Evaluation of a constitutive model for clays and sands: Part II - clay behaviour

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech., 2002; 26:1097–1121 (DOI: 10.1002/nag.237)

Evaluation of a constitutive model for clays and sands:Part I – sand behaviour

Juan M. Pestana1,*,y, Andrew J. Whittle2 and Lynn A. Salvati3

1University of California, Berkeley, CA, U.S.A.2Massachusetts Institute of Technology, Cambridge, MA, U.S.A.

3University of California, Berkeley, CA, U.S.A.

SUMMARY

This paper evaluates the performance of a generalized effective stress soil model for predicting the rateindependent behaviour of freshly deposited sands, while a companion paper describes model capabilitiesfor clays and silts. Most material parameters can be obtained from standard laboratory data, includinghydrostatic or one-dimensional compression, drained and undrained triaxial shear testing. A compilationof data on compression behaviour allows for estimation of compression parameters when this type of datais not available. Extensive comparisons of model predictions with measured data from undrained triaxialshear tests shows that the model gives excellent predictions of the transition from dilative to contractiveshear response as the confining pressure and/or the initial formation void ratio increases. A parametricstudy of drained response shows that the model describes realistically the variation of peak friction angleand dilation rate as a function of confining pressure and density when compared with an empiricalcorrelation valid for many sands. The proposed formulation predicts a unique critical state locus for bothdrained and undrained triaxial testing which is non-linear over the entire range of stresses and is inexcellent agreement with recent experimental data. Overall, the model provides excellent predictions of thestress–strain–strength relationships over a wide range of confining pressures and formation densities.Copyright # 2002 John Wiley & Sons, Ltd.

KEY WORDS: undrained strength; peak friction angle; stress–strain relationships; sand behaviour; failure

1. INTRODUCTION

MIT-S1 is a constitutive model developed to predict the rate independent behaviour ofuncemented sands, clays and silts. Complete details of the model formulation are presented in aprevious paper [1] and only a brief summary is discussed here. Key features of the MIT-S1 soilmodel include: (1) the shear behaviour is described by a single anisotropic bounding surfacewhich is a function of the effective stresses and current void ratio; (2) density hardening of thebounding surface is controlled by the compression behaviour of freshly deposited soilsrepresented by the limiting compression curve (LCC) [2], while rotational hardening accounts

Received 22 August 2001Copyright # 2002 John Wiley & Sons, Ltd. Revised 11 February 2002

yE-mail: [email protected]

Contract/grant sponsor: National Science Foundation; contract/grant number: CMS-9623979

*Correspondence to: Juan M. Pestana-Nascimento, Geoengineering group, Department of Civil and EnvironmentalEngineering, 440 Davis Hall-1710, Berkeley, CA 94720-1710, U.S.A.

for the evolution of anisotropic properties; (3) small strain non-linearity in shear and stress–strain response in unload–reload cycles [3] is described through a perfectly hysteretic formulation;and (4) large strain shearing is controlled by an isotropic frictional failure criterion [4]. Themodel has been extended recently to simulate the cyclic and dynamic response of cohesionlesssoils with particular emphasis on the description of potentially liquefiable materials [5].

Description of the behaviour in shear is accomplished using an anisotropic bounding surfaceresembling a distorted lemniscate with radial mapping used to describe the behaviour ofoverconsolidated materials (cf., Figure 1). The bounding surface and the large strain failurecondition are generalized to the three-dimensional space using the Matsuoka–Nakai (three-invariant) criterion as discussed by Pestana [3]. As a result of soil anisotropy and the proposedgeneralization, constant mean effective stress sections of the bounding surface change from‘nearly a circle’ at the tip of the surface to the rounded triangular Matsuoka–Nakai locus at theorigin. Furthermore, the aperture of the cone circumscribing the bounding surface is densitydependent and controls ‘peak conditions’, such as peak friction angles in drained tests. Themodel uses a new compression model to describe the hardening of the bounding surface inisotropic (or K0) compression. For sands, the model describes a behaviour that ranges fromnearly incompressible (only elastic compression) and dependent on the initial formation densityat small stresses to highly compressible and independent of formation density at large stresses asshown in Figure 2. The model is based on the existence of the LCC, where the behaviour isprimarily controlled by particle crushing. At intermediate stresses, the behaviour is described bya transitional regime, which is primarily dependent on particle angularity and gradation as wellas initial formation density. Similarly to a previous model prototype, the model uses a perfectlyhysteretic formulation to describe the small strain non-linearity observed in compressionand shear. The new model proposes significant changes in the constitutive laws to describenon-linearity in shear, resulting in a more realistic prediction of shear modulus reduction incycles of increasing strain amplitude (cf., Figure 3). The model also introduces a variablePoisson’s ratio, m; which better describes the variation of the lateral earth pressurecoefficient, K0; as a function of stress history in contrast to formulations which use a constantm; predicting identical values of K0 for loading or unloading stress histories at a givenrecompression ratio (mean stress ratio) R:

-0.20

0.00

0.20

0.40

0.0 0.20 0.40 0.60 0.80 1.0

She

ar S

tres

s, (

σ' v-σ

' h)

/2α′ Bounding

SurfaceLoadingSurface

Current Stateof Stresses

ImagePoint

Mean Effective Stress, p'/ '

RadialMapping 0.95

0.750.5

0.25

12

3

= p/α

Critical State ConeMatsuoka-Nakai

generalization

π Plane

ProyectedSections of the

Bounding Surface

ασ σ

σ

Figure 1. Bounding surface plasticity formulation and generalization to non-triaxial space.

Copyright # 2002 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2002; 26:1097–1121

J. M. PESTANA, A. J. WHITTLE AND L. A. SALVATI1098

The remainder of this paper will focus on the response of freshly deposited sands while acompanion paper will illustrate model capabilities for clays and silts. Freshly deposited sandsmust be distinguished from ‘in situ’ sands for which previous stress history and fabric are notknown and cementation effects may be present. Current density of ‘in situ’ cohesionless soils canbe accurately estimated from high-quality sampling and simulated in the laboratory throughreconstituted specimens using a variety of preparation procedures [6]. In contrast, variations inthe ‘in situ’ conditions resulting from ageing and cementation cannot be reliably simulated usingreconstituted specimens [7]. Uncertainties in past stress history and ‘in situ’ fabric pose majorproblems for parameter selection and accurate assessment of initial state conditions and it isconsidered beyond the scope of this work.

First Loading

ρcVoi

d R

atio

,e (

log

scal

e)

p'ref

Mean Effective Stress, p' (log scale)

p'b

p'

Current State (e, p')

Limiting Compression Curve, LCC

e

Unloading

1

log e = - ρc log (p'/p'ref

)

p'rev

erev

1.0

Figure 2. Conceptual compression model for sands (after Reference [2]).

