Damage and recovery of calcite rocks deformed in the ...

Damage and recovery of calcite rocks deformed in

the cataclastic regime

A. SCHUBNEL1,2, J. FORTIN2, L. BURLINI3 & Y. GUEGUEN2

1Lassonde Institute, University of Toronto, 170 College Street, Toronto,

Ontario, Canada, M5S 3E3 (e-mail: [email protected])2Laboratoire de Geologie, Ecole Normale Superieure,

24 rue Lhomond, 75005 Paris, France3Experimental Rock Deformation Laboratory, ETH, Sonneggstrasse 5,

8092 Zurich, Switzerland

Abstract: Compressional and shear wave velocities have been measured during the experi-mental deformation of Carrara marble and Solnhofen limestone in the cataclastic regime,both in dry and wet conditions at room temperature. Measurements were performedunder hydrostatic conditions (up to 260 MPa confining pressure and 10 MPa pore pressure)during triaxial loading (at the constant strain rate of 1025 s21) as well as during differentialstress relaxation. During a full cycle, our results show that the seismic velocities firstincrease as effective mean stress increases. However, when the stress onset of cataclasticdeformation was reached, elastic velocities showed rapid decrease due to stress-induceddamage in the rock. During stress relaxation tests we observed an increase of elastic vel-ocities with time, which suggested a fast ‘recovery’ of the microstructure. A substantialand rapid drop in the velocities occurred when reloading, suggesting that the previous‘recovery’ was only transient. Subsequent relaxation tests showed other marked increasesin velocities. These experimental results suggest that during the deformation of low-porositycalcite-rich rocks, dilatant (crack opening and frictional sliding) and compaction micro-mechanism (pore closure) compete. Evolutions of elastic properties (mainly sensitive tocrack density) and macroscopic volumetric strain (more sensitive to porosity) are thereforenot systematically correlated and depend on the strain rate, the solid stress conditions and thepore pressure.

Interest in the brittle–ductile transition hasincreased considerably in recent years, in largepart due to the fact that the maximum depth ofseismicity corresponds to a transition in thecrust and in the upper mantle from seismogenicbrittle failure to aseismic cataclastic flow, i.e.from localized to homogeneous deformation.The mechanics of the transition depends bothon some extrinsic variable (state of solid stress,pore pressure, temperature, fluid chemistry andstrain rate) and intrinsic parameters (crack anddislocation density, modal composition of therock or porosity, for example). The deformationmechanisms operative during the transitionoccur on scales ranging from microscopic tomacroscopic, and have a profound influence onthe spatio-temporal evolution of stress and defor-mation during the earthquake cycle, as well as inthe coupling of crustal deformation and fluidtransport.

Limestones and marbles, even the ones withvery low porosity like Carrara marble, havebeen studied widely in the past as they canundergo a brittle–plastic transition at roomtemperature for confining pressures easily attain-able in the laboratory (Robertson 1955; Paterson1958; Heard 1960; Rutter 1974). Such a beha-viour is probably due to the fact that calciterequires relatively low shear stresses to initiatetwinning and dislocation glide, even at roomtemperature (Turner et al. 1954; Griggs et al.1960). In consequence, many previous studieshave already documented the mechanical beha-viour of both Carrara marble (Fredrich et al.1989, 1990) and Solnhofen limestone (Robertson1955; Heard 1960; Byerlee 1968; Edmond &Paterson 1972; Rutter 1972; Fisher & Paterson1992; Renner & Rummel 1996; Baud et al.2000). These studies have demonstrated that poro-sity change and failure mode are intimately

From: BRUHN, D. & BURLINI, L. (eds) 2005. High-Strain Zones: Structure and Physical Properties.Geological Society, London, Special Publications, 245, 203–221.0305-8719/05/$15.00 # The Geological Society of London 2005.

related. In the cataclastic flow regime, Baud et al.(2000) showed that the pore space may eitherdilate or compact in response to an applied stress.

In the present study, compressional and shearwave velocities have been measured duringboth hydrostatic and triaxial experiments per-formed on Carrara marble and Solnhofen lime-stone at room temperature. Our new set of datashow that during cataclastic deformation,elastic wave velocities show large variations.Damage (a decrease in the effective elastic prop-erties) and apparent recovery of those propertiesare transient phenomena that depend mainly onthe the stress history, the pore pressure and thestrain rate. Elastic properties, macroscopicstrain and post-deformation microstructuralanalysis were put together and may have impli-cations for the understanding of fault gougebehaviour and, thus, direct consequences on the

understanding of fault zones and the earthquakecycle.

Experimental set-up

Here we describe for the first time the triaxial cellinstalled at the Laboratoire de Geologie of EcoleNormale Superieure (ENS) (Paris, France). Thisis a prototype that was designed and constructedby the company Geodesign, based in Roubaix,France.

Description of the vessel

The Geodesign triaxial cell can reach a confiningpressure of 300 MPa (Fig. 1) and the confiningmedium is oil. The confining pressure is servo-controlled with an accuracy of 0.1 MPa. Axialload is achieved through an auto-compensated

Fig. 1. Schematic diagram of the triaxial high-pressure cell installed at the Laboratoire de Geologie of Ecole NormaleSuperieure of Paris (France).

A. SCHUBNEL ET AL.204

(i.e. that does not move when confining pressurevaries) hydraulic piston that can be servo-controlled in either strain or stress. Maximumload is 90 tons, which corresponds to 717 MPafor a 40 mm-diameter sample. An internal loadcell, manufactured by AMC automation,enables measurement of the applied load directlyon top of the sample with an accuracy of approxi-mately 1 MPa. Minimum strain rate is 1026 s21

and is monitored externally through twoLVDTs (Linear Variable Differential Transfor-mers) placed on top of the piston, outside thevessel. Minimum axial loading rate is0.01 MPa s21 and is servo-controlled using twopressure transducers.

Both confining and axial pressure systems aredriven by the same hydraulic ram (0–35 MPa)along with two intensifiers: 35–300 MPa forthe confining pressure, 35–100 MPa for theaxial pressure. Pore pressure is driven by twoprecision volumetric pumps manufactured byMaxitechnologies. Pore fluid is introduced intothe sample through hardened steel end piecesplaced on the top and bottom of the rocksample. Maximum pore pressure in the systemis 100 MPa. Both pumps can be controlledeither in pressure (0.01 MPa precision), in flow(minimum flow is 0.1 h21) or in volume (pre-cision is approximately 0.005 cm3).

