Interpretacion de Resultados de Laboratorio Ocr

7/28/2019 Interpretacion de Resultados de Laboratorio Ocr

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Results of the International Workshop on Constitutive Relations for Soils /Grenoble /6-8 September 1982

Predictions of the results of laboratory tests 3.1

on a clay using a critical state model

G.T.HOULSBY & C.P.WROTH

Oxford University, UK

D.M.WOODCambridge University, UK

ABSTRACT: An elastic-plastic model for describing the stress-strain behaviour of a clayis presented. The model is based on the Modified Cam-Clay model, but includes anadditional yield surface to model failure of overconsolidated samples on a Hvorslev sur-face. The model is used to predict the behaviour of a kaolinite in a number of repeatedloading tests, based on the results of monotonic triaxial compression and extension

tests. Special account is taken in the model of the possibility of the clay beingunsaturated.

1 INTRODUCTION

The authors were invited to participate inthe International Workshop on ConstitutiveBehaviour of Soils, to be held at Grenoblein September 1982. The purpose of the Work-shop was to allow a critical comparison to

be made of various mathematical models ofmechanical behaviour of soils. Carefullycontrolled laboratory tests on both recon-stituted clay and sand were conductedat the University of Karlsruhe; some of thetest results were -provided to theparticipants as "input" tests for fittingof their models and establishment of soilproperties. The remaining tests ("output"tests) are those for which the participantswere expected to make predictions, which

had to be compared in detail with the ex-perimental data at the Workshop.In view of the proven success of the con-

cepts of Critical State Soil Mechanics, andin particular the family of Cam-Clay models,to describe the data from high quality testson reconstituted clays, it was decided toaccept the invitation and to use these con-cepts as the basis of the predictions calledfor by the organisers of the Workshop.A similar exercise took place at the

Symposium on Plasticity and GeneralisedStress-Strain Behaviour of Soils, held atMcGill University, Montreal, in May 1980.The predictions made on that occasion, anda full description of the assumptions madeand methods used, were given by Wroth &Houlsby (1980). In that event the ModifiedCam-Clay model was used for predicting the

behaviour of two natural clays and ofreconstituted kaolin.It is well recognised that one defi-

ciency of this model is that it leads tooverestimates of the strengths of heavilyoverconsolidated clays which are betterdescribed by the Hvorslev failure cri-

terion. Consequently, for this Workshop,the Modified Cam-Clay Model was adaptedto incorporate the Hvorslev failurecriterion. In order to avoid possible con-fusion and misunderstanding this new modelis called the Roscoe-Hvorslev model. It isfully described in a later section of thispaper.A further minor variation of the model

had to be introduced to allow for the con-sequences of the samples of clay not being

fully saturated, and being tested withoutback pressure. Initial perusal of the dataof the input tests suggested that this hadoccurred; subsequently, it was found thatthe data could only be fitted well by theRoscoe-Hvorslev model (RHM) if allowancewas made for incomplete saturation. Themethod adopted to allow for partial satu-ration is described later.Because it was considered that none of

the existing family of Critical State

models adequately represents the behaviourof sand, no predictions for the outputtests on sand have been attempted.

2 IMPORTANT FEATURES OF SOIL MODELS

The role of a mathematical model for the

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description of soil behaviour is twofold:firstly, as a simple quantitative frameworkagainst which soil behaviour may be asses-sed and, secondly, as a complete quali-tative model for analysis and design. Inits first role the model is of use both injudging the quality and consistency ofdata, and in predicting the character andtrends of behaviour of a soil under a

dation and time dependent behaviour of thesoil skeleton)7. Anisotropy8. Certain well-established empirical

relationships which should be reproducedby the model, e.g. the ratio of undrained

strength cu to overburdena iv may be expressed as a function

of overconsolidation

in a

variety of circumstances. For this purposethe model must represent the essentialfeatures of soil behaviour in as simple amanner as possible.A simple model is also desirable for the

second role; this is because,numerical calculation (such as one carriedout by means of the finite element method),an excessively complex model may be diffi-cult to implement and expensive to use.Very often the quality of the availabledata about the actual soil from the field

All the above features are included inthe model below, except that:1. Only the time effects due to primary

consolidation are considered (i.e. thosewhich arise through the flow of waterwithin the soil skeleton), whereas creep,secondary consolidation, aging and cemen-tation are excluded; thus the behaviour ofthe soil skeleton is rate and time inde-pendent ;2. The model is isotropic, so that cer-

tain effects due to the reversal or rota-will not warrant the additional precisionof calculation achieved using a complexmodel. If, however, the model is to beused for mathematical calculations, it isimportant that it should be properlyfounded in the theory of continuum mechanics and should be internally consistent.A simple model will require few para-

meters for the description of a soil, andphyseach of these should have a

ical significance so that an assessment caneasily be made of its importance and thelikely results of any changes in its value.The parameters should be measurable

tion of the principal stress axes are notincluded.The Roscoe-Hvorslev model was chosen for

use in the prediction workshop since it isa development of a well-established andunderstood model (the Modified Cam-Claymodel) which the general requirements of simplicity and completeness, which

for the economic solution ofreal problems. The new model is suitablefor the analysis of tests on both normallyconsolidated and highly overconsolidatedclays, but is not thought to be suitablefor sands.

directly in a small number of simple testsBoth the purpose and the limitations of

the model should be well understood. The 3 THE ROSCOE-HVORSLEV MODELuse of one model should lead to

an understanding of its range of applica-bility and of the types of problem forwhich it may be useful. The trends ofbehaviour predicted by the model underextreme conditions should be explored andtheir importance assessed.Some essential features of soil beha-

viour under conditions of monotonic loadingwhich should be included in a model are

The name of the model is taken from thetwo separate surfaces, the Roscoe surfaceand the Hvorslev surface in (q,p*,V) space

by Atkinson & Bransby (1978).The Roscoe-Hvorslev model (referred tohereafter as RHM) consists of the ModifiedCam-Clay model (MCCM) (Roscoe & Burland1968) with the additional feature of theHvorslev failure criterion. The latter is

listed below. The importance lies not inthe fact that these features should be

a case of the better-known MohrCoulomb failure criterion in which the

included but that the model should repro-duce well-established experimentallydetermined behaviour.1. A two (or more) phase material