0.0

0.2

0.4

0.6

0.8

1.0

0.0001 0.001 0.01 0.1 1

Shea

r M

odulu

s R

educt

ion,G

sec

/ G

max

Shear Strain, γ % (log scale)

Strain

Stress

Hysteretic cycles of

increasing amplitude

R = p'O

/p'

B

Unloading

K0

(lo

g sc

ale)

Reloading

O

A

Mean Stress Ratio, R (log scale)

KA

KONC

1

Figure 3. Perfectly hysteretic model formulation describing the small strain non-linearity in shear [3].


EVALUATION OF CONSTITUTIVE MODEL: PART 1 – SAND BEHAVIOUR 1099

The following sections show comparison of element level predictions with measured responseof Toyoura sand under monotonic loading in triaxial space and illustrate the dependence ofsand response on density and confining stress. Measured response in other modes of shearing(e.g. plane strain, hollow cylinder) is only available for a small range of confining stresses, andthey will be reported in a separate paper. Toyoura sand is a fine, poorly graded quarzitic sand,which is the standard cohesionless soil reported in the Japanese soil mechanics literature overthe last 20 years. Table I shows the average index properties of Toyoura sand compiled from avariety of sources [6, 8–11]. Toyoura sand consists of sub-angular particles with mean diameterD50 ¼ 0:16–0:20 mm and a uniformity coefficient Cu ¼ 1:3–1.7. Although there are slightpervasive differences in the engineering and mechanical properties of Toyoura sand reported bydifferent researchers, no comparative studies of the different batches have been reported to date.This paper relies primarily on undrained data presented by Ishihara [6], drained resultspresented by Miura [9, 10] and small strain measurements by Tatsuoka and co-workers [11, 12].Table II summarizes the input material parameters for Toyoura sand used in the model tocompare to the results of drained and undrained triaxial tests.

2. DETERMINATION OF MATERIAL CONSTANTS

Application of the model for freshly deposited sands requires the determination of 13 inputparameters, most of which can be determined from standard types of laboratory tests onspecimens that are prepared by techniques of pluviation or undercompaction, thus ensuringspecimen uniformity. Other methods of laboratory specimen preparation, such as moisttamping, are specifically excluded from consideration as they introduce non-uniformities. Someof the tests include isotropic or K0 compression to large confining stresses which, althoughavailable in many research institutions, are not standard equipment used in practice. As a result,considerable effort has been devoted to compile good-quality compression data on a variety ofsands such that estimates of compression parameters could be obtained through correlationswith properties such as angularity or grain size distribution [2]. Pestana and Whittle [1] proposeda procedure to select the remainder material parameters providing the most efficient use of datafrom a limited number of laboratory tests, and it is illustrated in the following sections.

Table I. Average index properties of Toyoura sand.

Property Toyoura sand

Mineralogy/principal constituents Quartz,Feldspar, magnetite

Grain description (shape) Sub-angularSpecific gravity of solids, Gs 2.65Mean particle size, D50 (mm) 0.16–0.20Coefficient of uniformity, Cu ð¼ D60=D10Þ 1.3–1.7Maximum dry density, gmaxðkN=m3Þ 16.15–16.45Minimum dry density, gminðkN=m3Þ 13.13Maximum voids ratio, emax 0.98Minimum voids ratio, emin 0.58–0.61Range of void ratio ðemax2eminÞ 0.40



2.1. Compression behaviour

In general, the model requires one-dimensional ðK0Þ or isotropic compression data for specimensloaded to high confining pressures, s0c; corresponding to the LCC regime. These pressures aretypically in the range, s0c ¼ 10–50 MPa for siliceous sands [2]. The tests can be performed ineither a high-pressure triaxial cell or a floating ring oedometer apparatus. The test should ideallybe performed at a constant rate of strain (or deformation), include one or more unload–reloadcycles, and should measure void ratios to an accuracy of De ¼ 0:005: The compressionbehaviour at high stresses is independent of initial void ratio and is described by the LCC. Thislocus is linear in logðeÞ � logðpÞ space and is controlled by parameters rc (slope) and pref

(reference mean effective stress at a unity void ratio) as shown in Figure 4. Based on availablecompression data, the slope of the LCC for Toyoura sand is approximately 0.37 ð�0:03Þ with areference stress, pref=pat; of 55. Parameter y controls the transitional regime from low stresses tostresses in the LCC regime and has been found to depend on angularity and material gradation.Figure 4 illustrates the determination of input parameter y ð� 0:20Þ from the observedbehaviour in the transitional regime for Toyoura sand. Please note that the batches of Toyourasand used by Miura and co-workers have slight variations in gradation and therefore an averagevalue of y is obtained. Based on quality compression data available in the literature, thefollowing correlations have been determined for siliceous sands:

p0ref=pat �

15=D0:550 ; angular particles;

50=D0:550 ; rounded particles:

(ð1aÞ

y � 0:10Cu þ a ð1bÞ

Table II. MIT-S1 input material parameters for Toyoura sand.

Test type Parameter/symbol

Physical contribution/meaning Toyourasand

Hydrostatic or 1-D rc Compressibility of sands atlarge stresses (LCC regime) 0.370

Compression test pref=pat Reference stress at unity void ratio for the H-LCC 55.0(Triaxial, oedometer) y Describes first loading curve in the transitional regime 0.20

K0NC K0 in the LCC regime 0.49K0-oedometer m0 Poisson’s ratio at load reversal 0.233or K0-triaxial o Non-linear Poisson’s ratio 1.00

1-D unloading stress pathUndrained/ f0

cs Critical state friction angle 31:08drained triaxial in triaxial compressionShear Tests: f0

mr Peak friction angle as a function of 28:58np formation density at low stresses 2.45

OCR ¼ 1; CIDC m Geometry of bounding surface. 0.55Undrained stress paths

OCR ¼ 1; CIUC os Small strain ð50:1%Þ non-linearity 2.5in shear

c Rate of evolution of anisotropy 50Stress-strain curves

Resonant column Cb Small strain stiffness at load 750bender elements reversal



where a ¼ 0:0; 0.05, 0.15, 0.25 for rounded, subrounded, subangular and angular particles,respectively.