The main advantage of the ENS triaxial appar-atus is its 34 electric feedthroughs that can allowsimultaneous measurements of seismic velocitiesin several directions, as well as other propertiesacross the sample (for example local strains).Finally, a thermocouple enables the temperaturein the oil chamber to be monitored.

Sample set-up and preparation

Four samples (two of Carrara marble and two ofSolnhofen limestone) were cored perpendicularto the bedding. All samples measured 80 mm inlength and 40 mm in diameter. The end surfaces

were rectified and polished to ensure homo-geneous loading and minimum friction duringtesting. The porosity of each sample wasmeasured using a double saturation technique.Four parallel flat surfaces were machined alongthe cylinder of the sample at 908 to oneanother. Compressional and shear wave vel-ocities were measured in each direction underdry atmospheric pressure conditions. Averageinitial porosity and velocity values of Carraramarble and Solnhofen limestone used in thisstudy are summarized in Table 1.

During an experiment, axial and radial strainswere measured directly on the sample’s surfaceusing strain gauge pairs (TML FLA-20,Tokyosokki). Each of which was mounted in a1/4 Wheatstone bridge; strain measurementaccuracy was approximately 1026. Volumetricstrain was calculated using strain gauges accord-ing to the formula 1v ¼ 1a þ 2 � 1r, where 1a isthe axial strain measurement from longitudinalstrain gauge, 1r is the radial strain measurementfrom horizontal strain gage and 1v the calculatedvolumetric strain.

P, SV and SH elastic wave velocities weremeasured perpendicular to compression axis,along diameters of the sample, using couples ofsource–receiver lead-zirconate piezoceramictransducers (PZTs) (Fig. 2). PZTs (PI255, PI cer-amics, 1 MHz eigenfrequency) were glueddirectly onto the flat surfaces of each sampleand positioned with approximately 0.5 mm accu-racy, whilst the distance between opposite(paired) PZTs from which the velocities werecalculated was measured within 0.01 mm. Com-pressional PZTs were 5 mm diameter discs,1 mm thick; Shear PZTs were plates (5 � 5 �1 mm). Pulse was generated by a Sofranelsource generator (up to 370 V at 1 MHzfrequency).

For each velocity measurement, more than 200waveforms were stacked on a digital oscillo-scope, in order to increase the signal/noise

Table 1. Composition, initial porosity, grain size and elastic wave velocities atatmospheric pressure under dry conditions of Carrara marble and Solnhofen limestoneused in this study

Carrara marble Solnhofen limestone

Composition .99% calcite .99% calciteInitial porosity (%) 0.5 4.5Grain size (mm) c. 150 c. 5Mean initial, Vp (km s21) 5.92 5.64Mean initial, Vs (km s21) 3.23 3.05Incompressibility, K (dynamic, GPa) 57.5 50.4Poisson ration, n (dynamic) 0.29 0.29Initial P wave anisotropy (%) ,1 c. 2

DAMAGE AND RECOVERY OF CALCITE ROCKS 205

ratio. In such conditions, the absolute velocityerror bar was of the order of a few per cent, butrelative error in between two consecutivemeasurements was lowered to 0.5% using adouble-picking technique. An example ofobtained waveform recordings is shown inFigure 2b. Inside the vessel, the sample wascovered with a Neoprene jacket that insulated itfrom the confining oil.

Experimental procedure

Two experiments (So#02 and Ca#01) werecarried out under dry conditions at a confiningpressure of to 200 MPa on Solnhofen limestoneand Carrara marble, respectively. Two additionalexperiments (So#03 and Ca#02) were carried outin the saturated regime in which rock sampleswere first subjected to 15 MPa confining pressureand 10 MPa pore pressure for a minimum of 2days in order to attain full saturation. Confiningpressure was then raised up to 260 MPa, andthe sample was left overnight for pore pressureequilibration before the triaxial loading cycle.

All triaxial cycles performed in this study werestrain-rate controlled at a constant strain rate of1025 s21. During each experiment several relax-ation tests were also performed in order to inves-tigate possible recovery of the sample. Duringthe stress relaxation tests, the piston positionwas locked and the differential stress decreaseddue to deformation of the sample. These relax-ation tests lasted from 1 h to several days in thecase of saturated samples Ca#02 and So#03.

Experimental results

Figure 3 illustrates stress–strain curves for thefour tests. Plastic deformation was reached forsmaller differential stress in Carrara marblethan in Solnhofen limestone. Hardening

Fig. 2. (a) Schematic view of prepared sample with attached strain gauges and PZT sensors. Elastic wave velocitieswere measured along diameters. All samples measured 80 mm in length, 40 mm in diameter. An example of obtainedwaveforms for SH waves is shown in (b).

Fig. 3. Differential stress v. axial strain in the four testspresented in this paper. Ca#01 and Ca#02 wereexperiments performed on Carrara marble in dry(Pc ¼ 200 MPa) and wet (Pc ¼ 260 MPa, Pp ¼ 10 MPa)conditions, respectively. So#02 (dry, Pc ¼ ¼ 200 MPa)and So#03 (wet, Pc ¼ 260 MPa, Pp ¼ 10 MPa) wereexperiments performed on Solnhofen limestone.


coefficient increases with confining pressure andfigure shows that several relaxation tests wereperformed during the testing of each sample.

During the four experiments, strains, stressesand elastic wave velocities were continuouslymeasured. This section presents a non-exhaustive compilation of our results in termsof coupled evolution of elastic wave velocitiesand volumetric strain v. effective mean stress,differential stress and time (Elastic wave vel-ocities were measured perpendicular to themain compressive axis, and S wave velocitiesrefer to the average between Sh and Sv wavemeasurements).

Volumetric strain and elastic wave

velocities v. effective mean stress

We will refer to onset of dilatancy C0 and onset ofcompaction C� (as defined by Wong et al. 1997)as the stress onset (in differential stress for agiven effective mean stress) at which macro-scopic inelastic dilatancy and compaction werefirst observed. The onset of cataclastic dilatancyC�0 is defined as the stress value at which cata-clastic deformation turned from macroscopicallycompactive to dilatant.

