstrainrate resultsin elasticstrain tional criterion.Then favorablyorientedcracksslideand load
largerthan the critical,which leadsto materialdegradation their tips giving rise to damageincrease(equation20). The
and stressdrop.No significantmaterialhealingis expected differencebetweenthe frictional strengthof prefaultedsurafterstressdrop,andthe appliedconstant
strainratedoesnot facesand the strengthof the intactrock is givenby the excess
stressthat is neededto increasethe damagefrom its initial
producesignificantconsequentstress.
value to critical.That stressdifferenceis rate dependent;since
it can be calculatedreadily from the model,it constrainsthe
rate coefficient Cd. In the limit that the strain remains near-
criticalfordamagegrowth(•-•,--> 0), thetimeforfracture
is
infinite,but the strengthis friction-like.In this casea smooth
surfacewill evolvealongwhich the damagewill approachthe
critical level (t•-->l). These featuresare exploredbelow analytically, and illustratedby numerical examples.Our main
concernhereis to estimatethe modelparameters
•, and Cd
whichgovernthestyleof damageevolution.
Theparameter
•
may be estimatedfrom differenttypesof rock mechanicsexperiments;the parameterCdis lesswell constrained.
5.1. Friction and the Onset of Damage
One of the beststudiedrock propertyis the frictionangle.
We relatethe criticalstraininvariantratio•, to the friction
0 0strain
rate gcr=rlO/•
r strain
rate
[.l,
max
Shearmodulus(g)
Figure2. Phaseplanewith evolutionof viscoelastic
damage
materialsubjectedto a constantstrainrate.
angle q) by consideringthe critical shear stressfor MohrCoulombsliding:
'1:= tan((p)lJ
n,
where On is normal stress. Consider a saw-cut interface be-
tween two intact blocksin a friction experimentcardedout
LYAKHOVSKY ET AL.' DISTRIBUTEDDAMAGE, FAULTING, AND FRICTION
Coefficient
L
-
L
L
L
withelastic
moduli•, go)andusingHook'slaw,themodifiedinternal
friction
•) associated
withinitiation
ofthedam-
of friction
0.2 0.4 0.6 0.81
0
0
2
L
34510
L
L
LL
27,643
ageprocess
is givenfromthestress(equation38) as
L
-•-
c) -0.3
v=0.2
..•
•
-0.6
- '
--
. (39)
•0-•/3()•0/g0)
2+4)•0/g0
+2-r(3)•0/g0
+2)
2
'
TakingthePoisson's
ratioof therockscloseto 0.25 (•=g0),
thecorresponding
rangeof variationof •) is foundfrom(39)
ø•25
to be between-0.7 and-1. Our previousestimatebasedon
Byerlee'slaw for axial symmetriccompression
experiments
(37) falls within this range.
5.3. Onset of Acoustic
-1.8
0
I
I
I
I
I
I
I
I
10
20
30
40
50
60
70
80
90
Friction angle
Emission
The emissionof acousticsignalsduringcompressive
failure
experiments
beginsat the onsetof dilatancy,and this activity
accelerates
in proportion
to therateof dilatancywhichis often
observedtogetherwith localizationof deformation[Scholz,
1990 and referencestherein]. Figure 5a shows observed
acousticemission(AE) data of Sammondset al. [1992], ob-
Figure3. Modifiedinternalfriction•)as a function
of friction tained during a deformationalexperimenton Darley Dale
for different Poisson ratios v=0.2, 0.25, and 0.3.
sandstone
witha nominal
strain
rateof 10-5s-• andconfining
pressureof 50 MPa. After a roughlylinearelasticloadingin
the first 2500 s (axial stressup to 220 MPa) thereis a steep
underconfiningpressure.Exceptfor the interface(and per- risein AE associatedwith first yieldingand onsetof cracking.
hapsthinadjacentboundarylayers),thesamplehasnegligible Material degradationthen leadsto a secondyielding,involvdamage.Stressesare transmitted
elasticallyto the interface, ing dynamicinstabilityand abruptstressdropat 3700 s (peak
and the corresponding
straincan be calculatedusingHook's stressof about 290-300 MPa). The observedfirst and second
law and conditionsof triaxial compression(Eli=E22>E33<0).yieldingpointscan be usedto estimatethe modelparameters
The conditionfor fault slip is then [Agnonand Lyakhovsky, • and Cd.In our framework,the damageof the sample,initially assumed
equalto zero(ct=0),startsto increaserapidlyat
1995]
the
first
yielding
when the deformationexceedsthe critical
-4/
,
(37) value•. Calculateddamageevolution,obtainedfrom equation(20), is similarto the experimental
risein AE rate(Figure
5b). The calculations
give lossof convexity,or dynamicstress
where
drop,within a finite periodof time and allow us to estimate
sin(tp)
the secondmodelparameterCd. We obtaina goodfit to the
•0-•/2q2
()•e/g
e+2/3)
2+1
q- 1- sin(tp)/3
'
Physically
(37) meansthatthemodelparameter
•)is some
modificationof the internalfriction.Figure 3 showsthe de-
pendency
of themodified
internal
friction(•) onthefriction
18
ß Candoro limestone
for threevaluesof •1•) corresponding
to valuesof Poisson
ratio0.2, 0.25, and0.3. Thusfor Westerlygranitewith friction
ß Granite
angle•p-30ø [Byerlee,1967],equation
(37) gives•--0.8.
ß Barea sandstone
The result varies little for different rocks with Poisson ratio
values between 0.2 and 0.3.