2.2. Perfectly hysteretic elastic response

Elastic shear wave velocity data, Vs; are used to define the maximum elastic shear modulus,Gmax; and thus control the elastic parameter Cb: Selection of Cb by this procedure requires thedetermination of the small strain Poisson’s ratio, m00 and will be described later. Measurementsof Vs are available from a variety of sources including (but not limited to): (a) resonant columndevices [13,14,39] and (b) triaxial tests with piezo-crystals or bender elements embedded in theend-platens [15,16]. Although these tests are not considered ‘standard’ tests, they are available atmany research laboratories. Among them, the bender elements are becoming increasinglycommon in most geotechnical laboratories. Alternatively, high-quality measurements ofvolumetric response during unloading can be used directly to assess the elastic bulk modulus(and therefore Cb) and do not require a priori determination of the Poisson’s ratio [2] and isshowcased in Figure 5. For Toyoura sand, values of Cb determined from both methods agreereasonably well and range from 700 to 850.

The model also requires measurements of lateral stresses during a cycle of one-dimensionalðK0Þ swelling and reloading, to a maximum overconsolidation ratio, OCR ¼ 5–10. Reliable dataof this type are not generally available for sands and require either a special instrumentation of arigid walled oedometer device [17], or high-quality feedback control in automated triaxial tests[18,19]. In MIT-S1, the small strain elastic Poisson’s ratio, m00 is determined from the initial slope

0.40

0.50

0.60

0.70

0.80

0.90

1.00

10 100 1000

1 10 100

c = 0.370

1V

oid

Rat

io,e

(lo

g sc

ale)

Mean Effective Stress, p'/pat

IsotropicLCC

MODELSIMULATIONS

Line0.100.200.30

MEASURED DATA: Toyoura Sand

Hydrostatic Compression(Miura, 1979;

Miura el al. 1984)e0Symbol

0.830.770.59

Mean Effective Stress, p' (MPa)

p'ref

Dr (%)

~25%

~50%

~75%

~100%

* approximate

ρ

θ

Figure 4. Determination of compression model parameters for Toyoura sand (after Reference [2]).



of the effective stress paths during one-dimensional swelling (i.e. m00 ¼ 1=ð1þ Ds0v=Ds0hÞ; where

Ds0v=Ds0h is the initial slope of the observed effective stress path after load reversal and Ds0v; Ds

0h

are the changes in vertical and horizontal stress, respectively). Figure 6 shows one-dimensionalunloading from LCC states for Pennsylvania sand, giving an average value of m00 � 0:233 ð2Gmax=Kmax ¼ 1:30Þ: It is assumed that this behaviour is representative for Toyoura sand and isused for the remainder of this work. For uncemented materials, the expected range of the elasticPoisson’s ratio at load reversal, m00; is narrow with typical values ranging from 0.15 to 0.25, whileseveral researchers [20, 21] recommend a constant value of 0.20. Jamiolkowski and co-workers[22] report values of Poisson’s ratio at low stresses ranging from 0.12 to 0.15 based on localstrain measurements from 0.20 to 0.25 based on external measurements of axial strain. As can

0.01 10

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Rec

over

ed V

olum

etri

c St

rain

, ∆ε

(%)

Mean Effective Stress, p'/pat

MEASURED DATA; Toyoura Sand

Isotropic CompressionMiura et al., 1984

'rev = 16.8 MPa

erev = 0.662

MODELSIMULATIONS

CbSymbol

7008501000

σ

Figure 5. Estimation of volumetric stiffness parameter, Cb; from measured data in isotropic unloading.

0.00

0.20

0.40

0.60

0.00 0.20 0.40 0.60 0.80 1.00

σ' h / σ

' vmax

= µ'0

Measured Data: 1-D unloadingPennsylvania sand (Hendron, 1963)

0.25

0.30

0.20

1

Overconsolidation Ratio, OCR1.5234568

Model ParametersK

0(LCC) ~ 0.49

µ'0~ 0.233, ω ~ 1.0

σ'v /σ'vmax

Figure 6. Determination of material parameters K0 (LCC), m00 and o from measured data in1-D unloading.



be seen in Figure 6, unloading along a linear strain path (i.e. constant ratio of applied strainincrements) does not generate a linear effective stress path. As a result, the value of K0 ð¼s0h=s

0vÞ is a non-linear function of stress history, OCR [23, 24]. Non-linearity of the effective

stress path is apparent at s0v=s0vmax40:5 ðOCR52Þ and can be well described by the variable

Poisson’s ratio with parameter o ¼ 1:Conventional triaxial equipment can obtain reliable shear data in the range 0:054ea410%

[25,40]. However, the model requires more accurate measurements of non-linear behaviour inthe small strain range, 0:0014ea40:05%; which can be obtained from internal (local) strainmeasurements [25, 26–29]. Tatsuoka and co-workers [12, 11, 29] report high-quality static localstrain measurements of shear modulus at small strain levels, g50:001%, for dense Toyourasand, e0 � 0:657; at s0c=pat ¼ 0:80 which tie-in with measured values of Gmax: The model wascalibrated with available measurements from a drained, plane strain compression test on denseToyoura sand, e0 ¼ 0:657; and Figure 7 suggests that a value of os ¼ 2:5� 0:5 gives an accuratedescription of the small strain non-linearity at g40:05%. Several researchers have suggestedthat the effects of grain size distribution on the shear modulus reduction curve and dampingproperties (affected primarily by parameter os) are minor [30, 31] and may indicate similarvalues of os for other clean uniform sands. Recent work on the description of soil non-linearityfor dynamic site response analyses suggests that the input parameter os increases with increasesin the coefficient of uniformity Cu [5].

2.3. Elastoplastic shear stress–strain response

The model requires data at relatively large strains (ea ¼ 20–30%) where the model assumes thatshearing occurs at constant volume [32], with shear strength controlled by the friction angle

MIT-S1 ModelPlane Strain

0

5

10

15

20

25

30

0.050.040.030.020.010.00

1

54

3

2

Dev

iato

ric

Stre

ss, σ

' v-σ

' h (

kPa)

Shear Strain, γ (= εv- εh) (%)

s =

Note: e0.05

= e at 'c /pat = 0.05

Measured DataToyoura Sand

Tatsuoka et al., 1993'c ~ 78 kPa; e

0.05= 0.657σ

σ

ω

Figure 7. Estimation of material parameter os from small shear strain measurements.