Figure 4 shows the coupled evolution of volu-metric strain and elastic wave velocities as a

Fig. 4. Coupled evolution of volumetric strain and elastic wave velocities as a function of effective mean stress[(s1 þ 2s3)/3 2 Pp] for experiments performed on Carrara marble. The elastic wave velocities and volumetric strain ofthe experiment Ca#01 (dry, Pc ¼ 200 MPa) are reported in (a) and (c), respectively, and those of experiment Ca#02(wet, Pc ¼ 260 MPa, Pp ¼ 10 MPa) are reported in (b) and (d) respectively. On the right-hand side of the diagram arereported the conditions at which the measurements were taken. Hydro, hydrostatic; dev, under differential stress. C 0

indicates the onset of cataclastic dilatancy.


function of effective mean stress for experimentsperformed on Carrara marble. The effectivemean stress is P ¼ ½(s1 þ 2s3)=3 � Pp�, wheres1 stands for the axial stress, s3 is the confiningpressure and Pp the pore pressure; P is the firststress invariant and allows the representation ofthe hydrostatic and the differential stressestogether on the same plot. The bar on the right-hand side of Figure 4 represents which part ofthe plot corresponds to hydrostatic stress andwhich to differential.

Figure 4a & b illustrate, respectively, thevolumetric strain and Vp measured during theexperiment Ca#01 (dry, Pc ¼ 200 MPa). Up to275 MPa effective mean stress, the rock mechan-ical response was elastic and the volumetricstrain increased linearly. Static effective com-pressibility of Carrara marble in the dry regimewas equal to 70.4 GPa (Fig. 4a). When onset ofcataclastic dilatancy C0 was reached (at275 MPa), the rock started to dilate. Inelasticdilation was also accompanied by non-negligiblestrain hardening. Figure 4b shows that during thelinear elastic phase, P wave velocity exhibitsa slight increase due to crack closure. At200 MPa, P wave reached a plateau value equalto 6.3 km s21 which corresponds to the Voigt–Reuss–Hill average reported for P wave velocityin calcite non-porous aggregates (see forexample Birch 1961). Macroscopic dilatancywas associated with a large linear decrease in Pwave velocity that actually started a littlebefore C0. One per cent of macroscopic dilationresulted in a decrease of more than 20% in Pwave velocities. Therefore, elastic wave velocitymeasurements appear more sensitive to dilatancythan macroscopic volumetric strain. Very inter-estingly, Vp increased when the differentialstress was removed.

Figure 4c & d show the volumetric strain andVp and Vs, respectively, v. effective mean stressin experiment Ca#02 (wet, Pc ¼ 260 MPa,Pp ¼ 10 MPa). Comparison of Figure 4a & cshows that the effect of water on the static effec-tive elastic incompressibility of Carrara marble issmall. Effective compressibility increased to73.6 GPa (þ5%). This is compatible with aninitially small and mainly crack-related porosityof Carrara marble (c. 0.5%). The onset of dila-tancy C0 was reached at 340 MPa effectivemean stress. Strain hardening of the samplewas less marked than during the dry experiment,whereas macroscopic dilation was much larger.Cataclastic dilatancy was also associated withlinear decrease of both P and S wave velocities(Fig. 4d). Total decrease of P wave velocityreached 20% after more than 2% dilation. Com-paring this experiment with the previous dry

experiment shows that P wave velocity under-goes a very similar reduction, although thesample dilated more and the strain-hardeningcoefficient was smaller. This could simply bedue to a saturation effect.

The sample was left for relaxation for morethan 3 days. During those three days, differentialstress slowly decreased, and both P and S wavevelocities showed significant increases. Afterremoving the confining pressure, the elasticwave velocities decreased drastically as newlyformed cracks re-opened. P wave velocityreached a final value of only 3.3 km s21 at5 MPa effective confining pressure. Once takenout of the pressure vessel, the sample showedless than 10% seismic anisotropy and no macro-scopic localization band.

Figure 5 shows the coupled evolution of volu-metric strain and elastic wave velocities as afunction of effective mean stress for experimentsperformed on Solnhofen limestone. In Figure 5a& b, respectively, volumetric strain and P wavevelocities are represented for experiment So#02(dry, Pc ¼ 200 MPa). The rock mechanicalresponse was first elastic and the static effectivecompressibility of dry Solnhofen limestone wasequal to 50.3 GPa (Fig. 5a), which is comparableto what has been previously observed by Baudet al. (2000). At 290 MPa effective mean stress,the onset of compaction C� was reached andthe rock started to compact inelastically. Theonset of cataclastic dilatancy C�0 was attainedat 335 MPa. Whether the sample was compactingor dilating, cataclastic deformation was associ-ated with significant strain hardening. Figure 5bshows that in the elastic regime, P wave velocityincreased linearly and reached a maximum valueof 5.75 km s21. Compaction was not accom-panied by an increase in velocity, but conversely,by a decrease. P wave velocity increased instan-taneously when the differential stress wasunloaded, whereas unloading of the hydrostaticstress was accompanied by a large decrease inP wave velocity due to crack opening. Whenretrieved, the sample showed no strong elasticwave anisotropy (less than 5%) as well as nomacroscopic localization band.

Figures 5c & d show, respectively, the evol-ution of volumetric strain and of P and Svelocities as a function of effective mean stressin experiment So#03 (wet, Pc ¼ 260 MPa,Pp ¼ 10 MPa). Comparing Figure 5a & c, it ispossible to observe that the presence of waterincreased significantly the static effectiveelastic compressibility of Solnhofen limestone(62.5 GPa, þ12%). Water had a smaller increas-ing effect on P wave velocities, as can be seenwhen comparing Figure 5b & d, which is


compatible with the fact that, contrary to Carraramarble, the initial porosity in Solnhofen is quitelarge (c. 4.4%) and mainly equant.

