Handin ei al. [1963], for example, found that the density of a sandstone and a limestone increased as they were. stressed. Formerly with U.S. Geological Survey.
Yon. 71, No. 16
IouRNAL OFGEOPHYSICAL I•ESEARCtI
AUGUST 15, 1966
Dilatancy in the Fracture of Crystalline Rocks W. F. BnxcE
U.S. GeologicalSurvey and Departme• of Geologyand Geophysics MassachusettsInstitute o] Technology,Cavibridge
B. W. Pxc•D•a, Js.• Illinois Institute of TechnologyResearchInstitute, Chicago C. Scso•z
Depa•'tment of Geologyand Geophysics MassachusettsInstitute of Technology,Cambridge
Volumechangesof a granite, a marble, and an aplite were measuredduring deformation in triaxiaI compressionat confining pressureof as much as 8 kb. Stress-volumetricstrain. behavior is qualitatively the same for these rocks and a wide variety of other rocks and concretestudiedelsewhere.Volume changesare purely elastic at low stress.As the maximum stress becomesone-third to two-thirds the fracture •ress at a given pressxxre, the rocks become aliistent;that is, volume increasesrelative to elasticchanges.The magnitude of the dilatancy, with a few exceptions,rangesfrom 0.2 to 2.0 times the elastic volume changesthat would have occurredwere the rock simply elastic. The magnitude of the dilatehey is not markedly affectedby pressure,for the range of conditionsstudiedhere. For granite, the stressat which dilatancy was first detected was strongly time dependent; the higher the loading rate the higher the stress.I)ilatancy, which representsan increase in porosity,was traced in the granite to open crackswhich form parallel with the direction of maximum compression. INTRODUCTION
In the study of deformationof materials one
usuallyfocusesattention on the linear rather than on the volumetric
strains.
With
metals
deformed in the ductile range this is justified, for volumechangesare typically small enough to be ignored.In the compression of a block or twistingof a hollow tube of, say, copper, changes in cross-sectional area or length may amountto a hundred times or more the change in volume,and, whereasmost of the linear
change is permanent, almostnoneof the volume change is. In fact, it is very difficultto detect permanent changesin densityin ductilemetals caused by deformation.
For rocks,a numberof observations suggest thatthesituation maybe quitedifferent. Appreciable permanent volumechanges, of bothsigns, havebeennoted.Handin ei al. [1963], for
to failure in confinedcompressiontests. Similar behavior was observedfor sand [Borg ei al., 1960] when confiningpressurewas fairly high. Of course this was not surprising, for these materials all had high porosity initially (10 to 20%), and some compactionwould be expected from a high effective confining pressure. The oppositeeffect,dilaiancy,• accompanied fracture of these materials when confiningpressurewas very low. Again, this was not surprising, for dila!ancyis a well-knowncharacteristicof granular aggrega!essuchas densesand.]Evidentlyat low pressure the limestone and sandstonerespondedto shearingstressmuch the way dense sand did, that is, by slidingalong rough intergranular surfaces[Handin ei al., 1963]. Scatteredobservationssuggestthat these are not the only geologicmaaterialswhich are di2 We will use the term 'dilatehey' to mean the
example, foundthat the densityof a sandstone increaseof volume relative to elastic changes,
anda limestone increased as theywere.stressed caused by
deformation. We follow Mead [1925]
and use the term for any sort of solid, not just 'granular masses,'as in the original usage [Reynolds, 1901].
Formerly withU.S. Geological Survey. 3939
3940
BRACE, PAULDING, AND SCHOLZ
werecarriedout with theseques. latant.Bridgman[1949]foundthat two dense, experiments ratherductilerocks,soapstone and calcitemar- ticns in mind. ble, becamedilat. ant at fractureat room presEXPERIMElXlTAL PROCEDURE sure,andHandinet al. [1963]showed thai a coarse-grained dolomite wasdilatantevenunder Volumetricstrainsthat accompanydeforma. in twoways.Bridgman confining pressure of severalkilobars. Robert- tionhavebeenmeasured son[1960]foundthat marbledeformed at sev- [19•] immerseda samplein a fluid-filledconeralkilobarspressure increased in volumeby as tainer and followedchangesin the fluidlevel. Handinei a•. [1963] observed changes in pore much as 2%. had Amongthe brittle rocks,Bridgm•n[1949'] volumeof a rock specimen.Both methods foundveryslightvolume increase for a diabase the disadvantageof low sensitivitybut theadof vol. and Matsushima[1960b]reporteda smallbut vantageof being a direct measurement pronounced increase relativeto elasticvolume ume change.In addition,they still gaveaccurate changes for a graniteanda quartzmonzonite,resultseven if deformationof the samplewas inhomogeneous. as they werefracturedin compression. Taken together,these observations suggest The method used here dependson measure. thai the porosityof many rocks,particularly ment of the principal linear strains.The parl wera thosewith initially very low porosity,increases of the specimenwhere the measurements slightly during deformation,even under con- made had a cylindricalshape,so that the prin. finingpressure. If this is generallytrue, it may cipal strains were the axial strain .