A review of the experimental evidence suggests that creep in brittle rock at low temper- ature is due to time-dependent cracking. A transient creep law is derived ...
•OUIINALOFGEOPHYSICAL RESEARCH
VOL. ?3, NO. 10, MAY 15, 1968
Mechanismof Creepin Brittle Rock C. I--I. SCHOLZ1
Departmento• Geologyand Geop.hysics, Massachusetts Institute o• Technology Cambridge, Massachusetts02139
A review of the experimentalevidencesuggeststhat creepin brittle rock at low temperature is due to time-dependent cracking. A transient creep law is derived from the mechanism of time-dependent cracking i• an inhomogeneous brittle material. The behavior is describedas a Markov processwith a stationary transition probability that is obtained from experimentalobservationsof static fatigue in glasses.The model is comparedwith experimental observationsand found to predict the observed stressdependenceof creep.
INTRODUCTION
When rock is subjected to high deviatoric stress,time-dependentstrain occursin a characteristic manner. This behavior, which occurs even in brittle rock at low temperature and pressure,is usually studied in the laboratory by subjectingrock to a single component. of stressthat is held fixed, whereasstrains are meas.uredas a function of time. Time-dependent deformation
under
these conditions is known
as creep.Figure I showsa typical creep,curve for such an experiment.Becauseof the nature of the curv'e,creepis usuallydescribedas occurring in three.stages[Ro,bertson,1964]. Immediately after the load is applied, creepstrain that decays off swiftly occurs.Creep strain at, this stageis proportionalto the log of time, and the behavioris known as transientor primary creep. The amount of creepthat occursdependson the
by catastrophicfailure on a fault acutely inclinedto the directionof maximumcompression.
,The creepbehaviordescribedaboveis very similarto that observedfor metals,polymers, and many other plastic and viscoelasticmaterials. Because.of this, creep of rock is often
describedin terms of viscoelastic rheological modelsand is interpretedwith respectto mechanisms that are known to occur in metals.
Creep.of •brittlerock at low temperatureand pressureis, however,typified by someunique
characteristics that do not occurin the plastic metals and are not describedby the usual rheologicalmodels.Matsushima[19'60.]found in creep tests on granite in compression that creep strain lateral to the direction of com-
pressionwas larger than creep strain in the
longitudinaldirection.I-Ie also found that lateral creep rate increasedmuch more rapidly with stress and rock type. At fairly low stresses, stress than did longitudinal creep, rate. I-Ie transient creep may account for most of the concludedthat the samples were cracking. time-dependentstrain of brittle rock, i.e., rock Robertson [1960] also concludedthat fracturthat fails by catastrophicbrittle fracture. Trans- ing is an important mechanismof creep.in rock. ient creepis often followed,particularly at high In experimentswith Solenhofenlimestone,he stresses,by deformation at a constant strain found that the densityof the samplesdecreased rate, which is termed secondaryor steady-state significantlyduring testing. creep.If secondarycreepis allowedto continue, Both results suggestthat the volume of the eventually creep beginsto accelerate(tertiary rock increasesduring creep. The inelastic be-
creep.)and fracture occurs.Suchcreepfracture
havior
in
which
volumetric
strain
increases
in brittle rock appears to be similar in all re- relative to what would .be expectedfrom elasspectsto that observedin a conventionalfrac- ticity is known as dilatancy and has been exture test. In compression, for example,it occurs tensivelystudiedin constantstrain rate experiments on rock by Brace et al. [1966]. They also suggestedthat dilatancy was producedby • Now at Seismological Laboratory, California small-scale cracking. This idea was confirmed Institute of Technology, Pasadena, California 91109. and expandedby Scholz[1968a]. In that study, 3295
3296
C. I-I. SCHOLZ
experiments were done at the same constant Fracture
strain rate. To obtain a complete theory of microfracturing, however,we must considerthe effectsof 'bothstressand time on the process.In
Tertiar
SecondaryCreep•
this study we shall extend the basic model into the time domain and derive a creep,equation for rock.