Figure 5c shows that the onset of compactionC� was reached at 310 MPa effective meanstress. Strain hardening was more importantthan during the dry experiment. However, nomacroscopic cataclastic dilatancy was everobserved. Figure 5d shows that, at first, compac-tion was not associated with any variation inelastic wave velocities. However, P and S wave

velocities both started to decrease at 360 MPaeffective mean stress. The decrease was smallerthan during the dry experiment. The samplewas then left to relax for 3 days. As differentialstress slowly decreased, the sample compacted.P wave velocities increased to finally reach thevalue of 6.3 km s21, i.e. the reported Voigt–Reuss–Hill average for P wave velocity in non-porous calcite aggregates. Such behaviour wasalso observed on S waves. The sample wasthen reloaded. Velocities started to decrease

Fig. 5. Coupled evolution of volumetric strain and elastic wave velocities as a function of effective mean stress[(s1 þ 2s3)/3 2 Pp] for experiments performed on Solnhofen limestone. The elastic wave velocities and volumetricstrain of the experiment So#02 (dry, Pc ¼ 200 MPa) are reported in (a) and (c), respectively, and those of experimentSo#03 (wet, Pc ¼ 260 MPa, Pp ¼ 10 MPa) are reported in (b) and (d), respectively. On the right-hand side of thediagram are reported the conditions at which the measurements were taken. Hydro, hydrostatic; dev, under differentialstress. C� stands for the onset of cataclastic compaction, while C�0 indicates the onset of cataclastic dilatancy.


again at 370 MPa. Inelastic compaction startedwhen the maximum effective mean stress pre-viously reached during the first cycle wasretrieved, i.e. at 410 MPa. The sample was leftagain for relaxation and after 3 days velocitiesrecovered completely to their maximum value.These results will be discussed more extensivelyin the next section.

When hydrostatic stress was unloaded, P and Svelocities started to decrease as cracks wereopening again. Once removed form the pres-sure vessel, the sample showed no strong elasticwave anisotropy (less than 5%) or macroscopiclocalization band.

Differential stress relaxation and

elastic recovery

During relaxation, the piston position waslocked. Differential stress Q ¼ (s1 � s3)decreased due to the sample deformation, whilepore pressure and confining pressure were keptconstant. Samples were left relaxing from 1 hin the case of dry experiments to several daysin the case of saturated ones. Because of themachine design, the piston could not be perfectlymaintained in position and therefore, some expo-nentially decreasing small amount of axial straincontinued while the relaxation tests were per-formed (as can be seen in Fig. 7c) – in all thefollowing plots, axial, radial and volumetricstrains were measured using strain gauges. As aconsequence and to be perfectly right, defor-mation taking place in the sample was not onlycomposed of true relaxation, but also of a smallcomponent of creep. However, in the rest ofthis paper, we will refer to those tests as relax-ation phases anyway. Seismic implications ofthese results will be discussed in the next section.

Results obtained while Solnhofen limestoneand Carrara marble were relaxing in the dryregime (So#02 and Ca#01) are shown inFigure 6. In both experiments, relaxation tookplace after approximately 5% axial strain.Figure 6a shows the evolution of P wave velocity(plain symbols) and differential stress (emptysymbols) v. time. On the figure, the differentialstress scale on the right is inverted for easierreading and the timescale is linear. Solnhofensample So#02 was left for relaxation for 1 h(bottom of Fig. 6a). We can observe that: (1) Pwave velocity reached a minimum value whendifferential stress was maximum; and (2) Pwave velocity increased as differential stresswas decreasing. Total P wave velocity increaseduring relaxation was greater than 3%. Differen-tial stress and P wave velocity curves are very

well anti-correlated, so that the stress depen-dence of P wave velocity is clearly demonstrated.

In the same way, Carrara marble sampleCa#01 was left for relaxation for 45 min (top ofFig. 6a) after approximately 5% axial strain.Again, P wave elastic velocity increased whiledifferential stress decreased, and total P wavevelocity increase this time was no larger than2%. Stress dependence of P wave velocity isclearly observed. More surprising was thatwhen differential stress increased again to thepeak value prior to relaxation, P wave velocitiesimmediately resumed to their prior minimumvalues. It can therefore be concluded thatduring cataclastic deformation of both Solnhofenlimestone and Carrara marble in dry conditions,P wave velocity exhibits an ‘elastic like’ beha-viour, i.e. reversible. Figure 6b shows the evol-ution of volumetric strain in both samplesduring the same relaxation tests. Time is in

Fig. 6. Solnhofen limestone and Carrara marblerelaxation experiments in the dry regime (So#02 andCa#01). In both cases, relaxation took place afterapproximately 10% axial strain. Figure 6a shows theevolution of both P wave velocity (plain symbols) anddifferential stress (empty symbols) v. time. In (a) thedifferential stress scale on the right-hand side is invertedfor easier reading and the timescale is linear. (b) Showsthe evolution of volumetric strain (measured using straingauges) in both samples during the same tests. Time is inlogarithmic scale.


logarithmic scale. During relaxation, Solnhofensample So#02 compacted, while Carrara marblesample Ca#01 showed no volumetric change.Hence, elastic wave properties are shown to bemuch more sensitive to small stress variationsthan the macroscopic strain.

Results obtained during two different relax-ation tests performed on wet Carrara marblesample Ca#02 are shown in Figure 7. The firstand second relaxation events took place afterapproximately 2.5 and 10% axial strain, respect-ively. Figure 7a shows the evolution of both Pwave (empty symbols to be read on the left-hand scale) and S wave (plain symbols to beread on the right-hand scale) v. time. We cansee that during both tests P and S wave velocitiesincreased monotonically with time in a logarith-mic scale. Increase in velocities was greaterduring the second relaxation phase, and reachedapproximately 6% and 3% for P and S wave,

respectively. This was certainly due to largerprior axial strain (c. 10%). Consequently,damage in the rock had been more importantbefore the second test started.

Evolution of differential stress v. time isshown in Figure 7b. During both relaxationphases, differential stress decreased monotoni-cally in the semi-log space. Again, the slopewas steeper during the second relaxation, i.e.when the rock had experienced more damage.Figure 7c shows the evolution of axial strainand volumetric strain v. time during both tests.It is important to remember that at such higheffective confining stress, Carrara marble poro-sity is close to zero and mainly crack-related. Thisexplains why, although the sample shortened sig-nificantly (approximately 0.4% in 3 days), volu-metric strain remained small. The evolution ofvolumetric strain can be schematically describedas follow: a first phase where the rock continued

Fig. 7. Carrara marble sample Ca#02 relaxing in the wet regime. First and second relaxation phases took place afterapproximately 2.5 and 10% axial strain, respectively. (a) shows the evolution of both P wave (empty symbols to be readon the left-hand scale) and S wave (plain symbols to be read on the right-hand scale) elastic velocities v. time.Evolution of differential stress is shown in (b) while (c) shows the evolution of both axial strain and volumetric strain(measured using strain gauges). Time scales are logarithmic.


to dilate, an equilibrium phase where compactionand dilatancy balanced, finally followed by acompaction phase.