(material properties must be expressed in

terms of2. Both non-linear response and irre

coverable3. Plastic dilation or compression4. Failure conditions (e.g. Mohr-

Coulomb or Hvorslev)5. The influence of consolidation history6. Time effects (both primary consoli-

cohesive component of strength is not con-stant, but is an exponential function ofthe current water content of the clay.The RHM has three separate and distinct

ingredients: (i) the Modified Cam-Claymodel, (ii) the Hvorslev failure criterionand (iii) the simulation of partial saturation. It is described in this sequence inthe three following sections of the paper,

3.1 The Modified Cam-Clay model

The theory of perfect plasticity is founded

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on e ypo es s a , ur ng p as cdeformation of a material, the strainincrements are functions of the absolutestresses. The theory evolved from carefulobservations of the behaviour of ductilemetals whose response to loading is typi-fied by an initial phase of elastic,recoverable deformation until a yieldcondition is reached; after yield, asecond phase of behaviour occurs which isa combination of elastic and plasticirrecoverable deformation. In general, theplastic deformations are greater by anorder of magnitude than the accompanyingelastic ones.The study of a typical stress-strain

curve for a soil in which the stress iscycled one or more times, for example, aone-dimensional consolidation test on aclay, reveals qualitatively similarbehaviour. However, there is one essentialdifference and that is that soils, asopposed to metals, experience irrecoverablevolumetric as well as irrecoverable shearstrains. The classical theory of plasticitydeveloped for describing the large strainresponse of ductile metals, has had to beextended to account for plastic volumetricstrains in order to become applicable tosoils.Making use of this analogy with metal

behaviour, and coupled with the experimen-tal evidence for a critical state for

deforming soil, a family of elasto-plasticstress-strain models has been developed atCambridge. One such model, known as Modi-fied Cam-Clay, initially proposed byBurland (1967) has been used extensivelyin finite element computations. A qualita-tive description of this model follows,and the mathematical details are providedin Appendix A.In detail, the typical behaviour of a

soil specimen in a triaxial compression

test can be characterised by the curve of

F g.1(a). If the spec men s loaded alongthe path OA and then unloaded along ABC,the path ABC can be approximated as elas-tic behaviour, with the strain representedby OC being a permanent, irrecoverableplastic strain denoted by eP. On reloading,the specimen behaves essentially elastic-ally along CDE (displaying a small amountof hysteresis) until at point E it experi-ences the previous maximum deviator stress;it yields and undergoes further plasticstrain. Point F denotes failure (a uniquecondition for this particular test);failure must be distinguished from yieldwhich is a progressive phenomenon andwhich may occur at any point along theprimary loading curve OAEF{depending uponthe exact loading sequence followed in thetest.

Analogous behaviour will be displayed bya soil specimen if tested in a consolido-meter, as shown in Fig.1(b), which is anunconventional plot of effective pressureagainst specific volume (V) (V=l+e, wheree is the voids ratio). If this diagram isrotated clockwise through 90 to giveFig.1(c), the more usual plot of consoli-dation is obtained. .It should be realised that for a speci-

men that is in the overconsolidated state,represented by point C in Fig.1(c), itspreconsolidation pressure p^, given bypoint A, is the current yield stress forfurther consolidation; it is the pressureat which further plastic volumetric strainsbegin to occur.Now consider a soil specimen that has

been isotropically normally consolidated(with o[ = c?2 = O3) to point A in Fig.2 (a) ,and then allowed to swell isotropically tosome state along the unloading curve ABC.If the specimen were then subjected to avariety of different effective stresspaths, it is assumed that there would be a

well-defined region of stress states for

3

(a)

Behaviour in shear

/

(b) (c)Behaviour in consolidation

Fig.l. Typical behaviour of soil specimen in consolidation and shear tests

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Direction

of plastic

increment

P',vp

(b)

w c e spec men wou rema n e as c(The stressthe data ofnormaldeviator stress q

used for representingtests are the mean

stress p'= c'-a'.

= (cr^+a +ap andThe corresponding

strain variables are the volumetric strainv2

3(e

l+e2+e3 and the deviatoric

1 e-).) This region is bounded by ayield curve. If the stresses applied to thespecimen take state outside the currentyield curve, it will yield and experienceboth plastic volumetric and plastic shear

Consider a second specimen that has beennormally consolidated to G and then allowedto swell to some point on the unloadingcurve GHI. Associated with this specimen isa larger yield curve, Fig.2(b), but onethat has in practice approximately the sameshape as that for the first specimen.The sizes of the yield curves are dicta-

ted by the points A and G, which lie on thenormal isotropic consolidation curve. Thechoice of yield curve appropriate for anyspecimen depends on the maximum consoli-dation pressure (i.e. the pre-consolidation

The shape of the yield curve, Fig.3(a),is assumed to be elliptical; this choice isbased on considerations of energypated plastically within the specimen(Roscoe & Burland 1968; Houlsby 1981). Onesemi-axis of the ellipse BA is fixed by the

consolidation history relevant to the

Fig.2. Yield curves for specimens withconsolidation

specimen The other semi-axis BX, is givenby the assumption that the point X, is onthe line of critical states q Mp

fof

failure states of normally consolidatedspecimens The critical state line isassumed to be parallel to the normal con-solidation line in the logarithmic plot ofFig.3(b).

q

(a )

Failure line(Critical states)

Yieldsurface

Rc

nv

v=r

Failure line(Critical states)

p'=I

NormalConsolidation

Gradient-X*

(b)

Fig.3. Details of yield surface for Modified Cam-Clay model

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As shown n F g.3(b), both consol dat onand swelling lines are considered asstraight in jlnV-Hnp1 space. This repre-sents a very minor alteration of the modelfrom previously published descriptions ofModified Cam-Clay, for which the lineswere taken as straight in V - np' space.Consequences of the changes are:

1. The numerical values of the parametersA* and K* (see below) are reduced by afactor approximately equal to V from the

original parameters A and k;2. The bulk modulus becomes truly pro-

portional to pressure rather than dependingalso upon volume (see Appendix A);3. Evidence in support of this change is

given by Butterfield (1979), and a furtheradvantage is a slight simplification of themathematics of the model. For a specimenundergoing small the new versionof the model is identical to the old oneThe complete description of the model

requires five parameters to specify theshape and size of the yield locus for asoil specimen at a given pressure and

volume, as well as theproperties of the material. The parameters

incremental bulk modulus, derivedfrom the local gradient of the swellingline, can be written

K P'/K* . (1)

The byassuming a constant shear modulus, G, sinceany assumption of variation of G withpressure can result in a model which isthermodynamically unacceptable.Nov*, suppose the state of stress experi-

enced by tHe specimen is at point J inFig.2(b) close to the relevant yield curve.If the stress increment JKL is applied tothe specimen, the increment JK will cause

increments only, whereasthe increment KL will cause bothand plastic strain increments. As thespecimen yields at K, the yield curve isexpanded as the specimen undergoes consoli-dation. The plastic volumetric strainincrement that occurs from K to L is given

by a hardening law,P

v(A* K*)

P 'CP 'c

. . .(2)

1. X*, the gradient of the consolidationline in nV-np' space. This is relateddirectly to the conventional compressionindex Cc' for a remoulded material byA * = CJ/(2.303V);

2.

cK* the gradient of the swelling line,

similarly related to the swelling indexCs' by k* /3. M, the gradient of the critical state

line in q Pi This is related to theangle of shearing resistance


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wh ch s der ved from the normal consoldation line,

3.2 Hvorslev's fa lure cr ter on

An(V/V ) X* n(p '/p'). . (3)The out by Hvorslevin the mid-1930's consisted of stress con

(The dot notation of plasticity theory hasbeen adopted to indicate a small (time-independent) increment.) The details ofthis derivation are not given here but canbe found in Schofield & Wroth (1968). Theimportant point is that no additional soilparameter is required.The flow rule governing the of the

plastic strain increments is given by thecondition of normality. This conditionstipulates that if the associatedstrain increments vP and e^ are plotted onthe same axes as the stresses in Fig.2(b),the vector KM of plastic strain incrementis normal to the yield curve at K. Thegradient of the curve at K is known fromits elliptical shape, so that the ratio

can be calculated and, since vP is

already established (from the hardeninglaw), eP can be evaluated.Consider now- a specimen undergoing a

conventional undrained triaxial compression, with initial state as shown by point

A in Fig.4. The condition of the test issuch that the total volume must stay con-stant, i.e. that the path in Fig.4(a)coincides with a line of constant voidsratio, that is, constant volume.The initial response of the specimen must

be elastic as the point A within thecurrent yield curve, Fig.4(b). There canbe no change of p' duringmation as the point A in Fig.4(a) is constrained to lie on both the elastic swelling line and the constant voids ratioline. When the stress pathreaches the point B, the specimen yields.Beyond this point, both plastic and elas-tic volumetric strains occur, of equalmagnitude but opposite sign so as to main-

tain the total volume constant. The ratioof the plastic volume change to theplastic shear strain is calculated as

, and the test proceeds to failureat point D. The effective stress path ABCDis as shown in Fig.4(b). The total stresspath for a triaxial compression test (withconstant cell pressure) is given by AE.The pore pressure at any stage of the testis simply the difference in mean stress

trolled shear tests on two reconstituted clays. His results and findingswere published in 1937 in his doctoral

in German, and published in Englishmany years later (Hvorslev 1969).His failure criterion has been confirmed

since for strain controlled triaxial testson clays by other workers, e.g.presented by Parry (1960), as interpretedby Schofield & Wroth (1968). The criterionwill be discussed in this paper in termsof and is presenteddiagrammatically in Fig.5.In Figs 5(a) and 5(b), the conditions at

failure are plotted in terms of the vari-


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f

(a)

f

i

(b)

q fl

Hvorslevsurface

Roscoesurface

V

(d)

/

(e)

9

e

/

(C)/

e

/

c

(f)

/

c

Fig.5. Hvorslev's failure criterion

The gently curving line DFG has a

relatively complex algebraic form, and as

a first attempt to incorporate the

Hvorslev criterion into the family of

Cam-Clay models, it has been assumed for

simplicity to be straight. Furthermore,

for convenience, normalisation has been

done by means of the isotropic preconsoli

dation pressure p'c rather than by the

equivalent (one-dimensional) pressure p'

which affects the arithmetic but not the

principle involved. The result of these

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assumptions is the reduced form ofFig.5(f).Because Hvorslev's original tests were

stress-controlled he studied conditions atfailure and he could not study post-peakbehaviour. This is now possible withstrain-controlled triaxial tests. Supposea drained test with mean normal effectivestress p1 kept constant is conducted on anoverconsolidated specimen with initial

state at L in Fig.6. The response of thespecimen is expected to be of the formgiven in the diagram with a quasi-elasticstage LM, failure observed at M, and sub-sequent work-softening as the specimendilates, sucks in water and approaches acritical state condition at N.The post-peak behaviour MN causes real

difficulties both experimentally and numeri-cally. The specimen is almost certain tobehave non-homogeneously and to develop a

narrow rup ure zone un ergo ng n enseshear (and softening) while the surroundingclay marginally unloads elastically. Thisform of behaviour has been discussed morefully by Schofield & Wroth (1968 8.5).On the macroscopic scale the dilation

and increase in water content (or specificvolume) of the saturated specimen whichoccurs in the rupture zone will not beobserved at the boundaries of the specimen.In order to represent this phenomenon in arealistic and simple way, the microscopicsoftening is neglected. The consequence isthat the specimen is simulated as havingan elastic, perfectly plastic response PQR,as illustrated in Fig.7(b). This means thatthe Hvorslev failure surface is beingtreated as a yield surface with a non-associated flow rule.In reality, this simplification means

that the post-peak drop in strength is


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a p/x

associated

/X

Associated

/c

/

(a) (b)