•, andthe havewidespread significance. For onething,if circumferentialstrain .•. Strain gageswereused rocks are generallydilatant during fracture, to measurestrain; the axial gageshad various lengths,and the circumferentialgagesextended generalfracturetheory,particularlyonewhich to very nearly half of the full circumference. ofthe predictshow strengthdepends on pressure._it Then,aslongasthe circularcrosssection present,none of the existingtheories,suchas specimenremainedcircularor becameelliptical thoseof Mohr or Griffith,is basedon a model duringthe deformation,vohunechange, this will have to be taken into account in a
which increases in volume as it deforms. For
of the specimenwas
another thing, dilatancyleads,under certain circumstances, to changes in porepressure,and, The present method had the advantage of if dilatancyis a fairly generalphenomenon, we might expectwidespreadeffects,suchas those high sensitivity;volumechangesof 10-'were readilydetectable. However,it alsosuffered from discussed by Mead [1925] and Franl• [!965]. Faulting is a highly localized Althoughthe aboveobservations suggested a a disadvantage. and (1) couldhardly be expected fairly general phenomenon, it seemedworth deformation, while to explore the matter further. For one to hold once individual fractures formed. Therefromthe thing, it was not clearto what extentthe con- fore,the volumetricstrainscalculated fining pressurewould suppressdilatancy.Also, linear strainsjust beforefractureof the rocks therewasa possibilitythat part of the observed were not accurate. Compression tests. Dogbone-shaped specidi!atancy,particularly of brittle rocks, might be due to the vertical splitting which often ac- mens were used for the compressiontests companiescompressirefailure (see Plate 1, [Brace, 1964b] to avoid end effectsandto verticalsplitting,insofarasthismight Griggs[1986], for example).This matter could eliminate photobe resolved,either by the use of dogbone- be due to end effects.Three-dimensional shapehasreshapedspecimensor by carryingout the ex- elasticanalysisof this specimen perimentunder high confiningpressure;verti- cently shown [Hock, !965] that stressconcenarenegligible in thecentralcylindrical cal splitting disappearsin either case [Brace, trations
1964b].And, finally,it was of someinterestto
section and that the stresses there are very
determinehow the magnitudeof volumechange nearly uniform. at fracture varies with the stress state. Is there
a 'critical void ratio' [$co•, 1968, p. $10] at fracture for a particular rock? _i number of
Electricresistance straingages weremounted, as described elsewhere [Brace,!964a],in •he centersectionof the specimens. The jacketing
DILATANCY
IN THE FRACTURE
OF CRYSTALLINE
ROCKS
3941
material which covered gages andcenter section The axial directionof the specimenshere was was room-temperature vulcanizing rubber(Gen- the 1-direction of the previous studies. The eralElectric ItTV 102). Theexperimental technique is the sameas described in an earlierstudy [Brace,19645] witha few modifications. A constantloading rateof about10 bars/see wasachieved by use of a Vanguardelectrichydraulicpump,the
calcitemarble, of unknownsource,was nearly pure calcite and appearedisotropic.Grain size was about 0.2 min. The aplite occurs in the mines of the central Witwatersrand
of South
Africa and is known locally as 'chert dyke.' It containsabout 63% oligoclase, 27% quartz, and 10% biotite. The feldspar is highly altered. output ofwhich wasmetered by a tIPH presDensityis 2.614g/cm'; grain sizeof the ground surecontrolvalve. The loadingmachineusedin the previous massis about 0.040mm and of the phenocrysts testshada rather low springconstant(0.6 to about 0.10 min. The rock appearsisotropic. 1.0X 10• dynes/era).Inasmuchas the samples OBSERVATIONS
hada springconstant of 1.0 X 10• dynes/cm or more,intrinsicbehaviorof the rock could
Tests a• a•mosphericpressure. Stress-strain curvesare shownin Figure !a for compression began to fall.To improvethissituation, which of Westerly granite at atmosphericpressure. wasparticularlytroublesome in tests at room The compressirestrength C at atmospheric pressure, a stiffening element[Paulding, 1965] pressurehad previouslybeenfound to be 2.3 --wasincorporated into the loadingmachine.This 0.1 kb, and, in this test, about 95% of this consisted of a very stiff, simplesupportedbeam stresswas applied and then removed, giving whichwas so placed that it carried a m_ajor the curves shown. C is the value of the axial notbeobserved oncethe load-deformationcurve
partoftheloadimposed by the machineonthe
stress --(•, at fracture.