5.2. Three-Dimensional Faulting Experiments
The modifiedinternalfriction• mayalsobe estimated
using resultsof faultingexperiments
under3-D strainfields
givenbyReches[1983].Figure4 displays
empiricalrelationshipsbetweenthe first and secondstressinvariantsfor the
firstyieldingin thoseexperiments.
Thedataappearto be well
approximated
by therelation
].
J2= rJ2
(38)
2
4
6
First invariant (J•, kbar)
Followingthenotationof Reches[1983],thestress
invariants
areJ1=(•1+(•2+(•3,
J2=(•(•2+(•(•3+(•2(•3
. Most of the experimen- Figure 4. Experimentalrelationsbetweenfirst (J]=(•+c•2+c•3)
talpointsfor differentrocktypescanbefittedby equation
(38) and second(J•=G•tJ2+tJ•tJ3+tJ2tJ3)
stressinvariantsat the first
with theempiricalcoefficient
r=0.20-0.27.Assuming
thatthe yieldingunder3-D stressfield for differenttypesof rocks
by equation
initialrocksamples
havenegligiblelevelsof damageuntilthe [afterReches,1983], andtheirapproximation
firstyielding(i.e., thattheybehaveas linearHookeansolids (38).
27,644
LYAKHOVSKY ET AL.' DISTRIBUTED DAMAGE, FAULTING, AND FRICTION
(b) 2.5E-4
(a)
300
2.0E-4
,
200 •
[ First•I•-•
I yielding
I 7- - - --]-
,-•,•1.5E-4
/
5000
/
/
/
5.0E-5
1000
o
2000
3000
-
/
•
/ :.........
......
•....
[•,.•
......
[,,•
....-,o
o
5000
4000
250
200
•'
-
• 1.0E-4
100
/
350
yielding
I/•l - 300
-
10000
/
I
Secona
L iI
Time (s)
0.0E+0
0
1000
2000
3000
4000
Time (s)
Figure5. (a) Observed
acoustic
emission
(solidline)andaxialstress
(dashed
line)duringdeformational
ex-
periment
onDarleyDalesandstone
witha nominal
strainrateof 10-5s-• andconfining
' pressureof 50 MPa
(Modifiedwithpermission
fromNature[Sammonds
et al., 1992];copyright
Macmillan
Magazines
Limited).
(b)Calculated
damage
evolution
(dcfldt
- solidline)andaxialstress
(dashed
line)bythedamage
rheology
model
with•)=-0.75andCd=0.5
S-].
experimental
results
of Sammonds
et al. [1992]with•=-0.75
andCd=0.5S-•. Thevalueof themodified
internal
friction
is againin goodagreementwith the previousestimates.
portedand the frictionangleis givenby the angleof saw-cut
samples.Figure 6 showsthe resultsof saw-cutWesterly
granitesamplesafterSteskyet al. [1974],whicharein a good
agreementwith Byedee's [1967] friction law for Westerly
granite
5.4. Intact Strength
Most experimentson fractureor intactstrengthof rocksdo
not recordAE and the first yielding is not defined.Only the
secondyieldingis reported.However,thesedata also can be
usedto estimateCd if the strain rate during the loadingis re-
'r = 05 + 0.6o•
(kbar).
Usingthe previousestimate(•=-0.8) for Westerlygranite
basedon the Byerleefrictionlaw, damageevolutionis simu-
latedfor strainrateof 2.7'10
-s s-• [Stesky
et al., 1974]and
differentvaluesof Cd=l, 3, and5 s-]. Threelinesshownin
Figure 6 representmaximum differentialstress(stressenvelopes)versusconfiningpressuresimulatedfor thethreediffer-
2000
1800
entCd.Thecurve
forCd=3s-• fitswelltheexperimental
data
_©Sawcut
of Steskyet al. [ 1974] andgivesanotherestimateof the dam-
o Intact
1600 -
,
age rate constant.
• •
It appearsthat the parameter• is well constrained
and
varieslittle with differenttypesof rocksand loadingconditions.The valuesof the damagerate constantCd varyby an
orderof magnitudebasedon a limitedrangeof experiments
with a similar strain rate of about 10-s s-l. Thus additional
Cd =1s-
1400 -•
1200
-
Cd =3s-
1000 •-
- • •'
Cd mss-
Byerlee
(1967)
constraintsfor Cd with different strain rates are needed.Some
800
of thosemay comefrom fitting simulatedseismicitypatterns
of thetypediscussed
in paper2.
600
400
6. Model Implications
200
6.1. Necking of Thin Plate
The large-scaleextensionof thin sheetsmay, undercertain
conditions,generatefurtherinstabilitiesby the formationof
fault zonesor local necks.First we providean expressionfor
Confiningpressure(MPa)
the
directionof thenecktracefor plasticmaterial.Expressions
Figure6. Frictional
stress
forsawcut
series
(solidcircles)
and
intactseries(opencircles)forWesterlygranite[afterStesky
et for a perfectlyplasticmaterialweregivenby Storenand Rice
al., 1974].Theheavysolidlineshowsthefrictionlawof Byer- [1975] and were used by Agnon and Eidelman [1991] for
rifts.To maintaina constantlengthand
lee [1967].Theheavydashed
linegivestheyieldingstress
for analysisof continental
themodifiedinternalfriction•)=-0.8. Thin solidanddashed rigidblocks,theneckmustform alonghorizontaldirections
of
linesgivethesimulated
yielding
stress
forCa=1,3, 5 s-l.
zero extension,or velocitycharacteristics,
symmetricalabout
0
100
200
300
400
500
LYAKHOVSKYET AL.:DISTRIBUTED
DAMAGE,FAULTING,AND FRICTION
27,645
where Si and S2 are slip directionsand (p is a friction angle.
