A lab-testing based geomechanical characterization of metamorphic rocks focusing on post-failure behavior

Rock Engineering and Rock Mechanics: Structures in and onRock Masses – Alejano, Perucho, Olalla & Jiménez (Eds)

© 2014 Taylor & Francis Group, London, 978-1-138-00149-7

A lab-testing based geomechanical characterization of metamorphicrocks focusing on post-failure behavior

I. Pérez-Rey, J. Arzúa, J. Barbiero & L.R. AlejanoDepartment of Natural Resources and Environmental Engineering, University of Vigo, Spain

G. WaltonDepartment of Geological Sciences and Geological Engineering, Queen’s University, Kingston, Canada

ABSTRACT: The stress-strain response of some intact metamorphic rocks (amphibolite, gneiss and marble)has been studied based on around 60 compressive uniaxial and triaxial strength tests. The results have beeninterpreted to obtain peak and residual strength, and post-peak parameters. Particular attention has been givento the study of dilation. The dilation angle of these intact rocks has been fitted to recently developed models forplastic shear strain and confinement-stress-dependent dilation. Even if the dilative response of these rocks hasshown to follow similar trends to those observed in other hard rocks, in the case of foliated ones the peak dilationangle attain is not as high as that of more homogeneous (granite, marble). This is attributed to the fact that thefinal fracturing of the samples partially follows already existing weakness planes, which show less dilation thannewly developed shear bands.

1 INTRODUCTION

1.1 Background

The stress–strain behavior of rocks has beenresearched extensively with respect to elastic andpeak-strength behavior. Nonetheless, a number ofpapers have recently been published focusing on thepost-peak portion of the stress–strain curve. In partic-ular, these studies have focused on evolving failurecriteria and dilation response of sedimentary rocks(coal, limestone, sandstone, mudstone, etc . . .) andigneous rocks (granite).

In order to improve this database and providea more complete view of post-failure behavior ofrocks, aseries of uniaxial and triaxial tests of differ-ent metamorphic rocks have been carried out withstress control, volumetric strain measurements andloading–unloading cycles (Fig. 1).

1.2 Dilatancy

Dilatancy is the volumetric expansion of a material,for instance rock, particularly after failure. Semi-nal researchers in rock mechanics remarked that thisparameter was difficult to calculate, due to the diffi-culties in controlling the stress-strain response afterpeak. In order to understand the dilation pattern ofa rock sample, multiple volumetric strain measuredcompression tests at various confinement levels arerequired; ideally, these tests should include someloading–unloading cycles.

Detournay (1986) was probably the first authorsuggesting that dilation could be dependent on the

Figure 1. Stress–strain curve with loading–unloadingcycles for atriaxial test on a Carrara marble specimen.

irrecoverable strain suffered by the rock. Alejano andAlonso (2005) put forward a confinement stress andplasticity dependent model of dilatancy, based onfitting triaxial test results on samples of sedimen-tary rocks by Farmer (1983) and Medhurst & Brown(1998). One of the main advantages of this model isthat it only depends on one parameter.

Later, Zhao and Cai (2010) and Arzúa & Alejano(2013) showed that the above mentioned approach didnot fit results of tests on igneous rocks. Zhao and Cai(2010) adjusted a model to various dilatancy resultsof different rock types which fit quite well, but at theexpense of needing 9 parameters of unclear physicalsignificance.

Within this framework, the authors have consid-ered the interest of performing more tests in some

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metamorphic rocks in order to extend the dilationdatabase, with the aim of better understand this param-eter and eventually try to propose simpler and/or moreaccurate models to capture dilatant behavior.