Results obtained during two relaxation testsperformed on wet Solnhofen limestone sampleSo#03 after approximately 2.5 and 10% axialstrain, respectively, are shown in Figure 8.Figure 8a shows the evolution of P wave(empty symbols to be read on the left-handscale) and S wave (plain symbols to be read onthe right-hand scale) v. time in logarithmicscale. During the first relaxation, increase in vel-ocities was more or less linear with time. Elasticwave velocity increase was also more importantduring that first test than during the second relax-ation, and variation in velocities reached morethan 25% (from approximately 5 to 6.3 km s21

in the case of P waves). During the second relax-ation, behaviour resembled much of what hasbeen seen previously on Carrara marble: P andS wave velocities increase monotonously and

velocity variations reached approximately 6 and3% for P and Sh waves, respectively. Figure 8cshows that compaction was more important(almost 0.3%) during the first relaxation thanduring the second relaxation (0.1%). However,decrease in differential stress was comparable(c. 150 MPa) and monotonous in both cases, asseen in Figure 8b.

Comparing this figure with Figure 5 indicatesthat compaction in Solnhofen limestone can beassociated both with an increase or decrease inelastic wave velocities. At low strain rates (typi-cally during a relaxation test) compaction is slowand wave velocities are increasing. At fast strainrates (typically during triaxial loadings) compac-tion is much faster but wave velocities decreasedue to increasing damage in the rock. This obser-vation highlights that dilatancy and compactionare transient during cataclastic deformation,probably because their associated mechanismsare in competition. As a consequence, and

Fig. 8. Solnhofen sample So#03 when relaxing in the wet regime. First and second relaxation phases took place afterapproximately 2.5 and 10% axial strain, respectively. (a) shows the evolution of both P wave (empty symbols to be readon the left-hand scale) and S wave (plain symbols to be read on the right-hand scale) elastic velocities v. time in alogarithmic scale. Evolution of differential stress v. time in logarithmic scale is shown in (b), while (c) shows theevolution of both axial strain and volumetric strain (measured using strain gauges).


because elastic properties are much more sensi-tive to cracks than to pores, macroscopic strainand elastic wave velocities are not systematicallycorrelated. Such observation is clear throughoutFigures 5–7, and seems to be quite common inporous rocks as it has also been reported recentlyin high porosity sandstones by Fortin et al.(2004). When the fully compacted state isreached, P and S wave velocities should attaina maximum value of approximately 6.3 and3.2 km s21, respectively, which are the reportedVoigt–Reuss–Hill averages for calcite non-porous aggregates. This stage was apparentlyreached for sample So#03 and from then, itsbehaviour resembles that of Carrara marble.This observation is clear when comparingdata from the second relaxation tests inFigures 7 and 8.

Microstructural analysis

Figure 9 presents micrographs obtained in thescanning electron microscopy (SEM) of theintact Solnhofen sample (Fig. 9a & b), asample deformed dry So#02 (Fig. 9c & d) anda sample deformed wet So#03 (Fig. 9e & f).The compressive axis is vertical on all the pic-tures. The initial porosity of the intact rockappears to be quite heterogeneous, with bothequant and elliptical pores. Grain boundariesare serrated and hardly visible, even at highmagnification, as in Figure 9b. When deformedin the dry regime (Fig. 9c & d), the initial poro-sity disappeared and was replaced by a largedensity of small intergranular cracks located atgrain boundaries. In Figure 9d, grain boundariesappear to be opened. Cracks are short (a few tensof micrometres) with large apertures and aspectratios (between 1021 and 1022). Furthermore,cracking is well distributed and non-localized,and shows no evidence of anisotropy in its distri-bution and orientation pattern. Such observationis in agreement with post-deformation elasticwave velocity measurements. In the sampledeformed in the wet regime (Fig. 9e & f),initial porosity decreased even more drasticallyand almost no spherical pores can be seen.Crack density is much lower than in the drycase. Cracks are still located at grain contactsbut their aperture appear much smaller (compar-ing with Fig. 9d). One might also argue that someevidence of dissolution can be seen at severalgrain contacts. Such features, however, mighthave been pre-existent in the rock. Figure 10presents two optical micrographs of Solnhofensamples So#02 and So#03 obtained in cross-polarized light. Irrespective of whether defor-mation occurred in the dry or wet regime,

calcite crystals exhibit extensive twinning thatdoes not pre-exist in the intact rock samples.

Figure 11 presents two optical micrographs ofCarrara marble sample Ca#02. The sample wasdeformed in water-saturated conditions, and onboth pictures the axis of compression is vertical.The microstructures of the deformation are quiteheterogeneous and takes place both on theintergranular and intragranular scales. Asobserved in Solnhofen limestone, many cracksare located at grain boundaries and are generallynever longer than one or two times the grain size.Voids initiating at the triple junction betweengrains show evidence of frictional sliding alonggrain boundary. On the intragranular scale,mechanical twins are present in every singlegrain at a very high density on both pictures.Bent and kinked twins are also present, as canbe seen in Figure 11a, and as was alreadyreported by Fredrich et al. (1989). Twin geome-tries are quite complex, with two and sometimesthree sets of active twins in a single grain (seeFig. 11b). In general and depending, of course,on every crystal orientation, twins seem to beoriented more or less vertically. Geometriesstrongly suggestive of interaction betweenbrittle and plastic deformation mechanisms canbe particularly well observed in Figure 11a asthe bending of twins often nucleates an intragra-nular crack. Tullis & Yund (1992) have reportedsimilar observations on feldspar aggregatesundergoing cataclastic deformation at a tempera-ture of 300 8C. Again, microstructural obser-vations suggest a complex interplay betweenplastic deformation and microcracking.

Features observed under the optical micro-scope and the SEM indicate that deformationmechanisms active during the experimentsincluded: twinning, microcracking and frictionalsliding along fractured grain boundaries and,possibly, pressure solution (when saturated).Macroscopic strain and elastic properties evol-ution depend on the interplay of each of thesemechanisms. As previously reported in Carraramarble by Fredrich et al. (1989), twinningactivity and dislocation glide in calcite grainsare likely to be largely responsible for cracknucleation at grain boundaries. Indeed, becauseof different crystallographic orientationsbetween neighbouring crystals, stress concen-trations due to differential strain may facilitatecrack nucleation and trigger frictional slidingalong fractured grain boundaries. Because ofstress concentration effects, granular flow wouldtend to concentrate around large pores and inter-play of several mechanisms (twinning, crackopening and frictional sliding) would thereforebe needed to induce macroscopic compaction.