Fig.7. Assumed Roscoe-Hvorslev yield surface

being ignored. Should the introduction of

the Hvorslev failure surface prove to be auseful development in modelling soil beha-viour, but this deficiency turns out to beimportant, then it would be possible toadopt a different flow-rule which allowsdilation with a consequent amount of sof-tening as the specimen approaches thecritical state.The introduction of the Hvorslev surface

(taken as straight in the section shown inFig.6(f)) requires only one additional

parameter for complete specification ofthe model. This is because the criticalstate line forms the outer edge of thesurface? in Fig.7(a) the point X which isthe crown of the ellipse forming the MCCyield surface establishes the position ofthe end of the straight line forming thesection of the Hvorslev surface. The addi-tional parameter chosen is the positive(dimensionless) constant a, where theabscissa of point Y in Fig.7(a) is given

by -ap'x,or by -ap'c/2.

3.3 Incomplete saturation

Many natural clays are not fully saturatedhaving gas present in the pore fluid. Evenif the gas is mostly dissolved in the porewater, the pore water will be relativelycompressible, and it cannot be assumed tobe incompressible in comparison with the

soil skeleton (as is usual for a saturatedsoil). For such a case, an undrained testwhen no pore fluid is allowed to cross theboundary of the specimen is not synonymouswith a constant volume test.It is difficult in preparing specimens

of reconstituted clay to ensure full satu-ration, and the only guarantee of achievingit is to apply sufficient back pressure,

until the pore pressure parameter B equals

unity. It was evident from careful scrutinyof the input test data that some of thespecimens of clay were not fully saturated.In order to model these data, a simplemethod had to be devised to simulate theeffects of incomplete saturation. Theproblem only arises in simulating thebehaviour of soils in conventional undrai-ned tests, because the increased compres-sibility of the pore fluid will have noeffect on a fully-drained test. .

In the undrained tests (without backpressure) each specimen was assumed at

first to behave as fully drained. Speci-mens subjected to triaxial compression, orwhich yield at low overconsolidation ratiosin triaxial extension, undergo compressivevolumetric strains. At a certain compres-sive strain, vsat* it was assumed that allgas bubbles within the specimen were com-pressed to zero volume and the sample wasthen assumed to behave in the usual un-

drained manner undergoing zero volumetricstrain.In the absence of back pressure it would

be expected that the bubbles might reappearwhen the pore pressure falls to a lowvalue. This value was arbitrarily set asatmospheric pressure, so that whenever thelateral total stress becomes equal to thecell pressure it is again assumed that thesample behaves as fully drained, i.e. itis assumed that the sample can develop

only positive, never negative, pore pres-sures with respect to atmospheric.The above procedure attributes zero

stiffness to any bubbles in the pore watertreating them as infinitely compressibleuntil a sudden point at which they arereduced to zero volume, and the pore fluidcan be taken as incompressible.


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SELECTION OF MATERIAL PARAMETERS

The Roscoe-Hvorslev model used in thisprediction exercise involves six material

, each of which has a clearphysical meaning. In this section each of

s p e ween AnV an An p s approx -mately linear for any constant stress ratiopath, and that the gradient is independentof the stress ratio. The gradient of thisline is termed X* (cf. the use of X for thegradient in V An p )

these parameters is defined, physicalsignificance noted and the means fordetermining it

The value of X* may be obtained from thatpart of a consolidation test which exceedsthe initial preconsolidation pressure. Themost readily available tests are usuallyone-dimensional oedometer tests, in which

4.1 The Critical State stress ratio M V and av are measured. For a normally

The concept of Critical States is centralto the model. The critical state isapproached asymptotically by a normallyconsolidated or lightly overconsolidatedsample as it is sheared monotonically, andrepresents a state in which the soil canbe sheared indefinitely with no furtherchange of effective stresses orvolume. It is found empirically that, atthe critical state, the stress ratio q/p'is essentially independent of p', andthis critical state stress ratio is deno-ted by M (capital y) .The value of M is sometimes found to

differ between triaxial compression andextension tests, and this effect could beincorporated within the model by adoptinga non-circular generalisation of the yieldlocus in the octahedral plane. For sim-plicity a circular generalisation (i.e.equal M in compression and extension) isadopted here.The value of M is most easily deter-

mined by plotting the stress paths ofshear tests on normally consolidatedsamples in p'-q space and determining thestress ratio at the end of the test (i.e.at very large strain). Adopting this pro-cedure for tests 1.1 and 1.2 yields avalue* of 0.74 in compression and 0.78 inextension. The value of 0.74 has been usedfor all predictions, which is a fairlytypical value for M, which usually fallsin the range 0.7-1.2. The value of M isdirectly related to an angle of frictionat constant volume (p'cv* e.g.compression

6 sin


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. Introduc ng a var at on of Gwith pressure results, however, in non

Su s u ng e va ues o p , q, pc an Mfor the failure points of tests 1.4.2 and

conservative elasticity if the bulk modu 1.4.4 results in a value of a = 0.141,lus does not depend on shear stress. Simi which compares well with data for otherlarly, the introduction of G dependent on clays.pc' means that the model becomes one ofcoupled elastic-plastic behaviour, whichis considerably more complicated than 5 EVIDENCE OF INCOMPLETE SATURATIONuncoupled plasticity. A constant shearmodulus is therefore used in the model for At the start of the exercise of fittingsimplicity. the input data to the Roscoe-HvorslevThe value of the shear modulus was model, the event was to plot the

chosen from the gradient of the few results of the consolidation tests in theloading increments on overconsolidated conventional manner, and the results ofsamples. It is obtained by plotting the the triaxial tests in terms of the varitests in e-q space with the gradient ables p', q, V and e.being equal to 3 G. A value of 7000 kN/m The results of input test 1.1, which iswas chosen. This may be compared with the an undrained compression test oncorrelation G 75 cu suggested by Wroth etal. (1979); taking a typical test (input

an isotropically normally consolidatedsample, are plotted as individual crosses

test 1.1) with cu 90 kN/m2 would indi

cate G of the order of 6750 kN/m For thein Fig.8. The effective stress path isevident in Fig.8(a). All evidence from

tests at lower pressures, a lower value of other undrained tests on saturated samplesG might be more appropriate, and for very of clay is that the effective stress pathsmall values of shear strain a higher curves to the left immediately at thevalue of G should be used. start of the test at point A. If, however,

the sample behaves initially as though itwere drained, it would have an effective