/•Transient.Creep
ANALYSIS
ea: Iogt Time
t
Fig. 1. An idealized creep curve for rock, showingthe several stagesthat are usually recognized.
the small crackingevents,whichwe refer to as microfractures,were studied directly by detecting and analyzing the elastic waves that
they radiate. By comparingthe number of microfractures
that
were detected with
ob-
served macroscopicstrains it was found that dilatancy can be entirely attributed to micro-
fracturing.Watanabe[19'6,3]did a similarstudy and found that creep strain was also,directly proportional to microfracturing activity for granite under compression.Mogi [1962] obtained the same result in creepexperimentsin bending,as did Brown [1965] in uniaxial tension. Gold [1960] also found that cracking activity was directly proportional to creep strain in ice.
Experimentalwork has demonstrated, therefore, that the primary mechanism of creepin britile rock at low temperatureand pressure is microfracturing. Each microfracture contributesan incrementof strain to the body, resulting in time-dependentdilatancy. In con-
junctionwith the earlierstudy [Scholz,1968a], we also developeda statisticaltheory of rock deformation in which rock was treated as an
General. Rock is a polycrystallineaggregate generally of several anisotropic phases. If a uniform stressis applied to such an inhomogeneousmaterial, the local stressat a point will not, in general,be the sameas the appliedstress but will vary in some complexway throughout the body.The diversecracksand poresthat exist in rock will produce additional fluctuations of the stressfield. This inhomogeneityis the basic factor governing the microfracturing of rock [Scholz, 1968a]. Without a precise knowledge of the irregularities producingthe inhomogeneity,
however,we cannotpredict the state of stress throughoutthe body. In view of this difficulty we choseto developour basicmodelalongstatistical lines.Supposea singlecomponentof uniform stressis applied to the body. To describethe distributionof the corresponding componentof local stressa (we disregardfor simplicityother componentsof local stressthat may be present), we definethe probability that a is somevalue in termsof a probabilitydensityfunction•(a; •). The stressprobability function •(a; •) is in general a function of the applied stress•. We also proposed that each small region would fracture when the local stressa exceedsS, the local strength. Variation of strength is treated as variation of stress.Normally, suchfractures will not causefailure of the entire body because they will tend to propagateinto adjacentregions, where the stressis lower, and will be arrested. Each small fracture, however, contributesan increment of local displacementthat in total
inhomogeneous brittle material. The theory was found to predict successfully the observed produces the inelastic stress-strain behavior of microfracturingan'd dilatancy. The above ob- the body. As the body is loaded,the stressin servationthat creep.is alsoproducedby microfracturing indicatesthat we did not take into considerationall aspects of microfracturing, however. The theory only consideredmicro-
each small region tends to increasewith the applied stress, so that eventually there is a finite probability that a exceedsS, and microfracturing will begin to occur. If the stressis fracturingthat is producedasstressis increase,d, increasedstill further, microfracturingactivity and it ignored the time dependenceof micro- will accelerate until eventuallythe formationof fracturing. In that study the influenceof time some grossinstability results in failure of the on the processcould be neglected,becauseall entire body.
MECHANISM
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In that model, we assumedthat the local strength S is constantin time. It is well known, however,that the strengthof most brittle mate-
For many different materials, for which different reaction mechanismshave been proposed, the static fatigue behavior is very similar.
rials, including rocks, is time dependent. If such a material.is subjectedto a constantload it will, in general, fracture after some time interval. This weakeningin time is known as static fatigue. It has been sho.wnfor a wide variety of materials that static fatigue is due to stresscorrosion;i.e., when a brittle material is stresse'd in a corrosiveenvironment,the high tensile stressesat the tips of cracksaccelerate
Alongthe lineso.fthis argument,in fact, Mould and Southwick [1'959] attempted to define 'universal'static fatigue law.
static fatiguewill producetime-dependent micro.fracturing, The primary corrosiveagent responsiblefor
on that scale.