f0cv ¼ f0

cs: For Toyoura sand, the friction angle, f0csð¼ f0

cvÞ mobilized at large shear strains intriaxial compression is reported as f0

cs ¼ 318 [6, 33], while Miura and co-workers [8, 10] reportf0cs ¼ 32:5–33:58: This difference is typical of variation in engineering properties for different

sources of Toyoura sand reported in the literature.Parameter f0

mr represents the aperture of the bounding surface at a reference void ratio, e ¼ 1while ‘np’ describes the increases in the aperture with increasing density (i.e. decreasing voidratio). These parameters control two important features of sand behaviour: (a) predictions ofpeak friction angles in drained shear tests at low confining pressures (� 0:02–0:05 MPa) and (b)the location of the critical state line (CSL) (unique for both drained and undrained tests).Parameter m describes the slenderness of the bounding surface for a given aperture at the originand control the shape of the undrained effective stress paths. Pestana [1, 3] proposed twooptions for estimating the parameters controlling the response in shear. The first option requiresthe calibration through simulation of the changes in measured peak friction angles as a functionof stress and formation density. The second option uses the effective stress–strain response fromundrained shear tests at two formation densities. For this option, the parameters are selectedthrough a short parametric study to match values of the mean effective stress at critical stateconditions, p0

failure ðe1Þ; p0failure ðe2Þ; for undrained shearing at two different void ratios ðe1; e2Þ: A

large increase in backpressure may be necessary to avoid cavitation of the samples in undrainedtesting of relatively dense soils at low stresses. Figure 8 compares model simulations of a CIUCtest on a dense specimen of Toyoura sand with e0 ¼ 0:735 and s0c=pat �¼ 30 for three values ofm ¼ 0:40; 0.55, 0.70 (all cases assume c ¼ 50; discussed later). The corresponding boundingsurface aperture f0

m for each m value was determined to satisfy the known critical stateconditions, and is shown in Figure 8. The best fit to the measured undrained effective stress pathcorresponds to m ¼ 0:55� 0:05: Another undrained test at the same confining pressure but at adifferent density ðe0 ¼ 0:833Þ was used to determine parameter ‘np’. Note that only the large

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

qq

m = 0.40 0.70

Dev

iato

ric

Stre

ss, σ

' v-σ'

h (M

Pa)


31˚ = φ 'cs

CIUC Test Data: Toyoura Sand(Ishihara, 1993). 'c = 3 MPa

0.55

e0Sym.

0.7350.833

0 5 10 15 20

0.55

0.40

0.70m =

Axial Strain, εa (%)

Parameters '

m* m

45.60.4040.50.5537.70.70

* Initial voidratio= 0.735

MIT-S1 Model

'mr

= 28.5

np = 2.45

σ

φ

φ

Figure 8. Estimation of input parameters m and np for Toyoura sand.



strain failure condition, and not the stress-path or stress–strain curve, was required for this test.Using the undrained critical state conditions from tests at e1 ¼ 0:735 ðp0

failure ¼ 29:0� 0:5Þ ande2 ¼ 0:835 ðp0

failure ¼ 10:0� 0:25Þ; the following parameters were determined, np ¼ 2:45� 0:10and f0

mr ¼ 28:5� 0:258:Figure 9 compares model simulations for different values of the evolving anisotropy

parameter with the measured behaviour of the same undrained triaxial compression test. Thestress–strain curve in the range 2%5ea520% is significantly affected by rotation rate. A lowvalue of c (e.g. 12.5) will result in a characteristic curve exhibiting a pseudo-yield stress (atea � 2%) followed by a slow strain hardening and reaching critical state conditions at strains inexcess of 20%. On the other hand, for rotation rates c550; the stress–strain response is muchstiffer with continuous strain hardening and a more rapid transition to critical state conditionswhich are reached at smaller strains ðea ¼ 10–20%). For rotation rates, c550–100 there areonly marginal differences in the predicted stress–strain behaviour for both drained andundrained shear tests. For Toyoura sand, c ¼ 50� 10 accurately represents the measuredbehaviour of dense sand and it is used throughout.

3. UNDRAINED BEHAVIOUR DURING TRIAXIAL COMPRESSION

This section evaluates model predictions of undrained shear tests on freshly deposited Toyourasand reported by Ishihara [6] over a wide range of confining pressures, 0:01 MPa4s0c43:0MPa; and formation densities, 0:7354e040:920: Figure 10 compares the predicted andmeasured undrained effective stress paths and shear stress–strain behaviour of dense specimensðei ¼ 0:735Þ at four different confining pressures, s0c � 0:1; 1.0, 2.0 and 3:0 MPa: Goodagreement is expected between model simulations for the CIUC tests at s0c ¼ 3:0 MPa; since this

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

qD

evia

tori

c St

ress

, σ' v-

σ' h

(MPa

)


12.5

10025

50

200

31˚ = 'cs

= ψ

MIT-S1 Model Simulations

0 5 10 15 20

25

50


100200

12.5 = ψ

Measured Data:CIUC Tests Toyoura Sand(Ishihara,1993)

'c = 3 MPa; e0= 0.735

φ

σ

Figure 9. Estimation of input parameter c for Toyoura sand.



test is part of the database from which the input parameters were determined. The modelpredicts that all samples having the same pre-shear void ratio will have the same undrainedstrength at large strains. The model accurately predicts the undrained stress paths (and hencepore pressure development) and describes the transition from strongly dilative behaviour(negative shear induced pore pressures) experienced by specimens consolidated at low stresses(e.g. s0c ¼ 0:1–1 MPa) to moderately contractive behaviour at high stresses (e.g. positive porepressures at s0c ¼ 2:0–3:0 MPa). Stress–strain curves and pore pressure-strain curves predictedby the model are in good agreement with measurements for tests at s0c ¼ 1:0 and 2:0 MPa andare in excellent agreement with the data at the lowest stress, s0c ¼ 0:1 MPa: The modelconsistently overestimates the shear stiffness for ea51–2% for tests at confining pressures,s0c > 1:0 MPa: This may be partially attributed to small differences in the small strain propertiesof Toyoura sand reported by the different researchers and/or a consistent bias in modelparameters. The results published by Ishihara focused on medium to large strain conditions anddid not include sufficient small strain data to revise the selection of parameters Cb or os:

Figure 11 compares the predicted and measured behaviour for a similar series of CIUC testsfor medium dense Toyoura sand, ei ¼ 0:833: Again, the model accurately predicts the transitionfrom strongly dilative behaviour (negative pore pressures) experienced by the specimensconsolidated at low stresses (e.g., s0c ¼ 0:1 MPa) to fully contractive behaviour for the specimen

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Dev

iato

ric

Stre

ss, σ

' v-

σ ' h (

MPa

)

Mean Effective Stress, p (MPa)

31˚ = 'csMIT-S1 Model Predictions

Test Used to Calibrate Material

Parameters

0 5 10 15 20


Measured Data: CIUC Tests Toyoura Sand (Ishihara, 1993)

