Dilatancy, compaction, and failure mode in Solnhofen limestone

Transcription

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 105, NO. B8, PAGES 19,289-19,303, AUGUST 10, 2000
Dilatancy, compaction, and failure mode in Solnhofen
limestone
Patrick Baud
Departmentof Geosciences,
StateUniversityof New York at StonyBrook
Alexandre
Schubnel
Laboratoirede G6ologie,D6partement
Terre,Atmosphere
et Oc6an,EcoleNormaleSup6rieure,
Paris
Teng-fongWong
Departmentof Geosciences,
StateUniversityof New York at StonyBrook
Abstract. Failuremodeis intimatelyrelatedto porositychange,andwhetherdeformation
occursin conjunction
with dilatationor compaction
hasimportantimplications
on fluid
transportprocesses.
Laboratorystudieson the inelasticandfailurebehaviorof carbonate
rockshavefocusedontheveryporousandcompactend-members.
In thisstudy,experiments
wereconducted
onthe Solnhofenlimestoneof intermediate
porosityto investigate
the
interplayof dilatancyandshearcompaction
in controllingthebrittle-ductile
transition.
Hydrostatic
andtriaxialcompression
experiments
wereconducted
onnominallydry samples
at confiningpressures
up to 435 MPa. Two conclusions
canbe drawnfrom our new data.
First,shear-enhanced
compaction
canbe appreciable
in a relativelycompactrock.The
compactiveyield behaviorof Solnhofenlimestonesamples(with initial porositiesaslow as
3%) is phenomenologically
similarto thatof carbonate
rocks,sandstone,
andgranular
materialswith porosities
up to 40%. Second,compactivecataclastic
flow is commonly
observed
to be a transientphenomenon,
in thatthe failuremodeevolveswith increasing
strainto dilatantcataclasticflow andultimatelyshearlocalization.It is therefore
inappropriate
to view stress-induced
compactionanddilatancyasmutuallyexclusive
processes,
especiallywhenlargestrainsareinvolvedasin manygeologicalsettings.
Several
theoreticalmodelswere employedto interpretthe micromechanics
of the brittle-ductile
transition.The laboratorydataon the onsetof shear-enhanced
compaction
arein reasonable
agreementwith Curranand Carroll's [ 1979]plasticporecollapsemodel.In thetransitional
regime,theStroh[ 1957]modelfor microcrack
nucleation
dueto dislocation
pileupcanbe
usedto analyzethetransitionfrom shear-enhanced
compaction
to dilatantcataclastic
flow. In
the brittlefaultingregimethe wing crackmodelprovidesa consistent
description
of the
effectof grainsizeonthe onsetof dilatancyandbrittlefaulting.
1. Introduction
It is importantto havea fundamentalunderstanding
of the
mechanicsof the brittle-ductiletransitionin many geologic
problems.The deformationmechanismsoperativeon scales
rangingfrom the microscopicto macroscopic
have profound
influence on the microstructurepreserved in naturally
deformed rocks, the limits on the state of stress in the
lithosphere, the spatiotemporalevolution of stress and
deformationduringthe earthquakecycle, and the couplingof
crustaldeformationand fluid transport.It is now recognized
that the brittle-ductile
transition
is associated with a broad
regime of complex deformationand failure mode. The
mechanicsof the transition is sensitively dependenton
extrinsicvariables(that includeconfiningand porepressures,
temperature,strain rate, and fluid chemistry)as well as
intrinsicvariables(that include modal composition,grain
size,andporosity).
Previousstudieshave demonstrated
that porositychange
and failure mode are intimately related. On one hand,
dilatancy is universally observed as a precursorto the
inceptionof shearlocalization
in the brittlefaultingregime
[Brace, 1978]. On the other hand, plasticflow (associated
with crystalplasticityand diffusivemasstransfer)doesnot
involve any volumetric change [Paterson, 1978]. In the
transitional regime of cataclasticflow (manifestedby
homogeneously
distributedmicrocracking)the scenariois
morecomplicated
sincetheporespacemaydilateor compact
in response
to an appliedstressfield.
In relativelyporoussilicaterocks[e.g.,HandinandHager,
1957; Hadizadeh and Rutter, 1983; Shimada, 1986] a
transition
of
the
failure
mode
from
brittle
fracture
to
cataclasticflow can be inducedat room temperatureby an
Copyright
2000by theAmericanGeophysical
Union.
Papernumber2000JB900133.
0148-0227/00/2000JB900133509.00
increasein pressureonly. In the cataclastic
flow regimethe
porespaceundergoessignificantinelasticcompaction
while
the rock strainhardens[Edmondand Paterson,1972]. The
19,289
19,290
BAUD
ET AL.' COMPACTION
transitionpressuredecreaseswith increasingporosity and
grainsize [Wonget al., 1997]. In contrast,relativelycompact
granite, gabbro, dunite, and eclogite remain in the brittle
fracture regime when tested at room temperature and
confining pressuresup to 3 GPa [Shimadaet al., 1983]. It
seems that even at the maximum confining pressure
accessible in the laboratory, elevated temperaturesare
necessaryto activate crystal plasticity processesso that
silicate rocks of relatively low porosity can undergothe
transitionto cataclasticand ultimately crystal plastic flow
[Tullis and Yund,1992;Hirth and Tullis, 1994].
In calcite rocks the brittle-ductile
different
attributes
from
silicate
transition has somewhat
rocks.
Limestones
mechanisms
are different
from
those
on
end-members
LIMESTONE
350
6OO
300
200
435
5OO
100
400
5O
300
and
marbles,eventhe oneswith very low porositylike the Carrara
marble, undergo the brittle to plastic transition at room
temperature for confining pressuresaccessiblein the
laboratory[Robertson,1955; Paterson, 1958; Heard, 1960;
Rutter, 1974]. This is probably due to the fact that calcite
requiresrelatively low shearstressesto initiate mechanical
twinning and dislocationslip even at room temperature
[Turner et al., 1954; Griggs et al., 1960]. The transition
occursat a pressurethat decreases
with increasinggrain size
[Fredrich et al., 1990]. In Carrara marble, appreciable
dilatancyoccursevenwhenshearlocalizationis inhibitedand
the samplestrainhardens[Fredrichet al., 1989;Fischerand
Paterson, 1989]. That dilatancy occurs while the compact
marble undergoes cataclastic flow suggests that the
deformation
SOLNHOFEN
200
lOO
o
o
1
2
3
4
5
Axial strain (%)
in the
Figure 2. Representative
mechanicaldatafor selectedtriaxial
compactivetype of cataclasticflow observedin a porous compression
experiments.
Differentialstressis plottedversus
axial strain.
sandstone
or chalk [Teufelet aL, 1991].
Previous experimental studies of the brittle-ductile
transition
have
focused
the
two
with
relatively low or high porositiesthat are associatedwith
purelydilatantor compactivecataclastic
flow. In a rockwith
porosityintermediatebetweenthesetwo end-members,
strain
hardeningand cataclasticflow are expectedto arisefrom the
complexinterplayof dilatant and compactivedeformation
mechanisms.In this transitionalregime it is unlikely that
compactionand dilatancy can be consideredas mutually
exclusive deformation processes,especially when large
strainsare involved as in many geologicalfield settings.
500
400
Walsh (1965): calcite aggregate
_wifh
2.8%
spherical
pores
••
300
crack closing-
200
pressure
_
Indeed,the interplayof compaction
anddilatancyis a central
featurein manyconstitutive
modelsfor porousgeomaterials
[e.g.,Fossumet al., 1995].
In an attemptto elucidate
the interplayandpartitioning
of
dilatancy and compaction we conducted detailed
measurementsof the stress and strain, as well as their
influence on the failure mode of the Solnhofen limestone
lOO
•)
iI
.
