Three-dimensional drained behavior of normally ... · Three-dimensional drained behavior of normally consolidated anisotropic kaolin clay Pongpipat Anantanasakula,n, Jerry A. Yamamuroc,1,

The Japanese Geotechnical Society

Soils and Foundations

Soils and Foundations 2012;52(1):146–159

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Three-dimensional drained behavior of normally consolidatedanisotropic kaolin clay

Pongpipat Anantanasakula,n, Jerry A. Yamamuroc,1, Poul V. Ladeb

aSchool of Civil, Mining and Environmental Engineering, University of Wollongong, NSW 2522, AustraliabDepartment of Civil Engineering, The Catholic University of America, Washington, DC 20064, USAcSchool of Civil and Construction Engineering, Oregon State University, Oregon, OR 97331, USA

Available online 7 February 2012

Abstract

A series of consolidated-drained true triaxial tests with a constant mean effective stress and constant Lode angles during shear is

performed on cross-anisotropic kaolin clay. All tests are performed in a fully automated true triaxial testing apparatus. The relative

magnitude of the intermediate principal stress, expressed in terms of the b-value, and initial cross-anisotropy show significant influence

on the stress–strain, strength, and shear band formation characteristics of the clay. Shear bands form and appear to cause failure in all

true triaxial tests performed, except in triaxial compression. The initiation and development of shear bands is observed to take place,

when the clay undergoes volumetric contraction. The lower strength exhibited in the shear bands is caused by alignment of the clay

particles. This shear band mechanism is different from that observed in granular materials, in which the lower shear strength is reached

due to dilation in the shear bands.

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Keywords: Anisotropy; Normally consolidated clay; Shear strength; Shear banding; Stress–strain behavior; True triaxial test (IGC: D6)

1. Introduction

In geotechnical engineering problems, most loading con-ditions result in fully three-dimensional stress states, in whichthe values of the major principal stress (s1), intermediateprincipal stress (s2), and minor principal stress (s3) aredifferent. Such stress states usually apply to soil elements in

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g author. Formerly: School of Civil and Construction

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ss: [email protected] (P. Anantanasakul).

the ground that show initial or inherent cross-anisotropy dueto preferred orientation of the soil particles developed duringthe deposition of the soils. In such deposits, the axis ofmaterial symmetry is typically parallel to the direction ofdeposition and therefore vertical.Past research has indicated significant influence of the

relative magnitude of the intermediate principal stress andinitial cross-anisotropy on the three-dimensional stress–strain, volume or pore pressure change, and strength char-acteristics of soils. The relative magnitude of the intermediateprincipal stress may conveniently be expressed by means ofthe b-value

b¼s2�s3s1�s3

ð1Þ

where 0rbr1. The parameter b is zero for triaxialcompression, in which s2 ¼ s3, and is equal to unity fortriaxial extension, in which s2 ¼ s1.Very few studies have investigated the effects of b-value

on the behavior of normally consolidated (NC) clay usingtriaxial apparatuses that allowed independent control of all

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P. Anantanasakul et al. / Soils and Foundations 52 (2012) 146–159 147

three principal stresses. Shibata and Karube (1965), Ladeand Musante (1978), Kirkgard and Lade (1993), andPrashant and Penumadu (2005) performed undrained truetriaxial tests on laboratory prepared and natural clays andreported that the values of undrained modulus increased,principal strains-to-failure in the direction of the majorprincipal stress ðe1f Þ decreased, and the pore pressurechanges at failure ðDuf Þ, in general, increased with increas-ing b-value. The effective friction angles increased fromb¼0.00 and remained relatively constant throughout therange of increasing intermediate b-values and decreasedslightly toward b¼1.00. The failure surfaces in the octahe-dral plane were curved and circumscribed the Mohr–Coulomb failure surfaces, when fitted to the friction anglesin triaxial compression.

Nakai et al. (1986) performed drained true triaxial testswith a constant mean effective stress during shear. The valuesof secant modulus increased, and the major principal strains-to-failure decreased with increasing b-value. The volumetricstrains were contractive during shear, and no consistentpattern with b-value was observed. The variation of effectivefriction angles with b-value generally showed the same trendas observed in the aforementioned undrained tests. In theoctahedral plane the failure surface was curved and againcircumscribed the Mohr–Coulomb failure surface, whenfitted to the strength in triaxial compression.

In contrast to these observations on the shape of thefailure surface in the octahedral plane, Pearce (1970, 1971)and Yong and McKeys (1971) observed that the effectivefriction angles from undrained true triaxial tests wereindependent of b-value. The failure surfaces in the octahe-dral plane were observed to agree well with the Mohr–Coulomb failure criterion.

Very few studies on the effects of initial cross-anisotropyon the three-dimensional behavior of NC clay have beenperformed, and not all results are in agreement. Kirkgardand Lade (1993) performed undrained true triaxial tests onintact specimens of cross-anisotropic San Francisco BayMud. Tests with Lode angles (y) from 01 to 1801 werecarried out. When the results of tests with similar b-valueswere compared, the values of secant modulus and porepressure changes at failure were greatest when 01oyo601and smallest when 1201oyo1801. The effective frictionangles for 01oyo1201 were, in general, 5–71 larger thanthose for 1201oyo1801. The effective failure surface in theoctahedral plane was also curved and circumscribed theMohr–Coulomb failure surface for 01oyo1101. For highervalues of y however, the effective failure surface was eitherinside or coincided with the Mohr–Coulomb failure surface.

Callisto and Calabresi (1998) and Callisto et al. (2002)presented results of drained true triaxial tests on lightlyoverconsolidated natural cross-anisotropic Pisa clay. Inthese tests the specimens were K0-consolidated, and themean stress and Lode angles were kept constant. Thevalues of secant modulus were observed not to increasewith increasing b-value, but rather with y. The volumetricstrains were close to zero and contractive during shear.