2 LABORATORYWORK

2.1 Test set-up

The rock mechanics laboratory at the University ofVigo uses a servo-controlled press able to carry outcompression tests while maintaining a constant con-fining stress. By measuring the displaced hydraulicfluid in the Hoek’s cell required to maintain confiningpressure, the lateral component of volumetric straincan be determined.Axial strain is measured by LVDTsfixed to the lower platen of the press.With this data ofaxial and volumetric strain for every stress level it ispossible to compute radial strain.The complete stress–strain curves derived,as this one presented in Figure 1,can then be used to characterize rock behavior.

2.2 Rocks tested

In this studywe have tested threemetamorphic rocks:agarnet amphibolite fromTouro (La Coruña, Spain), anorthogneiss from Noia (same province) and Carraramarble (Italy).

The garnet amphibolite is a compact, brownish togreenish grey, somewhat banded (Fig. 2 left handside), medium size grained (0.1 to 0.2mm) meta-morphic rock formed through recrystallization underconditions of high viscosity and directed pressure. Itis composed mainly of amphibole minerals (actino-lite) and plagioclase feldspar. It also contains notablequantities of quartz, garnets and pyroxenes. The meta-morphism has somewhat flattened and elongated themineral grains to produce abanded texture (in-betweenschistose and coarse grained, see Figure 3, left handside). The material was recovered in an old copperTouro mine, located some kilometres west of Santiagode Compostela.

The orthogneiss is a very compact high-graderegional metamorphic rock, derived from granite andcomposed by quartz, feldspar (alkaline and plagio-clase), mica (mainly biotite) and small quantities ofhornblende. It is medium to coarse-foliated rock (Fig-ure 2, central sample), characterized by discontinuous,alternating light and dark layers, the former usuallyhaving a coarsely granular texture (0.3 to 0.5mm)while the latter, may be more foliated (Figure 3, righthand side). The material was recovered in an aggre-gate quarry, two kilometres south of Noia (La Coruña,Spain).

The Carrara marble is a metamorphic rock formedalmost entirely by calcite grains. Most grains in thetested samples studied are closest to approximately0.1mm in size. The texture is granoblastic, so it canbe described as formed by equigranular crystals whichadopt a polygonal morphology, typical in the prod-ucts of thermal metamorphism such as this rock. It

Figure 2. Rock samples to be tested. First, amphibolite,second orthogneiss and third, marble.

Figure 3. Thin plate sections with crossed polarization(upper part), and with visible light (lower part) of amphi-bolite fromTouro (left hand side) and orthogneiss from Noia(right hand side). Remark the foliation in the amphibolite(amphibole colored crystals) and the gneiss (black Biotite).

therefore shows no banding or any relevant weaknessdirection (Fig. 2 right hand side).

2.3 Testing

We have tested between 12 and 20 samples of eachof the above mentioned rocks.All of these tests hadconfining pressures in the range of 0 to 12MPa. Dueto the gneiss is and extremely strong and brittle rock(UCS> 250MPa), only 13 tests were performed, withconfinements only between 0 and 6MPa.

As indicated in section 2.1, the radial strain is cal-culated using axial and volumetric strain by means ofequation 1.

Typical graphical results for confined strength test-sare shown in Figure 4. In these graphs it is alsodepicted how to obtain the most relevant geotechni-cal parameters: peak strength (σpeak

1 ), residual strength(σres

1 ), elasticYoung’s modulus (E), Poisson’s ratio (ν)and drop modulus (M ).

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Figure 4. Stress–strain graph for confined compressiontests on metamorphic rock samples, including amphibolite,orthogneiss and marble. The first curve shows how to obtainthe values of peak and residual strength (σpeak

1 and σres1 ),

elastic Young’s modulus (E), Poisson’s ratio (ν) and dropmodulus (M).

It should be noted that because of the brittlenessof gneiss (and also amphibolite to a lesser extent)it was difficult to control the failure of the samplesand achieve a stable transition from peak to residualstrength (Fig. 4). This suggests that the rate of strengthloss observed could be a function of the press strainvelocity, and not of the rock itself. Even if the strengthdata in this portion of the tests cannot be consideredreliable, however, the axial–radial strain relationshipsshould still be representative.