ß
I
_
(withaninitialporosity
of 3%). Althoughmanyaspects
of the
brittle-ductile
transitionand creepbehaviorin this rockhave
beenstudied[Robertson,
1955;Heard, 1960;Byedee,1968;
Edmond and Paterson, 1972; Rutter, 1972; $chmid et al.,
ß
0.2
crack porosity:
~ 0.2%
0.4
0.6
0.8
Volumetric strain (%)
1
1977; Fredrich et al., 1990; Rennet and Rumreel,1996;
FisherandPaterson,1989],to ourknowledgethe compactive
behaviorwas not investigated
systematically.
Motivatedby
the laboratory
data,we havedeveloped
theoretical
models
(involving
various
modes
of
grain-scale
fracture
and
crystal
Figure 1. Mechanicaldatafor a hydrostaticexperimentup to
to analyzethe micromechanics.
We alsodiscuss
450 MPa. Volumetric strain is plotted versus confining plasticity)
of themicromechanical
modelsontheeffect
pressure.The linear part of the curve is fitted by linear theimplications
andgrainsizeon the brittle-ductile
transition
in
regression.Closing crack pressureand microcrackporosity of porosity
are indicated.
calcite and silicate rocks.
BAUD ET AL.' COMPACTION
,1 - 2vef
• = 0.32
SOLNHOFEN
LIMESTONE
19,291
jacketed sample in orthogonaldirections.The volumetric
/
•
-
350
directions,respectively.This formula neglectssecond-order
//
-
strainwascalculated
usingtherelation
ev= ell+ 2%_,where
• IIand%_arethestrains
measured
intheaxialandtransverse
contributionsof strainsto the volume change that may be
/
appreciable
at relativelylargestrains.
0.5
o
2.2. Experimental Procedure
The jacketed sampleswere stressedin the conventional
triaxial configuration at room temperature.Kerosene was
used as the confining medium. The experimentswere
performedat confining pressuresranging from 10 to
435 MPa. The axial load was measured with an external load
i
200
-0.5
cell with an accuracy of 1 kN. The displacementwas
measuredoutsidethe pressurevesselwith a displacement
transducer
(DCDT) mountedbetweenthe movingpistonand
the fixed upper platen. The axial displacement
was servocontrolledat a fixed rate (corresponding
to a nominalstrain
rateof 1.3x 10'5s']).
The load, displacement,and strain gauge signalswere
-1
/,
,
0
, i I , , , ,
1
, , , , I
2
, , • • I ] , • • I
3
4
5
acquired
bya 14-bitA/Dconverter
ata sampling
rateof 1 s']
with resolutions
of 0.3MPa, 1 gm and 10's, respectively.
Owing to electrical noise and sample shortening,actual
Axial strain (%)
uncertainty
in strainwas2 x 10'3 (whencalculated
fromthe
DCDT
signal)
and
10
'5
(when
measured
directly
by
the
strain
were available
compressionexperiments.Axial strain is plotted versus gauges).Acousticemission(AE) recordings
transducer
(PZT-7, 5 mm diameter,1 MHz
volumetric strain (for the nonhydrostaticpart of the via a piezoelectric
experiments).
The confiningpressure(in MPa) is indicatedon
eachcurve.For reference,the linear portionis fitted with a
line thatcorresponds
to a Poisson'sratioof 0.34.
435
6OO
2. Mechanical
350
Data
2.1. Sample Material and Preparation
Solnhofenlimestoneis composedof 99.9% of calcite.We
determined
the totalporosityof eachof our samples
thatwere
coredfrom the sameblock. After measuringthe weight and
the volumeof the samplewe calculateditsporosity(assuming
that it was entirelycomposedof calcite).We foundthat the
total porosityrangesfrom 2.7% to 3.3%, with an arithmetic
meanof 3.0%. Our blockis lessporousthansamplesusedin
most previousstudies(with total porositiesranging from
3.7% to 5.9%), exceptfor Heard's [1960] samplesthat had
1.7%of porosity.
Wealsotlsedthesaturation
technique
to
determinethe interconnected
porositiesin selectedsamples,
whichwere foundto be -1.5%, indicatingthat the limestone
containsa largeproportionof unconnected,
vugulartype of
pores.We did not determinethe averagegrainsize,but it was
reportedto be -5 gm in previousstudies[Robertson,1955;
Heard, 1960; Edmond and Paterson, 1972; Fredrich et al.,
1990; Renner and Rumreel, 1996; Fisher and Paterson, 1989;
Rutter, 1972].
Rocksampleswere coredto a diameterof 18.1 mm andcut
to a lengthof 38.1 min. The samplesweredriedin vacuumat
80øC for severaldays. They were first jacketedwith a thin
copperfoil, and heat-shrinkpolyolefinetubingswere then
used to separatethe samplefrom confiningpressure.The
longitudinaland transversestrainswere measuredby electric
resistancestraingauges(BHL Constantanfoil FAE-50-12-S6
for measurements
of strainsup to 5%). A hydrostatic
pressure
of 50 MPa was first appliedto each sampleto "season"the
copperjacket, and then the straingaugeswere gluedon the
5OO
300
•
400
200
.•.__ C*'
300
lOO
5o
200
•
lOO
0
0.5
1
1.5
2
compression
experiments.Volumetricstrainis plottedversus
meanstress.The confiningpressure(in MPa) is indicatedon
each curve. For reference,the hydrostatis shownas heavy
dashedline. The criticalstressstatesfor dilatancyC', onsetof
shear-enhanced
compactionC*, and transitionfrom shearenhancedcompactionto dilatancy C*' are indicatedby the
arrowsfor experimentsperformedat confiningpressuresof
50, 300, and200 MPa, respectively.
19,292
BAUD ET AL.: COMPACTION
SOLNHOFEN
LIMESTONE
longitudinal
resonantfrequency)positioned
on theflat surface
of one of the end plugs.However,duringall the seriesof
experimentsthat we did, no significantAE activity was
observedover the backgroundnoiseof our laboratory.The
AE datawerenot of usein thisstudy.
samplevolumedecreased
monotonically
with deformation
up
to a finite strainof 5% (Figure3). Shearlocalizationwas not
evidentin the deformedsamples.As shownin Figure4, the
triaxialcurvesfor a givenconfiningpressurecoincidedwith
the hydrostatup to a criticalstressstate(indicatedby C* for
the dataat 435 MPa confiningpressure),
beyondwhichthere
was an accelerateddecreasein volume in comparisonto the
2.3. Stress,Strain, and PorosityChange
hydrostar.At stresslevels beyondC* the deviatoricstress
We will adoptthe convention
that compressive
stresses field provided significant inelastic contributionto the
and compactive
strainsare positive,and we will denotethe compactivestrain, and this phenomenonis referredto as
compaction"[Curran and Carroll, 1979;
maximumandminimum(compressive)
principalstresses
by "shear-enhanced
Wong
et
al.,
1997].
The differentialstressat C* is negatively
o.1ando.3,respectively.
Figure1 shows
theevolution
of the
volumetric
strainfor a hydrostatic
experiment
upto 450MPa correlatedwith the confiningpressureand meanstress(Table
of confining
pressure.
Thehydrostatic
response
wasnonlinear 1).
regime(100MPa 5 o.35 300MPa)the
up to a pressure
of- 200 MPa, beyondwhichthe stress-strain In thetransitional
curve becamelinear with a slope corresponding
to a evolution of volumetric strain showedthree distinct stages
(Figure 5). Initially, the triaxial and hydrostaticdata for
compressibility
of•3ef
f -- 1.6x 10'll Pa'l.
