Scientia Ir

Scientia Iranica� Vol� �� No� �� pp ��

c� Sharif University of Technology� April ��

Behavior of Gravely Sand

Using Critical State Concepts

S�M� Hosseini�� S�M� Haeri� and D�G� Toll�

A series of consolidated undrained triaxial tests were performed in order to understand the

behavior of a gravely sand� The material was selected from Tehran alluvium and is classi�ed

as gravely sand in the Uni�ed Soil Classi�cation System� Critical state concepts were used for

interpretation of the behavior of the soil� The critical state line in q � p� space was reasonably

unique� However� it was not possible to de�ne a unique critical state line in e�ln p� space� The

overall scatter� in the critical state line for the gravely sand studied� was � �� in terms of

void ratio� Two reasons can be identi�ed for this scatter� The �rst reason is the inevitable error

in void ratio calculation and the second is that the behavior of this material� even at a critical

state� is fabric and structure dependant� On the basis of these tests� it has been concluded that�

for this material� a critical state zone with upper and lower limits can be de�ned� Nevertheless�

a unique boundary surface can be de�ned by presenting the data in normalized stress space�

INTRODUCTION

The main objective of this paper is to examine thecritical state concept for gravely sands� Previousstudies have� generally� examined the critical state onuniformly graded sands or silty sands� The soil testedhere is gravely and more widely graded when comparedto those that have been reported in the literature�

The concept of a critical void ratio has beenused to explain the behavior of soil since �� Critical void ratio has two denitions in the literature�Casagrande �� dened critical void ratio as the voidratio at which sand does not change in volume whensubjected to shear� Taylor � � dened critical void ratioas the void ratio at which prevention of volume changeleads to no strength change� These two denitionsare consistent� as the rst denition applies to drainedconditions and the second to undrained conditions�Roscoe et al� �� showed that the concept of a criticalvoid ratio is valid for clays as well as for sands� Thepaper by Roscoe et al� �� was the starting point for

�� Corresponding Author� Department of Civil Engineering�Sharif University of Technology� P�O� Box ��Tehran� I�R� Iran�

�� Department of Civil Engineering� Sharif University ofTechnology� P�O� Box �� Tehran� I�R� Iran�

�� School of Engineering� University of Durham� Durham�UK�

the development of critical state soil mechanics� Theconcept of the critical state is illustrated� schematically�in Figure ��

There have been discussions in the literatureconcerning the ultimate behavior of sands� to di�erentiate between �critical state� and �steady state��Researchers who worked on the behavior of soil usingcritical state soil mechanics have dened the criticalstate as follows� �In a drained test� the critical voidratio state can be dened as that ultimate state ofa sample at which any arbitrary further incrementof shear distortion will not result in any change ofvoid ratio� In an undrained test� the sample remainsat a constant void ratio� but� the e�ective stress� p��will alter to bring the sample into an ultimate state�such that the particular void ratio� at which it iscompelled to remain during shear� becomes a criticalvoid ratio� �� The following shorter denition wasgiven by Schoeld and Wroth �� The critical stateis an ultimate condition where a soil element willcontinue to deform without further change in stressesand volume��

It is normally assumed that the critical state isone in which the initial fabric and structure of the soilis destroyed or rearranged to produce a random fabric�Poulos �� suggested that such a state of shearingunder constant stress and volume can be temporarilyachieved �a quasi steady state� but that it is not the

�� S�M� Hosseini� S�M� Haeri and D�G� Toll

Figure �� Schematic diagram showing de�nition ofcritical state�

true ultimate state� Continued shearing will developan oriented �ow structure when a true state of shearing under constant stresses and volume is achieved�Poulos �� indicates that these can only be identiedwhen shearing occurs at a constant velocity and� hence�denes the steady state as the following� �The steadystate of deformation for any mass of particles is thatstate in which the mass is continuously deformingat constant volume� constant normal e�ective stress�constant shear stress and constant velocity��

At critical state� volume change and shear stressare constant but the fabric coinciding with this statemay be random or oriented� Poulos emphasizes that�in the steady state� the structure is oriented anddeformation velocity is constant� Considering thesedi�erent denitions of critical state and steady stateas mentioned above� one can conclude that the denitions of both concepts are similar� based on constantvolume� constant e�ective mean stress and constantshear stress� However� Poulos emphasizes the constantvelocity and structure orientation for steady state� Hestates that� if the velocity of deformation were zero ornot constant� the specimen would not be in the steadystate condition�

Some researchers di�erentiate between criticaland steady states according to the testing method�Critical state has� generally� been observed on drained�

strainratecontrolled tests� Steady state has beenobserved in undrained tests� usually on loose �contractive� samples �� Examples of the steady stateof deformation have been observed in undrained testson saturated loose sand during liquefaction� Chu ��suggests that� for contractive soils� the critical stateis the ultimate state in a drained test and the steadystate is the ultimate state in an undrained test� However� Poulos �� stated that the steady state could beachieved in both drained and undrained tests�

The critical state was used in this paper� dened as the state observed in undrained triaxial testswhen constant deviator stress and mean e�ective stresshave been achieved� This denition may representa �quasi steady state� condition� such that� withfurther shearing� an oriented �ow structure would bedeveloped� However� as Poulos �� states� it may notbe possible to get to the steady state in triaxial tests�where there are limitations on the applied maximumstrain�

CRITICAL�STEADY STATE FOR SANDS

Critical state and steady state concepts have beeninvestigated for a range of geotechnical materials �e�g��sand� silty sand� clay and soft rock�� Early workfocused on clays� but during the last decade� attentionhas been given to sands� The following pieces of workare of particular signicance�

Been et al� �� examined the critical state for ane to medium� uniformly graded quartzite sand� Theyshowed that the critical and steady states are equal toand independent of a stress path� a sample preparationmethod and initial density� The overall scatter in thecritical state line for the sand studied was in the rangeof � �� in terms of void ratio� at a stress level lessthan �� kPa� They suggested that for real sands withmore variable intrinsic properties� the scatter might bemuch larger�

Riemer and Seed �� examined the e�ect of severalfactors on the apparent position of the steadystateline� They studied Monterey � � sand� which is a uniform� clean and ne beach sand� Their results showedthat the stress level to which samples were consolidatedprior to shearing could a�ect the undrained strength�However drainage condition did not appear to a�ectthe steady state relationship� From test results insimple shear and triaxial �compression and extension��it appeared that the stress path itself was not theimportant factor� but� that the mode of deformation�i�e�� strain orientation� could have a signicant impacton the e�ective stress condition achieved at largestrains�

Verdugo and Ishihara �� carried out triaxial testson Toyoura sand under both undrained and drainedconditions� The results indicated that the quasi steady

Behavior of Gravely Sand ��

state was slightly a�ected by the initial mean stress�whereas the steady state was una�ected by the initialmean stress� Furthermore� the locus of the ultimatestate achieved through drained conditions of loading�was shown to be coincident with the steady state lineevaluated by means of undrained tests� They suggestedthat for a homogeneous soil mass� the steady state lineis a unique reference line una�ected by the type of soildeposition and initial state condition of density andpressure�

Chu �� examined critical state and other similarconcepts for Sydney sand� a uniformly graded quartzsand� He suggested that the critical state curve isindependent of the initial void ratio of the soil� Theother states i�e�� the phase transformation state� thecharacteristic state and the steady state� were closelyrelated to the critical state� For contractive soils� allstates became the same and described the behavior atthe ultimate �or failure� state� For dilative soils� thephase transformation state and the characteristic statewere the same�

Fourie and Papageorgiou �� examined the steadystate for Merriespruit gold tailings� Four particle sizedistributions of Merriespruit tailings were tested todetermine the in�uence of the nes content on theposition of the steady state line� Test results showedthat tailings with the greatest percentage of nes gavea steady state line that plotted above all the others�They pointed out that it is impossible to dene voidratio at steady state conditions to the accuracy oftenquoted in the literature� It is preferable to accept thatsome error is inevitable and to present a condencezone with upper and lower limits to the steady stateline�

Yamamuro and Lade �� examined steady stateconcepts for Nevada silty sand� which included particlesbetween �� mm and �� mm� The sand wascombined with �� nonplastic Nevada nes� Resultsfrom undrained tests with di�erent initial void ratiosproduced di�erent steady state lines� Unique steadystate lines may� therefore� not always exist for siltysand�

Mooney et al� �� examined the uniqueness of thecritical state for a sand� Their results indicated that therelationship between shear stress� q� and mean e�ectivestress� p�� at critical state� was unique� However� therewas not a unique critical state void ratio for a givenmean e�ective stress�

Test results reported by Konrad �� on twosands� showed that the mean steady state line �Fline�� commonly used for the evaluation of liquefactionpotential� was not uniquely related to void ratio fora given sand� Rather� the tests revealed that initiale�ective conning stress also in�uenced the undrainedbehavior of granular material� Furthermore� the testsconrmed the existence of upper and lower limits of

the F line� which were dened as UF and LF lines�respectively�

It seems that� for current stateoftheart� severalbasic questions� previously mentioned by Been et al� ��have not yet been answered� These questions includethe following�

a� Is the Critical�Steady State unique for any particular sand� including silty sand and gravely sand� ordoes it depend on the initial fabric� structure andstress path of the test� Are the e�ects of thesefactors the same for all sands� including gravelysand�

b� From a practical standpoint� are the critical andsteady states the same�

c� What is the general shape of the critical �steadystate over a wide range of void ratio and stresses�

d� How precisely and repeatedly can the critical�steady state be measured�

This paper adds to the range of sands for whichthe critical state�steady state has been investigated�particularly towards a coarser �gravely� soil� and givesa few directions toward nding the answers to some ofthese questions�

EXPERIMENTAL PROGRAM

In order to investigate the behavior of gravely sandand to dene the critical state for this material� a totalof � triaxial tests were performed under isotropicallyconsolidated undrained conditions� The tests werecarried out under ve di�erent conning pressures�between � kPa and �� kPa� Tests were performedon samples compacted to two relative densities� ��and �� A summary of the tests performed is shownin Table ��

Test Material

Material for this research was selected from Tehranalluvium� Tehran alluvium is highly variable in gradingand degree of cementation� The range of cementationis from completely uncemented to strongly cementedand the range of gradation is from gravel and bouldersin the north of the deposit to clay and silt in thesouth� This research was performed on the alluviumfrom a region of Tehran where its gradation is in therange of gravely sand� The grain size distribution ofthis material is shown in Figure and is comprisedof �� nes� �� sand and �� gravel� The shape ofthe particles is subrounded� The properties of gravelysand� which is classied as SWSM in the Unied SoilClassication System� are tabulated in Table �


Table �� Summary of the testing program�

Relative Density Void Ratio Critical State Condition

Con�ning After After E�ective Deviatoric

Test No� Pressure� Consolidation Consolidation Mean Stress Stress

�kPa� �� ec p� �kPa� q �kPa�

CU� � ��

CU � ��

CU� � ��

CU� � ��

CU� ��

CU� ��

CU� ��

CU� ��

CU� ��

CU��

CU��

CU� ��

CU��

CU��

CU��

CU��

CU��

CU��

CU��

CU � ��

CU � ��

CU ��

CU � ��

CU � ��

CU � ��

CU � ��

CU � ��

CU � ��

CU � ��

Table �� Physical properties of gravely sand�

E�ective

Grain Size

D��

Median

Grain Size

D��

Coecient

of Uniformity

Cu

Coecient

of Curvature

Cc

Speci�c

Gravity of Solids�

Gs

Maximum

Void Ratio

emax

Minimum

Void Ratio

emin

�� mm � mm � ��

Test Procedure

A standard triaxial apparatus was used for testing thegravely sand samples� Samples� �� mm in diameterand �� mm in height� were prepared in split moldsmounted on the lower platen of the apparatus� Sampleswere tamped into the mold in eight equal layers toachieve the desired void ratio� The material formingeach layer was rst dried and weighed and ��

water �equal to the optimum moisture content of thematerial� was added� The samples were saturatedby �ushing with carbon dioxide gas before �ushingwith deaired water� The process of saturation wasfollowed by increasing back pressure until a B valuegreater than �� was obtained� During this process� an e�ective stress of approximately �� kPa wasmaintained on the sample� After saturation� samples were isotropically consolidated and were sheared


Figure �� Grain size distribution of the gravely sand�

under strain control at a strain rate of �� perminute�

Membrane Penetration

Triaxial testing devices are employed for a major portion of the engineering investigations on the behaviorand strength characteristics of soil� However� for coarsegrain soils� as pointed out by many researchers �� during undrained and drained tests on saturatedsamples� penetration of the �exible rubber membraneinto peripheral voids changes and in�uences pore pressure and volume change measurement� respectively�The membrane penetration e�ect depends on factorssuch as e�ective conning pressure� mean grain size�relative density and membrane thickness� Consolidatedundrained triaxial tests have been performed in thisresearch� Therefore� membrane penetration has ane�ect on the volume change measurement at the consolidation stage and on pore pressure at the shear stage�At the consolidation stage� the equation� proposed byBaldi and Nova �� for volume change correction� wasused as below�

Vc � V� ��V � Vm� ��

Vm ��

dgDV�

��

�dg

Emtm

��

At the shear stage� membrane penetration may havean in�uence on pore water pressure� Ansal andErken �� formulated a correction procedure to calculate the e�ect of membrane compliance on porepressure� They showed that pore pressure measurement can be corrected by an amplication factor�which depends on grain size distribution� membrane�exibility� conning pressure� soil compressibility� relative density and sample diameter� Although basesoil� in this research� contains coarse grain material�it also has su�cient ne grains� which can cover

voids on the peripheral surface of the sample� Onthe other hand� membrane penetration usually has asignicant e�ect on pore pressure when the soil is ata loose to medium density state� However� in thisresearch� the samples were tested at medium to highrelative densities� Two membranes were used duringthe triaxial tests in this research because the rstmembrane was usually damaged during the compactionof the soil inside the mold� during sample preparation�An increase in thickness decreases the �exibility ofthe membrane� resulting in the decreasing e�ect ofmembrane penetration� Using �� mm samples is agood defense against the membrane compliance e�ectbecause increasing sample diameter causes a decreasein the membrane compliance e�ect� With referenceto the aforementioned discussion� the membrane e�ecton pore pressure development during shear has beenneglected�

Void Ratio Calculation

Calculation of void ratio forms the basis for the analysisof test results using critical state concepts� Void ratiowas calculated by using a dimension of samples� Thevoid ratio was dened as�

e �V

Vs� ��

In which V and Vs are� respectively� the total volumeafter consolidation and volume of solids in the specimen� For a cylindrical sand specimen of height h�diameter d and with a dry weight� ms� and specicgravity of solids� Gs� Equation � can be rewritten as�

e ��d�hGs

�ms� ��

Vaid and Sivathayalan �� have determined that themaximum error in void ratio in this method is�

�e � �� e�

� �c

c��h

h��ms

ms��Gs

Gs

��

where �c� �h� �ms and �Gs are� respectively� errorsin circumference� height� mass and specic gravity measurements� Vaid and Sivathayalan �� assumed thatthe error in specic gravity for sand was zero� However�in this research� because gravely sand includes a widerange of grain size and is not uniform in material� onecannot assume �Gs to be equal to zero� For this study�the error in measurement of circumference ��c�� height��h�� weight ��ms� and specic gravity ��Gs� hasbeen taken to be equal to �� mm� �� mm� � g and�� respectively� Using Equation �� the maximumerror in the calculation of the void ratio is in the orderof ��


STRESSSTRAIN AND PORE WATERPRESSURE DEVELOPMENT

Stressstrain behavior� pore water pressure development and stress paths for samples with di�erent initialvoid ratios� consolidated under two e�ective conningpressures �� kPa and �� kPa�� are shown in Figures �and �� As can be observed from these gures� evensamples with similar void ratios� consolidated at thesame conning pressure� show di�erent stress paths�

Figure �� Stress�strain and pore water pressuredevelopment for con�ning pressure �� kPa�

stressstrain behavior and pore water pressure development� For example� test nos� CU� and CU� in Figure ��void ratios �� and �� respectively� and test nos�CU � and CU in Figure � �void ratios �� and�� respectively� show completely di�erent behavior�even though the void ratios are very similar� Sometests show inverse behavior� for example� in Figure ��test no� CU � shows strength higher than that of testno� CU � even though the void ratio of CU is lowerthan that of CU �� Similar trends in soil response could

Figure �� Stress�strain and pore water pressuredevelopment for con�ning pressure �� kPa�


be observed under other conning pressures� This canbe seen more clearly in Figure �� which shows thestress paths of the tests conducted at all conningpressures on samples prepared at two di�erent relativedensities� The stress path of the tests on samplescompacted at relative densities of around �� and�� are shown in Figures �a and �b� respectively�These results show that the behavior of the studiedmaterial is dependant on some other parameters� inaddition to the void ratio and conning pressure� Thetest condition and sample preparation techniques wereexactly the same for all the tests� However� it ispossible for the fabric and structure of the samples�which are unknown� to be di�erent� Although theoverall distribution of particle sizes and the overallvoid ratio were strictly controlled� the arrangement ofparticles of di�erent size and the distribution of voidswithin each sample could not be controlled� Chu etal� � �� presents evidence from Verdugo that samplenonuniformity� in terms of grain size distribution�results in completely di�erent behavior� This is also

Figure �� Stress paths of all the tests�

consistent with the fact that small changes of grain sizedistribution can alter the critical state line of sandssignicantly� The material studied in this researchis not a uniform soil� It includes a wide range ofparticle sizes� from less than �� mm up to �� mm�Therefore� the samples may contain di�erent particlearrangements� di�erent void distributions throughoutthe soil mass� di�erent interparticle forces and di�erentorientations of individual particles� This fabric andstructure heterogeneity can a�ect stressstrain� porewater pressure development and the stress path of thematerial under triaxial loading�

Critical State in q � p� Space

Deviatoric stress and pore water pressure at the endof the tests become almost constant� as can be seenfrom Figures � and �� This suggests that the criticalstate was reached at the end of each test� The valueof q and p� at the end of each test is shown in q � p�

space in Figure �� As can be observed� the state ofstresses at the critical state can be represented fairlywell by a straight line passing through the origin inq � p� space� The slope of this line� M � equals ��This is equivalent to a critical state angle of friction��

c� of ��

Critical State in eln �p�� Space

The paths followed by tests in eln �p�� space are shownin Figure �� The changes in p�� resulting from porewater pressure changes� can be seen in this gure� Itcan be seen that the tests starting from low initialconning pressure tend to move to the right �dilative��but� most of the tests at a higher initial conningpressure� after an initial movement to the right� thenmove to the left �contractive�� However� some of thetests at a higher initial conning pressure� still show

Figure �� Critical state of tests in q � p� space�


an overall movement to the right� although the trendat the end of the test is to the left�

The critical state line should separate these twotypes of behavior� the tests to the left of the criticalstate line showing dilation and those to the right ofthe critical state line showing contraction� However� ascan be observed from Figure �� the end points of thetests show scatter in eln �p�� space� This can be seenmore clearly from Figure �� which only shows the voidratio against p� at a critical state� This scatter canbe attributed to either error in void ratio calculationand�or variation in soil fabric and structure� Asdiscussed in the previous section� the maximum error invoid ratio calculation is � �� The di�erence betweenthe upper and lower limits� in Figure �� represents avariation of � �� Therefore� although errors in voidratio may be one factor in the scatter shown in Figure ��it is likely that the initial fabric and structure� whichform during the remolding of the samples� may alsohave an e�ect� Apparently� in a heterogeneous soil

Figure �� Paths followed by tests in e�ln p� space�

Figure � Critical state limits�

mass� initial soil fabric and structure a�ect the criticalstate line�

This suggests that a unique critical state linecan only be dened for a uniform homogeneous soil�This is consistent with the suggestion by Verdugo andIshihara �� who have shown that the steady state lineis una�ected by the initial fabric� as long as the soilmass is homogeneous� Therefore� for a well gradedmaterial� such as gravely sand� where heterogeneity isinevitable� the critical state should be dened as a zonewith upper and lower limits� as shown in Figure �� Thisfact is consistent with the conclusions presented byother researchers �e�g�� who dened the upperand lower limits for the critical state� The critical statelimits for gravely sand may be characterized by thefollowing equations as their upper and lower limits ineln p��

e � �� lnp� �upper limit��

e � �� lnp� �lower limit��

Therefore� for gravely soils� the mean e�ective stress atcritical state� p�c� can vary between �p

�

c�max and �p�

c�min�corresponding to the upper and lower limits of thecritical state zone� depending on the initial fabric andstructure�

NORMALIZED STRESS PATHS ANDBOUNDARY SURFACES

For comparing data from di�erent tests� it is convenient to normalize the e�ective stress paths� Threemethods have been used for presenting data as thenormalized stress path in the literature� Atkinsonand Bransby � �� describe these three methods� Inthe rst method� p�e� which is e�ective mean stresscorresponding to the void ratio on the normal consolidation line� is used for normalizing the stress path�This method is not suitable for granular material�because it is di�cult to identify the position of thenormal consolidation line for granular material� In thesecond method� p�c� which is the e�ective mean stresscorresponding to the void ratio on the critical stateline� is used for normalization� Since a unique criticalstate line cannot be dened for this material� it is notpossible to simply dene a unique p�c for any void ratioto be used in normalization of the stress paths� Forthe present paper� a renement of the second methodis used to overcome the di�culty of the critical statebeing a zone rather than a unique line� Rather thancalculating the value of p�c from an equation for thecritical state line� the value of p�c will be used� andobserved for each test� i�e�� the mean stress at the endof each test� This means that� by denition� the stresspath will reach p��p�c � � at the end of the test�


Figure �a shows the stress paths in normalizedp��p�c � q�p�c space� As can be observed� the stresspaths reach di�erent values of qc�p

�

c� Therefore� anormalized deviator stress� dened as q��p�c � M��which was used by Sladen and Oswell � �� can beexamined herein as well� Using this method� it followsthat all normalized stress paths end at the �� point�which represents the critical state �Figure �b�� Thisallows a direct comparison between the behavior ofgravely sands having slightly di�erent M values� Itcan be seen that the normalized stress paths give aclear� consistent picture of the behavior of gravely sand�The critical state is dened by a single point �p��p�c �� q�p�c��M � �� Samples with initial p��p�c valuesgreater than �� show overall contractive behavior�Although normalized stress paths initially move to theright� they are then bounded by a unique surface �theRoscoe surface� and they follow this surface to reachthe critical state� Samples with low p��p�c values showdilative behavior and move to the critical state fromthe left� Samples starting from very low p��p�c values

Figure � Normalized stress paths�

�� seem to reach a unique boundary surface �theHvorslev surface�� which they then follow to reach thecritical state� Samples at intermediate values of p��p�c�� tend to move directly towards the critical statewithout reaching the boundary surfaces on either side�but the pattern of the stress path direction is stillconsistent within this region�

The third form of normalization is to use v�� asexplained by Atkinson and Bransby � �� The datain normalized q�p� � v� space is shown in Figure ��a�It can be observed that end points of the tests showscatter in this space� because the critical state is azone rather than a unique line and values of M areslightly di�erent for each test� Therefore� a renementis introduced in this space to bring all normalized stresspaths to the �� point� to identify the critical statecondition� The data is presented in q�p��M againstv�� space� is the value of v� in the critical statefor each test� The data has been plotted in thisspace and is shown in Figure ��b� Irrespective ofdi�erent behavior in stressstrain and eln p� space� a

Figure �� Normalized stress paths�


unique boundary surface can be dened� As with thep�c normalization� one can see samples on the denseside of the critical state �values of v�� less than�� reach a limiting surface �the Hvorslev surface�and see them following that surface down to get tothe critical state� Those in a loose state �v�� also reach a limiting surface �the Roscoe surface�and follow that surface up to reach the critical state�Intermediate values of v�� tend to movedirectly towards the critical state without reaching theboundary surfaces on either side� but the pattern ofthe stress path direction is still consistent everywherewithin this region�

CONCLUSION

An extensive series of laboratory tests have beencarried out to understand the behavior of gravely sandand to examine critical state concepts for this material�The material studied in this research was a wellgradedmaterial and included a wide range of particle size�from less than �� up to �� mm� For this reason�the samples prepared for triaxial tests are likely tobe heterogeneous and can have a di�erent initial soilfabric and structure when prepared at the same voidratio� The results suggest that variations in fabricand structure do a�ect the behavior of gravely sand�Samples at the same void ratio showed signicantlydi�erent behavior� The critical state line� dened inq��p� space� for the gravely sand did not show too largea scatter� However� in eln �p�� space� the test resultsdid not show a unique critical state line� In this space�a critical state zone with upper and lower limits wasdened� Nevertheless� normalizing the behavior� basedon the mean stress at the end of each test� allowedthe denition of a unique boundary surface� Thenormalized plot showed a clear� consistent pattern ofbehavior� with samples on the dense side of the criticalstate demonstrating dilative behavior �bounded by theHvorslev surface� and those on the loose state showingcontractive behavior �bounded by the Roscoe surface��Plots of q��p�c �M� � p��p�c and q�p��M � v�� havebeen used to normalize the data to demonstrate thisunique behavior� The data showed a unique boundarysurface in both normalized spaces�

ACKNOWLEDGMENTS

The triaxial tests were performed in the Soil MechanicsLaboratory of Sharif University of Technology �SUT��with nancial support from the Deputy of Research andTechnology at SUT� which is acknowledged� The rstauthor gratefully acknowledges the nancial supportprovided by the Ministry of Science� Research andTechnology of the Islamic Republic of Iran for hisgraduate studies� including the visit to Durham Uni

versity� He also acknowledges the partial funding madeavailable for his visit to Durham University� providedby the British Council�

NOMENCLATURE

e void ratio

v specic volume! v � � � e

q deviatoric stress! q � ��

� � ��

�

��

� e�ective major principal stress

��

�e�ective minor principal stress

p� mean e�ective stress! p� � ��

��

��

p�c mean e�ective stress correspondingto critical state line at a dened voidratio �Figure ��

p�e mean e�ective stress corresponding tonormal consolidation line at a denevoid ratio �see Figure ��

slope of critical state line in vlnp�

space �see Figure ��

v� specic volume on the reference sectionat p� � � KN�m� �see Figure ��v� � v � ln p�

v� corresponding to p�

c �see Figure ��! � v � ln p�c

V� initial volume of the sample

Vm volume change due to membranepenetration

Vc volume of the sample afterconsolidation

M slope of critical state line in stressspace

Figure �� Schematic diagram showing de�nition ofsymbol and notation�


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