Most o.f the static fatigue studies have been done on homogeneous materials,suchas glasses. An inhomogeneousmaterial will react differently. Each small region (which can be considered homogeneous) will undergostatic-fatigue, but the resulting behavior of the entire body the corrosion reaction there so that the cracks will be time dependent microfracturing, resulttend to lengthen.After a period of time at a ing in creep.In the light of the basicempirical sustained stress level, a crack will reach the similarity of the static fatigue of brittle matecritical Griffith length and will propagateun- rials and Charles' conclusionson the silicates, stably. For a material to which the Griffith we can approximate the static fatigue behavior criterionapplies,such as glassin tension,this of eachregionin an inhomogeneous body by the will causefract.ureof the specimen.In an in- known behavior of homogeneousmaterials. homogeneous mediumsuchpropagatingcracks Glassis sucha homogeneous system,and abunwill be arrested by fluctuationsin the stress dant data are available for i•. This approach fieldexactlyasdescribed above,andmany thou- is justified on the grounds that co.rrosioninsandsmay occurbeforefractureof the body as volves the Si-O bond and there is little differa whole. In an inhomogeneous medium,then, ence between a crystalline silicate and a glass
static fatigue of silicate materials is water
[Charles,1959; le Roux, 1965]. Althoughseveral different types of corro,siv'e reactionshave been proposed [le Roux, 196.5; Stuart trod Anderson,1953], Charleshas studied the static fatigue of a number of silicatesand oxidesand found them to be quite similar in behavior. On the basis of his results he argues that static fatigue of silicatesis due to hydration of the silicon-oxygen bond.Becauseof the generalnature of this reactionhe suggests that all silicate materials shoul'dexhibit similar static fatigue behavior.In a moredetailedstudyle Roux confirmed that,this reactionis responsible for static fatigue in fusedsilica. Although the detailed reaction mechanismis
In static fatigue studies,the material is held at constant stress and data are generally reported as the mean fracture time (t). For most of the materials that have been studied a good
approximationfor the stressdependenceof over five to seven orders of magnitude in time is given ,by
ae*rs'-*'
(1)
where S* is the 'no corrosion'strength, •r is st.ress,and a and b are constants.This equation agreeswell with the data for E glass [Schmitz and Metcalf, 1966], alkali glass [Mould and Southwick,1959], and pyrex, soda glass,lead glass,and fused silica [Glathart and Presto•, 1946]. Equation 1 is obviously not valid over all ranges.Since S* is defined as the strength as t --> 0, (1) cannot hold for small t. It genunknown for most materials and would be exerally fits the data for t > 10-• seeand is quite pected to depend on the environment and the goodwithin the range of interest here [1 < t compositionof the stressedmedium, the gross 10ø]. It is also important t'o note that (1) does form of the time and temperaturedependence not hol'dat low stresses,say below 0.2 to 0.4S*. of strength might be expectedto be the same This can be seen qualitatively, since at lo,w for a wide variety of materials, owing to the stressesthe normal crack blunting tendency of basic thermodynamic similarity of the stress the chemical reaction should tend to counter the corrosionproe.ess.A review of the published stressinduced lengtheningand to inhibit the experimentaldata strongly supports this view. mechanism.The approximation that we shall
C. H. SCHOLZ
3298
assume for the stress dependence of static fatigue is shown in Figure 2. The range of ap-
plicability of (1) is shownby the solidpart of the curve.
Charles [1959] and l½ Roux [1965] have studied the temperature dependenceof static fatigue and found that over a considerable range of temperature the behavior could be given by
l•= c••
(•)
where E is the activation energy of the corrosion reaction, k is Boltzmann's constant, T is the absolutetemperature, and c dependson the material
and its environment.
Equations i and 2 agree well with all the available data on static fatigue within the range of interest
that
we are conceme'd with
This hasbeendemonstrated by the experimental observationsof the proportionality of strain and crackingactivit.y. .3. Each region acts independently. 4. Each region can only fail once. Adopting these assumptions,let the probability that the stressin a regionis from a to a -tda at time t be given by /•(a; if, t) d•, where /•(a; if, t) is the probability densityfunction of stress.In the casewheret = 0, this is the timeindependentfunction )•(a; •). The probability la(a) dt that a region at stressa will fracture duringthe followingtime interval dt is relatedto the meanfracturetime by
•(•r) dt = dt/(t}
here.