Figure
2 shows
thedifferential
stress
(o.l- o.3)asa function
of axialstrainin selected
experiments
conducted
at confining
pressures
(o.3)ranging
from50 to 435 MPa.Exceptfor the
experiment
at the lowestconfiningpressure
of 10 MPa, both
theaxialandradialstraindatawereacquired.
Thevolumetric
straindatafromtheseexperiments
areplottedasa function
of
the axial strain in Figure 3 and versusthe mean stress
(o.1+ 2o.3)/3in Figure4. Forreference,
thehydrostar
is also
shown(asthedashedline)in Figure4.
At confining pressuresup to 50 MPa the mechanical
responseand failure mode are typical of the brittle faulting
regime.The differentialstressattaineda peak,beyondwhich
strain softening was observed (Figure 4). In some
experiments
the unstablefailure resultedin breakageof the
straingages.The peakstressshowsa positivecorrelation
with
the confiningpressureand mean stress(Table 1), and the
failed samplesshow macroscopic
failure orientedat-30 ø
with respectto the o.1 direction.As indicated
for the
experimentat confiningpressureof 50 MPa, the onset of
dilatancyC' can be identifiedon Figure4 as the pointwhere
the volume of the triaxially compressedsample became
greater than that of the hydrostatically compressed
counterpart
at the samemeanstress.This impliesthatat stress
levelsbeyondC' the deviatoricstressfield inducedthe pore
spaceto dilate. The differentialstresslevel at C' showeda
positivepressure
dependence
(Table 1).
volumetric
strain as function of mean stress coincide. The
secondstagewas manifestedby shear-enhanced
compaction,
with the onset marked by the critical stress state C*.
However, these samples in the transitional regime
consistently
switchedfrom compactionto a third stageof
dilation after they had undergonecertain amountof strain
hardening.As indicatedin Figure4 for the experimentat 200
MPa confining pressure,the critical stress state at the
transitionfrom compactiveto dilatantcataclastic
flow will be
denotedby C*'.
Visual examinationof all the deformedsamplesfrom this
transitionalregime(up to the maximumaxial strain
did not reveal any signsof shearlocalization,eventhougha
transient phase of strain softening was observedin the
experimentat confining pressureof 100 MPa. While the
differentialstressat the onsetof shear-enhanced
compaction
in this transitional regime shows a negative pressure
dependencethat is comparable to the C* data for
o.3> 350MPa,thedifferential
stress
at C*' shows
a slightly
positivecorrelationwith mean stressand confiningpressure
(Table 1).
3. Discussion
Our laboratory data have important implicationsin the
interpretationof compactiveand dilatant processesin the
of the laboratory
At confiningpressures
o'3_>350MPa the mechanicalcrust.However,quantitativeextrapolations
responseand failure mode are typical of the compactive data to crustal settings would require a fundamental
cataclastic
flow regime.The slopesof the differentialstress- understandingof the micromechanicsof the brittle-ductile
axial strain curve were nonnegative(Figure 2), and the transition, which cannot be formulated without a realistic
Table 1. Compilationof MechanicalData for the SolnhofenLimestone
Confining
C'
Pressure, 0.1' 0.3'
MPa
MPa
10
25
35
50
200
225.5
245
C'
(0.1+20.3)/3'
MPa
92
112
132
PeakStress
0.1' 0.3'
MPa
326
360
409
428
PeakStress
(0.1+ 20.3)/3'
MPa
120
145
177
197
C*
0.1= 0.3'
MPa
C*
(0.1q-20.3)/3'
C ,t
0.1' 0'3'
MPa
MPa
C ,t
(0.!+ 20.3)/3
,
MPa
....
....
....
....
100
200
250
300
....
....
....
....
375
354
336
315
225
318
362
405
475
509
'510
526
258
370
420
475
350
395
435
....
....
....
270
231
179
440
470
495
-
-
BAUD
ET AL.' COMPACTION
Pc = 200 MPa
400
-
,
shear-induced
dilation
I
SOLNHOFEN
LIMESTONE
19,293
suchcompactivedeformationmay be transientin nature.As
mappedin the stressspace(Figure 6), the yield stresses
for
shear-enhancedcompaction in Solnhofen limestone are
initially describedby a compactiveyield envelope with
negativeslopethat expandswith strainhardening,gradually
evolving to a dilatant yield envelopewith positive slope.
Although it has not been explored systematically,this
intriguingphenomenon
of compactionasa transientprecursor
leadingto dilatancyseemsto be not very uncommon.In the
literature,Schocket al. [1973] reportedphenomenological
behaviorin the Lance sandstone(with 7.5% porosity)that is
very similarto our observations
for Solnhofenlimestone.For
comparison,we determinedfrom their stress-strain
curves
valuesof C*, C*', C', andbrittlestrengthand plottedthe data
in Figure7.
It is therefore inappropriate to view stress-induced
compactionand dilatancyas mutually exclusiveprocesses,
especially when large strains are involved, as in many
geologicalfield settings.Our experimentswere conductedup
350
shear.
enhanced
compac
300
250
I
,.•
no shear-induced
to a maximum axial strain of-5%.
It is conceivable that more
complexbehaviorwouldoccurif deformation
wasto develop
to even larger strains. The brittle-ductile transition is
.... •,.... , .... I ....
200
associatedwith a broad spectrum of highly complex
0.4
0.5
0.6
0.7
0.8
0.9
1
deformationmechanisms,failure modes,and fluid transport
processes.
While dilatantcataclasticflow may be a transient
precursorfor the inceptionof shearlocalizationand brittle
Figure 5. The mean stressis plotted as a functionof faulting, shear-enhancedcompactionmay also evolve to
volumetricstrainfor an experiment
performed
at 200 MPa of dilatantcataclasis.Extrapolationof our laboratorydatato the
confiningpressure.The sampleshowedcompactiveand temporalandspatialscalesof geologicsettingswould require
dilatant behavior. For reference,the hydrostatis shown as a fundamentalunderstanding
of the micromechanics
of the
heavydashedline.
brittle-ductiletransition,which may involve the complex
interplayof microcracking,crystal plasticity,and diffusive
masstransferprocesses
in Solnhofenlimestone.
conception
of theporegeometryanddefectstructure.
We will
first use the mechanicaldata to infer the pore structureof 3.2. Pore Structure of Solnhofen Limestone
Solnhofenlimestoneand to developtheoreticalmodelsto Inferred From Elastic Properties
analyze different stages of the brittle-ductiletransition.
Relevant micromechanical
models cannot be formulated
Specifically,
a wingcrackmodelwill be employed
to analyze
the developmentof dilatancyand brittle faulting,a plastic
pore collapsemodelfor shear-enhanced
compaction,
and a
dislocationpileup model for the transitionfrom shearenhancedcompactionto dilatancy.Thesemodelselucidate
the damagemechanics
as well as the importantinfluencesof
porosityandgrainsizein the brittle-ductile
transition.
withouta realisticconception
of the poregeometryanddefect
structure.Many geometricattributesof the porespacecan be
inferredfrom the elasticbehavior.Accordingto Walsh[1965]
the hydrostatin Figure 1 is characteristic
of the response
of a
rockporespacemadeup of two typesof cavities:microcracks
of relativelylow aspectratiosandrelativelyequantpores.The
progressiveclosure of the microcracksunder increasing
pressurewas manifestedby the nonlinearityobservedduring
3.1. Interplay of Compaction and Dilatancy
initial pressurization.
At and abovea pressureof- 200 MPa
During Cataclastic Flow
the microcrackswere closedand the hydrostatbecamelinear.