specimen. This wasequivalentto placinga stiff spring in parallelwith the specimen. In this way thespringconstantof the machinewasraisedto
In this experimenl, --a, was increasedat a rate of about 100 bars/secup to about half the fracture stress, at which the rock began to creepnoticeably.Load was then held constant
about10 timesthat of the rock specimen.With thisarrangement,collapseof the specimenat fracturewasnot nearly as sudden,and a number of stress-strain
curves
were
obtained
at
roompressurewhich showed a distinc.t maximum.The stiffentugelement was not used in theconfinedcompressiontests. An extra 'stiffcuing'of the machineunder these conditionsis provided by friction at the high-pressureseals andby the compression of the confiningmedium when thesamplestartsto fail. Rockss•udied. With the objectivesgiven
for several minutes at increments
of stress of a
few hundred bars, giving the steps in the two curves. These steps became broader as stress was increased.Finally, while the rock was still intact, --•, was reduced,again in increments of a few hundred
bars. This time the rock did
not creep, as it did on the ascendingloop. As the stressreachedzero, the rock had a permanent
circumferential
strain
but
little
or
no
permanent axial strain. Values along the two stress-linear strain curveswere used in calculatingvolume change according to (1), giving a curve of stressvolumetric strain. The ratio --.e•/e, for the
above, threerockswere chosenfor study. One, • medium-grainedcalcite marble, was very ductfie evena%low confining pressure. The two otherrockswere brittle at all pressuresat- ascendingloop is alsoshownin Figure la. tatned. Oneof these,a quartz-oligoc!ase aplite With increasingstress,volumefirst decreased wasfine-grained and flinty and had extremely (sharply at first and then nearly linearly), high strength. The otherwasWesterlygranite; reached a •ninimum, and, finally, increased. thestrength of this rock was low enoughso When stresswas removed,a positivevolmnetric that behaviorcould be studied over a wide strain,P, remained. range of pressure. The granitehad appreciable By analogy with measurements of linear crack porosity [Brace,1965],whereas the aplite compressibility, the part of the ascending stresshadessentially none. volumetric strain curve up to the end of the
Theparticularsampleof Westerlygranite near-linear region representselastio behavior.
has been described before [Brace, 1964b, 1965]. At least,if stresswascycledwithin this region,
3942
BRACE,PAULDING,AND SCHOLZ
15
6
02
04
06
0
O
STRAIN
02
04
-
,
,
15
MARBLE
0 25
kb
0
04
08
12
04
STRAIN. -,aV/V•,o -,aL/L.
-4
Fig. 1. Stress-strain relations.Left-handcurve,whichshowsstressversusvolumetricstrain, was calculated from the curves of stress versus axial and stress versus circumferential strain,
whicharetracings of •YIZ-recorder records madeduringthe experiments. Symbols areexplained in the text.
DILATANCY IN THE FRACTURE OF CRYSTALLINE I:tOCKS
nopermanent linearor volumetric slrainremained. A measureof the elasticstrain the
specimen should haveat fractureis givenby
3943
Valuesof C, C', D, and E for the granite, marble,and apliie are collectedin Table 1. The uncertaintyin C is a few per cent,in C' 10 to
extending thislinearpart of the stress-volume- 15%,andin E about10%.Asnotedabove,D is tric straincurve up to the fracture stressC. a very rough measureof the total dilation at Thevolumetric strain at this stressis calledE. fracture,with an uncertaintyof at least a Theaxialstressat which the siress-volumetric factor of 2. strainleavesthe linearregion,calledCq can be Testsunder confiningpressure.Tests were located approximately. For this specimen it was madeat pressures of up to 8 kb. Typicalresults 12 --+ 0.1 kb, or about C/2. The maximum for the threerocksareshown in Figureslb, c, departure of the stress-volumelric straincurve d, and e. The curvesof stress-linearstrain were from the line representingelastic behavior, traceddirectlyfromX/Z-recorder records;then, whichis a net volumeincrease,is calledD. using (1), volumetricstrahawas calculatedfrom
TABLE1. Failure Conditions ofGranite, Marble,andAplite
Cisstress difference, F andS refertofast-orslow-loading rates, f andi tofractured andintact,respec-
tively.For othersymbolsseetext and Figure 1.
Loading Final
Rock
Rate
Granite
F F F F S F S
(Figure la)
(Figure lb)
F F F
F
F F F
(Figure lc)
Form f i i i f i f
i i i
C,
D
E
kb
kb
kb
C'/C 10-s
10-a
D/E
0 0 0 0 0.50 0.50 0.51
!. 3 0,9 1.2 1.1 1.4 2.2 1.5
2.27 2.18 2.28 2.28 4.39 6.30 4.26
0.55 0.40 0.50 0.50 0.30 0.35 0.35
!. 75 1.77 1.45 1.40 3.0 3.5 2.4
2.6 2.3 2.4 2.8 0.5 3.8 0.8
i
!.00 1.00 1.50
1.50
3.3 3.9 5.2 5.2
10.6
0.50
f i i
1.62 2.0 2.0
5.0 6.25 6.5
10.7 • 11.8
8.05 9.28 10.5
0.40 0.40 0.50
4.5 4.1 3.5 3.9 1.4 13.4 1,9
9.0 9.8 4.0
4.4 4.8 5.7
2.0 2.0 0.7
0.45
10.4
7.2
1.4
*
S
f
2.34
3.8
10.5
0.35
0.55
24.2
6.6
3.7
S S
f f
3.06 4.00
4.7 6.0
14.5 16.1
0.30 0.35
15.7 7.8
8.0 9.0
2.0 0.9
S S S S
f f f f
5.10 5.30 6.50 8.00
8.3 9,7 11.7 13.3
17.5 18.6 19.5 22.2
0.45 0.50 0.60 0.60
6.2 4.7 3.9 4.6
10.2 12.8 13.5 !0.9
0.6 0.4 0.3 0.4
f
0.49
1.3
2.60
3.0
>2.0
2.8 4.5 5,4 5.4
5.95 7.2 7,5 13.8
0.45 0.63 0.72 0.40
1.0 0.6 0.! 0.4
4.0 4.7 4,3 6.8
19.8 20,7
0.50 0.68
2.6 0.9
12.0 17.0
S
f
Marble (Figure ld)
S S
f f
Aplite
S
f
(Figure le)
Pressure, C',
S
S S S S
f f f f
S S
f f
S
f
4.11
0 0.25
0
0 0.77 0.81 1.43
2.38 2.83 3.20
5.6
0.2 0.9
2.6
9.4
10 14
17.2
*
0.35 13.8 12.0
0.46 0.45 D,4.5 1.30 0.65 16 0.50 >6
6.05 0.45
15.3 0.61
!. 4
1.2
1.3 •3.5 2.0 8
3.8
2.5 13.5
0.4
0.3 0.1 0.02 0.1
0.2
0.2 0.1
*Afault formed directly under strain gage and destroyed itbefore therock rafted; hence maximum strain
could notbe measured.