The ratio S• to S2 gives the angle 0 betweenthe fault plane
andaxisof maximumcompression
as
S,)2- tan2(0)
1-sin((p)
•2-2J
-loll+
sin((p) (40)
Frictionalsliding is characterizedby p=l, and the angle betweena fault traceand the axis of maximumcompression
can
be expressed
throughthe Coulombcriterionas 0--+_(45ø-(p/2).
Simple calculationsshow that equation(40) reproducesexactly thisanglefor p= 1.
Figure 8a shows0 for a perfectlyplasticplate and a brittle
plate with t¾ictionangle(p=30ø, 40ø, 50ø (equation40). Also
Figure7. Direction
of necktracein a thinplatewithrespect shownare resultsof numericalsimulations
with the present
to the axisof maximumcompression.
model of damageevolutionfor a material with the modified
internalfriction•)=-0.8. This corresponds
to a frictionangle
(p=40 ø for Poissonratio v=0.25 (seeFigure 3). Each numerithe principalincremental
strainaxes.For incompressible
cal calculationstartsfrom randominitial damagedistribution.
plasticplates,the direction0 betweenthe neckand the axis of Withtime,•hedamage
increases
andformslocalized
zones
of
maximumcompression
(Figure7) is definedby theratiop
betweentherateof shortening
ϥandextension
œ2
tan2(0)= p= ---.
brittle material. These results, and additional simulations dis-
E2
The orientations
of preferredfaultsin a brittlematerialwere
determined
by Reches
[1983]numerically
andanalytically
for
differentcases.For p<0 theorientation
of a faultplanewith
respectto thecoordinate
systemof theprincipalstrainaxesis
givenfromequation(27) of Reches[ 1983]
S]- x/•'
(1
l'
2 ' +sin((p))•
(a)
cussedin paper 2, illustratethat our damagerheologymodel
is suitablefor the studyof the evolutionof fault branchingand
otherstructuralirregularities.
6.2. State Dependent Friction and Nonlinear Healing
Following and confirming the pioneeringexperimentsof
Rabinovicz[1965] on metals,studiesof rock friction provide
evidencethat the static friction increasesslowly with the duration of stationary contact [Dieterich, 1972]. Dieterich
(b)
50
ß
very high damage(Figure 8b) with orientationdependingon
the parameterp. The valuesbasedon the numericalsimulations fit well the predictionof the fault plane orientationin
Damage (numerical)
Plastic(Storen& Rice, 1975)
40 ...... Brittle
(Reches,
1983)•
cD30
.--'lq•30øt'
I/
10
0
0 0.20.40.60.8 1
P=-œ]/œ2
[E2
Figure8. (a) Faultzoneorientation
in plastic
material
(heavy
line),brittlematerial
withfriction
angle(p=30
ø,
40ø,and50ø (dashedlines),and modelof damageevolutionof a materialwiththemodifiedinternalfriction
•)=-0.8(squares
withvertical
bars).
Theassumed
•)corresponds
to{hefriction
angle
(p--40øforPoisson
ratio
v=0.25(seeFigure3). (b)Numerical
simulation
of localized
highdamage
zones
in a thinplateunder2-D
loadingwithp=-œ•/œ2=
1.Bandsof connected
damage
zones
havedeveloped
atanangleof about25øtothe
principalstressdirectionœ2.
27,646
LYAKHOVSKY ET AL.: DISTRIBUTED DAMAGE, FAULTING, AND FRICTION
[1979, 1981], Ruina [1983] and othersinterpretedresultsof
laboratoryfriction experimentsinvolvinghold times of the
pullingmechanismandjumpsin slidingvelocitiesin termsof
rate- and state-dependent
friction. As was mentionedin the
introduction,the RS friction, like our model, providesa conceptual framework incorporatingall important stagesof an
earthquakecycle. It is thereforeuseful to compareresults
basedon our model predictionswith laboratorymeasurements
of RS frictionalparameters.
In contrastto laboratoryfrictional experiments,our model
does not have sliding surfaces,but rather damaged zones of
weakness.Nevertheless,a comparisonof our model results
with laboratoryRS (andotherfrictional)datais useful,sinceit
allowsus to adaptour modelto macroscopicsituationsinvolving variousfaultingphenomena.
Following equation(20), material healing startswhen the
deformation
is lessthancritical(•<•). As discussed
in the
contextof Figure3, for zeroinitialdamagethe coefficient
•
may be estimatedfrom the frictionof the material.Once this
coefficientis fixed, substitutingthe effective elastic moduli
(13) into (37) we may calculatethe frictionangleas a tinction
of the initial damage (Figure 9). This is not the same as the
static friction that is measuredin laboratoryfriction experiments,but both coefficientshave a similar physicalsense,and
they are expectedto be proportionalto each other [Savageet
al., 1996].
Dieterich [1972] reporteddetailedresultsof frictionalexperimentswith differentnormal stressand hold times up to
10'ssandfittedthestatic
friction
withtheequation
g.,= go+ A,log10
(1+Bt),
(41)
wheret istheduration
oftheholdtimein seconds,
g%0.6-0.8,
A•0.01-0.02,andB--l-2 s-•. Theresults
wereinterpreted
as
i
i
i
i
i
i
i
i
'Onset
ofhealing'
0.9
!