4.5 Critical specific volume at unit stress path (in a conventional test withr constant cell pressure) of gradient 3. It

is considered to be most significant thatOne parameter is needed to locate the iso the path is initially at gradient 3 fromtropic normal consolidation line, or the A to B, and at B it undergoes a markedcritical state line in V-pi this is change to the curved path BCD, as positiveusually chosen to be the volume excess pore-pressures develop. It was onon the critical state line corresponding the of this evidence that it wasto unit pressure 1 kN/m2, and denoted byr (capital gamma).

believed that the test could not have beenconducted with back pressure, and that the

It is only required if absolute values sample in question was not fully saturatedof volume (or water content) are Similar from the other input testsneeded. In this exercise the volumetric was observed and two other examples arestrains (related to the of shown in Figs 9 and 10. Consequently

volume) are asked for, so that enquiries were made at that time (Marchthe value of r is irrelevant and it has 1982) and it was confirmed that back presnot been determined. sures had not been used in the series of

input tests

4.6 Hvorslev surface intercept aThe single set of basic soil properties

assumed to apply to all the tests was

Those samples with an OCR greater than 2selected as described in section 4 and

or so (the exact value will depend on thetype of test chosen to bring the sample tofailure) will be expected to fail in a

Table 1. Selected values of soil properfor Roscoe-Hvorslev model.

brittle manner on the Hvorslev surfaceSince the surface is chosen to pass through M Critical state stress ratio 0.74the point (pc'/2, Mp^/2) in p'-q space itcis only necessary to determine one para

X* Gradient of normal consolidation line 0.1225

meter in order to locate the surfaceThis is conveniently taken as a (see

k* Gradient of swelling lineG Shear modulus

0.01657000 kN/m

Fig.7(a)) where the surface is described r Critical volumeby: at unit pressure Undetermined

q +M

1 + a(P' + apA/2) ..(5)

a Hvorslev surface intercept 0.141


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q kN/m2

300

200

100

01100 1200 300

p'kN/m

(a)

2

B

q kN/m

300

2

200

100


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occurred whereupon the sample s modelledby the undrained path BCD, asymptoticallyreaching the critical state D after infi-nite shear strain. Note that in Fig.8(b)the computed result shows a small increase

T e resu s are sp aye n F g. ; IJrepresents the drained phase, JKthe undrained phase, and K represents the state at which the stressesreach the Hvorslev failure surface There

in stiffness at B when the abrupt changefrom drained to undrained behaviour isassumed to occur. For a fuller discussionof the problem of mnsaturation, seeAppendix B.Input test 1.4.1 is an undrained test on

a sample with an initial isotropic over-consolidation ratio of 2, shown in Fig.9.The value of vsa-j- adopted was 0.25%, whichcorresponds to the volumetric strain ex-perienced by the sample during the initialdrained phase from E to F. This phase isentirely within the yield surface so thereis zero plastic volumetric strain, and thetotal volumetric strain consists solely ofthe

after the sample is assumed to behaveperfectly plastically K to L. It should beremembered that the Hvorslev surface hasbeen selected to go through the failure(i.e. the peak) point, so that the computedresult for K in Fig.10(a) does not repre-sent a match but adata.

of the experimental

component.

5.1 Summary of input tests

Each test was analysed as fully saturatedthroughout, and also as partially satur-ated by an amount to give the bestThe optimum values of vsat are listed

in Table 3.At point F the sample becomes ii ii

and behaves in an undrained manner. Thestress state is still within the

yield locus, so that the sample is assumed

Table 3. Optimum values of parameter representing partial saturation in input tests

to behave in an (isotropic) elastic mannerwith the consequence that p' remains con-stant and the effective stress path FG isparallel to the q-axis. At point G, thestress state reaches the Roscoe yield sur-face, the sample yields and approaches thecritical state at H. In this case thegeneral character of the stress path iswell represented, but the stress-straincurve EFGH in Fig.9(b) is a poor match tothe experimental data.Input test 1.4.2 was on a heavily over-

consolidated sample with OCR =8.8. Thevalue of vsat was also very small (0.55%).

PokN/m2

Pc'kN/m2

Optimum

vSatemax

1.1 400 400 0.012 0.161.2 400 400 0.03 -0.11.3 400 400 0.01 3 cycles

0.007 ^ -0.0064

1.4.1 300 600* 0.0025 0.161.4.2 68 600 0.0055 0.161.4.3 101 200** 0.0017 0.161.4.4 22 200 0.006 0.16

* *

with pc'Better fit with pc' 175

q kN/m2

q kN/m

100

50

100

i I

50

L+ + + +

0 50p' kN/m

2 100 0 10 e 207.(a) (b)

Fig.10. Input test 1.4.2

X


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6 OUTPUT TESTS xoct kN/m2

Having established the importance and in-fluence of the incomplete saturation of theclay samples of the input tests, a majorproblem arises in making predictions forthe output tests as the degree of partialsaturation of each sample (which willalmost certainly vary, as was the case ofthe input tests) is unknown and unspecified

It was decided to make predictions forthe two following cases for each output, (a) full saturation, and (b) incom-

plete saturation, with a value of vsa^ =0.01. This figure was chosen as a roundnumber close to the average of the optimumvalues found for the input tests.The predictions are given in the form of

plots as specified in the following Figs11-24.