Mechanical analysis

In this section we analyse our results in the lightof previous studies performed on Solnhofenlimestone and Carrara marble.

Damage and dilatancy in carbonate rocks

Onset of crack propagation and frictionalsliding. Figure 12 compares the onsets ofdilatancy for both Solnhofen limestone and

Fig. 9. Series of secondary electron SEM images of (a) and (b) intact, (c) and (d) deformed dry (So#02), and(e) and (f ) deformed wet (So#03) Solnhofen samples. The compressive axis is vertical on the pictures. Scale is 20 mmin (a), (c) and (e), and 5 mm in (b), (d) and (f).


Carrara marble determined in this study withthose reported by various authors in the literature(see Table 2). In Figure 12, it appears clear thatmaximum compressive stress s1 at onset of dila-tancy evolves linearly with confining pressure.The onset of dilatancy is also much smaller inCarrara marble than in Solnhofen limestone.

Using the well-established wing crack model(Ashby & Sammis 1990; Fredrich et al. 1990;Baud et al. 2000) has proved to be of particularhelp in understanding the onset of crackpropagation in rocks under compression. Inaxisymmetric compression, and for a randomlyoriented distribution of cracks, the stress con-ditions for a tensile wing crack to propagate inmode I from a sliding crack tip are (see, forexample, Lehner & Kachanov 1996):

s1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ m2

pþ mffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ m2p

� ms3 þ

ffiffiffi3

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ m2

p� m

KIcffiffiffiffiffipl

p

(1)

where s1 and s3 stand for the axial stress andconfining pressure, respectively. m is the internalfriction coefficient of the material, KIc the criticalintensity factor (or fracture toughness) for acrack to propagate in mode I and l is the initiallength of the pre-existing crack or default. Insuch a simple model, but widely used in geome-chanics, the slope of the dilatancy envelopedepends solely on m. Assuming that m is con-stant, the dilatant envelope is a straight line (inthe space (s1, s3)) whose origin depends onthe surface energy of the crack and on itslength. According to Figure 12, best fit ofequation (1) gives a value for the internal frictioncoefficient equal to 0.54 and 0.56 for Solnhofenlimestone and Carrara marble, respectively.Such results are comparable with formervalues obtained by Ashby & Sammis (1990)(m ¼ 0.55) and Baud et al. (2000) (m ¼ 0.53)for Solnhofen limestone, and very close to the

Fig. 10. Optical micrographs obtained in cross-polarized light of Solnhofen samples (a) So#02 and(b) So#03 deformed in the dry and wet regimes,respectively. Compression direction is vertical in allphotographs. Note the twins oriented at a high angle tothe compression direction in the largest grains. Thesetwins developed during the experiments, as theundeformed material does not contain this type of twin.

Fig. 11. Optical micrographs in (a) reflected light and(b) cross-polarized light of deformed Carrara marblesample Ca#02. In both photomicrographs, the axis ofcompression is vertical. In (a) V points to a void at agrain triple junction. In (b) the arrows point towardsevidence of slip in the geometry of a grain boundary (topright-hand corner)


value found by Fredrich et al. (1990) (m ¼ 0.5)in Carrara marble. The best fit for KIc=

ffiffiffiffiffipl

pwas

found to be 124 MPa for Solnhofen limestoneand 27 MPa for Carrara marble. Since bothrocks are composed of 99% calcite, KIc shouldbe equal in both rocks. In our case, assumingthat KIc ¼ 0.2 MPa m1/2, as reported byAtkinson & Avdis (1980), we find an initialcrack length of 2l � 3 mm and 2l � 70 mm forSolnhofen limestone and Carrara marble,respectively. The initial crack length ratio isapproximately 20 when the grain size ratiobetween Solnhofen limestone and Carraramarble is about 30. These results are compatiblewith our microstructural analysis. In the case ofSolnhofen limestone, initial porosity also seems

to be a key issue and dilatancy occurs at lowerstress with increasing porosity. Such an obser-vation has already been reported by Baud et al.(2000) in terms of peak stress and could beexplained by the fact that initial damage in therock is likely to scale with initial porosity.

Cataclastic deformation and cracknucleation. Calcite is well known to requirerelatively low shear stresses to initiate twinningand slip, even at room temperature: Turneret al. (1954) and Griggs et al. (1960) obtainedvalues of the order of 145–216 MPa for f-slipand r-slip dislocation glide, respectively. Inaddition, e-twinning in calcite crystal has beenreported to be triggered for even smaller stresses.

Solnhoffen limestone, 5, 4.4%,

this study, DRY

Solnhoffen limestone, 5, 4.4%,

thisstudy, WET

Renner and Rummel,1996;

Solnhofen limestone, 5, 3.7%

Carrara marble,

this study, 250, 0.5%

Baud et al.,2000


Renner and Rummel,1996;


Edmond and Paterson,1972;


Frederich et al,1989;

Carrara Marble, 230, 0.7%

Axia

l str

ess σ

1 (

MP

a)

Confining pressure σ3 (MPa)

0

200

400

600

800

1000

0 50 100 150 200 250 300 350

brittle cataclastic

Fig. 12. Comparison in s1–s3 space of the onsets of dilatancy for Solnhofen limestone and Carrara marble, asreported in the literature and this study.

Table 2. Compilation of stress values at which the onset of dilatancy C0 and threshold of macroscopic dilatancyC� 0 were observed in Carrara marble and Solnhofen limestone, respectively, during our experiments

Confining pressure Pc Pore pressure Pp Effective mean stress P Differential stress Q(MPa) (MPa) (MPa) (MPa)

C0 Carrara marbleCa#01 200 dry 275 225Ca#02 260 10 340 270C�0 Solnhofen limestoneSo#2 200 dry 335 405


As shown by our microstructural analysis, duringcataclastic deformation, cracks seem to nucleatepreferably at grain contacts and boundaries.Baud et al. (2000) suggested that microcrackingcould be due to dislocation pile-ups. In such amodel, if the dislocations pile-up at an obstacle(for example at a grain boundary), tensile stressconcentrations nucleate a crack (Stroh 1957;Wong 1990).