,
-•
•
0.8
,
,
[ Fast
loading"
=-47
(Zhold
=
0.7
-2
-1.5
-1
-0.5
0
0.5
Strain invariantratio (•)
Figure10. Materialrecovering
in stationary
contactfrominitial damage0•0to CZhold
dueto normalstress.
Theincrease
of
static[¾iction
is proportional
to thelogarithm
of theholdtime
duration.
pendson the damageitself.This imposesthat for healingthe
damagerate coefficientis proportional
to the exponentof the
currentlevel of c•. Thus we substitute
a function[br the parameterCa in equation(20) to incorporatedifferent coefficientstbr degradation
andhealingin theform
• ={C,
exp(•-•-2)I2(•-•0)
for
•-<•0,(42)
act
COI2(•-•0
) /br
•>•0
representingenlargementof the real contactarea with time
due to indentationcreeparoundgeometricalasperities.Using
whereCdis constantdescribingthe rateof degradation
andC•
ou.rpreviousassumptionon the relationbetween0• and g• and
andC2 are constants
describingthe rateof healing.With this
employingequation(20) tbr material healing under normal
modificationthe equationfor healinghasthe logarithmicsostressC•n,the damagetheologymodelpredictslinear increase lution
of the static friction with time. This relation. cannot fit the ex-
perimentaldata, and it leadsto a quickerincreaseof g.•than
the logarithmiclaw. This suggests
that the rate of healingde-
c•(t) = o•o -C 2 In(10)x
/ C•I•l
/
xlog,,,
1-•2-2
exp i2(•-•o)t
.
(43)
Thus a damage decrease(healing) starts from some initial
value oq•(Figure 10) which is critical value for any type of
0=-0.8
0.8
v=0.25
detbrmation
•. Normalcompression
reducesthe actualstrain
invariantratiobelow• andprovides
conditions
for healing.
Accordingto (43), the healing is logarithmicin time in
agreementwith Dieterich [1972, 1979] and followingworks.
Comparing(43) with equation(41) for staticfriction,and using the relationbetweendamageand friction(Figure9), we
may suggestthat the coefficientC21n(10)should be of the
0.6
0.4
same order as A (i.e., C2•-10-2), and the relation
C•/C2exp{[oqJC2112}
is of the sameorderas B.
0.2
0
0
0.2
0.4
0.6
0.8
Damage(o0
Figure9. Variationof thefrictionasa functionof damage
Miao et al. [1995] reportedexpeimentalresultsshowing
time changesof Young's modulusduring the healing of
crushedrocksalt.In thoseexperiments,
Young'smodulusincreasesrelativelyfast at the beginningof the process.After
2000-3000min theevolutionratesignificantly
decreases,
producing a logarithmic-likerelationshipbetween Young's
modulusand densificationtime [Miao et al., 1995, Figure
10]. Our model unifies this observed behavior with the ex-
perimental
resultsof Dieterichon statedependent
friction.
LYAKHOVSKY ET AL.: DISTRIBUTED DAMAGE, FAULTING, AND FRICTION
7. Discussion
We have describeda damage rheologymodel based on
thermodynamic
principlesand fundamentalobservations
of
rock deformationin situ and in the laboratory.The model has
manyrealisticfeaturesof 3-D deformation
fieldswhichcanbe
summarized
as follows
7.1. Strength Degradation and Healing
27,647
(equation42) for damageevolutionduringmaterialdegradation and healing.The modifiedevolutionlaw provideslogarithmichealingwith time (equation43) in agreementwith the
experimentalresults.
The damagetheologyof the presentwork is used,together
with additionaldevelopments,
in a follow-uppaperwherewe
simulatethe coupledevolutionof regionalearthquakes
and
faults. A numberof potentialimprovementsto our damage
rheologymodelshouldbe exploredin parallel.
A stateof stresscorresponding
to strain•>•1 leadsto materialdegradation,
with a rateproportional
to thesecondstrain
invariantmultipliedby (•-•).
Conversly,
when•<•,, the
Acknowledgments.We thankJ. Lister,Z. Reches,L. Slepyanfor
usefuldiscussions,and A. Rubin, H. Schreyer,D. McTigue for constructivereviews.A. Agnon acknowledgessupportfrom the BinariohalScienceFoundation,grant BSF-92344. V. Lyakhovskywas supportedby the Golda Meir Foundation.Y. Ben-Zionwas supportedby
the SouthernCaliforniaEarthquakeCenter(basedon NSF cooperative
agreementEAR-8920136 and USGS cooperativeagreement14-08-
sameprocessresultsin materialstrengthening.
At eachtime
the existingvalue of the damageparameterreflectsan integratedhistoryof the damageprocess.The valuesof the damagerateconstants
Cd and Ci in equation(42) definethe dura0001-A0899).
tion of the rock memoryfor positive(degradation)and negative (healing)damageevolution,respectively.
Infinitely large
Cd and C i correspondto zero memory,in which case the References
modelgivesideal elastoplastic
behavior.Infinitelysmall Ca Agnon, A., and A. Eidelman, Lithosphericbreakup in three dimen-
andC• giveHookean
elasticbehavior.
sions:Necking of a work-hardeningplasticplate,J. Geophys.Res.,
96, 20,189-20,194,
7.2. Process Zone
Positivedamageevolution startsat low loadingwhen the
strainbecomes
critical•, andit produces
gradualdamagein
a "processzone" aroundcompletelydamaged("destroyed")
regions.Becauseof the finite size of the processzone, our
modeldoesnot have the unphysicalstresssingularitiesof the
ideal classicalcrack solution.The equationsof stresshave a
regularsolutionat everypoint, and they incorporate
fracture
zoneshavinga finite rate of growth.Suchprocesszonesare
observedin manyexperiments
with rocksand designmaterials [e.g.,Lockneret al., 1991] and their existenceoftengovernstherateand trajectoryof thefailureevolution.