2.4 Computing the dilation angle

Dilation angle was determined using the generalizedapproach proposed by Vermeer and De Borst (1994):

Figure 5. Stress–strain relationships of a compressive teston a rock sample. In the lower graph, the irrecoverable strainlocus (relating the plastic components of the volumetric andaxial strains) is presented, together with the formulation onwhich it is based.

This formulation is valid for the interpretation oftriaxial tests, plane strain conditions and true triaxialsituations. The plastic parameter selected is the plasticshear strain:

Consequently, to calculate the dilation angle it isnecessary to obtain the incremental plastic strains.Thisis achieved by decomposing the total strains into theirelastic and plastic parts:

This is depicted graphically in Fig. 5.For the axialstrain one can plot straight lines —with slopes equalto the Young’s modulus— from arbitrary stress–axialstrain curve values and then to read the plastic axialstrain in the abscissa axis.

Toobtain an accurate value for volumetric strain, theuse of unloading–loading cycles allows the determina-tion of the irrecoverable volumetric strain locus, whichlinks the points of minimum strain in every cycle, asFig. 5 illustrates. Once this irrecoverable strain locusis plotted, the techniquemimics that of the axial strain:a straight line is plotted from the total to the irrecov-erable volumetric strain locus, parallel to the nearestcycles.The volumetric plastic strain corresponds to thepoint on the irrecoverable strain locus, as shown inFig. 5.

Repeating this process several times for every test,it is possible to obtain pairs of values (εp

v–εp1). Then,

using eqs. (1 to 4), a point cloud was obtained fordilation as a function of γp.

It should be noted that some authors subtract thevalue of γp associated to crack damage (usually small)from the obtained values, as the irrecoverable strainsprior to this point are likely related to separate mecha-nisms. This has not been done here, since the authorsconsider the effect relatively negligible.

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Table 1. Best fit parameters for peak and residual strengthparameters for the 3 tested rocks according to Hoek–Brownand Mohr–Coulomb failure criteria.

σc

Rock Strength MPa m s a R2

Hoek–BrownAmphibolite peak 113.5 13.47 1 0.5 0.639

res 113.5 3.11 0.048 0.6 0.877Gneiss peak 255.0 29.21 1 0.5 0.765

res 255.0 6.25 0.004 0.68 0.792Marble peak 94.3 5.5 1 0.5 0.983

res 94.3 4.4 0.004 0.66 0.969

c φ

Rock Strength MPa o R2

Mohr–CoulombAmphibolite peak 17.17 52.6 0.980

res 4.49 41.2 0.983Gneiss peak 30.92 61.6 0.996

res 1.13 58.1 0.997Marble peak 25.6 33.2 0.998

res 1.3 46.2 0.998

3 RESULTS

3.1 Strength

To study the strain–softening strength response of therock samples, peak and residual Hoek–Brown (H–B)and Mohr–Coulomb (M–C) failure criteria have beenfitted to the peak and residual strength values derivedfrom tests.Whereas peak strength was fitted with thetraditional H–B criterion (a = 0.5 and s = 1) typical ofintact rock, the residual one has been fitted to the gen-eralized H–B criterion, which provided better results.These fits, together with the original test data, areshown in Fig. 6.

Triaxial tests were only performed for up to 12MPa(only up to 6MPa with reliable results for gneiss),so the curvature of the enveloping failure is not welldefined. This is probably why H–B and M–C fit-ted well with the results. The values of the obtainedstrength parameters and the observed regression coef-ficients are presented in Table 1.

Attention should be paid to how the strength dropoccurs mainly in the cohesive component of all therocks tested. For marble, friction strengthening isobserved, while friction kept almost constant forgneiss.Also note that the amphibolite results are morescattered showing lower regression coefficients.

For the sake of briefness,estimates of elastic param-eters (Young’s modulus and Poisson ratio) and dropmodulus are not presented in this text.