Dense ; ei= 0.735

Material Parameter Database

0 5 10 15 20

-2.0

-1.0

0.0

1.0

2.0


Exc

ess

Pore

Pre

ssur

e,

∆u

(MPa

)

0.1

1.0

2.0

3.0

Nominal Confining Stress (MPa)

φ

Figure 10. Comparison of predicted response for CIUC tests on dense Toyoura sand.



consolidated at s0c ¼ 3:0 MPa: The following features are observed: (a) for tests performed atlow confining pressures ðs0c ¼ 0:1 MPaÞ; the model predicts a substantial increase in the meaneffective stress (i.e. negative shear-induced pore pressures) with continuous strain hardening andmaximum undrained strength at axial strains ea � 15% (corresponding to critical stateconditions); (b) as the confining pressures increase, the model predicts a decrease in thegeneration of negative pore pressures and (c) at s0c ¼ 3:0 MPa (close to the LCC regime), thebehaviour becomes fully contractive with maximum shear stress mobilized at ea � 1–2%(predicted eaf ¼ 1:5%) and a post-peak reduction of shear stress until critical state conditions arereached at large shear strains. It is not surprising that the model matches the measured shearstrength at large strains, as these data were used in parameter selection. However, the results inFigure 11 show that the model is able to predict accurately the complete effective stress paths,stress–strain and pore pressure-strain behaviour for medium–loose sand from low strainsthrough high strains over the full range of confining pressures. Similarly to the results presentedin Figure 10, the model consistently overpredicts the shear stiffness for strains 51–2% and mayindicate a consistent bias in material parameters.

Figure 12 shows a series of four tests on loose samples of Toyoura sand (ei ¼ 0:915–0.917,Dr � 16%) at low confining pressures which range from s0c ¼ 0:01 to 0:1 MPa: For these cases,MIT-S1 tends to underpredict the peak shear stress (by 15–20%) but model predictions are in

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

q (MPa)31˚ = 'cs

Dev

iato

ric

Stre

ss, σ

' v-σ'

h (M

Pa)


MIT-S1 ModelPredictions

Large StrainConditions

0 5 10 15 20



Dense ; ei = 0.833

0 5 10 15 20

-1.0

0.0

1.0

2.0

3.0


Exc

ess

Pore

Pre

ssur

e,

∆u (

MPa

)

0.1

1.0

2.0

3.0

Nominal Confining Stress (MPa)

φ

Figure 11. Comparison of predicted response for CIUC tests on medium–loose Toyoura sand.



relatively good agreement with the measured stress path and the large strain conditions (within5% at ea � 0:5%). The model predicts peak shear stress conditions at axial strains, ea40:5%;followed by softening to a minimum mobilized resistance at ea � 1–2% and hardening tocritical state at ea ¼ 10%: On the other hand, the measured data show a softer initial responsewith strains at peak in the order of ea � 2% followed by a strain-softening region extending toea � 12–14% with further strain-hardening for ea > 15% (cf. Figure 12). The measured stress–strain curves show no equilibration of shear stresses, indicating that critical state (or steadystate) conditions may have not been reached yet. This discrepancy may be due to one or more ofthe following factors: (1) small variations of void ratio (i.e. De ¼ 0:005) in this range ðe0 ¼ 0:92Þresult in large changes in the ultimate undrained strength (since large strain conditions areuniquely determined by current void ratio (as shown in Figure 13); (2) peak undrained strengthmay be affected by initial (or inherent anisotropy) due to the formation/deposition process or(3) non-uniformities or localization of deformations within the specimens. The modelpredictions of the stress–strain response show the same trends as the measured response butthe rate of softening and subsequent hardening is much faster than those measured in the lab.This is believed to be a small shortcoming of the model resulting from the power law used in theshear stiffness, which predicts significantly higher initial shear stiffness at small values ofconfining stress. Figure 13 shows the effect of small changes in the initial void ratio (i.e. 0.915–0.940) on the undrained response of loose Toyoura sand. Predicted peak undrained strengthincreases only slightly with a reduction in initial void ratio. In contrast, the shear stress at largestrain conditions}also referred by some as residual strength}changes over one order ofmagnitude for such a small change in void ratio and is consistent with results reported in theliterature.

4. DRAINED BEHAVIOUR DURING TRIAXIAL COMPRESSION

This section illustrates model simulations of drained shear tests on freshly deposited Toyourasand for a wide range of confining pressures and densities. Figure 14 shows a parametric studyillustrating the effects of the initial void ratio, ei ¼ ð0:60; 0:70; 0:80; 0:90; 0:95Þ on simulations of

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.00 0.02 0.04 0.06 0.08 0.10 0.12

31˚ =φ 'cs

Dev

iato

ric

Stre

ss, σ

' v-σ'

h (M

Pa)



0 4 8 12 16 20 24



Loose ; ei~ 0.917

Figure 12. Comparison of predicted response for CIUC tests on loose Toyoura sand.



drained triaxial compression tests for Toyoura sand, at a confining stress, of s0c ¼ 0:1 MPa: Theresults show the following:

(a) As the initial void ratio decreases, there is an increase in peak friction angle anddilation rates, as well as a decrease in the strain required to mobilize the peak strength

0.00

0.05

0.10

0.15

0.00 0.05 0.10 0.15



31˚ = 'cs

0 5 10 15


0.94

CIUTC Tests'c = 0.1 MPa

0.935

0.930

0.925

0.920

0.915

Initial Void Ratio

φ

Dev

iato

ric

Stre

ss, σ

' v-σ'

h (M

Pa)

σ

Figure 13. Effect of small variations in the initial void ratio on the undrained response of loose Toyourasand.

0.0

1.0

2.0

3.0

4.0

5.00 5 10 15 20 25 30

CIDC Tests'c /pat= 1

'cs = 31o0.90

0.60 = e i


0.95

0.70

0.80

0.0

1.0

2.0

3.0

4.0

5.0-20-15-10-5.00.05.010

Volumetric Strain, ε (%)vol

Contraction

0.70 0.60

Dilation

0.90

0.80

e i= 0.95

-20.0

-15.0

-10.0

-5.0

0.0

5.00 5 10 15 20 25 30

Vol

. Str

ain,

εvo

l (%

)


0.60

0.80

0.90

0.70

0.950.50

0.60

0.70

0.80

0.90

1.00

1 10 100

E

Void R

atio,e

Mean Effective Stress, p'/ pat

Initial States

CSL

σ ′/σ

′–1

vh

σ

φ

σ ′/σ ′–1v

h

(a)

(b)

(c)

(d)

Figure 14. Parametric study on the effect of initial void ratio on the drained response in triaxialcompression.