6.1 Summary of output tests

\001e

0015i

Each was modelled as fully saturatedthroughout and also as partially saturatedwith vsat = 0.01 (i.e. experiencing a totalcompressive volumetric strain of 1% beforebecoming fully saturated).The test conditions are listed in

Table 4.No predictions are made for output test

2.0 since the specifying the test

Fig.11. Predicted results test 1.1fully saturated

test is a shear test at constant volume.None of the input tests provides adequateinformation about the shape of the yieldlocus very close to the origin. Tests in

were inadequate. It is thought that sincethis output test is referred to as corresponding to input test 3.0, it involved aninitial stress condition of approximately

. However, unlike input test 3.0,which is essentially a consolidation testat rate ratio (and hence

to\

would be expected, and is observed,follow a constant stress ratio path), this

Table 4. Conditions of output tests

TestkN/m2 kN/m2

Selected test conditions

1.1 400 400

1 .2 .1 400 400

1 . 2 . 2 400 400

1.2.3 400 400

1.3 300 600

25 cycles of strainex from 0.01 to - 0.0164 cycles of shear stressq from +59 to - 64 kN/m2

64 cycles of shear stressq from +80 to - 90 kN/m2

64 cycles of shear stressq from +100 to -122 kN/m2

Extension test toe 0.15

Notes: (1) For an undrained test

(2) q3

l

7iToct

p'kN/m2

\z

001 0 0005 0-01 0015Fig.12. test 1.1

fully saturated


15/23

Toct kN/m2

H -0-01

e

0015 i

e

0-04

*IE

0 0 2

-0-02-

-0-04

10 100

Fig.13. Predicted results test 1.1unsaturated

Fig.15. Predicted results: test 1.2.1fully saturated

' kN/m2

200-

100-

1 1 1 1 . I 1

001ei

0-005 0-01 0-015

e

004

*IE

0-02

0

-0-02

-0-04

10 100log


Fig.16 test 1.2.1unsaturated


16/23

e*IE

eIE

004 0-04

0 0 2 0 0 2

0 log N

- 0 0 2 -002

-004 -00/.

10 100

log N

Fig.17. Predicted results: test 1.2.2 Fig.19. Predicted results: test 1.2.3

fully saturated fully saturated

eIE

0-04

*

IE

0 - 0 2

log N log N

-002

-0-04

-0-02

-004

Fig.18. Predicted results: test 1.2.2unsaturated

Fig.20. Predicted results: test 1.2.3unsaturated

114


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Toct kN/m2

1

Ioct kN/m2

e

Fig.21. Predicted results test 1.3fully saturated


p'kN/m2

400p'kN/m

2

400

2 0 0 200

0-05 0-1 ei0-15 0 H0 0 5 +0- 1 e0-15Fig.22. Predicted results: test 1.3

fully saturatedFig.24. Predicted results: test 1.3

unsaturated


18/23

this region of stress space are difficultto carry out and little information aboutthe general behaviour of clays in thisregion is available.

>

It is certainly not sensible to extra-polate the elliptical yield locus (principally valid on the "wet" of critical)or the Hvorslev surface to points close tothe origin. There is some evidence that atvery low effective stress levels thestrength of the clay is determined by ten-

cracking, when zero normalstress occurs on any plane.If the test does indeed start at virtu-

ally zero mean normal stress and precon-solidation pressure then the only predic-tion that can be made is that all stresseswill remain virtually zero throughout this

This result would, however, be veryto the exact between the

, and any minute variation fromconstant volume could result in any of avariety of stress paths. Problems of un-saturation are also liable to introducesignificant uncertainties at very lowstress levels.

APPENDIX A

BRIEF SPECIFICATION OF THE MODEL

The assumptions embodied in the model aregiven in detail in sections 2 and 3 of thepaper. This appendix contains only amathematical specification.

A.1 FUNDAMENTALS

The model is elastoplastic in concept.Within the yield locus the material

behaves elastically with a bulk modulusproportional to pressure and a constantshear modulus. The yield locustwo

ofOne section is elliptical in

q~P' (Toct~P') space and involves an associated flow rule and work hardening linkedto the plastic volumetric Theother section is a straight line in q-p(Toct-p') space and involves perfectlyplastic behaviour with a non-associatedflow rule.

The yield surfaces in the model are

i

The model makes use of sotrop c hardening, with the parameter pc' which estab-lishes the size of the above loci beinglinked to the plastic volumetric strain.The relationship is most convenientlygiven in incremental form:

Pi

vP U

K*)C

PI . . . (A3)c

Plastic potentials may be defined forthe model. For the Roscoe surface the flowrule ispotential is

so that the plastic

P' (P'-Pj)M 0 . . . (A4)

For the Hvorslev surface a non-associatedflow rule with zero dilation is used, sothat the plastic potential is given by:

q qc 0 (A5)

where qc is a constant dependent on thecurrent stress state.The model does not make use of yield

(although at the critical statewhich forms the transition from the Roscoeto Hvorslev surfaces there is a corner inthe yield locus).Coaxiality of strain rate and stress is

assumed during plastic behaviour.If the Hvorslev surface is not included

in the model, it is then able to predictsoftening on the "dry" side of thecritical state. This process is thought tobe unstable, with localisation into narrowshear bands occurring. The use of theHvorslev surface in effect precludes thissoftening by modelling the formation ofnarrow shear bands by a perfectly plastic,non-dilatant behaviour.The model fulfils all the requirements

of a physically meaningful and mathemati-

cally consistent model in continuummechanics. As a consequence of the abovethe model should generate well posedproblems subject to appropriate boundaryconditions. Ill conditioning could ofcourse occur if certain combinations ofconditions are imposed, for example, arapid rate of loading resulting in"undrained" conditions linked to entirelydeformation-controlled boundary conditionsmay result in ill conditioning if a

bulk modulus of the pore fluid is notadmitted.