The specimens’ volume changes did not show any consis-tent pattern with either b-value or y. The failure surfacewas observed to, again, circumscribe the Mohr–Coulombfailure surface in all cases, except triaxial compression aty¼01 and 1201. The failure surface in this study showedmuch larger curvature when 01oyo1201 and thereforehigher friction angles than those reported by Kirkgard andLade (1993) for similar values of y.Very few reports of shear bands accompanying failure of

NC clay are available in the literature, and the effects ofthe intermediate principal stress on the initiation anddevelopment of shear bands in NC clay have not beenconclusive. Georgiannou and Burland (2001) presentedexperimental results showing that shear bands developedin triaxial compression tests on NC clay under drained andundrained conditions, whereas the data by Liu (2004)indicated they did not appear in all cases. Both studiesreported shear bands in all triaxial extension tests.Kurukulasuriya et al. (1999) and Peric and Hwang(2002) observed shear bands in all undrained plane straincompression tests on NC clays. In contrast to thesefindings, no shear bands were observed in the similar typesof tests performed by Vaid and Campanella (1974). Noshear bands have been observed or reported in three-dimensional studies on NC clay using true triaxial appa-ratuses (Shibata and Karube, 1965; Lade and Musante,1978; Nakai et al., 1986; Kirkgard and Lade, 1993; Callistoand Calabresi, 1998; Prashant and Penumadu, 2005).The present research was undertaken because very few

data on the effects of b-value and initial cross-anisotropyon the behavior of and shear band formation in NC clayare available, and not all are in agreement. Presentedherein is a laboratory study of the influence of theintermediate principal stress and initial cross-anisotropyon the stress–strain, volume change, shear band formation,and strength characteristics of NC kaolin clay underdrained conditions. A series of consolidated-drained triax-ial tests with independent control of all three principalstresses was performed. Since cubical specimens are tooshort to avoid interference with the stiff end plates, tallrectangular prismatic specimens that avoid interference ofthe cap and base with the development and directions ofshear band formation were used. Such specimens havebeen shown to allow freely developing shear bands insands (Wang and Lade, 2001; Lade and Wang, 2001). Inaddition, soft latex rubber membranes providing minimalresistance to development of shear bands were used.

2. Experimental procedure

2.1. Soil tested

The kaolin clay used in this study was commerciallysupplied by Unimin Corporation, New Canaan, Connecti-cut. It consisted of approximately 68% clay-sized particles(particle sizeo0.002 mm) and 32% silt-sized particles(0.002 mmoparticle sizeo0.075 mm). Atterberg limit tests

P. Anantanasakul et al. / Soils and Foundations 52 (2012) 146–159148

indicated a liquid limit of 47 and a plastic limit of 30,suggesting a plasticity index of 17. Skempton’s activitycoefficient of the clay was 0.25 and the specific gravity was2.63. According to the USCS Classification System, thekaolin clay was classified as ML.

2.2. Specimen preparation

Dry kaolin powder was mixed with water at a watercontent twice the liquid limit. The produced slurries wereconsolidated in a double draining consolidation tank madeof stainless steel at a vertical pressure of 200 kPa. Prior toconsolidation, a vacuum of 100 kPa was applied to thetank, purging air bubbles trapped inside the slurries. Theinside of the consolidation tank was highly polished andcoated with a thin layer of vacuum grease to minimizefriction between the tank and the clay.

Clay particles, which are platy in shape, reorientedduring the slurry consolidation such that their long axeswere perpendicular to the direction of consolidation,creating particle bedding planes in the clay perpendicularto the direction of consolidation. As such, the clay wascross-anisotropic with the axis of material symmetryparallel to the direction of slurry consolidation. In thisstudy, a coordinate system: x, y, and z, is used tocharacterize the orientation of particle bedding planes,and the axis of material symmetry is denoted as the z-axisin Fig. 1.

Cylindrical clay blocks 190 mm in height and 260 mm indiameter were produced. The blocks had an average watercontent of 38%, void ratio of 1.02, and unit weight of17.70 kN/m3. The clay blocks were then cut into prismaticsections, and these were individually stored in sealedplastic bags and kept in a humidity control room. Theclay sections were trimmed using a fine wire saw toproduce prismatic rectangular specimens 73 mm wide,73 mm long, and 170 mm tall, and these were installed inthe true triaxial apparatus. Specimens are referred to as‘‘vertical’’ and ‘‘horizontal’’ if they are trimmed along theheight parallel and perpendicular to the z-axis as shown inFig. 1.

Fig. 1. Vertical and horizontal specimens. Axis of material symmetry is

perpendicular to particle bedding planes and denoted by z-axis.

2.3. Testing apparatus

The true triaxial apparatus used in this study was amodified version of the one described by Shapiro andYamamuro (2003) and is shown in Fig. 2. This apparatuswas modified to accommodate tall prismatic rectangularspecimens and to allow for full automation. The truetriaxial apparatus consists of an oversized triaxial cell fittedwith a square cap and base, between which prismaticrectangular specimens are located. An external pressurecylinder controlled by an automatic pressure regulatorapplies vertical deviator loads to specimens through apiston rod. A rigid connection between the piston rod andcap enables measurement of vertical deformations duringconsolidation and prevents rotation of the cap, thusallowing uniform specimen deformations in the verticaldirection.Deviator loads in a horizontal direction are applied by a

horizontal loading system that is a self-reacting load frameand consists of two loading plates compressible in thevertical direction. One loading plate is attached tothe rectangular face of an internal pressure cylinder, whilethe other is attached to a backing plate. The internal

Fig. 2. True triaxial apparatus: (a) cutaway view of cell, and (b)

photograph of horizontal loading system.