$Pressure vessel leaked andexperiment hadtobeterminated before fracture.
BRACE,PAULDING,AND SCHOLZ theselinear strainsat closelyspacedintervals. A smooth curve of stress-volumetric strain was
drawnthroughthe calculated points.A plot of the ratio of linear strains versus stress is also shown.
Pressureis known to about 25 bars and the
stresses to about5%. 6' andC' arestressdifferences,that is, total axial compressive stress minus pressure.
Experiments wereconducted at two loading rates.For the sampleshownin Figure15, the highrate (100 bars/sec)wasapplied;loadwas heldconstant at intervals,asin theatmospheric pressuretests describedabove (the stepswere omitted in tracingthe curvesin Figure 15). For the other samples(Figureslc, d, and
Strain , --AL/L.
Fig. 2. Linear compressibilifies of granite. Curvesare tracingsmade from XY-recorderrec.
ordsshowing the relationbetweenconfining pres. and strain in axial (•,) and circumferential loadwasappliedat a lowerrate (about10 bars/ sure (ee) directions.Primed valuesrefer to rockwhich sec).Theserateshavean uncertaintyof about has been stressedto within about 20% of the breaking stress,and unprimed values referto
Someof the specimens werefractured;others virgin rock. were recovered intact
when the stress-strain
these crackswhen open pressure-linear s•ain curves are neededfor three perpendicular difracture stress. rections.In eachcurvean interceptoais fom•d The principal new feature of the tests done (Figure2) by projectingthe linear part ofthe under confiningpressureis the disappearance curve back to the strain axis. The sum of curvesappearedto level off. The stressat this point was probablywithin a few per cent of the
of the curved toe of the stress-strain curves.
This again is in accordancewith generalelastic behavior of rocks under pressure. The permanent strains of the marble were so large that elasticchangescouldnot be measured with any accuracy. (In the table, E was calculatedby usingthe mean linear compressibility of Danby marblefrom Brace [1965]). Measurements o/ linear compressibility.
interceptsœorthe three directionsgivesthe volumeof the rock at atmospheric pressure occupiedby opencracks.The curvesin Figure :2still haveslightcurvatures at the top,sothat the volume obtainedin this way is somewhat less than the true value.
One sampleof Westerly granite wasstressed in compressionto a stresswell under C. Both linear strains were measured,and permanent Walsh [1965b] and Brace [1965] showedthat changeof volumenoted.Linear compressibiliiies porosity in the form of cracks in rocks could of the sample were next measured,usingthe be found rather accurately from observation samestrain gagesand the techniquedescribed in of linear compressibilityin three directions. a previousstudy [Brace, 1965]. Crack porosity This method was used here to see whether
was calculated.Next, the samplewas stressed to a somewhathigherlevel,newvolumechanges 1, couldbe accounted for as cracks,and, fur- werenoted,and then the linear compressibilities thermore,to determinethe orientationof any were remeasured.This was repeatedat succesnewly formed cracks. sively higherstresses, throughC' •o just under If a rock is compressedunder hydrostatic C. The resultsare given in Table 2; the curves pressure,linear strain is relatedto pressureas of pressure-linearstrain for the next to thelast shownin Figure2, for example.The upper part cycleare givenin Figure 2. of the curvesis nearly linear,whereasthe lower In Table 2, --a, is the maximumstress impart is strongly curved. In the lower part, posedduringthe particularcycleand c' isthe change cracksare beingclosedby the pressure;when point of departureof the stress-volume permanentvolumechanges, suchas P in Figure
the curve becomeslinear, all crackshave been closed. To obtain the volume associated with
curvefromlinearityduringthat cycle;p isthe permanent volume increase remaining after
DILATANCY
IN TIIE
FRACTURE
OF CRYSTALLINE
ROCKS
3945
TABLE 2. CyclicTestson WesterlyGraniteat Atmospheric Pressure CrackPorosity,10-8
Cycle 0 1 2 3 4 5
-- •,,
c',
Axial
Circ.