Recentstudieshaveemphasized
thattherearefundamental As shown in Figure 1, one can then use Walsh's [1965]
differencesin the mechanicaland transportpropertiesof analysisto inferthatthemicrocrackporosityis -0.2% for our
compactive
anddilatanttypesof cataclastic
flow asobserved Solnhofenlimestoneand that the aspect ratios of the
in high-andlow-porosity
silicaterocks,respectively
[Wonget microcracksare of the orderof 10'3 or less.
al., 1997;Zhu and Wong,1997].However,therehavebeena
The linear portion of the hydrostat representsthe
paucityof datafor addressing
the question:
At whatporosity hydrostaticresponse
of a rock with pore spacemadeup of
will thetransitionfromdilatantto compactive
cataclastic
flow relativelyequantpores,with a porosityof-2.8% (equalto the
occur?A value of-5% was suggested
by Brace [1978], and difference between the total and microcrack porosity).
here we demonstratethat shear-enhanced
compactionand Additionalinformationon the poregeometrycanbe obtained
porecollapsearereadilyobservedin a limestone
thathasan by comparisonof the elasticmoduli with theoreticalresults
initialporosityas low as 3% andporesizethat is probably for an elastic mairix embeddedwith dilute concentrationof
sphericalpores[Walsh,1965; Walshand Brace, 1966].With
comparable
to theaveragegrainsize(-•5 I.tm).
This implies that pore collapse is a fairly common the Reuss assumptionof uniform stress the effective
deformation
process
thatcanoccurevenin rocksconsidered compressibility
13ef
f forsuchanidealized
porous
material
is
to be relativelycompact.However, it shouldbe notedthat linearlyrelatedto the porosity
19,294
BAUD ET AL ' COMPACTION
6OO
[]
500
SOLNHOFEN LIMESTONE
C'
[]
peak stress
ß
•,
C*
C*'
dilatant cataclastic
flow
.
brittle fracture
400
ß sh
[]
action
[]
ß
300
dilatancy
ß
[]
ß
200
100
,
0
,
,
0
,
I
,
I
I
1O0
I
J
I
I
200
I
I
I
300
!
;
[
,
J
400
i
,
,
,
I
500
Mean stress (MPa)
Figure 6. StressstatesC', C* and C*' andbrittlestrengthare shownin the P-Q spacefor our Solnhofen
limestonedata.Threeregimesof inelasticand failuremodescan be identified:brittlefracture,dilatant
cataclastic
flow, andshear-enhanced
compaction.
=1+3 (l-v)
q)
2 (1-2v)1+•
(la)
where13and v are the intrinsiccompressibility
and Poisson's
ratio, respectively,of the solid (calcite) grain. The Reuss
averagesfor the intrinsicelasticmoduliof a calciteaggregate
at 300 MPa pressure [Sintntonsand Wang, 1971] are
13=1.379x 10']l Pa'l andv=0.335. Assuming
a porosity
{ = 2.8%, equation(la) gives an effective compressibility
13eff
= 1.5x 10'l• Pa'• thatisveryclose
toourmeasured
value
of 1.6 x 10'l• Pa'•.
band[Tapponierand Brace, 1976; Wong,1982;Fredrichet
al., 1989].As shownin Figure8b, a conceptual
modelwidely
used to analyze such micromechanical
processesis the
"slidingwing crack"[e.g., Horii and Nentat-Nasser,
1986;
Ashbyand Samntis,1990; Kernenyand Cook, 1991]. The
model considersthe growth of "wing cracks"that initiate
from tensilestressconcentration
at the tips of preexisting
cracksundergoingfrictionalslip. The fracturemechanicsis
such that increasingthe stresscausesthe wing crack to
propagatealonga curvedpath and ultimatelyreacha stable
orientation
parallelto the directionof c•]. With the
accumulationof such anisotropic damage distributed
The hydrostaticdatathereforesuggestthat the Solnhofen
throughoutthe rock, it will ultimatelyattaina criticalstateat
limestonehas -0.2% of microcrackporosityand -2.8% of
whichthe multiplicityof crackscoalesceto developa shear
quasi-spherical
pores (Figure 8a). A similar conclusionis band.
obtained
if
we
consider
the
other
elastic
moduli.
The
theoretical value for the effective Poisson's ratio of an elastic
matrix embeddedwith dilute concentration
of sphericalpores
is [Walshand Brace, 1966]
The slidingwing crackmodelconsiderssourcesof tensile
stressconcentration
that are locatedat the tips of preexisting
cracks
(withinitiallength
2a andoriented
at angle¾to c•l).
The applied stressesinduce a shear traction on the crack
plane,andif thisresolvedsheartractionis sufficiently
highto
overcomethe frictional resistancealong the closedcrack,
v
2 (7-5v)
frictionalslipresultsin tensilestressconcentrations
at thetwo
tipswhichmayinducewing cracksto nucleateandextendout
Substituting
in v = 0.335and{ = 2.8%,wehaveVeff=0.34. of the initialplaneof themainslidingcrack(Figure8b).The
For comparison,
the linearlyelasticportionof the axial and
30-v2X5v1)•.
Veff
=l+--
(lb)
driving
forceischaracterized
bythestress
intensity
factor
KI
volumetric
straindatahasa slopeof 1- 2reft=0.32(Figure at thetip of the putativewing crack.With increased
loading,
3), which implies that the effective Poisson'sratio of the
it will attainthecriticalvalueKi½,at whichpointa wing
limestone
(whenall themicrocracks
hasbeenclosed)is given
byVef
f = 0.34,in agreement
withthemodelprediction.
3.3. Developmentof Dilatancyand Brittle Faulting:
Wing Crack Growth and Coalescence
Dilatancyin brittle rock has been observedto arise from
crack nucleatesand propagatesalong a curvedpath to
ultimatelyreacha stableorientation
parallelto thedirectionof
13'
1.
If the onsetof dilatancyC' is identifiedwith the initiation
of wingcracksandif therockis assumed
to containrandomly
orientedcracks,then the wing cracksshouldfirst nucleate
intragranularand intergranularcrackingwith a preferred fromthosesliding
cracks
oriented
at ¾= 1/2tan'l(1/•t),
where
orientationfor propagation
that is parallelto the maximum [t is the frictionalcoefficient.The principalstresses
at the
principalstresscy]. In the strain softeningstage, onsetof dilatancyare given by [Cotterelland Rice, 1980;
microcracking
activity localizesalong a macroscopic
shear Horii andNentat-Nasser,1986;AshbyandSamntis,1990]
BAUD ET AL ßCOMPACTIONSOLNHOFENLIMESTONE
19,295
Lance sandstone
1600
dilatant
cataclastic.
fiow
/
1400
1200
ß
1000
C*
800
600
/
4O0
dilatancy
.
2OO
ß
• ,
,
,
,
,
,
,
,
,
I
,
,
,
,
,
,
,
,
,
I
,
,
,
,
,
,
,
,
,
I
o
0
500
1000
1500
Mean stress (MPa)
Figure
7.Stress
states
C',C*,andC*'areshown
intheP-Qspace
forLance
sandstone.
Thestress
values
were evaluatedfrom dataofSchocket aL [1973].
--
•/1+g2+g(53+
qC• Kic .
(2)
C4=4•cos¾/(41
+Ix2-•).
If one
specifies
thematerial
parameters
Do,Kic/Ora)
1/2, and
oftheprincipal
stress
(5•asa function
Toanalyze
thepeakstress,
weadopted
Ashby
andSammis' It, thentheevolution
of damage
D ata fixedconfining
stress
(53canbecalculated
[1990]
two-dimensional
(plane
strain)
model
formathematical
convenience.