Roscoe surface

P' (P'-Pc') +3M2 0,

(Al)A.2 INCREMENTAL RELATIONS

Hvorslev surface Elastic behaviour

q+M1+a

+ 0 (A2) by:


19/23

p'P -eK*

(A6)

pq = 3G e (A7)

Behaviour on Roscoe surface

The conventional assumption of additiveand plastic strains is made. Using

the associated flow rule and the conven-tional formulation:

/unloading on the dry side of critical area particular area where a detailed under-standing of the principles of the modelare required. The precise numerical formu-lation used is important as the criticalstate is approached if numerical diffi-culties are to be avoided. The details ofa subroutine depend too strongly on theconstraints of the calling program to givea suitably general routine.For isotropic loading the model simpli-

to the incremental form:

e. .IDP A

3f 9g3a. _ 3a . .kl in

akl'

which results in

. . . (A8) P' < Pc'

P1

p' > 0P '

v

V

K*P'P'

A*P'p' P' P 'c

. .(A15

. .(Al6

0 *V _ | 0 p'P p'

20 1

+ A 0 3G_ q

( 2P -Pc'f

(2p'-pJ)M

(2p'-pJ)|f

M"

p'

q/

(A9)Closed form for the triaxial test may beobtained but is rather lengthy.

APPENDIX B

The work hardening equationUNDRAINED TESTS ON KAOLIN

P(A K*)

C

P 'c. . .(A10)

is combined with the consistency conditionobtained by differentiation of the yieldlocus:

P(2p-pc) +qq

M P Pc. . .(All)

Comparison with the appropriate term from(A10):

Alp' (2p'-p-)2 .(A12)

then yields the value of the hardeningparameter A:

A(A K*)

P'PJ (2p'-p '). . .(A13)

Behaviour on Hvorslev surface

Similarly the behaviour on the Hvorslevsurface may be determined as:

0 0 V P' 0 p' 0 0 p*s 1 + X 0 1 + X M 1 e 0 3G_ q -r1+a 1 q

. . .(Al4)

where, since perfect plasticity isassumed, the value of the scalar multiplier A may be determined, forby Hill's method (Hill 1951).In writing a FORTRAN subroutine to

implement the above calculations, greatcare must be exercised in attention to

in the model. Criteria for loading

Any soil test, whether performed in thelaboratory or the field, can be consideredas a miniature boundary value problem. Atriaxial test is no exception. An initiallycylindrical sample is subjected to a uni-form overall axial deformation through therigid end platens, and to a uniform lateral

total pressure by means of the cell fluidacting on the encapsulating rubber mem-brane. Only in certain circumstances willthe boundary conditions result in the soildeforming as a single homogeneous elementand it will not necessarily be correct toassume that conditions of stress and strainwithin the sample are uniform. Test pro-cedures can be chosen in order to try andencourage the development of uniform con-ditions : for example, lubricated end platens

and short samples may be desirable; a slowrate of testing will give the pore pressureswithin the sample an opportunity to reachequilibrium in drained or undrained tests.It is also important to appreciate that

the so-called undrained test,tionally interpreted as a constant volume

conven

is really a constant overall massthe term isomassic might be appro

priate to it. Assuming no leakageoccurs, once the drainage valves are closed,

no mass of soil or pore fluid can move intoor out of the sample, but this does notprevent the sample from changing in volume.If the pore fluid is not incompressible,for example, because the soil is not satu-

, then a soil sample can change involume as it is sheared, even though thedeformations and stresses within the sampleremain uniform. On the other hand, even if

117


20/23

the overall sample does not change nvolume, it is quite possible for localchanges in volume to occur within thesample, as a consequence of end restraintor rate of resting, as shown by the analy-ses of Carter (1982). A true constantvolume test path, the so-called isochoric,is best achieved as a drained test in adevice such as a true triaxial apparatusin which a combination of deformations can

9

be imposed on a sample, such that nochange in the volume occupied by the soilskeleton can occur, always assuming thatthe true triaxial sample can itself betreated as a single element of soil.When a new testing procedure is to be

used to study the undrained behaviour ofsoil, whether it uses a new pore pressureprobe or merely a new rate of testing, itis important to check that the resultsproduced by this new procedure are notinconsistent with existing knowledge ofsoil behaviour, or at least not withoutsome good explanation. There are only fourparameters to measure in the undrainedtriaxial test: the total radial stress(cell pressure), the total axial stress(cell pressure plus deviator stress), thepore pressure and the axial strain.The background of critical state soil

mechanics makes it clear that the behaviourof soils is best understood in terms ofeffective stresses, and that significantpatterns are more likely to emerge whensoil response is studied in terms ofeffective stresses and volumetric pack-ings, which are limited parameters, ratherthan in terms of stress:strain curves,shear strain being an unlimited parameter.The best way to examine the results of anundrained test is therefore to plot theeffective stress path in a convenientstress space, such as deviator stress q:

mean effect ve stress p'. Plott ng q, p'or pore pressure u against strain isunlikely to show up problems, particularlyif the test has been taken to large strains;the significant changes in effective stressoccur at small deformations.The effective stress paths for an un-

drained compression test and an undrainedextension test performed by Kuntsche onkaolin at Karlsruhe are shown in Fig.Bl.For comparison, two pairs of effectivestress paths for undrained compression andextension tests performed by Nadarajah(1973) on kaolin in Cambridge are shown inFig.B2. Some points of dissimilarity maybe noted:

(i) Nadarajah's compression and exten-sion paths are broadly of the same shape;Kuntsche's are very different. For aninitially isotropic material the effect ofa small increase or decrease in deviatorstress should be essentially identical,there can be no theoretical explanationfor the difference;

(ii) Nadarajah finds a lower ratio of qto p' at failure in extension than in com-pression and lower values of q at failurein extension than in compression; Kuntschefinds a higher strength in extension thanin compression and the same value of q/p*.The total stress paths for Kuntsche's

tests have been shown in Fig.Bl. It isclear that, for the first part of the com-pression test and for virtually the wholeof the extension test, zero pore pressurewas recorded. Kuntsche measures pore pres-sures by means of a rigid porous probeextending from the base to the centre ofthe sample. Evidently the average porepressure recorded by this probe is zero.Perhaps it provides a drainage path forpositive pore pressures near the centre ofthe sample to compensate for negative pore