7.3. Aseismic Deformation, Strain Localization, and
Seismic Events
1991.
Agnon,A., and V. Lyakhovsky,Damagedistributionand localization
during dyke intrusion,in The Physicsand Chemistryof Dykes,
edited by G. Baer and A. Heimann, pp. 65-78, A.A. Balkema,
Brookfield, Vt., 1995.
Ambartsumyan,S.A., RaznomodulnayaTeoria Uprugosti (StrainDependenceElasticTheory), 320 pp., Nauka,Moscow, 1982.
Anderson,E.M., The Dynamics of Faulting, 183 pp., Oliver and
Boyd, Edinburgh,Scotland,1951.
Andrews,D. J., Rupture propagationwith finite stressin antiplane
strain,J. Geophys.Res., 81, 3575-3582, I976.
Andrews, D. J., Mechanics of fault junctions, J. Geophys.Res., 94,
9389-9397, 1989.
Andrews, D. J., and Y. Ben-Zion, Wrinkle-like slip pulse on a fault
betweendifferentmaterials,J. Geophys.Res., 102, 553-571, 1997.
Atkinson, B.K., and P.G. Meredith, The theory of subcriticalcrack
growth with applicationsto minerals and rocks, in Fracture Mechanicsof Rock. Academic,San Diego, Calif., editedby B.K. Atkinson,pp. 11-166, 1987.
Aviles, C.A., C.H. Scholz, and J. Boatwright, Fractal analysisapplied
to characteristicsegmentsof the San Andreasfault. J. Geophys.
When damage increases,values of the effective elastic
Res., 92, 331-344, 1987.
moduli decrease,and at some point the elastic energymay
Baer, G., Mechanismsof dike propagationin layered rocks and in
lose its convexity.In that situationthe slope of the stressmassive,poroussedimentaryrocks.J. Geophys.Res., 96, 11,911strainrelationis negative.The strainlocalizesin a high dam11,929, 1991.
age zone with zero to negativeeffectivemoduli, and it may Barenblatt, G.I., The mathematicaltheory of equilibrium cracks in
brittle fracture, in Advancesin Applied Mechanics,pp. 55-129,
becomeunboundedif loadingcontinues.The detbrmationpreAcademic,San Diego, Califi, 1962.
cedingstrain localizationis stableor with negligibleenergy
loss to seismic emission, while the deformationfollowing Bazant,Z.P., and L. Cedolin, Stabilityof Structures.Elastic,Inelastic,
Fractureand Damage Theories,984 pp., Oxford Univ. Press,New
strain localizationis abrupt or seismic.Thus our model acYork, 1991.
counts for aseismic deformation, seismic events, and the
transitions between these two modes of failure.
Simplifiedi-D and 2-D versionsof the damagerheology
Ben-Avraham, Z., and V. Lyakhovsky, Faulting processalong the
northernDead Sea transform and the Levant margin, Geology, 20,
1143-1146, 1992.
Ben-Zion, Y., and J.R. Rice, Earthquake failure sequencesalong a
modellead to analyticalresultsincorporatinga varietyof decellular fault zone in a three-dimensionalelastic solid containing
formationalphenomena,such as strain localization(equation
asperityand nonasperityregions,J. Geophys.Res., 98, 14,10914,131, 1993.
30) and stick-slipbehavior(equation36). A practicalversion
of the general formulation provides a basic expression Ben-Zion,Y., and J.R. Rice, Slip patternsand earthquakepopulations
alongdifferentclassesof faultsin elasticsolids,J. Geophys.Res.,
(equation 20) for the evolution of damage in terms of two
100, 12,959-12,983,
1995.
modelparameters:
a criticaldeformation
• separating
mate- Brace, W.F., and D.L. Kohlstedt, Limits on lithosphericstressim-
rial degradationfrom healing,and a constantCd governingthe
posed by laboratory experiments,J. Geophys. Res., 85, 62486252, 1980.
rate of damage evolution. We have attemptedto constrain
theseparameterswith relevantlaboratoryfrictionand acoustic Budiansky,B., and R.J. O'Connell,Elasticmoduli of a crackedsolid,
lnt. J. Solids Struct., 12, 81-97, 1976.
emission data (Figures 4-8). Additional constraints are
Burridge,R., and L. Knopoff,Model and theoreticalseismicity,Bull.
needed,especiallyfor Cd. A variant of the basic damage
Seismol. Soc. Am., 57, 341-371, 1967.
evolutionlaw, motivatedby the time-dependent
frictionmeas- Byerlee,J.D., Frictionalcharacteristics
of graniteunderhigh confining
urements of Dieterich [1972], contains two different forms
pressure,J. Geophys.Res., 72, 3639-3648, 1967.
27,648
LYAKHOVSKYET AL.:DISTRIBUTEDDAMAGE,FAULTING,AND FRICTION
Carlson,J. M. and J. S. Langer, Mechanicalmodelof an earthquake, Kachanov,M., On the conceptof damagein creepand in the brittlePhys.Rev.A, 40, 6470-6484, 1989.
elasticrange,Int. J. Damage Mech., 3, 329-337, 1994.
Chai, H., Observationof deformationand damageat the tip of cracks King, G., The accommodation
of large strainsin the upperlithosphere
in adhesive bonds loaded in shear and assessment of a criterion for
of the Earth and other solidsby self-similarfault systems:The
fi'acture, Int. J. Fract., 60, 311-326, 1993.
geometricaloriginof b-value,Pure Appl. Geophys.,121, 761-814,
1983.