After Shapiro and Yamamuro (2003).


pressure cylinder and backing plate are attached to oneanother by two tie-bars. An automatic pressure regulatorcontrols the pressure in the internal cylinder. This hor-izontal loading system is supported by two rails mountedonto the cell base plate. One loading plate pushes, whilethe opposing plate is pulled inward when the pressurecylinder is inflated. The loading plates are laminatedstructures. Each plate is made of alternating stainless steeland balsa wood laminae. The compressibility of balsawood permits the vertical deformation of the loadingplates, which is necessary to avoid interference with thecap. The compression frame attached to the piston rodforces the loading plates to deform at the rate of verticaldeformation of the specimen. To accommodate tall speci-mens, more balsa wood and stainless steel laminae wereadded, and the heights of the backing plate and therectangular piston face of the horizontal loading systemwere increased to match the loading plates. The tie-barsare still positioned at the mid-height of loading plates. Thebacking plate and rectangular face of the piston providerigid vertical supports to the loading plates, thus allowinguniform deformations in the horizontal direction. Whenthe horizontal loading system is not activated, conven-tional triaxial compression and extension tests may beperformed.

The cell pressure is supplied by an automatic pressureregulator through an air–water chamber. De-aired waterwas used as the cell fluid. A layer of silicone oil was placedabove the water in the air–water chamber to prevent airfrom dissolving into the cell water and possibly de-saturating the specimens in long-term tests.

The true triaxial apparatus applies three independentprincipal stresses to specimens. The principal stress direc-tions are fixed: the major principal stress (provided by thecell pressure and vertical deviator load) acts in the verticaldirection, the intermediate principal stress (provided by thecell pressure and horizontal deviator load) is applied in thedirection of the horizontal loading system, and the minorprincipal stress (provided by the cell pressure) acts in theother horizontal direction.

Vertical deviator loads were measured by an internal loadcell embedded in the specimen cap. Horizontal deviatorloads and cell and pore pressures were determined frompressure transducer readings. Volume changes were deter-mined by a volume change device similar to that describedby Lade (1988). Vertical deformations of the specimens weremeasured by an LVDT. In the s2-direction, horizontaldeformations were measured by two submersible LVDTs.Deformations of the specimens in the s3-direction weredetermined from the volume changes and deformations inthe s1- and s2-directions. Signals from the transducers wereamplified as necessary and filtered to reduce signal noise.

2.4. Automatic test control

A custom Visual Basic computer program used in con-junction with the apparatus was employed to perform the

tests in a fully automated manner. The computer programconsists of one master unit and three other units; namely, ananalog-to-digital (A/D), a proportional–integral–derivative(PID) controller, and a digital-to-analog (D/A) unit. Themaster unit manipulates the other three units and executessteps necessary to perform the desired test. It obtains user’sinputs (e.g., specimen dimensions, strain rates, and types ofconsolidation and shear) and initiates the control process bystarting the A/D unit that continuously receives input signalsfrom the aforementioned transducers and computes thecurrent stresses and strains on the specimen.To follow the desired stress path, the master unit assigns

target stresses and strains to the PID controller unit, whichat any time (t) attempts to minimize the differences orerrors ðDðtÞÞ between the measured stresses and strains andthe targets by outputting corrective reactions ðRðtÞÞ to theapparatus. The PID controller unit calculates RðtÞ-valuesaccording to the formula

RðtÞ ¼ PDðtÞþI

Z t

t�wDðtÞdtþD

dDðtÞdt

ð2Þ

The parameters P, I, and D are the proportional, integral,and derivative gains, and w is an appropriate time interval.The proportional part determines the corrective reactionswith respect to the current errors, whereas the integral partdetermines the corrective reactions based on the sum oferrors over a time period of w, and the derivative partcomputes the corrective reactions based on the rates, atwhich the errors have been changing. The P, I, and D

parameters that allowed for optimum control of theapparatus were determined from trial-and-error.The D/A unit then converts the corrective reactions into

voltage signals, and these are sent to the appropriateautomatic pressure regulators. As such, the stresses andstrains are readjusted and a real-time feedback controloperation is formed.

2.5. Specimen installation

Filter paper slots cut in a vertical style were attached tothe specimens’ faces in the s3-direction. Parts of the slotscovered the filter stones located on the sides of the cap andbase, providing drainage from the specimens’ to thevolume change device. The cap, specimens, and base weresurrounded by soft latex rubber membranes 0.3 mm thick.Pressure from the o-rings placed on the latex rubbermembranes around the cap and base sealed the specimensfrom the cell water. The horizontal loading system wasinstalled and the rest of the triaxial cell was assembled.Double layers of greased latex membranes 0.3 mm thickwere provided on the cap, base, and both horizontalloading plates to reduce friction between the specimensand the rigid boundaries in the s1- and s2-directions.The filter paper slots and drain lines were saturated using

the standard CO2 method. Gaseous CO2 was introducedthrough the bottom drain line, pushing the air in the filterpaper slots and spaces between the membranes and specimens


out through the top drain line. The gaseous CO2 was thenreplaced by de-aired water introduced through the bottomdrain line. A constant back pressure of 100 kPa wasapplied to the specimens through the volume changedevice. B-values measured after the application of backpressure in all tests were 0.98 and higher, indicatingsatisfactory degrees of saturation.

2.6. Testing program

The specimens were consolidated at an isotropic effec-tive stress of 250 kPa. Prior to shearing all specimens weretherefore normally consolidated. Ten true triaxial testswith a constant mean effective stress ðp0 ¼ ðs01þs

02þs

03Þ=3Þ

equal to 250 kPa and constant values of Lode angle (y¼01,201, 401, y, 1801) were performed. The Lode angle ismeasured counterclockwise in the octahedral plane fromthe major principal stress direction coinciding with the z-axis as shown in Fig. 3(a).