Table Bl

MaterialCambridge kaolin(Nadarajah (1973)

Karlsruhe kaolin(Kuntsche 1982)

Index

Sample

Testingprocedure

LLPLGs

72.1%40.4%2.63

mix at 160%

ID to 195 kN/m2

over 8* daysfrom 103 kN/m2 in 69 kN/m2/I2h stepsthen 24 h rest(typically 3$-4i days)B values 0.97-0.99

load controlled: 2.25 kgf/h reducingto 0.45 or 0.22 kgf/h

(typical initial strain rate 0.125%/h

LLPL

50%16%

Gs 2.72

mix at 100%

to 80 kN/m2

over 7 daysto 400 kN/m2 in 1 step

B values 0.97-0.99

strain controlled: 12%/h


21/23

pressures developing near the base.It is worthwhile to compare the procedures

used by Nadarajah and Kuntsche for theirtests: these are summarised in Table Bl.The kaolins being tested are rather differ-ent. The plasticities are about the samebut the Cambridge kaolin has a much higherliquid limit. Nadarajah's isotropic consoli-dation procedure was much slower thanKuntsche's and one may wonder whether theKarlsruhe kaolin is completely consolidatedat the start of the tests. A condition ofinitial under-consolidation could well giverise to a tendency for continuing pore pres-sure development later in the tests, thusobscuring the attainment of critical states.The final isotropic consolidation, from80 kN/m2 to 400 kN/m2 in one step, couldconceivably leave the sample initially non-uniform as a result of the Mandel-Cryereffect.

The rate of testing used by Kuntsche isinitially typically about 100 times fasterthan that used by Nadarajah (who was con-ducting the tests with careful load control)

and th s h gh rate of test ng may have ledto inaccurate observation of pore pressures.For an isotropic sample deforming uniformly,an excessive rate of testing should, how-ever, lead at worst to an effective stresspath with no initial change in effectivemean stress (6p'=0, 6u=6q/3) as for theelastic material but not a path with anincrease in p'.Effective stress paths predicted by Cam-

Clay models for undrained tests have thesame shape, independent of consolidationpressure. Using volume change due to lack ofsaturation as an explanation of the resultsthat are shown in Fig.Bl, an estimate of thelack of saturation can be made by fittingcurved undrained effective stress pathsthrough as much as possible of the pathsobtained experimentally; these curves arealso shown in Fig.Bl. The compression andextension paths expected for kaolin con-

solidated to 400 kN/m

2

are shown. Much ofthe compression path is fitted by usingthe undrained effective stress path for asample consolidated to 440 kN/m2.

tii

/

i

Karlsruhe kaolin

Fig. Bl.


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compression Cambridge kaolin

oo

oo

oo

oi

kNAn2 500

o

o

o

o

Fig. B2

finV

2

In p1

for change of p1

4 0 0 4 4 0 kN/m2

ZnV/Vo \* in p ' / p '

X* 0.1225

V 1.665 at p'

400 550 kN/m2

400 kN/m2


23/23

T e norma compress on o e c ay canbe described by the expression (Fig.B3):

Jin V/V0 = X* In p'/p0'

with A* = 0.1225 and V = 1.665 at p' =400 kN/m2. A change of p' from 400 to440 kN/m2 implies a volumetric strain of6v = 1.17% (cf. 1.2% used in predictionexercise), a change in specific volume6V = -0.0194 and a saturation Sr = 0.97.The last part only of the extension test

is fitted by using the undrained effectivestress path for a sample consolidated to550 kN/m2. A change of p' from 400 to550 kN/m2 implies a volumetric strain of6v = 3.90% (cf. 3% used in prediction exer-cise) , a change in specific volume 6V =-0.0637 and a saturation Sr = 0.90.

-Stra n n Geotechn cal Eng neer ng,p.592-627.

Wroth, C.P., M.F.Randolph, G.T.Houlsby &M.Fahey 1979. A study of the engineeringproperties of soils with particularreference to the shear modulus. Univer-sity of Cambridge: Technical Report(Soils TR75).

REFERENCES

Atkinson, J.H. & P.L.Bransby 1978. Themechanics of soils: an introduction tocritical state soil mechanics. London:McGraw Hill.

Burland, J.B. 1967. Deformation of softclay. University of Cambridge: PhD thesis.

Butterfield, R. 1979.A natural compressionlaw for soils (an advance on e-log p').Gôtechnique 29:469-480.

Carter, J.P. 1982. Predictions of the non-homogeneous behaviour of clay in thetriaxial test. Gôtechnique 32:55-58.

Hill, R. 1951. The mathematical theory ofplasticity. Oxford University Press.

Houlsby, G.T. 1981. A study of plasticitytheories and their applicability tosoils. University of Cambridge: PhDthesis.

Hvorslev, M.J. 1969. Physical propertiesof remoulded cohesive soils. Vicksburg:US Army Engineer Waterways Experiment

Station, Translation 69-5.Nadarajah, V. 1973. Stress-strain proper-ties of lightly overconsolidated clays.University of Cambridge: PhD thesis.

Parry, R.H.G. 1960. Triaxial compressionand extension tests on remoulded satur-ated clay. Gôtechnique 10:166-180.

Roscoe, K.H. & J.B.Burland 1968. On thegeneralised behaviour of 'wet' clay. In

m

J.Heyman&F.A.Leckie (eds.), Engineeringplasticity, p.536-609. Cambridge

University Press.Schofield, A.N. & C.P.Wroth 1968. Criticalstate soil mechanics. London: McGrawHill.

Wroth, C.P. & G.T.Houlsby 1980. A criticalstate model for predicting the behaviourof clays. In Workshop on Limit Equili- *brium, Plasticity and Generalised Stress

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