(cf. Figure 14(a) and (b)). The model predicts initial contractive behaviour for obliquitiesless than the critical state (Note for f0

cs ¼ 318; ðs0v=s0hÞcs � 3:12Þ:

(b) For void ratios higher than those at the critical state, ei > ecs; the samples exhibitcontractive behaviour (i.e. reduction in volume, cf. Figure 14(b) and (d)). Peakconditions of stress and obliquity (as well as critical state conditions) are achieved atlarge strains in contrast to undrained shear tests at high void ratios (cf. Figure 12).

(c) Drained shearing of dense specimens ðei50:8Þ shows minimal change in volume up to thepeak friction angle after which there is a significant increase in volume accompanied bystrain softening to the critical state conditions at large strains. The critical state locus fordrained shearing coincides with results presented previously for undrained tests (cf.Figure 14(d)).

Figure 15 illustrates the effect of confining pressure on model predictions of drained triaxialcompression tests on Toyoura sand, at a specified initial void ratio, ei ¼ 0:80: Note that in orderto obtain a specimen with ei ¼ 0:80 at a stress level of s0c ¼ 10 MPa; the specimen must havebeen formed at a void ratio near the maximum void ratio emax ¼ 1:00: The results show thefollowing:

(a) Peak friction angles are mobilized at relatively low strains ðea � 1%Þ at low confiningpressures and initial void ratios lower than those at the critical state. Post-peak softeningleading to critical state conditions at large strains (cf. Figure 15(a)) is accompanied byincreases in the volume (dilative behaviour) (cf. Figure 15(c) and (d)). It should be noted

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.00 5 10 15 20 25 30

CIDC Testse i= 0.80

φ 'cs = 31o

520

50

10

Axial Strain, ε(%)a

0.10.5 1

100

2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0-10-5.00.05.010

Volumetric Strain, ε (%)vol

Contraction

0.50.1

Dilation

5

1

'c /pat= 10

50

20

100

2

-8.0

-4.0

0.0

4.0

8.00 5 10 15 20 25 30


100

0.5

5

0.1

10

12

50

20

0.50

0.60

0.70

0.80

0.90

1.00

1 10 100

Void R

atio, e

Mean Effective Stress, σ'/ pat

LCC

CSL

Initial State* Tests stopped at 40%

σ ′/σ

′–1

vh

Vol

. Str

ain,

εvo

l (%

)σ ′/σ ′–1

vh

σ

(a)

(b)

(c)

(d)

Figure 15. Parametric effect of confining stress on the drained response in triaxial compression.



that ‘global’ volume measurements during drained shear tests on dense specimens willgenerally underestimate the void ratio at critical state due to localization of deformation(large changes in volume in a relatively reduced zone of the specimen). In addition, thesoftening predicted as a result of the increase in void ratio should be viewed only as anindicator, since this behaviour will depend on the localization process as well as boundaryconditions (triaxial vs. plane strain)

(b) As the confining stress increases, the peak friction angle (i.e. s0v=s0h) and dilatancy rate

decrease and the strain to peak increases. At large confining pressures and therefore voidratios higher than those at the CSL, the peak stress and critical strain conditions coincide ataxial strains exceeding 40–50%, accompanied by large compressive volumetric strains(cf. Figure 15(c) and (d)).

4.1. Peak friction angles and dilation rates

Figure 16a shows MIT-S1 predictions of the peak friction angle as a function of the formationvoid ratio and effective confining pressure. Similar to the state parameter approach [34], themodel predicts variations in peak friction angle as a function of both confining pressure andformation void ratio. Figure 16(a) shows that for a given formation void ratio, e0; there is acritical confining pressure s0cr=pat; at which the peak friction angle approximates that at thecritical state (i.e., f0

p ! f0cs). The value of s

0cr=pat increases as e0 decreases, which is conceptually

similar to the empirical interpretation of Bolton [35] and Vesic and Clough [17]. Figure 16compares MIT-S1 predictions of peak friction angle, f0

p; and dilation rates, ð�devol=devÞmax; as afunction of confining stress at failure and relative density for triaxial shearing with the empiricalequations proposed by Bolton [35]:

ðf0p � f0

csÞTX ¼ 3Ir8 ð2aÞ

�ðdevol=devÞmax ¼ 0:3Ir ð2bÞ

30.0

32.0

34.0

36.0

38.0

40.0

42.0

44.0

46.0

48.0

1 10 100

Peak

Fri

ctio

n A

ngle

, φ'

0.60

Mean Effective Stress at peak conditions (p'/pat)

0.70

0.80

0.90

1.00

MIT-S1

Empirical(Bolton, 1986)

'cs = 31o

e0 =

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1 10 100

Dilation R

ate, (-dεvol / dε

v )

0.60 = e0

0.70

0.80

0.90

1.00

Dilation Ratesat peak shear stress

p

φ

(a) (b)

Figure 16. Predicted peak friction angles and dilation rates for shear in triaxial compression.



where devol and dev are the changes in volumetric and axial (i.e. vertical) strains in a compressiontest, respectively. Parameter Ir is the relative dilatancy index and can be written in terms of therelative density, Dr; and the effective confining pressure, s0c:

Ir ¼ Drð10� ln½s0c=pat�Þ � 1; 04Ir44 ð3aÞ

Dr ¼ ðemax � eÞ=ðemax � eminÞ; 04Dr41 ð3bÞ

where emax; emin are the maximum and minimum formation void ratios, respectively (forToyoura sand, emax � 0:98; emin � 0:60). Bolton empirical equations have been demonstratedto apply to many sands, including Toyoura sand and thus provide an indirect validation of theproposed model. The empirical equations for f0

p are generally in good agreement with modelpredictions for s0c=pat > 1:0: However, the empirical expressions give higher values of f0

p at lowconfining pressures ð� 0:05 MPaÞ where the model shows f0

pðe0Þ approaching an upper limit(i.e. f0

p varies by less than 10% for s0c=pat51:0). Sture and co-workers [36] have measuredrelatively large peak friction angles at extremely low values of confining stress ð50:005 MPaÞand they are not described by this model. This observation may indicate a potential shortcomingof the model at such small confining stresses.