In addition, two triaxial compression tests, one on avertical specimen and one on a horizontal specimen, wereperformed with a constant effective cell pressure of250 kPa. These additional tests were performed to adddata points for these two important test conditions and tostudy the details of shear banding, which occurred in thesoftening regime in triaxial compression tests.

Since the principal stress directions of the true triaxialapparatus are fixed, specimens were installed in the appa-ratus in three different orientations such that the specimenaxes of material symmetry coincided with the s1-, s2-, ands3-directions, as shown in Fig. 3(b). The half-space inFig. 3(b) can be divided into three sectors. In Sector 1,01oyo601, s1 is parallel to, and s2 and s3 are perpendi-cular to the z-axis. Here the b-values are computed from

b¼sy�sx

sz�sx

ð3Þ

Fig. 3. (a) Visualization of Lode angle in octahedral plane, and (b)

rectilinear stress paths in octahedral plane employed in true triaxial tests.

In Sector 2, 601oyo1201, s2 is parallel to, and s1 and s3are perpendicular to the z-axis. Here the b-values arecomputed from

b¼sz�sx

sy�sx

ð4Þ

Finally, in Sector 3, 1201oyo1801, s1 and s2 areperpendicular to, and s3 is parallel to the z-axis. Herethe b-values are computed from

b¼sx�sz

sy�sz

ð5Þ

The values of b in each sector varied from zero, when therectilinear stress path coincided with a major principalstress direction (true triaxial compression), to unity, whenthe rectilinear stress path was on a minor principal stressdirection (true triaxial extension).The specimens were loaded under conditions of strain

control in the vertical direction and stress control in thehorizontal directions. The computer control programobtained a constant vertical strain rate by readjusting thepressure in the external cylinder through the automaticpressure regulator. The major principal stress was deter-mined, and the pressure inside the internal horizontalcylinder and the cell pressure were readjusted so the Lodeangle or b-value and the mean stress were constant. Thecomputer control program updated the dimensions ofspecimens in real time to allow for accurate calculationsof the stresses and strains. All tests were continued to wellafter peak failure. After the tests were stopped, the triaxialcell was disassembled and the horizontal loading systemwas detached. The latex membranes surrounding thespecimens were cut, and photographs of the specimenswere taken. The specimens’ dimensions after testing andshear band inclinations were carefully documented.

2.7. Determination of strain rate

The proper vertical strain rate for the true triaxial testswas determined from specimens with the smallest verticalstrain-to-failure and lowest hydraulic conductivity. Todetermine the condition of lowest hydraulic conductivity,isotropic consolidation tests at an effective confining stressof 250 kPa were performed on specimens with all threeorientations. The volume change versus time relations ofthese tests indicated that the hydraulic conductivity waslowest for the specimen with the z-axis parallel to the s3-direction. Judging from the stress–strain results presentedby Lade and Musante (1978) and Wang and Lade (2001),the strains-to-failure of b¼0.67 tests were assumed to bethe smallest of the four b-values considered. Four trial truetriaxial tests with b¼0.67 on horizontal specimens whoseaxes of material symmetry coincided with the s3-direction(y¼1601) were performed at axial strain rates of 0.01%,0.005%, 0.001%, and 0.0005%/min. In each test, shearingwas halted and the stresses were held constant, when thevertical strain was 0.25%. The drain lines to the volume

Fig. 4. (a) Stress difference–major principal strain, and (b) volumetric

strain–major principal strain relations for 01oyo601 (Sector 1).


strain–major principal strain relations for 601oyo1201 (Sector 2).


change device were then closed, creating undrained condi-tions. The pore pressures increased and stabilized withinabout two hours. The drain lines were then opened, andthe test continued. The same procedure was repeated atvertical strains of 2.5% and 6%. The stabilized porepressure changes were found to decrease with decreasingstrain rate and increasing vertical strain. The pore pressurechanges at a vertical strain rate of 0.001%/min were veryslightly larger than those at a strain rate of 0.0005%/min.When the vertical strains were 0.25%, 2.5%, and 6%, thepore pressure changes at a strain rate of 0.001%/min were7%, 4%, and 2% of the cell pressure.

An axial strain rate of 0.001%/min was thereforeadopted for the true triaxial testing program. Althoughthis axial strain rate resulted in partially drained conditionsat the beginning of shear, it was still chosen becausepractically drained conditions were obtained at higherstrains, and lowering axial strain rate did not significantlydecrease the pore pressure changes. This strain rate alsoallowed the true triaxial tests to finish within a period oftime, in which the apparatus functioned reliably.

During the production tests, pore pressure changes werealso monitored in selected true triaxial tests with otherb-values using the same procedure. The monitored porepressure changes were practically equal to or smaller thanthose obtained in the trial tests. In the subsequent devel-opment, fully drained conditions were assumed, when thespecimens were sheared under a vertical strain rate of0.001%/min. The effective stresses were s0i ¼ si�uback, i¼1,2, 3, where si are the total stresses and uback ¼ 100kPa isthe back pressure held constant throughout the testing.

Additional isotropic compression tests with effectivestresses up to 250 kPa on a prismatic rectangular alumi-num specimen and true triaxial tests with p0 ¼250 kPa on arubber specimen were performed to determine necessarycorrections that accounted for the stiffness and volumetriccompliance of filter paper slots and the deformation oflubricating layers. The filter paper slots, end lubricatinglayers, and latex membranes employed in these tests weresimilar to those used during the production testing. Therubber and aluminum specimens’ dimensions were identi-cal to those of the actual clay specimens. These correctionswere then made during the computations of the measuredstresses and strains.