Chinnery, M.A., Secondaryfaulting, 1, Theoreticalaspects,Can. J.
Earth Sci., 3, 163-174, 1966.
Locknet,D.A. and J.D. Byerlee Developmentof fi'actureplanesdurß creep in granite, •n
ß 2',,d Confere••ce on Acou
Chinnery, M.A., Secondm'yfaulting, 2, Geologicalaspects.Can. J.
•ng
stc
i E'i- s
Earth Sci., 3, 175-190, 1966.
siotz/Microseismic
Activity in Geological Structuresand MateriChudnovsky,A., W.-L. Huang and B. Kunin, Effectof damageon faals, pp. 1-25, TransTech., Clausthal-Zelleffeld,Germany,1980.
tiguecrackpropagationin polystyrene,Polym.Eng. Sci., 30, 1303- Lockner,D.A., and T.R. Madden, A multiple-crackmodelof brittle
1308, 1990.
fracture, 1, Non-time-dependent
simulation,J. Geophys.Res. 96,
Coleman,B.D. and M.E. Gurtin, Thermodynamicswith internal state
19,623-19,642, 1991a.
variables,J. Chem. Phys.,47, 597-613, 1967.
Locknet, D.A., and T.R. Madden, A multiple-crackmodel of brittle
Cox, S.J.D., and C.H. Scholz,Rupture initiationin shearfractureof
fracture. 1. Non-time-dependentsimulation;2. Time-dependent
rocks:an experimentalstudy,J. Geophys.Res., 93, 3307-3320,
simulation.J. Geophys.Res.96, 19,643-19,654, 1991b.
1988.
Lockner,D.A., J.D. Byerlee,V. Kuksenko,A. Ponomarev,
and A. SiCowie,P., C. Vanneste,and D. Sornette,Statisticalphysicsmodelfor
dorin, Quasi-staticfault growth and shearfractureenergyin granthe spatio-temporalevolution of faults, J. Geophys. Res., 98,
ite, Nature, 350, 39-42, 1991.
21,809-21,821, 1993.
Lomakin, E.V., and Yu.N. Rabotnov,Sootnosheniya
teorii uprugosti
deGroot,S.R., and P. Mazur, NonequilibriumThermodynamics.
510
dlya isotropnogoraznomodulnogo
tela (Constitutiverelationships
pp., North-HollandNew York, 1962.
for the strain-dependentelastic body), Iz.v. Akad. Nauk SSSR
Delaney,P.T., D.D. Pollard,J.I. Ziony, and E.H. McKee, Field relaMekh. Tverd. Tela, 6, 29-34, 1978.
tions between dikes and joints: Emplacementprocessesand pa- Lubliner,J., On the thmxnodynamic
foundationsof nonlinearsolids
leostress
analysis,J. Geophys.Res.,91, 4920-4938, 1986.
mechanics,Int. J. Nonlinear Mech., 7, 237-254, 1972.
Dieterich,J.H., Time-dependentfriction in rocks,J. Geophys.Res., Lyakhovsky,V., The effectiveviscosityof a microfracturedmedium,
77, 3690-3697, 1972.
(in Russian) no. 4, 94-98, 1988. (English translation,Izv. Acad.
Dieterich,J. H., Modelingof rockfriction,1, Experimental
resultsand
Sci. USSR Phys. Solid Earth, Engl. Transl., 24(4), 318-320,
constitutive
equations,
J. Geophys.Res.,84, 2161-2168, 1979.
1988.)
Dieterich,J. H., Constitutivepropertiesof faultswith simulatedgouge, Lyakhovsky,V.A., Applicationof the multimodulus
modelto analysis
in Mechanical Behavior of Crustal Rocks:The Handin Volume,
of stress-strainstate of rocks, (in Russian) no 2, 89-94, 1990.
Geophys.Monogr. Ser., vol. 24, pp. 103-120, AGU, Washington,
(Englishtranslation,Iz.v.Acad. Sci. USSRPhys.SolidEarth, Engl.
D.C., 1981.
Transl., 26(2), 177-180, 1990.)
Dugdale,D.S., Yielding of steelsheetscontainingslits,J. Mech. Phys. Lyakhovsky,V.A., and V.P. Myasnikov, On the behaviorof elastic
Solids, 8, 100-104, 1960.
media with microdisturbances,(in Russian) no 10, 71-75, 1984.
Ekland, I., and R. Temam, ConvexAnalysisand Variational Prob(Englishtranslation,Izv. Acad. Sci. USSRPhys.Solid Earth, Engl.
lems,390 pp., Elsevier,New York, 1976.
Transl., 20(10), 769-772, 1984.)
Fitts, D.D., NonequilibriumThermodynamics,173 pp., McGraw-Hill, Lyakhovsky,V.A., and V.P. Myasnikov,On the behaviorof viscoeNew York, 1962.
Freund, L.B., Dynamic Fracture Mechanics, 563 pp., Cambridge
Univ. Press, New York, 1990.
Gibbs, J.W., The Scienttfic Papers, vol. 1, Thermodynamics,New
York, 1961.
lastic medium with microfracturessubjectedto extensionand
shear, (in Russian) no 4, 28-35, 1985. (English translation,/z.v.
Acad. Sci. USSRPhys.Solid Earth, Engl. Transl.,21(4), 265-270,
1985.)