The key damageparameter
in thismodelis the usingequation
(3). In the brittlefaultingregimethe stress
crack
density
D = •(l +acos¾)2N,•,
where
l isthelength
ofthe attainsa peakbeyondwhichinstabilitysetsin.
thecalculation
for different
valuesof fixed(53
wingcrack
andNAisthenumber
ofsliding
cracks
ofuniform Repeating
orientation
¾ per unit area initiallypresent.Beforewing allowsone to map out the brittlefailureenvelopein the
cracksnucleate
thelengthl-0 andtherefore
theinitialdamage principal
stress
space.
To a firstapproximation
thisfailure
envelope
forthewingcrackmodel[HoriiandNemat-Nasser,
is givenby DO=x(acos¾)2N,•.
Withtheprogressive
development
of dilatancythe remotelyappliedprincipal 1986;Ashbyand Sammis,1990; Fredrichet al., 1990;
stresses
evolvewithdamage
in accordance
withequation
(17) Kemeny
andCook,1991
] canbedescribed
bya linearrelation
of Ashbyand Sammis[1990],writtenherewith our sign
convention
(compression
is positive):
(51
=A(I't'D0)(53
+B(It'Oo)KIc/•a' (4)
(51
If triaxialcompression
datafortheonsetof dilatancy
and
C4(40/D0-1)
peak
stress
follow
thelinear
trends
described
byequations
(2)
and(4),thentheslopes
andintercepts
ofthetwosets
ofstress
data providefour constraints
for inferringthe three
= +l +d-oo4D/Do
-1
CI l-x/-•'
(53
D/ DO- 1+ 0.1/ cos¾
C4 KIc
parameters
Do, KIc/Ora)
1/2, and•. These
inferred
parameters
andthefit of ourlimestone
datatoequations
(2)
and(4) areshown
in Figure
9. Ourestimate
of g - 0.53is
(3)
closeto Ashby
andSammis'
[1990]estimate
of 0.55onthe
basisof Heard's[1960]data.However,our estimates
of
KIc/Ora)
1/2=97MPaandDo= 0.25arehigher
than
their
values
ofKic/(Xa)
•/2=68MPa
(corresponding
toKIC=0.6
where
MPamu2anda - 25Itm)andDo= 0.15,respectively.
Heard
[1960]presented
a limited
setofpeakstress
data,
andsince
he
did not measurethe volumetricstrain,the onsetof dilatancy
cannot be determinedfrom his data. Ashby and Sammis
[1990]assumed
thatwingcrackinitiation
inHeard's[1960]
experiments
wasassociated
withtheonset
of nonlinearity
in
and
the stress-strain
curves.In comparison,
our estimates
were
basedon a moreextensivesetof dataon bothpeakstressand
19,296
a)
SOLNHOFEN LIMESTONE
b)
x,
a)
1.
1,
1.1.1.
%
Figure8. Conceptual
models
of porespace
andthreemicromechanical
processes
in Solnhofen
limestone.
(a)
Theporespace
is composed
of microcracks
andquasi-spherical
pores.
(b) Theslidingwingcrackmodelis
applicable
atlowconfining
pressure
wheredilatancy
andbrittlefracture
wereobserved.
(c)Plastic
collapse
of
isolated
spherical
poresis responsible
forshear-enhanced
compaction
at relatively
highmeanstresses.
(d)
Microcracking
induced
by dislocation
pileupresults
in thetransition
fromshear-enhanced
compaction
to
dilatant cataclastic flow.
onsetof dilatancyof samples
froman identicalblock.Ashby
andSammis[1990]choiceof preexisting
cracklength(2a =
50 gm) is --- 10 timesthe averagegrain size of Solnhofen
limestone.
If we takethe moreplausibleassumption
thatthe
slidingcracklengthis comparable
to the average
grainsize,
then 2a- 5 gm and our data suggest that
concentration
can readilybe built up at impinginggrain
contacts,
Zhanget al. [1990]modeled
the porecollapse
that
arisesfromgrain-scale
brittleprocesses.
While thisHertzian
fraciure
modelhasbeendemonstrated
to bewidelyapplicable
to poroussiliciclasticrocks [Wong et al., 1997], our
mechanical
dataandpreliminarymicrostructural
observations
Kic- 0.27MPam1/2,which
isin reasonable
agreement
with indicatethattheplasticporecollapsemodelis morerelevant
experimentally
determined
value for calcite.For example, to compaction
in Solnhofen
limestone.
The compactive
yield
Atkinson
andAvdis[1980]reported
a valueof 0.19MPam•/2 stresses
(Figure6) aresignificantly
lowerthanthepredictions
of the HertzJan fracture model for a siliciclastic rock of
for crackpropagation
alongthe(1011)orientation.
porosityand grain size comparableto thoseof Solnhofen
limestone.
Furthermore,
thesestress
levelsarecomparable
to
3.4. Onsetand Developmentof Shear-Enhanced
levels
required
for
the
initiation
of
dislocation
slip
in
calcite
Compaction:PlasticPoreCollapse
[Turneret al., 1954;Fredrichet al., 1989].
Two end-members
have been developedto model the
We saw earlier that after closure of the microcrack
inelasticcompaction
of porousrock. For a rock in which porosity,the pore spaceis mainly composedof isolated,
crystalplasticityprocesses
(suchasmechanical
twinningand quasi-spherical
pores.CurranandCarroll [1979]considered
dislocation
slip)initiateat relativelylow stress
levels,Curran a dilute concentration
of sphericalpores,with dimensions
and Carroll [1979] modeledthe stress-induced
compaction sufficientlysmall that one can modelthe representative
thatarisesfromplasticcollapse
of isolatedspherical
pores. element
volumeasa thick,concentric
spherical
shell(Figure
For a relatively porous rock in which tensile stress 8c) of inner radius a and outer radiusb. To relate the local
BAUD ET AL ßCOMPACTION
SOLNHOFEN
500
LIMESTONE
19.297
are several typographicalerrors in their paper. For the
reader's convenience,we have summarizedthe mathematical
development
andcorrected
theerrorsin theappendix.
If theelasticmoduliarespecifiedandtheprincipalstresses
are normalizedwith respectto Y (or k), thena family of yield
envelopescan be calculatedusing equation(6) for different
valuesof initial porosity(Figure 10a). The differentialstress
at the onsetof plasticporecollapsedecreases
with increasing
mean stressand porosity.We have usedthe Reussaverages
400
300
•
200
a) 0.35
Ashby and Sammis[f990]
Do=0.25;
g=0.$3;
Ktc/(•
a//2=97
MPa
lOO
0.3
....
o
0
I
....
10
I
,
f
I
,
20
I
f
,
,
30
,
I
,
,
,
,
40
I
50
i..1..2•o
0.25
....
60
.
% (MPa)
'"-,.
',
•
Solnhofen
limestone
,,
• = 2.8%
0.2
Figure 9. Ashby and Sammis' [1990] model was used to
interpret our peak stress and onset of dilatancy data at
confiningpressuresof 10, 25, 35, and 50 MPa. The axial
stressat the peak is plottedas a functionof the confining
pressure,and the linear fit is also shown with parameter
0.15
'
50%
:- -5'T
'.
',
•\,
0.1
',.,
values as indicated.
,.
'•
0.05
geometryto the total porosity•, the radii a and b are chosen
I
such
that• = (a/ b)3.
A Reuss condition
0
I
i
I
0.5
i
1.5
is assumed for the local stress state: the
j •lk
outersphericalsurfaceof the representative
elementvolume
is subjectedto a stressfield identical to what is remotely
applied to the macroscopiccontinuum.The applied stress
field induces stress concentrationin the vicinity of the
sphericalpore, and plasticflow occursif the local stressfield
.......
..
Y = 975 MPa
•=2.8%
'-..
o/jsatisfies
a specified
yieldcondition.