3. Experimental results

3.1. Stress–strain behavior

The principal stress differences ðs1�s3Þ and volumetricstrains ðevÞ are plotted versus the major principal strainsðe1Þ in Figs. 4–6 to show the effects of b-value in the threesectors of the octahedral plane. Failure points, defined aspoints of maximum effective stress ratio ðs01=s

03Þ, are

denoted by the solid symbols in the figures. In drainedtests, points of maximum effective stress ratio are alsopoints of maximum principal stress difference.

In triaxial compression tests (b=0.00), the stress differ-ences gradually increase over rather large intervals ofmajor principal strain. Smooth failure with uniformstresses and strains is attained in each triaxial compressiontest. The stress–strain curves first gradually descend afterfailure, followed by a rather abrupt decrease in slope.For the triaxial compression test of Sector 1, smooth peakfailure occurs at e1=18.2%, as indicated by the solidsymbol in Fig. 4, while the abrupt decrease in strengthoccurs at e1¼22.2%, i.e., slightly beyond the frame of thediagram. Shear bands were observed to initiate slightlyafter the abrupt decrease in slope in the softening regime.This was more clearly seen in the two triaxial compression


strain–major principal strain relations for 1201oyo1801 (Sector 3).Fig. 7. (a) Stress difference–major principal strain, and (b) volumetric

strain–major principal strain relations for b¼0.00 tests.




tests with constant effective confining pressure (not shownin these diagrams).

For tests with b40.00, failure is reached at smallervalues of major principal strain. Peaks in the stress–strainresponse are followed by significant reduction in strength.The stress–strain curves then level off or increase slightly insome tests. Shear bands were observed in these tests.Except for the test with y¼1601 (b¼0.67 in Sector 3),the stress–strain curves for b¼0.35 and 0.67 are practicallyidentical up to major principal strains of 3–4% in all threesectors. Thereafter, the curves associated with the b¼0.67tests deviate in that they approach failure at lowerstrengths. The stress–strain curves associated with theb¼1.00 tests, except for the test at y¼1801, are similarto those for the b¼0.35 and 0.67 tests, but tend to reachfailure at smaller shear strengths.

The effects of initial cross-anisotropy on the stress–strainbehavior of the clay can be studied by comparing the resultsof tests from different sectors but with similar b-values.Figs. 7–10 show the stress–strain and volume change curvesof tests with the same b-values from each of the three sectors.From these comparisons, it is evident that the stress differencesincrease with strain most rapidly at small strains in Sector 1,and least rapidly in Sector 3. At large strains, the stress–straincurves of all tests, except the one associated with b=1.00, ofSector 1 increase at slower rates than those of Sector 2.

The volumetric strains are contractive during shear forall values of y as shown in Figs. 4(b), 5(b), and 6(b).Independent of b-value, contractive volumetric strainsduring shear with constant Lode angles and under con-stant mean effective stresses have also been reported forNC and lightly overconsolidated clays by Nakai et al.(1986) and Callisto and Calabresi (1998). In the presentinvestigation, the stress paths during shear are located inthe same octahedral plane with p0 ¼250 kPa. It follows that

only the deviatoric components of the stresses change onthis plane. Therefore, the volumetric strains are due onlyto the deviatoric components of the stresses. The volu-metric strain–major principal strain relations are verysimilar, but appear to be ordered such that slightly morecontraction is observed with increasing b-value in each ofthe three sectors. Close observation in Figs. 7(b)–10(b)indicates that the volumetric strains are least contractive inSector 1 and most contractive in Sector 3. These patterns,however, vanish at larger strains, since the volumetricstrain curves practically merge to a single curve. As shown





Fig. 11. Variation of strains-to-failure in direction of major principal

stress with b-values.

Fig. 12. Variation of values of secant modulus evaluated at 50% of stress

differences at failure with b-values.


in Figs. 4(b), 5(b), and 6(b), the specimens exhibit positivevolumetric contraction rates ð_ev=_e1Þ at failure in all truetriaxial tests.

Additional tests in Sector 1 with b-values of 0.00, 0.35,0.67, and 1.00 were performed to investigate the repeat-ability of the adopted experimental procedure. The resultsof these additional tests are also plotted in Figs. 11, 12, 16,and 17 using the crossed square symbols. Discussions onthe results of these tests are presented in the Repeatability

of true triaxial tests section.The strains-to-failure in the direction of the major

principal stress are plotted versus b in Fig. 11. The majorprincipal strain-to-failure for the b¼0.00 test of Sector 1 is

larger than those of Sectors 2 and 3. In general, the strains-to-failure decrease sharply with b-values from 0.00 to 0.35,decrease slightly for b¼0.67, and remain relatively con-stant toward b¼1.00. For b¼0.35, the strains-to-failure ofthe three sectors are practically identical. For b¼0.67 and1.00, the strains-to-failure are largest in Sector 3 andsmallest in Sector 1. Finally, changes in b-value have moreinfluence on the strains-to-failure at small b-values than athigh b-values.The values of secant modulus (Es) are evaluated at the

shearing states corresponding to 50% of stress differencesat failure. The results of these analyses are presented inFig. 12. The values of secant modulus are largest in Sector1 and smallest in Sector 3. The values of secant modulus,in general, increase by 5–10 MPa with b-values from 0.00to 0.67, and decrease by about 2.5 MPa for b¼1.00. Sector2 is an exception, in which the value of the secant modulusfor b¼1.00 is about 1 MPa higher than that of b¼0.67.The variation in secant modulus with b-value closelyfollows the pattern of variation exhibited by the stressdifferences at failure with b-value in all three sectors.The strains in the intermediate (e2) and minor (e3)

principal stress directions are plotted versus e1 in Figs.13–15. The intermediate principal strains are expansive for

Fig. 13. (a) Intermediate principal strain–major principal strain, and (b)

minor principal strain–major principal strain relations for 01oyo601

(Sector 1).