Total
kb
kb
e,
•o
•, + 2•0
1.0 1.3 1.6
400 400 420 430 430 450
590 580 600 640 810 1160
0.69 1.03 1.38 1.72 2.11
1580 1560 1620 1710 2050 2770
p
Total- 1580 0 -20 40 130 470 1190
10-8 50 100 160 370 1200
cycle whenstresswasremoved. Interceptsfrom haviorbegins,volumebeginsto increaserelative When granitewas stressed pressure-linear straincurvesare shownfor the to elasticchanges. axial and circumferentialdirections together withthe total, which is the axial plus twice the circumferential. This total contains for, say, thenex•tto the last cycle,not only any newly formedcracksbut also cracks present in the virginrock.The latter must be subtracted;for the circumferentialdirection (Figure 2) this amounts to subtractingoa from oa'. In Table 2 this is done by subtracting initial crack porosity,1580 X 10-', from the total at each cycle.The last iwo columnsin the table contain
a value higher than C' (about 1.0 kb) a permanent increase in voIume p was detected
(Table 2). As stresswas increasedup to the fracturestressC, the rock expandedmore and more relative to the elastic changes,until it reachedthe maximumD just as fracture occurred.
The va!ues C and C' for the granite are plottedin Figure 3 in the form of principal stresses.•rsis the maximum and •r• the minimum
compression.Data from several other studies
permanent volumetricstrain causedby uniaxial are added.Theseincludetestsusing•he large compression and the volume of the rock at the dogbone-shaped specimensdescribedin •Brace endof eachcyclewhichis associatedwith cracks. [1964b] on a new sampledesigndescribedby Theuncertaintyin thesetwo numbersis about Mogi [1966]. With one or two exceptions the 20%; sensitivityis about 50 X 10-5 stressesat fracture lie qui•e closeto a smooth curve; the scatterat any onepressureis about Dxsc•zss•oN 5%. Stressesat fracture from fast and slow
The stress-volumetric strain behavior of the
testsshowno consistent differences; both seem threerocksstudiedhas certain generalcharac- %ogroupabout the sameaveragecurve.This teristics. Initially, behavioris elastic.The linear in line with the rather smalleffectof loading partof the curvehasa reciprocalslopecloseto rate on strength reported elsewherefor silicate thelinearcompressibility of the rock; they rocks[$erdengec•iand Boozer,1961; Matsushould be identicalfor perfectelasticity.:For shima,1960b]. example, the reciprocal slopefor the graniteat The s•ressescorresponding to C' appear 4.11kb (Figure lc) is 0.69 mb-•; measured be stronglytime-dependent(Figure 3); all the valuesof linear compressibility for this rock points for the fast-loadingrate fall along one (extrapolated from values at 3 and 5 kb in line, and those from the slower tests group Brace [1965])are0.68,0.68and0.67mb-•, for aroundanotherline. Th•s groupingis shownin threeperpendicular directions. For the aplite anotherway: the valuesof (he ratio C'/C are (Figure le) the inverseslopeis 0.62mb-•; doseto 0.35for the slowloadingrate (Table 1) measured linearcompressibility of this rock in and close to 0.45 for the fast loading rate.
theaxialdirection of thespecimens is 0.62rob-' There is no distinct changein this ratio with at a pressureof 2.4 kb.
pressurefor the three rocks.
At thestress C', whichis the endof thelinear In contrastto the nearly constantratio region andthe point at whichnonelastic be- at a given loadingrate, the quantity D varies
BRACE, PAULDING, AND SCHOLZ
3946
-CT.• , kb 30
20
25
15
I0
5
0
./
/,.
..//'
d
,//oo
4
kb
//
//' o
///•/ o////. ß
/
o//,.// Fig. 3. Principalstresses at failure for granite.The solid pointsare stresses at failure,the open circlesstresses at the onsetof crack growth. Circles refer to present results,trianglesto testsmade on small dogbone-shaped specimensof a new design [JFfog% !966], and squares to tests made on dogbone-shaped specimensdescribedby Brace [1964b]. Fast and slow refer to loading rates; fast is about 100 bars/secand slow about 10 bars/sec.
widely,evenfor testsat the samepressure, and here. With the exception of the carbonate even when normalizedwith respectto elastic rocks, the ratio D/E has about the same the volume changes(the ratio 2)/E in Table 1). rangeas for the presentmaterials,although Additional work needs to be done to determine methods of measuringvolume changesvaried whether this is a real feature of rocks or whether widely. Bridgman's values of D are probably it is due solely to local inhomogeneityof the small becausehis sampleswere unjacketed. The linear strains. In any event, the range of quartziteand the diabasespecimens weredogseemsto be about the sameat low and at high bone-shaped as in the presentstudy,but pressure;there is no marked tendencyfor 2) to causethe strain gageswere too smallto givea decreasewith pressure.D/E is highest for correctaverageof co,the calculatedvaluesofD marble, the most ductile rock, and lowest for given in Table 3 may be on either sideof the the aplite. •rue one. For concrete,the volume changes If we may extrapolalebeyondpresentresults, showncontainan additionalfactor;thematerial the amount of dilatancy may eventually compacts underpressure, so that 2) is reduced crease.The curvesfor the onsetof cracking somewhat.This effect may becomelargefor (Figure 3) would appearto intersectlhe curve fairly porousrocks,as shownby Handine• for failure at pressuresbeyond those attained [1963]. The uncertaintyin the valuesof D here.Beyondthis intersection we might expect given by Matsushimais quite large,as the no dilatancyto accompanyfailure. lengthof straingageusedin lhe linearstrain Other measurementsof volum• change. measurement was not reported.The values Other measurements of volumechangeduring shownmay be larger or smallerthan thelrue de]orma•ionof nonporousrocksand of concrete value.Also,someof the materialsin Table are collectedin Table 3 for comparisonwith were recovered intact, so that the valueof presentresults.RobertSOh'S [1955,1960]meas- given is at best ,• lower limit. The volume uremenlsof permanenivolumechangefor Solen- changes givenin Table 4, althoughof themine hofen limestone and two marbles are collected
signasthepresentresults,aresomewhat higher,
in Table 4.