The results of maximum dilation rate exhibit analogous features to those described for f0p;

including: (a) small variation of dilation rates at low stresses ðs0c=pat51:0Þ; (b) a transitionalregime up to the critical confining stress with monotonic decrease in maximum dilation ratesand (c) zero dilation for f0

p ¼ f0cs (at s

0c=pat > s0cr=patÞ: The model predicts that the maximum

rate of dilation occurs after the peak strength is mobilized, which results from the particularchoice of the plastic potential function. However, the differences in the dilation rate at peakstrength and the maximum dilation rate are only significant for confining stresses, s0c=pat51:0;and formation void ratios, e050:7 (cf. Figure 16(b)). The dilation rates predicted by MIT-S1give very close agreement with empirical expressions for s0c=pat > 1:0:

Figure 17 compares MIT-S1 predictions of the peak friction angle and dilation rate in planestrain compression tests with similar empirical expressions:

ðf0p � f0

csÞPS ¼ 5Ir8 ð4Þ

The model predictions are qualitatively similar in plane strain and triaxial compression,however, there are significant differences between the predicted and empirical values of f0

p inplane strain compression. MIT-S1 gives consistently smaller values of peak friction anglesthan Bolton’s empirical expression for nearly all cases shown. The peak and critical statefriction angles for shear in plane strain compression are typically 3–48 higher than intriaxial compression or extension tests. The predicted ei � f0

p relation for triaxialcompression and plane strain compression are nearly parallel (for a given confining stress)for approximately all formation void ratios. Figure 17(b) shows that dilation rates at peak are ingood agreement with the empirical expression. Figure 17(b) also compares dilation rates forboth triaxial and plane strain compression. The differences between the two modes of shearingare for practical purposes negligible as previously suggested by Bolton [35].

4.2. Measured drained response

Model performance can be evaluated by comparing model predictions (based on calibrationswith undrained tests) with results from drained triaxial tests performed on Toyoura sand byMiura and Yamanouchi [8]. The tests were performed at approximately two different void ratios



ei ¼ 0:61 and 0.83 with confining pressures ranging from 2.45 to 49:0 MPa: The loose Toyourasand ðei ¼ 0:83Þ exhibited contractive tendencies over the entire range of confining stressestested. Figure 18 shows that MIT-S1 model predictions are qualitatively in good agreement; thedeviatoric stress and the volumetric strain increase continuously until a maximum value isreached. At the lower confining pressures (i.e. s0c ¼ 2:45 and 4:9 MPa), MIT-S1 predictionsmatched the deviatoric stress and the volumetric strains very well over the entire test. Atconfining pressures up to 29:4 MPa; MIT-S1 model predictions match the tests resultsquantitatively at axial strains less than 10%, but at higher axial strains, the predictionsunderestimate the deviator stress and volumetric strain measured. At the highest confiningpressure ðs0c ¼ 49:0 MPaÞ; the model underpredicts the deviatoric stress to about 25% axialstrain, and then overpredicts deviatoric stress at axial strains higher than 25%. The denseToyoura sand ðei ¼ 0:61Þ exhibits contractive tendencies for all of the tests except for thetest performed at the lowest confining pressure ðs0c ¼ 2:45 MPaÞ: Figure 19 shows that theMIT-S1 model predictions correctly capture this transition from dilative to contractivebehaviour. As with the tests on the loose Toyoura sand, the model predictions match the testresults well up to axial strains of 10% for all of the confining pressures below 49:0 MPa: Ataxial strains greater than 10% the model predictions of the deviatoric stress and volumetricstress are lower than the actual test results. One reason for the discrepancy in measuredand predicted results may be a consistent difference in the engineering properties of Toyourasand used by the two researchers. In addition, the behaviour in the LCC regime is significantlyaffected by rate effects, which will lead to a further reduction in void ratio at a givenconfining stress as discussed by Pestana and Whittle. The current model uses a rate-independentsmall strain formulation and therefore does not only qualitatively capture the response asthis level of strains. Model predictions shown in this section were generated using the sameinput parameters determined earlier with the exception of the critical state friction angle,which was changed to 338 to match the value reported by Miura and co-workers. The remainingparameters are identical as the ones for the tests reported by Ishihara. Although these

32

34

36

38

40

42

44

46

48

50

52

54

1 10 100

0.60

0.70

0.80

0.90 = e0

Peak

Fri

ctio

n A

ngle

, φ′

Mean Effective Stress at peak conditions, p'/pat

MIT-S1

Bolton, 1986

Plane Strain

( φ 'cs)PS

~ 34.4o

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

1 10 100

0.60 = e0

0.70

0.80

0.90

1.00

Dilation Rates atPeak Friction AngleMIT-S1 predictions

Triaxial

Plane Strain

pD

ilation Rate at peak, (-dε

vol / dεv )

(a) (b)

Figure 17. Predicted peak friction angles and dilation rates for shear in plane strain compression.



samples were prepared by wet tamping (and therefore produce significant fabric effects), it isvery difficult to assess how much of the fabric is preserved after a relatively large increase inmean effective stress. As a result, all the samples were assumed to start from isotropicconditions.

5. CRITICAL STATE CONDITIONS

Figure 20(a) shows excellent agreement between the predicted location of the critical state line(in log e� log p space) and the measured undrained response for Toyoura sand presented byIshihara [6]. The critical state line, CSL, forms a non-linear locus over the range of the measureddata and highlights the limitation of the earlier state parameter approach [34]. Figure 20(b)shows the CSL in a linear space for void ratio and stress. A linear regression of the results for

0.0

5.0

10.0

15.0

20.0

25.0


2.45

4.90

7.35

9.80

σc

0.0

20.0

40.0

60.0

80.0

100.0

120.0


14.7

19.6

29.4

49.0

0.0

5.0

10.0

15.0

20.00 10 20 30 40 50 0 10 20 30 40 50

Vol

umet

ric

Stra

in, ε

vol (

%)

Axial (natural) strain, εa (%)

Measure Data: Toyoura Sand CIUD TestsMiura and Yamanouchi, 1975

9.807.354.902.45 (MPa)

Symbol

2.45

4.907.359.80

0.0

5.0

10.0

15.0

20.0

Axial (natural) strain, εa (%)

14.7

19.629.449.0

c (MPa)


49.029.419.614.7 (MPa)

Symbol

Dev

iato

ric

Stre

ss, σ

' v-σ'

h (M

Pa)

σc

σc

σ

σc σc

Figure 18. Effect of confining stress in the drained behaviour of initially loose Toyoura sand.



the stress range 0.5–3:0 MPa yielded practically identical results to those predicted by the MIT-S1 model. Figure 21 shows the critical state line for both undrained and drained triaxialtesting described earlier. At low stress levels, the slope of the CSL is very small, resulting inlarge changes in undrained shear strength (at critical state) for a small perturbation in voidratio. In contrast, at large stresses, the critical state line is, for most practical purposes,parallel to the LCC. This observation is consistent with results presented by other researchers[37]. There is a critical void ratio, ecrit; for which undrained shearing will cause collapse of asand specimen with zero residual strength. The MIT-S1 model predicts ecrit � 0:94 forToyoura sand, while Ishihara [6] reports ecrit � 0:930–0.935. The undrained stress–strainbehaviour can be described qualitatively by the state index, Is; as proposed by Ishiharawhere:

Is ¼ ðecrit � eÞ=ðecrit � ecsÞ

0.0

5.0

10.0

15.0

20.0

25.0


2.45

4.90

7.35

9.80

σc σc

σc

σc

0.0

20.0

40.0

60.0

80.0

100.0

120.0


14.7

19.6

29.4

49.0

-5.0

0.0

5.0

10.0

15.00 10 20 30 40 50 0 10 20 30 40 50


9.807.354.902.45c (MPa)

Symbol

2.45

4.907.359.80

-5.0

0.0

5.0

10.0

15.0


49.029.419.614.7 (MPa)

Symbol

14.719.629.4

49.0

(MPa)

Dev

iato

ric

Stre

ss, σ

' v-σ'

h (M

Pa)

Vol

umet

ric

Stra

in, ε

vol (

%)

Axial (natural) strain, εa (%) Axial (natural) strain, εa (%)

σ cσ

Figure 19. Effect of confining stress in the drained behaviour of initially dense Toyoura sand.



where ecs is the void ratio at the CSL at the current stress level. The characteristics of theundrained shear behaviour can be related to Is as follows:

State Behaviour

Is50 Entirely collapsible behaviour}no residual strength

05Is51

Partially collapsible behaviour}with or without peak conditions occurringbefore critical state conditions

Is > 1 Peak and critical state conditions occurring simultaneously at large strains

Figure 21 includes measured critical strength data reported by Miura [10] in the range5–90 MPa obtained from drained triaxial tests. Model predictions slightly overestimate the voidratio measured at critical state conditions (i.e. overestimate the effective stresses at a given voidratio) in these tests. However, much better agreement is obtained using a critical state frictionangle f0

cs ¼ 338 (as opposed to 318; selected in Table II). The CSL is unique for drained andundrained tests but its location depends on the mode of shearing [38].

0.6

0.7

0.8

0.9

1.0

0.1 1 10 100

Voi

d R

atio

, e (

log

scal

e)

CSL

Measured Data: Toyoura SandUndrained Triaxial Compression

(Ishihara, 1993)

0.70

0.75

0.80

0.85

0.90

0.95

1.00

0 5 10 15 20 25 30 35

Voi

d R

atio

, e

MeanEffective Stress, p'/pat

CSL

ModelPrediction

Linear fit for0.5 to 3.0 MPa range

(a)

(b)

Figure 20. Comparison of predicted and measured behaviour at critical state for Toyoura sand forundrained tests.



6. SUMMARY

This paper evaluated the performance of the MIT-S1 model for predicting the rateindependent behaviour of Toyoura sand. Model evaluation focused on the predicted effects ofconfining stress and density on the undrained and drained behaviour of freshly deposited sand.Extensive comparisons with measured data from undrained triaxial shear tests show thatthe model gives excellent prediction of the transition from dilative to contractive shearresponse as the confining pressure increases for a wide range of formation densities. Modelsimulations of drained behaviour show that the model gives a realistic description of thevariation of peak friction angles and dilatancy rates as a function of stress level and densityand is in good agreement with measured data for Toyoura sand. The MIT-S1 modelpredicts a unique Critical State Line, CSL, for drained and undrained shear but isdependent on the mode of shearing. The critical state conditions form a non-linearlocus in log e� log p0 space, which is in excellent agreement with the recent experimental datafor sands. At low stresses, the slope of the CSL is small and is slightly larger than thecompressibility of the soil in this regime. As the confining stress increases, the CSL becomesnearly parallel to the LCC, which is conceptually similar to the observations on normallyconsolidated clays.

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.1 1 10 100 1000 104

Voi

d R

atio

, e

'cs

= 33˚

Loose

Mean Effective Stress, p'/ pat


0.1 1.0 10 100 1000

Dense

0.011.30

Measured Data: Toyoura SandDrained Compression Tests

Miura et al. 1984

Miura andYamanouchi 1975

Measured Data: Toyoura SandUndrained Compression Tests

Ishihara, 1993

H-LCC

CSL ( φ'csTX= 31˚)

φ

Figure 21. Evaluation of the critical state line from drained and undrained compression tests.



ACKNOWLEDGEMENTS

The authors would like to acknowledge INTEVEP, S.A. and the Gilbert W. Winslow Career DevelopmentChair at MIT for supporting this research. Additional support was provided by the National ScienceFoundation, CAREER award CMS-9623979. The authors thank Professor Debra Laefer and Dr CharlesC. Ladd for providing careful review of earlier versions of the manuscript.

APPENDIX A: NOMENCLATURE

The following symbols are used in this paper:

Cb material constant describing small strain ‘elastic’ bulk moduluse; ei; e0 current, initial and formation void ratio, respectivelyG;Gmax current and maximum (i.e. at reversal) shear modulus, respectivelyh material constant controlling irrecoverable plastic strains in unload–reload cyclesK; Kmax tangent and maximum bulk modulus (i.e. at stress reversal), respectivelyK0 lateral earth pressure coefficient for zero lateral strainK0NC K0 value for normally consolidated clays and sands in the LCC regimeLCC limiting compression curvem material parameter describing slenderness of the bounding surfaceOCR overconsolidation ratio during 1-D swellingnp material constant describing change of bounding surface shape as a function of

current void ratiopat atmospheric pressurea00; a

00i size of load surface and load surface at first yield, respectively

db dimensionless distance of the current load surface to the corresponding LCCea; g axial and shear strain, respectivelyf0cs critical state friction angle at large strains in triaxial compression tests

f0mðeÞ maximum friction angle describing bounding surface shape

f0mr material constant defining maximum friction angle, f0

m at e ¼ 1m00 Poisson’s ratio at stress reversal controlling 2Gmax=Kmax

y material constant describing transitional regime for freshly deposited sandsrc soil compressibility in the LCC regimerr parameter defining the current elastic volumetric stiffnesss0v; s

0h; s

0c vertical, horizontal and confining effective stress, respectively

s0vmax maximum vertical effective stressp0ref ; s

0vref reference mean and vertical effective stress at unity void ratio defining location of

H-LCC and K0-LCC, respectivelyp0rev mean effective stress at stress reversal point

s0 effective stress tensoro; os material constants describing variable Poisson’s ratio and small strain non-

linearity in shear, respectivelyc material constant controlling rate of rotation of bounding surface



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Evaluation of a constitutive model for clays and sands: Part II - clay behaviour

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