(Sector 2).


b=0.00 and become more contractive for higher b-values.This is due to increasing values of the intermediateprincipal stress. Values of b that result in no intermediatestrain (plane strain condition) are about 0.42, 0.35, and0.26 for Sectors 1, 2, and 3, respectively. The intermediateprincipal strains are least contractive in Sector 1 and mostcontractive in Sector 3.

In tests with b40.00, noticeable changes in the slopes ofthe intermediate principal strain curves take place shortlyafter failure and the curves level off thereafter. During thechanges in the slopes of the intermediate strain curves, thestress–strain curves undergo reduction in strength and leveloff. The test with b=1.00 in Sector 3 is an exception. Inthis test s1 ¼ s2 are both parallel with the bedding planesand, due to the symmetry around the vertical material axis,the strains continue at similar magnitude well beyond peakfailure. Shear bands caused by s1 and s2 formed at thesame time in this test. Although s2 decreases during thereduction of strength after failure, it still causes the speci-men to deform due to sliding of the soil blocks along theshear plane parallel to the s1-direction, and the computedintermediate principal strains continue to increase asshown in Fig. 15(a).

The minor principal strains are expansive for all testsand become more expansive with increasing b-value. Theminor principal strains are least expansive in Sector 1 andmost expansive in Sector 3. The triaxial compression tests(b¼0.00) are exceptions, since the minor principal strainsare identical.

3.2. Strength characteristics

In Fig. 16, the principal effective stresses at failure of theten true triaxial tests are plotted in the left half of theoctahedral plane with a mean effective stress of 250 kPa.The clay is cross-anisotropic, and the stress points atfailure are projected, around the axis of material symmetry(vertical axis), into the right half as shown in the figure.The effective friction angles (j0) are determined from thepoints of maximum stress ratio and are plotted versus b inFig. 17. The experimental failure surface and effectivefriction angles are compared with the Mohr–Coulombfailure criterion and with Lade’s failure criterion (Lade,1977), which are also plotted in Figs. 16 and 17. TheMohr–Coulomb failure criterion without cohesion can beexpressed in terms of the major principal effective stress



(Sector 3).

Fig. 16. Experimental failure surface in octahedral plane with p0 ¼250

kPa. Mohr–Coulomb and Lade’s failure surfaces corresponding to

strength of b¼0.00 test of Sector 1 are also plotted for comparison.

Fig. 17. Variation of experimental effective friction angles with b-values.

Effective friction angles predicted by Mohr–Coulomb and Lade’s failure

criteria with b-values are also plotted for comparison.


and minor principal effective stress

s01�s03

s01þs03

¼ sinj0 ð6Þ

A value of j0=23.811 represents the effective friction angleof the test with b=0.00 of Sector 1. The trace of the Mohr–Coulomb failure criterion in the octahedral plane is shaped

like an irregular hexagon symmetric about the principalstress axes. The Mohr–Coulomb failure surface on theoctahedral plane is straight in each sector because it doesnot take the intermediate principal stress into account. Thepredicted friction angles are constant and independent ofb-value as shown in Fig. 17.Lade’s failure criterion is expressed by means of the firstðI1 ¼ s01þs

02þs

03Þ and third ðI1 ¼ s01s

02s03Þ effective stress

invariants

I31I3�27

� �I1

pa

� �m

¼ Z1 ð7Þ

in which pa¼100 kPa is the atmospheric pressure, theparameter m indicates the curvature in meridian planes,and Z1 gives the opening angle of the failure surface. Bothm and Z1 are dimensionless model constants. Since all testswere performed in one octahedral plane, it is not possibleto determine the value of m. However, from the results ofthe triaxial compression test in Sector 1 the ratio I31=I3 isfound to be 35.1.The trace of Lade’s failure criterion in the octahedral

plane is shaped like a rounded triangle that is symmetricabout the principal stress axes. Lade’s failure criteriontakes the intermediate principal stress into account, so thefailure surface in the octahedral plane of each sector iscurved. According to this criterion, the friction anglesincrease by about 51 for b-values from 0 to 0.5 anddecrease by about 31 when the b-value increases towardunity. In comparison, the experimental failure surface inthe octahedral plane of each sector is curved, showing thefact that the intermediate principal stress or b-valueinfluences the failure of the clay. The friction anglesare lowest when b¼0.00 and increase by about 31 up tob¼0.35. They then remain relatively constant up tob¼0.67 and decrease slightly by about 11 as b approachesunity. The Mohr–Coulomb failure criterion under-pre-dicts, whereas Lade’s failure criterion over-predicts thestrengths and friction angles in all tests performed, exceptin the triaxial compression test of Sector 1. The experi-mental failure surface and j0–b relations are bounded by


the Mohr–Coulomb failure surface to the inside and byLade’s failure criterion to the outside. The effects of initialcross-anisotropy have not been erased by the shearingprocess and are still pronounced at failure. When testsfrom different sectors but with similar b-values are com-pared, the effective friction angles in Sector 2 are largestwhile those in Sector 3 are smallest.

3.3. Shear bands

Shear bands, that is the localization of deformation in thinzones of intense shearing, were observed in the hardeningregime for all tests with b40.00 and in the softening regimefor b¼0.00. Shear bands parallel to the s2-direction traversedthe specimens, and these were separated into blocks by theshear bands of very small thickness. In some tests the slidingblocks interfered with the top and bottom end plates. Thisresulted in the slight increases of the stress–strain curves afterthey leveled off, as shown in Figs. 4(a), 5(a), and 6(a). The testwith b¼1.00 in Sector 3 is an exception, in which shear bandssimultaneously formed in two directions. Two shear bands,both parallel to the s1-direction formed diagonally, criss-crossing the specimen. A single shear band also formedparallel to the s2-direction in this test.