Intheirelastic-plastic
model, Curran and Carroll [1979] adoptedthe von Mises
yield criterion
= =r/45,
0.8
PLASTIC
PORE
COLLAPSE
•
Curran and Carroll (1979)
..... Gurson
(1977)
•
0.6 '
o
0.4
'-
I
.
' •ß••plastic
Fully
nitial
tt
where
J2=2[(ø•
'•022)2
+ (022
' 033
)2+ (033
' ø•)21/6
2
+ o•2 + o•3 + o23• isthesecond
invariant
ofthedeviatoric
stresstensor.The parameters
k and Y correspond
to the plastic
yield stressesfor pure shear and uniaxial tension (or
compression),respectively.For a porousrock subjectedto
conventional
triaxialcompression,
plasticyield initiatesat the
circular
perimeter
of thespherical
porethatisparallel
too3 if
the remotelyappliedstresses
satisfythiscondition:
t
!
!
i
!
i
,
00
0.5
,
,
I
1
,
•
•
•
I
•
1.5
,
,
,
I
2
,
,
,
,
2.5
(c;a
+ 203)13Y
Figure 10. (a) Initial yield surfaceassumingmatrix material
governedby von Mises yield criterioncalculatedusingthe
modelof Curran and Carroll [1979]. Numbersabovecurves
indicatethe porosity.All the curvesare calculatedwith the
9RP2+ sP(3P-Q)+(u+ V+ 3W)(P-Q/3)2
+2WPQ=k2 =y2/3,
-
(6)
Reussaverages
for elasticmoduliof calcite:
K = 7.25x 10s
MPaandg = 2.69x 10• MPa.Thesolidcurvewascalculated
usingthe porosityof the Solnhofenlimestoneafter closureof
the microcracks(i.e., qb= 2.8%). (b) Evolutionof the yield
where
P isthemeanstress
(o• + 203)/3,Q isthedifferentialsurfaceof Solnhofenlimestonewith plasticporecollapse:the
stress
o• -03, andthecoefficients
R, S, U, V, andW depend solid curveis for initial yield calculatedusingthe modelof
on the elasticmoduli and the ratio a/b ( and thereforethe Curran and Carroll [1979], and the dashedcurveis for total
porosity).Althoughthesecoefficients
wereexplicitlyderived plasticyieldingcalculatedusingthe modelof Gutson[1977]
by Curran and Carroll [1979], it shouldbe notedthat there with the yield stressY = 975 MPa.
19,298
BAUD
ET AL.: COMPACTION
for the elasticmoduliof calcite,but as notedby Curran and
Carroll [1979], the yield envelopehas a relativelyweak
dependenceon the elastic moduli. Except for slight
differencesin the curvatures,our curvesin Figure 10a are
nearly the same as theirs calculatedusing different elastic
moduli.
SOLNHOFEN
LIMESTONE
resultantenhancementof compactionwere analyzedby
Curran and Carroll [1979] usingfinite elementsimulations.
Ultimately,the solid part of the hollow spherewill have
yielded completely.Analytic approximations
for the stress
statesat whichthis final stageoccurswere derivedby Green
[1972] and Gurson [1977]. According to Carroll and
The experimentaldata for the onset of shear-enhanced Carman's [ 1985] finite elementcalculations,Gurson's[1977]
compaction
(Figure6) canbe fittedwith the theoreticalyield relation gives better approximationat low porosities.His
envelopefor plasticporecollapseif an uniaxialyield stressof relation for total plastic yield of a porous spherecan be
k = 563 MPa (or Y = 975 MPa) is assumed
(Figure11). For expressedin terms of the applied mean and differential
reference,the criticalresolvedshearstresses
for f slip in stresses
calciteis 216 MPa, andfor r slip it rangesfrom 145 to 185
MPa [Turneret al., 1954;Griggset al., 1960].Althoughour
+ 2qb
cosh = (1+ qb
2),
(7)
inferred value of k is higher than these yield stresses
measuredfor specificslip systemsof calcite,we considerit
likelythatthemacroscopic
yieldwouldrequirea significantly
where Y is the initial yield stressunderuniaxialloading(as
higherstresslevel. The plasticpore collapsemodelrequires
definedfor equation(5)) and qbis the currentporosity.It
homogeneous
plasticflow to occur in the proximityof the
shouldbe notedthatthe total yield conditiondoesnot depend
pore surface.As elaboratedby Paterson [1969], the von
on the elasticmoduli or initial porosity.As shownin Figure
Mises requirementof five independentslip systemsas a
10b, the attainment of total yield requires a significant
necessary
conditionfor thistype of macroscopic
flow implies
increaseof differentialstresssubsequent
to initial yield. Our
that multipleslip systems(someof which are not favorably
mechanicaldatado not suggestthat this final stageof plastic
oriented)needto be activated,and strainhardeningwould
porecollapsewas achievedin our experiments
on Solnhofen
takeplacevery rapidly.
limestone.
It is well knownthat e-twinningin calcitecanbe activated
at very low stresses.When calcite crystals are favorably
orientedfor twinning,then the combinationof twinningand
one of the slip modeswould providethe equivalentof five
independentslip systems[Paterson, 1969, 1979]. Indeed,
microstructural
observations
of Fredrich et al. [1989] suggest
that fully plastic flow in Cartara marble (at a differential
stressof 329 MPa and confiningpressureof 300 MPa) can be
achievedby the combinationof twinning and r slip. Our
preliminary microstructuralobservationson a Solnhofen
sampledeformedin the fully compactiveregime(at confining
pressureof 435 MPa) indicatethat pervasivepore collapse
hadoccurredwith somewhatlimitedtwinningactivity.
If we focuson the deformationin the vicinity of a pore,
then the plastic zone expandswith increasingdifferential
stress.The progressiveexpansionof the plasticzone and the
3.5. Transition From Shear-EnhancedCompaction
to Dilatancy' Microcracking Induced by
DislocationPileup
The plasticcollapsemodel focuseson the deformationin
the vicinity of the quasi-spherical
pores.However,as density
of the dislocations
increases
duringthe strainhardening,their
movement is inhibited by obstacles that include grain
boundaries,second-phase
particles,and nearbydislocations.
As a result,a pileup will occurin which mutuallyrepulsive
forcesbetweenneighboringdislocationsare counteracted
by
the externallyappliedshearstress(Figure 8d). Microcracking
may be inducedif the tensilestressconcentration
aheadof the
dislocation pileup is sufficiently high [Zener, 1948].
Nucleationand propagationof suchmicrocracksinducethe
porespaceto dilate.
If we considera slip planethat is subjectedto a resolved
shear
stress
x,, dislocations
canmoveif thisstress
exceeds
the
0.6
Y
lattice "friction" stressx If the dislocationspileup at an
obstacle,then the stressconcentrationaheadof the pile up
may nucleatetensile cracking if the stressessatisfy the
followingcondition[Stroh,1957; Wong,1990]:
975 MPa
{=2.8%
•
0.5
I
and
Carroll
(1979)
• ßCurran
Solnhofen
limestone
data
0.4
(8a)
0.3
where L is the lengthof the dislocationpileup. If the rock is
undertriaxial compression,
then the resolvedstressattainsits
0.2
maximum
valueof x, = (o1-o3)/2 fora slipplaneoriented
at 45ø to the principalstresses.Hence the crack initiation
conditioncanalternativelybe writtenas
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(c•+ 2%)13Y
Figure 11. Comparisonbetweenexperimentaldata and the
modelof Curran and Carroll [ 1979], plottedin the P-Q space
normalizedby the yield stresstakenas 975 MPa.
•/-•xKIC
øl- ø3=2--•• +2•:.œ
.