Accordingly,we assumetha• static fatigue of and from (3), we have small regionswithin an inhomogeneous silicate material follows the form, combining (1) and la(O-) dt = • exp [--E/KT (•),
(t) - (1/•) cxp [•/•
+ •(s* - •)]
(•)
-- b(S* -- (r)] (4)
The volumetric creeprate ;x is thengivenby
where (r is the local stressand • is a constant.
;x=,
We will now incorporate the above behavior
I(o-;a. t)u(o-) ao-
(•)
in•o the basic model and calculate the micro-
We are treating the caseof creep in compression. 'Compressivestressis taken as negative. For creep in tension the limits of integration wo.uldbe --• and T•, where T • is the local tensilestrength. 1. Each region undergoesstatic fatigue and Sinceeachregionfails only once,the number follows the general empirical approximation, of available regions in each stress range deequation 3. creases a• a rate 2. When a region fails, i• contributesan
fracturing behavior in time. This is characterized by consideringthat each of the small regions within the heterogeneous body have the followingproperties:
averageincrementV to the volumetricstrain.
o•i(o-; a.t) = -I(o-;a.t)u(o-) which integratesto give
[(o-;(•, t) -- [(o-;if, O) exp[--,u(o-) t]
(7)
where )•(a; •, 0), the initial density function, equals the time-independentdensity function )•(a; •) definedearlier. Equation 4 allows us to write
Log Mean Fracture Time ,
Fig. 2. The assume4microscopicstatic fatigue behavior of rock. The range of applicability of
equation I is indicated by the solid part of the curve.
au(o-)= - •,u(o-)•oIf we assumethat the initial distribution )•(a; •,
O) - M, a constantoverS* _• a _• 0 and zerofor • •_ O, the creeplaw is given by
MECHANISM
OF CREEP
IN BRITTLE
ROCK
3299
function of stress/(a; •) will look somethinglike the curve shownin Figure 3. The curve is truncated to the left of S* since instantaneous M
r/•- [exp (--t/(t)o) -- exp (--t/(t)s.)] (8)
frac-
ture will occur in those regions in which the stress exceedsthat value. Cottrell [Davis and Thompson,1950] has shown that, since exp (--t•(a)t) is approximatelya step function, creep could be viewed as an advance of this step
where(t)s.-• = t•(S*)= ft exp[--E/KT] and (t)o-• = •(0) = 1•exp[--E/KT- S*I. In the alongthe a axis. The advancebeginsat S* and rangeof practical interest, (t)s, phys.Res., 73(4), 1447, 1968c. Stuart, D. A., and O. L. Anderson, Dependence of ultimate strength of glass under constant load on temperature, ambient atmosphere, and time, J. Am. Ceram. Soc., 36(12), 416, 1953. Walsh, J. B., The effect of cracks on the uniaxial elastic compression of rocks, J. Geophys. Res., 70(2), 399, 1965a. Walsh, J. B., The effect of cracks on the compressibility of rocks,J. Geophys.Res., 70(2), 381, 1965b.
Walsh, J. B., The effect of cracks in rocks on Poisson's ratio, J. Ge.ophys.Res., 70(20), 5249, 1965c.
Watanabe, H., The occurrenceof elastic shocks during destruction of rocks and its relation to London, 1948. the sequence of earthquakes, in Geophysical Mould, R. E., and R. D. Southwick,Strengthand Papers Dedicated to Professor Kenso Sassa, static fatigue of abradedglassunder contro.lled pp. 653-658, Tokyo, 1963.
ambient conditions,2, Effects of various abra-
sions and the universal fatigue curve, J. Am. Ceram. $oc.,42(12), 582, 1959.
(Received August 30, 1967; revised November 9, 1967.)