On the other hand, microstructural obser-vations reported by Fredrich et al. (1990) andourselves in this paper showed that interplaybetween twinning and crack nucleation areintense during deformation. At least one sourceof cracking might be twin intersections, as seenin Figure 11, that could depend inversely ongrain size, as is suggested when comparingtwin density between Figures 10 and 11.

However, two paradoxes can be pointed out.First, we observed throughout our experimentsthe decoupling of macroscopic strain and elasticproperties. For example, during our experimentCa#02 (Fig. 4c & d), dilatancy was observed tobe very large (2%) with small effects on elasticproperties. Second, and as noted originally byBaud et al. (2000), the data suggest a slight depen-dence of C�0 on pressure, whereas dislocationpile-up models or twin–crack interactionmodels predict it to be pressure independent (forinitiation shear stress). Figure 12 seems tosuggest that if cracks can be nucleated by dislo-cation pile-up or by twinning, damage anddilatancy is nevertheless due to frictional slidingon these newly formed crack surfaces.

Compaction mechanics and recovery in

Solnhofen limestone

Effect of initial porosity and water. In theirstudy of compaction and failure modes inSolnhofen limestone, Baud et al. (2000) per-formed experiments in the dry regime only, onsamples with an average porosity of 3% and an

average grain size of 5 mm. As previously men-tioned, our experiments were performed on ablock whose initial porosity was 4.4%, both indry and wet conditions. Four additional exper-iments were performed on samples coming fromthe same block (Schubnel 2003), in the wetregime, at the State University of New York inStony Brook. Table 3 compiles the values ofeffective mean stress P and differential stress Qat which the onset of compaction C� (as definedby Wong et al. 1997) were observed in that block.

A comparison with data previously obtainedby Baud et al. (2000) is plotted in Figure 13. Itshows that water enhances compaction greatly.At a given confining pressure, decrease in differ-ential stress at the onset of compaction can be aslarge as 150 MPa. This has already been shownby Rutter (1986) and could be explained inthree different ways: (1) the presence of waterincreases single crystal plasticity; (2) pressuresolution takes place and accounts for the earlyobserved compaction; and (3) water has a lubri-cating effect.

Variation in porosity also seems to have asmall reduction effect on the onset of compac-tion: C� for experiment So#03 falls a littlelower than the compactive envelope previouslyobtained by Baud et al. (2000). The latter obser-vation is confirmed by a recent general study per-formed on the compaction of several carbonaterocks (Vajdova et al. 2003).

Baud et al. (2000) and Vajdova et al. (2004)modelled the behaviour of limestones usingCurran & Carroll’s (1979) model of porousvoid compaction. In this model, compactionoccurs when a pressure-independent plasticyield criterion, sY (or von Mises criterion), isreached. Carroll (1980) predicted that initialyield in hydrostatic compression occurs at themacroscopic pressure:

P� ¼2

3sY 1 �

2mf

2mþ sY(1 � f)

� �(2)

Table 3. Compilation of stress values at which the onset of compaction C� was observed in Solnhofen limestone.In our block of Solnhofen limestone, initial porosity was 4.4%. The superscripts � indicates experiments that havebeen performed in the Rock Mechanics Laboratory of SUNY at Stony Brook

Solnhofen Confining pressure Pc Pore pressure Pp Effective mean stress P Differential stress Qlimestone (MPa) (MPa) (MPa) (MPa)

C�

So#12� 160 10 225 225So#2 200 dry 290 270So#07� 210 10 270 210So#3 260 10 310 180So#11� 390 10 410 90


where m is the shear modulus and f the initialporosity. Baud et al. (2000) concluded that, inorder for Curran & Carroll’s (1979) model toprovide a reasonable fit of Solnhofen limestone’scompactive envelope, sY needed to be larger(c. 975 MPa) than experimentally determinedvalues for single-crystal plasticity in calcite.When water is present, we found the value ofsY to decrease to approximately 600 MPa,which, in any case, is still much larger thanexperimentally determined values for single-crystal plasticity in calcite. It is interesting toadd that investigating the compaction of dryTavel limestone (9% porosity, 5 mm averagegrain size), Vajdova et al. (2004) showed thatCurran & Carroll’s (1979) model overestimatedthe yield criterion in the same way. In light ofthis paradox, one can suggest that because thevon Mises criterion requires five independentslip systems (Paterson 1969) for homogeneousplastic flow to occur in the proximity of a poresurface, the plastic yield criterion sY was likelyto be larger than the one observed for singleslip systems. However, and as shown onFigure 10, twinning of calcite was observed.Paterson (1969, 1978) showed that whene-twinning in calcite is activated, the combi-nation of twinning and only one slip mode isequivalent to five independent slip systems.The problem remains therefore unanswered buttwo hypothesis can be drawn: (1) plasticdeformation mechanisms like twinning anddislocation glide are grain size sensitive; and(2) operative mechanisms also involve

pressure-dependant frictional processes. Theobservation of evidence for frictional slidingalong grain boundaries in our microstructuralanalysis and the observed effect of water on com-pactive yield stresses may point towards thelatter solution.

Recovery of elastic properties. Cyclic loadingin the brittle regime is characterized generallyby a large hysteresis between loading and unload-ing. However, there is an almost elastic regionwhen changing the state of stress, with verylittle hysteresis and almost constant P and Swave velocities. Our experimental observationsof P and S wave evolutions in low-porositycalcite rocks deforming in the cataclastic regimeshowed that such a ‘deadband’, typical of hyster-esis cycles where the velocities remained constantduring loading and unloading, does not exist. Onthe contrary, the apparent elastic properties ofthe rock showed a high stress dependency andvaried immediately when the stress state waschanged.

The simplest option to explain apparent recov-ery of elastic properties is to assume that it is partlydue to back-sliding on cracks. If so, the onset ofback-sliding on a crack is simply given by:

Dt ¼ mDsn (3)

where Dt and Dsn correspond to the variations inshear and normal stresses on the crack surfaceafter the loading changed direction. In the caseof an axisymmetric unloading, Kachanov

0

100

200

300

400

500

600

0 100 200 300 400 500

C' (dilatancy env)C*(compaction env.)

Peak stress (failure env.)

C*' (cataclastic dilatancy env.)