This may indeedbe a characteristic of carbomle
The ralio C'/C rangesfrom 0.35 lo 0.60 and thus appears to be similar to the values found
rocks;the marblesludiedhere(Table1) had a larger D than the silicate rocks. However,
DILATANCY IN Tile FRACTURE OF CRYSTALLINE ROCKS
3947
TABLE 3. VolumeChanges in CrystallineRocksandConcrete SeeTable I for symbols. Pressure,
C,
kb
kb
Material
D
C'/C
10-3
D/E
0.25 0.50 1.0 1.5
5.3 5.4 5.6 5.9
Calcitemarble
0
0.51
0.6
!.0
2.1
[Bridgman,1949] Sample re-
Soapstone
0
0.65
0.6
0.4
0.6
[Bridgman,1949]Samplefrac-
Diabase
0
2.3
0.6
0.4
0.1
[Bridgman,1949]Samplefrac-
Quartzmonzonite
0
2.0
0.6
2.0
2.3
[Matsushima,1960b] Sample
01ivinebasalt
0
1.5
0.5
3.0
1.5
[Matsushima,1960b] Sample
0.55
4.9
0.4
0.5
0.2
[Matsushima,1960c] Sample
1.02
6.7
0.5
3.0
0.6
[Matsushima,1960c] Sample
1.50
9.1
0.4
1.0
0.2
[Matsushima,1960c] Sample
2.60
10.4
0.6
1.0
0.3
[Matsushima,!960c] Sample
3.20
12.1
0.5
1.5
0.2
[Matsushima,1960c] Sample
3.80
13.5
0.5
3.0
0.3
[Matsushima,1960c] Sample
4.40 •
12.1
0.5
3.0
0.5
[Matsushima,1960c] Sample
4.6!
0.5
1.0
0.2
[Brace, unpublished]Sample
11.2
0.5
2.0
0.2
[Brace, unpublished]Sample
Dolomiterock
36 27 25 16
Remar 'ks
[Handin etal., 1963]D includes elastic changesand is the value at ez : 0.14. Initial porosity was in the form of cracks and was about 0.035. covered intact. tured.
tured.
fractured.
fractured.
I Granite
fractured.
fractured.
fractured.
fractured.
fractured.
fractured.
Quartzite
fractured.
0
fractured.
0.60
fractured.
Diabase
0
5.15
0.6
1.9
0.8
[Brace, unpublished] Sample
Concrete,
0 0 0 0.0!0 O.022 0.027 0.036
0.13 0.15 0.14 0.19 0.25 0.25 0.30
0.5 0.5 0.4 0.6 O. 5 0.4 0.4
0.78 1.0 1.2 1.7 1.7 1.3 2.5
1.8 2.0 2.4 2.4 1.4 0.9 1.7
[Richart et al., 1929] Values given when samples were badly cracked but still intact. Confining 'pressure'
1:2.1:2.5
fractured.
appliedby spiralreinforcing.
partofthedifference maybe dueto effects asso- fromresults gathered fromtheliterature(Table
ciated withunloading [Paterson, 1963];anom- 3). First,with a fewexceptions, fractureis ac-
alous length changes accompany ire removal oœ companied by an increasein porositywhich pressure in confined compression experimentsvariesfrom 0.2 to 2 times the elasticvolume withlimestone• The measurements of volume decrease the materialwouldhave had were i• charge givenin Table4, having beenmade perfectlyelastic.This is apparentlytrue for afterpressure wasremoved, arenot'strictly brittleaswellasductilerocks,for rockswhich comparable with thosein TablesI and 3.
are looseas wellas compact [Brace,1964b, In spiteof these uncertainties, two'conclu-1965],andfor fractureat atmospheric pressions canbedrawn fromthepresent workand sureas wellas underconfining pressure of up
3945
BRACE, PAULDING, AND SCHOLZ TABLE 4. PermanentVolume Changesin Limestoneand Marble
Elasticvolume changes, E, calculated assuming linearcompressibility of Solenhofen limestone tobe
0.82 mb-x and of the marblesto be 0.44 mb-L Data from Robertson[1955, 1960].