Intersection of shear bands with the s3-faces resulted inoffsets of the specimen geometry. A typical example ofshear bands and offsets is shown in Fig. 18. For all testswith b40, the presence of shear-band offsets on thes3-faces is usually observed at 0.25–0.5% after failure inthe e1-direction. Since the horizontal loading plates prohi-bit direct observation of the specimens on the s2-faces, theexact points of shear band initiation in the specimens arenot known. Shear bands may initiate earlier and maycoincide with the failure points. It is likely that thedevelopment of shear bands in tests with b40.00 is thecause of peak failure.

Fig. 18. Shear band developed parallel to s2-direction and offset of

specimen geometry on specimen’s s3-face.

In these tests shear bands prevent the stress–straincurves from further ascending and from attaining failurestates, in which the stresses and strains are uniform.Therefore, shear banding occurred in the hardening regimeand the development of shear bands was the reason forfailure. Thus, the smooth peak failure corresponding tohomogeneous behavior was not reached in the tests withb40.00. This may explain why the experimental failurepoints of tests with b40.00 are located inside Lade’sfailure surface in all sectors (Figs. 16 and 17).In similar tests on sand, Lade (2003) showed that shear

banding reduced the measured friction angles in the mid-range of b-values, and this reduction in friction angles waspredicted from the Single Hardening Model (Kim andLade, 1988; Lade and Kim, 1988), in which the failurecriterion in Eq. (7) was employed. This failure criterion isused to predict the occurrence of shear banding in sand(Lade, 2003). It anticipates the peak failure to be smoothand serves to pull the stress–strain relation up such that thefailure created by shear banding can occur in the hard-ening regime. This failure criterion is therefore not correctfor the middle range of b-values where shear bandingoccurs in the hardening regime, but it serves an importantrole as target for the condition of smooth peak failure,which is never reached due to shear banding. The details ofprediction of shear banding and consequent lower frictionangles for some b-values are further discussed by Lade(2003). Thus, it is plausible that the measured frictionangles for the cross-anisotropic kaolin clay may be pre-dictable from the Single Hardening Model that employsthe failure criterion in Eq. (7).In all tests performed, shear bands are observed, when

the specimens undergo volumetric contraction. The speci-mens continue to undergo volumetric contraction evenbeyond failure, while the stress–strain curves show reduc-tion in strength. This finding does not agree with thegeneral perception that shear bands initiate and develop ingranular materials that dilate at the onset of shear banding(Wang and Lade, 2001; Desrues and Viggiani, 2004). Ingranular materials, volumetric dilation causes the voidratio to increase and the materials to become weaker, thusallowing shear bands to initiate and develop. Volumetriccontraction, however, would have the opposite effect as itreduces the void ratio and causes the materials to gainstrength. This makes it more difficult for shear bands toinitiate and develop through the materials, causing thesmooth peak failure mechanism to be more likely.Since shear bands are observed in all tests, and the

initiation points are observed when the clay undergoesvolumetric contraction, the mechanism for shear bandingin NC clay appears to be different from the one discussedabove for granular materials. This may be due to the shapeof clay particles, which are small and platy. After initiationof a shear band, the clay particles in the zone of maximumshear stress ratio ðtmax=s0Þ may reorient into a preferredorientation and start to slide along each other, face-to-face, and parallel to the direction of maximum shear stress

Fig. 19. Variation of shear band inclination angles versus b-values.

Predictions by Coulomb, Roscoe, and Arthur’s criteria are also plotted

for comparison.


ratio, as explained by Skempton (1985). The spacesbetween the clay particles in the zone of sliding maydecrease, and a portion of the pore water is squeezed out.At larger displacements, this process may result in aslickensided plane or shear band that progressively growsand extends across the specimen, as discussed by Skempton(1985) and Georgiannou and Burland (2001). Electronmicrographs of shear bands are presented by Lupini et al.(1981) to show the alignment of platy clay particles intoshear bands, along which the residual strength develops. Thestrength of the specimen decreases as the shear banddevelops fully, and the specimen is separated into two blockssliding relative to one another. The clay may reach thecritical state, in which shear deformations in the sliding zonecontinue with a constant volume and a constant shearresistance. The sliding friction corresponds to friction anglessmaller than those measured at peak strength. All measureddeformations in the s1- and s3-directions are caused by thesliding of the two blocks, and no deformations occur in thes2-direction. The strains as well as the stresses are no longeruniform. The observed behavior is not a constitutivecharacteristic, but rather a softening response of the speci-men under a discontinuity of the deformation field.

The shear band inclination angles (Y) relative to thes3-direction are plotted versus b for Sectors 1, 2, and 3 inFig. 19. The measured shear band inclination angles arecompared with the theoretical predictions put forth byCoulomb, Roscoe (1970), and Arthur et al. (1977).Coulomb’s criterion is given in the following

YCoulomb ¼ 45þj0

2ð8Þ

Roscoe’s criterion assumes that the angle is governed bythe angle of dilation (c) at failure

YRoscoe ¼ 45þc2

ð9Þ

The dilation angle at failure is determined from thedilatancy rate at failure ð_ev=_e1Þf

c¼ sin�1 �_ev=_e1� �

f

2� _ev=_e1� �

f

" #ð10Þ

Arthur’s criterion suggests that the shear band inclinationangle depends on both the effective friction angle andangle of dilation at failure, i.e.,

YArthur ¼ 45þj0 þc

4ð11Þ

The measured shear band inclination angles are veryconsistent, varying between 561 and 581. They are notnoticeably influenced by either b-value or the initial cross-anisotropy. The measured shear band inclination anglesagree well with Coulomb’s criterion in all sectors. Roscoeand Arthur’s criteria, however, under-predict the anglesby 8–151.