C*' C*

C* C*'

Effective mean stress P (MPa)

Dif

fere

nti

al s

tres

s Q

(M

Pa) cataclastic dilatancy

compactivebrittle

failu

re

dilata

ncy

Solnhoffen limestone, φ~3%, DRY, Baud et al., 2000

Solnhoffen limestone, φ~4.4%DRY

WET

Cataclastic envelopes for ε = 10-5.s-1

+H2 O

.

Fig. 13. Comparison of our results to those previously obtained by Baud et al. (2000). P is the effective mean stressand Q the differential stress.


(1982a, b) showed that the macroscopic envelopeof the ‘dead band’ for a family of randomlyoriented cracks is a hyperbola in the (s1, s3)space. At effective confining pressures higherthan 200 MPa, and assuming a friction coefficientof approximately 0.5 for calcite, frictional back-sliding on crack surfaces cannot be initiatedbefore axisymmetric extension state is reachedand, therefore, back-sliding cannot account forany observations of recovery.

However, if we assume that the friction coeffi-cient is a state variable (as in the framework ofRate and State Friction; see, for example,Marone 1998), recovery can be due to contactstrengthening. Under contact loading, particlesurface may deform inelastically (for exampledue to dissolution of crack asperities or visco-elastic interpenetration of crack surfaces) andcohesion between contacting surfaces mayincrease. In such a way, strengthening derivesfrom growth of contact area (Dietrich &Kilgore 1994) via chemically assisted mechan-isms (Frye & Marone 2002). In the laboratory,contact healing has been widely observed andwas shown to depend on the logarithm ofcontact duration (e.g. Beeler & Tullis 1997),which has also been observed in our experiments.However, in our relaxation experiments, infer-ring internal friction coefficient cannot bestraightforward and further analysis would beneeded to quantify the exact variations of frictioncoefficient with time.

Conclusions and implications for

fault zones

In the cataclastic regime, our experimentalresults shows that during the deformation oflow-porosity calcite-rich rocks, dilatant (crackopening and sliding, void creation) and compac-tive (pressure solution, pore closure) microme-chanisms are not exclusive and may interact. Inthe case of an initially porous rock likeSolnhofen limestone, the evolution of apparentelastic properties (mainly sensitive to crackdensity) and macroscopic volumetric strain(more sensitive to equant porosity) is not system-atically correlated and depends on the strain rate,the solid stress conditions and the pore pressure.Our data suggest that twinning activity and dislo-cation pile-up in calcite polycrystals participatein the nucleation of cracks at grain boundaries.Macroscopic compaction occurs when frictionalsliding is triggered on those boundary cracks.

During our experiments, damage and recoveryof P–S elastic wave velocities were observed tobe transient phenomena that are associated withdistinct elastic wave velocity variations. In ourexperiments, these variations could be largeand fast, especially in the presence of fluid(.10% in a couple of days, at room tempera-ture). Crack nucleation and propagation wasshown to be responsible for increasing damage,while recovery of elastic properties is thoughtto be due to re-strengthening of grain contacts.

Diffe

ren

tia

l S

tre

ss Q

(M

Pa

)

Effective Mean Stress P (MPa)

0

100

200

300

400

500

600

0 100 200 300 400 500

TR

AN

SIT

ION

?

Tri-X

BRITTLE

FAILURECATACLASTIC DEF.

catac. dilatancy (Soln.)

catac. dilatancy (Carr.)

dilatancy

(Soln.)

dilatancy

(Carr.)

compaction (Sol. +H2 0)

compaction (Sol. DRY)

decreasing grain sizeincreasing crack density

decreasing porositysolid phase change ?

Fig. 14. Schematical deformation map in the principal stress space for calcite rocks according to our experimentalobservations. In the principal stress space, we recall that the triaxial loading path has a slope equal to 3.


Recent in situ velocity monitoring field studies(Crampin et al. 2002) where temporal changesof both P and S velocities were observed beforeearthquakes and volcanic eruptions could beexplained by similar mechanisms.

As fault gouges are highly complex materials,experimentalists need to find good analogues forsimple and relevant laboratory experiments.Figure 14 summarizes schematically the defor-mation map in the principal stress space forcalcic rocks according to our experimental obser-vations. In this space, we recall that the triaxialloading path has a slope equal to 3. In the cata-clastic domain, we see that the dilatant and com-pactive envelopes are overlapping. The observedmacroscopic strain was shown to be a function ofboth strain rate and pore pressure, and the domi-nant mechanism is selected depending on its kin-etics. On one hand, increasing mean stress seemsto favour porosity decrease (and possible solid-phase changes like the calcite–aragonite phasetransition). On the other hand, increasing differ-ential stress is generally accompanied by grainsize reduction and an increase in the crackdensity.

One of the main parameters controlling theearthquake cycle is known to be not only thedifferential stress but also the pore pressure(Rice 1992). Owing to the new fracturenetwork and increase in fault gouge per-meability, poro-elasticity predicts a viscousrelease of the pore pressure during the co- andpost-seismic phases. Such variations in porepressure suggest that the stress path a gougematerial experiences during the seismic cycle islikely to be much more horizontal than regulartriaxial paths in the PQ plot. This last obser-vation highlights the limitations of the ‘regulartriaxial’ stress path when trying to understandfault gouge behaviour or the brittle–ductile tran-sition. Recent analysis of the seismic cycle byMiller (2002) and Shapiro et al. (2003) are point-ing towards that way. As a consequence, webelieve it is necessary to perform experimentsalong different stress paths (for example fluidpressurization) in order to elucidate the naturalfailure processes and their interplay. Furtherexperimental investigations of the cataclasticregime using acoustic emission localizationtechniques, could, potentially, provide insights.

A. Schubnel would like to thank warmly ProfessorR.P. Young for supporting this research under anNSERC grant and for his many instructive comments.This work was initiated when L. Burlini was a visitingscientist in the Laboratoire de Geologie of ENS Paris.The authors would also like to thank Professor B. Evansand an anonymous reviewer, whose comments helped

improve this paper greatly. Also to be thanked areV. Vajdova, W. Zhu and T.-F. Wong for very useful andinstructive discussions when A. Schubnel performedadditional experiments in the Rock Physics Laboratoryat SUNY Stony Brook under an NSF/CNRS co-operationprogram. This work also benefited from discussions withmany scientists. Among them, we would like to thank par-ticularly B. Thompson, K. Mair, J. Hazzard and PierreBesuelle. G. Marolleau and T. Descamps were indispensa-ble for their technical help on the rig.

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