Pressure,
Material Solenhofenlimestone
Danby marble
Rutland marble
kb 0.29 0.35 0.57 0.75 1.00 2.00 3.00 4.00 2.50 4.00 3. O0 4.00
6',
--(e,)rr, a= 0.047 0.150 0.51 0.16 0.196 0.313 0.114 0.217 0.097 0.134 O.073 O.029
kb
d-AV/Vo
3.6 3.6 3.8 3.8 4.2 5.2 6.1 6.7 2.5 2.9 2.3 3.0
0.047 0.123 0.223 0.089 0.112 0.107 0.020 0.043 0.010 0.02! O.004 --0.004
-q-(zxV/Vo)/E 16 41 72 29 32 25 4 8 9 16 4 3
to 5 kb. Second,this dilationbeginsat a stress possibility wasby studyof thelinearcompressione4hird to two-thirds of the fracture stress. bility of ourspecimens. The particularratio seems,from the present Asnotedabove, linearcompressibility changes results for granite, to be strongly time-de- when a material becomes cracked. Furthermore, pendent--thelowerthe loadingrate the lower if the cracking is in onedirection, linearcomthe stressat whichnonelasticchangescom- pressibilitycorresponding to that directionwill mence.
It wouldbe of someinterest'to exploreboth these effects in more detail. The existence of a critical void ratio at fracture can neither be
proved nor disprovedby the presentresults. Perhaps one of the more direct methods of measurement of D, suchas wasusedby Handin
bemoststrongly affected. Using themethod suggesiedby Walsh [1965b],we can measure the porosityassociated with newcracks,the crack porosity.
The results of thisanalysis for onesample of Westerlygraniteare givenin Table 2. As compressirestresswasincreased, initial linearcompressibilityin the circumferentialdirectionbe-
et al. or Bridgman,mightyieldsuflqciently accuratevaluesof D to test sucha possibilit, y. ganto change markedly(Figure2), whereas the The ex!entof dependence of C' onloadingrate valuein the axialdirection wasrelatively un-
also is of considerableinterest and needs to be
more carefullystudied.Ct may also turn
affected. Changes beganat a stressof 1 kb, whichis C• forthissample. Thissuggested that,
to be influenced by environment; thisis sug- beginningat C; cracksformedin the axialdigestedby stress-strain curvesgivenby Boozer rection.Furthermore,the cracksthat formed
ei a!. [1963].
Physical character o/ihe volume changes.It has beenknown for sometime that fracture in
were open,and the crack porositydetermined
fromlinearcompressibilities ofthesamples after stresswas removedwas within experhnent,•l
uniaxialcompression, particularly of concrete,error of the permanentincreasein volumefound is precededby crackingwhichstartsat about from curves of stress-volumetric strain taken
halfthefracturestress. Thiswasdetected by duringa deformation cycle(compare lasttwo ultrasonic techniques [Jones, 1952],by special columns of Table2). Thusthe volumechanges opticalmethods [Berg,1950],by straingsges that we had noted during deformation ap-
[B!akeyandBeres/ord, !955],andby directob- parently can be traced to open axial cracks. servation of sectioned material[Hsu et al., Of course,formationof closedcrackscouldhave !963]. it was of interestto see whetherthe preceded or accompanied this, for linearcomincrease in porositywhichwehadobserved could pressibility isunaffected by closedcracks. betraced to cracking. Onewayof testing this Formationof open axial cracksat a fraction
DILATANCY IN THE FRACTUREOF CRYSTALLINEROCKS
39•9
of themaximumstressis alsosuggested indi-
rectly by variationof compressional wavevelocity in axialandradialdirections as a func-
(•)
tionof stress.Observations for rocks [Tocher,
1957;Matsushima, 1960a] and for concrete
[Jones, 1952]areall quitesimilar;velocity in theaxialdirectionis hardly changedby stress, whereas velocityin a radial directionbeginsto decrease at about half the fracture stress.It
(b)
maydrop10to 20%. The actualformationof cracksduringdeformation is very difficultto observe.Nevertheless, it seems worth while to considerthe form which
newcrocksmight have, and the ways in which theyznightoriginate.Three possibleforms are
(c)
shown in Figure 4.
The simplestpossibility (Figure 4a) is the isolated axial crack; it might form alongcleav-
Fig. 4. Crackgeometry.Three typesof open ages withingrainsor at grainboundaries. Open axial crackare givenwith referenceto principal
axialcracksmight alsoform at the junctionsof threegrain boundariesor three pre-existing cracks, two of which are inclined and one is
stress directions.