Lyakhovsky,V.A., and V.P. Myasnikov, Relation betweenseismic
Hansen, N.R., and H.L. Schreyer,A thermodynamicallyconsistent
wave velocityand stateof stress.Geophys.J. R. Astron.Soc., 2,
framework for theories of elastoplasticitycoupled with damage,
429-437, 1987.
Int. J. Solids Struct., 31, 359-389, 1994.
Hansen, N.R., and H.L. Schreyer, Damage deactivation.J. Appl.
Mech., 62, 450-458, 1995.
Hoek, J.D., Dyke propagationand arrestin Proterozoictholeiticdyke
swarms, Vestfold Hills, East Antarctica, in The Physics and
Chemistryqf Dykes, edited by G. Baer and A. Heimann, pp. 7995, A.A. Balkema, Brookfield, Vt., 1995.
Hoff, N.J., The necking and rupture of rods subjectedto constant
tensileloads,J. Appl. Mech., 20, 105-108, 1953.
Huang, W.-L., B. Kunin, and A. Chudnovsky,Kinematicsof damage
zone accompanyingcurved crack, Int. J. Fract., 50, 143-152,
1991.
Hull, J., Thickness-displacement
relationshipsfor defo•Tnationzones,
J. Struct. Geol., 10, 431-435, 1988.
Ida, Y., Cohesiveforce acrossthe tip of longitudinalshearcrack and
Griffith'sspecificsurfaceenergy,J. Geophys.Res., 77, 3796-3805,
1972.
Irwin, G.R., Elasticityand Plasticity,in Handbuchder Physik, vol. VI
editedby S. Flugge,pp. 551-590, Springer,Berlin, 1958.
Jones, R.M., Stress-stainrelation for materials with different moduli in
tensionand compression,AIAA J., 15, 62-73, 1977.
Ju, J.W., Isotropic and anisotropicdamage variablesin continuum
damagemechanics,J. Eng. Mech., 116, 2764-2770, 1990.
Kachanov,L.M., On the time to rupture under creep condition(in
Russian), Iz.v.Acad. Nauk SSSR,Otd. Tekh. Nauk, 8, 26-31, 1958.
Kachanov. L.M., Introduction to Continuum Damage Mechanics,
135 pp., MartinusNijhoff, Dordrecht,Netherlands,1986.
Kachanov,M., Elastic solidswith many cracksand relatedproblems,
in Advancesin Applied Mechanics,vol. 30, editedby J. Hutchinsonand T. Wu, pp. 259-445, AcademicPress,1993.
Lyakhovsky,V.A., and V.P. Myasnikov,Acoustics
of rheologically
nonlinearsolids,Phys.Earth Planet. Inter., 50, 60-64, 1988.
Lyakhovsky,V., Y. Podladchikov,and A. Poliakov,Rheological
modelof a fi'acturedsolid,Tectonophysics,
226, 187-198,1993.
Lyakhovsky,
V., Z. Ben-Avraham,
andM. Achmon,The originof the
Dead Searift, Tectonophysics,
240, 29-43, 1994.
Lyakhovsky,V., Z. Reches,R. Weinberger,and T.E. Scott,Nonlinear
elasticbehaviorof damagedrocks,Geophys.J. Int., 130, 157-166,
1997.
Malvern, L.E., Introduction to the Mechanicsof a ContinuumMedium, 713 pp., Prentice-Hall,EnglewoodCliffs, N.J., 1969.
Miao, S., M.L. Wang, and H.L. Schreyer,Constitutivemodelsfor
healingof materialswith applicationto compactionof crushedrock
salt,J. Appl. Mech., 62,450-458, 1995.
Mosolov,P.P., and V.P. Myasnikov,Variational methodsin flow theory of visco-plasticmedia, (in Russian),Prikl. Math. Mekh., 29,
468-492, 1965.
Myasnikov,V.P., V.A. Lyakhovsky,and Yu.Yu. Podladchikov,Nonlocal modelof strain-dependent
visco-elastic
media,(in Russian),
Dokl. Acad. Sci. USSR, 312, 302-305, 1990.
Nishihara,M., Stress-strainrelationof rocks,DoshishaEng. Rev., 8,
32-54, 1957.
Onsager,L., Reciprocalrelationsin irreversibleprocesses,
Phys.Rev.,
37, 405-416, 1931.
Okubo,P.G., and K. Aki, Fractalgeometryin the San Andreasfault
system,J. Geophys.Res.,92, 345-355, 1987.
Palmer,A. C., andJ. R. Rice, The growthof slip surfaces
in the progressivefailure of over-consolidated
clay, Proc. R. Soc. London,
Ser. A, 332, 527-548, 1973.
LYAKHOVSKYET AL.: DISTRIBUTEDDAMAGE,FAULTING,AND FRICTION
Papa, E., A damagemodelfor concretesubjectedto fatigueloading.
Eur. J. Mech. A Solids, 12, 429-440, 1993.
Papageorgiou,A. S., and K. Aki, A specificbarrier model for the
quantitativedescriptionof inhomogeneous
faultingand the prediction of stronggroundmotion,II, Applicationsof the model,Bull.
Seismol. Soc. Am., 73, 953-978, 1983.
Prigogine,I., Introductionto Thermodynamics
of Irreversible Processes,! 15 pp., Thomas,Springfield,Illinois, 1955.
Rabotnov,Y.N., Creep Problemsin StructuralMembers,822 pp.,
North-Holland, Amsterdam, 1969.
Rabotnov,Y.N., Mechanicsof DeformableSolids, (in Russian),712
pp., Science,Moscow,1988.
Rabinovicz, E., Friction and Wear of Materials, pp. 97-99, John
Wiley, New York, 1965.