(Sb)
It is plausiblethat the dimensionsof the dislocationpileup
and slidingcrackboth scaleas the grainsize, andthereforeas
BAUD ET AL.: COMPACTION SOLNHOFEN LIMESTONE
19,299
a first approximation,one may take L -• 2a (Figures8b and
8d) and usethe resultsfrom our wing crackanalysisto infer
Francois and Wilshaw [1968] and Wong [1990], the
extensionof a crack nucleatedby dislocationpileup is
sensitively
dependent
onpressure,
eventhoughthenucleation
thatKic/(r•L)
1/2•68MPa.Substituting
thisintoequation
(8b), we seethat the first term on the right-handsideis -187
processitself is pressure-independent.
Last, our preliminary
MPa,whichis significantly
lessthanthe øl-ø3 valuesfor microstructural
observations
of a sampledeformed
to beyond
C*'(Figure
6).Hence
equation
(8b)requires
thatthestress C*' (at the confiningpressureof 200 MPa) indicateextensive
is comparableto the critical resolved shear stress for microcracking,
aswell asmechanical
twinning,andtherefore
the interplaywith dislocation,
microcracking,
and twinning
translational
slipincalcite.
If indeed
'Ifcanbeapproximated
by experimentallydeterminedvalues6f the criticalresolved activitiesshouldbe integratedin future refinementof the
shearstresses
for the r andf slip systems[Turneret al., 1954;
Griggs et al., 1960], then the C*' data are bracketedby
theoreticalestimatesfrom equation(Sb) (shownas solid lines
in Figure 12).
The apparent agreementsuggeststhat the interplay of
dislocation and microcrackingactivities can explain the
transition from shear-enhancedcompaction to dilatancy.
However, it should be noted that such a comparisonhas
certain limitations.First, the shear stressxr as originally
envisionedby Stroh [1957] representsthe lattice friction
stress,which is probablyvery small in calcite. Hence the
relatively high values here should be interpreted as
generalizedresistancestressesthat includelattice friction as
well as other stressesthat result from complex interactions
with various obstacles present in the polycrystalline
aggregate.Second,the model is for the initiationof dilatant
cracking,butthe overalldevelopment
of dilatancyin the rock
as a whole would require the cumulativedilation due to
crackingexceedthe concomitantcompactiondue to plastic
pore collapse.Hence equation(Sb) only providesa lower
bound
on the critical
stress at the transition
from
shear-
enhancedcompactionto dilatancy, which probably occurs
afterthe crackshavepropagatedover certaindistances.
Third,
the data suggesta slight dependenceof C*' on pressure,
whereasthe modelpredictsthat the differentialstresslevel is
pressure-independent
(equation (8b)). As elaborated by
micromechanical
model.
3.6. Effect of Porosity and Grain Size
on the Brittle-Ductile
Transition
We have focusedon porositychangeand failure mode as
two key aspectsof the phenomenology
of the brittle-ductile
transition.A comparisonof the Solnhofenlimestone(Figure
6) and Lance sandstone (Figure 7) shows that the
phenomenologicalbehaviors in these two rocks of
intermediateporositiesare qualitativelysimilar.In eachcase
the brittle-ductiletransitionis associatedwith a complex
interplay of dilatant and compactive deformation
mechanisms.
While
these
similarities
underscore
the
importantcontrolthat porosityexertson the phenomenology
of the brittle-ductiletransition,it shouldalsobe kept in mind
that the micromechanical
processes
are actuallyvery different
in the silicate and calcite rocks.
While dilatancyandcompactionin silicaterocksarisefrom
various modes of grain-scalecracking operativeat room
temperature,our micromechanicalanalysesemphasizethe
importanceof crystal plasticity as well as microcracking
mechanisms
in the limestone.
The
same mechanisms
are
activatedin calcite rocks of different porositiesand grains
sizesto resultin fundamentallydifferentfailure modes.This
complex connection between micromechanics and
macroscopic
failure modecanbe illustratedby comparisonof
Solnhofen limestone with Carrara marble.
Figure 13 summarizesthe critical stressesand failure
modesof the relativelycompactCarraramarble [Fredrich et
al., 1989]. On the one hand,the overall behaviorin the brittle
faultingregimeis qualitativelysimilarin Carraramarbleand
the moreporousSolnhofenlimestone.Applyingequations(2)
and (4) to dataof Fredrich et al. [ 1989] on onsetof dilatancy
and peak stress, we obtained values of la = 0.55,
600
215 MPa (Turner et al., 1954; f slip)
$$0
; ½q=185
MPa(Griggs
etal.,1960;
rslip)
[]
500
Kic/(r•a)
•/2=27MPaandDo= 0.3forCarrara
marble.
As
450
expectedthe friction coefficientsin the limestoneand marble
• ½q=
145MPa(Tumer
etal.,1954,
r slip)
aresimilar.
Thefracture
toughness
parameter
KI½/(r•a)
•/2 is
400
Microcracking nucleated by
dislocation pileup
350
K=I(;
L)•n=68.6
MPa
300
0
100
200
300
400
500
(• + 2o'=)/3
(MPa)
Figure 12. Predicted values of differential stress for the
nucleationof tensile crack induced by dislocationpileup
(solid lines).For reference,experimentallydeterminedvalues
for r slip andf slip systemsin calcite[Turner et al., 1954;
Griggs et al., 1960] are also shown. For comparison,the
critical
stresses C*'
for the transition
from shear-enhanced
significantlylower in coarse-grained
Carraramarblepossibly
becausethe cracklengtha is expectedto scalewith grainsize.
Such effects of grain size on strengthand brittle-ductile
transitionhave been discussedin somedetail by Fredrich et
al. [1990]. While the wing crack model may idealize the
complicatedmechanisms
that are actuallyinvolved in brittle
faulting, it providesa semiquantitativedescriptionthat is
particularlyusefulfor a comparativeanalysis.It shouldalso
be noted that porosity also has influence on the brittle
strength.In Figure 14 we compileall the publisheddata for
Solnhofenlimestone(with initial porositiesranging from
1.7% to 5.9%). There is a generaltrendfor the peak stressto
decrease
with increasingporosity.
On the otherhand,two significantdifferences
betweenthe
compaction
to dilatantcataclastic
flow areplottedin theP-Q
marble and limestone were observed in the cataclasticflow
space.
regime.First,inelasticcompaction
was absent,and only
19,300
SOLNHOFEN LIMESTONE
Carrara
marble
fully plastic
250
[]
peak stress
I
C'
I
fully plastic
2OO
dilatant
cataclastic
flow
150
brittJe
fracture
[]
100
m
50
....
I
50
....
I
,
1O0
,
,
•1•
•,,
150
I,,,,
200
I
....
250
I
....
300
I
....
350
I
400
Mean stress (MPa)
Figure13. Stressstateat theonsetof dilatancyC' (in bothbrittlefaultingandcataclastic
flow regimes)and
brittlestrength
areshownin theP-Q spacefordataonCarratamarblefromFredrichet al. [1989].
microstructural
observations
usingtheopticaland
dilatancywas observed
in the relativelycompact
Carrara systematic
transmission
electron
microscopes
should
be pursuedto
marble.As documented
by Fredrichet al. [1989],dilatant
cataclastic flow in Carrara marble is manifested by
elucidatethis question.
mechanical
twinninganddislocation
slipthatmaypromote
or
inhibit microcrackingat different stress and pressure 4. Conclusion
conditions.