axial (Figure 4b). Under axial compression,strain, in the manner suggestedby Walsh slidingon the two inclinedsurfacesis accom- [1965a,b] in hisanalysisof elasticbehaviorof modated by openingof the axial crack.By still rocks under uniaxiai stress. a thirdpossibility(Figure 4½), whichis not unStagesin the bri•le fracture of rocks. From relatedto the second,axial cracksmight form study of brittle fracture of various rocks, at eitherend of a pre-existingcrack or grain Brace [1964b] suggested that a typical axial boundarywhich is inclined to the ax•s. Axial stress-axialstrain curve leadingup to brittle cracks havebeenshownto form in this way fracture might consistof four parts (Figure [Bracea•d Bombolakis,1963; Hock andBienia- 5a). Parts I and II representedelastic beivski,1966]in plasticsand glass.In rocks,the havior; the existenceof a recognizabletoe,
newaxialcracksmight form alonga suitably oriented grainboundaryor •ithin a grain. Severalobservations suggestthat the first of thesepossibilities is unlikely. IsoIaled axial cracks oughtto openandclosecompletely, with
regionI, depended on whetherthe rockinitially containedcracks.In region III, grain boundariessupposedlyloosenedthroughoutthe rock. Growth of cracks from grain boundariesand formation of faults were believed to occur in
nohysteresis. Hysteresisin stress-straincurves a smallregionof nearly constantstress,region
ispronounced [Paulcling, 1965], however,and IV. It is of interest to re-examine this scheme in the light of presentfindings. at smallstresses (Table 2). In addition,the In the presentstudy, we focusedattention axialstrainsat fracturedepart from purely on volumetricrather than linear strains; for elastic behavior(Figuresla throughe) by an comparison, a schematic axial stress-volupermanent changesin volume are detectedeven
appreciable amount. This would not occur if all metric strain curveis shown,togetherwith the newcrackswere simply isolatedaxial cracks. linearstress-strain curve,in Figure5. Someof Either the secondor the third mechanism the results given above can be restated in re-
shown in Figure4 seems likely.Unfortunately,lation to these stress-strain curves. Note that microscope techniques areasyet inadequate to stresshere refers to the stressdifference(to in provewhich actually does occur. But either the axial direction;if (r, is total axial stress,
mechanism wouldgivehysteresis duringa cycle eraequals[(r,]minusconfining pressure. of increasing and decreasing stressand either We believe,as before,that behavioris primechanism mightshowan apparentpermanent marilyelasticin regionsI and II. At least,no
3950
BRACE, PAULDING, AND SCIIOLZ
elasticdistortion of the grainsand frictional
sliding continues, along withformation ofopen
•
axial cracks as described above. /FRACTURE, c
ß
•z • / // ,
Formation of openaxialcracks was
parentlyoverlooked in the earlierstudy, in whichrockswhichhadbeenstressed to within
regionIII were examined microscopically.
The.remayhavebeenseveral reasons forthis. For onething,newopenaxialcrackscould not
easilybe detected if they werealonggrain
boundaries. In Figure 4aand4c,forexample,
bothinclined andaxialcracks mightbeat grain boundaries; sincethe crackwallsof anopen
axialcrackare probablyno morethana few micronsapart, both axial and inclinedcracks wouldappearequally'detached' in a polished
section. Anothersource of this discrepancy
Strain
, --AV/Vo, -AL/Lo
Fig. 5. Idealized stress-straincurves.(a) Stress versuslinear (axial) strain; (b) stressversusvolu-
metricstrain.Behaviorin eachof the fourregions, I, II, III, and IV, is discussed in the text.
may havebeenthe sectioning procedure used in the earlierstudy.It is almostimpossible not to introducecracksduringsectioning of a rock.
Cracksfrom this sourcewouldprobably be randomlyoriented,and this might obscure pattern of crackingdue to stress.
Formationof openaxial cracksnear theends
of closed inclinedcracksalongwhichsliding occurs(whichwe postulateoccursin region permanent changesare observed if stress is
III) suggests behaviorsimilar to that observed in some simpler systems.Brace a•d Bombo-
cycled within these limits. Walsh [1965a] showed that two separate thingshappenin regionII. First, the mineralgrainsdistortelastically. Second,grainsor parts of grainsshift
studiedcrackgrowthphotoelastically. Inclined crackswerecut in glass,.and plasticspecimens,
lakis[1963]andHoekandBieniawski [1966]
which were then loaded to failare in uniaxial slightly under the applied stressand slide or biaxial compression. Two featurescharacter-
relativeto oneanother. Thisleadsto hysteresisizedall of the experiments: newcrackgrowth [Cook,a•d Hodgson,1965] and.othereffects. was in an axial direction,and initial crack Walsh[19650]alsosixowed that,although slid- growth did not lead to failure. The load at failing occurs,no hysteresisshouldbe found in a ure of a specimen was oftenmany timesthat curve of stress-volumetricstrain if the volu- requiredto initiate cracking.This behavior is metricstrainis calculated from(1). Thismeans qualitative,lythe same as what we have obthat sucha curveportraysonlytheelasticdis- servedhere.Boththe rocksand the glass and tortionof the grains and,therefore, theinverse plasticspecimens illustratean interesting slope oughtto beequalto thelinearcompressi-peet of failurein compression, namely,thai bility. We showedabovethat this was true for crack growth in compression does not imseveral compression testsunderconfining pres- mediatelyproducea •nechanical instability as sure.ThroughregionII, then,behavioris still it doesin tension.Cracksgrow and then that of virgin rock. comestable,givingan effectwhichmighibe The beginningof regionIII coincides with calledcrackhardening. The risingstress-strain
thepointwe haveherecalledC'.Assuggestedcurve throughout region III showslhat the in Figure5, thispointcanbemoreeasilylo- stressrequired•o make a crackgrowactually catedin a stress-volumetric strain curve than
increases after somecrackgrowthhasoccurred.
in thestress-linear straincurves, although the This is shownquantitatively for granitein former isderived fromthelatter.In region III, Table2; thestress to cause newcracking, here
I)ILATANCY
iN TtIE Fi:tACTUI