3.2 Dilatancy results

By means of the dilatancy computation approach pre-sented in section 2.4, (γp − ψ) points clouds havebeen produced for each type of rock at every available

Figure 6. Peak and residual strength tests results and fits toHoek–Brown and Mohr–Coulomb failure criteria for the 3studied metamorphic rocks.

confinement levels (results from multiple tests areaggregated for each confinement level). Results areillustrated in Fig. 7, which shows dilation angle depen-dencies as suggested by Alejano & Alonso (2005). Inparticular, the dilation angle is controlled first by con-fining pressure (when σ3 grows, ψ diminishes), andsecond, by plastic shear strain (i.e. as γp increases,ψ decays).

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Figure 7. Dilation results and Zhao & Cai model fits.

It is worth mentioning that the wide dispersion oftest results is far larger than those observed in gran-ites (Arzúa & Alejano, 2013) or coal (Medhurst &Brown, 1998), particularly for the foliated amphiboliteand gneiss samples. This can be attributed to the moreheterogeneous nature of these rocks. Since 3 to 4 testswere performed for each rock and stress level, evenif there is wide dispersion, we consider the averagevalues to be reasonably representative.

The obtained results were compared with the modelproposed by Zhao &Cai (2010). This requires lab datato be fitted to eq. (5) for every confinement stress,which relates ψ with γp:

Parameters a, b and c control the shape of thecurve.Although each of them affects the whole curve,they primarily influencejust one aspect of shape. amainly controls the peak dilation, b controls peak dila-tion location and c controls the decay rate. Table 2shows the best fit parameters for dilation angle–plasticparameter curves for the 3 rocks studied at different

Table 2. Best fit parameters for dilation angle–plasticparameter curves for the 3 rocks studied at differentconfinements.

σ3

Rock MPa a b c R2

Amphibolite 2 40.00 0.90 0.312 0.2734 39.59 0.89 0.400 0.7166 34.00 0.70 0.590 0.189

10 30.00 0.62 1.047 0.45312 34.42 0.49 0.674 0.351

Gneiss 2 35.00 50.00 0.156 0.5074 30.19 32.00 0.170 0.3306 27.64 22.04 0.209 0.719

Marble 0 45.82 8.42 0.011 0.1551 63.36 2.44 0.715 0.8512 58.08 4.33 0.838 0.8624 52.67 3.48 0.651 0.8836 47.10 4.28 0.885 0.8118 42.83 5.71 1.151 0.936

10 41.74 4.25 0.663 0.92312 45.53 3.68 0.926 0.912

confinements, as fitted to eq. (5) with MATLAB. Thefits roughly capture the general trends, but not the fullyrange of variability, which is indicated by some lowregression coefficients shown in Table 2. The fittingof the b parameters is particularly complex, since thereare few data for the lower values of γp. Fig. 7 illustratesthe fitted points cloud (lines) for each rock and for eachconfining pressure.

The amphibolite is strongly foliated,the marble hasa homogeneous grain structure and the gneiss canbe considered intermediate (homogeneous regions ofwhite minerals with some dark foliated bands). At afirst glimpse, Fig. 7 suggests that marble dilates morethan gneiss and this rockmore than amphibolite. How-ever, decay rates seem higher for marble than for theother two foliated rocks. This is consistent with thefabric of these rocks (Fig. 8).

The parameters a, b and c also showed confiningdependence, as depicted in Fig. 9, according to:

where σ3 is expressed in MPa.The dilation behaviour of a rock is fully charac-

terised by a1, a2, a3, b1, b2, b3, c1, c2 and c3 (see Table3). Using these coefficients it is possible to comparevariations in fitted parameters at different confiningpressures. To obtain the fitting of this values startingfrom the data inTable 2, some outlier values have beenremoved, in particular b and c values for amphibolite.Only three points were available for gneiss. These fit-ting problems may be attributed to the scarcity andvariability of data, but also to the fact that Zhao &Cai (2010) model does not provide a perfect fit to thedilatant behaviour of these rocks.