While thesecrystalplasticityprocesses
induce
In this study,we havedemonstrated
that shear-enhanced
local tensile stressconcentration
that nucleatewing cracks,
and porecollapseare readilyobserved
in the
twin and slip bandsmay alsoact as barriersthat arrest compaction
microcrackpropagation.In Solnhofenlimestoneour Solnhofenlimestonethat hasan initial porosityas low as 3%
preliminary
microstructural
observations
indicate
thattherole andporesizeof theorderof 10 gm.As a functionof pressure,
in thisrelatively
of twinning
seems
to besecondary,
especially
in samples
that threeprimaryfailuremodeswereidentified
limestone.
At relativelylow pressures
(<_50 MPa),
hadundergone
plasticporecollapse.
This is in agreement compact
as a precursor
to the occurrence
of
with RoweandRutter's[1990]observation
thata relatively dilatancywas observed
brittle
faulting.
At
relatively
high
pressures
(>
350
MPa)
the
highstress
(-300MPa)isrequired
toactivate
twinning
inthis
flow associated
with shearfine-grained
limestone.
Mechanical
twinning
arises
fromthe samplesfailed by cataclastic
compaction
and strainhardening.
At intermediate
homogeneous
shearof successive
planes
of atoms
thatcauses enhanced
shear-enhanced
compaction
wasobserved
during
oneportion
of thelattice
to become
a mirrorimageof the pressures,
the
initial
stage
of
strain
hardening
but
only
as
a
transient
other.The natureof the twinningprocess
is suchthat it is
thatultimatelyledto dilatantcataclastic
flow.This
confined
withina singlegrain.Consequently,
thelocaltensile precursor
stress
concentration
atthetipof a twinbandshould
scalewith intriguing phenomenon implies that stress-induced
anddilatancyshouldnot be viewedasmutually
the spatialextentof the shearstrain(andtherefore
the compaction
dimensions
of the twin bandand grainsize),andtherefore exclusiveprocesses,especiallywhen large strains are
wingcracknucleation
or crackarrestis lesslikelyto be involved.
induced
by twinningactivityin the relatively
fine-grained Severaltheoreticalmodelswere employedto interpretthe
Solnhofen limestone.
micromechanics of the brittle-ductile transition in Solnhofen
Comparison
of ourdatawithresults
for siliciclastic
Second,
Carraramarble(Figure13)underwent
a transition limestone.
to fully plasticflow(withnegligible
volumechange)
at a
rocksunderscores
that sincecrystalplasticityof the calcite
hasa predominant
rolein limestone,
theHertzian
stresslevelof 329 MPa (at a confiningpressure
300 MPa) grains
thatis significantly
lowerthanourinferred
valueof Y (-975 fracture model is not appropriatefor analyzing the
phenomena.
Our laboratory
dataontheonsetof
MPa)forSolnhofen
limestone.
In Carrara
marble,
fullyplastic compaction
compaction
arein reasonable
agreement
with
flow involvesthe cooperative
mechanisms
of mechanicalshear-enhanced
Curran
and
Carroll's
[1979]
plastic
pore
collapse
model
if a
twinninganddislocation
slip [Fredrichet al., 1989].In
of Y= 975 MPa is assumed.
In the
contrast,
theroleof mechanical
twinning
is relatively
limited relativelyhighyieldstress
regimethe Stroh [1957] modelfor microcrack
in Solnhofenlimestone,andthereforeplasticflow hingeson transitional
pileupcan be usedto analyze
theactivation
of multiple
dislocation
slipsystems
thatrequire nucleationdue to dislocation
the
transition
from
shear-enhanced
compactionto dilatancy.
a significantly
higherstresslevel [Paterson,
1969].More
SOLNHOFEN LIMESTONE
19,301
500
[]
ß
ß
o
4OO
0
[]
l&
E]O
0
0
og
3OO
ß
ß
ß
ß
O
E1
o
15;1
m
2OO
100
0
20
40
60
H©ard,'1960,{ = 1.7%
This study, { = 3.0%
Rsnnsr and Rummsl, 1996, { = 3.7%,R = 5pm
Robertson, 1955, { = 4.1%, R = Spin
Byedee, 1968, { = 4.8%, R = 10prn
Rennsr and Rurnmsl, 1996, { = 5.5%, R = 5pm
Frederich st al., 1990, { = 5.5%, R = 6prn
Ruttsr, 1972, { = 5.8%
Edmondand Paterson,1972, { = 5.9%, R = 10prn
80
100
120
140
160
Confining pressure (MPa)
Figure14.Compilation
ofpeakstress
dataforSolnhofen
limestones
of different
porosities
andgrainsizes.
In thebrittlefaultingregimethewingcrackmodelprovides
a
consistent
description
of the effectof grainsizeon the onset
of dilatancyand brittlefaultingin Solnhofen
limestone
and
S=•2g2a2(8
A'Ba
- 4BC
+4A'B'a
2- 2B'Ca
Cartara marble.
+4A'C'-•
2CC'
6A'C
I
+•
a
Appendix:Initiation of PlasticCollapse
of a SphericalPore Induced
by Triaxial Compression
(A2b)
a
2
U=2g2a412B
2 2BC'2B,2a
+ 8BB'a + •
+
3
FollowingCurran and Carroll [1979], we considera
hollow sphereof homogeneous,
isotropicmaterial.The
spherical
shellhasinnerandouterradiia andb, respectively,
andit is subject
to thefollowingboundary
conditions:
(1) the
innerboundary
is tractionfreeand(2) ontheouterboundary,
a
C'26B'C
!
+ 4B'C' + •
2a2
+
a
(A2c)
thetraction
isgiven
byti = ri/nj , where
n.istheoutward
unit
normal
totheouter
surface'of
thesphere,
andtheapplied
6B
2a
+-3C,2
2g2a2[
2+6BC'a
stress
is T•]= o],T•_
2 = c•3,T33= c•3, withalltheoff-diagonalV=•elementsbeingzero.
Thematerialresponse
is elastic-plastic.
Beforetheonsetof
plasticyield,the solidpartof the sphereis isotropic_and
linearlyelastic,with moduligivenby thetwo Lam6constants
3. and •t. Curranand Carroll [1979] considered
the initial
2
+ 12BC+
67')
.
(A2d)
plasticyield to obeythe Drucker-Prager
criterion,which
I/V= 292C2
(A2e)
depends
onthemeananddeviatoric
stresses.
Herewe focus
on plasticyield due to crystalplasticitythat is pressure- The primedsymbolindicatesdifferentiationwith respectto
independent.
Hencethe Drucker-Prager
criterionreduces
to the radial distancer, and the functions,4, B, and C are given
thevonMises
criterion:
0?/2=k asdefined
inequation
(5) by
of thetext.For our case,plasticyield initiatesat the circular
perimeter
of the spherical
pore(ontheX1-X2 plane)when
theappliedstresses
satis• thiscondition:
{Re]
+2c•3)
2+S•3½
]+2o'3)+
Uo'
•
A(r)
= -• D2
+ D3•'r2
+ D5
+ D6
5r' (23.
+71-07
B(r)=
D1+3 2 +D3
77-
(A3a)
(A3b)
withtheparameters
R, S, U, V, andW aredefinedby
(A2a)
2it2
(2A'2a2-2A'Ca-C
2)
C(r)=
3½
+[t•3 5r5 ½•+7•t)
D3r
+D4 (A3c)
19,302
SOLNHOFEN
LIMESTONE
He wouldlike to thankYvesGu6guen
for thehospitality
inwhichtheparameters
D•, D2,D3,D4,D5andD6areinturn Sup6rieure.
and many useful discussions
on this topic. This researchwas
partially supportedby the Office of Basic Energy Sciences,
Departmentof Energy,under grantsDE-FG-02-94ER14455and
definedby
16•t(19X
+14[tXb7-a7)(A4a)
D•= (2•.
+7[t)a7b7A
DE-FG02-99ER14996.
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Dilatancy, compaction, and failure mode in Solnhofen limestone

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