3.4. Repeatability of true triaxial tests

Additional tests in Sector 1 with b-values of 0.00, 0.35,0.67, and 1.00 were performed to investigate the repeat-ability of the adopted experimental procedure. The strains-to-failure in the major principal stress direction, values ofsecant modulus at 50% of stress differences at failure,effective stresses at failure, and effective friction angles ofthese additional tests are also plotted in Figs. 11, 12, 16,and 17 using the crossed square symbols. The stress–strain,volume change, intermediate principal strain, minor prin-cipal strain curves, and variation in shear band inclinationangles are, however, not plotted for the sake of clarity.The stress–strain, volume change, intermediate principal

strain, and minor principal strain curves of these addi-tional tests are very close to those of the presented tests inSector 1. As shown in Fig. 11, the strains-to-failure of theadditional tests are close to those of the tests presentedearlier. The observed experimental scatter is smaller than7%. In Fig. 12, the experimental scatter for the secant


modulus is also small, being within 8% of the presentedtests in Sector 1. The effective stresses at failure andeffective friction angles are also close to those of thepresented tests as shown in Figs. 16 and 17, with experi-mental scatter smaller than 4%. The measured shear bandinclination angles of the additional tests are practicallyidentical to those of the representative tests.

Since no additional tests were performed in Sectors 2and 3, the same magnitudes of experimental scatterobserved in Sector 1 are assumed for those in Sectors 2and 3. The observed experimental scatter is relativelysmall, suggesting that the adopted experimental procedurecan produce test results with high degrees of repeatability.This, in turn, indicates that the observations made in thepreceding sections regarding the stress–strain, volumechange, shear band, and strength characteristics of thenormally consolidated anisotropic clay are accurate.

4. Conclusions

A series of drained true triaxial tests with constant meaneffective stress of 250 kPa and constant Lode angles wasperformed on normally consolidated cross-anisotropicspecimens of kaolin clay using a true triaxial apparatus.Additional triaxial compression tests were performed withconstant effective confining pressure of 250 kPa on verticaland horizontal specimens. The results allowed for investi-gation of the effects of b-value in each of three sectors inthe octahedral plane as well as initial cross-anisotropyon the stress–strain, volume change, strength, and shearband formation characteristics of normally consolidatedkaolin clay.

In triaxial compression (b¼0.00) tests, the stress differ-ences gradually increased over large intervals of strain.Smooth peak failure with uniform stresses and strains wasattained in the triaxial compression tests with b¼0.00. Thestress–strain curves then gradually descended after failure.Shear bands were observed in the softening regime in thetriaxial compression tests.

The stress–strain relations for the tests with b40.00increased over smaller intervals of vertical strain. Pointedpeaks were reached and followed by significant reductionin strength. Shear bands were observed in the hardeningregime in these tests. Independent of the Lode angle, themajor principal strains-to-failure sharply decreased with b-values from 0.00 to 0.35, slightly decreased for b¼0.67 andremained relatively constant for b¼1.00. The values ofsecant modulus generally increased with increasing b-valueto b¼0.67 and then decreased toward b¼1.00. Thevolumetric strains were contractive during shear for alltests and became slightly more contractive with increasingb-value. The intermediate principal strains were expansivefor b¼0.00 and became contractive for higher b-values.The minor principal strains were expansive for all tests andbecame more expansive with increasing b-value.

For each constant b-value, the stress–strain curves, ingeneral, increased with vertical strain most rapidly in Sector 1

of the octahedral plane and least rapidly in Sector 3.The major principal strain-to-failure for the test withb¼0.00 of Sector 1 was larger than those for Sectors 2and 3. For b¼0.35, the major principal strains-to-failure ofthe three sectors were practically identical. Although verysimilar, the major principal strains-to-failure were largest inSector 3 and smallest in Sector 1 for b¼0.67 and 1.00. Thevalues of secant modulus were generally largest in Sector 1and smallest in Sector 3. The volumetric strains were leastcontractive in Sector 1 and most contractive in Sector 3. Theintermediate principal strains were, in general, least compres-sive in Sector 1 and most compressive in Sector 3. The minorprincipal strains, in general, were least expansive in Sector 1and most expansive in Sector 3.The failure surface in the octahedral plane was curved,

showing the influence of the intermediate principal stresson failure of the clay. The Mohr–Coulomb failure criterionunder-predicted the strengths and friction angles, whereasLade’s failure criterion over-predicted these quantities. Theeffects of initial cross-anisotropy were not erased by theshearing process and were still pronounced at failure. Thestrengths and friction angles in Sector 2 were, in general,largest, while those in Sector 3 were smallest.The initiation and development of shear bands caused

failure in all tests, except in triaxial compression. The shearbands initiated while the clay underwent volumetric con-traction. The measured inclination angles of shear bandsvaried between 561 and 581. They were not noticeablyinfluenced by either b-value or the initial cross-anisotropy.The measured inclination angles of shear bands agreed wellwith Coulomb’s criterion in all sectors.The mechanism of shear banding in the clay was

observed to be different from that in granular materials.In clays, a lower strength is obtained, when the platyparticles align face-to-face with each other and are orientedin the direction of maximum shear stress ratio ðtmax=s0Þ.This is different from granular materials, in which thelower strength is caused by dilation of the materialsresulting in looser soils with lower friction angles.

Acknowledgments

This research was supported by the National ScienceFoundation (NSF) through Grant CIMM 0244354. Thefinancial support is gratefully acknowledged. The authorswould like to thank Professor Scott A. Ashford, Dean ofthe College of Engineering, Oregon State University, forhis support to the first author during the period this study.

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Documents

Three-dimensional drained behavior of normally ... · Three-dimensional drained behavior of normally consolidated anisotropic kaolin clay Pongpipat Anantanasakula,n, Jerry A. Yamamuroc,1,