Reches,Z., Faulting of rocks in three-dimensional
strain fields, II,
Theoreticalanalysis,Tectonophysics,
95, 133-156, 1983.
Reches,Z., Evolutionof fault patternsin clay experiments,Tectonophysics,145, 141-156, 1988.
Reches, Z., and D.A. Lockner, Nucleation and growth of faults in
brittlerocks,J. Geophys.Res.,99, 18,159-18,173,1994.
Rice,J. R., Spario-temporal
complexityof slip on a fault,J. Geophys.
Res., 98, 9885- 9907, 1993.
Rubin, A.M., Propagationof magma filled cracks,Anu. Rev. Earth
Planet. Sci. 8, 287-336, 1995a.
27,649
Sedov,L.I., Variational methodsof constructingmodelsof continuos
media,in IrreversibleAspectsof ContinuumMechanics,pp. 1740 Springer-Verlag,New York, 1968.
Segall,P., and D.D. Pollard,Nucleationand growthof strike slip
faultsin granite,J. Geophys.Res.,88, 555-568, 1983.
Sornette,D., P. Miltenberger,and C. Vanneste,Statisticalphysicsof
fault patternsself-organized
by repeatedearthquakes,
Pure Appl.
Geophys.,142,491-527, 1994.
Stesky,R.M., W.F. Brace,D.K. Riley, and P.Y.F. Robin, Frictionin
faultedrockat hightemperature
andpressure,
Tectonophysics,
23,
177-203, 1974.
Storen,S., and J.R. Rice, Localizedneckingin thin sheets,J. Mech.
Phys. Solids, 23, 421-441, 1975.
Swanson,P.L., Subcriticalcrack growth and other time and environment-dependent
behaviorin crustalrocks,J. Geophys.Res., 89,
4137-4152, 1984.
Turcotte, D.L., Fractals and fragmentation,J. Geophys.Res., 91,
1921-1926, 1986.
Valanis, K.C. A theory of damagein brittle materials,Eng. Fract.
Mech., 36, 403-416, 1990.
Walsh,J.B., The effectof crackson the uniaxialelasticcompression
of rocks,d. Geophys.Res.,70, 399-411, 1965.
Ward, S.N., A syntheticseismicitymodel for SouthernCalifornia:
Cycles,probabilities,and hazard,J. Geophys.Res., 101, 22,393-
22,418, 1996.
Rubin, A.M., Why geologistsshouldavoidusing"fracturetoughness"
R., Z. Reches,T.S. Scott,andA. Eidelman,Tensileprop(at leastfor dikes), in The Physicsand Chemistryof Dykes, edited Weinberger,
ertiesof rocksin four-pointbeamtestsunderconfiningpressure,
by G. Baer and A. Heimann,pp. 53-65, A.A. Balkema,Brookfield,
pp. 435-442, in ProceedingsFirst North AmericanRock MechanVt., 1995 b.
ics Symposium,Austin,Tex., 1994.
Rudnicki, J.W., and J.R. Rice, Conditions for the localization of deformationin pressure-sensitive,
dilatant materials,J. Mech. Phys. Weinberger,R., G. Baer, G. Shamir,and A. Agnon,Defo•Tnation
Solids, 23, 371-394, 1975.
Ruina, A., Slip instabilityand statevariablefrictionlaws,J. Geophys.
Res., 88, 10,359- 10,370, 1983.
Robertson,E.C., Relationshipof fault displacementto gouge and
brecciathickness,Min. Eng., 35, 1426-1432, 1983.
Robinson,E.L., Effect of temperaturevariationon the long-termrupture strengthof steels,Trans.ASME, 174, 777-781, 1952.
Sammonds,P.R., P.G. Meredith,and I.G. Main, Role of porefluidsin
the generationof seismicprecursorsto shearfracture,Nature, 359,
228-230, 1992.
Savage,J.C., J.D. Byeflee,and D.A. Lockner,Is internalfrictionfriction?,Geophys.Res.Lett., 23, 487-490, 1996.
Scholz,C.H., The Mechanicsof Earthquakesand Faulting, 439 pp.,
CambridgeUniv. Press,New York, 1990.
Schreyer,H.L., and M.K. Neilsen, Analytical and numericaltestsfor
loss of material stability, lnt. d. Num. Methods Eng., 39, 1721-
bands associatedwith dyke propagationin porous sandstone,
MakhteshRamon,Israel,in The Physicsand Chemistryof Dykes,
editedby G. Baer and A. Heimann,pp. 95-115, A.A. Balkema,
Brookfield, Vt., 1995.
Yukutake, H., Fracturingprocessof graniteinferredfrom measurementsof spatialand temporalvariationsin velocityduringtriaxial
deformation,
J. Geophys.Res., 94, 15,639-15,651,1989.
A. AgnonandV. Lyakhovsky,Instituteof EarthSciences,
The
Hebrew University, Givat Ram, Jerusalem,91904 Israel. (e-mail:
vladi•cc.huji.ac.il.)
Y. Ben-Zion, Departmentof Earth Sciences,University of
Southem Califomia, Los Angeles, CA 90089-0740. (e-mail:
benzion•terra.usc.edu)
1736, 1996a.
Schreyer,H.L., and M.K. Neilsen, Discontinuousbifurcation states
for associatedsmoothplasticityand damagewith isotropicelasticity, lnt. d. SolidsStruct., 33, 3239-3256, 1996b.
(ReceivedOctober29, 1996;revisedJune4, 1997;
acceptedJune 30, 1997.)