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Figure 8. Pictures of two failed specimens of every testedrock (amphibolite, gneiss and marble) and sketch of failurepattern observed in some specimens of foliated rocks.

Figure 9. Graphs of parameters a, b and c against con-fining stress for the tested rocks and fitting of parametersa1, a2, a3, b1, b2, b3, c1, c2 and c3.

Table 3. Plastic shear strain parameters of the Zhao & Cai’smodel, for the 3 studied and 3 other rocks.

Rock a1 a2 a3 b1 b2 b3 c1 c2 c3

Quartzite 63.1 11.9 2.8 5.83 6.2 6.7 0.14 1.14 1.23Granite 29 28.1 4.7 8.3 17.5 1.5 0.014 0.07 0.71Marble 41.9 28.9 3.45 4.5 3.97 0.06 8e-3 0.5 0.29Gneiss 24.8 19.3 3.15 11.4 72.3 3.18 0.08 0.05 0.48Amphibolite 22.8 21.9 10.2 1e-4 1.08 15.1 1e-4 0.21 0.53Coal 20.0 35.6 0.89 10.47 26.58 1.31 0.15 17.5 0.82

The obtained coefficients are within a reasonablerange (Table 3). Using these values, it is possibleto compare variations in fitted parameters at dif-ferent confining pressures. Some fits associated forparticular tests are also shown in Fig. 7.

4 CONCLUSIONS

The main aim of this study is to contribute to thedatabase on post-failure stress–strain behavior ofmetamorphic foliated and non-foliated rock samples.

Peak and residual strength were estimated foreach test and dilation angle behavior was computedthroughout the deformation process. As previouslyobserved, this angle was shown to be dependent onboth confining stress and plastic shear strain.

Test results were fitted to a previously proposedmodel (Zhao & Cai (2010)) with moderate success.The average results obtained are similar to thosereported in literature for analogous rocks, since ourdilation angle results were located between those for aharder rock (quartzite) and a softer rock (coal).

Failure in a foliated rock may follow pre-existingweakness bands, so dilation does not rise as high as inmore homogeneous grain size rocks (marble). How-ever, once attained the dilation peak (associated withthe breaking of the asperities of new fractures) thedecay is stronger in marble.

ACKNOWLEDGMENTS

The authors thank the Spanish Ministry of Scienceand Technology for financial support under ContractReference Number BIA2013–44690–P, which fundeddevelopments in this paper. This project was partiallyfinanced by means of ERDF funds of the EU.

REFERENCES

Alejano, L.R. & Alonso, E. 2005. Considerations of thedilatancy angle in rocks and rock masses. InternationalJournal of Rock Mechanics & Mining Sciences 42(4):481–507.

Arzúa, J. & Alejano, L.R. 2013. Dilation in granite dur-ing servo-controlled triaxial strength tests. InternationalJournal of Rock Mechanics & Mining Sciences 61:43–56.

Detournay, E. 1986. Elasto-plastic model of a deep tunnel fora rock with variable dilatancy. Rock Mechanics & RockEngineering 19: 99–108.

Farmer, I.W. 1983. Engineering behavior of rocks. 2nd ed.London: Chapman & Hall.

Medhurst T.P. & Brown E.T. 1998.A study of the mechanicalbehaviour of coal for pillar design. International Journalof Rock Mechanics & Mining Sciences & GeomechanicAbstracts 35(8): 1087–105.

Vermeer, P.A. &De Borst, R. 1984. Non-associated plasticityfor soils, concrete and rock. Heron 29(3): 64 pp.

Zhao, X. G., & Cai, M. 2010. A mobilized dilationangle model for rocks. International Journal of RockMechanics & Mining Sciences 47: 368–384.

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